ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS VOLUME 68
EDITOR-IN-CHIEF
PETER W. HAWKES
Laboratoire d Optique Electronique du Centre National de la Recherche Scientifique Toulouse, France
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Advances in
Electronics and Electron Physics EDITEDBY PETER W. HAWKES Laboratoire d Optique Electronique du Centre National de la Recherche Scientifique Toulouse, France
VOLUME 68 1986
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CONTENTS PREFACE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Distance Measuring Equipment and Its Evolving Role in Aviation ROBERTJ . KELLYand DANNYR . CUSICK I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. General Background to Distance Measuring Equipment Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. The DME/N System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV . Systems Considerations for the New DME/P International Standard . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Einstein-Podolsky-Rosen Paradox and Bell’s Inequalities W . DE BAERE 1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. The Einstein-Podolsky-Rosen Argumentation . . . . . . . . . . . . . . . 111. The Bell Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV . Experimental Verification of Bell’s Inequalities . . . . . . . . . . . . . . V . Interpretation of Bell’s Inequalities and of Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
1 3 118
171 236 237
245 246 273 304 315 325 327
Theory of Electron Mirrors and Cathode Lenses E . M . YAKUSHEV and L . M . SEKUNOVA I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1. Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. Electron-Optical Properties of Axially Symmetric Electron Mirrors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV . Mirror Electron Microscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V.CathodeLenses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VI.Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
350 372 397 414 415
INDEX. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
417
V
337 338
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PREFACE
In Volume 57 of these Advances, an article by H. W. Redlien and R. J. Kelly was devoted to “The microwave landing system: A new international standard,” and I am assured that this has been heavily used by the aviation community. Since then, precision distance measuring equipment (DME/P) has come into widespread use as an integral and critical part of the MLS, a development in which R. J. Kelly has been deeply involved. I am therefore delighted to be able to publish his detailed account with D. R. Cusick of this new addition to the instrumentation that enables modem aircraft to land and take off so safely. In view of the importance of the subject and the authors’ concern to make their account at once complete, self-contained, and yet readable, this article dominates the volume and I have no doubt that members of the aviation world will be grateful for so clear and full a document. In 1935 A. Einstein, B. Podolsky, and N. Rosen published a paper in Physical Review entitled “Can the quantum-mechanical description of physical reality be considered complete?” This analysis of the orthodox interpretation of the quantum theory revealed a paradox, now universally known as the EPR paradox. This in turn has generated a voluminous literature, among which a paper by J. S. Bell is something of a landmark, adding Bell’s inequalities to the scientific vocabulary. The fiftieth anniversary of the EPR paper seemed a good occasion to survey this work, and I am extremely happy that W. De Baere, himself an active contributor to the debate, has agreed to put all these complex ideas into perspective. The point at issue is a difficult one, but is vitally important for the understanding and future development of quantum mechanics. The final article is devoted to two common elements of systems of chargedparticle optics that have been neglected in comparison with round lenses, prisms, and multipoles. These elements, mirrors and cathode lenses, have been regarded as difficult to analyze owing to the fact that some of the usual mathematical approximations that are fully justified in lenses and prisms can no longer be employed. Considerable progress with this problem has been made in the group led by V. M. Kel’man, co-author of the standard Russian book on electron optics, in the Nuclear Physics Institute in Alma-Ata. This work is available in translation since most of it appeared in the Zhurnal Tekhnicheskoi Fiziki (Soviet Physics: Technical Physics), but it is not widely known and I hope that this connected account in a widely accessible series will help to remedy the situation. The authors have taken considerable trouble to make their article self-contained, and thus reference lists of key formulas, for aberration coefficients in particular, are included. vii
...
Vlll
PREFACE
As usual, I conclude by thanking all the authors and by listing forthcoming articles.
Monte Car10 Methods and Microlithography Invariance in Pattern Recognition
K . Murata H. Wechsler
PETERW. HAWKES
ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS, VOL. 68
Distance Measuring Equipment and Its Evolving Role in Aviation ROBERT J. KELLY AND DANNY R. CUSICK Bendix Communications Division Allied Bendix Aerospace Corporation Towson, Maryland 21204
I. INTRODUCTION For almost 40 years, distance measuring equipment (DME) has been a basic element in world air navigation. Increasing needs of automation in aviation-and the increasing levels of sophistication that resulted-fueled its evolution, significantly improving its capabilities, while the soundness of its principles guaranteed its wide use. As a result of this continuing evolution, a new version of DME is now part of the new international standard for landing systems, the microwave landing system (MLS). Because of its persistent presence, DME has played a dominant role in the development of air navigation, and its principles are common to other parts of the air traffic control system (ATC). This article discusses these basic principles by placing them in perspective with other types of air navigation systems as part of a broader, integrated discussion of navigation requirements. Specifically, the discussions relate to the needs of the air traffic control system and to the dynamic flight characteristics of the aircraft operating within that system. To expose the diverse requirements that the DME element must satisfy in the modern and evolving National Airspace System, substantial background information is provided on the elements of the ATC system and on the implementation of automatic flight control systems. Two main types of DME have been standardized by the International Civil Aviation Organization (ICAO). The DME that was standardized ;n 1952 and that has been in general use during the past 40 years is referred to as the DME/N. The new precision DME, to be used with MLS, is known as DME/P. The ICAO standardized it in 1985.
1 Copyright @ 1986 by Academic Press, Inc. All rights of reproduction in any form reserved.
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ROBERT J. KELLY A N D D A N N Y R. CUSICK
The DME principle is simple. Distance information is computed in the aircraft by measuring the round-trip time between interrogations from an airborne transmitter and replies to those interrogations from a gound beacon. Because the DME has proven itself to be a robust technique, its use and implementation will continue into the next century. This chapter views both DME/N and the new DME/P from a broad systems engineering perspective. Like many other navigation aids DME is characterized in terms of a signal format, accuracy (error budgets), coverage (power budgets), radio frequency assignment(channel plan), the measurement technique used to extract the DME navigation parameter (range), and the output guidance information rate. A DME must also satisfy certain levels of signal integrity and availability. The signal format is the blueprint of the system; it ensures that the DME ground and airborne elements will be compatible. Range accuracy must facilitate safe arrival of aircraft at their destinations. The ground facility must provide a usable signal that covers the volume of airspace where navigation information is required. Well-conceived radio frequency (rf ) channel plans assure that ground facilities can be geographically located so that their individual service volumes unambiguously “combine” to provide navigation guidance throughout the navigable airspace. The techniques used to make navigation parameter measurements must be simple so that ground and airborne equipment designs are reliable and economical while achieving the necessary accuracy and coverage. Finally, the guidance output information rate must be consistent with the aircraft flight control system’s (AFCS) dynamic response so that it will not limit the aircraft’s maneuvers. The DME system discussions in this article will follow the functional theme outlined herein. Emphasis will be directed toward systems considerations rather than toward hardware/software implementations or flight test data analysis. Section I1 provides the background and historical perspective necessary to view DME in the broader context of air navigation and its specific role in the National Airspace System (NAS). Section I11 is devoted to DME/N, and Section IV addresses the most accurate version of DME, the DME/P. The navigation applications discussed in this article are restricted to the air traffic control system as it is defined by the International Civil Aviation Organization and as it is implementedin the United States.Therefore,only the domestic enroute, terminal, and approach/landing phases of flight are addressed. Enroute and approach/landing scenarios are defined using only the conventional takeoff and land (CTOL) aircraft type. However, DME is not restricted to these applications or to CTOL aircraft. Both helicopter and remote area operations, for example, come under the broad envelope of DME applications.
DISTANCE MEASURING EQUIPMENT IN AVIATION
3
11. GENERAL BACKGROUND TO DISTANCE MEASURING EQUIPMENT APPLICATIONS A . Basic Operational Scenario and Introduction to the Navigation Services
A commercial jet aircraft leaving Chicago’s O’Hare International Airport for New York’s LaGuardia Airport will fly along a national airway system divided into enroute and terminal areas. During the enroute portion of the flight, the aircraft flies airways defined by a network of straight-line segments or radials. These radials are generated by very-high-frequency omni range (VOR) and DME navigation aids that provide direction and distance information to known ground locations. Using these radials, the pilot can navigate to the destination terminal area. Arriving in the LaGuardia terminal area, the aircraft enters a region of converging tracks and high-density traffic. Position accuracy becomes increasingly more important as the aircraft approaches its runway destination, particularly when visibility is reduced. Here, the enroute guidance supplied by the VOR/DME radials is replaced with approach and landing guidance such as that generated by the MLS. The three-dimensional guidance provided by MLS is of sufficient .accuracy to guide safely the aircraft to a landing, even under zero-visibility conditions. Accurate range data are essential during both phases of flight. The guidance aid which generates this ranging information is DME. It provides range data accurate to 0.2 nautical miles (nmi) along the enroute airways. Within the terminal area, its range error decreases to 100 feet or less for the landing guidance applications. DME has evolved from the radar beacon concept to satisfy the navigation needs of both the civilian and the military aviation community. This evolution is reviewed in Subsection B. The general principles of aircraft position estimation, as applied to the radio navigation aids used in the NAS, are summarized in Subsection C. Both bearing and range measuring systems are treated. In order to place the DME in its proper operational context, other navigation aids that operate in conjunction with it must be discussed. In Subsection D the role of VOR, ILS, MLS, and the ATC surveillance system (secondary radar) in the overall ATC system are briefly discussed. Also included is the Tactical Air Navigation System (TACAN) upon which the military currently depends for its enroute navigation. In this system, DME plays a central role in a signal format that is arranged to contain both range and bearing information. Both military (TACAN) and civil (VOR/DME) enroute navigation
4
ROBERT J. KELLY AND DANNY R. CUSICK
systems are combined on the ground in the configuration called VORTAC. For the United States NAS and many of the European ICAO member states, the principal ATC enroute airways are based on range and bearing information from VOR/DME or VORTAC facilities. For the precision phases of flight conducted in the terminal areaapproach, landing, missed-approach, and departure- the principal aids are ILS and MLS. A high-precision DME is the range-providing element of the microwave landing system. When used with the instrument landing system (ILS), another version of DME/N provides the range information normally generated by the ILS marker beacons (which "mark" a few specific points along the straight-line approach path). DME, VOR, ILS, MLS, and TACAN are navigation aids (navaids) that provide the primary navigation parameters of range and bearing. These navaids can be used directly for navigation, or they can be integrated into the aircraft by an area navigation system known as RNAV. Navigation derived from RNAV is becoming increasingly important and has led to the development of RNAV routes in addition to the VORTAC routes. Subsection D briefly describes RNAV. Table I summarizes the navigation services. With the exception of MLS, they have been in use for the past three or four decades. It is essential to understand the interaction of navigation signals with the Aircraft Flight Control System (AFCS) in order to comprehend a navigation system. To assist in this understanding, background information on aircraft guidance and control is given in Subsection E. Accuracy performance comparisons between VOR/DME, DME/DME, and TACAN, using RNAV to calculate the position fixes, are given in Subsection F. TABLE I PRESENT NAVIGATIONAL SERVICES Service VHF omnidirectional range (VOR) Distance measurement, DME/N Air traffic control surveillance system" Interrogator Transponder Instrument landing system Localizer Glide slope Marker MLS Azimuth and elevation DME/P
Frequency band (MHz) 112-118 960-121 5 1030 1090 108-112 328-335 15 5031-5091 (Same as DME/N)
Includes mode S and the traffic alert collision avoidance system (TCAS).
DISTANCE MEASURING EQUIPMENT IN AVIATION
5
B. History of the DME Principle
This section introduces the principal elements of DME, the airborne interrogator, and the ground transponder, by tracing the early history of their development. Both the DME and the ATC Secondary Radar were derived from the radar beacon concept developed during World War 11. All of the techniques described herein were operational in the early 1940s (Roberts, 1947). Conventional radar operates by sending out high-energy pulses of radio waves. The radar receiver detects the echo of the transmitted energy that is reflected back from a target. The elapsed time between the transmission of the pulse and the reception of the echo is a measure of the range (or distance) to the reflecting target. Further, by use of a suitable antenna design, the radio energy is concentrated into a narrow beam so that the echoes are received only when the radar is ‘‘looking’’ at targets residing within the antenna’s beamwidth. By proper coordination of the motion of the antenna and the sweep of an intensity-modulated cathode-ray tube (CRT) display device, a plan view (like a map) of the reflecting targets in the region of the radar set is traced out. This map-type display, which is illustrated at the top of Fig. 1, is called the plan-position indicator (PPI). Thus the target’s position is identified by a “bright spot” on the PPI. The radial trace through the spot is the target’s bearing relative to the radar’s reference. The distance from the center of the PPI to the bright spot is proportional to the target’s range. I . The Beacon Concept Conventional radar pulse transmissions are reflected by targets of different sizes regardless of their individual importance to the radar user. Sometimes echoes are too weak to be observed, as are those from a small airplane at a great distance. Under other circumstances, strong echoes from buildings or natural features such as mountains may mask weaker echoes from aircraft. Furthermore, the exact location of a geographical point on the ground may be of importance to a radar-equipped aircraft even though the terrain gives no distinguishing echo. In all such cases, radar beacons find their use as cooperative aids in pinpointing specific targets. A radar beacon is a device that, upon reception of the original radar pulse, responds with its own transmitter to radiate a strong signal of one or more pulses which is easily distinguished from the weaker radar echoes from surrounding objects. The beacon, therefore, is a device which amplifies the echo. The beacon transmitter need not be very powerful to be able to give a reply much stronger than the target echo itself. Most importantly, the beacon, when placed upon the ground, can provide an accurate reference point which can be used to determine an aircraft position relative to the reference point.
6
ROBERT J. KELLY AND DANNY R. CUSICK NORTH
INTERROGATOR
e
= BEARING R = RANGE
DIRECTIONAL ANTENNA
RECEIVER
TRANSMITTER BEACON
FIG. 1. The beacon concept.
In a typical beacon application (Fig. l), the scanning narrow beam emanates from the aircraft. The position of the beacon on the ground (its range and bearing relative to the aircraft) can be determined by viewing the airborne PPI display. The ground beacon’s antenna is designed to be omnidirectional, that is, to receive signals equally well from all azimuth directions. The beacon
DISTANCE MEASURING EQUIPMENT IN AVIATION
7
bearing is derived from the narrow scanning beam while the “round-trip” delay of the radar transmission and its reply from the beacon is proportional to the beacon’s range to the aircraft, resulting in a unique measurement of position. The measurement of range and bearing constitute one of the most fundamental procedures in determining a position “fix” by a radio navigation aid. It is the basis, for example, by which guidance is obtained to fly the VORTAC routes. The process by which a radar set transmits a signal suitable for triggering the beacon is known as interrogation; the corresponding beacon transmission is termed the reply. Radar beacons which reply to interrogations are called transponders and the radar set used to interrogate a beacon is called an interrogator. The provision of a special character to the pulsed signals of either the interrogator or the transponder is called encoding. A device that sets up such a coded signal is called an encoder; a device that deciphers the code at the other end of the link is called a decoder. Although the transponder reply is like an echo, it differs from one in several significant respects. The strength of the response is independent of the intensity of the interrogating signal, provided only that the interrogating signal exceeds a predefined minimum threshold intensity at the beacon receiver. Also, the response frequency is different from that of the interrogation frequency. The response signal pulse may also differ from the interrogation signal in form. It may even consist of more than one pulse, the duration and spacing of the pulses being an arbitrary design choice. Thus echoes from a desired beacon on the ground can be distinguished from others on the ground not only in terms of pulse amplitude (nearer beacons have larger amplitude returns) but also in terms of radio frequency and pulse spacing (code). All of the important components necessary to form a beacon system are illustrated in Fig. 1. Roberts (1947) is a complete treatise on the systems-engineeringconsiderations of radar beacon designs. The complete downlink/uplink cycle starts with the generation of a trigger pulse in the airborne radar interrogator. This trigger initiates the transmission of an interrogating signal and, after a predetermined delay (called the zero mile delay),starts the cathode-ray tube sweep on the indicator. On the ground, the beacon transponder receiver detects the interrogating signal and generates a video signal that is passed to the decoder. The decoder examines the video signal to see if it conforms to an acceptable code. If not, it is rejected. If accepted, a trigger pulse is passed to the blanking gate that prevents the receiver from responding to any further interrogations for a time sufficient to permit the complete coded reply to be transmitted. This time, called the reply dead time, varies generally from about 50 to 150 ys. Because the beacons require a reply dead time, there is a limited number of interrogations (called
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ROBERT J. KELLY AND DANNY R. CUSICK
the traffic load) that can be accommodated by the transponder (see Section 111,C). The transponder decoded output, a single pulse, is delayed for a time which when added to the reply code length equals the “zero mile delay” used in the interrogator. The delayed signal is then encoded with a reply code and transmitted equally in all directions on a different rf frequency. At the airborne radar antenna the reply is detected by the interrogator receiver, decoded, and displayed to the operator on the PPI indicator. If the interrogator is “on top of” the transponder, i.e., if the range were zero miles-there would be no propagation delay through space, and the transponder reply signal’s arrival time would equal the zero time delay. The PPI trace would then show a “spot” at the center of its display. This system was developed almost 50 years ago and forms the basis of the secondary surveillance radar (SSR) system and the DME. If the airborne element is the transponder with nondirectional antenna and the interrogator with its directional antenna is placed on the ground, the configuration is the basis of the SSR system. Range and bearing and, thus, a position fix on the aircraft are made available to the air traffic controller on the ground. On the other hand, if the beacon is located on the ground and the PPI scope in the aircraft is replaced by an automatic range tracking circuit, the range to the fixed beacon is displayed to the pilot. This is the DME concept (see Fig. 2). Interestingly enough, the principal components-interrogator transmitter and receiver, encoder, decoder, transponder dead time, echo suppression, transponder receiver, and transmitter -remain the principal elements of today’s DME system. Only elements of performance such as accuracy have been improved over the years in response to more demanding applications. Since range is the only parameter of interest in DME, bearing information is sacrificed in exchange for use of simple omnidirectional ground and airborne antennas. In addition, a higher data rate is possible since the ground and airborne equipments are always “looking” at each other. An aircraft “range-only’’ position fix can be obtained by measuring the range from two separated ground transponders. See Subsections C and D,5 on scanning DME. 2. History of the Beacon Concept
The radar beacon was invented in 1939 by a group at the Bawdsey Research Station of the Air Ministry in the United Kingdom (Williams et al., 1973). It was developed in response to a military need, and its invention was not immediately made public for reasons of security. During the war, radar beacons were used by Germany, Japan, and the Allies. Because these countries
9
DISTANCE MEASURING EQUIPMENT IN AVIATION
TRIGGER AIRBORNE INTERROGATOR
4 ENCODER
DECODER
4
t
TRANSMITTER
RECEIVER
a
\
AIRCRAFT SKIN
4
El GROUND BEACON
FIG.2. DME operation.
veiled the invention in secrecy, there were no pre-World War I1 nonmilitary uses of beacons. The initial purpose of the beacon was for Identification of Friend or Foe (IFF), where appropriate transponder responses to coded interrogations indicated that a particular military vehicle was either a “friend” (correct code response) or “foe” (incorrect response). It was soon discovered that beacons could be used for offensive purposes by helping to locate target areas. That is, they were used not only for identification but for navigation as well. One such system which employed lightweight ground beacons was the Rebecca-Eureka system. It included an airborne interrogator called “Rebecca” and a ground beacon called “Eureka.” Originally designed by the Telecommunications Research Establishment (TRE) for British use, the Rebecca-Eureka system was later adopted by the United States Army and used in many operations. The Rebecca-Eureka system operated at a radio frequency of 200 MHz. Table I1 shows its channel plan, which is the forerunner of the DME channeling concept (Roberts, 1947).
10
ROBERT J. KELLY AND DANNY R. CUSICK TABLE I1 REBECCA-EUREKA CHANNEL PLAN Interrogation channels (MW
Response channels (MW
A 209 B 214 C219 D 224 E 229
209 214 219 224 229
Characteristics Only cross-channel used; e.g., interrogation on A, response on B, C, D, or E
3. Development of the D M E Concept
As remarked in Dodington (1980), “it must have occurred to many people that the addition of a relatively simple automatic tracking circuit would enable an interrogating aircraft to constantly read its distance with respect to a ground beacon.” However, it was apparently not until 1944 that such a device was actually built (by the Canadian Research Council), operating in the Rebecca-Eureka band of 200 MHz. In November 1945, the Combined Research Group at the U.S. Naval Research Laboratory built and demonstrated such a system operating at lo00 MHz. It was largely based on work that the Hazeltine Corporation had been doing on the Mark V IFF, at 9501150 MHz. This system, however, did not use precise frequency-control techniques. A crystal-controlled DME system was developed at ITT Nutley Laboratory in 1946 (Dodington, 1949) and had a large impact on the final DME channel plan selected by ICAO. Whether the DME had crystal control or not was not simply a hardware question. It was most importantly a signal format question involving the efficiency of radio frequency spectrum use. The use of precise frequency control permitted more DME channels to be accommodated within a specified frequency bandwidth. The development of DME from 1946 to 1959 was very complex, involving the U.S. military, Congress, industry, and national pride and will receive only a cursory review. The reader is referred to several excellent references (Dodington, 1980; Sandretto, 1956;Rochester, 1976)for details on this subject. As described in Sandretto (1956), the air-navigation and traffic-control system in the United States grew as the product of necessity and without being planned. It began about 1919 with the installation of 4-course radio ranges for furnishing guidance to aircraft along defined routes. Later, other navigational aids were added, such as markers (both low and high frequency), to indicate points along the airways. Still later, low-approach equipment was added. When a controlling agency was established on the ground, it was equipped with slips of paper to designate aircraft, and utilized limited knowledge of airspeed, heading, and winds to
DISTANCE MEASURING EQUIPMENT IN AVIATION
11
predict their future positions. Reliance was placed on communicationsfrom the aircraft to the ground controller to learn of the aircraft’s actual positions. Communications were furnished largely through the high-frequency equipment owned and operated by the airlines.Teleprinter communicationswere later added between the various control centers.
In reviewing this system, it is clear that it was slow and inefficient. Because of this situation, several companies (ITT, Hazeltine, RCA, and Sperry) proposed schemes to alleviate this problem through implementation of a comprehensive navigation and traffic control system. An ITT system called NAVAR was based upon the following concepts developed in 1945 (Busignies et al., 1946): (1) means whereby the pilot can know his position in three dimensionsso he can navigate to his destination; (2) means whereby a ground authority, capable of regulating the flow of air traffic, can know the position of all aircraft; (3) means whereby the ground authority can forecast the future positions of all aircraft; (4) means whereby the central authority can issue approvals to pilots of aircraft to proceed or to hold. When the present ATC system is discussed later, it will be interesting to note how closely it approached ITT’s original planning concepts. The first navigation elements of NAVAR were the DME and an associated bearing device. They were demonstrated for three weeks in October, 1946, at Indianapolis where the U.S. played host to the Provisional International Civil Aviation Organization (PICAO). A major result of the Indianapolis demonstrations was that ICAO later decided that the future short-range navigation system would be p-8, i.e., range-bearing [as opposed to GEE, which was the British VHF hyperbolic navigation system (Colin and Dippy, 1966)l. Further, it was decided that the DME part would be located at 1000 MHz, not 200 MHz. It was argued that the p-8 system was the natural choice since man is born with the knowledge of left and right and learns about distance when he learns to walk (Sandretto, 1956). Meanwhile, subcommittees for both the Radio Technical Commission for Aeronautics (RTCA) and the ICAO began working on standards for a coordinated navigation system, including DME. During 1946, RTCA Special Committee SC-21 developed a DME channeling plan at L band (the 1000-MHz region) that ignored crystal control (RTCA, 1946).Its 13interrogation channels and 13 reply channels were spaced at a relatively wide separation of 9.5 MHz based on using free-running oscillators whose stability was to be k 2 MHz. Later in the summer of 1947, RTCA organized committee SC-31 on Air Traffic Control. This committee’s
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ROBERT J. KELLY AND DANNY R. CUSICK
report (RTCA, 1948a) stated that The Navigation Equipment is a transmitter-receiver having multiple channels. This equipment, with associated ground equipment:
(1) Provides distance and bearing information for navigation. These data, when used in conjunction with a computer, will allow the pilot to fly any desired course. (2) Provides precise slope, localizer, and distance information for instrument approaches. (3) Provides information for airport surface navigation to enable the pilot to taxi his aircraft. (4) Provides air-ground aural communication of a reliable and status-free type. ( 5 ) Provides a situation display in pictorial form which enables the pilot to monitor traffic conditions in his vicinity or receive other pertinent data such as holding areas, airlane locations, and weather maps from the ground. (6) Provides suitable output to allow the aircraft to be automatically flown, either enroute or during final approach and landing.
In retrospect this appears to be the first attempt to define the ATC system as it exists today. The SC-31 committee’s navigation system is further described to have “the airborne navigation equipment and its directly associated ground elements wholly contained in the 960-1215 MHz band. The airborne navigational equipment for the ultimate system will have accuracy of k0.6“ in bearing and & 0.2 nautical mile or 1% of the distance, whichever is greater.” As will be discussed in Subsection F, none of the bearing systems finally placed in operation achieved this & 0.6” accuracy. However, DME/N did achieve a performance of 0.2 nautical mile and-with the recent development of DME/P-has surpassed that goal by almost twentyfold. Although the equipment proposed by the SC-31 committee for the ultimate or target system was very dissimilar to the NAVAR system, it constituted a coordinated system with essentially the same elements that had been described by ITT. Committee SC-31 had brought together the thinking of the Air Force and the Navy as well as the Civil aviation industry. By participating in the committee’sactivities, each group learned of the work that had been done in the other groups. As proposed by SC-31, only the DME and the Secondary Surveillance Radar remained in the 960-1215 MHz band. The bearing information (VOR) was assigned to the VHF band while the aircraft landing systems ended up at VHF for ILS and C-band (in the 5000-MHz region) for MLS (see Table I). That is, through the consensus of committee participation, the NAVAR and other proposals were discarded. In early December, 1947, another RTCA Committee (RTCA, 1948b),SC-40, was convened for the purpose of studying two channeling techniques under development for DME. One of these was the multiple radio frequency channel, crystal-control technique, and the other the pulse-multiplex system which was similar to that proposed by SC-21. The committee made its report in December of 1948 with a recommendation that
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both crystal-control and pulse-multiplex techniques be included in the channel plan. The envisioned DME system would interrogate on ten crystal-controlled channels at the low end of the band and reply on ten crystal-controlled channels at the high end of the band. Channels were spaced 2.5 MHz apart and used 10 pulse codes in multiples of 7 ps, starting with 14 ps. Thus, there were 10oO combinations of frequency pairs and codes, of which it was proposed to pair 100 with the VOR and the ILS localizer (see Table VI). 4. DME Versus TACAN
The U.S. Civil Aviation Agency (CAA)-the forerunner of the Federal Aviation Administration-adopted the SC-40 plan as its official position and convinced ICAO to adopt it. The first ICAO Channel Plan, given in Tables V and VI, is essentially the same as that proposed by SC-40. However, as noted in Dodington (1980), there was little enthusiasm for this system outside the CAA, both within and outside the US. The main reason was that the US. military was developing an entirely different system in the same frequency band. When the SC-40 system was finally abandoned in 1956, there were only 340 airborne sets in existence. The trouble started shortly after World War 11, when the CAA began to install the recently developed VOR and DME units that were expected to form the nucleus of the emerging “common” system. The Navy found VOR/DME to be unsatisfactory for aircraft carrier operations because of the large size and complex siting requirements of the relatively low-frequency VOR antenna. They contracted with the ITT’s Federal Telecommunications Laboratories in February, 1947, to design an alternative system that could be used effectively at sea. By 1951,the Navy project had not only acquired the name TACAN but also the support of the Air Force, which had originally endorsed VOR/DME. The Air Force was converted to TACAN by its promise of greater flexibility and accuracy and its integration of the bearing and range elements into a single unit and, accordingly, a single frequency band. Thus, scarcely before the SC-40 “common system” document had circulated, civil and military agencies were following separate courses of action. Rochester (1976) provides a complete history of the VOR/DME/TACAN controversy. It was resolved in August, 1956,by discarding the SC-40 system and replacing it with a TACANcompatible DME. Part of the solution involved a fixed pairing of 100TACAN channels one-for-one with 100 VOR/ILS channels. This proposal by Dodington (1980) thus allowed a single-channel selector for VOR/ILS/DME and removed the objections of the airlines which were concerned that blunders would result from VOR bearing information received from one station and DME from another.
14
ROBERT J. KELLY AND DANNY R. CUSICK
The new enroute navigation system, christened VORTAC, retained the VOR bearing component but replaced civil DME with the distance component of TACAN. Each VOR station, instead of being collocated with a DME, was now collocated with a TACAN beacon (which, of course, also provides the DME service). Figure 3 shows the general VORTAC arrangement. At the ground station, the VOR central antenna is housed in a plastic cone that supports the TACAN antenna. Leads to the TACAN antenna pass through the middle of the VOR antenna, along its line of minimum radiation, and thus do not disturb the VOR antenna pattern.
1
I
I
I I -
i'
I
CHANNEL SELECTOR
TACAN
I I
I
I
L - _M l L l-T l 7
CHANNEL
- - - -----
1 SELECTOR
CHANNEL SELECTOR
TACAN ANTENNA
'
VOR
I VOR
ANTENNAS
NORTH NORTH
BEARING
70 m.BEARING TO BEACON NOTE: W E AND VOR CHANNELS ARE PAIRED
FIG.3. The VORTAC system.
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Civil aircraft read distance from the TACAN beacon and bearing from VOR transmitter. Military aircraft read both distance and bearing from the TACAN beacon. Both types of aircraft therefore fit into the same ATC network. Although the CAA and DOD had thus resolved their differences, the system defined by ICAO in 1952 was still the SC-40 format. Most of ICAO’s member countries did not implement it and, in fact, some took the opportunity to implement a hyperbolic system such as DECCA, the British low-frequency navigation system. After more years of debate, ICAO finally adopted the TACAN-compatible DME in 1959, and the VORTAC system has remained intact since that time. The evolution of the international DME standards will be detailed in Section 1II.A.
C. Principles of Air Navigation Systems
In this section basic concepts of navigation, position fixing, and dead reckoning are defined and applied to air navigation. I . Dead Reckoning, Fixing, and Navigation Systems
The fundamental task of a navigator is the selection of a route to be followed by his aircraft. To do this the navigator must know his aircraft’s present position and destination before he can choose a route and, from that, a course and heading. All position-determination schemes can be classified as dead reckoning, position fixing, or a combination of both. A position, once known, can be carried forward indefinitely by keeping continuous account of the velocity of the aircraft. This process is called dead reckoning. Because velocity v is a vector, it has a direction and magnitude (speed)u. From a given starting point p o and a velocity vo, one can determine a new position p after time Ato by stepping off a distance Ad = vot, in the direction of the velocity vo. At position p l , the process is repeated to find the next position p z using v1 and Atl, and so on. Unfortunately, direction and speed cannot always be precisely measured, particularly if direction is derived from aircraft heading only and effects of winds are approximated. Thus, on the average, the errors of dead reckoning increase in proportion to the length of time it is continued. Ordinarily, such an elementary dead reckoning process is seriously in error after an aircraft has traversed a few hundred miles without any indication of position from some other source. The Doppler radar navigator is an example of a good and more complex dead reckoning system (Kayton and Fried, 1969). It can guide an aircraft across the Atlantic Ocean and have a position error at its destination of less than 10 nautical miles. In
16
ROBERT J. KELLY A N D DANNY R. CUSICK
recent years, the Doppler radar navigator has been replaced by the inertial navigation system (INS) for transatlantic flight navigation. A position fix, in contrast to dead reckoning, is a determination of position without reference to any former position. The simplest fix stems from the observation of a recognizable landmark. In a sense, any fix may be thus described; for example, the unique position of stars is the recognizable landmark for a celestial fix. Similarly, some radio aids to navigation provide position information at only one point, or relatively few points, on the surface of the earth. In this case the landmarks are the physical locations of the ground stations. To obtain a position fix using radio navaids such as DME or VOR requires measurements from at least two different ground facilities;for example, the use of two differentDME stations (DME/DME) requires measurementsfrom two different ground stations or, for DME/VOR, two different measurements (range and bearing) from the same location. A single measurement yields only a locus of position points; two measurements are necessary to identify the point corresponding to an aircraft’s position with respect to the surface of the earth. Since a fix is based upon essentially “seeing” landmarks whereas dead reckoning is more associated with a computational process, it is “natural” that most electronic aids to navigation are based on the determination of fixes. Dead reckoning can be combined with position fixing to realize a more accurate navigation system. These configurations are referred to as multisensor navigation systems in the literature (Fried, 1974; Zimmerman, 1969). As will be shown in Subsection E, dead reckoning and position fixing complement each other; each provides an independent means of checking the accuracy of the other. Where position fixing is intermittent, with relatively long intervals between fixes, dead reckoning is appropriately considered the primary navigation method. In these systems, position fixing constitutes a method of updating the dead reckoning calculations such as in the “aiding”of an INS by a VOR and DME position fix, as shown in Fig. 4. Notice how the position fix corrected the dead reckoning measurement. Britting (1971) gives an in-depth discussion of INS. This correcting combination is extremely important in modern air transports which may have the navigation guidance autocoupled to the flight control system (Subsection C,3).A major concern is controlling the aircraft so that it maintains its desired course in the presence of wind shears and wind gusts. Small tracking errors are achieved by having the aircraft tightly coupled to the guidance system. Forcing the aircraft to follow the guidance signal closely requires low-noise navigation sensors. The INS is such a low-noise device, but, as mentioned above, it has a tendency to drift from its initial setting with time (about 1 nmi/h). These drift errors are corrected periodically by
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17
NORTH
t
NORTH
\
$ = HEADING
GT
--
-TRACK ANGLE R RANGE V AIRCRAFT GROUND VELOCITY
FINAL APPROACH WAY POINT
-
-
FLIGHT PATH CONSTRUCTION DESTINATIONS .WAY POINTS
position-fixing radio navaids. With position-fix aiding, the INS becomes the desired accurate windproof navigation system. If fixes are available continuously or at very short intervals, position fixing becomes the primary navigation method. Rapid position fixing constitutes the most common primary navigation system currently in use over the continental US. (see Subsection D,5 on RNAV). Its popularity arises from the low acquisition and maintenance cost of the airborne element. In some cases, dead reckoning measurements are necessary to augment the rapid position-fix information. Such a case arises when excessive noise from the VOR/DME sensors causes the aircraft flight control system to have undesirable pitch and
18
ROBERT J. KELLY AND DANNY R. CUSICK
roll activity. The aircraft may also follow the sensor noise because it is tightly coupled to the guidance signal (high signal gain). Unfortunately, reducing the noise by uncoupling from the guidance signal (low signal gain) can degrade the aircraft’s transient response so that it cannot closely follow the course in the presence of wind turbulence.’ On the other hand, a stable aircraft transient response requires accurate rate information (velocity) to generate the necessary dynamic damping so that the aircraft will not oscillate about the desired course. Since low-noise rate information cannot always be derived from the position-fix information, accurate rate information may be derived from dead reckoning measurements. An accelerometer output, for example, can be integrated once to provide the velocity. In this case dead reckoning measurementsfrom inertial sensors aid the VOR/DME position-fix guidance. This augmentation is called inertial aiding. In the context of the above discussion, both velocity data as well as position data are necessary to control an aircraft accurately along its desired flight path. Complementing position-fix data with dead reckoning information and vise versa is a notion which will be encountered again in detail in Section V. It is one of the most important solutions to the problem of obtaining accurate navigation, particularly during the approach/landing applications. As stated in Pierce (1948), navigation does not consist merely of the determination of position or the establishment of a compass heading to be followed. Navigation requires the exercise of judgment; it is a choice of one out of many courses of action that may lead to the required result based on all available data concerning position, destination, weather, natural and artificial hazards, and many other factors. Therefore, there can be no electronic navigational system-only electronic aids to navigation. An aircraft may be automatically guided along a predetermined course by the use of equipment that performs the dead reckoning function, or may be made to follow a line of position known to pass through a desired objective. Neither of these achievements constitutes navigation by equipment. The automatic devices simply extend the control exercised by the navigator in time and space. A navigation system comprises: (1) Sensors or navigation aids that generate primary guidance parameters; (2) a computer that combines the guidance parameters to obtain the aircraft position; (3) flight path commands (the path to be followed) that, when contrasted with the aircraft’s present position, generate the steering commands that define needed corrections in flight path; and (4) a flight control system, which causes the aircraft to correct its flight path while remaining stable. Figure 4 functionally describes the principal sensor and position calculation elements for a two-dimensional(2D) lateral guidance navigation system. Three-dimensionalflights would include Alternatively, filtering the sensor output data is not always a viable solution because the added phase delays may reduce the aircraft flight control system’s stability margin.
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19
the aircraft’s altitude which would be used to derive the vertical guidance component. A navigation system provides output information in a variety of forms appropriate to the needs of the aircraft. If the information is to be directly used by the pilot, it involves some type of display; other outputs can be steering signals sent directly to the autopilot. In some applications such as manual flights along the VORTAC radials, the sensor outputs-range and bearing-are sent directly to the displays without additional data transformations and computations. Subsection C,3 will describe how the outputs of the navigation system are used to “close the loop” with the aircraft to form the total aircraft flight control/guidance system. Only position-fix navigation systems will be treated. Bearing “only” navaids, such as the airborne direction finder (ADF) using nondirectional beacons (NDB),and not discussed in this article (see Sandretto, 1959). 2. Principles of Navigation Parameter Measurement: Range and Bearing
Electronic navigation aids are defined as those systems deriving navigationally useful parameters-for example, range and bearing- through the application of electronic engineering principles and electronic technology.’ Following the partition developed in Sandretto (1959), these systems are of two general classes: the path delay and the self-contained. Path delay systems are characterized by employment of at least one radio transmitter and at least one radio receiver. The transmitter emits energy that travels to the receiver. The navigational parameter is derived by measurement of the delay incurred in the transmission. All path delay systems are based on the assumption that radio wave propagation is rectilinear and that its velocity of propagation is constant. Therefore, parameters such as phase delay or pulse arrival time delay between a reference and a distant object can be measured and made proportional to range or bearing. The self-contained systems consist of devices that sense certain natural phenomena and, with the aid of computers, derive navigational parameters. The INS is such a system. Path delay systems are characterized by the various types of position lines which they produce. There are only two fundamental types: the single-path and the multiple-path system. Figure 5 notes the two main variations of each of these types. The single-path system measures absolute transmission time and produces circular “lines of position”- that is, a single-path navigation system determines aircraft range. DME is such a system. The terms navigation aid (navaid)and navaid sensor are used synonymouslyin this article. In the system context, a radio navaid is a sensor which includes both the ground and airborne elements and thus includes cooperative techniques such as DME. Traditionally,MLS, ILS, and VOR are characterized by a signal in space (ground element) and a sensor (airborne element).
20
ROBERT J. KELLY AND DANNY R. CUSICK NAVAIDS USED I N ATC SYSTEM
SELF-CONTAINED SYSTEMS MEASUREDELAYOVER TRANSMISSION PATH POSITION FIX
I SATELLITE
SINGLE PATH (CIRCULAR LOP)
4
FI
I
OBSERVES NATURAL PHENOMENA & COMPUTES
FlFl RECKONING
I + ’
MULTIPLE PATHS PATH DELAYS COMPARED
.CELESTIAL FIXES (NOT USED I N SYSTEM)
LOP
-
LINES OF POSITION
.INS
mzri:R
RNAV
NAVIGATOR
.AIRDATA
TRUE HYPERBOLIC
ONE-WAY
DME
I
FOR EQUAL DELAY (RADIAL LOP)
(HYPERBOLIC LOP)
VOR RADAR BEARING ILS MLS ANGLE .TACAN BEARING
‘UNDER DEVELOPMEN7
FIG.5. ATC navaid family tree.
The multiple-path system measures difference in, or compares, transmission times. Since the locus of all points having a constant difference from two other points is a hyperbola, these systems should produce lines of positions which are hyperbolas. Many of these mutliple-path systems are so instrumented that they can only determine when the differencein transmission time is zero (that is, when the times are equal). In this case, the line of position is a hyperbola of zero curvature, or a striaght line. That is, the locus of all points that are equidistant from two other points is a straight line (the bisector). These simplified multiple-path systems are known as radial systems; those with nonzero time delays are called true hyperbolic systems. Radial systems measure aircraft bearing. Examples are VOR and the TACAN bearing signal. True hyperbolic systems include DECCA, LORAN, and OMEGA (Kayton and Fried, 1969). In summary, multiple-path systems must use transmissions which differ in either time or frequency. a. Single-Path Systems. The simplest system in concept is the single-path,
one-way system; it employs a transmitter at one point and a receiver at
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another. Transmission occurs over a single path between the two points. The delay in transmission over this path is the important element in securing the navigational parameter. The parameter which is delayed, for example, could be the phase of a subcarrier whose frequency is maintained very accurately. The receiver similarly employs an oscillator matched in phase to the transmitter subcarrier. The transmitter-receiver phase difference, therefore, is a measure of the time and, therefore, the distance taken by the wave to traverse the transmitter-to-receiver distance. Such systems are called coherent systems. They require very stable references such as atomic clocks and are not used in the navigation systems discussed in this article. A second type of single-path system employs a round-trip path as shown in Fig. 6. Energy from the transmitter illuminates a “reflector.” A reflector may be passive, such as an airplane hangar, or active, such as a radio beacon. The reflected energy is returned to the originating transmitter location where its delay (time of arrival) is compared with the initial reference time. Conventional radar applies the passive reflector concept, while DME and the ATC secondary radar are based on the beacon idea. The two-way, single-path system overcomes the costly reference problem associated with the one-way, single-path system because only one reference oscillator with greatly reduced frequency stability characteristics is necessary to achieve a coherent system. Thus, the stringent frequency stability requirements for coherent signal processing need be maintained only over the equivalent of several round-trip delay periods. Coherent signal processing is desirable when high accuracy performance is necessary. It permits both the amplitude and the phase information in the reflected echo to be used in determining the target range and velocity. However, in systems where the accuracy requirements are modest, such as the DME application, noncoherent signal processing is sufficient. Frequency stability requirements can then be reduced even further. Noncoherent systems measure only the time delay of the return signal’s pulse envelope, as illustrated in Fig. 6. In these systems the frequency spacing of the rf channel usually dictates the frequency stability requirements. As mentioned in Subsection B, the conventional airborne radar is not an effective navigation aid because it cannot easily identify ground reference points. Position-fix navigation using beacons can easily identify the ground reference landmark via the channel plan using a unique radio frequency and/or pulse code. Because radar-to-beacon and beacon-to-radar transmissions are each one way, the signal power varies as the inverse square of the range, rather than as the inverse fourth power, as do ordinary radar echoes. This means that the range of a radar beacon may be doubled by increasing the radar transmitter power or the receiver sensitivity fourfold, whereas a 16-fold increase would be
22
ROBERT J. KELLY AND DANNY R. CUSICK
+ d = DISTANCE BETWEEN RADAR AND REFLECTOR c = VELOCITY OF LIGHT
At =
9 ;PULSE LEADING EDGE DELAY MEASUREMENT
TRANSMITTED PULSE ENVELOPE
ECHO OR
W E IS NON COHERENT RANGE SYSTEM (PULSE ENVELOPE)
COHERENT RADAR MEASURES BOTH AMPLITUDE AND PHASE TO OBTAIN TARGET RANGE AND RANGE RATE
MEASUREMENT
PHYSICAL REFLECTOR
RECEIVER
4 - - - - d d
-
MEASUREMENT RECEIVER
REFLECTOR (BEACON) TRANSMITTER
FIG.6. Two-way single path systems
required to double the range of ordinary echoes. Consequently, beacon systems are relatively modest in size and require less transmitter power than conventional radar. b. D M E : A Single-Path Two-way Navaid. For the above reasons, the DME is based upon the beacon principles. Ground transponders are used to mark and identify known positions on the ground so that the aircraft can determine its position relative to them. Such a system is cost-effective because the user requires only the purchase of the airborne unit, while the ground equipment is
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purchased and maintained by local, state, or federal governments. The system design permits low-cost airborne equipment having small weight and size for two reasons. One, the codes and different radio frequencies permit omnidirectional antennas, and two, the transmitter power requirements have to overcome only the inverse square of the range propagation loss as noted above. System costs are further reduced because the two-way, single-path system does not require a coherent reference. That is, the operational accuracy (0.2 nautical miles for DME/N and 0.017 nmi for the DME/P) can be achieved with simple noncoherent time-of-arrival “leading edge” measurements as indicated in Fig. 6. In addition, since the system does not have to be coherent, the uplink radio frequency can be different than the downlink without degrading the system, thus permitting a flexible channel plan while contributing to the low cost of the airborne equipment. Finally, the beacon concept permits the use of an omnidirectional antenna on the ground, allowing 360” azimuth coverage in the horizontal plane with low transmitter power at a sufficient data rate. Aircraft can then obtain range guidance at all points necessary for enroute, terminal, and landing applications. c. Radial Navaids. Radial systems used in the ATC are radio navaids providing bearing information. They generate a zero delay time by having coincident paths (same ground antenna). This means that the two path signals must be coded differently: A reference signal is required to establish zero time delay while a second signal provides the variable delay. Figure 7 illustrates the principle of the radial system used in VOR and TACAN. The rotating cardioid pattern generates a sinusoidal signal, which is a delayed version of the reference signal. In order to separate the reference from the signal, it must vary in time or have a different subcarrier modulation f ~ r m a t . ~ Radial systems used in ATC system are VOR, TACAN (bearing), ILS, and MLS (angle).The VOR and TACAN derive bearing by measuring phase delay between the reference and the signal, which are diverse in subcarrier modulation formats. In MLS (angle) this bearing information is proportional to a time delay. In ILS the signals are diverse in frequency, and bearing is derived from the amplitude differences of the two signals.
d. Radio Navaid Position-Fix Conjigurations. Radio navaids whose locus of points define a circle are called p systems, while those whose locus of points define radials are called 8 systems. The four systems used in the ATC system are summarized in Fig. 8 in terms of the p/8 navigation parameters. Not In terms of performance,it is noteworth that two-way systems have their reference signal in the aircraft. The reference clock (delay) is “turned on” with the interrogation transmissionand is “turned off” with the reply signal. The elapsed time on the clock is proportional to the range. Unlike radial systems, the reference element in two-way range systems is not corrupted by propogation effects and receiver noise.
24
ROBERT J. KELLY AND DANNY R. CUSICK NORTH
GROUND STATION RADIATES BOTH REFERENCE AND ROTATING CARD10
CARDIOID ANTENNA PATTERN ROTATES AT FREQUENCY f
INFORMATION SIGNAL (CARDIOID)
FIG.7. Principle of radial navaid for VOR and TACAN.
shown in Fig. 8 is the VOR/DME configuration, which is not widely implemented in the enroute application. The ATC enroute and terminal navaids can determine position fixes by the intersections of radials and circles (see Fig. 9). In the 2 D navigation application, the slant range from the ground station (DME or VOR) is projected onto a plane tangent to the earth. This plane is illustrated later in Fig. 10. Thus the lines of position shown in Fig. 9 are assumed to also lie in the tangent plane. In ILS, the position fixes are determined by the intersection of a sphere (DME), a vertical plane along the runway center line (localizer),and a lateral plane perpendicular to the localizer plane (glide path). For MLS, the point in space is the intersection of two cones (azimuth and elevation scanning
25
DISTANCE MEASURING EQUIPMENT IN AVIATION AIR -
DMEIP IIA)
DMEIP
GROUND NOTE: NOT SHOWN IS A VORlDME CONFIGURATION
FIG.8. Air-derived p/B navaids in the ATC system.
beams) and a sphere (DME/P). Wax (1981) demonstrates how several different position-fix techniques can be integrated into an algorithm to estimate an aircraft’s position. All the navigation aids described above are air-derived systems; that is, the relevant navigation parameter is calculated in the aircraft. The Secondary Surveillance Radar is an example of a ground-derived system; that is, the relevant information is derived on the ground and then transmitted to the aircraft via a communications link. The global positioning system (GPS) is a new position-fix system which provides three-dimensional position and velocity information to users anywhere in the world. Position fixes are based upon the measurement of transit time of rf signals from four of a total constellation of 18 satellites (Milliken and Zoller, 1978). The exact role that GPS will play in the future NAS has not yet been defined. One of the tasks of RTCA Special Committee 159 is to define that role (RTCA, 198%). As stated by the Department of Defense/Department of Transportation (DOD/DOT) (1984), the DOD intends to phase out its Air Force TACAN systems beginning in 1987. They will be replaced with GPS. 3. Additional Navigation System Considerations
The purpose of this section is to complete the general discussion of navigation principles by describing how a navigation system is “connected” to the aircraft to form a closed-loop control system. The central notion represents
26
ROBERT J. KELLY AND DANNY R. CUSICK
n VO R IDME OR TACAN
VOR/VOR
FIG.9. Position fixes determined by p / B intersection.
an aircraft in flight as a time-varying velocity vector. It has a direction in which it is going and a speed at which it is going there. The time integral of this velocity vector is its flight path. The navigation system’s role is to generate position and velocity corrections so that the aircraft’s velocity vector can be altered to follow the desired flight path in an accurate and timely manner. To continue the conceptual development of navigation principles further requires a coordinate system to reference the desired flight path and the navigation system’s output signal. Since the aircraft’s guidance system is useful only if the aircraft closely tracks the desired flight path, accuracy performance definitions are necessary to compare the AFCS’s tracking performance. This subsection briefly describes coordinate reference frames and accuracy considerations as they apply to a closed-loop navigation system. An aircraft in flight has a position and velocity at some altitude above the earth at some instant in time. In order to specify these aircraft states, a reference coordinate system needs to be defined. The subject of navigation coordinate frames is a vast and difficult topic and will not be pursued in any depth here. For example, seven reference systems, useful to navigation are defined in Chapter Two of Kayton and Fried (1969). One of these systems, geodetic spherical coordinates, has wide application in the mechanization of dead reckoning and radio navaid systems. It gives the aircraft lateral position in terms of longitude I and latitude CD on an ellipsoidal (earth), where I and @ are measured in degrees. The coordinate system is used
a. Coordinate System.
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27
in current RNAV equipment with the waypoints also being defined in A and (D. The vertical dimension is given by altitude h above the reference ellipsoid. See Appendix D of RTCA (1982). Navigation system designers assume that the earth‘s surface can be approximated by an ellipsoid of rotation around the earth’s spin axis. In this representation it is postulated that, when the earth cooled from its molten mass, its surface assumed contours of equipotential gravity. The resulting gravitation field g is very nearly perpendicular to the earth’s surface. It is called the “apparent” gravity and is the vector difference between the Newtonian field of gravitational attraction and the centrifugal forces due to the earth‘s rotation. The reference ellipsoid is chosen to minimize the mean-square deviation between the normal to the ellipsoid and g. In order to keep the narrative simple and to focus on the essential ideas, the normal to the ellipsoid will be assumed to lie along g (the direction that a “plumb b o b falls). Use of the geodetic coordinates which are given in degrees allows the lateral position (say A. and (Do) of the aircraft above the earth to have the same coordinates (Ao, (Do) on the earth’s surface. The aircraft’s altitude is its height above the reference ellipsoid as measured along the ellipsoid’s normal (i.e., approximately along g). Enroute navaids provide guidance so that the pilot can fly between two waypoints specified in terms of A and (D. For example, assume that the waypoints were at the same height. Steering commands would be generated by the navigation system to guide the aircraft along the geodetic bearing and distance. The aircraft’s trajectory would trace an arc following the earth’s curvature at a constant attitude, as shown in Fig. 10(a). For short-range navigation (50 to 100 nmi for enroute and 20 nmi for approach and landing), the lateral position can be referenced to Cartesian coordinates instead of to the geodetic coordinates. As a result both the calculations and the geometric picture of the flight path are greatly simplified. This is accomplished by using a tangent plane to approximate locally the ellipsoid, as shown in Fig. lqa). It can be shown that, for short-range navigation, the length of the aircraft’s ground track on the earth’s surface will not deviate significantly from the length of the flight path’s projection onto a plane tangent to the earth’s surface near the derived waypoints. At a distance of 50 nmi from the tangent point the difference in length is about 16 ft. Measuring the aircraft’s height as the perpendicular distance to the tangent plane would, however, deviate significantly in an operational sense from that height measured along the normal to the earth’s surface. At 50 nmi the altitude difference is about 1300 ft, which is unacceptable when compared with the lo00 ft aircraft separation distance defined for the ATC route structure. Thus, for short-range enroute and terminal area navigation, lateral guidance can be defined with respect to the tangent plane. However, the
ROBERT J. KELLY A N D DANNY R. CUSICK
28
A/C FLYING CONSTANT ALTITUDE
\ _c--
TANGENT PLANE
9
(a)
AIRCRAFT CM
a
i 9
FIG.10. Earth reference coordinate system.
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vertical guidance must be measured along the normal to the earth’s surface using, for example, a barometric altimeter. For enroute navigation it is the aircraft’s altitude directly above the earth‘s surface which is operationally important, not the aircraft’s height above the tangent plane reference point. In the final approach landing application the operationally significant coordinate system is referenced to the tangent plane which contains the runway. Thus the aircraft’s position can be given in xyz Cartesian coordinates. The decision height at the terminus of the final approach is measured with respect to the glide path intercept point (GPIP) on the runway (see Subsection D,4). The elevation angle glide path is chosen so that the aircraft will be safely above the hazardous obstacles along the approach path. This article, then, will define a very simple Cartesian coordinate system that permits description of the most essential concepts of navigation, guidance, and control. The Cartesian reference frame is defined at some point on the earth’s surface (e.g., airport runway). As shown in Fig. 10(b) this fixed Cartesian reference frame is called an earth-fixed reference. A tangent plane is defined through this point. Two of the Cartesian coordinates lie in the tangent plane, one of which lies parallel to the meridian (North). A third coordinate lies along the earth’s gravity vector and is normal to the tangent plane. It defines the aircraft altitude above the tangent plane. However, as long as the aircraft’s altitude is determined consistent with the enroute, terminal area, and approach/landing, then the earth-fixed reference will accurately describe the aircraft flight path. In other words, the tangent plane altitude calculation must be corrected for the earth’s curvature during enroute flight missions. The earth-fixed reference,just like the geodetic reference, permits the navigation problem to be separated into aircraft lateral positions and altitudes. Making a second assumption that the earth is fixed in space allows an inertial frame of reference to be defined on the earth’s surface; that is, unknown acceleration effects are neglected (McRuer et al., 1973). This inertial frame, which is accurate for relatively short-term navigation, guidance, and control analysis purposes, permits a description of aircraft motion in terms of inertial sensor^.^ It does have practical limitations for very long-term navigation (worldwide). With respect to this earth-fixed coordinate system, the aircraft’s center-ofmass (c.m.) can now be specified by a position vector with components (Xa,x,Za).In addition, three orthogonal axes exist within the aircraft. They are the pitch, roll, and heading axes of the airframe. The origin of this bodyfixed triad “sits” on the tip of the aircraft position vector, which is coincident with the aircraft’s c.m. Aircraft equipped with INS can establish a local vertical using Schuler tuning (Schneider, 1959), thereby forming the required inertial reference. Accurate position data can then be obtained from the platform stable accelerometers.
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ROBERT J. KELLY AND DANNY R. CUSICK
The attitude of the aircraft is defined by the Euler angles (a, /?,and y ) through which the earth-fixed axis must be rotated to be coincident with the airframe body-fix axis. Understanding aircraft navigation techniques cannot be confined to just the motion of the aircraft’s c.m. Attitude must also be considered, because the flight path traced out by the motion of the aircraft c.m. is controlled by changes in the aircraft’s heading, roll, and pitch. The navaid guidance signal is also referenced to the earth-fixed system so that the steering commands will be in the correct form to modify the aircraft’s velocity vector when necessary. Subsection E will show that the AFCS for conventional fixed-wing aircraft has two orthogonal channels: a lateral channel and a longitudinal (vertical) channel. Since the earth-fixed reference frame is an orthogonal system, as described above, the lateral navigation problem can be separated from the vertical navigation problem as a consequence of this simplification. This is an important distinction because later it will be shown that the lateral enroute navigation control laws are similar in form to the lateral (azimuth) approach/landing control laws. With this simplifying division of the navigation problem, the narrative and the examples given in the balance of this article will be selected with respect to the lateral navigation application, wherein the aircraft’s altitude is assumed to be constant. When the aircraft’s altitude and TRUE N O R T H
4
= HEADING
e= BEARING SIDE SLIP ANGLE = 0
FIG.11. Definition of true track angle during coordinated flight.
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time are included as steering commands, the navigation problem becomes a 3D or a 4D problem, respectively. The following terms-bearing, heading, ground track velocity, and track angle-are important for the understanding of lateral navigation discussions. As shown in Fig. 11,the bearing 8 is the angular position of the aircraft c.m. to its intended destination as measured clockwise from a reference such as true North. Track angle tjT is the true angular direction of the aircraft’s ground velocity. The heading tj is the angular direction that the longitudinal axis of the aircraft is pointing. Air speed is the speed of the aircraft in the direction of its heading through the air mass. From these definitions, three distinctions in the aircraft flight path can be given. If range to a destination can also be determined, then a desired flight path (course) can be defined as a connected time series of intended bearing and range points. The aircraft’s actual flight path would be a connected time series of true track angles and true ranges, while the indicated flight path is a time series of range and track angle measurements. As part of a recurring theme in this article, track angle or ground velocity is the critical parameter for enroute navigation. When accurate navigation is necessary, it can be determined from a series of position measurements or it can be derived from accelerometers on board the a i r ~ r a f t Guidance .~ and control are simply the corrections of the velocity vector in such a manner that the aircraft follows its intended flight path. Clearly, if air speed and heading are known, then, as indicated in Fig. 11, a simple but less accurate method is available; one uses the wind vector and the air speed vector to obtain the ground speed velocity.
b. The Aircraft Flight Control/Guidance System. The task of a navigation system is not only to derive the aircraft’s position, but also to generate commands to change its velocity vector. In order to accomplish these changes, its output signals are fed into the aircraft flight control system, thereby forming a “closed loop” composed of the aircraft and the navigation system. This “closed loop” is called the aircraft flight control/guidance system. If the navigation system correction signals are sent to a display, then the loop is
’
The central concept of INS consists of a platform suspended by a gimbal structure that allows three degrees of rotational freedom (Maybeck, 1979).The outermost gimbal is attached to the body of the aircraft so that the aircraft can undergo any change in angular orientation while maintaining the platform fixed with respect to, for example, the earth-fixed coordinate frame. Gyros on the platform maintain a desired platform orientation regardless of the orientation of the outermost gimbal. Thus, the platform remains aligned with respect to a known reference coordinate system. Accelerometers on this stabilized platform can then provide aircraft acceleration with respect to that known set of reference coordinates. After local gravity is subtracted, the resulting signals can be integrated to yield aircraft velocity and acceleration. An alternative to the gimbaled implementation is the strap-down inertial system (Huddle, 1983).
32
ROBERT J. KELLY AND DANNY R. CUSICK
-
(a) MANUAL-COUPLED FLIGHT
-
1
AIRCRAFT POSIT1ON
AIRCRAFT A
COURSE DEVIATIONS ~
NAVIGATION SYSTEM
.
DESIRED FLIGHT PATH
NAVAID SENSOR ENROUTE DME VOR APPROACH & LANDING I LS M LS
(b) AUTOCOUPLED FLIGHT AIRCRAFT POSITION STEERING
DESIRED FLIGHT PATH
FIG.12. Principal aircraft flight control/guidancesystems.
closed through the eyes and hands of the pilot, as shown in Fig. 12(a). When the pilot closes the loop, the configuration is called a manual-coupled flight control system (FCS). Similarly, if the guidance signal is sent to an autopilot, then the configuration is called an auto-coupled FCS, as shown in Fig. 12(b). Guidance then involves the instrumentation and control of the six aircraft coordinates. For the 2D problem, the guidance system provides corrections to the aircraft’s velocity vector by changing its bank angle and heading, such that the aircraft’s c.m. follows the desired 2D flight path. The aircraft’s c.m. velocity is called the ground track velocity. When altitude corrections are included, the aircraft velocity is a 3D vector composed of the ground track velocity plus its vertical speed. The navigation system outputs for the autocoupled 2D enroute application are lateral steering commands. For manual-coupled flight, the visual display will present bearing and range guidance. To maintain the intended
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flight path the guidance corrections are displayed as (1) range along the straight-line segments between selected waypoints, and (2) bearing or crosscourse deviations (left/right) normal to the straight-line segments. The guidance correction signals for autocoupled flight are steering commands expressed in terms of cross-course deviations. The cross-course corrections are coupled to the lateral channel of the autopilot. Some aircraft are instrumented to adjust their power (thrust) to maintain a range schedule along the flight path. For the straight-in final approach and landing application, the aircraft velocity vector lies in and points down an inclined plane called the glide path. The navigation system, or more accurately, the landing guidance system, shown in Fig. 12, is now an ILS or MLS. Both use DME and both are described in Subsection D,4. For manual flight on final approach, landing guidance is displayed as left/right indications on the course deviation indicator (CDI),which are crosscourse deviations about the extended runway center line. Vertical deviations about the inclined plane are displayed on up/down indications on the CDI. The CDI is also used in enroute applications. Lateral steering corrections in autocoupled flights are sent to the lateral channel of the autopilot, and vertical corrections are sent to the autopilot’s longitudinal or vertical channel. Fixed-wing, conventional take-off and land aircraft will be considered in the closed-loop aircraft/flight control system discussions and analyses. The other major aircraft types, short takeoff and land (STOL) and vertical takeoff and land (VTOL), will not be considered. However, the ATC navaids summarized in Subsection D (especially MLS and its DME/P) can and do provide navigation guidance to both the STOL and VTOL aircraft types. c. Navaid Accuracy Considerations. In navigation, the accuracy of an estimated or measured position of an aircraft at a given time is the degree of conformance of that position with the true position of the aircraft at that time. Because of its statistical basis, a complete discussion of accuracy must include the following summary of the statistical assumption made and the measuring techniques used. A navigation system’s performance must be evaluated in the context of the total system, aircraft plus navigation system. Several measures of a navigation system’s performance require that the aircraft flight path remain inside its designated air lanes, and that the pilotage error of the aircraft about the indicated navigation signal as well as its attitude be maintained within prescribed limits. Keeping the aircraft within its air lane is directly related to the navigation system’s bias accuracy; while maintaining a small tracking error and prescribed attitude are direct functions of the closed-loop systems’ bandwidth, autopilot gain, and the navigation system’s noise error. Understanding the performance of the closed-loop guidance system requires an
34
ROBERT J. KELLY A N D DANNY R. CUSICK
understanding of the navigation system’s error mechanisms. The navigation system, in turn, is a configuration of navaid sensors, a computer, and output signals that drive an autopilot or a cockpit display, as shown in Fig. 4. Determining the performance of the navigation system, therefore, also entails determining the performance of the navaid. This section discusses the necessary considerations. The navaid sensor measures some physical phenomena such as time delay of the electromagnetic radiation. As indicated in Fig. 4, the sensor measurements are then transformed to a coordinate system where they are combined by the navigation system to estimate the aircraft’s position and velocity vector. The aircraft’s position, as determined by the navigation system, will differ from its true position due to errors in the sensor measurements, assumptions in the earth coordinate system reference model, errors in computation, and, finally, departures in the mathematical model implemented and the real world. This difference is called the RNAV system error, as shown in Fig. 13. The figure also indicates an additional error, namely the flight technical error, which combines with the previously addressed Navaid system error to produce the total system error or tracking error. The flight technical error (or autopilot error for autocoupled flight) is the difference between the indicated flight path and the desired flight path. The “flight technical error” (FTE) refers to the accuracy with which the pilot controls the aircraft as measured by his success in causing the indicated aircraft position to match the indicated command or desired position on the display. FTE error will vary widely, depending on such factors as pilot experience, pilot workload, fatigue, and motivation. Blunder errors are gross errors in human judgment or attentiveness that cause the pilot to stray significantly from his navigation flight plan, and are not included in the navigation system error budget. Blunder tendency is, however, an important system design consideration; their effects are monitored by the ATC AIC TRUE /FLIGHT PATH
A/C TRUE
/ POSITION
ERROR
ERROR
OESl RE D COURSE
RNAV ROUTE WIDTH
RNAV FLIGHT PATH
WHERE RNAV SAYS A/C IS
FLIGHT TECHNICAL ERROR
DESIRE0 A/C POSIT ION
FIG. 13. RNAV position error.
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controller. Roscoe (1973) discusses flight test results which attempted to measure flight technical errors and blunder distributions. To complete the picture, the total system error (tracking error) consists of the navigation system error and the flight technical error; it is the difference between the aircraft’s actual position and the desired flight path. An example of a navigation system error budget is given in Table V and illustrated in Fig. 40 in Subsection F. A navigation system can be the sophisticated configuration of sensors, as shown in Fig. 4, or it can be simply the navaid measurements themselves, as in VOR/DME, where bearing and range are directly displayed to the pilot. As discussed previously, the intended aircraft’s position (c.m.)as well as its measured position are determined relative to the earth-fixed coordinate system. The position errors can be defined in at least two differentways: They can be referenced directly to the earth-fixed coordinates or they can be referenced to the aircraft’s intended flight path, which, in turn, is referenced to the earth-fixed system. To keep the discussion focused on the important concepts, only the two-dimensional plane used in lateral navigation will be treated. (The vertical component, altitude, is simply a one-dimensional measurement, orthogonal to the lateral plane.) In the first method, the error components are traditionally defined in terms of a location error (bias) and a dispersion error (variance). When the error components are defined along the coordinates of a Cartesian system, the position error dispersion (variance) is given in terms of the circular error probability (CEP). The CEP is the radius of a circle such that the probability is 0.5 that an indicated position will lie within the circle (Burt, et al., 1965). The center of the circle is chosen at the center of mass of the probability distribution, which is the desired destination point of the aircraft. For a biased system, the center of the circle will be displaced from the aircraft’s desired destination. The second method measures 2D navigation system performance in terms of its error about the intended flight path. It is the method used in this article to evaluate the accuracy performance of navigation systems. As shown in Fig. 13, the error normal to the indicated flight path is called the cross-track (XTRK) error, while the error along the indicated flight path is called the along-track (ATRK) error. These error components are thus total system errors or tracking errors. Referencing the aircraft’s true position against the desired flight path is the natural method of specifyingnavigation errors. The cross-track error must be consistent with the width of the enroute air lanes or RNAV routes, while the along-track error must not degrade the aircraft longitudinal separation criteria from other aircraft. This cross-track/along-track theme is also present in the straight-in approach and landing application where, for example, the
36
ROBERT J. KELLY AND DANNY R. CUSICK
cross-track errors are the lateral azimuth errors. In this case, the allowed errors must be consistent with the runway width, while the along-track errors must be consistent with the runway touchdown zone. In the simple VOR/DME navigation application, the VOR error is the cross-track error, while the DME error is the along-track error. The aircraft path deviations-cross-track (lateral), along-track, and vertical-are assumed to be orthogonal random variables. Since most of the operational accuracy questions in this article address only one of these components, the statistical problem is one dimensional. For example, the 2D navigation application usually addresses only the cross-track corrections. Therefore, the 95% confidence limits are one-dimensional statistics using the variance and probability distribution of the cross-track deviation measurement. On the other hand, when analyzing a 3D navigation flight path, additional considerations arise because the error surface can be envisioned as a tube in space. In this case, the 95% probability error contour corresponding to the tube’s cross section is determined using a bivariate distribution of the lateral and vertical flight path deviations. [See Table 2.1 in Mertikal et al. (198511. To summarize the tracking error or total system accuracy is a combination of the autopilot accuracy and the navigation system accuracy. The accuracy performance of navigation systems is specified in terms of 95% probability. This means that for a single trip along a defined flight path (e.g., between two waypoints) 95% of all position measurements must lie within the specified XTRK error allowance. This implies that out of 100 flights, 95 of them must satisfy the specified XTRK error allowance. In other words, the measurement defines a confidence interval and its confidence limits are compared to the accuracy standard. It is a confidence interval because the desired flight path is an unknown constant parameter which the navaid is estimating. The computed 95% confidence interval is a random variable which covers the desired flight path 95 out of a hundred times. The confidence interval is used to evaluate one-dimensional guidance signals (e.g., XTRK deviation). Discussion will now center on the navaid error components which are a subset of the navigation system components. The navaid sensor measurement differs from the true navigation parameter due to variations to the phenomena (e.g., time delay errors induced by multipath) and due to errors in measurement and computations (instrumentation errors).Navaid errors fall in three separate categories: outliers, systematic, and random. Outliers are the results of miscalculations, electrical power transients, etc., which can be detected by comparing a new measurement with respect to the three or four standard deviation values of past position estimates (see Section IV,D). Systematic errors obey some deterministic relation, such as a circuit temperature dependency, or are the result of faulty calibration procedures. They
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usually can be removed when the deterministic relation is known or the calibration procedure is corrected. In other words, if an ensemble of navigation instruments were known to produce nonzero average errors, simple recalibration or computation could nullify these effects. No generality is lost, therefore, in assuming zero mean ensemble errors. Random errors are unpredictable in magnitude and they are best described by statistical methods. It is these errors which are addressed by the navaid accuracy performance standards. Other accuracy types, such as predictable, repeatable, and relative accuracy are defined in DOD/DOT (1984). Predictable and repeatable accuracies are included in the error components described next and in Subsection F. Relative accuracy does not apply to the applications addressed in this article. In summary,it is assumed that both the outliers and systematicerrors have been removed and that only the random errors remain. This means that although any particular navaid can have a bias AxB,the average bias of the ensemble of navaids is zero, i.e., E[AxB] = 0, where E is the expectation operator. Further, each navaid has an equipment variance (noise)component a; that is assumed to be identical for each member of the class of navaids. Therefore, as with the navigation system as a whole, the navaid errors may be specified in terms of bias and noise (variance). The assumptions above are important, and they are discussed further in the next section. Once equipment errors have been accounted for, the only remaining major error component is site-dependent errors such as multipath. Unlike equipment noise errors, which vary with time but are essentially independent of aircraft motion or position, site-dependent errors remain unchanging for long periods of time, while exhibiting extreme variations as the aircraft changes position. Thus site-dependent errors are said to be spatially distributed. A time-varying error is generated when the aircraft flies through the spatially distributed interference fields. It is implicitly assumed that the variance component associated with site-dependent errors and with aircraft motion is detected by all radio navaids having simple signal processing algorithms. The degree to which this variance component is reduced depends upon the level of output data filtering applied.
d. Error Budget Considerations. The purpose of an error budget is to allow facility planning by local, state, and federal governments, including the obstacle clearance needed to define enroute airways, terminal approaches, and runway clearance areas. They are also used to prepare equipment procurement specifications.Error budgets must be simple if they are to be useful. Also they must be simple because there usually is no database to substantiate anything more than simple statistical procedures. It is for this reason that the root-sum-square (RSS) calculation procedure is used extensively throughout
38
ROBERT J. KELLY AND DANNY R. CUSICK
the navaid industry to estimate system performance. The purpose of this section is to identify the assumptions upon which RSS calculations are valid. Even though these assumptions may only be partially justified, the RSS procedure is applied anyway because there is no other alternative short of directly adding the maximum excursion of the error components. Direct addition of the error components is not statistically or economically defensible. The true navaid system accuracy lies somewhere between these two calculation procedures. The problem is how to use ensemble statistics which were obtained by averaging over all terminal areas, equipment classes, and environmental conditions to predict the performance of a given aircraft on a single mission to a given runway. An estimate of the total system error is the mean square error (MSE) or second moment. If uncorrelated ensemble statistics can be used and the ensemble average of each error component is zero, then the value of each component in the error budget (including the bias errors) can be given by their respective variance. Under these conditions, the MSE can be easily calculated by simply summing the variances and extracting the positive square root. The MSE is thus the variance for the total system. In the literature, this calculation is known as the root-sum-square. The difficulty with the RSS approach is that the random variables which represent the slowly changing drift errors may be correlated during flight missions to the same airport. To understand the problem, consider the error due to temperature effects. The equipment design engineer can determine the maximum temperature drift errors from the specified operating temperature range. Civil navigation equipments are usually designed to operate over temperature ranges between 70 and - 50 "C. These excursions usually have daily and seasonal cycles whose average values would be nearly zero when the equipments are calibrated for the average temperature of a geographic area. This means that, although the drift errors are a nonstationary process, they will have a near zero average over a long time interval. This point can be used to advantage because it implies that repeated flights by a given aircraft to the same runway (for example) will at worst only be partially correlated. In any event, the ensemble average over all conditions and missions will be nearly zero. Although the value of the ensemble variance is less than the peak drift error, it is common practice to use the peak drift value for the drift error variance component. Thus the error budgets for some geographic locations may be pessimistic when the ensemble variance of the temperature drift errors are added with the other error component variances. There is generally no statistical correlation computation problem with the noiselike errors. Physically the bias error component and the noise errors are associated with separate error source mechanisms. They can be viewed in the frequency
DISTANCE MEASURING EQUIPMENT IN AVIATION
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domain as separate spectral components and thus as separate random variables. The low-frequency component when grouped together are called the bias error, while the sum of the high-frequency components is called the noise error. Both spectral groups, bias and noise, are defined as random variables having zero means and variances equal to otiasand 0.; In principle, each flight mission can be considered as a statistical experiment where each biaslike error component AxB is selected from separate batches, each having a zero ensemble mean E [ A x B ]and variance otias.Before each flight mission, one error component is drawn from each batch and is directly added with samples from the noise component (a;) batches. The result should, with high probability, be within the total specified system error allowance. In the next section the discussions above will be further refined to include quantitative definitions of the time average, ensemble averages, and the RSS calculations. e. Statistical Dejnitions and Root-Sum-Square Applications. Navaid output data can be viewed as a combination of a static or quasi-time-invariant component (bias) and a dynamic or fluctuating component (variance). The dynamic component or noise term can be further partitioned into spectral components, corresponding to errors, which (1) aircraft can follow and are called scalloping course errors or “bends”; (2) can affect the aircraft’s attitude; or (3) have no effect on the AFCS. (The degree of data filtering performed determines whether the noise component of a navaid disturbs the aircraft position or its attitude or both.) DME/P and MLS angle guidance systems define the noise error components in terms of these spectral components. ILS defines the course errors in terms of bias and noise with no distinction made between the spectral groups comprising the noise. Most navaids, such as VDR, DME/N, and TACAN, relax the definition error further and make no distinction between bias and noise. For these navaids the accuracy is described by a single value equivalent to the mean square error. The MSE describes in a rudimentary way the general intensity of random data. Before defining MSE, two types of averaging processes, time averages and ensemble averages, will be defined. The accuracy performance of a class of navaids is determined by first calculating the time average on one member of the class. To do this M time sample measurements x ( t i )are taken over a short period of 10 to 40 seconds, where i = 1, 2,. . .,M. From these measurements, M error terms A x ( t i )are calculated. Using the M error terms, the time-average sample error AX^, the time sample variance S i , and the time sample mean square error
40
ROBERT J. KELLY AND DANNY R. CUSICK
are determined. Figure 14(a) illustrates these concepts for a series of range measurements from a s i n g l a M E interrogator/transponder combination. The positive square E is called the RMS error (Bendat and Piersol, 1971) and is = ,/when the measurements are uncorrelated. To obtain the ensemble average of the short interval time averages, time measurements are repeated on (N - 1)additional units. The ensemble average is then determined from the N sets of measurements. Let
where j is the jth navaid and the bar indicates the ensemble average calculations. The sample variance of the bias measurements is I
N
In this discussion, both N and M are sufficiently large that the sampling errors are negligibleand thus represent the population statistics for a given type of 2 navaid, i.e., AxB N E[AxBj], S;,, 2: dbiasr S i s' o i , and MSE = E[Axs]. Further, it is assumed that all the bias measurements are uncorrelated, i.e., E[AXB~,AXB~] = E[AXBj12. AS assumed above, E[AxBj] = 0 and the noise component of each navaid in the ensemble is equal, that is, a i j = dj?,k = o i . Therefore, with these assumptions and as shown in Fig. 14(b), the ensemble mean-square error is simply MSE = nj?,+ otias.Its positive square root is defined as the root sum square, RSS = ,/=. The distinction in this article between RMS and RSS is that when E[AxBj] = 0, the RSS is only a sum of variances. With the above-defined measurement procedures, it is now possible to illustrate how both a navaid error budget and a navigation system error budget are constructed using the RSS procedures. Let A X AR, and A F be different error components of a navigation system such as RNAV. The XTRK component of the sensor error could be represented by AYwhile AR and A F could represent the RNAV computation error and the XTRK flight technical error, respectively. Assuming a linear model, the system error AS is given by AS = A Y + AR + AF. Assume further that these error components are zeromean, uncorrelated random variables; then the ensemble mean square is
BIAS
NOISE
which itself is the standard The total system error is simply the RSS = deviation for the total system as indicated in Fig. 14(b).
41
DISTANCE MEASURING EQUIPMENT IN AVIATION X AXIS PROBABILITY DENSITY FUNCTION
b
DME MEASUREMENTS
ARIANCE (NOISE)
DME GROUND STATION
@
2 r
Y AXIS MEASUREMENTS ON A SINGLE INTERROGATOR /TRANSPON DER COMBINATION
(a) TIME AVERAGE MEASUREMENT
E [ A X j ] = O MSE = E [ A X j
=
1.2 '
' ' ' '
N NAVAID UNITS
DME ENSEMBLE BIAS ERRORS
j
=
f]=
UN2+ UBfAs
1, 2 ' . ' ' . N NAVAID UNITS
RSS = DME ENSEMBLE MEAN SOUARE ERRORS
(b) ENSEMBLE AVERAGE MEASUREMENT
FIG.14. Range error definitions.
42
ROBERT J. KELLY AND DANNY R. CUSICK
For the DME/N, VOR, and TACAN area navigation systems, the total system error is the single number, RSS = (a: + ... + a;)”’, where a:, a:, . ..,a: are the individual variances for each of the P error components. In ILS system error analysis, the RSS of the bias variance components are calculated separately from the RSS of the noise variance components, as indicated in Table 111. For DME/P and the MLS angle guidance, the error components are defined in Sections E,7 and IV,C,1 and in Table 111. Combining error components on an RSS basis is almost universally adopted in the literature (RTCA, 1984, 1985a; DOD/DOT, 1984). They implicitly assume that the error components satisfy a linear model and that they are uncorrelated, zero-mean, random variables. The linear model assumption further implies that there is no coupling between the navigation system error (NSE) and the flight technical error (FTE). In fact, there is a small nonlinearity between these terms because of the NSE changes with aircraft position, which, in turn, is a function of the FTE. This nonlinearity is negligible, and the NSE and FTE can be combined on an RSS basis if the pilot has no a priori knowledge of the NSE (e.g., severe VOR course bends). The validity of the uncorrelated error component assumption depends upon several factors. Clearly the error components are uncorrelated over the ensemble of all flight routes. For repeated flights over the same route the sitedependent errors may be correlated while the equipment bias errors depend upon the nature of the drift mechanism. Bias errors are assumed to be constant over the duration of a single mission, however. In any event for a given facility installation the site and equipment are selected and configured such that the system accuracy for a single mission is achieved. In addition, the FAA periodically has flight inspections to check a facility’s performance. Since the exceptional sites are treated on an individual basis to ensure flight safety, and since the statistical distributions are not known anyway, the uncorrelated assumption is retained because of the simplicity of the RSS calculation. Simple addition of the error components would unrealistically limit the approach path and skew the facility implementation economics. Error budgets are defined in terms of 95% confidence limits, rather than 20 limits, because the two specifications are essentially equivalent only for Gaussian random processes. The probability that a given event will occur is the important quantity not the 20 value. The probability of exceeding specified limits is the only meaningful measure by which air-lane route widths, decision windows, touchdown foot prints, and obstacle clearance surfaces can be operationally defined. Although each error component can be given in terms of a sample time variance, S i , the calculation of the 95% confidence interval requires knowledge of the underlying probability distribution function which is, in general, not known. In practice this lack of knowledge is circumvented by
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estimating the confidence intervals using the 2.5 and 97.5 percentile limits as determined by the measured data. The overall system error is then determined by the RSS of the 95 percentile value of each error component. The confidence limits of the combined measured error samples is then compared to the appropriate accuracy standard. Figure 83 illustrates the measurement methodology for the DME/P. In particular, it shows how the 95% confidence limits are determined from the system flight test results and the individual navaid component test results. Caution should be exercised when invoking the central limit theorem to approximate the total navigation system error distribution. [See Fig. 2 in Hsu (1979).] Based upon the above definition, it should be emphasized that the term “bias with 95% probability” means that, over the ensemble of navaids, 95% of the equipment bias measurements lie between the 2.5 and 97.5 percentile limits. The specified bias error value is the upper and lower percentile limits, as noted in Fig. 14. Figures 43 and 44 in Subsection F show how errors are combined to yield position-fix RSS accuracy estimates. All error components defined in Sections I11 and IV of this article are 95% confidence level specifications. One operational basis for using the single RSS value is that guidance obtained from enroute navigation sensors for manual flight is usually filtered extensively by the flight control system. Errors varying at a very high frequency are readily filtered out in the aircraft equipment, leaving only lowfrequency error components. Since this error component can be tracked by the aircraft, it can, when combined with the bias component, displace the aircraft from its intended flight path. Thus, the single RSS value is a useful measure of how well the aircraft will remain in its designated air lane. This rationale does not apply to autocoupled approaches. Biaslike errors again must be controlled so that the aircraft will remain in its air lane. The variance of the noise errors cannot always be reduced by heavy filtering if certain AFCS stability margins are to be maintained. Therefore, the unfiltered portion of the noise may induce aircraft pitch and roll motions (control activity) that may be unacceptable to the pilot (see Subsection E). As noted earlier, the single RSS specification is not adequate for the approach and landing application. Two numbers are defined for the ILS accuracy standard, the mean course error, crbias, and the “beam bends” noise, crN. The noise term is specified separately to limit the control activity and to limit, for example, lateral velocity errors away from runway centerline. (Note that an ILS localizer or MLS azimuth bias error always directs the aircraft toward the runway centerline, whereas a low-frequency noise error component may direct the aircraft away from runway centerline.) With the advent of the MLS, accuracy standards made a significant departure by expanding the traditional notions of bias and noise to include the spectral content of the navigation signal. With these new definitions it is
44
ROBERT J. KELLY AND DANNY R. CUSICK
intended that the MLS angle and DME/P accuracy standards will more accurately represent the qualities needed to achieve successful landings. The lateral error components for the enroute, terminal, and landing phases of flight are summarized in Table 111, Subsection D,2. The table indicates the increasing level of sophistication in the error component definitions. Clearly, the magnitude, nature, spectral content, and distribution of errors as a function of time, terrain, aircraft type, aircraft maneuvers, and other factors must be considered. The evaluation of errors is a complex process, and the comparison of systems based upon a single error number could sometimes be misleading. As stated earlier, the purpose of an error budget is to estimate the accuracy performance of a proposed navigation system hardware (and software) implementation. The idea is to select the 95% confidence interval for each component of the navigation system, such that the RSS of the total combination is within the 95% probability error limits specified for a given application. If the assumed conditions are correct, then there is high confidence that 95% of the aircraft’s position measurements will be within the allowed error limits during a specified segment of the aircraft’s Bight mission.
4 . Summary The basic notions of a navigation system were defined with particular emphasis on the DME navaid. DME, a position-fix navaid, was contrasted with dead reckoning techniques. DME can be used in a navigation system in two ways. If it is the primary method of navigation, then its data may require inertial aiding so that its guidance corrections do not affect the AFCS dynamic response. However, if dead reckoning is the primary navigation aid, then DME may assist it by periodically correcting its drift errors. It was shown that the navigation parameters follow a common theme whether the application is enroute navigation or aircraft approach and landing guidance. The desired flight path and indicated flight path coordinates can be defined for both applications with DME playing a central role. Error components for the approach and landing application are smaller and are more refined than in the enroute navigation application. D. The Air TrafJic Control System and Its Navigational Aids
This section describes the principal elements of the Air Traffic Control System, thereby helping to illuminate the operational role of DME in the context of the enroute, approach, and landing phases of flight.
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1. Goals of the National Airspace System
The National Airspace System (NAS) is a large and complex network of airports, airways, and air traffic control facilities that exists to support the commercial,private, and military use of aircraft in the United States [Office of Technology Assessment (OTA), 19823. The NAS is designed and operated to accomplish three goals with respect to civil aviation: (1) safety of flight; (2) expeditious movement of aircraft; and (3) efficient operation.
These goals are related hierarchically with safety of flight as the primary concern. The use of airport facilities, the design and operation of the ATC system, the flight rules and procedures employed, and the conduct of operations are all guided by the principle that safety is the first consideration. In the U.S., the FAA has defined a single nationwide airway system shared by all users of the NAS based on strategically located elements of VOR, DME, and TACAN navaids (Fig. 15). These facilities are geographically placed so as to provide for continuous navigation information along the defined airways. The airways serve as standard routes to and from airports throughout the U.S. DME provides the means to determine the position of the aircraft along a given route and to aid in verifying arrival at intersecting airways. The airways are divided into two flight levels. Below 18,000 feet mean sea level (MSL), the airways that run from station to station make up the system of VOR, or “VICTOR,” airways. These airways can be successfully navigated utilizing VOR bearing information only. The use of DME is not required but
FIG.15. Depiction of how air routes are constructed from VOR (and TACAN) defined radials.
46
ROBERT J. KELLY AND DANNY R. CUSICK
I
FIG.16. Excerpt from low-altitude enroute chart depicting the VICTOR airway system in Southern Virginia.
certainly simplifies the process of position determination along an airway. VICTOR airways are depicted on low-altitude navigation charts (Fig. 16), marked with a letter V and a number, e.g., V23. VOR stations in these airways are located an average of 70 miles apart. The high-altitude routes, or jet routes, are defined for navigation at altitudes of 18,000 to 45,000 ft and are utilized by high-performance turbine powered aircraft. These routes are identified on high-altitude navigation charts by the letter J followed by a number, e.g., J105. Unlike the VICTOR airways, DME, or its equivalent, is required for navigation along the jet routes. In addition to the VICTOR and JET airways the FAA (Federal Aviation Administration, 1983d) recently authorized random route using area navigation (RNAV). VOR, DME, and TACAN derived position information is also utilized for terminal area navigation (Fig. 17).
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VORlDME or TACAN 1 RWY 14
FIG. 17. A terminal instrument approach procedure utilizing a DME defined arc with position fixes along the arc defined by VOR radial intersections. Missed approach procedure (to BOAST intersection)is defined by a DME arc.
The FAA provides planning and advisory services to guide the aviator in making use of the NAS under either of two basic sets of rules-visual flight rules (VFR) and instrument flight rules (1FR)-which govern the movement of all aircraft in the United States. Similar visual and instrument flight rules are in force in foreign countries that are members of ICAO. In many cases, ICAO rules are patterned on the US.model. In general, a pilot choosing to fly VFR may navigate by any means available to him; visible landmarks, dead reckoning, electronic aids such as the VORTAC system, or self-contained systems on board the aircraft. If he intends to fly at altitudes below 18,000 ft, he need not file a flight plan or follow prescribed VOR airways, although many pilots do both for reasons of convenience and safety. The basic responsibility for avoiding other aircraft rests with the pilot, who must rely on visual observation and alertness-the “see-and-avoid” principle.
48
ROBERT J. KELLY A N D DANNY R. CUSICK
In conditions of poor visibility or at altitudes above 18,000 ft, pilots must fly under IFR. Many also choose to fly IFR in good visibility because they feel it affords a higher level of safety and access to a wider range of ATC services. Under IFR, the pilot navigates the aircraft by referring to cockpit instruments (e.g., VOR and DME for enroute flights) and by following instructions from air traffic controllers. The pilot is still responsible for seeing and avoiding VFR traffic when visibility permits, but the ATC system will provide separation assurance from other IFR aircraft and, to the extent practical, alert the IFR pilot to threatening VFR aircraft. The distinction between VFR and IFR is basic to ATC and to the safe and efficient use of airspace since it not only defines the services provided to airmen, but also structures the airspace according to pilot qualifications and the equipment the aircraft must carry. Much of the airspace below 18,000 ft is controlled, but both VFR and IFR flights are permitted. The altitudes between 18,000 and 60,000ft are designated as positive control airspace; flights at these levels must have an approved IFR flight plan and be under control of an ATC facility. Moreover, the aircraft must be equipped with VOR and DME in order to fly the VORTAC routes or other approved navigation aids if using RNAV. Airspace above 60,000 ft is rarely used by any but military aircraft. The airspace around and above the busiest airports is designated as a terminal control area (TCA), in which only transponder-equipped aircraft with specific clearances may operate regardless of whether using VFR or IFR. All airports with towers have controlled airspace to regulate traffic movement. At small airports without towers, all aircraft operate by the see-and-avoid principle, except under instrument meteorological conditions. Figure 18 is a schematic representation of the U.S. airspace structure. Aircraft flying under IFR, on the other hand, are required to have a radio and avionics equipment allowing them to communicate with all ATC facilities that will handle the flight from origin to destination. Figure 19 illustrates a hypothetical VORTAC and RNAV route from takeoff to touchdown. 2. The ATC System
The third major part of the National Airspace System offers three basic forms of service: navigation aids (including landing), flight planning and inflight advisory information, and air traffic control. The essential feature of air traffic control service to airspace users is separation from other aircraft. The need for this service derives from the simple fact that under IFR conditions the pilot may not be able to see other aircraft in the surrounding airspace and will therefore need assistance to maintain safe separation. Figure 20 represents the four functional elements of the ICAO
z=
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4
60,OOOFT.
A
(FL600)
:ONTINENTAL CONTROL AREA
t
49
45,000 FT. (FL450)
TRANSPONDER WITH ALTITUDE ENCODING PoS'TIVE REQUIRED
18,000 FT. MSL
14.500 FT MSL
FIG.18. (FAA).
Airspace structure. AGL,above ground level; MSL, mean sea level; FL,flight level
standardized air traffic control system: radar, VHF/UHF communications, enroute navigational aids, and landing aids. The ATC navigation service is, of course, a subject of interest in this article because it intimately involves DME. In the U.S. the Department of Transportation (DOT), through the Federal Aviation Administration (FAA), operates these four systems plus a primary radar system to fulfill its statutory responsibilities for airspace management. Controllers use these systems to provide navigation services, separation between aircraft, and ground-proximity warnings. Pilots require these systems for navigation and to receive ATC services. Specifications for these systems
50
ROBERT J. KELLY AND DANNY R. CUSICK
ZNM SECONDARY AREA
4NM
I
t
IAWP VORIDME INWP
FAWP
IAWP = I N I T I A L APF’ROACH WAVPOINT INWP = INTERMEDIATE APPROACH WAVPOlNl FAWP = F I X E D APPROACH WAYPOINT MAWP = MISSED APPROACH WAVPOINT
0
RUNWAY THRESHOLD
_ _ ---_ I ,
DEPARTURE AIRPORT
.C..I.....^. AZIDMEIP
AREA
AREA
I AZIDMEIP
FIG. 19. Hypothetical VORTAC and RNAV f l i g h t paths and route structure.
such as frequency and power output are determined by both national and international standards. These “service volume” standards are absolutely necessary. Unacceptable interference in the air-ground or ground-air links of these systems could deny or distort information essential to the controller or pilot in providing or receiving safe ATC service. Sections III,C,2 and 3 describe the procedure for ensuring interference-free DME service volume. The systems presently in use for ATC are: (1) A means for the controller to detect and identify aircraft using the primary or secondary radar;
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OTHER AIRCRAI
USED BY CONTROLLER
FIG.20. The ATC system.
(2) A common enroute navigation system (VORTAC); (3) Precision approach and landing guidance (MLS/ILS); (4) Nonprecision approach guidance (e.g., VOR/DME); (5) Facilities for direct controller-to-pilot communications.
Note how closely this ATC system satisfies the four requirements stated by ITT in 1945.See the Institute of Electrical and Electronics Engineers (1973)for an in-depth review of the ATC system. a . Enroute and Terminal Area IFR Route Structures. In addition to aircraft separation, the pilot under IFR conditions requires navaid guidance to avoid obstacles as he descends from the enroute structure to the terminal area and executes a precision or nonprecision approach. The following discussion illustrates how the navaid accuracy performance requirements are consistent with the airway route structure. The narrative will be in the context of area navigation (RNAV) because the VOR/DME radials are really special cases of the more general RNAV route. RNAV and its present status in the NAS is reviewed in Subsection D,6. It will be assumed that the primary navigation guidance is derived from position-fix radio navaids such as VOR/DME, TACAN, or DME/DME. In addition to the navaid guidance, flight procedures and obstacle clearance surfaces must be defined so that aircraft can safely fly along the
52
ROBERT J. KELLY A N D D A N N Y R. CUSICK
enroute airways and maneuver in the terminal area. Enroute procedures are given in FAA (1975b, 1984)while terminal area procedures are documented in FAA (1976) and ICAO (1982). Enroute airways and terminal area approach paths are defined by a series of waypoints connected by straight-line segments. The enroute airways are usually specified by VORJDME radials with waypoints defined directly over the ground stations or at the intersection between the radials. A waypoint may be identified in several ways, i.e., by name, number, or location. Waypoint location is necessary in the computation of navigation information and to minimize pilot workload during time-critical phases of flight. At a minimum, enough waypoints are provided to define the current and next two legs in the enroute phase of flight, and to define an approach and missed approach. In some mechanizations, waypoints are entered into the equipment in terms of their latitude and longitude. In other equipment a waypoint’s location may be specified in terms of a bearing and distance from any place whose position is itself known to the pilot. Recalling earlier discussions in Subsection C, lateral guidance using path deviation correction signals is the principal means of enroute and terminal area navigation. Vertical guidance is typically not derived from path deviation signals; aircraft separation and obstacle clearance is achieved by measuring height with a barometric altimeter. Although the most inexpensive and therefore most implemented form of lateral guidance is essentially bearing information using NDB or VOR, the trend is toward bearing and range position-fix navigation. From the position fixes, deviation signals about the desired course called steering commands are input to the aircraft flight control system. In particular, for the RNAV application, the lateral navigation information displayed to the pilot commonly takes the form of bearing and distance from present aircraft position to the defined waypoint and deviation from a desired track proceeding to or from the waypoint. As noted in Subsection C,2, when the displayed guidance permits lateral position-fix navigation the process is called 2D navigation. In summary, the most utilized form of area navigation is 2D lateral guidance. The role assigned to vertical information as derived from the barometric altimeter is minimum descent altitude alerts and aircraft descent rate. When vertical deviation signals are derived from the altitude data, then RNAV guidance is called 3D navigation. To avoid collisions with other aircraft, precipitous ground terrain, and airport obstacles, the enroute airways and terminal approaches are separated both laterally and vertically. Lateral separation is defined by primary and secondary protection areas, as shown in Fig. 19. The full obstacle clearance is applied to the primary area, while for the secondary area the obstacle clearance is reduced linearly as the distance from the prescribed course is
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increased. The widths of the air route are based upon the accuracy of the navaid guidance, the flight technical error, and the RNAV equipment errors. Blunder errors are monitored by enroute and airport surveillance radars and are not included in the flight technical error component. The minimum authorized flight altitude is determined by the obstacle clearance, which is the vertical distance above the highest obstacle within the prescribed area. For precision approaches this distance is based upon a collision risk probability of lo-’ over the entire flight mission. The minimum flight altitude may equal the obstacle clearance or it may be raised above it under certain conditions. If there are no obstacles then the obstacle clearance is the height above the ground. A displacement area is defined about each waypoint. It is a rectangular area formed around the plotted position of the waypoint. The dimensions of this area are derived from the total navaid system alongtrack and across-track error values, including the flight technical errors (see Subsection F). In other words, the waypoint displacement area defines the accuracy with which an aircraft can reach a waypoint and is directly related to the airway route width or terminal area approach path width. In the final analysis, it is the displacement area of the approach path which defines the minimum descent altitude. For the enroute airways, the lateral guidance protected area is k4 nmi about the route center line and the secondary area extends laterally 2 nmi on each side of the primary area, as shown in Fig. 19. The required vertical separation is lo00 ft below 29,000 ft altitude and 2000 ft above 29,000 ft altitude. The minimum required altitude (MRA) is lo00 ft and is equal to the obstacle clearance. In general, the minimum enroute altitude (MEA) is equal to the MRA. However, as noted earlier, the MEA may be raised above the MRA when, for example, the air route is over mountainous terrain. For the mountains in the eastern United States MEA = MRA 1000 ft; over the Rocky Mountains, MEA = MRA + 2000 ft. Terminal area approach paths are also protected by lateral and vertical separation distances. When in the terminal area, a series of position fixes define the approach path. They are initial approach fix (IAF), intermediate approach fix (INAF), final approach (FAF), missed-approach fix (MAF), and the runway fix (RWYF). The path between each pair of fixes is called a segment; e.g., the intermediate segment is between the INAF and the FAF. About each segment there is defined a protected primary area and a secondary area. Terminal protected areas are 2 nmi about the route center line and the secondary obstacle clearance areas extend laterally 1 nmi beyond the primary area. FAA Handbook 8260.3A, called the TERPS (FAA, 1976), prescribes criteria for the design of instrument approach procedures. Although it does not at the present time contain special criteria for the design of RNAV
+
54
ROBERT J. KELLY A N D DANNY R. CUSICK
procedures, most of the TERPS criteria are applicable to RNAV procedures with minor modifications. Guidelines for implementing RNAV within the NAS are given in AC 90-45A (FAA, 1975), which is oriented toward VOR/DME. Future revisions to AC 90-45A will reflect the more general configurations of multisensor RNAV as defined in RTCA (1984).In applying TERPS to a RNAV procedure, the term “fix”isequivalent to “waypoint”;thus “final approach fix (FAF)” becomes “final approach waypoint (FAWP).” Therefore, an RNAV instrument approach procedure may have four separate segments. They are the initial, the intermediate, the final, and the missedapproach segments. The approach segments begin and end at waypoints or along-track distance (ATD) fixes, which are identified to coincide with the associated segment as shown in Fig. 19. The RNAV instrument approach procedure commences at the Initial Approach Waypoint (IAWP).In the initial approach, the aircraft has departed the enroute phase of flight and is maneuvering to enter the intermediate segment. The purpose of the intermediate segment is to blend the initial approach into the final approach and provide an area in which aircraft configuration, speed, and positioning adjustments are made for entry into the final approach segment. The intermediate segment begins at the Intermediate Waypoint (INWP) and ends at the FAWP. For a standard procedure, the INWP is about 8 nmi from the runway with a minimum segment length of 3 nmi. During the final approach segment, the aircraft aligns itself with the runway and begins the final descent for landing. The final approach segment beings at the final approach waypoint or along-track distance fix and ends at the missed-approach point, normally the runway threshold waypoint. When it is not the runway threshold, it is an ATD fix based on a distance to the runway waypoint. The optimum length of the final approach segment is 5 miles. The maximum length is 10 miles. The final approach primary area narrows to the width of the FAWP displacement area at the runway threshold. A missed-approach procedure is established for each instrument approach procedure. The missed approach must be initiated no later than the runway threshold waypoint. The obstacle clearance for the initial approach segment is 1000ft and thus matches the MEA of the enroute airway. Its secondary area obstacle clearance is 500 ft at its inner edge, tapering to zero at the outer edge. The optimum descent gradient to the intermediate fix is 250 ft/mi. Obstacle clearance for the intermediate segment is 500 ft. Because the intermediate segment is used to prepare the aircraft speed and configuration for entry into the final approach segment, the gradient should be as flat as possible. The optimum descent gradient in this area for straight-in courses should not exceed 150 ft/mi. The
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obstacle clearance for the final approach segment begins at the final approach waypoint and ends at the runway or missed-approach waypoint, whichever is encountered last. Associated with each final approach segment is a minimum descent altitude (MDA). It is the lowest altitude to which descent shall be authorized in procedures not using a glide slope. Aircraft are not authorized to descend below the MDA until the runway environment is in sight, and the aircraft is in a position to descend for a normal landing. As mentioned earlier, sometimes the MDA must be raised above the obstacle clearance. Conditions which necessitate raising the MDA are precipitous terrain, remote altimeter setting sources, and excessive length of final approach. The required obstacle clearance for a straight-in course is 250 ft. The decision height (DH) minimum is also 250 ft when the visibility is 1 nmi. The above final approach discussions are directed toward nonprecision approaches, as contrasted with precision approaches. Precision approaches require a descent path from which vertical deviation correction signals can be derived, as described in Subsection D,4. The MDA and DH are determined by the size of the final approach displacement area and the obstacle clearance. Although typical nonprecision approach heights are 400 ft and above, the lowest authorized minimum is 250 ft. Precision approaches routinely achieve decision height at 250 ft and below because the final approach displacement area is smaller. Precision navaids and cockpit displays achieve smaller displacement areas because (1) positive vertical guidance is required (i.e., a glide path as well as azimuth guidance); (2) the navaid guidance errors are smaller; and (3) the display sensitivities are increased to reduce the flight technical errors. For example, the full-scale sensitivity of the CDI is L 1.25 nmi at 20 nmi and is +350 ft at runway threshold. The full scale sensitivity is always designed to be within the obstacle clearance surface. With smaller displacement areas the obstacle clearance for precision approaches can be satisfied more easily because the projected footprint of its displacement area is smaller. Consequently, with precision approaches, lower decision height minima are more readily achievable to airports or runways with smaller obstacle-free surfaces. The principal distinction between nonprecision and precision approaches as used by the FAA is that a combined azimuth and elevation guidance (glide slope) is required for precision approaches (FAA, 1976). The term nonprecision approach refers to facilities without a glide slope, and does not imply an unacceptable quality of course guidance. It is the reduction of the final approach displacement area which is important to achieving lower minima. Positive vertical guidance is not necessary down to 250 ft because adequate obstacle clearance is provided. At that altitude the pilot has visible cues and is in the region of “see and avoid.” He can correct the attitude and position of the
56
ROBERT J. KELLY AND DANNY R. CUSICK
aircraft to accomplish a manual flare and have a safe landing. Nonetheless, the pilot is more comfortable executing nonprecision approaches down to higher minima such as 400 ft. It is clear that precision approaches, with their accurate descent path, provide additional margins of integrity and pilot confidence. Table 111 summarizes the lateral route width accuracy requirements for the radio navaids servicing the enroute, terminal area, and the approach/landing application. TABLE I11 LATERAL ACCURACIES IN NAS"
Route type
XTRK system error (95% prob.)
Navaid XTRK error (95% prob.)
Includes FTE
No FTE
Route width
+ 4.0 nmi
Random
k 3.8 nmi
f3.0 nrni
J/V
& 2.8 nmi
+4 nmi + 4 nmi
Random JIV
+ 2 nmi Runway displacement area
k 2 nmi
f1.7 nmi
(k0.7 nmi) Runway displacement area
kO.5 nmi
ILS Cat I
k455 ft
& 455 ft
Bias 45 ft Noise 46 ft
ILS' Cat I1 approach
f400 ft Obstacle clearance
75 ft bias (runway width)
25 ft Noise
k27.5 ft 12.7 ft
ILSd Cat 111 landing MLS' Cat I1 approach
k 7 5 ft (Runway width) +400 ft
k 27.5 ft
Bias Noise
+ 11.5 ft
+75 ft
+ PFE +PFN +BIAS kCMN
21.7 ft
MLS" Cat 111 1anding
+75 ft (Runway width)
+ PFE +PFN +BIAS *CMN
20.0 ft 11.5 ft 1O.Oft 10.4 ft
Enrouteb Random JJV Terminalb RNAVb Non precision approach
+
95% Prob. (runway width)
+
27.5 ft About runway centerline
a J = Jet routes, V = VOR routes, random error; runway length 10,000 ft RTCA (1984). ICAO (1972b); FAA (1970). ICAO (1972b); F A A (1971, 1973). ICAO (1981a).
=
RNAV; FTE
+
+loft
12.5 ft 10.8 ft 11.3 ft
= flight
technical
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Adding a third dimension of vertical guidance to the two-dimensional RNAV systems can achieve significant operational advantages. Briefly, a 3D RNAV capability permits altitude change by following vertical routes (tubes) of known dimensions; thus vertical guidance is available for stabilized descent in instrument approach procedures using computed glide path information. In some cases, the computed glide path can make it possible to safely eliminate obstacles from consideration. To provide vertical guidance during ascent or descent, the 3D RNAV equipment compares the indicated altitude with the desired altitude and presents the computer correction instrumentally. In prescribing obstacle clearance for 3D RNAV, it is useful to think of the vertical route as being the center of a tube of airspace. The lateral dimension of the tube is the width of the RNAV route as described in Table 111. The vertical dimension of the tube is sufficient to contain the combined 3D RNAV vertical errors. The longitudinal dimension of the tube is limited only by its operational use. For example: The distance required to climb 10,000 ft at a climb angle of 2", or the descent from the final approach waypoint to the missed approach waypoint at a descent angle of 3". For obstacle clearance, it is necessary to consider only that portion of the tube which is at and below the designed vertical flight path. An aircraft is protected from obstacles when no obstacles penetrate the tube from below. The along-track error also has significance in vertically guided flight. When an aircraft is ahead or behind its assumed position, it will be either above or below its intended path. The possibilities for a hazardous incident are analyzed using the total navigation system error and the distance to an obstacle. For the approach/ landing application, the hazard probabilities are small and the obstacles are known fixed objects, e.g., towers, hangars, etc. Although the probability of exceeding the route width of an enroute airway is only 95%, the lateral obstacle is in effect another aircraft in a parallel route; thus the joint probability of a collision is very small, especially when ATC aircraft separations are maintained using ground radar control. The longitudinal separation using the ground radar (ARSR) is 3 to 5 nmi. Without radar it is 5 nmi with DME and the faster aircraft is in front of the slower aircraft (Kayton and Fried, 1969). The principal ideas associated with terminal approaches are worth restating. They are: (1) The collision risk probability establishes the obstacle clearance; (2) the total navigation system error determines the approach displacement area; and (3) the height of the displacement area for a given approach path is determined by the obstacle clearance. Thus the displacement area is lowered until it equals the obstacle clearance. For a nonprecision approach the displacement area is larger than the precision approach displacement area. Therefore, its MDA or DH will generally be higher than that required for a precision approach.
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ROBERT J. KELLY A N D D A N N Y R. CUSICK
6. Surveillance Radar. The primary radar is intended to detect all aircraft by processing echoes of ground facility transmissions-no airborne equipment is required. There are two types. The first, air route surveillance radar (ARSR), operates in the 1300-1350 MHz band and normally has a range of 200 nmi. The airport surveillance radar (ASR),in the 2700-2900 MHz band, has a range of 60 nmi. ARSR systems have peak power outputs of 1.0 to 5.0 MWatts, and ASR systems from 400 kW to 1.0 MW. The power output for a particular site is determined by site conditions, i.e., terrain, elevation, and the proximity of adjacent sites. Bearing and range of the aircraft are displayed to the controller on a plan position indicator (PPI)called a plan view display (PVD).The primary radar permits the controller to see any aircraft within the coverage area, regardless of whether or not the pilot is using the ATC system. The ATC primary surveillance radar is a self-contained system whose purpose is to maintain aircraft separation by dicect communication with the pilot when the aircraft is carrying a radio. Since the ARSR and ASR do not involve equipment on board the aircraft, international standardization is not required. The secondary radar, as noted in Subsection B, or air traffic control radar beacon system (ATCRBS), is a system wherein the airborne element is the transponder and the ground element is the interrogator. The ATCRBS power output is 200-2000 W and has a nominal 250 nm range when used in enroute applications. The secondary radar ground-air link, or interrogator, operates at 1030 MHz and activates the 1090 MHz aircraft air-ground link, or transponder. The transponder reply among the 4096 available codes generally includes an identification code and an altitude code. Transponder replies are received by the ground interrogator, and heading and ground speed are calculated by tracking devices. Heading and ground speed are displayed on the PVD along with the aircraft identification number and altitude. A rotating directional ground-based interrogator transmits approximately 400 interrogations per second. The cooperative ATCRBS is one of the keys to advanced ATC automation. The ATC system is devloping additional automated features to economically meet the high demand for services in the 1990s. In the U.S. there are about 300 ATC radars in use, over 100 ARSRs and about 200 ASRs. All ATC primary radars are equipped with the ATCRBS secondary radar system. Unlike the primary radar system, the secondary radar is an ICAO standard; its “Standards and Recommended Practices”(SARPs) are given in ICAO, (1972b, para. 3.8). 3. Navigation Aids
Aircraft users require short-range navigation aids to probide enroute guidance over the defined airways under IFR conditions. The VORTAC
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routes were established to provide a simple, common system that is easy to use, thereby reducing pilot workload. DME/N and VOR provide the guidance with RNAV augmenting the VORTAC route structure by providing random routes. Table IV places the VORTAC system in perspective by comparing the number of units in use with all the air and marine navigation aids in use throughout the world. The table was excerpted from Dodington (1984). The accuracy and coverage of navigational aids are determined by the enroute and terminal area operational requirements. Systems deriving bearing information (VOR and TACAN) must offset the effects of multipath and siting,which are the major sources of error (Subsection F). Signal coverage is a design problem only because adjacent channel effects must be considered (see Section III,C,3). In other words, it is easy to obtain coverage with a single facility, but it takes special care to eliminate spillover into other channels so that coast-to-coast service can be provided using a limited number of channel frequencies. The next several subsections describe the VOR, TACAN, ILS, and MLS angle systems. The DME/N and DME/P are treated in Sections 111and IV of TABLE IV MAINRADIONAVIGATION SYSTEMSUSEDTHROUGHOUT THE WORLD' Users System
Frequency
Omega Loran-C Decca ADFb ILSb
10-13 kHz 90-1 10 kHz 70-130 kHz 200-1600 kHz 75,108-1 12, 329-335 MHz 108-118 MHz 150,400 MHz 960-1215 MHz 960-1215 MHz 1030, 1090 MHz 5031-5091 MHz
VORb Transit Tacan DMEb SSRb MLSb
Number of stations 8
33 chains 50 chains 5000 1600
2000 5 satellites 2000 1000
Air
Marine
10,600 2000 1000 200,000 70,000
7000 60,000 30,000 500,000
200,000
-
-
38,000
17,000 80,000 100,000 Implementaticm about to' start
Estimated number of U.S. aircraft: 224,000 General aviation (Quinn; 1983) 2500 Air carrier 20,800 Military aircraft Dodington, 1984. Signal format standardized by the International Civil Aviation Organization. a
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ROBERT J. KELLY AND DANNY R. CUSICK
this article. These navaids are viewed from a system level viewpoint. Each system is characterized by its signal format, accuracy, coverage, channel plan, and data rate. Two other system level characteristics are needed before the navaids used in the ATC system are complete. They are integrity and availability. Integrity is a measure of the truthfulness of the signal in space. It is ensured by a ground monitor system, which shuts the facility down if certain critical signal characteristics exceed predetermined limits. Integrity is quantified by the reliability (mean time between outages, MTBO) of the ground equipment and its monitor. A navigation system must be available to the pilot, especially under IFR conditions. Although a facility which has been shut down has good integrity, it is not available. The system availability A is given by A = MTBO/(MTBO + MTTR), where MTTR is the mean time to repair. Integrity and availability are critical to the successful operation of approach/landing systems. Radio navaid accuracy performance comparisons are made in Hogle et al. (1983). Integrity and reliability comparison of civil radio navaids are made in Braff et al. (1983). a. VOR.
The very-high-frequencyomnidirectional range is the ICAO international standard for providing short-range enroute bearing information. Paragraph 3.3. of ICAO (1972b) is the ICAO SARPs for VOR. As noted in Subsection C , it is a radial determining navigation aid, and along with the automatic direction finder (ADF) (see Table IV) is the most popular system currently in use. Since it is a radial system, it must have a reference signal and a signal which carries the bearing information. The reference signal is frequency modulated (FM) at 30 Hz on a 9960 Hz subcarrier. A cardioid antenna pattern rotates 30 times per second (Fig. 21), generating 30 Hz amplitude modulation (AM)in the receiver, and thus provides the bearing information. The airborne receiver then reads bearing as a function of the phase difference between the FM reference signal and the AM modulated signal. The system is graphically described in Fig. 2 1. The VOR estimates the aircraft bearing in the same manner as the TACAN; it employs a phase detector. Early implementations of phase detectors were electromechanical devices using resolver phase shifters which were rotated by a servo motor until a peak or null was obtained (Hurley et al., 1951). The bearing display was connected directly to the phase shifter shaft. Today’s systems are all-electronic digital implementations. The VOR ground transmitter radiates continuous-wave signals on one of 20 channels between 108 and 112 MHz (interleaved with ILS localizer frequencies) and 60 channels between 112 and 118 MHz. Transmitter power is from 50 to 300 W.
61
DISTANCE MEASURING EQUIPMENT IN AVIATION CHANNEL A f, = 9960 Hr
AIRBORNE RECEIVER
PHASE DETECTOR
e^
CHANNEL B CHANNEL B CHANNEL A
PAlTERN ROTATES
FIG.21. Principles of VOR.
Because of the frequency used, it is difficult to get an antenna with good vertical directivity. Consequently, significant radiation strikes the ground in the vicinity of the transmitter which, upon reflection, interferes with the desired radiating signal causing a bearing error. This error, which is sometimes 2" to 3" in magnitude, appears as course variations along the radials of the VOR station. Several approaches to fixing this problem have been tried, including elevating the antenna to get it farther away from reflecting objects and using a large counterpoise (ground system) under the antenna. Both of these methods have been somewhat successful, but the development of Doppler VOR (DVOR) and Precision VOR (PVOR) offered better solutions. Severe constraints, however, were placed upon these solutions. They were not to affect the service of the already existing airborne equipment and, if possible, should improve the system's overall performance. Doppler VOR accomplished this by simply reversing the roles of the reference signal and the information signal. Because their phase relations remain the same, they allow a standard airborne receiver to operate without
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ROBERT J. KELLY AND DANNY R. CUSICK
modification. With this modification, the effective aperture of the bearing information signal increased more than 10:1 (antenna diameter of 44 ft); thus, the effects of site error could theoretically have a tenfold reduction. Experiments have confirmed this; site errors of 2.8" were reduced to 0.4". Steiner (1960)describes DVOR. The second solution used the same idea as TACAN; that is, it modified the ground antenna to produce multiple lobes, thereby creating a two-scale system. The coarse information was 30 Hz and the multilobe antenna (11 lobes) generated a 330 Hz harmonic for the fine information. It could potentially obtain and 11:l reduction not only in site errors but also in airborne receiver instrumentation errors (Winick, 1964; Flint and Hollm, 1965).The primary reasons the FAA selected the DVOR solution and did not pursue the PVOR further were that the conventional VOR receivers could not decode the multilobe harmonic and DVOR sufficiently reduced site errors to less than 0.5". Since a DVOR installation is a rather extensive and expensive process, it is reserved for the most difficult sites, where the simpler techniques do not suffice. Given the deficiencies of VOR, many airlines are making use of the scanning DME as their primary enroute aid to form a p-p navigation system based on RNAV principles. VOR/DME is then used as an alternative if acceptable DME/DME geometric configurations cannot be found (see RNAV, Subsection D,6). b. TACAN. TACAN range and bearing information defines a natural p-8 system. Unlike VOR/DME, which uses two different rf transmissions, TACAN provides range and bearing information using a single rf emission. TACAN is not an ICAO standard. Range information is derived within the TACAN system in the same way as for DME: A transponder responds to an interrogation. It uses the same frequencies, pulse characteristics (coding, form), and interrogator pulse rates as DME/N. Bearing information is determined at the interrogator in a manner that is functionally identical to the VOR phase detector. That is, it generates both a reference signal and bearing information signal and is, therefore, a radial system. Because DME is a pulse system, the TACAN bearing information is equivalent to a sampled data VOR system. Moreover, TACAN bearing information is a two-scale system similar to precision VOR. The coarse bearing is 15 Hz and the fine bearing is the ninth harmonic or 135 Hz. Bearing information is obtained with the following additions to the DME signal format:
(1) A circular antenna array replaces the simple omnidirectional DME antenna. It generates a cardioid pattern in space and sequentially rotates it at 15 Hz. At a fixed point in the coverage sector, the DME pulses appear to be
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amplitude modulated at 15 Hz. A second pattern composed of 9 lobes spatiallymodulates the cardioid pattern, which generates a 135 Hz sinusoid or ninth harmonic at a fixed receiver in space. (2) The 15 Hz reference signal is generated by a North reference burst which is composed of 24 pulses, the spacing between pulses being alternately 12 and 18 ps. When decoded in the airborne equipment they become 12 pulses spaced 30 ps apart. This pulse train is referenced once per revolution of the cardioid antenna pattern. (3) The 135 Hz reference signal is generated 8 times per revolution of the cardioid antenna pattern. It consists of 12 pulses spaced 12 ps apart. (4) The transponder operates at a constant duty cycle of 2700 pps. When interrogations are lacking, pulse pairs with random spacing are generated (called squitter pulses) so that the output rate is kept constant at 2700 pps. This constant duty cycle permits the sampled 15 and 135 Hz sinusoids to be reconstructed by zero-order hold (ZOH) circuits in the aircraft with complete fidelity. It also permits AGC to be generated in the aircraft such that the 15 and 135 Hz AM modulated signals can be detected without distortion. In the airplane the 15 Hz reference burst provides a time reference mark from which the phase of the 15 Hz information signal can be determined. Similarly, the 135 Hz reference burst provides a time reference mark from which the phase of the 135 Hz information signal can also be determined. After reconstruction from the samples, the information signal is sent to the 15 and 135 Hz phase detectors where the operations are functionally identical to that used in the VOR receiver. The 135 Hz phase detector accurately estimates the aircraft's bearing; it is, however, ambiguous over each 40" sector (there are nine of them). The 15 Hz phase detector resolves the ambiguity by determining in which of the 9 sectors the actual bearing angle lies (see Fig. 22). Because TACAN utilizes a multilobe spatial antenna pattern, it can employ a two-speed phase detection system which is less susceptible to siting effects and equipment error than is conventional VOR. TACANs accuracy is comparable to the PVOR. The ninth harmonic reduces the bearing error by about 4 of those for the 15 Hz signals. TACAN's accuracy is discussed further in Subsection F. 4 . ApproachfLanding Aids
A guidance system for approach and landing is simply a precise, lowaltitude navigation aid with the additional accuracy and reliability needed for landing aircraft under reduced visibility. The standard system now in use is the Instrument Landing System; its SARPs are defined in paragraph 3.3 of ICAO (1972b). The Microwave Landing System SARPs are defined in paragraph 3.11 of ICAO, (1981a);it is scheduled to replace ILS in 1998. ILS and MLS permit precision approaches at or below a 250 ft weather minimum
64
ROBERT J. KELLY AND DANNY R. CUSICK HORIZONTAL ANTENNA PATTERN MAGNETIC NORTH
15 RESOLUTlONSlSECOND
WEST 270'
SOUTH 180'
---
HORIZONTAL PATTERN COMPONENT THAT GENERATES 15 Hz SIGNAL COMPOSITE PATTERN THAT GENERATES 15 Hz AND 135 HZ SIGNALS
TIME WAVEFORM AND REFERENCE BURSTS
15 Hz MODULATION
-b
1/15 SECOND
. .
I.
.ANTENNA PATTERN ORIENTATION SHOWN AT THE TIME OF NORTH REFERENCE BURST LET THE NORTH REFERENCE BURST DEFINE TIME 0. THE RECEIVED SIGNALS ARE THEN 15Hz : -SIN r m o n t - 8 ) 1 % ~ -SIN ~ : rz701~t-se) WHERE IS THE BEARING FROM THE STATION. AUXILIARY REFERENCE BURSTS ARE TRANSMITTED EACH TIME A 135 Hz PATTERN PEAK POINTS EAST
e
FIG.22. Principles of the TACAN bearing measurement.
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using a barometric altimeter. If in addition a middle marker or DME measurement is included, the minimum can be reduced by 50 ft to 200 ft. Nonprecision approaches do not require positive vertical steering commands; minimum descents down to 250 ft are permitted depending upon the obstacle clearance. Before describing ILS and MLS, this section will discuss the landing operation itself, using MLS as an example. The landing process consists of curved or segmented flight paths for noise abatement and the transition to the final centerline approach (Fig. 23). The decision heights where the pilot must be able to “see to l a n d are 200 and 100 ft ceilings-Categories I and 11, respectively. The flare maneuver, touchdown, and rollout complete the landing. MLS must support each of these maneuvers under IFR conditions. The elevation element, typicallly located 861 ft from threshold and offset from the centerline by 250 ft, provides elevation guidance to the decision height of Category I and I1 operations and to threshold in Category I11 (zero ceiling). The approach-azimuth element supports lateral guidance to the decision height in Categories I and I1 operations and to touchdown and rollout in Category 111. The flare maneuver is typically performed manually for
FIG.23. The landing operation with MLS.
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ROBERT J. KELLY AND D A N N Y R. CUSICK
Categories I and I1 by visual reference, but, for Category 111, positive azimuth and vertical control are required to touchdown. The Category I1 “window” is centered at the 100 ft decision height. Its height is f 12 ft around the indicated glide path and its width is the lateral dimensions of the runway (typically f 7 5 ft). If the aircraft is not in the window, the pilot will execute a missed approach. If the aircraft passes through the Category I1 window within specified variations of pitch, roll, speed, and lateral drift velocity, then the probability is high that a successful landing can be achieved when the pilot commences the flare maneuver. The decision height must be determined within f 5 ft. On the other hand, no Category I window is necessary because the pilot can maneuver his aircraft to the correct position and attitude before he executes his flare maneuver (FAA, 1970). MLS facilitates the autoland maneuver with its increased accuracy, which is extended to lower elevations further down the landing path than provided by ILS. Flare to touchdown requires the height above the runway and, in MLS, it may be provided by a combination of the approach elevation angle and DME prior to runway threshold and then by transitioning to the radar altimeter in the vicinity of runway threshold. Terminal area IFR procedures for MLS are currently under development by the FAA. a. ZLS.
The Instrument Landing System is a radial system providing guidance for approach and landing by two radio beams transmitted from equipment located near the runway. One transmitter antenna, known as the localizer, forms a single course path aligned with the runway center line. The other transmitter, the glide slope, provides vertical guidance along a fixed approach angle of about 3”.These two beams define a sloping approach path with which the pilot aligns the aircraft, starting at a point 4-7 mi from the runway. Instrument landing systems operate on frequencies between 108.1 and 111.9 MHz for localizers, and between 328.3 and 335.4 MHz for glide slopes. Both localizer and glide slope are paired on a one-to-one basis. The service volumes for an ILS are bounded by wedges of an 18 or 25 nmi radius from the localizer antenna. Lateral guidance (Fig. 24) is provided by the localizer located at the far end of the runway. Two identical antenna patterns, the left-hand one modulated by 90 Hz and the right-hand one by 150Hz are provided. The vertical needle of the airborne display is driven right by the 90 Hz signal and left by 150 Hz signal, and centers when aircraft is on course. Vertical guidance (Fig. 24) is generated by the glide slope antenna located at the side of the approach end of the runway. A 150 Hz amplitude-modulated signal is provided below course and 90 Hz modulation is provided above
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LOCAL1ZER
A
A\
-
ANTENNA
RUNWAY
FIG.24. Principles of ILS.
course. The horizontal needle of airborne display is driven up when the amplitude of the 150 Hz signal exceeds that of the 90 Hz signal and is driven down when the reverse occurs. The needle is horizontal when the aircraft is on course. (The “cross-pointer” is a course deviation indicator combining a vertical localizer needle and a horizontal glide slope needle.) Along-course progress is provided by fan markers that project narrow beams in the vertical direction and operate at 75 MHz. The middle marker is placed at the 200 ft decision height; the outer marker about 5 miles out. The outer marker corresponds to the final approach fix. The outer marker can be viewed as the final approach fix. Since DME may be used as a replacement for the markers, 20 DME channels are provided for hard pairing with ILS localizer frequencies. ILS’s limitations are detailed in Redlien and Kelly, (198 1). b. MLS. Like ILS, the angle portion of MLS is an “air-derived’’ radial system where signals are radiated from ground antennas in a standard format and then processed in an airborne MLS receiver. The angle information is derived by measuring the time difference between the successive passes of highly directive narrow fan-shaped beams, as shown in Fig. 25. Range information, required to obtain the full operational benefits of the MLS angle data, is derived from precision DME, which Section IV describes in detail. MLS operation is detailed in Redlien and Kelly, (1981).
68
ROBERT J. KELLY A N D DANNY R. CUSICK
m .-dI
TIME-ANGLE
4-
AIRCRAFT RECEIVER
FIG.25. MLS angle measurement technique.
MLS is a highly modular system and it may be implemented in simple configurations. The ground systems may contain approach azimuth, the “bearing” facility, approach elevation, back azimuth, and DME/P, as shown in Fig. 26. The signal format is time multiplexed; that is, it provides information in sequence on a single-carrier frequency for all the functions(azimuth, elevation, basic, and auxiliary data). The format includes a time slot for 360” azimuth guidance with provisions for growth of additional functions. The angle guidance and data channel plans provide 200 C-band channels between 5031 and 5091 MHz. Narrow fan-shaped beams are generated by the ground equipment and scanned electronically to fill the coverage volume. In azimuth, the fan beam scans horizontally and has a vertical pattern that is shaped to minimize illumination of the airport surface. In elevation, the arrays are designed to minimize unwanted radiation towards the airport surface, thereby providing accurate guidance to very low angles. Unlike ILS, the MLS conforms to a single accuracy standard equivalent to that required to perform fully automatic landings. A ground-to-air data capability is provided throughout the angle guidance coverage volume by stationary sector coverage beams that are also designed to
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77ms SEQUENCE
BASIC DATA NO. 2
FIG.26. MLS signal format.
have sharp lower-sidecutoff characteristics. This capability is used to transmit the identity of each angle function and to relay information (basic and auxiliary data) needed to support simple and advanced MLS operations. The wide angle capability of MLS is graphically displayed in Fig. 27. Because proportional guidance is available over a +40" azimuth sector, RNAV curved and segmented approaches down to the 200 ft decision height can be executed. Shown in the figure are the airborne elements which comprise the MLS RNAV equipment configuration. RTCA Special Committee 151 is currently defining the minimum operational performance expected of the airborne equipment intending to perform MLS/RNAV approaches (RTCA, 1985a). For approaches below the 200 ft decision height, the MLS data will bypass the RNAV box and be utilized directly by the aircraft flight control
70
ROBERT J. KELLY AND DANNY R. CUSICK MLSRCVR
I
A2 DMEP FLIGHT DIRECTOR
I
RNAV
CONTROL PANEL
I
W
FIG.27. MLS avionics for segmented approaches
system. RTCA (1985a) recommends that the RNAV error contribution be limited to 35 ft (95% probability) for MLS area navigation applications. c. Nonprecision Approach. Aircraft equipped with the appropriate enroute navaids are permitted to fly final approaches down to MDAs which are consistent with the obstacle clearance.The nonprecision approach procedures additionally require that the final approach displacement area be 20.7 nmi, as indicated in Table 111. Over 30% of the runways with nonprecision approaches use on-airport VOR (DOD/DOT, 1984). See the discussion in subsection D.5, which addresses the role that the scanning DME may play in nonprecision approaches. 5. Communications
Radio communications between controllers and pilots take place on VHF frequencies between 118.0 and 136.0 MHz, and on UHF frequencies between 225.0 and 400.0MHz. ATC communications are necessary to ensure the safe, orderly, and expeditious flow of air traffic.Chapter 4 of ICAO (1972b)defines the ICAO standard for communication systems.
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6. Special Radio Navaid Techniques
This section discusses two techniques which enhance the operational performance of DME and other navaids in the ATC system. They are RNAV and the “scanning DME.” RNAV enhances enroute navigation and nonprecision approaches by providing point-to-point navigation along line segments other than the VORTAC radials. Scanning DME is a single interrogator that obtains the range to several different ground transponders by using frequency hopping or scanning frequency techniques. From these data a very accurate position fix is determined. a. RNAV. Since the introduction of VORTAC, air navigation has been improved by providing a means for unlimited point-to-point navigation. This improved method of navigation, which utilizes a computer to determine courses that need not lie along VORTAC radials, is called Area Navigation. It permits aircraft operations on any desired course within the coverage of various stations providing referenced navigational signals, i.e., DME or VOR, or within the limits of self-contained airborne systems, such as the INS. Sensor inputs to the RNAV may be p-p, 8-8, or p-8. As stated in Quinn (1983), there are three general operational concepts in area navigation: (1) two dimensional (2D), involving only horizontal movements; (2)three dimensional (3D), in which vertical guidance is combined with horizontal; and (3) four dimensional (4D), in which time of departure and arrival are combined with horizontal and vertical movements for complete aircraft navigation. At present, 2D area navigation is the most widely used version. Many pilot-related operational issues regarding 3D and 4D approach procedures have been and still are under study (Jensen, 1976; RTCA, 1985a). The conventional implementation of2D RNAV divides the flight course into a series of waypoints connected by straight-line segments (see Fig. 19). The waypoints are stored in the RNAV computer, and are identified by suitable coordinates such that DME/DME or VOR/DME measurements can generate the steering commands to guide the aircraft along the straight-line segments.In the more simple implementations, the position fixes derived from VOR and/or DME measurements are sometimes aided by inertial sensors so that a waypoint turn can be anticipated and thus minimize the error incurred in capturing the next flight path segment (Tyler et al., 1975). In the early 1970s, the FAA established a high-altitude RNAV airway structure. As stated in RTCA (1984) these airways were generally aligned to avoid all special-use airspace and coincide with regional traffic flows. They consequentlyoffered little in mileage savings over VOR airway structures. The RNAV structure did not take into consideration the center-to-center traffic flow that had evolved. Since preferential procedures are usually tied in to the VOR system, the high-altitude RNAV airway structure was not used to any
72
ROBERT J. KELLY AND DANNY R. CUSICK
great degree after a short period of interest in the early 1970s. In 1978, the RNAV airway structure was significantly reduced, and in early 1981 all published high-altitude RNAV routes were revoked. By the end of the 1970s, it became apparent that the potential economy and utility of RNAV could best be realized by random area navigation routes rather than by the previous grid system. Without a fixed route structure to use in normal flight planning, pilots and air traffic controllers developed an informal system in which pilots requested “RNAV direct destination” routing from controllers after they were airborne and beyond the airport terminal area. Such direct clearances are granted when possible. Figure 19 illustrates such a requested direct destination route. These procedures reflect the original intent of the RNAV concept wherein the “R’in RNAV signifies “random.” An operational evaluation by the FAA showed that there were no adverse effects in using latitude and longitude coordinates for domestic routes having direct random route clearance (FAA, 1981). Advisory circular 90-82 (FAA, 1983d) was issued, authorizing random routes for RNAV equipped aircraft having 1and (D coordinates and flying above 39,000 ft. The evident benefits in the use of RNAV are a reduction in flight time obtained by following the shortest or best route between origin and destination, and reduced fuel consumption. Other benefits include more efficient use of airspace, reduced pilot workload, and weather avoidance. A status review of RNAV is given in Quinn (1983),where an informal 1981 FAA study estimated that the largest number of RNAV avionics units was VOR/DME-based (over 33,000 units). About 20,000 of these units accepted 4 or 5 waypoints. Costs range from $2,000 to $20,000. The balance of the units included Omega/VLF (6,400 units) with costs ranging from $20,000 to $80,000; LORAN-C (700 units) with costs ranging from $2,000 to $20,000;and expensive INS units ($100,000 and above). It is estimated that today about 20% of the civil fleet has RNAV. The RTCA minimum standards on airborne RNAV equipment using VOR/DME imputs are documented in RTCA (1982). As shown in Fig. 4,RNAV outputs can be displayed to the pilot for manual coupled flight or they can be coupled directly as steering commands to the autopilot. Autocoupledflightsusing RNAV steeringcommands result in small tracking errors (Fig. 13).This means that the optimal maneuvers necessary to fly the desired course can be chosen using, in some cases, a flight management system (DeJonge, 1985), which minimizes fuel consumption. Moreover, since the pilot has confidence in his RNAV coupled AFCS, he can reduce his workload. He is free to fix his attention on other cockpit concerns. For manual coupled flights, the RNAV inputs can be as simple as a position-fix VOR/DME or DME/DME, with little or no inertial aiding (Bryson and Bobick, 1972). Autocoupled systems use VOR/DME or DME/DME with inertial aiding for dynamic damping to reduce control
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activity (ROLL) and tracking errors. In the most sophisticated RNAV implementations, INS is used as the primary navigation input with radio aiding by VOR/DME or DME/DME [Karatsinides and Bush, 1984; Bobick and Bryson, 1973; Zimmerman, 1969). Minimum operational peformance standards for these multi-sensor RNAV systems have recently been published in RTCA (1984). With over 750 VORTAC stations currently in service in the continental U.S., VOR/DME signal coverage is not a problem in flying defined air routes above the minimum enroute altitude. It is, however, incomplete for aircraft that regularly operate at low altitudes (above 2000 ft) in offshore and mountainous areas in the eastern part of this country. For direct requested RNAV routes, signal coverage is a factor which must be considered. According to Goemaat and Firje (1979),RNAV coverage throughout the 48 states above 10,OOO feet using VOR/DME is essentially complete. Coverage criteria were based upon the FAA Standard Service Volume (SSV) (see Section 111,C). RNAV coverage using DME/DME was more complete than VOR/DME because a line-of-sight criterion could be employed instead of the more restrictive SSV. For example, 86%of the continental U.S. (CONUS) had dual DME coverage at 3000 ft above ground level. Phasing out the current airway structure and converting to a more flexible system of area navigation is a process that will require many years to complete. At present, the FAA is upgrading VORTAC stations to solid-state equipment at a cost of roughly $210 million (fiscal year 1980 dollars). At the same time, ihe question of adopting a new navigation technology to conform to new international standards is scheduled for consideration by the ICAO. The issue is not so much selection of a single new navigation system to replace VORTAC as it is a question of adopting procedures for worldwide navigation using RNAV, which will be compatible with several possible technologies. Nevertheless, the national airspace plan has been updated to ensure the continuation of VORTAC service until 1995. Present indications are that these systems will remain prime elements in a mix of future navigation system plans which extend to beyond the year 2000. Subsection F compares the accuracy performance of DME/DME, VOR/DME, and TACAN, using an RNAV computer, and discusses additional RNAV system considerations.
b. Scanning DME. A position fix can be determined from the range measurements obtained from two DME ground stations. The intersection of the circular locus of points representing each DME range measurement defines the airborne receiver position. This position fix technique is referred to as DME/DME. In areas that have suitable DME/DME coverage the potential accuracy of dual DME offers a significant improvement over
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ROBERT J. KELLY AND DANNY R. CUSICK
VOR/DME. The degree of accuracy improvement is dependent upon the geometry between the interrogator and the selected DME ground stations. In principle these measurements can be made using two or more interrogators, requiring provisions for channel selection and synchronization of the range measurements from the individual interrogators. In one implementation which complements an INS (Karatsinides and Bush, 1984), two DME measurements are processed every 5 s. The “scanning” DME, however, performs all these tasks in a single interrogator. The “scanning” DME innovation provides the potential for rapid position fixes (position update rate) consistent with the AFCS control laws used in many jet transports. The idea is to use rf frequency-hop techniques to scan the spectrum of usable ground transponders in a given geographic area. The scanning DME, as defined in Aeronautical Radio, Inc. (ARINC) (1982), has two frequency scanning modes, a directed frequency foreground mode and a background mode. The tuning source may designate from one to five stations for geometry optimization, with one of these stations available for cockpit display. These stations are designated as foreground stations and are placed in a loop which cycles through the full list at least once every 5 s. All five selected stations are identified by separate digital words containing both the rf channel frequency and the distance information. A second loop, called the background loop, provides the acquisition and output of data obtained during the background scan mode. When in the background mode the interrogator scans through the entire DME channel spectrum, excluding those channels already designated in the foreground loop. For any station where squitter pulses are received during the background scan, the interrogator computes the distance to that station. Both the distance measurement and the channel frequency are given to the aircraft navigation systems utilizing the DME data. The ARINC characteristic states that the initial scan of the background channels should be completed in a maximum of three minutes. This full spectrum scan time is based upon 20 channels, occupied by a ground transponder with a 70%reply efficiency and a minimum reply signal strength of - 87 dB m. In effect, the scanning DME has six time slots available every second. Five are utilized for obtaining the range to the designated foreground stations while the sixth, or “free scan” time slot, obtains range data on channels in the background loop whenever stations are present. As required by ARINC (1982), the data output rate (on the ARINC 429 bus) shall not fall below six outputs per second and the measurement age of the data output should not exceed 0.2 s. This means that each time slot (or data output) per frequency can occur at least once per second. Since the interrogation PRF is limited to 30 pulses per second, this implies that each time slot has 3 to 4 replies available for
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filtering before being sent to the 429 data bus, assuming a 70% reply efficiency. As the aircraft proceeds along its desired route, foreground stations are discarded and replaced by background stations having better position fix geometries. Algorithms have been developed that select foreground channels to be used in determining a position fix. The advantage of the ARINC 709 characteristic for scanning DME is its short dwell time per channel (0.067 s.), which permits rapid position fixes. Because of the short dwell time, the range to 5 or more ground transponders can, in principle, be used by an RNAV computer to determine the aircraft’s position. Using special signal processing algorithms (e.g., regression analysis (Latham, 1974) demonstrated achievable 100 ft CEP. However, since RTCA (1984) requires only a cross-track error of 3.8 nmi for enroute RNAV systems, there is no motivation to reduce the errors to less than that obtained with a two-station DME fix. In general, the noise is reduced only about 50% for a five-station fix. Most of the error reduction is achieved by removing the bias contribution from each ground station. Because of the above considerations some airframe manufacturers use the five foreground stations as follows: position fix (2),ILS (l), display (2). They then use the background stations to ensure that the facilities chosen for their planned navigation flight route are operating correctly. As the geometry of the foreground stations used to obtain the position fix deteriorates (geometric dilution of precision, GDOP), these foreground stations are replaced by the appropriate background channels. Station selection algorithms are under development that pick not only stations within signal coverage but also provide acceptable position-fix geometries whenever they exist (Ruhnow and Goemaat, 1982).This means the stations should be offset to the left and right of the flight path. Subsection F discusses the DME/DME GDOP considerations. Goemaat and Firje (1979) analyzed the number of major airports within CONUS in which an aircraft can make a landing approach down to 400 ft. (nonprecision)using scanning DME. They concluded that 22% of the airports examined provided DME/DME approach capability on at least one major runway (5000ft in length or more). This coverage will increase with the implementation of DME/P as the FAA begins its deployment of the planned 1200 MLSs. In some systems a flight management computer system (FMCS) contains a navigation database for all navaids (DME and VOR) in the flight area. The selection process used in the Boeing 737-300 FMCS is detailed in Karatsinides and Bush (1984). Failure modes for navigation systems which use INS to aid DME/DME position-fix navigation are still under investigation. For example, the Boeing 737-300 FMCS, in the absence of inertial inputs, switches into a second mode and works on DME information only, with some
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degradation of the navigation performance. When DME fails temporarily, the system continues in a crude inertial dead-reckoning mode. 7 . Summary In this section, today’s ATC system was described, both from the operational and navaid equipment viewpoint. The narrative highlighted the roles of the conventional DME/N and of the new DME/P. For enroute navigation the DME/N is one of the central elements to the VORTAC system of air lane radials. Today it participates as one of the principal inputs to RNAV. Tomorrow, it is expected that the DME/N will have an even bigger role in the enroute air structure and in nonprecision approaches where the scanning DME, because of its high accuracy, can replace the VOR bearing information as a principal input to the RNAV. With respect to the approach/landing applications, DME/Ns role is expanding, where it will supplement the ILS middle markers and will provide three-dimensional guidance at those locations where middle markers cannot be installed because the necessary real estate is not available. Today and in the future, DME/P plays a critical role in MLS. This role will become even more visible and necessary as all the MLS operational features are exercised (1) by increasing demand for more certified Category I1 runways, (2) by coordinate conversionsto achieve curved approaches and (3)by providing MLS service to more than one runway using a single ground facility. E. Principles of Air Navigation Guidance and Control
The purpose of this section is to present the principles upon which the guidance and control of aircraft are based, with particular emphasis on the DME as it is used in the enroute (VOR/DME) and the approach/landing phases of flight (MLS). For the most part, the material will be qualitative with emphasis on the four or five key concepts associated with aircraft control systems. It is not intended that the material represent, even in summary form, the current thinking of today’s AFCS engineers. These concepts are necessary to understand the requirements imposed upon the output data characteristics of a radio navaid. An aircraft follows the desired flight path when the AFCS control law, which requires position and velocity information as inputs, is satisfied. The ideal or natural solution is to directly measure the position and velocity deviations of the aircraft from its intended path. Radio navaid and referenced accelerometers make these measurements directly; other inertial sources are indirect measurements. Radio navaids are desirable sources for this information because they are less expensive and more easily maintained than referenced accelerometers.
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There are, however, constraints imposed upon the radio navaid when velocity data as well as position displacement information are derived directly from the navaid guidance signal. The two applications of interest in this article, enroute navigation and approach/landing guidance, have widely different requirements in terms of output signal noise and information bandwidth, the most restrictive being the approach/landing application down to 100 ft decision height and lower. At these weather minima imperfections in the approach/landing guidance signal usually require that the velocity or the rate information be complemented by inertial sensors. This article develops the background necessary to explain the complementation process, and extends it to include the concepts of path following error (PFE) and control motion noise (CMN) which form the basis of the DME/P and MLS angle accuracy specifications.
I . Feedback-Controlled Aircraft- General Principles Aircraft use enroute navaids to fly between terminal areas and, once in the terminal area, landing guidance aids are used to complete their flight. These navaids can be coupled to the AFCS manually (through the pilot) or automatically via an autopilot. In either case, the guidance loop is closed and a feedback control system is created. Feedback control systems tend to maintain a prescribed relationship between the output and the reference input by comparing them and using the difference as a means of control. Such systems play a dominant role in flight control because they suppress wind disturbance and reduce the phase delay of the corrective signals and the effects of nonlinearities. For example, in the presence of disturbances such as wind gusts, feedback control tends to reduce the difference between the reference input (the desired flight path) and the system output of the actual aircraft position. When the pilot is operating in clear air, information about the attitude of his aircraft, its position, and velocity with respect to objects on the ground and other aircraft are fed back to him through his eyes. Under these VFR conditions, a navaid is not necessary because the pilot navigates using visual “landmarks.” The feedback system is then a special case of that shown in Fig. 12, where the block marked nauaid is replaced with a straight line and the display is eliminated. When visual information is unavailable, the pilot can use his cockpit instruments to obtain pitch, roll, heading, air speed, rate of turn, altitude, and rate of descent and, when combined with considerable skill, may reach his destination. In general, however, he does not have the information to obtain the guidance needed to follow a prescribed path over the ground, much less for approach/landing maneuvers in mixed air traffic.
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It is the purpose of the navigation system to help the pilot follow his desired flight path by providing electronic “landmarks” under IFR conditions. This desired flight path can be stored, for example, in computer memory. In some cases, only several numbers need be preprogrammed; this is possible when the flight path is simply a straight-line segment connecting two known enroute waypoints or is a line segment connecting the ILS outer marker with the decision height during a landing approach. Information from the navaid “tells” the pilot where the aircraft is. The difference between the desired flight path and the actual aircraft position is the correction (or error) signal. The pilot flies along his desired flight path by maneuvering his aircraft toward an indicated null position on his display. This null seeking process is mechanized using a negative feedback control system, as shown in Fig. 28. In accordance with control system terminology, the aircraft is the controlled element, the navaid generates a measurement relative to a known reference, and the controller is a human pilot or autopilot which couples the correction signal to the aircraft. The correction signal is maintained near its null position by changing the direction of the aircraft’s velocity vector by controlling the attitude of the aircraft. Inducing a roll angle command causes the aircraft to turn left or right from its flight path. Similarly, a change in its pitch angle causes the aircraft to fly above or below its intended flight path. An early paper which discusses the general problem of automatic flight control of aircraft with radio navaids is by Mosely and Watts (1960). As stated in Graham and Lothrop (1955), “the human pilot himself represents an equivalent time lag of 0.2-0.5 second, therefore, he is seldom able to operate at ‘high‘ gain without instability (overcontrolling). For this reason the pilot is generally unable to achieve the accuracy and speed of response of an automatic system. He represents, however, a remarkable
-
STEERING CONTROLLER (AUTOPILOT)
AIRCRAFT (CONTROLLED ELEMENT)
a
POSITION ESTIMATE A
-
r(t)
t
AIRFRAME ATTITUDE SENSORS
c
+
OBSE&VATlON
at)
RADIO NAVAID (SENSOR)
.
‘(t?;A;pdN’
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variable-gain amplifier and nonlinear smoothing and predicting filter. He can stabilize and control a wide variety of dynamics. Memory and rational guesswork often play an effective part in deciding his control actions.” An autopilot can be used to overcome the limitations of the human pilot. Autopilot feedback control provides speed of response and accuracy of control. These particular advantages are enhanced by high gain. Unfortunately, the combination of fast, high-gain response will adversely affect dynamic stability. If time lags are present in the control system, high gain can cause the system to hunt or oscillate. It also usually increases the susceptibility of the system to spurious signals and noise. For these reasons, an autopilot, unlike the human pilot, usually gives satisfactory performance in only a narrow range of system parameters. As noted in Graham and Lothrop (1955), design of an automatic feedback control system for an aircraft begins with a thorough understanding of the inherent mechanics of aircraft response to its controls. The conventional airplane has at least four primary flying controls-ailerons, rudder, elevator, and throttle-which produce forces and moments on the airframe.6Actuation of these controls gives rise to motions in the six output variables: roll, pitch, heading, forward velocity, sideslip, and angle of attack. These motions (in particular the aircraft attitude change) redirect the velocity vector (whose time integral is the flight path), as shown in Fig. 29. Qualities of an aircraft which tends to make it resist changes in the direction or magnitude of its velocity vector are referred to as stability, while the ease with which the vector may be altered to follow a given course are referred to as the qualities of control. Stability makes a steady, unaccelerated flight path possible, and maneuvers are made with control. 2. Aircraft Guidance and Control Using Navigation Systems
As shown in Fig. 29, the aircraft’s center of mass (c.m.) is positioned on the tip of a position vector r(t). Its time derivative dr(t)/dt is the aircraft velocity v. Its components are measured relative to the same earth reference coordinate system as the position vector. The aircraft’s attitude (pitch, roll, and heading)is measured relative to a translated replica of the ground coordinate system which rides on the tip of the moving position vector. That is, the rotation of the aircraft’sairframe coordinate system (Fig. 29) is defined by the Euler angles relative to the translated earth-fixed coordinates. Similarly, the guidance signal after a suitable coordinate transformation in the airborne navaid is defined with respect to the earth-fixed coordinate frame. Since the aircraft’s position as measured by the navaid and the pilot’s desired position are in the In the cockpit, rotation of the pilot’s wheel moves the ailerons.Pushing the wheel’s column back and forth raises and lowers the elevators.
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ROBERT J. KELLY A N D DANNY R. CUSICK A/C ATTITUDE COORDINATE SYSTEM PITCH =
ep
ROLL = 0 HEADING = 0
AIC POSITION VECTOR
REFERENCE SYSTEM
LATERAL CH-ANNE L COMMAND DEVIATION SIGNAL
FIG.29. Aircraft attitude, position, and velocity coordinate definitions.
same coordinate system, then their difference generates the correct deviation signal. The AFCS for a CTOL aircraft is a two-axis system having lateral (roll) and longitudinal channels. The longitudinal channel is further partitioned into a vertical mode (pitch) and a speed control (thrust) mode. The input signals to these channels depend upon the deviation signal’s relation to the aircraft’s velocity vector. For example, assume as shown in Fig. 29 that the aircraft’s velocity vector v(t,) is in the horizontal plane (tangent plane). Then the component of the deviation signal which is assumed to be also in the horizontal plane and perpendicular to v(t2) will enter the lateral channel of the autopilot. It is important to note that the aircraft, for the practical reasons given below, does not maintain its commanded course using angular correction signals. Each navaid sensor must eventually have its output signal transformed to the rectilinear coordinates of the earth-fixed reference. Although an AFCS can be designed to receive guidance signals in any coordinate system, rectilinear coordinates are preferred because the flight mission is always defined in terms of linear measure (feet, meters). In the aircraft landing application, the aircraft must ultimately fly through a decision window having linear dimensions so that it will land in the touchdown zone located on the runway. This is why the MLS angle accuracy standards (ICAO, 1981a) are given in linear measure and not in angular measure (degrees).
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Similarly, in the enroute and terminal area application, the flight paths (air lanes) are defined and separated in terms of linear measure such that the aircraft cannot “bump” into each other. In both cases, it is the linear deviations from the intended course in feet or nautical miles at each point in space, which are important for aircraft navigation. Guidance data can be derived from any navigation sensors as long as they are transformed into the appropriate rectilinear coordinate system of the autopilot channels. Although not immediately obvious, even straight-in MLS (or ILS) approaches, as illustrated in Fig. 30, require this transformation. In the literature the process is usually called course softening or desensitization. Lateral conversion is obtained by simply multiplying the azimuth guidance angles by the range to the antenna site, using information provided by the DME/P element of the MLS ground equipment. In Fig. 31, the aircraft and the navigation aids are placed in their automatic feedback control system configuration. The block diagram functionally represents one of the two autopilot channels. The intended flight path is the input signal command with the output being the actual aircraft path. The outer loop, also called the guidance loop, contains the aircraft and flight control systems, the ground reference navigation aid (MLS, ILS, DME, TACAN), the airborne sensor, and coordinate transformer. A display device which allows the pilot to monitor or use the measured deviation of the aircraft from the selected flight path completes the system. Unlike today’s ILS, which supports straight-in approaches only, the MLS guidance loop must be prepared to accept any of several types of input signals as itemized in Fig. 31. In addition to the guidance loop, there is a series of inner loops involving the feedback of airframe motion quantities such as attitude (roll and pitch) and attitude rate which serve to control the aircraft. These are called control loops in Fig. 31. As with any closed-loop control system, the AFCS guidance loop has specific gain, bandwidth, and stability properties based on the components of the loop. Separate guidance loops exist for each of the aircraft channels
5
APPROACHING AIRCRAFT WITH ITS COORDINATE SYSTEM
I-:tf(
R = DMEIP
AY
R A 6 AZ
FIG.30. Simple straight-in approach coordinate system.
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ROBERT J. KELLY AND DANNY R. CUSICK CURVED APPROACH PATH WAYPOINTS FOR SEGMENTED APPROACH PATH mSELECTEDGLIDE PATH AND RADIAL
A I R C R A F T. -[NEAR ._...-..-. I
FROM - DEVIATION COMMAND
Y
INPUT SIGNAL
4-
(COMMANDED @OEVIA!IOtFROM POSITION) -f
FLIGHT GAIN +CONTROL SYSTEM
-
POSITION (FEET)
AIRCRAFT 3
COORDINATE TRANSFORM
t
AIRBORNE ANTENNA L
-
RADIO NAVAID
mentioned above. For CTOL aircraft the guidance loop bandwidth is less than 0.5 rad/s for the lateral autopilot channel and less than 1.5 rad/s for the vertical (pitch) mode of the longitudinal channel. 3. Qualitative Description of the Aircraft Control Problem
It is important in the following discussions to understand the distinctions between the navaid output observations, the navigation system steering commands, control law, and control action. These distinctions will be clarified below using as an example the system shown in Fig. 28. Illustrated in the figure is a functional description of an aircraft/guidance system which uses inertially aided position-fix measurements from a radio navaid. The navigation system is composed of the radio navaid sensors, airframe attitude sensors, aircraft position estimator, and the steering command summer. The problem is to control the aircraft such that its flight path r(t) follows the commanded path c(t).How closely r(t) follows c(t)is determined by how timely and accurately the aircraft velocity vector d r / d t can be changed. Automatic flight control systems are easily comprehended using the notation and bookkeeping of the state-variable formalism (Schmidt, 1966).In
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this formalism, drldt is related to r and the airframe attitude states through a system of first-order differential equations. Integrating dr/dt then yields the aircraft flight path. Controlling the aircraft trajectory proceeds according to the following sequence: (1) Radio navaid sensors (e.g., VOR/DME) measure the aircraft position state observable z(t), which contains the aircraft true position states r ( t ) plus random disturbances. (2) Estimating techniques on board the aircraft attempt to extract r ( t ) from z(t). The resulting estimate r ( t ) can be obtained by suppressing the random disturbances using a simple low-pass filter or a sophisticated technique such as a Kalman filter. ( 3 ) The estimate r(t) is compared with c ( t )to determine if control action is required. If they are equal, no added control is necessary. That is, the steering command remains unchanged, which, in turn, does not change the aircraft control surfaces (e.g., ailerons or elevators). (4) A “control law” is used to calculate the control action necessary to make the actual flight path r ( t ) at time t + At equal the command c(t + At). The “control law” must also use the velocity vector dr/dt to ensure that r(t) is stable and does not oscillate around c(t). The velocity input dr/dt can be obtained from r ( t ) directly, or it can be derived from the airframe attitude sensors. ( 5 ) Application of control action in accordance with the “control law” is achieved by changing, for example, the aileron position to bank the aircraft SO that its c.m. position r ( t ) will move to follow the command c(t).
Using the above sequence to fix the ideas contained in Fig. 28 permits the terms guidance and control to be succinctly defined. Guidance (McRuer et al., 1973) is the action of defining the velocity relative to some reference system to be followed by the aircraft. Control is the development and application of appropriate forces and moments that (1) establish some equilibrium state of the aircraft and (2) restore a disturbed aircraft to its equilibrium state. Control thus implicitly denotes that r ( t ) will be a stable flight path which is achieved using both position displacement feedback, r(t), and rate feedback, dr/dt. When performing an analysis, it is the deviations about the intended flight path c ( t )which are important. In this case, a perturbation model can be used which, in many applications, may reduce a nonlinear AFCS to a system of linear first-order differential equations. The following discussions assume that the controller is an autopilot. The key issues associated with a radio navaid and its interaction with AFCS can be clearly presented using the lateral guidance loop as a discussion
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ROBERT J. KELLY AND DANNY R. CUSICK
example. In the following the terms A+, A$, and Ay are deviations from the desired attitude or course. Further the notation A j will represent the velocity of the deviation signal at Ay. The detailed mechanism for turning an aircraft onto a desired flight path proceeds as follows. A lateral position correction signal Ayc (ft) is transformed into a roll command A4,(deg) which causes the aircraft ailerons to deflect a given number of degrees. Relative to position changes in the aircraft c.m., this action occurs rapidly (< 1 s). The moment generated by the air flow across the deflected ailerons causes a change A 4 in the aircraft attitude about its roll axis. After about 1 s A 4 = A$c. Slowly the aircraft velocity vector begins to rotate such that the aircraft’s c.m. departs laterally from its previous flight path until Ay = Ayc. During this time the aircraft heading $ is also changing. A widebody jet will reach this new equilibrium state in about 10seconds. This process continues until the aircraft is on the desired flight path as represented by a new velocity vector with track angle $T. The feedback loop which tracks attitude changes A$ are called control loops; those which track the c.m. changes Ay are called guidance loops. Thus the time constants of the control loops are faster than the time constants for the guidance loops. The accuracy with which the null-seeking loops track the attitude and position commands is dependent on loop gain, which, in turn, must be chosen to ensure that they remain stable over the range of input parameter changes. An unstable control loop would lead to uncontrolled attitude variations, which, unfortunately, would not be related to the steering input command signals. Under such conditions, the aircraft cannot be controlled to stay on the flight path. In the following discussions it will be assumed that the control loops are stable. Attention will now focus on the requirements for a stabilized guidance loop. A small tracking error, sufficient guidance loop stability margin, and good pilot acceptance factors are the critical performance indices necessary to achieve a successfulautocoupled guidance. Each of these points will be treated separately. Clearly, an autocoupled aircraft must accurately follow the desired track angle in the presence of steady winds, gusts, and wind shears. To correct the differencebetween the desired course (input) and the aircraft track (output), the guidance system must obtain the position and velocity of the aircraft and determining its heading. Assume a pilot wants to turn the aircraft to follow a new course which intersects his present course to the right. His navigation system will provide a displacementcorrection signal Ay to the autopilot which in turn generates a roll command to bank the aircraft. If only the position displacement signal were used, it is apparent that a stable approach could not be achieved, because the aircraft would continuously turn as it approaches the new course and overshoot it. That is, the air-
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craft’s c.m. would be accelerated toward the new course in proportion to the distance off course as if it were constrained to the course by a spring. Without damping, the c.m. would oscillate continuously about the new course. To prevent this unstable maneuver, a term related to the rate of closing toward the new course must be introduced. That is, to use the above analogy, the spring requires damping. Such a “rate” term A j is available either by differentiatingdirectly the position displacement A y with respect to time or by using the relative angle between the aircraft track angle and the new course bearing. In either case, a signal proportional A j can be derived to provide dynamic damping. An equally important situation arises when the aircraft is subject to steady winds and wind turbulence. Because an aircraft has a response time inversely proportional to its guidance loop bandwidth, presence of wind should be detected as soon as possible. Inserting a rate signal A j in addition to Ay into the autopilot anticipates the wind disturbance so that the aircraft’s attitude can be adjusted to counter the effects of wind force and moments which may push it off course. These examples qualitatively demonstrate why both position and velocity information are necessary for stable flight path control. To achieve the tracking accuracy necessary to guide the aircraft through the Category I1 window on a final approach places additional demands on the feedback signals. The aircraft must be tightly coupled to the guidance signal, which, in engineering terms, means that its guidance loop gain must be large. High guidance loop gain, however, is synonymous with high guidance loop bandwidth. A wide guidance loop bandwidth is necessary so that the highfrequency components of the wind gusts can be counteracted by the AFCS. This is desirable to decrease the influence of wind, wind shears, and gusts which tend to cause the aircraft to depart from its intended approach and landing path. On the other hand, tight coupling is undesirable if the guidance signal is noisy or has guidance anomalies caused by multipath. In this case, the tightly coupled aircraft will tend to follow the anomalies and depart from its intended approach and landing path. Regardless of whether the departures from the intended path are because of winds, shears, and gusts or due to anomalies in the received signals, they are tracking errors. With this background the discussion will move on to examine the performance of the radio navaid. Unlike wind disturbances, which can be attenuated by increasing the loop gain, guidance signal noise will not be attenuated because its source is not in the forward portion of the loop. Increasing the loop gain (or bandwidth) amplifies the effects of the noise. The situation is further compounded when the rate feedback signal is derived from the guidance signal, since the differentiating process accentuates the high-frequency components of the noise.
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Reducing the guidance loop gain is not always a solution because the effects of wind gusts and shears may increase. Removing the higher-frequency components from the wind disturbance means that they are not available as correction signals. The consequence is larger tracking errors. Narrow loop bandwidths (low gain) will also degrade the aircraft's transient response, perhaps to a point which is unacceptable to the pilot. Heavy filtering of the navaid output signal does not alleviate the above problems. The data filtering process introduces phase delays on the guidance signal that may erode the AFCS stability margin, which is only about 60" to 90" to begin with. That is, an additional phase delay of 60" to 90" will destabilize the system. There is yet a third problem. To maintain a sufficient phase margin, the navaid data filter bandwidth must be much wider than the guidance filter bandwidth. Guidance signal noise may disturb the roll and pitch control loops and also the control surfaces (ailerons and elevators). Since the wheel is coupled to the ailerons and the column is coupled to the elevators, unwanted control activity may also induce unwanted wheel and column motion. During an autocoupled approach, excessive control activity can cause the pilot to lose confidence in the AFCS. Nevertheless, although these noise components cause airframe attitude fluctuation and control surface activity, their spectral content lies outside the guidance loop bandwidth. The induced motions are too rapid to influence the direction of the aircraft's c.m. and therefore do not afect the aircraf's Jlight path. How are the above noisy navaid issues resolved? For some autocoupled enroute applications using VOR/DME and for some ILS approaches, the solutions include (1) doing nothing and accepting the unwanted control activity, (2) decreasing the loop position gain and accepting larger tracking errors, (3) using inertial aiding for dynamic damping, (4)reducing the control activity by reducing the autopilot gain and the navaid filter bandwidth. Clearly the solutions involve trade-offs between tracking error, stability margin, windproofing, control activity, and costs. A more sophisticated solution is the complementary filter.' Subsection E,6 describes the complementary filter approach used for more demanding ILS applications. MLS system designers took the approach that the radio navaid designs must begin with a low-noise output signal. The MLS angle and the DME/P standards, which are the most recent navaid standards approved by ICAO,
' A complementary filter blends the good high-frequency characteristics of the inertial sensors with the good low-frequency characteristics of the radio navaids. The overall combination will then have the desired wideband frequency response with low sensor noise and minimum data phase lag. A complementary filter permits the radio navaid output data to be heavily filtered if necessary.
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attempt to correct the deficiencies of the previous systems by purposely defining low-noise accuracy standards even in the presence of multipath. Despite these more stringent requirements, the MLS signal-in-space may require some degree of inertial aiding or complementary filtering to provide the high dynamic damping and the position accuracy necessary for achieving successful Category I1 approaches, Category I11 landings, and automatic landings under VFR conditions.
4. Radio Navaid Data Rate In all AFCSs the data sampled guidance signal must eventually be reconstructed back into an analog signal since, in the final analysis, the aircraft moments and wind forces are analog by nature. For sampled data systems such as MLS, DME, and TACAN, the position samples must be provided at an adequate data rate so that the signal can be faithfully reproduced. Undersampling can generate an additional tracking error component and can induce an additional phase delay after the signal is reconstructed. For example, guidance deviations caused by wind gusts must be accurately reconstructed so that the control loops can apply the appropriate compensating corrections in a timely manner. Reconstruction circuits (e.g., a zero-order hold, ZOH) introduce phase delay in addition to the data smoothing filter. In order to provide a low-noise guidance signal with a bandwidth that satisfies the requirements developed in Subsection E,3, the navaid data rate must be high enough to (1) minimize the signal reconstruction errors and the reconstruction circuit phase delays, and (2) allow data smoothing while still providing enough independent samples to reconstruct the signal. It can be shown that if thebutput data filter satisfies a six-degree phase delay criteria, then it will also satisfy the data reconstruction requirements for independent samples (Ragazzini and Franklin, 1958).Therefore, only the six-degree phase delay filter criterion need be satisfied, as discussed in Subsection E,7. The second requirement is that the data rates must be high enough to permit filtering of the raw data samples if the CMN is excessive. For a given noise reduction factor g, the second data requirement can be derived from Eq. (40)in Section IV,D,l. It is data rate = f , = 29' (noise BW), where (noise BW) = (a/2)wfi, for a single-pole filter. For example, if the raw CMN is 0.14, then to achieve 0.05" CMN at the input to the AFCS requires that g 2 = 8. Assuming mfi, = 10 rad/s, then the data rate must be at least 40 Hz. Figure 37 illustrates the concepts described above. The data rate relation applies when the noise source has a broad power spectral density such as the MLS azimuth function. If the radio navaid sensor cannot provide an adequate number of independent samples, some form of inertial ordering will
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be necessary. Example implementations are given later in Fig. 38.’ The need for small tracking errors and wide-bandwidth navaid filters that minimize phase delay points toward data rates which are about 20 times the loop guidance bandwidth. 5 . Functional Description of AFCS The conclusions in Subsections E,3 and E,4 placed qualitative accuracy, bandwidth, and data rate constraints on the output data of radio navaid sensors. With these constraints an aircraft guidance system can be designed that is accurate, responsive, and stable. In this section, the key issues will be revisited using the mathematical formalism of control theory. This elementary review will serve to fix clearly the ideas treated earlier and to indicate how the aircraft and its navigation system can be analyzed quantitatively. For the moment, the AFCS functional description will be general, applying either to the lateral or to the vertical channels. Figure 32 is the basic control system, where K is the autopilot gain, H,(s) and H2(s) represent the transfer functions of the autopilot, aircraft, and the aircraft dynamics. F(s) is the transfer function of the sensor, s is the Laplace transform variable, and the input/output functions, c(t) and r(t), are scalar quantities. In an ideal control system the controlled output r ( t ) should followthe commanded input c(t) with the smallest mean-square tracking error e(t)’, where e(t) = c ( t ) - r(t). This is the optimum meaning of accuracy, fast response, and stability. The intended purpose of the AFCS is to keep the aircraft as close as possible to its intended course c(t), even in the presence of wind gusts d(t) and with an imperfect radio navaid having error n(t). Let R(s) and C(s) be the Laplace transforms of r ( t ) and c(t), respectively. For the time being, let the sensor error source n(t) = 0 and the disturbance d ( t ) = 0; then, from Fig. 32,
Equation (1) is the closed-loop transfer function of the aircraft’s guidance loop. The 3 dB bandwidth of R(s)/C(s)is the loop guidance bandwidth w 3dB; the forward transfer function is KH,(s)H,(s), and the open-loop transfer function is KF(s)H,(s)H,(s). For radio navaids, F(s)is equivalent to a low-pass filter. As stated in Subsection E,3, the 3 dB bandwidth of the navaid sensor transfer function F(s) should be much larger than the loop guidance bandwidth w~~~ for two reasons. One, the phase delays of the corrective signal’s components should not erode the phase stability margin of R(s)/C(s).
* Accuracy and data rate limitations are among other considerations that may preclude GPS from being used as a primary navaid in approach/landing applications.
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WIND DISTURBANCE
d(t)
GAIN K
AIRCRAFT
FIG.32. Generalized block diagram of attitude controlled A/C and navaid.
Two, the spectral component of c ( t ) and d ( t ) should be faithfully reproduced so that e(t)2 is maintained within design limits. When F(s) cannot satisfy these requirements, the radio navaid may require some form of inertial aiding. An important design consideration is the effect of sensor disturbances upon the tracking error. The AFCS should respond to the desired signal c(t), but not respond to extraneous noise sources. Let N(s)be the Laplace transform of n ( t ) the error term of the navaid sensor. In Fig. 32, let c(t)=O and d ( t ) = O ; then the closed-loop transfer function R(s)/N(s)for the noise source is identical to Eq. (1). Again, the navaid sensor determines the guidance system performance. Unfortunately, the characteristic that F(s) should possess for small tracking errors, when combined with an imperfect navaid, is in conflict with those required for the command signal c(t). For the command signal, the bandwidth of F(s) should be greater than co3dB, while, for the navaid error source n(t), the bandwidth of F(s) should be small. In other words, the spectral content of c ( t ) should be unaltered while that of n(t) should be suppressed when the variance of n(t) is excessive. An important property of a closed-loop AFCS transfer function is its capability to suppress wind disturbances. Again the essential design consideration is high loop gain. In Fig. 32 let D(s)be the Laplace transform of d ( t ) and let c(t) = n ( t ) = 0. Using Fig. 34 the closed loop transform for a wind disturbance is R(s) - D(s)
H,(s) 1 + KF(s)H,(s)H*(s)
(2)
The effects of negative feedback on d ( t )can be compared with c ( t )and n ( t ) by inserting the wind disturbances on the input side of H(s). The transfer functions for all three sources are now equal to Eq. (1) if D(s) is transformed
90
ROBERT J. KELLY A N D D A N N Y R. CUSICK
into D(s)/KH,(s).Wind disturbances are therefore reduced by large autopilot gain. In summary, the loop bandwidth, w 3 d B , must be sufficiently large to reproduce c ( t ) faithfully and attenuate d ( t ) , but not to the extent that n(t) introduces excessive tracking errors. After choosing w3 dB, the navaid noise can be filtered to a level consistent with the desired control activity and windproofing using a complementary filter. The formalism above implicitly assumes the control loops needed for dynamic damping. The tracking error e ( t ) thus contains three components: the effects of the band-limited signal c(t), the wind disturbance d(t), and the radio sensor error n(t). a. Lateral Channel Control Law. The control law for the lateral channel will now be developed. The lateral channel is chosen because essentially the same control law is used in the enroute, terminal, approach, and landing phases of flight. As stated in Subsection E,3, it is necessary that the bank angle command be proportional to the lateral deviation from the desired flight path. Moreover, in order for the flight path to be stable, the bank command must also be proportional to the rate of change of the path deviation. Consequently, the bank angle control law assumes the form (3) A 4 c = K , A y +- K + A j Not included in Eq. (3) is integrated feedback, which removes any dc offset errors. Control law gain coefficients are chosen to ensure a given stability margin while maintaining an operationally acceptable tracking accuracy. It is an involved analysis using root locus techniques (see Bleeg, 1972,Appendix B). Although the derivation of these coefficients requires extensive analysis, once they are determined, 03dB can be derived using simple relations such as those given in Fig. 33(b). This is fortuitous because the guidance loop bandwidth (03 dB) is critical to the navaid filter design. The control loop's performance, on the other hand, will not succumb to a simple analyses; a full-blown computer simulation is necessary. At this point, the design engineer must make several decisions. From what sensor should the lateral path deviation A y and its velocity A j be measured? Ideally, one would want to obtain A y from the ground reference navaid, and using that position measurement, derive the velocity data needed for dynamic damping. Relative to inertial aiding this is the most economical approach. However, as noted in Subsection E,3 there are several important considerations depending upon which phase of flight is being addressed. As expected, the autocoupled final approach/landing phase has the most stringent requirements. Anomalies in A y and A j must be limited such that unwanted roll commands A$c as expressed by Eq. (3) are limited to less than 2" (95% probability) (paragraph 2.1.4, Attachment C, ICAO, 1972b; Kelly, 1977) and Fig. 38. It is the opinion of many fight control engineers that the 2" roll
DISTANCE MEASURING EQUIPMENT IN AVIATION
91
requirement for autocoupled flight also applies to the terminal area and enroute applications. Control law considerations for MLS are discussed in Section II,E,6. For the enroute application, a wide variety of control law implementations are available to the user, depending upon whether he flies the VORTAC radials or uses RNAV routes, and whether he files manual or has his RNAV outputs coupled directly to the autopilot. When flying manual, the displayed VOR and DME raw data may be adequate because the pilot is more forgiving when he is in control of the aircraft. However, when the aircraft is in the autocoupled mode, autopilot tracking errors are viewed more critically by the pilot because he is not in direct control of the aircraft. Thus, DME and VOR position data for the autocoupled mode may require velocity inertial aiding. Anomalous attitude or control surface activity under no wind conditions is disconcerting to the pilot and may result in negative pilot acceptance factors. On the other hand, the pilot expects control activity when he encounters air turbulance. Small tracking errors in the presence of wind and good pilot acceptance factors are desirable because pilot work loads can be reduced and fuel consumption can be minimized. Only when the pilot can use the autocoupled mode with confidence will these goals be achieved. Although some ILS landing operations require the position data to be complemented with inertial aiding (see Fig. 38c), the major concern is the accuracy of the velocity information. Only the velocity information considerations will be treated below. In order to address explicitly which variable can be used to provide the velocity feedback components, the physics and geometry of how an aircraft negotiates a lateral turn must be analyzed. Figure 33 illustrates such an analysis, where a small-signal perturbation model has been developed, i.e., c ( t ) - r ( t ) = Ay when the desired path has changed from c ( t ) = 0 to c(t) = Ay. The necessary variables can be determined using Eqs. (1)-(5) in Fig. 33(b). They are derived by successive integrations beginning with the equation of motion for a coordinated turn [Eq. (I)]. Equations (1)-(5) illustrate how the velocity vector is changed so as to achieve a desired flight path. For ease of exposition the equations of motion in Fig. 33, assume that the direction of the aircraft’s c.m., ICIT, equals $; i.e., the effects of wind are negligible. Also assumed is the complete decoupling between the roll and directional axes. Since o3dB is small compared with the natural frequency of the bank angle transfer function 6/OCN 1. The actuator dynamics have also been neglected. The variables available for dynamic damping are: (1) A j as derived from Ay [Eq. (4)] (2) bL as integrated from 4, where CEq. (113
6 is
measured by a vertical gyro
ROBERT J. KELLY AND DANNY R. CUSICK
92 a
AIRBORNE RECEIVER
P. 0
e.g VORIDME
i
”,
@ ‘BANK ANGLE (DEGREES) $ = AIC HEADING IDEGREESI
GROUND REFERENCE NAVAIDS e.9. VORIDME
p w =WIND GUST
V, = AIC AIRSPEED (FEETISECOND)
9
=
GRAVITY I32 FEETISECOND’)
$
=
LAPLACE INTEGRATION OPERATOR
W
=
-
IDEGREESl
WIND SPEED IFEETISECOND)
S = LAPLACE DIFFERENTIATION OPERATOR
NOTE: A y IS I N THE HORIZONTAL PLANE NORMAL TO c(tl
FIG.33. (a) Lateral channel perturbation model.
(3) $ (heading), as measured by the directional gyro or radio compass rEq. (3)1 (4) $T (track angle), as determined from v and measured by referenced accelerometers. Under wind conditions $T # $ at point A in Fig. 33. $T = tan-’(j/i), where v = From v the feedback correction term A$T is calculated, where is the difference between the commanded track angle and the true track angle $T. (See Fig. 11.)
Jm.
Assuming that A j cannot be derived accurately from the radio navaid, A$T is the ideal input for dynamic damping, since it is based upon the true tracking angle of the aircraft. However, the determination of At)T requires an expensive INS platform or its equivalent in strap-down inertial sensors. Less accurate, but perhaps adequate, tracking performance can be obtained from the heading $ and the bank angle 4 to obtain the damping signals. In the 1950s and 1960s, heading feedback was the solution used in many autopilots for approach and landing operations. However, when the aircraft is subjected to a low-frequency crosswind, the tendency of the airplane is to
DISTANCE MEASURING EQUIPMENT IN AVIATION
ASSUME
(1)
va
=
mv2 =
Ra
93
7 mg Tan A@
BUT V = A $ X R,
THEREFORE
t mg
FIG.33. (b) Physics of a lateral turn.
"weathercock" in the wrong direction. This produces a heading change which is not related to +T (Bleeg et al., 1972). In general
$*=*,+*=*,+-
:sd
4dt
(4)
where $, is the initial wind heading transient, 4 is a function of time, and the wind disturbance is injected as indicated in Fig. 33. Thus, using $ as a damping term, is not an ideal solution in the presence of steady wind and low-frequency wind shears. Before the advent of INS, the means to obtain dynamic damping and windproofing was to measure the bank angle directly and to perform a long-term constant integration, as indicated by
94
ROBERT J. KELLY AND DANNY R. CUSICK
Eq. (4). A solution was to incorporate the operation called lagged-roll, as indicated in Fig. 33. However, pure integration was not desirable because the airplane may be mistrimmed; i.e., it may have a steady roll angle which could prevail throughout the approach.’ Another approach to obtaining dynamic damping is to combine heading with a rate signal derived from the imperfect navaid sensor in a complementary filter. The heading signal has good highfrequency response and correspondingly poor low-frequency wind response while the navaid rate signal has good low-frequency characteristics. Such an example is given in Brown (1983) for the ILS application. Because the exposition which follows will be simpler and clearer, $ will be used to obtain A j with the full understanding of the limitation mentioned above. The use of K , A$ = K+A j will allow an estimate to be made in the tracking error as generated by the sensor noise errors. Equation (3) now becomes
A4= = K , A y
+
(5)
The block diagram in Fig. 33 is obtained by starting with Eq. (5) and using Eqs. (2)-(5) of Fig. 33. The blocks with dotted inputs and outputs are the alternate feedback terms A j and A&. Wind effects which push the aircraft c.m. off course can be included by a summing junction in Fig. 33. For a steady crosswind, the input to the summing junction is a step function. The resulting path deviation A y would be a ramp function. A wind shear would be represented by a ramp function at the summing junction input. The composite transfer function using only heading feedback is given in Fig. 34. The analysis assumes that the sensor output data filter bandwidth is large compared with the guidance loop bandwidth (look ahead to Fig. 38). As shown in the figure, the transfer function is a low-pass filter of second order, which is given in standard form in terms of the damping factor t and the natural frequency wo to emphasize the important performance factors. Also given is the expression relating the loop guidance bandwidth w3dB to w o . By choosing K , and K,, the loop guidance bandwidth wJdBand t can be adjusted to suit the enroute, terminal, approach, and landing phases of flight. For the approach/landing phase 5 z 0.5-0.7 and w3dB2: 0.1-0.2 rad/s. Note that if K , = 0 (no damping, t = 0) then the guidance loop is conditionally stable; i.e., it will become unstable with any input. This condition dramatizes the need for velocity information in the aircraft guidance loop. As shown in Fig. 34, increasing or decreasing K y changes w 3dB as mentioned several times in Subsection E,3. If rate feedback were incorporated as shown in Fig. 33(a), then wo = 4and 5 = 3(Kj/K,)wo. Using I) in the control loop to maintain a constant aircraft heading will cause the aircraft c.m. to drift off course in the presence of wind. To avoid drift errors in the approach/landing application the aircraft “decrabs”just before touchdown.
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95
NATURALFREQUENCY OF THE GUIOANCE LOOP
STANOAROIZEO TRANSFER FUNCTION
Jc
LOOP GUIDANCE BANOWIOTH w3db =wo
KII, IS IN '$)/
t 2,
+
2-4
t 2+ 4
OJ/
NOTE: THE COEFFICIENTS VARY SLOWLY WITH TIME AS A FUNCTION OF THE AFCS GAIN SCHEDULING
KylSIN O@/FT W o IS
1-2
I N RAOIAN / SEC
FIG.34. Transfer function of lateral channel perturbation model.
Using certain simplifying assumptions, the desired tracking error can be easily calculated. The mean-square tracking error e ( t ) = c ( t ) - r ( t ) is defined by
S'
-
e(t)2 = lim e 2 ( t ) d t= T - ~ -~T T
where E(s) is the Laplace transform of e(t). If the input command signal c(t), the wind distance d(t), and the radio navaid errors n ( t ) are zero mean uncorrelated random variables, then 3
2
=
C(t)++ d(t)++ n(t)+
(7)
96
ROBERT J. KELLY AND DANNY R. CUSICK
where each component on the right is the tracking error contribution due to c(t),d ( t ) , and n(t), respectively. Data sampled reconstruction effects have not been included. (See Subsection E,4.) Note that inertial aiding component errors are assumed to be negligible. Assume in the following that the bandwidth of c(t)is well inside w 3dB, then c(t)gwill be small. Assume also that the autopilot gain is sufficient to suppress the effects of d(t), then under these assumptions, e(t)’ z n(t)+. Since the guidance loop transfer function H ( s ) has been derived in Fig. 34 using heading as the damping term, the tracking error due to sensor error perturbations can now be calculated. Using Eq. (6)
where E [ s ] = H ( s ) N ( s )and N ( s ) is given by (9) below. Let the sensor noise be represented by a Gauss-Markov process (Brown, 1983), then its power spectral density is
where the sensor filter bandwidth 0, >> W j d B and 0,’is the noise power at the output of the sensor. Then N(s) = m
/
(
S
+ w,)
where, for example, in the autocoupled approach/landing application w , = 10 rad/s. Performing the integration in Eq. (8) using the integrals evaluated in James et al. (1947), the tracking error is
-
e(t)2= o ~ K , u / w , K , ( 5 7 . 3 )
(10)
Thus, although the tracking error due to a sensor with noise errors is proportional to autopilot gain K,, it is also inversely proportional to K,. For a given noise power spectral density S,(w) and aircraft speed u, the tracking error can be reduced by decreasing the ratio K,/K,. Since there are two degrees of freedom, K , is selected to achieve the desired dynamic damping, while K, is adjusted to obtain the desired tracking error. The above analysis assumed that the noise error of the heading feedback was negligible. If the navaid error n ( t ) has a bias component, it would be added to Eq. (10) on a root-sum-square basis as indicated in Subsection C,3. Equation (10) has a simple physical meaning when w, >> w J d B . The noise power spectral density given in Eq. (9) becomes S , N 2a,’/w,, a constant. Using the relations for K , and K, in Fig. 34 and letting = 1/& then
DISTANCE MEASURING EQUIPMENT IN AVIATION
wo = w3dB and Eq. (10) becomes = Snw3 dB/(2
97
a>
The tracking error is simply that portion of the noise power residing inside the guidance loop bandwidth which is intuitively satisfying. A similar analysis could be performed to determine the effects of wind gusts on tracking accuracy. Equation (9) can also be used to simulate wind gusts. The above discussion was meant only to introduce the basic uses of AFCS design and to show why it is necessary to have radio navaids which can provide wide-band guidance (about 10 x w3dB)with a low-noise output. A more complete discussion of the lateral channel as it applies to the approach and landing maneuver is given in McRuer and Johnson (1972). In the next section it is shown how the complementary filter permits the designer to achieve system stability without regard to the time constant introduced by heavily filtering the noisy radio navaid output. 6. The Notion of the Complementary Filter
In the preceding sections, the problem of using a radio navaid sensor with a noisy output continually figured highly in all the discussions. The problem was how to filter the noise without introducing excessive data delays. It was suggested in the earlier narrative that the solution involved the use of inertially derived position data which would augment the filtered radio navaid data. Also discussed were ways of providing dynamic damping using inertial sensors such as vertical gyros to replace rate signals derived from the radio navaid position fixes. The problem now is how to augment the position data fixes with dead reckoning position data obtained from an inertial sensor. The idea is that the radio navaid can obtain good low-frequency data after filtering while the inertial sensors (accelerometers) have good high-frequency characteristics (i.e., small high-frequency noise components). Drift errors are typically the source of the poor low-frequency performance exhibited by inertial sensors. Radio navaids have goods-frequency response because ground field monitors ensure that the bias errors are maintained within acceptable limits. The technique of combining the two sensor outputs to yield a wide-band filter without phase delays, signal distortion, or excessive noise error is called the complementary filter. The concept assumes each source has the same signal, but the measurement noise from each source is different. That is, both the radio navaid and the inertial sensor are measuring the same navigation parameter, only their measurement noises are different. The development below follows Brown (1973). Let s(t) be the position of the aircraft in flight, and let nl(t) be the inertial sensor noise and n2(t)be the radio navaid noise. Configure the filter as shown
ROBERT J. KELLY AND DANNY R. CUSICK
98
in Fig. 35(a). The output is the estimate $(to = z(t). The Laplace transform of the output is
Z ( S )=
S(S)
+ N,(s)[l - G(s)] + N ~ ( s ) G ( s )
Signal term
Noise terms
(1 1)
From Eq. (11)the signal term is not affected by the choice of G(s) while the noise terms are modified by the filter. The point here is that the noise can be reduced without affecting the signal in any way. If, as in the navigation problem, the two sensors have complementary spectral characteristics, then G(s) can simply be a low-pass filter and the configuration in Fig. 35(a) will attenuate both noise sources. Finding the optimum cutoff point of the lowpass filter G(s) is an optimization problem which can be solved using the Wiener filter theory. The filter configuration in Fig. 35(a) can be rearranged into the feed-forward configuration shown in Fig. 35(b). An equivalent feedback configuration is discussed in Gelb and Sutherland (1971). The feed-forward implementation of Fig. 5(b) provides some insight into how the complementary filter works. The goal is to eliminate the signal in the lower leg by substitution thereby producing a new input signal nl(t) - n2(t).A filter design is chosen to obtain the best estimate of nl(t) in the presence of a distrubing signal n2(t).The estimate n,(t) is then subtracted from the sensor output s(t) nl(t), yielding an estimate of s(t). Intuitively, the filter mechanism is clarified if n,(t) is predominantly high-frequency noise and n2(t) is predominantly low-frequency noise. In this case, 1 - G(s) is a high-pass filter which passes nl(t) while rejecting n2(t). The Kalman filter gain optimization for the case where aircraft range is determined from a radar and the double
+
FIG.35. (a) Basic complementary filter. (b) Feed-forward complementary filter configuration.
99
DISTANCE MEASURING EQUIPMENT IN AVIATION
integration of a strap-down accelerometer is noted in Maybeck (1979). The Kalman filter overcomes the limitations of the Wiener filter which is only optimum for stationary random processes. The configuration in Fig. 35(d) forms the basis of the inertially aided navigation system, which is the cornerstone of many modern flight management systems currently in use on new jet transports. Figure 36 (Brown, 1983; Bobick and Bryson, 1970; Gelb and Sutherland, 1971; Karatsinides and Bush, 1984) is the state vector representation of Fig. 35(b), where multiple sensors are used as inputs. It employs a linear Kalman filter which assumes that the difference between the inertial and the aiding sensor errors is small and therefore justifies the use of a linear filter. Thus the aided INS is a dead reckoning system which uses redundant navigation information from other sensors to compensate for its own error sources. The comparison of the complementary filter with the Kalman filter is given in Brown (1973) and Higgins (1975). It should be noted that the complementary filter can also be used to obtain the dynamic damping feedback by complementing radio navaid rate data with an inertial source such as the track angle error The output of the complementary filter is multiplied by the appropriate gain constant and inserted into the control law defined by Eq. (3). For example, a velocity complementary filter output would be A j c . The state vector configuration of the complementary filter is known in the literature as a “multisensor navigation system” (Fried, 1974; Zimmerman, 1969). The concept is very general. Although the example in Fig. 36 used the INS as the primary sensor, depending upon the application, radio navaid can also serve as the primary sensor. 7. Concepts of PFE and CMN
In this section the concepts of path following error (PFE) and control motion noise (CMN) are introduced together with the rationale for defining the guidance errors in terms of these concepts. They are generalizations of the traditional measures of error, namely, bias and noise. These concepts, TRUE VALUES + INERTIAL ERRORS BEST ESTIMATE OF POSITION & VELOCITY
INS
I
I\
AIDING SENSORS
KALMAN FILTER
INERTIAL ERRORS
TRUEVALUES MEAS ERROR
t
FIG.36. State vector complementaryfilter configuration.
100
ROBERT J. KELLY AND DANNY R. CUSICK
originally defined in Kelly (1974), are the accuracy terminology used in specifyingthe MLS angle and the DME/P signal in space. These concepts are very general and apply to all navaids as well as MLS. The MLS angular error or DME/P range error is the difference between the airborne receiver processed sampled data output and the true position angle at the sampling time. The MLS guidance signal is distorted by ground/airborne equipment imperfections and by propagation-induced perturbations such as multipath and diffraction. To fully delineate the signal-inspace quality, these perturbations are viewed in the frequency domain, that is, as an error spectrum." This representation allows the interaction between MLS errors and the AFCS to be understood in terms of the parameters which define a successful landing. Associated with the signal error spectrum is the idea of the AFCS frequency response. The frequency response of the aircraft dynamics can be divided into three major spectral regions-a low-, middle-, and high-frequency region. Guidance error fluctuations in the low-frequency region fall within the aircraft's guidance loop bandwidth and, since they can be tracked, cause physical displacement of the aircraft. The effect of these lateral or vertical displacements is measured in terms of unsuccessful landings. The middle frequencies result in attitude changes (roll and pitch) and in control surface activity whose motions are too rapid for the aircraft to follow. They also include induced wheel (lateral channel) and column motions (longitudinal channel). Control surface motions (ailerons and elevator) can lead to component wear-out and fatigue, whereas the column and wheel motion affect pilot acceptance factors. Figure 37 summarizes the azimuth control activity criteria used by the airframe industry. One measure of these pilot acceptance factors is the Cooper Rating (Hazeltine, 1972). Although subjective and not universally accepted, it is the only test currently in wide use for measuring these effects.The high-frequency regime, which begins around 10 rad/s, does not affect the aircraft guidance system or pilot and passenger comfort measures. Figure 37, which was taken from Neal (1975),illustrates the lateral channel spectral response of a typical transport aircraft. Note that the roll and aileron responses cannot be approximated by a simple filter as can the guidance loop response (Fig. 34). Based upon the above considerations, ICAO defined PFE and CMN in the MLS angle and DME/P SARPs (ICAO, 1981a, 1985). The SARPs definitions for DME/P are: PFE-that
portion of the guidance signal error which could cause aircraft
l o The error spectrum is defined as the Fourier transform of the flight test error trace. The flight test error trace is the difference between the MLS measurement and the actual position of the aircraft.
DISTANCE MEASURING EQUIPMENT IN AVIATION
NAVAIO
OUTPUT OATA
SENSOR
OlGlTAL FILTER
ZERO OROER ANALOGUE
BW 'WF IL
BW-fJZ
101
OELAV
-
OUTPUT :WJdbT,
w3dB = GUIDANCE LOOP BANOWIOTH
'
W3db
wFIL
fr
fJz
AZIMUTH CONTROL ACTIVITY CRITERIA COOPER RATING: LATERAL ACCELERATION: ROLL ATTITUDE VARIATION: CONTROL WHEEL VARIATION: AILERON DEFLECTION: CONDITION:
0.01
1
COOPER 11, PLEASANT TO FLY 0.04 0
2 1'
i2'
2 20
+5 2'
+5 So
15.OOO FT RUNWAY, AT THRESHOLD
0.10 0.8 1.0 SIGNAL E ~ O R SPECTRUM FREOUENCY IRADISECI __*
2.0
10
FIG.37. (a) Output data filter and data reconstruction unit. (b) Typical transport aircraft power spectral response.
102
ROBERT J. KELLY AND DANNY R. CUSICK
displacement from the desired course and/or glide path. PFE frequency regions are 0-1.5 rad/s for elevation and 0-0.5 rad/s for azimuth. For the angle SARPS the PFE is partitioned into 2 additional components, pathfollowing noise (PFN) and bias. PFN is the PFE component with the bias removed. CMN-that portion of the guidance signal error which could affect aircraft attitude angle and cause control surface, wheel, and column motion during coupled flight, but which does not cause aircraft displacement from the desired course and/or glide path. The frequency regions for CMN are 0.5-10 rad/s for elevation and 0.3-10 rad/s for azimuth. Clearly, an excellent guidance signal would result if all frequency components above the path-following error regime were suppressed by a smoothing filter placed at the output of the navaid receiver. However, this cannot be accomplished without introducing a large phase lag in the higher path-following frequency components with the resulting adverse effects on AFCS stability discussed in Subsections E,3 and E,4. The MLS community chose a 10-rad/s, single-pole filter that induces less than a 6" phase lag for a 1-rad/s path-following signal component. In other words, the navaid filter should be 2 10 times the guidance loop bandwidth. The penality is, of course, that the higher nonguidance (i.e., control motion) frequency components become inputs to the AFCS:As shown in Fig. 37, the MLS and DME/P SARPs permit all MLS angle receivers and DME/P interrogators to band limit their output data to 2 10 rad/s using a single-pole filter or its equivalent. Based on the 0.2 rad/s guidance loop bandwidth shown in Fig. 37, the MLS data can be filtered (down to 2 rad/s) while still maintaining the I6" phase lag. A measurement methodology is also defined to determine if the MLS angle and DME/P equipment satisfy the PFE and CMN error specifications(Kelly, 1977; Redlien and Kelly 1981). Using the definitions of PFE and CMN above, the question of obtaining rate information from the radio navaid can be revisited. The control law in Eq. (3) requires knowledge of the bank-angle transfer function, the pseudoderivative transfer function, and most importantly the power spectral density of the MLS azimuth signal before the unwanted control activity can be estimated (see Fig. 37). An example of an MLS control law proposed for autocoupled approaches in the terminal area (Feather, 1985) is
A4c = 0.026 Ay + 0.49 A j Complementary filters using simple inertial sensors (i.e., non-INS) were used to ensure that the anomalies in Ay and A j did not cause A4c to exceed the 2" limit. MLS must be operational for several years before a statistical representation of the MLS signal can be defined. Only then will the required degree of inertial aiding for each aircraft type be accurately known. Based upon flight
DISTANCE MEASURING EQUIPMENT IN AVIATION
103
test data collected during the MLS development program, there is reason to believe that the MLS error signal at many airports may approach an essentially flat power spectral density function. Multipath bursts, when they exist, may be of short duration. If this is the case, then at the very least complementary rate filter designs, when required, will be easier to implement in MLS-equipped aircraft. MLS control laws, as well as the tracking error and control activity criteria, are currently under development not only for the autocoupled mode, but also for flight director and manual-coupled flight. These development programs are directed at approach and landing operations in the terminal area. Critical issues are the following questions: What are the operational benefits of a curved approach? Given that the autopilot error is less than 30 ft at the Category I1 window in the presence of steady wind and wind turbulance, is an autopilot error larger than 100 to 150 ft acceptable in the terminal area? Similar control law design considerations are relevant to the enroute application for the "cruise mode." One control law currently in use for aircraft speeds of 500 knots at 30,000 ft is A4c = (0.0025 deg/ft)Ay + (1.6deg/deg)A+,. Using 5 = and the equation in Fig. 34, the guidance bandwidth for this control law is 0.04 rad/s. In this case, the position deviation Ay can be derived from VOR/DME or DME/DME. The dynamic damping is derived from the track angle deviation A+T as determined from an INS. The roll anomaly for the above cruise mode control law will be well within 2". In summary, navaid sensors (not only MLS) should satisfy the following three criteria or inertial aiding may be required:
l/fi
(1) PFE, when combined with either the flight technical error or the autopilot error, should permit the aircraft tracking accuracy to be achieved. (2) Filter bandwidth should not introduce more than 6"lag or more than 1 dB of gain at the guidance loop bandwidth. (3) The CMN level must not include more than 2" rms unwanted roll activity for autocoupled flight. The MLS guidance signals satisfy the first two criteria. Only the third criterion must be analyzed when developing MLS control laws for the terminal area and the approach/landing phases of flight. If criterion 3 cannot be satisfied, then a control law implementation having one of the forms illustrated in Fig. 38 may be necessary. 8. Interaction of DMEIP Signal with AFCS
In the ICAO SARPs, the DME/P accuracy components are described in terms of PFE and CMN. The purpose of this section is to provide the technical basis for specifying the range measurement in these terms rather than the more conventional ones of bias and noise.
104
ROBERT J. KELLY AND DANNY R. CUSICK
(a) CONVENTIONAL
COMPLEMENTARY
2 K%zc drp 9
(b) INERTIALLY DERIVED RATE COMPLEMENTARY FILTER
-
RADIO NAVAID
,
AY
b COMPLEMENTARV
Ayc
FILTER
AV INERTIAL.
INERTIAL DATA h
i
(GI
COMPLEMENTARY FILTER
FIG.38. Alternate control law implementations.
Depending upon the aircraft’sorientation with respect to the MLS ground facility, azimuth, elevation, and DME signals can be inputs to any of the two autopilot channels when properly converted to linear path deviations. The precision DME accuracy standard must ensure that the DME/P has approximately the same operational accuracy as the angle MLS in specific portions of the approach path. For example, two-dimensional (constant altitude) curved and segmented approaches may require that the DME/P
DISTANCE MEASURING EQUIPMENT IN AVIATION
105
accuracy be equivalent to azimuth guidance cross-course deviations. On the other hand, for most straight-in approaches, range data will not be required in either the lateral or vertical channels of the AFCS. As mentioned earlier, CMN has a negative influenceupon pilot acceptance factors. Resolution of this difficultyrequires the CMN to be at least limited by the system design to a maximum of 0.1" everywhere in the MLS coverage sector while utilizing an airborne output data filter bandwidth which limits the phase delay to 5" or 6" (see Section IV,D). It is against this angular error requirement that the DME/P CMN accuracy standard (when it is applicable) is derived. Since the range information is already linear, it is specified to be equivalent to the angular error of 0.1" at all points within the coverage volume. Future tests are required to determine whether 0.1" is an adequate CMN upper limit. The situation is different for the PFE component. At the MLS entrance gate, the PFE can be large because the aircraft separation standard is large ( rfi 2 nmi); however, as the aircraft proceeds toward runway threshold, the guidance error must tighten up so that the aircraft can comfortably fly through the 100-ft decision height window. Thus, unlike CMN, the PFE can degrade with increasing distance from runway threshold by a 10:1 factor. Figure 39 graphically illustrates the differences between PFE, intended course, and autopilot error. It should be emphasized again that for DME/P, CMN is only a consideration when the application requires vertical or lateral guidance using the DME/P, e.g., curved approaches. It is not critical for example in determining the 100-ft decision height. MLS INDICATED POSITION
/ /
INTENDED COURSE
/
,
'<
/
PATH FOLLOWING ERROR
AIRCRAFT ACTUAL POSITION
AUTOPILOT ERROR
FIG.39. Definition of autopilot error and path following error.
ROBERT J. KELLY AND DANNY R. CUSICK
106
9.
Summary
This section developed the background necessary to understand why an aircraft radio navaid for use in the NAS should provide low-noise, highinformation-rate output data. Aircraft guidance and control in the enroute navigation and the approach/landing application requires accurate position and velocity data with small phase lag at several samples per second. If the radio navaid is unable to achieve these position and velocity data requirements, then the radio navaid position fix data will require some form of inertial aiding. The aiding may be as simple as gyro attitude measurements to obtain dynamic damping or it may be as sophisticated as a complementary filter. Also introduced in this section were the concepts of PFE and CMN. They are new terms which form the basis of the MLS angle and the DME/P accuracy standards.
F. DME, TACAN, and VOR Enroute Navigation Accuracy Performance Comparisons The accuracy performance of three position-fix systems, DME/DME, VORTAC, and TACAN, are compared in this section. The results will show that the DME/DME position-fix system is by far the most accurate method of providing enroute guidance in the ATC system,particularly when operating in a strong multipath environment. I . General Error Analysis
An RNAV system using position-fix sensors has the following error components: (1) ground element instrumentation error,
(2) airborne element instrumentation error, (3) site effects (multipath), (4) RNAV instrumentation errors, (5) flight technical error.
Two analyses are presented. First, position-fix accuracy obtainable along a VORTAC radial is examined. Navigation along such radials represents the standard mode of flight utilized in the past and will continue to be utilized for some time to come. Second, since the use of RNAV is expected to grow rapidly in the future, a general RNAV route will be used as a means to compare the position-fix performance of four different configurations of sensors: VOR/DME, DVOR/DME, TACAN, and DME/DME. In the discussions
107
DISTANCE MEASURING EQUIPMENT IN AVIATION
below it is assumed that the slant range (DME) has been converted into ground range using the aircraft height measurement. The ground instrumentation errors are defined by the FAA for VOR and DME (FAA, 1982).The airborne sensor instrumentation errors are defined by equipment manufacturers, who may elect to comply with the FAA’s TSO (FAA, 1966) or the ARINC characteristics (ARINC, 1982), depending upon the customer’s operational requirements. Potentially the largest radio navaid error sources are multipath reflectionsin the vicinity of the ground station site. Bearing determination systems are more sensitive to enroute or terminal area site effects than a range determining system. This point will be justified and explored further under Subsection F,2. The RNAV instrumentation errors include computation and resolution errors as well as the course selection error (CSE). The final error component is the flight technical error and, as noted in Subsection C, it is the difference between the intended flight path and the indicated flight path as displayed in a manual-coupled AFCS. For autocoupled flight, the flight technical error is called the autopilot error and for the purposes here is considered to be negligible. As discussed in Subsection C,3 the error components are the meansquare-error (MSE) values. Figure 40 functionally illustrates these errors. Table V lists those error components as they apply to the four VORTAC sensors: DME, TACAN, VOR, and DVOR. The analysis was based upon the most elementary RNAV flight path, a VORTAC radial. In general, VOR/DME RNAV performance accuracy is determined at a waypoint or at any point along the RNAV route by using the concept of the
I
-
-L I
P
t I
AIR 0.1 nm.
I
L
-------------- 4 RNAV SYSTEM I N A IR C R A F l
FIG. 40. Two-dimensional RNAV error budget.
ALONG TRACK
108
ROBERT J. KELLY AND DANNY R. CUSICK TABLE V NOMINAL ERRORBUDGETSFOR VORTAC NAVAIDS' Error component (95% prob.)
(1)
TACAN bearing
VOR
DME/N
DVOR
Airborne Ground
0.11" 0.1 1"
1" 1"
0.1 nmi 0.08
1" 1"
Instrumentation only errors [RSS of (1) & (2)]
0.16"
1.4"
0.13 nmi
1.4"
Linear instrumentation error at 50 nmi Convert (3)
0.14 nmi Cross track
1.75 nrni Cross track
0.13 nmi Along track
1.75 nrni
Site error 5.4" 0.04 nmi 1.91" 1 = - 10 dB for DME and TACAN 2 = - 20 dB for VOR and DVOR; p = multipath-to-direct signal ratio Subtotal 1.92" 5.58' 0.14 nrni RSS (3), (5) Subtotal linear 1.67 nrni 4.85" nmi 0.14 nmi Convert (6) at 50 nmi
0.53"
CSE
1.6"
1.6"
1.6"
Flight technical error
2.0"
Total system error RSS (31, (51, (71, (8) RNAV computation Total linear system error at 50 nmi Convert (9)
3.2"
2.0" 6.1"
0.5 nrni 2.7 nmi Cross track
1.5" 1.28 nmi
0 0 0.14 nmi
2.0
0.5 nmi
0.5 nmi
0.5 nmi
5.3 nmi Cross track
0.52 nmi Along track
2.6 nmi Cross track
2.97"
a Assuming air carrier quality airborne elements and a manual coupled VORTAC flight path radial.
tangent point. It is the point from which a line perpendicular to the RNAV route centerline passes through a specified VOR/DME location. The tangent point distance is the distance from the VOR/DME to the tangent point, as shown in Fig. 41. The along-track distance (ATD) is the straight-line distance from the tangent point to the waypoint. The waypoint displacement area is calculated using Fig. 41 and Eqs. (1) and (2) in Fig. 42. In Fig. 42, Do is the tangent point distance. The flight geometry used for the Table V comparison is an aircraft positioned 50 nmi along a VORTAC radial. It is the same course range used by the FAA (Del Balzo, 1970) to define the VOR and RNAV course widths. The bearing angular errors and the range errors can be transformed into the crosstrack and along-track errors using Eqs. (1) and (2) in Fig. 42 with a = 0. Thus
ALONG TRACK DISTANCE (ATD)-O
a N
\
4
D = d and the tangent point distance Do = 0. For a VORTAC radial, the along-track error is simply the DME range error, while the cross-track error is the bearing error component of the VOR, DVOR or the TACAN. The error component values listed in the table were determined as follows. The RNAV computation errors, the CSE, and the flight technical errors are the values used in RTCA (1982).The equipment errors are the nominal values used in the air carrier class of equipment. Only the multipath site errors which can comprise the largest error component require further discussion. Specular multipath can be characterized by three parameters, the multipath amplitude to direct signal ratio p, the separation angle 4 between the multipath and the desired signal, and their relative rf phase. Only p and the separation angle will be considered in estimating the multipath error component; the relative rf phase will be considered to be zero. In order to proceed, it is necessary to determine the site multipath level p for the different systems. Its value depends upon the geometry, dimensions, and material of the reflectors in the vicinity of the ground station antenna. It is assumed that only the rf frequency dependence of the lateral dimension of the first Fresnel zone is significant in determining the relative difference between the DME and VOR reflection coefficients (Kelly, 1976; Redlien and Kelly, 1981). Assume for a given site that the VOR reflection coefficient pVoR= -20 dB. Further assume that the reflection coefficient of the given VOR reflector increases for DME signals to pDME = - 10 dB. The multipath geometry is given in Fig. 43.The error analysis below derives the multipath error for the DME, TACAN, VOR, and DVOR using - 10 and -20 dB for pDME and pVOR, respectively. The results were used in Table V.
110
ROBERT J. KELLY A N D DANNY R. CUSICK
6,j= VOR SYSTEM ERROR 6~ d
= DME SYSTEM ERROR
-
AIRCRAFT DISTANCE FROM TANGENT POINT
Do =TANGENT POINT DISTANCE
E
[&I
= 0 6 2 = VOR ERROR VARIANCE
-
= DME ERROR VARIANCE E 16~21 U R ~
+ GCJ 0 2 C O S Z ~] 1’2 hRAND 60 ARE INDEPENDENT RANDOM VARIABLES
(1) ~ X T K Z =I ~ R SIN% Z
NOTE
(2) “ A T K ~ = [ ~ CRO~S ~ Q+ 020g2 S I ~ 2 a ] 1 / 2
(3) 0 = (4)
[ Doz + d2 ]
”‘
cos a = d
TOTAL SYSTEM ERRORS ARE
(51 u X T K Z (TOTAL) = o X T K Z +
uRNAVZ + oFTEZ
(6) UATKz(TOTAL) = UATKZ+ (JRNAVZ
\
-\
\ \ \
\
\ \ \
COURSE
VORTAC STATION
FIG.42. Geometry of DMEjVOR error equation computation.
In summary, Table V states that the DME along-track errors for the VORTAC geometry are negligible when compared with the bearing errors of either the DVOR, VOR, or the TACAN. The TACAN accuracy performance is comparable to the DVOR under the conditions assumed for the multipath. Note that, because of the site error, the VOR would not satisfy the 4 nmi enroute accuracy requirement noted in Subsection D. In this case, a DVOR would be required for that site. As shown in the table its total system error is only 2.6 nmi and is well within the 4 nmi error limit.
DISTANCE MEASURING EQUIPMENT IN AVIATION
ER = REFLECTED SIGNAL, ED= DIRECT SIGNAL EC = COMBINED SIGNAL p=ERIED
@
= SEPARATION ANGLE
NORTH HANGER, HILL, OR OTHER REFLECTOR
20 DEGREE
FIG.43. Multipath geometry and error mechanism.
111
112
ROBERT J. KELLY A N D DANNY R. CUSICK
The RNAV will now use DME/DME, VOR/DME, or TACAN as input sensors to estimate an arbitrary course instead of the VORTAC radial discussed above. Also, the outputs of the RNAV are assumed to be coupled to an autopilot and the flight path is preprogrammed. With these assumptions, the model shown in Figure 40 is modified with CSE E 0 and the flight technical error is negligible. The along-track and crosstrack errors can be calculated for any RNAV course AB using the equations and geometry of Fig. 42 for a DME/VOR configuration or Fig. 44 for the DME/DME configuration. The DME/DME position-fix configuration is given in Fig. 44 assumes a scanning DME interrogator. Since the scanning interrogator is common to both the DME measurements, the errors of the DME/DME are correlated. Most importantly, it is clear from Eq. (1)in Fig. 44 why the station selection algorithms described in Subsection D,4 choose only those sites where 8 is more than 30" but less than 150". In this range, the cross-track error degradation will only be double the optimum value at 90". The DME/DME position error equations used in Fig. 44 are derived in Braff (1966). The geometry and equations in Figs. 42 and 44 will be used to compare the RNAV performance when DME/DME, VOR/DME, TACAN, and DVOR/DME are used as sensor inputs. Let the course line A B = 70 nmi, b = 50 nmi, D, = 20 nmi, and D, = 15 nmi as defined in Fig. 44 for the DME/DME calculation. Assume the same course line AB for the VORTAC calculations in Fig. 42 and assume the VORTAC station location is identical to the DMEl ground station. Figure 45 graphically displays the computer simulation results for an aircraft traversing the RNAV course AB. The simulation used the equations in Figs. 42 and 44 and the error components were those used in Table V. As shown in the figure, the DME/DME was the most accurate, and, as expected, the DVOR and the TACAN had equivalent performance. Because of the assumed site error defined in Fig. 43, the VOR/DME configuration was unacceptable because it exceeded the k4 nmi enroute RNAV course width. Figure 46 gives the position of the aircraft relative to the ground transponder coordinates in terms of DME/DME range measurements. Note that the ambiguity in they coordinate has to be resolved from either one more DME station measurement or from some other reference. The equations are derived in Porter et al. (1967). 2. Approximate VORTAC Multipath Error Relations
In this section, the approximate multipath error relations are derived for the DME, TACAN, VOR, and DVOR navaids. The method of analysis will treat both the DME range element and the VOR/TACAN bearing element
DISTANCE MEASURING EQUIPMENT IN AVIATION
hR1 ,,6 d
=
DME ERROR FOR STATION 1 A N D SCANNING DME INTERROGATOR
=
DME ERROR FOR STATION 2 A N D SCANNING DME INTERROGATOR
=
AIRCRAFT DISTANCE ALONG RNAV COURSE AB
E
[6R,2]
E
[ 6R22] =
= OR12 UR22
=
DME 1 ERROR VARIANCE
=
DME 2 ERROR VARIANCE
R1
=
DME 1 MEASUREMENT
R2
=
DME 2 MEASUREMENT
p,,
[&R1 &R2 OR1 OR2
=
STATION 1
1=
CORRELATION BETWEEN ,6 ,
+ ,,6
STATION 2
FIG.44. Geometry of DME/DME error equations computations.
113
114
ROBERT J. KELLY AND DANNY R. CUSICK 1
m
l5O0 0 0
7
----
14
21
28
35
42
49
56
63
70
DISTANCE ALONG COURSE AB (nm) ALONG COURSE CROSS COURSE
FIG.45.
Position fix accuracy performance of DME, VOR, and TACAN.
using the same mathematical formulation. Let the desired DME pulse eD(t) be represented by a half-sine wave of period z. Since the DME pulse width between the half-voltage points of the leading and trailing pulse edges is 3.5 ps, then z = 10.5 ps. The DME pulse is therefore eD(t) = EDsin(271ft)
for 0 I t I 2/2,
where T
=
l/f
(12) for 212 < t I z eD(t)= 0 Let the multipath disturbance 6n be a delayed version of the direct signal with separation angle 4. The disturbance is given by
6n = ERsin(2nfi+ 4)
(13)
ED and E R are the peak amplitudes of the direct signal and the multipath signal, respectively. For DME/N the leading edge of the pulse is detected at the half-voltage point, which corresponds to t = 2/6. The leading edge time of arrival error 6t for small disturbances 6n is
where the derivative is evaluated at the half-voltage point to = ~ / 6 The .
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POSITION F I X COORDINATES ( x , y I USING MEASURED R 1 A N D R 2 A N D G R O U N D STATION COORDINATES
x =
R1
2
-R2
2
4x,
Y
+
--
-1
b 1
X1
+ x AXIS FIG.46. Geometryfor determiningposition fix of DME/DME equippedaircraft relative to ground transponder coordinates.
result is
where p = E,/E,. Using Eq. (15) the maximum disturbance 6tM occurs at 4 = x/6:
The distance error 6RDM,in feet is
6~,,,
1
=- x
1096t,
1
=-
2 2 4 p T log ft The estimated peak distance error in the presence of -10 dB multipath
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ROBERT J. KELLY AND DANNY R. CUSICK
(p = 0.32) using Eq. (17) is 103 ft. The exact relation is given in Sec-
tion IV,B,4. With suitable modifications,Eq. (13) can be used to derive the phase error for the VOR system. The phase error is evaluated at the zero crossing, namely, to = 0. Using Eq. (13) the zero crossing error 6t, is
6to = (p sin 9)/2nf
(18)
To convert to bearing error in degrees, multiply Eq. (18) by 57.3 (27cf). The bearing error in electrical degrees is 88VOR = ( 5 7 . 3 ) ~sin 9
(19)
where 360 electrical degrees equals 360 spatial degrees. For the geometry of Fig. 43 and - 20 dB multipath, the VOR bearing error 6OvO, = 5.4". The exact equations are given in Flint and Hollm, 1965. The reason bearing measurement systems have less accuracy than range measurement systems can be explained as follows: Bearing errors are proportional to the effective beam width of the ground station radiation pattern. The VOR rotating cardioid antenna pattern has a 3 dB beam width of about 133". DME errors in general do not increase with range R; however, the bearing error 68 when expressed in a linear measure 6Y does increase with range by definition (6Y = R68). In order to offset this linear increase, the bearing ground station effectivebeam width must be drastically reduced so that 68 is small to begin with. This cannot be accomplished in any practical way. However, as stated in Subsection D, the multipath performance of VOR can be improved using techniques which in effect reduce the antenna beam width. One approach which culminated in the DVOR effectivelyincreased its antenna aperture by a factor of 10 (Steiner, 1960). Thus, its effective beam width was reduced about 10:1, with an equivalent reduction in multipath error. For the multipath configuration shown in Fig. 43, the DVOR would have about a 0.53" error, thus yielding a 10: 1 improvement over the VOR implementation. To understand the error mechanism of TACAN, it is necessary to revisit the two-scale concept. If the TACAN were implemented using only the 15 Hz cardioid, then the error would be identical to the VOR. However, the error is reduced by a factor of 9 when the nine-lobe antenna pattern which generates the 135 Hz (ninth-harmonic) component is considered. The key point is that both the 15 and 135 Hz reflected signals are delayed by the same amount as they traverse path Q + R, as shown in Fig. 43. The processing of the received data consists of determining the phase of the first harmonic (15 Hz component) in the received pattern relative to the first-harmonic time reference markers and the phase angle of the ninth harmonic (135 Hz component) in the received pattern relative to 9th harmonic reference markers.
DISTANCE MEASURING EQUIPMENT IN AVIATION
117
The function of the first harmonic is to determine which 40" sector the measured phase of the ninth harmonic falls. Thus, the phase measurement on the 15 Hz channel component constitutes a coarse measurement while the 135 Hz channel component constitutes the fine measurement. To determine the multipath improvement which can be expected, Eq. (19) will be modified to reflect the fact that 360 electrical degrees equals 40 spatial degrees. Thus, the TACAN bearing ~ T A , is given by
6 6 ,= ~ 68vo,/9 ~
6 . 4 sin ~ 4
(20) For the configuration described in Fig. 43, the TACAN bearing error is 66,,, = 1.92".This value was used in Table IV. The exact equation for the TACAN bearing error is given in Sandretto (1958).Introducing the multilobe radiation pattern can be viewed as reducing the effective beam width of the TACAN source from 132"to about 15". In summary, the site error is reduced by 6 of that for the VOR for equal reflection coefficients. An important consequence of the TACAN multilobe antenna pattern is that both the site and instrumentation errors are reduced. As stated in Subsection D, the DVOR reduces the site errors, but not the instrumentation errors. As a consequence, the instrumentation errors for TACAN in Table V are reduced by a factor of 9 relative to the VOR errors. TACAN ground facilities have to be sited very carefully because a multipath level of, for example, - 6 dB would induce an error of 30" and thus could cause a large ambiguity by jumping to the wrong 40"sector. This is one of the known drawbacks of systems like TACAN which use a two-scale measurement system. The DVOR and TACAN bearing errors would have equivalent DME linear errors at distances of about 3100 and 11,000 ft, respectively. On the other hand, the new MLS 1" azimuth antenna has an equivalent DME linear error at 12.9 nmi. The MLS ICAO standard requires 20-ft accuracy at a distance of 15,000 ft. Interestingly enough, the same formulation can be used to derive the MLS angle error equation (Kelly, 1976; Frisbie, 1976). Finally, both Eqs. (14) and (18)can be used to approximate the rms receiver noise performance or diffuse multipath if 6n represents the standard deviation of the random error source. In this case p = l/(s/6n), the inverse of the voltage signal-to-noiseratio or signal-to-interference voltage ratio. In all of the formulation above the reference signal was not included in the analysis. Such an analysis is not required because for VOR the reference signal is embedded in an FM format which is not sensitive to amplitude disturbances. In the case of TACAN, the reference signal is derived from a series of reference pulses whose maximum timing errors, as calculated using Eq. (16), is less than 103 ft for the - 10 dB multipath level. The corresponding electrical phase error would be negligible. Note that, depending upon the reflector distance N
118
ROBERT J. KELLY AND DANNY R. CUSICK
used in Fig. 43, the DME multipath induced error may be zero if the pulse time is more than 1 ps. In this case, there is no error because the multipath occurs after the pulse leading edge measurement point.
3. Additional Accuracy Considerations Additional comments will now be made with respect to the DME/DME position-fix system. The error described in Fig. 44 for the DME/DME can be reduced if more DME ground facilities are tracked. Test results in Latham (1974) indicated that the use of regression analysis on the data taken with at least 8 to 9 ground stations in a flight run resulted in a DME/DME position error of 100 ft CEP. Latham (1974) collected data from 17 different DME/N ground stations in southern New England. The measured bias error from these stations had an ensemble standard deviation of 400 ft. The ensemble average was 200ft and, because the ensemble average of the ground station was assumed to be zero (Subsection C), this average error was attributed to the airborne interrogator and the distribution geometry of the ground stations. The standard deviation of the range noise from each station was measured to be about 200ft. The RSS error for the 17 ground stations investigated, including the airborne bias, was
+
RSS = -J(400)2+ (200)2 (200)2 = 490 ft which reinforces the earlier statement that for many ATC navaids it is the bias error which predominates. If the distributions in the above study are assumed to be normally distributed, then the 95% probability of the DME measured system error is about 2 x 490 = 980 ft. This is consistent with the new DME/N error standard of 0.2 nmi (1215 ft). A very general treatment of the accuracies achievable from multiple range measurements is given in Groginsky (1959) and Wax (1981). 111. THEDME/N SYSTEM Section I1 of this article has presented the origins of the DME concept together with the evolving role of the DME within the ATC system. In addition, the characteristics of the information provided by any navigation sensor (accuracy-noise and bias, data rate, and bandwidth) for use by manual and automatic flight control systems were detailed and justified. Section 11,therefore, serves as the foundation for the discussions that follow by not only justifying the importance of DME within the ATC system but by justifying the need for the design constraints imposed on the DME as well. In this section the focus is on details of the DME/N system, which provides the
DISTANCE MEASURING EQUIPMENT IN AVIATION
119
reader with a knowledge of how the system is put together and why it is uniquely qualified for its role in the ATC system. Today’s DME system represents a refinement of the beacon concept introduced in Section II,B. It is an internationally defined method of determining aircraft slant range to known ground locations by measuring the elapsed round-trip time between the transmission of an interrogation by an airborne interrogator and the receipt of a reply to that interrogation from a ground-based transponder. The characteristics of the DME system are defined by the International Civil Aviations Organization in the form of Standards and Recommended Practices (ICAO, 1972b). The scope, structure, and evolution of the DME SARPs is the subject of Subsection A. In Subsection B, an overview of the DME/N system and its major components is presented. The emphasis is both on how the system is put together and on the functional elements that comprise the ground and airborne equipment. In Subsection C, the emphasis shifts to the details of the DME/N signalin-space. Specifically addressed are the quantitative aspects of accuracy, coverage, the channel plan, and ground system capacity. The goal of the section is not only to detail the basic system characteristics but to stress the interrelationships between each. A . DME Standards and Recommended Practices
In December, 1982, the ninth meeting of the All Weather Operation’s Panel (AWOP) of the International Civil Aviation Organization completed technical standards for the precision distance measuring equipment, DME/P. Also approved were the revision and updating of the conventional DME (DME/N) technical standards (ICAO, 1972b). After minor review by AWOP10 in September, 1984, the technical standards were adopted by the ICAO Air Navigation Council on December 6, 1984, and became applicable as amendments to Standards and Recommended Practices of the ICAO Annex 10 paragraph 3.5 on November 21,1985 (ICAO, 1985). From the first draft of the DME SARPs put forth in 1950 to the 1985 version, this document has evolved in its level of specification reflecting increased operational need (e.g., more channels, improved accuracy) as well as the available technological capability. This 1985 revision is the latest improvement to a system whose role in the ATC system has been expanding for the last 40 years. The present document defines three types of DME ground facilities: DME/N (narrow), DME/W (wide), and DME/P (precision). Each of these systems operate in the L band of the frequency spectrum between 960 and 1215 MHz. DME/N is the primary ranging system used in aviation today. In conjunction with VOR or TACAN bearing or with other DMEs, it is used to
120
ROBERT J. KELLY AND DANNY R. CUSICK
obtain position fixes for enroute and terminal area navigation. DME/W, operationally equivalent to DME/N, has a less restrictivespectrum constraint on the transmitted waveform. This rarely used system, to be phased out by January, 1987, was intended to serve as an inexpensive means of providing DME service in isolated regions where frequency protection of the service volume is not a concern. DME/P serves as the MLS range element and provides precision range information (accuracy on the order of 100 ft) having characteristics suitable for approach and landing operations. The DME/P standards represent the newest addition to the ICAO SARPS. The purpose of the SARPs is threefold. First, it defines the required minimum performance of the DME system in terms of accuracy, coverage, and capacity. These specifications define a system having accuracy and coverage capable of satisfying enroute and terminal area navigation needs with the capability of a single facility to serve 100 aircraft simultaneously. Second, the SARPs define the system characteristics necessary to ensure compatibility among ground and airborne equipments of different manufacture. These characteristics include: (1) Signal format: Characteristics of the signal format are specified that include the transmitted pulse shape, the zero mile delay, and the calibration technique. (2) Ground facility signal power-in-space/receiver sensitivity: The SARPs specify the minimum power densities to be provided by the ground facility at the extremes of its defined coverage volume. The size and shape of the coverage volume are determined by the governing authority based on operational need. Ground facility receiver performance is based on received power density at the antenna. Interrogator radiated power requirements and receiver performance are required to be consistent with the ground system specifications. Note that the SARPs avoid specifying implementation details such as the transmitter power, cable losses, antenna characteristics, and receiver performance, leaving the details of how the requirements are satisfied to the equipment designer. (3) Signal spectrum, stability, and rejection: The SARPs specify radiated signal spectrum characteristics, carrier frequency stability, and off-channel rejection characteristics.These characteristics provide a basis for the development ground facility channel assignment criteria. (4) Channel plan: The channel plan defines interrogator/reply frequency pairs and the associated pulse-pair spacings or codes. In addition, the plan explicitly provides for the “hard pairing” of DME channels with other navigation aids, namely VOR, ILS, and MLS. Finally, the SARPs serve as the basis for the development of ground equipment procurement specifications, frequency planning, and field deploy-
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ment by ICAO member countries. It serves as the basis of airborne equipment specifications as well. It must be emphasized that the SARPs are not equipment specifications. Rather, they are an internationally agreed document that defines the system characteristics to a level that ensures compatibility and performance among units of differentmanufacture and national origin. Traditionally, the emphasis of the SARPs has been on standards pertaining to system performance and the ground equipment with airborne specificationsplaying a lesser role. It is the ground transponder that can better bear the burden of increased size, weight, power consumption, and cost incurred in adhering to strict performance standards. Specifications pertaining to the interrogator are limited to those absolutely necessary to ensure performance and compatibility, thereby reflecting the primary goal of lightweight, small-size, low-power, and low-cost airborne equipment. Detailed airborne characteristics have been and will most likely remain in the domain of the various user groups such as ARINC, RTCA, and EUROCAE. Some countries incorporate the ICAO SARPs DME specification into their national aviation standards in a way that provides equal or more stringent performance than the SARPs. In the US., the DME/N characteristics have been included in the VOR/DME National Standard (FAA, 1982), while the corresponding ground equipment specification, for the FAA's second-generation VORTAC, is found in FAA (1978). For DME/P, the national standard is defined in FAA (1983b) while its equipment specification is detailed in FAA (1983~). In the US., airborne equipment specifications have been defined by three different organizations, each addressing different, but perhaps overlapping, user communities. The FAA requirements for DME/N performance are in the form of a Technical Service Order (TSO) (FAA, 1966). The DME/P TSO has not been developed at this time. Although compliance with the TSO is not mandatory for DME equipment utilized in US. airspace, a user can more easily obtain the required aircraft certification for operations in conditions where a DME is required (e.g., above 18,000 ft under IFR conditions) if TSO'd equipment is utilized. The Radio Technical Commission for Aeronautics publishes Minimum Operational Performance Specifications (MOPS) pertaining to both DME/N and DME/P interrogators (RTCA, 1978, 1985b). Usually, the MOPS forms the basis of the FAA's TSO requirements. RTCA (1985b) reflects the latest improvements to the SARPs that apply to DME/N and represents the first detailed specification of the new DME/P interrogator. Airborne equipment for use by the airlines typically conforms to specification that are more stringent than those defined by the RTCA MOPS. Such specifications are developed and published by Aeronautical Radio, Inc.
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ROBERT J. KELLY A N D DANNY R. CUSICK
(ARINC) and are called ARINC Characteristics. ARINC (1982)defines airline requirements for DME/N, while the DME/P requirements remain to be defined. In Europe, the European Organization for Civil Aviation Electronics (EUROCAE) establishes both ground and airborne equipment specifications (EUROCAE, 1980). As in the U.S., EUROCAE is currently developing DME/P equipment specifications. The present ICAO SARPs begin with a system description which defines not only the function of the DME system but its relationship to other systems, particularly VOR, ILS, and MLS. Next, system characteristics concerning coverage, accuracy, channel plan, capacity, and facility identification are presented. Detailed characteristics of the ground element (transponder) and the airborne element (interrogator) are presented in seperate sections. Guidance material is provided at the end of the SARPs “white pages.” These “green pages” are not specifications but rather recommendations concerning system details such as channel assignment criteria and example ground and airborne equipment power budgets. 1 . Evolution of the DME Standards
The first official notice by ICAO concerning DME was a recommendation by its Special Radio Technical Division in late 1946 for the development of distance measuring equipment (ICAO, 1947). In January, 1949, ICAO recommended in Section 2 of “Aid to Final and Approach and Landing” that, along with requirements for inner, middle, and outer markers, a U H F distance measuring equipment be added to the ILS at the earliest possible date. At the same time, it required that a DME become an integral part of the ILS not later than January 1, 1954 (ICAO, 1949). In the same year, ICAO recommended that the DME become a basic component of the VOR at the earliest practical date. In May, 1950, the first draft specification for DME was published in Attachment B of Annex 10 of the SARPs for Aeronautical Telecommunications. As mentioned previously in Section II,B of this article, the channel plan was essentially that proposed by RTCA SC-40, having 10 frequencies and 10 codes for a total of 100 channels (Table VI). The ground transponder transmitter frequency stability was limited to & 200 kHz, implying that the oscillators need not be crystal controlled. The radiated spectrum constraint, so important in DME today, was to be determined at a later date. Table VII summarizes the principal SARPs requirements at various critical times during its evolution and includes the upgraded DME/N and new DME/P parameters added with the 1985 revision. Figure 47 shows the
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TABLE VI THEDME CHANNEL PLANAS PROPOSED I N THE FIRSTDRAFTDME SPECIFICATTON INCLUDEDI N THE SARPs OF MAY,1950'
Mode
Pulse Spacings Interrogation (ps)
Reply (PSI
14 21 28 35 42 49 56 63 70 77
17 70 63 56 49 42 35 28 21 14
A B C D E F G H I J
Interrogation/Reply Frequency Pairs Interrogation Band Reply Band (960-986 MHz) (1188.5-1215 MHz) Channel
Frequency
Channel
Frequency
1 2 3 4 5 6 I 8 9 10
963.5 966.0 968.5 971.0 973.5 976.0 978.5 981.0 983.5 986.0
00 10 20 30 40 50 60 70 80 90
1188.5 1191.0 1193.5 1196.0 1198.5 1201.0 1203.5 1206.0 1208.5 1211.0
'The plan utilized 10 interrogation/reply frequency pairs and 10 pulse codes for a total of 100 channels.
evolution of the pulse-shape specification over the same time period. Compliance with the upgraded DME/N standards, identified in the SARPs by a dagger, is required after January 1, 1989. Attention should be given to the continual tightening of the specifications with each revision, as well as the increased requirements imposed on the airborne equipment with the latest revision. These changes reflect both the growing need for enhanced performance from the DME as well as the availability of improved technology.
TABLE VII Specification
1950 (Draft)
1952
1960
1972
1985 DME/N
DME/P
System level Coverage
200 nrni enroute REC 200 nrni (enroute) 360" up to 60,000 ft
System accuracy
None
REC (0.5 nrni or 3%)
Channel plan
10 freq 10 codes
10 freq 10 codes
F
REC 200 nmi (enroute) 360" up to 75,000 ft REC (0.5 nrni or 3%)
REC 200 nrni (enroute) 360" up to 75,000 ft
X - 126
X-Y 252
REC (0.5 nrni or 3%)
22 nmi and at least that of MLS azimuth coverage k0.2 nmi; 95% prob Standard 1-100 ft PFE, 60 ft CMN Standard 11-40 ft PFE, 40 ft CMN
200 nmi; same as equip. associated with VOR, ILS
x,y, w, (352 channels)
h)
*
Interrogation PRF 30 pp/s
30 PP/S
Aircraft capacity
50
None
100
100 or peak traffic
FA Mode: 40 pp/s; IA mode: 16 pp/s 100 or peak traffic
360"; 4 dB gain
None
None
6" vert. beam 0.002%
*
+0.002%
Same as DME/N
< 1.6 p, 200-300 ns partial rise time Same as DME/N
30 PP/S 100
x, Y,w,x-200 channels
Transponder
6 dB gain +fdipole gain k 200 kHz
360"; 4 dB gain +fdipole 6" vert. beam 200 kHz
360"; 4 dB gain f dipole 6" vert. beam
Pulse rise time
None
<0.3 ps
2.5 k 0.5 ps
NOM 2.5 ps < 3 ps
< 3 ps
Pulse width
None f0.2
2.5
None
<0.5 ps
3.5 t 0.5 /.is NOM 2.5 ps < 3.5 ps
3.5
Pulse fall time
3.5 f 0.5 ps 2.5 f 0.5 ps
Antenna
Transmitter freq. stability
+ 0.2 ps
+
*0.002%
++dipole
0.5 ps
2.5 ps NOM < 3.5 ps
Same as DME/N
N
Pulse spacing
f 0.5 p s
f 0.5 p s
f0.25 p s
f0.5 ps
Effective radiated power
> 2 kW
>8 kW
> 8 kW
-83 dB W/m2
Transmission rate
None
Receiver freq. stability Transponder sensitivity
k 100 kHz
None (REC 2700 pps) f 100 kHz
None (REC 2700 pps) f 60 kHz
-82 dBm at receiver
-90 dBm at receiver
Dynamic range Squitter rate Frequency rejection Recovery time
None 300 50 dB at 2 MHz away None
Decoder rejection Decoder accept. Time delay accuracy
None <0.5 p s
None
f 0.5 p s (REC f 0.10) -89 dB W/m2 (under all conditions)
Same as DME/N
REC 2700 pps +60 kHz
REC 2700 pps
Same as DME/N
f 0.002%
Same as DME/N
-95 dB m at receiver
-95 dB m at receiver
- 103 dB W/mz
None None
None > 700 pps
-22 dB W/m2 = 700 pps
50'dB at 10 MHz away 8 p s after 60 dB pulse None None f 500 ft
80 dB; >900 kHz 8 p s after 60 dB pulse None None k 500 ft
None >700 pps 80 dB; >900 kHz 8 p s after 60 dB pulse None None f 500 ft
IA Mode: -86 dB W/m2 FA Mode: - 75 dB W/m2 Same as DME/N < 1200 pps
80 dB; >900 kHz 8 p s after 60 dB pulse -75 dB at f 2 p s f 1 p s (1 dB degrad.) f 500 ft
- 89 dB W/m2 7 nmi coverage limit -75 dB W/m2 <7 m i - 70 dB W/mz MLS app. ref. datum -79 dB W/mZ 8 ft above runway
Same as DME/N Same as DME/N Same as DME/N Same as DME/N Standard 1-33 ft PFE, 26 ft CMN Standard 11-16 ft PFE, 16 f: CMN
TABLE VII (continued) Specification
1950 (Draft)
1952
I960
1972
1985
DME,”
DMEjP
Time delay
None
115 ps
50 ps
50 p s X channel 56 ps Y channel
Depends on channel code
Same as DME/N
Time delay measurement
None
2nd pulse at 50%
2nd pulse at 50%
2nd pulse at 50% pt
1st pulse at 50% pt
1st pulse at virtual origin
Reply efficiency
67%
67%
70%
70%
70%
FA Mode: 80% IA Mode: 70%
Dead time
133 ps
10 ps
60 ps NOM
60 ps NOM
60 p s NOM
Same as DME/N FA Mode has priority
Spectrum
None
- 18 dB at 1 MHz away - 3 0 dB at 10 MHz away
-60 dB at 800 kHz away in 500 kHz BW -65 dB at 2 MHz away in 500 kHz BW
200 MW at 800 kHz away in 500 kHz BW 2 MW at 2 MHz away in 500 kHz BW
200 MW at 800 kHz away in 500 kHz BW 2 MW at 2 MHz away in 500 kHz BW
Same as DME/N
Monitor
None
Time delay XT power (REC) RX sensitivity (REC) Pulse spacing (REC) Frequency (REC)
Time delay XT power (REC) RX sensitivity (REC) Pulse spacing (REC)
Time delay XT power (REC) RX sensitivity (REC) Pulse spacing (REC)
Time delay XT power (REC) RX sensitivity (REC) Pulse spacing (REC)
Time delay (PFE) XT power RX sensitivity Pulse spacing
Transmitter stability
f400 kHz
& 400 kHz
+400 kHz
- 100 kHz
+
Same as DME/N
Pulse rise time
None
<0.3 ps
2.5 f 0.5 ps
< 3 ps
< 1.6 p s 200-300 ns partial rise time
Pulse width
None k0.2 ps
Pulse fall time
None
2.5 5 0.2 ps <0.5 ps
3.5 f 0.5 ps 2.5 f 0.5 ps
Interrogator 100 kHz
2.5 ps NOM ( < 3 PS) 3.5 f 0.5 ps
3.5
0.5 ps
Same as DME/N
2.5 p s NOM (< 3.5 ps)
2.5 ps NOM ( < 3.5 ps)
Same as DME/N
5 L
Pulse spacing
k1 p s
+1 ps
T0.5p s
k0.5 p s
Spectrum
None
- 14 dB at 1 MHz away - 30 dB at 10 MHz away
90% in 0.5 MHz BW
90% in 0.5 MHz BW
T0.5p s (REC 0.25 p s ) 90% in 0.5 MHz BW
Jitter interrogation Yes
Yes
Yes
Yes
Yes
Yes
Effective radiated power
None
None
None
None
None
None
Time delay measurement
None
2nd pulse at 50% pt
2nd pulse at 50% pt
2nd pulse at 50% pt
1st pulse at 50% pt
1st pulse at virtual origin
Time delay
115ps
115 p s
50 p s
50 p
Depends on channel code
Same as DME/N
Time delay accuracy
None
None
None
None
0.17 nmi (REC)
IA Mode: 100 ft PFE, 50 ft CMN FA Mode: 50 ft PFE, 33 ft CMN, Standard I 23 ft PFE, 23 ft CMN, Standard I1
Receiver sensitivity
None
None
None
None
- 89 dB W/m2
Consistent with uplink specifications
Dynamic range
None
None
None
None
- 18 dB W/m2
Same as DME/N
Receiver selectivity
None
None
None
None
-42 dB at 900 kHz away (REC)
Same as DME/N
Decoder rejection
None
None
None
None
-42 dB at T 2 p s
Same as DME/N
Decoder accept.
<1ps
None
None
None
None
None
Same as DME/N Same as DME/N
Abbreviations: PROB, probability; REC, recommended specifications; BW, bandwidth; NOM, nominal; XT, transmitter; RX receiver.
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ROBERT J. KELLY A N D DANNY R. CUSICK
w 1.OA
w 1.OA
5 0.9A f L:0.5A
5 0.9A k d
: $
0.5A
m
a
m
!i
9 0.1A
5
9 0.1A
\ _I
TIME <2.5m 1952 PULSE SHAPE I S ESSENTIALLY RECTANGULAR WITH FAST RISE TIME AND WEAK RELATIVE ADJACENT CHANN€c RAQCATED POWER LIMITATION.
1 VIRTUAL ORIGIN
1960 PULSE SHAPE IS ESSENTIALLY GAUSSIAN WITH AN ABSOLUTE ADJACENT CHANNEL RADIATED POWER LIMITATION.
T*
PARTIAL RISE TIME
200-300nr
1985 DME/N PULSE SHAPE AND SPECTRUM REOUIREMENT REMAIN UNCHANGED. PARTIAL RISE TIME AND LINEARITY OF LEADING EDGE ARE SPECIFIED FOR THE D M E P APPLICATION. USE OF A WIDE SPECTRUM PULSE NECESSITATES A REDUCTION IN RADIATED POWER TO MEET THE ABSOLUTE ADJACENT CHANNEL LIMITATION.
FIG.47. Evolution of the D M E transponder transmitted pulse shape.
B. A Functional Description of the DMEIN System
Today’s DME system is based on the beacon concept introduced in Section II,B and is composed of the following components: (1) Interrogator encoder and transmitter (2) Transponder receiver and decoder (3) Transponder encoder and transmitter (4) Interrogator receiver and decoder ( 5 ) Predetermined zero mile delay (6) Common pulse arrival time measurement technique (7) Interrogator range tracking and display unit.
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Recall that the use of the ground based beacon has the following obvious advantages:
(1) In traditional radar, as the range (R) increases, the power of the echo decreases by a factor of l/R4. Since the ground beacon serves as a repeater, the power of the transponder reply only decreases by a factor of 1/R2. Consequently, the radiated power requirements in DME are substantially less than those necessary in traditional radar ranging applications. (2) Unambiguous range information is provided to a known location on the ground which can have physical significance such as an airport, the stop end of a runway, or a waypoint in the VICTOR airway or JET route system. (3) The interrogation and response frequencies are different. Hence, a beacon receiver will not be confused by its own transmissions. (4) The use of an omnidirectional antenna at the ground beacon provides distance information simultaneously to a multiplicity of aircraft. All of the above aspects will become evident in the discussions that follow. In this section, a qualitative description of DME/N operation, components, and system considerations is provided. Detailed discussions are deferred to Subsection C. In Subsection B,1, the operation of the DME/N system along with a description of the major functional components that comprise the ground and airborne equipment is presented. In Subsection B,2, those aspects of the DME that make it a viable navigation aid are described with the major goal of illustrating the interrelationship of the major system considerations. In Subsection B,3, the problem of automatically acquiring and tracking the desired beacon replies is introduced. Finally, the general trends apparent in DME designs today are reviewed in Subsection B,4. I . Operation and Components of the DME/N
The operation of the DME/N system can be concisely summarized as follows. The airborne unit (interrogator) interrogates a single ground facility that transmits a reply following a known calibrated fixed delay (the zero mile delay). The airborne unit then computes the slant range to that facility by measuring the elapsed time to the receipt of the transponder reply. The measured range is then provided to the pilot and other aircraft systems as required. Each interrogation consists of a pulse pair where the spacing of the pulses (or code) together with the rf frequency define the operating channel thereby allowing the interrogations to be addressed to a specific ground facility. Similarly, the transponder reply consists of a pulse pair having a code and rf frequency corresponding to the channel in use thereby allowing the airborne unit to distinguish desired ground facility transmissions from those of other facilities within line of sight of the interrogator.
130
ROBERT J. KELLY AND DANNY R. CUSICK
The DME interrogator and transponder are similar in that both must identify valid DME signals using the three discriminates of pulse width, frequency, and code. In DME, the rf frequency and pulse-pair spacing or code of the DME signal (interrogation and reply) designate the operating channel. The width of the DME pulse is used to discriminate a valid pulse from those due to noise which are of short duration. The interrogator and transponder differ in function. An interrogator must accurately measure the time delay between the transmission of the interrogation and receipt of the reply. It does so by starting a clock coincident with the transmission of each interrogation and stopping the clock upon reception of the reply. On the other hand, the transponder must accurately generate a fixed time delay between receipt of the interrogation and the transmission of a reply. This is done by starting the time-delay clock upon receipt of an interrogation, and after a fixed time transmitting a reply such that the roundtrip propagation delay is accurately extended by a fixed amount. A ground transponder will “hear” many interrogations; those from aircraft utilizing the transponder and from other aircraft within line of sight that are utilizing other DME beacons. To minimize the load on the beacon and to ensure unambiguous range information to the user, the transponder must only respond to those interrogations intended for it-those having rf frequency and pulse code corresponding to the assigned channel. Although a reply to an interrogation will only be transmitted by the desired ground facility, the airborne unit will “hear” the desired ground station and others that are within its line of sight. Consequently, the interrogator must sort out the reply to its own interrogation from the thousands of pulse pairs that it may actually receive. It does so by identifying those beacon transmissions that are consistently synchronous with the interrogations. Those pulse pairs which do not have the proper rf frequency and pulse codes are eliminated from consideration as valid replies by the interrogator signal processor. In addition to reply pulse pairs, the transponder radiates squitter and a facility identification signal. Both consist of pulse pairs that are identical to those transmitted in response to interrogations. Squitter consists of random transmissions of on-channel pulse pairs radiated at a rate of 700 to 3600 pulse pairs per second. The intertransmission times are randomized to prevent synchronization with replies to interrogations. Squitter is utilized by many interrogators to determine appropriate automatic gain control (AGC) settings and, as noted in Section II,D, squitter serves as the source of the TACAN bearing information which is conveyed by modulating the amplitude of the beacon transmissions. When using any navigation facility, it is important that the pilot know with certainty the source of the displayed navigation information. DME, VOR,
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ILS, and even MLS ground facilities identify themselves by transmitting a 3- or 4-character identifier in Morse code that corresponds to the facility identifier shown on air navigation charts. For example, receipt of the Morse code identifier “ D C A uniquely identifies the VORTAC facility located at Washington National Airport in Washington, D.C. The DME identification signal consists of on-channel pulse pairs sent at a periodic rate of 1350 pulse pairs per second. Appropriate circuitry in the airborne equipment converts the periodic train of decodable pulse pairs into an audible tone. The periodic sequence is keyed on and off in order to form the dots and dashes of the Morse code identification signal. During the key-down period of an I D transmission, squitter transmissions and replies are inhibited, thereby providing a clear identification signal to the pilot. Both airborne and ground DME receivers must handle signal levels from the weakest levels which occur when the aircraft is at the limits of the ground facility coverage (either above, below, or beyond) to the strongest which can occur when the aircraft is on the ground near a DME facility. Up to an 80 dB variation in signal strength can be expected with levels from - 90 dB m on the low end to - 10 dBm on the high end. The dynamic range required of the interrogator receiver is operationally quite different from that of the transponder. Since the interrogator is only concerned with replies from a single source (i.e., the desired ground facility), a receiver having a dynamic range on the order of 10 dB will suffice. Once the signal level being received from the desired facility is known,” changes of the level can be tracked by automatic gain control circuitry. As noted above, squitter from the desired transponder often serves as the basis of AGC settings. Such is not the case for the transponder. Since the transponder does not know the amplitude of the next pulse to be received, conventional AGC implementations cannot be utilized, and a wide dynamic range receiver is required. Consequently, logarithmic amplifiers or log/linear hybrid configurations are typically used in the receiver IF to provide the required dynamic range. The mechanisms which affect the accuracy of the range information are the same for both the interrogator and transponder. The major error mechanisms are simple, controllable, and readily identified. The accuracy and stability of the clocks, counters, and/or shift registers used to measure the round-trip time in the interrogator and to generate the reply delay in the transponder directly affect the accuracy of the range measurement. As the clock count-to-elapsed time relationship changes (i.e., oscillator drift due to temperature, age, etc.) from that assumed when performing the range computation, a bias error will l 1 The DME system is designed such that the squitter from the desired ground station is the most persistent signal. The AGC voltage is derived on the assumption that the most persistent signal is the desired signal.
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ROBERT J. KELLY AND DANNY R. CUSICK
result. Since digital timing techniques are typically used, quantization noise error is inevitable. These quantization, errors are reduced by utilizing highfrequency clocks. Finally, range measurement errors may also exhibit a dependence on pulse amplitude and rise time. Although such errors are insignificant in the DME/N application, they are of great concern in the design of DME/P equipment (see Section IV,D). In order to prevent the radiation of erroneous navigation information, the ground facility is actively monitored in real time by what is in essence a highquality interrogator. The monitor system interrogates the transponder and measures those parameters that affect system accuracy and coverage: mean reply delay, radiated power, receiver sensitivity, frequency stability, and transmitted pulse code. Because mean reply delay can affect safety of flight, it is a primary parameter. Primary alerts shut down a ground facility or switch in redundant equipment if available. Shutdown occurs when the reply delay error exceeds predefined limits: nominally 500 or 250ft when the ground facility is associated with a landing aid such as ILS. Radiated power, receiver sensitivity, frequency stability, and transmitted pulse code are not considered hazardous and are therefore designated secondary parameters. Secondary alerts cause an indication to be provided to a control point (such as a Flight Service Station) when variations exceeding allowable limits are noted by the monitor. The remaining components of the DME system are the ground and airborne antennas. Ground facilities typically provide horizontal coverage over a full 360". The vertical lobe of the ground antenna must include the minimum and maximum altitudes of the facility's service volume-for example, from the horizon to an altitude of 45,000 ft at 130 nmi for the standard high-altitude service volume defined by the FAA. Pattern shaping to improve ground system coverage near the horizon results in gains of 9 to 11 dB in the vertical plane with an upper coverage limit on the order of 60". This upper limit results in a "cone of silence" in the airspace above the station. An aircraft requires range information independent of its direction of flight with respect to the ground facility and throughout roll and pitch variations of up to f30". Typically, the airborne antenna is a 4-wave monopole providing nearly uniform gain laterally around the aircraft and maximum gain of 5 db i in the aircraft's vertical plane. In the power budgets presented later, a gain of only 2 dB i is assumed. This value represents a lower bound on the antenna gain that can be expected over nominal flight attitude and aircraft-to-ground facility geometries and therefore determines the minimum received signal level. For example, consider the common occurrence of an aircraft in level flight at the limits of the service volume. In this case, the ground facility lies along a line at a shallow angle with respect to the longitudinal axis of the aircraft (approximately 3" below the nose when the
DISTANCE MEASURING EQUIPMENT IN AVIATION
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aircraft is at an altitude of 30,000 ft and 100 nmi from the facility) yielding a gain on the order of 2 dBi. Typical vertical patterns for both the ground facility and the aircraft are shown in Fig. 48 (Office of Telecommunications, 1978). 2. An Overview of DME System Considerations As a viable navigation aid, DME/N satisfies the following system considerations: (1) It provides coverage and accuracy that satisfy operational requirements for enroute and terminal area navigation. In its most accurate configuration,it provides position fixes suitable for nonprecision approaches. (2) It has the capacity to service 100 aircraft desiring distance service from the same ground transponder. (3) It provides unambiguous distance information to known ground locations. (4) It will service the user demands beyond the year 2000, providing over 200 channels that can be assigned in a way that minimizes mutual interference and favors simple inexpensive hardware implementations in the aircraft. Service volumes are defined with range/altitude coverage characteristics that reflect the intended use of the facility, i.e., high-altitude enroute, lowaltitude enroute, and terminal. The coverage of a single facility is limited to line-of-sight operations, a characteristic of the L-band portion of the radio spectrum utilized by DME.' The minimum available coverage is determined by the radiated power, receiver sensitivity,and antenna characteristics of both the ground and airborne units. Enroute facilities are geographically placed so as to have overlapping service volumes, providing for continuous navigation along defined airways throughout virtually all of the United States. Terminal facilitiesare associated with airports and are intended to satisfy the need for short-range navigation in the terminal area. Some terminal facilities are associated with landing aides such as ILS and serve as replacements for marker beacons. Today the system accuracy of DME/N derived range information is on the order of 0.2 nmi (95%probability). By defining a common calibration method, system accuracy is preserved among equipments of different manufacture. Specifically, the 4 amplitude point on the pulse leading edge serves as the reference point for declaring pulse arrival. All time measurements, both for range and pulse code, are referenced to the 5 amplitude point of received and transmitted pulses and are thus independent of pulse rise time and amplitude. Line of sight is defined as the height above an obstacle equal to the first Fresnel zone radius. (See Appendix C of BrafT, 1966.)
134
ROBERT J. KELLY A N D DANNY R. CUSICK ZENITH
AIRCRAFT BOTTOM MOUNTED h 14 MONOPOLE VERTICAL ANTENNA PATTERN (MAXIMUM GAIN 5.0dBi) 90
80
70
60
50
40
30
20
10
DO
-10
-20
-30
-90
-80
-70
-60
-50
-40
FREE SPACE VERTICAL ANTENNA PATTERN MODEL FOR TACAN GROUND FACILITY (MAINBEAM GAIN 7.4dBi)
FIG.48. Typical vertical plane antenna patterns for DME ground and airborne elements.
DISTANCE MEASURING EQUIPMENT IN AVIATION
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The ground beacon has a finite capacity to process interrogations, a consequence of the dead time that results from the time required to identify a valid interrogation and transmit the reply pulse pair following the appropriate fixed delay. Valid interrogations arriving during this dead time are ignored and result in missed range measurements at the interrogator. This effect defines an important system performance parameter known as the reply efficiency. It is simply the ratio of the number of interrogations from a single interrogator to the number of replies generated by the ground beacon in response to the interrogations. The reply efficiency decreases as the total number of interrogations increases, thus limiting the number of aircraft which can be accomodated by a single ground beacon. The DME/N system is designed to provide a 70% reply efficiency or better in the presence of 100 aircraft. In order to serve all 100 aircraft and satisfy the minimum reply efficiency requirements, both the beacon dead time and interrogation rate must be limited. Dead times are at least 60 ps while the interrogation rate for a single interrogator is limited to 30/s for track and 15O/s for search. Computations of the peak load capacity typically assume that 95% of the interrogators utilizing the beacon are in track while the remaining 5% are in search. The reply efficiency is of prime importance during acquisition when the interrogator is attempting to sort out replies to its own interrogations from all other beacon transmissions. Once the desired signal has been acquired, the minimum range information update rate determines the acceptable beacon reply efficiency, a value which is typically less than that required for successful acquisition. Three features of the DME/N system ensure the availability of unambiguous distance information to a known ground location. First, ground beacons are assigned distinct channels characterized by an airborne interrogation frequency/pulse code and a corresponding beacon reply frequency/pulse code. Second, recall that range information is derived by identifying those beacon transmissions (replies) which are consistently synchronous with the interrogator transmissions. In order to prevent synchronization with other interrogators utilizing the beacon which can lead to false range indications, the time between consecutive interrogations is randomized (jittered). Finally, the Morse code identifier transmitted by the beacon allows the pilot to verify that the desired station is indeed the one in use. 252 DME/N channels are defined in the L-band portion of the radio spectrum at frequencies from 960-1215 MHz. Each channel consists of an upand downlink frequency pair (63 MHz apart) with adjacent channels spaced on 1 MHz intervals. Pulse codes are utilized to allow multiple channels on a single frequency.
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ROBERT J. KELLY AND DANNY R. CUSICK
Since the field deployment of DME is necessarily composed of many overlapping service volumes, due consideration must be given to minimizing the potential for mutual interference. This is accomplished through a combination of equipment performance specifications and the appropriate geographicseparation of the ground facilities.Together these techniquesfavor simple, inexpensiveairborne implementations while accomodating a sufficient number of facilities to satisfy air navigation needs. Today’s DME system is designed to place the burden of signal integrity in the hands of the ground transponder. In the airborne interrogator, simple modulators generating square pulses are permitted while rejection of signals and adjacent frequencies is accomplished with simple filters. To allow for this airborne simplicity,strict controls must be placed on the uplink spectrum, and more sophisticated adjacent frequency signal rejection techniques (such as the Ferris discriminator) must be utilized. The four key aspects of interference protection are summarized below. The frequency rejection characteristics of the ground transponder must ensure that off-frequency interrogations will not result in undesired replies and unnecessarily degrade the ground system reply efficiency. Since essentially square pulses can be used by the airborne interrogators, simple IF filters do not provide adequate attenuation of adjacent channel signals. Hence, an alternate means of rejecting an offfrequency pulse is necessary. Adequate frequency rejection is provided in the ground transponder receiver by comparing the output levels of a narrow and a wide band IF filter. The narrow-band filter, with a typical bandwidth of 200 kHz, will pass nearly 90% of the energy contained in an on-frequency pulse while an off-frequency pulse is attenuated. The output of the wide-band filter (typically 800 kHz or larger) is needed to make the range measurement since it preserves the highfrequency information contained in the pulse leading edge. A received pulse is declared on-frequency when the output level of the narrow-band filter exceeds a predetermined wide-band filter output level. In DME engineering circles, this circuit configuration is called the Ferris discriminator, named after its originator, Hal Ferris (Dodington, 1949). Rather than simply attenuating the off-frequency pulse as is done in the airborne equipment, the Ferris Discriminator “identifies” the off-frequency pulse, allowing it to be excluded from further processing by the ground transponder receiver. It does so by making use of the fact that 90% of the energy in a DME pulse is confined to within 0.5 MHz of the carrier frequency, a consequence of the fact that a relatively wide (3.5 ps) pulse is used. This technique readily provides the equivalent of 80 dB of frequency rejection, a level greater than that which can be provided by a simple 800 kHz I F filter. a. Ferris Discriminator.
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b. Uplink Pulse Spectrum. To accomodate the use of a simple filter in the interrogator, the spectrum of the uplink pulse is strictly controlled. Specifically, the effective radiated power (ERP) in the first adjacent channel is limited to 200 mW and to 2 mW in the second adjacent channel. This is in contrast to the simple relative adjacent frequency signal level requirement imposed on the downlink waveform. The implication is that any ERP may be utilized as long as the power in the adjacent bands is maintained below the specified absolute levels. This limitation requires careful shaping of the uplink pulse so that adequate signal power can be provided to achieve the desired coverage while simultaneously providing a rise time consistent with system accuracy requirements. c. Decoder Rejection Characteristics. Pulse codes are utilized to provide for multiple channels on a single interrogator/reply frequency pair. Thus, although the spectral resource is limited, more facilitiescan be accomodated in a given geographic area. To obtain a true increase in the number of available channels, off-code signals must be rejected by both the ground and airborne equipment is effectively as off-frequency signals. This is achieved by imposing strict decoder tolerances on both the airborne and ground equipment. These tolerancesrequire that pulse pairs having spacingsdeviating more than & 2 ps be rejected.This ensures that off-code interrogations do not result in undesired replies and that off-code replies received by an interrogator are excluded from consideration as valid replies. d. Channel Assignment Procedures. Channel assignment procedures are utilized to guarantee that the uplink desired-to-undesired( D / U )signal ratio at all points within a defined service volume falls within the defined signal rejection capabilities of the airborne equipment. This is accomplished by appropriate geographic separation of facilities. Co-frequency/co-code assignments are made such that an 8 dB D / U or greater is provided. Adjacent frequency assignments (both same and different codes) are made such that when the airborne frequency rejection characteristic and ground system radiated spectrum are assumed, a post filter D/U of + 8 dB or better is again obtained. On-frequency/off-code assignments are on equal footing with first adjacent frequency assignments-a result of the defined decoder characteristics. This method of frequency protection ensures correct interrogator AGC operation since the desired signal will always be stronger than any decodable undesired signal. It is clear that much effort and analysis has been devoted to defining equipment characteristics and channel assignment criteria to minimize interference between DME equipments. In addition, interference protection from secondary radar systems operating in the 1030 and 1090 MHz
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ROBERT J. KELLY A N D DANNY R. CUSICK
portions of the L band is afforded by utilizing the immediately adjacent DME channels only when absolutely necessary. However, such protection cannot be afforded from other systems utilizing the same frequency band, especially those operating on an essentially uncontrolled basis. The Joint Tactical Information Distribution System (JTIDS) is one such system which is permitted only limited operations within the L band (9601215 MHz). It is being developed in the US. and NATO countries to provide the Allied Forces the level of command, control, and communications required to ensure effective and affordable military operations in the 1980s and 1990s. JTIDS is a secure,jam-resistant, high-capacity, digital communications system. It is a spread spectrum system transmitting information in pulsed form using both pseudorandom coding of each transmitted pulse and frequency hopping across the entire L band. It will be utilized on airborne, shipboard, and ground platforms. If JTIDS operations were allowed to occur “anywhere/anytime,” the level of JTIDS signals received in both ground and airborne equipment are clearly uncontrolled and can, for example, be as high as -36 dBm when civilian aircraft is in close proximity to a JTIDS equipped aircraft. Such high signal levels can degrade the performance of DME interrogators, with the degradation manifesting itself as a change in the acquisition sensitivity. This is often due to the JTIDS signals “capturing” the AGC of the interrogator and thereby desensitizing the receiver. With the possibility that JTIDS can interfere with not only DME ground and airborne equipment but the secondary radar systems as well, the FAA, NTIA, and DOD are participating in an electromagnetic compatibility (EMC) study to determine the effects of JTIDS on all L-band ATC and navigation systems, particularly DME/N and DME/P. Once the effects are determined, necessary restrictions, such as limits on the transmitted pulse rate and power as well as geographic separation requirements, will be imposed to ensure the continued performance of all the L-band systems. An earlier JTIDS-EMC study is documented in Office of Telecommunications (1978). The results of the most recent effort are expected to be published in 1987.
3. Interrogator Range Tracking Considerations As noted earlier, the DME interrogator determines the range to the ground transponder by measuring the time between transmission of its interrogation and the receipt of the reply to that interrogation. However, the reply to the interrogation is simply one of many ground facility transmissions that will be “heard” by the interrogator following an interrogation. Hence, before the range to the facility can be determined, the interrogator must identify which, if any, of the received pulse pairs is the reply to its own interrogation.
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When the interrogator is first tuned to a new channel, it must locate the “time slot” in which the desired replies are actually occurring. This process (mathematically known as correlation) is referred to as SEARCH and in modern interrogators takes some 2 s when a valid reply signal is present. Once the reply is identified, the interrogator can determine the slant range to the station and the slant range rate and therefore track the desired replies. In other words, the interrogator now knows where to find the desired replies (to within a few p s ) on subsequent interrogations. This process is referred to as TRACK. The search or acquisition process identifies the correct “time slot” by finding the interval that contains more replies per interrogation than would result from the reception of the randomly transmitted squitter. Historically, search was done by a range gate, or time window, that was slowly (6 nmi/s) moved out from zero range to a maximum range of say 200 miles. If, at any one interval during this process, the range gate received a statistically significant number of replies in excess of that which would be expected from random ground station squitter, then the range gate would be stopped and tracking initiated on the basis that synchronous replies were occurring in that time (or range) interval. With this technique, acquisition of the synchronous range could take 20 to 30 s. When tracking, the gate would be made to follow the reply, moving in a direction dependent upon the position of the reply within the gate, i.e., either early or late. A popular circuit used to mechanize the function described above was the phantastron (Kelly, 1963), whose output generated a delay which moved the early/late range windows. The delay was generated by a voltage picked off of a potentiometer driven by a servomotor whose error signal was derived from the early/late range window. Many DME interrogators designed in the late 1950s and early 1960s used an improved two-scale system which were variations of the Meacham Range unit developed by Bell Telephone Laboratories (Chance et al., 1949). In modern interrogators, range correlation is accomplished by dividing the covered range into cells. Associated with each cell is a counter where the counter value represents the confidence that a synchronous reply is falling in that cell. Following each interrogation, those cells in which valid replies occur have their associated counters incremented. Those cells where no reply occurs have the associated counters decremented. The counter corresponding to the synchronous range will attain a higher value than all the others. This technique can be implemented by utilizing a random access memory (RAM), where each RAM location corresponds to a range cell. A specific implementation is described in Cusick (1982). Figure 49 graphically represents the search process. Assuming a 200 nmi maximum range, the maximum elapsed time between an interrogation and a reply is approximately 2500 ps. Hence, the interrogator need only “listen” for
140
ROBERT J. KELLY AND DANNY R. CUSICK SQUITTE R
INTERROGATIONS
F
I
OWN REPLIES SYNCHRONOUS TO INTERROGATIONS
1
-SEC
5th
11-1
II I I I
d t h
LESS THAN TRANSPONDER DEAD TIME
:I
I
RANDOM OCCURRENCE OF THREE CONSECUTIVE SOUITTER
REPLIES TO OTHER AIRCRAFT
I
2 5 O O / l S z 200 nmi
*
NOTE: EACH MARK REPRESENTS A PULSE PAIR. RANGE MEASUREMENTS ARE MADE ON THE FIRST PULSE.
FIG.49. DME search process.
its reply for a period of 2500 p s following the transmission of an interrogation. Since the ground transponder is transmitting an average of 2700 pulse pairs/second, each interrogation sweep will indicate an average of six to seven randomly spaced pulse pairs. If each sweep is compared, there will be very little correlation between the times at which the pulse pairs are received from sweep to sweep except for the desired replies, which will be highly correlated. For the example shown, the valid reply would have a count of five after six sweeps as compared to a maximum count of three in the other cells. Therefore, depending upon the search algorithm utilized, the desired reply could be identified after four or five sweeps. The design of acquisition and track algorithms is an interesting and important aspect of DME interrogator design. Although details of the many possible implementations and the analysis of their performance goes beyond the scope of this article, some general comments can be made. The algorithm must minimize the acquisition time when a valid reply signal is present. The algorithm must with certainty acquire and track the
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correct transponder reply. It must never acquire and continue to track the wrong range, as this could lead to a hazardous situation. In doing so, the following aspects of the DME environment must be accomodated: (1) Interrogation rate limits: Clearly, high interrogation rates result in short acquisition times since the interrogator can collect more reply samples for correlation in a shorter period of time. However, search interrogation rates are limited to a maximum of 15O/s in order to minimize the loading of the ground facility. (2) Reduced reply efficiency: As has been discussed, the interrogation load on a DME beacon reduces the reply efficiency as viewed from the interrogator. The reply efficiency can be as low as 50% under certain loading conditions and equipment configurations (FAA, 1982). The implication for the acquisition/track algorithm designer is that the valid reply is present only on a probabilistic basis, making it more difficult to separate from the randomly occurring squitter. (3) Aircraft motion: As the aircraft moves, the desired replies will move from cell to cell. During acquisition, high slant range velocities may cause desired replies to appear as uncorrelated replies on subsequent interrogations. Hence, provision must be made for retaining accumulated range cell confidence as replies traverse the range cell boundaries.
4 . Hardware Implementation Considerations
A review of various hardware DME realizations during the past four decades will not be undertaken in this article. However, four design and component trends are apparent. First, since the 1970s transmitters and receivers have evolved from vacuum tube designs to all-solid-state designs. Second, in the middle 1960s, analog circuits were replaced by digital circuits, especially in the video, timing, and logic circuits. All electro-mechanical ranging circuits (phantastrons, range potentiometers, and servo loops) have been replaced by digital configurations using counters, clocks, and shift register delay lines. Third, with the advent of the microprocessor, all control functions and data processing functions are performed by computer software programs. Finally, the most recent innovations have incorporated custom LSI designs. C. The DMEIN “Signal in Space”: Accuracy, Coverage, Channel Plan, and Capacity In the previous section, a qualitative description of the DME/N system was presented, illustrating both how the system works and why it is put
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ROBERT J. KELLY AND DANNY R. CUSICK
together as it is. In this section, the emphasis shifts to the details of the DME/N signal in space. Four topics are fundamental to the definition of the DME/N “signal in space”: accuracy, coverage, channel plan, and capacity. As is evident from earlier discussions, these topics are interrelated in that the selection of a system parameter related to one cannot be done without due consideration of the effect on the complete system. The purpose of this section‘is to detail these fundamental aspects of DME while stressing the inherent complex interrelationships. Subsection C,1 reviews current DME/N accuracy requirements and how they are achieved. Subsection C,2 discusses various aspects of DME/N system coverage. Subsection C,3 defines the details of the DME channel plan in terms of its structure, frequency-assignment considerations, and the “hard pairing” of the DME channels with the VHF navaid frequencies. Finally, Subsection C,4 presents details concerning the computation of ground beacon capacity. 1. Accuracy
The accuracy requirements imposed on the DME/N system reflect its intended use. They satisfy basic enroute and terminal area navigation needs when utilized in conjunction with bearing information derived from VOR and TACAN ground stations. For the purpose of facility planning and equipment procurement specifications, DME/N system errors must be partitioned into instrumentation errors and site-dependent errors (i.e., those resulting from multipath and random pulse interference or garble). Detailed error budgets, such as those to be presented for DME/P in Section IV of this article, are traditionally not defined for DME/N. For the DME/N, only the total system error and the equipment error components are specified. Thus, site-dependent errors represent the margin after the equipment error components have been removed from the total system error. As stated in Section II,c, the error components are assumed to be zero mean random variables and hence can be combined on a root-sum-square basis. Further, no distinction is made between noise and bias errors; only a single number is specified. The error limits are interpreted on a 95% probability basis. Table VIII illustrates a partitioning of the DME/N system range errors among the instrumentation and site-dependent components. The values shown are based on SARPs ground equipment and system error specifications and SARPs, ARINC, and RTCA airborne specifications (ICAO, 1985; ARINC, 1982; RTCA, 1985b). The SARPs ground equipment error limits are more stringent when the facility is associated with a landing aid such as ILS.
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TABLE VIII BUDGET (95% PROBABILITY) DME/N SYSTEMERROR Specification (nm)
Instrumentation Ground (SARPs) Airborne RSS total System error (SARPs) Margin for site-dependent errors (multipath,garble)
SARPs
RTCA
ARINC
0.08 0.17 0.19 0.20 0.07
-
-
0.17 0.19 0.20 0.07
0.10 0.13 0.20 0.15
The ground system instrumentation error is 0.04 nmi when the DME is associated with a landing aid (ILS). Total instrumentation error and margin for site-dependent effects are modified accordingly. Ground and airborne instrumentation error components are those defined in the new DME/N SARPs (ICAO, 1985) and the new RTCA MOPS(RTCA, 1985b).
Instrumentation errors are specified for received signal power levels that are at minimum sensitivity and thus include receiver noise effects. Other factors affecting instrumentation error were noted in Subsection B,1. Sufficient margin remains for the site-dependent components since: (1) Multipath errors will typically not exceed 0.04 nmi-a result of the fact that multipath-to-direct signal ratios can be expected to be less than - 6 dB since the aircraft is above most of the severe multipath. (2) Garble-induced errors, which result from extraneous pulses that are coincident with the ranging pulse (first pulse of an interrogation or reply pair), are negligible. This is a result of the frequency rejection characteristics of the ground and airborne equipment as well as the channel assignment methodology that ensures an 8 dB post IF filter D / U in the airborne equipment. That is, extraneous pulses will always be below the amplitude point of the desired pulse and hence can contribute little to the range error.
It must be emphasized that the site-dependent components of the DME/N range errors (i.e., multipath and garble) are identical to those that must be considered when assessing the performance of the I A mode of DME/P. A more detailed analysis of these effects is presented in Section IV of this article. Two system level definitions are key in ensuring that the system error budgets are achieved: the shape of the envelope of the transmitted rf pulse and the calibration technique. These are detailed in Figs. 50 and 51.
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ROBERT J. KELLY AND DANNY R. CUSICK
A 0.9A
0.5A 0.1A 0.05A I
a b
c
d
e
f
g h
TIMEMeasured w i t h a wideband linear detector. Note: The characteristics of the DME pulse (shape and spectrum) are defined w i t h respect t o the radiated signal-in-space and d o n o t include the effects of receiver f iItering.
Pulse envelope:
Leading edge
@
0.5A: Reference point for all ranging and delay measurements.
Pulse Amplitude (A):
Peak amplitude of pulse envelope
Pulse duration (h-a):
Pulse duration utilized in determination of average transmitted power.
Pulse rise time (d-b):
Rise time as measured between the 10% and 90% amplitude points on the pulse leading edge. (d-b) 2 3ps
Pulse width (f-c):
Time interval between the 50% amplitude points on the leading and trailing edges. (f-C)
= 3.5 * 0 . 5 ~ ~
Pulse decay time (g-e): Decay time as measured between the 90% and 10% amplitude points on the pulse trailing edge. (g-e) 5 3.5~s
FIG.50. DME/N transmitted pulse specifications.
2. Coverage A single nationwide airway system shared by all users of the National Airspace has been defined with the use of VOR, DME, and TACAN radio navigation aids. These airways, described earlier in Section II,D, serve as standard routes to and from airports throughout the U.S. The navaid facilities (VOR, DME, TACAN, or combinations thereof) are geographically placed so
DISTANCE MEASURING EQUIPMENT IN AVIATION TRANSMITTED
INTERROGATION
INTERROGATION PULSE PAIR
PAIR RECEIVED A T TRANSPONDER
TRANSMITTED REPLY PULSE PAIR
REPLY CODE 12 us (XI 30us I Y I
INTERROGATION CODE 1 Z U S (XI
t b-
DOWN LINK PROPOGATION DELAY (6.18 pslnmil
&$::
TOTAL ROUND TRIP
Range(nmi) =
145
REPLY PAIR RECEIVED AT INTERROGATOR
PROPOGATION up LINK DELAY
TRANSPONDER DELAY IZERO RANGE REFERENCE)
(6.18 pslnmil
DELAY MEASURED BY INTERROGATOR
Total Round Trip Delay-Transponder 12.36psl nmi
-4
Delay
The ‘12 amplitude point (i.e., 6dB below the pulse peak) on the pulse leading edge is the reference pont for all time measurements: range, transponder delay, and code. Hence, range measurements are independent of pulse shape. Range measurements are based on first pulse timing Transmitted RF interrogation pulse pair is processed by interrogator receiver
to begin range timing. Bias errors due to delays in the processing path are minimized since the received reply pulse pair is processed by the same receiver path. The transponder begins transmission of the reply pair such that the desired delay results between the ‘/z amplitude point on the leading edge of the first interrogation pulse received and the same point on the first pulse of the reply pair.
FIG.51. DME/N calibration and range coding.
as to provide for continuous navigation information along the defined airways. The volume of airspace served by a single DME ground facility is influenced by a large number of factors: antenna characteristics, propogation path variation, ground cover, natural and manmade obstructions, equipment characteristics such as transmitter power and receiver sensitivity of both the ground and airborne equipment, and the interrogation load on the ground beacon. Since the L-band portion of the radio frequency spectrum is utilized, the maximum coverage is limited to line of sight. The factors which determine the coverage of a single DME facility are discussed below. a. Standard Service Volumes.
The volume of airspace to be served by a single DME ground facility is required by the SARPs to be consistent with the coverage of the facility with which it is paired, i.e., VOR, TACAN, ILS, or
146
ROBERT J. KELLY AND DANNY R. CUSICK
MLS azimuth. In the United States, the FAA has chosen to define “standard service volumes” to satisfy this requirement. The standard service volume (SSV) is characterized by a defined volume of airspace and the signal characteristics (e.g., signal level and frequency protection) that are guaranteed within that volume. Each service volume definition reflects operational need and user experience with various types of ground facilities. Three service volumes applicable to DME/N are defined: terminal, low altitude, and high altitude. Each service volume is pictorially depicted in Figs. 52 and 53. Collectively they connect the airspace up to 60,000 ft (enroute jet routes) down to the 200 ft decision height (final approach). The shape of the lower edge reflects the effects of signal attenuation at low elevation angles, the combined result of the freespace antenna pattern and signal reflections from the ground. In each case, 360”azimuth coverage is provided, with vertical coverage limited to 60” above the horizon (not shown), thereby defining a “cone of silence” in the area above the station. The SSVs reflect the airspace about a given facility that is subjected to flight inspection prior to commissioning the facility for operational use (FAA, 1980,1982;ICAO, 1972a).Local terrain and obstruction conditions that result in inadequate system performance are restricted in use, with public notification via an appropriate notice to airmen (NOTAM). Where terrain conditions are such that operational performance is achievable outside the SSV, and an operational need for expanded service is required, an extended service volume (ESV) may be defined. ESVs typically consist of air routes defined along a VOR or TACAN radial that extends over water or flat terrainareas where adequate signal power is assured at long ranges and low angles relative to the ground facility due to the lack of signal obstructions. The same signal standards and flight check certification procedures applicable to SSVs are required prior to commissioning of the ESV. b. Power Budgets. Power budgets play a role similar to that of error budgets. They identify both ground and airborne equipment losses, propogation losses, and antenna characteristics to ensure that the up- and downlink signal in space is adequate for the accurate determination of range. In essence, they allow the determination of the coverage provided by a single DME ground facility. The coverage problem consists of two basic parts: the downlink and the uplink. First, to ensure that the ground facility will respond with high probability to an interrogation, the airborne interrogator must provide a signal level at the ground facility that exceeds the beacon receiver sensitivity. The beacon receiver sensitivity is defined as the signal level that results in 70% reply efficiency from an unloaded beacon. Second, the ground facility must
147
DISTANCE MEASURING EQUIPMENT IN AVIATION STANDARD LOW ALTITUDE SERVICE VOLUME
STANDARD HIGH ALTITUDE SERVICE VOLUME
100 nrni 40 nmi
c18,OOO ft. 45,000 h . 4
-1,000
h.
DEFINITION OF THE LOWER EDGE OF THE STANDARD LOW AND HIGH ALTITUDE SERVICE VOLUMES
W
CI
400
0
0
5
10
15
20
25
30
35
40
DISTANCE TO THE STATION IN nrni
FIG.52. Low- and high-altitude Standard Service Volumes (SSVs) as defined in the U.S. VOR/DME/TACAN National Standard (AC-00-31A). All elevations are with respect to the station elevation (AGL).
148
ROBERT J. KELLY AND DANNY R. CUSICK STANDARD TERMINAL SERVICE VOLUME
25 nmi
-12,mft.
DEFINITION OF THE LOWER EDGE OF THE STANDARD TERMINAL SERVICE VOLUME
1000
0 0
5
10
15
20
25
DISTANCE TO THE STATION IN nmi
FIG.53. Terminal Standard Service Volume as defined in the U.S. VOR/DME/TACAN National Standard (AC-00-31A). All elevations are with respect to the station elevation (AGL.)
provide a signal level at the airborne interrogator that exceeds the airborne receiver sensitivity. The airborne receiver sensitivity is the minimum signal level at which the interrogator can acquire and track the desired beacon replies when the beacon reply efficiency is 70%. Clearly, trade-offs exist in the selection of radiated power and receiver sensitivity on both the air-to-ground (AIG)and ground-to-air (GIA) links. An increase in the radiated power on one end of the link allows a decrease in the sensitivity of the receiver on the other end of the link and vice versa. Since weight, size, power consumption, and cost are inherent constraints in avionics
DISTANCE MEASURING EQUIPMENT IN AVIATION
149
design, the burden of high transmitter power and sensitive receivers has traditionally fallen on the ground equipment. The DME SARPs as well as the US. National Standard define the minimum uplink power density available within the standard service volume as well as the minimum power density required at the ground facility to obtain 70% reply efficiency. With these details defined, the interrogator designer is free to select transmitter power and receiver sensitivities to obtain the desired performance (i.e., range coverage). The power density specifications and typical airborne transmitter/receiver parameters are noted in Table IX. TABLE IX SUMMARY OF GROUND SYSTEMRADIATEDPOWERAND SENSITIVITY REQUIREMENTS, AIRBORNE RECEIVERSENSITIVITYREQUIREMENTS,AND TYPICAL TRANSMITTER POWERLEVELS Ground-to-air “available signal power” requirements‘ SARPs US. National Std
-89 dB W/m2 -86 dB W/m2 -91.5 dB W/m2
All points in defined coverage volume Below 18,000 ft Above 18,000 ft
Beacon receiver sensitivities (70% reply efficiency) SARPs US. National Std
- 103 dB W/m2 - 101.5 dB W/m2 - 104.5 dB W/m2
SARPs RTCA (1978) ARINC RTCA (1985b)
- 89 dB W/m2 -80dBm -90dBm -83 dBm
All facilities-input to antenna DME facilities-input to antenna TACAN facilities-input to antenna
Airborne receiver sensitivity requirements Input to antenna Input to receiver terminals Input to receiver terminals Input to receiver terminals
Typical transmitter power levels Ground beacon (US.) Terminal Low altitude High altitude
100 W 1000 W 5000 W
Interrogator US. National Std
2 8 0 W EIRP 2 800 W EIRP
RTCA
250 W 2250 W 21000w
ARINC Useful relations
Free-space propogation loss (dB) = - 37.8 - 2010g(f&) - 20 lOg(R,,J Conversion of power density (dB W/m2) to isotropic power (dB m) P(dBm) = 68.5 - 2010g(fM,) + P(dB W/m2) All levels are provided on a 95% time-of-availability basis.
Below 18,000 ft Above 18,000 ft Below 18,000 ft Above 18,000 ft
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ROBERT J. KELLY A N D D A N N Y R. CUSICK
Signal level specifications in terms of power density (dB W/m’) ensure independence from implementation details. The ground system designer must select an appropriate combination of transmitter cabling and antenna characteristics to provide the required power density in space. No assumptions regarding the airborne installation (i.e., net antenna-cable gain) utilizing the ground facility are required. The airborne equipment designer must select antenna, cable, and receiver characteristics that provide the desired performance in the presence of the minimum available signal power density. Examples of power budgets that reconcile the ground and airborne equipment losses are noted in Table X. c. The Signal Level in Space.
The previous discussions have defined the standard service volumes and the minimum signal levels to be provided within those service volumes. For a particular equipment configuration, the prediction of the resulting signal levels within the service volume is a complex and nonexact task. The discussion below is neither rigorous nor complete, but is intended as an introduction to the problem of predicting facility coverage. For operations in free space, away from all reflecting surfaces and in a nonchanging medium (atmosphere), the computation of the available signal in space can be accomplished when the transmitter power, antenna gains, cable losses, and distance between the transmitter and receiver are known. This problem is described in Fig. 54. The gain of an antenna G is defined as the ratio of the power density at a point in space caused by a specific antenna to the power density at the same point caused by an ideal lossless isotropic antenna where each antenna has the same input power. The antenna gain will vary as a function of its horizontal and vertical angles from antenna boresight. The isotropic antenna is one which is perfectly omnidirectional. The effective capture area A is that area (square meters) which when multiplied by the power density at the antenna location results in the power produced at the antenna output port. For an isotropic antenna, the capture area A is A2/47c, where A is the wavelength of the transmitted signal. Note that the physical cross section of an antenna can be much smaller than its capture area. For example, the capture area of a +-wavelength dipole is 1.64 (A2/4x),while the physical cross section may be 100 times less. The relationship of antenna gain to the capture area A (Silver, 1949) is A
= (A2/4n)G
(21)
Hence, as it should be, the gain of an isotropic radiator is unity. For the DME application the antenna radiation pattern, G is assumed to be omnidirectional in the horizontal plane and is assumed to vary only as 0 in the vertical plane.
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DISTANCE MEASURING EQUIPMENT IN AVIATION TABLE X EXAMPLE GROUND AND AIRBORNE DME/N POWERBUDGETS' Airborne receiver SARPs
Low alt and terminal
High altitude
Power available at antenna airborne
-82.0 dB m
-79.0 dB m
- 84.5 dB m
Antenna gain
+ 2.0 dB
+ 2.0 dB
+ 2.0 dB
Aircraft cable loss
- 3.0 dB
- 3.0 dB
- 3.0 dB
Signal available at receiver terminals
-83 dBm
- 80.0 dB m
- 85.5 dB m
Receiver noise video noise figure 12 dB; IF bandwidth 0.8 MHz Signal-to-noise ratio (video)
-106dBm 23 dB
-106dBm 26 dB
-106dBm 20.5 dB
Beacon receiver SARPs
US National Standard Low alt and terminal High altitude
Power available at ground antenna
-96 dBm
-94.5 dBm
-97.5 dB m
Antenna gain
+ 8.2 dB
+ 8.2 dB
+4.5 dB
Cable loss
- 1.5 dB
- 1.5 dB
- 1.5 dB
Signal available at beacon receiver terminals
- 89.3 dB m
- 87.8 dB
-94.5 dB
Receiver noise video noise figure 6 dB; I F bandwidth 0.8 MHz
-112dBm
-112 dBm
-112dBm
22.7 dB
24.2 dB
17.5 dB
Signal-to-noise ratio (video)
Notes: (1) Power density (dB W/m2) = power (dB m) - 7; (2) SIN (video)= SIN (IF) + 10 log (IF noise bandwidth/video noise bandwidth); (3) These budgets are intended as examples only. Other combinations of antenna/cable coupling are possible and should reflect installed conditions.
That is, G, = GT(8), where 8 is the angle that a given ray makes with the horizon for the ground antenna. Similarly, let 6, be the angle that a given ray makes with the aircraft pitch axis for the airborne antenna; i.e., G, = G,(8,). Let PT equal the power radiated by an isotropic radiator. The power radiated in any direction is PT/471 watts per steradian. Assume that the isotropic radiator is replaced by an antenna with gain GT(6) over isotropic. The power radiated in that direction is P&(8)/4n watts/per steradian.
152
ROBERT J. KELLY A N D DANNY R. CUSICK GT
GR
P i = TRANSMITTER POWER (WATTS)
LT = TRANSMITTER CABLES LOSS GT = TRANSMIT ANTENNA GAIN (WATTSMTATT) R = TRANSMITTER TO RECEIVER DISTANCE (METERS)
GR = RECEIVER ANTENNA GAIN (WATTSMTATT)
LR = RECEIVER CABLE LOSS PR = RECEIVED POWER (WATTS)
FIG.54. Definition of parameters for the analysis of a transmit-receive link
Placing a receiver antenna with effective capture area A at distance R from the transmitting antenna subtends a solid angle of AIR2 steradians. Thus the power at the receiver PR is PR = P,GT(B)A/4ZR2
(22)
The effective receiving antenna capture area is A
= (A2/47r)G,(8,)
m2
(23)
Let LT and L R represent the cable losses between the antenna and the transmitter or receiver, respectively, as shown in Fig. 54. Hence, the received power at the input to the receiver is simply the product of the power density at the antenna, the capture area of the receiving antenna, and the cable losses:
The term (A/47rR)’ is the space or propagation loss and is proportional to the
DISTANCE MEASURING EQUIPMENT I N AVIATION
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inverse square of the range. At L band, the propagation loss is conveniently expressed as free-space propagation loss (dB) = -37.8 - 20 log(&Hz)
-
20 log(R,,i)
(25)
If beacons and interrogators operated in free space, the above discussion would provide the means to predict the useful range of operations that could be achieved with an interrogator/beacon pair whose operational parameters-antenna characteristics, transmitter power, and receiver sensitivities-were known. But they do not. Operations are conducted near a spherical Earth from which radiated signals are reflected and scattered and in an atmosphere that has pronounced and not always predictable effects. These factors must be considered in predicting the useful range of the DME system. The signal at the interrogator receiver is the sum of the direct path signal and reflected signals from the ground (assuming that neither signal path is blocked by obstructions and that long delay reflections are absent). This results in a received signal which can deviate significantly from the direct path signal (Fig. 55). The reason is simply that the two signals are related in phase and add vectorially. When the direct and reflected signals are in phase, the received signal will be larger than the free-space value. When they are 180" out of phase, the received signal is the difference between the direct and reflected path signal levels and hence the received signal level is reduced below the freespace value.
FIG.55. Geometry for signal strength calculation in the presence of ground multipath.
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ROBERT J. KELLY A N D DANNY R. CUSICK
Ground reflection effects can be approximated using Fig. 55 as follows: Let the direct signal be
where o is the rf carrier frequency and cpD is the path delay. Let the reflected signal be
where pantis the magnitude of the reflected ray at the transmitter antenna, pRis the ground reflection coefficient, I$ is the reflection coefficient phase, and cpR is the reflected signal path delay. The above assumed that the antenna gain GT(6D) = 1 for ZD and CT(6R) = pan,for i f R . RD and RR are the path lengths for the direct and reflected signals, respectively. A pattern propagation factor I FI can be defined which permits the effects of ground reflections to be incorporated into the range equation. IF1 is the ratio of the resultant signal at the point in question to the value it would have been under free-space conditions. Using Fig. 55
where
and AR = RR - RD N 2hi h,/R
Note that h , and h, have been measured from the tangent plane shown in Fig. 55. For a flat earth, h , = hT and h, = hR. The range Eq. (24) becomes
h = PTGT(6)GR(e)A)IF12(~/4~R)2
(27)
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where GR(OA)is equal for both the direct and reflected signals. Equation (27) applies both to the DME up- and downlinks. The phase change on reflection 4 and the reflection coefficient pRdepend on, in addition to the factors already noted, the electrical properties and the degree of roughness of the reflecting surface. The roughness of the surface determines the resemblance of the surface to a mirror. A surface which appears smooth to long-wavelength radiation may be rough for short-wavelength radiation. For the case of a single reflection, IF1 varies between (1 p ) and (1 - p). When pRis nearly unity, that is, the reflecting surface is mirrorlike (little or no attenuation occurs), and pantN 1, IF1 will vary between 0 and 2 times the freespace value. On the other hand, for p << 1, IF1 will approximate the free-space value. When AR is small, pR 2: 1 and 4 = 180°, then Eq. (26) becomes (F12 = ( 4 ~ h , h , / Rand ) ~ Eq. (27) becomes
+
This condition can occur when the aircraft is close to the ground during the landing phase of flight. Since the received power decreases as l/R4, a careful antenna design is warranted as discussed in Section IV,C,4 for the DME/P siting considerations. It is clear that if the system parameters of transmitter power, antenna characteristics, and the reflection parameters of 4 and pR are known, the effects of reflections can be accounted for and the received signal level computed for a specified receiver/transmitter geometry. Although the system parameters are known, the parameters which determine the characteristics of the reflected signal are subject to wide variation since the ground and its covering vegetations change with season, weather, and wind. In order to make prediction of the signal level in space under a wide range of conditions, a statistical treatment of the reflecting characteristics and atmospheric effects is required. Effects of the reflecting surface and atmospheric losses must be treated as random variables whose density functions must be convolved to determine the distribution of the power density in space. This complexity requires the use of computer models to perform the computations accurately. A computer-based propagation model, developed by the Institute for Telecommunication Sciences, was developed to perform just this function (see Johnson and Gierhart 1978a,b, 1980, 1983). The model is applicable for ground/air telecommunication links operating at frequencies from 0.1 to 20 GHz for aircraft altitudes below 300,000 ft and is utilized by the FAA to
ROBERT J. KELLY AND DANNY R. CUSICK
156 100 I-
w
Y
LL
0
80
: 260 3
0 I-
z
-40 -I (1
-a x
3 20
t
5a 0
0
20
40
60
80
100
120
140
160
180
200
220
DISTANCE FROM THE DESIRED STATION I N nmi
FIG.56. Signal strength in space for a SO00 W TACAN beacon primarily utilized for highaltitude enroute navigation. This graph does not show the upper coverage limit, i.e., the signal strength in the “cone of confusion” above the station.
DISTANCE FROM THE DESIRE0 STATION I N nmi
FIG.57. Signal strength in space for a lo00 W DME beacon primarily utilized for lowaltitude navigation. This graph does not show the upper coverage limit, i.e., the signal strength in the “cone of confusion” above the station.
define standard facility performance for a given set of transmitter power levels and antenna characteristics. Since the DME system is a safety service, signals must be highly reliable, and thus conservative assumptions are made as inputs to the propogation model. Figures 56, 57, and 58 illustrate typical results of the propagation model for actual DME ground facilities.
157
DISTANCE MEASURING EQUIPMENT IN AVIATION
0
20
40
60 80 100 120 140 160 DISTANCE FROM THE DESIRED STATION IN nmi
180
200
220
FIG.58. Signal strength in space for a 100 W DME beacon primarily utilized for terminal area navigation. This graph does not show the upper coverage limit, i.e., the signal strength in the “cone of confusion” above the station.
Additional such “Power Available” curves are provided in FAA (1980) and Johnson and Gierhart (1980). Note, for example, in Fig. 56 that the -79 dBW/m2 contour equals the free-space loss at 140 nmi. At 70 nmi the free-space loss should have been only -73.5 dBW/m2 but it remains at -79 dBW/mZ, thus reflecting the above ideas. That is, propagation losses are a random phenomenon and can only be described statistically. The signal strength curves shown in Figs. 56,57, and 58 are in terms of 95% time of availability. This means that the instantaneous signal strength will be greater than or equal to that value 95% of the time. Similarly, the D/U ratios that comprise the channel assignment rules are also defined on a 95% availability basis. The curves in Figs. 56, 57, and 58 contain many propagation effects not included in the Fig. 55 analysis. One of these effects, radio beam bending caused by atmospheric refraction, requires comment. The atmosphere above the earth has an index of refraction which decreases with increasing altitude. This decreasing refractive index causes radio beams near the earth’s surface to bend downward, as indicated in Fig. 59a. Therefore, the distance to the radio horizon is greater than the geometric horizon distance. In order to perform intuitively simple calculations, it is desirable to transform these curved radio beams into straight-line transmission paths. The transformation is accomplished by subtracting the radius of curvature of the refracted beam from the earth‘s curvature (Johnson and Gierhart, 1978a). The result is an effective
158
ROBERT J. KELLY AND DANNY R. CUSICK
earth radius K a = (4/3)a, where a is the actual earth radius. As noted in Fig. 59b, the radio beams now plot as straight lines. The geometry used in Fig. 55 was based upon an equivalent 4/3 radius earth. As long as the receiver is within the radio horizon, the propagation losses are simply the straight-line losses. Beyond the radio horizon, the radio beam is in the diffraction region; therefore, an additional diffraction loss factor must be included in the power budgets. The diffraction loss factors are 1.6 dB/nm and 2.8 dB/nm at 1000 and 5000 MHz, respectively.These considerations are important when performing cochannel protection ratio calculations in Subsection III,C,3,b.
d . Uplink/Downlink Design Considerations. DME is a two-link system in that both the up- and downlinks must operate successfully for range information to be obtained by the interrogator. Specifically, the signal at the ground beacon must have sufficient strength to result in a reply efficiency that is adequate for acquisition to occur with the available signal power at the interrogator. The signal level at the interrogator that results in acquisition of the desired signal, determined by considering the performance of both the upand downlinks, is referred to as the beacon-coupled acquisition sensitivity (Office of Telecommunications, 1978). It may be different from a bench measurement of interrogator sensitivity, since the interrogator transmitter power and ground facility receiver performance must be taken into account. The analysis method described below illustrates how the appropriate values of interrogator transmitter power and receiver sensitivity can be chosen to ensure that an interrogator will acquire uplink replies to its downlink transmissions. Using Eq. (24), the ratio of the power at the beacon receiver terminals PRb to the interrogator transmitter power PTiis R‘
b
- = GTGRLTLRJFJ2 pTi
The ratio of the power at the interrogator terminals PRito the beacon transmitter power PTbis
Since the interrogating signal and the reply signal traverse the same paths, the right-hand sides of Eqs. (28) and (29) are equal, and the ratios can be equated, yielding
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159
INTERFERENCE REGION
REFRACTION
RAY CURVATURE OVER SPHERICAL EARTH OF RADIUS a
INTERFERENCE REGION
WITH REFRACTION
RAYS WHEN USING EOUlVALENT4/3 RADIUS EARTH.
RADIATION PATHS WITHIN~RADIOHORIZON HAVE STRAIGHT LINE RAYS RADIATION PATHS BEYOND RADIO HORIZON IDIFFRACTION REGION1 HAVE l.BdB/nm ADDITIONAL LOSS P L-BANDA N 0 2.8dBInm B C-BAND. RADIO HORIZON DISTANCE dlnml = d1.51h (FEET)
FIG. 59. Definitions of radio horizon and geometric horizon.
The power at the interrogator receiver terminals therefore differs from that received at the beacon receiver terminals by PTb(dBm)- PTi(dBm). Equation (30) is applied as follows. The ground beacon transmitter power and receiver sensitivities are specified by the FAA (1982)and listed in Table IX. For the low-altitude service volume PRb= -94 dBm and PTb= 60 dBm. Furthermore, as shown in Table IX, the FAA provides a signal in space of - 79 dBm, (- 86 dBW/m2) with 95% availability for the low-altitude service volume. Using Table X, the power at the interrogator receiver input is PRi= - 80 dBm. The manufacturer must design a receiver with at least a - 80 dBm to detect and measure the signal at the limits of the low-altitude service volume. Inserting PRb= - 94 dBm, PTb= 60 dBm, and PRi= - 80 dBm into
160
ROBERT J. KELLY AND DANNY R. CUSICK
Eq. (30) yields PTi= 46 dBm = 40 W [the MOPS (RTCA, 1978) requires at least 50 W]. Thus at least a 40 W transmitter is necessary to ensure that the downlink is balanced with the uplink. A calculation to determine the required transmitter power for the highaltitude service volume proceeds as follows. From Table IX the uplink signal in space is - 91.5 dBW/m2 or - 84.5 dBm. Therefore, the receiver design must successfully detect PRi= -85.5 dBm (ARINC requires -90 dBm). Using Table IX, PTb= 67 dBm and PRb= -96 dBm for the TACAN beacon. The required interrogator transmitter power is 446 W. 3. Channel Plan
The purpose of a channel plan is to allocate the available spectral resource in an efficient manner while satisfying the operational requirements of the system. In the case of TACAN/DME, the spectral resource is the 960-1215 MHz portion of the L band, and the operational requirement is to provide unambiguous high-quality range data along all points of defined airways. To be complete, the channel plan concept must include the following: i. Channel Dejnition. Defines the frequency separation that accomodates the pulse shapes and signal processing bandwidths necessary to achieve system accuracies. ii. Channel Assignment. Provides a required number of ground facilities in an manner that precludes or at least minimizes mutual interference. iii. Channel Pairing. Simplifies selection of co-located stations, i.e., VORDME, VORTAC, ILS-DME, and MLS-DME, thereby minimizing pilot workload and blunders.
Accommodating the above involves a multitude of system considerations. For example, channel frequency spacing impacts equipment transmit and receive frequency stability requirements-the closer the channel spacing in frequency, the more stable the equipment must be to prevent interference due to drift. In contrast, accuracy considerations dictate the receiver bandwidth requirements which in part determine susceptibility to off-frequency extraneous signals. In essence, this defines the channel assignment criteria. Channel assignment criteria are then made manageable by strict spectral limitations imposed on the transmitted waveforms. a. TACANIDME Channel DeJnition. The TACANIDME system operates in the 960-1215 MHz (L-Band) portion of the radio spectrum. This band, allocated for aeronautical radionavigation, is reserved on a worldwide basis for the use and development of airborne electronic aids to air navigation and any directly associated ground-based facilities [National Telecommunication
DISTANCE MEASURING EQUIPMENT IN AVIATION
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and Information Administration (NTIA), 19843. TACAN/DME interrogation and reply frequencies are defined on 1 MHz increments between 962 and 1213 MHz, leaving 2 MHz guard bands at both ends of the band. Frequencies immediately below the 960-1215 MHz band (942-960 MHz) are allocated for use in mobile and fixed communications as well as broadcast functions. Frequencies immediately above the 960-1215 MHz band (12151240 MHz) are allocated for radiolocation and radionavigation-satellite (space-to-earth) functions (e.g., the NAVSTAR global positioning system operates in this band). The basic TACAN/DME channel plan defines 252 distinct operating channels numerically designated 1-126. A channel is defined by an interrogation/reply frequency pair and a pulse code, either X or Y. The interrogation frequency is the transmission frequency of the airborne equipment, while the reply frequency is the transmission frequency of the ground equipment. These two frequencies are 63 MHz apart, with 63 MHz serving as the IF frequency in typical TACAN/DME equipment. The use of a 63 MHz I F takes advantage of the interrogation/reply frequency separation in that the transmit frequency can be used as the local oscillator in the receive channel, thereby reducing the number of crystals required in early DME equipment and simplifying the frequency synthesizers utilized in modern DME equipment. The code of the channel refers to the spacing of the pulse pairs as utilized by the interrogator and the ground beacon. This “encoding” of interrogation and reply transmissions allows for multiple channels on a single frequency. The channel plan utilizes 126 interrogation frequencies on 1 MHz increments between 1025 and 1150 MHz inclusive. Channel numbering is sequential with increasing interrogation frequency. Associated with each interrogation frequency are two channels formed by the X and Y codes. Each channel is uniquely distinguished by beacon reply frequency and the pulsepair code used for interrogation and reply. X channels 1-63 utilize reply frequencies between 962 and 1024 MHz, while the corresponding Y channels utilize reply frequencies between 1088 and 1150 MHz. X channels 64-126 utilize reply frequencies between 1151 and 12 13 MHz, while the corresponding Y channels utilize reply frequencies between 1025 and 1087 MHz. Note that, unlike the X channels, the Y channels reuse frequencies within the interrogation frequency block. This basic channel plan structure is summarized in Fig. 60. Although the DME/P channel plan is not detailed until Section IV,A,4, the interrogation and reply frequencies for the newly defined W and Z coded channels, used exclusively for DME/P, are shown for future reference. This channel pairing has been adopted by ICAO for facilities supporting operations in the international air navigation service and also serves as the basis for all frequency planning and assignments in the U.S. national airspace
ROBERT J. KELLY AND DANNY R. CUSICK
162
I
960 MHz
1213 M H r
- '.
96; M H r
1025 MHr
1088 M H r
11215 MHz
1151 MHz
/
0 '
H0
" , ,1 - 6
'.
1
\
'.
I
0 '
64-
'0 0
0
/ ' 0 ,yO
1 -
-. 7-562 1
"'
30-119Zc
-
,
0
KW:
INTERROGATION FREQUENCIES
1-1
REPLY FREQUENCIES
CH = C H A N N E L NUMBER (1-1261 I = I N T E R R O G A T I O N FREOUENCY IMH7. R = REPLY FREQUENCY (MHzl
X CHANNELS
Y CHANNELS
I = 1025 t (CH-11
I = 1025 + (CH-1 I
< CH G 63 < C H < 126
R=I-63
1
R=It63 12 us
64
INTERROGATION CODE
12 us
REPLY CODE
50 us
TRANSPONDER REPLY D E L A Y
R=It63 R=I-63
36 us 30 vs 56 vs
NOTE: W A N D Z CODED CHANNELS ARE USED EXCLUSIVELY FOR DME/P A N D ARE CODE M U L T I P L E X E D ON THE X A N D Y FREQUENCIES RESPECTIVELY
FIG.60. TACAN/DME channel plan with interrogation reply frequency pairing, codes, and transponder reply delays.
DISTANCE MEASURING EQUIPMENT IN AVIATION
163
system (NAS).In the International channel plan, channels 1-16X are reserved for national use. Additionally, channels 1-16Y, 60-69X, 60-79Y, and 124126Y are to be utilized on a secondary basis to afford frequency protection to secondary surveillance radar systems such as ATCRBS and MODE S, as well as the newly defined TCAS. These systems utilize 1030 MHz as the ground radar interrogation frequency (airborne interrogation frequency in TCAS) and 1090 MHz as the airborne transponder reply frequency. In the United States, TACAN/DME channels 17-59 and 70-126 are designated for use in the national airspace system, while channels 1-16 and 60-69 are reserved for tactical military operations.
b. TACANIDME Channel Assignment. Several basic principles govern TACAN/DME ground facility channel assignment. These principles, as described earlier in Subsection B,2, make optimum use of the frequency and code rejection characteristics of the ground and airborne equipment as well as the radiated pulse spectrum to maximize the number of facilities that can be accommodated in a geographic area. Ground facility channel assignments are made in a way that provides a minimum uplink free-space desired-to-undesired ( D / U ) signal ratio at all points within a desired facility’s service volume. When making a frequency assignment, each facility must be considered a desired facility with all others treated as undesired facilities. Only when the D / U requirements are satisfied for all facilities can an assignment be made. The specific free-space D / U criteria is chosen such that an effective (postprocessing) D / U of 8 dB is achieved in the interrogator when its frequency and code rejection capabilities are considered. In this case, the undesired signal level is the peak power from another ground facility. Thus, the interrogator designer is guaranteed that, within the defined operational service volume, the desired signal will always have a minimum 8 dB advantage over all undesired beacon transmissions. The designer is free to utilize this discriminate as desired. Frequency assignment rules are based on both the spectrum of the ground system transmitted waveform and the airborne interrogator receiver frequency rejection characteristic as summarized in Table XI and Fig. 61. Ground system radiated spectrum specifications limit the radiated power in adjacent frequency bands to an absolute level of 200 mW on the first adjacent channel and 2 mW on the second adjacent channel. This implies that the transponder design may utilize any ERP level so long as (1) the pulse shape specifications are satisfied, (2) the minimum power densities are provided at all points within the coverage volume, and (3) the absolute spectrum requirements are satisfied. Thus, for a facility with a 43 dBW ERP, the first adjacent channel will be 50 dB down with respect to the on channel signal. The
164
ROBERT J. KELLY AND DANNY R. CUSICK
-50dB for 43dBW ERP -47dB for40dBW ERP -3718 for 3OdBW ERP
RADIATED SIGNAL
11 ADJACENT CHANNEL
f0
ON CHANNEL FREllUENCY
1st ADJACENT FREOUENCY 0
Assumes 50 d B Rejection in Interrogator
2nd ADJACENT FREQUENCY
oAtsumer 7 0 dB Rejection in Interrogator
FIG.61. Ground system frequency assignment criteria and related interrogator frequency rejection assumptions.
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TABLE XI SPECTRUM CONSTRAINTS ON rn DME TRANSMITTED WAVEFORM Adjacent frequency band relative to transmit frequency
Maximum ground system radiated power"
Power with respect to peak on-channel power
0.55-1.05 MHz 1.75-2.25 MHz
200 mW 2 mW
-23 dB -38 dB
,I Radiated power = total radiated energy in band/pulse duration; pulse duration is the time between the 5% points on the leading and trailing edges.
interrogator frequency rejection characteristics, whether achieved by IF filtering or by the use of a Ferris Discriminator, are such that the undesired main lobe ground transmission is attenuated at least 50 dB. RTCA (1978) specifies that all interrogators satisfy this requirement. For undesired ERP levels (P,,) >43 dBW, the frequency assignment process is governed by the main lobe of the radiated signal. This results from the fact that even after the 50 dB rejection by the interrogator receiver, the main lobe of the undesired signal will be larger than the undesired sidelobe signal in the interrogator receiver. This dictates the need for a free-space DIU > -42 dB to achieve the required post processing 8 dB DIU protection. The required DIU is achieved by physically separating the two beacons until D/U > -42 dB. For undesired ERP levels <43 dBW, it is the sidelobes of the radiated signal which dominate in the interrogator receiver. In this case, a free-space DIU = 1 - P,, is required to ensure the desired 8 dB post-processing DIU. Note that the two ground facilities are physically separated until the 200 mW sidelobe is 8 dB below the desired signal. Reducing P,, does not mean that the facilities can be brought closer together. Defining DIU = 1 - P,, is just a means to make the D/U measurement. For example, suppose the channel assignment conditions are satisfied at a separation distance of 50 nmi with P,, = 40 dBW, then reducing the power to 26 dBW, for example, does not reduce the separation distance. For second adjacent frequency assignments, the same 8 dB post-processing DIU criterion applies. This results in an allowable free-space DIU of - 62 dB for undesired ERP values >63 dBW and a preprocessing DIU = - 19 - P,, (P,, in dBW) for P,, <63 dBW. The additional 20 dB in each of the various criteria reflects the 20 dB decrease in the adjacent channel power. Until the adoption of DMEJP, there was only one code (X or Y) associated with each uplink frequency. There was no need to consider cofrequencyloff-code assignments except for air-to-air interference. The assignments assumed that the interrogator decoder rejection was 8 dB with a
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ROBERT J. KELLY AND DANNY R. CUSICK
+ 6 p s aperture. However, in order to provide 200 DME/P channels for hard pairing with the 200 MLS angle guidance channels, additional pulse codes (W and Z ) were introduced. The W channels are formed by code multiplexing with X channel frequencies, while the Z channels are formed by code multiplexing with the Y channel frequencies. Channel plan details specific to DME/P are discussed in Section IV,A,4. The W and Z codes can be used on a restricted basis after Jan. 1, 1989, and for general use after Jan. 1, 1995 [All Weather Operational Panel (AWOP), 19821. With the addition of multiplexed codes, channel assignment criteria and interrogator pulse code rejection characteristics must be revised accordingly. Ground facility code assignment criteria for co-frequency assignments require a -42 dB free-space DIU based on 50 dB interrogator decoder rejection of received signals with pulse spacings more than 2 p s from the nominal code spacing. For the off-code second adjacent channel, the required interrogator decoder rejection is 70 dB, where the additional 20 dB arises from the fact that the second adjacent power cannot exceed 2 mW. With these design principles, the uplink assigned codes are on an equal footing with respect to the uplink frequency assignments. That is, cofrequencyloff-code channels are rejected as effectively as co-code/offfrequency assignments. The new SARPs airborne decoder tolerances for both the DME/N and DME/P decoder apertures have been tightened to & 2 ps. Because Ferris Discriminators are necessary, DME/P interrogators reject offfrequency channels at least 50 dB. Therefore, in DME/P, the off-frequency channels have the same rejection characteristics as the off-code channels. Note that the term “rejection,” as used here, refers to the ability of the interrogator signal processing to ignore off-code and off-frequency signals during acquisition and in determining the appropriate AGC levels. It does not refer to the ability of the signal processing to handle garble-induced range measurement errors and reply efficiency degradation when desired and undesired signals are coincident in the interrogator receiver. The channel assignment process explicitly considers only the assignment of beacon reply frequencies. Air-to-ground frequency protection is provided by appropriate specifications imposed on the ground equipment in terms of off-code and off-frequency signal rejection. These specifications require that ground transponders be equipped with both: (1) A Ferris Discriminator which will inhibit the generation of replies for all signals 900 kHz and more removed from the desired channel nominal frequency; and (2) A decoder which will inhibit the generation of replies for all signals less than 75 dB above the minimum receiver sensitivity level with the received pulse pair 2 p s removed from the nominal channel code spacing.
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The above results are summarized in Table XII, Column A. The protection criteria defined in Column A are applicable until Jan. 1, 1989. After Jan. 1, 1989, the protection criteria listed in Column B will become effective. The criteria change after 1989 includes not only added emphasis on off-code rejection and improved interrogator performance; it also changes the protection criteria for off-frequency assignments. After 1989 the post processing D/U 2 8 dB for off-frequency assignment will be abandoned. The cochannel criterion of 8 dB remains in effect. As shown in Column B, the first adjacent frequency co-code criterion is no longer (1 - P,) dB but has been relaxed to -42 dB. This means that a new facility can be physically moved toward an undesired facility (with P, = 25 dBW, for example) until the freespace D/U = -42 dB. Then the 200 mW adjacent spectal sidelobe will not be 8 dB below the desired signal level at the output of the IF Filter. In fact, the adjacent sidelobe of the interfering signal will be 10 dB above the desired signal. Therefore, after 1989, Ferris Discriminators or their equivalent will be required in DME/N interrogators as well as in DME/P interrogators. The frequency assignment process for all navaids within the National Airspace System-TACAN/DME, VOR, ILS, MLS, and DME/P-is clearly a complex task where the assignment criteria are interrelated due to the collocation of various facilities and the “hard pairing” of various channel plans. Given the enormous number of navaids, the use of ITS propagation models for determining the desired and undesired signal levels within the TABLE XI1 PROTECTION RATIO D/U (dB) Type of assignment
A (before 1989)”
B (after 1989)
8 0
- 42
Cofrequency Same pulse code Different pulse code
8
First adjacent frequency Same pulse code Different pulse code
-w” + 7)
Second adjacent frequency Same pulse code Different pulse code
-(Pu -(Pu
-(P”
-
-
1)*
19)
+ 27)
- 42
- 75 - 75 - 75
a The DjU ratios in Column A protect those DME/N interrogators operating on X or Y channels. Column A applies to decoder rejection of 8 dB and a 6 ps aperture. * Puis the peak effective radiated power of the undesired signal in dB W. The frequency protection requirement is dependent upon the antenna patterns of the desired and undesired facility and the ERP of the undesired facility.
168
ROBERT J. KELLY A N D DANNY R. CUSICK
defined service volumes, the need to maximize the number of facilities which can be accommodated within the NAS, and the fact that frequency assignments for new facilitiesmust satisfy the specified D / U relationships with all existing navaids, the complexity of the channel assignmentprocess dictates the need for computerized assignment procedures. Such procedures have been developed. These and other details concerning navaid assignment are found in Hensler (1984) and FAA (1980). c. TACANIDME Channel Pairing with Other Navaids. In practice, TACAN/DME stations may be collocated with VOR facilitiesto provide both range and azimuth information to civil aircraft. Similarly, TACAN/DME stations may be collocated with ILS installations to provide terminal area range, azimuth, and glide slope information to properly equipped aircraft. To facilitate the collocation of these equipments, the VOR/ILS (VHF) channels are “hard paired” with the TACAN/DME (UHF) channels. This “hard pairing” between the VHF and DME facilities, summarized in Table XIII, is used to advantage to reduce pilot work load. Typical installations provide for automatic DME channel selection consistent with the manual selection of the VHF facility by the pilot. VOR/ILS localizer facilities use frequencies in the 108-118 MHz band with channels spaced on 50 kHz increments. Forty ILS localizer (azimuth) channels are defined between 108.10 and 111.95 MHz on each odd 100 kHz pair (i.e., 108.10, 108.15, 108.30, 108.35, . .. , 111.90, 111.95). The remaining frequencies in the VHF band are assigned for VOR. ILS glide slope facilities TABLE XI11 TACAN/DME AND VHF NAVAIDCHANNEL PAIRING SCHEME ~
~~
~
TACAN/DME channel 200 NAS channels 17-59 X & Y 70-126 X & Y 18-56 X & Y even (40channels) 17-55 X & Y odd (40 channels)
x
57-59 &Y 70-126 X & Y (120 channels)
VHF Navaid 200 VHF frequencies 108.0-1 17.95 MHz 50-kHz spacing ILS localizer: 108.10-111.95 MHz odd 100 kHz and next higher 50 kHz (ie., 18X with 108.10, 18Y with 108.15,...) VOR: 108.0-111.05 MHz even 100 kHz and next higher 50 kHz (i.e., 17X with 108.00, 17Y with 108.05,.. .) VOR: 112.0-1 17.95 MHz SO-kHz spacing (i.e., 57X with 112.0, 57Y with 112.05,...)
Note: ILS localizer frequencies are non-sequentially paired with ILS glide slope frequencies (329.15-335.0 MHz, 150 kHz spacing).
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use the 328.6-335.4 MHz band with assignments beginning on 329.15 to 335.0 MHz on 0.15 MHz increments. Localizer and glide slope frequencies are also hard paired. Even-numbered X and Y TACAN/DME channels 18-56 are hard paired with the 40 ILS localizer frequencies. The pairing is in sequence with increasing channel number and frequency: 18X with 108.10 MHz, 18Y with 108.15 MHz, etc. Odd-numbered X and Y TACAN/DME channels 17-57 are hard paired with the VOR frequencies between 108.0 and 112.05 MHz. The remaining TACAN/DME channels in the NAS (58,59,70-126) are hard paired with the remaining VOR frequencies 112.1-117.95 MHz in 50 kHz increments. Complete listings of the VHF/UHF/MLS frequency channeling and pairing can be found in NTIA (1984).
4 . System Capacity The ground beacon has a finite capacity to process interrogations, a consequence of the dead time that results from the time required to identify a valid interrogation and transmit the reply pulse pair following the appropriate fixed delay. Valid interrogations arriving during this dead time are ignored and result in missed range measurements at the interrogator. This important system performance parameter, introduced in Subsection B and referred to as reply efficiency, is simply the ratio of the number of interrogations from a single interrogator to the number of replies generated by the ground beacon. The reply efficiencydecreases as the number of interrogations increases and thereby affects the number of aircraft which can be accomodated by a single ground beacon. As noted earlier, ground facilities must serve a minimum of 100 aircraft while maintaining a reply efficiency of at least 70%. Figure 62 illustrates the relationship of beacon dead time to the received interrogation pulse pair and the transmitted reply. In the example shown, the dead time is 60 ps, for both the X and Y pulse codes, and is easily implemented by inhibiting the decoder output. Thus, the receipt of a valid second pulse during the dead time period will not result in a reply. Note that the actual dead time may be longer than 60 p s if receiver blanking is utilized during ground facility transmissions. At sites where long delay multipath is present, such as in mountainous terrain, additional dead time is introduced by the use of a retriggerable blanking gate. This gate, which may be up to 250 p s in duration, is intended to inhibit replies to interrogation echoes on the basis of an amplitude test. The direct path interrogation triggers the blanking gate and sets a threshold level to which subsequent interrogations are compared. Interrogations arriving during the blanking time having amplitude less than the direct path signal are ignored while those having higher amplitudes result in replies. The assumption
170 1st PULSE
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ROBERT J. KELLY AND DANNY R. CUSICK X CHANNEL
2nd PULSE
t
t
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1st PULSE
1r t PULSE
1
MARGIN 10us
REPLY 12us
50pr REPLY DELAY
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1 REPLY 30 us
56pr REPLY DELAY I
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t
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t
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REPLIES TO INTERROGATIONS WITH DECODES DURING THE DEAD TIME PERIOD ARE INHIBITED B 1011s MARGIN ALLOWS ADEQUATE TIME FOR THE COMPLETE
TRANSMISSION OF THE SECOND REPLY PULSE
Fic;. 62. Dead time definitions for [)ME N X and Ycodes.
is simply that interrogations that arrive shortly after a previous interrogation and are of lesser amplitude are probably due to echoes. Replies to the echoes would appear as an interrogation load and further degrade the facility capacity. The reply efficiency (RE) is a function of the interrogation load (A) and the effective dead time (7)and is given by
That is, the reply efficiency is simply the “free time “-to-” total time” ratio for a given interrogation load. It is a special case of the more general equation [Eq. (36)] applicable to DME/P ground equipment shown in Section IV,B,S.
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As an example, consider 100 aircraft, each with 2 interrogators having an average interrogation rate of 30 Hz. When a 60 p s dead time is assumed a 74% reply efficiency results. This exceeds the minimum acceptable value of 70% and implies that 100 aircraft can be readily served by a single ground station. At sites where a 250 p s retriggerable blanking gate is utilized, a 40% reply efficiency results under the same traffic loading conditions. The traffic load must be reduced in order to keep the reply efficiency above 70%. Hence, such sites will have less capacity than sites that do not utilize the retriggerable blanking gate.
IV. SYSTEMS CONSIDERATIONS FOR THE NEWDME/P INTERNATIONAL STANDARD The Precision Distance Measuring Equipment (DME/P) standard is based upon the same principle as DME/N, which provides distance information in the aircraft by measuring total round-trip time between interrogations from an airborne transmitter and replies from a ground transponder. The difference is that, unlike DME/N, DME/P provides accuracy and data rates that will support approach and landing operations in conjunction with the MLS angle data. The DME/P functions have been successfully integrated with those of DME/N and represent the newest addition to the ICAO DME specification. Kelly (1984) provides a concise summary of the DME/P concept adapted by ICAO. The material in this part is an extension of that paper. Subsection A serves to introduce the basic DME/P concepts by reviewing the history of the DME/P standards development, reviewing the operational requirements for precision range information, and describing the operation of the DME/P system and its associated channel plan. Subsection B provides detailed discussions of the DME/P technical characteristics and an analysis of the major sources of performance degradation: noise, multipath, and garble. Subsection C combines this information to provide system level error and power budgets. Implementation considerations for both the ground and airborne equipment are discussed in Subsection D, while Subsection E summarizes the results of DME/P field tests. A . An Overview of the D M E / P System
The purpose of this section is to introduce the basic DME/P system concepts, beginning in Subsection A,1 with the history of the development of the DME/P concept and the international standard.
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The first priority in approaching the task of defining a precision DME was to establish the DME/P operational requirements (OR). The OR is simply the definition of the user requirements that the DME/P must satisfy so that measured range data in conjunction with the MLS angle data can lead to a successful aircraft approach and landing. In addition, the DME/P OR had to ensure interoperability with existing DME/N system. Subsection A,2 briefly summarizes the OR, which contain an interpretation of the operational need in terms of the range accuracy values which must be achieved in various regions of the MLS coverage volume. DME/P is a two-pulse/two-mode system. As in DME/N, pulse pairs are utilized for interrogation and reply transmissions where the code or pulse spacing identifies the operating channel. Two modes are provided: the initial approach, or IA mode, which functions identically to the DME/N and the final approach, or FA mode, which provides the accuracy required for the final approach and landing operations. System operation and the rationale for its selection is detailed in Subsection A,3. The DME/P channel plan, described in Subsection A,4, is simply a subset of the DME/N channel plan. Although the same interrogation/reply frequency pairing is utilized, additional pulse codes have been defined to accommodate the “hard pairing” with the MLS angle channels. The plan is configured to handle estimated aircraft traffic densities well beyond the year 2000 by minimizing the potential for DME self-interference or garble. 1. Historical Background
The ICAO Seventh Air Navigation Conference, April, 1972, established the All Weather Operations Panel (AWOP) as the designated body to develop SARPs for a new nonvisual precision approach and landing guidance system (MLS) (Redlien and Kelly, 1981). While several states included proposals for precision DME as part of their submissions on MLS, AWOP determined that its initial obligation was the preparation of SARPs for the angle and data functions of the MLS. This decision recognized that the MLS could be implemented, if necessary, with conventional distance measuring equipment as specified in ICAO (1974),to satisfy early operational uses. The realization of the full benefits of MLS, which required standardization of a precision DME (DME/P) capability, was addressed accordingly as a second priority. The eighth meeting of AWOP (AWOP-8) in March, 1980 (AWOP, 1980), resulted in adoption of SARPs (ICAO, 1981a) by the panel for the angle guidance portion of the MLS, which were accepted by the Communications Divisional Meeting in Montreal, April, 1981 (ICAO, 1981b). In conjunction with the final activity in developing the angle SARPs, the panel began its consideration, in a very limited way, of DME/P SARPS in
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September, 1979, at its Working Group M (WG-M) meeting in Seattle.I3 At this meeting, WG-M agred that the DME/P specificationscould be integrated into the existing text of Annex 10. At the March, 1980, AWOP-8 meeting, it was decided to review and update the DME/N standards where necessary. It was during this meeting that the first “updated” DME/P concept was offered in the form of a technical paper (Kelly and LaBerge, 1980). A complete analysis and design was provided for range guidance for CTOL, STOL, and VTOL operations. The paper presented, for the first time, a proposed accuracy standard (including error budgets, power budgets, and the concept of system efficiency) and the signal processing functions necessary to achieve the proposed standard. The concept it described was a synthesis of ideas developed and field tested by investigators under contract with the Federal Aviation Administration in the early 1970s.Chronological details are given in Kelly and LaBerge (1980). Since the AWOP-8 meeting, DME/P became the principal activity of AWOP. The chronology of events was as follows: (1) Rio de Janeiro, Brazil, September, 1980, Working Group M-4 (WGM4) Meeting-Operational requirements for DME/P were developed and the Traffic Loading/Channelization, Multipath, and System Concepts Subgroups could, after a thorough examination of suggested techniques, arrive at a recommendation to the WG-M concerning the best DME/P concept. (2) Amsterdam, The Netherlands, January, 1981, System Concepts Subgroup Meeting-Power budget and error budgets were normalized to permit the assessment of the various DME/P technique suggestions. (3) London, England, February, 1981, Traffic Loading/Channelization Subgroup Meeting-A standard traffic loading model was defined and its use to predict uplink and downlink loading effects was agreed upon. (4) Montreal, Canada, April, 1981, Informal WG-M meeting-WG-M was briefed on the progress of the three subgroups. ( 5 ) Amsterdam, The Netherlands, August, 1981, Traffic Loading/ Channelization Subgroup Meeting-The channel plan was developed. Ralph Vallone of the U.S. was the originator of the new DME Channel Plan. (6) Munich, Federal Republic of Germany, September, 1981, Systems Concepts Subgroup Meeting-DME/P interoperability with DME/N was addressed. The original concept described in Kelly and LaBerge (1980)carried with it an error component that would be incurred, if a DME/N interrogator were to obtain distance service with a DME/P transponder. The error arose in l 3 Membership in WG-M included Australia, Brazil, Canada, France, International Air Transport Association (IATA), International Federation of Airline Pilots’ Associations (IFALPA), Italy, Japan, The Netherlands, U.S.S.R., U.K., US., and W. Germany.
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ROBERT J. KELLY AND DANNY R. CUSICK
this concept because it was thought that the ground transponder could not be made to differentiate reliably between DME/N and DME/P interrogations, and means to attempt such a distinction were intentionally omitted. Since the pulse time of arrival decision threshold for the two applications must be different, a bias error will be generated in mixed operations because, without prior information, the ground transponder zero-range delay can only be calibrated with respect to one or the other threshold. Three new system concepts were informally submitted to the Concepts Subgroup for evaluation, each with a unique method for resolving the interoperability error. All concepts contained the notion of a low-threshold precision mode and a - 6 dB threshold nonprecision mode. They were: (a) A two-pulse system concept (France/United Kingdom) much like the system proposed in Kelly and LaBerge (1980), except a pulse rise-time recognition circuit in the transponder was proposed to differentiate between DME/N and DME/P interrogators. The integrity of the pulse recognition circuit became an issue. (b) A three-pulse system concept (Federal Republic of Germany/United States) using a conventional pulse pair followed 2 0 0 ~ slater by a single precision mode pulse; DME/N and nonprecision DME/P range measurements were to be made on the conventional DME pulse pair; precision range measurements would be performed on the third pulse. This concept was viewed as a large departure from the present DME system. (c) A two-pulse/two-mode concept (France), which was the technique unanimously chosen by the subgroup (Carel, 1981);Josef Hetyei of France was the originator of this concept. (d) Italy and Japan proposed alternative pulse-shaping implementations which are applicable to the two-pulse/two-mode technique. (7) Paris, France, September, 1981, AWOP WG-MS-The Working Group accepted the recommendation of the System Concepts Subgroup and initiated the development of DME/P SARPs predicated on the twopulse/two-mode technique. (8) Moscow, USSR, March, 1982, AWOP WG-M6-A first draft of the DME/P Transponder SARPs was developed. (9) Rome, Italy, September, 1982, AWOP WG-M7-The final draft of the DME/P SARPs was developed by the group, including revisions to the DME/N SARPs and the DME Guidance Material. (10) Montreal, Canada, November, 1982, AWOP-9-DME/P Draft SARPs approved by full AWOP-9 panel meeting.
With an approved SARPs in the wings, work began in earnest on the formulation of airborne and ground equipment specifications. In the United
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States, a ground facility equipment specification (FAA, 1983C)for a complete MLS system including DME/P was released in conjunction with a request for proposal in April, 1983. This led to a contract award by the FAA in January, 1984, to the Hazeltine Corporation and represents the first production procurement of MLS equipment (208 systems) by a government body. E-Systems of Salt Lake City, Utah, would build the DME/P ground facility. In the avionics arena, the Radio Technical Commission for Aeronautics convened Special Committee 149, whose primary charter was to update the existing DME Minimum Operational Performance Standards (RTCA, 1978) to reflect important changes in the SARPs and to add the DME/P to the avionics standard. Only after final completion of the DME/P MOPScould the avionics industry truly begin the development, design, and marketing of DME/P airborne equipment. 2. DME/P Operational Requirements The DME/P is an integral part of the microwave landing system for aircraft approach, landing, and missed approach operations. Thus, the DME/P system must be capable of providing high-accuracy range information in a potentially severe multipath environment such as that encountered during landing operations. The accuracy required (100 ft-30 m) for such operations is at least an order of magnitude better than that provided by the present conventional DME system (DME/N). The operational requirements defined by ICAO state that the DME/P must provide high-quality guidance to a range of 22 nmi from the MLS ground facility and be fully compatible and interoperable with conventional DME/N. This implies that (1) the ICAO-defined (ICAO, 1974) radiated pulse spectral characteristics, applicable to DME/N, must be satisfied; (2) a conventional interrogator may be used with a DME/P transponder to obtain at least the accuracy expected with a conventional transponder; and (3) a DME/P interrogator may be used with a conventional transponder to obtain at least the accuracy expected with a conventional transponderinterrogator combination. In addition, the OR states that, when operationally required, the DME/P distance information shall permit, in combination with the appropriate MLS angle and data information, the functions in Table XIV to be performed. The two-pulse/two-mode technique selected by ICAO as the basis for DME/P implementation satisfies these requirements. This technique is discussed in the next section.
ACCURACY NECFSARY
TABLE XIV OPERATIONAL REQUIREMENT (95% PROBABILITY)
TO SATISFY DME/P
Distance from ref. datum (nmi)
Function
PFE
CMN
Segmented approach" Extended runway centerline At 40" azimuth
20
f250m +375 m
68 m 68 m
Segmented approachb Extended runway centerline At 40" azimuth
5
k85 m k127 M
34 m 34 m
5
f800m f400 m
NA NA
0.3
+30 m +l5m
NA NA
0
f30m +12m
18 m 12 m
20 to 0
f250m
NA
0
f30 m f12m
12 m 12 m
+12m
30 m
Marker replacement Outer marker Middle marker
0.57
30 m decision height' determination (100 ft) 3" Glide path (CTOL) 6" Glide path (STOL) Flare initiation over uneven terraind 3" Glide path (CTOL) 6" Glide path (STOL) Autopilot gain scheduling' Flare maneuver-CTOLI elev.-STOL
with MLS flare
CTOL high-speede rollout/turnoffs Departure climb and missed approach VTOL approachh Coordinate translations'
Runway region 0105 0.5 to 0 -
+100m f12m f12to f30m
68 m 12 m 12m
a DME/P accuracy approximately corresponds to the azimuth function PFE at a distance of 20 nmi from the MLS reference datum both along the extended runway centerline and at an azimuth angle of 40".Also the DME/N error at the limits of MLS coverage is consistent with the DME/N 0.2-nmi system accuracy. The CMN is the linear equivalent of the 0.1" azimuth angle CMN. PFE corresponds to azimuth angular error; CMN is approximately the linear equivalent of the 0.1" azimuth angle CMN. Equivalent to radar altimeter performance for 3" glide path. The 30-m PFE corresponds to a 1.5-m vertical error for a 3" elevation angle. Flare initiation begins in the vicinity of the MLS approach reference datum; MLS elevation and DME provide vertical guidance for automatic landing when the terrain in front of the runway threshold is uneven. Sensitivity modification or autopilot gain scheduling requirements are not strongly dependent on accuracy. It is intended that this specification apply when vertical guidance and sink rate for automatic landing are derived from the MLS flare elevation and the DME/P. 9 The rollout accuracy requirement reflects system growth potential. The requirement is to optimize rollout deceleration and turnoff so as to decrease runway utilization time. It is intended to assure the pilot that the aircraft is over the landing pad before descending. 'Figures in the table are typical of a VTOL application, where MLS coordinates are translated to center of landing pad.
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Based on the accuracy requirements in Table XIV, two accuracy standards were defined in the SARPs to accomodate a variety of applications in an economical manner. Standard 1 is intended to satisfy the accuracy requirements of CTOL applications where “flare height” is obtained by radar altimeter. Standard 2 is intended to satisfy the accuracy requirements of STOL and VTOL applications as well as operations utilizing the MLS flare function. For both standards, accuracy degradations with distance and azimuth angle are specified. These accuracies, defined on a 95% probability basis, are summarized in Table XV. No degradation is permitted with elevation angle but a 1.5:l degradation is permitted in azimuth. Between the lateral limits at f40”,the accuracy is permitted to degrade linearly with increasing range. 3. The Two-PulselTwo-Mode System The two-pulse/two-mode technique is the method selected by ICAO as the basis of DME/P system implementation. It provides high accuracy in a final approach mode (FA) and full interoperability with DME/N in an initial approach mode (IA). In the final phases of the approach and landing operation, the major source of accuracy degradation is due to multipath rather than interrogator and transponder instrumentation errors. Multipath perturbations, which always occur later in time than the desired signal, are minimized by thresholding the received pulse at a point which has not been significantly corrupted by multipath. Therefore, the necessary accuracy for DME/P is achieved by thresholding low (about 17 dB below the pulse peak) on the leading edge of a fast rise-time pulse. This is in contrast to the 50% level thresholding and slow rise-time pulses used in DME/N. By achieving enhanced accuracy in this manner, questions are raised concerning interoperability with DME/N. First, the use of a low threshold level impacts transmitter power in the sense that more power is required to obtain an adequate threshold-to-noise level throughout the coverage v01ume.l~At the same time, a fast rise-time pulse is desired, which increases the spectral width of the radiated pulse. These two requirements are in conflict with the need to satisfy the adjacent channel spectrum requirements noted earlier. Hence, both the pulse rise time and the maximum range at which the low-threshold technique can be used must be restricted. Second, conventional interrogators are calibrated assuming that transponder reply delay timing is l4 A low threshold-to-noiseratio can cause pulse arrival-timemeasurement anomalies which induce noisy range error measurements. At threshold-to-noise levels of 12 dB or more, these effects are insignificant when, in addition, a pulse-width discriminatoris employed. When these conditions are satisfied,the magnitudeof the receiver noise errors is simply inversely proportional to the signal-to-noisevoltage ratio.
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ROBERT J. KELLY AND DANNY R. CUSICK TABLE XV ALLOWABLE DME/P ERRORS (95% PROBABILITY)' Location
PFE
CMN
20 to 5 nmi from MLS approach reference datum
+250 m reducing linearly to +_85m
68 m reducing linearly to 34 m
5 nmi to MLS approach reference datum
f85m reducing linearly to +30 m k85 m reducing linearly to i12m k100rn
18 m
34 m
+30m
18 m
+12m
12 m
At MLS approach reference datum and through runway coverage Throughout back azimuth coverage volume
Std.
68 m
68 m " At distances from 5 nmi to the MLS approach reference datum and throughout the back azimuth coverage the IA mode may be used when the F A mode is not operating.
based on the 50% threshold level. However, transponder reply delay initiation based on low-level thresholding of a conventional DME pulse will be different from that based on the 50% points. Thus, if a conventional interrogator were to interrogate a DME/P transponder which used only the low-level threshold technique, an interoperability error (actually a bias) could occur. The two-pulse/two-mode technique solves the interoperability problem by providing two modes of operation, a wideband final approach mode and a narrowband initial approach mode. The F A mode utilizes the lowthresholding technique in both the interrogator and transponder for ranges to 7 nmi from the transponder. By limiting the FA mode coverage range to 7 nmi, the adjacent channel radiated power constraints on the transponder can be satisfied while an adequate threshold-to-noise ratio (for low-threshold pulse detection) is maintained throughout this region. Beyond 7 nmi, the IA mode is used, and its operation is identical to that of DME/N. In addition, a transition between the two modes is provided in the 7-8 nmi region. Therefore, in the 1A mode, no DME/N interoperability bias error occurs because both the
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interrogator and the transponder threshold on the 50% pulse levels. Further, use of the 50% threshold allows an adequate threshold-to-noise ratio to be maintained to the coverage limit of 22 nmi. The interrogator selects the mode, F A or IA, according to the measured range from the ground facility. The interrogation mode is characterized by the pulse code (i.e., pulse-pair spacing). The transponder reply delay timing is then based on the proper threshold point for the type of interrogation received. Within a channel assignment, the transponder reply code remains the same regardless of the type of interrogation. Note that both the DME/P and the revised DME/N SARPs require range measurements on the first pulse, thereby avoiding the effects of first pulse multipath echoes which can disturb range measurements predicated on second pulse timing. The operation of the DME/P system is summarized with a specific example (an X-channel ground facility) in Fig. 63. An aircraft located 8 to 22 nmi from the transponder interrogates with the IA mode interrogation code of 12 ps at a rate of 40 Hz for search and 16 Hz for track. In the 7-8 nmi region, an “approaching” interrogator transmits in both IA and F A modes to effect a smooth transition to the FA mode by 7 nmi. In the event that no FA mode replies are received, the interrogator will remain in the IA mode with an appropriate warning to the pilot and aircraft systems utilizing the range data.
FIG.63. DMEjP two-pulse/two-mode system operation using X-channel codes.
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ROBERT J. KELLY AND DANNY R. CUSICK
Within 7 nmi of a DME/P transponder, the aircraft obtains precision range information with FA mode interrogations at 40 Hz. These interrogations are identified by the X-channel code of 18 ps. Receipt of this code at the transponder causes the incoming waveform to be thresholded low. The FA mode replies use the same code as the IA mode replies: 12 ps for the X-channel example. In the FA mode, both the interrogator and the transponder process received signals in a 3.5-4.5 MHz bandwidth. The wide bandwidth is required, for reasons to be noted in Subsection B, to ensure that the pulse leading edge is essentially linear in the threshold region. As was discussed in Section II,E, aircraft control requires on the order of five data samples each second. Clearly, with an FA mode interrogation rate of 40 Hz, more than enough data are available. Thus, raw range measurements can be filtered to reduce the effects of noise, multipath, instrumentationinduced measurement errors, and to compensate for mising data. DME/P IA and F A mode functions can be included in an interrogator package also having the conventional DME/N capability. On the other hand, some users may elect to implement only the IA mode such that the combination would yield, in effect, an enhanced DME/N having increased accuracy and all of the additional DME/P channels. 4. DMEfP Channel Plan
The channelization scheme for DMEfP makes use of the same frequency/code concept used in the TACAN/DME channel plan detailed in Section III,C,3 and summarized in Fig. 60. The major difference is the use of additional pulse codes necessitated by (1) the need to uniquely identify IA and F A mode interrogations and (2) the need to accommodate “hard pairing” of DME/P with the 200 MLS angle channels. The DME/P-related additions to the existing channel plan are such that they provide an optimum pairing between MLS C-band angle and L-band range channels. They do not interfere with present DME services while minimizing the potential for DME garble interference. A major concern in the development of the channel plan was the traffic loading effects of candidate channel plans on pulse garble interference, which is directly proportional to the number of pulse-code-multiplexed channels that are defined on each frequency. The selected channel plan minimizes these interference effects by spreading the number of channels over a large number of frequencies without incurring any meaningful degradation of existing or planned DME/N enroute services. As a result, the channel plan can accomodate channel assignments in extremely dense environments such as those postulated for the Northeast and Southwest United States. Furthermore, the selected channel assignment plan was assessed against a worst-case traffic loading environment.This assessment
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assumed that the system efficiency would never be less than SO%, given an F A mode transponder reply efficiency of 80%. Another consideration was to pair the uplink codes with the uplink frequencies so that all codes are spaced at least 12 ps apart. Existing DME/N interrogators with f 6 ps decoder apertures can therefore successfully operate without interference from the new channels. Associated with each DME/P channel are two interrogation codes and a single beacon reply code. The two interrogation codes allow the transponder to discriminate between IA and FA mode interrogations as described in the previous section. Four codes are defined: W, X, Y, and Z. Interrogation/reply frequencies for DME/P X and Y channels utilize the same pairings as the TACAN/DME channel plan. Additionally, the IA mode interrogation and reply codes are identical to those used in the TACAN/DME channel plan, thereby ensuring interoperability of TACAN/DME equipment with DME/P equipment. Interrogation/reply frequencies associated with DME/P Z channels utilize the same frequency pairs as the Y channels, while the W channel frequency pairs correspond to the X channel structure. The channel plan structure is summarized in Table XVI. All of the DME/P channels are “hard paired” with the 200 MLS angle channels,as summarized in Table XVII. The angle channels are in the C-band portion of the radio spectrum between 5031 and 5091 MHz in 300 kHz increments and are numerically designated 500-699. Note that the even channels 18-56 X and Y correspond to the TACAN/DME channels that are TABLE XVI DME/P CHANNELPLAN Pulse pair spacing (ps)
Number Channel designation
Pulse code
channels
18-56 even
WE
20
18-56 even
X b
20
17-56/80-119
Yb
80
17-56/80-119
Z‘
80
of
Operating mode
Inter
Reply
IA FA IA FA IA FA IA FA
24 30 12 18 36 42 21 27
24 24 12 12 30
‘W channels are pulse-code multiplexed on the X frequencies.
30 15 15
Reply delay (ps) 50 56 50 56 56 62 56 62
DME/N X and Y channels are fully compatible with DME/P X and Y IA mode channels. Z channels are pulse-code multiplexed on the Y frequencies.
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ROBERT J. KELLY AND DANNY R. CUSICK TABLE XVII DMEjP MLS CHANNEL PAIRING DME/P Channels 200 channels
MLS Channels 200 channels
18-56 X and W even 17-56 Y and Z 80-119 Y and 2
500-699 5031.0-5090.7 MHz, 300 kHz spacing
18-56 X and W even (40 channels)
Channels 500-539 5031.0-5042.7 MHz (i.e., 18X with 500, 18W with 501,. ..) Channels 540-619 5043.6-5066.7 MHz (i.e., 17Y with 540, 1 7 2 with 541 ,...)
17-56 Y and Z (80 channels)
80-119 Y and Z (80 channels)
Channels 620-699 5067.0-5090.7 MHz (k, 80Y with 620, 802 with 621 ,...)
“hard paired” with the 40 ILS channels. This correspondence is intended to accommodate a single DME/P at joint ILS-MLS sites while maintaining the “hard pairing” of DME with ILS and DME/P with MLS. B. Pulse Shape, Time-of-Arrival Determination, and the Major Sources of Peformance Degradation This section details the technical characteristics of the DME/P (pulse shape and time-of-arrival estimation) that are the keys to obtaining the desired ranging accuracy in the presence of noise, multipath, and garble. A new pulse waveform, discussed in Subsection B,1, was defined to accommodate the accuracy specification while at the same time satisfy the existing DME/N adjacent channel interference specification. Subsection B,2 describes the method of pulse time-of-arrival determination. It takes advantage of the new pulse shape to achieve accuracy in the presence of severe multipath. Three sources of error in DME/P which must be accommodated in order to achieve the desired performance are receiver noise, multipath, and garble. Effects of receiver noise are primarily determined by the received signal level (i.e., signal-to-noise ratio) and the receiver signal processing. These effects are detailed in Subsection B,3. Multipath, the single most important threat to DME/P accuracy performance, is treated Subsection B,4. It is shown that the multipath effects are controlled by selecting the critical parameters associated
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with pulse rise time and time-of-arrival technique so that they match the runway geometry and obstacle clearance zones. Garble, the subject of Subsection B,5, degrades two major performance parameters: system efficiency and accuracy. Unlike DME/N, garble becomes a critical issue in DME/P due to the use of wide bandwidth receivers, and the multitude of DME equipments operating simultaneously in the environment. System efficiency is a new DME/P concept which influences many aspects of the system design, specifically the interrogator data filter. Accuracy degradation occurs due to coincidence with extraneous pulses. Methods of determining the level of accuracy degradation which occurs for a given garble environment are described. 1. DME/P Pulse Shape
Three factors govern the selection of the DME/P pulse: (1) The waveform must be compatible with DME/N ground and airborne equipment. (2) The pulse must have a fast rise time leading edge in order to provide timing measurements of the required accuracy in the presence of noise, multipath, and garble. (3) The DME/P waveform must satisfy the same spectrum constraints imposed on the DME/N waveform as noted in Section 111 of this article. ICAO (1985)provides a template for the DME/P waveform where the key parameters are the 10-90% rise time, 3 amplitude width, partial rise time, and partial rise time linearity. The same specifications apply to both the interrogator and transponder equipments. Although it is noted below that the IA and F A mode requirements differ, only a single waveform satisfying the FA mode specifications need be generated, thereby simplifying modulator design. The IA mode pulse is specified to have a 10-90% rise time 1600 ns or less and a nominal width as measured at the $-amplitude points of the leading and trailingedges of 3.5 ps. [RTCA (1985b) specifies that the rise time be I1200 ns for the reasons noted in Subsection D,2. The 1200ns maximum rise time specification will be used in this article.] These requirements also satisfy the DME/N waveform specifications, thereby maintaining interoperability between the conventional and precision DME systems. Three additional requirements, detailed in Fig. 64, are imposed on the FA mode waveform. First, the 5-30% leading edge rise time, referred to as the partial rise time, is specified to be 200-300 ns. This segment of the pulse leading edge is where time-of-arrival measurements are made when operating in the FA mode. The specification is key to achieving high accuracy in the presence of multipath as discussed in Subsection B,4.
184
ROBERT J. KELLY A N D DANNY R. CUSICK
28--
24--
L I M I T I +20%OF THE MEASURED PARTIAL RISE TIME SLOPE)
22--
-ae 2 0 - Y
RISE TIME SLOPE1
------
BOUNDARY FOR PULSE TURN-ON TRANSIENTS
8
--
RISE TIME SLOPE
MEASURED PULSE
-me-
o r - d -------r 1%----
:
VIRTUAL ORIGIN L I M I T 1OOOnr
I
1
0
100
I
I
200 300 PARTIAL RISE TIME Ins)
I
400
I 500
FIG.64. The DME/P pulse: FA mode requirements and the cos/coszwaveform.
DISTANCE MEASURING EQUIPMENT IN AVIATION
185
Second, a linearity requirement is imposed on the partial rise time segment of the pulse leading edge and requires that the instantaneous slope at all points in this segment may not deviate by more than 20% from the slope of a straight line passing through the 5% and 30% points.15 This ensures that the virtual origin of the pulse can be accurately determined (assuming the use of the delay-attenuate-compare (DAC) time-of-arrival estimator, as described in Subsection B,2) and that calibration of the DME/P airborne and ground equipment is independent of DAC parameters and pulse rise times. Finally, the instantaneous magnitude of any pulse turn-on transient that occurs prior to the virtual origin of the transmitted pulse must not exceed 1% of the peak pulse amplitude. In addition, any such transient must not occur more than 1OOOns prior to the virtual origin. The reason for such a specification is as follows. First, a transient signal prior to the pulse can prevent a pulse from being detected by the DAC circuit. Limiting the amplitude of the turn-on transient eliminates this possibility. Second, such transients can be declared pulse arrivals by the DAC. Limiting the duration of the transient to a maximum of lo00 ns allows the resulting false pulse to be discarded by a pulse-width discriminator. Wide-band signal processing is necessary to preserve the shape of the fast rise time DME/P pulse and therefore achieve the up- and downlink accuracy. Thus, unlike DME/N, Ferris Discriminators or their equivalent are required in both the interrogator and the ground transponder to provide the means of identifying and rejecting off-frequency DME pulses. As in DME/N, the downlink spectrum requirements are relative and are easily satisfied.The most severe restriction is on the uplink, where the radiated power is limited to absolute levels in the adjacent channels. As noted in Section 111,B,2,any ERP can be utilized by a ground facility provided that the radiated power is limited to 200 mW in the first adjacent frequency channel and 2 mW in the second. Thus, a fast rise time pulse (as required in the DME/P application) having greater spectral width than the conventional pulse requires that the ERP must be reduced such that the stated limitations on adjacent frequency radiated power are satisfied. On the other hand, the radiated signal power must result in a threshold-to-noise ratio throughout the coverage volume such that the effects of receiver noise do not adversely impact ranging accuracy. These conflicting goals, fast rise time versus coverage, are a major reason for the inclusion of two modes of operation (IA and FA) into the DME/P signal format. A pulse shape which satisfies the requirements described above is the cos/coszpulse. Figure 64 depicts this pulse for partial rise times at the extremes l 5 The intersection of the line connecting the 5% and 30% points of the pulse leading edge with the time axis is known as the virtual origin.
186
ROBERT J. KELLY AND DANNY R. CUSICK
of the allowable limits. The pulse consists of a leading edge which is a portion of a cycle from a sinusoidal function, a trailing edge which is a portion of a cycle from a sinusoidal function which has been squared, and stretched at the peak to provide the desired $-amplitude width. The characteristics of this pulse, for the range of allowable partial rise times, are summarized in Table XVIII. In addition, the maximum uplink ERP which can be used based on the adjacent channel specifications is defined. Note that the ERP limitations for the first and second adjacent channels are nearly equivalent. In this sense, the cos/cos2 pulse is nearly optimum. 2. Pulse Time-of-Arrival Estimation Technique
To measure range in the interrogator or to begin the reply delay timing in the transponder, the DME pulse must be detected and its time of arrival precisely determined. Envelope detection is used in DME; the phase information is discarded. Methods for estimating the pulse time of arrival (TOA) must satisfy the accuracy specifications appropriate to the aircraft class subject to power
TABLE XVIII SUMMARY OF cos/cos2 ~ L S ECHARACTERISTICS 5-30% partial rise time (ns) Maximum deviation from linearity 10-90% rise time (ns) 90-10% fall time (ns) f amplitude width (ns) 5%-5% pulse duration-t(ns) Equivalent energy rectangular pulse duration (terr) Relative first adjacent channel energy (W,lW, dB) Relative second adjacent channel energy W21WdB) Max ERP" allowed by first adjacent channel limit (dB m) Max ERP allowed by second adjacent channel limit (dB m)
200 5% 800 3000 3500 5800
250 5% 1000 2919 3500 5800
300 5% 1200 2677 3500 5800
2650
2615
2615
-28.4
-31.9
- 34.5
-48.6
-50.1
-51.9
54.9
58.4
60.9
55.0
56.6
58.4
= a Computation of maximum ERP: (W/w)(f/ferr)p&, where energy in ith adjacent channel, i = 1, 2; W = total energy in pulse; p:,,, = maximum average power permitted in ith adjacent channel; t = 5%-5% pulse duration; and ref( = duration of equivalent energy rectangular pulse.
DISTANCE MEASURING EQUIPMENT IN AVIATION
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budget considerations. Trade-offs are then necessary for satisfactory performance taking the following into account: (1) noise, (2) lateral specular multipath, (3) signal amplitude and rise-time variations, (4) threshold-to-noise anomalies, and ( 5 ) circuit parameter variations. In general, emphasis on noise performance is counterproductive to good multipath resistance. Matched filter designs are not necessary because, as will be shown in Subsection B,3, the system noise performance when widebandwidth signal processing is utilized is well within the system error budgets given in Subsection C,2. Therefore, wide-bandwidth signal processing is preferred so as to obtain the necessary multipath discrimination. This means, given the constraints described earlier, the TOA measurement must be made on the pulse leading edge. Since the operating environment produces high levels of ground multipath interference, the leading edge threshold technique must not be sensitive to signal-strength-dependent errors. A technique which satisfies these conditions for the FA mode is the delayattenuate-compare circuit. This circuit compares a delayed version of the pulse to an attenuated version of the same pulse. Pulse arrival is declared when the delayed signal exceeds the attenuated signal. A delay of 100 ns and an attenuation of 5 to 6 dB will result in a nominal threshold level 15 to 18 dB below the pulse peak when a 1000 ns rise time cos/cos2 pulse is utilized. These values represent a compromise between good multipath performance and the avoidance of noise anomalies. In any case, the delay and compare parameters must always be chosen such that the threshold crossing occurs during the partial rise time; that portion of the received pulse between the 5 and 30% amplitude levels having guaranteed rise time and linearity. The importance of these factors will become evident below. The key characteristic of the DAC is that for a linear leading edge pulse, the threshold time relative to the start of the pulse is independent of the pulse rise time and amplitude. It is for this reason that the DME/P precision pulse shape specification requires that the pulse leading edge be linear in the threshold region. Thus rise-time variations within the SARPs specification and amplitude variations that result from signal power and propagation effects do not induce TOA estimation errors when a DAC processor is used. General design relations for DAC are given in Fig. 65, as well as an example design. Just as in DME/N, pulse detection in the IA mode is based on the 50% ( - 6 dB) pulse amplitude level using a peak-amplitude-find (PAF) circuit (sometimes called a +-amplitude find circuit). In this technique the pulse is delayed (e.g., 2000 ns) until the peak is found and a threshold, 6 dB down from the peak, is generated (Fig. 66). Since multipath is generally not a critical factor within the IA mode service volume, the higher threshold level provides a
188
ROBERT J. KELLY AND DANNY R. CUSICK
I
I
DECODE TIME ITD) FOR ARBITRARY PULSE
TD:p(T~-7)'Ap(T~)
= 0 : p (1- 7 <Ap
DAC OUTPUT (LOGIC SIGNAL)
DEFINITION OF LINEAR LEADING EDGE PULSE
1
P It) =
1
DECODE TIME FOR LINEAR LEADING EDGE PULSE
0
50
150
200
250 TIME (nr)
:iR
'D=&
AT
RELATIVE THRESHOLD OF DELAYED LINEAR LEADING EDGE PULSE
100
{
T (1-A) TR
RELATIVE THRESHOLD OF ATTENUATED LINEAR LEADING EDGE PULSE
I
It1
= 1;p (t- 7 ) > A p (t)
300
350
400
450
PULSE LEADING EDGE SHOWN I S LINEAR WITH A SLOPE CORRESPONDING TO THE 200 ns 0.05-0.30 RISE TIME SPECIFICATION. THIS CORRESPONDS TO A N 800 nr TOTAL RISE TIME TR .
7 = DAC DELAY TIME, 100 ns TD = DAC DECODE TIME, 200 ns = 7 /1-A RELATIVE THRESHOLD LEVEL OF ATTENUATED PULSE = TII1-A) TR = 0 . 2 5 RELATIVE THRESHOLD OF DELAYED PULSE = A T / I l - A ) TR = 0.125
FIG. 65. Delay-attenuate-compare design relations
500
DISTANCE MEASURING EQUIPMENT IN AVIATION
DELAY
+
MAX
ATTENUATE
189
4
COMPARE
4
t
J
FIG.66. The +-amplitude find circuit.
satisfactory means to obtain greater range coverage than the low-level threshold technique required in the F A mode. Most importantly, the interoperability error is eliminated because the &litude level is the same calibration point used in DME/N.
a. Transponder Calibration. Calibration of the IA mode of the DME/P transponder is identical to the procedure used for DME/N described in Section III,C,l. Although the situation is more complex, the calibration of the FA mode of the DME/P transponder using the DAC time-of-arrival circuit is amazingly straightforward for waveforms which are linear in the partial rise time region. To appreciate this, note that the SARPs define the transponder delay using the notion of the “virtual origin.” The virtual origin is the point at which the straight line through the 30% and 5% amplitude points on the pulse leading edge intersects the zero amplitude axis. Except for a constant offset given by the DAC detection time equation, the DAC declares the arrival of the pulse virtual origin. Thus, the delay between the interrogation pulse and the reply pulse for a DAC detection circuit is the time interval between the virtual origins of the two pulses. Remarkably, the DAC parameters in the interrogator, transponder, and the external test sets which calibrate them can all have different values, and the round-trip delay will be measured correctly. The measured delay will be correct even if uplink and downlink pulse rise times are different (Fig. 67). All this is a consequence of the linear partial rise time assumption and is the motivation for the partial rise time linearity specification in the SARPs. For any combination of DAC parameters in the range of 100 ns f 10% delay and 6 L 1 dB attenuation, the nonlinearity error for a +20% slope variation is less than 10 ft (3 m). It should be emphasized that other F A mode TOA measurement techniques are not excluded as long as their equipment calibration procedures satisfy the virtual origin concept.
190
ROBERT J. KELLY AND DANNY R. CUSICK
DAC DECISION TIME DAC DECISION TIME ROUND TRIP DELAY = T l i T + T D ~ ; n y + T R i p L Y DAC DECISION TIME ROUND TRIP DELAY = ITlNT
- TDI + TDC) I TDELAy + ITREpLy - To2
t
TD,l
DAC DECISION TIME ROUND TRIP DELAY = TINT + TDELAY + TREpLy DAC DECISION TIME ROUND TRIP DELAY = VIRTUAL ORIGIN ROUND TRIP DELAY
FIG.67. Calibration of the DMEIP.
3. Receiver Noise Receiver noise, which perturbs the leading edge of the received DME pulse, is a source of pulse time-of-arrival errors that translates into range errors at the interrogator. These errors are considered part of the ground and airborne instrumentation errors in that the minimum received signal levels must be accommodated within the equipment instrumentation error budgets. The purpose of this section is to define the effects of receiver noise on DME/P range measurement accuracy. The thermal noise component in the receiver IF is given by IFnoise= kTB
where kT
=4
mW/Hz ( T
x
= 290
K) and B
(32) = IF
noise bandwidth
(W.
The signal-to-noise ratio in the receiver video (i.e., ratio of the peak signal to the rms value of the video noise) is then S/N(video) = S/N(IF)
+ lolog
IF noise bandwidth - (receiver noise figure) (33) video noise bandwidth
The receiver noise figure is typically assumed to be 6 dB for DME/P ground
DISTANCE MEASURING EQUIPMENT IN AVIATION
191
equipment and 9 dB for the less expensive airborne receiver (see Subsection C,3 on power budgets). The FA mode of the DME/P utilizes the delay-attenuate-compare circuit to determine the time of arrival of a received pulse. A pulse width discriminator is utilized to discard pulses in the receiver video that are due to noise and have insufficient width to be considered valid DME pulses. Discarding pulses of less than 2 ps will result in satisfactory discrimination. Figure 68 summarizes the performance of the DAC circuit over a wide range of signal-to-noise ratios for two I F bandwidths (3.5 and 4.5 MHz) and two different DAC attenuation parameters (5 and 6 dB). The time-of-arrival errors increase rapidly as the SNR drops below 15 dB. Such behavior is expected since 15-17 dB is approximately the level at which the received pulse is thresholded by the attenuated signal. Similarly, the probability of pulse detection decreases rapidly as the SNR drops below 10 dB. Note that the reply efficiency would drop off as the square of the detection probability, since both pulses of the pulse pair must be received by the ground (or airborne) unit for a valid range measurement to be made. The following fact is evident from the plots: The signal levels required to reduce the noise-induced time-of-arrival errors to an acceptable level exceed those required to obtain an adequate reply efficiency. This is in contrast to DME/N, where the accuracy requirements are readily satisfied for signal levels that provide 90% or greater reply efficiency. Figure 69 shows the PFE and CMN components of the noise-induced time-of-arrival errors for SNR values that are typical of those encountered in operation. Contrary to intuition, the performance improves when the I F bandwidth is increased from 3.5 to 4.5 MHz. This is attributed to the fact that the wider bandwidth allows for more rapid signal transitions. Hence, pulse edges, including noise pulses, are better defined and the desired pulse leading edge becomes easier to distinguish from the noise pulses particularly when a noise pulse occurs just before the valid pulse leading edge. Note that the 1 MHz increase in the IF bandwidth results in only a 1 dB decrease in video SNR. For both DME/N and the IA mode of DME/P, pulse time-of-arrival determination is accomplished with the $amplitude find circuit. Figure 70 summarizes the performance of the +-amplitude find for an I F bandwidth of 800 kHz and a 2 p s pulse width discriminator. As in the F A mode case, the time-of-arrival errors increase rapidly as the SNR drops. Similarly, the pulse detectability decreases with decreasing SNR. For SNRs greater than 30 dB, the range errors are well accommodated by the instrumentation error budgets, as will be shown in Subsection C,2. Note that at the point where the reply efficiency begins to drop, the range errors are on the order of 100 ft and hence are negligible when compared to DME/N range accuracy requirements.
192
ROBERT J. KELLY AND DANNY R. CUSICK 840
5
700
I" c
0600
e
f m
-: LL
E"
a w
100 0 0
20 30 40 PEAK VIDEO-TO-RYSNOISE RATIO (dB)
10
-.-----....... ...... KEY
~
IF BANDWIDTH 3.5 mnz 3.5 mnz 4.5 mnz 4 . 5 ~ ~
DELAY 1Wnr 100 nr 100 nr 100 M
50
60
ATTENUATION 5 dB 6 dB 5 d0 6 dB
FIG.68. Noise performance of delay-attenuate-compare time-of-arrival detection circuit. A 2 ps pulse-width discriminatorwas utilized.
o
s
l
o
g
CONTROL MOTION NOISE (FEET-=% PROBABILITY)
g
g
PATH FOLLOWING ERROR (FEET-%% PROBABILITY)
L
w
\o
1wl-
-z 6
E $0 w v)
2
90 80
70
500
- - 450
-
Gi 400 350
6 0 - 8300 50-
P250
P
B
40-
f
30- z 1 5 0
>
=8
gn
200 U
2010-
g,,
50
0 -
0
+
-40 t
t i m a
$30
E
\
-‘\.
‘FWN -\-
----------__
--\
FIG.70. Noise performance of the f-amplitude find time-of-arrival detection circuit for an IF bandwidth at 800 kHz and a 2 p pulse-width discriminator.
4. Multipath
The single most important DME/P approach and landing performance index is the system’s resistance to pulse leading edge distortion by multipath. Without a multipath solution, the DME/P would not be viable. Interestingly enough, using the solution of low thresholding on the pulse leading edge, the multipath error component can be constrained to a value comparable to the equipment instrumentation errors. Multipath was not a concern with the DME/N because the accuracy allowance is 10 to 50 times that allowed for the DME/P. Moreover, for the enroute application, the multipath has a long delay time (which affects principally the second pulse) rather than the short
DISTANCE MEASURING EQUIPMENT IN AVIATION
195
delay multipath (less than 1 p s ) encountered in the approach and landing environment.
a. Specular Multipath ( F A Mode). Specular multipath is produced by scatterers which are sufficiently large (e.g., a substantial fraction of a Fresnel zone) to yield a reflection level within about 10 dB of the direct signal. Example sources include sizable buildings, wide-body aircraft, and vehicles near the ground facility. Specular multipath is characterized by four parameters: the relative strength of the reflected to the direct signal (MID ratio), the relative delay of the reflected to the direct signal, the relative rf phase of the reflected and direct signals, and the rate of change of relative phase (scalloping frequency). Given these parameters and the pulse time-of-arrival estimation technique, the specular multipath effects in the vicinity of the runway threshold can be calculated. Figures 71 and 72 estimate the multipath performance when a DAC is utilized to determine pulse time of arrival. The results are consistent with the accuracy specification (Subsection C,2) for the FA mode. The ground and airborne pulse arrival time measurement circuits are designed to prevent large specular multipath errors along the extended runway center line. This is accomplished in a manner similar to the angle components of the MLS (Redlien and Kelly, 1981). That is, the pulse arrival time decision point is chosen to be less than the shortest multipath delay time associated with large obstacles around the runway. The multipath delay times are determined by the runway obstacle clearance surfaces. Obstacles such as hangars, which are located so that they conform to the obstacle clearance surfaces, typically generate large multipath reflections, having delays in excess of 300 ns. Figure 72 is a plot of constant-delay ellipses about the runway. Such plots can be used to estimate the multipath delay that results from an obstacle at a given position relative to the aircraft, runway, and ground facility. Computer simulations and experimental measurements have shown that representative signal processors (such as the DAC with a 3.5-4.5 MHz bandwidth) will not be susceptible to multipath with delays in excess of 300 ns, as noted in Fig. 71. Multipath environment studies and field measurements (Evans, 1982; Evans and Swett, 1981) suggest that high-level multipath with delay less than 300 ns will occur at less than 20% of the runways. In addition, if the multipath delays are less than 300 ns, most of these reflections will be less than -3 dB. Under this -3 dB multipath assumption, the errors will be bounded at about 33 ft (10 m) for DAC parameters of 100 ns delay and 5 to 6 dB attenuation values. This error component was included in the system error budget (Subsection C,2) proposed for the 100 ft (30 m) system accuracy standard.
100.0
v)
2
g
80.0 60.0
-60.0 -80.0 -1-0
. '
-
- . - -
-
1
-
1
-
1
-
~
.
.
-
.
-
- -
PULSE SHAPE, COS/COs RISE TIME: 800 NS DAC DELAY: 100 NS DAC AlTENUATION: -6dB MID: -6 dB
NO IFFILTER
.
-
.
' . '
'
I
*
I
I
1
-
'
1
*
'
-
MULTIPATH ERRORS WITH N O IF FILTERING DELAY
PEAK ERROR
TIME DELAY FOR PEAK NEG. ERROR
TIME DELAY FOR PEAK POS ERROR TIME DELAY FOR NO ERROR (DAC DECODE TIME)
T TD-
I-A
0
p = MULTIPATH-TO-DIRECTAMPLITUDE RATIO A = DAC ATTENUATION 7 DAC DELAY TD = DAC DECODE TIME
-
NOTE: THESE EQUATIONS DO NOT APPLY WHEN IF FILTERING IS USED. THE ERROR CURVES ARE MODIFIED AS SHOWN ABOVE.
FIG.71. pulse.
Multipath errors as a function of delay and phase for DAC processing of cos/cosz
The spatial frequency content of the multipath interference field contributes to the CMN component of system error. As the aircraft moves through the interference field, the relative phase of the direct and reflected signals varies at a rate (scalloping frequency) dependent upon the geometry and the aircraft velocity. Hence, the range errors are time varying and therefore contribute to the CMN component of the system error. The highfrequency components (> 10 rad/second) can be reduced by the interrogator data filter (see Subsection D), while nothing can be done to attenuate those
197
DISTANCE MEASURING EQUIPMENT IN AVIATION SCALLOPING FREQUENCIES LESS THAN 10 RAD/SEC FOR REFLECTORS WITHIN THESE LIMITS
2500 2ooo Y
f
=r
1000
zc
500
o w
gf
z'c Y
0
2
5
0 -500
-1000 -1500 -2000 -2500
-1250
0
1250
2500 3750 5OOO 6250 7600 8750 DISTANCE ALONG CENTERLINE I N FEET
loo00
1126
NOTE: SCALLOPING FREQUENCY LIMITS ASSUME 200 FT/SEC AIRCRAFT VELOCITY ALONG RUNWAY CENTERLINE
FIG.72. Contours of constant multipath delay for aircraft-to-transponderdistance of 10,oOo ft.
components of the error signal falling within the data filter bandwidth. Figure 72 illustrates, for the given geometry, the region where the multipath error components will be filtered. Kelly and LaBerge (1980) discusses additional multipath control techniques which are applicable to those few sites where multipath exceeds - 3 dB in the region where the scalloping frequency is less than 10 rad/second.
b. Specular Multipath (ZA Mode). Multipath performance for the IA mode may be different from that encountered in the FA mode because the TOA measurement threshold is much later on the pulse and thus the system is more sensitive to long-delay multipath (Fig. 73). The critical range is 7 nmi because the error allowance is most stringent at that range. Fortunately, however, the approach geometry is such that the aircraft is above most of the severe multipath. Studies indicate that at most runways along the runway center line extended (< 20"), the MID is less than - 6 dB. More severe conditions may exist on the wide radials (30"-40"). For these rare cases, special multipath control measures are available which will minimize the multipath interference at wide angles (Kelly and LaBerge, 1980). c. Difuse Multipath (FA and IA Modes). Diffuse multipath is produced by a "collection" of small objects (i.e., small relative to the Fresnel zone radius in at least one dimension) which individually yield very low level reflections (e.g.,
ROBERT J. KELLY AND DANNY R. CUSICK
198
-160.0
@=18o,MID=-6dB
-240.0
-320.0 -400.0
-
0.0
"
300.0
-
l
'
600.0
l
.
900.0
'
.
1200.0
l
'
l
.
1500.0 1800.0 DELAY (nr)
'
*
2100.0
l
*
2400.0
l
.
2700.0
3OW.O
6 RELATIVE PHASE OF REFLECTED SIGNAL MID = RATIO OF REFLECTED SIGNAL LEVEL TO DIRECT SIGNAL LEVEL FIG.73. IA mode multipath performance for a 1200 ns rise time cos/cosz pulse.
20 dB below the direct signal level), but which may be significant as a group due to the large number and disposition of such scatterers. Examples include boulders, posts, approach lights, ocean waves, trees, and small buildings or aircraft. As viewed by the receiver, diffuse multipath in flight (test) error traces appears noiselike and will be reduced by the interrogator output filter. Analytical and experimental work (Evans, 1982; Evans and Swett, 1981) indicates that the diffuse multipath component will be satisfactorily bounded by the error budget allocations. Flight test traces have shown diffuse multipath error components to be less than 10 ft (3 m) (Edwards, 1981).
5 . Garble-System Eficiency and Accuracy Eflects The DME/P system accuracy requirements must be satisfied in the presence of pulses other than just the desired interrogation and reply pulse pairs. The interference caused by the extraneous pulse environment is referred to as garble and, in contrast to DME/N, is an important system consideration in DME/P. There are two sources of garble interference of interest, DME and non-DME. DME garble is simply due to the effects of other DME equipment operating at or near the same frequency in the surrounding environment. Non-DME components are, for example, those caused by other L-band systems such as the ATC Radar Beacon System and the Joint Tactical Information Distribution System. The effects of these non-DME garble sources are beyond the scope of the present discussion. The effect of the DME extraneous pulse environment is twofold, involving both accuracy and system efficiency. Accuracy degradation occurs when an
DISTANCE MEASURING EQUIPMENT IN AVIATION
199
undesired pulse is coincident with a desired pulse, resulting in a distorted pulse leading edge. Such coincidence can occur on both the up- and downlinks. The occurrence of garble-induced range errors is a probabilistic event and primarily adds to the CMN component of the system error. Since the DME/P receiver processing parameters (i.e., bandwidth, time-of-arrival determination, pulse-width discrimination, etc.) are selected to satisfy other considerations such as multipath performance, the garble-induced error component must be controlled by the interrogator data filter and channel assignment procedures. The total garble error for the up- and downlinks combined must be less than 20 ft (6 m) for both the CMN and PFE components. System efficiency is defined as the percentage ratio of valid replies processed by the interrogator in response to its own interrogations. It is an important parameter because system accuracy must be achieved when all of the expected reply samples are not present. It is degraded when coincidence with an undesired pulse prevents detection of the desired pulse or when an interrogation falls in the transponder’s dead time and cannot result in a reply. The interrogator’s output filter must therefore be designed such that the effects of the missing data are minimized.
a. Downlink DME Garble. Downlink garble occurs when valid interrogations at the ground transponder are coincident with interrogations from other aircraft and transmissions from other ground facilities,resulting in either a loss of signal or a pulse time-of-arrival measurement error. The garble can be on-channel interference from aircraft in the same service volume or it can be interference from aircraft whose interrogations are within the bandwidth of the serving transponder. The garble effect is then a function of the number of interrogations, and the corresponding distribution of interrogation frequencies and signal amplitudes received at the transponder. b. Uplink DME Garble. Uplink garble occurs when valid replies at the interrogator are coincident with transmissions from other transponders and interrogators and results in either a range measurement error or a missed reply at the interrogator. The garble can be interference from any transponder whose frequency is within the bandwidth of the interrogator, including those on the same frequency, but with different pulse coding. This undesired groundto-air loading is a function of the number of transponders in the vicinity of the interrogator and the corresponding distribution of reply frequencies and signal amplitudes received at the interrogator. c. Transponder Reply EfJiciency. As in DME/N (Section III,C,4), the DME/P transponder has only a finite capacity to process received interrogations. Recall that this results from the fact that each interrogation that results in a reply creates a period of time during which arriving interrogations
200
ROBERT J. KELLY AND DANNY R. CUSICK
will not result in replies. This inhibit time is the result of dead time gates, reply transmission gates, receiver blanking gates, and, indirectly, the priority determination technique for replies, squitter, and facility identification pulse pairs.
d. Interrogator Processor EfJiciency. The interrogator signal processor efficiency is the ratio of the number of replies processed by the interrogator to the number of interrogations in the absence of garble and transponder dead time effects. This efficiencydepends on the receiver noise level and, as shown in Subsection B,3, will exceed 95% for SNR values that are adequate to satisfy the system accuracy requirements. The interference rate due to garble (i.e., the rate at which range measurements are in error or missing as a result of garble effects) is dependent upon the channel assignment plan, traffic loading, and the ground transponder and interrogator receiver bandwidths. Garble effects are not a critical factor in the DME/N system because of the use of narrow-bandwidth receivers, which limit the garble susceptibility to co-frequency and first adjacent frequency extraneous pulses. DME/N system efficiency is therefore primarily dependent on the transponder reply efficiency, rather than on the up- and downlink garble effects, and the accuracy degradation due to garble is negligible when compared with the DME/N accuracy requirements. In DME/P, the garble problem becomes significant because of tighter accuracy standards imposed on the IA mode and the use of wide-bandwidth receivers and low-level thresholding in the FA mode. As one might expect, the F A mode reflects a greater susceptibility to garble in terms of accuracy and system efficiency than the IA mode. These factors have been accommodated in the DME/P system definition and, under the channel assignment constraints detailed in Section III,C, do not require special consideration by the using authority. Worst-case traffic loading for an interrogator in the FA mode occurs at the extremes of coverage, that is, at the 7-8 nmi transition region at a 2000 ft altitude and the runway rollout region, where the signals received at the ground equipment are at the minimum operational values. In the IA mode, the worst environment will occur at the 22 nmi coverage limit, where not only are the down- and uplink signal levels at low values, but the interrogator is at the extremes of the frequency protected service volume. Garble environments can be estimated by considering the geographic separation of DME ground facilities, the number and location of aircraft using each facility, the location of the victim ground and airborne equipment, transmitter powers, and antenna patterns. Although the interference environment is composed of downlink and uplink components, only one of the links dominates in terms of effect. For example, if the desired interrogator is at a low
20 1
DISTANCE MEASURING EQUIPMENT IN AVIATION
altitude (final approach at 2000 ft), then the downlink is critical because not only is the ground transponder exposed to all of the aircraft present in a large volume about the transponder, but the received signal level will be at or near the transponder sensitivity value. The uplink effect is minimized since, at a low altitude, the number of transponders within the line of sight of the victim interrogator is reduced. Table XIX defines one of the DME extraneous pulse environments used by the ICAO to develop the DME/P channel plan and the 50% system efficiency specification. These pulse interference rates and amplitude levels represent the worst-case downlink environment for the FA mode at 7 nmi. This analysis is summarized in Koshar and Smithmeyer (1982), where, in addition, garble environments at other critical points in the service volume such as runway threshold and at 22 nmi are provided.
e. Estimating System EfJiciency. System efficiency is the combined effect of downlink garble, ground transponder dead time, uplink garble, and interrogator signal processor efficiency. Since each of these efficiency components are statistically independent, they can be computed individually and then combined to yield the overall system efficiency. The effect of a single component is defined as the percentage ratio of valid replies processed by the interrogator in response to its own interrogations, assuming all other components are not present. The system efficiency is then the product of the individual components which must exceed the 50% SARPs requirement. Figure 74 defines a procedure for calculating the system efficiency,making use of the data shown in Fig. 75. The “detection probability” data (PDET) were
TABLE XIX INTERROGATIONLOADING AT THE EL MONTE, CALIFORNIA, DME/P AIR-TO-GROUND TRANSPONDER FOR A DESIREDINTERROGATOR LQCATED7 nmi FROM THE BEACON AND AT 2000 ft ALTITUDE ~~
~
~
~~
Number of interrogations per second” Relative frequency
DIu
DJU
DJU
DJU
I+30
I+20
I+14
I + 9
Cofrequency k1 MHz + 2 MHz k3 MHz Total individual pulses
2848 6264 4736 6208 40,112
2112 2432 2016 2356 17,960
1264 1560 704 960 8276
672 840 432 540 4968
DJU I + 6
DJU
DJU
I0
2-6
368
112
80
208 360 1872
112 120 688
160 ~
Number of interrogations greater than or equal to an undesired-to-desiredsignal power ratio (DJU of 30,20, 14 dB, etc.) are represented in each column. a
202
ROBERT J. KELLY A N D DANNY R. CUSICK
ik p,, = ith pulse (of pulse pair) down-link garble effiimncy from kth adjacent dunnel; k = 0 is the "on" channel frequency.
prePly = ($a8 equations 35 and 36).
p:p
= uplink garble efficiency
Nota:
where fk = ka adjacent frequency garble rate 7 = Intaractionwindow about desird pulw (lour)
FIG.74. Computation of system efficiency.
derivea by computer simulation of the effects of different interfering DME pulses. Range gate widths and decoder apertures were assumed to be 2 1.5 p. Figure 76 illustrates the time interval z during which interactions with a desired pulse can occur. As an example, assume that the downlink garble rate on the first adjacent frequency is fi= 10 kHz and that the D/U = 10 dB. From Fig. 75, PDET = 0.65. Therefore, P&
=
1 - (1 - e-f")(l - PDET) 1 - (0.1)(0.35) = 0.96
(34) This is but one of the many possible components of the system efficiency. To determine the system efficiency, one need only repeat the computation for all other components and combine them as shown in Fig. 74. The computation of reply efficiency for the DME/P transponder is more complex than that already noted for DME/N. This is due to the fact that a single transponder is providing service to both IA and F A mode users. The IA and FA mode reply efficiencies can be estimated using =
DISTANCE MEASURING EQUIPMENT IN AVIATION
203
FIG.75. Graph depicting the detection probability determined by computer simulation of a garble source of given frequency and D / U given that the undesired pulse having random rf phase falls at a random position in a 10 ps window about the desired pulse. Assumes a 4 MHz IF bandwidth, DAC parameters of 100 ns delay and 5.5 dB attenuation, and 2 ps pulse-width discriminator.
where R,A is the FA channel reply efficiency, W is the number of FA channel interrogations/second, R,, is the IA channel reply efficiency, N is the number of IA channel interrogations/second, T,, is the FA inhibit interval due to earlier FA interrogation, TWNis the IA inhibit interval due to earlier FA interrogation, TNwis the FA inhibit interval due to earlier IA interrogation, and T ” is the IA inhibit interval due to earlier IA interrogation. T,, and T N N are simply measures of the effect (or dead time) an interrogation has on its own channel. TwNand TNwindicate the effect (or dead time) of an interrogation on the opposite channel. For the DME/N transponder, TNw= TWN= 0, and Eq. (36) reduces to the classical reply efficiency relation given in Section III,C,C Typical values of T, TwN,TNw, and TNNare 80,86,50, and 76 p s for the Y channel, assuming a 60 p s transponder dead time and a 10 &transmitter pulse blanking time.
f. Estimating Garble Errors. The range error component due to garble is a complex function of the extraneous pulse environment and the receiver processing parameters, particularly the interrogator data processing al-
ROBERT J. KELLY AND DANNY R. CUSICK
204
OUTLIER WINDOW DESIRED PULSE RANGE GATE OR DECODER APERTURE
GARBLE PULSE
TIME
-+
APPLIES TO ANY PULSE OF A TWO PULSE GROUP
GARBLE INTERVAL INTERACTION 7=
tour
4
FIG.76. Garble interaction interval. A garble pulse in the interval will cause an error or missed pulse. A garble pulse outside the interval has no effect.
gorithm. As already noted, the garble environment is characterized by the pulse arrival rate and the undesired signal level of the extraneous pulse on each frequency about the on-channel frequency. The occurrence of a garble event on a given interrogation depends on the probability that an undesired pulse is positioned close enough to the desired pulse to result in an error or missed reply. Assuming Poisson arrivals at rate f and an interaction interval z, the probability of a garble event appearing on a single interrogation is 1 - e p f r . The nature of the garble event, that is, whether the result is a range measurement error or missed reply, depends on the relative undesired pulse amplitude, phase, frequency, and position with respect to the desired pulse. The signal processing parameters, that is, the IF bandwidth, Ferris Discriminator, pulse width discriminator, and track gate, all influence the outcome of a garble event. Figure 77 illustrates an ungarbled DME pulse. Figure 78 is an example of where an on-frequency garble pulse caused a missed pulse detection (top figure) while the bottom figure resulted in a loo0 ns error. The effects of adjacent frequency garble are depicted in Fig. 79. The overall effect of garble events on the range accuracy is highly dependent on the interrogator data filter algorithm (Subsection D,2). The garble-induced errors occur at random times in the sequence of range measurements and typically will have large amplitudes relative to other
205
DISTANCE MEASURING EQUIPMENT IN AVIATION lo
T
0
_-
.10
_DAC DETECT TIME: 8993.0 ns
DECLARED PULSE
-20-
-m-. a, : ;
:
; : ; : ;
::
: I :
:
:
:
I
DELAV.ATTENUATE-COMPARE
I
PULSE WIDTH DISCRIMINATOR TRACK GATE 0
m
uylo
6Ooo
8wo
lmao
12M)o
1uylo
1 m
TIME Ins1
FIG.77. Processingof ungarbled DME pulse. Pulse time of arrival indicated by rising edge of “ D A C inside track gate that is associated with valid pulse-width discriminatoroutput.
sources of measurement error. Hence, garble errors appear as impulse noise. Linear filters are good for removing the effects of random noise errors (such as those due to receiver noise) from the range information. However, the memory of the filter causes the effect of a single impulse error to be seen over many subsequent samples. Hence, nonlinear filtering techniques, such as slew rate limiting, noted in Subsection D, are useful in minimizing the effects of garble errors. Additionally, the data filter must have the ability to “coast” through missed data samples due to garble and transponder reply efficiency effects while satisfying the accuracy requirements. There are two approaches to predicting the accuracy degradation due to garble. One can simulate a given multisource garble environment and record the results. However, the simulation must be completely rerun if any of the garble conditions or signal processing parameters is changed. A more rigorous and useful approach is to generate (by simulation) data defining the error statistics as a function of D / U and relative frequency. The functional relation between the error and the garble rate is then determined using procedures similar to those outlined in Figs. 74 and 75. In this way, the statistics of the raw error can be estimated for a general garble environment. As is evident, predictions of the magnitude of the garble error component that are generally applicable are difficult if not impossible to make due to the complexity of the phenomenon. In general, garble effects must be experimentally determined via bench measurements or studied with the aid of a time domain computer simulation. Figure 80 shows the error statistics (mean, standard deviation) when the garble pulse has a D / U level ranging from - 30 to + 30 dB and rf frequency of
206 100
--
-lo-Y
3
t
-20-L
NO DECODE
k
5
DECLARED PULSE ARRIVAL
.m--40,:
I
:
! : ; :
I
;
I
:
-I
DELAY-ATTENUATE-COMPARE WLSE WIDTH DISCRIMINATOR TRACK GATE
0
2000
4000
6000
8000
TIME
ioom
12000
14000
16ooo
14Wa
1-
In4
UNDESIRED PULSE: D I U I S 3 dB ONFREOUENCV SAME RF PHASE 3000 ns I N FRONT OF DESIRED PULSE
1°T 0
--
-lo-DECLARED PULSE
-20-
-1000 ns ERROR
-30.-
-I
DELAY.ATTENUATE-COMPARE
-
PULSE WIDTH DISCRIMINATOR TRACK GATE 0
2000
4000
MMO
8000
loo00
12000
TIME Ins1 UNDESIRED PULSE. DIU I S 3 dB ONFREOUENCY SAME RF PHASE
1000 ni IN FRONT OF DESIRED PULSE
FIG.78. Processing of garbled (on-frequency)DME pulse.
0 to 4 MHz relative to the desired signal. Pulse detectibility data for the same conditions were shown in Fig. 75. The results at a particular DIU and frequency are simply statistics of multiple samples taken for various relative rf phases and garble pulse positions in a 10 ps window about the desired pulse. A garble pulse falling outside the 10 ps window was found to have no effect on the measurement of range with the desired pulse.
207
DISTANCE MEASURING EQUIPMENT IN AVIATION
,; ;+;:/!: rn
0.-
; n
-10-
t
-30 .20--
Y
3 ~
3
+266.5 ns ERROR OF TRACK GATE
-30-40
: ; : I :
; : ; : I ;
I
DELAY-ATTENUATECOMPARE
”
PULSE WIDTH DISCRIMINATOR TRACK GATE 0
aooo
4& 4000
wbo woo
8000 TIME
lwoo
lZOw 1ZOw
14wo
16Mo
~4ooo
1wlm
hi1
UNDESIRED PULSE: D/U IS 3 dB ZMHZ OFF FREOUENCY FREOUENCY
3wo ns IN FRONT OF DESIRED PULSE
-
DELAY-ATTENUATE.COMPARE
“ “ I
PULSE WIDTH DISCRIMINATOR TRACK GATE
0
2000
4wo
M)o
8ow
10000
lzwo
TIME Ins1 UNOESIREDPULSE: D I U l S 3 d B 2MHZOFFFREQUENCY 1000 n i IN FRONT OF DESIRED PULSE
FIG.79. Processing of garbled (off-frequency)DME pulse.
The data presented in Fig. 80 are applicable when a garble event is present on each interrogation. In an actual environment garble events occur at random during the sequence of interrogations. These random events are taken into account by using the pulse detectibility data in Fig. 75. The statistics of the raw (unfiltered) garble-induced error component (mean and standard deviation) resulting from N sources (source i is characterized by specific DIU, rf frequency, and arrival rate) can be determined from the data given
208
ROBERT J. KELLY AND DANNY R. CUSICK 400
350
300 250
200 150 100 50 0 -50 -100 -30
-30
-25
-20
-15 -10 -5 0 DESIRED-TO-UNDESIRED
5 10 15 SIGNAL RATIO (dB)
20
25
30
-25
-20
-15
-10 -5 0 DESIRED-TO-UNDESIRED
5 10 15 SIGNAL RATIO (dB)
20
25
30
KEY
GARBLE FREQUENCY
a . . -
---------
....-.,..........-
---_----.---
0 mH, 1 mH,
2 mH,
3 mH, 4 mH,
FIG.80. Graphs depicting the statistical effect determined by computer simulation of a garble source of given frequency and D/Ugiven that the undesired pulse having random rf phase falls at a random position in a 10 p s window about the desired pulse. Assumes a 4 MHz IF bandwidth, DAC parameters of 100 ns delay and 5.5 dB attenuation, and a 2 p s pulse-width discriminator.
DISTANCE MEASURING EQUIPMENT IN AVIATION
209
in Figs. 75 and 80 using the following equations: N
with the constraint that N
2pi=1 i= 1
(39)
The symbols utilized are defined as follows: N is the number of garble sources of a given D / U , rf frequency, and arrival rate f;ui, the mean error component due to source i as determined from Fig. 80; a:, the error variance due to source i as determined from Fig. 80; and pi,the percentage of interrogations in which an error due to garble source i is present. It is the product of two terms: (1 - e-fr), the probability that a garble pulse falls in the interval r; PDET, the probability that the desired pulse is detected given the presence of a garble pulse in the interval T as obtained from Fig. 75. Finally, f is the pulse arrival rate; while z is the interaction interval; 10 p s for the data shown in Figs. 75 and 80. Although the filtered error component cannot be accurately determined using the above procedures, the magnitude of the raw error resulting from the garble is determined. The utility of this computation is in comparing the relative effects of different garble environments on D M E / P performance. Figure 81 shows the predicted raw error standard deviation for a pulse rate of 16,200 pulses/s. Figure 82 shows the filtered garble error statistics for the same case that results when an a-8 filter with a 40 ns slew rate limit is used as the interrogator data filter (see Subsection D,2). These results were derived by direct simulation of the garble environment. The garble error calculation procedures outlined above are summarized by the following examples. Assuming an on-frequency garble error source with D / U = 10 dB and a rate of 16,200 pulses/s, what is the standard deviation of the raw range error? From Figs. 75 and 80, p1 = 0 ns, a1 = 160 ns, and PDET = 0.67. The percentage of interrogations in which a garble source induces an error is calculated using Eq. (39). That is, p1 = (1 - e-0.16)(0.67)= 0.1. The raw noise error is then determined from Eq. (38): a, = = = 51 ns. In Fig. 81, the complete simulation resulted in a 55 ns noise error. Therefore, the generalized procedure described here permits a close approximation to those estimates obtained from a complete simulation of the garble environment.
Jm
ROBERT J. KELLY A N D D A N N Y R. CUSICK
210
-30
-25
-20
-15
-10
-5
5
0
DESIRED-TO-UNDESIRED KEY -
-.-.--
---------.............-... --- --- --- ---
10
15
20
25
30
SIGNAL RATIO IdB)
GARBLE FREQUENCY ~
0 mH, 1 mH,
2 mH,
3 mH, 4 mH,
FIG.81. Standard deviation of range error that results for unfiltered measurements made in the presence of a 16,200 pulse/second garble source of specified D/U and frequency.
The filtered error for this example can be estimated from Fig. 82. The filtered standard deviation is only about 55.5 ns. The efficiency of the DME/P signal processing, particularly the data filter shown later in Fig. 91, is clearly evident by this example, where the garble noise error component was reduced almost by a factor of 10. 6. Summary
This section has shown how the performance required of the DME/P is achieved in the presence of noise, multipath, and garble by taking advantage of a fast rise time pulse, the delay-attenuate-compare circuit for pulse timeof-arrival determination, and a wide IF bandwidth that preserves the shape of the received DME. Summarized below are the major points presented in this section. The fast rise time pulse (1000 ns), called cos/cosz, satisfies not only the need for DME/P to be fully interoperable with DME/N but the strict spectral limitations imposed on the DME ground stations as well. The delayattenuate-compare detector simply takes maximum advantage of the fast rise time pulse to enhance the performance of the DME/P in the presence of multipath.
211
DISTANCE MEASURING EQUIPMENT IN AVIATION 15.00 13.50 12.00 10.50
c
9
20 O
K
c
0
9.00 7.50
6.00
z
4.50
k
3.00
a
1.50 0.00 -20
-16
-12
-8
-4
0
DESIRED-TO-UNDESIRED KEY -.-.-. ---------................-
--- --- --- ---
4
8
12
16
20
SIGNAL RATIO (dB)
GARBLE FREQUENCY 0 mH, 1 mH, 2 mH,
3 mH, 4 mH,
FIG.82. Standard deviation of range error that results for filtered range measurements made in the presence of a 16,200 pulse/second garble source of specified D/U and frequency. Interrogator data filter is an a-p filter using outlier limit detection and slew-rate limiting. a = 0.125, fi = 0.0078, outlier limit = 300 ns, slew-rate limit = 40 ns. Data derived by direct computer simulation of the garble environment.
Accuracy and system efficiency degradation due to receiver noise was shown for both the IA and FA modes. As will be shown in Subsection C , the noise-induced errors are readily accomodated within the IA and FA instrumentation error budgets when the minimum required signal levels are available. For the important FA mode, the effects of receiver noise were shown for a variety of DAC parameters and I F bandwidths. Of particular interest is the fact that when a pulse-width discriminator is used, increasing the I F bandwidth from 3.5 to 4.5 MHz will reduce the noise-induced range errors. Additionally, it is clear that accuracy, and not reply efficiency, is the limiting consideration. The pulse shape, pulse detection technique, and the wide I F bandwidth are the keys to multipath immunity of the DME/P FA mode. Unlike DME/N, where strong multipath can result in errors of 200 or more feet, FA mode errors are negligible for reflected pulse delays in excess of 350 ns while the maximum error is less than 33 ft for short delay multipath. In the IA mode, the aircraft is only exposed to low-level (6 dB or more below the desired)
212
ROBERT J. KELLY AND DANNY R. CUSICK
multipath, and the resulting errors are readily accomodated within the accuracy requirements. Finally, due to its wide bandwidth and tighter accuracy requirements, DME/P exhibits a greater susceptibility to garble than DME/N. Although difficult to analyze, the effects of garble have been shown to be both predictable and manageable. Prediction of the garble effects requires a knowledge of the rate, rf frequency, and levels of the pulses that comprise the garble environment, the particular signal processing in use, and a database of error and pulse detectibility data that can only be obtained by time domain computer simulation or accurate laboratory measurements. The effects of errors and missing data are minimized by appropriate data processing techniques like those to be discussed in Subsection D. C . D M E / P Signal in Space: Accuracy and Coverage
The purpose of this section is to bring together the detailed performance discussions of the previous sections and thereby define DME/P accuracy and coverage requirements. First, Subsection C,1 defines the DME/P error components, path following error, and control motion noise, which describe the effect of the error on aircraft flight control system performance. Additionally, methods of measuring these components are described. Subsection C,2 presents the system error budgets for Standard 1 and Standard 2 DME/P indicating that the system accuracies can be achieved for specified values of site-related errors and instrumentation errors. Subsection C,3 presents example DME/P power budgets where the values shown reflect the parameters required to obtain the desired signal coverage. Subsection C,4 treats one of the elements necessary to achieve the desired vertical coverage, the siting of the DME antenna. I . DMEIP Error Measurement Methodology
The DME/P system accuracies, like the MLS angle system accuracies, are specified in terms of path following error and control motion noise. These parameters describe the interaction of the DME/P guidance signal with the aircraft in terms directly related to aircraft position errors and flight control system design. The PFE is that part of the error signal having frequency components less than 1.5rad/s. These low-frequency components can displace the aircraft from its intended course over the ground. The portion of the error signal having frequency components between 0.5 and 10 rad/s is the CMN. CMN components fall above the aircraft guidance loop bandwidth and thus are too rapid for the aircraft to follow. However, the CMN components of the error
DISTANCE MEASURING EQUIPMENT IN AVIATION
213
signal may induce aircraft control activity such as aileron and cockpit wheel motion as well as aircraft roll variations. This unwanted control activity affects the pilot’s confidence in both the guidance system and his flight control system during autocoupled approaches. Performance evaluation of the DME/P system in terms of PFE and CMN is summarized in Fig. 83. The range error signal is generated by forming the difference between the range output of the interrogator and the reference range. The PFE and CMN components are determined by processing the error signal with the appropriate digital filters. Digital implementation of a PFE and a CMN bandpass filter are shown in Figs. 84 and 85. For the purposes of determining compliance with the accuracy standard, the PFE and CMN components are evaluated over any 10 s measurement interval of the F A mode flight error record taken within the DME/P coverage limits. The measurement interval for the IA mode is 40 s. The 95% probability requirement is interpreted to be satisfied if the PFE and CMN components do not exceed the specified error limits for a total period that is more than 5% of the evaluation interval. Determination of the 95th percentile of the error data is illustrated in Fig. 86. This same technique can also be utilized to determine the transponder and interrogator instrumentation error components where the overall system range error is replaced with transponder delay measurements or interrogator range measurements. 2. DMEIP System Error Budgets
As noted earlier in Section III,C,l, error budgets serve to aid facility planning and equipment procurement specification by identifying, quantifying, and combining the instrumentation and site-dependent errors that determine overall system performance. These budgets are critical to the specification of the DME/P due to the tight accuracy constraints. An allowance in the budget must be made for each of the error components discussed earlier-multipath, garble, and noise-as well as the ground and airborne equipment instrumentation errors. Receiver noise effects are incorporated by satisfying the instrumentation error requirements at the minimum received signal power levels. Table XX is the error budget recommended for guidance by ICAO. The itemized components are to be satisfied on a 95% probability basis. Allowances for the FA mode Standards 1 and 2 as well as the IA mode are shown. To be useful, an error budget must have a simple procedure for combining the individual components. If the error sources are additive, independent random variables with zero mean, they can be combined on an RSS basis. Although these conditions are not always fulfilled for DME/P, experience
214
ROBERT J. KELLY A N D D A N N Y R. CUSICK DME/P INTEROGATOR
RAW DISTANCE
PATH FOLLOWING
ESTIMATES
OUTPUT
ERROR F I L T E R O O
-
POSITION REFERANCE
CONTROL
,
wg* 1.5
w2-
PATH FOLLOWING FILTER
'10
0.5
~-
~~~
W2 StW 2
CORNER FREQUENCIES
DME IP
ERROR
MOTION NOISE FILTER W
RECEIVER OUTPUT FILTER GUIDANCE
PATH
-@
*Wo=O64Wn
ur2 I
S2 + ZSW, +W:
CONTROL MOTION NOISE FILTER
* * THIS CORNER FREQUENCY M A Y BE LESS
[*]
THAN 10 RADlSEC FOR SOME APPLICATIONS.
PATH FOLLOWING FILTER (LOW PASS) CONTROL MOTION NOISE FILTER
----I
I
I
I
I
I
w,
Wo
I \
\
RADlANSlSEC
4
3
FILTER CONFIGURATION AND CORNER FREQUENCIES OUTPUT OF PFE OR CMN FILTER
T-SLIDING WINDOW MLS-DMEIP measurement methodology. E: error specification: T: region t o be evaluated: T1, T 2 T3 : time intervals that error exceeds specification For ground equipment t o be acceptable in t h i s region, the following inequality h w l d be true: (T1 + T2 + T j + -.) / T 5 0.05.
_...
FIG.83. MLS-DME/P measurement methodology. E : error specification T region to be evaluated. T,, T2, T3,...: time intervals that error exceeds specification. For equipment to be 0.05. acceptable in this region, the following inequality should be true: (TI T2 + T3 + ...)/ T I
+
215
DISTANCE MEASURING EQUIPMENT IN AVIATION DEFINITIONS H(S) = ANALOG FILTER TRANSFER FUNCTION RATIO OF THE LAPLACE TRANSFORM OF THE OUTPUT SIGNAL TO THE LAPLACE TRANSFORM OF THE INPUT SIGNAL D I G I T A L FILTER TRANSFER FUNCTION RATIO OF THE 2-TRANSFORM OF THE OUTPUT DATA TO THE 2-TRANSFORM OF THE INPUT DATA
H(2) =
'n
=
INPUT DATA SAMPLE AT TIME n
Yn
=
OUTPUT DATA SAMPLE A T TIME n
ANALOG PFE FILTER 5.49
H(S) =
S2 + 4.698 + 5.49 DIGITAL PFE FILTER-FA MODE (40 Hz DATA RATE)
HE)
=
7.9762 X
Yn
=
7.9762 X
lop4 ( X n + 2XnP1 + Xn-2 ) + 1.887Yn-1 - 0.8902Yn-2
D I G I T A L PFE FILTER-1A MODE (16Hz DATA RATE) H(Z0 = 4.5942 X
Yn
=
4.5942 X 10-3 (X,
+ 2XnP1 + X,-2
1 + 1.7288YnP1
- O.7472Ynp2
FREQUENCY RESPONSE 0
-4 -8
-*
-12
LU
-16
U
n
3
z W a
=
-20
-24 -28
-32 -36 -40
0
2
4
6
8 10 12 14 FREQUENCY (RADIANS/SEC)
16
FIG.84. Analog and digital implementations of PFE filter.
18
20
ROBERT J. KELLY AND DANNY R. CUSICK
216 DEFINITIONS
H(S) = ANALOG FILTER TRANSFER FUNCTION RATIO OF THE LAPLACE TRANSFORM OF THE OUTPUT SIGNAL TO THE LAPLACE TRANSFORM OF THE INPUT SIGNAL H(2) =
DIGITAL FILTER TRANSFER FUNCTION RATIO OF THE 2-TRANSFORM.OF THE OUTPUT DATA TO THE 2-TRANSFORM OF THE INPUT DATA
Xn
=
INPUT DAJA SAMPLE AT TIME n
Yn
=
OUTPUT DATA SAMPLE AT TIME n
ANALOG BANDPASS CMN FILTER H(S) =
1 . 1 0 S+O.5
S+10
DIGITAL BANDPASS CMN FILTER-FA MODE (40 Hz DATA RATE) H(2) = 0.11094 =
Yn
I
(22- 1 )
(2’ - 1.76432 + 0.7671)
0.1 1094Xn - 0.1 1094Xn + 1.7643YnP1
- 0.7671 YnP2
DIGITAL BANDPASS CMN FILTER-1A MODE (16 Hz DATA RATE) H(2) = 0.2404 Yn
=
I
0.2404Xn
+ 0.4958)
(2’ 2:0;::!
I
- 0.2404Xn-2 + 1.4808Ynp1
- O.4959Ynp2
FREQUENCY RESPONSE
0
-3
c
-12
-15
0
3
6
9
12 15 18 21 FREQUENCY(RADIANS/SEC)
24
27
FIG.85. Analog and digital implementationsof bandpass CMN filter.
30
217
DISTANCE MEASURING EQUIPMENT IN AVIATION
(a)
ao
480.0
240.0
960.0 12w.o 1440.0 SAMPLE NUMBER
720.0
1680.0
1920.0
2160.0
2400.0
1m.o 90.0 INPUT TO SLIDING WINDOW PROCESSOR ABSOLUTE VALUE OF CMN
Gi 80.0
:
7013
400 POINT/lO SE SLIDING W I N D 0
4
5
60.0
(b)
OUTPUTOF SLIDING WINDOW PROCESSOR: 95th PERCENTILE OF DATA IN WINDOW
50.0
5,
s 5
40.0
30.0
zao 10.0
0.0
ao
720.0
480.0
240.0
960.0
1440.0
12w.o
1680.0
2160.0
1920.0
2400.0
SAMPLE NUMBER
. ,
1w.o
,
,
,
I
,
,
1 95th PERCENTILE &j 00.0
z Z
70.0
Y
60.0
+
50.0
z Y
(C)
c
'
-
"
'
.
,
'
I
. J
.-
.-
.
5= 40.0 --
all.lrLL
I.VI.IDL"
FIG. 86. Determining the CMN component of DMEjP range error. (a) CMN determined by filtering interrogator range error with CMN filter. (b) Processing of CMN with 400 point (10 s) sliding window to determine 95th percentile of CMN data in window relative to 0 ns error. (c) Output of sliding window processor.
218
ROBERT J. KELLY AND DANNY R. CUSICK TABLE XX DME/P ACCURACY SPECIFICATTONS” (METERS-95% Standard 1
PROBABILITY) Standard 2
IA mode
Error source
Error component
PFE
CMN
PFE
CMN
PFE
CMN
Instrumentation
Transponder Interrogator RSS Subtotal Downlink specular multipath Uplink specular multipath Nonspecific (diffuse)multipath Garble (DME) Garble (non-DME) RSS Subtotal Combined instrumentation and site-related error components not allowed to exceed system accuracy standard
10 15 18 10 10 3 6 1 15.6
8 10 13
5 7 8.6
5 I 8.6
15 30 33
10 15 18
8 8 3 6 1 13.2
3 3 3 6 1 8.0
3 3 3 6 1 8.0
31 31 3 6 1 53
20 20 3 6 1 29
30
18
85
34
Site related
System
12
12
’The figuresfor “nonspecular multipath and for “garble” are the RSS totals of the uplink and downlink components. The pulse waveform is assumed to have a cos/cosz shape and a 1200-nsrise time. The error used for specular multipath is the peak value obtained with an MID = - 6 dB for IA mode and MID = - 3 dB for FA mode. Interrogator instrumentation error includes CMN component due to transponder reply efficiency. indicates that reasonable estimates can be obtained in this manner. In practice, the combination of site-related components with the instrumentation errors cannot exceed the overall system errors noted earlier in Subsection A,2. Thus, if the specified error components are not individually exceeded, it can be expected that the overall system performance will be achieved. 12.3 NM
--------_____
HORIZONTAL
FIG. 87. DME/P vertical coverage.
0.85’-
1-
DISTANCE MEASURING EQUIPMENT IN AVIATION
219
3. System Coverage and Power Budgets
A DME/P ground facility is required by the SARPs to provide range information throughout a coverage volume (in azimuth) that is at least as large as that provided by the MLS azimuth facility with which is paired. Typically, 360" DME azimuthal coverage to 20 nmi from the MLS reference datum will be provided. Vertical coverage is summarized in Fig. 87. Note that coverage is to be provided down to 8 ft above the runway surface, ensuring the availability of range information following touchdown. As is the case with DME/N, the signal in space to be provided by the transponder is specified in terms of power density (dB W/m2),thereby keeping the specification independent of transmitter, antenna, and cabling considerations. Specifications are similar for the transponder sensitivity requirements. Table XXI summarizes the relevant signal-in-space, sensitivity, and transmitter requirements for DME/P ground and airborne equipment. TABLE XXI SUMMARY OF GROUND AND AIRBORNE RADIATED POWERAND RECEIVER SENSITIVITI. REQUIREMENTS Transponder requirements (SARPs)" Uplink peak ERP at interrogator antenna
- 89 dB W/m2
At any point within the defined coverage volume beyond 7 nmi of transponder antenna At any point within the defined coverage volume within 7 nmi of transponder antenna At MLS approach reference datum At 8 ft above the runway surface at the MLS datum, or the farthest point on the runway centerline within line of sight of the antenna FA mode IA mode
- 75 dB W/m2 - I 0 dB W/m2 -79 dB W/m2
Downlink sensitivity at transponder antennab
-75 dB W/mZ -86 dB W/m2
Interrogator requirements (RTCA)' FA Transmitter power Receiver sensitivity at input terminal
250 W -73 dBm
1A 50 W
-73 dBm - 83 dB m
At 7 nrni range At 22 nmi range
SARPs requires interrogator performance to be consistent with ground system specifications. ' FAA (1983~)requires - 89 and - 79 dB W/m2 for the IA and FA modes, respectively. RTCA interrogator requirements assume a 2-dBi antenna gain and a 3-dB cable loss in determining receiver input power.
220
ROBERT J. KELLY A N D D A N N Y R. CUSICK TABLE XXII CTOL AIR-TO-GROUND POWERBUDGET (DME/P)
Power budget items Peak interrogator transmitter power (dB m). Aircraft antenna gain (dBi) Aircraft cable loss (dB) Peak effective radiated power (dBm) Ground and lateral multipath loss (dB) Path loss (dB) Polarization and rain loss (dB) Received signal at transponder antenna (dB m) Ground antenna gain (dBi) Pattern loss (dB) Cable loss (dB) Received signal at transponder (dB m) Receiver noise video (dB m) (NF = 6 dB) IF bandwidth is 3.5 MHz IF bandwidth 0.8 is MHz Signal-to-noise ratio (video) (dB)*
22 nmi
7 nmi
Reference datum
Rollout
54 0 -4 50 -5 - 125 -1 -81 8 -4 -3 - 80
54 0 -4 50 -3 -115
-69 8 -2 -3 - 66
54 0 -4 50 -4 - 107 0 -61 8 -5 -3 - 61
54 0 -4 50 - 17 - 103 0 - 70 8 -5 -3 - 70
- 106
- I06
- 106
40
45
36
- 112 32
-1
Power density (dB W/m2) = power (dB m)- 8.
* S/N(video) = S/N(IF) + 10log(IF noise bandwidth/video noise bandwidth). As noted earlier in Section 111,C,2,power budgets play a role similar to that of error budgets. They are a means to identify, quantify, and combine the ground and airborne equipment losses, propagation losses, and antenna characteristics to ensure that the up- and downlink signal in space is adequate to satisfy all relevant accuracy and system efficiency requirements. Tables XXII and XXIII are the DME/P CTOL power budgets as defined in AWOP (1982).The permitted ground system ERP is based on the use of an 800 ns cos/cos2 pulse. The calculations assume that the aircraft antenna is not shielded by the aircraft structure. In addition, free-space propagation loss is assumed. This is an accurate method of loss computation due to the short ranges (< 22 nmi) involved. However, the effect of ground multipath that results in signal fading is an important consideration that will be addressed in the next section. It is important to note that the receiver sensitivity (in the case of both ground and airborne equipment) is the minimum signal which satisfies both the system efficiency and accuracy requirements. However, in contrast to DME/N, the sensitivity of a given link is limited by the CMN component arising from the receiver noise rather than the system efficiency. This is clear from the discussions of Subsection B,3.
22 1
DISTANCE MEASURING EQUIPMENT IN AVIATION TABLE XXIII CTOL GROUND-TO-AIR POWERBUDGET (DME/P)
Peak effective radiated power (dB m)” Ground and lateral multipath loss (dB) Antenna pattern loss (dB) Path loss (dB) Monitor loss (dB) Polarization plus rain loss (dB) Received signal at aircraft (dB m) Aircraft antenna gain (dBi) Aircraft cable loss (dB) Received signal at interrogator (dB m) Receiver noise video (dB m) (NF = 9 dB) IF bandwidth is 3.5 MHz IF bandwidth is 0.8 MHz Signal-to-noise ratio (video) (dB)b ~
a
~
22
7
nmi
nmi
55 -5 -4 - 125 -1 -1 -81 0 -4 - 85
55 -3 -2 -115 -1 -1 -6 1
Reference datum
0 -4 -71
55 -4 -5 - 107 -1 0 - 62 0 -4 - 66
- 103
- 103
Rollout 55 - 17
-5 - 103
-1
0 -71 0 -4 - 75
- 103
- 109
32
24 ~
37 ~
28 ~
Power density (dB W/mZ) = power (dBm) - 8. S/N(video) = S/N(IF) + 10log(IF noise bandwidth/video noise bandwidth).
It is also important to note that the IA mode is not the limiting consideration; rather it is the FA mode rollout maneuver for which consideration must be given. This is primarily because the*-amplitude find (or PAF) circuit is utilized for pulse detection in the IA mode as opposed to the - 20 dB thresholding DAC utilized in the FA mode. Additionally, a low SNR results from the 22 dB signal attenuation due to ground multipath and antenna pattern behavior at the low coverage angle. 4 . DMEIP Antenna Considerations
The DME/P antenna should be located laterally as close to the runway centerline and near the MLS azimuth function antenna site as possible, consistent with obstacle clearance criteria. The relation of the distance L in back of the runway stop end where a DME antenna of height H must be sited is L = 61 + 50H,where L and H are in meters. Locating the DME/P antenna at the azimuth site simplifies the coordinate conversion computation when MLS is used to determine three-dimensional position information as in MLS area navigation operations. Additionally, a near centerline location keeps the range information meaningful when unconverted information is utilized to determine decision height and rollout distance.
222
ROBERT J. KELLY AND DANNY R. CUSICK TABLE XXIV SUMMARY OF SIGNAL ATTENUATION DUE TO GROUND MULTIPATH AS A FUNCTION OF PHASE-CENTER HEIGHT Phasecenter height (m) 4.6 9.1
Signal attenuation" (including ground multipath) 15.2 m
2.4 m (rollout)
Antennato-threshold distance (m)
- 8 dB -3.0 dB
-22 dB - 19 dB
3048 4513
Relative to free-space peak signal.
Siting of the DMEIP antenna in the vertical dimension is dependent on runway length, runway profile, and local terrain. These siting factors are particularly important when it is necessary to assure that adequate signal power density is available in the runway threshold region or near the runway surface. Table XXIV gives the expected attenuation for a 7 ft (2.1 m) vertical aperture onmi antenna. This subject and the antenna design are covered in more detail in Kelly and LaBerge (1980). D . DMEIP Signal Processing and Transponderllnterrogator Implementation considerations
The previous sections have identified all of the components required to successfully implement DME/P ground and airborne equipment as well as how these components affect system performance. This section is intended to unify these concepts by first summarizing the signal processing functions that are key to the DME/P and then illustrating the ground and airborne equipment configurations at the functional level. It must be emphasized that many of the signal components required for DME/P are not unlike those utilized in DMEIN, as described in Section I1 of this article. Hence, the discussions below will detail only those aspects which are unique to DMEIP.
I . Summary of DMEIP Signal Processing functions Several signal processing techniques are critical to the successful implementation of DME/P and are common to both the ground and airborne equipment. As in DMEIN, these functions are receiver bandwidth and dynamic range, signal identification and validation, and pulse detection.
DISTANCE MEASURING EQUIPMENT IN AVIATION
223
Although the signal processing circuits described in this section may appear to represent one hardware concept among many, it must be emphasized that these concepts are fundamental and necessary in one form or another to achieve the required accuracy in most approach and landing situations. Figure 88 summarizes all of the functional components by illustrating signal flow through the DME/P interrogator. Figure 89 summarizes the operational environment, with an emphasis on the pulse interference environment, which necessitates the signal processing concepts addressed below. a. Receiver Considerations- Bandwidth and Dynamic Range. In order to achieve the desired accuracy in the FA mode, the receiver must faithfully reproduce the sharp linear leading edge of the received pulse in the vicinity of the threshold point. To do so requires an IF bandwidth on the order of 4 MHz in contrast to the 400-800 kHz bandwidths typical of DME/N equipment. As noted in Subsection B,4, the use of wide-bandwidth processing and the delayattenuate-compare time-of-arrival detector ensures excellent multipath immunity. On the other hand, the exposure to noise and garble effects is increased. In the IA mode, an 800 kHz I F bandwidth is all that is required due to reduced accuracy requirements and a less stressful multipath environment. The narrow bandwidth, combined with the +-amplitude thresholding of the received pulse, results in better immunity to garble and receiver noise eirects. The dynamic range requirements of the DME/P airborne and ground receivers are no different from those defined for DME/N in Section III,B.
b. Signal ldentijcation and Validation. As in DME/N, the DME/P signal processing must identify and validate received DME signals before they are utilized to generate replies or for range measurement at the interrogator. The same characteristics of the DME signal, pulse width, rf frequency, and code, are used to perform this function. Recall that the purpose of the pulse-width discriminator is to discriminate a valid pulse from those due to noise which are of short duration. This function takes on a more important role in the implementation of the DME/P F A mode. In the presence of noise, the delay-attenuate-compare circuit will indicate many pulse arrivals. If a noise-induced pulse occurs near enough to a desired pulse (e.g., within the interrogator track gate or on the leading edge of the pulse itself), it may be misconstrued as a valid pulse and errors may result. This is, in part, due to the use of a wide-bandwidthIF and a low threshold level relative to the peak of the pulse. However, because of the wide receiver bandwidth, the average duration of the noise bursts are much shorter than the nominal 3.5 p s DME pulse width. Therefore, a pulse-width discriminator is
Combined Desired Signal with Interferance
Low Pan Equivalent DME Signal and Interference
Estimated Time of Arrival
,
IcL Zero Error Reference
m T Limiter
Data
PAF
-=a-
Valid Code
6of Estimate db Arrival Time
+
Decoder 12r2us
I
I A Mode Filter
IFBW800kHz Valid on.channel signal I A OutputbFA output
A transmitter interrogation sample turns the range counter "ON," The synchronized reply turns the counter "OFF." The number of counts is proportional to the range.
FIG.88. Interrogator signal processing functions.
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225
15
F A Mode Wideband 4.5 MHz Filter
FIG.89. DME/P interference environment.
employed to reject the noise pulses and identify the received DME pulse. A pulse-width criterion of 2 p s or more is sufficient to discriminate between valid DME pulses and noise-generated pulses in both the IA and FA modes. As in DME/N, only pulses that are on the assigned channel frequency must be considered for further processing. In the DME/P, the channel frequency separation of 1 MHz is less than the FA mode receiver bandwidth. Thus, pulses on at least the first and second adjacent frequencies will be indistinguishable in the receiver video from those on the desired frequency. Consequently, unlike DME/N, some variant of the Ferris Discriminator (see Section III,B) must be used not only in the ground transponder but in the airborne interrogator as well. A received signal which satisfies the pulse width and frequency tests must also have the proper pulse spacing for the operating code. Requirements for a pulse pair decoder are the same as those for DME/N equipments. Note that the DME/P ground transponder decoder is more complex than that utilized in conventional DME equipment. It must distinguish not only between channel codes (e.g., X and 2 )but also between IA and FA mode interrogation codes as well, ensuring the use of the proper processing when generating the reply. c. Pulse Time-of-Arrival Determination. As in DME/N, the arrival time of the DME pulse must be determined in both the airborne and ground equipment. As noted in Subsection B,2, pulse time of arrival in the FA mode is
226
ROBERT J. KELLY AND DANNY R. CUSICK
determined using the delay-attenuate-compare circuit, with nominal pulse thresholding 18 to 20 dB below the pulse peak for DAC parameters of 100 ns delay and attenuations of 5 to 6 dB. As in DME/N, the IA mode utilizes a *-amplitude find circuit to determine pulse time of arrival. 2. Transponder/InterrogatorImplementation Considerations
As is evident, the functional components of the DME/P transponder and interrogator are much like those required in DME/N and discussed earlier in Section II1,B. The major differences are the inclusion of the FA mode channel and the need for strict attention to the effects of each on system accuracy. a. Transponder. Functionally, the DME/P transponder, illustrated in Fig. 90, can be viewed as two separate transponders, one which responds to F A mode interrogations and one which responds to IA mode interrogations. However, in order to reduce cost, hardware implementations will most certainly integrate functions common to the IA and F A modes. For the X channel example shown, the IA mode transponder responds to the receipt of the 12 p s code. A 12 p s coded reply is transmitted 50 ps after the receipt of the interrogation. The reply delay timing is adjusted to 50 ps between the +-amplitude point on the leading edge of the first interrogation pulse to the same point on the leading edge of the first reply pulse. The FA mode transponder responds to interrogations having a code of 18ps. A reply is then generated 56 ps after the arrival of the first pulse with a reply code of 12 ps. The reply delay timing is adjusted to 56 p s between the virtual origin of the first interrogation pulse and the virtual origin of the reply pulse, as discussed in Subsection B,2. As in DME/N, the DME/P transponder must transmit squitter and the Morse code identification signal. Squitter is provided by the DME/P ground system to ensure compatibility with DME/N interrogators, some of which
TRANSMITTER
a
PRIORITY LOGIC
5011% DELAY
\
RECEIVER
DECODER X-CHANNEL I A MODE 1211s
FIG.90. DME/P transponder.
5611s DELAY
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227
utilize the squitter signal amplitude for performing automatic receiver gain control. The priority logic determines the type of transmission to be made when conflicts occur between the transmission of squitter pairs, ID pairs, and IA/FA mode replies. FA mode replies, due to their critical nature, will always take priority over all other ground system transmissions, while IA mode replies and squitter transmissions are suppressed during the identificationkey down time. An integral part of the transponder function is the need to ensure the PFE instrumentation accuracy of the FA and 1A mode reply delays in an environment of parameter variations with time and temperature. Tbis is accomplished in real time via automatic delay stabilization (ADS), a technique also utilized in modern DME/N ground transponders. The ADS function is performed by simply measuring the reply delay in both the FA and IA modes just as an interrogator measures range. Since the ADS is at zero range, the ADS measurement corresponds to the transponder delay associated with the assigned channel. If this estimate differs from the desired value, the transponder delay is adjusted to compensate accordingly. The ADS corrects all errors that vary slowly with time for a fixed signal level. It cannot, however, correct rapidly time varying errors which contribute to the transponder CMN component, such as receiver noise and clock quantization effects, nor can it correct for components which induce delays that are amplitude or rise-time dependent such as those which arise due to the log IF receiver. The key to understanding the transponder instrumentation error is the realization that the mean reply delay is the important parameter rather than the reply delay which occurs on a single interrogation. The reply delay on a single interrogation may deviate from the desired value, for reasons which will be discussed below. Such deviations are of two types: those which exhibit a slowly changing magnitude over a period of many interrogations and those which change on each interrogation. Those which change slowly over time contribute to the PFE component of the instrumentation error. Those which change with each interrogation contribute primarily to the CMN component of the error. Transponder instrumentation errors arise primarily from the following sources. ADS. The transponder reply delay is controlled by the ADS. The ADS measures the reply delay, estimates the mean reply delay, and corrects the reply delay accordingly. Errors in this correction result directly in errors in the transponder mean reply delay, ultimately translating to range errors as viewed from an interrogator. Hence, the accuracy of the transponder can be no better than the accuracy of the ADS error estimates and the resolution of reply delay corrections.
228
ROBERT J. KELLY A N D DANNY R. CUSICK
Reply Delay Clock Rate. The accuracy of the transponder delay is determined in part by the accuracy of the clock used to time this delay. Assuming that a digital clock is used, the reply delay will have a small random component which changes with each interrogation due to the quantization effectsof the clock, the magnitude of which is directly proportional to the rate. That is, the higher the clock rate used for reply delay timing, the smaller the quantization error. Log Amplijiers. Log amplifiers are recommended for use in each of the I F channels in order to provide the necessary dynamic range receiver. However, log amps typically exhibit an imperfect log-linear amplitude transfer curve in the form of amplitude ripples about the desired response curve. These ripples distort the pulse leading edge such that pulses of different amplitudes will exhibit different detection times. In contrast to DME/N, the magnitude of these deviations is of concern in the DME/P application. The error can be reduced either by appropriate design methods, which may be costly, or by utilizing received pulse amplitude information and compensating the reply delay according to a stored calibration table. IF Filter. When narrow-bandwidth filters are utilized, such as in the IA mode channel of the transponder, the delay variation with pulse rise time can be significant. Specifically, rise-time variations across the population of interrogators must be considered in the design if the received pulse rise time can be expected to be different from that used for system calibration. For example, a five-pole 800 kHz bandwidth Butterworth filter will have an 11 ns delay variation for input pulses having rise times between 800 and 1200 ns, the range of rise-time values permitted by interrogator specifications. Bessel filter implementations will have negligible delay variations. Since a wide bandwidth is utilized in the FA mode, delay variation with pulse rise time (within the range allowed by the specifications)is not a concern. Finally, other factors, such as circuit parameter drifts in the rf, IF, and digital circuitry, as well as the reply delay clock oscillator frequency, potentially affect the transponder accuracy. However, such slowly varying drifts are compensated by the ADS.
b. Transponder Monitor. Since the DME/P transponder is utilized during critical approach and landing maneuvers where the quality of the data directly impacts flight safety, the importance of a reliable, fail-safe monitoring system cannot be over emphasized. The structure of the DME/P monitor parallels that of the DME/N monitor described in Section II1,B. However, it must monitor both the IA and F A modes of the ground system. The primary parameters reflect the status of the system accuracy and coverage and are the mean reply delay (PFE), the
DISTANCE MEASURING EQUIPMENT IN AVIATION
229
radiated power, the reply efficiency, and the receiver decoder aperture. As in DME/N, a primary parameter that exceeds its predetermined limit causes the transponder to be removed from service, or redundant circuitry (if available)is utilized to keep the system in an operational mode. Such transfers or shutdowns must occur in less than 1 s to minimize the time that erroneous guidance is provided to aircraft utilizing the DME/P ground facility. The facility may remain on the air in the IA mode in the event that the failure is isolated to FA mode circuitry. Secondary parameters are those for which a maintenance action only is required and are pulse-pair output rate, pulse partial rise time, and CMN. Since these parameters are not critical to flight safety, only an indication to the ground facility need be provided in the event that a parameter exceeds its limits. c. Interrogator. Aside from the inclusion of two operating modes and improved accuracy, the DME/P interrogator functions in the same way as its DME/N counterpart. It must (1)identify and validate received pulse pairs; (2) derive range measurements from replies which are synchronous with the interrogations; and (3) process the measured range data for output to displays and the aircraft flight control system. Identification and validation of received pulse pairs is identical to the techniques utilized in the ground transponder. Identification of valid pulse pairs that are synchronous with interrogations is identical to the process used in DME/N and described in Section 111,B,3.The raw range measurements are filtered by an appropriate data filter that reduces the effects of the measurement errors due to instrumentation, noise, multipath, and garble and accommodates missing data conditions for system efficiencies down to 50%. This filtering of the range data is typically not done in DME/N airborne systems and is therefore unique to DME/P applications. The error mechanisms in the DME/P interrogator are identical to those noted earlier for the transponder, with the exception of the automatic delay stabilization function. The ADS is replaced by a similar concept where timing is begun in the interrogator by the interrogation pulse, which is processed using the same circuitry used to process the replies. In this way, slowly varying circuit parameters common to both paths are compensated. However, amplitude and rise-time-dependentdelay variation cannot be so compensated and must be handled separately.
d. Signal Acquisition. For the IA and F A mode, the interrogator must acquire and validate the reply signal within 2 s. After loss of the tracked signal, the interrogator must provide a suitable warning after 1s for the FA mode and 10 s for the IA mode. Signal acquisition is defined to be based on the IA mode.
230
ROBERT J. KELLY AND DANNY R. CUSICK
For DME/P, a more sophisticated range correlation and acquisition scheme is required than that described for DME/N. Not only must the interrogator acquire and track in the IA mode, it must automatically begin the transition to FA mode processing 8 nmi from the transponder. To accomplish this transition, the interrogator must acquire confidence in the FA mode while simultaneously tracking in the IA mode. Once FA mode confidence is acquired, FA and IA mode data are blended to prevent discontinuities in the range output due to differing bias errors in the two modes. That is, an increasing proportion of FA mode data is used to generate the output range information. Once the interrogator moves within 7 nmi of the transponder, the IA mode interrogations are stopped and only the FA mode is used to ge:ierate range information. In the absence of FA mode data, tracking may continue in the IA mode provided that suitable indication of this condition is provided to the pilot and aircraft systems utilizing the DME/P range data. e. Interrogator Data Processing Algorithm. The SARPs accuracy specifications, as well as the error budgets discussed in Subsection C,2, assume that the higher-frequencynoise contributions are limited by an interrogator output data filter. Depending upon the user’s application, additional filtering for noise reduction can be used provided the dynamic response of the aircraft’s flight control system is not adversely affected. For a linear filter, the noise reduction factor is given by = (2 x
sample rate (output filter noise bandwidth)
>”’
(40)
and is applicable when the noise spectrum is flat from dc to the sampling frequency. The 2a range noise level is thus reduced, by filtering, to l/g of its initial value. Large noise reduction factors correspond to narrow-bandwidth filters. For a single-pole filter, the noise bandwidth is n/2 x (3 dB bandwidth). For example, a 10 rad/s single-pole filter utilizing a 40 Hz sample rate has a noise reduction factor of 2.8 and a phase delay of 4” at 1.5 rad/second. In addition to data smoothing with adequate transient responses, the data processing algorithm should perform two other functions that will enhance the quality of the range data: prediction of missing data samples and limiting of spurious measurement values (e.g., those due to garble) which would otherwise deviate significantly from their predicted values. When the system efficiency is less than loo%, replies will not be received for all interrogations. To “coast” through periods of missing data, the data filter algorithm replaces the missing samples with predictions based on past position and velocity data. A filter with velocity memory is the a-j? filter (Kelly and LaBerge, 1980; Benedict and Bordner, 1962).An a-j? filter which satisfies CTOL FA mode applications has a = 0.125 and j? = 0.0078. The noise
DISTANCE MEASURING EQUIPMENT IN AVIATION
23 1
reduction factor for this filter is 3.3 and it will satisfy the “not-to-exceed aircraft flight control phase delay requirement of 6” (i.e., 4.6” at 1.5 rad/s). Assuming that the output of the data filter is close to the true range, significant deviations (greater than 300 ns) from the predicted range are then physically unrealistic in terms of aircraft response and can therefore be removed. This process is called “outlier rejection.” If the rejected sample is replaced by a predetermined value, the entire algorithm is called a “slew-rate limiter.” The outlier window width and the slew-rate limit value are determined by the error budget associated with the user application and the data filter noise reduction factor. Acceptable values are a 300 ns outlier limit with a 40 ns slew-rate limit. These parameters, together with the above specified a-8 filter coefficients, permit the DME/P to accurately track all CTOL maneuvers. Figure 91 summarizes the a-B filter algorithm which incorporates outlier rejection and slew-rate limiting.
f. Interrogation Rate Jitter. It is common practice in DME to “jitter” the time between interrogations to preclude the occurrence of synchronous interference between interrogating aircraft. This jittering process has the additional effect of reducing the aliased spectrum peaks of the range error frequency response such that the sampled frequency components outside the filter pass band are uncorrelated. This means that sinusoidal errors, such as those that can be generated by multipath having nonzero scalloping frequency,appear noiselike to the filter and are therefore attenuated according to the filter noise reduction factor given in Eq. (40). A jitter value of f50% of the intersample time is adequate to remove the aliased peaks, as illustrated in Fig. 92. E. System Field Tests
The feasibility of the DME/P concept was confirmed by a comprehensive FAA field test program. Figure 93 is an example flight error trace recorded in June, 1980, as part of this test program. It illustrates many of the accuracy concepts discussed earlier. The ground transponder was a modified Cardion conventional 100 W terminal area DME. The airborne interrogator was a modified version of a Bendix enroute DME. This equipment constituted the range element of the FAA’s basic wide MLS which was contracted to Bendix. The ground station was installed on Runway 22 at Wallops Station Flight Center, Chincoteague, Virginia, December, 1979. A laser ground tracker was used within 5 nmi of the runway threshold, and a beacon tracker was used at ranges greater than 5 nmi. The airborne antenna used on the FAA’s Convair CV-880 was a standard L-band omni type.
232
i f (data sample available) then
Fk = uk - Yk else
4
(missing data) then
if
YK
Fk = 0 end i f
A i'k I
--- - -- _+_a % ? _
__
&--
-
OUTLIER REJECTION AND SLEW R A T E LIMITER
---
--_
41
yk
Uk
~
RAN GEMEASUR EMENT
Ek
=
DIFFERENCE BETWEEN MEASURED A N D PREDICTED RANGE AUGMENT ED BY OUT LlER iSLEW R AT EiMlSSlNG D A T A ALGORITHM
Vk
=
VELOC IT Y ESTIMATE
Xk
=
POSITION ESTIMATE
Yk
=
PREDICTED POSITION
a,C
=
F ILT ER COEFFICIENTS
T
-
INTER-SAMPLE TIME
la1
=
OUT LIER REJECTION L I M I T
Ibl
~
SLEW R AT E L I M I T
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(2-a) (i-aI2 i a2iz-al+zP i i - a i ~ap re-p-zai T~ a0 i4-0-2ar FIG.91. Block diagram of a-B filter with outlier rejection, slew-rate limiting, and missing data capability. The optimal filter provides the maximum noise reduction while providing the best transient tracking capability to a unit ramp input.
DISTANCE MEASURING EQUIPMENT IN AVIATION
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A detailed description of these field tests, which verified the validity of the precision mode concept, is found in Edwards (1981). F. Conclusion This article has described the internationally standardized radio aid to navigation known as “Distance Measuring Equipment” and its evolving role in support of air navigation in the world today. Based on the beacon concept developed in the 1940s, DME was first standardized as an international system by ICAO in the early 1950s.In conjunction with VOR and TACAN, DME has since become one of the primary navigation aids in the U.S. The most recent revision of the DME standard provides for two forms of the DME system. The DME/N serves the needs of the enroute navigation, while the newly adopted DME/P standard provides the high data rates and accuracy necessary for approach/landing operations in conjunction with MLS. VOR and TACAN, together with DME, have defined the basic air route structure in the U.S. with aircraft flying station to station along inter-
ROBERT J. KELLY AND DANNY R. CUSICK
234
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connecting radials. To put the DME system in its proper perspective, this article has described the enroute navaids, VOR and TACAN, and the landing aids, ILS and MLS, showing the role of the DME as an integral component of these systems. With the availability of economical airborne computing facilities, aircraft need no longer fly along VOR radials. Instead, with an RNAV computer, arbitrary flight paths are possible. In an RNAV environment, the navigation sensor data, be it DME, VOR, or TACAN, are used to compute the aircraft's position. The superior position fix performance of a multi-DMEbased RNAV clearly emerges from the alternative position fixing techniques based on VOR and TACAN. The basic concepts of aircraft navigation, guidance, and control were introduced by defining how the information from the navigation sensors is utilized to execute these functions.In doing so, the characteristics of the sensor
DISTANCE MEASURING EQUIPMENT IN AVIATION
235
output data that affect the performance of an aircraft in both manual and autocoupled flight were defined. This article has described the conventional DME system-DME/Ndetailing its primary components and its evolution. The most important aspects of the DME/N signal in space, accuracy, coverage, the channel plan, and service capacity, were discussed. The standardization of DME/N has been achieved through the definition of a handful of system parameters, particularly the pulse detection and calibration methods. It has been shown that, through a combination of equipment characteristics, namely frequency rejection, decoder rejection, and channel assignment rules, an adequate number of facilities can be accommodated without mutual interference to serve the enroute, terminal, and approach/landing navigation needs of an expanding and dynamic aviation industry. Finally, this article addressed the newest member of the DME family, the DME/P. DME/P, which provides slant range accuracy on the order of 100 ft at runway threshold, is to be an integral part of the MLS. Only with DME/P can the full capabilities of MLS (i.e., full 3D guidance within the airport environment) become reality. The newly defined ICAO standard represents the successful integration of the precision DME function with that of DME/N. Its signal format provides for full interoperability between DME/N and the initial approach mode of DME/P, eliminating the need for dual equipment on the ground and in the air. In addition, the DME/P waveform satisfies the strict spectral requirements imposed on ground facilities and therefore minimizes frequency compatibility issues. The discussions have emphasized those factors which make DME/P a viable system, even in an environment characterized by strong multipath, 50% missing data, and an ever-increasing number of undesired pulses. Careful selection of the pulse shape (cos/cos2),the IF bandwidth (4 MHz), and the pulse detection technique (delay-attenuate-compare), ensures immunity to multipath in nearly all foreseeable runway environments. Use of even simple data filter algorithms, incorporating such features as outlier rejection and slew-rate limiting, will minimize the effects of missing data and accuracy degradations due to the presence of undesired pulses. Finally, the prevalence of high-speed digital technology has rendered obsolete the analog and mechanical timing/tracking techniques of the past, ensuring the availability of economicalground and airborne equipment exhibitinga 10-foldimprovement in instrumentation error. When combined, these factors yield a precision DME system that is compatible with its DME/N counterpart and that provides the accuracy required to support complex operations in the terminal area, including automatic landings. In summary, the DME system is one whose capability and utility will well serve the aviation community into the next century.
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ROBERT J. KELLY AND DANNY R. CUSICK
LISTOF ACRONYMS AiC AFCS AGC ARINC ASR ATA ATARS ATC ATCRBS AWOP
Aircraft Aircraft flight control system Automatic gain control Aeronautical Radio, Inc. Airport surveillance radar Air Transport Association Automatic Traffic Advisory and Resolution Service Air traffic control Air traffic control radar beacon system All Weather Operations Panel
CMN
Control motion noise
Diu DAC DME DOD DOT
Desired-to-undesired signal ratio Delay-attenuate-compare Distance measuring equipment Department of Defense Department of Transportation
EUROCAE
European Organization for Civil Aviation Electronics
FA mode FAA FAR
Final approach mode Federal Aviation Administration Federal Air Regulation
GPS
Global positioning system
IA mode ICAO IF IFR ILS INS
Initial approach mode International Civil Aviation Organization Intermediate frequency Instrument flight rules Instrument landing system Inertial navigation system
Mi0 MLS Mode S
Multipath-to-direct signal ratio Microwave landing system A digital data link system (formerly DABS)
NAS NOTAMs
National airspace system Notices to Airmen
PAF PFE
Peak amplitude find Path following error
rf RNAV RSS RTCA
Radio frequency Area navigation Root-sum-square Radio Technical Commission for Aeronautics
SARPs SNR SSR
Standards and recommended practices Signal-to-noise ratio Secondary surveillance radar Standard service volume
ssv
DISTANCE MEASURING EQUIPMENT IN AVIATION TACAN TCAS
Tactical air navigation system Traffic alert and collision avoidance system
VFR VOR VORTAC
Visual flight rules Very high frequency omnirange transmitters A TACAN colocated with a VOR station
237
ACKNOWLEDGMENTS The authors wish to dedicate this chapter to Jonathan and Anne Kelly and Woodrow and Christopher Cusick. We wish to thank C. LaBerge, W. Reed, and A. Sinsky of the Allied Corporation-Bendix Communications Division for their helpful technical comments and G. Jensen and K. Jensen for their much-needed editorial comments. Finally, we would like to thank M. Morgan and M. ODaniell for their efforts preparing the manuscript and Dick Neubauer for the excellent graphics.
REFERENCES Aeronautical Radio, Inc. (ARINC) (1982). “Mark V Airborne Distance Measurement Equipment -Characteristic 709- 5.” ARINC, Annapolis, Maryland. All Weather Operations Panel (AWOP) (1980).“Report of the All Weather Operations Panel8th Meeting,” Doc. 9310, AWOP/8. ICAO, Montreal, Canada. A11 Weather Operations Panel (AWOP) (1982). “Report of the All Weather Operations Panel9th Meeting,” Doc. 9400, AWOP/9. JCAO, Montreal, Canada. All Weather Operations Panel (AWOP) (1984). “Report of the All Weather Operations Panel10th Meeting,” Doc. 9449, AWOP/lO. ICAO, Montreal, Canada. Bendat, J., and Piersol, A. (1971). “RANDOM Data; Analysis and Measurement Procedures.” Wiley (Interscience), New York. Benedict, T., and Bordner, G. (1962). Synthesis of and optimal set of radar track-while-scan smoothing equations. IRE Trans. Autorn. Control AC-7,No. 4. Bennett, J., and Hung, J. (1971). Applications of statistical techniques to landmark navigation. J . Inst. Nauig. 17, No. 4. Bleeg, R., et al. (1972).“Inertially Augmented Automatic Landing System: Autopilot Performance with Imperfect ILS Beam,” FAA Final Rep. No. FAA-RD-72-22. Federal Aviation Admin., Washington, D.C. Bobick, J., and Bryson, J. (1970). Updating inertial navigation systems with VOR/DME information. AIAA. 11, No. 10. Braff, R. (1966). “VORTAC Error Analysis for Helicopter Navigation: New York City Area,” FAA Rep. No. RD-66-46. Federal Aviation Admin., Washington, D.C. Braff, R. (1972).The effect of geometry on area navigation system errors. J. Znst. Nauig. 19, No. 1. Braff, R., Shively, A,, and Zeltser, M. (1983). Radio navigation system integrity and reliability. Proc. ZEEE 71, No. 10. Britting, K. R. (1971).“Inertial Navigation System Analysis.” Wiley (Intercience), New York.
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ROBERT J. KELLY AND DANNY R. CUSICK
Brown, R. (1973). Integrated navigation systems and Kalman filtering: A perspective. J . Inst. Nauig. 19, No. 4. Brown, R. (1983). “Introduction to Random Signal Analysis and Kalman Filtering.” Wiley, New York. Bryson, A., and Bobick, J. (1972). Improved navigation by combining VOR/DME information and air data. J. Aircr. 9, No. 6. Burt, A,, et al. (1965). “Mathematical Considerations Pertaining to the Accuracy of Position Location and Navigation Systems,” AD-629-609. Stanford Res. Inst., Menlo Park, California. Busignies, H., Adams, P., and Colin, R. I. (1946). Aerial navigation and trafiic control with navaglobe., NAVAR, Navaglide, and Navascreen. Electr. Commun. 23(2). Carel, 0.(1981).“Proposal for a DME/P Employing a Two Pulse Format,” ICAO All Weather Operations Panel-System Concepts Subgroup, Working Pap. No. M-10. ICAO, Munich, West Germany. Chance, B., Hughes, V., MacNichol, E., Sayne, P., and Williams, F. (1949). “Electronic Time Measurements,” MIT-Radiat. Lab. Ser., Vol. 20. McGraw-Hill, New York. Colin, R. I., and Dippy, R. J. (1966). The hyperbolic navigation system. IEEE Trans. Aerosp. Electron. Syst. AESZ, 475-487. Cusick, D. R. (1982). “A Microprocessor-Based Precision DME Interrogator for the Microwave Landing System,” Proc. Micro-Delcon ’82, University of Delaware. IEEE Computer Society Press. DeJonge, M. (1985).“Flight Management Systems: Their Role is Steadily Increasing,” ICAO Bull. ICAO, Montreal, Canada. DelBalzo, J. (1970). Inertial systems and area navigation in the U.S. domestic airspace. J. Inst. Nauig. 17, No. 4. Department of Defense/Department of Transportation (1984). “Federal Radionavigation Plan,” DOT-TSC-RSPA-84-8. DOD/DOT, Washington, D.C. Dodington, S. H. (1949). Crystal control lo00 MHz for aerial navigation, Electr. Commun. 26(4). Dodington, S . H. (1980). Development of la00-MHz distance measuringequipment. IEEE Trans. Aerosp. Electron. Syst. AES16(4). Dodington, S. H. (1984). Civil navigation aid in ITT. Electr. Commun. 58, No. 3. Edwards, J. (1981). “DME/P Flight Test Data,” ICAO All Weather Operations Working Group M-4, Background Inf. Pap. 2, ICAO, Paris. European Organization for Civil Aviation Electronics (EUROCAE) (1980). “Minimum Performance Specification for Ground Distance Measuring Equipment,” ED-31. EUROCAE, Paris. Evans, J. E. (1982). “L-Band DME Multipath Environment in the MLS Approach and Landing Region,” Proj. Rep. ATC-116. MIT Lincoln Laboratories, Cambridge, Massachusetts. Evans, J. E., and Swett, P. H. (1981).“Results of L-Band Multipath Measurements at Operational U.S. Airports in Support of the MLS DME-P,” Rep. ATC-109. MIT Lincoln Laboratories, Cambridge, Massachusetts. Farrell, J. (1976). “Integrated Aircraft Navigation.” Academic Press, New York. Feather, J. B. (1985). “Guidance Studies for Curved, Descending approaches using MLS,” Douglas Aircraft Company, Rep. No. Cl-E87-TN-386, Long Beach, California. Federal Aviation Administration (FAA). (1985). “Airman’s Information Manual. Part 1. Basic Flight Manual and ATC Procedures (issued quarterly). U.S. Govt. Printing Office, Washington, D.C. Federal Aviation Administration (FAA) (1966). “Technical Standard Order: Airborne Distance Measuring Equipment, TSO-C66A, Part 37 of Federal Aviation Regulations.” FAA, Washington, D.C.
DISTANCE MEASURING EQUIPMENT IN AVIATION
239
Federal Aviation Administration (FAA) (1970). “Criteria for Approving Cat. I and Cat. I1 Landing Minimum for FAR 121 Operations,” FAA Advis. Circ. 120-29. FAA, Washington, D.C. Federal Aviation Administration (FAA) (1971).“Automatic Landing System,” FAA Advis. Circ. 20-57A. FAA, Washington, D.C. Federal Aviation Administration (FAA) (1973). “Criteria for Approval of Cat. I11 A Landing Weather Minima,” FAA Advis. Circ. 120-28A. FAA, Washington, D.C. Federal Aviation Administration (FAA)(1975a).“Approval of Area Navigation Systems for use in the US. National Aerospace System,” AC-90-45A. FAA, Washington, D.C. Federal Aviation Administration (FAA) (1975b).“Flight Procedures and Airspace,” FAA Handb. 8260.19. FAA, Washington, D.C. Federal Aviation Administration (FAA) (1976).“United States Standard for Terminal Instrument Procedures (TERPS),” FAA Handb. 8260.3B, 3rd ed. FAA, Washington, D.C. Federal Aviation Administration (FAA) (1978). “VORTAC Equipment Replacement and Facility Modernization,” FAA-B2678A. FAA, Washington, D.C. Federal Aviation Administration (FAA)(1980). “Frequency Management Engineering Principles: Geographical Separation for NAVAID Frequency Assignments,” Order 6050.5B. FAA, Washington, D.C. Federal Aviation Administration (FAA)(1981).“Operation Free Flight,” FAA Rep. No. FAA-AT81-1. FAA, Washington, D.C. Federal Aviation Administration (FAA) (1982). “US. National Standard for the Very High Frequency Omnidirectional Radio Range (VOR)/Distance Measuring Equipment/Tactical Air Navigation (TACAN) Systems,” AC 00-31A. FAA, Washington, D.C. Federal Aviation Administration (FAA) (1983a). “United States Standard Flight Inspection Manual,” 3rd ed. FAA, Washington, D.C. Federal Aviation Administration (FAA) (1983b). “Microwave Landing System Interoperability and Performance Requirements,” FAA-STD-022b. FAA, Washington, D.C. Federal Aviation Administration (FAA) (1983~). “Microwave Landing System Ground Equipment/Precision Distance Measuring Equipment (DME/P),” FAA-E-2721/3b. FAA, Washington, D.C. Federal Aviation Administration (FAA)(1983d).“Random Area Navigation Routes,” FAA Advis. Circ. AC 90-82. FAA, Washington, D.C. Federal Aviation Administration (FAA) (1984). “Procedures for Handling Airspace Matters,” FAA Handb. 7400.2C. FAA, Washington, D.C. Flint, R. B., and Hollm, E. (1965). VOR evolutionary system improvements in the US. IEEE Trans. Aerosp. Nauig. Electron. ANEIZ(1). Fried, W. (1974). New developments in radar and radio sensors for aircraft navigation. IEEE Trans. Aerosp. Nauig. Electron. AESIO( 1). Frisbie, F. (1976). “MLS Performance in Rising Terrain,” London BIP No. 12 AWOP Working Group A, ICAO. Gelb, A,, and Sutherland, J. R. (1971). Software advances in aided inertial navigation systems. J . Inst. Nauig. 17, No. 4. Goemaat, M., and Firje, E. (1979). Analysis of VOR/DME signal coverage over the continental United States. J. Inst. Nauig. 26, No. 2. Graham, D., and Lothrop, R. C. (1955). Automatic feedback control and all-weather flying. Aeronaut. Eng. Rev., October. Groginsky, H. (1959).Position estimation using only multiple simultaneous range measurements. IRE Trans. Aerosp. Nauig. Electron. ANE-6(3). Gum, S., and Walters, B. (1982).“An Invitation to Fly: Basics for the Private Pilot.” Wadsworth Publishing, Belmont, California.
240
ROBERT J. KELLY AND DANNY R. CUSICK
Hazeltine Corporation (1972). Sperry Rand Report on “MLS Data Format: Analog Computer Studies,” Rep. 10926, DOT-FA72WA-3804. Hazaltine Corporation, Greenlawn, New York. Hensler, T. C. (1984). “Upgraded MLS Channel-Assignment Model.” Electromagnetic Compatibility Analysis Center/IIT Res. Inst., Annapolis, Maryland. Higgins, W. (1975). A comparison of complementary and Kalman filtering. IEEE Trans. Aerosp. Electron. Syst. AES-11, No. 3. Hogle, L., Markin, K., and Bradley, J. (1983). Navigation systems performance versus civil aviation requirements. Proc. IEEE 71, No. 10. Honold, P. (1971).“Secondary Radar.” Heydon & Son, New York. Hopkins, J. (1981). Integrated satellite navigation and strapdown attitude and heading reference for civil air carriers. J. Inst. Nauig. 28, No. 3. 1981. Hsu, D. (1979). Long-tailed distributions for position errors in navigation. Appl Stat. 28, No. 1. Huddle, R. (1983).“Inertial Navigation System Error Model Considerations in Kalman Filtering Applications.” Control and Dynamic Systems, Vol. 20. Academic Press, New York. Hurley, H. C., Anderson, S. R., and Keary, H. F. (1951). The CAA VHF omnirange. Proc. I R E 39(12). Institute of Electrical and Electronics Engineers (IEEE) (1973). Special issue on aeronautical communications. I E E E Trans. Commun. COM-21, No. 5. International Civil Aviations Organization (ICAO) (1947). “Final Report-First Session,” Special Radio Technical Division of the Provisional ICAO. ICAO, Montreal, Canada. International Civil Aviations Organization (ICAO) (1949). “Standards and Recommended Practices for Telecommunications and Radio Aids to Air Navigation.” ICAO, Montreal, Canada. International Civil Aviations Organization (ICAO) (1972a). “Manual on Testing of Radio Navigation Aids,” Doc. 8071,3rd ed., Vol. 1. ICAO, Montreal, Canada. International Civil Aviations Organization (ICAO) (1972b). “International Standards and Recommended Practices- Aeronautical Telecommunications Annex 10,” 3rd ed., Vol. 1. ICAO, Montreal, Canada. International Civil Aviations Organization (ICAO) (1981a). Microwave landing system (MLS) characteristics. In “International Standards and Recommended Practices- Aeronautical Telecommunications Annex 10,” 3rd ed., Vol. 1, Sect. 3.11, Chapter 3. ICAO, Montreal, Canada. International Civil Aviations Organization (ICAO) (1981b).“Report of the ICAO Communications Divisional Meeting,” Doc. 9341. ICAO, Montreal, Canada. International Civil Aviations Organization (ICAO) (1982). “Procedures for Air Navigation Services, Aircraft Operations,” Doc. 8168-0PS/611, 2nd ed., Vol. 2. ICAO, Montreal, Canada. International Civil Aviations Organization (ICAO) (1985).UHF distance measuring equipment. I n “International Standards and Recommended Practices- Aeronautical Telecommunications Annex 10,” 4th ed., Vol. 1, Sect. 3.5, Chapter 3. ICAO, Montreal, Canada. James, H., et al. (1947). “Theory of Servomechanisms,” MIT-Radiation Lab. Ser., Vol. 25. McCraw-Hill, New York. Jensen, R. (1976). Effects of 3-D RNAV approach procedural complexity. J. Inst. Nauig.22, No. 2. Johnson, M. E., and Gierhart, G. D. (1978a). “The IF-77 Electromagnetic Wave Propagation Model,” DOT/FAA/ES-83. US. Department of Commerce, Institute for Telecommunication Sciences, Boulder, Colorado. Johnson, M. E., and Gierhart, G. D. (1978b). “Applications Guide: Propagation and Interference Analysis Computer Programs (0.1 to 20 GHz),” FAA-RD-77-60. U.S. Department of Commerce, Institute for Telecommunication Sciences, Boulder, Colorado. Johnson, M. E., and Gierhart, G. D. (1980).“An Atlas of Basic Transmission Loss for 0.125 to 15.5
DISTANCE MEASURING EQUIPMENT IN AVIATION
24 1
GHz,” FA-RD-80-1. US. Department of Commerce, Institute for Telecommunication Sciences, Boulder, Colorado. Johnson, M. E., and Gierhart, G. D. (1983). “Computer Programs for Air/Ground Propagation and Interference Analysis(0.1 to 20 GHz),”FAA-RD-73-103.U.S. Department of Commerce, Institute for Telecommunication Sciences, Boulder, Colorado. Karatsinides, S., and Bush, R. (1984). “Navigation Processing of the Flight Management Computer System for the Boeing 737-300,” Proc. 40th Annu. Meet. Institute of Navigation, Cambridge, Massachusetts. Kayton, M., and Fried, W. (1969). “Avionics Navigation Systems.” Wiley (Interscience), New York. Kelly, R. J. (1963).“A New Light-Weight DME,” Proc. 10th Annu. East Coast Conf. IEEE Aerosp. Navig. Electron. Kelly, R. J. (1974). “System Design and Flight Test Results of the Bendix/Bell CAT II/III Elevation Approach Guidance Function,” Proc. AIAA Mech. Control Flight Conf., AIAA Pap. 74-909. Am. Inst. Aerosp. Admin., Anaheim, California. Kelly, R. J. (1976) Time reference MLS multipath control techniques. J. Inst. Naoig. 23,No. 1. Kelly, R. J. (1977). Guidance accuracy considerations for the microwave landing system. J . Inst. Naoig. 24, No. 3. Kelly, R. J. (1984). System considerations for the new DME/P international standard. IEEE Trans. Aerosp. Electron. Syst. AE-520,No. 1, 1-24. Kelly, R. J., and LaBerge, E.F.C. (1980). Guidance accuracy considerations for the microwave landing system precision DME. J. Inst. Naoig. 27,No. 1. Kelly, R., Redlien, H., and Shagena, J. (1978). Landing aircraft under poor conditions. IEEE Spectrum 15(9). Koshar, A,, and Smithmeyer, J. (1982). “MLS Channel Plans and Traffic Loading,” DOT/FAA/RD-81-113. Federal Aviation Admin., Washington, D.C. Kuo, B. (1962). “Automatic Control Systems.” Prentice-Hall, Englewood Cliffs, New Jersey. Latham, R. (1974). Aircraft positioning with multiple DME. J. Inst. Nauig. 21, No.2. Latham, R., and Townes, R. (1975).DME errors. J. Inst. Nauig. 22,No. 4. McRuer, D., and Johnson, W. (1972).“Development of Approach Control System Requirements with Application to a Jet Transport,” NASA CR-2023. Natl. Aeronaut. Space Admin., Washington, D.C. McRuer, D., et al. (1973). “Aircraft Dynamics and Automatic Control.” Princeton Univ. Press, Princeton, New Jersey. Maybeck, P. (1979). “Stochastic Models, Estimation, and Control,” Vol. 1. Academic Press, New York. Mercer, J. (1954). A quantitative study of instrument approach. J . R. Aeronaut. Soc. 58, No. 518. Mertikal, J., et al. (1985).Treatment of navigational accuracies: Proposals for the future. J. Insr. Nauig. 32, No. 1. Milliken, R., and Zoller, C. (1978). Principles of operation of NAVSTAR and system characteristics. J. Inst. Nauig. 25,No. 2. Moseley, F., and Watts, C. (1960). Historic paper on automatic radio flight control. IRE Trans. Aeronaut. Nauig. Electron. ANE-’I(1). National Telecommunication and Information Administration (NTIA) (1984). “Table of Frequency Allocations and Other Extracts From: Manual of Regulations and Procedures for Federal Radio Frequency Management.” U.S. Department of Commerce, Washington, D.C. Neal, G. (1975) MLS-navigation, guidance, and control. J. Inst. Naoig. 22,No. 4. Office of Technology Assessment (OTA) (1982).“Airport and Air Traffic Control System,” DTAST1-175. OTA, Washington, D.C.
242
ROBERT J. KELLY AND DANNY R. CUSICK
Office of Telecommunications (1978). “EMC Analysis of JTIDS in the 960-1215 MHz Band,” Vols. 1-7. US. Department of Commerce/Office of Telecommunications, Washington, D.C. Pierce, J. A. (1948). Electronic aids to navigation. Adu. Electron. 1,425. Porter, W. A., ODay, J., Scott, R., and Volq R. (1967). “Modern Navigation Systems,” Engineering Summer Course. University of Michigan, Ann Arbor. Quinn, G. (1983). Status of area navigation. J. Inst. Nauig. 30. Radio Technical Commission for Aeronautics (RTCA) (1946). “Report of SC-21 Meetings.” RTCA, Washington, D.C. Radio Technical Commission for Aeronautics (RTCA) (1948a).“Air Traffic Control,” RTCA DO12, Spec. Comm. 31. RTCA, Washington, D.C. Radio Technical Commission for Aeronautics (RTCA) (1948b). “Final Report of SC-40,” Doc. 121/48/CO 24. RTCA, Washington, D.C. Radio Technical Commission for Aeronautics (RTCA) (1963). “Standard Performance Criteria for Autopilot/Coupler Equipment,” Doc. DO-1 18 (SE-79). RTCA, Washington, D.C. Radio Technical Commission for Aeronautics (RTCA) (1978). “Minimum Performance Standards Airborne DME Equipment Operating Within the Radio Frequency Range of 960-1215 MHz,” Doc. DO-151A. RTCA, Washington, D.C. Radio Technical Commission for Aeronautics (RTCA) (1982). “Minimum Operational Performance Standards for Airborne Area Navigation Equipment Using VOR/DME Reference Facility Inputs,” RTCA DO-180. RTCA, Washington, D.C. Radio Technical Commission for Aeronautics (RTCA) (1984). “Minimum Operational Performance Standards for Airborne Area Navigation Equipment using Multi-Sensor Inputs,” RTCA DO-187, RTCA, Washington, D.C. Radio Technical Commission for Aeronautics (RTCA) (1985a). “Minimum Operational Performance Standards for Airborne MLS Area Navigation Equipment,” RTCA Pap. 532-85/SC151-79,5th Draft. RTCA, Washington, D.C. Radio Technical Commission for Aeronautics (RTCA) (1985b). “Minimum Performance Standards Airborne DME Equipment Operationwithin the Radio-Frequency Range of 960-1215 MHz,” Doc. DO-189. RTCA, Washington, D.C. Radio Technical Commission for Aeronautics (RTCA) (1985~).“Minimum Aviation System Performance for the Global Position System,” Spec. Comm. SC-159. RTCA, Washington, D.C. Ragazzini, J., and Fanklin, G. (1958).“Sampled Data Control Systems.” McGraw-Hill, New York. Redlien, H., and Kelly, R. (1981). Microwave landing systems: The new international standard. Adv. Electron. Electron Phys. 57,311. Roberts, A. (1947). “Radar Beacons,” MIT-Radiat. Lab. Ser., Vol. 3. McGraw-Hill, New York. Rochester, S. I. (1976). “Take-off at Mid-Century: FAA Policy 1953-1961.” Federal Aviation Admin., Washington, D.C. Roscoe, S. (1973). Pilotage error and residual attention: The evaluation of a performance control system in airborne area navigation. J. Inst. Nauig. 20, No. 3. Ruhnow, W., and Goemaat, J. (1982).VOR/DME automated station selection algorithm. J . Inst. Nauig. 29, No. 4. Sandretto, P. C. (1956). Development of TACAN at federal telecommunications laboratories. Electr. Commun. 33(1). Sandretto, P. C. (1958). “Electronic Aviation Engineering.” International Telephone and Telegraph, New York. Sandretto, P. C. (1959). Principles of electronic navigation systems. IRE Trans. Aeronaut. Nauig. Electron. ANEX(4). Schmidt, S . (1966).Application of state space methods to navigation problems. Adv. Control Syst. 3.
DISTANCE MEASURING EQUIPMENT IN AVIATION
243
Schneider, A. (1959). Vector principles of inertial navigation. IRE Trans. Aerosp. Nauig. Electron. ANE-6(3). Silver, S . (1949). “Microwave Antenna Theory and Design,” MIT-Radiat. Lab. Ser., Vol. 12. McGraw-Hill, New York. Steiner, F. (1960).Wide-base doppler VHF direction finder. IRE Trans. Aerosp. Nauig. Electron. ANE-1. Tanner, W. (1977). Inertially aided scanning DME for area navigation. J . Inst. Naoig. 24, No. 3. Tyler, J., Brandewei, D., Heine, W., and Adams, R. (1975). Area navigation systems: Present performance and future requirements. J . Inst. Nauig. 22, No. 2. Wax, M. (1981). Position location from sensors with position uncertainty. IEEE Trans. Aerosp. Electron. Syst. AES19, No. 5. Williams, C., Bowden, B., and Harris, K. (1973). Pioneer award for contribution in field of secondary radar. IEEE Trans. Aerosp. Electron. Syst. AES-9, No. 5. Winick, A. B., and Hollm, E. (1964). “The Precision VOR System,” IEEE International Convention, New York. Zimmerman, W. (1969). Optimum integration of aircraft navigation systems. IEEE Trans. Aerospace Electron Syst. AFS-5(5).
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ADVANCES IN ELECTRONICS A N D ELECTRON PHYSICS, VOL. 68
Einstein-Podolsky-Rosen Paradox and Bell’s Inequalities W. DE BAERE Seminarie voor Wiskundige Natuurkunde Rijksuniversiteit Gent B-9000 Gent, Belgium
1. INTRODUCTION
Quantum mechanics (QM) is a statistical theory that makes predictions about the outcome of measurements on microphysical systems. The peculiarity of this physical theory is, according to the standard, or Copenhagen, intepretation of QM, that its statistical predictions are claimed to be inherent and cannot be explained further by completing our knowledge by some supplementary physical parameters. This inherent statistical feature of QM is so strange as compared with all known (classical) theories before the advent of QM, that resistance to accept such a counterintuitive world view has always been present in the scientific community, from the early days of QM. On the contrary, classical theories give us a rather comfortable feeling just because our most intuitive notions of physical systems (e.g., having definite properties of their own, being localized and separable from the rest of the world) are somehow inherent in this classical kind of theory. Moreover, the dogmatic claim of the proponents of the Copenhagen School, that at the atomic level physical events are no further analyzable in more detail than is allowed by QM (e.g., the impossibility of a space-time picture or analysis of individual events) has resulted in a critical attitude by a large number of scientists,among whom are de Broglie, Schrodinger, Einstein, and Bohm. In the words of Stapp (1982123, the origin of the uneasy feeling of those people with the present state and orthodox interpretation of quantum theory (QT) may be understood as follows: “Some writers claim to be comfortable with the idea that there is in Nature, at its most basic level, an irreducible element of chance. I, however, find unthinkable the idea that between two possibilities there can be a choice having no basis whatsoever. Chance is an idea useful for dealing with a world partly known to us. But it has no rational place among the ultimate constituents of Nature.” 245 Copynght @ 1986 by Academic Press, Inc. All rights of reproduction in any form reserved.
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The most important critical quantitative analysis of QM appeared in 1935 when Einstein, Podolsky, and Rosen (EPR) (Einstein et al., 1935) made a thorough analysis of the orthodox interpretation of QT, based on a Gedankenexperiment with correlated systems. Starting from a realistic conception of nature, according to which physical systems may have independent intrinsic properties (whether observed or not), EPA came to conclusions which contradicted QM (the EPR paradox), from which they inferred that QM was not a complete theory. This conclusion was at the origin of the famous Bohr-Einstein debate over the interpretation of QT, the end of which is now, 50 years later, apparently not yet in sight, in spite of claims to the contrary. An important step forward in the clarification of the EPR problem came from a development by Bell (1964), who used the ideas of EPR to set up a simple inequality, the Bell inequality (BI), which has to be satisfied if EPR are correct. There are several reasons for which this BI is interesting. First of all, it is violated by QT in appropriate circumstances. Furthermore, the original EPR issue about the completeness(a rather vague concept which is not easy to define)of QT has shifted to the question whether or not QT is compatible with locality and separability. Finally Clauser et al. (1969) showed that this issue of basic importance may be decided on the empirical level. Hence, as a result of these developments initiated by Bell, the original EPR Gedankenexperiment has become since then within the reach of experimental realization. The result of most of the existing correlation experiments agree very well with QT and violate the BI, from which it is concluded that on the quantum level nonlocality is now a basic property. However, it appears that, due to various criticisms, this conclusion is not yet final. It is the purpose of this article to present a critical review of the problems related to the EPR argumentation and to the Bell inequalities. As a complement to this review we advise the reader to consult other books or reviews on these subjects, e.g., Hooker (1972),Belinfante(1973),Scheibe(1973), Jammer (1974),Clauser and Shimony (1978), Selleri and Tarozzi (1981),Selleri (1983b),and DEspagnat (1983, 1984). In the following we will use the terms “EPR paradox,” “EPR problem,” and “EPR argumentation” interchangeably,without bothering about which is preferable. 11. THEEINSTEIN-PODOLSKY-ROS~ ARGUMENTATION A . Original Version
EPR made their analysis on a pair of correlated physical systems for which they showed that, according to some set of particular definitions and
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hypotheses, a physical system can have both position and momentum, in contradiction to the QM postulates, according to which noncommuting observables cannot be determined or known simultaneously. The main definitions in EPR are those of “complete theory” and of “element of physical reality.” As a necessary requirement for a complete theory they adopt the following definition:
(Dl) Complete theory: “Every element of the physical reality must have a counterpart in the physical theory.” As to the concept of “element of physical reality” which is contained in (Dl), EPR use the following definition, which they find reasonable enough for their purpose. (D2) Element of physical reality: “If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity.” To illustrate the meaning of these definitions, consider a quantummechanical system S in one dimension x, described by a wave function $(x). If $(x) is an eigenfunction of an operator A with eigenvalue a
444 = M x )
(1)
then measurement of A will give with certainty the value a. Moreover, from the preparation of the state of the ensemble of systems S, this result is known even without measuring A, i.e., without disturbing S. According to (D2)there exists in this case an element of physical reality corresponding to the physical quantity A. Using (Dl) this element of physical reality must have a counterpart in the underlying theory, QT. This must then be the case for every other physical observable quantity C with operator C for which Eq. (1) is valid C W ) = c$(x)
(2)
i.e., [A, C] = 0. Hence, according to (Dl) the only allowed elements of physical reality within orthodox QT are those corresponding to a complete set of commuting observables if the ensemble of systems S is described by an eigenfunction of one of them. If, on the contrary, the operator C corresponding to the physical quantity C does not commute with A: [ A , C ] # 0, then one has W(X)
# clCl(4
(3)
i.e., the result of measuring C on this state is not known without disturbing S. Certainty about the value of C, among the different possible values, will in this case only be obtained by actually measuring C, i.e., by disturbing S. Hence, as long as C is not measured actually, with C there does not correspond an
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element of physical reality of S , according to (D2). Up to now all this is consistent with orthodox QM. However, at this point EPR remark that, in order to criticize the completeness of QT, it is sufficient to find at least one particular situation for which all the above is not valid. This means that, if it can be shown that A and C are elements of physical reality of S, even if [ A , C ] # 0, then the alleged completeness of QT would no longer be tenable. Now, to show that this actually happens in some particular cases is the intention of the EPR reasoning. In particular by their argumentation they try to show that Heisenberg’s uncertainty principle may be violated. To this end EPR consider two correlated systems S , and S, which are supposed to interact only in a finite time interval (0, T ) .Suppose that for t > T the wave function $(x1,x2)of the combined system ( S , , S , ) is known completely. At the basis of the EPR argumentation, which leads to their contradictory conclusions, are furthermore the following three hypotheses:
(H 1) Einstein locality (separability), or the validity of the relativity principle: in any reference frame physical influences between nonseparated physical systems S , and S2(ie., between interacting systems) should propagate with a maximal velocity c, the velocity of light. If S , and S2 are separated, then no mutual influence at all should exist. In both cases, the idea of action at a distance or influences propagating with infinite velocity is considered as an absurdity and empirically incorrect. As a consequence, a measurement on one of two separated systems, say S , , cannot influence at all the outcome of a measurement made on the second system S , , if the two measurements are separated in space. Einstein, in his proper and direct way of expressing his intuitive insights in such fundamental matters, expressed his view on locality as follows (Einstein, 1949):“But on one supposition we should, in my opinion, absolutely hold fast: the real factual situation of the system S, is independent of what is done with the system S , , which is spatially separated from the former.” (H2) Universal validity of Q M : In particular it is supposed that the process of state vector reduction applies when a certain measurement is made, or when information about a system is gained. (H3) Counterfactual dejiniteness (counterfactuality)(Stapp, 1971) is valid: This is essentially the assumption that, if on a certain system S , an observable A is measured with the result a, it does make sense or one is allowed to speculate about what would have been the result if, instead of A , another (eventually noncompatible) observable B were measured on precisely the same system S, in exactly the same individual state [which is not contained in the quantum formalism, but may be contained in some future more fundamental
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hidden variable theory (HVT)]. Hence the above hypothesis amounts to assuming that physically meaningful conclusions may result from combining actual measurement results with hypothetical ones. Suppose then that at t > T we measure the physical quantity A on S , , with eigenvalues a , , a , , . . . , and corresponding eigenfunctions u , ( x , ) , u,(x,), . . . : = akuk(xl)
(4)
According to standard QM rules we have to write II/(x,,x2) in terms of the wave function U k ( x 1 ) $(xl, x2)
=
uk(x1)$k(x2) k
(5)
with $ k ( x z ) the expansion coefficients. If the result of the measurement is ak, then the prescription of the reduction of the wave function says that I&,,x 2 ) reduces to one single term, namely uk(xl)II/k(xz).In QM this must be interpreted to mean that, after measurement of A on S , , S , must be described quantum mechanically by uk(x1) and S, by II/k(x2). Again, up to this point everything is quantum mechanically correct. I think that at this point it is important to note that, in order to understand the origin of the EPR paradox, the foregoing conclusion is correct only if the measurement of A on S , has actually been carried out. If not, the wave function $(xl, x,) does not reduce to one single term; in particular S, is not described further by $ k ( x Z ) . Now, EPR would propose to measure, instead of A, another quantity B with eigenvalues b , , b,, . . ., and corresponding eigenfunctions u 1(x 1 u 2 (x 11, . * * BUk(X1) = bkUk(X1) (6) 19
Instead of Eq. (5) one should now write $(xl,xZ)
=~uk(x1)(Pk(x2) k
(7)
This means that if B is measured actually (say at t = t o ) with the result bk,then $ ( x l , x 2 )reduces again to one single term, namely uk(xI)(Pk(xz),and after this measurement any further predictions for measurements on S , should be derived from its wave function (Pk(X2). In the foregoing we have stressed the condition that the measurements were actually carried out. It is important at this point to note that the EPR reasoning violates what is allowed by standard QT: Indeed, if A and B are incompatible, then either their values cannot be measured or known simultaneously, or A and B are measured simultaneously but on different systems, prepared analogously by different sources. Now it is precisely this violation of QM which EPR allow themselves in their argumentation, namely, as they state it: “If, instead of this, we had chosen another quantity, say B,
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having the eigenvalues bl, b,, ...,and eigenfunctions ul(xl), uz(xl), u3(x1),.. ., we should have obtained,. . ..” When discussing the Bell inequalities in Section 111, we will see, however, that this counterfactual reasoning, “If, instead of this,. .,we had chosen . ..we should have obtained ...” lies also at the origin of the disagreement between QM predictions and the BIs. Furthermore we will see (Sections I11 and IV) that the BIs are violated empirically in favor of QM. We will also make plausible there that it is the breakdown of counterfactuality (H3), rather than of locality (Hl), that is at the origin of the disagreement between BI and QT. Hence, at this point it already seems reasonable to suspect that it is incorrect and unjustified to speculate about the outcome of an already-carried-out experiment if some parameters of the total experimental arrangement were changed. In fact, on closer examination, it appears that it is reasonable to assume that, once an experiment is carried out, a subsequent experiment can never be done again under exactly the same conditions as those which were responsible for the specific outcome in the actual experiment. Therefore, it is to be expected that the result of any argumentation, like the EPR, which violates at some stage some QM principle, will lead to conclusions which are contradicted by QM (see below). Before continuing the original EPR reasoning to its end, let us remark that, even before choosing the alternative to measuring B, there already exists a conflict with the initial assumption, namely that for t > T the wave function is represented by the pure state, Eq. (5). Let us consider therefore the consequences of the hypothesis of locality (H1)and of the universal validity of QT (H2) and show that these hypotheses lead to contradictory statements. Indeed, for simplicity suppose for a moment that the sum on the right-hand side of Eq. (5) contained only a finite number of terms, and let us introduce time
Then the measurement of A = uk at c = t o > T reduces J/(x1,xZrt) to uk(x1, tO)+k(XZ, to). Now, because of the absence of any mutual influence between S1 and S, from t = T on, it may be concluded that S2 is also described by +&, t) for all t satisfying T < t c t o . This means that a statistical ensemble of couples (Sl,S,) for t > T may be decomposed into a series of subensembles described by ul(xl,t)+l(xZ,t),uZ(xl,t)+z(xZ,c),...; i.e., the original ensemble should be described by a mixture of states uk(x1, t)+k(xz, t), which is in contradiction to the original assumption of the ensemble being described by the pure state, Eq. (5). There is also an observable difference between the two possible descriptions (Furry, 1936a; Selleri, 1980; Selleri and Tarozzi, 1981). In the literature, the last description is by means of so-called
EINSTEIN-PODOLSKY -ROSEN PARADOX
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“state vectors of the first type” (corresponding to proper mixtures), while the first way is by means of “state vectors of the second type” [i.e., Eq. ( 5 ) ] (corresponding to improper mixtures) (DEspagnat, 1965). QM observables which have different values in the two schemes are sometimes called “sensitive observables” (Capasso et al., 1973; Baracca et al., 1976; Bergia and Cannata, 1982; Cufaro Petroni, 1977a,b; Fortunato et al., 1977). For later reference we shall call the above argumentation, which uses only hypotheses (Hl) and (H2), the modified or simplified EPR argumentation. From the assumed legality of measuring B instead of A on S,, EPR conclude: “We see therefore that, as a consequence of two different measurements performed upon the first system, the second system may be left in states with two different wave functions. On the other hand, since at the time of measurement the two systems no longer interact, no real change can take place in the second system in consequence of anything that may be done to the first system. This is, of course, merely a statement of what is meant by the absence of an interaction between the two systems. Thus, it is possible to assign two diflerent wave functions (in our example, $k and rp,) to the same reality (the second system after the interaction with the first).”This conclusion contradicts the QM formalism and constitutes what is known as the EPR paradox. Note again that here the spirit of QM is violated because it is supposed that the wave function represents the system itself. Inasmuch at the time of EPR much attention had been paid to whether or not physical systems (viewed as “particles”) followed individually a welldetermined path (a world line) in space-time, in the remaining part of their paper EPR are concerned with showing by an explicit example of the above conclusion that one and the same physical reality may be described by eigenfunctionsof the two noncommuting operators P and Q: [P, Q ] # 0. To this end EPR consider as the wave function describing the couple (S,, S,)
1
m
vwX19
xz) =
expCi(x1 - X Z + XO)P/51 dP
(9)
-m
with xo being a constant. For observable A they choose the momentum PI = h/i a / a x , of S , with eigenvalues p and eigenfunctions u,(x 1) = exp(ipx 1 lh)
(10)
Because of the continuity of p , Eq. (5) takes the form
1
m
$(XI,XZ)
=
up(x1)$p(x2)dp
(1 1)
-m
where the wave function for S, $Jx2)
= exPC - i(x, - xo)p/hl
(12)
W.DE BAERE
252
is an eigenfunction of P2 = h/i a/ax, with eigenvalue -p. Applying the above general results of the EPR analysis we have to conclude the following. Suppose the measurement of Pl on s, gives the result p. Then $(x1,x2) in Eq. (11) reduces to ~ , ( x , ) ~ ~ ( x ,This ) . allows us to conclude that P2 = - p without disturbing in any way S, (S, and S, are separated by assumption). Hence P, = --p is an element of reality for S,. For the observable B EPR take the coordinate Q , of S,: B = Q1,with eigenvalues x and eigenfunctions ux(xl) = 6(x, - x). $(x,,x,) may now be written
with (Px(x2)=
rm
expCi(x - x2
= hd(x - x,
+ xg)
+ XO)P/hl
dP (14)
which is an eigenfunction of Q , with eigenvalue x2 = x + xo. Applying again the EPR criteria we have: If the measurement of Q1 on S, has the result Q1 = xl, then $(xl, x,) reduces to u,(xl)~,(x2).This allows EPR to conclude that x2 = x xo without in any way disturbing (again by assumption) S,. Hence, x2 = x xo is also an element of reality for S,. Now [P2,Q2] = h/i # 0, and from this particular example EPR conclude that the same physical reality, S,, may be represented by two wave functions which may be eigenfunctions of two noncompatible observables P2 and Q 2 , in contradiction with what is allowed by orthodox QT. Hence they conclude that QT is not complete, because in its formalism either position or momentum is contained, whereas according to EPR both should be contained in it as elements of physical reality. But remember the crucial role played by the hypothesis of counterfactuality (H3) in the EPR argumentation. The particular situation above considered by EPR (measurement of P, or Q1 on one member of two coupled systems) is up to now only a kind of Gedankenexperiment. However, recently Bartell (1980b) has shown how the EPR proposal can be transformed into a real experiment. It has been shown that inequalities of the Bell type (Section 111) can be constructed, on the basis of which the validity of the EPR hypotheses (Hl) and (H3) may be verified. However, we consider it as not justified, as Bartell argues, that violation of the proposed inequalities should be considered as evidence for the existence of nonlocal influences. Indeed, as we will show in Section V, locality or the relativity principle may be saved by assuming a breakdown of counterfactuality (H3) as a more plausible origin of an eventual violation.
+
+
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EINSTEIN-PODOLSKY -ROSEN PARADOX
B. E P R Paradox in the Bohm Version Instead of measuring two noncompatible observables A = P and B = Q which have a continuous spectrum of eigenfunctions, Bohm (1952a)proposed to consider two systems S , , S , having the property of possessing spin 3. Instead of the original EPR Gedankenexperiment, this proposal has afterwards turned out to be realizable experimentally. In this case $ ( x , , x , ) has to be replaced by x ( x l , x z ) $ ( a l , a z ) , with $(u1,a2)being the spin part of the wave function and the Pauli matrices, a,,a, representing the spins of S, ,S, . Suppose the couple (S, ,S,) is prepared somehow in the singlet spin state: $(u1,a2)=I+s) = 100). For simplicity it may be supposed that we have a spin-0 system S(0)which decays according to S(0) + Sl(# S,(i). If this process ends at T, then for t > T, S1 and S2 may be supposed to be separated or at least no longer influencingeach other. From elementary QM it follows that, if angular momentum is conserved in the decay process, 100) may be written
+
I$J = 100)
= (1/Jz)(lu:Y)Iub3
-b .I">":).!I
(15)
in terms of basis vectors II&) for S , and IU:~) for S2 in which S , , S 2 have spin projections & 1 (in appropriate units) along an arbitrary unit vector a. Let us now illustrate the EPR argumentation on this conceptually simple example of correlated systems with only dichotomic variables. Take as a physical observable A of s,:A = a, z = q Z= an operator for the measurement of the spin projection m,,(z, t) along the unit vector z . The eigenvalues are f 1, and the respective eigenvectors, lu$!), Iui!!). An alternative option is to take as an observable B for S,: B = a, y = olY= an operator for the measurement of the spin projection ms,(y, t) along the unit vector y. The eigenvalues are again k l , and the respective eigenfunctions, It&)>, IuY2). EPR option ( I ) : Measure A = a, z, with the result m,,(z, t ) by sending S , through a Stern-Gerlach device SG, ( z ) with its analyzer direction (i.e., its magnetic field) along z . According to Eq. (15) we should write for this case
-
-
-
loo) = ( l / J z ) ( ~ u : ~ ) ~ u : ? )- ~ u ~ ! ! ) ~ u : ~ ) ) (16) From Eq. (16) it is seen that m,,(z, t) = + 1 and m,,(z, t) = - 1 will be found
with equal probability = 3. Suppose that at t = to the result obtained is ms.(z, t o ) = + 1. According to QT, 100) reduces to l u ~ ! ) l u : ? ) .It follows then that for t 2 to measurement of msz(z,t 2 to) will give - 1 with certainty without disturbing S, . According to the EPR criterion, this means that with msz(z, t 2 t o ) = - 1 there corresponds an element of physical reality and this must have a counterpart in a complete theory.
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W. DE BAERE
EPR option (2): Applying the counterfactuality hypothesis (H3), one can argue that one could equally well have measured the quantity B = a, * y with the result ms,(y, to), by replacing SG,(z) by SG,( y). Instead of Eq. (16) we must now start from
loo) = (l/fi)(luy2)lup)
- luy?)lu$3)
(17)
Again msl(y, to) = + 1 and ms,(y, t o ) = - 1 will be found with equal probability 3. Suppose again that at t = to we would have found msl(y, to) = - 1. This measurement may now be considered as a state preparation of S2 for t 2 to: In fact 100) is reduced by the measurement process to the single term 1ur?)1u1‘+‘). As a result of this process, the result ms,(y, t 2 t o ) = + 1 is then predicted with certainty, again without disturbing S2. Hence with this value for ms,(y, t 2 to) there must also correspond an element of physical reality and this must again have a counterpart in a complete theory. Now this is certainly not the case in orthodox QT because of the noncompatibility of oZty and 02*: [02,, 02,] # 0. Hence the EPR conclusion that QT is incomplete. As already mentioned, this incompletness conclusion may be avoided by insisting that for this to be true, ms2(z,t) and ms,(y, t) should be available at the same time. This way out of the problem is anticipated by EPR; indeed they offer it themselves at the end of their analysis: “One could object to this conclusion on the grounds that our criterion of reality is not sufficiently restrictive. Indeed, one would not arrive at our conclusion if one insisted that two or more physical quantities can be regarded as simultaneous elements of reality only when they can be simultaneously measured or predicted. On this point of view, since either one or the other, but not both simultaneously, of the quantities P and Q can be predicted, they are not simultaneouslyreal. This makes the reality of Q and P depend upon the process of measurement carried out on the first system, which does not disturb the second system in any way. No reasonable definition of reality could be expected to permit this.” However, their remark in the last sentence of this quotation can be circumvented rather easily by extending their definition of an element of reality; indeed each measured value should be considered as representing an element of reality not only of the object system but of the combined system, object plus measuring apparatus. However, the simplified EPR argumentation in the Bohm version is again much more difficult to reject. To see this, let us apply the EPR reasoning on each of the couples ( S , , S 2 ) (Selleri and Tarozzi, 1981): (1) Measure mS,(a,tO)= + 1 at an instant t = to when S,,S2 are separated (located) so far that it may be supposed that they no longer influence each other (principle of separabilty). (2) From the standard QM formalism (assumed to be universally valid
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EINSTEIN-PODOLSKY -ROSEN PARADOX
by EPR), it follows then that at this moment the state 100) reduces to lu:!+))lu:?), which describes henceforth (Sl,S,) for t 2 t o . In particular, as a result of measuring ms,(a, to), the state describing S, is now luf!) for t 2 t o . (3) Because of the supposed separability and absence of any influence, this must also have been the case for t < to (at least for those times where separability holds); the state vector of S , must have been lu:?) and mS2(a,t < t o ) = - 1. (4) From msI(a, t) + ms2(a, t) = 0 (conservation of the component of the total spin along an arbitrary direction) it follows further that mS1(a,t < to>= -1 and that the state vector of S , is It&!) for t < t o (notice that this is already in conflict with the usual QM formalism, according to which it is only after ms2(a, to) = 1 is measured by SGl(a), thus for t 2 to that the state vector of S , is 1.)~': (5) Making the same reasoning for each couple ( S , , S,), one arrives at the contradictory conclusion that, the original ensemble was not described by the pure state, Eq. (15), but by a mixture of states Iu::')lu:?) and I u ! , ! ! ) l u : ~ ) with equal probability (see also Table I).
+
TABLE I SG,(a) IN PLACE, SG2(a)NOT IN PLACE: EPR PARAWX Time
SI
s2
State vector of (Sl,S2):100)
t < to
msl(a,t < t o ) = ? m,,(a, t < t o ) = -ms,(a, t
to
< to)=?
ms,(a,to)is measured: S, moves through
No measurement on S,, but SGI(a)(physical, local interaction) m,,(a, t o ) = -m,,(a, t o ) predicted Reduction of (00): if t L to State vector becomes Iu:?!) by nonlocal ms,(a,to)= + 1:100> reduces to influence Iu~!)tu~?!)for (s,,s,) Inferences: m,,(a,t = to) = - I (separability,locality) m,,(a, t < to) = + 1 c- m,,(a,t < to) = - 1 (conservation of angular momentum) state vector = 1)~':; state vector = Iu;!!) f =
1
I
I
(Sl,S2jdescribed by mixture of Iub:' )I ub? ) and 1 ub? )I ub:' ) = observably different from (00)
EPR Paradox
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W. DE BAERE
In order to show the contradiction in a quantitative way, let us introduce the triplet state
I$,)
= 110) = (1/&)(lu:Y)lu:9
+ lu:?)lu:Y))
(18)
Then one may write lu:Y)lu:3
= (l/JZ)(lW
+ 110))
( 194
lu:?)iu:Y)
= (l/Jz)(lOo) - 110))
(19b)
such that the original ensemble of couples ( S , , S , ) should now be a superposition of singlet and triplet spin states, instead of a pure singlet state. In fact there are observable differences between the two kinds of descriptions (Selleri and Tarozzi, 1981). In the pure state one has
( o o ) J ~ ~ o =o )0
(20)
Obviously there is something that has to be wrong in the above reasoning. Noting that no contradiction arises if one does not go outside the strict QM framework [Bohr's resolution of the EPR paradox (Bohr, 1935a,b)]; Selleri and Tarozzi (1981) arrive at the conclusion that it is step (3) in the EPR reasoning, namely the locality hypothesis (Hl), which is incompatible with QM, and which is responsible for the above contradiction.
C . Critical Analysis of the Original EPR Argumentation When the EPR paper appeared in 1935, it was already clear that the quantum formalism was very powerful and successful in the treatment of physical problems at the atomic level. As a consequence, it had become the daily working tool of a large majority of physicists who did not bother about fundamentals. Yet the original EPR argument was considered a serious challenge of the completeness of QT, especially by those interested in the soundness of its basis. In this section we will review the main reactions immediately after the EPR argumentation was published. We will see that most of the early reactions emphasized the difference between the definition of state of a physical system in EPR, which is that of classical mechanics (CM), and that adopted in QM. In CM the state is supposed to be a representation of the system itself, while in QM the state represents, at least in the Copenhagen interpretation (Stapp,
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1972),merely our knowledge about the system. Also attention is drawn to the fact that measurement of noncompatible observables requires mutually exclusive experimental arrangements. This may be seen as an indirect criticism of EPRs implicit hypothesis of counterfactuality (H3). Kemble (1935) was one of the first to defend the orthodox viewpoint against EPR. He argues that, if it were true, as EPR contend, that “it is possible to assign two different wave functions ... to the same reality ...,” the QM description would be erroneous. According to Kemble’s view, the EPR problem may be avoided by sticking to “. . . the interpretation of quantum mechanics as a statistical mechanics of assemblages of like systems.” Hence in the above EPR contention, one of the wave functions describes the future statistics of one subassemblage, while the other describes the future of another, different, subassemblage. And which wave function is to describe the subassemblage depends on the choice of what will be measured on one of two correlated systems in the original assemblage. Kemble concludes that there is no reason to doubt the completeness of QM on the grounds advanced by EPR. Once it is accepted that the wave function does not describe the intrinsic state of a system itself, but represents only the regularities between future observations on a system, the original EPR problem on the completeness of QM disappears. According to Wolfe (1936), the origin of the EPR troubles lies in an unjustified extrapolation of ideas on “physical reality,” which are certainly satisfactory and reliable in classical physics, to the domain of QM. EPRs own specific example of a system having both position and momentum illustrates this point. In CM both may indeed be determined and known simultaneously, and states in CM accordingly contain this information. However, in QM only one of the two may be known, and this is again reflected in the quantum state. According to Wolfe, the quantum state only represents our knowledge about a system. Hence Wolfe concludes that “Viewed in this light, the case discussed by Einstein, Podolsky, and Rosen simply dissolves. Two systems have interacted and then separated. Nothing that we do to the first system after this affects the “state” of the second. But ..ieasurements on the first system affect our knowledge of the second and therefore affect the wave function which describes that knowledge. Different measurements on the first system give us different information about the second and therefore different wave functions and different predictions as to the results of measurement on the second system.” The reply of Bohr (1935a,b) goes along similar lines. After explaining in detail his general viewpoint, called “complementarity,” Bohr shows that the QM formalism is mathematically coherent and that the limitations on the measurement of canonically conjugate observables are already contained in the formalism. According to Bohr, the origin of the EPR contradiction is
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“. .. an essential inadequacy of the customary viewpoint of natural philosophy for a rational account of physical phenomena of the type with which we are concerned in quantum mechanics.” Bohr continues to explain in detail that the experimental arrangements and procedures for measuring noncompatible observables, in particular x and p , are essentially different and mutually exclusive. Choosing one arrangement means the renunciation of knowledge of the variable measured by the other. Otherwise stated: Measurement of one variable means an uncontrollable reaction on the system which definitely prohibits the measurement of the other variable. Hence Bohr concludes that only if two or more observables can be known or measured simultaneously,can they have simultaneous “reality” status. This was also the point made by Ruark (1935). According to Bohr then, it is therefore clear that the original EPR argumentation is invalid for noncommuting observables A and B. Also EPR were apparently aware of the above criticism because of their final remarks towards the end of their paper (Section 11,B): “... one would not arrive at our conclusions if one insisted that two or more physical quantities can be regarded as simultaneous elements of reality only when they can be simultaneously measured or predicted.” They rejected this solution to their problem on the ground that “This makes the reality of P and Q depend upon the process of measurement carried out on the first system, which does not disturb the second system in any way. No reasonable definition of reality could be expected to permit this.” However, the last viewpoint is not correct because it must admitted that any measurement result is somehow characteristic for elements of reality of both the instrument and of the physical system, and not only of the system itself. Hence, according to Stapp (1971), it appears that EPR violates “. . . Bohr’s dictum that the whole experimental arrangement must be taken into account: The microscopic observed system must be viewed in the context of the actual macroscopic situation to which it refers.” For a recent discussion of the debate between EPR and Bohr, see de Muynck (1985). An interesting discussion of the EPR criticism against QM has been given by Furry (1936a,b). According to Furry (1936a) the essence of Bohr’s reply to EPR is that “one must be careful not to suppose that a system is an independent seat of “real” attributes simply because it has ceased to interact dynamically with other systems.”However, one should be warned that such an extreme positivistic standpoint may also be criticized. Indeed, it may well be assumed that both the apparatus and the system itself have their own “real” and independent (in the sense of not being influenced instantaneously by the rest of the world) attributes of which any measurement result is but a representation. It will be shown in Section V,B that such a viewpoint is perfectly consistent with the Bell inequalities and the hypothesis of Einstein locality (H l), provided the hypothesis of counterfactuality (H3) is abandoned.
EINSTEIN- PODOLSKY-ROSEN PARADOX
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Furry then studies in a quantitative way the observable differences between the above assumption of EPR that the QM wave function describes the “real” attributes of the system itself and the orthodox QM standpoint that applies the usual QM formalism and denies that physical systems possess independent real properties. Consider then two correlated systems S1,S 2 which are described by a QM state vector l$(xl,x2)). According to von Neumann (1955; see also Baracca et al., (1974), l $ ( x l , x2)) may in general be written
Moreover in Eq. (22) all &’ are different, and also all P k . The state vector Eq. (15) or (16),corresponding to the Bohm version of the EPR paradox with correlated spin-) systems in the singlet state, may be considered as an illustration of Eq. (22), for which moreover the expansion on the right-hand side is not unique. From the one-to-one correspondence between & and p k in Eq. (22) it follows that if L = & is measured on S1,then R = Pk is predicted on S2 and vice versa. Furry then gives a definite form to the above-mentioned opposite viewpoints of systems possessing real attributes and of standard QM in the following way: Method A assumes that, as a result of the interaction which causes the correlation between S , and S 2 , S, made a transition to one of the states I q A k ( x , ) and ) S2 made a transition to the corresponding state 15Pk(x2)).Also it is assumed that the probability for transition to JCP,,(X,)> is wk. Note that the assumptions of Method A are precisely the conclusions of the modified EPR argumentation (Sections II,A and I1,B) according to which a mixture of states 1 q A k ( x l )1)t p k ( x 2 ) )should describe an ensemble of correlated systems (S,, S2), instead of a pure state as in Eq. (8) or (22),which was the assumption from the outset. Method B, on the contrary, uses only standard QM calculations based on the pure state, Eq. (22). Furry considers then four types of questions which may be answered by using either Method A or Method B. The particular type of question for which the two methods give different answers is the following.Suppose observable M is measured on S1with eigenvaluesp and eigenstates1 &(x1)) and observable S is measured on S , with eigenvalues c and eigenstates 147,(x2)). The question
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is: If M = p on S , is measured, what is the probability that S = (r will be measured on Sz? According to Method A the probability is
and according to Method B
IT
~
(rl,(xz)l 5pk(XZ)>
1 k
wkl
<%k(x
1
)I$p(x
I
1)>
(25)
It is seen that the difference between Eqs. (24) and (25) consists in the appearance of interference terms in Eq. (25) which are not present in Eq. (24). The origin of these terms is usually ascribed to the influence of the measuring apparatus on the internal conditions within the system, whose future behavior has therefore been influenced. We will see in Section III,F in a direct way that this must happen with each system that passes some apparatus. Moreover there does not exist a way by which Method A can be made consistent with Method B. Hence, Furry claims to have shown mathematically the inconsistency between standard QM and the EPR viewpoint that physical systems may have objective properties which are independent of observation. However, we will show below that this is not entirely correct. It is only correct if Furry’s assumptions of his Method A are accepted. The link with the original EPR argumentation is obtained by consideringa particular example for which the expansion of Eq. (22) is not unique. Furry then applies his Method A to each of the expansions, which amounts to assuming that in each case the system S1 or Sz made a transition to a definite state for which a certain observable has some well-defined value. As in the case of EPR, contradiction with QM occurs when both observables are noncompatible. To illustrate these points clearly, Furry analyzes a thought experiment of the EPR type and shows that the final conclusions are in conflict with Heisenberg’s uncertainty relations. Furry (1936b) concludes that “. . . there can be no doubt that quantum mechanics requires us to regard the realistic attitude as in principle inadequate.” In defense of the realistic attitude of EPR we should remember, however, what has been said at the start of the discussion of Furry’s contribution. Also it may be questioned whether Furry’s Method A is identical to that which Einstein had precisely in mind. Indeed, the eigenstates l q A k ( x l ) ) of the observable L, which Furry ascribes to subensembles of systems S 1 , are QM
EINSTEIN-PODOLSKY -ROSEN PARADOX
26 1
states. This means that, for the subensemble described by l q A k ( x l ) (selected ) by retaining only those S1 for which measurement of L is A&, each subsequent measurement of L will again give the result 1,.However, if another observable M is subsequently measured instead of L, then only the probabilities ~ ( $ , , ( x , ) ~ q A k ( x l for ) ) ~ obtaining z the value p are known. Now the idea of Einstein was probably that, once the state of S, is known completely, it should be possible to predict with certainty all possible measurement results at one instant. Perhaps EPR’s only weak point was to assume that all these possible measurement results also correspond to actual results; i.e., they assumed implicitly the validity of counterfactuality (H3). We will see in Section V that, on account of the empirical violation of the BIs either Einstein locality (Hl) or counterfactuality(H3) has to be abandoned. In the latter case this means that, among all possible measurement results, there is only one that has physical sense, namely the one that is actually obtained in a real measurement. It is seen that the difficulty with considerations on states of individual quantum systems is that there does not exist any formal scheme within which such problems can be discussed. Tentatively this may be done along the following lines. Suppose we represent the individual state of each quantum system S at each instant t by a set of (field) functions { d ( x , t ) } ( i = number of field components).In the spirit of Bohr, we shall consider the individual state of the apparatus, say a Stern-Gerlach device SG(a) with analyzer direction a, as equally important for the determination of the measurement result, e.g., the spin projection ms(a, t) along a at time t. This means that SG(a) too should be represented by a similar set {tl,&a)(x, t)}. Measurement of the spin projection ms(a, t) of S along a should then formally be represented by the mapping
In this formal scheme the “elements of reality” corresponding to the particular value rn,(a, t) should therefore be represented by { a i ( x ,t ) } for S and by {cL$~(,,)(x, t ) } for SG(a); i.e., rn,(a, t ) should not be considered as an attribute of S alone, but of S and SG(a) together. Hence, within such a scheme, a reconciliation between the realistic standpoint of EPR and the orthodox or Copenhagen standpoint should be possible (see also de Muynck, 1985). D . First Attempt at an Experimental Discrimination between EPR and Standard QM
We have seen above that the simplified EPR argumentation leads to the conclusion that an ensemble of correlated systems S,, Sz should be described by a mixture of product states, instead of a pure state as required by QM. The
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quantitative analysis of Furry (1936a)(Section I1,C)has shown very clearly the difference between the predictions based on both kinds of description. Subsequently, Bohm and Aharonov (1957) were the first to suggest that, eventually, the original situation of correlated systems, described in standard Q M by means of a pure state, could evolve spontaneously to a situation which should be described by means of a mixture of product states of the individual systems when the separation becomes sufficiently large. In the special case of correlated photons it is assumed that their polarizations are opposite, and, in order to retain the QM rotational symmetry, that the polarization directions are uniformly distributed in an ensemble of coupled systems. In order to discriminate between the two alternatives Bohm and Aharonov discuss an experiment (Wu and Shaknov, 1950)in which correlated photons y1 and y2 originate from the annihilation of an electron and a positron, each of which is supposed to be at rest. Under these conditions the state of the couple (yl, y 2 ) has total spin and parity J p = 0-(Kasday, 1971)and according to QT one can write either = ( l / f i ) ( I R l ) l R Z ) - IL1)ILz))
(27)
or = (1/JZXlX,>lY2) - IYl)lX2))
(28) for the pure state vector of the coupled photons. In Eq. (27) the notation is such that IR1)(R2)(ILl)lL2)) represents photons moving in opposite directions (e.g., along z) which are right (left) circularly polarized. Similarly, in Eq. (28), IXl)l Y,) represents two photons which are linearly polarized in, respectively, the x and the y direction. Hence, according to standard QM, measurement of one kind of polarization on y1 immediately gives information on the corresponding polarization of y 2 , because of the correlation in Eq. (27) or (28). Now, because such ideal polarization measurements on individual annihilation photons do not yet exist, this kind of one-to-one correlation between individual photons is not verifiable. Therefore, to distinguish between the two alternatives, one has to proceed in an indirect way, namely by means of the process of Compton scattering, which also depends on the photon polarization. Suppose then that y1 and y 2 , with momenta pyl and py2= -py,, are Compton scattered at points A and B with final momenta and ptz (Fig. 1). In Bohm and Aharonov (1957)two different situations are considered, namely for the scattering planes (pyl,pi,) and (pyz,pi2)being perpendicular or parallel. In each case coincidence counts of photons are registrated for which the scattering angle 8 is the same for both photons. The ratio of the number of coincidence counts in both cases is called R. It is this experimentally determined ratio R, for an ideal angle of 8 = 82", which is compared with the l~ylyz)
PI,
EINSTEIN-PODOLSKY -ROSEN PARADOX
263
Y FIG.1. Compton scattering of correlated annihilation photons.
predictions following from three assumptions:
(Al) Standard QT is universally correct, i.e., for large as well as for small separations between y,l y2 ; (A2) Standard QT applies only for small separations, e.g., when the wave packets overlap. For those separations for which this is not the case, it is assumed that the photons are already in QM states which are circularly but oppositely polarized about their direction of motion; (A3) The same as (A2), but now it is assumed that each photon y1 is already in a state of linear polarization in some arbitrary direction, and the other, y2, is in a state of perpendicular polarization. All directions are equally probable. The theoretical predictions as calculated by Bohm and Aharonov (1957) for the different assumptions, taking into account the characteristics of the experimentalconfiguration (e.g., finite experimental solid angle), are as follows RA1 = 2.00,
R,, = 1.00,
RA3 = 1.5
(29)
These values are to be compared with the experimental result R = 2.04 f.0.08 of Wu and Shaknov (1950). From this it is evident that in this case standard QM gives the correct result. More recent, similar experiments on annihilation photons (Bertolini et al., 1955; Langhoff, 1960; Section IV,A) give further evidence for the correctness of the QM description of correlated and widely separated photons; hence the above hypothesis of Bohm and Aharonov Lie., assumption (A2) or (A3)] is not correct. The relevance of the experiment of Wu and Shaknov with respect to the EPR paradox has been criticized by Peres and Singer (1960). They argued that photons, being massless, have their spin always along their direction of propagation, and that the spin components orthogonal to this direction do not have physical meaning because these quantities are not gauge invariant. However, Bohm and Aharonov (1960) subsequently showed that these arguments were invalid because they were based on an incorrect interpretation of photon polarization in QT.
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E. Attempts to Resolve the EPR Paradox Margenau (1936) apparently was the first who tried to resolve the EPR paradoxical conclusions by emphasizing that the assumed validity of the QM process of state-vector reduction may be the origin of the troubles. Following EPR,Margenau remarks that, if this process is correct, then “. . . the state of system 1, which, by hypothesis, is isolated from system 2, depends on the type of measurement performed on system 2. This, if true, is a most awkward physical situation, aside from any monstrous philosophical consequences it may have.” However, note again that from this statement it is seen that Margenau too accepts implicitly the validity of counterfactuality (H3). Margenau then argues with respect to the EPR problem that “. . .if it be denied that in general a measurement produces an eigenstate, their conclusion fails, and the dilemma disappears.” However, we have already shown above that the EPR contradictions are most striking when (H3) is not used at all. In this respect, we will discuss below (Section II,F) some interesting recent proposals for a direct experimental verification of the correctness of the process of statevector reduction in the case of correlated systems, or of the validity of locality or the relativity principle. According to Breitenberger (1965) there can be no question of a real paradox, neither in the original EPR situation nor in the later Bohm version, because the contradictory conclusions result from statements which are unverifiable (i.e., counterfactual) and therefore are devoid of physical sense. It is also noted that the EPR reasoning is based implicitly on the assumption of the knowledge of a precise value of a conserved physical quantity, the importance of which was first emphasized by Bohr (1935b). Moldauer (1974)re-examines the EPR argumentation from the standpoint that physical theories, in order to be verifiable, should deal with reproducible phenomena. It is argued that under this requirement the EPR conclusion of incompleteness of QT does not apply, although it is admitted that nonreproducible events, such as the decay of a radioactive atom, are objective observations which endow physical reality to such events. EPR claim that these events too should be described in a complete theory. According to Moldauer such inherently nonreproducible events (e.g., no ensemble of radioactive atoms can be prepared which decay at precisely the same instant) ought not to be the object of theoretical investigation. Zweifel (1974) uses Wigner’s theory of measurement to resolve the EPR paradox. This theory is based on the idea of the existence of some kind of interaction between the system under investigation and the mind of the observer. It is argued that S , and S , are interacting with each other, because both interact with the mind of some observer via the Wigner potential. The observed correlation between S , and S, is then viewed as a direct consequence
EINSTEIN- PODOLSKY -ROSEN PARADOX
265
of their interaction with the mind of the respective observers. It is argued that both observers must exchange information in order to be sure that their measurements on S , and S2 concern two systems which have a common origin, and this exchange should be responsible for the correlation. The physics of the EPR paradox has also recently been re-examined by Kellett (1977). It is argued that the EPR argumentation against the completeness of Q T is unsatisfactory on two grounds: (1) The EPR argument is invalid as it stands because it is based on a Gedankenexperiment which is physically unrealizable [however, see Bartell, 1980b, Section II,A and Section 11,F); (2) The basic EPR assumptions are equivalent to assuming that the QM description of physical systems is independent of any observation or measurement. It is argued that the present experimental evidence for the violation of the BI (Section IV) rules out these basic assumptions, hence make invalid the EPR argumentation itself. Kellett admits, however, that the concept of electron “in itself‘’ is not necessarily meaningless and that individual electrons may well exist. The difficulty is that up to now no theory for such individual systems exists and that questions pertaining to such systems are not relevant to the state vector II/ of QM. The completeness claim of Q M is considered as being the claim that all information that is necessary to any future observation is already contained in I). Hence, I) is a summary of all possible outcomes of future measurements. Nevertheless, in spite of admitting the existence of individual systems, according to Kellett “. . . it is clearly the EPR definition of physical reality that is at fault.” An approach to the EPR paradox that sounds rather strange consists in assuming that influences are not only propagating into the future, but equally well into the past (retroactivity). Adherents of such a view are, e.g., Costa de Beauregard (1976, 1977, 1979), Rietdijk (1978, 1981, 1985), and Sutherland (1983). However, it seems very difficult to grasp the physical significance, if any, of the idea that events or processes that already happened in the past could have been influenced by events that happen just now. The only significance the present author may recognize in such a picture is within a completely deterministic evolution of the world: What happens in the future depends on what happened in the past, and in this way both the future and the past are related. Yet most of us will prefer the picture that all events which will happen later on are influenced by events that happened earlier, and not the other way around. Of course, in such a scheme, knowledge of the present would also provide knowledge of the past, however without making it necessary that influences are propagated backwards in time toward the past. Also Sutherland (1985) has recently shown that such a model is implausible,
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because a new paradox arises from the requirement that a nonlocal influence of a measurement on S,, exerted on the measurement on S,, should be independent of whether S, is inside or outside the forward light cone of S, . Another approach to the EPR problem is that of Destouches (1980), in terms of De Broglie's theory of the double solution. It is shown that within this framework a more general and more satisfactory theory than QT may be set up, satisfying the following set of conditions: (1) the results of QM are recovered, (2) the Bell inequality is violated, (3) the EPR paradox disappears and (4) there do not exist retroactive influences in the sense of Costa de Beauregard (1976,1977,1979), of Rietdijk (1978,1981,1985), or of Sutherland (1983). Other attempts to resolve the EPR paradox are in terms of the density matrix formalism of Jauch (1968) and of Cantrell and Scully (1978). In both cases it is claimed that within this formalism a satisfactory answer to the EPR problem may be given. However, Whitaker and Singh (1982) remark that in this way one is implicitly using the ensemble interpretation of QT, inside which no paradoxical conclusions appear (see also Ballentine, 1970,1972).It is stated that the EPR paradox requires a resolution only within the Copenhagen interpretation (CI) (Stapp, 1972) of QM, because this maintains that it gives a complete (probabilistic) description of individual systems. The EPR paradox has been discussed in different QM interpretations in Whitaker and Singh (1982). Muckenheim (1982) presents the following solution to the Bohm version of the EPR problem. It is based on the allowance of negative probability distributions. It is assumed, in the EPR spirit, that each of the correlated spin4 systems S,, S, has a definite spin direction on its own, e.g., s and -s. Hence for each system it is accepted that spin components along different directions have simultaneous reality, even if these correspond to noncompatible observables, although it is admitted that these cannot be observed simultaneously. Call w,(a,s) the probability for obtaining the result s* = ki along a, with w,(a,s) + w-(a,s) = 1. It is assumed further that w+(a,s)w_(a,s) = (const)a s, such that w,(a,s) = 3 k a s. With s2 = 2, it follows from a s = ( a / 2 ) c o s 8 that the probabilities w+ may become negative. This model allows Miickenheim to reproduce all QM results. For a single system it is found that
-
-
-
(s(a,s))
=
-
s.
p(s)[w+(a,s)s+
+ w-(a,s)s-]dR
=0
(30)
with p(s) = (47c-',a s = &/2)cos0, dC2 = sinOd8dz and s+ = -l while +, for correlated systems the result for the correlation function is P(a,b) = (s(a,s)s(b,s)) = -$a
-b
(304
EINSTEIN-PODOLSKY -ROSEN PARADOX
267
in accordance with standard QM. However, no explanation is given of the physical significance of the negative probabilities. In an interesting paper de Muynck (1985) investigates the relation between the EPR paradox and the problem of nonlocality in QM. In this paper it is discussed, along the lines of Fine (1982a,b) (Section V,C) whether the EPR pr6blem and the violation of the BIs by recent experiments has anything to do with a fundamental nonlocality or inseparability at the quantum level. In this respect it is interesting that de Muynck makes a clear distinction between unobserved, objective EPR reality and observed reality that is described by QT. Because of the absence of any experimentally verifiable consequence on the basis of this EPR reality, the EPR analysis itself is incomplete and on a metaphysical level. Remember also that similar points were stressed when pointing to the necessity of introducing counterfactuality as well in the EPR reasoning (Section II,A,B) as in the Bell reasoning (Section 111,C). In fact it cannot be denied that on the quantum level the result of what would have happened if, instead of an actual experiment, one had carried out another one, with other macroscopic apparatus parameters, is completely unverifiable. de Muynck repeatedly stresses not to forget the central lesson of QM (and of Bohr) that it is impossible to gain knowledge without taking into account the measuring arrangement. However, by remarking that the existence of an objective reality is not excluded at all by QM, de Muynck tries to show that “. .. the positions of Bohr and Einstein with respect to reality are less irreconcilable than is often taken for granted.” Hence, in this article no extreme standpoints are taken and it is a virtue that a reconciliation between EPR and Bohr is strived for. As de Muynck remarks: “. . . Bohr’s completeness claim,. . . restricts the possibility of defining properties of a system . . .,”and: “Einstein, indeed, may be right in pointing at the possibility that we might obtain knowledge about a system exceeding the quantum mechanical knowledge.” Hence, “. .. the Einstein-Bohr controversy is not a matter of principle. It is just about the domain of application of quantum mechanics.” Other proposals for a resolution of the EPR paradox and of the problem of locality in relation with the Bell inequalities will be discussed in Section II1,G. A generalization of the EPR paradox by Selleri (1982) and the resulting new Bell-type inequalities will be mentioned in Section III,D,6.
F. Recent Proposals for Testing the Validity of Einstein Locality In Section II,D we have seen that the agreement of the experimental results of Wu and Shaknov (1950) with Q M (and also with the results of more recent similar experiments) was used by Bohm and Aharonov (1957) to rule out the
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hypothesis that a pure state vector evolves spontaneously into a mixture of product states (known as the Bohm-Aharonov hypothesis).Such an evolution is indeed expected if Einstein locality (Hl) is supposed to be valid. Hence the conclusion that (Hl) must be wrong. In Section IV,A we will see that further evidence supporting this conclusion comes from atomic cascade experiments suggested by the Bell inequality, such as the very convincing recent experiment of Aspect et a!. (1982). However, a drawback of this kind of experiment is that the validity of the hypothesis of counterfactuality (H3) has to be assumed in order to justify the combination of results obtained in at least two different experiments. If one neverthelesscontinues to stick to nonlocality,then it appears that the EPR paradox is avoided by introducing another paradox, namely the violation of the relativity principle. Now, eliminating paradoxes by introducing new ones is certainly not a recommended method to solve physical problems. Moreover Fine (1982a,b) and others have recently shown (Section V,B,l) that locality is irrelevant for the derivation of the BI. Therefore, there is some ground to suspect that the resolution of the EPR paradox, which consists in the rejection of the existence of separated entities and a finite speed of propagation of influences,is not the correct one. Bearing in mind that the process of state-vector reduction plays a crucial role in the EPR paradox, we will discuss below two recent proposals of Gedankenexperiments for verifying (non)locality. Although it was not the purpose of EPR to criticize the QM formalism, it will appear that the EPR reasoning may be used to criticize the CI of Q M (knowledge of the result of an experiment is sufficient to determine the subsequent state vector)as it is applied to the case of two correlated systems S , and S,. Remember that Margenau’s resolution of EPR (Margenau, 1936) consisted precisely in the rejection of state-vector reduction. In fact in the following we shall assume unlimited validity of the relativity principle (i.e., no action at a distance) and ask what may be wrong in the use of the CI of the quantum formalism to two correlated systems S , and S , . We will try to make plausible that it is indeed the reduction of the state vector for the system ( S , , S,) which may be subjected to criticism.
I . Poppers’s New E P R Experimenl As a first example in which state-vector reduction in the case of correlated systems may be seriously in trouble, we mention Popper’s recent EPR experiment. Recently Popper (1985) proposed a new version of the classic EPR Gedankenexperiment. As in the original EPR paper, he is mainly concerned with showing that physical systems may have both position and momentum at
EINSTEIN-PODOLSKY -ROSEN PARADOX
*L
BL
I
TY
I
269
*R
FIG.2. Setup for Popper's new EPR experiment.
the same time, and hence follow a (classical) trajectory. However, we will use this proposal to criticize state-vector reduction in the case when only one measuring device is in place. Popper's new proposal is essentially as follows (Fig. 2): 0 is the origin of correlated systems ( S , , S,) which move in opposite directions: S, moves to the right towards a screen A, which has a slit of extension Ay, while S, moves to the left. Because of their common origin, S, and S, may be represented by a state vector which reflects their correlation I$S1S$O))
=
s
14Y(1))l -d,YfZ) = - Y 9
x (d,ycl'; -d,y(,) = - ~ ' " ~ $ ~ , s , ( t o ) )dy'"
(3 1)
It follows that if at t = t o S , is observed at (d,y(')), then at this moment the state of Eq. (31) reduces to Id,y('))l -d,y'" = -y(')). This means that at t = to the position of S , will be (- d , - y'") if this were actually measured. In particular, those systems S , passing through the slit Ay will be diffracted (to be observed at 4)as a result of the Heisenberg relation Apy x h/Ay; with the knowledge of the position of S, with an accuracy Ay corresponds an uncertainty Apy of its corresponding momentum component. Now, because of the correlation, the same must happen with S,; indeed, the resulting statevector reduction provides us with the knowledge of the position of s, with an accuracy of the same order Ay. Hence there must be, according to standard QM, an uncertainty Apy of the momentum, which means that S, will be diffracted too (to be observed at screen B,), even when the screen A, with a slit Ay is not in place. If this were actually the case, this would be a direct confirmation of the validity of state-vector reduction in the case of correlated and widely separated physical systems, and as a consequence also a strong argument for the existence of action at a distance. Thus the physical interaction of S, with the slit in A, would be responsible for both the diffraction of S , and, by a nonlocal influence, for a diffraction of S, at the imagined screen A,. On account of separability and locality, however, diffraction of S, is not expected
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W. DE BAERE
to occur (at least as long as A, is not in place);to all observations of diffraction on 4 should correspond observations on B, in a small region around the x axis. In a quantitative analysis of Popper's proposal, taking into account the initial (inevitable) uncertainties at the source, Bedford and Selleri (1985) have shown that the above experiment is possible in principle under certain conditions. With annihilation photons, for example, the uncertainty at the source is such that an observed effect on S , cannot be related to the observed diffraction of S , . In the case of photon emissions in opposite directions in the center of mass, however, with sufficiently large mass of the emitting source, a relation between both diffractions should be possible. Indeed, these authors have shown that the angular deviation between the momenta of S, and S2 in the laboratory system is inversely proportional to the mass of the source, hence in principle may be made arbitrarily small. 2. Double-Slit Experiment with Correlated Systems
As a second example of a situation in which troubles with state-vector reduction in the CI of QM may be expected, we discuss a double-slit experiment connected with an EPR-type situation (Fig. 3) (De Baere, 1985). 0 is the source of two correlated systems S,, S,. S, moves towards a screen A , which contains two slits s1 and s2 . S, moves toward a screen A , where its position can be observed. To the right of A, we have a third screen 4 where S, may be observed, in the case S , has passed A,. Suppose further that all relevant parameters (distances OAR,OA,, slsz, momentum of S,) are such that on 4 an interference pattern is observed in the case that 0 is the source of systems S, only. The question we are interested in then is the following: If S, is observed at the screen A, at about the moment that S , reaches A,, will the subsequent interference pattern at 4 subsist or will it disappear (as demanded
"i FIG.3. Double-slit arrangement with correlated systems.
EINSTEIN-PODOLSKY -ROSEN PARADOX
27 1
by the CI)? Again, the argumentation of Bedford and Selleri may be used here to conclude that, with each impact of S , on A,, there must correspond a unique position of impact of S , on A,. To answer the above question, let us look at the QM description of this situation. The correlated couple ( S , , S,) is again described by the state vector of Eq. (31). Suppose we observe the impact of S, at A, at t = t o . At this moment the state vector I$s,s,(to)) reduces to Id, y(l) = -y(”)1 -d, y‘”). If it happens that y(,) = y:) = -s then we may say that observation of S , on A, has indirectly determined the slit s, through which S, will move. More precisely, if a counter were placed behind the slit sl,S1 would have been observed by it. In any case (counter behind s, in place or not), by the observation of S, on A, the CI assures that at t = to the state vector for S, is Id,y(’) = s). Having determined in this way the slit s, through which S , will pass, standard QM predicts that the corresponding part of the incident S, beam will give rise to a diffraction pattern at &. The same can be said for observations of S, at A, for which y(’) = yji’ = +s. For those cases, statevector reduction steers the corresponding s, systems in the state vector Id, y“’ = - s) (which corresponds in the usual terminology to those systems S , which “pass” slit s,), which again will give rise to a diffraction pattern at 4 . For all other observations of S , at A,, the original state vector will reduce to a state Id, y ( ’ ) # fs), which will correspond to absorption (or at least an impact) of the corresponding systems S , on the left side of screen A,. In any case, these systems will not pass either through slit s, or s,, and hence cannot contribute to the final pattern observed on &.All this can be visualized by imagining that photographic plates are mounted on the appropriate sides of screens A,, A,, and &. Therefore, one may expect that, under the validity of the QM process of state-vector reduction (as used by EPR and in the CI), the pattern observed at 4 will be a superposition of two diffraction patterns, each corresponding to observations of systems S, at positions s; and s; . On the contrary, under the conditions of Einstein locality and separability, a measurement on S, will not disturb or influence the conditions within S, at the moment it interacts with the screen A,. It follows that the interference pattern at 4 will subsist in this case, independently of what is done with S,. If this turns out to be the case experimentally, then this would be evidence for our conjecture that in the case of correlated systems S , , S,, observation on S , does not steer S, in some QM state, unless an appropriate measurement apparatus is present. At the same time this would constitute a resolution of the EPR paradox (see also Table 11). However, we want to remark here that for single systems the so-called noresult Gedanken experiment of Renninger (1960) may be used to gain information about the system (and its wave function) without subjectingit to a measurement apparatus.
272
W. DE BAERE
TABLE I1 SG,(a) I N PLACE,SG,(a) IN PLACE:EPR PARADOX Time t
< to
t
=
to
t 2 to
S,
S,
State vector of (Sl,S2):100) m,,(a, t < t o ) = ? m,,(a, t < to) = -m,,(a, t < t o ) = ? m,,(a, t o ) is measured: S, moves through SG,(a) (physical, local interaction) Reduction of 100): if m,,(a, to) = + 1: 100): reduces to 1)~’: for S , -+ State vector for ( S , , s , ) : lu!,:’)Iu~?!)
ms,(a, t o ) is measured: S, moves through SG,(a) (physical, local interaction)
Inferences ms,(a,t 2 to) = 1 No inference about S, for t < to
+
m,,(a, t o ) = - 1: 100) reduces to for S ,
Iub!!)
ms,(a, t 2 t o ) = - 1 No inference about S, for t < t o 3 No EPR Paradox
Considering the double-slit experiment with a source of single quantum systems, then according to Feynman et al. (1965) this is “. . . a phenomenon which is impossible, absolutely impossible, to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the only mystery. We cannot make the mystery go away by explaining how it works.” Perhaps the proposal made in this section with a source of correlated quantum systems may help to get some insight in this mystery by allowing for the possibility, via coincidence measurements on A, and 4 , to explain or to follow how the pattern on 4 is built up. Let us note in this respect that recently Wootters and Zurek (1979) studied in a quantitative way, in the context of information theory, Bohr’s complementarity concept and Einstein’s version of the double-slit experiment (with a source of single photons), in which both the path and the interference pattern of the photons are attempted to be observed, it was found that the more information one has about the path, the less one has about the interference pattern. Subsequently Bartell (1980a) made a proposal of simpler realizable systems than that of Wootters and Zurek for observing intermediate waveparticle behavior. It is based on Wheeler’s reformulation of the Einstein version of the double-slit experiment (Wheeler, 1978). G . Conclusion
We have seen that the EPR argumentation [using (Hl), (H2) and (H3)], which resulted in the claim of incompleteness of QM, has been criticized
EINSTEIN-PODOLSKY -ROSEN PARADOX
’
273
rather convincingly, mainly on the grounds of the impossibility in principle of knowing simultaneously noncompatible observables. We have made plausible that this amounts to the invalidity of the hypothesis of counterfactuality (H3), i.e., that one is not allowed to combine in an argumentation actual measurement results and hypothetical ones (supposed to be carried out on the same systems in exactly the same individual states). A disagreement on the observational level exists between QM and what we called the simplified argumentation [using only (Hl) and (H2)]. The discussion above has shown that the problem of QM nonlocality and nonseparability deserves further critical analysis, both on the theoretical and the experimental level. According to a new approach to the EPR paradox (Section II,F), it may be interesting to question the validity of the CI of the QM formalism, especially in its application to correlated but widely separated physical systems.
111. THEBELLINEQUALITIES
A . Introduction
The status of the EPR paradox, after the early criticism, remained unchanged for about 25 years. Several reasons may be given for this. First of all, EPR were not able to elevate their objections to QM from the purely epistemological level to the empirical level. This was done for them by Bell (1964). Moreover, QM predicted with great success and accuracy all results in atomic and molecular physics, and for every proposed new and realizable experiment a definite and unambiguous answer could be given. Hence, there was no practical need at all to supplement this successful formalism by means of hypothetical or hidden variables (HVs). Also, attempts to derive QM from classical ideas (Fenyes, 1952; Weizel, 1953a,b) were in general not very convincing because they did not lead to clear, verifiable differences with QM. As a result of all this, opinion had settled that Bohr’s (1935a,b) reply to EPR was convincing and final. Another important reason for this was the existence of the von Neumann theorem. Indeed, already in 1932 von Neumann (1955)had proved mathematically, starting from a number of plausible assumptions, that no deterministic theory for the individual system, based on the idea of dispersion-freevariables, is able to reproduce all statistical predictions of QT. However, since Bohm (1952b,c) proposed a concrete counterexample of von Neumann’s theorem, and Bell (1966) and Bohm and Bub (1966b) showed that one of the underlying assumptions, namely the linearity hypothesis, was not necessarily valid in all HV theories, the interest in the fundamentals of QM
274
W. DE BAERE
was again awakened. Also, at about the same time, Bell (1964)made his famous analysis of the EPR paradox which consisted in translating the EPR ideas in precise mathematical form and deriving a simple inequality, the Bell inequality. The importance of this BI stems from the fact that it made very clear the conflict between QM and the EPR point of view, and that Clauser et al. (1969)showed that by generalizing the BI the whole issue could be brought to the experimental level. B. Von Neumann’s Theorem
The essentials of von Neumann’s proof of the impossibility of dispersionfree states are as follows (von Neumann, 1955; Albertson, 1961; Bell, 1966; Bohm and Bub, 1966a; Capasso et al., 1970; Jammer, 1974, pp. 265-278). The starting points are a number of hypotheses: (vN1) To each observable R there corresponds in a one-to-one way a Hermitian operator R in Hilbert space. It is assumed that R is hypermaximal, i.e., that its eigenvalue problem is solvable; (vN2) With the observable f ( R )there corresponds the operator f ( R ) ; (vN3) If to the observables R, S , . . . (not necessarily simultaneously measurable) there correspond Hermitian operators R, S, . . ., then to the observable R + S + ... there corresponds the operator R + S + ” ’ ; (vN4) Hypothesis of linearity: If R, S , . . . are arbitrary observables and a, b, .. . real numbers, then the following relation holds between expectation values (uR
+ bS + ...) = u ( R ) + b ( S ) + ...
(32)
with
(R)
(33)
= CWn($nIRI$n) n
if the ensemble is described by the pure states I$,)
with probabilities w,.
The dispersion A R of an observable R in an ensemble may be defined by (AR)’ = ( ( R - ( R ) ) ’ )
=
( R 2 ) - (R)’
(34) A dispersion-free ensemble is then defined by the condition AR = 0 for all R, i.e. (35) A further definition introduced by von Neumann was that of the homogeneous or pure ensemble. Any partition of such an ensemble results in a series of subensembles, which, by definition, have the same statistical properties as the original one. < R 2 ) = ’,
VR.
EINSTEIN-PODOLSKY -ROSEN PARADOX
275
On the basis of the above postulates and definitions, von Neumann was able to prove his theorem on the impossibility of reconstructing QM from any kind of theory which starts from dispersion-free ensembles, usually identified with a HVT. In the words of von Neumann “we need not go any further into the mechanism of the ‘hidden parameters,’ since we now know that the established results of quantum mechanics can never be re-derived with their help” (von Neumann, 1955, p. 324). The proof proceeds by showing that the assumption of dispersion-free ensembles leads to inconsistencies. In the particular case of spin 3,this proof runs as follows. Consider a dispersion-free ensemble of spin-3 systems corresponding to a definite HV 1.For such an ensemble, each observable has a well-defined value when measured. Consider in particular the observables R = ox, S = oy,and T = R + S = ox + ay,with a,, oy,and a, the Pauli matrices. Then the results of measuring R, S, and T on each element of the ensemble may be represented by r(A), s(A), t(A). Hence
(R)
= r(4,
( S ) = s(4,
( T ) = t(4
(36)
and, according to (vN4), one should have (R
or
+S) =(T)=(R) +(S) t(2) = r ( 4 + s(1)
(37)
Now, because of the fact that eigenvalues of R, S, and T are, respectively, 1, f 1, and Eq. (37) cannot be true, from which follows von Neumann’s impossibility theorem. Challenged by these assertions, Bohm constructed in 1952 (Bohm, 1952a,b) a concrete counterexample of von Neumann’s theorem. In Bohm’s model, which is a reinterpretation of QM, the notion of quantum potential is introduced, and position and momentum are treated as hidden variables. Subsequently, Bell (1966) and Bohm and Bub (1966a) criticized the general validity of von Neumann’s hypotheses, in particular the hypothesis of linearity (vN4). Bohm and Bub (1966a) came to the conclusion that only a limited class of HV theories was excluded by von Neumann’s theorem [in fact, only those satisfying the von Neumann assumptions (vNl)-(vN4)]. Moreover, Bell (1966)explicitly constructed a physically reasonable HV model for spin-4 systems, which did not satisfy (vN4). Bell showed (Selleri, 1983b, pp. 49-52) that, although (vN4) is correct for standard QM states, it is unjustified to require it to be valid for all conceivable HV models. Other impossibility proofs for HV theories were set up by Gleason (1957), by Jauch and Piron (1963), and by Kochen and Specker (1967). All these are apparently more general than von Neumann’s proof and are not based upon the criticized hypothesis (vN4).The proof of Jauch and Piron has in turn beer,
276
W. DE BAERE
refuted by Bohm and Bub (1966b). The general problem with proofs of this kind is not their mathematical correctness but the physical relevance of their basic assumptions. For this reason it may be stated that no such proof will ever be able to rule out a priori all possible HV models. For more details we refer to the already mentioned papers, Clauser and Shimony (1978)and Jammer (1974, pp. 296-302).
C . The Original Bell Inequality 1 . Derivation and Discussion
In 1964 Bell re-examined the Bohm version of the EPR argumentation. Bell investigated the properties of any local deterministic HV theory which satisfies the requirements of EPRs realistic conception of reality. The main result is the famous Bell inequality, which has to be satisfied by any such theory. The fact that the BI is violated by QM constitutes Bell's theorem: No local deterministic HVT can reproduce all results of QT. To prove this, a couple of correlated spin-+systemsS,($), &(+) in the singlet state is considered, and it is assumed that the internal state of each system may be represented, because of their correlation, by some common set of timedependent parameters (HVs),collectively denoted by A(t). Suppose that s, and S2 move in opposite directions toward Stern-Gerlach devices SG,(a) and SG2(b)which measure the spin component m,, along unit vector a and of mS2 along unit vector b. Bell then remarks that, under the validity of Einstein locality (separability) (H l), the value of m,, is independent of band the value of ms2is independent of a. Furthermore, these values are assumed to depend on the HVs A(t), representing the individual state of S, and S 2 . This suggests the existence of functions msl(a,A(t)), ms2@,A(t)) which completely determine the spin projections (whether measured or not). Hence the basic assumption is that at any time t , S , and S, have correlated properties which are definite and locally determined (in the above sense),in agreement with the EPR idea of elements of physical reality. To simplify notation we shall write A(a,A) = m,,(a, A)
=
k 1,
B(b,A) = m,,@,A)
=
k1
(38)
and adopt the convention that the spin projections are k 1 instead of i-3 (in units of h). The central quantity in Bell's reasoning is the correlation function A(a, A)B(b,A ) p ( l )dA
(39)
277
EINSTEIN-PODOLSKY-ROSEN PARADOX
in which A represents the whole HV space, and p(A) a normalized distribution function of the HVs il (40) Now, the QM prediction for the correlation function P(a, b) is (Peres, 1978)
. -
P(a, b) = (OOJa, aa, b100)
= -a
-b
(41) In Eq. (41), 100) is the singlet state vector of Eq. (15) and a, a,@, b are the operators for the spin projections along a and b. In particular, one has P(b,b) = - 1. Hence, if the HV expression in Eq. (39) reproduces this result, then one should have that B(b,A) = - A @ , A) and Eq. (39) becomes P(a,b) = -
I
A(u,il)A(b,A)p(A)dA
.
.
(42)
Bell then goes on to reason along the lines of EPR and considers the hypothetical situation that instead of analyzer direction b one had chosen another direction c. Now, it is seen immediately that at this point,just as in the EPR argumentation, the validity of counterfactuality (H3) is needed to allow such reasoning. As remarked while discussing the EPR argumentation, QM forbids, however, in principle such a reasoning because the spin projections B(b,A) and B(c,A) are incompatible quantities, and hence cannot be known simultaneously because of the practical impossibility of measuring them together in one single measurement. We believe that this is a central lesson from the successful QT, that one necessarily has to incorporate in any future theory. Simultaneous measurements of incompatible observables was studied long ago, e.g., by She and Heffner (1966), Park and Margenau (1968), and recently by de Muynck et al. (1979) and by Busch (1985).In de Muynck et al. it has been shown that a simultaneous measurement of such observables causes an unavoidable mutual disturbance of both results; i.e., the value of B(b,A) will depend on whether it has been measured alone or together with B(c,A) (after having given an operational meaning to such combined measurements). Otherwise stated, the joint probability distributions will not reproduce the marginal distributions. We will see in Section V,B that this is precisely a point in the derivation of the BI which is severely criticized. Yet, to make physical sense of the above counterfactual hypot!iesis (Lochak, 1976; De Baere, 1984a,b), it may be assumed that in similar correlation experiments (which may be carried out either simultaneously at different places, or subsequently at the same place) the HVs are distributed according to the same HV distribution p(A). Under this assumption one may
278
W. DE BAERE
write then, according to Eq. (42)
h lA
A(a, A)A(c,A)&)
P(a, c ) = -
dA
(43)
Subtracting Eq. (43) from Eq. (42) one obtains P(a, b) - P(a, c) = -
= --JA
Because A(a,A)
=
CA(a,4A(b,4- 4, A)A(c,4 l P ( 4
&a,
(44)
& 1, B(b,A) = f 1, it follows that
I
IP(a, b) - P(a, c)l I
or
W ( b ,4L-1 - A@, AMc, AllP(A)dJ.
[1 - A(b, A)A(c,A)]p(A)dA
(45)
+ P(b,c)
(46)
IP(a,b) - P(a,c)l I 1
This is the original form of Bell's inequality, which expresses in very simple mathematical terms the consequences of Einstein locality (H 1) and counterfactuality (H3). Equation (46) is the inequality that each local, deterministic HVT in which counterfactuality is valid has to satisfy. It is easily seen that Eq. (46)is violated by the QM prediction in Eq. (41).To this end it suffices to take the following configuration of spin analyzer directions (Fig. 4)
O,, = O,,
= O,,
f 2 = 60"
(47)
for which P(a,b) = P(b,c) = -P(a,c) = -3
(48) Inserting Eq. (48) into Eq. (46), one gets the contradiction 1 5 3 . From this result follows Bell's theorem: No local deterministic HVT is able to reproduce
FIG.4. Configurationof orientations for which the BI is violated by QM.
EINSTEIN-PODOLSKY -ROSEN PARADOX
219
all predictions of QT. We will see in Section IV that most correlation experiments verify QM, and hence violate the BI, from which it is concluded almost generally that the locality hypothesis (H 1) is invalid. In other words, it is believed that any acceptable HVT which does not violate QT a priori should be nonlocal. However, we have seen that Bell implicitly assumes the validity of counterfactuality (H3) as self-evident. From the above conclusion of nonlocality, it may be argued then that perhaps the importance of counterfactuality (H3) is being overlooked. It may be stated, therefore, that locality may be saved if counterfactuality is abandoned. 2. Wigner‘s Version of Bell’s Theorem
Wigner (1970) considers two correlated spin-$ systems S,($), S,($) in the singlet state and assumes that the HVs A determine the spin components in any number of directions. Wigner’s argument uses only three directions a, b, and c. For a given 1, and under the condition of locality, the HVT gives the predictions A(a,A) = a, = & 1, A(b,A) = b, = k 1, and A(c, A) = c, = k 1 for the spin components of S , and predictions B(a,A) = a, = & 1, B(b,A) = b, = L- 1, and B(c,A) = c, = & 1 for S,. Let us denote then by p(a,, bl,cl;az,b,,c,) the probability for having the configuration (al,b,, c,; a,, b,, c,) of spin components for S , and S, . In the singlet state one must have that a, = -a,, b, = -b,, c, = -cl. Again QM forbids from first principles that the above characterization be verified experimentally. Indeed, Heisenberg’s uncertainty principle forbids the simultaneous measurement, hence knowledge, of spin components such as a,, b,, c, for S, or of a2,b2,czfor S,. Refusing to accept this basic truth will again necessarily lead to contradictory conclusions, as we will see. Also, and for the same reason, only one couple of values such as, e.g., a,, a, or b,,c, will represent values that correspond to actual results, while all the other combinations such as b , , a , , etc. are then hypothetical. Hence, as in all other derivations of Bell-type inequalities, the validity of counterfactuality is already built in implicitly. To see that the above HV scheme contradicts QM, let us derive probabilities from p(a,, b,, c,; a,, b,, c,) which are experimentally verifiable and, hence, for which QM makes specific predictions. Consider then the probability p ( a , + ,b 2 + )for the results a, = 1, b, = 1
+
+
p ( a , + , b , + )= ~ ~ ( ~ , + , ~ , ~ ~ = ~ -l C J~ ~ z - ~ ~ z + ~ ~ c1
and thus one should have from Eq. (50)
which is not satisfied if c bisects oab. Wigner’s argumentation has been criticized by Bub (1973), who shows that the assumption of a probability distribution for the values of incompatible observables leads to inconsistencies in the case of single spin-4 systems. Freedman and Wigner (1973) reply that Bub’s criticism is based on a misinterpretation of Bell’s locality postulate.
D. Generalized Bell Inequalitiesfor Dichotomic Variables It has been remarked that a weak point in the derivation of the original BI is the requirement that the condition P(a,a) = - 1 has to be satisfied exactly. Now, no experimental setup is perfect; so that this condition will never be realized in practice. The solution to this problem led to the first derivation of the generalized Bell inequality (GBI) by Clauser et al. (1969). It has been argued, furthermore, that neither the restriction to a deterministic scheme nor the use of the HV concept is necessary for the derivation of generalized forms of the BI. It appears that even the concept of locality may be defined in different ways with respect to Bell’s inequalities (Eberhard, 1978; Rastall, 1981; Bastide, 1984; Stapp, 1985). As a result, a large number of generalized Bell inequalities (GBI) has been constructed since the appearance of the original BI, the most well known of which we will review below.
EINSTEIN-PODOLSKY -ROSEN PARADOX
28 1
I . Generalization by Clauser et al. Clauser et al. (1969) were the first to derive a GBI and to show how it may be transformed so as to allow direct experimental verification. Consider again the correlation function P(a, b) defined by Eq. (39). With the notation and conventions of Section II1,C one may write the following inequality for IP(a, b) - P(a, b’)l IP(a,b) - P(a, b’)J I
= =
=
I I
IA(u, l)B(b,A) - A(a, l)B(b‘,l ) l p ( l )d l I A k , WW,4lC1 - B(b,W ( b ’ ,m(4 dl C1 - B(b,
1-
WW, 4 l p ( 4 dl
B(b,l)B(b’,l ) p ( A )d l
(53)
If the criticized condition A(b,A) = - B ( b , l ) were valid, Eq. (53) would lead immediately to the original BI, Eq. (46). However, suppose that we now have
P(u’,b) = 1 - 6,
0I 6 I 1
(54)
for some directions a‘ and b, and such that 6 differs from zero for a’ = -b. Writing A = A+ u A - , with A+ = (A1 A(a’,A) = &B(b,A))
(55)
it may be shown that
With this result the right-hand side of Eq. (53) may be written B(b,WV’,M 4 d l =
P(a’, b’) - 2
J1,-
A@’, /Z)B(b’,&(A)
dl
IA(a’,A)B(b’,I)(p(A)dA = P(u‘,b’) - 6 = P(u’,b‘)
+ P(u’,b) - 1
(57)
282
‘i= W. DE BAERE
+
450
2
-a‘
FIG.5. Analyzer directions for which the GBI is violated by QM.
From Eqs. (53)and (57) it follows that IP(u, b) - P(u, b’)l
+ P ( d , b) + P(u’,b’) I2
(58)
This is the GBI, also called the Clauser-Horne-Shimony-Holt or CHSH inequality, which again is violated by QM; e.g., for %,b = %,,./3 = Oarb= %arb, = 45” (Fig. 5) one has the contradiction 2& I2. The GBI of Eq. (58) may be transformed to an inequality which contains only directly available experimental quantities for correlated optical photons yl, yz, originating from an atomic cascade. In this case the Stern-Gerlach apparatuses SG,(a), SGz(b) are replaced by linear polarization filters Pl(a), Pz(b). Normally A(a) = + 1 would correspond to the detection of y , which passed P,(a), and A(a) = - 1 would denote nondetection. However, because of problems connected with small photoelectric efficiencies for optical photons, another convention must be introduced: A(a) = - 1 means emergence from P,(a) and A(a) = - 1 nonemergence. With this convention, P(a, b) is an emergence correlation function, and in order to relate it to experimental data one has to introduce the supplementary assumption that the probability for coincidence detection of y,, y, is independent of a, b. If, furthermore, the following experimental counting rates are introduced: R(a,b) = rate of coincidence detection of y1,y2 with Pl(a),P,(b) in place; R, = R,(a) = rate of detection of y , with P,(b) removed, assumed independent of a; R, = R,(b) = rate of detection of yz with P,(a) removed, assumed independent of b; then Eq. (58) may be transformed to IR(u,b) - R(u,b’)l + R(u’,b) + R(u’,b’) - R , - R , I 0
(59)
For R(a,b) = R(a - b) and relative polarizer orientations la - bl = la - b‘1/3 = la’ - bl = la’
-
b’l
= ~p
(60)
283
EINSTEIN-PODOLSKY -ROSEN PARADOX
Eq. (59) simplifies to 3R(q) - R ( 3 q ) - R , - R , I 0
(61)
Defining
Ncp) = C3R(cp)- R(3cp) - R ,
-
(62)
R,l/Ro
with Ro being the rate of coincidence detection of yl, y z with Pl(a), P,(b) removed, Freedman and Clauser (1972)arrived at a lower bound - 1 for A(cp). Hence, instead of Eq. (61) one may write finally -
1 I [3R(cp) - R(3cp) - R1 - R , ] / R o I 0
(63)
Now, it may be shown that if the QM prediction for the correlation function is of the form r + scosncp, then there is maximal violation of the upper limit of Eq. (63) for ncp = n/4 and of the lower limit for ncp = 3n/4. In the case of correlated cascade photons one has n = 2, and these angles become, respectively, cp = n/8 and 3n/8. Inserting these two values for cp into Eq. (63),and noting that R(9n/8) = R(n/8),one has
- 1 5 [3R(n/8)- R(3n/8) - R1
0
(64)
[ 3 R ( 3 ~ / 8-) R(n/8) - R1 - R , ] / R o I 0
(65)
- R,]/Ro I
and -1I
By subtracting Eq. (65)from Eq. (64),both inequalities may be combined to one single inequality
a
(R(22.5")/R0- R(67.5")/R0(- I 0
(66)
which no longer depends on R , , R , . Actually it is Eq. (66) which has been tested in most cascade-photon correlation experiments. 2. The Proof by Bell
In his own derivation of the GBI, Bell (1972) starts from the idea that the respective Stern-Gerlach apparati SG, (a)and SG,(b) carry their own hidden information, which may be represented formally by HVs A,, & with domains A,, I \ b and normalized distributions pu(Aa),p b ( & ) satisfying JAa
and n
pa(Au)
d l a = 1,
la
E Aa
(67)
284
W. DE BAERE
The measurement results are now A(u,A,&) = f 1,O and B(b,A, &) = the value 0 assigned to A or B if S1 or S2 is not detected. For the correlation function P(a,b) we now have
+
- 1,0,
with
44 4= B(b,A) = and
1. 1.,
N u , A,&)pa(Aa)
(704
B(b,A, i b ) p b ( & ) d&
(70b)
I&&i)I I 1, lB(b,i)l I 1 (71) If for SGl(a),SG2(b)alternative settings a’ and b’ are considered, then one may write P(a, b) - P(a, b’) =
=
I
[A(a,i)B(b,A) - A(a,A)B(b’,i ) l p ( l )dA
b
A(a,A)B(b,A)[ 1 f A@’,i)B(b’,i ) ] p ( i )d i
A(a,i)B(b’,i)[l & A(a’,i)B(b,A ) ] p ( A ) d l (72) -
or, because of Eq. (71) IP(a,b) - P(a, b’)l I
I
k A(a’, A)B(b‘,i)]p(i) d A
+
c1
[l
jAf
A(a’,i)B(b,41p(A)dA
< 2 f [P(a’,b’) + P(a’,b)]
(73)
which finally may be written
+
IP(a,b) - P(~,b’)l IP(a’,b’)
+ P(~’,b)lI 2
(74) This form of the GBI is similar to that first derived by Clauser et al. (1969),
EINSTEIN- PODOLSKY-ROSEN PARADOX
285
Eq. (58). It is seen that, as a result of the conditions in Eq. (71), the GBI of Eq. (74) has to be satisfied by any stochastic HVT for which locality and counterfactuality are assumed to be valid. 3. The Proof by Stapp In his version of Bell’s theorem, Stapp (1971,1977) considers an ensemble of N correlated spin4 systems S,(i), S2($) moving toward Stern-Gerlach devices SG,(a), SG,(b). We will review this version in some detail because it is claimed and believed (DEspagnat, 1984) that the concept of HV is not needed. However, we will see that this is not correct, because the HVs I may be considered as a formal representation of some elements in Stapp’s argumentation. It is assumed that each device has two alternative settings: a, a’ for SG,, b, b’ for SG,. Call Aj(a,b) the prediction of a hypothetical, more complete theory for the measurement of the spin component of the jth system S , along a, and likewise for Bj(a,b). If Aj(a,b) and Bj(a,b) are actually measured, then Stapp supposes that the theory predicts definite values for the numbers Aj(a,b’), Bj(a,b’),Aj(a’,b), Bj(a’,b), Aj(a‘,b‘),and Bj(a’,b‘) which correspond to alternative settings (a, b’), (a‘,b) and (a’, b’) of the Stern-Gerlach devices. Moreover, it is assumed that these numbers would have been the measurement results if, instead of the couple of actual settings (a,b), one of the three alternative possibilities were chosen. This assumption is the essence of what Stapp calls the hypothesis of counterfactual definiteness, and he further assumes that QT also makes correct predictions for these hypothetical measurement results. Now, if Aj(a,b) and Bj(a,b) are measured, then according to QT Bj(a,b’) can never be measured simultaneously with Bj(a,b) (unless b and b’ are parallel). Hence, if the values Bj(a,b’),j = 1,. . .,N are to be used to define a correlation function whose value is determined in a real experiment, then the hypothesis of counterfactuality is equivalent to the following assumption. Suppose that in one actual experiment with settings (a,b) we have obtained N couples of results Aj(a,b),Bj(a,b), j = 1 , . .. ,N and from another actual measurement with settings (a,b’) we have obtained the series of results Aj(a,b’),Bj(a,b’). Counterfactuality then assumes that between the [ ( N / 2 ) ! I 2 permutations which make the series Aj(a,b‘), j = 1 , . .. ,N coincide with the series Aj(a,b), j = 1,. ..,N there is at least one permutation such that all internal conditions (which are supposed to be responsible for each actual result) within S, and S, coincide exactly in both series. It is readily seen that these “internal conditions” may be identified with the former concept of HV 1. Hence, although HVs do not appear explicitly in Stapp’s reasoning, they are implicitly contained in it. Note that only when the
286
W. DE BAERE
above condition is satisfied does it have sense physically to consider what would have been the result if instead of an actual apparatus setting, one had chosen another possible one. Suppose then further that such a reasoning is allowed. Consider then the series of results Aj(u,b),
j = 1,. ..,N
Bj(u,b),
Aj(u,b’),
Bj(u,b’),
Aj(U’,b),
Bj(U’,b)
A ~ ( u b’), ’,
j = 1,. . ., N
.. ,N = 1,. .. ,N
j = 1,.
B~(u’, b’),
j
(754 (75W (754 (754
If the first series are actual results, then the other three are hypothetical, but, on account of the assumed validity of counterfactuality, may be assumed to be equivalent to actual results in some subsequent experiment. Under this condition, then, Stapp introduces locality by the following requirements: j = 1,. . .,N
Aj(u,b) = Aj(u,b’) = Aj(u), A ~ ( u b) ’ , = A ~ ( u b’) ’ , = A~(u’),
Bj(u,b) = Bj(u’,b) = Bj(b), B~(u, b’) = B~(u’, b’) = Bj(b’),
. . ,N = 1,. . . ,N j = 1,. . .,N j = 1,.
j
(764 (76b) (764 (764
The relevant correlation functions are then defined by (for N sufficiently large) 1
N
1
P(u, b) = N j = 1 Aj(u)Bj(b) 1
(774
c N
P(u’,b) = - Aj(U’)Bj(b) N j=1 1
P(u, b’) = -
1N Aj(a)Bj(b’)
Nj=l
1
1N
P(a’,b’) = Aj(U’)Bj(b’) N j=1
(774
Stapp shows then that Eqs. (77a-d) are not compatible with the Q M prediction of Eq. (41) for all conceivable settings a, a’, b, b’. Indeed, for O,, = O”, eaZb= 135”, Oob, = 0”,and Oarb,= 45” one arrives at the contradiction f i I 1. Stapp concludes that the locality conditions of Eqs. (76a-d) have to be false. Recently Stapp (1985) has been defending his counterfactuality hypothesis further. According to Stapp there are two different concepts of locality, both
EINSTEIN-PODOLSKY-ROSEN PARADOX
287
due to Einstein. The first is the one that is used in Einstein’s relativity theory, according to which no physical signal can travel faster than light. The second is the idea that events separated in space may in no way disturb each other, and this should be incompatible with QM. However, both concepts must necessarily have something to do with the propagation of influences, and it is not clear why both propagation speeds should be different, at least not in a unified world view. In Section V we will see, on the contrary, that many arguments point to the validity of locality in both the above senses. 4 . The Proof by Selleri
Selleri (1972) has given a very simple proof of the GBI by using the result [Eq. (53)] of Clauser et al. (1969) IP(a, b) - P(a, b’)(= 1 -
I I
B(b,A)B(b’,A)p(A)d I
(78)
Noting that on both sides of Eq. (78) the sign may be changed and that the right-hand side does not depend on a, one may also write IP(a’,b)
+ P ( d ,b’)l = 1 +
B(b,A)B(b’,A)p(A)d A
(79)
Adding Eqs. (78) and (79), one immediately gets the GBI of Eq. (74). 5 . The Proof b y Clauser and Horne
An alternative to the proof by Bell (1972) of the GBI which is valid for deterministic, as well as for stochastic HV theories, is given by Clauser and Horne (1974). Assume analyzer orientations a and b, and suppose measurements are made on N couples (Sl,S2).Call N , ( a ) and N,(b) the number of counts at detectors D,and D, behind the respective analyzers. Call N,,(a, b) the number of coincidence counts. If N is large enough, then one may write for the probabilities of these counts
p12(a,b) = N,,(a, b ) / N
(804
The internal conditions within the correlated systems S , , S , are again represented formally by means of HVs I with a normalized distribution p ( l ) . Clauser and Horne then assume that knowledge of these HVs and of the
288
W. DE BAERE
analyzer directions determines only the probabilities pl(a, A), pz(b,A), and p12(a,b, A) for counts to be registrated at Dl, at D,,and at both D1 and D,. Now, here locality is introduced by assuming that pl(a, A), pz(b,A) do not depend on b and a and that
4= P l h 4P,(b, 4
(81) Equation (8 1) is known as the Clauser-Horne factorability condition. Also Plz(a9
Pl(4 = P2(W =
Pl&,b) =
I I I
Pl(44 P ( 4d l
(82a)
A M 4d l
(82b)
Plz(a,b,A)P(A)dA
(824
P&
To prove the GBI, Clauser and Horne consider two alternative orientations a, a’ for the first analyzer and two alternatives b, b’ for the second one. From the inequality - X Y Ix y - xy’
+ x ’ y + x’y’ - X’Y - yx I
0
(83)
which is valid for real numbers x , x’,y , y‘, X , Y satisfying 0 I x , x’ 5 X , 0 5 y , y’ IY, it is concluded that - 1 5 p,,(a,b,A) - Pl,(4b‘,A) -P1(U’,4
+ Pl2(af,b,4 + P1z(a’,b’,4
- pz(b,A) 5 0
(84)
Multiplying Eq. (84) by p ( l ) and integrating over A, one gets -
1 5 P l 2 ( 4 b) - P l 2 ( 4 b’) + P1&’,
4 + P l Z b ’ , b’) - P 1 ( 4
-P
2 W
I0
In the above inequality the probabilities may be replaced by the observed counting rates &(a‘), R2(b),R(a,b), etc. defined in Clauser et al. (1969)(Section III,D,l), to obtain R(u,b) - R(u,b’) + R(u’,b) + R(u’,b’) &(a’)
+ R2@)
I1
(86)
which is essentially the same as Eq. (59). As already remarked when discussing the derivation by Clauser et al., to compare Eq. (86) with atomic cascade experiments necessitates, as long as no
EINSTEIN-PODOLSKY-ROSEN PARADOX
289
perfect polarizers exist, the introduction of a supplementary assumption. This is the so-called “no-enhancement’’ assumption, according to which the probability for photon detection with a polarizer in place, e.g., pl(u,A), is not larger than the corresponding probability pl(A) with the polarizer removed
0I p1(a,A) I p1(A) s 1
(874
0 I pz(b, A) I p#)
(87b)
s1
for all I, a, and b. To save locality, the validity of Eqs. (87a,b) in the existing cascade experiments has recently been seriously criticized (Section V,C). The general validity of the factorability condition [Eq. (Sl)] may be criticized (Selleri and Tarozzi, 1980; Selleri, 1982) by making the following remarks. In the above scheme, the correlation function P(a, b) may be written as p(a?b) =
h
PAa+ A)PZ@+? 4- PI@+ 9
-Pk-
9
A)PZ(b+
9
4+ P l k -
9
9
4Pz(b-
A)P,(b-
7
2
4 M A )&
f
with pl(a*, A) the probabilities for A(a,A) = f 1. Now, suppose that A is just a short-hand notation for the set of HVs A’,A“, . . .: A = (A’,A”,. . .). Then one may write
s
P(a, b) = ql(a,A‘, I“,. . .)q,(b, 2,il”,. . .) x p ( X , A”,. . .) dA’ d l ” . . .
(89)
and, integrating Eq. (89) formally over A’, the new expression should also satisfy the factorability condition, i.e.,
s
P(a, b) = rl(a,I”,.. .)rz(b,A”, . . .)p(A“, ...) dI“ dA”’ . . .
(90)
and one should have that rl(u, A”,. . .)r,(b,A“,. . .) =
jql(..r,A!!,. ..)qz(b,Z,i”, .. .)p(A’, A”,. .. ) d X
(91)
290
W. DE BAERE
Dividing Eq. (91) by its a derivative, one obtains r
with
r\(a,I’’,
. . .) = dr,(a,I”,. . .)/da
q\(a, I’,A“, . . .) = dql(u, I’,I”,.. .)/da
(93)
It is seen that the left-hand side of Eq. (92) is independent of b, while its righthand side depends on b. This cannot be true in general; hence the same applies to the factorability condition [Eq. (81)] on which Eqs. (88) and (90) are based. 6 . The Proof by Eberhard and Peres
Like Stapp, Eberhard (1977) and Peres (1978) claim to give a proof of the GBI which starts only from the principle of locality and does not need to introduce the HV concept nor determinism. The validity of counterfactuality is considered as self-evident or at least a “rather natural way to thinking.” To give an outline of the simple proof, we shall use the same notation as in Section III,D,3 and start from Eqs. (77) for the correlation functions and from Eqs. (76), which express the requirement of locality. Eberhard’s derivation is based on the remark that for the j t h event one has Xj
=
Aj(a)Bj(b)+ Aj(~’)Bj(b) + Aj(a)Bj(b’)- Aj(~’)Bj(b’) I 2
(94)
because Aj(a’)Bj(b’) is the product of the other three terms, and Aj = f 1 and Bj = 5 1. It follows from Eq. (94) that 1 -C
N j
Xj = P(u, b) + P ( d , b) + P(a, b’) - P ( d , b’) I2
(95)
which is the generalized Bell inequality. From the violation of Eq. (95) by QT, Eberhard concludes that Eqs. (76) are not correct, hence requiring nonlocality, because he unconditionally assumes counterfactuality to be correct. Eberhard also gives an alternative formulation of Bell’s theorem: It amounts to saying that, for certain relative orientations of four analyzer directions a, a’, b and b, it is impossible to find four sets of actual spin measurement results {Aj(a,b),Bj(a,b ) } , {Aj(u’,b), Bj(a’,b)}, {Aj(a,b’),Bj(a,b ’ ) } , and { Aj(a’,b’), Bj(a’,b‘)}for which Eq. (76) and hence the GBI Eq. (95)are valid.
EINSTEIN-PODOLSKY- ROSEN PARADOX
29 1
However, according to us, in this reformulation of Bell‘s theorem the invalidity of Eq. (95) does not imply the breakdown of locality but rather the breakdown of counterfactuality. A proof of the GBI along similar lines has been given by Peres (1978). Remembering that the derivation of the GBI is the result of combining one series of actual data with three series of hypothetical ones, Peres distinguishes two possible attitudes with respect to the violation of the GBI Eq. (95) by QM: “One is to say that it is illegitimate to speculate about unperformed experiments. In brief, ‘Thou shalt not think.’ Physics is then free from many epistemological difficulties. For instance, it is not possible to formulate the EPR paradox.” At this point Peres refers to the EPR statement “If ... we had chosen another quantity . .. we should have obtained .. .,” which is according to him “The key point in the EPR argument.” The second attitude is: “Alternatively, for those who cannot refrain from thinking, we can abandon the assumption that the results of measurements by A are independent of what is being done by B.” Here A and B stand for SG,(a) and SG,(b) in our notation. 7 . Other Bell-Type Inequalities
We have seen that the hypothesis of locality (Hl) and of counterfactuality (H3) leads in various ways either to the original BI [Eq. (46)] or to the GBI [Eq. (58)l.Although the theoretical and experimental disagreement between these inequalities and QT (Section IV) is already sufficient to discard one of these underlying hypotheses, various people have extended the BI further by constructing what we shall call Bell-type inequalities. In most of these generalizations Eq. (46) or (58) is somehow contained. Pearle (1970) considers n possible analyzer orientations a,, a,,. . . ,an for one measuring device and n orientations b,, b,, . ..,bn for a second one. In terms of a HV A with a normalized distribution p(A), and dichotomic measurement results A(a,,A), B ( b j ,A) = 5 1, a correlation function P(a,, b j ) is defined in the usual way as P ( a i ,b j ) =
Starting from the identity
A(a,,A)B(bj,A)p(A)dA
(96)
292
W. DE BAERE
one easily derives the inequality
+ P(a,,b,) + ... + P(a,,b,)
P(a,,b,)
I 2n - 2
+ P(a,,b,)
(98a)
or n
1 CP(ai,bi) + P(ai+ ,, bill I 2n - 2 + P(a,, b,)
(98b)
i= 1
For equal angles between adjacent vectors in the sequence a,, b,, a,,. Eq. (98b) simplifies to
(2n - I)P(a,b ) I (2n - 2)
+ P(a,, b,)
. .,b, (984
in which a, b represent any two adjacent vectors. Applying the approach of Wigner (1970),DEspagnat (1975) considers an ensemble, for which it is somehow possible to predict the number n(rl,r2,. . ., r m ) of systems having dichotomic values r , , r , , . . .,rm for observables R,, R,, . . .,R,. A correlation function P(ri,r j ) is defined as the mean value of the product rirj. D’Espagnat is then able to derive the general inequality
On the basis of locality and counterfactuality Herbert and Karush (1978) derived the following set of Bell-type inequalities
-(m
-
1) I mP(8) - P(m8) - (m - 1) I 0
-n I nP(8) - P(n8) - n
+ P(0) I 0
(100)
with m representing any odd integer and n any even integer. Another method to generate Bell-type inequalities (Selleri, 1978)is to start from the inequality CaA(a.1)
+ /3B(b,A)+ yA(c,A)]’
2
1
(101) with a,& y = k 1, A(a,A) = f 1, etc., and B(a,A) = -A(u, A). It may be shown that Eq. (101) generates inequalities such as
+ P(a,c) + P(b,c) I 1
(102a)
P(u, b ) - P(u,c) - P(b,c) I 1
(102b)
-P(a,b)
+ P(a,c) - P(b,c) I 1
( 102c)
-P(a,b)
-
P(a,c) + P(b,c) I 1
(102d)
P(a,b)
Garuccio (1978) generalizes this method by considering the inequality [aA(a, A)
+ /?A@, A) + yA(c, A)]’
2 min( k a
k /3 k y)’
(103)
293
EINSTEIN-PODOLSKY -ROSEN PARADOX
in which a, fl, and y are positive, but otherwise arbitrary, and a 2 With these conditions min(_+a_+
a
_+
Y )= ~ (-c(
B 2 y > 0.
+ + y)‘
( 104)
and the following Bell-type inequality may be deduced P(u, b )
+ xP(u,C ) + yP(b,C) I 1 + x - y
(105)
with a = y//? and y = y/a. Other equivalent inequalities may be derived, such that instead of Eqs. (102a-d) one may write
P(u,b)
+ xP(u,C ) + yP(b, c) I 1 + x
y
(106a)
+ x -y -P(u, b ) + xP(u,c)- y f ( b , c ) I 1 + x - y P(u,b ) - xP(u,C ) + yP(b,C ) I 1 + x - y
(106b)
-
P(u,b ) - xP(u,c)- y f ( b , c ) I 1
-
(106c) (106d)
All these inequalities are again violated by QM for certain values of u, b, c. From Eqs. (106a-d) follows an inequality with four correlation functions: From Eqs. (106a) and (106b) we have P(u, b )
+ IxP(u,c)+ yP(b,c))I 1 + x - y
(107a)
and from Eqs. (106c) and (106d), replacing x by x’, y by y’, and c by c’
+
- P ( u , ~ ) Ix’P(a,c’)- y’P(b,c’)l I1
+ X ’ - y’
(107b)
Adding Eqs. (107a) and (107b), one obtains
+ yP(b,c)[ + Ix’P(u,c’) - y ’ f (b,c’)l 2 + x + x’ - y - y’
IxP(a,C )
I
(108)
in which 1 2 x 2 y > 0, 1 2 x‘ 2 y’ > 0. From numerical calculations for variable parameters x, y, x’, y’ it follows that maximal violation of Eq. (108) occurs for x = y = x’ = y’ = 1; i.e., the original GBI is the strongest inequality with respect to the QM singlet state. A systematic derivation of all Bell-type inequalities for dichotomic measurement results has been given by Garuccio and Selleri (1980). They use the probabilistic formulation of Clauser and Horne (1974), in which the correlation function is given by Eq. (88): %(a, 4q2(b,4
P ( 4d i
(109)
294
W. DE BAERE
with
and - 1 Iq,(u,A.) < 1, - 1 < q , ( b , l ) I 1. Suppose we have analyzer directions a,, a,,. . .,a,,, for S1 and analyzer directions b, ,b,, . ..,b, for S,. The intention of Garuccio and Selleri is then to derive inequalities, for all possible linear combinations of correlation functions, of the type
1cijP(ai,b j ) I M
(111)
i,j
in which cij are real coefficients. The result of their study is that appropriate upper bounds M are given by
in which ti = & 1, yli = f 1. Another method to construct all generalizations of the BI has been proposed by Froissart (1981).The method appears to be very general because it covers not only an arbitrary number of orientations of each measuring device, but also an arbitrary number of correlated systems. By introducing a “quality factor” for two-systems inequalities, Froissart shows that the original Bell inequality has the best quality factor. This is consistent with the conclusion of Garuccio (1978). Other Bell-type inequalities have been derived by Selleri (1982) as a result of a generalization of the current versions of the EPR paradox. The original EPR definition (D2) of an “element of physical reality” has been criticized by Selleri because of its insistence on predictability “with certainty.” Selleri argues that, on account of the existence of the Wigner-Araki-Yanase theorem (Wigner, 1952; Araki and Yanase, 1960; Yanase, 1961), it is in principle necessary to have imperfect measurements, if Q T is to be correct. Therefore, it is not justified to base an argumentation on the possibility in principle to predict some physical observable with certainty. Selleri proposes some new, generalized criteria for elements of reality and separability, which are based on the possibility of predicting probabilities pi for measurement results ui of an observable U on objects belonging to a subensemble E’ of an ensemble E, without disturbing the objects. In this case Selleri speaks of a physical property l l ( U ; p , , p , , . . .) of the subensemble E’. Furthermore, it is said that, if another subensemble E“ of E is considered, E‘ and E” are separated, if the physical property l l ( U ; p , , p , , . . .) of E‘ does not depend on what property ll( V; q l , q 2 , .. .) is measured on E“.
EINSTEIN-PODOLSKY -ROSEN PARADOX
295
On the basis of these generalized definitions, Selleri then derives the following Bell-type inequalities IP(a,b) - P(a,b’)l
+ IP(a’,b) + P(a’,b’)l + 81P(a,a)l I 10
P(a, b) = - (P(a,a)la.b
(113) ( 1 14)
Combining Eqs. (1 13) and (1 14), and choosing all analyzer orientations such that maximal violation of the resulting inequality occurs, one arrives at the inequality
2 4
+ 8 I10/IP(a,a)l
(115)
which is violated if IP(a,a)l > 0.923. E. Generalization of the Bell Inequality to Arbitrary Spin
Until now we have considered only generalizations of Bell’s inequality which were concerned with two-state systems or with two-dimensional subspaces of many-state systems for which only dichotomic measurement results are possible. In recent years, however, it has been shown that the conflict between QM and all the above-mentioned Bell-type inequalities is not restricted to spin-$ systems, but may be extended to systems with arbitrary spin s. The first consideration of correlated systems on which nondichotomic quantities A(a, A), A(a’,A), B(b, A), and B(b’,A) may be measured came from Baracca et al. (1976). It was shown that, if values M, and M, exist such that IA(441I MI,
t’@’,4l 4 M ,
IB(b,A)I I M , ,
lB(b’,A)l IM ,
(116)
then the following generalization of the GBI [Eq. (74)] exists: IP(a, b ) - P(a, b’)l
+ /P(U’,b’) + P ( d , b)l I 2M2
(117)
with M 2 = M1M2
However, inequality (1 17) is apparently not a very strong one for the following reason. If A(a,A), B(b, A), . .. are the results of measuring the observables j a, j b,. . . for the spin projections of systems S,, S , along a, b,. . ., then Baracca et al. (1976) show that in the singlet state one has
- -
P(a,b)=(001j-aj-b100) = - * j ( j + 1)a.b
(1 19)
and that this QM result agrees with Eq. (1 17) for all possible orientations a, a’, b, and b’ for j > 5.
296
W. DE BAERE
Yet it appears that observables exist, without having a direct physical meaning such as spin projections, for which the QM predictions violate Eq. (117). Interesting spin-s Bell-type inequalities which are more successful with respect to their violation by nondichotomic observables that have a clear physical meaning were constructed by Mermin (1980). The basic inequality from which Mermin starts is slms,(a)
+ ms,(b)I 2 -ms,(a)ms,(c) - ms,(b)ms,(c)
(119a)
in which mS,(a),m,,(b), and m,,(c) denote spin components of S , along arbitrary directions a, b, and c, s the spin value, and
S1)* als, ms,(4> = ms,(a)ls, ms,(a)>
.
( 120)
etc., with Sf1) a the operator corresponding to the spin projection of S , along a, and Is, ms,(a)) the corresponding state vector. Assuming perfect correlation between S1, S2,for which mS,(a)= - ms2(a),and taking expectation values of both sides of Eq. (119), one has
-
s(OO11(S(’). a - S(’) b)llOO)
-
2 (OO1S(’) as(’) clOO)
+ (001S(’)- b S 2 )- C
l W
(121)
in which (Edmonds, 1957) 1
+S
1
l o o ) = ~ m m s , = - s
Is,ms,(a))ls,ms2(a) = -ms,(a)>(- l)s-msJe) ( 122)
Mermin shows then that for the above expectation values the following results hold true:
-
(001S(” * as(’) b100) = -$(s
+ l)a - b
(1 23a)
and
with direction y perpendicular to the plane (a, b), and the spin components m and m‘ both taken along a. If it is assumed that a, b, and c are coplanar and a and b make an angle 8 + n/2 with c (Fig. 6) then Mermin’s GBI for spin s takes on the form
This inequality has to be satisfied for all 8 if locality is supposed to be compatible with QM. However, it may be shown that Eq. (124) is violated by 8
297
EINSTEIN-PODOLSKY -ROSEN PARADOX *
C
FIG.6. Relative orientiations a, b, and c for which Mermin’s spin-s version of the GBI applies.
values in the range
0 < sin0 < sine,
=
112s
(125)
If x = sin 8, then explicit expressions of Eq. (124) and corresponding values 8, for low spin values are as follows: spin 3:x 2 2 x,
0,
spin 1: 2x2 - x 4 2 x,
e, = 38.170
spin 3: 5x2 - 6x4 + 3x6 2 $x,
0,
= 24.08”
spin 2: 20x2 - 42x4 + 48x6 - 20x8 2 5x,
0,
=
= 90”
17.58”
In the classical limit s + co, the above contradiction disappears and ms,(a), etc., become simply components of a classical angular momentum along a. It may be shown further that in this limit the inequality (119) reduces to s sin 0 2 $ 9 sin 8 and hence is always satisfied. Further work on spin-s Bell-type inequalities has been pursued by Bergia and Cannata (1982), Mermin and Schwarz (1982), and Garg and Mermin (1982, 1985).
It may be observed that all possible BIs and GBIs can be classified according to how they are derived in a deterministic local HVT [in which the results A(a,A),B(b,A) are fixed] or a probabilistic one [in which only probability functions pl(a,, A ) , p 2 ( b i , A) exist]. The relation between both kinds of approaches has been discussed by Garuccio and Selleri (1978) and by Garuccio and Rapisarda (1981). F. Direct Proof of QM Nonlocality
Because of the crucial importance of the issue of locality, it is interesting to note that recently Rietdijk and Selleri (1985) claimed to have proven “that the locality hypothesis is in any case untenable if the predictions of quantum mechanics are all correct.” We shall give below a physical outline of their argumentation.
unpolarized beam
A
I.$>'
No transmission 7
2
To illustrate how the process of state-vector reduction, applied to an ensemble of correlated systems S,, S,, strongly suggests the existence of nonlocal influences, we shall consider the following four steps: Step I. (Fig. 7) An unpolarized beam of spin-$ systems S,($) moves in the + z direction towards a polarizer Pl(+x) with its spin analyzer in the + x direction. Suppose that only systems polarized along this direction are transmitted; i.e., the spin state of the transmitted beam is described by the state vector luili ). A second polarizer P,( - x) will subsequently stop all transmitted systems. Step 2. (Fig. 8) Insert a third polarizer P3(a)between Pl(x)and P,(-x). Now we have luilj)
= cos(8/2)lu!y)
luil?) = sin(8/2)lu;y)
- sin(8/2)lubl))
( 126a)
+ COS(~/~)~U:~!)
(126b)
which means that a fraction c0s2(8/2) of the beam transmitted by Pl(x)will pass P3(a).From Iu!?) = cos(8/2)lui11)
+ sin(8/2)lui1!)
Iub'?) = -sin(8/2)1ui1i)
+ COS(~/~)~U~~!)
(127a) (127b)
it follows that P,( - x) will transmit a further part sin2(8/2); i.e., P,(a) is responsible for the fact that from the beam transmitted through Pl(x)a part cos2(8/2)sin2(8/2) will now pass Pl(-x), instead of zero in Step 1. Hence, it may be concluded that the passage of S, through some apparatus (e.g., a
FIG.8. Unpolarized beam moving toward three polarizers.
EINSTEIN-PODOLSKY-ROSEN PARADOX
299
?”
FIG.9. Two correlated systems moving toward two polarizers with opposite analyzer directions.
polarizer) changes by a local interaction the internal physical conditions within S , . Step 3. (Fig. 9) Consider now a source which produces a beam of spin-* systems S,($) and S,($) in the singlet state IOO), such that S , moves toward a polarizer P,(a), and S2 toward a polarizer in the opposite direction, P2( - a) (Fig. 3). With each S , that passes P,(a), there corresponds an S2 that passes Pz( -a). From the state vector of Eq. (15), it follows that the beams that are transmitted through P,(a) and P2(- a) are described by the state vector Iu;J)luh’l). From Step 2 it follows that the internal conditions within each S2 that has passed P2(- a) are changed by a local interaction between S , and
PZ(- 4. Step 4. (Fig. 10) The same situation as in Step 3, but now P2(- a) has been removed. Quantum mechanically we have the same description as in Step 3: The statistical properties of the couples ( S , , S,) for which S , passed P,(a) must again be calculated by means of the reduced state vector Iub?)lub”l). Now we have that S , has changed locally by passing through P,(a), and apparently this process has steered the correlated system S2 such as to become a member of an ensemble described by a well-defined state vector, namely, Iuh?). The difference with Step 3 is that there a local physical interaction was responsible for this steering, while here this is lacking. Rietdijk and Selleri argue that the only way to avoid the conclusion that these systems S, were already described by lub2!) before S , passed through P,(a) is to assume that
FIG. 10. Two correlated systems; only one polarizer is present.
300
W. DE BAERE
precisely at this moment the internal conditions within S, are changed nonlocally (i.e., by action at a distance), so as to become described by luL2!). In this way state-vector reduction in the case of correlated systems strongly suggests a nonlocal behavior at the quantum level. G . Recent Developments
An understanding of the origin of the BIs or of the nonlocality of QT in terms of Einstein’s own unified field theory has been attempted by Sachs (1980) and by Bohm and Hiley (1981). It is well known that Einstein’s attempts to construct a fundamental theory of matter are based on an extension of his general relativity and on a representation of matter and radiation by a continuous field, obeying deterministic nonlinear field equations. It was hoped that the probabilistic and alleged nondeterministic predictions of QT could be derived somehow from such a deterministic and classical scheme. It is argued by Sachs that in his own approach Einstein finally rejected the concept of “Einstein locality,” in the sense of the existence of physical systems which could be supposed to be described independently of anything else in the world, in particular of measuring devices. Hence, according to Einstein, the idea of separability of physical systems should not be considered as a fundamental one (Einstein, 1949). Sachs then goes on to discuss the conflict between QT and the BI within his own version of the above deterministic scheme, in which a maximum velocity c (the velocity of light) for the propagation of physical influences is built in, and which reproduces QT in the linear limit. A similar attempt at understanding the Q M nonlocality along the same lines comes from Bohm and Hiley (1981). After recalling that Einstein rejected the “. . . fundamental and irreducible feature of the quantum theory, i.e., nonlocality,” and describing Einstein’s views on locality, these authors propose a way how nonlocality could follow from Einstein’s approach. Remembering that in Wheeler and Feynman’s absorber theory of radiation (Wheeler and Feynman, 1949) a photon is emitted when there is somewhere matter to absorb it, Bohm and Hiley suggest a similar mechanism in the case of correlated photons. If the analyzer directions are a and b, then it may well happen that these circumstances help to determine the way the photons leave the source. In this way they are led to assume that the distribution of HVs will no longer be represented by p(A), but instead by p(a, b, A). And this would lead to an HV framework in which the usual BIs can no longer be derived. Cramer (1980) constructs a QM version of the Wheeler-Feynman absorber theory (Wheeler and Feynman, 1949) as an explanation for the nonlocality which follows from the theoretical and experimental violation of the BI. It is shown that within this approach there can be a nonlocal
EINSTEIN-PODOLSKY -ROSEN PARADOX
30 1
communication between measurement devices separated in space, so as to produce a violation of the BIs. In recent years an interesting connection has been revealed between the existence of a joint probability distribution for the values of some observables and the validity of the BI. As in Clauser and Horne (1974) (Section III,D,5) let us introduce the experimental probability distributions plz(a, b), p , ,(a, b’), plz(a’,b), plz(a’, 0 p l ( 4 pda‘), p,(b), and p,(b’) for spin projections or polarizations of systems separated in space S,, S, along directions a, a’ and b,b. Then Fine (1982a,b)has proved that in any theory (with or without HVs, deterministic or stochastic, local or nonlocal) whose framework provides a joint probability distribution p(a,a‘, b, b’) for the values a *, a;, b +, b+ ( = f1) of the respective dichotomic observables, such that the above experimental distributions are obtained as marginals of p(a,a’, b, b’), the GBI [Eq. (85)] should hold: -1
Plz(a,4 - Pl2@,6’)
+ Plz(a’,b) + P,&’,b’)
-P l ( 4
-PAN I 0 (128)
It may be shown that by appropriate interchanging of the parameters in Eq. (128) seven similar equivalent inequalities exist. Writing for the previously defined correlation functions P(a, b) P(a,b) = P I Z ( Q + , ~ + ) Plz(a+,b-) - P I Z ( ~ - , ~ + +) ~ i z ( a + , b + )(129) and similarly for the other correlation functions P(a, b‘), P(a‘, b), and P(a’,b’), one may arrive (de Muynck, 1986)at the GBI [Eq. (58)l in the form given by Clauser et al. (1969) and by Bell (1972). These results are very interesting because it appears that for the derivation of the BIs the locality hypothesis does not play any role. In this respect it may be worthwhile to mention that Edwards (1975) and Edwards and Ballentine (1976) have constructed nonlocal HV models which satisfy the BI. And in contrast with this Scalera (1983) and Caser (1984b)(see below) have developed local models which violate the BI. These examples seem to support the thesis of the irrelevance of the locality condition with respect to the BIs. As a first illustration of such models with anomalous properties, we note that Scalera (1983) constructed a kind of local classical helix model for two correlated photon beams moving in opposite directions. Each beam is assumed to be represented by a continuous helix-shaped ribbon. If a photon passes some polarizer with its analyzer direction along a, it means that the associated ribbon passes undistorted, but somehow the directions f a are marked on it. If a photomultiplier is to receive and to count such a photon, it is assumed that the energy contained within two marked directions is integrated. A correlation between the two ribbons produced at the source originates by
302
W. DE BAERE
their being marked in the same direction. Scalera then shows that the BIs may be violated by this model. Another local model that violates the BIs has been constructed by Caser (1984b).The peculiarity of this model is that it starts from the assumption that the state of a measuring apparatus depends on some or all previous measurements made by means of it. It is argued that QM is compatible with such a memory effect. By showing quantitatively that the model satisfies all physical requirements of locality, however without being identical to Bell’s definition, it appears that the BIs can be violated. That the violation of the BIs is not a peculiar property of correlated quantum systems has been shown by Aerts (1982a, 1984).If one considers two macroscopic systems which are correlated because of the fact that they are not completely separated (e.g., there are some connecting tubes or wires), then results of measurements made on them will also be correlated, and the GBI may be shown to be violated. Hence, according to Aerts, the problem is to separate physical systems completely, whether quantum mechanical or classical. If it were possible really to produce such separate systems, then the GBI would be verified and QM should turn out to be wrong in describing such systems. Hence, Aerts (1982b) concludes that the present QM is unable to describe separated systems. For this reason, a more general scheme than QM is proposed that encompasses both CM and QM (Aerts, 1983). To explain the observed violation of the GBI, hence the alleged nonlocality, one may attempt to set up concrete physical models. This has been done by Caser (1982), who remembers that, according to Mach’s principle, the inertial properties of matter are determined by the distribution of far-away matter in the form of stars, etc. In the same way, Caser argues, it may be that, although the magnetic field H of a Stern-Gerlach analyzing device is different from zero only in the region of the device, the vector potential A (H = V x A) is nonvanishing in a much larger region, which may include the source of correlated spin4 systems. Now from the existence of the Aharonov-Bohm effect (Aharonov and Bohm, 1959), it follows that the behavior of a quantum system depends on the local values of A. Hence, it may be, according to Caser, that the values of A at the region of the source influence the distribution of HVs in such a way that one has to write p(a,b, A) instead of p(A) (a,b being the orientations of the fields H in both SternGerlach devices). It follows that Bell’s theorem is no longer valid under these conditions, and it is shown explicitly how such a model can reproduce the results of QM. In a subsequent work Caser (1984a) shows that in an anisotropic space, in which two measuring devices act in a different way on physical systems, a local HVT may reproduce the QM results for spin correlation experiments. Moreover it is shown that, unlike the argumentation of Clauser et al. (1969),
303
EINSTEIN-PODOLSKY -ROSEN PARADOX
no supplementary assumption about the detection probabilities is needed when the HVs lie in the plane of the analyzer directions. A further illustration of the disagreement between Q M predictions and the BI when the relevant observables do not all commute pairwise, has been given by Home and Sengupta (1984). They show that the violation of the BIs is not only characteristic for systems having correlated components separated in space. It is argued that instead one may equally well consider an ensemble of single systems and measure the compatible pairs (A(a),B(b)), @(a), B ( b ) ) , (A(a’),B(b)),and (A(a’),B(b’))on the individual members of the ensemble, of course under the condition that only one pair is measured at a time. Adopting a specific labeling scheme which transforms nondichotomic results to dichotomic ones, they are able to derive Bell-type inequalities of the form - P(a, C) + P(b, C ) P(b, a) I 1 (130) or IP(a,b) + P(a, b’) + P ( d , b) - P ( d ,b’)l < 2 (131)
+
with the usual definition [Eq. (39) or (77)] of the correlation functions. They consider then an ensemble of atoms (e.g., alkali atoms), which have a single valence electron in the 2p1,, state. In such a state the total angular momentum is J = +,the orbital angular momentum is L = 1, and the spin of the valence electron is s = +. If m,, rnL, and m, denote the components of the respective angular momenta along some arbitrary direction a, then one may write for m, = ++
1.1
= +,m, = $) =
( ~ / J I )=(l,m, ~ ~= 1)ls I L= + , m , - +> -IL
=
l , m L = O)ls = +,m, = ++))
(132)
Now Home and Sengupta take as observables and their operators
-
.
A(b):L b,
.
B(b):a b,
A(a):L a,
B(a):a a,
-
-
A(c): L c,
-
B(c):a c
(133)
and calculate P(a, c), P(b,c), and P(b,a) P(a,c) = (+,+1L . a a - c l + , + ) = $c0s2(e,/2) - $sin2(8,/2) -
3
(134a)
P(b,c) = (+,+lL*ba-c1+,3) =
+os2(el
-
e,/2) - $sin2(6,/2)
- -
-
5
(134b)
P(b,a) = (f,+IL b a a[+,+) = $cos2
o1 - 71
(1 34c)
304
W. DE BAERE
In Eqs. (133) and (134), a, b and c are coplanar unit vectors and 8, = Oab, 8, = Oat. The left-hand side of Eq. (130) becomes
-5 + 4
~ 0 ~ 2 +cos2(ez/2) 0 ~
+ $cosz(e, - 02/2)
(135)
which for, e.g., 0, = 7c/6 and 0, = 2n/3 becomes 4, such that the BI [Eq. (130)] is violated. Franson (1985) remarks that the recent experiment of Aspect et al. (1982) (Section IV,A) does not exclude local theories in which a measurement result may be known only some time after the measurement event has occurred. Under these circumstancesit should be possible that information between two measurement devices could be exchanged with subluminal velocities. In a recent paper Summers and Werner (1985) show that the violation of the BIs is also a general property of a free relativistic quantum field theory. Use is made of the Reeh-Schlieder theorem (Reeh and Schlieder, 1961; Streater and Wightman, 1964), according to which any local detector has a nonzero vacuum rate. It is shown that the vacuum fluctuations in free-field theories are such that the BIs are maximally violated. It is concluded that all physical and philosophical consequences with respect to the violation of the BI apply also in free quantum field theories. Some proposals for a resolution of the (non)localityproblem have already been discussed in Section II,E,F. Other recent investigationson that issue with respect to the BI will be reviewed in Section V, where the significance of the BIs and their violation will be considered from a critical point of view. A number of papers which may be of help in the study and clarification of the problems raised by EPR and the BIs are those of Berthelot (1980), Bub (1969), Corleo et al. (1975), Flato et al. (1975), Gutkowski and Masotto (1974), Gutkowski et al. (1979), Liddy (1983,1984), Rastall (1983,1985), and Stapp ( 1979).
Iv. EXPERIMENTAL VERIFICATION OF BELL’SINEQUALITIES In this section we will give a brief survey of the results of existing experiments which were intended to verify the GBI [Eq. (86) or (66)] of Clauser et al. (1969) (Section III,D,l) and at the end we will mention some new proposals. For more details on the experiments themselves and for a thorough discussion of the experimentalproblems connected with verification of the GBI [Eq. (86) or (66)], we refer to the review papers of Paty (1977), Pipkin (1978), Clauser and Shimony (1978), or to the original papers themselves.
305
EINSTEIN-PODOLSKY -ROSEN PARADOX
In their paper Clauser et al. (1969) also discussed the inefficiency of the then-available data on polarization correlation of photons (Wu and Shaknov, 1950; Kocher and Commins, 1967) with respect to an experimental discrimination between QM and HVT via their GBI [Eq. (58) or (63)]. They proposed a concrete new experiment which would allow the verification of Eq. (63). The idea of Clauser et al. was to optimize the experiment of Kocher and Commins (1967),i.e., to choose optimal relative polarizer orientations of 22.5" and 67.5O, to increase polarizer efficiencies, and also to make observations with one polarizer and then the other removed. Under these conditions one will get sufficiently reliable data to be used in the GBI or CHSH inequality (66) lR(22.5")- R(67.5")]/RoI I
(136)
They actually made two proposals: One was for the J = 0 -+ J = 1 + J = 0 electric-dipole cascade, while the other was for the J = 1 J = 1 J = 0 cascade. In terms of the relative angle cp between the two polarizer orientations and of the half-angle 8 of the cone in which the photons are detected, the QM predictions for the polarization correlations are for both cases --+
R(cp)IRo = &f
--+
+E m f i +4
+ $(&;I
- E:)(Efi
R i / R o = ( ~ f l+ ~ 1 ) / 2 ,
- Efi)F1(8)COS243
i
=
1,2
(137)
(138)
In Eqs. (137) and (138) and E\ (i = 1,2) are the efficiencies of the ith polarizer for light polarized parallel and perpendicular to the polarizer axis and, further, for the cascade J = 0 + J = 1 J =0 -+
+
Fl(8) = 2G:(8)[G2(8) iG3(8)]-1 G,(8) = - cos 8 + sin2 8 - 3cos38)
(139) ( 140a)
G2(@ = 8 - +(sin28 + 2) cos 8
( 140b)
G3(8)= 4 - cos 8 - ~
( 140c)
a(+
C O 0S ~
For the second cascade J = 1 --+ J = 1 -+ J = 0, the relations in Eq. (140) remain valid, but in Eq. (137) -Fl(8) should be replaced by
+
F2(8) = 2G:(8)[2G2(8)G3(8) iG:(8)]-'
(141)
Assuming that E: and E: are vanishingly small and takings1 = ~ f =i it may be seen that Eq. (137) may be violated in both cascades for experimental parameters satisfying the relation
J Z e ( 8 ) + 1 > 2 / ~ ,, ,
j = 1,2
(142)
306
W. DE BAERE
A , Cascade Photon Experiments 1 . Experiment of Kocher and Commins (1967) This was the first correlation experiment with cascade photons. The source was the 6 IS, excited state of Ca. Via a two-step cascade J = 0 + J = l+J=O (4 lP1) 4,= 4227 (4 lS0) (143) (6 IS,)
4
4
towards the ground state, two photons y1 and y2 are emitted. Because the total angular momentum of y1 and y2 is zero, the QM transition probability is proportional to (el .e2)', with e, the polarization vector of photon y i . The results were in agreement with this prediction: For perpendicular polarizer settings no coincidences above background were measured, while there was a high coincidence rate for parallel polarizer settings. However, the experiment was not appropriate for a verification of the GBI [Eq. (136)] because the efficiencyof the polarization was not high enough and only the angles cp = 0" and 90" were considered. 2. Experiment of Freedman and Clauser (1972) In this experiment correlated photons y1 and y2 were obtained in the following J = 0 + J = 1 + J = 0 cascade in calcium (4p2 IS,)
.
=
5513 A
* (4P4S Pl)
I,,,
= 4227
A
,(4s
IS,)
The polarizer efficiencies had the following values
~ f =i 0.97 f 0.01, ~ f =i 0.96 f 0.01,
E:
= 0.038
E:
= 0.037
f 0.04 0.004
(144)
The half-angle 6, within which the photons were detected, was 8 = 30". In different runs of the experiment the results R(22.5")/R0= 0.400 f 0.007 and R(67.5")/R0 = 0.100 f 0.003 were determined. Denoting by dexp,and, ,a the experimentally determined and theoretically predicted values for the left-hand side of Eq. (136), then it is found that aexpt= 0.300 k 0.008, which violates the BI of Eq. (136), but is in agreement with the Q M prediction , ,a = 0.301 0.007. This was the first clear evidence for the validity of Bell's theorem which excludes all local HV theories.
+
3. Experiment of Holt and Pipkin (1973)
J
Here the source was the spin-0 isotope Ig8Hg of which the J 1 + J = 0 cascade was used to produce photons y1 and y 2
=
= 1+
307
EINSTEIN - PODOLSKY -ROSEN PARADOX
The polarizer efficiencies were as follows: = 0.910
0.001,
E:
<
E ; = 0.880
k 0.001,
E!
<
E;,
(145)
and the half-angle 0 = 13". In this case one has dexpt= 0.216 k 0.013, in agreement with the GBI 6 I0.25, but contradicting the Q M prediction 6,M = 0.266. At the time this experiment was carried out, this result was exciting because now HVT was favored over QT. However, as reported by Clauser and Shimony (1978), the origin of the disagreement may lie in an improper mounting of some components, such as lenses, in the experimental arrangement of Holt and Pipkin. Some indication for this came from a repetition of this experiment by Clauser (1976a,b). 4 . Experiment of Clauser (1976a,b)
Stimulated by the previous surprising results, Clauser repeated the experiment of Holt and Pipkin with only minor changes, such as the use of a spin-0 isotope 'O'Hg instead of 19'Hg. The experimental parameters were
&;I= 0.965,
~ f =i 0.972, e = 18.60
E:
= 0.01 1
E:
= 0.008
(146)
The result was hexpt= 0.2885 k 0.0093, violating the GBI 6 I 0 . 2 5 , but in agreement with the Q M prediction: 6,, = 0.2841. It is seen that these results do not support the conclusion of Holt and Pipkin.
5 . Experiment of Fry and Thompson (1976) The experiment consisted in the study of the J cascade in the spin-0 isotope '"Hg (7 3s,) 4, = 4358 A, (6 3p1)' 7 1
=
2537
= 1 +J =
1+J
=0
4 (6 lS0)
The experimental parameters were as follows:
&i = 0.98 f 0.01,
E:
= 0.02
f 0.01,
E:
+ 0.02 f 0.005
&;
= 0.97
0 = (19.9 f 0 . 3 ) O
k 0.005 (147)
W.DE BAERE
308
The result Bexp1 = 0.296 k 0.014 was again in agreement with - 0.007, but violated the GBI: 6 I 0.25.
+
BQM
= 0.294
6. Experiment of Aspect et al. (1981)
Use is made of the following J = 0 + J = 1 -P J = 0 cascade in calcium I,, = 4227 A (4p2 Is,) 17, = 5513 44s2 IS,) (4S4P Pl )
5
There were several improvements with respect to the results of the previous experiment of Fry and Thompson (1976). First of all the excitation rate was more than ten times greater. Hence the statistical accuracy was much better than ever before. Correlation measurements for the full 360” range of relative polarizer orientations were in agreement with the Q M predictions,for various distances (up to 6.5 m) between the source and the polarizers. In addition this means that, up to these distances, the Aharonov-Bohm hypothesis (Section II,D) of a pure state evolving spontaneously towards a mixed state is not correct. The experimental parameters and results are
&i = 0.971 6;
0.05,
= 0.968 & 0.005,
E!
= 0.029 & 0.005
E:
= 0.028
& 0.005 ( 148)
F(9) = 0.984 Bexpt
= 0.3072
Ifr 0.0043,
B Q M = 0.3058
& 0.002
In order to avoid the Clauser-Horne assumption of rotational invariance (Section III,D,4), the more general GBI [Eq. (86)] - 1 I S = [R(a,b) - R(a, b’)
+ R(a’,b)
+ R(a’,b’) - &(a’) - R,(b)]/R, I 0
(149)
was tested for relative orientations = eazb = eaPb* = gab!f 3 = 22.5”
(150) The result Sexpl= 0.126 & 0.014 was in good agreement with the QM value SQM = 0.1 18 k 0.005. $,b
7 . Experiment of Aspect et al. (1982)
In all previous experiments the polarizer directions were fixed during each run of a correlation measurement. In these cases it could be criticized that perhaps somehow these static directions influenced the way the source emits the correlated photons, such that Bell’s locality conditions [Eq. (76)] would no longer be valid: This would then be a plausible explanation for the observed
EINSTEIN-PODOLSKY -ROSEN PARADOX
309
experimental violation of the GBI. Hence, Bell (1964) argued that, in order to prevent such a possible influence, it would be of crucial importance to carry out “. . . experiments of the type proposed by Bohm and Aharonov, in which the settings are changed during the flight of the particles.” The experiment of Aspect et ul. (1982)was a refinement of that of Aspect et al. (1981), in which it was attempted to realize these requirements. However, they were fulfilled only partially, because the switching was by means of an acousto-optical device which operated periodically each 10 ns. Yet the two switchesoperated independently of each other, so that it is plausible to assume that their global effect on the measurements on both sides was uncorrelated. Also, as the distance between source and polarizers was up to 6.5 m, the condition of measurements separated in space was very well satisfied. The version of the GBI that was verified in this experiment was
R(a’,b’) R(a’,co) R(co,b ) + R(co’, I 0 00’) R(co’,co) R(co, 00)
(151)
which is similar to Eq. (149). In Eq. (151) the symbol co corresponds to coincidence measurements in which the respective polarizer is removed, e.g., R(oo’, 00) means that the polarizers with orientations a’ and b are removed. Again the relative orientations of Eq. (150) were chosen, and an average value S,,,, = 0.101 t 0.020 was determined from two runs. This corresponds with the predicted value S,, = 0.112 and violates S I0 by five standard deviations. The results of this experiment are at this moment considered as the strongest evidence against all local realistic HV theories. 8. Experiment of Perrie et al. (1985)
The most recent correlation experiment with cascade photons is that of Perry et al. (1985),who used a source of metastable atomic deuterium D in the state 2s. In this case, as a result of a return to the ground state lS, the two correlated photons are emitted simultaneously (in contrast with the former cascade experimentswhere emission is consecutive due to the finite lifetime of an intermediate state). This implies that, if the sites of observation are equidistant from the source, the condition of events separated in space is fulfilled. For the polarizer transmission efficiencies it is found that E: = E: = 0.0299 k 0.0020 = C; = 0.908 k 0.013, and with these values the QM prediction (137) for R((p)/R,is well verified.
310
W. DE BAERE
Furthermore, one has that dexp,= 0.268 & 0.010, in agreement with the prediction d,, = 0.272 f 0.008, but violating the form of Eq. (136) of the BI by two standard deviations. It is seen that the above-mentioned experiments by Aspect et al. (1981, 1982) provide up to now the clearest evidence against local HV theories. However, the main reason for carrying out the present experiment was to eliminate a recent criticism (Section V,C) of the current interpretation of the results of the former two-step cascade experiments. In essence this criticism amounts to the idea that absorption and subsequent re-emission of photons in the source could be appreciable and significant, such that the conclusion of complete agreement with QM and nonlocality would not be justified. Now it appears that with a source of D(2S) this problem is negligible, and from the above results, confirming QM, it would follow that the only remaining loophole for concluding nonlocality would be the “no-enhancement” hypothesis (Section III,D,5).
B. Positronium Annihilation Experiments Another source of correlated photons is the annihilation of an electron and a positron at rest. From the experiment of Wu and Shaknov (1950), it followed that the spin parity J p for the ground state of positronium is J p = OK. If spin parity is conserved during the annihilation, then according to QM the correlated photons should be described by the state vector
I$>
0,- I V l I X ) 2 )
= (1/Jz)(lX>lI
(152)
if both photons move opposite in the z direction. In Eq. (152) I X ) , represents a state of photon y,, which has the property of being linearly polarized in the x direction, and similarly for the other states. If it were possible to measure, by means of ideal polarization analyzers, the polarization of y1 along a (lz)and of y2 along b (lz),then it can be shown that QM predicts for the correlation function P(a, b) the result P(a, b ) = P(rp) = - cos 2rp
(153)
with rp the angle between a and b. Now, as the above dichotomic measurement results are of the type considered by Bell, the GBI [Eq. (74)] should also be valid for the above situation, provided locality holds IP(a, b ) - P(a, b’)l
+ IP(a’,b’) + P(a’,b)l I2
(154)
Comparing with the QM prediction [Eq. (41)] for the spin-9 case, it is seen immediately that Eq. (153) violates Eq. (154), too.
EINSTEIN-PODOLSKY -ROSEN PARADOX
31 1
However, the problem with annihilation photons is that, although the efficiency for detection is very good, the efficiency of linear polarizers is by far insufficient. Therefore, information about the correlation between the planes of polarization of the respective photons can only be obtained indirectly, e.g., via measuring the polarization-dependent joint distribution for their Compton scattering. From this reliance on QM and from the fact that supplementary assumptions are needed (see below), in order to allow for a comparison between QM and local HV theories via the GBI, this type of correlation experiment is considered less reliable than cascade-photon experiments.
I . Experiment of Wu and Shaknov (1950) The first experiment with photon annihilation was that of Wu and Shaknov, whose results were already used by Bohm and Aharonov (1957) to disprove the assumption that each of the two correlated photons was already in a well-defined quantum state, before any measurement is carried out (Section 11,D). In this experiment coincidence rates between the Compton-scattered photons are registrated for two different arrangements: The first rate, r L , is for perpendicular scattering planes (Fig. 1, Section II,D) while the second rate, r l l ,is for parallel scattering planes. QM predicts a value of 2 for the ratio r = r l / r , which agrees very well with the experimental result r = 2.04 k 0.08. However, the GBI [Eq. (154)] cannot be verified directly because it is not possible, from this kind of experiment, to obtain a clear answer about the precise polarization of each photon of the pair. The information one gets is only statistical.
,
2. Experiment of Faraci et al. (1974) A more recent experiment with photon annihilation was that of Faraci et al. Referring again to Fig. 1 (Section II,D), let us call 0, the angle between pyl and pyI,0, the angle between pi2 and py2,and cp the angle between the planes (Pt,,P,,) and (Pi2,PY2). From the method of partial-polarization analysis (Snyder et al., 1948),it follows that for the coincidence rate W ( 4 ,B,, cp) we have
+ 2sin2 0, sinZ0,[(2k - l)sin2 cp + 1 - k]}
(155)
in which K O is the wave number of the annihilation photons, K i the wave number of yi scattered in direction pii with scattering angle 0,, a, = K J K , K , / K i , k = 1 for the pure Q M state [Eq. (152)], and k = i f o r a mixture of
+
312
W. DE BAERE
products of photon states, such as l
~
~
l
l
~
~
2
~
l
~
~
2
l
~
~
As a measure for W(O1, 8,, cp), the number of coincidence counting rates N ( 8 , , 02,cp) of Compton-scattered photons was observed and the ratios N(8,8, 90°)/N(0,8, Oo) and N(60",60°,rp)/N(60",60°,Oo) determined for various values of 8 and cp. The results were within the limit permitted by the BIs, and disagreed with the QM predictions. Furthermore, the ratio N(60",60°,90")/N(60°,60",Oo)was determined as a function of the difference in the flight paths of the two photons. It was found that there was a disagreement with QM and that the data allowed for a description in terms of a mixture of product states. 3. Experiment of Kasday et al. (1970, 1975) In order to use two annihilation photons for verifying the GBI [Eq. (154)], Kasday et al. make the following two supplementary assumptions. The first is that it is possible in principle to construct ideal analyzers for linear polarization. If results 1 and - 1 are assigned if a photon is polarized parallel or perpendicular to the polarizer axis, then the QM prediction for the correlation function P(a,b) for annihilation photons is given by Eq. (153). Now, such ideal polarizers do not exist, so that recourse has to be made to Compton scattering of the respective photons. The second assumption in snnihilation experiments is then that the results that would have been obtained with ideal polarizers and those obtained via Compton scattering are correctly related by QM. This means that one may use the Klein-Nishina formula for Compton scattering, which depends on the plane of polarization of the incoming photon. The quantity measured by Kasday et al. was the coincidence rate W ( q )= W(8, = 90°,8, = 90°,cp), for which the QM prediction is
+
W V ) = 1 - M,M,P(cp)
(156)
In Eq. (156) MI and M 2 are instrumental factors and P(cp) is the correlation function [Eq. (153)] for ideal polarizers. The results of this experiment were in agreement with QM and the GBI was violated. 4 . Experiments of Wilson et al. (1976)and of Bruno et al. (1977)
The experiment of Faraci et al. was repeated with different experimental equipment by Wilson et al. (1976) and by Bruno et al. (1977). The results of both experiments agreed very well with the predictions of QM, and no influence of the difference in the flight paths of the two photons was observed.
l
~
EINSTEIN-PODOLSKY -ROSEN PARADOX
313
C. Proton-Proton Scattering Experiment Following a suggestion by Fox (1971), Lamehi-Rahti and Mittig (1976) studied low-energy S-wave proton-proton scattering at 13.2 and 13.7 MeV. The idea was to investigate the spin correlations between the protons in the final state of the interaction. In the ideal case it should again be possible to determine the spin components of both correlated protons along directions a and b.Now, as in the case of annihilation photons, such ideal analyzers do not exist, and the correlation function P(a,b) as defined by Eq. (39) cannot be determined directly. Therefore, other processes with the correlated protons must be used, which depend on their polarization state. The extra process which is used here is the interaction of the protons with carbon foils C , and C, (Fig. 11). Detectors A, A' are placed behind C1,and detectors B, B' behind C,. It is assumed that plane (B,B') coincides with the reaction plane (pl, p,) and that 6 denotes the angle between the planes ( A ,A') and (p,, p,). In the experiment, a correlation function P(@, defined as
is determined. In Eq. (157), N A B is the number of coincidence counts in detectors A and B, etc. As in the annihilation case, the link with the GBI [Eq. (154)] can again be established only at the cost of introducing similar supplementary assumptions. It is found that with these assumptions good agreement is obtained with QM, and that the GBI is violated. However, a drawback of this experiment is that there is no separation in space between the measurements.
FIG. 11. Experimental setup in proton-proton scattering.
314
W. DE BAERE TABLE 111 LISTOF CORRELATION EXPERIMENTS
Type of experiment Atomic cascade photons
Annihilation photons
Proton-proton scattering
Source Ca Ca '"Hg '"Hg '''Hg Ca Ca D
Authors
Results agree with:
(1) (2) (3) (4) (5) (6) (7) (8)
Kocher and Commins (1967) Freedman and Clauser (1972) Holt and Pipkin (1973) Clauser (1976a,b) Fry and Thompson (1976) Aspect et al. (1981) Aspect et al. (1982) Perrie et al. (1985)
QM QM
(1) (2) (3) (4) (5)
Wu and Shaknov (1950) Faraci ef al. (1974) Kasday et al. (1975) Wilson et al. (1976) Bruno et a2. (1977)
QM
(1) Lamehi-Rahti and Mittig (1976)
HVT? HVT
QM QM QM QM QM HVT QM QM QM QM
In fact, it is one of the extra assumptions that this does not influence the mutual dependence of the measurements. The conclusions of the above correlation experiments are summarized in Table 111. D . Recent Proposals for New Experiments
As a complement to the current EPR experiments, a number of new experimental tests for Einstein locality or for the BIs have been proposed recently. Roos (1980) considers the interaction e+e- -+ K ° K o and shows that if locality holds then some decays K,K, have to be observed, which is excluded in the QM scheme if charge conjugation C and parity P are conserved. Selleri (1983a) and Six (1984) also discussed the observable difference between the Q M description of the .Ip = 1-- state of the K ° K o system and the description based on a mixture of product states [which follows from the simplified version of the EPR argumentation (Section ll,A,B)]. Selleri showed that, because of the absence of interference terms in the latter description (Furry, 1936a; Section II,C), certain correlated observations on the K ° K o system are predicted to be 12% lower than predicted by QM. Tornqvist (1981) investigated the reaction e+e- -+ A1\ -+ n-pn'p and considered the decay distribution W(p,+, pz-) of n- and TL+ in the final state. It appeared again that there is an observable difference between the predictions
EINSTEIN-PODOLSKY -ROSEN PARADOX
315
of both the above kinds of description: According to QM
- + ap,+ ‘PnWP,+,P,-) - + @/3)P,+ ‘PnW(P,+,P,-)
1
(158)
whereas were Einstein locality valid, one should have, instead of Eq. (1 58) 1
(1 59)
In Eqs. (158) and (159) a is some parameter depending on Q M S- and P-wave amplitudes. Up to now none of these proposals has been tested in real experiments.
v.
BELL’SINEQUALITIES EXPERIMENTAL RESULTS
INTERPRETATION OF AND OF
A . The Current Interpretation As a result of the almost general agreement with QM of those experiments which were set up to discriminate between Q M and all local HV theories (Section IV), the prevailing opinion has settled that it has now been decided, on rather safe empirical grounds, that nonlocality is a basic feature of Nature at the quantum level. The origin of this state of affairs is probably that in almost all derivations of the BIs or similar inequalities (Section III,C,D), the importance of the locality hypothesis has not only been overestimated, but in many of them it is considered as the only condition. Therefore, when reviewing these derivations, we have called attention to the equal importance of the hypothesis of counterfactuality, which is made either implicitly or explicitly and then considered as self-evident. We will see below that the neglect of this last hypothesis is at the origin of the main criticism of the nonlocality interpretation of the BIs. The large attention that is paid to the locality hypothesis is further reflected in various statements. According to Stapp (1975) “Bell’s theorem is the most profound discovery of science,” and Vigier (1979) is of the opinion that, with respect to the standard predictions of QM, “Its experimental issue might prove as important for the future evolution of physics as the negative result of the Michelson-Morley experiment.” Vigier continues that “. . . the theoretical discussion started by Bell has shown that a positive experimental test of the truth of quantum-mechanical measurement predictions in the case of correlated particles.. . has far-reaching consequences which.. . imply a destruction of the Einsteinian concept of material causality in the evolution of Nature, since they would establish the physical reality of nonlocal interactions
316
W. DE BAERE
between spacelike separated instruments of measurements.. . .” In the view of Bohm and Hiley (1981), “It seems clear from both the theoretical analysis and from the experiments that some kind of nonlocality is a fact ...,” and on account of the results of Aspect et al. (1981, 1982), Rohrlich (1983) concluded that the “local hidden variables theory is dead.” A very long list of similar statements may certainly be found in the literature. Remember also that recently Rietdijk and Selleri (1985) claimed to have proven nonlocality (Section 111,F). Furthermore, various people, considering nonlocality as a fact, have been constructing rather detailed models for its explanation (Vigier, 1979; Halbwachs et al., 1982; Section 111,G). Whether or not the alleged nonlocality property may be used for superluminal signaling is still a matter of debate (Hall et al., 1977; Herbert, 1982; Ghirardi and Weber, 1983). B . Criticism of the Current Interpretation and Alternative Interpretations
I . Joint Probability Distributions and Bell’s Inequalities Although for many people nonlocality is now an established fact, there are others who on no account can accept such a conclusion, because it violates the relativity principle. However, in Sections III,C,D we have seen that the counterfactuality hypothesis is as important as the locality hypothesis for the derivation of all the BIs. Einstein locality can therefore be retained by abandoning counterfactuality. The first criticism along these lines came from de la Peiia et al. (1972). These authors remark that, on account of Bell’s theorem, an eventual nonlocal HVT reproducing all results of QM requires the instantaneous propagation of signals and can therefore not be Lorentz invariant. The alternative, the acceptance of the ultimate and irreducible random character of physical events as expressed by the quantum formalism, is according to de la Peiia et al. equally unacceptable. Therefore, they re-examine the BI and show under what conditions the BIs are to hold. Their criticism directs against the simple derivation of the BI by noting the following: Because all terms in the derivation of the BI [e.g., in Eq. (44) or in Eq. (53)] use the same HV A for the quantities A(a, A), A(a’,A), B(b,A), B(b’,A), they must necessarily refer to four measurements carried out on the same pair ( S , , S2). Now, if A(a, A) and A(a’,A) are not compatible, then these cannot be known or measured simultaneously, at best measured by subsequent measurements. Because of the large empirical success of QT (which incorporates in its formalism this restriction), this must simply be admitted as true.
317
EINSTEIN-PODOLSKY -ROSEN PARADOX
Now, de la Peiia et al. remark that under these conditions (of subsequent measurements) it may happen that the original density p ( l ) , effective for A(a, A), may be changed by the measurement of A(a, A). In a recent paper Rietdijk and Selleri (1985)(Section III,F) have shown that A is indeed changed by the measuring process. According to de la Peiia et al., when such a local change happens, this does not require in any way a nonlocality in the theory. The result of their analysis is a modified Bell inequality
IP(a,
b) - c(a’,b)
+ Pb(a, b’) + &(a‘, b’)l
2
( 160)
in which P,(a’,b), P,(a, b’), and &,(a’, b’) may be called conditional correlation functions. For example, C(a’,b) uses the results A@’) and B(b) after A(a) has been determined previously. The same applies to Pb(u,b’) and to P,,(a’, b’). With this new interpretation of the BI, it is seen that the QM predictions for the various correlation functions, for the singlet spin state, are as follows (de la Peiia et al., 1972; De Baere, 1984b)
P(u, b) =
(161a)
-COS
c(a’, b) = -cos
6,b cos eao#
Pb(a, b’) = - cos 60, cos e b b , eb(a’,b’) = - cos 6,b cos eaa,cos e b b ,
(161b) (161c) (161d)
Inserting these predictions into Eq. (160), one obtains the inequality lcos 6,b[cos
oaa!(1
+ cos
ebb?)
+ cos 6bbf - 1115 2
(162)
which is satisfied for all possible orientations. de la Peiia et al. conclude: (1)the original BIs [Eq. (46) or (58)] are valid only if the measuring process does not affect the distribution of the HVs A (which is not the case; see above) and all measurement operators commute [which is not the case in actual experiments (Section IV)]; (2) the modified BI [Eq. (160)] applies under the assumption that the measurement process affects the distribution p(A) and it is not necessary that all the operators commute. In further work Brody and de la Peiia Auerbach (1979) show that the BI already holds under the condition of “joint measurability.”According to these authors Peres’ conclusion that “Unperformed Experiments Have No Results” (Peres, 1978) is equivalent to well-known properties of the state vector for correlated systems, and that in correlation experimentswe “. ..merely observe the persistence of a correlation when there is no interaction available to destroy it.” It is shown very clearly that, just as in later work by Fine (1982a,b), the only condition necessary to derive the BIs isjoint measurability of the four quantities A(a), A(a’),B(b), and B(b‘), the first two being measured on S , , the
318
W. DE BAERE
last two on S,. Hence, what experiments with respect to the BI tell us is whether or not these four quantities can be measured simultaneously. According to Brody and de la Peiia Auerbach, then, “We see that four independent series of data allow us to determine whether the system under study possessesjoint measurability or not. This point, we believe, is central to an understanding of Bell’s inequality and yet has been almost universally ignored.” Furthermore, “In other words, the Bell inequality requires a condition of joint measurability of spin projections along two different directions on the same particle; it is this fact, rather than any nonlocality of hidden variables, which helps us to explain why the inequality is inconsistent with quantum theory.. .,” and “We conclude that the Bell inequality tells us nothing about hidden variables, not even about their nonlocality.. ..” For an interesting discussion, from this standpoint, of the most important derivations of the BIs we refer to two recent papers by Brody (1985a,b).The present author is also of the opinion that the above points are basically correct and that they amount to the nonvalidity of counterfactuality, as suggested at various places in this paper. Similar criticisms against the BI and Bell’s theorem were raised by Bub (1973), Lochak (1976), and Lehr (1985). Bub shows that the Wigner procedure of taking a probability measure over the six-dimensional subspace of phase space [which leads to the Wigner inconsistency [Eq. (52)] in the case of two correlated s p in 3 systems], leads to the same inconsistencies in the case of a single spin-4 system. The point is again that it is assumed that there exists a distribution for the values of incompatible observables. Lochak‘s criticism is against the use of one single distribution p(A) for the HVs 1 in all the correlation functions figuring in the BI. It is shown that this amounts to the implicit hypothesis that the values of two noncompatible observables may be determined via one single measuring instrument or via two mutually compatible instruments. This, however, is not allowed in the Q M formalism and hence it is not a surprise that Q M predictions violate the BIs. Also, Bell’s theorem that, on account of this violation, all local theories are impossible is necessarily incorrect, just because it is based on the above incorrect implicit assumption. We have seen (Section 111) that this implicit hypothesis (equivalent to the counterfactuality hypothesis) is used in any proof of the BIs. An interesting clarification of the significance of the BIs came from Fine (1982a,b) and from de Muynck (1986) (Section 111,D). Fine’s results are valuable because they show that the problem at issue, locality, is irrelevant for the BIs. This has been illustrated in Section III,D by various nonlocal HV models which satisfy the BIs and by local models which violate the BIs. Moreover, it appears that the only requirement for the validity of the BIs is the
EINSTEIN-PODOLSKY-ROSEN PARADOX
319
existence of a joint probability distribution (JPD) for the simultaneously measured values of the relevant observables [i.e., A(a),A(a’),B(b),and B(b’)], and such that the observed distributions, figuring in the BIs [i.e., p12(a,b),etc.], are recovered as marginals. Some points in the argumentation of Fine have been criticized by Eberhard (1982) and by Stapp (1982a)and have been replied to by Fine (1982~). As stressed by various authors (de la Peiia et al., 1972;Lochak, 1976;Brody and de la Peiia Auerbach, 1979;Fine, 1982a,b;de Muynck, 1983;Rastall, 1983; de Muynck and Abu-Zeid, 1984; Kraus, 1985a,b; De Baere, 1984a,b; Brody, 1985a,b),if this possibility of simultaneous measurement is not fulfilled the BIs may be violated. According to de Muynck and Abu-Zeid (1984) “If this interpretation is correct, then the EPR-type experiments do not constitute a test of the Bell inequalities because only two of the four observables are measured jointly in these experiments. In order to relate such experiments to the Bell inequalities we would have to construct a J P D of the four observables that are involved, from the probability distributions of the pairs of observables measured in the EPR experiments. Even if such a construction is possible mathematically the physical significance of merging the outcome of difSerent experiments into one single probability statement is completely obscure in a theory dealing only with measurement results. In orthodox quantum mechanics the joint measurement of incompatible observables is held to be impossible. J P D s of incompatible observables are judged to be meaningless. For this reason the Bell inequalities cannot be derived in this theory, as necessary properties to be obeyed by all EPR experiments.” In recent work (De Baere, 1984a,b,c)the above points have been illustrated by concrete examples. As a first example one may consider the following decay chain of a spin-zero object S(0)
S(0) + Si(1) + SL(1) -P S,(+)
+ S,(+) + S,(+) + S,(+)
(163) For this situation one may choose as compatible observables the spin projections of S , , S,, S, ,S4 along, respectively, the unit vectors a, b, a’,b’, with results denoted by A,(a),A,@), A,(a’),A4(b’) if one prefers locality to be valid, or by A,(a;a’,b, b’), A,@; a, a’, b’), A3(a’,a, b, b’), A,(&; a, a’, b) in a nonlocal scheme. Using the Eberhard-Peres procedure (Eberhard, 1977; Peres, 1978; Section III,D,6), one arrives in both schemes at the GBI [Eq. (74)]. Now, in order to calculate the correlation functions between spin-component measurements on s,,. . ., S, one has to consider the following state vector for the situation of Eq. (163)
I$>
=
loo> = (l/~)Clu:)lu:)Iu3)Iu4) -(1/$)(lu: > b 2 ) + lu’>lu: ))(1/$) x (lu:)lu:)
+ Iu3)Iu4+))+
lu’)lu2)lu:>lu4+>1
(164)
320
W. DE BAERE
The results for the relevant correlation functions are P(U, b) = (0()1U1 * aU2 * bJOO) = 3 C O S
(165a)
~ ( u , b ’ ) =
(165b)
P(a’,b) = <001U3 *a’U2 *b(OO)=
(16%)
-$COSe,.b
P(u’,b) = ( ~ ~ u , ~ a ’ u , ~ b=+cost&,. ~00)
(165d)
Inserting (165a-d) into the GBI [Eq. (74)] gives 13cos 6,b
+ jcos &,* + 13cOs oath! - jcos
8,Pbl
<_
2
(166)
which is satisfied by all possible directions of the spin analyzers. Another example (De Baere, 1984c) is the decay of a spin-4 system into three distinguishable spin-4 systems
sc+,
+
Sl(9
+ S2C9 + S,(+)
(167)
such that all orbital angular momenta are zero. In this case one may consider the simultaneous measurements of spin components A,(u), A,@), A,(c) of S,, S,, S,. One of the relevant Bell-type inequalities is IP(u, b)
+ P(u,c ) + P(b,c) - 11 I 2
(168)
and for the QM predictions of the respective correlation functions one has P(U, b) = - ( F i - 3F:) COS 6,b P(u,C) =
-
2Fl(FO/JT
P(b,C ) = 2F,(Fo/JT
-
(169a)
+- F1/3)cos O,,
(169b)
F1/3)Cos 8bc
( 169c)
In Eq. (169) Fo and Fl are the so-called fractional parentage coefficients (Rose, 1957),which appear in the QM description of the process in Eq. (167). It may be shown that the correlation functions of Eq. (169) never violate the BI [Eq. (168)l. The example of the QM correlation functions [Eq. (161)] always satisfying the conditional BI [Eq. (160)] is a further illustration of the points discussed above. Now, we have seen that no one of the existing correlation experiments fulfills the condition of joint measurability, because the correlation functions figuring in the BIs [Eqs. (46) and (58)] are determined in separate experiments. Recently it has been argued (de Muynck and Abu-Zeid, 1984; De Baere, 1984a,b,c;Kraus, 1985a,b; Todorov, 1985a,b,c)that an interesting alternative interpretation may be given to the violation of these BIs. The argumentation is based on the idea that HVs 1are representations of the state of individual systems, or, equivalently, for their internal physical conditions. It has been
EINSTEIN-PODOLSKY - ROSEN PARADOX
321
shown that, for the original BIs to be valid, it is necessary that in different (similar) experiments these physical conditions are somehow reproduced, or, in technical terms, that the respective HV distributions p(A) remain the same. Under this condition, then, it appears that one is allowed to suppose that spin components A(a,A), A(a’,A) are known for one single individual state A. It is seen that this is Stapp’s counterfactuality hypothesis, which basically contradicts QM. Now, the fact that the BIs for such situations (i.e., separate experiments) are violated may well point to a new basic property of Nature which we have called the nonreproducibility of physical situations at the HV or quantum level. It follows also that Einstein locality may be maintained. The concrete examples discussed above may be considered as supporting such a viewpoint, because precisely there the reproducibility hypothesis does not have to be made. With respect to this picture, let us note that Feingold and Peres (1980) studied the probability that Bell’s locality conditions [Eq. (76)] are violated. They concluded that in at least ,/? - 1 = 41% of the cases the result of a spincomponent measurement on one of two correlated systems must depend on what is measured on the other system. In our picture this result would correspond to a measure of the nonreproducibility of physical situations. 2. First- and Second-Kind States and Bell’s Inequalities
Another interpretation of the BIs (Capasso et al., 1973; Baracca et al., 1976) is that these allow for a discrimination between a description of ensemblesof correlated systemsin terms of pure states (also called states of the second kind, describing an ensemble then called an improper mixture) or in terms of product states (also called states of the first kind, describing ordinary or proper mixtures). Observables which are able to discriminate between the two kinds of description are called sensitive observables (Capasso et al., 1973; Baracca et al., 1976; Cufaro Petroni, 1977a,b; Fortunato et al., 1977; Section 11,A). 3 . Angelidis’ Criticism on the Clauser-Horne Factorability Condition
In Section III,D,S we discussed Selleri’s arguments (Selleri and Tarozzi, 1980; Selleri, 1982) against the general validity of the Clauser-Horne factorability condition [Eq. (82)]. Another criticism of this condition came from Angelidis (1983), who criticizes the general validity of the Clauser-Horne version [Eq. (63)] of the GBI, (A(cp)l I 1, for all values of cp and for all local HV theories. Remember that such theories were defined by Clauser and Horne (1974) (Section 111,D,5) as those for which there exist probabilities
322
W. DE BAERE
pl(a, i), pz(b,A), plz(a,b, A) and a unique normalized HV distribution ~ ( 2 )If. the factorability condition of Eq. (82) holds, if p ( l ) does not depend on a, b, and
if p l ( 4 = pl,pz(b) = p2,p12(a,b)= p 1 2 ( b - bl), then IA(cp)I 5 1 has to be valid for any cp. The point Angelidis makes is that the above Clauser-Horne universality claim is incompatible with what he calls conservation of angular momentum, expressed by P l ( 4 - PZ@)
=0
or
for any a and b. From this condition Angelidis concludes that Pl(a94 - p,(b,A) = 0
hence = Pl(b,4 = Pl(4
(1 72a)
4= Pz@, 4= P 2 ( 4
(172b)
P h 4 and similarly P2(%
But then, on account of the Clauser-Horne factorability condition [Eq. (Sl)] one should have that PlZ(4 b, 4= P1(4P2(4
(173)
is independent of a, b, which is physically impossible and contradicted by observation. As expected with such controversial matters, Angelidis’argumentation has been criticized vigorously recently by a number of authors (DEspagnat, 1984; Garg and Legett, 1984; Barut and Meystre, 1984; Horne and Shimony, 1984; Cushing, 1985). Most of them point to a logical error in Angelidis’ argumentation: Namely, in order to infer Eq. (171) from Eq. (170), it is necessary that Eq. (170) be valid for arbitrary square-integrable functions p(A). However, as Horne and Shimony (1984) note, in Eq. (170) p(A) is not arbitrary at all. Moreover, Garg and Legett remark that in the most recent correlation experiments by Aspect et al. (1981,1982) it is not IA(cp)I I 1 which is used, but instead the inequality (151) which is free from Angelidis’ criticism. It follows that, according to the above authors, nonlocality is unavoidable, In his replies Angelidis (1984a,b) claimed that the above critics have misunderstood his argumentation.
EINSTEIN-PODOLSKY-ROSEN PARADOX
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4 . Nonmeasurable Sets and Bell’s Inequalities
Another attempt to save locality and to resolve the apparent problem of nonlocality came from Pitowski (1982a, 1983), who constructed a local deterministic model that reproduces the QM statistical predictions for correlated spin-3 systems. The model introduces spin functions s(x) for any unit vector x with the property s(-x) = -s(x). It is supposed that the precise form of these spin functions is not known. Introducing some particular assumptions about the way the set {xls(x) = *} is distributed over the sphere 1x1 = 1, Pitowski proves mathematically that the observed coincidence frequencies, compatible with QM, are reproduced. The idea behind it is that there exist distributions of spin values that are nonmeasurable in the mathematical sense. The above approach is based on an extension of the concept of probability to nonmeasurable sets. This approach has been criticized subsequently by Mermin (1982) and Macdonald (1982) on the grounds that from Pitowski’s model Bell-type inequalities may be derived which are violated by the QM predictions. In his reply Pitowski (1982b) stresses that these inequalities are based also on the counterfactuality hypothesis (besides the hypothesis of locality) and hence do not invalidate his proposed model. This model is interesting because it shows clearly that it is not impossible to build local deterministic models which reproduce the results of QM for EPRlike situations with correlated systems. However, the physical significance of sets of points which are nonmeasurable in a mathematical sense is not very clear. Based on these ideas Gudder (1984) has shown further that in a generalized probability theory the concepts of reality, locality, and HVs can be made compatible with QM for systems with spin. C . Criticism of Supplementary Assumptions
We have seen (Sections III,D,5 and IV,A,B,C) that, in order to be allowed to use the available experimental data for the verification of QM or of the BIs, it has always been necessary to introduce some plausible, but not directly verifiable, supplementary assumptions. The number of such assumptions is least in cascade photon experiments and for this reason their results are considered as the most reliable. Generally, these additional assumptions have to do with the relatively low efficienciesfor optical photon detection and with the so-called “no-enhancement’’condition, expressed by Eq. (87). Because these hypotheses are by no means trivial, and on account of the lack of any indication for their correctness, the current interpretation of the
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correlation experiments with cascade photons has been seriously criticized during the past years (Marshall et al., 1983a,b; Marshall, 1983, 1984; Santos, 1984; Selleri, 1984, 1985; Garuccio and Selleri, 1984; Marshall and Santos, 1985). Local HV models were proposed which either violated at least one of the additional hypotheses (mainly the condition of no-enhancement) or even did not need them. The first who showed the crucial importance of the no-enhancement assumption were Clauser and Horne (1974). They constructed a local HV model that reproduced all QM results for correlation experiments, based on the (according to them) physically implausible idea of enhancement. Furthermore, Marshall et al. (1983a) proposed “. .. a very plausible model.. . with enhancement features, which, though not giving identical predictions to the quantum-mechanical model, fits the experimental data very closely.” In particular this is illustrated for the available data of Aspect et al. (1981). Marshall et al. (1983b) and Selleri (1984) also called attention to the necessity of absence of collective effects in cascade-photon experiments in order to supply reliable data. Such an effect may be, e.g., the resonant rescattering of yz in the atomic cascade due to absorption by other atoms in the ground state and subsequent re-emission. These authors calculated that in concrete experiments such resonant scattering should be very important. As a result of these developments, it is concluded that the evidence for nonlocality, coming from the existing cascade experiments (mainly from those of Aspect et al., 1981, 1982), cannot yet be considered as conclusive. It is insisted that further experimental work be done. Results of such work has been reported very recently by Aspect and Grangier (1985), who show, on theoretical and experimental grounds, that the above resonant scattering has been overestimated by Marshall et al. (1983b) and Selleri (1984). It appears then that in the experiments by Aspect et al. (1981, 1982) such collective effects are negligable. Moreover, the most recent data by Aspect and Grangier (1985) again clearly violated the BIs, even without subtraction of accidental coincidences. Furthermore, it is worth noting that in the recent correlation experiment by Perrie et al. (1985) (Section IV,A) the process of absorption and re-emission of photons does not play a significant role, yet the results confirm QM. Another possible origin of the apparent nonlocality has been given by Pascazio (1985). It is remarked that a further supplementary assumption, which is always made implicitly, is that the coincidencemeasurements on both photons occur locally. Now, Pascazio argues that this is not the case because the electronic equipment on both sides of the arrangement may be considered as a source of nonlocality. The argumentation is accompanied and supported by a simple model.
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VI. CONCLUSIONS In this work we have given a survey of past and recent work on the problems related with the (original and simplified) EPR argumentation and the Bell inequalities. These problems have been considered mainly as being the completeness of QM and QM nonlocality. We have seen that both topics (EPR, BI) are related because they start from common hypotheses, namely locality (H 1) (explicitly) and counterfactuality (H3) (implicitly). From a compilation of existing correlation experiments (Section IV), testing QM and the BIs, during the last decade, it may now be concluded almost with certainty (however, see Section V,C, for recent criticism on the validity of supplementary assumptions and of the data) that these BIs are violated and QM confirmed. If this is correct, then the crucial problem is to know how this evidence has to be interpreted, or to bring about its real significance. According to the current view (Section V,A), opinion has apparently settled that locality is responsible for the violation of the BI and is incompatible with the QM framework. Hence QM, and any future theory with it, should be basically nonlocal. This conclusion originates certainly from the disproportion importance attributed to the hypotheses (Hl) and (H3). However, there exists a relation between the hypothesis of counterfactuality (H3) and the assumption of the existence of a joint probability distribution (JPD) of observables which are, in the QM framework, incompatible (Section V,B,l). The important point now is that if such a JPD is assumed, the BIs are satisfied automatically and the converse is also true. It follows that from this point of view the whole issue of locality is irrelevant with respect to the BIs and that Bell’s theorem does not apply. The lesson from the empirical violation of the BI is then clear: (1) Any future (HV?) theory, superseding QT, should not contain in its formalism a prescription for constructing JPDs for incompatible observables; (2) (H3) is invalid. It follows immediately that the original EPR argumentation, which leads to the incompleteness statement of QM, is also invalid. However, this does not necessarily mean that Einstein’s concept of elements of physical reality is wrong, nor his idea of an independent reality with its own intrinsic properties. We have seen that a small extension of the original EPR ideas may realize a reconciliation between EPR and Bohr (Section 11,F). At first sight it would seem then that the BI should be deprived of irs original importance (the issue being the fundamental property of locality). However, that is not the case. This may be seen by trying to give a physical interpretation of the violation of (H3). In each verification of the BIs (Section IV) one combines individual measurement results coming from at least
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two different experiments (corresponding to different experimental parameters). We have shown (Section III,C,D) that, in order to be allowed to do so, (H3) should be valid. Now, this is apparently not true, and we have argued that a physical reason for this may be that on the quantum level the conditions (tentatively represented by HVs A) which determine the choice between several possibilities (e.g., spin up, spin down) in a real experiment, do not always reproduce themselves in the future, and hence are not interchangeable. Formally, one could say that one should write p(A.,t) instead of p(A). And this “nonreproducibility” would be as interesting and astonishing as nonlocality. In this way one may be led to the wholeness idea of Bohm (1980), in which the world does not contain separated systems and is eventually evolving from a starting point to an end point, in such a way that the propagation of influences are in agreement with the relatively principle, and hence are not instantaneous. As to the locality problem, this should be decided in single experiments such that the predictions depend in a clear way on whether locality is valid or not. Indeed, this cannot be done when combining results from diferent experiments, because then it is apparently the existence of a JPD or (H3) which is at issue. It is argued in Section II,F that the simplified EPR paradox [based on the validity of locality (Hl) and of QM (H2)] could be avoided by assuming a breakdown of the CI of the QM state-vector reduction (also suggested by Margenau, 1936) in the case of correlated systems when only one measuring device is present. Some candidates of experiments for verifying this are proposed there. We shall conclude by stating that the locality problem has not yet been solved completely, and that further theoretical and experimental work should be done. Perhaps it may be rewarding not to have too much confidence in the alleged universal validity of QM, and try to invent situations where QM may eventually get into trouble. At any rate, the claim of the proponents of the orthodox viewpoint, that there is no basis at all for a choice between different possibilities, seems to be too conservative and a stumbling block for future research. This is also reminiscent of universality claims of classical physics towards the end of the last century. Seen in this historical perspective, it is not prudent to advance and defend extreme standpoints (Section V,A). This holds especially with respect to the issue of locality, because other secure pillars of modern physical theory, Einstein’s relativity theories, are founded on it. Let us end finally then with the following careful statement of Dirac (1976): “. . . I think it might turn out that ultimately Einstein will prove to be right, because the present form of quantum mechanics should not be considered as the final form. There are great difficulties.. . in connection with the present quantum mechanics. It is the best that one can do till now. But, one should not suppose
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that it will survive indefinitely into the future. And I think that it is quite likely that at some future time we may get an improved quantum mechanics in which there will be a return to determinism and which will, therefore, justify the Einstein point of view.” ACKNOWLEDGMENTS The author thanks Prof. T. A. Brody, Prof. F. Selleri, Dr. W. M. de Muynck, and Dr. J. Kriiger for clarifications and/or discussions about some topics treated in the article, and the N.F.W.O. Belgium for financial support.
REFERENCES Aerts, D. (1982a).Example of a macroscopic classical situation that violates Bell inequalities. Lett. Nuouo Cimento SOC.Ital. Fis. [2] 34, 107. Aerts, D. (1982b). Description of many separated physical entities without the paradoxes encountered in quantum mechanics. Found. Phys. 12, 1131. Aerts, D. (1983). Classical theories and nonclassical theories as special cases of a more general theory. J. Math. Phys. 24,2441. Aerts, D. (1984).The missing elements of reality in the description of quantum mechanics of the E.P.R. paradox situation. Helu. Phys. Acta 57,421. Aharonov, Y., and Bohm, D. (1959). Significance of electromagnetic potentials in the quantum theory. Phys. Rev. 115,485. Albertson, J. (1961). Von Neumann’s hidden-variable proof. Am. J. Phys. 29,478. Angelidis, T. D. (1983). Bell’s theorem: Does the Clauser-Horne inequality hold for all local theories? Phys. Rev. Lett. 51, 1819. Angelidis, T. D. (1984a).Angelidis responds. Phys. Rev. Lett. 53, 1022. Angelidis, T. D. (1984b). Angelidis responds. Phys. Rev. Lett. 53, 1297. Araki, H., and Yanase, M. M. (1960).Measurement of quantum mechanical operators. Phys. Reo. 120,622. Aspect, A., and Grangier, P. (1985). About resonant scattering and other hypothetical effects in the Orsay atomic-cascade experiment tests of Bell inequalities: A discussion and some new experimental data. Lett. Nuouo Cimento Soc. ItaL Fis. [2] 43,345. Aspect, A,, Grangier, P., and Roger, G. (1981). Experimental tests of realistic local theories via Bell’s theorem. Phys. Rev. Lett. 47,460. Aspect, A., Dalibard, J., and Roger, G .(1982).Experimental tests of Bell’s inequalities using timevarying analysers. Phys. Rev. Lett. 49, 1804. Ballentine, L. E. (1970).The statistical interpretation of quantum mechanics. Rev. Mod. Phys. 42, 358.
Ballentine, L. E. (1972). Einstein’s interpretation of quantum mechanics. Am. J. Phys. 40, 1763. Baracca, A., Bergia, S., Bigoni, R., and Cecchini, A. (1974).Statistics of observations for “proper” and “improper” mixtures in quantum mechanics. Riv. Nuovo Cimento SOC.Ital. Fis.[114,169. Baracca, A., Bergia, S., Livi, R., and Restignoli, M. (1976).Reinterpretation and extension of Bell’s inequality for multivalued observables. Int. J. Theor. Phys. 15,473.
328
W. DE BAERE
Bartell, L. S. (1980a). Complementarity in the double-slit experiments: On simple realizable systems for observing intermediate particle-wave behavior. Phys. Rev. D 21, 1698. Bartell, L. S. (1980b). Local realism and the Einstein-Podolsky-Rosen paradox. On concrete new tests. Phys. Rev. D 22, 1352. Barut, A. O., and Meystre, P. (1984). Rotational invariance, locality, and Einstein-PodolskyRosen experiments. Phys. Reu. Lett. 53, 1021. Bastide, C. (1984). The different non-localities in the Einstein-Podolsky-Rosen paradox. Phys. Lett. A 103A, 305. Bedford, D., and Selleri, F. (1985).On Popper’s new EPR-experiment. Lett. Nuouo Cimento SOC. Ital. Fis. [2] 42,325. Belinfante, F. J. (1973). “A survey of Hidden Variable Theories.” Pergamon, Oxford. Bell, J. S. (1964).On the Einstein Podolsky Rosen paradox. Physics(Long Island City, N . Y.)1, 195. Bell, J. S. (1966). On the problem of hidden variables in quantum mechanics. Rev. Mod. Phys. 38, 447. Bell, J. S. (1972). Introduction to the hidden variable questions. In “Foundations of Quantum Mechanics” (B. DEspagnat, ed.), Course 49, p. 171. Academic Press, New York. Bergia, S., and Cannata, F. (1982). Higher-order tensors and tests of quantum mechanics. Found Phys. 12, 843. Berthelot, A. (1980). Comments on determinism, locality, Bell’s theorem and quantum mechanics. Nuovo Cimento SOC.Ital. Fis. B 1111 57B, 193. Bertolini, G., Bettoni, M., and Lazzarini, E. (1955). Angular correlation of scattered annihilation radiation. Nuovo Cimento 2,661. Bohm, D. (1952a).“Quantum Theory.” Prentice-Hall, Englewood Cliffs, New Jersey. Bohm, D. (1952b). A suggested interpretation of the quantum theory in terms of “hidden” variables. I. Phys. Rev. 85, 166. Bohm, D. (1952~).A suggested interpretation of the quantum theory in terms of “hidden” variables. 11. Phys. Rev. 85, 180. Bohm, D. (1980). “Wholeness and the Implicate Order.” Routledge & Kegan Paul, London. Bohm, D., and Aharonov, Y.(1957).Discussion of experimental proof for the paradox of Einstein, Rosen, and Podolsky. Phys. Rev. 108,1070. Bohm, D., and Aharonov, Y. (1960). Further discussion of possible experimental tests for the paradox of Einstein, Podolsky and Rosen. Nuovo Cimento 17,964. Bohm, D., and Bub, J. (1966a). A proposed solution of the measurement problem in quantum mechanics by a hidden variable theory. Reu. Mod. Phys. 38,453. Bohm, D., and Bub, J. (1966b).A refutation of the proof by Jauch and Piron that hidden variables can be excluded in quantum mechanics. Rev. Mod. Phys. 38,470. Bohm, D., and Hiley, B. J. (1981). Nonlocality in quantum theory understood in terms of Einstein’s nonlinear field approach. Found. Phys. 11,529. Bohr, N. (1935a). Quantum mechanics and physical reality. Nature (London) 136,65. Bohr, N. (1935b). Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 48,696. Breitenberger, E. (1965). On the so-called paradox of Einstein, Podolsky and Rosen. Nuouo Cimento [lo] 38,356. Brody, T. A. (1985a). The Bell inequality. I. Joint measurability. Preprint. Found. Phys. (to be published). Brody, T. A. (1985b). The Bell inequality. 11. Locality. Preprint. Found. Phys. (to be published). Brody, T. A,, and de la Peiia Auerbach, L. (1979). Real and imagined nonlocalities in quantum mechanics. Nuouo Cimento SOC.Ital. Fis. B [l 13 54B,455. Bruno, M., DAgostini, M., and Maroni, C. (1977). Measurement of linear polarization of positron annihilation photons. Nuovo Cimento SOC. Ital. Fis. B [I 11 40B, 143.
EINSTEIN-PODOLSKY -ROSEN PARADOX
329
Bub, J. (1969). What is a hidden variable theory of quantum phenomena? Int. J. Theor. Phys. 2, 101. Bub, J . (1973). On the possibility of a phase-space reconstruction of quantum statistics: A refutation of the Bell-Wigner locality argument. Found. Phys. 3,29. Busch, P. (1985). Indeterminacy relations and simultaneous measurements in quantum theory. lnt. J . Theor. Phys. 24,63. Cantrell, C. D., and Scully, M. 0.(1978).The EPR paradox revisited. Phys. Rep. 43,499. Capasso, V., Fortunato, D., and Selleri, F. (1970). Von Neumann's theorem and hidden-variable models. Riv. Nuouo Cimento SOC.Ital. Fis. [l] 11, 149. Capasso, V., Fortunato, D., and Selleri, F. (1973). Sensitive observables of quantum mechanics. Int. J. Theor. Phys. 7 , 319. Caser, S . (1982).Objective local theories and the symmetry between analysers. Phys. Lett. A 92A, 13. Caser, S. (1984a). On a possible role of the vector potential in the Einstein-Podolsky-Rosen paradox. Phys. Lett. A 102A, 152. Caser, S. (1984b). On a local model that violates Bell's theorem. LPTHE ORSAY&4/51, Nov. Clauser, J. F. (1976a). Experimental investigation of a polarization correlation anomaly. Phys. Rev. Lett. 36, 1223. Clauser, J. F. (1976b). Measurement of the circular-polarization in photons from an atomic cascade. Nuouo Cimento SOC.Ital. Fis. B [111 33B, 740. Clauser, J.F., and Horne, M. A. (1974). Experimental consequences of objective local theories. Phys. Rev. D 10,526. Clauser, J. F., and Shimony, A. (1978). Bell's theorem: Experimental tests and implications. Rep. Prog. Phys. 41, 1881. Clauser, J. F., Horne, M. A., Shimony, A., and Holt, R. A. (1969).Proposed experiment to test local hidden-variable theories. Phys. Reu. Lett. 23,880. Corleo, G., Gutkowski, D., Masotto, G., and Valdes, M. V. (1975). Are Bell's type inequalities sufficient conditions for local hidden-variable theories? Nuouo Cimento SOC.Ital. Fis. B 25B, 413. Costa de Beauregard, 0. (1976). Time symmetry and interpretation of quantum mechanics. Found. Phys. 6,539. Costa de Beauregard, 0. (1977).Time symmetry and the Einstein paradox. Nuouo Cimento SOC. Ital. Fis. B [ l l ] 42B, 41. Costa de Beauregard, 0.(1979).Time symmetry and the Einstein paradox. 11. Nuouo Cimento SOC. Ital. Fis. B [111 51B, 267. Cramer, J. G. (1980). Generalized absorber theory and the Einstein-Podolsky-Rosen paradox. Phys. Rev. D 22,362. Cufaro Petroni, N. (1977a).On the observable differences between proper and improper mixtures. I. Nuouo Cimento SOC.Ital. Fis. B 1111 40B, 235. Cufaro Petroni, N. (1977b).On the observabledifferences between proper and improper mixtures. 11. Nuouo Cimento SOC.Ital. Fis. B [ll] 40B, 381. Cushing, J. T. (1985). Comments on Angelidis's universality claim. Phys. Rev. Lett. 54,2059. De Baere, W. (1984a). On the significance of Bell's inequality for hidden-variable theories. Lett. Nuouo Cimento SOC.Ital. Fis. [2] 39,234. De Baere, W. (1984b). On conditional Bell inequalities and quantum mechanics. Lett. Nuouo Cimento SOC.Ital. Fis. [2] 40,488. De Baere, W. (1984~).Evidence for nonreproducibility of physical situations at the hidden variable level as the origin of the violation of Bell's inequality for the singlet state. Unpublished, University of Ghent. De Baere, W. (1985). EPR paradox considered as a criticism of the state vector reduction in the
330
W. DE BAERE
quantum mechanical description of correlated separated systems. In “Proceedings of the International Conference on Microphysical Reality and Quantum Formalism.” Reidel, Dordrecht, Netherlands. (to be published). de la Peiia, L., Cetto, A. M., and Brody, T. A. (1972). On hidden-variable theories and Bell’s inequality. Lett. Nuouo Cimento SOC.Ital. Fis. [2] 5, 177. de Muynck, W. M. (1983). Objectivity, nonlocality and the Bell inequalities. Phys. Lett. A 94A, 13. de Muynck, W. M. (1985). On the relation between the Einstein-Podolsky-Rosen paradox and the problem of nonlocality in quantum mechanics. Preprint, Eindhoven University of Technology. de Muynck, W. M., and Abu-Zeid, 0. (1984). On an alternative interpretation of the Bell inequalities. Phys. Lett A IOOA, 485. de Muynck, W. M., Janssen, P. A. E. M., and Santman, A. (1979).Simultaneous measurement and joint probability distributions in quantum mechanics. Found. Phys. 9,71. de Muynck, W. M. (1986). The Bell inequalities and their irrelevance to the problem of locality in quantum mechanics. Phys. Lett. A 114A, 65. DEspagnat B. (1965). “Conceptions de la Physique Contemporaine.” Hermann, Paris. DEspagnat, B. (1975). Use of inequalities for the experimental test of a general conception of the foundation of microphysics. Phys. Reo. D 11, 1424. D’Espagnat, B. (1983). “In Search of Reality.” Springer-Verlag, Berlin and New York. DEspagnat, B. (1984). Nonseparability and the tentative descriptions of reality. Phys Rep. 110, 201. Destouches, J. L. (1980). Usingde Broglie’sphysical waves to solve the Einstein-Podolsky-Rosen paradox. Nuouo Cimento SOC.Ital. Fis. B [113 58B, 73. Dirac, P. A. M. (1976). In “Directions in Physics” (H. Hora and J. R. Shepansky, eds.). Sidney. Eberhard, P. H. (1977). Bell’s theorem without hidden variables. Nuouo Cimento SOC.Ital. Fis. B [11] 38B, 75. Eberhard, P. H. (1978). Bell’s theorem and the different concepts of locality. Nuouo Cimento SOC. Ifal. Fis. B Ell] 46B, 392. Eberhard, P. H. (1982). Constraints of determinism and of Bell’s inequalities are not equivalent. Phys. Rev. Lett. 49, 1474. Edmonds, A. R. (1957). “Angular Momentum in Quantum Mechanics.” Princeton Univ. Press, Princeton, New Jersey. Edwards, J. (1975). Sufficient conditions for objective local theories. Nuouo Cimento SOC.Ital. Fis. B [11] 29B, 100. Edwards, J., and Ballentine, L. (1976). Sufficient conditions for objective local theories. 11. Nuovo Cimento SOC.Ital. Fis. B [11] MB, 91. Einstein, A. (1949). Autobiographical notes. In “Albert Einstein: Philosopher-Scientist’’ (P. A. Schilpp. ed.), p. 85. Tudor Publishing Company, New York. Einstein, A,, Podolsky, B., and Rosen, N. (1935).Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 41, 777. Faraci, G., Gutkowski, D., Notarrigo, S., and Pennisi, A. R. (1974). An experimental test of the EPR paradox. Lett. Nuouo Cimento SOC.Ital. Fis. [2] 9,607. Feingold, S. J., and Peres, A. (1980). How often are Bell’s inequality premises violated? J . Phys. Math Gen. A 13, 3187. Fenyes, I. (1952). Eine warscheinlichkeitstheoretische Begrundung und Interpretation der Quantenmechanik. Z. Phys. 132, 81. Feynman, R. P., Leighton, R. B., and Sands, M. (1965).“The Feynman Lectures on Physics,” Vol. 111. Addison-Wesley, Reading, Massachusetts.
EINSTEIN-PODOLSKY-ROSEN PARADOX
331
Fine, A. (1982a). Hidden variables, joint probability, and the Bell inequalities. Phys. Rev. Lett. 48, 291. Fine, A. (1982b). Joint distributions, quantum correlations and commuting observables. J . Math. Phys. 23, 1306. Fine, A. (1982~).Comments on the significance of Bell’s theorem. Phys. Reo. Lett. 49, 1536. Flato, M., Piron, C., Grea, J., Sternheimer, D., and Vigier, J. P. (1975). Are Bell’s inequalities concerning hidden variables really conclusive? Helv. Phys. Acta 48, 219. Fortunato, D., Garuccio, A,, and Selleri, F. (1977). Observable consequences from second-type state vectors of quantum mechanics. Int. J . Theor. Phys. 16, 1. Fox, R. (1971). Low-energy proton-proton scattering as a test of local hidden-variable theory. Lett. Nuovo Cimento Soc. ftal. Fis. [2] 2, 565. Franson, J. D. (1985). Bell’s theorem and delayed determinism. Phys. Rev. D 31,2529. Freedman, S. J., and Clauser, J. F. (1972). Experimental test of local hidden-variable theories. Phys. Rev. Lett. 28, 938. Freedman, S. J., and Wigner, E. (1973). On Bub’s misunderstanding of Bell’s locality argument. Found. Phys. 3,457. Froissart, M. (1981). Constructive generalization of Bell’s inequalities. Nuooo Cimento Soc. Ital. Fis. B 64B,241. Fry, E. D., and Thompson, R. C. (1976). Experimental test of local hidden-variable theories. Phys. Rev. Lett. 31,465. Furry, W. H. (1936a). Note on the quantum-mechanical theory of measurement. Phys. Rev. 49, 393. Furry, W. H. (1936b). Remarks on measurements in quantum theory. Phys. Rev. 49,476. Garg, A,, and Legett, A. J. (1984). Comment on “Bell’s theorem: Does the Clauser-Home inequality hold for all local theories?” Phys. Reo. Lett. 53, 1297. Garg, A,, and Mermin, N. D. (1982). Bell inequalities with a range of violation that does not diminish as the spin becomes arbitrary large. Phys. Rev. Lett. 49,901. Garg, A., and Mermin, N. D. (1985). Farka’s lemma and the nature of reality: Statistical implications of quantum correlations. Found. Phys. 14, 1. Garuccio, A. (1978). Generalized inequalities following from Einstein locality. Lett. Nuooo Cimento Soc. Ital. Fis. [2] 23, 559. Garuccio, A,, and Rapisarda, V. A. (1981). A comparison between deterministic and probabilistic local hidden-variable theories. Lett. Nuovo Cimento Soc. Ital. Fis. [2] 30,443. Garuccio, A,, and Selleri, F. (1978). On the equivalence of deterministic and probabilistic local theories. Lett. Nuooo Cimento Soc. Ital. Fis. [2] 23, 555. Garuccio, A., and Selleri, F. (1980). Systematic derivation of all the inequalities of Einstein locality. Found. Phys. 10,209. Garuccio, A., and Selleri, F. (1984). Enhanced photon detection in EPR type experiments. Phys. L e f t . A 103, 99. Ghirardi, G. C., and Weber, T. (1983). Quantum mechanics and faster-than-light communications: Methodological considerations. Found. Phys. 12, 1 1 71. Gleason, A. M. (1957).Measures on the closed subspaces of a Hilbert space. J . Math. Mech. 6,885. Gudder, S. P. (1984). Reality, locality, and probability. Found Phys. 14,997. Gutkowski, D., and Masotto, G. (1974). An inequality stronger than Bell’s inequality. Nuovo Cimento Soc. Ital. Fis. B [ l l ] 22B, 121. Gutkowski, D., Masotto, G., and Valdes, M. V. (1979). On the sufficiency of Bell’s conditions. Nuooo Cimento Soc. Ital. Fis. B [ l l ] SOB, 323. Halbwachs, F. Piperno, F., and Vigier, J. P. (1982). Relativistic Hamiltonian description of the classical photon behavior: A basis to interpret aspect’s experiments. Lett. Nuooo Cimento Soc. Ital. Fis. [Z] 33, 311.
332
W. DE BAERE
Hall, J., Kim, C., McElroy, B., and Shimony, A. (1977).,Wave-packet reduction as a medium of communication. Found. Phys. 7,759. Herbert, N. (1982). FLASH-A superluminal communicator based upon a new kind of quantum measurement. Found. Phys. 12,1171. Herbert, N., Karush, J. (1978). Generalization of Bell’s theorem. Found. Phys. 8,313. Holt, R. A., and Pipkin, F. M. (1973). Harvard University preprint. Home, D., and Sengupta, S. (1984). Bell’s inequality and noncontextual dispersion-free states. Phys. Lett. A 102A, 159. Hooker, C. A. (1972). The nature of quantum mechanical reality: Einstein versus Bohr. In “Paradigms and Paradoxes”(R. G. Colodny, ed.), p. 67. Univ. of Pittsburgh Press, Pittsburgh, Pennsylvania. Home, M. A., and Shimony, A. (1984). Comment on “Bell’s theorem: Does the Clauser-Horne inequality hold for all local theories?” Phys. Rev. Lett. 53, 1296. Jammer, M. (1974). “The Philosophy of Quantum Mechanics.” Wiley, New York. Jauch, J. M. (1968). “Foundations of Quantum Mechanics.” Addison-Wesley, Reading, Massachusetts. Jauch, J. M., and Piron, C. (1963).Can hidden variables be excluded in quantum mechanics? Helv. Phys. Acta 36,827. Kasday, L. R. (1971). Experimental test of quantum predictions for widely separated photons. In “Proceedings of the International School of Physics ‘Enrico Fermi”’ (B. d‘Espagna1, ed.), p. 195. Academic Press, New York. Kasday, L. R., Ullman, J. D., and Wu, C. S. (1970). Bull. Am. Phys. SOC.[2] 15,586. Kasday, L. R., Ullman, J. D., and Wu, C. S. (1975). Angular correlation of Compton-scattering annihilation photons and hidden variables. Nuovo Cimento SOC.Ital. Fis. B [ll] 25B, 633. Kellett, B. H. (1977). The physics of the Einstein-Podolsky-Rosen paradox. Found. Phys. 7,735. Kemble, E. C. (1935). The correlation of wave functions with the states of physical systems. Phys. Rev. 47,973. Kochen, S., and Specker, E. P. (1967). The problem of hidden variables in quantum mechanics. J . Math. Mech. 17,59. Kocher, C. A,, and Commins, E. D. (1967). Polarization correlation of photons emitted in an atomic cascade. Phys. Rev. Lett. 18, 575. Kraus, K. (1985a). Einstein-Podolsky-Rosen experiments and macroscopic locality. In “Open Questions in Quantum Physics” (G. Tarozzi and A. van der Merwe, eds.), p. 75. Reidel, Dordrecht, Netherlands. Kraus, K. (1985b). “Quantum Theory, Causality, and EPR Experiments,” Preprint. Center for Particle Theory, University of Texas, Austin. Lamehi-Rahti, M., and Mittig, W. (1976). Quantum mechanics and hidden-variables: A test of Bell’s inequality by the measurement of the spin correlation in low-energy proton-proton scattering. Phys. Rev. D 14,2543. Langhoff, H. (1960). Die Linearpolarisation der Vernichtungs strahlung von Positronen. Z. Phys. 160,186. Liddy, D. E. (1983). On locality, correlation and hidden variables. J . Phys. A: Math. Gen. 16, 2703. Liddy, D. E. (1984). An objective, local, hidden-variables theory of the Clauser-Horne experiment. J . Phys. A: Math. Gen. 17,847. Lochak, G. (1976). Has Bell’s inequality a general meaning for hidden-variable theories? Found. Phys. 6, 173. Macdonald, A. I. (1982). Comment on “Resolution of the EPR and Bell Paradoxes.” Phys. Rev. Lett. 49, 1215.
EINSTEIN-PODOLSKY -ROSEN PARADOX
333
Margenau, H. (1936). Quantum-mechanical description. Phys. Rev. 49,240. Lehr, W. J. (1985). On the hypothesis that quantum mechanics has a nonlocal structure. Nuovo Cimento Soc. Ital. Fis. B WB, 185. Marshall, T. W. (1983).The distance separating quantum theory from reality. Phys. Lett. A 99A, 163. Marshall, T. W. (1984).Testing for reality with atomic cascades. Phys. Lett. A 100A, 225. Marshall, T. W., and Santos, E. (1985). Local realist model for the coincidence rates in atomiccascade experiments. Phys. Lett. A 107A, 164. Marshall, T. W., Santos, E., and Selleri, F. (1983a). Collective effects in the atomic-cascade experimental tests of Bell inequalities. Lett. Nuovo Cimento SOC.Ital. Fis. [2] 38, 417. Marshall, T. W., Santos, E., and Selleri, F. (1983b).Local Realism has not been refuted by atomic cascade experiments. Phys. Lett. A 98A, 5. Mermin, N. D. (1980). Quantum mechanics vs local realism near the classical limit: A Bell inequality for spin s. Phys. Rev. D 22,356. Mermin, N. D. (1982).Comment on “Resolution of the EPR and Bell Paradoxes.” Phys. Rev. Lett. 49, 1214. Mermin, N. D., and Schwarz, G. M. (1982).Joint distributions and local realism in the higher-spin Einstein-Podolsky-Rosen experiment. Found. Phys. 12,101. Moldauer, P. 0. (1974). Reexamination of the arguments of Einstein, Podolsky, and Rosen. Found. Phys. 4, 195. Miickenheim, W. (1982). A resolution of the Einstein-Podolsky-Rosen paradox. Lett Nuouo Cimento Soc. Ital. Fis. [2] 35,300. Park, J. L., and Margenau, H. (1968). Simultaneous measurability in quantum theory. Int. J. Theor. Phys. 1,211. Pascazio, S . (1985). Experimental tests of Bell inequalities. Are all local models really excluded? Phys. Lett. A 111A, 339. Paty, M. (1977). The recent attempts to verify quantum mechanics. In “Quantum Mechanics, a Half Century Later” (J. Leite Lopes and M. Paty, eds.), p. 261. Reidel, Dordrecht, Netherlands. Pearle, P. M. (1970). Hidden-variable example based upon data rejection. Phys. Rev. D 2, 1418. Peres, A. (1978). Unperformed experiments have no results. Am. J . Phys. 46,745. Peres, A. and Singer, P. (1960). On possible experimental tests for the paradox of Einstein, Podolsky and Rosen. Nuouo Cimento 15,907. Perrie, W., Duncan, A. J., Beyer, H. J., and Kleinpoppen, H. (1985). Polarization correlation of the two photons emitted by metastable atomic deuterium: A test of Bell’s inequality. Phys. Rev. Lett. 54, 1790. Pipkin, F. M. (1978).Atomic physics tests of the basic concepts in quantum mechanics. Adu. At. Mol. Phys. 14,281. Pitowski, 1. (1982a).Resolution of the Einstein-Podolsky-Rosen and Bell paradoxes. Phys. Rev. Lett. 48, 1299. Pitowski, I. (1982b).Pitowski responds. Phys. Rev. Lett. 49, 1216. Pitowski, I. (1983).Deterministic model of spin and statistics. Phys. Rev. D 27, 2316. Popper, K. (1985).Realism in quantum mechanics and a new version of the EPR experiment. In “Open Questions in Quantum Physics” (G. Tarozzi and A. van der Menve, eds.), p. 3. Reidel, Dordrecht, Netherlands. Rastall, P. (1981).Quantum mechanics, locality and completeness. Nuovo Cimento SOC. Ital. Fis. B 1113 62B, 175. Rastall, P. (1983).The Bell inequalities. Found. Phys. 13, 555.
334
W. DE BAERE
Rastall, P. (1985). Locality, Bell’s theorem, and quantum mechanics. Found. Phys. 15,963. Reeh, H., and Schlieder, S. (1961). Bemerkungen zur Unitaritatsaquivalenz von Lorentzinvarianten Feldern. Nuouo Cimento 22, 1051. Renninger, M. (1960). Messungen ohne Storung des Messobjects. 2. Phys. 158,417. Rietdijk, C. W. (1978). Proof of a retroactive influence. Found. Phys. 8,615. Rietdijk, C. W. (1981). Another proof that the future can influence the present. Found. Phys. 11, 783. Rietdijk, C. W. (1985). On Nonlocal Influences. In “Open Questions in Quantum Physics” (G. Tarozzi and A. van der Merwe, eds.), p. 129. Reidel, Dordrecht, Netherlands. Rietdijk, C. W., and Selleri, F. (1985). Proof of a quantum mechanical nonlocal influence. Found. Phys. 15, 303. Rohrlich, F. (1983). Facing quantum mechanics reality. Science 221, 1251. Roos, M. (1980). Tests of Einstein locality. HU-TFT-80-5, Helsinki preprint. Rose, M. E. (1957). “Elementary Theory of Angular Momentum.” Wiley, New York. Ruark, A. E. (1935). Is the quantum-mechanical description of physical reality complete? Phys. Rev. Lett. 48,466. Sachs, M. (1980). Bell’s inequalities from the field concept in general relativity. Nuouo Cimento Soc. Ital. Fis. A 58A, 1. Santos, E. (1984). Microscopic and macroscopic Bell inequalities. Phys. Lett. A 101A, 379. Scalera, G. C. (1983). On a local hidden-variable model with unusual properties. Lett. Nuouo Cimento SOC.Ital. Fis. [2] 38, 16. Scheibe, E. (1973). “The Logic Analysis of Quantum Mechanics.” Pergamon, Oxford. Selleri, F. (1972).A stronger form of Bell’s inequality. Lett. Nuouo Cimento SOC.Ital. Fis. [ 2 ] 3, 581. Selleri, F. (1978).On the consequences of Einstein locality. Found. Phys. 8, 103. Selleri, F. (1980). Quantum correlations and separability. Lett. Nuouo Cimento SOC.Ital. Fis. [2] 21, 1. Selleri, F. (1982). Generalized EPR-paradox. Found. Phys. 12,645. Selleri, F. (1983a). Einstein locality and the K ° K o system. Lett. Nuouo Cimento SOC.Ital. Fis. [2] 36,521. Selleri, F. (1983b). “Die Debatte um die Quantentheorie.” Friedrich Vieweg & Sohn, Braunschweig/Wiesbaden. Selleri,F. (1984). Photon scattering with Doppler-shift in the atomic-cascade experimental tests of Bell inequalities. Lett. Nuouo Cimento SOC.Ital. Fis. [2] 39,252. Selleri, F. (1985). Local realistic photon models and EPR-type experiments. Phys. Lett. A 108A, 197. Selleri, F., and Tarozzi, G. (1980). Is Clauser and Horne’s factorability a necessary requirement for a probabilistic local theory? Lett. Nuouo Cimento SOC.Ital. Fis. [2] 29, 533. Selleri, F., and Tarozzi, G. (1981). Quantum mechanics reality and separability. Riu. Nuouo Cimento SOC.Ital. Fis. [l] 4, 1. She, C. Y., and Heffner, H. (1966). Simultaneous measurement of noncommutable observables. Phys. Rev. 152, 1103. Six, J. (1984). Test of the non separability of the K ° K o system. In “The Wave-Particle Dualism’’ (S. Diner, D. Fargue, G. Locak, and F. Selleri, eds.), p. 391. Reidel, Dordrecht, Netherlands. Snyder, H. S., Pasternak, S., and Hornbostel, J. (1948). Angular correlation of scattered annihilation radiation. Phys. Rev. 73,440. Stapp, H. P. (1971). S-matrix interpretation of quantum theory. Phys. Rev. D 3, 1303. Stapp, H. P. (1972). The Copenhagen interpretation. Am. J . Phys. 40, 1098. Stapp, H. P. (1975). Bell’s theorem and world process. Nuouo Cimento SOC.Ital. Fis. B [I I] 29B, 270.
EINSTEIN-PODOLSKY PROSEN PARADOX
335
Stapp, H. P. (1977).Are superluminal connections necessary? Nuooo Cimento SOC.Ital. Fis. B [I 11 4OB, 191. Stapp, H. P. (1979). Whiteheadian approach to quantum theory and the generalized Bell’s theorem. Found. Phys. 9, 1. Stapp, H. P. (1982a).Bell’s theorem as a nonlocality property of quantum theory. Phys. Rev. Lett. 49, 1470. Stapp, H. P. (1982b). Mind, matter, and quantum mechanics. Found. Phys. 12,363. Stapp, H. P. (1985).Einstein locality, EPR locality, and the significancefor science of the non-local character of quantum theory. LBL-20094. In “Proceedings of the International Conference on Microphysical Reality and Quantum Formalism.” Reidel, Dordrecht, Netherlands (to be published). Streater, R. F., and Wightman, A. S. (1964). “PCT, Spin and Statistics, and All That.” Benjamin, New York. Summers, S. J., and Werner, R. (1985).The vacuum violates Bell’s inequalities. Phys. Lett. A llOA, 257. Sutherland, R. I. (1983). Bell’s theorem and backwards-in-time causality. Int. J. Theor. Phys. 22, 377. Sutherland, R. I. (1985).A corrolary to Bell’s theorem. Nuouo Cimento SOC.Ital. Fis. B [ l l ] 88B, 144. Todorov, N. S. (1985a).On Bell’s argument. Ann. Fond. L. de Broglie 10,49. Todorov, N. S. (1985b). The Einstein-Podolsky-Rosen and Bell arguments revisited. I. On the validity of certain critiques against the current interpretation of Bell’s argument. Ann. Fond. L . de Broglie 10,207. Todorov, N. S. (198%). The Einstein-Podolsky-Rosen and Bell arguments revisited. 11. Restrictive character of the entire system of Bell’s axioms and unfalcifiability of the EPR standpoint. Ann. Fond. L. de Broglie 10, No. 4. Tornqvist, N. A. (1981). Suggestion for Einstein-Podolsky-Rosen experiments using reactions like efe- + AA + n-pn-p. Found. Phys. 11, 171. Vigier, J. P. (1979). Superluminal propagation of the quantum potential in the causal interpretation of quantum mechanics. Lett. Nuouo Cimento SOC. Ital. Fis. [2] 24,258. von Neumann, J. (1955). “Mathematical Foundations of Quantum Mechanics.” Princeton Univ. Press, Princeton, New Jersey. Weizel, W. (1953a).Ableitung der Quantentheorie aus einem klassischen, kausal determinierten Modell. Z. Phys. 134,264. Weizel, W. (1953b). Ableitung der Quantentheorie aus einem klassischen, kausal determinierten Modell. 11. Z. Phys. 135, 270. Wheeler, J. A. (1978). The “past” and the “delayed-choice” double-slit experiment. In “Mathematical Foundations of Quantum Theory” (A.R. Marlow, ed.), p. 9. Academic Press, New York. Wheeler, J. A,, and Feynman, R. P. (1949). Classical electrodynamics in terms of direct interparticle action. Reo. Mod. Phys. 21,425. Whitaker, M. A. B., and Singh, I. (1982).Interpretations of quantum mechanics and some claimed resolutions of the EPR paradox. J. Phys. A : Math. Gen. 15,2377. Wigner, E. P. (1952).Die Messung quantenmechanischer Operatoren. Z. Phys. 133, 101. Wigner, E. P. (1970).On hidden variables and quantum mechanical probabilities. Am. J . Phys. 38, 1005.
Wilson, A. R., Lowe, J., and Butt, D. K. (1976).Measurement of the relative planes of polarization of annihilation quanta as a function of separation distance. J. Phys. G 2,613. Wolfe, H. C. (1936). Quantum mechanics and physical reality. Phys. Reu. 49, 274. Wootters, W. K., and Zurek, W. H. (1979). Complementarity in the double-slit experiment:
336
W.DE BAERE
Quantum nonseparabilityand a quantitative statement of Bohr’s principle. Phys. Rev. D 19, 473. Wu, C. S., and Shaknov, I. (1950). The angular correlation of scattered annihilation radiation. Phys. Rev. 71, 136. Yanase, M. M. (1961). Optimal measuring apparatus.Phys. Rev. 123,666. Zweifel, P. F. (1974). Measurement in quantum mechanics and the EPR paradox. Int. J . Theor. Phys. 10,67.
ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS,
Theory of Electron Mirrors and Cathode Lenses E. M. YAKUSHEV AND L. M. SEKUNOVA Institute of Nuclear Physics Academy of Sciences of Kazakh SSR SU-480 082 Alma-Ata, USSR
I. INTRODUCTION Electron mirrors and emission systems are used in various electronoptical devices: electron mirror microscopes (Oman, 1969; Luk’yanov et al., 1973), cathode ray tubes (Zworykin and Morton, 1954; Zhigarev and Shamaeva, 1982), image converters (Artamonov et al., 1968; Butslov and Stepanov, 1978),brightness amplifiers (Kasper and Wilska, 1967),and so on. In practice it is of a great interest to make use of a mirror for correction of spherical and chromatic aberrations of electron lenses (Zworykin et al., 1945; Septier, 1966; Slowko, 1975). Application of a mirror in a multipassage mass spectrometer (Fortin and Baril, 1972; Noda, 1963) is worthy of noting. A mirror is applied for monochromatization of electron beams, e.g., in an ion microanalyzer (Leleyter and Slodzian, 1971), in an ion detector of the mass spectrometer (Vasil’ev et al., 1974), and in an emission microscope (Vorob’ev, 1959; Gruner et al., 1971). Recently a new field of ion mirrors application comes to light, namely, nonmagnetic time-of-flight mass spectrometry (Mamyrin et al., 1973; Wechsung et al., 1978; Daumenov et al., 1981; Gohl et al., 1983; Berger, 1983). A peculiarity of this field is the use of a substantially nonstationary flux of charged particles-an extremely narrow (in the axial direction) ion packet, which, while traveling from the source to the detector, splits up, due to the differences in velocities of ions having different masses, into several fragments. Herewith the mirror is to perform spatial focusing of these fragments in the radial and, primarily, in the axial directions. It should be noted that one has also to deal with the nonstationary flux problem in ordinary emission and scanning microscopes and image converters if one investigates the processes the duration of which becomes comparable with the time of particle flight in the system considered. 337 Copynght @ 1986 by Academic Press, Inc All nghts of reproduction In any form reserved
338
E. M. YAKUSHEV A N D L. M. SEKUNOVA
When stating the general theory of electron mirrors and emission systems, we shall take into account the exclusive variety of physical situations where these elements are used. We shall also consider some concrete applications of the theory.
11. EQUATIONS OF MOTION
It is characteristic for both the electron mirror and the cathode lens that in the space occupied by the electric magnetic field and the particle flux there is, as a rule, a rather narrow specific region where the particles' velocity is close to zero, and the electric field is quite different from zero. In electron mirrors particles are reversed in this region. In cathode lenses one of the equipotential surfaces of this region is the cathode surface emitting particles. Herewith, if in mirrors the field is analytic everywhere including the neighborhood of the specific region, in cathode lenses the field has a discontinuity on the cathode surface. However, for theoretical study of electron mirrors and cathode lenses this difference is not substantial, if the field on the internal side of the cathode surface is assumed to be the analytical continuation of the external field. It provides a possibility of studying properties of the appropriate cathode lens as well when investigating particle motion in the analytical field of the mirror and choosing one of its equipotentials as an emitting surface. In a cylindrical coordinate system, r, I), z , where the z axis coincides with the symmetry axis of the field, the electric field is described by a scalar potential cp = cp(r,z), and the magnetic one by a vector potential A having components A , = A , = 0, A , = A,@, z). The equations of motion in this field for a particle having charge e and mass m are as follows:
i 2
+ + r2$2 i 2
mr2$
=
2e m
--('
+ erA, = -eC,
+ E)
(2) (3)
where C , is a constant, and E is the total energy of a charged particle expressed in terms of a potential. The potential cp is normalized in such a way that it equals zero at the place in space where the velocity of a chosen particle equals zero. Then the energy of any other particle in this place, due to the spread of particle velocities, may differ by E.
ELECTRON MIRRORS A N D CATHODE LENSES
339
The quantity A , is taken to be equal to zero on the symmetry axis of the system. Using the complex variable
r = reio
(4)
where
and t is the time, we can rewrite the set of equations (1)-(3) as follows:
1 i 2+ cp k2
-_
..
1 k
= lrr*
-E
k2 + -(2C, + A,r)TA , 4
(7)
where k C,
=
J
q
= (i/k2)(ri* - r * i )
and an asterisk (*) denotes the complex conjugate. Equations (6) and (7), together with Eqs. (5) and (9) describe the motion of a charged particle in a stationary axially symmetric field. Here Eq. (7) represents energy conservation for a particle, whereas Eq. (9) is the conservation law for the azimuthal component of its generalized momentum. With the methodology of the static electron-optic system investigation developed by Scherzer (1937), Glaser (1952), Sturrock (1955), the law of energy conservation for a particle is used in order to eliminate time of flight from the equations of motion and to introduce the coordinate of the main optical axis z as an independent variable. This transformation implies certain assumptions. First of all, the particle flux is assumed to be stationary, and the time-of-flight characteristics of the electron-optical system are eliminated from further consideration. The other assum tions are the results of the fact that the structures like d G and*J (the prime denotes differentiation with respect to z ) appear in the equations. Because of the presence of these structures, when linearizing the equations, one has to adopt both the requirement of smallness r, and the following conditions of the paraxial approximation E/(P
<< 1,
Ir'l << 1
(10)
In doing so, mirror and emission systems are not considered, because the
340
E. M. YAKUSHEV AND L. M. SEKUNOVA
inequalities (10) can not be satisfied for them in the neighborhoods of the reversal points of electron trajectories or in the near-cathode region. When constructing the theory of electron-optical mirrors Recknagel (1937) used Eqs. (6) and (7) without eliminating time of flight. His method assumes solely small deviations of electrons from the axis, allowing whatever large angles of electron trajectory slopes in the neighborhood of a reversal point, where particle energies are close to zero. Generalizing the concept of paraxial approximation in such a way, he obtained the equations giving the first approximation, then found the expression for the correction term and showed that it was third order with respect to the first approximation. The method by Recknagel was then developed further by Zworykin et al. (1945) and Kas'yankov (1966). They derived the final formulas which had complicated forms and required rather serious computational difficulties to be overcome. The insufficiency of the theory sets one seeking for new ways of mirror study. Bernard (1952), in particular, proposed using v = & as an independent variable. Hahn (1971) used the azimuthal angle characterizing the particle output from the Larmor coordinate system as such a variable. Slowko and Mulak (1973) used the model representation according to which particle reflection takes place directly in the zero equipotential of the field obeying the law of incident and reflected angle equality, and the entire field of the mirror represents an ordinary electron lens. However, one could not manage to obtain the finished theory of electron mirrors as complete and accessible for practical applications as the theory of electron lenses. Up to now necessary practical calculations have been carried out, mainly, without using aberration representations-either via direct integration of the equations of motion (e.g., Regenstreif, 1947), or using the transformation proposed by Recknagel (1937) allowing one to overcome computational difficulties (Lafferty, 1947; Berger and B a d , 1978). In other papers (Kel'man et al., 1971, 1972, 1973a,b, 1974a,b; Sekunova and Yakushev, 1975a,b; Sekunova, 1977; Daumenov et al., 1981; Bimurzaev et al., 1983a,b) a new method of theoretical study of electron mirrors allows one to overcome analytically the above-mentioned mathematical difficulties and to obtain final expressions for aberration coefficients up to third order in the forms of dependences on the principal optical axis coordinates. A . Trajectory Equations
Consider two synchronously moving particles with equal specific charges, e/m, with their initial conditions somewhat different (Fig. 1). One of these
particles, 1, moving along the axis z with E = 0, will be called the guiding particle. Its position at the given time instant t will be denoted by (.Then from
341
ELECTRON MIRRORS AND CATHODE LENSES
FIG. 1. Synchronous motion of (1) guiding and (2) arbitrary particles in the field of an electron mirror.
Eq. (7) the precise equality follows:
i=a
k r n
(11)
+(c)
where +(z) = cp(0,z)is the potential distribution along the mirror axis, is the same function of the argument and a is the sign factor characterizing the direction of the particle's travel along the z axis. The position of the other particle, 2, at the same time instant twill be characterized by the radius vector, p = q + r, drawn to it from the guiding particle. Then the equality takes place:
c,
z=c+q
(12)
Any particle 2 can be associated with a great number of particles, each moving along the axis with E = 0. As a guiding particle choose the particle 1 from the set for which the identified equality holds: fJ
= i/lil
(13)
where i is the axial component of the velocity for particle 2. This condition implies that synchronization of motion of these particles is produced over the time instant of their reflection. Using Eqs. (11) and (12), we introduce a new independent variable and a dynamical variable q into Eqs. (6) and (7), eliminating from them t and z. It implies introducing the movable coordinate system, r, $, u. associated with the guiding particle, the time being measured by the distance covered by the guiding particle along the z axis. On performing these transformations, we obtain
c
342
E. M. YAKUSHEV AND L. M. SEKUNOVA
Herein
where B ( z ) is the distribution function of the magnetic field induction along the principal optical axis. B ( i ) is the same function of the argument i
The concept of a guiding particle implies that y equals zero for it. It means that there is a particular solution of Eq. (15) y = y(i), and its magnitude is small for small E and r. We shall use just this solution. As no limitation is used when deriving Eqs. (14) and (15), they are valid for any values of r and E. We shall seek the solution of these equations for the region close to the principal axis of the mirror and for small values of the energy spread. Equations (14) and (15) are put down in such a form that admits a solution by the method of successive approximations under specified conditions. The fact that these equations are derived without limitations for the trajectory slope does not mean that the limitation is removed altogether. As r is required to be small, large slopes may be allowed only at small sections of the trajectories, in particular, in neighborhood of the reversal points. We shall assume that, excluding these sections, everywhere, and also beyond the field, the trajectory slope with respect to the principal optical axis is the value of the order of smallness such as r. To define geometrical aberrations of third order and chromatic aberrations of order rc, it is necessary to preserve in the trajectory equations terms of order not higher than third in r and of first order in rc. We shall stick to this requirement further on. Using the known representation of the field potentials near the symmetry axis of the system
ELECTRON MIRRORS AND CATHODE LENSES
343
and the expansion
we reduce Eqs. (14) and (15) to the form
with
Here
The primes mean differentiation with respect to the independent variable i. The right-hand side of Eq. (23) is of third order in r and of first order in rE, as the constant C , is the value of second order in r; q is of second order in r and of first order in E. The estimation of the order of C 2 follows from Eq. (18) if one takes into account that lr’l is small far from the reversal points. The fact that lr’l increases without limit at reversal points does not contradict the smallness of C , , as q5(i)= 0 in this point. To substantiate the order of r], we consider Eq. (24) in more detail. It is necessary to choose such a particular solution of this equation for the value of r] which equals zero for the guiding particle. So, due to the linearity the equation under consideration, the order of r] will be the same as the order of its right-hand side. This part may be reduced to the following: s(3)
=
-
4arbrt‘ - aQ,,r,’
-
a
Q;r2 d i
-
3kBC2 + E
(29)
because in the approximation considered
(4r’r*’ + a Q 2 r 2 ) ’= $Q;r2 Here the subscript a refers to values associated with the point which is far from the reversal region. From Eq. (29) it follows that the right-hand side of Eq. (24) remains everywhere the value of second order in r and of first one in E.
344
E. M. YAKUSHEV AND L. M. SEKUNOVA
Thus, the smallness of the quantities S'3) and d3)necessary for application of the method of successive approximations is guaranteed either by the smallness of 4 which is valid in the reversal region, or by the smallness of lr'l rather far from this region. This is the difference between Eqs. (23)and (24) and the corresponding equations for electron lenses in which the smallness of the right-hand sides is provided solely by the smallness of the trajectory slope, and so they are not suited for analyses of mirror systems. In order to solve the sets of equations (23)and (24),we shall find at first the quantity U determining r in the first approximation, setting S(3)= 0 in Eq. (23). Substituting U into the right-hand side of Eq. (24), we determine q. Then, substituting U and into the right-hand side of Eq. (23), we obtain a linear nonhomogeneous differential equation of third order for the correction term x z=r-U
(31)
With S'3' = 0 we have
4,''
+ $$'U' + $Q2U = 0
(32) The equation has a singularity at [ = zk determined by the equality + ( Z k ) = 0. The point is a first-order pole for the coefficients at U' and U, because it is assumed that the electric field differs from zero in the vicinity of electron trajectory reversal points. In accordance with the theory of appropriate equations (Smirnov, 1974), choose two linearly independent real partial solutions of Eq. (32), one of which, p ( 0 , is the analytic function, and the other, g ( l ) , can be represented as
(33) where q(5) is the analytic function. On substituting the function g in Eq. (33) into Eq. (32), instead of U we obtain S(l)=4 ( t ) r n
4q" + &$'q'
+ $(34"+ $ k 2 B 2 ) q= 0
Let us specify the initial conditions at the point 5
(34)
= zk
(35)
Pk = qk =
Then from Eqs. (32) and (34) it follows
The subscript k denotes the quantities referring to
4(Zk)
= 0.
ELECTRON MIRRORS AND CATHODE LENSES
345
Note that, using Eqs. (32) and (34), we can find the derivativesfrom p and q of any order at = z,; they may be needed for numerical integration of these equations. The particular solutions, p and g , are related as follows: - P’d = 4LP
&(Pd
(38)
We shall use these solutions as the fundamental set of solutions of Eq. (32). Then the general solution of this equation may be represented in the form U
= UP
+ bg
(39)
where a and b are arbitrary, complex, in general, constants of the first order. On substituting this solution into Eq. (26), instead of r, and taking into account Eqs. (18) and (38), we obtain
d3)= - u u * ( ~ P + ’ ~2Q2p2) - (a*b + ab*)(4p‘g’+ 2 Q 2 ~ g )
+ ia(a*b - ab*)$kB+; - bb*(c$g’2+ 2Q2g2)+ E
(40)
We shall seek the solution of Eq. (24) in the form q
= eqq
+ aa*vl + (a*b + ab*)q2+ bb*q3 + ia(a*b - ab*)q4
(41)
Then Eq. (24) decomposes into a set of equations the solutions of which are
1
v3
=
24?l2-
(45)
The quantities qrp,ql - q4, and their derivatives are finite everywhere. According to Eq. (41), it provides the second order for the functions v and 1’. On substituting into the right-hand side of Eq. (23), determined by Eq. (25), instead of r and v, their approximate values, in accordance with Eqs. (39) and (41) and using the relations in Eqs. (18), (31), (32), and (38), we obtain a linear nonhomogeneous differential equation for defining x, whose particular
346
E. M. YAKUSHEV AND L. M. SEKUNOVA
solution will be sought in the form
x
+ a2b*x2 + aa*bx3 + a*b2x4 + abb*xs + bZb*x6+ &(ax7+ bxs)
= a2a*x1
(47)
In doing so, the differential equation for x decomposes into a set of equations where
cfq;
+ &YxJ + $Q2xj = Si3),
j = 1,2,. . . , 8
Solving Eq. (48) by the method of variation of parameters, we find
Equations (31), (39), and (47) together with Eq. (57) determine r as a function of [. Now in order to find r and $ as functions of [ based on Eqs. (4) and (9,it is necessary to eliminate time t from Eq. (5).Using Eqs. (1 l), (20),and (22),we write, neglecting the excess terms o=*-o--Ic
where
When deriving Eq. (58), the integration constant is set equal to zero; however, this fact does not restrict generality of consideration, because the constants a and b are arbitrary. It follows from Eqs. (4), (31), and (58) = rei(+e-K)
=
u+x
(61)
ELECTRON MIRRORS AND CATHODE LENSES
347
hence
re'(^-') = U
+ x + ilcU
Transform Eq. (60), substituting U and q into it in accordance with Eqs. (39) and (41) IC = a a * q
+ (ab* + a*b)K, + bb*K3 + E I C ~+ i(a*b - ab*)K5
(63)
Herein
Using Eqs. (39), (41), (47), and (63), we write in a parametric form the equations of a charged particle trajectory in the field of a mirror
These equations determine particle trajectories in an electron mirror throughout their paths including reversal points. They are valid up to terms of third order of smallness in r and of first order in re.
348
E. M. YAKUSHEV AND L. M. SEKUNOVA
+
In order to obtain the explicit dependences of r and on z, it is necessary to define = c(z)from Eq. (70) and to substitute it into Eq. (69). The solution of Eq. (70) with respect to ( will be defined by the method of successive approximations. Up to terms of second order in r and of first order in E
c
i=
-
rl(4
(71)
On substituting this equality into Eq. (69) and expanding the entering functions of the argument z - q(z) in series in q, which is necessary for an analysis of aberrations, we find = rein = aP(z)
+ b d z ) + x(z) - c
+ i[ap(z) + bg(z)] K ( Z ) -
(
+
0 -
a m + bg’(z)lrl(z)
kB(z) rl(z)) 4 m
where the angle R = - 8 characterizes a particle going out of the plane = 8(z) rotating about the z axis, and
+
The functions p(z), g(z),~ ( z ~) ,( z and ) , K ( Z ) entering Eq. (72) were defined earlier as the functions with argument [. Thus, Eq. (72) describes charged particle trajectories in the rotating coordinate system r, R, z as the explicit dependence of rand R on z. Equation (72) is valid everywhere, excluding the neighborhood of trajectory reversal points because the values of the derivatives u’(c), x’((), and 8’(c) increase without limit there, and, due to this, the expansion becomes invalid. However, if an image is formed far from the reversal point, the limitation mentioned is unimportant. To determine a concrete trajectory, it is necessary to express the arbitrary constants a and b via the initial conditions. When the initial conditions are specified far from the reversal point, the arbitrary constants can be found by Eq. (72). When the initial conditions are specified near the reversal point (e.g., when calculating the trajectories in an electron emission/mirror microscope), only Eqs. (69) and (70), also valid in this region, are suitable for determining the arbitrary constants. From Eq. (72) it follows that in the first approximation the particle trajectory is described by the dependence u = U(z),where U ( z )is the solution of the equation
~’(c),
+
+
~ ~ ( z ) U ” ( Z+’(z)U’(z) ) +QZ(z)U(z)= 0
(74)
This equation fully coincides with the paraxial equation for axially symmetric electron lenses; however, in contrast with lenses, it cannot describe the entire
ELECTRON MIRRORS AND CATHODE LENSES
349
trajectory in a mirror. Near the reversal point z = zk, both Eqs. (74) and (72) become invalid. This circumstance is insignificant when observing an image situated, just as the object itself, far from the reversal point. B. Time of Flight
In the derived trajectory equations, Eqs. (69), (70), or (72), the explicit dependence of particle coordinates on time of flight is eliminated. Let us show that the information on time-of-flight characteristics of electron-optical systems is contained in q. Indeed, due to the synchronous motion of an arbitrary particle and the appropriate guiding particle, their time of flight from the instant of reflection, corresponding to 5 = zk, to any other instant, corresponding to 5,will be equal, according to Eq. (1l), to
During this time any synchronously moving particle of the set reaches its plane, z = const. These planes, as seen from Eq. (12),do not coincide, and their parallel displacements relative to each other are determined by the dependence of q on the initial conditions for the entire particle set. On the contrary, all the particles of the set intersect some specified plane, z = const, at various instants of time. The spread of time instants A t of the intersection of the chosen plane by these particles characterizes time-of-flight aberrations. To define the spread, we rewrite Eq. (75) taking account of Eq. (12) in the form
Here where
represents the time of flight for the guiding particle, and the quantity
means the total time-of-flight aberration. Equation (79) is rigorous. If one restricts oneself by the terms of order not higher than second in r and of first
350
E. M. YAKUSHEV A N D L. M. SEKUNOVA
order in E, then for the region far from the reversal points of electron trajectories we obtain a simple and physically clear equality At = --=w
kv%
--
B
kv%
[&qs
+ aa*ql + (a*b + ab*)q2
+ bb*q3 + i(a*b - ab*)q4]
(80)
determining time-of-flight aberrations in any plane z = const.
111. ELECTRON-OPTICAL PROPERTIES OF AXIALLY SYMMETRIC ELECTRON MIRRORS The previous section presented the method of calculation of axially symmetric mirrors, allowing one to overcome mathematical difficulties associated with a breakdown of classical paraxiality conditions [Eq. (lo)] for charged particle trajectories in the neighborhood of reversal points. It gives one a possibility to carry out investigations of electron mirrors and to find the expressions for aberration coefficients in a form like that for electron lenses, i.e., depending on the field distribution over the mirror axis. Below, the electron-optical properties of a mirror in a paraxial approximation are investigated, aberration classification is produced, and the aberration coefficients are calculated. The object plane, z = z,, and the image plane, z = zb, are assumed to be located out of the mirror field; hence, there is a possibility to study the electron-optical properties of a mirror using Eq. (72). The z coordinate enters the equation as an independent variable. All the functions considered in this section are functions of this coordinate; the prime denotes differentiation with respect to it. It is the peculiarity of electron mirrors that there are two branches of a trajectory- the direct one, corresponding to the particles incident on the mirror, and the reversed one, corresponding to reflected particles. It is evident that for the values a, b, 0 given, Eq. (72) describes one of the two branches of a trajectory-the direct or the reversed. For every branch there is its own sign of c and, generally speaking, its own meanings of the arbitrary constants a and b. We refer B, a, b to the incident particle before its reflection. After reflection these meanings change and turn out to be 5,Z, 6. Then the following equalities hold:
u = u(z, a, a, b), yI = q(z, 5,a,b),
= x(z,
a,a, 6) -
2 = K ( Z , 5,a, b )
(81)
ELECTRON MIRRORS AND CATHODE LENSES
351
Herein and after, the bars refer to the quantities or the functions related to the reversed branches of trajectories. Let us establish the connection between the quantities characterizing the direct and the reversed branches of a trajectory. According to the identity in Eq. (13), 5 = - 6.We shall find the relation between the constants ii,band the constants a, b using continuity of a particle trajectory and of its velocity in the neighborhood of a reversal point. A s ~ ( 2 ,7 ) 0, i ( z k )= 0 in this point, it follows from Eq. (61) that u(z,) = U(z,), u(z,) = U(z,). Hereof, using Eq. (39), the relations in Eqs. (1 1) and (33), and taking into account for differentiability of the functions p , q at the reversal point, we obtain: up, = Zp,, bak&q, = bCk+;q,. Thus the constants before and after reversal are connected by the relations -
5=--o,
Z=a,
b=-b
(82)
These relations, together with Eqs. (81), give a possibility, based on Eq. (72), to write the equation for the reversed branch of a trajectory for specified initial conditions corresponding to the direct branch of this trajectory.
A . Determination of the Constants a and b
We shall determine the constants a and bin terms of the coordinate u, and the trajectory slope ub in the object plane. The rectilinear section of the trajectory of a particle going out of the object plane z = z, may be represented in the form
Here c1 and jl are complex variables of first order determined by the initial conditions in the object plane
The subscript a denotes the quantities referring to the object plane. The initial value of SZ, is set to be equal to zero (u, = r,); however, this fact does not restrict the generality of consideration because the system is axially symmetric. R is a constant value and is equal to
352
E. M. YAKUSHEV AND L. M. SEKUNOVA
On the other hand, based on Eq. (72) for the same section of the trajectory, we can write u = (a
+ x p - ubq, + ioic)p(z)+ (b + xg - ubq, + ibic)g(z)
(87)
In this expression the quantities xp, xs,q p ,qg are connected with the functions ~ ( zand ) q(z) by the relations
As follows from Eqs. (47) and (57)
The quantities qp,qg may be found by proceeding from the conditions for q and q’ in the object plane: qa = paqp gaqg; qb = p:qp g:qg. Thus we obtain
+
+
It is worth noting that the values x p ,xe, qp,qg are the constants independent of the position of the object plane z = z,. Now, on comparing Eqs. (83) and (87), we may obtain
(95)
In doing so it is enough to substitute into x p , xg, qa, and ic, instead of the constants a and b, their approximate values alp: and jllg:, respectively. Thus, Eqs. (94) and (95) with account of Eqs. (84) and (85) determine the constants a, b in terms of the initial conditions for the object plane with accuracy up to third order.
ELECTRON MIRRORS AND CATHODE LENSES
353
B. Mirror Properties in the First Approximation
Using the developments made in the previous section, the trajectory equation can be written as u = rein = ap
+ bg + x - u'q + iu
( -K
:$q)
for the direct branch, and
for the reversed one. At first, let us consider mirror properties in the first approximation. For this purpose, in Eqs. (96) and (97) we restrict ourselves by terms of first order in r and set E = 0. Then, on taking into account Eqs. (73), (82), (84), (85), (94), and (95), the equation of the trajectory in this approximation takes the form
Herein, with the double sign (f), the upper refers to the direct branch of a trajectory, while the lower refers to the reversed. This notation will also be used further on for trajectories in a mirror. Let us determine the cardinal elements of an electron mirror. For this aim consider several specific mirror trajectories. It is clear from Eq. (98), provided that
ub in the meridional plane R family of trajectories
=0
= raPb/Pa
(99)
of a rotating coordinate system there is a r
= rbp(z)/pb
(100)
the direct and the reversed branches of which coincide (Fig. 2). Depending on the function p(z), these trajectories may or may not intersect the z axis. By analogy with light optics, we shall assume that such trajectories determine the curvature center of an electron mirror located in the point zc of intersection of the mirror axis and the continuations of the out-of-field rectilinear sections of the trajectories. Out of field P(4
= (z - Z
dPb
(101)
354
E. M. YAKUSHEV AND L. M. SEKUNOVA
Iz=zk
FIG.2. Trajectories determining the center of curvature of an electron mirror (Sekunova, 1977): (a) divergent mirror with repulsive potential & = -q$, (b) convergent mirror with LM is an equivalent optical mirror. repulsive potential do = -0.185 4d.
So the curvature center of the mirror will be at the point z = zc belonging to its principal optical axis. When satisfying the condition 4 2
( 102)
= (ra/sa)sh
Eq. (98) describes the trajectories lying in the meridional plane R = 0 and symmetric with respect to the principal optical axis (Fig. 3) r = +rbg(z)/gb ( 103) Intersection of trajectory out-of-field section continuations with the z axis determines the point z = zB playing the role of the electron mirror vertex. The value of zB may be found if the function g(z) is known, because out of field g(z) = (z - zB)gh
(104)
The constant R introduced above [see Eq. (86)] and equal to R
= palpb
- 9.19:
= ZB - zc
(105)
=-
has a sense of the curvature radius of an electron mirror. With oR 0 the mirror is convergent, while with OR < 0 it is divergent. Note that, according to
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ELECTRON MIRRORS AND CATHODE LENSES
,r
@a
I
I I
FIG.3. Trajectories determining the electron mirror vertex. Notation as for Fig. 2.
Eq. (38), there is a relation between the electric field intensity in the reversal point of a trajectory and the curvature radius of an electron mirror
R
=
4;/2PhgbJ,
(106)
The trajectories initially parallel to the principal optical axis (Fig. 4) in the plane !2 = 0 of the rotating coordinate system may be represented in the form
It follows from Eqs. (101),(104),(106), and (107) that both for the electron and the light-optical mirror the following relations hold: The focal distance is
f = R/2 the position of the focal plane z
= zF is
ZF = +(ZB
( 108)
determined by the equality
+ zc)
(109)
while the principal plane passes through its vertex. It follows from Eq. (98) that the positions of mutually conjugated planes z = z , and z = zb are determined by the relation Pagb
+ Pbga =
(110)
356
E. M. YAKUSHEV A N D L. M. SEKUNOVA
I
+f--i
z=zFI
!z=&
IZ=tb
FIG.4. Trajectories determining the electron mirror focus. Notation as for Fig. 2.
On incorporating Eqs. (101), (104), and (109, this relation is reduced to the form
which is also characteristic of light-optical mirrors. Magnification of an electron mirror equals M
=f/(za
- zF)
(112)
It is necessary to take into account that, due to the presence of a magnetic field, the image plane is turned with respect to the object plane by the angle y = *b
- *a
= --
C . Aberrations In order to find the mirror aberrations, we substitute Eqs. (94) and (95) for the constants a and b into Eq. (97) and write the latter for a rectilinear section
ELECTRON MIRRORS AND CATHODE LENSES
357
of a trajectory out of field. Subdividing the equation obtained according to the orders of the values, we represent it as
ii=U+Au
(1 14)
where is the paraxial approximation for a rectilinear section of a trajectory determined on the basis of Eq. (98), and Au is a correction term of third order characterizing the image distortion (aberrations) in an electron mirror. On taking into account Eqs. (84),(85), (88), and (98),the expression for Au takes the form
- U'(z)V(z)
+ A(z) - iu(z)o(z)
(115)
where = (lip - X,)P(Z)
+ (lig + xg)s(z)
o=lC-E
(1 16)
(117)
Using Eqs. (47), (49)-(56), (82),(88), (90), (91), (94), and (95), we represent A as follows:
+ a(afl* + a*fl)A,p + aflp*A3g + aa*pA4p + fl(ap* + a*p)ASg + flzfl*Atjp + ia(ap* - a*fl)A,g + ip(afl* - a*P)A8p
A(z) = aza*Alg
+ aEA9S + BEAlOP Constant quantities from A1 to Al0 are defined by the expressions
(118)
358
E. M. YAKUSHEV A N D L. M. SEKUNOVA
and do not depend on initial conditions, because the plane z = za is located beyond the mirror field, where the integrands of these expressions equal zero. Involving Eqs. (63)-(68) and (82) represents the function o as the sum 0 = a a * o l + flp*02 + i(afl* - a*fl)o3 + E 0 4 ( 129) where
1
ok
0 4
__ B’qp dz
=-
(133)
JZk
are constants independent of initial conditions. Taking into account that beyond the field qba ?;a
= =
lI24a>
=
vko
=0
1 r2 3
1 --pa
via
I2
>
via
=
-3pbgi>
(134)
as well as involving Eqs. (39), (41), (63), (82), (94), (95), and (1 18), we rewrite Eq. ( 1 15) as the expansion
3 59
ELECTRON MIRRORS AND CATHODE LENSES
-".>I"
+ m{[A9gb + k(2qpa 4a ~b
gb - iu4:}
There is a number of correlations between the quantities entering the last expression. On comparing Eqs. (120) and (123) it is clear that € 7 3 2 = sbA5 (136) As shown below, the following relations hold
[A3g:
+
k(2+ ;)]
- [A4&
A7gb---2 R
~
4
Phsb
+ f($ +
-a - 0 1
F)]
=1
(137) (138)
The left-hand side of Eq. (137), involving Eqs. (121) and (122), may be represented as follows:
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E. M. YAKUSHEV AND L. M. SEKUNOVA
On differentiating the equations
- 4 ~ ”- ‘4Q 2 p’ = -4d2 - $Qzg’
241; - 4‘11 =
(141)
241; - 4 ’ 1 3
(142)
we obtain
+ 4‘ql - 4”11 = -+Q;p’ 241; + 6’1;- 4”13 = -$Q;g’ 241;
(143) (144)
taking into account that
( 4 ~ ’+ ’ $QzP’)’ = $Qh2 (4s’’ + bQzgZ)’= $Qh2
(145) (146)
because p(z) and g(z) satisfy Eq. (74). Multiplying Eq. (143)by q 3 ,and Eq. (144) by ql ,subtracting the first from the second, and dividing the obtained equation by 2&, we find
- 1;13)I’
(147) Using the resulting equation, we carry out the integration in Eq. (140) directly proving Eq. (137), if we take into account Eqs. (106) and (134). To verify Eq. (138), substitute A7 and q4 in the forms in Eqs. (125) and (46) into its left-hand side. The twofold integral entering the resulting expression is carried out by parts, taking into account Eqs. (145) and (106) C f i ( ~ i &
= (Q;/~&)(P’v~ - g21i)
( 148)
This expression is rewritten realizing that p satisfies Eq. (74)
On carrying out the latter integration by parts, we find, after eliminating the uncertainty occurring at the point Zk
ELECTRON MIRRORS AND CATHODE LENSES
36 1
The right-hand side of Eq. (138), determined by Eq. (130), may be reduced to the same form, if q1 is substituted into Eq. (138) in the form of Eq. (43), and double integration is carried out by parts. Equation (139) is verified in the same way. Taking into account Eqs. (136)-( 139), we reduce the expansions of Eq. (135) to the form
where 1 5 1
=-
JZ
=
+ 4Q;p2p’ + 32p”(W2 + aQ2p2)]dz
2’pCQ4~3
1 9 -CQ4p2g 84kPaga I z k f i
+ 4 Q W g + 32~”(4p’g’+ aQ2pg)l dz
ok J , = 16pbz~ ~ ~ & ( B ” p4B’p’
+
J
(152) (153)
+ 8Bp”)dz
ok 6 -
(157)
362
E. M. YAKUSHEV A N D L. M. SEKUNOVA
The expansions in Eqs. (135) and (151) differ by the circumstance that there is no double integrationin the latter. We have eliminated them by the same way used for verifying Eqs. (137)-( 139). Using Eqs. (84), (85), (101), (104), (105), and (109), we rewrite the expansion in Eq. (151) as follows
Equation (162) determines the aberration of an electron mirror in an arbitrary plane, located beyond the field. If this plane is the plane of the Gaussian image, z = z b , then using Eqs. (108)-(112), we may simplify the expressions for abberrations
Equation (163) determines the image aberration in an electron mirror when a limiting diaphragm is absent. If an incident ray is determined by the coordinates of an object point and by the coordinates rD and R, of the point of its intersection with the plane of the diaphragm z = z D ,which also located beyond the field, then the following relation is valid uD=
D e i R ~=
r,
+ uh(zD - z,) = r, + ubZDa
(175)
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E. M. YAKUSHEV AND L. M. SEKUNOVA
Using it, we write expression for the image aberrations in a mirror with the limiting diaphragm
where
With E = 0 Eq. (176) gives geometrical aberrations of a mirror in a form quite analogous to the case of geometrical aberrations of electron axially symmetric lenses. This allows one to use for electron mirrors the classification of aberrations adopted for electron lenses. According to this classification, G D is the coefficient of spherical aberration, FD is the coefficient of isotropic coma, while f , is the coefficient of anisotropic coma, C, is the coefficient characterizing astigmatism, c, is anisotropic astigmatism, D D is the image curvature coefficient, and ED and e, are the coefficients of isotropic and anisotropic distorsions, respectively. The coefficients at c/4a give one the possibility to determine the mirror chromatic aberrations. K I D represents the coefficient of chromatic aperture aberration, the coefficient K 2 , characterizes the change of the image scale, and the coefficient k, denotes the change of the image turning angle. The coefficients f,, c,, e,, and k , differ from zero only in the presence of a magnetic field.
ELECTRON MIRRORS AND CATHODE LENSES
365
D . Time-o f-Fligh t Focusing
Investigation of time-of-flight focusing of nonstationary fluxes of charged particles is one of the basic problems when calculating and designing a large number of electron-optical and ion-optical devices, in particular, time-offlight mass spectrometers. Now we study this problem theoretically, in the general case, i.e., without imposing any restriction, besides their rotational symmetry. The time of flight of a particle from the initial plane z = z, to an arbitrary plane z = const after reflection, according to Eqs. (11) and (12), equals
When deriving this equality, a change of the sign of has been taken into account in the particle reversal point at i= zk, and the notation q, = ~(2,) has been adopted. The planes z = z, and z = const are assumed to be located beyond the mirror field. In this case Eq. (188) takes a simpler form t = to
+ (d/k&)(?,
+ 'I)
(189)
is the time of flight of a particle along the z axis with E = 0 between the planes considered, z = z, and z = const, before and after reflection, 4a denotes the potential of the field-free space. The second term of the sum in Eq. (189) determines time-of-flight aberration. We shall find the structure of this aberration, taking account the quantities of first order in E and of second order in I , substituting q in the form of Eq. (41)into Eq. (189) and using Eqs. (81) and (82). t
-
to
=(
+
~ / 4 a ) ~ t e (a/k&){aa*Cqt(za)
+ V~(Z)I
+ (ab* + a*b)Cqz(z,) - qz(z)l + ia(a*b - ab*)C~4(z,)+ G'4(Z)I + bb*C%(Z,) + %(Z)1) (191) where Dra(Zu,Z) = Dre = ( D / ~ ) & C V J Z J
+ V,(Z)I
(192)
is the time-of-flight energy dispersion of the first order. Let us show that the particle time-of-flight focusing in energy may be achieved with a mirror and deduce the equation of focusing. As follows from
366
E. M. YAKUSHEV AND L. M. SEKUNOVA
Eq. (42), the function q,(z) in a field-free space may be represented as being linearly dependent on z 110 = (1/24cz)(z - zT)
(193)
Note that the quantity zT does not depend on the presence of the magnetic field in a mirror or on the choice of the point z, beyond the field and is determined only by the electric potential distribution along the z axis. Taking into account Eq. (193), Eq. (192) takes the form
The first-order condition of time-of-flight focusing will be understood as the independence of the time of flight of particles on their spread in initial energy, i.e.,
Then it follows that 2,
+
ZB
= 22,
(197)
This expression means that time-of-flight focusing is achieved if the plane z = zar from which particles are emitted, and the plane z = zB, where they are registered, are located symmetrically with respect to the plane z = z T , determined, according to Eq. (194), by the electric field distribution over the principal optical axis of a mirror. The plane z = zT will be called the principal plane of time-of-flight focusing. In mirrors, time-of-flight focusing of particles with a small spread in energy is realized based on the fact that, when reflecting, the higher the particle energy, the longer the path it covers in the reflecting field. Assuming that the principal plane z = zT is located before a mirror, in the field-free space we rewrite Eq. (190) as follows:
where
ELECTRON MIRRORS AND CATHODE LENSES
367
is the time interval between the instants of intersection of the plane z = zT by a particle before and after its reflection. This interval will be called the timefocusing interval. Its dependence on the particle mass determines the amount of a mirror time-of-flight mass dispersion dTo
D,, = mam
1 2
=-T O
It is a characteristic feature of the principal plane z = zT of time-of-flight focusing that the time-focusing interval To does not depend, to a first approximation, on the initial energy of the particles. It means that particles with equal specific charges, elm, and with a small spread in energy traversing the principal plane simultaneously before reflection, will also traverse the plane simultaneously after reflection. The time-focusing interval Todetermines the effectivedrift distance L
It is worth noting that both L and zT depend only on the mirror electric field. If the position of the principal plane of time focusing is known, the position of two mutually conjugated, in the sense of time-of-flight focusing, planes, z = z, and z = zB is determined by Eq. (197). The latter will be called the equation of time images, an object/image understood as the structure of substantially nonstationary flux (of a short current pulse) in the plane of the object/image. The mirror time magnification, M,, determined as a ratio of the current pulse duration (At)Bin the image plane, z = z B ,to the initial duration of the same pulse, (At),, in the object plane, z = z,, is always constant, due to the stationary nature of the electric and magnetic fields forming a mirror, and equals
It means that to a first approximation a mirror adequately transmits the time structure of the nonstationary charged particle flux from the object plane to the image plane. As it is seen from Eqs. (197)and (198),the object and the image are shifted, relative each other, in time by To,independent of the values of z, and zg. Now we shall find the position of the mutually conjugated planes, z = z1 and z = z 2 , for which both space and time-of-flight focusing are realized. Setting into the equations of an image, Eqs. (111) and (197), z, = z, = zl, zb = zB = z2, and solving these equations with respect to z l and z 2 , we obtain 21,2
= zT
kJ(zT
- zB)(zT
- zC)
(203)
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E. M. YAKUSHEV A N D L. M. SEKUNOVA
When this equality is satisfied, the correct point-by-point space and time image of an object is formed in the same plane, i.e., space-time-of-flight focusing of the nonstationary flux of charged particles is realized. For an electrostatic mirror having a specific distribution of the field along its axis, only a pair of the mutually conjugated planes satisfying the conditions of space-time-of-flight focusing, Eq. (203),may be found. When a magnetic field is present, one may find a set of such pairs due to the fact that the quantities zB and zc depend now both on electric and magnetic fields. Note that, as for the possibility of realizing space-time-of-flight focusing of a charged particle nonstationary flux, a mirror is a unique element in the class of the axially symmetric fields considered. Now let us consider the second-order time-of-flight aberrations, tracing the motion of two particles with initial energies E = 0 which simultaneously traverse the plane z = z , located before a mirror. Let one of them move along the z axis and, after reflection in a mirror, reach the plane z = const in a time interval t o . Then the time of flight up to this plane for the other particle remote from the z axis, due to the presence of aberrations determined by Eq. (19l), will differ somewhat from to t
= to
+ (At)!”
(204)
where
+ + + $(ab* + a*b)pbgb(z - z,) + 2ia(a*b - ab*)q, + bb*gbZ[zB- +(z, + + R I , ] }
(At)!2)= ( ~ ~ / k f i ) { ~ ~ *-p&z, b ~ [ Z~) c R I , ]
Z)
(205)
and
When deriving these expressions, it was taken into account that the initial and final planes, z = z , and z = const, are located beyond the field of a mirror. In this case, according to Eqs. (101) and (104), the functions p ( z ) and g ( z ) are approximated by the straight lines p = pb(z - zc) and g = gb(z - zB), and Eqs. (43), (44), and (45) for the quantities q l ,q,, q3 are simplified; the quantity q4, determined by Eq. (46), becomes constant. Represent the structure of geometrical time-of-flight aberrations as the dependences on a radial deviation of particles ra and on a slope of their trajectories r: and r,& given in the initial plane z = z,. For this purpose, we use the expressions in Eqs. (84),(85), (94),and (95) determining the parameters
ELECTRON MIRRORS AND CATHODE LENSES
369
+ ir,$k. Then, combining in a proper way the quantities of second order, we find
a, b and the relation uh = rk
(At):2) = T,(ri2 + r:$i2)
where T -
-b, ff
-k R f i
-
zJ21,
+ T2r,rh + T,r:$i + T,r;
(208)
+ (za - z,)~Z,
The aberration associated with the coefficient TI describes the deformation of the time structure of the charged particle's nonstationary flux emitting from the point located in the z axis and in the plane z = z,. The aberration of this sort may be called the time-of-flight spherical aberration. The coefficient T4 determines the time deformation of a structure of the flux entering the mirror field parallel to the z axis. An aberration of this sort depends only on the particle coordinates in an object plane, and so it may be called the timeof-flight distortion. The coefficients T2,T3 determine mixed time-of-flight aberrations, and in pure electrostatic fields T3 = 0. If the first-order time-of-flight focusing in an energy E is available as well as the initial spread in energy which is relatively great, it is of interest to calculate time-of-flight aberrations of second order and above. In order to calculate these aberrations, it is necessary to consider the particle motion along the z axis and to define, with a required accuracy, the quantity q = q ( ( ) as the dependence on the initial energy E. Then for the space free of field the total time-of-flight chromatic aberration may be written as follows
It is possible to calculate chromatic aberrations with any specified accuracy, as the equations of motion of axial particles can be integrated strictly at arbitrary values of E. For axial particles (at r = 0) Eq. (15), with the
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E. M. YAKUSHEV A N D L. M. SEKUNOVA
use of Eqs. (12) and (17), reduces to an extremely simple form
1:'s
allowing solution by the method of separation of variables
=j
:
+
dz V
k
d
m
where the quantity q k satisfies the equality
4(zk + qk) +
(216) Equation (215) combined with Eq. (216) determines the required dependence 9 = V(Z,&). In order to make Eq. (215) more convenient for carrying out the analysis of time-of-flight aberrations, it is necessary to fulfill certain transformations in it. We rewrite it using Eq. (216) as follows: =
The expression under the radical on the right-hand side of the equation will be rewritten in the form
4(z+ q k ) - 4 ( z k + q k ) = 4(z)[1+ pL(z,q k ) ] where p is a small quantity. Expanding the functions 4,4(z + v k ) , + ( z k
+
(2 18) q k ) in
a power series in q k , we obtain
It is seen from this expression that the function p is continuous and differentiable everywhere including the singularity z = zk. The value of p = pk at the singularity will be found after eliminating the uncertainty of the 0/0 type
In the space free of field the quantity p becomes constant, pa = &Ida.Recall that the object and its image are located beyond the field of a mirror. This provides a possibility of solving Eq. (217) with respect to 9. On carrying out this operation and involving Eqs. (218), we find
ELECTRON MIRRORS AND CATHODE LENSES
371
This equation, combined with Eqs. (213), (216), and (219), determines the overall time-of-flight aberration (At), with any specified accuracy with respect to the spread in energy in the nonstationary flux of particles. So, restricting ourselves by the values of second order, on the basis of Eq. (221), one may obtain (At), = (&/4Pt, + Wi2’
(222)
where D,, is the energy dispersion determined by Eq. (195), and (At)L2’ = (e2/&)TV
(223)
is the time-of-flight chromatic aberration of second order, and
(224) with PI
=
(4‘ - 4;M
p2
=
(4“ - 4;1/4
(225)
The second-order total time-of-flight aberration of an electron mirror is composed of geometrical aberrations [Eq. (208)] of four kinds and of chromatic aberration [Eq. (2231 of one kind (At)‘,’ = (At)!,)
+ (At)!”
(226)
In the case for which space-time-of-flight focusing is realized, the coefficients of geometrical and chromatic time-of-flight aberrations in the image plane take the form
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E. M. YAKUSHEV A N D L. M. SEKUNOVA
IV. MIRROR ELECTRON MICROSCOPE
An electron mirror possesses a remarkable feature-it creates an image of its reflecting electrode. For the first time this effect was noticed by Hottenroth (1936, 1937). Investigating experimentally electron-optical properties of electrostatic mirrors, he noticed that, with a solid reflecting electrode, tiny roughness of the electrode surface is observed in a mirror image. The mirror ability to create a magnified image of the surface of its electrode reflector allows one to use it as an original electron microscope like a shadow lightoptical microscope. The scheme of a mirror electron microscope (MEM) is shown in Fig. 5. The main part of the microscope is the mirror M (in the simplest case: aflat sample, a cathode and a diaphragm with an aperture, and an anode). An electron beam emerges from the source S and, retarded in the field of the mirror objective, reflects at a very short distance from the sample surface K. When near a sample a slow electron beam is modulated by microfields produced by micrononhomogeneities of the sample surface, which may be of electric, magnetic, or geometrical character. After changing the direction of its motion, the beam again traverses the objective field, which now acts upon it as accelerating one.
FIG.5 . Scheme of the simplest mirror microscope: (a) of the direct construction (the incident and the reflected beam axes coincide), (b) with separation of the incident and reflected beams by virtue of the magnetic field B.
ELECTRON MIRRORS AND CATHODE LENSES
373
On the screen P the reflected beam creates a brightness picture which represents the magnified image of the nonhomogeneities of the sample surface. It is the advantage of the device that in it a sample does not undergo bombardment by an electron beam. As a result, the sample is not destroyed, its subcharging is reduced to minimum, there is no sample heating, and no secondary emission of its surface or other undesirable phenomena. MEM is extremely sensitive to the micrononhomogeneitiesof the surface under study, because in the region occupied by fields associated with these micrononhomogeneities which are very close to the sample surface, the velocities of particles belonging to the beam incident on the sample are extremely small, and even insignificant disturbances of the field in the near-cathode region lead to noticeable redistribution of the velocities in its components. In the neighborhood of the reversal point the character of the particle motion is such that its trajectory is parallel to the surface under consideration, and it still raises the device sensitivity because of an increase in the time of interaction between the beam and microfields. If the potential difference between the sample and the cathode of the source of illuminating electrons (bias voltage) varies, it is possible to study the surface microfields of the sample at various distances from its surface. Owing to the merits mentioned above, the application field of MEMs is rather wide (Luk’yanov et al., 1973). An image in a mirror electron microscope operating in a shadow regime is formed in a different way than in a microscope of a transmission type. In a mirror microscope an image represents a brightness picture formed by a reflected beam in which the density of electron current is redistributed as a result of its interaction with micrononhomogeneitiesof the cathode surface. Conformity between the object micrononhomogeneities and the received image is of a rather complex character. The contrast theory explaining this conformity is constructed based on the calculation of the current density of the reflected beam of electrons moving in the field of a mirror objective disturbed by the nonhomogeneities of the cathode surface in any cross section. As the region of action of micrononhomogeneitiesin the direction perpendicular to the cathode surface is small compared with the objective field extent, one may agree with Wiskott (1956) that the field disturbance caused by them, is concentrated in a narrow near-cathode region, and that the entire additional pulse is imparted to the electron at its reversal point. The problem of contrast calculation then involves two tasks. The first is to define the value of an additional pulse which is imparted to the electron in the disturbed field (a microfield) near the cathode. Here, in the absence of micrononhomogeneities on the cathode surface, the field may be considered as homogeneous. The field disturbance is determined by a concrete character of micrononhomogeneities of various types on the cathode surface. Calculations are based on determination of electron motion in such a field. The study by Wiskott (1956)
3 74
E. M. YAKUSHEV A N D L. M. SEKUNOVA
underlying the theory of a mirror microscope was the pioneering work. Later much effort was devoted to this problem. The most comprehensive papers to be mentioned are those by Barnett and Nixon (1967-1968) and Sedov (1970). The second task is to calculate the electron trajectory in the basic field of a mirror objective (in a macrofield). In doing so it is necessary to take into account that in the reversal point of an electron, due to a disturbance, the initial conditions for its motion in the reversed direction are changed. When interpreting the image contrast, the calculation of trajectories in a macrofield is no less important than the calculation of disturbances in a microfield. It is known (Sedov et al., 1971) that disappearance of the contrast and even its inversion are possible for certain geometries of a mirror objective and for appropriate parameters of illuminating electron beams. However, the complete studies of trajectories in the macrofield of a mirror objective are not enough. Therefore, the image deformation associated with geometrical aberrations of electron mirrors have not been studied, and theoretical investigations in a paraxial approximation have been carried out only in application to the simplest designs of the MEM objectives (e.g., Luk’yanov et aZ., 1968; Artamonov and Komolov, 1970; Schwartze, 1965-1966). Here the geometrical optics of MEMs operating in the shadow regime is considered taking into account the mirror’s geometrical and chromatic aberrations of third order in r and of first order in rE without imposing any restriction on its design but the rotational symmetry of the field forming a mirror. In doing so it is taken into account that, equally with electrostatic fields, magnetic fields can take part in forming a shadow image, and the cathode surface (a mirror reflecting electrode), in general, has a form of the surface of an axially symmetric body. Electron-optical properties of a mirror, including its aberrations, were dealt with, as a whole, in Section 111. However, a mirror operating in the MEM regime has a peculiarity that it creates on a screen an image of one of its electrodes-a cathode, i.e., the object located in the neighborhood of the reversal points of electron trajectories. Therefore, the formulas obtained in Section 111, to deduce which it is assumed that both the object and the image are located beyond the mirror field, cannot be applied for calculation of mirror microscope aberrations. However, such calculations may be made using the equations deduced in Section I1 that determine the trajectories of charged particles in the field of an electron mirror taking into account the terms proportional to the initial energy spread of the particles and the terms of third order with respect to a particle’s deviation from the axis of field symmetry. An electron mirror is considered to be the entire optics of a mirror microscope, except for the system of separation (when it is available) of the direct and the reversed beams, as well as the emission system specifying the illuminating beam of electrons. Then the source of illuminating electrons is understood as
ELECTRON MIRRORS AND CATHODE LENSES
375
the electron beam crossover formed by the emission system. The position of an electron source and its size are assumed to be the position of the crossover and its size that are formed by the emission system in the absence of a mirror field. Such definition of a source gives one the possibility of separating investigations of the mirror electron-optical properties and those of the emission system. We shall assume that there is a field-free region between the mirror and the illuminating system, and it is there that the microscope screen is located. Introduce the cylindrical coordinate system r, $, z , whose origin is set into the point of intersection of the z axis and the cathode surface, while the positive direction of the z axis is chosen along the direction from the cathode to the screen. The potential on the cathode surface is assumed to be constant and equal to zero. Under these conditions the equalities hold o=-l,
z,=o
which are to be introduced when using the formulas deduced in Section 11. The energy of illuminating electrons will be characterized by the quantity cp + 5, 5 now being associated not only with the initial energy spread E of the illuminating particle beam, but also with the bias voltage 6 between the cathode reflector and the cathode of the emission system specifying the flux of illuminating electrons
(=&+S
(233)
To investigate the mirror electron-optical properties in the MEM regime we shall use both Eq. (72), determining the explicit dependences of I and $ on z in a region far from the cathode surface, and the parametric dependences of the trajectory [Eqs. (69) and (70)], which are also valid in the near-cathode region. The constants a and b entering these equations are to be expressed in terms of the coordinates of the point on the cathode surface and the coordinates of the point of an electron source. In doing so, it is necessary to keep in mind the specificity of the MEM operating in the shadow regime, the essence of which is that when illuminating monoenergetic electrons by a homocentric beam, the information at a certain point on the cathode surface is transmitted by a unique trajectory (a reflecting trajectory) which passes by this point at the smallest distance (in the limit, it is a tangent of the cathode surface at this point). Every reflecting trajectory is associated with the appropriate point on the cathode surface by definite conditions: First, it intersects the normal to the cathode surface drawn through this point, and, second, in the place of their intersection the reflecting trajectory and the normal are mutually perpendicular (Fig. 6). If the source of illuminating electrons has a finite size and emits electrons with an energy spread, then every point belonging to the
376
E. M. YAKUSHEV AND L.M. SEKUNOVA
FIG.6. Scheme determination of the connection between a point on the cathode and the reflecting trajectory. ANB, cross section of the cathode surface by a meridional plane; A'N'B', projection of the reflecting trajectory section on this plane; NN', normal to the cathode surface.
cathode surface is associated not with one reflecting trajectory, but with a set of trajectories (a reflecting beam of electrons), each of which satisfies the specified conditions. A . Determination of Arbitrary Constants
In order to establish the connection between the constants a and b referring to the reflecting trajectory and the coordinates of the point on the cathode corresponding to this trajectory, it is necessary to consider the trajectories traversing within close proximity of the cathode surface. Then, on account of aberrations, one should keep the terms up to and including third order in r. Let the cathode surface be an equipotential; then its equation near the axis may be approximated by the paraboloid of revolution = (4i/44;)uku?
(234)
where U f = rkei$*;rk,$k are the coordinates of the point on the cathode surface. The equation of the normal to this surface will be
NOWwe shall find the point of intersection of this normal and the trajectory. Using Eq. (62),we write re'*
=
[u(()+ x ( ( ) + i ~ ( ( ) ~ ( ~ ) ] e ' @ ' ~ '
(236)
Setting Eqs. (235) and (236) equal, and taking into account that z = [ + '1, we
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ELECTRON MIRRORS AND CATHODE LENSES
obtain
c
The value = incorresponding to the intersection point will be found from the condition of perpendicularity of the normal and the reflecting trajectory, which, in the required approximation, may be written as follows:
(4;:/44;)[uku*’(c)
+ u:v’(c)15=5,
(238)
=
Then, using the equality U = ap(c) + b g ( c ) and g(c) = q ( c ) m , as well as the initial conditions of Eq. (35) for p ( c ) and q(c), we obtain = $4;r)(Ukb*
From this equality it is clear that and (239) give
+ Uzb)
(239)
enis the fourth-order value. Equations (237)
When deducing these expressions, due to the smallness of equations are adopted: p(in) = 1, ei8(‘n)= 1
g(cn) =
+ Wn),
a? 4’(cn)
O(5,)
=
= 4;?
- kBk&$5/24;,
en, the following
q ( c n ) = q(O) = q k ,
42 = x(i,)
=0
Equation (240)connects the constants a, b of the reflecting trajectory and the coordinates of the corresponding point on the cathode surface. The relation between these constants and the coordinates rs, Rs,zs of the source point may be determined on the base of Eq. (72).Extrapolating the rectilinear section of the trajectory direct branch from a certain plane z = z, located beyond the mirror field to the source plane z = zs, we obtain us = IseQ - aps
+ bgs + x s - ( a ~ h+ bgb)qs + i(aps + bgs1k.a
(241)
where ps
= (zs - zdpb, Ka
g s = (zs = K(Za),
-
Zs)gb,
x s = x(za)
v ~ s=
~ ( 2 , )+ q’(za)(zs - za)
+ ~ ’ ( z , ) ( z-s z,)
the points z = zB,z = zc determining, as before, the positions of the vertex and the center of curvature of an electron mirror on the z axis. On solving the set of equations, Eqs. (240) and (241) with respect to a, b with accuracy up to values
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E. M. YAKUSHEV A N D L. M. SEKUNOVA
of third order, we obtain
- x s - 4aPS
+ BSS)~,l
(243)
Here a, p are values of first order in r (244)
a = uk
B = (l/gs)(us
- Psuk)
(245) As can be seen from the expressions in Eqs. (242) and (243), to a first approximation one may set a = a, b = /?.This approximation is enough to calculate the values I],x,K to an order not higher than first by Eqs. (41), (47), and (63). Thus, Eqs. (69) and (70) [or (72)] combined with Eqs. (242) and (243), completely determine the charged particle trajectories in the field of a mirror objective under specified conditions of the object point and of the point of an electron source. B. Electron-Optical Properties of MEM in the First Approximation
Now we shall investigate the properties of a mirror microscope in the first approximation. In it, the equation of a reflecting trajectory in the region far from the cathode surface, according to Eqs. (72) and (242)-(245), will be T @(@I
=
ukP(z) f [ g ( z ) / g s l ( u s
- Psuk)
(246)
Recall that the functions p(z),g(z) describe the trajectories lying in the meridional plane of the rotating coordinate system and determining the cardinal elements. In the field-free region these functions are approximated by straight lines and, according to Eqs. (101), (104), (105), (108), and (109), are as follows:
dZ)= ( z - zF + f)pb
(247)
g ( z ) = ( z - zF - f)gb
(248)
ELECTRON MIRRORS AND CATHODE LENSES
379
where f is the focal distance, zF is the position on the z axis of the focal plane z = zF.Just as in light optics, the position of the mirror curvature center, z c , and the position of its principal plane, z B , are given by the relations zc = ZF - f
(249)
+f
(250) In the screen plane, z = z b , which is the plane of a shadow image, Eq. (246), written for the reversed branch of the trajectory, gives z g = ZF
where b - r be i[#b+@(Zb)l P b = ( z b - ZF + f ) P b , g b = ( z b - Z F - f ) g b . Equation (251) means that it is the correct image of the cathode surface that is formed in the screen plane for the illuminating electron point source. Indeed, if such a source of electrons is found on the mirror axis (r, = 0),then Eq. (251) becomes
Hence, it follows that there is a linear conformity between the points of the cathode and the microscope screen. The microscope magnification is constant in all directions and equals = rb/rk = P b
+ (Ps/gs)gb
(253)
From the expression in Eq. (208) it also follows that, in the presence of a magnetic field, the image plane is rotated with respect to the object plane around the z axis by the angle (254) The infinity sign in the upper limit of the integral means that the latter is calculated up to a certain arbitrary value z = z , belonging to the region free of field. Now we consider the case when the point source of illuminating electrons is off the z axis (us # 0). In this case, from Eq. (251) it follows that the central point of the cathode (rk = 0,z = 0)corresponds to the point (q,,u b o ) of the plane of the microscope screen ObO
= -(gb/gs)%
(255)
The above conclusions on the adequacy of a shadow image, its rotation with respect to the object plane, and the microscope magnification also remain valid when the point source of illuminating electrons is displaced. However,
380
E. M. YAKUSHEV AND L. M. SEKUNOVA
establishing the point-to-point conformity between the object and the image, it is necessary to take into account the displacement of the image central point from the z axis caused by the displacement of the point source and determined by Eq. (255). Equation (253)gives one the possibility to determine the magnification of the mirror microscope in any position of the planes of the screen, z = zb, and the source, z = z,. It is common for these planes to coincide (z, = zb); the magnification of a microscope is M = 2(z, - zF f ) p b . If the source is at infinity (the illuminating beam is parallel), then M = 2(zb - zF)ph.In the case when the source is in the center of curvature of a mirror, M = (zb - z F - f)&. When the positions of a point source and the principal plane of a mirror are brought close together (z, + zB), the microscope field of view degenerates to a point, while its magnification tends to infinity. In this case an electron probe is formed which illuminates the only point of the cathode surface uk = u,/p,.
+
C . Image Forming Now we consider the reflecting trajectories in the presence of small disturbances near the surface of the mirror cathode. These disturbances may be caused both by (electric or magnetic) microfields on the cathode surface and by micrononhomogeneities of the surface relief. However, whatever the character of disturbances, it is common that after their action redistribution of electron velocities in its components takes place, and, as a result, the intersection of the proper reflecting trajectory and the device screen is changed. If one disregards the details of the disturbing process, the mathematical result may be characterized by some changes of the constants u and P in Eq. (246) for the reversed branch of the reflecting trajectory, writing the equation for this branch as follows:
Taking into account that the disturbance is concentrated in the neighborhood of the reversal point of the reflecting trajectory, we find from the condition of continuity of a trajectory in this point that Au = 0. In order to express the value Ab in terms of the particle velocity change caused by the disturbance, we find the dependence of /3 on the particle velocity in the reversal point of the reflecting trajectory. Taking into account values of third order, this dependence is determined by Eqs. (69) and (70), in which it is necessary to set [ = T k m ) . If we restrict ourselves by values of first order, this dependence may also be derived from Eq. (246), if we set i = T k m) On. differentiating Eq. (246) with respect to time in the neighborhood of the
ELECTRON MIRRORS AND CATHODE LENSES
38 1
reversal point, we find
Analogously, from the expression for the reversed branch of the reflecting trajectory [Eq. (256)], we find the value fl + Afl and, comparing it with Eq. (257), obtain Afl
= -L e i * k ( A i k
k4L
+ irkA$k)
where Aik and i k A$k are, respectively, the radial and azimuthal electron velocity changes undergone in the perturbation process. Then, caused by this disturbance, the displacement Aub of the point the reflecting trajectory traverses the microscope screen is determined by the equality 2 . At+, = y e r S (Aik k
k4k
+ irkA$k)(Zb - zF
-
f)gb
(259)
Note that the electron velocity, as a result of a disturbance, changes only its direction and, therefore (ik
+ At,)’ + r i ( $ k + A&)’ + (Aik)’
= i,’
+ li$i
(260)
where A i k is the electton velocity component normal to the cathode surface received by an electron in the perturbation process. If the disturbance is so small that the quantity [(Aik)’ ri(All/k)2 (Aik)2] is negligible compared with (if ri$:), then, instead of the expression in Eq. (260), we may write the equality
+
+
ik
Aik
+
+
rf$k
=0
(261)
implying the perpendicularity condition of the velocity change .vector Av = Aik + irkAIjk and the velocity component vector vll = i, + irk$k parallel to the cathode surface in the neighborhood of the reversal point. With this condition Eq. (259) takes the form
where y = arctan( - i k / r k $ k )
The quantities
ik
(264)
and rk$k entering the last equation may be found with the
382
E. M. YAKUSHEV AND L. M. SEKUNOVA
help of the trajectory equation [Eq. (69)]. Differentiating it with respect to time and taking into account that = f k& as well as Eqs. (242)-(245), we obtain in the neighborhood of the reversal point
i
rssin($s - $k
-
)]
0,) - 7kBk gsrk 2d)k
(265)
Then Eq. (264) takes the form
Thus, Eq. (262) combined with Eqs. (263) and (266) gives one the possibility to determine the deviation of the disturbed trajectory relative to the undisturbed one both in the magnitude and in the direction in the screen plane. Note that it is a very common case for the source of illuminating electrons to be on the axis (rs = 0), and the magnetic field to be absent (Bk = 0) in the near-cathode region, for the velocity of the reflecting beam in the place of reversal to have only a radial component (rk& = o), and for the velocity change to come about in the azimuthal direction ( A i , = 0). It is just this case for which the value Au is calculated for some concrete forms of nonhomogeneities of the sample surface in a number of papers (e.g., Sedov et al., 1968a,b; Sedov, 1970). It is evident that these calculations are also suitable for the case when the source of illuminating electrons is far from the axis; however, now Av denotes the change of the velocity component parallel to the cathode surface and not that of the radial velocity. In any case, if the velocity change Av, caused by nonhomogeneities of various types, is known, then Eq. (262) combined with the expression for magnification [Eq. (253)], gives one the possibility to calculate the contrast of a shadow image in any mirror. Here, besides electrostatic fields, a magnetic field may also take part in forming an image; in the general case, the cathode surface may be curved, and the source of illuminating electrons may be located outside the symmetry axis of the system. As follows from Eq. (262), the deviation of the disturbed trajectory from the undisturbed one, Arb, in the plane of the microscope screen equals
Arb = K Av/kJ& where
(267)
ELECTRON MIRRORS AND CATHODE LENSES
383
The last quantity has a sense of the microscope sensivity to relatively small changes of the electron velocity caused by the disturbing nonhomogeneities. This increase of sensivity is always desirable. However, one should keep in mind that such an increase involves deterioration of the microscope resolution, because in this case identification of the micro-objects located close to each other becomes difficult. In reality, if two identical objects are at a distance Ark < Arb/M, where M is the microscope magnification, then their shadow images are superimposed. When the resolution is high, the case seems to be optimal when Arb/M
2:
2
(269)
where I is the limiting resolution of the microscope.
D . Impact Parameter and the Circle of Secondary Emission The effect of disturbance is determined, to a great extent, by the distance h (see Fig. 6) between the cathode surface and the reflecting trajectory in the neighborhood of the analyzed point. Taking into account third-order values, this distance may be calculated on the base of Eqs. (12) and (234).In doing so one should, in general, substitute the value 5 = [, into Eq. (12) corresponding to the intersection of the reflecting trajectory and the normal to the cathode surface. However, according to Eq. (239),5, is of fourth order; therefore, in the adequate approximation one may write
where
While deducing the last relations, besides Eqs. (12) and (234), Eqs. (41)(46),(232),and (242)-(245), as well as the conditions in Eqs. (35),(36),and (37), imposed on the functions p , q near the cathode surface, are taken into account. For a point source of monoenergetic electrons ( E = 0,d # 0) Eq. (270) determines the surface which represents the set of the reversal points of reflecting trajectories. In the adopted approximation this surface is a paraboloid of revolution with the axis parallel to the z axis and intersecting the cathode surface at the point uk = u, = r,ei*". As seen from Eq. (270), the quantity h, in the general case, is not a constant for all points of the cathode surface. The only exception is when there is no magnetic field near the surface of the cathode (Bk = 0), and the source of illuminating electrons is located in
384
E. M. YAKUSHEV A N D L. M. SEKUNOVA
the plane z = z c , passing through the center of curvature of a mirror. If the negative bias 6 is given to the cathode, so that ( = E 6 < 0, then electrons never reach cathode surface.The minimal distance at which they can approach the surface equals ISl/4;. With positive bias (6 2 0) some part of the electrons reach the cathode, forming the circle of absorption. The coordinates of the circle center coincide with r, and II/. and are determined by Eq. (271),while its radius, as follows from Eq. (270),equals
+
s++k2Bzg: E
= 4gs(44;’p:
YI2
Note that the center position of the circle of absorption does not depend on the bias voltage 6.
E. Aberrations of a Shadow Image When deducing the formulas describing the first-order electron-optical properties of a mirror microscope operating in the shadow regime, the source of illuminatingelectrons was supposed to be pointlike, and the influenceof the terms of third order in r and of first order in rE on forming a shadow image was assumed to be small. The account of these effects determines aberrations of a shadow image. We shall consider, first, aberrations associated with a finite-sizedsource of illuminating electrons. Suppose that in the plane z = z, the source represents a circular spot of a radius p , , and its every point may be considered as a selfdependent point source of illumination. If the quantity us is given a sense of the coordinates of the spot center, the coordinates of any spot point upmay be written in the form up = us + peinp,and 0 5 0,5 2n, p 5 p , . In order to take into account the influence of finite-sized source on the microscope resolution, one should substitute up instead of us into Eq. (246). Then for the reversed branch of a trajectory we obtain 9 U + -us gs
uk
- -(pg,
gs
+ p,g) =
-
9 -peiRp
(273)
ss
From the equality obtained it is seen that after reversal the cross section of the reflecting beam by the plane z = const is a circle. Thus, single-valued conformity between the points of an object and its shadow image is broken in this case. In the screen plane the radius of the spot of diffusion equal to represents the aberration determining the smearing of the shadow image caused by the finite-sized source of illuminating electrons. The aberration
385
ELECTRON MIRRORS AND CATHODE LENSES
limits the microscope’s resolving power up to the value
For a specified aperture angle of the electron reflecting beam yp = psgb/gs,the dependence of microscope resolution on the positions of the source and the screen is determined by the equality
Now we turn to a consideration of geometrical aberrations of third order in rand of chromatic aberrations of the order of rE. Using Eqs. (72),and (242)(245),we represent the coordinates of the point of intersection of the reflecting trajectory reversed branch and the plane of a microscope screen, z = zb, located beyond a field, in the form rbeiRb=
u b
+ AM
(277)
where 0, gives the first-order coordinates of the point determined by Eq. (251), and AMis the total aberration in the plane of a shadow image
Bk
- i-a(aP*
8
-
+
+ Bsb)a - x s - 4aPS + /?s,)..l
- %Pb 9s
- bgb)wb + x b
+ i(aPb
-
Pgb)lcb
Using Eqs. (41), (47), and (63), in which one should set a = a,b Eqs. (82), (232), and (233), one may reduce Eq. (278) to the form
= /?,and
AM= A u ( ~ + ’ AM<
(279)
where the quantity AM(^) = a2a*L,
+ .(a/?* + a*B)L2 + aPP*L, + C~CC*PL,
+ /?(aP* + a*B)L, + PZP*L, + ia(aP* - a*P)L, + $(a/?* - a*P)L,
(280)
determines the total geometrical third-order aberration, and AM< = SaL,
+ 5PL1,
(281)
the aberration associated with the initial energy spread of illuminating electrons E and the bias voltage 6. The coefficients from L , to L,, entering the
386
E. M. YAKUSHEV AND L. M. SEKUNOVA
above two relations equal
Here M is the microscope magnification, and = (l/gs)(Pbgs - Psgb) The quantities from J1to J19are integrals which do not depend on the position of the illuminating electron source, z = z,, and of the microscope screen plane, z = Zb
ELECTRON MIRRORS AND CATHODE LENSES
J4 -- - 7 j "1 E ( Q 4 g 2
164k
0
fi
- 8Q;v3)dz
388
E. M. YAKUSHEV AND L. M. SEKUNOVA
Due to the fact that the integrands equal zero beyond the mirror field, the integration region is extended to infinity. There are a number of interconnections between the integrals entering the expression for geometrical aberrations
4;
-iJ4
4
a
4;
-(JS
4
a
- J 7 ) = qZs?;s
- q i s q 3 s = q 2 b q ; b - ?;by/3b
- J 6 ) = qlsq;s - q i s q 3 s = q l b q ; b - Y ] ; b r / 3 b
(312) (313)
The proof of the given interconnections may be made in the same way as done in Section I11 for substantiating Eqs. (1 37),(1 38), and (1 39). In a similar way we eliminate the double integration in Eqs. (293)-(311) associated with the presence of the quantities qq, ql, q,, q 3 ,q4 in the integrands. On carrying out these transformations and substituting the values u, /3 into Eqs. (280)and (281), according to Eqs. (244)and (245), we find, taking into account Eqs. (312)-(3 16)
+ ia1)u,2u: + ( A , + ia,)uku:us + ( A 3 + ia3)u,2ur + (A4 + ia,)u,usu,* + (AS + ia5)u:u: + + ia,)u:u,* AUg = + ia7)ukt + (A8 + ia8)ust;
Ad3) = (A,
(A6
(A7
(317) (3 18)
where
P2
- 3Mf1, gs
P3
-N f l , gs
(319)
ELECTRON MIRRORS AND CATHODE LENSES
389
(333)
390
E. M. YAKUSHEV AND L. M. SEKUNOVA
and
I,
jmf i
=7
164k
%s2(Q4p
+ 4 Q Y ) + 8p”(44d2 + Q2g2)]dz
(338)
0
+ 8q”(4$~g’~ + Q2g2)]dz
(340)
(343) (344)
ELECTRON MIRRORS AND CATHODE LENSES
39 1
Thus, in the general case, geometrical aberrations of a mirror microscope operating in the shadow regime are determined by twelve aberration coefficients, while chromatic aberrations are determined by four coefficients; their magnitudes depend both on the position of the illuminating electron source plane and on the position of the microscope screen plane. To calculate all these coefficients, together with a number of quantities characterizing a mirror in the first approximation-zF, &,pb,gb, as well as the quantities determined by the positions of the source and the screen
Ps
= ( z s - zF
Pb
= (zb
+ f)ph?
9s = ( z s - zF
- zF + f)Pb,
gb
= (zb
- zF
-
f)gb,
-
f)gb
it is necessary to calculate the quantities from I , to I,,, depending only on the distribution of the electric and magnetic fields along the mirror axis. If the magnetic field is absent, the coefficients from a, to a8 turn out to be zero, and the geometrical/chromatic aberrations are characterized by six/two aberration coefficients, whereas I , = 1 8 = I , = I,, = I , , = 0. As can be seen from Eqs. (317) and (318), most of the aberrations are caused by the fact that the point source of illuminating electrons is far from the mirror symmetry axis. If the source is on the axis (us = 0), then
A d 3 )= ( A , + ia,)u;u: AUc = (A7
(349) (350)
ia7)ukt
In order to investigate in more detail the image deformations associated with geometrical aberrations, we introduce, instead of the rotating cylindrical coordinate system, I , R, z, the rotating Cartesian coordinate system, x , y, z, the x axis being drawn through the source point. In this system us = x s , uk = x k + iy,, and A d 3 ) = A x + i Ay. Then from Eq. (3 17) it follows that Ax = A I X k ( X ?
+ A3(X;
+
+ Y,") - a l y k ( x i + Y:) + A Z ( x i + Y i ) & - y i ) x s - 2a3xkykXs
A5XkXf
AY = A l y k ( x f
+ A4XkXf
- a4ykx:
+ a5ykxf + A,X:
+ y.?) + a l X k ( X i + Y : ) +
(351)
+ y,')xs
+ 2A,xkykx, + a,(x; y,')xs + A4ykx: + A 5 y k x l + agxkxf + a 5 x i -
-
a4XkXf
(352)
Let a test net be drawn on the cathode surface, along the lines of its intersection with the planes parallel to x z and yz, and its projection on the plane z = 0 represent a square net with constant-size steps along the x and y axes. Using Eqs. (351), (352), (251), and (253), we consider the image of the net
392
E. M. YAKUSHEV AND L. M. SEKUNOVA
in the presence of this or that type of aberrations. If all the aberration coefficients are zero, the mirror gives an adequate, magnified image of the net
The image of the test net will be a square net with an M-fold increased step. Now we assume that all aberration coefficients but A, equal zero. In this case the quantities Ax and Ay determine deformation characterized by the aberration coefficient A,. The test net image in the presence of this deformation is shown in Fig. 7(a). Such a type of aberration represents isotropic distortion. Anisotropic distortion [Fig. 7(b)], occurring only in the presence of a magnetic field, is characterized by the coefficient a,. Deformations of these types are also inherent to electron lenses. All other types of geometrical aberrations characteristic of a shadow image have no analogue among aberrations of electron lenses. In Fig. 8(a) the image of a test net is given for the case when only one coefficient, A 2 , is not zero. The displacement of the net node image in the rotating coordinate system takes place in the direction of the point-source shift from the symmetry axis of a
FIG.7. (a) Isotropic ( A , # 0) and (b) anisotropic (al # 0) distortion of the image
ELECTRON MIRRORS AND CATHODE LENSES
393
FIG.8. Image deformationscaused by aberrationsdetermined by (a) the coefficient A 2 and (b) the coefficient a2.
mirror. In the perpendicular direction a similar node shift occurs in the presence of a magnetic field when all coefficients, besides u 2 , equal zero [(Fig. 8(b)]. Deformation of the test net image associated with aberrations characterized by the coefficients A , and a, is represented in Fig. 9. The presence of the coefficients A , and a, does not lead to an image deformation. Aberration associated with A, results in a uniform and isotropic change of the image scale, and aberration caused by a, gives a change of the image rotating angle. Aberrations caused by A, and u5 lead to the stretching of the image along a certain direction and to the straining across the image (Fig. 10).If all aberration coefficientsbesides A , and u6 equal zero, a certain shift of the entire image by Ax = A,X: and Ay = a6x$ occurs without changing the image scale or deformation. Note that, although the presence of geometrical aberration does not affect the resolution of a mirror microscope, taking them into account is necessary for a strict interpretation of a shadow image. Now we shall consider chromatic aberration of an electron mirror microscope associated with the bias voltage 6 and the spread in the initial
394
E. M. YAKUSHEV AND L. M. SEKUNOVA
FIG.9. The image deformations caused by aberrations characterized by (a) the coefficient A, and (b) the coefficient a,.
energy E of illuminating electrons. The quantity 5 entering Eq. (318) equals t = E 6. We shall consider the aberrations associated with 6 and E separately. First we set E = 0. Then the aberrations associated with the bias voltage in the rotating Cartesian system take the form
+
+ A8xS6 Ay, = A,yk6 + a7x,6 + a,x,6
Ax, = A7Xk6 - a7yk6
(355)
(356) Hence, the coefficient A , characterizes the change of the image scale; a 7 ,the angle of rotation; and A 8 and as, the displacement of the entire image along the x and y axes, respectively. Since the bias voltage is a constant, the aberrations associated with it do not cause image diffusion. If 6 = 0 while E # 0, Eq. (318) determines the diffusion of image points associated with the initial energy spread of the illuminating electrons. The value of the diffusion in
395
ELECTRON MIRRORS AND CATHODE LENSES
FIG.10. (a) The image scale change at A, # 0; (b) image rotation angle change at a4 # 0.
the rotating Cartesian coordinate system is determined by the equalities
+ A8Xs)E
(357)
+ a7xk + a8Xs)E
(358)
Ax&= (A7xk - a7yk A Y ~= ( A 7 y k
These equalities mean that, in the plane of a microscope screen in the presence of the initial energy spread of the electrons, an object point is depicted as a stroke whose length depends both on the position of the source point and that of the point on the cathode. The length divided by magnification determines the limiting resolution of a microscope associated with the initial energy spread &
= -4(A7x,
M
-
+ A ~ X S+)(A7yk ~ +
a7xk
+ a g x ~ ) ~ (359)
396
E. M. YAKUSHEV A N D L. M. SEKUNOVA
which may be achieved with a sufficiently small size of the illuminating electron source, if the value A, determined by Eq. (275) can be neglected. From Eq. (359) it follows that, in the general case, on the cathode surface there is only one point with coordinates
the shadow image of which is free of chromatic aberration. If the source is located on the mirror axis ( x , = 0), then this point is the central point of the cathode surface, xk = yk = 0. In order to make a shadow image achromatic throughout the entire viewing field of a microscope, a simultaneous satisfaction of three conditions A,
= 0,
U,
X, = 0
= 0,
(362)
is enough. For any distribution of fields along the mirror axis the quantity A , may be set equal to zero by a special choice of the microscope screen position, z = z b . This position will be found on the basis of Eq. (325), setting its righthand side equal to zero k2Bt ~-
zb - z B --
f
_+Ill Pa
+
I12
(S 84k2
-
Pb z, - zc -7 113 S a Zs - ZB - zF z, - zB
zs
+
j& g;
(363) zs - zC
(2,
- ZB)2'13
From Eq. (333) it is clear that the second condition, a, = 0, may be satisfied only by an appropriate choice of the field distribution along the mirror axis. In particular u, = 0 if B = 0. The third condition, x, = 0, is satisfied when the source of illuminating electrons is located on the mirror axis. Thus, all three conditions [Eq. (362)] may be satisfied simultaneously. However, realization of these conditions does not mean that the limitations for the initial energy spread of illuminating electrons is eliminated completely. There is a limitation caused not by the requirements of resolution, but by the necessity to obtain the image contrast which, evidently, dissappears if e / 4 ; >> 1, where 1 is a characteristic size of analyzed nonuniformities (of topography or microfields) in the direction perpendicular to the cathode surface. The influence of the initial energy spread on the image contrast is considered, in particular, in the work by Sedov and Nazarov (1972); however, it should be noted that the effects of image diffusion and contrast forming in the screen plane are propagated, as seen from Eqs. (262) and (318), in opposite directions. For
ELECTRON MIRRORS AND CATHODE LENSES
397
instance, when considering in an electrostatic mirror the meridional motion of a reflecting beam originating from a source located on the axis, these effects occur in mutually perpendicular directions. Note that when determining the electron-optical properties of a mirror operating in the regime analogous to a light shadow microscope, we do not impose the requirement of large magnification which is characteristic of microscopes. This allows one to apply the above-stated theory for all electronoptical devices using an electron mirror in the indicated regime -for example, in a mirror image converter (Artamonov et al., 1968)and a brightness amplifier with a mirror system (Kasper and Wilska, 1967). V. CATHODE LENSES
Cathode lenses are widely applied in electron devices of various types: electron guns, image converters, emission electron microscopes, and so on. In this connection, it is of great interest to investigate these lenses theoretically, including their aberrations. Study in this field, started with well-known papers by Recknagel(l941,1943) and Artzimovich (1944), have a long history, and, in total, there is a rather large number of appropriate works. Recently electrostatic cathode lenses were studied by Ignat’ev and Kulikov (1978), Dodin and Nesvizhski (198 l), Nesvizhski (1984),and Takaoka and Ura (1984). The works by Monastyrski and Kulikov (1976, 1978), Kulikov et al. (1978), Monastyrski (1978a,b), and Ximen Ji-ye et al. (1983) are devoted to the investigation of cathode lenses with combined electric and magnetic fields. However, the involved assumption on a finite (although small) value of the longitudinal component of the electron initial velocity contradicts the specific nature of a cathode lens in which, in principle, the electrons having small, even zero energies take part in image formation. In a cathode lens the particle initial energy E and the distance r from the symmetry axis are assumed to be small. Although these quantities are different in nature, the initial energy, if it is not zero, may be considered (Artzimovich, 1944)as values of second order in r. When stating the theory of cathode lenses, we shall follow this statement, and the values proportional to E are assumed to be of second order in r. Then the value orders will be determined by their powers of r; e.g., re is a third-order value. A . Determination of Arbitrary Constants
The origin of the cylindrical coordinate system (r, rl/, z ) used below will be placed in the central point of the cathode surface. The z axis will coincide with
398
E. M. YAKUSHEV A N D L. M. SEKUNOVA
the principal optical axis of the lens, while its direction is from the cathode to the anode. In doing so, in all the equations derived in Section I1 we set zk=o,
a-1
(364)
As long as the potential of the cathode surface is set equal to zero, the equation ofthis surface is cp(r,z) = 0
(365)
Here in the general case the cathode may be not flat. The radius of curvature of the central region of the cathode surface is related to the field distribution in the near-cathode region by the well-known relation
Rk = -24;/4: (366) It is assumed that Rk > 0 for the convex cathode and Rk < 0 for the concave. The charged-particle trajectory in the region far from the cathode surface is described by Eq. (72), which is valid with an accuracy up to values of third order. However, the constants a, b entering this equation are to be determined proceeding from the conditions on the cathode surface. In this case it is necessary to use the set of equations, Eqs. (69) and (70), describing the trajectory of this particle also depending on the variable ( in the near-cathode region, including its point of emission. First we define the value [ = [ k corresponding to the particle emission point. At this point
if = k2&Cos2
(367)
where & is the angle between the particle initial velocity and the z axis. On the other hand, on the grounds of Eqs. (1 1) and (12), we can write
(3681 From the last two equations we obtain
whence, neglecting the values of the order of smallness higher than the third, we obtain [k =
(E/4;)CoS2 $k
(370)
While deriving this, it is taken into consideration that, according to Eq. (24), r] is a second-order value. When determining the arbitrary constants a, b, it is necessary to take into account that the direction of the normal n to the cathode surface at the point of
ELECTRON MIRRORS AND CATHODE LENSES
399
particle emission is the natural direction specifying symmetry of the emitted particle angular distribution. So, when studying cathode lenses with curved emitting surfaces, it is reasonable to specify the initial conditions for particles with respect to the normal by two angles: Y k , O k (see Fig. 11). Y k is the angle between the normal o and the particle initial velocity v k . This angle can have any magnitude over its entire range: - 4 2 I y k I 4 2 . a k is the angle between the two planes, one of which contains the point of the particle emission and the z axis while another involves the vectors v k and n. It also can have all the magnitudes of its range: 0 I 0, I 211. The connection between the angles $ k , y k , a k has the form cos 8, = cos y k - ( r k / R k ) cos w k sin y k
(371)
When deriving the last equation, it has been taken into account that in the approximation considered the angle between the normal n and the z axis equals r k / R k . At the point of particle emission Eq. (69) gives rke-iN5k)
=
ap(ck)
+ b ( c k ) + x ( c k ) + iK(ck)[ap(ck) + bg(ck)l
(372)
Herein, it is assumed that the plane $ = 0 contains the point of emission. It does not restrict the generality of the subsequent conclusions because of the system’s axial symmetry. On differentiating Eq. (69) at this point with respect to time and using Eq. (1l), we obtain [ik
+ irk&
-ikrk6’(ck)m]e-ie(ck)
= k{ap’(ck) iCap(ck)
+k’(ck)
-k
X’(ck)
+ bg(ck)lK’(ik))
+ i[ap’(Ck) + bg’(ck)IIC(ck) (373)
It is necessary to solve Eqs. (372) and (373) with respect to the constants a, b. The solution is to be carried out with an accuracy up to values of third
FIG. 11. Scheme for determination of the initial conditions of motion in a cathode lens
400
E. M. YAKUSHEV AND L. M. SEKUNOVA
order. For convenience of calculation we shall transform these equations into the form (374)
In doing so Eq. (38) is taken into account. The quantities + k , r k $ k entering Eq. (375) can be expressed in terms of the particle’s initial energy E and angles of emission Y k ,
&sin The quantities e([,‘), i k $k
=k
Yk
sin o
k
(377)
g ( c k ) in the approximation considered can be written, using Eqs. (370) and (371), as follows:
Substituting Eqs. (370), and (376)-(379) into Eqs. (374) and (375) and producing the expansion in small values, we find
ELECTRON MIRRORS AND CATHODE LENSES
a=rk,
,!3-
2J.5
4;
.
40 1
Bk
elmksin y k - irk24;
As the constants a , b are now known, further, when investigating aberrations sufficiently far from the cathode surface, one can use a more simple equation [Eq. (72)] without involving the set [Eqs. (69) and (70)] which is valid everywhere including the near-cathode region. A study of trajectories in the region far from the cathode surface, where 4 ( z ) >> E, is of the greatest practical interest, as this region contains the place of smallest cross section of a beam in electron guns and the image plane in image converters and emission electron microscopes. B. Near-Axial Approximation
Cathode lenses can be used for solving problems of two kinds: first, to produce an image of the cathode surface and, second, to focus the electron flux emitted by the cathode surface into a spot of a small diameter-a crossover. In both cases the trajectory sections far from the cathode surface are of interest. Let us consider, first, near-axial trajectories, preserving in Eq. (72) the terms of first order in I and substituting into Eq. (72),instead of the constants a, b, their values a, p defined in the first approximation according to Eqs. (382). Then the trajectory equation becomes rein = U
= ap(z)
+ Pg(z) = rk
402
E. M. YAKUSHEV AND L. M. SEKUNOVA
We shall consider the cathode surface as the object plane. According to the definition of the function g(z) [Eq. (33)], g(0) = 0, because 4(0)= 0. Then, in accordance with Eq. (383), the image plane, z = zb, will be determined by the equation dZb)
=0
(384)
In this plane, as seen from Eq. (383), the cathode lens field forms a similar electron-optical image of the cathode surface. The cathode lens magnification is constant in all directions and equals =dZb)
(385)
It is necessary to keep in mind that the magnetic field of the cathode lens, in accordance with Eq. (73), leads to the rotation of the image about the z axis at an angle y b equal to
All the electrons emitted by the cathode surface can take part in the formation of the electron image of this surface in the approximation considered. The change of the emitted particle’s initial energy E from zero to its maximum value E,(O IE IE,,,), as well as the change of the initial angles of the particle emission over the entire range ( I r k 1 In/2, lWkl I274 causes neither deformation nor diffusion of a point-to-point image. Now we shall consider the peculiarities of the electron image formation in the presence of a limiting diaphragm. Introducing the latter becomes necessary in those cases when a cathode lens is used to receive the image of (geometrical,electrical, or magnetic) micrononhomogeneitieson the cathode surface, which deform the energy or the angular distribution of the particles emitted by every point of the cathode surface. In these cases the electron flux diaphragm is used for obtaining an image contrast. Visualization of the cathode surface potential relief is an example of a problem of this kind. In this case the particle’s initial energy E is considered as a function of coordinates rk, t,bk of the point on the cathode surface. When using a diaphragm, the portion of the electron flux emitted by every point of the cathode surface is removed by an operating diaphragm, and in the image plane z = zb of the cathode lens a brightness picture corresponding to the function E = & ( r k , $k) is formed. , which a Let the diaphragm be the equipotential surface p(r, z ) = 4 ( z D ) in coaxial (with the axis z ) round aperture of small diameter d is cut. The plane z = z D tangent to this surface will be called the diaphragm plane. The equation of the trajectory traversing the two points-of the object and of the
ELECTRON MIRRORS AND CATHODE LENSES
diaphragm-has
403
the form
where pD = P(Z,), gD = g(z,) # 0, and uD = rDeinDare the coordinates of the intersection point of the trajectory and the diaphragm (rD 5 4 2 ) . When using Eq. (387), one should keep in mind that now not all electrons emitted by the cathode surface take part in the formation of its electron image, because only electrons whose initial conditions, according to Eq. (383), satisfy the nonequality
can pass through the limiting diaphragm. In the general case, Eq. (388) determines the phase volume of the beam passing through the diaphragm. If the field of view of the emission system is determined by the electron emitted normal to the cathode surface (&sin y k = O), then, according to Eq. (388), for this field of view
(389)
From this inequality it follows, in particular, that the field of view reaches its maximum value if there is no magnetic field (Bk = 0) in the near-cathode region, while the diaphragm plane z = zD contains the ray crossing point satisfying the equation
(390) In this case, as follows from Eq. (388), the variation range E and yk of the beam passing through the diaphragm does not depend on the emitting point coordinates and is determined by the inequality P(zD) =
The problem the formation of the electron flux emitted by the cathode surface as a small spot (a crossover) is, as usual, solved using electric fields. As for an electrostatic cathode lens, it is of interest to determine the beam cross section in the neighborhood of the crossing point (z = zD)satisfying Eq. (390). Here the area of the beam cross section does not depend on the cathode size and can be made significantly smaller than the cathode surface area. Thus the crossing point, z = zD, determines the crossover position on the principal optical axis of an electrostatic cathode lens. In the presence of the magnetic
404
E. M. YAKUSHEV A N D L. M. SEKUNOVA
field the formation of a similar crossover may occur, if the magnetic field near the cathode surface equals zero (Bk= 0).
C . Aberrations of Spatial Focusing
In order to analyze aberrations of a cathode lens in the region far from the cathode surface, it is necessary to represent Eq. (72) as a power series depicting an explicit dependence of the trajectory on the initial conditions, rk, E, y k , a k , characterizing the particles leaving the cathode. In accordance with this, we substitute into Eq. (72)the functions ~ ( z )~ ,( z )and , JC(Z), according to Eqs. (41), (47), and (63), as well as the constants a, b, according to Eqs. (380) and (381), and then combine therein the terms of third order in powers of a and fl. Then, neglecting terms of orders higher than third and taking into account Eqs. (382) and (383), we find
(
u = rein = U - p ( z )
Bk + i7g(z) 24k
+ a 3 [ L l ( z )+ ill(z)] + C C ~ ~ ~ * +[ Li12(z)] ~(Z) + a 2 f l [ L 3 ( z )+ i13(z)] + af12[L4(z)+ i14(z)] + a f l P * [ L 5 ( z ) + i15(z)1 + f12fl*[IL6(z)
f ii6(z)i
+ ae[L,(z) + il,(z)l + P E [ L f j ( Z ) + iZ,(z)]
(392)
This equation describes the trajectories of particles in the region far from the cathode surface under the specified initial conditions on the cathode. When deriving Eq. (392), the following notations are used (393)
3
k2B: +-8&4; L4(4 =
~
-(%+
k4B:)y 644i2
JJP
+ (J,
--
ELECTRON MIRRORS AND CATHODE LENSES
405 (397)
+ 4Q;p‘) + 8p“(44p” + Q z p 2 ) ]dz
(410)
These integrals are obtained as a result of transforming Eqs. (42)-(46), (57), (64), and (65)-(68). Double integrals entering Eqs. (57), and (64)-(68) are reduced to single integrals via integration by parts just as was done in Section I1 for the proof of Eqs. (137)-(139).
ELECTRON MIRRORS AND CATHODE LENSES
407
It can be seen from Eq. (392) that aberrations of a cathode lens in an arbitrary plane z = const located sufficiently far from the cathode can be represented as the sum of the terms
Au = u - lJ
= A#(')
+ Au(3)
(422)
where A d 2 ) is the second-order aberration, while A d 3 ) is the overall thirdorder aberration. The second-order aberration in an arbitrary plane is described by the expression
A d 2 )= - p(z) + i7g(z) sin 2yk (423) 2+k kBk ) i k In the image plane of a cathode lens, z = z b ,this aberration takes, according to Eqs. (384) and (385), the familiar form
(
A@
-+Ok
&
= - M-eiWk
4L
sin 2yk
(424)
Diffusionof the point image caused by this aberration should be defined using the energy and the angular spread of the particles emitted by the cathode. It is worth noting that the magnitude of this diffusion referred to the emission system magnification does not depend on the magnetic field. We shall find the radius of the diffusion circle associated with the second-order aberration, setting E = E,, = n/4 in Eq. (424) Ar(2)= M&,,,/q5;
(425)
The overall third-order aberration of a cathode lens in the image plane is determined by the expression Au(3)
=
a3 [L1(zb)
+ i l l ( z b ) l + a 2 P * [ L 2 ( z b ) + ilZ(zb)]
+ a2fl[L3(zb) + iz3(zb)l + a P 2 [ L 4 ( z b ) + i14(zb)l + aflfl*[L5(zb) + i15(zb)l + P Z P * [ L 6 ( z b ) + i26(zb)l
+ a & [ L 7 ( Z b ) + i17(zb)l + P E [ L 8 ( Z b ) + i18(zb)l
(426)
According to Eq. (382), we substitute the values a, j? into this expression. Then, after reducing the like terms, we obtain
Aui3)= Gr&&eiWksin3yk + (5- if,)rk&e2i"ksin2yk
+ [2(~, + if,)]rkesin' + (c,+ O,)r,"&eiok sin y k + (c,+ icr)r~&e-imksinyk+ (E, + ie,)rf + ( K +~ikrm)&&ei"'*sin ~ ykcos2y k + ( K ~+, ik,)&rk Yk
(427)
408
E. M. YAKUSHEV A N D L. M. SEKUNOVA
where
Here p;.gb,Bb,4b are the values of the functions p',g',B,#~ at z = zb. Equation (427) describes various kinds of deformation of a cathode surface image, when using a nondiaphragmed beam.
ELECTRON MIRRORS AND CATHODE LENSES
409
Introduce a limiting diaphragm, placing it into the plane z = zD.Using the first-order equation [Eq. (387)], we express the quantity b in terms of the coordinates of the cathode point and the diaphragm uD = rDeiRD
b = (uD - rkPD)/gD
(440)
Substitute this value of /3 and ct = rk into Eq. (426). Then, combining the terms in the appropriate way, we obtain Auk3'
= (CD
+ ibD,)uk~g + (FD - ifD)rkuk
+ [2FD + FDm + i(2fD + f D m ) l r k U D U g
+ (CD + DD + idD,)riUD + (CD + icD)rZug + (ED + ieD)r; (441) + (KID + ikDm)&UD+ ( K ~ +D ikD)&rk where
(443)
(446) (447)
410
E. M. YAKUSHEV AND L. M. SEKUNOVA
(453) (454) (455)
The coefficients G D , F D , C,, OD, E D , f D D , c D , e D determine the geometrical aberrations: spherical aberrations, coma, astigmatism, the field curvature, distortion, anisotropic coma, anisotropic astigmatism, and anisotropic distortion, respectively, also characteristic of ordinary electron lenses (Glaser, 1952).The coefficient K I D determines aperture chromatic aberration, while K Z D and kD are responsible for the changes in magnification and the angle of the image turning, respectively, associated with nonzero energies of the particles leaving the cathode. Chromatic aberrations of this kind are inherent in ordinary lenses as well. The other four kinds of geometrical aberrations (FD,, bDm,fDm,dD,) and one kind of chromatic aberration (kD,) are not to be found in ordinary lenses. Moreover, in combined cathode lenses they are also absent if there is no magnetic field (Bk = 0) on the cathode surface. As shown in Section V,B, if the magnetic field in the near-cathode region Bk = 0, and the diaphragm plane, z = zD, contains the crossing point where p D = 0, the value of the field of view of the emission system does not depend on the size of the aperture in the limiting diaphragm and can be made significantly greater. Satisfying the same conditions (Bk = 0,pD = 0)leads to a great simplification of the expressions in Eqs. (442)-(457) for the aberration
ELECTRON MIRRORS AND CATHODE LENSES
41 1
coefficients
However, the dependence of aberration coefficients on Bk,pDcan be used in order to decrease separate kinds of aberration. In this case, one should have in mind that such a decrease of aberrations can result in some limitation of the emission system’s field of view.
D . Time-of -Flight Aberrations of Cathode Lenses When designing the electron devices intended for investigation of rapid processes whose duration is comparable with the time of particle flight, a problem arises associated with the time structure of the analyzed chargedparticle flux. Therefore, investigation of the particle’s time of flight in emission systems becomes of primary importance owing to the wide application of these systems in various kinds of electron devices. The particle’s time of flight from the cathode surface to an arbitrary plane z = const, according to Eqs. (11) and (12), is determined by the expression
where [k is the value of corresponding to the instant the particle goes out of the cathode surface. This value is determined strictly-by Eq. (369), or approximately-by Eq. (370). Represent Eq. (459) as the sum of two integrals
412
E. M. YAKUSHEV A N D L. M. SEKUNOVA
Calculation of the first with an accuracy up to values of second order is given in Section II,B. We shall calculate the second integral using the smallness of [ k . In doing so values of third order in r will be neglected. In this approximation we can write
When deducing this relation, Eqs. (366), (369), and (371) are taken into account, and the values of order higher than second are neglected. Using Eqs. (76)-(80), (460), and (461) and taking into account Eqs. (380)-(382), we represent the particle's time of flight in the region far from the cathode surface, where 4 ( z ) >> E as follows t =T
+D
, m COS Y k
+ At(')
(462)
where
is the time of flight of an axial particle with zero initial energy E cathode to some plane z = const
= 0 from
the
is the first-order dispersion of the particle's time of flight along the component uI of their initial velocity normal to the cathode surface At(2) = T1(E/4)sin2yk-t T2lkmCOSc%ksinyk
+ T3rkJ&7sinw,sinyk+ T4r; + T5E/4
(465)
is the second-order time-of-flight aberration of cathode lenses in the plane z = const. The coefficients of these aberrations are
(467)
ELECTRON MIRRORS AND CATHODE LENSES
413
m
(470) k Is The quantities qs, ql,q 2 ,v 3 ,q4 entering the last expression are determined by Eq. (42)-(46). Equation (465) describes the time-of-flight aberrations of a nondiaphragmed flux of particles. In order to obtain the proper expression for the aberrations of a diaphragmed flux, one should carry out the transformations used earlier when obtaining Eq. (465);however, one should now substitute into Eq. (80)the value j3 determined by Eq. (440)instead of b. On making these transformations we obtain T,(z) = --
At‘2’ = T1Du D u g + 3 T 2 D r k ( U g
+
+ (i/2)T3Drk(Ug
- uD)
+ T4Drk2 + T 5 D E / 6 (471)
where
(473) (474)
As can be seen from Eq. (464),the first-order time-of-flight dispersion of a cathode lens is determined by the intensity of the electric field near the cathode surface. It depends neither on curvature of the cathode surface, nor on the electric field distribution or the presence of a magnetic field in the cathode lens. Let us consider the second-order time-of-flight aberrations. As seen from Eq. (47l), these aberrations are subdivided into a geometrical one, associated with the coefficients Tin, TZD, T3,,,T4D,and a chromatic one, determined by the coefficient T S D . The geometrical aberration associated with the coefficient Ti, determines the spread in time of flight up to the plane z = const of the monoenergetic particles emitted by the central point of the cathode surface (rk = O,uD # 0). The aberration of this kind may be called the time-of-flight spherical aberration occurring owing to the existence of the particle’s initial
414
E. M. YAKUSHEV AND L. M. SEKUNOVA
velocity component parallel to the cathode surface. The aberration associated with the coefficient T 4 D determines the spread in time of flight, provided Ti, = T,, = T3D= TsD = 0. The aberration of this kind depends solely on the coordinates of the point of particle emission, and so it may be called the time-of-flight distortion. The aberration associated with the coefficients T,, and T3, will be called the mixed geometrical time-of-flight aberrations. It is worth noting that in electrostatic cathode lenses T3, = 0. The analysis of the time-of-flight aberrations of a cathode lens have been carried out for the region located sufficiently far from the cathode surface. In some cases it is interesting to study the peculiarities of the time-of-flight aberration formation in a near-cathode region, where the expansion in Eq. (80) is invalid. In these cases one should use the strict relation of Eq. (459) or its expansion for a near-cathode region.
VI. CONCLUSION Let us emphasize several aspects of the theory developed which may be used for solving the modern problems of physical electronics that are not associated directly with axially symmetric electron mirrors. One of these aspects is a special choice of the basic set of particular solutions of the near-axial trajectory equation. This choice is based on the character of the equation singularity, and not on the initial conditions of particle motion. It allows one to consider in the same way the qualitatively different systems-electron mirrors, mirror microscopes, and emission systems. As the presence of a singular point (the particle reversal point) is the peculiarity of electron mirrors which is not associated with the type of field symmetry, the methodology developed for theoretical study of electron mirrors with rotational symmetry may be applied, with appropriate modification, to the systems which do not possess rotational symmetry. The concept of arbitrary and guiding particles is the other aspect. This idea, expressed by Eqs. (11)-( 18), may be extended, without serious difficulty, into arbitrary static electron-optical systems with curved axes and relativistic particles. Thus, the perspective for development of a general theory of charged-particle nonstationary flux focusing and flux control by virtue of static electromagneticfields is open. This problem refers to a poorly developed field of physical electronics which is located at the junction of static and dynamical electron optics. At the same time, in the techniques and practice of physical experiments the use of nonstationary fluxes is widened, in particular, owing to the implementation of lasers into electronics. The studies carried out here deal with this fundamental problem only partly.
ELECTRON MIRRORS AND CATHODE LENSES
41 5
ACKNOWLEDGMENTS We are greatly indebted to Dr. P. W. Hawkes for stimulating our writing this article. We also wish to express our gratitude to Prof. V. M. Kel’man for his support and constant cooperation.
REFERENCES Artamonov, 0.M., and Komolov, S. A. (1970). Radiotekh. Elektron. 15 (l),220. Artamonov, 0. M., Bolotov, B. B., Gerasimova, N. B., and Yudovina, G. A. (1968). Elektron. Tekhn. 4 (S), 267. Artzimovich, L. A. (1944). Izv. Akad. Nauk SSSR,Ser. Fiz. 8,313. Barnett, M. E., and Nixon, W. C. (1967-1968). Optik 26,310. Berger, C . (1983). J. Appl. Phys. 54,3699. Berger, C., and Baril, M. (1978). Rev. Phys. Appl. 13,271. Bernard, M. (1952). C. R . Hebd. Seances Acad. Sci. 234,606. Bimurzaev, S . B., Daumenov,T., Sekunova, L. M., and Yakushev, E. M. (1983a). Zh. Tekh. Fiz. 53, 524. Bimurzaev, S. B., Daumenov, T., Sekunova, L. M., and Yakushev, E. M. (1983b). Zh. Tekh. Fiz. 53, 1151. Bruche, E., and Scherzer, 0.(1934). “Geometrishe Electronenoptik.” Springer-Verlag, Berlin and New York (Russian transl.: Lenizdat, Leningrad, 1943). Butslov, M. M., and Stepanov, B. M. (1978). “Image Converters and their Applications in Research Investigations.” Nauka, Moscow. Daumenov, T., Sapargaliev,A. A., Sekunova, L. M., and Yakushev, E. M. (1981). Zh. Tekh. Fir. 51, 1137. Dodin, A. L., and Nesvizhski, M. B. (1981). Zh. Tekh. Fiz. 51,897. Fortin, M., and Bad, M. (1972). Rev. Sci. Instrum. 43, 1140. Glaser, W. (1952). “Grundlagen der Electronenoptik.” Springer-Verlag, Vienna (Russian transl.: GIITL, Moskva 1957). Gohl, W., Kutscher, R.,Laue, H. J., and Wollnik, H. (1983). Int. J . Mass Spectrom. Ion Phys. 48, 41 1. Gruner, H., Gaukler, K. H., and Mollenstedt, G. (1971). Optik 33,255. Hahn, E. (1965). Jenaer Jahrb (Jena),p. 107. Hahn, E. (1971). Wiss. Z . Tech. Univ., Dresden 20 (2). 361. Hottenroth, G. (1936). Z. Phys. 103,460. Hottenroth, G. (1937). Ann. Phys. (Leipzig) [S] 30,689. Ignat’ev, A. N., and Kulikov, Ju. V. (1978). Radiotekh. Elektron. 23,2470. Kachiarz, J., Jaworski, M., and Zarembinski, S. (1982). Optik 61, 157. Kasper, E., and Wilska, A. P. (1967). Optik 26,247. Kas’yankov, P. P. (1961). Opt. Spektrosk. (Leningrad) 11,765. Kas’yankov, P. P. (1966). Tr. Opt. Inst. im. Vavilova 33, N162. Kel’man, V. M., Fedulina, L. V., and Yakushev, E. M. (1971). Zh. Tekh. Fiz. 41,1489. Kel’man, V. M., Sekunova, L. M., and Yakushev, E. M. (1972). Zh. Tekh. Fiz. 42,2279. Kel’man, V. M., Sekunova, L. M., and Yakushev, E. M. (1973a). Zh. Tekh. Fiz. 43,1799. Kel’man, V. M., Sekunova, L. M., and Yakushev, E. M. (1973b). Zh. Tekh. Fiz.43, 1807. Kel’man, V. M., Sapargaliev, A. A., and Yakushev, E. M. (1974a). Zh. Tekh. Fiz. 44,928.
416
E. M. YAKUSHEV AND L. M. SEKUNOVA
Kel’man, V. M., Sapargaliev, A. A. and Yakushev, E. M. (1974b). Zh. Tekh. Fiz. 44,938. Kulikov, Yu. V., Monastyrski, M. A., and Feigin, Kh. I. (1978). Radiotekhn. Elektron. 23, 167. Lafferty, J. M. (1947). Proc. IRE 35,778. Leleyter, M., and Slodzian, G. (1971). Rev. Phys. Appl. 6,65. Luk’yanov, A. E., Spivak, G. V., Sedov, N. N., and Petrov, V. I. (1968). Izv. Akad. Nauk SSSR, Ser. Fiz. 32,987. Luk’yanov, A. E., Spivak, G. V., and Gvozdover R. S. (1973). Usp. Fiz. Nauk 110 N(4), 623. Mamyrin, B. A., Karataev, V. I., Shmikk, D. V., and Zagulin, V. A. (1973). Zh. Exp. Teor. Fiz. 64, 82. Monastyrski, M. A. (1978a). Zh. Tekh. Fiz. 48, 1117. Monastyrski, M. A. (1978b). Zh. Tekh. Fiz. 48,2228. Monastyrski, M. A., and Kulikov, Yu. V. (1976). Radiotekh. Elektron. 21,2251. Monastyrski, M. A., and Kulikov, Yu. V. (1978). Radiotekh. Elektron. 23,644. Nesvizhski, M. V. (1984). Radiotekh. Elektron. 29,972. Noda, T. (1963). J . Appl. Phys. 2, 128. Oman, R. M. (1969). Adv. Electron. Electron Phys. 26,217. Recknagel, A. (1936). Z . Tech. Phys. (Miinchen) 17,643. Recknagel, A. (1937). Z . Phys. 104,381. Recknagel, A. (1941). Z . Phys. 117,689. Recknagel, A. (1943). Z . Phys. 120,331. Regenstreif, E. (1947). Ann. Radioelectr. 2, 348. Scherzer, 0.(1937). Beitrage zur Electronenoptik, 33. Schwartze, W. (1965-1966). Optik 23,614. Sedov, N. N. (1970). Izv. Akad. Nauk SSSR, Ser. Fiz. 34,1529. Sedov, N. N., and Nazarov, M. V. (1972). Radiotekh. Elektron. 17,2391. Sedov, N. N., Luk‘yanov, A. E., and Spivak, G. V. (1968a).Izv. Akad. Nauk SSSR, Ser. Fiz. 32,996. Sedov, N. N., Spivak, G. V., Petrov, V. I., Luk’yanov, A. E., and Rau, E. I. (1968b). Izv. Akad. Nauk SSSR, Ser. Fiz. 32,1005. Sedov, N. N., Labutin, N. I., and Wolk, Ch, P. (1971). Prib. Tekh. Eksp. No. 5 , p. 211. Sekunova, L. M. (1977). Zh. Tekh. Fiz. 47,2030. Sekunova, L. M., and Yakushev, E. M. (197Sa). Zh. Tekh. Fiz. 45,723. Sekunova, L. M., and Yakushev, E. M. (197Sb).Zh. Tekh. Fiz. 45,732. Septier, A. (1966). Adv. Opt. Electron Microsc. 1,204. Slowko, W. (1975). Opt. Appl. 5 (2), 27. Slowko, W., and Mulak, A. (1973). Opt. Appl. 9 (2), 37. Smirnov, V. I. (1974). “Kurs vysshei matematiki,” Vol. 2. Nauka, Moskva. Sturrock, P. (1955). “Static and Dynamic Electron Optics.” Cambridge Univ. Press, London and New York (Russian transl.: Inostrannaya Literatura, Moskva 1958). Takaoka, A., and Ura, K. (1984). Optik 68, 107. Vasil’ev, M. A,, Tchenakin, S. P., and Tcherepin, V. T. (1974). Prib. Tekh. Eksp. No. 4, p. 224. Vorob’ev, Yu. V. (1959). Izv. Akad. Nauk SSSR, Ser. Fiz. 23,694. Wechsung, R., Hillenkamp, F., Kaufmann R., Nitsche R., and Vogt, H. (1978).Mikroskopie 34,47. Wiskott, D. (1956). Optik 13,463. Ximen Ji-ye, Zhou Li-wei, and Ai Ke-cong (1983). Optik 66, 19. Zhigarev, A. A., and Shamaeva, G. G. (1982).“Electro-ray and Photo-electron Devices.” Visshaya Shkola, Moskva. Zworykin, V. K., and Morton, G. A. (1954). “Television.” Wiley, New York (Russian transl.: Inostrannaya Literatura, Moskva, 1956). Zworykin, V. K., Morton, G. A., Ramberg, E. G., Millier, J., and Vance, A. W. (1945). “Electron Optics and Electron Microscope.” Wiley, New York.
Index A ADF, see Airborne direction finder ADF, see Automatic direction finder ADS, see Automatic delay stabilization Aeronautical Radio, Inc., 74, 75, 107, 122 ARINC Characteristics, 122 AFCS, see Aircraft flight control system AGC, 63 Aharonov-Bohm effect, 302 hypothesis, 308 Airborne direction finder, 19 Aircraft flight control system, 2 , 4, 30, 39 frequency response, 100 guidance loop, 81 stability margin, 86 Air navigation guidance and control, 76-106 Air navigation systems, principles of, 15-45 2D, 17-18 Airspace structure diagram of, 49 Air traffic control radar beacon system, 58, 198 Air traffic control system, 1, 48-58 and its navigational aids, 45-76 Secondary Radar, 5 special radio navaid techniques, 71-76 surveillance system, 3 All Weather Operations Panel, 119, 172 Along-track distance, 54, 108 Along-track error, 35-36 Angelidi’s criticism on Clauser-Home factorability condition, 321-322 Annihilation photons, 263, 31 1 Approach-azimuth element, 65 Approachilanding aids, 63-70 Appropriate equations, theory of, 344 Area Navigation, 71 ARINC, see Aeronautical Radio, Inc. ARSR, 57, 58 Aspect experiments (1981), (1982), 308-309 Astigmatism, 364, 410 anisotropic, 364, 410 ATC. see Air traffic control ATCRBS, see Air traffic control radar beacon system
ATD, see Along-track distance ATRK, see Along-track error Attitude of the aircraft, 30 Autocoupled guidance, 84, 90 Automatic delay stabilization, 227 Automatic direction finder, 59, 60 Autopilot, 79 autopilot feedback control, 79 Autopilot error definition, diagram, 105 Availability of a system, 60 AWOP, see All Weather Operation’s Panel
B Bawdsey Research Station, U.K., 8 Beacon concept, 5-9 diagram, 6 interrogation, 7 transponder, 7 Beacon performance curve, 158 Bell’s inequalities, 245-335 direct proof of QM nonlocality, 297-300 generalized BI for dichotomic variables, 280-295 original Bell inequality, 276-280 Wigner’s version of Bell’s theorem, 279280 recent developments, 300-304 Bell’s inequalities, experimental verification of, 304-315 cascade photon experiments, 306-3 10 Clauser experiment, 207 Freedman-Clauser experiment, 306 FryThompson experiment, 307-308 Holt-Pipkin experiment, 306-307 Kocher-Commins experiment, 306 Perry experiment, 309-310 positronium annihilation experiments, 3 10-312 proton-proton scattering, 3 13-3 14 recent proposals for new experiments, 3 14315 BI, interpretation of, 315-325 criticism of current interpretation, 3 16-321 current, 315-316 417
418
INDEX
Bell’s locality conditions, 286, 308 Bell-type inequalities, 291-295, 320 for dichotomic measurement results, 293 BI, see Bell inequality Boeing 737-300 FMCS, 75 Bohm- Aharonov hypothesis, 263 Bohm version of EPR paradox, 253-256 EPR option ( I ) , 253 EPR option (2), 254-256 Bohr-Einstein debate, 246 Bohr’s complementarity concept, 272 Bohr’s resolution of the EPR paradox, 256 Brightness amplifiers, 337 Bruno experiment, 312 C
CAA, see Civil Aviation Agency Cascade photon experiments, 306-310 cascade in calcium, 306 Cathode lenses, 397-416 aberrations of spatial focusing, 404-41 1 electrostatic, 397, 414 near-axial approximation, 401 -404 brightness picture, 402 time-of-flight aberrations of, 41 1-414 equation for, 412 geometrical, 414 spherical, 413 Cathode ray tubes, 337 intensity-modulated, 5 CDI, see Course deviation indicator CEP, see Circular error probability Channel pairing in TACANIDME, 168 Channel plan, 160-169 CHSH, see Clauser-Home-Shimony-Holt CI, see Copenhagen interpretation Circular error probability, 35 Civil Aviation Agency, 13 Clauser experiment, 307 Clauser-Home assumption of rotational invariance, 308 Clauser-Home factorability condition, 288, 321-322 general validity of, 289 Clauser-Home proof of GBI, 287-290 Clauser-Home-Shimony-Holt inequality, 281-283 Clauser-Horne universality claim, 322 Closed-loop guidance system, 33
Closed-loop transfer function, 88 closed-loop tranform for wind disturbance, 89 CMN, see Control motion noise Collision risk probability, 53 Coma, anisotropic, coefficient of, 364, 410 Coma, isotropic, coefficient of, 364, 410 “Complementarity,” 257 Complementary filter, 86, 97-99 Completeness, 246, 257, 265, 325 Complete theory definition of, 247 Compton scattering, 262, 311-312 of correlated annihilation photons, 263 Klein-Nishina formula for, 3 12 “Control law,” 83 lateral channel, 90-97 Control motion noise, 77, 87, 199 definition of, 102 and PFE, 99-103 CONUS, 75 Conventional takeoff and land aircraft type, 2, 80 Cooper rating, 100 Copenhagen interpretation, 245, 256 Correction signal, 78 Correlated photons state vector for, 310 Correlation experiments, 246 list of, 314 Correlation function, 276-277, 292-293, 303, 313 Counterfactuality, 248-3 18, passim Course deviation indicator, 33 Cross-track error, 35-36 CRT, see Cathode ray tube CTOL, see Conventional takeoff and land
D DAC, see Delay-attenuate-compare Dead reckoning, 15-19 Dead time, 169-170 De Broglie’s theory of the double solution, 266 DECCA, British navigation system, 15, 59 Decision height minimum, 55, 57 Delay-attenuate-compare time-of-arrival estimator, 185, 189, 195 design relations, 188 Density matrix formalism, 266
419
INDEX
DH, see Decision height Dichotomic variables, 253, 280-295 Dispersion-free ensemble, 274-275 Dispersion-free variables, 273 Distance measuring equipment, in aviation, 1 243 channel plan, 123 operation diagram, 9 range data, 3 range error, 3 scanning DME, 73-76 standards, 120-127 evolution of, 122-127 versus TACAN, 13- 15 Distortions, coefficients of, 364, 410 isotropic and anisotropic, 364, 410 DME, see Distance measuring equipment DMEiDME enroute navigation accuracy, 106-1 18 geometry of error equation computations, 1 I3 DMEiN system, 1, 12, 118-171, 200 accuracy, 142-144 system error budget, 143 calibration and range coding, 145 coverage, 144- 160 ground transponders, 227 operation and components, 129- 133 power budgets, table of, 151 “signal in space,” 141-171 system capacity, 169-171 DME navigation parameter, 2 DMEiP international standard, I , 12, 165, 171-235, see also Precision Distance Measuring Equipment accuracy specifications, table of, 218 antenna considerations, 22 1-222 channel plan, 180-182 decoder, 166 DME used with MLS, I error measurement, 212-213 garble-system efficiency and accuracy effects, 198-212 interaction with AFCS, 103-105 interference environment, 225 multipath, 194-198 operational requirements, 175- 177 accuracy necessary, table, 176 allowable errors, table, 178 pulse shape, 183- 186 pulse time-of-arrival estimation, 186- 189
receiver noise, 190- 194 signal in space, 212-222 signal processing and transponder-interrogator implementation, 222-23 1 system field tests, 231-233 transponder, diagram of, 226 two-pulseitwo-mode system, 177- 180 DMEIVOR geometry of error equation computation, 110 DODIDOT, 37, 42 Doppler radar navigator, 15 Doppler VOR, 61-62, 117 Double-slit experiment with correlated systems, 270-272 Einstein’s version, 272 setup for, 270 with source of single quantum systems, 272 Downlink/uplink cycle, 7 Drift distance, effective, 367 DIU
desired-to-undesired signal ratio, 137, 163 specific free-space DIU criteria, 163 DVOR, see Doppler VOR
E Eberhard-Peres procedure, 3 19 Eberhard-Peres proof of GBI, 290-291 Einstein-Bohr controversy, 267 Einstein locality, 248, 258, 271, 276, 298, 300, 321 consequences of, 278 recent proposals for testing validity of, 267272 double-slit experiment with correlated systems, 270-272 Popper’s new EPR experiment, 268-270 Einstein-Podolsky-Rosen paradox, 245-335 attempts to resolve, 264-267 critical analysis of original EPR argumentation, 256-261 Feynman and, 272 first attempt at an experimental discrimination between EPR and standard QM, 261-263 hypotheses counterfactuality, 248 Einstein locality, 248 universal validity of QM, 248 modified EPR argumentation, 251 original version, 246-252
420
INDEX
Einstein’s relativity theories, 326 Einstein unified field theory, 300 Electron lenses, 337 Electron mirror microscopes, 337, see also Mirror electron microscope aberrations, 356-364 time-of-flight spherical aberration, 369 conservation law for azimuthal component of its generalized momentum, 339 electron-optical properties of axially symmetric, 350-371 constants a and b mirror properties in first approximation, 353-356 energy conservation law for a particle, 339 equations of motion, 338-350 time of flight, 349-350 trajectory equations, 340-349 mirror time-of-flight mass dispersion, 367 time-of-flight energy dispersion of the first order, 365 time-of-flight focusing, 365-37 1 principal plane of, 366 space-time-of-flight focusing, 37 1 time-focusing interval, 367 trajectories determining focus, 356 Electron mirrors theory of, 337-416 Element of physical reality, 247, 261, 294 definition of, 247 Encoding and decoding, 7 Ensemble mean-square error, 41 EPR, see Einstein-Podolsky-Rosen EPR argumentation, see EPR paradox EPR-Bohr debate, 258 EPR problem, see EPR paradox Error analysis, 106- 112 Error budget, 37-39 bias error, 38 noise error, 38 from temperature effects, 38 time sample mean square error, 39 EUROCAE, see European Organization for Civil Aviation Electronics European Organization for Civil Aviation Electronics, 122 Expectation values, 296
Federal Telecommunications Laboratories, 13 Feedback-controlled aircraft, 77-79 feedback loop, 84 Ferris discriminator, 136, 165, 166, 167, 185, 204 Final approach fix, 53, 54 Final approach waypoint, 54 First- and second-kind states and Bell’s inequalities, 321 Fixing position, 15-19, see also Position fix Flight control system auto-coupled, 32 manual-coupled, 32 Flight technical error, 34, 42 Fractional parentage coefficients, 320 Freedman-Clauser experiment, 306 Fry-Thompson experiment, 307-308 R E , see Flight technical error
G Garble, 198-212 downlink DME garble, 199 garble source, 208 interaction interval of, 204 uplink DME garble, 199 Gaussian image, plane of, 363 Gauss-Markov process, 96 GBI, see Generalized Bell inequalities GDOP, see Geometric dilution of precision Generalization of Bell inequality to arbitrary spin, 295-297 Generalized Bell inequalities, 280-295 Clauser generalization, 281-283 proof by Bell, 283-285 proof by Clauser and Home, 287-290 proof by Eberhard and Peres, 290-291 proof by Selleri, 287 proof by Stapp, 285-287 Geometric dilution of precision, 75 Glide path, 33 Glide path intercept point, 29 Global positioning system, 25 GRIP, see Glide path intercept point GPS, see Global positioning system Guidance loop bandwidth, 90 Guidance output information, 2
F FAF, see Final approach fix Faraci experiment, 31 1-312 FAMP, see Final approach waypoint FCS, see Flight control system
H Hazeltine Corporation, 10 Mark V IFF, 10
42 1
INDEX
Heisenberg’s uncertainty relations, 260, 269, 279 Hermitian operator, 274 Hidden variable theory, 249, 273 hidden parameters, 275 Hilbert space, 274 Holt-Pipkin experiment, 306-307 HVT, see Hidden variable theory
Joint probability distributions and Bell’s inequalities, 316-321 Joint Tactical Information Distribution System, 138, 198 JPD, see Joint probability distribution JTIDS, see Joint Tactical Information Distribution System
K I IAF, see Initial approach fix IAWP, see Initial approach waypoint ICAO, see International Civil Aviation Organization ICAO Channel Plan, 13 Identification of Friend or Foe, 9 IFF, see Identification of Friend or Foe IFR, see Instrument flight rules ILS, see Instrument landing system Image converters, 337 Image curvature, coefficient of, 364 Impossibility proofs, 275 INAF, see Intermediate approach fix Inertial aiding, 18 Inertial navigation system, 16 Information theory, 272 Initial approach fix, 53 Initial approach waypoint, 54 INS, see Inertial navigation system Instrument flight rules, 47-48, 58 route structures, 51-57 Instrument landing system, 4, 39, 66-67 system error analysis, 41 Integrity, 60 MTBO, 60 Intermediate approach fix, 53 Intermediate waypoint, 54 International Civil Aviation Organization, I , 48 Seventh Air Navigation Conference, April, 1972, 172 Interrogator, 58 Interrogator performance curve, 158 Interrogator range tracking, 138- 141 INWP, see Intermediate waypoint ITT Nutley Laboratory, 10
J Jitter interrogation rate jitter, 231
Kalman filter, 83, 98-99 gain optimization, 98 Kasday experiment, 3 12 Klein-Nishina formula, 3 12 Kocher-Commins experiment, 306 L
Lagged-roll, 94 Lateral channel control law, 90-97 perturbation model, 92 transfer function of, 95 Lateral guidance, 52, 66-67 Lateral turn, physics of, 93 “Leading edge” measurements, 23 Leading edge time of arrival error, 114 Linear filter noise reduction factor, equation for, 230 Linearity hypothesis, 273, 275 Locality, 248, 250, 326, see also Einstein locality LORAN, 20, 59, 72 Lorentz invariant, 3 16 M
Mach’s principle, 302 Mark V IFF, 10 MDA, see Minimum descent altitude MEA, see Minimum enroute altitude Mean square error, 38 Mean-square tracking error, 95 Mean time between outages, 60 MEM, see Mirror electron microscope Mermin’s GBI for spin, 296-297 Microwave landing system, I , 3, 44 air-derived radial system, 67 angle accuracy standards, 80 control laws, 103 guidance loop, 81 guidance signals, 103 MLSIRNAV, 69 signal format, 69
422
INDEX
straight-in approaches, 81 Minimum descent altitude, 53, 5 5 , 57 Minimum enroute altitude, 53 Minimum Operational Performance Specifications, 121 Minimum required altitude, 53 Mirror electron microscope, 372-397 aberrations of a shadow image, 384-397 electron lens aberrations, 392 geometrical aberrations, 39 1 test net image, 392-393 arbitrary constants, 376-378 diagram of, 372 electron beam crossover, 375 electron-optical properties, in the first approximation, 378-380 image forming, 380-383 impact parameter and circle of secondary emission, 383-384 microscope magnification, equation for, 379 MLS, see Microwave landing system MOPS, see Minimum Operational Performance Specifications MRA, see Minimum required altitude MSE, see Mean square error MTBO, see Mean time between outages Multipassage mass spectrometer, 337 Multipath geometry and error mechanism, 11 1 “Multisensor navigation system,’’ 99 N NAS, see National Airspace System National Airspace System, I , 45 Navaids, see also Navigation aids accuracy consideration, 33-37 bias accuracy, 33 error budget, 41 error components, 36 guidance signal, 30 NAVAR, I 1 Navigation accuracy performance of DME, TACAN, and VOR, 106-1 18 Navigation aids, 4, see also Navaids Navigational services, table of, 4 Navigation parameter measurement, 19-25 DME, 19 INS, 19 range and bearing, 19-25 Navigation system error budget, 35, 42 NDB , see Nondirectional beacons Negative probability distributions, 266
“No-enhancement” hypothesis, 3 10, 323-324 Nondirectional beacons, 19 Nonlocality in QM, direct proof of, 297-300 Nonmagnetic time-of-flight mass spectrometry, 337 Nonmeasurable sets and Bell’s inequalities, 323 Nonreproducibility of physical situations, 32 1 No-result Gedunken experiment, 271 NOTAM, see Notice to airmen Notice to airmen, 146 NSE, see Navigation system error
0 Office of Technology Assessment, 45 OMEGA, 20, 59, 72 OTA, see Office of Technology Assessment Outlier rejection, 231-232
P Path following error, 99-103, 199 and CMN, 99-103 definition of, 100- 102 definition, diagram, 105 Perry experiment, 309-310 PFE, see Path following error Phantastron, 139 Photon annihilation, 31 1 Photon polarization, 262-263 PICAO, see Provisional International Civil Aviation Organization Plan-position indicator, 5, 58 Plan view display, 58 Polanzer efficiencies, 306-307 Popper’s new EPR experiment, 268-270 setup for, 269 Position fix, 15-19, 26 Position fix accuracy, I14 Position-fix aircraft guidance system, 78 Positive control airspace, 48 Positive vertical guidance, 55 Positronium annihilation experiments, 3 10-3 12 Bruno experiment, 312 Kasday experiment, 3 I2 Wilson experiment, 312 Wu-Shaknov experiment, 3 I1 Power budgets, 146 PPI, see Plan-position indicator Precision Distance Measuring Equipment, 171-235
423
INDEX
PRF, 74 Protection ratio DIU, 167 Proton-proton scattering experiment, 3 13-314 experimental setup, 313 Provisional International Civil Aviation Organization, 11 PVD, see Plan view display
Q QM rotational symmetry, 262 “Quality factor,” 294 Quantum potential, 275
R Radar, 5 Radial navaids, 23-24 Radial systems, 20 TACAN, 20 VOR, 20 Radio navaid data rate, 87 Radio navaid position-fix configurations, 2325 Radio navigation systems table of, 59 Radio Technical Commission for Aeronautics, 11, 12 RAM, see Random access memory Random access memory, 139 Rebecca-Eureka system, 9 table of. 10 Reeh-Schlieder theorem, 304 Relativity, 248, 316, see also Einstein’s theory of relativity Reply efficiency, 169 equation for, 170 Resonant scattering, 324 Retroactivity, 265 equipment, 27 RNAV, 4, 27, 34, 50-52, 107 flight paths, 50 instrumentation errors, 107 position error, 34 routes, 35 system error, 34 3D navigation, 52, 57 Rocky Mountains, 53 Root-sum-square calculation, 37-44 RSS, see Root-sum-square RTCA, see Radio Technical Commission for Aeronautics
RTCA, 1984, 42, 54, 69, 75 RTCA Special Committee 159, 25 Runway fix, 53 RWYF, see Runway f i x
S SARPs, see “Standards and Recommended Practices” SEARCH, 139 Secondary surveillance radar, 8, 12, 25 “See-and-avoid’’ principle, 47 Segment, 53 Selleri proof of GBI. 287 Separability, 271, 276, 300 SG, see Stem-Gerlach Short takeoff and land Slew-rate limiting, 231-232 Specular multipath, 109 Spherical aberration, coefficient of, 364, 410 Spin correlation experiments, 302 Spin projections, 277 Squitter, 130 signal amplitude, 227 SSR, see Secondary surveillance radar SSV, see Standard service volumes “Standards and Recommended Practices,” 58, 120
Microwave Landing System, 63 for nonvisual precision approach, 172 table of, 122, 124-127 for VOR, 60 Standard service volumes, 145, 147, 148 Stapp proof of GBI, 285-287 State of a physical system in EPR definition of, 256 State vectors of the first type, 251 State vectors of the second type, 251 State-vector reduction, 264, 269, 270, 271, 298 Stem-Gerlach device, 253, 261, 276, 285, 302 STOL, see Short takeoff and land Successive approximations, method of, 348 System efficiency definition of, 199 estimating, 201-203 T TACAN, see Tactical Air Navigation System Tactical Air Navigation System, 3, 13- 15, 5 1
424
INDEX
area navigation systems, 41 enroute navigation accuracy, 106-1 18 error definition for, 109 TACAN bearing measurement, diagram, 64 Tangent point, 108 tangent point distance, 108 TCA, see Terminal control area Telecommunications Research Establishment, 9 Terminal control area, 48 TERPS, 53, 54 Time images, equation of, 367 Time-of-flight aberrations, 349 Total system error, 35 TRACK, 139 Tracking error, see also Total system error Transponder, 58, 63 TRE, see Telecommunications Research Establishment True hyperbolic systems, 20 DECCA, 20 LORAN, 20 OMEGA, 20 Two-dimensional lateral guidance navigation system, 18
U Universal validity of QT, 248, 250 U.S. Naval Research Laboratory, 10
V Vertical guidance, 52, 66-67 Very-high-frequency omni range, 3, 60-62 Doppler VOR (DVOR), 61-62 enroute navigation accuracy, 106- I 18 Precision VOR (PVOR), 61-63 principles of, diagram, 61 “scalloped, ” 43 VFR, see Visual flight rules VICTOR airways, 45-46, 129 Visual flight rules, 47 von Neumdnn’s impossibility theoiem, 275
von Neumann theorem, 273, 273-276 VOR, see Very-high-frequency omni range VORIDME, 36, 54, 72, 108 error definition for, 109 navigation application, 36 VORIDME position-fix guidance, 18, 51 VORIDMEITACAN controversy, 13 VORIILS localizer facilities, 168 VOR/ILS/DME, 13 VORTAC,4, 14, 19, 44-50, 59,73, 112-118 flight paths, 50 multipath error relations, 1 12- 1 18 system diagram, 14 TACAN plus VOR, 4 VORTAC radials, 19, 44
W Wallops Station Flight Center, 23 I , 234 Wheeler-Feynman absorber theory of radiation, 300 Wholeness idea of Bohm, 326 Wiener filter theory, 98, 99 Wigner-Araki-Yanase theorem, 294 Wigner potential, 264 Wigner’s theory of measurement, 264 Wigner’s version of Bell’s theorem, 279-280 Wilson experiment, 312 Wind turbulence, 18, 85, 89 Working Group M meeting in Seattle, 173 Wu-Shaknov experiment, 31 1
X XTRK, see Cross-track error
Z “Zero mile delay,” 8 Zero-order hold, 63, 87 ZOH, see Zero-order hold
