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)]
(3.10)
= 2 cos(27rd/' + d
) where
is the insertion phase at frequency/ df, and
is the insertion phase a t / + df The group delay T,(/) is the insertion delay of the modulation envelope. In other words, group delay is the insertion delay of phase at modulation frequency df. Insertion phase d
-Bl = tan-'(fY tan Bl) - Bl (t) is N f cos 2-rrfj, where peak frequency deviation i s / = 4>f„. The signal V(t) for ^ l can be shown to contain spectral lines of frequency modulation at / + f and at / - /„ relative to the carrier, as illustrated in Figure 3.22(a). m (t) is then determined, which, from (3.112), determines 7(0, T) at t = 0 and 7(7, T) at / = 7. If the two signal sources have the same phase noise statistics, the Allan variance for either source is the measured variance divided by two. The Allan variance may be dependent to some extent on the time 7 between sample pairs separated by averaging time r. Multiple measurements of a (2, 7, t) are usually taken to be averaged. As a rule, the Allan variance of a source is quoted in terms of the normalized variance, expressed as 2 ,)|A for compressed pulses weighted for -36-dB sidelobes. With no weighting, the main response and sidelobe structure of a compressed pulse are degraded with band-edge quadratic-phase error as low as ) = ±iri% rad. On heavily e . The integrated response written in complex form becomes Z>4>) = j n c ^ ' c L r approaches ylR for small azimuth angle displacement 0, from boresight, where y is the linear cross-range displacement from boresight at range R. At first, it will be assumed that the PRF is high enough so that the echo signal can be approximated as a continuous signal. This signal is then processed by integrating it over a time interval -772 to +7/2. The integrated response in terms of cross-range displacement y for uniform illumination during -7/2 to 4-7/2 is expressed as r . is the squint angle of the real beam from directly side-looking. The maximum unambiguous cross-range length (.Rtp)^,, as for the side-looking SAR, must be greater than the illuminated cross-range length Rip. Therefore, for spotlight SAR, the required PRF for unambiguous sampling is (2v,|cos <#)//. An additional consideration regarding PRF occurs when the combination of large illuminated range extent and high platform speed is involved, as in the case of spaceborne SAR. Data foldover in the slant range is prevented by requiring a sufficiently low PRF so that echoes arriving from each pulse do not overlap those from the previous pulse. For illuminated range extent A A?,, the minimum PRI to avoid overlapping is 2ARJc. Maximum PRF is c/(2Ar? ). At the same time, the PRF must be sufficiently high to avoid undersampling of the illuminated cross-range extent. Selection of the PRF to meet these conflicting criteria will be discussed later, using the SEASAT spaceborne SAR as a design example. ( = - — = -t-£— f = +ipf2 through integration angle ip. The radar range to scatterer 1, for a small integration angle, varies from R - Sr to R + Sr at )'ik Amplitude and phase in the /th range cell of the synthetic range profile produced by the Ath burst of a pulse-to-pulse stepped-frequency waveform. (A/«) Amplitude function of angular frequency. ABCD Transmission line matrix. B, B' Amplitudes. £ g, Amplitude of transmitted waveform, received signal, respectively, at " ith frequency step or ith pulse for pulse-to-pulse stepped-frequency waveform. C Constant. C Characteristic function. C (-v) Characteristic function at t = -v. C{z) Fresnel cosine integral. D Dispersion factor; antenna directivity; constant. D„ Antenna diameter. D,j Magnitude of Ith Doppler cell of y'th range cell in an ISAR image. £ Received signal energy. Ei Incident electric RF field. E, Backscattered electric RF field. £(v, a) Image entropy obtained from a target data set corrected for estimates of initial values of target velocity and acceleration. u
(3.34)
The maximum phase deviation (S
71
2
LINE SOURCE
2
LOAD
Figure 3.6 Long-line insertion phase versus frequency. (From [3]. Reprinted with permission.)
assuming that Z,, Z , and Z are real and independent of frequency, and that /?/ is directly proportional to frequency. The result can be expressed as 2
0
tanlWU] =
(3.35)
y]4F
T
If we write (3.30) in terms of normalized resistances, this results in the expression F = T
r.n + 1 ' r, + r 2
where
(3.36)
72
(3.37] and (3.38) The quantities r, and r are the input and output VSWRs, respectively. Curves of constant phase deviation (50)™, are plotted in Figure 3.7 versus r, and r by using (3.36) and (3.35). 2
2
3.5 THE MATCHED-FILTER AND AMBIGUITY FUNCTION 3.5.1 Matched Filter The concept of a matched filter is a very general notion, common to many aspects of radar and other types of signal analysis. A filter matched, as defined below, to a given input signal can be shown to be an optimum filter for signal reception when the received signal is corrupted by additive white Gaussian noise. The filter is optimum in several senses. These include maximizing the output SNR and maximizing the accuracy of parameter estimation (for parameters such as delay, Doppler frequency, and signal amplitude). A filter matched to an input signal s (t) with spectrum S,{f) is defined in terms of the matched-filter transfer function H(f) and the corresponding impulse response function h(t) as follows. t
//(/) = GSXf )<;-»*'
(3.39)
and h(t) = GsXT-t)
(3.40)
where G is the fixed component of net gain through the filter, t is the fixed component of delay through the filter, and H(f) is the Fourier transform of h(t). The asterisk refers to the conjugate form. The basic relationships stated in (3.39) and (3.40) for unity gain and fixed time delay of zero through the filter are H(f) = SXf)
(3.41)
and Kt)
=
jK-i)
(3.42)
73
Figure 3.7 Loci of maximum phase deviation in terms of input and output VSWR. (From (3). Reprinted with permission.)
Complex quantities are implied throughout. The transfer function of a filter matched to any signal, except for a linear-phase-versus-frequency slope, is proportional to the conjugate of the spectrum of the signal. The matched-filter impulse response, except for a fixed delay, is proportional to the conjugate of the time inversion of the signal. Matched filters for radar are actually not normally designed to match an input signal. Instead, the match is made to the transmitted waveform, which remains constant, regardless of the target. It is possible to design filters matched to very wideband waveforms. A wellknown type of matched filter for high-resolution radar is the pulse-compression filter.
74
Matched-filter processing of a target echo signal can be thought of as a coherent summation of the reflected signal energy from each of the target's reflection points. The processed response for the entire target is the phasor summation of the individual matched* filter responses for all of the target's reflection points, which are generally spread in range over the target's range extent. The principle of matched-filter processing can refer to pulse-to-pulse coherent integration as well as to coherent processing of individual echo pulses.
1
3.5.2 Ambiguity Function Like the concept of matched filtering, the concept of an ambiguity function is also a very general notion, common to many aspects of radar signal analysis. A radar waveform's ambiguity function is probably the most complete statement of the waveform's inherent performance. It reveals the range-Doppler position of ambiguous responses and defines the range and Doppler resolution. The ambiguity function of the waveform s,(t) can be defined in terms of the crosscorrelation of a Doppler-shifted version Si(t) exp(j27r/o/) of the waveform with the unshifted waveform. From the definition of cross-correlation, we can write
j2
Xir, fo) = jjs (0e ''»'][*;(/ - T)]dt l
(3.43)
Rearranging the terms in the integral produces a common form of the ambiguity function \X(r,f )?, for which D
X(r, f ) = f__s,(t)s\(t - T)e' "»'d, J
D
(3.44)
It is common to refer to the absolute value of x(T,f ) as the ambiguity surface of the waveform. The shape of the ambiguity surface is entirely dependent upon waveform parameters. A normalized expression is obtained by requiring that D
f k.MPdr = 1 J
~"
(3.45) i
With this normalization, the magnitude of the ambiguity function has a value at (0, 0) of unity. Examples of ambiguity surfaces generated by (3.44) are shown in Figures 3.8 and 3.9. Level contours are illustrated for one pulse of two idealized pulse waveforms: the Gaussian-envelope monotone pulse (Fig. 3.8) and the Gaussian-envelope linear frequencymodulated (chirp) pulse (Fig. 3.9). Practical waveforms consist of continuous pulse trains. The ambiguity surfaces for these pulse trains reveal ambiguous responses in range and Doppler. Performance for specific surveillance applications can be understood in terms of unambiguous range-
75
A
11 1f. s Figure 3.8 Level contour of the ambiguity function of a Gaussian-envelope monotone pulse.
Doppler regions of operation determined by radar pulse repetition frequency (PRF), pulse duration, and pulse bandwidth. For high-resolution applications, we are also interested in the ambiguity surface of individual pulses of the pulse train. 3.53 Matched-Filter Response Function Closely related to the notion of the ambiguity function is the matched-filter response function of a signal or waveform. The filter's output signal spectrum, produced by an input signal s,(t) is
76
Figure 3.9 Level contour of the ambiguity function of a Gaussian-envelope chirp pulse.
W) = «(/)5,(/)
•
(3.46)
where H(f) is the filter's transfer function and S,(f) is the input signal spectrum. The temporal response of the filter to the signal *,(/) is , 'M = Kt) * *,(/) = j j i ( '-
(3.47) TMTWT
where the asterisk indicates convolution and h(t) is the impulse response of the filter. The convolved response •$„(/) can be thought of as the signal produced at the output of the matched filter shown in Figure 3.10 when the signal described by s,{t) passes
78
through the filter. When the input signal is Doppler-shifted, the convolved response, using (3.47), becomes *„('. /o) =
-
(3.48)
TMT^'I-'&T
From (3.42) for h(t) matched to s{i): h(t-r)
= s;[-(t-T)]
= s-(T-t)
(3.49)
Then, sJLt. f ) = £ / ( T - t) (r)t^dT D
Si
(3.50)
By rearranging terms, we have sM.fo) = j % ( r K ( r - t)e*"-"&T
(3.51)
Similarity to the ambiguity relationship, expressed by (3.44), is obvious when the input signal Sj(t) is taken to be the transmitted waveform s (t). The matched-filter response function of a waveform and its ambiguity function are terms that are sometimes used interchangeably. The input signal s,(l) to the matched filter of a high-resolution radar is likely to be the extended echo pulse produced by the superposition of the echo signals from multiple reflection points of the target when illuminated by a wideband pulse such as a short pulse or chirp pulse. Echo signal duration can be greater than the transmitted pulse duration for targets or target regions of large range extent. {
3.6 WIDEBAND MIXING AND DETECTION Mixers and detectors of many forms appear in numerous components of RF equipment, including radar systems. Mixers are used for frequency translation of RF input signals. In the typical case for radar, echo signals at frequencies occupying the input bandwidth of the radar's microwave receiver are translated to some lower IF. Detectors are used to convert RF pulses into video pulses by removing the RF signal, leaving only the pulse envelope. The term video detection should not be confused with the term target detection, which is a decision process. Target detection decisions may be based on the magnitude of individual or summed video-detected pulses. Since World War II, mixers and detectors have been designed using semiconductor diodes, originally called crystals, operating in the nonlinear region of their current-versus-
79
voltage curves. Much of their behavior for various applications in radar, including wideband radar, can be understood in terms of the nonlinear response of a diode to an applied voltage. Diode current versus voltage is illustrated in Figure 3.11. The output voltage of a diode is the voltage across the load Z , which is assumed here to be small relative to the diode impedance. In the forward conducting region, diode current / produced by an applied voltage V can be described by the series 0
o
-o
V
Figure 3.11 Mixer diode current versus voltage.
80
2
I = a + bV + cV + dV> + ...
(3.52)
where a, b, c, and d are constants. 3.6.1 Mixers The voltage V applied to a mixer diode is the sum of the voltages of two or more input RF signals to be mixed. The first term of (3.52) is the dc offset. This term contains no RF signals. The second term, because it is linear, contains only signals at the same frequencies as those of the input signal components of the applied voltage V. Higher order terms in (3.52) produce mixer products. The squared term is of interest for many mixing applications. Typically, a fixed-frequency local oscillator (LO) signal is mixed with a relatively lower level received signal to produce a difference-frequency output called the IF signal. A receiver that uses a mixer in this manner to translate a frequency band of signals down in frequency to a convenient IF band is called a superheterodyne (superhet) receiver. Today's technology makes it possible to translate wideband signals at all microwave and millimeter-wave bands of interest for high-resolution applications. Up-translation is also common for translating low-level reference waveforms up to the transmitted carrier frequency. Mixer performance is often analyzed by assuming that the diode is biased to operate primarily in a current-versus-voltage region represented by the third term of (3.52). (Bias is not actually needed to obtain the desired performance for most applications.) The diode is then said to be operating in its square-law region. Higher order terms produce mixer products containing generally unwanted signals that are filtered out. The first two terms, if present, are not of interest for mixing because they do not produce mixer products. In the square-law region, for two input signal voltages V, and V , the diode current represented by the third term is 2
/ = c(V, +
2
V,) = c(V] + 2V, V
1
2
+ V )
(3.53)
Only the product term of (3.53) is normally of interest in mixer applications. The other two terms contain second-harmonic frequencies of the two input signals, respectively, which are filtered out. Consider two input signals expressed as V, = B cos(27r/,r + ^ , )
(3.54)
Vj = tT cos(27r/ f + ipi)
(3.55)
and 2
81
where ift, and i// are the relative phases of V, and V , respectively. The product term of the square-law current of (3.53) produces a voltage across the impedance Z expressed as 2
2
0
sM = IZ = 2cV,V Z 0
2
(3.56)
0
with V, and V from (3.54) and (3.55), the product term becomes 2
s,(t) = 2cBB'Za[cos(27r/,r + ifi )\ x [COS(2TT/ ( + {
2
fc)]
(3.57)
By using the trigonometry identity for the product of two cosine functions, and after dropping the constants 2c and Z , 0
s,(t) = BB'
COS[2TT(/, -
f )t + 2
fa-
fa \ 2
(3.58)
+ BB' cos|27r(/, +f )t + fa + fa ] 2
2
Two input signals produce a mixer product, which is seen to contain frequencies equal to the sum and difference of the two input signal frequencies. The mixer output illustrated in Figure 3.12(a) for down-conversion to IF is filtered as shown in Figure 3.12(b) such that only the difference frequency signal appears at the output of the filter. Signals produced by other product terms of (3.52) are also filtered out. In the standard superheterodyne configuration, one of the mixer inputs is the LO signal and the other is the received signal. The output IF signal s (i) can be seen in (3.58) to be proportional to the amplitudes B and B' of the two input signals, respectively. Therefore, for a constant-amplitude LO signal, the IF signal is linearly related to the input RF echo signal amplitude. r
3.6.2 Quadrature Detection Quadrature detection is used in various types of coherent radar systems to recover received signal phase relative to the transmitted carrier. For high-resolution applications, such as for a chirp-pulse or short-pulse radar, amplitude and phase are required as a function of range delay, along a selected range-delay extent of the received response, relative to the fixed-frequency carrier of the transmitted pulse. In other applications, received amplitude and phase are required relative to the transmitted phase for each of a set of narrowband transmitted pulses spread over a wide band of discrete frequencies. Quadrature detection can be thought of as a mixing operation that translates the received signal to baseband to recover amplitude and phase in the form of quadrature components. A quadrature detector is illustrated in Figure 3.12(c). For quadrature mixing, both the reference LO signal and signal carrier are at the same frequency, except for
82
B'COS (2rtf,t+ y ) 2
BCOS(2nf,t+v,)
•{
•S(t) = BB' COS[2n(f,-f )t + ( v , - V a ) ] 2
+ BB' COS[2n(f,+f )t + ( v , + 2
(a) IF SIGNAL AT f, - i WITH PHASE V , 2
V
2
LO SIGNAL @ 'a.Va BANDPASS FILTER
RECEIVED SIGNAL
(b) n/2 REF.
- Q = BB' SIN ( , V
v ) 2
B' COS (2nft + y,) o LPF
• I = BB' COS ( v , - V ) 2
SIG. BCOS(2trft +
y )02
(c)
INPUT RF PULSE
DETECTED VIDEO DETECTOR DIODE
M
»•
LPF
Figure 3.12 Mixing and detection: (a) mixer square-law products of two sinusoidal input signals; (b) superhet mixer; (c) quadrature detector, (d) video detector.
83
Doppler shift. The output of the lower mixer in Figure 3.12(c), following low-pass filtering, is then represented by the first term of (3.58) with /, = / ; in effect, 2
sfr) = BB' cosM -
fr)
(3.59)
This signal is called the inphase (I) output of the mixer. A second mixer with the reference signal delayed by irtl rad of phase produces a quadrature (Q) output. The / and Q output video pair is called the baseband signal. A Doppler-shifted echo signal will produce a baseband signal at the Doppler frequency. The transmitted signal, for pulsed-Doppler radar, is amplitude modulated into discrete pulses at some pulse repetition interval (PR1). The Doppler shift then appears as a pulse-to-pulse phase shift. Figure 3.13(a,b) illustrates quadrature detection. Practical systems are likely to operate as shown in Figure 3.13(c), so that filtering and amplification can be done more conveniently at lower frequencies.
3.6.3 Quadrature Detector Errors Figure 3.14 illustrates a quadrature detector with an input signal at +/from the reference signal frequency/. Signal amplitude is A and signal phase is 2irfrelative to the reference phase. An ideal quadrature detector produces / and Q outputs x = A cos lirft and y = A sin lirft, respectively. An actual system will exhibit gain and phase imbalance between the two channels and bias in each channel. Output for gain imbalance d, phase imbalance S, /-channel bias p. , and Q-channel bias py, can be expressed as x
x = A cos lirft + p,
(3.60)
y = A(1 +d) sin[27r/Y+ S] + /x,
(3.61)
x
and
It has been shown [4] that the effect of gain and phase imbalance in a quadrature detector is to generate "images" in the spectral domain of the complex output signal. An input signal at +/has a main response at +/and an "image" response at -/. Consider, for example, the Fourier transform processing of the quadrature-detected signal in Figure 3.13(c) produced by a target signal that is Doppler-shifted from the carrier by f . The Fourier transform will produce a main response at +/ and an image response at - / . SAR and ISAR processing commonly involves a discrete Fourier transformation of baseband sample data sets. The result, when amplitude and/or phase imbalance exists, is the appearance of "ghost targets" in the presence of real targets in the high-resolution display, SAR map, or ISAR image display. D
0
0
84
REFERENCE LO@f
lo @
Q<
POWER AMPLIFIER
TRANSMIT
i
QUADRATURE DETECTOR
SIGNAL®'+ („
LOW-NOISE AMPLIFIER
RECEIVE
K"
(a)
REFERENCE (LO)
-ECHO DELAYS-TRANSMIT PULSE
TRANSMIT AND ECHO PULSE FOR ONE PRI
ECHO PULSE
/
NEXT TRANSMIT PULSE
-PRII CHANNEL. OUTPUT X
QUADRATURE DETECTED ECHO PULSE
•t — Q CHANNEL OUTPUT
(b)
Figure 3.13(a,b) Quadrature detection: (a) idealized coherent radar: (b) waveforms for quadrature detection.
From [4], the image power relative to that of the main response is (P/4 for gain imbalance d and &I4 for phase imbalance 5. Bias errors ft, and fi, produce dc responses of relative power (/VA) , where fi = /i] + Gain and phase imbalance tends to change with input frequency so that fixed values of d and S cannot be defined for quadrature detectors required to operate over wide input bandwidths. High-quality SAR and ISAR performance has, nonetheless, been demonstrated for instantaneous bandwidths beyond several hundred megahertz. Some high-resolution radar designs avoid the use of wide instantaneous bandwidth by transmitting narrowband monotone pulses in a pulse-to-pulse hopped-frequency mode over a wide band of discrete RF frequencies. Baseband detection can then be performed over only a relatively narrow IF bandwidth by hopping the transmitted and reference LO signal frequencies together. Measurement and correction of / and Q errors are then 2
85
LO@f.
BASEBAND SIGNAL @ f„
PWR. AMPL.
BPF
SOURCE f + f,
QUADRATURE DETECTOR
TRANSMIT @f « + «. SOURCE
RECEIVE @f + l
I + f, " 'o
IF AMPL.
SINGLE ANTENNA
DUPLEXER
D
LOW-NOISE PREAMP
BPF
TYPICAL f, = 60 MHz TYPICAL J = 10 GHz (C)
Figure 3.13(c) Practical design.
Ideal system: x = Acos 2ntt y = Asm 2ntt >4cos2rc(/+/)Nonideal system: x = Acos 2ntt + fi y = A(1 +d)sm
t
(2nft
+ &)+M
y
Effect:
Relative image power from gain unbalance is —
Relative image power from phase unbalance is —
2
Relative DC power from bias is (jij A)
Figure 3.14 Quadrature detector errors.
where fi = yjfi]
86
relatively simple because imbalances d and S and biases fi and /t, can be treated as constants. A suggested measurement approach to be summarized below is to analyze a large set of statistically independent samples x and y of / and Q data obtained by sampling baseband outputs produced, for example, by an input test signal slightly offset in frequency by / from that of the reference to the quadrature detector. Let true / = cos 27r/t and true Q = sin 2irft with amplitude A = 1. Then, by trigonometric identity, (3.60) and (3.61) can be written as s
2
x = I + ti,
(3.62)
y = G[Q cos S + I sin S] + n,
(3.63)
and
where G = 1 + d is the ratio of Q- to /-channel gain. /- and Q-channel biases are simply calculated as fi, = x and /x, = y, where a bar over the symbol indicates average value. To obtain expressions for amplitude and phase imbalances, we rewrite (3.62) and (3.63) as x =I +x
(3.64)
y = CQ + DI + y
(3.65)
and
for C = G cos S and D = G sin S. With C and D defined in this way, we can show from trigonometry that gain and phase imbalance can be expressed, respectively, as d = G - 1 = VC
2
2
+D - 1
(3.66)
and fi=tan-'^
(3.67)
where expected values of C and D are experimentally obtained from random samples of x and y. The quantities C and D are first expressed in terms of the statistical values x-x and y - y by writing (3.64) and (3.65), respectively, as / =x - x 2. The analysis below is based on unpublished notes by Barry Hunt of San Diego.
(3.68)
87
and Z = y-y
(3.69)
= CQ + Dl
Next we solve (3.68) and (3.69) for D in terms of Z, C, Q, and /. Then, by recognizing that the expected value TQ of the product of true quadrature outputs / and Q, respectively, for many random samples of x and y is zero, we can obtain the expected value of D as -_IZ_(x-X)(y-y) 2
I
(3.70) 2
I
By solving (3.68) and (3.69) for C in terms of Z, D, Q, and / and noting that the expected values ~P and Q are equal and expected values 7 and Q are zero, we obtain 2
(3.71) In summary, the set of statistically independent x, y values are processed as follows. 1. Calculate /-channel bias as the expected value fi = x and £)-channel bias as /x, = x
y. 2. Calculate the set of expected values P = (x - X) , Z (x - X)(y -J). 3. Solve for D from (3.70) and C from (3.71). 4. Determine imbalances d and 8 from (3.66) and (3.67). 2
2
2
- (y - y") , and IZ =
Once the constants d, 8, fi„ and //, are determined from analysis of the test signal data set, they can be incorporated in a preprocessor to convert baseband data represented by (3.62) and (3.63) into approximations to true / and Q values x = A cos 2 77/1 and y = A sin lirft, respectively. Collection of the independent data set can often be conveniently obtained in a number of ways as a radar system test or receiver system test without requiring direct access to the quadrature detector. 3.6.4 Square-Law and Linear Detection Detectors using microwave diodes operating in their square-law region are used in highresolution radars, as well as in radars generally, for envelope detection of processed responses at RF or IF. Square-law characteristics are approximated for low signal levels. Detected video output current in the square-law region is proportional to input RF power. Relatively flat response over octaves of bandwidth is possible with square-law detectors. Linear detectors operate in the linear-current-versus-voltage region of the diode by using high signal levels biased so that only positive swings conduct. The output envelope of
88
an ideal linear detector, following low-pass filtering, is represented by the second term of (3.52). Detector video output current in the linear region is proportional to RF voltage. Operation of a video detector is illustrated in Figure 3.12(d). 3.7 SELECTION OF LOCAL-OSCILLATOR FREQUENCY The LO frequency of a superheterodyne receiver for a wideband radar must be carefully selected so as to avoid responses to signals in the preselector bandwidth that are not related to the echo signal. These responses, called spurious responses, become more of a problem as percentage bandwidth increases. Selection of an LO frequency can be made on the basis of calculations for forbidden zones of the LO frequency that result in spurious responses in the IF passband. These spurious responses occur at frequencies equal to the differences in frequency between harmonics m and n of the unwanted signal and LO frequencies, respectively. Once the forbidden zones are located for the radar center frequency and bandwidth, it is possible to select an optimum LO frequency that at least avoids the low-order spurious responses. This in turn determines the center of the IF passband. Assume that the LO frequency f is chosen so that an echo signal at frequency / appears at an intermediate frequency / = \f-fw\- Spurious responses then occur for the following two cases [5]. w
;
Case
= n/Lo - mf
I:/-/lo
Case I I : / - /
(3.72)
= mf - nf^
L 0
where m and n are harmonic numbers starting with zero, and/* is the frequency of an unwanted signal within the preselector passband that results in a spurious output within the receiver's IF bandwidth. Forbidden LO frequencies for case / occur at
J
'"> TTT
•
(3 73)
-
n+1 The minimum LO frequency for the harmonic set (m, n) that will result in a spurious output can be seen to occur w h e n / a n d / ' are minimum (i.e., both are at the low end'of the preselector passband). The maximum LO frequency for spurious response will occur w h e n / a n d / " are maximum (i.e., both are at a high end of the passband). Therefore, for a receiver with a preselector band covering a frequency range of / , to /, + [}, forbidden LO frequencies produced by the case / conditions will lie between the values m
, , . , ft + f< /u)
(min) = —
f
m+ 1 =— / ,
(3.74)
89
and , . . f. + P + « ( / . + P) m+\, Ao(max) = — =
, m+ 1 „ +— 3 f
/
(3.75)
for the harmonic set {m, n) of the signal and LO frequencies, respectively. Forbidden LO frequencies for case II occur at mf'-f n-\
(3.76)
The minimum LO frequency that will result in a spurious output will occur for f at the minimum preselector frequency / , and the desired signal frequency / at / + B. The maximum LO frequency for spurious response will occur for/* at the maximum preselector frequency f, + /3 and the desired signal frequency / at /,. Thus, forbidden LO frequencies produced by the case II condition lie between the values mf
( /
+
{ m
l)f
/ „x = - ^> — ~ ' j A - - —~ — ' ~ — ? y / (m.n) f
L0
mi
n (3.77) 77^
and , ,
,
Ao(max) =
n>a+B)-f, n
X
(m - l)f, , mB =— j -
+—
(3.78)
No significant harmonics of undesired low-level signals exist above m = 1, but particular care must be taken to avoid (m, n) harmonics of (0, 1), (0, 2), (0, 3), and (0, 4). The z t T O t h harmonic (a dc component) of signal frequency may be present, regardless of signal level. Figure 3.15 indicates forbidden LO frequency choices for an experimental HRR radar with 600-MHz bandwidth and a center frequency of 3.2 GHz. For this radar, it was decided to use an LO frequency of /uo = 4.55 GHz. This resulted in an IF band of 1.05 to 1.65 GHz. The only possible spurious signals from this choice of LO frequency up to the fourth harmonics are seen to result from (m, n) values of (2, 1), (3, 2), and (4, 3). Spurious response, according to Lepoff [5], drops off at 10 dB per harmonic order. Unavoidable spurious responses can be suppressed by using balanced mixers. 3.8 DATA SAMPLING 3.8.1 Time-Domain Sampling Wideband radar signals are often sampled, then converted to digital quantities, before data processing for target detection or imaging. The type of sampling required depends
90
3
Qco-
o:0<2
oQ«
Hjuifoo Fot H
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91
on the type of waveform selected and its bandwidth. The simplest type of wideband waveform is a short pulse. To sample echo signals produced by a short-pulse radar without introducing ambiguity requires that the sampling rate meet the Nyquist criteria. Nyquist's sampling theorem states that if a signal has no frequency components above some frequency /, then the signal is completely determined by sample values of the signal separated in time by 1/(2/), extending over the signal duration. Normally, it is not feasible to sample the RF form of a radar echo signal at this rate. To reduce the required sampling rate, the signal is usually mixed down to an IF carrier or to baseband (zero IF carrier). A target response produced by a single short transmitted pulse is illustrated in Figure 3.16(a). Its spectrum is illustrated in Figure 3.16(b). Assume that pulse spectral components outside bandwidth 0 can be neglected. Maximum frequency components of the downconverted baseband version of the echo signal are then reduced to BJ2, as shown in Figure 3.16(c). When converted to baseband, the echo signal is composed of / and Q components, each having a spectrum as shown in Figure 3.16(c). Low-pass filtering of either the / or Q output need pass no frequencies higher than BI2. Sample spacing in time, therefore, must be equal to or less than \IB sec for each output. One sampled / and Q pair of real samples taken simultaneously is called a complex sample. The echo signal is unambiguously determined by B complex samples per second. This corresponds to a spacing of IIB, which is also the temporal resolution associated with the pulse. Baseband samples of the echo signal of Figure 3.16(a) are indicated in Figure 3.16(d). For a typical highresolution radar system, the transmitted pulses, unlike that represented in Figure 3.16(a), are likely to be coded wideband rectangular pulses of long duration compared to the processed response from point scatterers. The response of Figure 3.16(a) would then represent the processed HRR response. Even at baseband, the required sampling rate for high-resolution radar is high. For example, to achieve 0.5m resolution, the received signal bandwidth from (1.1) is
~ 2Ar,
3 x 10' "2x0.5
(3.79)
= 300 MHz Achieving unambiguous sampling for signals of this bandwidth requires a sampling rate of 300 x 10 complex samples per second. Sampled data needs to be digitized (converted from sampled analog data to digital values) and stored. To complicate the problem further is the reduction in dynamic range, which occurs when data sampled at high rates is digitized. Dynamic range is reduced because the sampling-time aperture available to quantize the sampled data into fine quantization levels is decreased to accommodate high sampling rates. Thus, sampling rate and quantization are traded off. Sampling, digitizing, and storing of large quantities of data at rates over 300 x 10" complex samples 6
92
AMPLITUDE AMPLITUDE / ENVELOPE OF /tvxTRANSMITTED
A
PUL5E
„
/
ENVELOPE OF ECHO SIGNAL; e.g., FROM AIR TARGET
.„
A/i/Vb
(a)
POWER
FREQUENCY
(b)
POWER
FREQUENCY
(c) SAMPLE AMPLITUDE OR Q SAMPLES OF THE BASEBAND ECHO SIGNAL
i
M—7*
'DELAY
-SAMPLING WINDOW -
Figure 3.16 Echo signal, signal spectrum, and sampling: (a) echo response from a single short transmitted pulse: (b) short-pulse echo spectrum at RF; (c) short-pulse echo spectrum at baseband; (d) timedomain sampling at baseband.
93
per second with over 6 to 8 bits of quantization is difficult and expensive. However, limited quantities of data sampled with 8 or more bits of quantization at above 10' samples per second can be digitized and stored at this writing, and the technology is now rapidly improving. Figure 3.17 approximately represents lines of constant difficulty for this technology. Two analog-to-digital (A/D) converters are required to digitize sampled baseband data, one to digitize the /-channel samples and one to digitize the (3-channel samples. Each converter is required to convert B real samples per second if the signal bandwidth is B.
3.8.2 Frequency-Domain Sampling The limitation of wideband radar signal processing associated with time-domain sampling can be circumvented for some applications by sampling in the frequency domain. Frequency-domain sampling can be carried out by collecting an / and Q sample pair at a selected range-delay position of the response for each pulse of sequences of transmitted pulses coded in frequency pulse to pulse. For example, each pulse of a sequence (called burst) of pulses could be transmitted at a carrier frequency that is shifted by a constant amount from that of the previous pulse. Complex samples collected at a target's range-delay position along the received responses from such a pulse sequence can be thought of as discrete frequency-domain samples of the target's reflectivity if the target reflectivity can be assumed to remain constant during the pulse sequence. While the pulses of the sequence are spread over a wide bandwidth B, sampling can be carried out on each narrowband response at baseband by shifting the LO for each transmitted pulse to maintain a constant IF to be mixed to baseband. Figure 3.18 illustrates pulse-to-pulse frequency-domain samples collected at baseband. The sampling rate at each range-delay sample position is at the radar's PRF, which is likely to be less than 20 KHz for most applications, even though target reflectivity data may be sampled at an RF bandwidth of typically 500 MHz. Thus, the dynamic range of the A/D converter can be large (e.g., 12 to 14 bits). Sampled data collected at discrete frequency steps over the total bandwidth B of each burst can be processed to form a target range profile with range resolution equivalent to that obtained by transmitting a single pulse of duration MB. Sample pairs can be collected at a single range-delay position if the target's range-delay extent is less than the receiver's pointtarget response at baseband. To determine the required frequency spacing for unambiguous sampling in the frequency domain, we must consider Nyquist's theorem for sampling in the frequency domain. This theorem states that the spectrum of a signal is exactly determined by samples of the spectrum separated by l/(2<5f) Hz when the signal is zero everywhere, except during the delay interval St. The delay interval St, associated with a highly resolved target that produces reflection over a range extent / is 2l/c. Thus, for unambiguous sampling of a target's reflectivity in the frequency domain, real frequency samples are required to be taken at intervals of
4™
/
A
"-Pi
L
o o
o o
o o
o m
o cm
(s/saidwvsvoaw) aivu
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z
Duo
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:
-
aidwvs
95
SAMPLE AMPLITUDE
/
DISCRETE SAMPLES AT BASEBAND OF A TARGET'S REFLECTIVITY VS. FREQUENCY (I OR Q)
P Figure 3.18 Frequency-domain sampling.
(3.80) Complex samples may be taken at A / = c/(2/). As an example, the frequency steps required for unambiguous frequency-domain sampling of a 300m target must be spaced by a maximum of
3 x 10' 2 x 300
(3.81)
= 0.5 MHz If range resolution is to be 0.5m, 600 complex frequency samples are required for a total bandwidth of 600 x 0.5 MHz = 300 MHz, the same bandwidth as that required to obtain 0.5m resolution for the time-domain sampling example of (3.79). Sampling target reflectivity in the frequency domain to obtain wideband rangeprofile signatures is analogous to the more conventional sampling of reflectivity in the time domain to obtain target velocity. Consider the synchronous detection in Figure 3.13 of the echo signals produced by a single target at velocity v, relative to the radar. The radar transmits monotone pulses, all at the same carrier frequency. Baseband signal frequency is the target Doppler frequency 2v,l\ from (2.50). The range of Doppler frequencies po that can be unambiguously sampled is the radar PRF l/T , where T is the PR1 at which complex samples of the target response are taken. Thus, the range of target velocities that can be determined unambiguously by complex sampling at the PRI T is (A/2)p = A/(2T ). This is analogous to the target range extent c/(2A/) from (3.80), which can be determined unambiguously by frequency-domain sampling at a frequency spacing of A/. 2
2
2
2
D
96
3.9 TRANSMITTED-FREQUENCY STABILITY REQUIREMENTS The generation of high-resolution radar images and Doppler measurements requires coherent processing of data collection' records of target echo amplitude and phase history. Generally the phase history is most critical. The presence of random phase noise in the data reduces the quality of the processed target imagery or other outputs by reducing target response and increasing the noise floor. Random phase noise on the data record is produced primarily by relatively short-term frequency fluctuations of the transmitted signal. Methods for evaluating the effect of short-term random frequency fluctuation on processed outputs and methods for specifying frequency stability are described in this section. An ideal stable oscillator produces a single constant frequency signal of constant amplitude. Phase advances linearly with time and no amplitude modulation exists. The real signal sXt) from an actual oscillator is represented as sM = [A + a (t)} cos[27r/r + <Mt)\
(3.82)
N
where / i s the steady carrier frequency, d^t) is the instantaneous phase modulation (PM) noise, and a 't) is the instantaneous amplitude modulation (AM) noise superimposed on the constant amplitude A. The instantaneous frequency of sjlj) from (3.2) and (3.82) is N
/(') =f+fM
(3.83)
where f (i) is the instantaneous FM noise. Of interest to radar designers is the relatively short-term frequency stability of the transmitter and reference oscillators of the radar that determine pulse-to-pulse phase deviation from the noise-free phase of the response. Noise-free phase is measured when > (t) = 0 for all /. Short-term frequency stability is often measured, analyzed, and specified in terms of the power spectral density of phase and frequency fluctuation
N
N
N
3. The notion of target data collection used here and elsewhere in this book does not preclude online and/or real-time processing. The term data collection is used to differentiate from data processing.
97
3.9.1 Effect of Frequency Fluctuation on Radar Performance Random frequency fluctuation of the radar's transmitted signal produces echo signal data records that are contaminated with cumulative phase noise. Cumulative phase noise is the random phase variation with time that accumulates during the time interval between transmit time t and echo time / + T f o r range delay r. Consider the simplified radar system in Figure 3.19(a). Assume that a pulse was transmitted at time t. An echo pulse from a point target at range delay T returns at time / + r, at which time the / and Q channels of the quadrature detector output signal are sampled. Quadrature detector output phase at time /, omitting any fixed phase, is ideally 2irft for carrier frequency / free of frequency noise. The actual CW reference frequency from (3.83), however, is / + fj,t), where f^t) is instantaneous frequency noise. Assume that the instantaneous phase and frequency of the transmitted pulse of the Figure 3.19(a) system is that of the reference. Frequency noise on the carrier then produces instantaneous cumulative phase noise, seen at the output of the quadrature detector according to the expression (3.84) where t ^ ( 0 is the instantaneous phase noise on the carrier at time f, and
4
G,, = Ae**""'\ i = 0, 1.2
,i-l
(3.85)
where A is a constant amplitude, if/ is the constant noise-free phase, and ux is the random phase error accumulated during delay r produced by random frequency error x,. The i
4. The DFT is referred to here and elsewhere for illustration. Other transforms such as the FFT would likely be used in actual processing.
98
Figure 3.19 Effect of radar frequency instability: (a) radar system; (b) data record G,; (c) cumulative phase noise.
99
random phase error vx, contains the phase constant v = 2 VT in radians per hertz. Random frequency error x, is the average frequency error during the delay r for pulse i. Random frequency error defined in terms of cumulative phase noise
M+f-W
(3
86)
5
It can be shown that the output of the DFT processor of Figure 3.19(a), which processes the data represented by (3.85), is a set of n discrete values for which the expected peak value and variance are, respectively, Peak E\Hix)\ = nAC,
(3.87)
and tr\ = nA^\
- Cj)
(3.88)
where C is the characteristic function defined as s
Q=exp[-(w)72]
(3.89)
for rms frequency deviation a of x . Variance is the phase noise floor and is proportional to signal power. Range- and/or Doppler-resolved echo signals from small targets can be hidden by the phase noise produced by larger nearby targets. The quantity a = vcr is the standard deviation of cumulative phase noise. Some sources of phase noise may be known directly without reference to frequency noise. For example, phase noise may be produced in transmitter tubes by the pushing of noise on the applied voltage or current. Table 3.1 lists phase sensitivity for several types of microwave tubes. Table 3.2 lists signal power loss C) and signal-to-noise floor C//<7H for various values of standard deviation of phase noise and number of pulses coherently integrated. Note from (3.87) with (3.88) that, for an ideal stable transmitter where cr-0} the peak value of the DFT-processed response is nA, as expected for lossless coherent integration of n discrete complex values, each with an amplitude of A. The expected peak output of the DFT drops off as rms cumulative phase noise w increases. For example, in the above example, let delay time t = 1,234 fxs (100 nmi). Assume we know that over this delay the rms frequency deviation cr of the transmitter is 100 Hz. The rms cumulative phase noise is then va= 2TTTO-, which becomes 0.78 rad (44 deg). The resulting peak response of the processed data from (3.87) is {
c
5. Details of this analysis appear in Chapter S for the more general case of pulse-to-pulse frequency-hopped waveforms. A normal probability distribution of random frequency error was assumed.
100
Table 3.1* Transmitter Phase Ripple Phase Ripple (rod)
Tube Type Klystron TWT CFA Triode Tetrode Amplitron
SV — is the fractional voltage ripple. SI
— is the fractional current ripple.
Peak E\tUx,)\ = nAe^"
= nA e"
030
= 0.74nA
(3.90)
which is about a 3-dB reduction from that for a perfectly stable transmitter. For a = 200 Hz, vcr= l.SS rad (84 deg). Peak response drops to 0.30/1/1, about a 10-dB reduction. Typically, rms cumulative phase noise would be specified to be 10 deg or less, which corresponds to about 0.1-dB loss. Actual coherent radar systems are not likely to carry out quadrature mixing at the transmitted frequency as indicated in Figure 3.19(a). A more typical system is indicated in Figure 3.20. The transmitter here is a coherent power amplifier that amplifies a gated RF segment of the reference signal. In other coherent radars the transmitter may be a phase-locked pulsed oscillator locked to an amplified version of the reference signal. For example, phase-locked pulse Gunn oscillators are used as transmitters in low-power millimeter-wave systems. CW systems also exist. Cumulative phase noise for coherent radars can, however, often be estimated from analysis of the frequency stability of the transmitter based on definitions (3.87) and (3.88) for the idealized radar of Figure 3.19(a).
101
Table 3.2 Signal Loss and Signal-lo-Noise Floor Produced by Random Cumulative Phase Noise Sid. Dev. of Phase Error ar, - vcr 0.01 0.02 0.05 0.1 0.2 0.5 1.0 2.0
S/ajt Ratio of Signal Power to Variance Signal Power Loss dB(Cj) Cf
1-Cj
-0.0 -0.0 -0.0 -0.0 -0.2 -1.1 -4.3 -17.4
0.000 2.500 400 101 26 4.52 1.58 1.02
0.9993
0.999 0.998 0.990 0.961 0.779 0.368 0.018
J = £ ' [ / % ) ] = n'A'CJ
Transmitter
Frequency references
7+ U
d (j B
x n x n: xn x n xn x n x n xn
n = 64
n = 128
n = 2S6
n = 5/2
58 52 44 38
61 55 47 .41 35
64 58 50 44 38
67 61 53 47 41 M
23
J2G 24-
32:
p9
,27
Duplexer
Preamplilier Downconverter
-
Quadrature detector
LPFs
s&h + A/D
Range-delay trigger
Figure 3.20 Conventional coherent radar.
702
The assumptions required are that (1) instantaneous phase of the reference signal to the quadrature detector is the same as (or differs by a constant phase from) the instantaneous phase of the stable oscillator from which reference frequencies // a n d / + / of Figure 3.20 are obtained, and (2) cumulative phase noise produced by frequency fluctuation of the transmitted signal dominates any phase (not thermal) noise produced by the duplexer and receiving system (including preamplifier, down-converter, and IF amplifier). Frequency fluctuation of the transmitted signal can be produced by instability of the frequency reference as well as that of the transmitter itself. Frequency instability for coherent transmitter systems may be dominated by that of the reference. For these situations, transmitter stability can be estimated indirectly from specifications or measurements of the reference. For phase-locked transmitter systems, instability may be dominated by the transmitter, in which case frequency instability of the transmitter needs to be known directly. High-resolution radar systems are likely to transmit FM chirp pulses or hoppedfrequency pulse sequences instead of a series of monotone pulses of the same RF frequency that were illustrated in Figure 3.19(a). Cumulative phase noise of received echo data for hopped-frequency and chirp-pulse radar can be estimated from the frequency stability of the stable frequency reference to the transmitted waveform's center or start frequency. Frequency stability of the wideband transmitted signal itself needs to be known for situations in which the transmitter is expected to contribute significant phase noise. Estimates or measurements of frequency stability can be taken at a convenient frequency, such as the center or start frequency. In summary, the two parameters that determine cumulative phase noise on sampled data were seen to be (1) the phase constant v for the radar and (2) the rms frequency deviation a of x defined by (3.86). Cumulative phase noise cr = w in the example following (3.89) was determined from the phase constant v and frequency rms noise a. The phase constant for the coherent system of Figure 3.19(a) was seen to be lirr for range delay T. (Chapter 9 will discuss coherent-on-receive systems where v differs from 27TT.) Once the cumulative phase noise is determined at a specified range delay for coherent systems, we can then calculate performance degradation in terms of processed signal loss and increased noise floor from (3.87), (3.88), and Table 3.2. Two common measures of frequency stability from which cumulative phase noise of a radar can be determined are (1) one-sided phase noise spectral density ££(/„) and (2) Allan variance cr\2, T, T), which is a time-domain measure of frequency noise. These will be described below. (
{
c
3.9.2 Frequency Stability in Terms of Power Spectral Density of Phase Noise Probably the most common measure of short-term frequency stability is the parameter represented as ££(/„), which refers to the one-sided power spectral density of phase noise.
103
To be more precise, i£(/„) is the relative power spectral density of phase noise in watts per hertz per watt of carrier power as a function of positive offset frequency/„ from the carrier. It is nearly always seen in terms of dB [££(/„)] versus offset frequency plotted on a log scale. Figure 3.21 is an example of !£(f ) for a phase-locked Gunn oscillator transmitter. The quantity i£(f„) is an indirect measure of phase noise modulation energy, which for small or suppressed AM can be obtained directly from the power spectrum observed on a spectrum analyzer. It is also referred to as phase noise or spectral purity, which is actually the inverse of £(f„). m
3.9.3 Phase and Frequency Noise Modulation The relationship of SE(f ) to phase and frequency noise modulation is understood by the application of FM theory for small modulation index. A signal at carrier frequency / phase-modulated at a single modulation frequency /„ is expressed as m
V(f) = A cos(27r/( + if> sin 2irfj)
(3.91)
where
m
0|
1
1
1
1
Figure 3.21 Phase noise power spectral density for phase-locked Gunn oscillator.
1
Figure 3.22(a,b) Power spectrum and phase noise power spectral density: (a) power spectrum of a sinusoidally modulated carrier for 4> <* \; (b) power spectral density of a phase noise modulated carrier.
705
(C)
Figure 3.22(c) Power spectrum and phase noise power spectral density: (c) one-sided phase noise power spectral density. 2<J ). m
2
Sideband-to-carrier power from Bessel analysis is ($/2) in radians squared (or watts per watt of carrier power) in terms of peak phase modulation in each sideband. The quantity ( # 2 ) corresponds to [//(2/„)] in hertz squared (or watts per watt of carrier power) in terms of the peak frequency modulation in each sideband. Now consider continuous rms phase noise modulation per hertz, denoted by P (f„). Peak phase noise per hertz at/„ is >/24> (/ ), so that the one-sided power spectral density relative to the carrier in watts per hertz per watt is 2
2
(
N
N
2
m
(3-92)
J =
where OJU/m) is the two-sided phase noise power spectral density in radians squared per hertz. The corresponding frequency noise modulation is F (J ). Peak frequency noise at /„ is yJ2F (f ), so that in terms of frequency noise modulation, N
N
m
m
106
( 3 9 3 )
^ = [ - w - } - ^
where /•*(/«,) is the two-sided frequency noise power spectral density in hertz squared per hertz. The symbol S (f ) is commonly used to denote the density
m
m
£(/.)
= ^J^
(3.94)
and
AJ M
By definition, power spectral density of phase noise is the Fourier transform of the autocorrelation function R(0), which is the expected value of the square of the magnitude of phase noise | ^ * ( i ) | . Likewise, S ^ ( / J is the Fourier transform of the square of the magnitude of frequency noise \M0f- Both S/f ) and Sy(/») are two-sided functions because \(f>M\ and |/«(/)| are both real. Figure 3.22(b) illustrates the two-sided density S / / J and Figure 3.22(c) illustrates the one-sided density ££(/.). Figure 3.23 is a phasor-diagram illustration of the relationship between i£(/„) and S+(f ). The two counter-rotating phasors represent the respective noise voltages relative to the carrier in a 1-Hz bandwidth at modulation frequency f each side of the carrier. Their phasor sum, for small modulation index and no amplitude modulation, is the phasor with a sinusoidally varying magnitude at the quadrature phase to the carrier that generates phase noise
2
m
2
m
m
m
m
m
m
m
S/f.)
=
= 2<£(/J
(3.96)
which can be seen to be consistent with (3.94). 3.9.4 Cumulative Phase Noise The quantity ££(/*„), though sometimes referred to as phase noise, is actually the relative sideband power spectral density associated with phase noise modulation. Although it is
107
Figure 3.23 Phasor diagram of a phase-modulated carrier.
convenient to measure, it does not explicitly define phase noise in radians. Of interest to the radar designer is cumulative phase noise, which is the instantaneous phase difference M' + r) - MD of (3.84). Variance of zero-mean cumulative phase is expressed as 2
a] = E{IM' + r) - <Mr)] }
(3.97)
Expanding (3.97) and assuming stationary statistics for which <$Jj + T) is equal, on the average, to
+ r)}
(3.98)
Assuming that phase noise deviation is ergodic, we can treat (3.98) as a time average. This allows us to express the terms d9(t) and
108
(3.99)
a] = 2[R(0) - R(T)]
Recall that the power spectral density-of a function is the Fourier transform of its autocorrelation function. The inverse relationship written in terms of S//„) is (3.100) By substituting (3.100) into (3.99) for R ( 0 ) and R(T), we can, after several steps, obtain (3.101) Finally, from (3.94), the above integral in terms of the one-sided density ir?(/J becomes (3.102) This expression allows us to determine cumulative rms phase noise a in radians at a range delay T based on the one-sided density £f?(/») of the transmitted signal. The integration of (3.102) tends to a limit for integration beyond /„ = 1/rfor typical 2?(/J data. c
3.9.5 Specifying Phase Noise Power Spectral Density When designing a radar, one needs to set limits for i £ ( / J that allow for meeting desired phase-noise performance. One practical approach is to define a straight dB[Sf?(/„)]-versuslog(/„) line below which phase noise is acceptable. Generation of this phase noise limitation can be carried conveniently by assuming a white-noise approximation to ££(/„). White frequency noise implies that the power spectral density of frequency noise is constant. Relative phase noise power spectral density from (3.93) with white noise becomes (3.103) Rewriting (3.102) in terms of the white noise of (3.103), we obtain
Rewriting (3.104) as a (sin x)lx function and integrating, we obtain
109
a\ =
(2NFT-f
K ft Consider the following example. A radar is to be designed to operate out to 100 nmi (l,234-/xs range delay). Cumulative rms phase error at this range is specified to be less than 0.7 rad (40 deg). An upper-limit straight-line plot of dB[i£(/J] versus offset frequency/, plotted with a log scale is desired in order to evaluate alternative transmitter designs in terms of their frequency stability. From (3.103) with (3.105), we obtain one-sided phase noise power spectral density for white noise in terms of specified cumulative phase noise o> as 2
=
(3.106)
For
T
I
[IM
(3.107)
d B [ 2 ( / J J = 10 log (10//l) 10
The straight-line upper limit plot of Figure 3.24 is obtained from (3.107). Figure 3.25 is a plot of cumulative phase noise computed from the general equation (3.102) for the above white-noise approximation of i£(/ ) = \0/fi. Cumulative phase noise can be seen to approach a limit for integration above/„ > Mr. m
dB [ # < / „ , ) ]
•50
-70 -90,
10
3
10
5
Figure 3.24 Calculated dB(3?(/„)) versus /» based on assumption of white noise.
110
40 r Delay = 1234 us (100 nmi) Delay = 123.4 us (10 nmi) Delay = 12.34 us (1.0 nmi)
1 1
mrTTn
111
n
• •' • M i u w n
nTi
1 1 1 1 1 11 1 1 1 1
ii
o
Max integration frequency in Hz Figure 3.25 Cumulative phase noise for white-noise approximation.
3.9.6 Frequency Stability in Terms of Allan Variance A second method for evaluating cumulative phase noise is from the Allan variance definition of frequency stability. The Allan variance for stable oscillators is a time-domain measurement commonly used for specifying long-term frequency stability, which often is not applicable for the relatively short delay times associated with radar. However, frequency deviation for some frequency sources is also available for averaging times as small as a millisecond, which is on the order of the PRI of many radars. The Allan variance [6] is an expression for frequency variance in terms of repeated samples of frequency deviation, averaged over a time interval T. The true variance (infinite number of samples) of a random variable x is expressed according to the relationship [7] 2
2
a* = E(x ) - E (x)
(3.108)
A sample estimate of variance called sample variance, defined for n sample values x, is expressed as
of
(3.109) The expected value of the variance, based on the sampled estimate, can be shown [8] to be related to the true variance according to the expression £[(*)>] =
(3.110)
Thus, the estimate of variance from (3.109) can be said to be biased by a factor {n - 1)/ n from the true variance. The estimated true variance in terms of sample variance is then
n- 1
(3.111)
E[(o)]]
Allan defines average frequency deviation during the interval from t to / + T as
At. r) =
<Mt
+ r) - <M)
(3.112)
2TTT
s
where, as for (3.86),
3T+T
T is averaging time T is sampling interval
Figure 3.26 Allan variance.
6. Center frequency denoted b y / i s not to be confused here with average frequency deviation/((, r) from center frequency.
1)2
a'\n, 7, r) = ^ { l l / O T . r ) )
2
(3.113)
If n — 2, (3.113) becomes
2
a\2, 7. r) = (1/2)|7(0. r) -f(T. T)}
(3.114)
This definition represents one estimate of frequency variance during time interval r, based on two samples / ( 0 , r) and 7(7, r) of the frequency deviation, separated by the sampling spacing 7. The measurements indicated in (3.114) are conveniently made by connecting the output of the two identical sources to be measured to a mixer followed by a low-pass fdter. A precise digital counter is used to measure the time r required for some set number of periods of the low-pass filter's output voltage. Average phase drift
cr\2, 7, r) 2
7
(3.115)
where / is the long-term average frequency of the source. The standard deviation (rms frequency noise) of a radar's transmitter frequency can be estimated by multiplying the fractional standard deviation a, of the frequency source with the transmitter's center frequency. This assumes that the only source of frequency fluctuation is the frequency source itself. Frequency multipliers or other transmitter components may further degrade spectral purity to some extent. Further degradation also occurs due to phase pushing and pulling, which is discussed below. The standard deviation of fractional frequency for several averaging times is quoted for a typical frequency synthesizer in Table 3.3. The value of 7 is not provided, but results are not expected to depend significantly on time between sample pairs.
113
Table 3.3 Fractional Frequency Deviation of a Typical Synthesizer T
°> 1.5 X 1.5 x 5 x 5 x 5 x 1x 5 x
10"' sec 10* sec 10"' sec 1 sec 10 sec 100 sec 24 hr
10-'° 10-" 10" I0" 10" 10-" 10IJ
10
3.9.7 Cumulative Phase Noise From Allan Variance From the fractional Allan variance we obtain the average rms frequency noise cr = aj about transmitted carrier frequency / . The corresponding cumulative phase noise of the transmitted frequency for noise-free frequency translation is a = va= lirraj
(3.116)
c
where *>is the phase constant 27TTat delay r. Consider the following application. A 94-GHz spaceborne radar is to provide SAR mapping of a planet's surface at ranges of up to 10*m. The frequency stability of the radar is expected to be established primarily by a stable oscillator for which the fractional frequency deviation is specified as in Table 3.3. It is determined that the rms phase noise needs to be held to less than 15 deg (0.262 rad) to achieve the desired image quality in terms of contrast, which is defined as the ratio of peak signal to phase noise floor. The problem is to determine if the synthesizer of Table 3.3 is adequately stable to meet the contrast requirements. To determine needed fractional rms frequency deviation, we refer to (3.116) to obtain cr^-^-z
(3.117)
6
Range delay at 10 m is 6.67 ms. For this delay and the phase noise criteria of cr = 0.262 rad, we obtain the maximum tolerable fractional frequency deviation as c
0
2
6
2
3
. = 6.7X10-
2TT(6.67 x 10' )(94 x 10')
which appears to be roughly the limit of the synthesizer performance of Table 3.3.
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3.10 FREQUENCY SYNTHESIZERS Modern frequency synthesizers make it possible for the radar designer to design radars that tune pulse to pulse over a wide band of discrete frequencies and operate with precisely defined but adaptable pulse or continuous waveforms of all forms.
3.10.1 Direct and Indirect Synthesizers Two basic classes of frequency synthesizers are the direct and indirect types. For wideband radar applications, the direct synthesizer has been of greater interest because it is able to switch from frequency to frequency in a short time as compared with typical radar PRIs. Indirect synthesizer outputs are generated from tunable oscillators that are phase locked in frequency increments related to a stable reference. The phase-locking process typically requires from 0.5 to 5 ms to establish a lock for each new frequency. The typical direct synthesizer, by comparison, can change frequencies in less than 1 LIS. Output frequencies produced with the direct method can be synthesized from a basic reference signal by selecting combinations of frequency addition, subtracting, multiplication, and division using mixers, multipliers, and dividers. Actually, the direct synthesizer was developed earlier. R. R. Stone of the U.S. Naval Research Laboratory (NRL) developed it in 1949 [9]. Hewlett-Packard (HP) built the 5100 series of direct synthesizers as part of their line of general-purpose test equipment in the early 1960s [101. The first ISAR images of ships and aircraft were generated in 1975 from radar data collected at the Naval Ocean Systems Center (NOSC) in San Diego using a pulse-to-pulse stepped-frequency radar waveform obtained using a 5100 series frequency synthesizer. It is worthy of note that the indirect synthesizer was developed after the direct synthesizer in order to reduce spurious response and phase noise, which were problems with the early direct synthesizers. The indirect synthesizer developed later was also less costly than the early add-and-divide type of synthesizer developed by Stone of NRL. Later, R. I. Papaieck developed a binary-coded-decimal (BCD) type of direct synthesizer [11], which combines most of the advantages of direct and indirect synthesizers. Both the add-and-divide and BCD direct synthesizer designs require a set of comb frequencies. Output frequencies for the add-and-divide design are synthesized from selections from 10 comb frequencies using divide-by-ten decade units. Output frequencies for the BCD design are synthesized from selections of two comb frequencies. Basic operation of the Stone and Papaieck types of direct synthesizers is illustrated in Figures 3.27 and 3.28, respectively. These are greatly simplified design examples, both producing only 1,000 frequency combinations. Each example provides synthesized output frequencies of 20.000 to 20.999 MHz in 1,000-Hz steps.
115
COMB FREQUENCIES I, — — 18.0 18.1
18.9 MHz
MATRIX SWITCH FOR COMB FREQUENCY SELECTION TO DECADES
l„,=2 MHz OUTPUT FREQUENCY (20.000 TO 20.999 MHz)
TO OBTAIN LOWEST TO OBTAIN HIGHEST DECADE OUTPUT FREQUENCY OUTPUT FREQUENCY COMB FREQUENCY SELECTIONS
»1 »2 13
18.0 MHz 18.0 MHz 18.0 MHz
DECADE #1 OUT DECADE »2 OUT DECADE # 3 OUT
(18t2)+10 = 2 (l»*2) + 1 0 » 2 (18*2) 120
OUTPUT FREQUENCY
20.000 MHz
TO OBTAIN 20.583 MHz
18.9 MHz 18.9 MHz 18.9 MHz
18.3 MHz 18.8 MHz 18.5 MHz
(I8.9.2)+10 = 2.08 (18.9.2.09)tI0 = 2.099 (18.9.2.099) =20.999
(I8.3.2) 10 i 2.03 (18 8 . 2 03) + 10= 2.083 (18.5.2.083) =20.583
20.999 MHz
20.583 MHz
T
Figure 3.27 Add-and-divide direct synthesizer example.
3.10.2 Add-and-Divide Design (Stone) Examples of comb frequency settings for three output frequencies are shown in Figure 3.27 for the add-and-divide type. A set of 10 comb frequencies for this example are spaced by the frequency increment A / = 0.1 MHz. This is the increment of the lowest order decade (decade #1). A value for the input frequency/„ was selected to result in the desired output base frequency of 20 MHz. Assume that by suitable filtering, only sum frequencies are allowed out of the mixers. The lowest frequency, 20.000 MHz, will then occur for/„ = 2 MHz when the block of n = 10 comb frequencies are selected according to the expression
116
^^CpUB
FREQUENCIES^^
(, = 37 MHz
f, = 38 MHz
t,„ = 20 MHz
x4
•A< 38
°A' 37
37
38
38 37
37 38
- BINARY DIGIT
(a) SIGNAL DECADE. 37
= 20 MHz
OUTPUT FREQUENCY (20.000 FOR SWITCHES AS SHOWN)
38
BCD DECADE #1 110 0
37
20.3 MHz
12 4 8
38
BCD DECADE #2 0 001
37
20.83 MHz
38
BCD DECADE #3 10 1 0
12 4 8
12 4 8
OUTPUT FREQUENCY (20.583 FOR BINARY CODES AS SHOWN)
(b) THREE CASCADED DECADES.
Figure 3.28 BCD direct synthesizer example: (a) signal decade; (b) three cascaded decades.
/ = / o + /A/. / = 0 , 1 , 2 = 18.0, 18.1
9
(3.118)
18.9 MHz
These selections are consistent with Stone's formula [9]: fo +/.
, — /in
(3.119)
Other comb and input frequency combinations can be found to produce the same output frequencies. Actual designs are based on practical considerations such as the ability to
117
filter spurious responses. Filters, which pass only the sum-frequency outputs of the mixers, are not shown in Figure 3.27. Additional decimal places for increments of output frequency selection are possible by adding more decades in the series. 3.10.3 Binary-Coded-Decimal Design (Papaieck) The BCD synthesizer of Figure 3 . 2 8 uses three BCD decades to synthesize the same 1,000 frequencies between 2 0 . 0 0 0 and 2 0 . 9 9 9 MHz, as shown for the add-and-divide example in Figure 3.27. This is done with the BCD design by using only two comb frequencies, selected according to the formulas /,=
1.8/„ + 1 0 A /
(3.120)
and JWD+10A/
(3.121)
where/„ is both the input frequency and the lowest output frequency, and A/is the singledecade frequency-step size. The two comb frequencies for our simplified design example are/o = 3 7 MHz and/, = 3 8 MHz with A / = 0.1 MHz. The BCD settings that result in the lowest output frequency of 2 0 . 0 0 0 MHz for a single decade are indicated in Figure 3.28(a). The settings correspond to the binary number 0 0 0 0 . This and other frequencies can be checked by following through the single decade with the switches set according to four-digit binary numbers associated with the output frequency to be selected. Filters following each of the mixers allow only difference frequency outputs. The filters themselves are not shown. Binary settings for an output frequency of 2 0 . 5 MHz for the single BCD decade, for example, are 1 0 1 0 in the 1, 2 , 4 , and 8 binary digits, respectively (left to right. Figure 3.28(a)). This corresponds to the binary number 0 1 0 1 , which is 5 in decimal form (i.e., 5 of the 0.1-MHz increments above the lowest output frequency, 20.000 MHz). The highest output frequency for decimal stepping of a single decade is 20.999 MHz for binary settings 1001, the binary number equivalent to the decimal number 9. (A single BCD decade could actually produce 1 6 frequency increments, 2 0 to 2 1 . 5 MHz, if it were not to be used in the decimal system.) Other frequency settings are possible by setting in the corresponding binary numbers. Multiple decades in series add further decimal places of output frequency selection increments. Suppose an output frequency of 2 0 . 5 8 3 MHz were desired, as was the case for the add-and-divide synthesizer in the first example. The cascaded BCD decade binary settings in Figure 3.28(b) can be seen from left to right to be 1100, 0 0 0 1 , and 1010, respectively, corresponding to binary numbers 0 0 1 1 , 1000, and 0 1 0 1 , respectively. In decimal form, these are the numbers, 3 , 8, and 5 , which result in the 0 . 5 8 3 MHz added to 2 0 . 0 0 0 MHz.
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3.10.4 Direct Digital Synthesizer At the time of writing, the direct digital synthesizer (DDS) is rapidly becoming the preferred method of frequency and waveform synthesis for many applications, including high-resolution radar. The DDS generates fully synthesized, digitally controlled output signals with precise control of frequency, phase, and amplitude. Frequency can be changed and returned to a previous frequency without losing coherence. There is no frequency addition or subtraction by mixing or frequency multiplication and division except as needed by the user to multiply or up-convert the synthesized frequency for a specific application. The DDS, unlike the add-and-divide and BCD synthesizers, can generate pulse waveforms, such as chirp pulses, as well as discrete frequencies. Figure 3.29 illustrates a basic DDS that includes a frequency control word register, a phase accumulator, read-only memory (ROM) or random access memory (RAM), a digital-to-analog (D/A) converter, and required auxiliary circuit elements. The low-pass filter and clock are selected by users for their specific applications. The synthesized output frequency is set by the frequency control word. Frequency resolution is the clock frequency divided by the size of the accumulator. For example, a 32-bit accumulator clocked at 40 MHz can set up frequencies as finely spaced as (40 x lO )/! = 0.0093 Hz. Output frequency is frequency resolution times the frequency control word. A frequency control word of 2 for the above example produces a frequency of [(40 x lO )/^ ] x 2 = 10 MHz. From Nyquist's theory, the maximum frequency is one-half the clock frequency. Practical systems synthesize frequencies up to about 45% of the clock rate. Contents of the frequency control word register are accumulated in the phase accumulator. When the control frequency remains constant, the output of the accumulator is a contiguous series of phase ramps, each ramp scaled from 0 to 2 ir radians. Ramp length is the accumulator size divided by the frequency control word size. For the above example, the number of phase words per phase ramp is 2 /2 = 4 phase words. Each ramp is one 6
32
30
6
32
10
32
M
Clock | Phase accumulator
Sine map in ROM or RAM
Frequency control
Digital to analog converter
Output
1111111
Figure 3.29 Direct digital synthesizer. (Courtesy of Sciteq Electronics, Inc., from "Frequency Synthesize! Strategies for Wireless," Microwave J., June 1993, p. 26. Reprinted with permission.)
119
cycle of the synthesized output signal. Thus, the above four phase words composing one phase ramp occur at the above synthesized frequency of (40 x ltfJAt = 10 MHz. The ROM or RAM maps the contiguous series of phase ramps into a contiguous series of single-cycle sinusoids at the control frequency. The D/A converter converts the digital amplitude words into a continuous analog signal at the control frequency. Finally, the low-pass filter removes the clock frequency and other aliases. Chirp waveforms can be generated by adding another accumulator called a frequency accumulator before the phase accumulator. The output of the frequency accumulator represents the needed instantaneous frequency to produce phase control words representing the quadratic phase associated with the chirp waveform. System phase and amplitude equalization is possible. Phase compensation can be performed by an adder between the phase accumulator and the memory. Amplitude compensation can be provided by a digital multiplier function between the memory and the D/A converter. Theory and design of direct digital synthesizers is covered in detail by Goldberg [12]. 3.10.5 Summary Each of the above synthesizer approaches can be extended to cover any desired range of output frequencies spaced by any desired frequency increment. For example, the output of a frequency synthesizer that can step in 0.1-MHz increments over a frequency band of 50.0 to 59.9 MHz can be translated to a frequency band of 3.0500 to 3.0599 GHz that steps in 0.1-MHz increments by mixing with a stable 3-GHz source and passing the sum signal. This signal can in turn be doubled in a frequency doubler to 6.1000 to 6.1 198 stepped in 0.2-MHz increments. Furthermore, the entire frequency selection process can be digitally controlled or programmed to produce any desired sequence of frequencies. 3.11 TRANSMISSION LINES FOR WIDEBAND RADAR Radar systems usually use either coaxial transmission line or waveguides to conduct transmitter power to the antenna and signals from the antenna to the receiver. Coaxial line size is selected for transverse electric and magnetic (TEM) propagation, which is nondispersive unless there are multiple mismatches along the line, as discussed above. However, for longer lines, loss increase with frequency, unless equalized, can become a source of signal distortion in wideband systems. Low-loss coaxial runs of up to above 100 feet, well matched at each end, allow radar bandwidths of 25% or more at frequencies up to the top end of the S-band (3.70 GHz). High-power radars operating above S-band nearly always require waveguides for transmission over more than a few feet. This is because of the reduced peak-power handling capability of the smaller diameter coaxial lines required at higher microwave frequencies to avoid losses associated with propagation at unwanted higher order modes.
120
Waveguides also have the advantage over coaxial lines in having lower loss and better impedance matching over wide bandwidths. Unfortunately, for wideband radar applications, waveguides are dispersive because they propagate in the dispersive transverseelectric (TE) and transverse-magnetic (TM) modes instead of the nondispersive TEM mode. Of most interest to radar designers is the rectangular waveguide propagating in its lowest order mode, TE, - The propagation constant in radians per meter of rectangular waveguides operating below cutoff frequency in this mode is expressed as 0
(3.122) assuming a lossless waveguide, where/is the frequency of propagation,/ is the waveguide cutoff frequency, and v is the propagation velocity of the medium inside the guide. Distortion produced by transmission through waveguides can be evaluated in terms of waveguide insertion phase characteristics. As an example, insertion phase versus frequency was calculated from (3.122) for the WR-284 S-band waveguide. The cutoff frequency of this guide, when propagating in the TE, mode, is 2.078 GHz. Phase linearity will be examined over a 400-MHz bandwidth from 3.05 to 3.45 GHz. Table 3.4 lists values, for several frequencies in this band, of (1) the propagation constant from (3.122), (2) linear phase (with slope equal to the difference in WR-284 phase constants at the two band edges divided by the bandwidth), and (3) deviation from linear phase. The bandedge phase deviation can be seen to be 6.8 deg/m when the phase reference is zero at the band center. Phase deviation from a linear best fit is one-half this amount, ±3.4 deg/m at the band edges. For a 30m WR-284 waveguide run, the effective phase deviation will be about 100 deg. The paired-echo theory discussed above for this deviation would predict extreme distortion of a short RF pulse containing frequencies from 3.05 to 3.45 GHz. Only 2m of waveguide would retain ±6.8-deg band-edge deviation. From Figure 3.3, this amount of phase deviation would result in time-sidelobe levels of -25 dB if no other sources of distortion existed in the radar. Distortion predicted by paired-echo analysis is 0
Table 3.4 Frequency Dispersion in WR-284 Waveguide
Frequency (GHz)
Propagation Constant (deg/m)
Linear Phase (deg/m)
Phase Deviation from Linear (deg/m)
3.05 3.15 3.25 3.35 3.45
2,679.1 2.840.8 2,998.7 3,153.2 3.304.8
2,685.9 2,842.3 2,998.7 3.155.1 3.311.6
-6.8 -1.5 0.0 -1.9 -6.8
121
pessimistic, however, because phase deviation in waveguide transmission lines from the linear phase mostly consists of what is called quadratic-phase error, which will be discussed later in relation to pulse-compression systems. As much as ir/4 rad of quadraticphase error (measured at the band edges) can be tolerated by unweighted signals before significant distortion occurs. Weighted signals are even more tolerant because weighting reduces the effect of phase deviation at the band edges where the deviation is highest. Some methods for reduction of waveguide-produced distortion are the selection of a smaller waveguide size to operate farther from cutoff frequency, the use of waveguide equalization filters, and FM slope adjustment (on chirp-pulse-compression radars).
3.12 WIDEBAND MICROWAVE POWER TUBES The design of most moderate to high-power wideband radar systems, like their narrowband counterparts, is built around the venerable transmitter tube, which has its origins in the 1940s. Fortunately, for wideband radar designers, power tubes are available with wide bandwidths at all microwave frequency bands. Radars tend to be divided into two categories based on transmitter type: coherent radars and noncoherent radars. Coherent radars transmit signals by power amplification of an input RF drive signal. In other words, phase coherence is maintained through the transmitter. Coded waveforms are generated at low power, and LO signals for the receiver can be coherently related to the transmitted signal. Noncoherent radars transmit signals from power oscillators (usually magnetrons). The frequency depends on the power oscillator characteristics as well as the applied voltage and current. Waveforms for noncoherent radars are normally limited to either monotone pulse or monotone CW. Klystron power amplifiers are inherently narrowband devices, but stagger-tuned, linear-beam klystrons may have up to 10% bandwidth. The klystron was invented in 1939 by W. W. Hanson, R. H. Varian, and S. F. Varian. The traveling-wave tube (TWT) is a linear-beam tube, some types of which can operate over very wide bandwidths. The TWT was invented in 1940 by R. Kompfner [13]. This amplifier is characterized by the continuous interaction of an electron beam with a helix for low- to moderate-power applications, and coupled-cavity or other slow-wave structures for high-power applications. Octave bandwidths are possible with helix TWTs, but at relatively low power. About 10% to 30% bandwidth is possible at higher power with coupled-cavity TWTs. The magnetron power oscillator was the device that made microwave airborne radar possible during World War II, and was at one time so closely associated with radar that common microwave ovens, because they use magnetron output power to heat food, were sometimes called radar ovens. The magnetron was invented in 1921 by A. W. Hull, but the device was not used for radar until 1939 when J. T. Randal and H. A. H. Boot of the United Kingdom invented the resonant-cavity traveling-wave magnetron, which operated at about 0.1m. Raytheon Company in the United States developed the means for highvolume production of magnetrons at low cost. This made it possible for the United States
122
and the United Kingdom to field thousands of microwave radars for surface and airborne platforms during World War II. Pulse magnetrons can generate pulses as short as SO ns, equivalent to a range resolution of about 25 feet. Frequency-agile coaxial magnetrons can produce pulse-to-pulse frequency-agile bandwidths of up to 400 MHz at frequencies within X-band (8.5 to 10.68 GHz). Figure 3.30(a) shows a cut-away view of a tunable coaxial magnetron. Slow tuning is done by adjustment of the tuning piston. A frequency-agile magnetron operates by MAGNETIC FIELD LINES r COUPLING SLOT ' r ANODE RESONATOR VANE /-CAVITY MODE ATTENUATOR ! "N^TEon MODE ELECTRIC FIELD LINES
I
IJ^Li^V V
V/ ^ NV * /4P^'. ' !\ ' t i l
i.&jH^L y N,[
T E
0 1 1 STABILIZING CAVITY
[
OUTPUT WAVEGUIDE
—
V ^ OUTPUT VACUUM \ - L - .VACUUM BELLOWS - INNER CIRCUIT MODE ATTENUATOR 1
y
l
TUNING PISTON
CATHODE
(a)
CAM
MOTOR
RESOLVER
SLOW TUNE \L< (BROADBAND) (i x
POTENTIOMETERS -WELDED BELLOWS V
WAVEGUIDE OUTPUT
, , L ANODE RESONATOR VANES TUNING J I ^CATHODE PLUNGER T E o MODE CAVITY L
l n
<W Figure 3.30 Tunable and frequency-agile magnetrons: (a) tunable magnetron; (b) frequency-agile magnetron. (From J. R. Martin, Varian Associates, The Frequency Agile Magnetron Story, pp. 9, 10.).
123
driving the piston with a motor through a bellows, as shown in Figure 3.30(b). Typical tuning rates are 70 Hz for a 60-MHz tuning range. (One cycle of tuning takes the frequency from one end of the tuning range to the other and back.) Maximum tuning rates are lower at wider tuning ranges. In some designs, a servo motor provides control of the magnetron's frequency. Other frequency-agile techniques exist. While they do not possess signals of particularly wide instantaneous bandwidths, frequency-agile magnetrons can provide a radar with the advantage of improved detection and electronic counter-countermeasures performance. The potential for target imaging with frequency-agile magnetrons will be discussed in Chapter 9. The crossed-field amplifier (CFA) can produce microwave power levels that are high enough for a long-range air search. It is similar in some ways to a magnetron, but the CFA is an amplifier, not an oscillator. Characteristically, CFAs are relatively lowgain devices, but they operate at high efficiency and 10% or more bandwidth is possible. Amplitron amplifiers, also called backward-wave CFAs, were developed first and have the capability of a slightly wider bandwidth than the forward-wave CFAs. Amplitude and phase distortion occur in wideband signals transmitted through highpower transmitter tubes. Because power amplifier tubes are typically driven into saturation, the amplitude tends to be flat across the pulse width. It is, therefore, the phase ripple during the pulse or from pulse to pulse that is of most concern. Phase ripple produced by the ripple of the current or voltage applied to the tube is listed for various tubes in Table 3.1, obtained from Cook and Bernfeld [14]. The reader is referred to Chapter 4 of Skolnik's Radar Handbook [15] for more details on radar transmitters.
3.13 WIDEBAND SOLID-STATE MICROWAVE TRANSMITTERS When solid-state components emerged in the 1960s, they not only provided low-noise characteristics, small size, and low power requirements, but their flat response and low phase ripple simplified wideband radar-receiving system design. Higher power solid-state devices for use as imbedded active phased-array elements have since been developed for radar surveillance in the lower microwave bands [16]. Consumer applications of millimeterwave high-resolution radars have been tested experimentally with phase-locked gallium arsenide and indium phosphide devices as transmitters at up to 1W of pulse power. The radar's transmitter, whether in the form of a high-power microwave tube amplifier or a phased array of embedded lower power solid-state amplifiers, is today a major cost driver and prime power user. One alternative that may greatly reduce size, weight, cost, and power requirements of future radar transmitters is the electron-beamactivated diamond switch. This device can be configured in the form of a solid-state transmitter that possesses characteristics of conventional pulsed microwave tube transmitters in terms of gain, bandwidth, duty cycle, and power, but may be much more attractive in terms of size, weight, cost, and efficiency. The device, at the time of this writing, is
124
in the early stages of experimental development and testing by the ThermoTrex Corporation of San Diego. 3.14 WIDEBAND ANTENNAS Simple microwave antennas, such as waveguide horns, slots, and dipoles, provide adequate instantaneous bandwidth with sufficiently low phase and amplitude ripple for most highresolution radar applications. When fed by these antennas, parabolic reflectors or lenses and various fixed-array antennas also can be designed to possess adequate instantaneous bandwidth. The problem comes with phase-steered arrays. A planar phase-steered array is steered by generating a phase slope across the aperture electronically to scan the beam off normal to broadside. Unfortunately, the insertion phase for any propagation path length / of the aperture and feed system is -lirllX. The phase slope for a given steering command is therefore affected by the signal frequency. Spectral components of a wideband transmitted waveform are radiated over a range of beam positions about the steered beam position at center frequency. The effect in both transmit and receive modes is reduced antenna efficiency and degraded beamwidth as the scan angle toward broadside increases. With equal-line-length feeds, the percentage bandwidth limitation at a scan angle of 60 deg, based on frequency scanning less than one-fourth the local beamwidth, is given by Cheston and Frank [17] in terms of beamwidth as % bandwidth = 2 x (beamwidth in deg)
(3.123)
for modulo 2 7rphase-shift scanning. The corresponding limit of range resolution as defined in (1.1) in terms of absolute bandwidth can be obtained from (3.123) from known aperture beamwidth and center frequency. From a time-domain viewpoint with equal-line-length feeds, the radar range-delay resolution is limited to the aperture fill time defined [17] as (3.124) for aperture dimension d and scan angle
125
focal plane arrays, where fixed beams are selected by activating the needed elements in the focal plane instead of steering a single beam by adjusting delay or phase of all elements in the aperture array. Another solution is to operate the radar with pulse-to-pulse frequencystepped waveforms to allow time between pulses to adjust phase shifters to compensate for frequency sensitivity. PROBLEMS Problem 3.1 What is the instantaneous frequency within the envelope a(t) of a chirp pulse defined as s(t) = a(t) exp(j27r(// + KtVl)}? Problem 3.2 What is the insertion phase and the group delay at 3 GHz through a 20m, air-filled, TEMmode transmission line? Problem 3.3 Derive the expressions for the inband phase delay and group delay through a pulsecompression filter that has a transfer function given by
Problem 3.4 a
A 300-m/s target observed with a 10-GHz Doppler radar begins accelerating at 5 m/s toward the radar at time t = 0. (a) What is the instantaneous frequency of the echo signal at / = 0 and t = 10 sec? (b) What is the instantaneous Doppler frequency shift in each case? Problem 3.5
The total transfer function of a 1 -GHz short-pulse radar receiver measured from receiving antenna terminals to display input has the following steady-state amplitude and phase characteristics over its pulse spectrum:
126
A(a>) = 1 + 0.02 cos(8 x 10-"w) 4>(u) = -10-" u - 0.02 sin(8 x 10""w) What are the paired-echo amplitudes and delay positions relative to the main response to a 10-ns echo pulse from a point target? Problem 3.6 What is the delay position of the main response seen at the short-pulse radar display of Problem 3.5 relative to the antenna terminals, based on (3.12) (expressed in angular frequency at) and on (3.21)? Problem 3.7 Show that the value c in (3.17) with respect to the number of cycles of ripple across bandwidth B of the transfer function A(co)e' is given by c = (number of cycles of ripple)//3. M
Problem 3.8 A short-pulse radar receiver with a 1-GHz center frequency has linear-phase response, but 20 cycles of sinusoidal amplitude ripple appear in the receiver transfer function over its 10% bandwidth. The amplitude of the ripple is 20% of the average amplitude response. What are the amplitudes and positions of the resulting paired echoes relative to the main response to a short pulse at a 1-GHz center frequency with 5% bandwidth? Problem 3.9 Compute the maximum allowable amplitude ripple in a network that has zero phase ripple and the maximum allowable phase ripple in a network that has zero amplitude ripple if the sidelobes of the output response for each network are to be at least 46 dB below the peak. Check the results with Figure 3.3. Problem 3.10 A short-pulse radar receiver is to use the transversal equalization filter shown in Figure 3.4 to reduce time sidelobes introduced by phase and amplitude ripple in the radar system, (a) How many divider outputs are required to cancel five prominent time sidelobes? (b)
127
If the input peak-to-sidelobe level for the highest sidelobe is 15 dB, what is the output peak-to-sidelobe level, assuming that the filter itself is distortion-free? Problem 3.11 A radar transmitter is connected to an antenna by a long transmission line. The input VSWR to the line varies from 1.0 to 1.4, and the output VSWR varies from 1.0 to 1.6 over the radar frequency band. What is the maximum possible phase deviation from linear? Assume zero loss. Problem 3.12 What is the maximum possible phase deviation from linear phase versus frequency produced by a long transmission line? Given are the following VSWR conditions: (a) Input VSWR = 1, output VSWR = 2; (b) Input VSWR = 2, output VSWR = 1; (c) Input and output VSWR = 2. Assume zero loss. Problem 3.13 Show that the expression for the video pulse that has a spectrum given by S(f) = rect(// B) (where rect(///3) = 1 for |(///3)| 1/2 and zero elsewhere) is 5(f) = (sin
ir/3t/(7rt))
Problem 3.14 Show that the expression for the RF pulse at carrier frequency / that has the spectrum S(f) equal to rect[(/-/)//J] is given by ., > sin irBt s(t) = e' f — 2
rrt
Assume that rect[(/ - f)ip\ = 1 for | ( / - ~f)ip\ < 1/2 and zero elsewhere. Problem 3.15 Show that the expression H(f) = recl( f//3) (rect(/7/7), defined as in Prob. 3.13) is the transfer function of a filter matched to a video pulse expressed as
128
sin irBt
Problem 3.16 A rectangular pulse at a carrier frequency / is expressed as 2
s(t) = rectCOe' "?' where rect(/) = I for |/| < 1/2 and zero elsewhere. Show that the normalized response of a filter matched to this pulse is given by h(t) = s(t). Problem 3.17 (a) Write the integral expression ^ r , 0) for the ambiguity surface of a rectangular pulse for f = 0. Let the rectangular pulse be represented by rect(/) = 1 for |/| < 1/2 and zero elsewhere, (b) Plot the graph of X(T, 0). D
Problem 3.18 A square-law detector is to be designed for envelope detection of microwave pulses of 10-ns duration at the half-power points. What are the approximate band-edge frequencies of the output Hlter that is matched to the detected video pulse? Problem 3.19 Two sinusoidal signals of voltages x, and x at frequencies f\ and / , respectively, are applied to a mixer operating in the square-law region. By using (3.53), show that the output spectrum contains the sum and difference frequencies of x and jt , and their second harmonics. Assume that x, = A cos lirft and x = B cos 27r/ /. 7
2
t
2
2
2
Problem 3.20 Show that the cubic and fourth-power terms of the current response to applied voltage cos lirft to a mixer produce the first and third harmonics, and the zeroth (dc), second, and fourth harmonics, respectively, of the input signal. Problem 3.21 J
A radar illuminates a lm target. The IF response is at 0.3V rms. The first target is replaced by a second target at the same range and the IF response goes up to 1.2V rms. Assume an ideal mixer and a linear receiver. What is the RCS of the second target?
129
Problem 3.22 An HRR radar uses a wideband square-iaw detector to detect target-range profiles at long range. A detected profile, when amplified and displayed linearly on a wideband oscilloscope, shows two major peaks at 2.2V and 1.1 V, respectively. What is the ratio of RCS at the corresponding two resolved target locations? Assume that the receiver is operating in its linear range, except for square-law detection. Problem 3.23 The radar system of Figure 3.13(a) illuminates a fixed target at range R. (a) In terms of R, what is the phase, relative to the reference signal, of the received signal represented by the / and Q outputs of the synchronous detector? (b) Express the individual / and Q outputs for output signal magnitude A. Assume that the normalized reference signal is exp(j27r//) and ignore all delay, except for two-way propagation delay 2R/c, Problem 3.24 Assume that an ideal (square-law) mixer is to translate to an IF those signals appearing within a radar's receiving system RF passband of 500 MHz, centered at 3.20 GHz. The LO frequency is at 4.55 GHz. What are the band edges of an ideal (rectangular response) bandpass filter at the down-converted IF output that provides nonspurious signal translation for spurious signal and LO signal harmonics below the second harmonic? Problem 3.25 Calibration of a coherent narrowband radar receiver is performed by collecting thermal noise l/Q data with the transmitter disabled and receiver input blocked. Noise is adjusted by receiver gain control to have peak values below the 5 V maximum levels for digitization. Averaged inphase and quadrature-phase video is found to be +0.1 V and -0.05V, respectively. During the operation, a 100-m target at some range R is seen at a video power level of / + Q = 0.13W. What is the apparent zero-velocity target size in meters squared produced by the / and Q bias at the same range with no target present? 2
2
2
Problem 3.26 Calibration data collected from the radar of Problem 25 is further analyzed to determine relative image power. The averaged value of the square of the bias-corrected inphase and quadrature-phase video voltage is found to be 1.51 and 1.43, respectively. The average value of the product of bias-corrected inphase and quadrature-phase video voltage is found
130
to be 0.23. What is the equivalent target size at range R of the images produced by gain and phase imbalance, respectively? Problem 3.27 A short-pulse radar is to obtain 4m range resolution, (a) What is the required sampling rate, in terms of complex sample pairs per second, to sample the baseband range profiles? (b) What is the required sampling rate, in terms of real samples per second, if the echo signals are square-law detected before sampling? Problem 3.28 Range-profile video data are sampled and digitized with a 4-bit A/D quantization. What is the maximum dynamic range in decibels that can be sampled in terms of relative signal along the range profile? Problem 3.29 A ship model 5m in length sits on a turntable in an anechoic chamber. Frequency-domain reflectivity measurements are to be made at small increments in angle as the model is rotated through 360 deg relative to the radar. What is the maximum frequency-step size and minimum number of steps required to be able to obtain 0.1m range resolution unambiguously at all rotation positions? Problem 3.30 A steady target at a fixed range exhibits a 15-deg sector of rms phase noise seen on a polar display of the l/Q output. Radar PRF is 5,000 pulses per second in a pulsed-Doppler mode, (a) What is the expected reduction in signal-to-thermal-noise ratio, due to phase noise, of the response to the target obtained by FFT processing of data obtained during each beam dwell of 10 ms? (b) What is the phase noise floor of processed data relative to the single-target response? Problem 3.31 The source of phase noise in a 94-GHz coherent radar system is the transmitter, which at an offset frequency of 1 kHz from the carrier has a one-sided power spectral density ^ ( / • ) of ~60 dBm. What is the signal-to-thermal-noise loss and signal-to-phase-noise floor produced by cumulative phase noise on echoes from targets at 25 nmi, assuming white noise for FFT processing of 128 pulses?
131
Problem 3.32 What is the standard deviation of frequency during an averaging time of 1 sec of an ideal 5.4-GHz transmitter driven from a 100-MHz frequency source multiplied up to the radar frequency? Assume that Table 3.2 is applicable for the frequency source. Problem 3.33 (a) What are the 10 comb frequency settings that must be made available to each decade of an add-and-divide synthesizer design that generates outputs of 10.0000 to 10.9999 MHz from a 1-MHz input? (b) How many decades are required? Problem 3.34 What are the binary digit settings in the order 8421 that will produce an output signal of 20.8 MHz from the synthesizer of Figure 3.28(a)? Problem 3.35 What is the control word in binary form for an 8-bit DDS clocked at 50 MHz that produces the nearest frequency to 10.5 MHz? What is the deviation from 10.5 MHz? Problem 3.36 Use the definition of group delay (3.12) and the expression (3.122) to show that the group delay per unit length of a rectangular waveguide operating below cutoff in the TE mode is given by l0
where f is the cutoff frequency and v is the velocity of propagation in the medium inside the guide. c
Problem 3.37 Fractional peak ripple voltage on the pulse modulator of a 10-GHz TWT amplifier is 1%. Assume that the TWT electrical length is about 15A and that several cycles of voltage
132
ripple appear during the transmitted chirp-pulse interval. What is the paired-echo sidelobe level? Problem 3.38 What is the delay dispersion through 30m of the waveguide of Table 3.3 over the 400MHz frequency range? Problem 3.39 A 30m x 30m planar array scans 60 deg off broadside in one dimension. Compare the maximum resolution possible based on (3.123) and (3.124). Assume that broadside beamwidth is Aid rad for aperture dimension d, REFERENCES [II MacColl, L. A., unpublished manuscript, cited by C. R. Burrows in "Discussion of Paired-Echo Distortion Analysis," Proc. IRE (Correspondence), Vol. 27, June 1939. p. 384. [2] Wheeler, H. A., "The Interpretation of Amplitude and Phase Distortion in Terms of Paired Echoes," Proc. IRE, Vol. 27. June 1939. pp. 359-384. [3] Reed, J., "Long-Line Effect in Pulse-Compression Radar," Microwave J.. Sept. 1961, pp. 99, 100. [4) Churchill, F. E., G. W. Ogar, and B. J. Thompson, "The Correction of I and Q Errors in a Coherent Processor," IEEE Transactions on Aerospace and Electronic Systems, Vol. AES-17, No. 1, January 1981. (5) Lepoff, J. H , "Spurious Responses in Superheterodyne Receivers," Microwave J., June 1962, pp. 95-98. |6) Allan, D. W„ "Statistics of Atomic Frequency Standards," Proc. IEEE, Vol. 54, No. 2, Feb. 1966, pp. 221-230. [7J Papoulis, A., Probability, Random Variables, and Stochastic Processes, New York: McGraw-Hill, 1965. p. 144. [8] Papoulis, A., Probability, Random Variables, and Stochastic Processes, New York: McGraw-Hill, 1965, p. 246. 19) Stone, R. R., Jr., and H. F. Hastings, "A Novel Approach to Frequency Synthesis." Frequency, Sept. 1963, pp. 24-27. [10] " H P Direct-Type Frequency Synthesizers, Theory, Performance and Use," Frequency Synthesizers, Hewlett-Packard Application Note 96. Jan. 1969. [II] Papaieck, R. J., and R. P. Coe, "New Technique Yields Superior Frequency Synthesis at Lower Cost," Electronic Design News, 20 Oct. 1975, pp. 73-79. [12] Goldberg. B. G., Digital Frequency Synthesizers, Englewood Cliffs, NJ: Prentice Hall, 1993. [13] Gilmour, A. S., Jr., Microwave Tubes, Dedham, MA: Artech House, 1986. (14] Cook, C. E. and M. Bernfeld. Radar Signals, New York: Academic Press, 1967, p. 395 (republished by Artech House in 1993). [15] Weil, T. A., "Transmitters," Ch. 4 in Radar Handbook, 2nd edition, M. I. Skolnik, ed., New York: McGraw-Hill, 1990. [16] Borkowski, M. T„ "Solid-State Transmitters." Ch. 5 in Radar Handbook, 2nd edition, M. I. Skolnik, ed.. New York: McGraw-Hill, 1990. [17] Cheston, T. C , and J. Frank, "Phased Array Radar Antennas," Ch. 7 in Radar Handbook, 2nd edition, M. 1. Skolnik. ed.. New York: McGraw-Hill, 1990, p. 7.51.
Chapter 4 High-Range-Resolution Waveforms and Processing 4.1 INTRODUCTION The genesis of wideband radar came about at the end of World War II when the peakpower limitations of microwave-transmitting tubes were beginning to manifest themselves. There appeared to be a growing gap between the requirements of long-range detection and high resolution. In order to achieve the high resolution, shorter pulses were employed with the result that less energy was being transmitted per pulse. The need for unambiguous range measurement prevented raising the PRF, so that increasing the peak power seemed to be the only available option. The dilemma began to be resolved when it was realized that range resolution need not be limited by pulse length. If the frequency of ihe carrier, which usually had been constant, were instead varied over some frequency bandwidth, this bandwidth would determine range resolution, according to (1.1) of Chapter 1, written as (4.1) where fi is the frequency bandwidth and c is the propagation velocity. In principle, the range resolution can be made arbitrarily small by transmitting a signal of large enough bandwidth. The pulse length can then be stretched as much as necessary to radiate the energy required to detect distant and small targets without losing resolution. Consequently, microwave power tubes can be operated at the relatively high duty factors at which they tend to be most efficient. This is even truer of the solid-state power sources that are now beginning to supplant thermionic tubes. Also, high operating voltages, which had previously been a source of unreliability and even danger in the operation of tubes, could now be kept within manageable bounds. One of Ihe pioneers of the new type of radar, 133
134
apparently imagining himself to be able to hear both a typical short pulse as well as the new FM one, wrote a Bell Laboratories memorandum entitled: "Not With a Bang, But With a Chirp!" (B. M. Oliver, Bell Laboratories, 1951). This was the first use of the term chirp to describe linear FM of pulses for pulse compression [1]. To this day, chirp radars remain an important class of high-resolution radar. However, more recently, wideband processing to achieve HRR is carried out by using a variety of waveforms in addition to linear FM within each transmitted pulse, as is done in chirp radar. Waveform selection for any radar design is closely tied to transmitter type. The simplest type of radar transmitter is probably the magnetron oscillator. Magnetron radars are called noncoherent radars because the transmitted signal is determined only by the oscillation characteristics of the magnetron. By contrast, coherent radar systems using power amplifiers, such as a TWT or klystron amplifiers, generate the transmitted signal by power amplification of an input RF reference waveform. We shall see in this and subsequent chapters that methods exist to collect wideband reflectivity data from targets using a number of categories of waveforms. Given below are some common types of radar categorized according to transmitter type and listed with likely waveforms to achieve HRR: • Fixed-frequency magnetron—short pulse; • Dithered magnetron—coherent-on-receive magnetron imaging (described in Chapter 9); • Wideband, CW power amplifier—discrete frequency coding and digital phase coding; • Low PRF, wide-instantaneous-bandwidth power amplifier—chirp pulse, phase-coded pulse, and stretch; • High PRF, wideband power amplifier—discrete pulse-to-pulse frequency coding. Table 4.1 lists six waveforms for providing HRR capability. The first four are briefly discussed further in this chapter. Then, chirp-pulse waveforms and associated pulse compression processing will be discussed in more detail. Pulse-to-pulse, stepped-frequency waveforms are discussed in Chapter 5. Not included in Table 4.1 is a class of waveforms referred to as impulse or ultrawideband waveforms. Their common characteristic is large fractional bandwidth. Pulses contain one-half to several cycles of RF, and bandwidth is usually greater than 1 GHz. Renewed interest has come about in recent years as transmitting devices have improved to the point where sufficient pulse energy appears possible at high enough average power for some military applications. Potential applications are target classification, clutter discrimination, and improved performance against stealthy targets. Research is still in the early stage. 4.2 SHORT-PULSE WAVEFORMS HRR using short transmitted pulses is possible with both coherent and noncoherent radars. In coherent systems, very short RF pulses have been generated by using even shorter
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video or RF pulses to drive ringing filters. The filter bandwidth is designed to correspond to that of the desired transmitted RF pulse, which then approximates the true impulse response of the filter. (See Fig. 4.1.) Resolution of three inches has been achieved for RCS diagnostics. Magnetron transmitters in noncoherent radars can be turned on and off rapidly enough to generate pulses as narrow as SO ns, which corresponds to about a 25ft resolution. An example of a search-radar application of short-pulse waveforms is the AN/FPS114 system developed at the NOSC in San Diego. In this radar, a fixed-frequency magnetron generates 110-ns pulses to provide adequate resolution for the location of individual small craft at sea from vantage points along suitable coastal sites for the purpose of missile test-range safety. 4.3 BINARY PHASE CODING Phase-coded waveforms consist of various forms of digitally controlled phase modulations on the transmitted carrier signal. The most common form is phased-reversal modulation (also called binary phase coding), wherein the phase of the carrier is switched between ±180 deg according to a stored digital code. The resulting echo signal can be processed by correlation with a stored reference of the code or by matched-filter processing. Both individual pulses and CW signals can be phase-coded. The range-delay resolution of these
BANDPASS FILTER (RINGING FILTER)
>
S H O R T RF PULSE RF PULSE ENVELOPE
POWER
IMPULSE POWER SPECTRUM
f
f IS CENTER FREQUENCY OF THE FILTER AND THE RF PULSE CARRIER FREQUENCY
B IS THE BANDWIDTH OF THE BANDPASS FILTER t, an/B IS THE RF PULSE WIDTH Figure 4.1 Generation of short pulses with a ringing filter.
137
waveforms is equal to the reciprocal of the bit rate. Figure 4.2 illustrates a common phasereversal modulation-demodulation technique for generating binary phase-coded waveforms from digital codes and converting the resulting echo signals into digital signals for matched filtering. Figure 4.3(a) shows a simple 3-bit binary code and the resulting phasereversal-modulated RF pulse. (Two cycles of RF per bit are illustrated. Actual waveforms would likely contain many cycles per bit.) Figure 4.3(b) illustrates the digital response to a single point target seen in one quadrature channel out of a digital filter, which is matched to the digitized version of the 3-bit transmitted waveform. Figure 4.3(c) illustrates the required processing of both quadrature outputs to produce compressed responses that are independent of the unknown phase delay of the target. Figure 4.3 illustrates the processing of an input signal from a single point target. An input signal, in general, is the range-extended response produced by reflections from the multitude of reflection points illuminated by the radar antenna beam. The echo signal DIODE CONTROL SIGNALS FROM DIGITAL CODE
REVERSAL MODULATOR
MODULATED CARRIER TO TRANSMITTER
CARRIER
ECHO SIGNAL IN
Y— DEMOD
Figure 4.2 Phase-reversal modulation and demodulation.
I TO DIGITAL MATCHED FILTER
138
+1 DIGITAL CODE
t -1
-1
PHASE-REVERSALMODULATED RF PULSE
Figure 4.3(a) Digital matched-filter processing illustrated for a 3-bit binary code: 3-bit code and waveform.
from a single point target is a delayed replica of the transmitted waveform reduced in size. The signal from a range-extended target can be thought of as containing many overlapped replicas of the waveform. The output response to an echo signal s,(t) produced by a transmitted binary phasecoded waveform can be determined by carrying out the discrete version of the convolution process: sJLt) = h(t) * s,0)
(4.2)
where />(/) is the matched-filter impulse response and the asterisk denotes convolution. A single coded pulse at carrier frequency/, code length n, and bit length t,, can be defined in complex form by the expression: (4.3) where b(t) = 1, iti S / < (i + l)f,
(4.4)
= 0, elsewhere and where fa = 0 or ir, according to the code sequence. Practical digital codes may be many bits in length. The Barker Codes [2] are called optimum codes because, for zero-
139
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t TRANSMITTED CODE
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Figure 4.3(b) Digital matched-filter processing illustrated for a 3-bit binary code: digital matched filtering
Doppler shift, the peak-to-sidelobe voltage ratio following matched filtering is ±n for all code lengths, where n is the number of bits in the coded waveform. Nine Barker codes have been developed with code lengths of up to 13 bits. An 8-bit phase-coded pulse waveform is illustrated in Figure 4.4 along with its matched filter and the output response to the echo signal from a single point target. (An actual waveform would likely contain many cycles per bit instead of the one cycle per bit shown for simplicity.) We can see that the compressed pulse width t, for this ideal representation is equal to the bit duration. Also, we can easily show that the response of the filter to an impulse is the time inverse of the coded waveform shown above the filter. The matched filtering of Figure 4.4 occurs at a carrier frequency, typically at an IF, in contrast with that for Figure 4.3, where matched filtering is performed digitally. The process of discrete convolution of the signal from a single point target is illustrated in Figure 4.5 in terms of the binary bits associated with the phase-reversal modulation of the binary phase-coded pulse of Figure 4.4. We can observe that the folding
140
Bl-POLAR I
-»t
l-CHANNEL DIGITAL MATCHED FILTER Bl-POLAR Q
-*• t
COMPRESSED I
Q-CHANNEL DIGITAL MATCHED FILTER
12 + Q2 COMPRESSED Q •
Figure 4.3(c) Digital matched-filter processing illustrated for a 3-bit binary code: quadrature processing.
aspect of convolution, as also seen from Figure 4.3(b), comes about because the first bit of the code arrives first. This is also illustrated in Figure 4.5, where we can see that the bit stream moves across the matched filter "first bit first." For simplicity, Figure 4.5 illustrates only one quadrature component of the convolution so that h(t) = s (-t), which is the time reversal of s (t) without the conjugate. The set of s,(t) values in the lower part t
t
142
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of Figure 4.S are the convolved values at each time displacement through the matched filter. For illustration, we show the individual products at shift 4, which are summed to produce the convolved value J„(4/|) = - 2 . The pulse-compression ratio of a binary phasecoded pulse is simply the code length n = 8. Binary phase-coded waveforms can be conveniently generated digitally. One disadvantage is their intolerance to Doppler shift. FM-derived polyphase codes [3] have the advantages of chirp-pulse waveforms, such as Doppler tolerance to be described below, while also lending themselves to digital generation and processing for narrowband applications. 4.4 CONTINUOUS DISCRETE FREQUENCY CODING Discrete frequency coding to improve resolution is a concept that was known early in the evolution of radar. A recent development uses the CW transmission illustrated in Figure
143
4.6(a), whereby the frequency is shifted from one time segment to another in a periodic manner. In this way, the problem of transmit-to-receive leakage associated with singlefrequency CW radars is reduced. The received signal at any instant is likely to be composed mostly of echoes at frequencies offset from the transmitted frequency. Further, because the waveform is coded in frequency, it is possible to measure the target range delay associated with a given frequency segment in a manner similar to that used for pulsed radar. Waveform segments are synthesized from a common stable oscillator and are thereby coherently related. Multiple-frequency echo responses are processed to produce HRR responses in the matched filter by first applying appropriate delays to each of the n frequency segments to align the segments in time. Coherent addition of the n aligned responses then produces an HRR response. Actual processing is likely to be carried out following down-conversion to a convenient IF or to baseband. The symbols f to represent frequencies of the waveform segments before up-conversion to the transmitted frequencies. This type of waveform possesses some aspects of pulse compression, but it is also related to synthetic HRR to be discussed in Chapter 5. A stepped-frequency version can be defined in terms of a contiguous repeating sequence of waveform segments, each of time duration T stepped A/ in frequency in n steps over some repetitive period nT . A single period of n frequency steps for a contiguous periodic waveform can be described in complex form by the expression 0
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144
where b(t)= l.iT, < / < ( / + 1)7,
(4.6)
= 0, otherwise The filter matched to this waveform will have an impulse response h(l) that is the timereverse order of the conjugate of one period of the waveform. The impulse response, shown in Figure 4.6(b), can be expressed in terms of the nonrepeating function: h(t) = s\(-t)
(4.7)
where b(-t) = 1, iT, <. t < (i + 1)7",
(4.8)
= 0, otherwise Figure 4.7 illustrates a four-frequency periodic waveform with its matched filter. The duration of the response to a point target will be shown to be less than that of each frequency segment. This is in contrast to that for the binary phase-coded pulse, where the compressed-pulse duration is equal to the bit size t\. The action of the matched filter illustrated in Figure 4.7 can be explained by assuming that the individual bandpass filter responses to an inband segment are zero beyond ±T from the peak response. The delay lines in Figure 4.7 produce alignment of these responses so that signals at all n frequencies appear simultaneously in the summation network each nT,-sec period, and remain for 27, sec. It is convenient to match step size A/to frequency segment duration T, by setting A/equal to 1/7V As we shall see below, this criteria ensures that the matched-filter response approaches zero beyond the unambiguous range window associated with the waveform. Figure 4.8(a) illustrates the convolution of a contiguous, periodic, stepped-frequency waveform with its matched filter. This corresponds to the response, normalized to unity, to a fixed point target at zero range. Only the discrete shift positions are shown in the figure. A response occurs when the signal is shifted in time to positions where its set of discrete frequencies overlap in time with the corresponding set of delayed response functions of the bandpass filter. The overlap for each frequency is assumed to last for 27", sec and occurs every nT, sec, as discussed above. The matched-filter response during the overlap time is the coherent summation of the aligned signals of the n frequency steps. The response in complex form to a point target is expressed as {
146
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h(t) = s,(t) folded
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*.(0 = a ( f ) X
c J j
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(4.9)
i-O
where a(f) is the envelope of the output responses of each of the n bandpass filters. The responses peak at / = 0, nT 2nT referred to the output of the delay lines and are zero beyond ±T| from the peaks. Taking the constant frequency Jo term outside the summation, we have u
{
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*.(0 = a ( / ) e * ' £ e
il,rt,v
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)
(4.10)
The remaining summation term is the expression for a Fourier series, except that the summation is over a finite number n of frequencies. An identity used to analyze the response of linear antenna arrays [4] is given by . Ba sin
/M T* g ** = ~exPMfi- D/2]« 6
By using this identity with B=n,p=i
and a = 2-rrAft, we obtain
(4.11)
147
SIGNAL ENVELOPES AT INPUT TO BP FILTERS BP FILTER AT f„ . I—2T,-
SIGNAL ENVELOPES AT OUTPUT OF BP FILTERS
DELAY OF3T, I—
2T,—|
2T.
SIGNAL ENVELOPES AT OUTPUT OF DELAY LINES - 1 T, | — A A SIN rtnAft
n = 4
0 nT, ENVELOPE OF OUTPUT FROM E NETWORK Figure 4.8(b) Convolution of a signal from a single point target with the discrete frequency-coded periodic waveform of Figure 4.7: signals associated with the convolution process.
148
.
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The quantity f + (n - \)Af/2 is the average frequency, which can be thought of as the carrier frequency. Expression (4.13) for the matched-filter response to the point target can be rewritten as 0
,
sin rmAft sin
.„ -
Trujt
where / is the carrier frequency. The sin imAftl(s\n irA/r) term, without the bandpass-filter function a(t) and for 7", = 1/A/ will have peaks with magnitude n at t = 0, ±7*i, ±27",, ± 3 7 , , . . . Polarity is positive at t = 0, and depends on the value of n at the other peak magnitude positions. Resolution at the 2/7r(-4 dB) points is equal to the reciprocal of the waveform bandwidth nAf. The bandpass filter response term a(t) is assumed to be zero beyond ±7", from the time of the peak of the response from each delay line. Peaks in s (t), because of the bandpass filter function, occur at intervals of nT,, where n is the number of frequency segments in one period of the periodic sequence. Signals associated with the convolution process for a point-target input signal are illustrated in Figure 4.8(b). The matched-filter response for periodic, discrete frequency-coded waveforms can be characterized as follows. a
1. Peak response to an input signal from a fixed point target is n times the input signal voltage level. 2. Responses occur at delay intervals of nT,, where n is the number of frequency segments of length 7,. 3. An unambiguous range-delay window of nT, sec is obtained for 7, = 1/A/. 4. Delay resolution is l/(nA/) at the -4-dB points. All of the above characteristics would be also seen in a series of DFTs produced from sampled data obtained at each frequency within each period of the same periodic waveform. The DFT method for stepped-frequency pulse waveforms, to be discussed in Chapter 5, is called synthetic processing. An advantage of the continuous, discrete frequency-coded waveform is its inherent unambiguous response in Doppler and the independent ambiguity control in range. Because individual frequency segments are generated from a single stable source (e.g., by way of
149
a frequency synthesizer), the echo signal can be down-converted to a single CW signal, which is then unambiguous in Doppler. Range ambiguities can be independently controlled, at least conceptually, by adjusting the period length. A more recent variant of this waveform, now under development, is a pseudoperiodic version in which the frequency code is changed from period to period. In this variant, no range ambiguities appear at all, and a true thumbtack ambiguity response exists. The practical advantages of this approach are being investigated for target imaging and detection. 1
4.5 STRETCH WAVEFORMS An important class of waveforms used for modern high-resolution mapping and target imaging falls under a category called stretch waveforms. The stretch concept provides HRR by processing returns from linear FM, large time-bandwidth-product pulses called FM ramps. Processing can consist of down-converting the return from each ramp with an FM reference signal of identical or slightly different FM slope. The resulting spectrum for each ramp corresponds to a range profile. The original concept, developed by Caputi [5], involved a narrow-bandwidth pulse-compression filter to process returns containing residual FM after mixing with a reference signal of slightly different FM slope. Other implementations for long-range target imaging and synthetic aperture radar transmit pulse trains of FM ramps or a contiguous series of linear FM ramps. Returns are deramped with a reference signal containing identical linear FM, and no additional pulse compression is performed. A simple illustration for using the stretch waveform to obtain an HRR signature of a target is illustrated in Figure 4.9 for the return from one transmitted FM ramp. Stretch and related types of processing for SAR applications are discussed in some detail by Curlander and McDonough [6]. Processing methods covered in this reference are referred to as deramp FFT processing, step transform processing, spectral analysis (SPECAN) processing, and polar processing. The stretch process is a variant of chirppulse compression, the original and still most common form of which is discussed below. 4.6 CHIRP-PULSE COMPRESSION The term pulse-compression radar refers to the transmission of relatively long coded pulses and the processing of echo signals into high-resolution responses in a pulsecompression receiver. The transmitted energy can be either phase-coded or frequencycoded as discussed above, and coding can be generated by either digital or analog techniques. For HRR applications, only the analog frequency coding called chirp (including stretch) has been used to any significant extent. The more general term, pulse-compression radar, is therefore often used to refer to the chirp-pulse type of pulse-compression radar. I. See footnote 2 of Chapter 5.
JiO
TRANSMIT
ECHO FROM TARGET OF 3 SCATTERERS
TRANSMIT & RECEIVE PULSES
LOCAL OSCILLATOR OUTPUT
MIXER OUTPUT
TARGET SIGNATURE
Figure 4.9 Stretch waveform and processing.
The chirp-pulse waveform will now be analyzed in more detail because many of the general characteristics of pulse compression may be observed in this simple waveform. The chirp radar concept is described in detail by Klauder et al. [7], and the discussion here is based mostly on Klauder's classic analysis. Some of the figures and plotted curves, as indicated, have been obtained from that reference. A chirp-pulse waveform is illustrated in Figure 4.10. Figure 4.11 illustrates the action of the pulse-compression network to compress a received point-target response, which is a delayed version of the transmitted chirp pulse. Delay-versus-frequency behavior of a pulse-compression network is shown in Figure 4.12(a), and the envelope of the resulting unweighted response is shown in Figure 4.12(b).
151
Figure 4.10 Chirp-radar transmitted waveform: (a) transmitter pulse envelope; (b) transmitted pulse frequency versus pulse duration; (c) transmitted pulse RF waveform. (From [7], p. 750. Reprinted with permission.)
ECHO IN (FROM POINT TARGET)
PULSECOMPRESSION NETWORK
Figure 4.11 Pulse-compression network and response to a chirped input pulse.
COMPRESSED PULSE OUT
152
Z
-3/A
-2/A
FREQUENCY—-
-1/A
0 TIME
1/A
2/A
3/A
Figure 4.1Z Chirp-pulse-compression network characteristics and compressed pulse. (From [7], p. 751. Reprinted with permission.)
Analytically, the transmitted signal from a chirp radar with frequency-versus-delay slope K over pulse duration 7", can be expressed in complex form as (4.15)
where / is the carrier frequency and < 1/2
= 0.
(4.16)
i f - > 1/2
Chirp waveform phase is the argument of s,(t) of (4.14) given as
(4.17)
153
It can be seen that the phase of the chirp waveform varies quadratically with the time advance or delay from the pulse center. Frequency-versus-time behavior of the chirp signal is its instantaneous frequency, which, as defined by (3.2) of Chapter 3, is obtained from (4.17) as 1
-
dib
( 4 1 8 )
Within the pulse of duration T , the instantaneous frequency given by (4.18) changes from / - KT,/2 to / + KTJ2. This linear frequency sweep, after Klauder et al. [7), is symbolized by A, and a quantity called dispersion factor is symbolized by D, where t
Z>=r,A
(4.19)
Dispersion factor D is also called the time-bandwidth product of the waveform. (It should be noted that an uncoded pulse of duration T although A = 0, has a time-bandwidth product of about unity by virtue of its bandwidth B = \IT\.) u
4.6.1 Analysis Based on Phase Equalization The most common form of a pulse-compression filter is a phase equalizer, which equalizes the quadratic-phase response of the chirp pulse. The phase equalizer is equivalent to a delay equalizer, which equalizes the slope of linear frequency versus time of the chirp pulse. The transfer function of a phase equalization network, with delay-versus-frequency slope P, is written as H{f) = e'^'-tf
(4.20)
The amplitude characteristic of the equalizer's transfer function will, for now, be assumed to be unity. Instantaneous insertion delay (group delay) is obtained by differentiating the phase of (4.20) with respect to frequency according to (3.12) of Chapter 3, from which we can write 2
r,(f) = - j ^ W - / ) ] = - *•(/-/>
4 21
<- )
Note from (4.21) that insertion delay is zero when instantaneous chirp frequency / equals the carrier frequency f. The total transfer function of an actual radar system will result in additional fixed and dispersive delay through transmission lines and other components. Fixed delay produces linear insertion phase versus frequency and does not affect the shape of the pulse-compression filter's output pulse. Additional undesired dispersive delay will be assumed, for now, to be small.
154
Refer to Figure 4.10(b), where the frequency-versus-delay slope of the transmitted chirp pulse is K = A/TV By referring to Figure 4.12(a), we can see that the delay-versusfrequency slope of the pulse-compression network is P = 7VA. If P = A'" over band A, we have a pulse-compression filter. Only this case is treated, which relates to the concept of a matched filter in terms of phase response. An expression will now be developed for the compressed echo pulse from a point target when the delay equalization filter's delayversus-frequency slope matches the time inverse of the frequency-versus-delay slope of the transmitted chirp pulse. The complex spectrum £,(/) of the input echo signal s^t) is defined as 1
S,(J) = FTfo(0]
(4.22)
where the notation FT refers to the Fourier transform. The spectrum of the output signal from the pulse-compression network is given by « / ) = //(/)£(/)
(4.23)
where / / ( / ) is the network transfer function. Fourier analysis allows the output response s„'t) to be defined as sjtl)
= *(/)«,(/)
(4.24)
where h(t) is the network impulse response. For the normalized response to a single point target, Sj(t) = s^t). The point-target response can therefore be written as s„(t) = A(fW,(/)
(4.25)
The asterisk in (4.25), as before, denotes convolution, defined for convolving the two functions h(t) and s,(t) as h(t)*s,'t)
=
~ T ) j , ( T ) d T = £_5,(T)/l(/ - T ) d T
(4.26)
The impulse response h(t) is the inverse Fourier transform of the network's transfer function H(f). Therefore, the impulse response associated with / / ( / ) , expressed in (4.20) for P = K'\ is written 2
h(t) = £ / j ' ( / ) e ' ' * d /
(4.27)
155
This integral can be evaluated by converting to a form given by h(l) = j ^ e ^
, + w + c ,
d/
(4.28)
= J^_cos(a/ + 2bf + c)df + j £ s i n ( o f + 2bf+ c)df 2
which is evaluated in the table of integrals by Gradshteyn and Ryzhik [8]. The integral, according to Klauder, is also evaluated in Campbell and Foster [9]. The result by either method can be expressed, after some manipulation, as /,(/) = ^pe***-"^
(4.29)
The complex matched-filter output, given by the convolution (4.25) defined in (4.26) and written in terms of a point-target input signal equal to the chirp waveform of (4.15) and matched-filter impulse response (4.29), becomes
s (t) = £ . J i ( r ) A ( / 0
(4.30)
,
_ - (Jr. I
IT,
- r)dr
e
i2^/T-Kr /2+ri-r)/-*ril-r)';2] J . (
1
J-V2
After rearrangement of terms, we have
:„(,) = ^e^'^'J^e^'dr
( 43
,j
The integral term can be integrated as follows. ST/2
*JA ei2
dT
J- 2 **" = j JV
(cos 2nKtT + j sin 2irKfr)dT
1 r*™
J
27rKti-r
+
jO
(4.32)
sin irKtT, irKt Because we assumed that A = KT and recalling from (4.19) that time-bandwidth product D is T,A, the output of the matched filter from (4.31) with (4.32) becomes U
156
(4.33) The response envelope of (4.33) is simVA)/!
(4.34)
(TTA)/
This is the familiar (sin x)/x form of expression with x = 'irk)t. The normalized power at the half-power points of the envelope of (4.34) occurs for |sin(irA)j|
(4.35)
(TTA)/
which is satisfied by (wA)/ = 1.39. The time interval on each side of the peak response, then, is 1.39
0.443
(4.36)
so that the half-power compressed-pulse duration is /i = 2 T | .
(4.37)
3 d B
0.886
Compressed-pulse duration at the 2/IT points is t, = 1/A, which is the Rayleigh resolution. We will show later that frequency weighting reduces the (sin x)lx sidelobes, illustrated in Figure 4.12(b), and slightly increases the compressed-pulse duration. The increase in pulse duration occurs because of the effective reduction in bandwidth produced by weighting. Significant conclusions are given below. 1. The amplitude of the compressed pulse is increased over that of the input pulse by -V/D and the new pulse duration is about 1/A. The corresponding radar range resolution for swept bandwidth A is c/2A. 2. FM exists in the compressed pulse, but in the reverse sense from that of the transmitted pulse. (Compare the instantaneous frequency of the output s (t) from (4.33) with that of the input s,(t) from (4.15).) 3. The (sin x)lx output, when unweighted, results in peak sidelobes of 13 dB below the main response. 0
157
4. The input and output pulse envelopes, although they are both time functions, are related in form by a Fourier transform. We can show analytically that SNR is maximum for a pulse-compression network for which K = LIT, = \IP. The transmitted chirp pulse can be generated either passively or actively, as will be discussed below. 4.6.2 Effect of Rectangular Pulse Shape The above analysis of pulse compression in terms of a phase-equalization filter relates to matched filtering. However, as discussed in Chapter 3, an actual matched filter is defined as a network for which the impulse response h{i) and transfer function H(f) are related to the waveform time and frequency functions s,(t) and S,(f), respectively, as follows. H(f) = S\(f)
(4.38)
mo = *;(-<) The equalization filter defined by (4.20) and (4.29) does not fully meet these criteria because the effect of the rectangular pulse shape was not taken into consideration. The spectrum S,(f) of the finite-length chirp signal s,(t), given by (4.15), is a complicated function involving complex Fresnel integrals. The spectrum derived by Klauder et al. [10] is expressed, with some rearrangement, as
S,(f) =
"^^''"''"[^ + J**) ~
- J(z>)I 5
(4-39)
where C(z) and S(z) are the Fresnel sine and cosine integrals. The arguments z and Zi are defined as 2
(4.40)
The transfer function of the phase-equalization network, defined in (4.20) for P = K'\ can be seen to meet the criteria H(f) = S',(f) insofar as its phase response is concerned, but its amplitude does not contain the complex Fresnel integral functions associated with the rectangular pulse shape. A normalized form of the absolute value of the spectrum 5|(/) is
758
=
^
{
(
C
f
o
)
"
C ( Z
'
) ) J
+
[ 5 ( Z 2 >
"
S ( Z l )
^
m
(
4
-
4
l
>
It can be shown that is a function only of the factor D and ( / - / ) / A . The calculated spectra for three values of D are shown in Figures 4.13, 4.14, and 4.IS. Rectangular bandpass characteristics are shown comparatively for each value of D. We can easily show from the definition of a matched filter that, in terms of magnitude, a matched-filter transfer function and the normalized input spectrum to which it is matched are identical. Figures 4.13, 4.14, and 4.13 show that a rectangular passband is approached for large values of D. For smaller values of D, the mismatch to an idealized chirp pulse will result in reduced SNR. Figure 4.16, obtained from Klauder, illustrates this. We can see, however, that the degradation of SNR is small, even for relatively low dispersion. Thus, the approximation of a rectangular transfer function (sharp cutoff filter of width A) is normally attempted in the design of pulse-compression filters for systems where the chirp signal is typically a linearly swept, constant-level pulse. Complicated transfer functions with amplitude characteristics, such as those shown in Figures 4.13, 4.14, and 4.1S, are not normally needed. Amplitude weighting of the frequency response, to be discussed below, is often superimposed on the approximately rectangular response, usually by means of a separate weighting filter to reduce time sidelobes. The mismatch between an ideal rectangular chirp signal and a phase-equalization . network that is band-limited by a rectangular filter transfer function can also be viewed intuitively. The spectrum of the ideal chirp signal defined by (4.15) contains both the unlimited spectrum of frequency components produced by the assumed rect(f/7|) envelope
0
0.2
0.4 (f-i)/A
0.6
0.8
1.0
Figure 4.13 Spectral amplitude of a rectangular chirp signal and magnitude of the transfer function of its matched filler for D = 10.125 compared to a rectangular bandpass characteristic; shape symmetric about the point ( / - f V A = 0 (From [7], p. 756 (rectangular passband added and symbols modified). Reprinted with permission.)
159
1.5
I
IN O
RE CTANGUL AR SSBAND
ID
II Q
1.0
OC ui o 3 K -J
0.5
is,(>)-
OL
= |H(f)|
2
< 0.2
0.4
0.6
0.8
1.0
(f-f)/A
Figure 4.14 Spectral amplitude of a rectangular chirp signal and magnitude of the transfer function of its matched filter for D = 60.5 compared to a rectangular bandpass characteristic. (From [7), p. 757 (rectangular passband added and symbols modified). Reprinted with permission.) 1.5 RECTA NGULAR PASSB AND ll Q
111
a
1.0
1A A A A
0.5
3 1-
\
IS
=|H(f)| ,/"A
Zj a. 2 <
0.2
0.4
0.6
0.8
(J-f)/A
Figure 4.15 Spectral amplitude of a rectangular chirp signal and magnitude of the transfer function of its matched filter for D = 120.125 compared to a rectangular bandpass characteristic. (From [7], p. 757 (rectangular passband added and symbols modified). Reprinted with permission.)
and the desired frequency components of the linear FM spectrum. For a chirp signal of large time-bandwidth product, the desired spectrum of the linear FM dominates. The chirp signal of (4.15) can be generated by driving a voltage-controlled oscillator (VCO) with a time-varying drive voltage that results in the desired linear frequency ramp. This method is sometimes called active chirp generation. A chirp signal can also be generated by driving a linearly dispersive filter with a very short video or RF pulse. This
160
0.1
0.7 ' 0
' 40
1
1
80 120 DISPERSION FACTOR, D
1 160
1
200
Figure 4.16 Degradation in SNR from ideal maximum when the rectangular chirp signal passes through a delay equalizer and a sharp cutoff filter of width A; the degradation decreases rapidly as the dispersion factor D increases. (From (7), p. 772. Reprinted with permission.)
method is called passive chirp generation. The required pulse-compression filter for passive chirp generation can be an identical filter, but arranged in the receiver system in such a manner, to be described below, as to provide the inverse of the delay-versusfrequency function of the filter when used as the chirp generator. For this situation, exact matched filtering is achieved, regardless of how small the dispersion D is. 4.6.3 Weighting The idealized transfer function of a phase-equalization filter, expressed by (4.20), for a chirp bandwidth A, was shown above to produce a compressed pulse having a half-power
161
width of /, = 0.886/A. The SNR is maximum when the pulse is processed with a matched filter. A rectangular bandpass characteristic, while limiting receiver noise, was shown to result in a slight mismatch, even when band edges are set at the edges of the chirp FM. An actual radar system is band-limited according to the transfer function of the entire radar system, including propagation effects that vary with frequency. Normally, we attempt to provide flat frequency response over the entire chirp bandwidth, except for the weighting filter. Weighting filters are designed to modify the flat response to provide desired tradeoffs between SNR, time sidelobes, and resolution performance. An example of the amplitude response of a (raised cosine) weighting filter is illustrated in Figure 4.17. SNR degradation, seen with a delay equalization filter, is shown in Figure 4.18 for the case in which the bandpass response is that of the Gaussian-taper filter. It is assumed here that no other band-limiting is applied. SNR, relative to that for a lossless matched filter without weighting, reduces toward zero for Gaussian filters tapered so that they introduce nearly no loss at the chirp-pulse band edges. SNR increases slowly for increased weighting to a maximum, then slowly decreases as band-edge loss is further increased. Weighting can greatly reduce time sidelobes of the compressed pulse with a penalty of slight degradation in resolution and SNR performance. We can see this for band-limited Gaussian weighting from Figure 4.19, and for band-limited Gaussian and three other types of weighting from Figure 4.20. Distortion produced by system amplitude and phase ripple near the band edges is also reduced. 4.6.4 Hardware Implementation
The above analysis for chirp-pulse compression is quite general and implementations differ greatly. Relatively narrowband pulse-compression radars normally use delay lines made of very thin strips of aluminum or steel to produce frequency-dispersed pulses. Dispersion occurs at so-called acoustic wavelengths. Longitudinal acoustic waves, propa1.6
/
1.4
z
3 0.8 0.6 0.2
\
/
1.0
0.4
\
/
1.2
"
\
/
" \
-
0 t_ A 2
1 i FREQUENCY -»-
t + — +
2
Figure 4.17 Weighting-filler amplitude response (raised cosine). (From [7], p. 782. Reprinted with permission.)
162
Figure 4.18 Degradation in SNR from ideal maximum when the rectangular chirp signal passes through a delay equalizer and a smooth, Gaussian-taper Alter, which introduces a loss at the band edges of the chirp signal. (From [7], p. 775. Reprinted with permission.)
BAND-EDGE LOSS, dB(L)
Figure 4.19 (a) Increase in half-power pulse width versus Gaussian filter band-edge loss of a chirp signal passed through a Gaussian-taper filter, (b) the relative amplitude between the maximum of the first adjacent sidelobe and the central maximum of the output signal following shaping by a Gaussian filter. (From [71, p. 776. Reprinted with permission.)
gated from one end of a properly designed strip, disperse nearly linearly over about a 10% bandwidth at typically 5 to 45 MHz. Such lines can be used for pulse compression by translating the radar echo signal to an IP signal, then transducing to acoustic vibrations at one end of the line. The compressed output signal is transduced back to a signal voltage at the other end of the line for display or threshold comparison in a detection circuit. Bandwidths for these types of devices are limited to less than about 20 MHz, hardly in
163
Figure 4.20 Pulse widening due to weighting. (From [7], p. 784. Reprinted with permission.)
the category for HRR radar as discussed in this book. A number of relatively narrowband devices have also been developed to use nondispersive acoustic media. In this case, the dispersion is achieved by the arrangement of electric-to-acoustic signal transducer lines on the surface of the medium. Early HRR Designs. Early work with HRR radar was carried out by taking advantage of the frequency-dispersive characteristics of waveguides, both to generate wideband FM chirp pulses and to compress the received echo signals. A disadvantage is the requirement for long lengths of heavy and bulky waveguide. Producing differential delay over chirppulse durations greater than about 0.05 fts requires unpractically long guide lengths. Smaller dispersive devices have been developed for HRR systems since the earlier use of waveguides. One such device is the folded-tape meander line (FTML) [11]. In this device, a conducting tape is folded back and forth onto itself with dielectric spacers between the folds. The entire line is immersed in dielectric. Ground planes are placed above and below the line. The number of folds, their length, and the spacing of the folds and ground plane combine to determine the desired dispersive delay characteristic. An example, as shown in Figure 4.21, was used in an early U.S. Navy experimental radar [12] to compress 0.3-/xs pulses to 2.0 ns. This FTML required an amplitude-equalization filter. A combined equalization and weighting filter, shown in Figure 4.22, was one type that was used [13]. Performance is shown in Figure 4.23. SAW Devices. More recently, various types of surface acoustic wave (SAW) devices (Fig. 4.24) and bulk acoustic wave devices have been developed, which are capable of timebandwidth products of more than 5,000 and bandwidths greater than 1,000 MHz. These are usually very small devices, which can be easily duplicated once they have been designed. The use of FTML devices and waveguides for phase equalization has now
164
Figure 4.21 Folded-tape meander line type of phase equalization filter.
become outmoded due to these later developments and the more recent development of the direct digital synthesizer to be described below. Active and Passive Chirp Generation. The transmitted chirp pulse can be generated passively or actively, as indicated in Figure 4.25(a,b). The choice of technique depends on several criteria. Passive generation is relatively reliable, and the pulse-expansion filter can be identical to the pulse-compression filter if the expansion is done at an IF and
165
Figure 4.22 Amplitude weighting and equalization filter. (From [13]. Reprinted with permission.)
then mixed upward to the transmitted RF. Hence, the identical filter will perform pulse compression in the receiver at the opposite IF sideband. Passive generation of the transmitted pulse, however, is not always easy to achieve for HRR, because losses are high and circuits may be too delicate for an impulse of sufficient power to achieve a reasonable output SNR. The passive method also lacks the advantage of the active chirp method,
166
S ~Z
| p
3
GAUSSIAN WEIGHTING PLUS IDEAL EQUALIZATION'
i—i—i
r
-
DISPERSIVE DELAY-LINE LOSS 20 [
ui
RESPONSE OF EQUALIZED AND WEIGHTED DELAY LINE
1.0 1.1
1.2
1.3
1.4
1.5
1.6
1.7
FREQUENCY (GHz)
Figure 4.23 Amplitude equalization and weighting-niter performance. (From [13]. Reprinted with permission.)
OUTPUT TRANSDUCER ARRAY
OUTPUT
NONDISPERSIVE DELAY MEDIUM INPUT - r f "
INPUT TRANSDUCER ARRAY
GROUND PLANE
Figure 4.24 Surface acoustic wave dispersive delay line.
167
VOLTAGECONTROLLED OSCILLATOR
DRIVER
*• TO TRANSMITTER
PRF TRIGGER (a) A C T I V E C H I R P G E N E R A T I O N
IMPULSE GENERATOR
J RINGING *1 CM F I L TT EERD
PULSE EXPANSION FILTER
+• T O T R A N S M I T T E R
PRF TRIGGER
(b) P A S S I V E C H I R P G E N E R A T I O N
Figure 4.25 (a) Active chirp generation and (b) passive chirp generation.
which permits adjustment of the chirp slope to correct for quadratic-phase distortion (to be discussed below) in waveguides or from other components. Active generation, on the other hand, tends to be less repeatable. A passive pulse-expansion filter impulse can be generated with either a video or RF pulse containing broadband energy spread over the bandwidth of the desired chirp signal. A video pulse and its spectrum are illustrated in Figure 4.26. Of interest is the flatness of the video pulse spectrum across the chirp bandwidth A centered at/, the chirppulse center frequency. Variations in power over the bandwidth A of the video pulse spectrum for a linear system translate to variations in power over the chirp-pulse delay extent. The required flatness can be estimated on the basis of paired-echo theory. For example, signal variation across A must be less than ±0.5 dB, based on Figure 3.3(b) of Chapter 3, to ensure that time sidelobes introduced are less than 30 dB below the peak response. Actual degradation would be less for a frequency-weighted system, because of the reduced effect of a deviation from flatness at the band edges. A ringing filter is often used to provide a better match between the video pulse and the pulse-expansion filter. More useful RF power is thereby available to the pulse-expansion filter than can be provided directly from the video pulse for a given pulse power level. Typically, the peak input voltage of the expansion filter is limited, but the filter is typically driven as hard as possible short of voltage breakdown in order to obtain an adequate output SNR. The ringing filter provides a 6-dB advantage in this regard over the direct impulse because its RF drive power for the same peak voltage is 6 dB higher than the unipolar video pulse drive power.
168
V(l)
NARROW VIDEO PULSE
RINGING FILTER OF BANDWIDTH A CENTERED AT 1
PULSEEXPANSION FILTER
• CHIRP PULSE
VIDEO-PULSE POWER
* FREQUENCY
Figure 4.26 Video pulse and video pulse spectrum for passive chirp generation.
4.6.5 Time Jitter Chirp-pulse radars designed for target imaging need to produce responses that can be time-aligned adequately for low-loss pulse-to-pulse coherent integration. Range delay of the data sample gate in a conventional system is established relative to the-main trigger. Any time jitter between the PRF trigger and the output of either the chirp VCO or pulseexpansion filter of Figure 4.25 will convert to pulse-to-pulse phase error in the data collection record. From Chapter 3 it was found that rms phase noise greater than about 10 deg begins to produce significant coherent processing loss. As an example, consider the time jitter requirements for a high-resolution radar that transmits 250-MHz chirp pulses generated at a 750-MHz intermediate frequency and upconverted to a center frequency of 10 GHz. Time jitter in the transmitted waveform corresponding to 10 deg (0.175 rad) is 0.175 rad IQ.tr x 10 x 10') = 2.78 x 10"' sec (about 3 ps). The required fractional frequency deviation at a PRF of 500 Hz is about 1.5 x 10"), which is easily achievable. More difficult is achieving less than 3-ps jitter with practical implementations of the driver or impulse generator of Figure 4.25. One approach for meeting the above time jitter requirements is to drive a passive pulse-expansion filter with a gated segment of the IF reference, as illustrated in Figure 2
169
4.27, in which the monocycle generator is shown substituted by a fast gating switch. Time jitter of the gated pulse is then determined by the IF reference independent of gating switch jitter. The spectral width of the gated pulse in this approach, however, needs to equal or exceed the bandwidth of the chirp waveform to be generated. The approximate spectral width of the gated pulse in terms of the number of its IF cycles n is f, /n, which is the reciprocal of the gated pulse width. Bandwidth associated f
Chirp Pulse
^>-^DupT|«-^
/
Point Tgt. Compressed Response @ IF / P o i n t Tgt. / Signal @ IF
A/WWlAr Compression Saw
LO
IF Rel. Alternative Galed IF Approach
PRF Trig.
LO
IF Ref.
t
I
t
RMO
IF Rel.
Fast Switch Waveforms
AAAAAAAAAAAr — I
-MyGated Pulse se / 3 IF
Figure 4.27 Coherent chirp-pulse generation.
Spectrum
170
with a gated pulse containing three cycles of the above 250-MHz chirp pulse at a 750MHz IF equals the 250-MHz spectral width needed to excite all components of the chirp pulse. However, achieving the required gate duration of fi/fa = 4.0 ns, though possible, is difficult at this time. 4.6.6 DDS Chirp Generation Basic limitations in the technology for analog passive and active chirp generation have in the past made it difficult to produce chirp pulses with the desired frequency linearity and pulse flatness, particularly for the large time-bandwidth products needed for highresolution radar imaging systems. The generation of adequately linear, large time-bandwidth-product chirp pulses using analog methods often requires frequency linearization, temperature control, and calibration procedures. The result is that these systems are complex and exhibit reliability problems. Application of DDS technology now makes it possible to generate chirp pulses that are nearly perfectly linear and repeatable. Furthermore, while the signal is in the digital domain, phase, frequency, amplitude, and on/off timing can be controlled with accuracy determined by a stable clock frequency. This allows adaptable compensation for phase and amplitude ripple of the radar system over the chirp bandwidth. Synthesized waveforms can be changed pulse to pulse and pulse-to-pulse jitter is minimal because on/off timing is determined by clock stability instead of by switch stability, as for the analog systems of Figure 4.25(a,b). Figure 4.28 is a block diagram of a DDS chirp synthesizer. The dual accumulator provides the function of dual integration of phase. A control word at the input to the frequency accumulator can be thought of as a fixed phase that is integrated to produce a linear-phase ramp, which, when integrated in the phase accumulator, produces the quadratic-phase ramp associated with linear FM. The input to the phase accumulator a frequency accumulator in the sense that it provides linearly ramped frequency control words to the phase accumulator instead of the selected frequency control word in the DDS of Figure 3.29. Accumulation of the selected frequency control words of Figure 3.29 produce the linear-phase ramps for the selected frequency. The accumulated frequency control words of the Figure 4.28 DDS produce the quadratic-phase ramps associated with linear FM. Device 1 -
Frequency accumulator
Phase accumulator
Phase adder
- Device 2 -
- Device 3 •
Memory (ROM)
DAC
Figure 4.28 DDS chirp generator. (Courtesy of Scitec Electronics, Inc., San Diego, California.)
171
QUADRATIC PHASE
ANGULAR FREQUENCY
Figure 4.29 Quadratic phase (shown as phase deviation from center frequency to the high end of the band over which pulse compression is to be carried out).
The adder following the phase accumulator provides for correction for radar system phase ripple and a multiplier (not shown) provides amplitude correction or control. Figure 4.28 is a simplified block diagram of a DDS chirp generator developed by Scitec Electronics, Inc., for high-resolution synthetic aperture radar. The clock rate is 500 MHz. Chirp bandwidth is 230 MHz. Sandia Laboratories of Albuquerque, New Mexico, tested a similar unit [14] for synthetic aperture radar application. 4.6.7 Quadratic-Phase Distortion The phase and amplitude ripple seen in the transfer function of a pulse-compression radar will produce distortion of the compressed signal in the form of paired echoes, as discussed in Chapter 3. Another type of distortion unique to chirp waveforms is called quadraticphase distortion, which is produced by any deviation from the quadratic-phase-versusdelay function for the matched condition. The distortion can be produced by either deviation from linear FM of the generated chirp signal or by dispersive components in the radar system (other than the phase equalizer), including RF transmission lines in the radar
172
system. Figure 4.29 illustrates deviation from linear phase versus frequency produced by unwanted dispersion in the radar system. We will now derive expressions that relate quadratic-phase error and delay mismatch for chirp-pulse systems. Tolerance to mismatch in terms of pulse widening and sidelobes will also be discussed. The ideal transfer function, given in (4.20) for phase equalization of a chirp waveform, is based on matching the delay-versus-frequency slope P to the inverse of the frequency-versus-delay slope K = A/7V From the definition (4.19) of dispersion factor, the ideal transfer function (4.20) can be rewritten in terms of chirp-pulse length T and dispersion factor D as x
«(/) = e x p j - ^ ( / - / )
2
(4.42)
This transfer function has the desired quadratic-phase characteristic, wherein phase, expressed in terms of angular frequency (radians per second), becomes #a>) = ^ ( w - 7 5 )
2
(4.43)
where a> is the instantaneous angular frequency and CJ is the center angular frequency. The desired instantaneous chirp delay, written in terms of angular frequency, becomes dS(oj)
T\
which is another form of (4.21). Similar expressions for undesired quadra'^-phase and equivalent delay error are
2
a)
(4.45)
and T,(«) = - ^ ^
=-rj(«-6i)
(4.46)
where Tl is a constant. We can see that the delay error expressed in (4.46) is of the same form as (4.44) for delay associated with a chirp matched filter. Angular frequency w in radians per second instead of frequency / in hertz will be used henceforth in connection with unwanted quadratic-phase error in order to distinguish from the desired phase and delay-versus-frequency characteristics of the chirp signal.
173
Chirp waveforms are often used with waveguide radar systems. The waveguide, if its group delay is assumed to vary linearly with frequency over the band of interest, produces quadratic-phase error that can be corrected by adjusting the transmitted FM slope. Quadratic-phase error, in radians, evaluated at the band edges +/, of the chirp pulse, from (4.45), becomes 1
(4.47)
where = U(a> -a>)\
A = 2M-f)\
(4.48)
l
The constant To of (4.45) written in terms of band-edge phase error then becomes
7T A 2
2
The magnitude of the resulting delay error at either band edge ±co„ from (4.46) and (4.48), is \TA*>,)\
= Tl\(oj - 1S)\
(4.50)
t
= r 7rA 0
By substituting for T\ from (4.49) into (4.50), we have 2d>(u> ) r
M w
'
) {
=
( 4 5 I )
riT
Total chirp delay error over the entire band A (in hertz) is 4
174
0
2
4
6
8
10
0
2
|2t>.)A|
4
6
8
10
|2t>.)A|
Figure 4.30 Pulse widening and amplitude loss of mismatched chirp pulse with weighting. (From [IS]. Reprinted with permission.)
weighted pulses, values of
(
' "'
) l =
200x^
, r x 4 0 0 x 10*
(
4
5
3
)
= 0.01 1 /AS Quadratic distortion produced by the 30m waveguide length could be equalized by increasing the chirp-pulse duration by 2|TJ(«,)| = 0.011 /us with the same chirp bandwidth, as indicated in Figure 4.31. The reader will recall that, because the instantaneous delay of the waveguide approximates a linear delay-versus-frequency function, the waveguide can be used for passive generation of HRR chirp waveforms, but impractical waveguide lengths are needed to obtain significant energy transmitted per pulse. 4.7 DIGITAL PULSE COMPRESSION The chirp pulse is an analog signal and pulse compression, described in Section 4.6.4 to convert target return signals into HRR profiles, was performed with analog hardware. At
175
DELAY VS. FREQUENCY OF PULSE-COMPRESSION SYSTEM (WITH WAVEGUIDE)
| 2 T > , ) | = 0.011 us \ SYSTEM (NO WAVEGUIDE) ORIGINAL CHIRP , 3.05
3.25
3.45
FREQUENCY (GHz)
Figure 4.31 Waveguide quadratic-distortion correction by FM slope adjustment (for slope error produced by 30m of WR-284 waveguide).
this point, the high-resolution target range profiles are sampled and digitized for further processing to perform target recognition, target detection, or target imaging. The pulsecompression process can also be performed digitally on echo data sampled at baseband. The advantages are reduced quantization noise at the output for a given number of bits quantized from the A/D converter and the potential for adaptive control of the matchedfilter transfer function, including weighting, to improve resolution and sidelobe performance. The reduction in output quantization noise occurs because of the increased signalto-quantization noise provided by the signal-processing gain associated with convolution. Pulse compression was described above in terms of the mathematical process of convolution. Likewise, digital pulse compression is also a process of convolution. A digitized version of the echo pulse at baseband can be convolved with a digitized version of the matched filter's impulse response to produce a digitized HRR response. Digital convolution can employ a DFT process equivalent to convolution. The process is sometimes called/aif convolution. Although more complex, it is faster than direct discrete convolution for large data sets because of efficiencies obtained by using the FFT algorithm. The DFT equivalent to convolution can be described in terms of the convolution theorem, which relates the convolution expression and its Fourier transform. This is a very important relationship for many areas of engineering and scientific analysis. It states that the Fourier transform of the convolution of one function with another is the product of the Fourier transform of the first function multiplied by the Fourier transform of the second function. The convolution theorem, expressed in terms of an input echo signal sfj) and the impulse response h(t) of the matched filter, is FTfaM * MO] = £ ( / ) » ( / )
(4.54)
176
Thus, convolution in the time domain can be carried out by multiplication in the frequency domain. The quantity S,(f) is the spectrum of the echo signal from one transmitted pulse. The transfer function / / ( / ) is the Fourier transform of the impulse response of the matched filter. Following each transmitted pulse, the return signal in each quadrature channel out of a quadrature mixer is sampled at or above the Nyquist rate, which is A complex samples per second for chirp bandwidth A. Discrete samples, called range data, are collected over some desired range window corresponding to the target range extent to be processed and then converted into digital quantities by an A/D converter. The result is a digital version of the input signal for one transmitted pulse. The digital version of the matched filter's transfer function (also called reference function) can be stored directly as a series of digitized complex pairs. This transfer function will remain constant for a particular chirp waveform but can be controlled to correct for radar system phase and amplitude ripple. Weighting can also be included. Pulse compression, regardless of the method, refers to convolution of the received echo signal, after appropriate down-conversion, with the impulse response of the matched filter to the transmitted chirp pulse. For analog pulse compression, the convolution process is accomplished by simply passing the echo signal through a physical matched filter and an appropriate weighting filter. For digital pulse compression, the convolution process could be carried out by convolving the digitized input range data for each transmitted pulse with a digitized discrete version of the matched-filter impulse response to the transmitted pulse. If fast convolution is to be used, the digitized range data are convolved as shown in Figure 4.32 by first transforming to the frequency domain, then vector multiplying with the digital version of the transfer function / / ( / ) , and, finally, transforming back to the time domain, which is then the compressed range data. Not shown are lowpass filters at the / and Q outputs of the quadrature detector, which pass the chirp bandwidth at video but reject the sum signal and harmonics. LOCAL OSCILLATOH
ECHO SIGNAL
QUADRATURE! MIXER
RANGELINE BUFFER SAMPLING AND A/D CONVERSION
FFT
•
MULTIPLIER
DIGITAL REFERENCE GENERATOR (TRANSFER FCT OF MATCHED FILTER TO TRANSMITTED PULSE)
Figure 4.32 Digital pulse compression.
FFT-<
DISCRETE VERSION OF ' COMPLEX RANGE PROFILE
777
Use of the convolution theorem for digital pulse compression is based on using the DFT as an approximation of the continuous Fourier transform. The DFT process transforms n discrete range values spaced by A/ in the time domain into n discrete values spaced by A/ = l/(«Af) in the frequency domain. In shorthand notation, the convolution response becomes DFT[^(/A7) * h(lAt)} = Si(jA/) x W(iA/)
(4.55)
Both functions sAl&t) and /i(/Ar) are periodic with the same period n&t. Because the DFT process is periodic, discrete versions of the input signal and impulse response function are required to be generated such that the resulting periodic response is a replica of the desired aperiodic result. Consider first the analog pulse compression process illustrated in Figure 4.33 for range compression of the signal produced by two point targets, which appear within an assumed radar range extent. The rectangular waveforms represent envelopes of the transmitted and received signals. An analog chirp waveform is represented in Figure 4.33(a). The input received signal is represented in Figure 4.33(b) along with the radar's matched-filter impulse response. Figure 4.33(c) represents the analog compressed rangedelay response produced by convolving the input signal with the matched-filter impulse response. Convolution for analog signals is carried out by passing the input signal through a matched filter realized in hardware—for example, a SAW device. Now consider the digital pulse-compression process illustrated in Figure 4.34. The transmitted chirp signal s (t) of Figure 4.34(a) and the received baseband signal s£t) and t
ENVELOPE OF TRANSMITTED CHIRP PULSE
»,(«)
- t ENVELOPE OF RECEIVED TARGET SIGNAL
h(t) ENVELOPE OF MATCHED-FILTER IMPULSE RESPONSE
RANGE EXTENT •„(») = s,(t) • h(t)
h* TGT #1 COMPRESSED TARGET SIGNALS
Figure 4.33 Analog pulse compression (example for two targets).
178
ENVELOPE OF TRANSMITTED CHIRP PULSE
s,(t)
Ii.L 2
2 ENVELOPE OF RECEIVED TARGET SIGNAL
s,(t) TGT #2,
SAMPLED RANGE EXTENT
r Q
H,
1
I 2 n,
t 2
ENVELOPE OF SAMPLED SIGNAL DATA
1
• • ••
1 1
• •• • •
1
tmill i
ENVELOPE OF DIGITIZED MATCHED-FILTER IMPULSE RESPONSE 1 1
•
0
n-1
SAMPLES ZEROS WINDOW OF DISCRETE FORM OF SIGNAL SPECTRUM
2
h(/A/)
±ui ITnli 11i-LI*
(c)
S,(IA()
ENVELOPE O F MATCHED-FILTER IMPULSE RESPONSE
I — i ^ . TGT #1
h(t)
SAMPLES U
/
I
A
#
H
(
i
A
f
1
• 111111 >
n-1
ZEROS
. '
;
WINDOW OF DISCRETE FORM OF MATCHED-FILTER • TRANSFER FUNCTION
Ii I |
n-1
n-1 DISCRETE VERSION OF THE CONVOLVED SIGNAL FFT-'[S,(IAf)-H(IAf)I
(e)
i n u n m m
' imiiiiiillmi
'*•••'"•'
*
n-1 Y ONE PERIOD OF C O M P R E S S E D DATA (RANGE PROFILE)
Figure 4.34 Fasi-convoluiion example for two targets.
matched-filter impulse response h'l) of Figure 4.34(b) are respectively identical to those of Figure 4.33(a.b). Figure 4.34(c) represents discrete versions of both the received signal and matched-filter responses. A common period length n must be set sufficiently large that the convolved result of one period does not overlap that of the succeeding period. This is achieved by applying the following rule to the discrete versions of both $,<0 and
„>r.
+ PW-*'>*_! Ar
(4.56)
179
where R and R, are the edges of the range window to be processed. Zeros are added to the signal-data samples and to the T,/Af samples of the impulse response function, as shown in Figure 4.34(c), to produce the common period of length n. At this point, the two resulting data sets of Figure 4.34(c) could be convolved to produce the compressed response. However, use of the convolution theorem carried out digitally by the FFT algorithm for the DFT, although not shown, is implied. The DFTs of J,(iAr) and h(iAt) can be defined, respectively, as follows. 2
2
S,(iAf) = £ s,(lAt) expl^- j ^ ' / j . 0 < i < n - 1
(4.57)
WAf)
(4.58)
and = £ WAf) e x p / - j—17 j , 0 < i < n - 1
where A/ = l/(nAr). The FFT algorithm calculates (4.57) and (4.58) for values of n =V
(4.59)
where y is an integer. Equation (4.59) imposes a second requirement on the selection of n when the convolution theorem is to be applied with the FFT algorithm to generate the compressed-range data. The first requirement, (4.56), applies whether or not the FFT type of DFT is used. A third requirement in the selection of n is that the sampling rate /, equal or exceed the Nyquist sampling rate, which is related to the chirp bandwidth A as /, > 2A
(4.60)
The last criterion can be met by taking complex samples at baseband, spaced by 1/A. Application of (4.57) and (4.58) by using the FFT algorithm produces the discrete versions of the signal spectrum 5,(iA/) and matched-filter transfer function H(iAf), illustrated in Figure 4.34(d). Next, these quantities are vector multiplied to form the frequency spectrum of the range-compressed output. The final step is to perform the inverse (frequency-to-time) FFT of the output frequency spectrum to obtain the output range-delay response. Of interest is the response made up of the first n discrete values £ = 0 through n - 1. This result, illustrated in Figure 4.34(e), replicates that of Figure 4.33(c) for analog pulse compression when the criteria expressed in (4.56), (4.59), and (4.60) are met. Table 4.2 lists minimum acceptable values of period length n versus both sampling interval At and the sum of the sampled signal extent plus chirp-pulse length in seconds. Convolution and correlation by using the FFT are described in more detail by E.O. Bringham [17].
180
Table 4.2 Minimum Acceptable Period Lengths for Discrete Convolution (Assuming Complex Sampling) Chirp-Pulse Length Plus Sampled Range-Delay Extent T,
(
Minimum Acceptable Period Length n Versus Sampling Interval, Af
2(K, - R,)
2
1
10 ns
16
20
10
5 4
50
Al (ns)->
2
20 ns
32
16
4
2
50 ns
64
32
16
4
100 ns
128
64
32
16
4
2
200 ns
256
128
64
32
16
4
500 ns
512
256
128
64
32
16
1 /is
1.024
512
256
128
64
32
2
4.8 DISTORTION PRODUCED BY TARGET RADIAL MOTION Up to this point, our analysis of methods for obtaining HRR performance from radar systems has assumed a stationary target. We now consider the effect of Doppler shift produced by target radial velocity, which reduces peak response and degrades resolution. The nature of this distortion is probably best studied from the viewpoint of the ambiguity function. Two ideal waveforms will be considered: the short monotone pulse and the linear FM (chirp) pulse. Expressions for the rectangular envelopes of the two waveforms
sM = \fc
rect( - ]
(4.61)
for the monotone pulse, and
s,0) = A / F
RCCT
( ¥)
J
exp(j2,7r/Y/ /2)
(4.62)
for the chirp pulse. The term rect(//T|) is defined by (4.16). The waveforms are normalized according to the expression 2
jjs,(f)| df = 1
(4.63)
This normalization results in an ambiguity surface of unit height at the origin. The ambiguity surfaces for the rectangular monotone and rectangular chirp pulses are determined using (3.44), together with (4.61) and (4.62). Results are expressed as follows.
181
l-nvr,)] 7r/ r,(i - |r|/r,) . M < r,
sin[ff/i,r,(i -
U
(4.64)
0
W>v"i
= 0, for the monotone pulse, and \xir,fo)\=
|r|\sin[7r(/:r+/ )(r,-M)]] T,j TT(KT + f )(Ti - M)
(4.65)
0
1
D
M>r,
= 0,
for the chirp pulse, where T is the delay relative to the origin and f is the Doppler shift produced by the moving target. Critical features of the ambiguity functions, (4.64) and (4.65), can be discussed with reference to Figures 4.35 and 4.36. In each case, the ambiguity surface extends from -7", to +7, in range delay and -°° to +<*> in Doppler. Doppler frequency response at zero delay points has (sin x)lx profiles for both the FM and monotone pulses. Also, the responses for both monotone and FM pulses are maximum at matched delay and Doppler shift points T = 0 and f = 0, respectively. Range-delay resolution is optimum at f = 0 and the response broadens as [f \ increases. A distinctive feature of the chirp-pulse ambiguity function is its range-Doppler coupling characteristic. A Doppler shift produces a range-delay shift in the response. Profiles normal to the Doppler axis for FM pulses maximize above the line f = - K T through the origin of the/ , rcoordinates. Profiles for the monotone pulses, by comparison, are maximized above t h e / = 0 axis. It is clear from Figures 4.35 and 4.36 that for either monotone or chirp pulses the pulse duration T, determines tolerance to Doppler shift. The response to a target observed with a monotone pulse degrades with target radial velocity. Resolution is reduced and sidelobes increase. The peak of the zero-Doppler response occurring at a given range delay is seen to go to zero at f = 1/7*|, and at that Doppler frequency the range-delay response bears no resemblance to the matched response at zero Doppler. By contrast, the chirp waveform is said to be Doppler-invariant or Doppler-tolerant. Location of the peak shifts with Doppler frequency, but the response remains relatively unaffected well beyond D
D
D
D
D
0
D
D
f =m. 0
A3 DISPLAY, RECORDING, AND PREPROCESSING OF HRR TARGET RESPONSES For simple viewing of a target's HRR profile generated by analog pulse compression, the output of the matched filter, such as a SAW device, can be envelope-detected and then displayed on a wideband oscilloscope activated by a range-delay trigger pulse. The detector and oscilloscope's phase and amplitude characteristics then become part of the total system
182
BASEBAND PULSE (FOR LARGE CARRIER FREQUENCY)
RECT
(f)
N
\
IX(T.« )I 0
| SIN n T , f | 0
|*<°'' >| = D
-Hurl
A
Figure 4.35 Ambiguity surface for rectangular monotone pulse.
transfer function. Distortion, in terms of decreased resolution and time sidelobes, occurs in the manner discussed for RF components in Chapter 3. However, wideband video detectors and oscilloscopes are available today with sufficiently flat amplitude response and low phase ripple to view target range profiles obtained with greater than 1-GHz bandwidth. Display can be achieved by connecting the wideband video output to the y-axis of a wideband oscilloscope. The horizontal sweep is set to move across the x-axis during the time interval associated with the range window to be observed. The result is an Ascope display of the target's range profile. A range-delay trigger pulse starts the range window. The horizontal sweep time sets the extent of the range window delay. Rangedelay jitter must be about an order of magnitude better than the range resolution; otherwise blur will appear on the A-scope display.
JS3
lz(o,f )l = D
Figure 4.36 Ambiguity surface for rectangular chirp pulse.
Jitter-free range-delay trigger pulses to track moving targets can be generated by the circuit shown in Figure 4.37. A stable oscillator, followed by shaping and divider circuits, generates the radar's PRF. A VCO, in the form of a second stable oscillator, is adjusted in frequency slightly above and below that of the first oscillator to generate a variable delay trigger. The delay is continuously adjusted to track the target as it moves in range. Manual range tracking is carried out by setting the VCO voltage drive so that the delay trigger starts the oscilloscope sweep just ahead of the arrival of the target's compressed response. An earlier version of a range tracker used a motor-driven phase shifter, as shown in Figure 4.38, to generate the delay trigger from a single fixed oscillator. HRR target range profiles, as viewed on an oscilloscope, have had some limited value. Early work in the late 1960s and early 1970s in San Diego at the Naval Electronics Laboratory (NEL) and the Naval Electronics Laboratory Center (NELC) demonstrated that air and ship targets were largely made up of individual backscatter sources. Targets were found to be easily tracked through severe land clutter by manually tracking the target's range profile as it "moved through" a clutter background that produced much higher return than the target. It was also apparent that the range-profile signatures were unique to target type within a limited range of target aspect angles. Sea clutter showed up as individual scatterers (called spikes), which appeared and disappeared with lifetimes on the order of three to five seconds. Recording or HRR target signatures and clutter was originally done by photographing the A-scope display. It was soon found necessary to develop a digital recording capability
184
JUULT STABLE OSCILLATOR e.g., 10 MHz
PULSE SHAPER
_n_
LRTUU 1 VOLTAGECONTROLLED OSCILLATOR e.g., 10 MHz ± «
MANUAL VCO CONTROL
AUTOMATIC VCO CONTROL
MAIN TRIGGER
*N
_n_
PULSE SHAPER
DELAY TRIGGER
AUTOMATIC TRACKER
Figure 4.37 Range tracker for HRR radar.
in order to obtain suitable data for analysis to determine target recognition potential. Later, clutter analysis was also carried out by using digitized data. The digitizing of short-pulse or pulse-compression data requires samples of the detected envelope of the range profile at range-delay intervals separated by an amount equal to or less than the duration of the compressed response. For a 500-MHz pulsecompression radar, for example, the compressed pulse, duration will be about 2 ns. This corresponds to sampling the detected video at a rate of 500 x 10 per second. Sampling and A/D conversion at these rates has recently become possible, but the degree of amplitude quantization is limited, as indicated in Figure 3.17. An early method used at NEL to circumvent the requirement for a high-speed A/ D converter employed a serial sampling system closely related to the design of wideband sampling oscilloscopes. The concept is to sample the target signature at the radar's PRF while advancing the sample position for each pulse. In this way, the entire signature is sampled during n radar pulses, where n then becomes the number of samples that make up the range window. The technique allows data to be collected with a high degree of amplitude quantization for those target-signature features that do not vary significantly during n radar pulses. Range tracking was carried out as described above. 6
18S
PULSE SHAPER
MAIN TRIGGER
STABLE OSCILLATOR e.g., 10 MHz
MOTORDRIVEN PHASE SHIFTER
PULSE SHAPER
-i-N
DELAY TRIGGER
MANUAL MOTOR CONTROL
Figure 438 Range tracker for HRR radar using a motor-driven phase shifter.
This serial sampling method was used to collect aircraft and ship signature data from a ground site at NEL. The technique was used to collect the first dynamic HRR signature measurements of ships and aircraft targets in motion. A block diagram of the sampling system is shown in Figure 4.39. Also shown in the figure is a second sampling mode that is able to collect samples from a selected modulating portion of the rangeprofile video signature. In both modes, only a small segment of the signature is sampled for each pulse. The serial sampling technique, therefore, "throws away" signal energy, which, if sampled and processed, could provide a higher output SNR. The problems of sampling, digitizing, and processing HRR signatures obtained in the time domain remain formidable for resolution less than about one-third of a meter. For this reason, frequency-domain sampling techniques have been developed (e.g., for stretch and synthetic range-profile generation) which provide increased resolution over that possible with present technology for direct sampling of the compressed pulses. Examples of HRR signatures are shown in Figures 4.40, 4.41, and 4.42. Figures 4.40 and 4.41 were obtained by photographing range profiles appearing on a wideband
186
TARGET ANGULARPOSITION — AND AGC DATA FROM RADAR
DETECTED HRR SIGNATURE FROM RADAR
SAMPLING UNIT IN INCREMENTALDELAY ADVANCE MODE
INTERFACE EQUIPMENT AND A/D CONVERTER
DIGITAL MAGNETICTAPE RCDR
RANGEPROFILESIGNATURE RECORDING MODE
SAMPLING UNIT IN FIXED t DELAY MODE
INTERFACE EQUIPMENT AND AID CONVERTER
DIGITAL MAGNETICTAPE RCDR
MODULATION RECORDING MODE
DELAY TRIGGER FROM RANGE TRACKER
Figure 4.39 Target signature and modulation recording (serial sampling).
CRT. Figure 4.42 was generated from serial samples obtained using the range-profile recording system of Figure 4.39.
7' 8' 10'
6'
Figure4.40 HRR signature of T-28 at S-band (I-ft resolution, nose aspect).
Figure 4.41 HRR signature of C-45 aircraft at S-band (1-ft resolution, tail aspect).
189
190
PROBLEMS Problem 4.1 /2
Show that H(f) = e'* is the correct expression for the transfer function of the matched filter to a Gaussian-shaped video waveform expressed as
Problem 4.2 A filter that is driven by an ideal impulse has a rectangular bandpass filter characteristic with bandwidth fi and center frequency /. Use the Fourier shift theorem to show that the complex expression for the normalized output signal is given by sin ir/3l exp flirft irt Assume
f>P-
Problem 4.3 What is the highest sidelobe level in decibels of the envelope of the output pulse of Problem 4.2? Problem 4.4 Determine the half-power temporal resolution of the envelope of the monotone Gaussian pulse expressed by
i(/) = e-°'e'*' 2 7
Assume resolution
1//.
Problem 4.5 A Gaussian-shaped waveform is represented by
191
(a) What is the duration of the pulse envelope in terms of cr at the half-peak points? (b) What is the range resolution associated with this RF pulse at the half-peak points for a - 2 ns? Assume resolution < l/f. Problem 4.6 (a) What is the achievable compression ratio of a 5-fis, 32-bit binary phase-coded pulse waveform? (b) What is the range resolution? (c) What is the waveform bandwidth? Problem 4.7 (a) Write the complex expression for the baseband form of the waveform illustrated in Figure 4.4. (b) Write the expression for its matched filter. Problem 4.8 Using a block diagram like Figure 4.5, show that the binary phase-coded Barker code ( + + + - + ) has a peak response of +5 and peak-to-sidelobe ratio of +14 dB. Problem 4.9 Show that as the number of frequency steps n in a contiguous, discrete frequency-coded waveform approaches infinity, the envelope of the matched-filter response near the peak approaches that of a compressed chirp pulse of the same bandwidth. Assume both waveforms are matched-filtered but unweighted and that the frequency-segment length T\ is equal to the reciprocal of the frequency-step size. Problem 4.10 Show that the pulse-compression ratio of an n-element discrete frequency-coded pulse following matched-filter processing is approximately n for large n when the frequencysegment length T| is equal to the reciprocal of the frequency-step size. 1
Problem 4.11 A radar is to be designed for 5-ft (1.524m) range resolution. What are the required clock rates to generate the discrete delay segments of (a) a phase-coded waveform, and (b) a 32-element, contiguous, stepped-frequency-coded waveform, where segment duration
192
equals the reciprocal of frequency-step size? (Either coded pulses or periodic CW waveforms may be assumed.) Problem 4.12 We want to use a periodic stepped-frequency-coded CW waveform for unambiguous resolution of isolated targets of up to 300m in length with 10m resolution. Assume uniformly stepped frequencies in each period with step size set equal to the reciprocal of frequency-step duration, (a) What is the total bandwidth required? (b) What is the frequency-step size if frequency-step duration is matched to target length? (c) What is the waveform period in number of steps? (d) What is the waveform period in seconds? Problem 4.13 A radar transmits 100-/is pulses, each with a linear FM of 250 MHz over the pulse duration. Compression is to be accomplished using stretch processing by first mixing the return signal with a delayed reference having an identical FM slope. What is the timebandwidth product of the signal before and after mixing? Assume a point target. Problem 4.14 A stretch waveform is used to obtain signatures of space objects from earth-based radar stations. The waveform consists of 100-/AS linear FM pulses with 500-MHz bandwidth. Return signals are processed as in Problem 4.13 by mixing with a delayed reference that is a replica of the transmitted waveform: What is the total bandwidth seen at IF when a 30m target is to be observed? Problem 4.15 A radar transmits monotone pulses of 5-/AS duration, (a) What is its approximate slantrange resolution following matched-filter processing? (b) If the radar were redesigned so that the same pulse envelope is frequency modulated with linear FM over 100 MHz, what is the possible new range resolution? (c) What is the time-bandwidth product in each case? Problem 4.16 Dispersion D for a chirp-pulse radar's waveform is 100 and the point-target compressed response width is 2 ns. What is the approximate FM bandwidth across the response width?
193
Problem 4.17 A pulse-compression radar transmits rectangular chirp pulses of 500-MHz bandwidth. What is the approximate slant-range resolution after Dolph-Chebyshev frequency weighting that results in 30-dB sidelobes? Problem 4.18 With reference to the MacColl paired-echo analysis, compute the allowable amplitude deviation in a pulse-compression radar system if the sidelobes of the output response are to be at least 46 dB below the peak. Assume no phase ripple. Amplitude deviation is defined here as (1 + aja ), expressed in decibels. Calculate from the equations, then compare with Figure 3.3(b) of Chapter 3. a
Problem 4.19 A pulse-compression filter for a radar has a time-bandwidth product of 80. Two methods of chirp generation are being considered: (1) active generation with a VCO that produces a rectangular-envelope chirp, and (2) passive generation using a dispersive filter of the same time-bandwidth product. Assuming equal losses and no weighting in each method, use Figure 4.16 to compare the optimum SNR performance. Problem 4.20 A 2-jjs chirp pulse with chirp slope K = 5 x 10" Hz/s undergoes pulse compression in a phase equalizer exhibiting a delay-versus-frequency slope of P = 0.2 x 10 s/Hz, followed by a Gaussian weighting filter of 100-MHz bandwidth at the -8-dB points. No other band-limiting is involved, (a) What is the chirp-pulse FM bandwidth? (b) What is the degradation in SNR from that of an ideal matched filter to the chirp pulse? (c) What is the compressed pulse duration at the half-power points? (d) What are the peak-tosidelobe levels? Use Figures 4.18 and 4.19. 13
Problem 4.21 The pulse-compression receiver of a radar is matched to its transmitted 10-/JS chirp pulse of 200-MHz bandwidth centered at 3.25 GHz. The only source of distortion is 60m of WR-284 waveguide, (a) What is the approximate band-edge phase deviation from the best linear fit, based on Table 3.4 of Chapter 3? (b) What is the equivalent chirp-delay error? (c) What is the fractional pulse widening and amplitude loss based on Figure 4.30?
194
(d) What new pulse length of the same bandwidth is required to equalize the quadratic error produced by the waveguide?
Problem 4.22 Show that if 4>(
ir,|
4
r,A
where IT) - T\\ is the allowable delay mismatch over the chirp bandwidth A of a pulse of length T,.
Problem 4.23 What is the maximum tolerable mismatch in microseconds, based o n ^ e criterion of Problem 4.22, for the active chirp generation of a 20-/ts pulse with a 200:1 compression ratio?
Problem 4.24 A chirp pulse is to be generated by an impulse to 100m of a WR-284 waveguide. The output of the waveguide is Filtered by a 400-MHz rectangular bandpass filter, band centered at 3.25 GHz. From Table 3.4 of Chapter 3 and the discussion of quadratic-phase distortion, what is the time-bandwidth product of the chirp pulse*?
Problem 4.25 Gaussian weighting following ideal, unweighted equalization is used to reduce time sidelobes seen with a 250-MHz chirp-pulse-compression radar. Rectangular chirp pulses are transmitted, (a) What is the half-power compressed-pulse width before weighting? (b) What is the half-power pulse width following Gaussian weighting to reduce peak sidelobes to 25 dB below the main response? (c) Assuming that no other band-limiting of the chirppulse spectrum occurs, what is the SNR loss following weighting? Use Figures 4.18 and 4.19.
195
Problem 4.26 What length of a WR-159 air-filled waveguide (f = 3.711 x 10' Hz) is required for chirppulse compression to produce 0.05-//S pulses of 500-MHz chirp bandwidth at 5.3-GHz center frequency? Compute delay based on the expression for group delay per unit length given in Problem 3.36 of Chapter 3. c
Problem 4.27 Digital pulse compression of return signals produced using a 1.5-/xs pulse-width chirp waveform is to be carried out over a sampled-range extent of 5 nmi (9,260m). The rangedelay sample spacing is 10 ns. Assume that fast convolution is to be used to convolve the sampled-range-delay extent of data obtained with each transmitted pulse with a digitized reference of the transmitted waveform, (a) What minimum common period length for the reference and signal is required in terms of the number of complex samples? (b) How many zeros will be added to the time-domain samples of the signal data? Problem 4.28 A 9.5-GHz pulse-compression radar transmits 10-/us chirp pulses. Resolution is 150m. A target approaches the radar at a radial velocity of 300 m/s. What is the apparent range shift produced by the target's Doppler shift? Problem 4.29 Detected HRR target range profiles are to be recorded digitally by using the serial sampling method described in Figure 4.39 (range-profile-signature recording mode). What is the maximum allowable incremental-delay advance required for unambiguous sampling of the range profile data collected by using a chirp-pulse radar of 500-MHz bandwidth? REFERENCES [1] Klauder. J. R., et al., "The Theory and Design of Chirp Radars," Bell System Technical J., Vol. XXXIX, No. 4, July 1960. p. 747 (footnote). (2) Cook, C. E., and M. Bemfeld, Radar Signals. New York: Academic Press, 1967, p. 245. (3] Lewis, B. L., F. F, Kretschmer, Jr., and W. W. Shelton, Aspects of Radar Signal Processing, Dedham: Artech House, 1986, pp. 9-14. [4] Kraus. J. D., Antennas. New York: McGraw-Hill, 1950, pp. 76-77. [5] Caputi, W. J., "Stretch: A Time-Transformation Technique," IEEE Trans. Aerospace and Electronic Systems, Vol. AES-7, No. 2, March 1971, pp. 269-278. [6] Curlander, J. C , and R. N., McDonough, Chs. 9 and 10 in Synthetic Aperture Radar Systems and Signal Processing, New York: John Wiley and Sons, 1991.
196
[7] Klauder, J. R., et al.. "The Theory and Design of Chirp Radars," Bell System Technical J., Vol. XXXIX, No. 4. July I960, pp. 745-808. (8] Gradshteyn. I. S., and I. M. Ryzhik, Tables of Integrals, Series and Products, 4th edition (translated from Russian), 1965, New York: Academic Press, p. 397. [9] Campbell, G. A., and R. M. Foster, Fourier Integrals for Practical Applications, New York: D. Van Nostrand Company. 1942. (10) Klauder. J.R., el al.. "The Theory and Design of Chirp Radars," Bell System Technical J., Vol. XXXIX, No. 4. July I960, p. 755. (11) Cook, C. E . and M. Bemfeld, Radar Signals. New York: Academic Press. 1967. pp. 476-483. (12) Maynard, J. H., and B. F. Summers, "An Experimental High-Resolution Radar for Target-Signature Measurements," Supplement to IEEE Trans. Aerospace and Electronic Systems, Vol. AES-3, No. 6, Nov. 1967, pp. 249-256. f 13} Wehner, D. R.. "Tailored Response Microwave Filter." IEEE Trans. Microwave Theory and Techniques, Vol. MTT-17. No. 2. Feb. 1969. pp. 115, 116. [14] Remund, B. L and C. R. Srivaisa, "A 500 MHz Phase Generator for Synthetic Aperture Radar Waveform Synthesizers," Technical Digest of the 1991 IEEE GaAs IC Symp., 20-23 Oct. 1991, pp. 349-352. [15] Cook, C. E.. and M. Bemfeld, Radar Signals, New York: Academic Press, 1967, p. 159. [16] Klauder, J. R.. et al„ "The Theory and Design of Chirp Radars." Bell System Technical J., Vol. XXXIX. No. 4. July I960, p. 795. [17] Bringham. E. O.. The Fast Fourier Transform, Englewood Cliffs, NJ: Prentice-Hall. 1974, pp. 198-222.
Chapter 5 Synthetic High-Range-Resolution
Radar
5.1 FREQUENCY-DOMAIN TARGET SIGNATURES Any signal can be described as either a time or frequency function. The echo signal from a range-extended target illuminated by a typical radar RF pulse is usually observed in the time domain. Its amplitude and phase versus frequency is the echo signal spectrum, which is a frequency-domain description of the signal. Because descriptions of a signal as functions of time and frequency are equivalent, the signal spectrum can be obtained from the time-domain response and vice versa. Thus, measurements of a target's echo signal in the time and frequency domains provide equivalent data for determining target reflectivity. Consider first a radar that transmits short monotone pulses. The target's reflectivity profile in range delay can be defined as its echo signal amplitude and phase-versus-range delay measured with respect to the carrier signal of the transmitted pulse. For pulsecompression radars operating at the same bandwidth and wavelength, the matched-filter output signal is approximately the same as that for a short-pulse radar. In either case, a time-domain measurement of reflectivity produced by a single transmitted pulse is generated nearly instantaneously. A continuous series of short RF monotone pulses transmitted at a fixed PRF can be defined as a Fourier series of steady-state frequency components with a frequency spacing equal to the radar's PRF. Rather than transmitting the continuous train of short pulses, assume that all of the equivalent steady-state frequency components were transmitted instead. Then the Fourier series of the received steady-state frequency components of the echo from a fixed target would appear in the time domain as a periodic set of identical range profiles of period equal to the radar's PRI. The profiles would be identical to those produced by the train of short pulses, assuming identical radar and target geometry parameters. Reflectivity equivalent to that measured from the train of short pulses could be obtained from measurements of the amplitude and phase of each of the received Fourier series of frequency components relative to the respective transmitted component. This set 197
198
of frequency-domain measurements of reflectivity is the spectrum of the time-domain echo pulse train. In practice, what we want is the HRR reflectivity profile of a target, not the periodic echo response. Therefore, frequency spacing need only be the reciprocal of the target's range-delay extent, instead of the reciprocal of the radar's PRI. Also, the duration of each transmitted frequency component need only be sufficient to produce an approximation of the steady-state echo response. This is achieved for a pulse duration that is somewhat greater than the target range-delay extent. As we will see in this chapter, if a series of RF pulses were transmitted stepped in frequency from pulse to pulse over a bandwidth p , the set of sampled echo amplitude and phase measurements made relative to each transmitted pulse can be transformed by the DFT process into the range-profile equivalent of echo amplitude and phase measurements obtained relative to a transmitted short RF pulse of bandwidth /?. \ Thus far, the term reflectivity has been used to refer to the amplitude and phase of the echo signal at a given viewing angle for a given set of radar parameters. Reflectivity\ in terms of RCS versus range delay could be measured by an ideal radar with calibrated square-law detection of the echo signal power S. Absolute RCS of the target, in principle, could then be determined by solving for a in terms of 5 and the other radar parameters of (2.7) in Chapter 2. Square-law-detected responses from a short-pulse radar, in this way, could be converted into target range profiles of target RCS versus r a n ^ delay. An uncalibrated but otherwise ideal radar using square-law detection would generate profiles for which the signal is proportional to absolute RCS. An envelope-detected range profile is illustrated in Figure 5.1(a). Actual rangeprofile signatures from real targets appeared in Figures 4.40, 4.41, and 4.42 of Chapter 4. Early work to assess the target classification potential of these signatures was carried out by using sampled data of the form illustrated in Figure 5.1(b). Algorithms for classification of these range-profile signatures required prealignment of input signatures to each of the selected reference signatures. This requirement was typically avoided by conversion of sampled range-profile data into the range-invariant spectral magnitudes, illustrated in Figure 5.1(c), by means of DFT processing. Later, it was found that equivalent data could be obtained, while avoiding HRR processing altogether, by collecting the echosignal-magnitude data over the same bandwidth used to obtain HRR profiles, and by transmitting narrowband pulses stepped in frequency from pulse to pulse. Although found to be useful for target recognition, the resulting target signatures in the form of spectral data could not be inverse transformed into range profiles, because phase information was lost in the video detection process to obtain magnitude. Obtaining a discrete frequency signature that is the frequency-domain equivalent to a time-domain signature requires the retention of amplitude and phase of the echo signal at each frequency. When this is done, the process is sometimes called synthetic range-profile processing, because the target's range profile is not measured directly. In this chapter, a technique will be described for obtaining target range-profile signatures synthetically by processing echoes resulting from narrowband transmitted pulses stepped or hopped in frequency.
199
FREQUENCY — » •
Figure 5.1 Frequency spectrum from samples of envelope-detected HRR profiles: (a) square-iaw-detecled range profile; (b) sampled data of range profile; (c) DFT of range-profile-sampled data (magnitudes).
Pulse-to-pulse stepped-frequency waveforms avoid certain practical design problems associated with pulse-compression waveforms. The transfer function of the entire system of an HRR pulse-compression radar from transmitter through receiver, including data collection, must possess the bandwidth associated with the desired resolution. This is often neither desirable nor required for search and track functions and entails additional cost and complexity. Synthetic HRR, in contrast, is a capability that can be achieved in surveillance and tracking radar that has narrow instantaneous bandwidth, but is frequencyagile, to perform target recognition and target imaging, as well as narrowband search and track functions. A form of synthetic range-profile generation will now be discussed that is applicable to coherent radar systems using stepped-frequency sequences. An early version of this
200
technique is described by Ruttenberg and Chanzit [1] for use with magnetron radars operating in a coherent-on-receive mode. 5.2 CONCEPT OF SYNTHETIC RANGE-PROFILE GENERATION The process for generating a synthetic range profile of an illuminated target in the radar beam can be summarized as follows. 1. Transmit a series of bursts of narrowband pulses, where each burst is a sequence consisting of n pulses stepped (shifted) in frequency from pulse to pulse by a fixed frequency step size A/. 2. Set a range-delayed sampling gate to collect and digitize one pair of / and Q samples of the target's baseband response for each transmitted pulse. 3. Store the digitized quadrature components of each of the n received signals from each transmitted pulse burst. Each stored burst of target signal data approaches the equivalent of the instantaneous discrete spectral signature of the target seen at the same bandwidth and center wavelength if the following conditions are met: (a) burst times are short relative to time associated with significant target aspect change, (b) the duration of the baseband response exceeds the target range-delay extent, and (c) the frequency step size is less than the reciprocal of the target rangwJelay extent. 4. Apply frequency weighting to each burst of data and corrections for target velocity, system phase and amplitude ripple, and quadrature sampling bias and imbalance errors. 5. Take an inverse discrete Fourier transform (IDFT) of the resulting set of n complex frequency components of each burst to obtain an n-element synthetic range-profile signature of the target from each burst. Repeat the process, if needed, for N bursts to obtain N slant-range profiles, one range profile for each burst. The stepped-frequency waveform removes the requirements for both wide instantaneous bandwidth and high sampling rates by sampling near-steady-state reflectivity as the target is illuminated at discrete frequencies stepped, pulse to pulse. A functional block diagram of a stepped-frequency radar is shown in Figure 5.2(a). The block diagram of a practical design is illustrated in Figure 5.2(b). Waveforms are shown in Figure 5.2(c). A series of Nstepped-frequency transmitted bursts is illustrated in Figure 5.3. The transmitted and reference waveforms are shown in Figure 5.4. Envelopes of RF signals are illustrated. Narrowband pulses are assumed. The process can be described analytically by considering a single point target with radial motion relative to the radar. A train of stepped-frequency pulses with resulting echoes from a moving target are shown in Figure 5.5. The point target is assumed to have velocity v, toward the radar and is at an initial range R when time is zero. For the analysis to follow, the burst number k in Figures 5.2, 5.3, and 5.4 will be dropped, because only one burst is analyzed. The transmitted waveform is x&t). The received waveform is y,<0The echo delay of the moving target is T(/)-
201
POWER AMPLIFIER
(a)
DUPLEXER
TARGET •
K
PASS GATE
MONOSTABLE MULTIVIBRATOR REFERENCE SIGNAL GENERATOR! PULSE GENERATOR
MIXER #1
TIME DELAY 1
m (t)
MIXER #2
lk
RANGE DELAY
SYMBOLS vt R **<«>
TARGET RANGE TRANSMITTED WAVEFORM RECEIVED SIGNAL REFERENCE WAVEFORM TO I CHANNEL REFERENCE WAVEFORM TO Q CHANNEL I CHANNEL MIXER OUTPUT Q CHANNEL MIXER OUTPUT SAMPLED I CHANNEL
a.
t
SAMPLE HOLD #1
TARGET VELOCITY
SAMPLED Q CHANNEL TARGET SPECTRAL-SIGNATURE ELEMENT SYNTHETIC RANGE-PROFILE ELEMENT FREQUENCY OF STEP I
m
ik<"ik>
SAMPLE - * 1 HOLD #2
m
ADC #1
8
'lk< .k>
ADC #2
I
STORE COMPLEX ARRAY
EVALUATE I DFT EACH BURST |H/k
Figure 5.2 Stepped-frequency radar system: (a) functional block diagram, (b) practical design, (c) waveforms.
202
PRF trigger
Complex samples to processor
Source
t t s/h and A/D
<8>
BPF
Power amplifier
fl*fl Quad, detector
Baseband signal
1 Source
Transmit Of I
Duplexer
Single , J Sii ^ * \ a antenna n
Receive
Q signal •fl Range-de lay trigger(s)
IF amplifier
BPF
HEK-
Low-noise preamp
Typical/, = 60 MHz Typical / / = 10 GHz + & / , / = 0 , 1 , 2
255
Typical A/ = 1.0 MHz
(b)
^
* • t
Reference • •
-Echo delay Transmit and echo < pulse at / ,
Synchronously detected echo pulse
-Transmit pulse
/ Next transmit pulse @ / , + A/
-PRI-
I channel output . . . . rQ channel output -
(c)
Flgure 5.2 (continued)
Echo^ pulse
—i
203
204
TRANSMITTED WAVEFORM
'o
«,
n n » n n n iv-fl—« SECOND BURST k = 1
FIRST BURST k = o
REFERENCE WAVEFORM
Figure 5.4 Stepped-frequency transmitted waveform and reference waveform. ui
TRANSMITTED PULSES
Q 3
A
/
t(t) H
<
R ,
ECHO PULSES
n /n T,
n., n n,,
N
3T, 3T,
2T,
n
(n - 1)T (n - 1)T, 2
nT, nT,
FREQUENCY OF TRANSMITTED PULSES
V
f,
Ui 3
a Ui
A
1—
1 = 0
1= 1
1
= 2
-M
• t
1= 3
Figure 5.5 Stepped-frequency waveforms and echo pulses.
One n-step burst of a stepped-frequency transmitted waveform is expressed as Xj(0 = B, cos(2ir/» + A), iT, S t £ iT + T, . . . = 0, otherwise 2
n
/ = 0to/i-l
(5.1)
205
where 0, is the relative phase and B is the amplitude of the ith pulse at frequency step i and frequency f,. The received signal is expressed as t
yXD
=
B'
t
a a l l v f A t
-
Til)]
+
0,),
iT
+
2
r(t)
u
= 0,
< t <
iT
2
+
7,
+
r(t)
(5.2)
•
otherwise
where B' is the amplitude of the ith received pulse at frequency step i. Range delay for the target, which has an initial range R at time t = 0 is
The reference signal is expressed as Z , ( / ) = B cos(27r/f + (?;), iT < t < iT + T 2
2
2
i = 0 to n - 1
(5.4)
where B will be assumed to be a constant. The resulting baseband mixer product output from the inphase mixer before low-pass filtering is m,(f) = A, cos[-2rr/T(/)].
=
0,
n
iT + r(t) < t < iT + T, + r(t) 2
2
. . . otherwise
i = 0 to n - 1 (5.5)
where A, is the amplitude of the mixer output at frequency step <'. The phase of the mixer output is ,p,(t) = - 2 i r / r ( 0
(5.6)
From (5.3), the mixer output phase of (5.6) is expressed in terms of target range and velocity as
This is the total echo phase advance seen from transmission to reception for the ith pulse. The mixer output m,(;) of (5.5) is low-pass-filtered and sampled at time / = S„ where 5, is advanced pulse to pulse to produce a sample near the center of each echo response at baseband. Assume that the sampling time is set according to the expression (5.8)
206
where r, is the receiving system transfer delay (to the center of the response at baseband to a zero-range point target). The phase of the sampled quadrature mixer output (5.7) then becomes „
X2R
2v,/
2R\
(5.9)
The sampled output from the inphase mixer is m, = A, cos ip,. The sampled mixer output from both / and Q channels can be represented as G = A, [cos ip + j sin i/rj. Written in complex polar form, the sampled output becomes t
t
i
G, = A e *
(5.10)
i
The position of the sampling delay position S within the duration of the received echo response for most frequencies and target velocities of interest does not significantly alter the sampled outputs G„ except for the effect of the shape of the response envelope before sampling. The sampled quadrature mixer output signals G, for each transmitted burst are samples of target reflectivity in the frequency domain. The n complex samples in each burst are Fourier transformed by the IDFT, or FFT equivalent,*TO a series of range-delay reflectivity estimates H . This series of complex quantities is also referred to as the target's complex range profile. The IDFT is expressed as t
t
»-i H, = £ G,e* , um
0<(Sn-l
(5.11)
w>
where n is the number of transmitted pulses per burst and / is the slant-range position. This form of the IDFT includes the gain n associated with coherent processing of the n stepped-frequency pulses. From (5.11) with (5.10), ""'
27T
tf, = Y A e x p U ^ e x p j — / < 1=o «
(5.12)
The normalized synthetic response, assuming A, = 1 for all /, is expressed as
w
' =I
e x
Pj(v
/ ,
+
^)
( 5 1 3 )
Equation (5.13) at range R and zero target velocity with fa from (5.9) becomes (5.14)
207
For frequency step size A / , / =f + i"A/, for which (5.14) becomes 0
J2m/-2nRAf
H, = e x p ^ - j 2 ^ j g exp j ^ — ( — ^ T
+ /j
(5.15)
The above expression can be simplified by using the following identity (from 4.11),' which has been used for phased-array antenna analysis [2]: 0a
.
sin —
M y j«p = p=0
£_ j«/2
e
(5.16)
e
*-4
n
For B = n, p = /, and a = 2iryln with -2nRAf
,
(5.17)
we obtain the synthetic response of (5.15) as / .„ , 2R\ sin Try /.n - I 2vy\ H, = exp/ - J 2 7 T / 0 - J — exp/j— — J 'sin — ' v
(5.18)
x
The magnitude of the complex form of the synthetic range profile represented by (5.18) is sin rry\ ny\
(5.19)
We can show that the envelope of the synthetic range profile described by (5.19) for IRIc = / in y of (5.17) is identical to the (sin i77iA/r)/sin irkft part of the matchedfilter response of the periodic discrete frequency-coded waveform of (4.14). In both cases, a stepped-frequency waveform was assumed, each with n frequency steps of A/ Hz per burst (called period in Chapter 4). Discrete values |/Y,| of the synthetic range-profile envelope of the response to a point ttrget are illustrated in Figure 5.6 along with the corresponding profile envelope. Responses I . The symbols a and ft as used here represent variables associated with this identity only, and do not refer I l o bandwidth, as is the case elsewhere in the text.
208
IHJ
D I S C R E T E VALUE O F SYNTHETIC RANGE PROFILE
n
PROFILE ENVELOPE
0 1
2
Figure 5.6 Synthetic range-profile response lo a fixed point target.
to a point target will maximize at y = 0, ± n, ± In, ± 3n, . . . The range p o t i o n nearest each of these peak responses will be referred to as / = l . Range positions corresponding to index l are given by 0
0
R =
dp 2nA/
+• n) c(l + 2n) 2nA/ ' 2nA/
c(
0
(5.20)
An unambiguous range length of c/(2A/) is evident. Discrete range positions in meters corresponding to indexes / = 0 to n - 1 within the profile envelope are determined by the choice of frequency step size. Sampling resolution can be defined as the range increment between any two adjacent discrete range positions along the profile. A set of n frequency steps produces n equally spaced range increments within the unambiguous range length c/(2A/), so that sampling resolution is expressed as
' 2lkf
Ar
=
(5.21)
It can be shown that for large n, the resolution defined as the range distance between the 2Jir points of the synthetic range-profile envelope given by (5.19) approaches the sampling resolution given by (5.21), which corresponds to the Rayleigh resolution c/(2A) measured between the 2/w(-4 dB) points of the sin A7ri7(7n)-shaped envelope of a single chirp pulse having a chirp bandwidth A. Frequency weighting to reduce sidelobes will degrade range resolution to c/(2fi), where J3 is the weighted bandwidth.
209
Equation (5.21) with nA/ = B can be seen to conform to (1.1) of Chapter 1, which expresses the fundamental dependence of radar range resolution on radar bandwidth. A hypothetical set of stepped-frequency echo signals at baseband produced by a multiplescatterer target and the resulting complex range profile are illustrated in Figure 5.7. 5.3 EFFECT OF TARGET VELOCITY Expressions (5.19) and (5.21) were obtained for the synthetic range profile of a target at a fixed range. The effect of target velocity can be examined by considering a point target at range R with velocity v, toward the radar. The expression for the synthetic range profile of the moving point target obtained from (5.13) with $ from (5.9) is
SAMPLED STEPPED-FREQUENCY D A T A , G,
PROCESSED SYNTHETIC RANGE PROFILE, H ,
Rfsre 5.7 Sampled stepped-frequency target data and processed synthetic range profile.
210
where
%
=
— n
li
-
(5.23)
lirfi
In terms of quadrature components, we have a-l
a-l ,
/Y = £ c o s ¥ , + j £ s i n I ' , 1
(5.24)
The phase of H, is £sin
(5.25)
= tan ]T cos
The magnitude of the synthetic range profile (5.24) was evaluated w parameters associated with a U.S. Navy experimental wideband radar at the NOSC. Results are shown in Figure 5.8 for a 256-step burst, and in Figure 5.9 we have results for a 25-step burst. The characteristic shift in peak response associated with moving targets observed with linear (up-chirp) FM-coded waveforms is apparent in these results. For targets with positive velocity toward the radar, the shift, as in Figure 5.9, is to an earlier time (i.e., less delay). The true target position is in, or near, range bin 1 for Figure 5.8 and range bin 25 foi Figure 5.9. The effect of target velocity shown in Figures 5.8 and 5.9 can also be predicted by referring to the ambiguity surface of an equivalent chirp pulse. In Figure 4.36, the peak response at r = 0 out of the matched filter to a chirp pulse of duration T becomes zero when Doppler frequency reaches fo = ±1/7;. A point target at velocity v, produces a Doppler frequency shift of lv,IX. The velocity that produces the first null response at the matched range-delay position for synthetic processing of an n-step burst at a PRI of Tj can therefore be expected to occur for (5.26) where nT corresponds to chirp-pulse duration 7V For the parameters of Figure 5.9, the first Doppler null response occurs for velocity 2
A
2/iT
2
c = 10.3 m/s 2n(/ + ,14/72)72 0
(5.27)
211
n 1 At Tj 0
= 256 - 3000 MHz = 1 . 0 MHz = 180 us
METERS/SECOND
i
0
1
20
1
40
1
60
1
80
1
100
1
120
1
1
140
160
1
180
1
200
1
220
1
240
RANGE CELL NUMBER
- figure S.8 Synthetic range profiles of a point target obtained with stepped-frequency processing for various velocities at the same range (256 frequency steps). (Courtesy of John A. Bouman, formerly of the Naval Weapons Center, China Lake, CA.)
for A approximated as c/(/ + nA//2). In Figure 5.9, we can see that the response of a stationary target has nulled at a range position where a 10-m/s target produces a peak response. Serious distortion does not appear in Figure 5.9 until a velocity of over 30 m/s b reached. Distortion seen in Figures 5.8 and 5.9 can be reduced by multiplying complex received data at each frequency step i by the complex factor 0
f r2v,/ 2/?\"n
(5.28)
213
where R and v", are estimates of target range and velocity, respectively. Methods for velocity correction of data collected from isolated ship and air targets will be discussed in Chapter 7. Range-Doppler coupling from Figure 4.36 is T = -f IK, where, for the stepped-frequency sequences, K = A//7V Range shift in terms of target velocity becomes D
c
c
T=
2v,flc
V,fT2
=
( 5
2 ~2~Jf7r ~~~Kf~
2
9
)
1
As an illustration, range shift from (5.29) for parameters of Figure 5.9 for a +50-m/s target is Sr = -1.462m. The number of range cells of shift Sr/Lr, to the nearest integer, with range resolution 0.30m as given by (5.21), is - 5 cells. It is can be seen in Figure 5.9 that the synthetic response to a point target at +50 m/s is shifted from range cell / = 25 for the target at zero velocity to about / = 20. Range-Doppler coupling becomes a problem when multiple targets at different velocities are to be located relative to one another in range or when single or multiple targets are to be located in absolute range. The problem is avoided by the use of pulseto-pulse hopped-frequency waveforms, to be discussed below. Examples of actual synthetic range profiles are shown in Figures 5.10, 5.11, and 5.12. The significance of synthetic range-profile processing can be illustrated by consider8.0 7.0 6.0 5.0 4.0
i
3.0
X ui
2.0
I
2
p
1.0 0.0
100
200
Figure 5.10 Synthetic range profiles of a fishing boat.
300
400 RANGE (ft) •
214
8.0 1
Figure 5.11 Synthetic range profiles of a moving small craft.
ing Figure 5.13. A stepped-frequency radar is shown observing a target using a pulse duration of 3 LIS, equivalent to about 1,500 feet in range. Radar resolution from (5.21) with 256 1-MHz frequency steps is about 2 feet. The target, although immersed in the resolution cell associated with the transmitted pulse, is resolved into individual scatterers by using the synthetic process. 5.4 HOPPED-FREQUENCY SEQUENCES
2
The terms hopped-frequency sequences and hopped-frequency waveforms will refer in this book to bursts of pulses for which frequency changes pulse to pulse in a prescribed or pseudorandom order other than linear frequency shift with time, as with steppedfrequency sequences. The transmitted frequency of the ith-frequency pulse of a hoppedfrequency burst, like that of a stepped-frequency burst, is given by /=/o+/A/
(5.30)
2. Waveforms consisting of hopped-frequency pulse sequences are a low-duty-cycle subset of a larger class of pseudorandom frequency-hopped waveforms investigated by Lance C. Martin of Martin Consulting primarily for high-PRF, high-duly-cycle radar surveillance. This class of waveforms provides both high coherent processing gain and large regions of unambiguous range/velocity target response. The analysis here was developed from discussions with Mr. Martin and from his unpublished notes.
215
Figure S.I2 Synthetic range profiles of a moving ship.
but where i for an n-pulse hopped-frequency burst is an integer from 0 to n - 1 selected in a prescribed or pseudorandom time order other than the linear order / = 0, 1,2 n -1 in a stepped-frequency sequence. As for stepped-frequency bursts, a hopped-frequency burst contains all of the frequencies from i = 0 to n - 1 with no repeating frequencies. The complex sample collected from the baseband echo response produced by the yth transmitted pulse of a hopped-frequency sequence for j = 0, 1,2 n = 1 can be expressed as Gj = Aje'*
(5.31)
where for an isolated target at range R,
Processing of hopped-frequency data is performed in the same way as stepped-frequency processing, except that the hopped-frequency data sequences are reordered in a linear frequency sequence before performing the IDFT. The synthetic range profile, obtained by processing (5.31) applied to a fixed point target, is identical to that given by (5.18)
2/7
and (5.19) for the stepped-frequency sequence. Synthetic responses to finite velocity targets, however, remain fixed in range with pseudorandom hopped-frequency sequences, in contrast to the shift seen in Figures 5.8 and 5.9 for the stepped-frequency sequences. This is illustrated in Figure 5.14 for the parameters of Figure 5.8. Note that the rangeDoppler coupling is absent. Also, the relatively Doppler-invariant behavior of linear chirp waveforms is absent with the hopped-frequency sequences. Short-range surveillance for applications such as vehicle navigation allows operation at high PRF without the range ambiguity associated with operation at high PRF at longer ranges. These systems can be designed for some applications to operate at a high enough PRF to produce one or more hopped-frequency sequences per antenna beam dwell in a search mode to unambiguously map fixed surface features and locate and track moving targets. In a typical application, the radar scans a selected azimuth sector containing multiple fixed and moving targets in a clutter environment. Complex data samples are n / A/ T 0
2
=256 = 3000 MHz = 1 . 0 MHz = 180 n s
Meters/second
-3.2 -2.8 -2.4 -2.0 -1.6 -1.2 -0.8 -0.4
/ V
A_
-0.0 10
20
30
40
50
60
Range cell number, /
Figure 5.14 Synthetic range profiles of a point target with hopped-frequency processing at various velocities at the same range (256 frequency steps).
2/8
collected from the baseband response at one or more range sample positions extending over the selected surveillance range. Range sample positions can be thought of as coarserange bins in which synthetic high-resolution segments will be generated. The sampling strategy for processing over multiple range sample positions is discussed below. Processed complex data samples collected in each coarse-range bin are summed over the selected surveillance range at multiple velocity bins to produce high-resolution range/velocity target responses during azimuth scanning, as illustrated in Figure 5.15. The antenna in this illustration is assumed to scan in azimuth at a fixed elevation near grazing angle. Targets for the instantaneous azimuth position 0illustrated in Figure 5.15 are shown at /?,, v, and R , v , respectively. Fixed clutter return is illustrated in portions of the zerovelocity range profile. Detected moving targets would, in a typical application, be entered into a track file for scan-to-scan update. Figure 5.16 illustrates transmitted and received signals for one hopped-frequency burst of n pulses. The two targets are at velocities v and v and start ranges (not shown) of R, and R , respectively. The range sampling interval is 7" . The transmitted-pulr?. duration is T, and baseband response width is 27",, which extends over four sampling intervals. Table 5.1 lists frequency, sampled phase, and needed velocity correction for sampled data 2
2
t
2
Figure 5.15 Azimuth scanning with hopped-frequency waveforms.
2
3
219
Baseband response (I or Q) for target 1 at +v 2
' Transmitted pulse at /g
Baseband J / for target 2response at - v (I or Q ) 2
.0
I I'l I I I I I I I I I
II I I I I I II I I I / / I'l I M ) I I I I I t'l
•Delay
W2
K-r
3
\})[(7\ \ 1111111 (~+\ 'I II I I I I I I I
i i i i i i i i i i i i/ /53
+~ Delay
I
27IT( | [| | | | | | | | | |))f+H^|I |I II| I III III III II IIMmN I I I I I I III 2
i-Delay
/28
(n-1) T
2
7^/^+4+ I I I I I I I I I I I \/fl I I I HI I I I I I I I I'l IZ+I^ IMIIIIIIII
•Delay
Figure 5.16 Hopped-frequency data collection (one burst).
collected in range sample position m i . where the baseband response to target I at range R, appears. In general, sampled complex data from a target at R, v, for pulse number j , range sample position m, and sample delay interval 7 , is 3
G = i4 expjJ-2ir/^-^0-7 -rm7 ) j ,
i
y
I
1
(5.33)
The velocity correction factor for the yth pulse is [ = exp j j-2ff/j^'( i7i +
mTy)
jJ
(5.34)
Velocity-corrected data for the jth echo pulse from targets at range R and velocity v, is
(Gj) = Aj exp j|-2w/!^J c
(5.35)
220
Table 5.1 Collected Hopped-Frequency Phase Data and Velocity Correction Factors for Figure S.16 (for Hypothetical Pseudorandom Frequency Sequence)
Pulse Number, j
Freq., f,
0
h
1
In
2
Argument of Velocity Correction Factor for Vi Velocity Bin
Sampled Phase.
-2,^-^W,)]
/„
- 2 i r / „ ^ - ^ ( 7 , + m,7i) j
-2irf ^(K
- 2 t t / ^ - ^ ( 2 7 , + m,T>) j
- 2 ^ , ^ ( 2 7 , + m,T ) j
- - ^ O T . + m,r,) j
- 2 7 r / ^ 0 T ! + m.T,^'
- ^ ( 6 3 7 , + m,r,) j
- 2 7 r / ^ ( 6 3 7 - , + m,^) j
u
- 2 * / ^
63
f»
- Z ^ ^ T l ) ]
-2*17
i2
+ m,T,) j
t
a
Note that the sample time for surveillance in (5.33) is represented as_/T, + mT> with delay mTt, in contrast to iT + r + 2R/c with delay T, + 2R/c in (5.8) for sampling the response from an isolated target at range R. Inverse DFT processing, after frequency-reordering the velocity-corrected hoppedfrequency sample data of (5.33), produces synthetic profiles expressed (with j = i) in the same way as for stepped-frequency data in (5.14), shown to be the high-resolution response of (5.19). Consider the processed response to targets 1 and 2 of Figures 5.15 and 5.16 and target 1 of Table 5.1. The peak response for target 1 will occur at range and velocity position R,, v, in Figure 5.15. Other targets at or near range R may also contribute to the baseband response sampled at range sample position m,. Responses to these targets, however, will be suppressed unless their velocity is at v,. The peak response for target 2 will peak at range and velocity position /? , v if velocity correction is made for velocity v. 2
r
t
2
2
2
5.5 RANGE-EXTENDED TARGETS The narrowband echo signal resulting from the summation of the individual responses from scatterers of an extended target is illustrated in Figure 2.7 of Chapter 2. The angulai
221
extent of summation of scatterers from extended targets is limited by the radar beamwidth. The range extent of the summation of scatterers from extended targets is limited by the duration of the point target response seen at the baseband output of the low-pass filters (LPFs) in Figure 5.2(b). It is convenient to assume that the receiving system bandwidth is matched to the transmitted-pulse duration T,. Then the range extent over which individual scatterers contribute to the amplitude and phase of the response at baseband is cT|, which is c/2 times the duration 27, of the triangular matched-filter response to a point target illuminated by a rectangular transmitted pulse. For receiver bandwidth > 1/7",, the range extent over which scatterers contribute to the baseband response reduces toward cT,/2, which is the range extent of the transmitted-pulse width. Design considerations for sampling and processing two types of extended targets will now be discussed: (1) a single extended but isolated target, such as an aircraft or a ship, and (2) extended surveillance target areas in the presence of clutter.
5.5.1 Isolated Targets To obtain the undistorted range profile of an isolated target, the complex sample of reflectivity collected at each frequency in the burst sequence must approximate that obtained from a steady-state signal with uniform target illumination. Stated differently, echo signals arriving from each of the multiple reflection points of the target must be summed in the receiving system, before sampling, with nearly uniform weighting across the target's range and azimuth extent. This condition is met for a receiver bandwidth that is much less than the reciprocal of the target's range-delay extent, assuming that the crossrange extent of the target is uniformly illuminated by the radar's antenna beamwidth. When the receiving system bandwidth is matched to transmitted-pulse duration 7",, distortion-free range profiles are approximated for a target range-delay extent that is much less than the duration 27 , of the triangular baseband response. For example, to image a 300m ship target without distortion requires a matched transmitted-pulse duration exceeding 300/c = 1 /is. The matched pulse duration should be several times longer than the range-delay extent of expected targets to approach uniform weighting. Target range extent also sets an upper limit on frequency step size. From the first expression of (3.81), the maximum frequency step size for a 300m ship target was shown to be c/(2 x 300) = 0.5 MHz. The azimuth extent of target illumination must also exceed 300m. 1
5.5.2 Surveillance Applications We now consider applications such as ground mapping and other surveillance functions, wherein targets of interest are not isolated in range, but extend continuously over relatively large ranges compared to, say, the range extent of a ship or an aircraft. Requirements for
222
frequency step size and matched pulse duration based on surveillance range extent for these applications tend to be impractical. Practical stepped-frequency and hopped-frequency sequences can, however, be used for surveillance of range-extended target areas by creating a set of fixed range-sample positions extending over the selected surveillance range as illustrated in Figure 5.17. T/his allows unambiguous sampling at each sampling position, with pulse duration and frequency step size determined by a small segment of the extended range. Overlapped unambiguous segments of processed data from the multiple range-sample positions are summed to form the extended synthetic range profile. The top of Figure 5.17 illustrates one channel of the baseband signal produced by one pulse of a sequence of stepped-frequency or hopped-frequency echo pulses. Individual components of the baseband signal appear as triangular / and Q responses, each corresponding to the convolution of the rectangular baseband echo signal from an individual scatterer with the rectangular matched-filter impulse response of an idealized low-pass filter. The baseband signal itself, which is the sum of responses for individual scatterers, is shown as the dash-line signal. Triangular components, though all shown for convergence as I or Q for target A alone at / I or Q (or target B alone at // I or Q (or target C alone at / ;
Baseband I or Q output Transmitted pulse I
I
(
1-
-^f^K
-+r?H 1
2
3
4
5
1—*- Range
6
Unambiguous range window A
A
B
AIL
> >—•
A AB
- Range
• Range C
A
B
• Range
Hi(4)
B
F,
• Range C
• Range
Figure 5.17 Unambiguous range sampling for extended surveillance regions.
223
positive, would actually be bipolar. The unambiguous range windows shown in Figure 5.17 are generated by selecting a frequency step size equal to 1/(2T|), where 2T, is the duration of the triangular / and Q signal, before sampling, for an ideal matched filter matched to the transmitted pulse of duration T,. Processed responses from stepped-frequency data collected at f to /„_, from range gates 2. 3, 4, 5, and 6 are illustrated as H{2), Hi~i), H/iA), H{5), and 77,(6), respectively. Synthetic high-resolution responses to individual scatterers can be seen to occur at range positions corresponding to their true ranges within the unambiguous range windows centered for illustration at their respective range-gate sample positions. Actual positions of scatterers seen in the synthetic range profile domain will depend on the IDFT process, but the relative range alignment of the range profiles HJi2) through H16) remain constant. Dotted responses represent multiple ambiguous IDFT outputs that fall outside the ambiguity windows associated with the five sample positions, and they will not contribute to the sum of the overlapped segments H&) through H/(6). Consider first the H{2) profile, which was obtained by IDFT processing of frequencydomain data collected at sample position 2. Sample position 2 can be seen to sample the contributions to the baseband response from scatterers A and B only. The amplitude and phase of the sampled data are the complex sum of triangular components produced by echoes from scatterers A and B, respectively. Contributions from both of these components of the baseband response can be seen in range-sample position 2 to be sampled below their peak response. Resolved scatterers A and B are can therefore be seen in HIT) at a reduced level. Scatterer A can be seen at its maximum value in H^i) processed from range-sample position 3 data, where scatterer B also appears larger because range-sample position 3 is closer to the range position of scatterer B and thus closer to the peak of its contribution to the baseband response. Range-sample position 4 samples the baseband response where / and Q components are determined by scatterers A, B, and C. None of the contributions to the baseband response for these three scatterers are near the peak of range-sample position 4. As a result all are seen to appear reduced in amplitude. Profiles H,(5) and Hffi) can be understood in the same manner. Individual profiles Hi\) through Hffi) can be seen to possess the following characteristics: (1) resolved scatterers associated with sampling at the five range-sample positions appear in range alignment; (2) although a given scatterer appears at different amplitudes in different profiles, the complex sum of the profile amplitudes //XI) through 7/,(6) will show the three resolved scatterers at approximately their correct relative amplitudes; (3) no foldover appears from responses outside the unambiguous range windows. It is concluded that contiguous synthetic high-resolution profiles of extended-range surveillance regions can be obtained by complex summation of overlapped unambiguous segments of processed data from multiple range-sample positions. Ambiguity in the form of foldover of responses is eliminated by selecting frequency step size A/ so that the unambiguous range-delay window 1/A/is at least the duration 27", of the baseband response to a point target following ideal matched-filter low-pass filtering. 0
224
Two distortion effects are noted. First, some discrete scatterer positions could be found where separate portions of the response, though not ambiguous, would appear at both edges of one of the ambiguity windows. The summed responses that form the contiguous extended range profile would then include two responses to the single scatterer. This effect could be eliminated by complex summing of processed responses jfrom H, segments in Figure 5.17 which are reduced slightly from c7",. Secondly, some distortion is produced by the nonlinear insertion phase that exists in practical low-pass filters. Sampled baseband data, when phase nonlinearity exists, will shift in phase with the sample position, which introduces phase ripple. However, since the summed contiguous profiles contain scatterer responses from five sample positions, samples taken near the peak, being of larger magnitude, will dominate the sum, thus reducing the distorting effect of phase nonlinearity. In an actual system, baseband responses will deviate from the triangular shape illustrated in Figure 5.17. Sample spacing T} could vary from as small as desired to as large as the pulse duration T . The penalty for small sample spacing is increased complexity. The penalty for larger spacing is reduced fidelity with severe degradation appearing when 7/3 approaches 2T,. {
5.5.3 Surveillance Example Hopped-frequency parameter selection will now be illustrated for a hypothetical allweather landing radar that displays to the pilot the runway, surrounding fixed structures, moving ground vehicles, and moving aircraft on the runway during final approach from 2 km to 75m from touchdown. The requirement for maximum two-dimensional surface resolution cell is set as a function of distance from touchdown in three stages: (1) 20m by 20m for distances between 2 km and 1 km, (2) 10m by 10m between 1 km and 500m, and (3) 5m by 5m between 500m and 75m. Moving ground targets are to be displayed unambiguously over at least ±40 m/s with velocity resolution cells smaller than 3 m/s. The maximum antenna dimension to be accommodated in the aircraft nose is lm. The display over a forward-looking sector of 7r/4 rad is to be updated at the rate of one update per second. (The above application and requirements are selected for illustration only and' are not based on an analysis of any actual landing system needs.) Pulse duration T, is,selected to be 0.5 /AS, which allows surveillance at a range as small as c7",/2 = 75m without attempting to receive while transmitting. Time interval 7j between sample positions following each pulse is 0.25 /AS, which corresponds to about four samples per matched-filtered baseband (coarse range) resolution cell. Unambiguous range profiles are obtained for step size A/ = 1/(27*,) = 1 MHz. As the runway is approached, bandwidth is increased by increasing the number of pulses per burst in three stages to approximately match processed slant-range resolution, with cross-range resolution provided by the antenna beamwidth. From (5.21), the number of pulses, before rounding to the nearest V where 7 is an integer, is
225
(5.36)
2A/Ar,
The wavelength is chosen to be 8.57 mm, which corresponds to a frequency of 35 GHz. The azimuth beamwidth, for the allowable lm aperture at this wavelength with the approximation <^ = XJd, equals 8.57 x 10~ rad (0.5 deg). The azimuth resolution is Ar = R
dB
r
2
2
D
3
8.57
x
IP'
77/4
3
= 1.09
x
10-
2
(5.37)
Doppler resolution could be as fine as \lt if multiple bursts of hopped-frequency echo data collected during dwell time t were processed for Doppler as a single sequence. The resulting velocity resolution would then be d
d
Av, =
—r—
= — = 0.39 m/s
(5.38)
corresponding to tJT = 546 pulses per dwell. Achieving the required 3-m/s velocity resolution requires coherent processing of a minimum number of pulses per dwell given by 2
„ . = 72 pulses, 27 Av, n
2
9 bursts from 2 km to 1 km 4 bursts from 1 km to 0.5 km 2 bursts from 0.5 km to 75m
(5.39)
It is assumed that weather penetration would be adequate at up to the maximum range of 2 km for this example. Also, aircraft altitude is assumed low enough that slant range and ground range are approximately equal.
226
Range processing for the above design would have been further complicated if the target scatterers migrated out of processed range resolution cells during the coherent integration time nT . Range migration for targets approaching the radar at velocity v, js 2
(5.4b) Ar, The worst-case situation is the 0.5-km to 75m stage, where n = 32 and Ar, = 5m. Migration, assuming a maximum closing speed between ground targets and radar of 150 m/s, is about 0.02 range cells. It is concluded that range migration is not an issue for this design. 5.6 RANGE-PROFILE DISTORTION PRODUCED BY RANDOM FREQUENCY ERROR The derivation of (5.19) for the synthetic range profile of a point target assumes jv^cise A/ frequency steps. Any frequency deviation from / = f + /A/ results in distortion. In Chapter 3, random frequency error JC, of a sequence of constant-frequency transmitted pulses was shown to produce cumulative random phase error, which reduces the expected peak value and increases the noise floor of the coherent sum of the corresponding sequence of received pulses from a point target. It will now be shown that analogous distortion mechanisms occur for stepped-frequency sequences. Pulse-to-pulse frequency deviation of transmitted stepped-frequency sequences will be shown to reduce the peaks of the processed target range profiles and introduce noise elsewhere along the profiles. In other words, the noticeable effects in processed data will be reduced SNR and contrast. Positions of peaks and nulls are not disturbed. Stepped-frequency processing will now be examined for the point target case to assess distortion caused by random frequency error and to determine the relationship of this distortion to radar range. To examine the effect for a point target, (5.10) with if/, from (5.9) for zero target velocity is rewritten as a
(
2R
-2irf,— -vx\
\
(5.41)
for A , = 1, where vx is the random cumulative phase error produced by random frequency error x in the ith frequency step for phase constant v. A random error JC, in frequency step i produces a random cumulative phase error in G, of -2ir(2Rlc)x,. For ideal coherent transmitter systems, the only frequency error is that produced by the frequency synthesizer. Thus, the phase constant associated with random phase error that accumulates during target range delay is v, = 2irQ.RIc). Reduction in the range-profile response produced by random phase error is derived by substituting (5.41) into (5.11) and solving for the peak response of the expected value t
t
227
E[H£x,)], which occurs at y = 0, ± n, ± 2n, ± 3n, . . . The noise-power floor is derived by substituting (5.41) into (5.11) and solving for variance
2
H
1. The magnitude of the peak expected value is ElHfa)] = nC at positions of peak |W,| f
(5.42)
and
2. The variance at the null positions is cr'mx,))
= nC}[l ~Cj]
(5.43)
where 2
/
i^a \
C =exp[--^J The expected ratio of peak signal to variance from (5.42) and (5.43) is /
S
(5.44)
n
1
where the peak signal power is 5 = (nCj) . The symbol C is the characteristic function of the random frequency variable x, and the symbol a, refers to the pulse-to-pulse standard deviation of the frequency-synthesizer output frequency. The quantity v,a, in (5.44) is the standard deviation (rms) of cumulative phase noise a of the instantaneous cumulative phase noise
c
228
band. The Allan variance approach is useful when it can be assumed that transmitted phase noise is primarily determined by the frequency stability of the radar's frequency reference and when Allan variance data for the frequency reference is available\for averaging times near the radar's PRF for pulse radars. Acceptable values of standard deviation of frequency error will now be calculated for a 3-dB loss to illustrate the effect on range performance. From (5.42) a 3-dB loss in response peaks occurs for Cj = 0.5, which from (5.44) with v, = 2w(2R/c) occurs for 2R v.a, = 2TT— a, = 0.8326 c
(5.46)
for which standard deviation in hertz is
a
-
=
0-8326 —2R 2n— c
( 5 4 7 )
In terms of range in nautical miles we obtain the criteria
a
'~
0.8326 2 x I852 x/?(nmi) 3 x lOWs)
(
'
10,732 ~ fl(nmi) The maximum tolerable random frequency error for several values of radar range are listed in Table 5.2. A more complete analysis would need to consider pulse-to-pulse phase noise produced by pushing the transmitter transfer phase by random pulse-to pulse variation of the pulse modulation (current, and voltage). Table 5.2 Tolerable Random Frequency Error Versus Radar Range (3-dB Loss) R (nmi)
10 100 1,000 10,000 100.000
1,073 107 11 1.1 0.11
229
5.7 RANGE TRACKING OF MOVING TARGETS
Synthetic HRR techniques can be used to generate synthetic range profiles of moving targets such as ships and aircraft. Range tracking is required to sample the received signal phase and amplitude from each transmitted pulse as the target moves either in or out of range from the radar. The sampling of quadrature mixer outputs m ( 0 and m' (t) of Figure 5.2(a) produces a pair of / and Q samples of each echo pulse at baseband. The phase determined from each / and Q pair is nearly independent of the sample location on the pulse, except that amplitude and therefore SNR decrease for sampling at the pulse edges, where response is reduced as determined by the transmitter-receiver bandpass characteristic. Synthetic HRR tracking requirements are, therefore, not as severe as those for real HRR, for which the delay trigger position, to prevent blurring on the A-scope display or degradation of the recorded data, is required to move from pulse to pulse with precision, corresponding to range jitter that is far less than the range-delay resolution. Incremental shifting of the sample position along the middle of stepped-frequency responses at baseband does not significantly affect the / and Q samples, and has proved to be a convenient way to adjust the delay trigger for target tracking. Increments of range shift may be a convenient fraction of the baseband response duration that is much greater than the synthetic delay resolution. u
a
Figure 5.18 is a block diagram of a tracking system used for experimental ISAR imaging tests for the U.S. Navy at the NOSC. A basic clock is set to generate a frequency that, when properly divided, results in convenient range readout increments. The NOSC tracker uses an 8.09-MHz clock rate to drive down-counters for both the main (transmitter) trigger and delay trigger. The result is a main-trigger count interval of 123.6 ns, which corresponds to a 0.01-nmi range increment. The main-trigger counter, following each count down to zero, is reset to a count n, corresponding to the desired PRF. For example, a reset count of n, =1,500 produces a PRF of 5.39 kHz, corresponding to an unambiguous range of 0.01 nmi x 1,500 = 15 nmi. A main trigger is generated and the counter is reset to 1,500 each time the counter is counted down to zero from 1,500. The delay-trigger counter is clocked at the same rate as the main-trigger counter, but the range reset count n, is controlled either manually or automatically by means of an up-down counter. Assume that we want to move the delay trigger in range delay toward that of a target of interest. The count n, set in the delay-trigger down-counter is then set to some value less than 1,500 that corresponds to the range delay of the target. Both the main-trigger counter and delay-trigger counter are driven by the same clock and are reset at the same instant. Thus, the range counter will count down to zero to generate a delaytrigger pulse to sample the target response before the next main trigger occurs. The delaytrigger reset count for close-in targets will approach zero. For targets near the maximum unambiguous range of 15 nmi, the reset count will be near the 1,500 count set in the main-trigger down-counter. For moving targets, the delay-trigger reset count changes continuously to maintain target tracking.
230
231
The delay-trigger reset count n, is produced by the up-down counter in the middle part of Figure 5.18. The count in this counter is adjusted up or down as needed for target acquisition and range tracking by manual or automatic control of the VCO, which oscillates at a nominal frequency of 8.09 MHz. The gross rate is controlled by the VCO rate switch. Automatic range tracking is achieved by controlling the VCO from a conventional earlylate gate range-tracking circuit. The end-of-sequence (EOS) burst input of Figure 5.18 prevents the count of the up-down counter to the range counter from changing during a burst. 5.8 DEGRADATION PRODUCED BY RANDOM FREQUENCY ERROR We now develop expressions for the expected values of peak and variance of the synthetic range-profile response for a single point target in terms of standard deviation trof frequency and associated radar parameters. The synthetic range profile is generated by the IDFT of the complex sampled data set G, of (5.41) obtained for a point target at range R illuminated by one burst of n pulses. The transmitted frequency f, contains random frequency error i,. Substitution for G, from (5.41) into (5.11) with / = / + /'A/ yields 0
W^,) = X
e
C
K
c
'^"
>lc->"'
(5.49)
Let -2nA/R
+I
(5.50)
Then with (5.50) and taking terms not including i outside the summation, we write (5.49) as ^ ^ e ^ ^ X e i ^ K -
(5.51)
The expected value [3] of fifa) is ElMxd)
= J l j l
•• -
£/7Xx,Kro. x
^-OAtodjt, . . . dx..,
(5.52)
where p(x , x *„_,) is the joint probability density of random frequency error XQ, X,. x,-,. Substitution of H^xi) from (5.51) into (5.52) results in the expression a
EiHix,)]=r
r ••r
j
^
A
^
A
^
"
^
p^,
x
^.od^d*,... d*.., (5.53)
232
For p(x,) independent of p(Xj) for all i and j except i = integration, we can write (5.53) as W
]
=
and exchanging summation apd
Iff., . j V ^ e ^ Y -
(5.54)
x P(x )p(x,) ... p(x„-,)dxadx, . . . dx„_, 0
where pfo) is the probability density function of frequency error x,. By carrying out the summation in (5.54) for i = 0, 1, 2 . . . . , / » - 1. we have 3
,
1
EMU)] = j V 4 ^ e ~ ^ e + » p ( * ) x J^pfx.jdx, x j^pfxjjdxi... J^p(jc,.,)dx,., jd^fl fyr*-k'—'^ >x )
+
P
t
x ^£p(xo)dx x £_p(x )dx,... £p(x„.,)dx„_, jdx. 0
2
(5.55)
e"
p(x..,)
x j^p(x )dxo x £p(x,)dx, . . . £p(x„_,)dx„-i jdx.,, 0
For all i, we can write
v
£pfx,)dx, = I
(5.56)
£lH,(x,)l = " f e ^ ^ e ^ T c-^pMdx,
(5.57)
Then (5.55) with (5.56) becomes
3. The index ( = 0 is indicated* but not multiplied out. in the hope of maintaining clarity.
23}
A normal distribution of frequency error with standard deviation a and zero mean will be assumed for p(x,) for all /. The standard deviation is expressed as [4] !
p(x) = — ^ = e - * * " ' er-^2 JT
(5.58)
Equation (5.57) with p'x) from (5.58) becomes
£[///(*.)] = Ye"'' *"' IA "' \
!/
— =e->"e' "°''dx ~tryJ2ir 7
J
i«o
i
(5.59)
Terms not including x can be brought outside the summation. Also omitting the subscript i inside the integral, we obtain (5.59) as E\HAx,)) = t^
'*' Y e'^"'' f — ^ - ' " e - ' ^ ' d x o-\J2ir
(5.60)
Equation (5.60) can be simplified by using the notion of a characteristic function. The] characteristic function of the random variable x takes on the form of the inverse Fourier transform of its probability density function p(x) [5]. The integral term of (5.60) can be viewed as containing the characteristic function of the zero-mean normal probability density, which for standard deviation a is C,(t) = FT-'IpWl = f — ^ e ^ e - ' '
4 2
' ' ^ = e"
V / 3
(5.61)
where the symbol t is used here because the inverse Fourier transform commonly transforms from frequency/to time t. Thus, in (5.61) for / = -**, the integral term in (5.60) becomes C,(-p) = f - 4 = e - * " e ~~o~y2ir i
w
' d r = e^"'"
(5.62)
Equation (5.60) with (5.62) becomes E\HHx,)\ = e ' V " * ' / £ e ' " V** /,_,2*\
The application of the identity (5.16) to (5.63) yields
»-l
(5.63) Iwi
234 -
fij^-e^e-^e^-^fr
(5.64)
which is the expected value of the synthetic range profile H, expressed as a function of the variance a of the frequency error. The peak response of the expected value of the range profile occurs at y = 0, ± n, ± 2/i, ± 3n which are the same positions as for the ideal response (5. 19). The expected value given by (5.64) of the peak at y = 0 becomes 1
hT
Peak £[ff/jt,)] = ne^' \-'''''
(5.65)
In terms of absolute value with C from (5.62), f
Magnitude of Peak E[H,(Xi)] = nC,
(5.66)
1
From (5.66), if the frequency variance a were zero, the magnitude of the pea^response of the expected value of the range profile would be n. With random frequency error present, the peak value is reduced to nC - n exp(-p
]
s
t
a\ = E\HUx,)] - E'lHixM
(5.67)
We are interested in the noise-power floor, which is the variance evaluated at null positions y = ±(1, 2, 3, . . . . n - I), where the expected value ElHfai)] is zero. Null positions, as well as peak-value positions, are unchanged by random frequency error. The variance for p(x ) independent of p(Xj) for all i and;' except / = j becomes t
°t
=
JU1-
• -fjHtoMpWpQt,)
• • .•/X*.-,)d 6dar . . . d*„_, JC
l
(5.68)
Proceeding as for the evaluation of the expectation E[H/(x,)), we obtain, at nulliposition y = I, the variance expressed as N
o-l = nC}(l-Cj)
(5.69)
The evaluation of the variance (5.68) to obtain (5.69) is much more involved than that of (5.52) to obtain the expected value (5.66) because the W/(JC,) term in (5.68) is the square of the summation of the IDFT of the n complex data samples for each burst. Squaring H / x i ) produces terms containing combinations of / and j instead of i only as in (5.55) for the expected value. This results in the Cj terms seen in (5.69). Details of the derivation
235
are not included here. It can be shown that the expected value and variance, derived above for synthetic range profiles generated from data containing phase noise produced by random frequency error, are the same for stepped-frequency, hopped-frequency, and constant-frequency sequences (A/ = 0). PROBLEMS Problem 5.1 A radar is to be designed to generate synthetic range profiles of 2m resolution for ship targets that are up to 300m in length, (a) What is the minimum required number of pulses per burst and maximum frequency step size? (b) What is the minimum pulse duration? Problem 5.2 (a) Show by using L' Hospital's rule that the peak value of \H of (5.19), derived on the basis of (5.11) for a point target, is equal to n. (b) Show that IDFT processing of data collected from a burst of n stepped-frequency pulses into a synthetic range profile results in a coherent processing gain of n, where coherent processing gain is defined as the ratio of output SNR of the synthetic response to input SNR. (c) What is the signal processing gain in decibels for lossless coherent processing of data produced from one burst of 256 stepped-frequency pulses? a
Problem 5.3 The detection range of the pulse-compression radar in Table 2.3 of Chapter 2 for video integration of 200 pulses was shown to be 54 km against a 1 -m target. What would be (he detection range for the same-size target if one burst of n = 200 stepped-frequency echo pulses were processed into a synthetic range profile for detection as a single response? Assume the same processed resolution so that target fluctuation is unaffected and all other parameters, except required S/N and signal processing gain, remain the same. 2
Problem 5.4 To what accuracy must target velocity be corrected to maintain tolerable synthetic rangeprofile distortion for a radar operating at a center frequency of 5.4 GHz with a 2,000-Hz PRF and using a stepped-frequency waveform of 128 pulses per burst? Use the criterion that uncorrected velocity must be within one-half the velocity for which the response nulls (first null along r = 0 on the ambiguity surface).
236
Problem 5.5 Rewrite the expression (5.8) for sample time S in terms of pulse duration T\ for sampling at the center of the stepped-frequency baseband responses to an isolated target of rangedelay extent much less than 7",. Assume that receiving system bandwidth B > \IT, and receiving system transfer delay is negligible? it
Problem 5.6 What are the maximum possible unambiguous range and velocity regions for a hoppedfrequency vehicle-navigation radar that operates at 77 GHz with a PRF = 25 kHz? Problem 5.7 A radar transmits stepped-frequency bursts of 128 pulses, each burst at a center frequency of 10 GHz for synthetic target-range profile generation. The standard deviation o cumulative phase noise was calculated to be 25 deg from (3.102) at a range delay of 1 rhs based on measurements of transmitter frequency stability at the center of the radar's frequency band. As another check, the fractional frequency deviation, defined in (3.115), was measured to be 5 x 1 0 ' at the center frequency of the transmitter for time interval r = 1 ms. What is the predicted synthetic range-profile distortion in terms of signal-to-thermal-noise power loss and signal-to-phase-noise floor, based on each type of measurement? Assume a stationary target near the maximum unambiguous range delay of 1 ms. r
Problem 5.8 What is the maximum allowable rms frequency deviation o\f,) in hertz during the interpulse time on the 8.09-MHz reference oscillator in Figure 5.18 to keep the sample gate within the St = ±0.1 /is (rms) central region of the delayed echo pulses? Compute oif,) at maximum unambiguous range delay T = 1/PRF. 2
Problem 5.9 Compare the results of Problem 5.8 to the allowable rms error in hertz of the 10-MHz reference oscillator in the range tracker of Figure 4.37 used for pulse-compression HRR to maintain 0.1-ns rms delay error for the same PRF. REFERENCES 11) Ruilenberg. K., and L. Chanzit, "High Range Resolution by Means of Pulse-to-Pulsr Frequency Shifting." IEEE EASCON Record. 1968. pp. 47-51.
237
(21 Kraus. J. D.. Antennas, New York: McGraw-Hill. 1950. pp. |3) Papoulis, A., Probability. Random Variables, and Stochastic p. 239. [4| Papoulis, A., Probability, Random Variables, and Stochastic p. 100. (5) Papoulis. A., Probability, Random Variables, and Stochastic pp. 153.159.
76-77. Processes, New York: McGraw-Hill. 1965. Processes. New York: McGraw-Hill. 1965. Processes, New York: McGraw-Hill. 1965.
Chapter 6 Synthetic Aperture Radar 6.1 INTRODUCTION
Synthetic aperture radar (SAR) is an airborne (or spacebome) radar mapping technique for generating high-resolution maps of surface target areas and terrain. The first experimental demonstration of SAR mapping occurred in I'M when a Urip map o/ a w.fion of Kry West, Florida, was generated by frequency analy>h of data collected al 5 tm wavrlrn^fhs frdrn a C-46 aircraft by a group from the University of Illinois (!J. .Some useful SAU references are Kovaly [2], Cutrona [3], Harger [4j, Hovanessian [5], and Curlander and McDonough [6]. SAR is used to obtain fine resolution in both the slant range and the cross range. Cross-range resolution refers to resolution transverse to the radar's line of sight along the surface being mapped. The term slam range refers to the line-of-sight range, as distinguished from the cross range. Resolution in the slant range to the radar is often obtained by coding the transmitted pulse, typically FM chirp coding. Cross-range resolution is obtained by coherently integrating echo energy reflected from the ground as the aircraft or spacecraft carrying the radar travels above and alongside of the illuminated area to be mapped. We define resolution in two ways: (I) the half-power extent of the response to a point target designated hr \, , and (2) the Rayleigh resolution designated Ar„ which is the 21 ir extent of the response to a point target. Other terms for cross-range resolution are along-track resolution, cross-path resolution, and azimuth resolution. The term synthetic aperture refers to the radar platform's cross-range travel distance over which reflectivity data collected from the illuminated surface is coherently integrated to obtain high cross-range resolution. r
a
Three versions of SAR will be discussed: side-looking SAR, spotlight SAR and Doppler beam sharpening. With the side-looking SAR, a fixed side-looking antenna, commonly called the real aperture, illuminates the surface of one or both sides of the SAR platform. The beam is usually pointed normal to the platform motion, but some systems squint the beam a fixed amount away from normal. Unless otherwise stated, the term side-looking SAR will refer here to unsquinted real antenna beams. 139
240 ,
j
—
The coherent integration length (synthetic aperture) of a side-looking SAR shrinks for echoes arriving from points closer to the radar and increases for echoes arriving from points farther from the radar. The effect for ideal processing is to produce constant cross-range resolution independent of range. Indeed, the maximum possible cross-range resolution is approximately equal to one-half of the real aperture's cross-range dimension. Figure 6.1 illustrates a side-looking SAR configuration. Signal energy collected during illumination of each range-resolved scatterer is made to arrive in phase at the output of the radar processor in order to realize the narrow beamwidth associated with the long, synthetically generated aperture. This is achieved by first correcting for all movement of the aircraft that deviates from straight-line motion. At this point, we have what is called unfocused SAR. Then, for focused SAR, the quadraticphase error produced by straight-line motion of the radar past each point of the area to be mapped is corrected. It is possible to achieve a second form of SAR, sometimes called spotlight SAR, illustrated in Figure 6.2, in which the real antenna squints off in azimuth to track a particular target area of interest over some azimuth angle ip. Here the cross-range resolution is limited, not by the size of the real aperture, as for side-looking SAR, but by tv£ target dwell time. Synthetic aperture length for small if/ can be thought of as the tangential distance that the radar travels while moving through the angle ift to the target. A third type of SAR is achieved by integrating echo energy as the antenna is scanned in azimuth. This is called Doppler beam sharpening. Here the resolution is limited by
BEAMWIDTH Figure 6.1 Side-looking SAR.
241
SPOTLIGHTED AREA
figure 6.2 Spotlight SAR.
the target dwell time determined by the real beamwidth and scan rate as surface scatterers are illuminated, during each dwell, at changing view angles produced by radar platform motion as in spotlight SAR. The effect is to sharpen the synthetic beam over that of the real antenna beam. Inverse synthetic aperture radar, to be discussed in detail in Chapter 7, can be explained in terms of SAR with reference to the spotlight form of SAR. After correction for unwanted deviation from straight-line motion and for quadratic-phase error, a spotlight SAR can be thought of as if the radar were flying a portion of a circle around the target irea. It is clear that, although the radar moves about the target, the same data would be collected if the radar were stationary and the target area rotated, which is precisely what occurs in ISAR. The aspect motion of the target relative to the radar is used to generate i radar map of the target, which is called the target image. Some fundamental characteristics of the side-looking SAR concept can be explained in terms of equirange and equi-Doppler lines on the earth's surface to be mapped by a moving radar platform above the earth. Consider the side-looking SAR illustrated with the coordinate system of Figure 6.3. Equirange lines on the earth's surface are the intersections with the earth's surface of successive concentric spheres centered at the radar. Points on each of these spheres are equidistant from the radar. Equi-Doppler lines on the earth's
Figure 6.3 SAR range-Doppler coordinates. (From Fig. I of C. Elachi, et al., "Spaceborne Synthetic-Aperture Imaging Radars: Applications. Techniques, and Technology," Proc. IEEE. Vol. 70. No. 10, Oct 1982, p. 1175. Reprinted with permission.)
surface are produced by intersections with the earth's surface of coaxial cones, which are concentric about the radar platform's flight line as the axis, and the radar position as the apex of the cones. Points on each of these cones appear at constant velocity relative to the radar. The zero-velocity cone is a plane perpendicular to the line of flight through the radar's position. The cones for maximum positive and negative velocity are straight lines on the flight axis extending ahead of and behind the radar, respectively. A flat-earth surface results in a coordinate system made up of the families of the concentric circles and hyperbolas shown in Figure 6.3. At any instant, the radar is able to view that portion of the range-Doppler coordinate system illuminated by the real antenna beam. The distribution of echo power from the illuminated area, as a function range delay and Doppler, is the SAR image for that area. The brightness of an image pixel is proportional to the echo power from the corresponding range-Doppler cell on the earth's surface. The mapping resolution is determined by the ability of Ihe radar to measure differential range delay and differential Doppler. Ideally, resolution is independent of radar range, but the image will degrade as thermal or other noise sources begin to determine pixel brightness at low echo signal levels. 6.2 REAL-APERTURE RADAR MAPPING Before we begin to develop SAR theory, let us first consider mapping with real-aperture, side-looking radar as illustrated in Figure 6.4. Here the cross-range resolution is directly
243
3-dB EDGES OF A TWO-WAY REAL-APERTURE ANTENNA BEAM A r , IS THE CROSS-RANGE \ RESOLUTION AT RANGE R /
>
FIfure 6.4 Real-aperture mapping radar operating in the side-looking mode.
obtained as a result of the narrow antenna beam produced by a long real antenna aperture operating at a relatively short wavelength. Resolution of a radar is commonly defined in terms of the extent, at the half-power points, of the one-way or two-way power response to a point target measured in range delay, Doppler shift, or angle of the target to the radar. Cross-range resolution of a realaperture radar can be defined as the cross-range extent measured between the half-power points of the one-way power response to a point scatterer, as illustrated in Figure 6.5. The total transfer power response, however, actually involves the two-way power gain of the antenna. An idealized radar (Fig. 6.6) using a linear superheterodyne receiver will generate an IF signal power level that is proportional to the echo signal power appearing at the antenna terminals. If we assume square-law detection, the detected IF signal voltage to the display will be proportional to echo signal power. The echo signal, given the above assumptions, is displayed as brightness by a linear display. Display brightness is then proportional to the echo signal power appearing at the output of the receiving antenna terminals. The real-aperture antenna of the radar functions as a power transfer function that operates twice on the signal to be displayed: once to transmit and once to receive. In each case, the power response function to a scatterer displaced in azimuth angle
244
ECHO POWER
BORESIGHT
Figure 6.5 Cross-range resolution associated with a real-aperture antenna.
it appears logical to define resolution in terms of the two-way half-power beamwidth of the antenna. This would make brightness proportional to G (d>), as indicated in Figure 6.6. The transfer response of an actual real-beam-mapping radar, however, may produce resolution corresponding more closely to its one-way antenna pattern. Cross-range resolution Ar of a real-aperture radar according to the one-way definition is the cross-range distance at some range R, corresponding to the one-way beamwidth of the antenna power gain pattern. Therefore, for small beamwidths, 2
r
A r> = /? x beamwid|h
(6.1)
The power gain patient for a cross-range antenna length / will now be determined fokan ideal, uniformly weighted antenna to illustrate a general approach, which shall be used later for discussing synthetic apertures. Figure 6.7 illustrates a line antenna of length /. Assume that the antenna continuously integrates incident radiated energy as though there were an infinite number of array elements, spaced infinitesimally close to one another along the line. Also assume far-field conditions for which all radiation from the boresight direction to the line antenna, from a source located at infinite range, will arrive at the same phase, which is taken to be zero phase. For radiation at wavelength A arriving from off-boresight angles, the arrival phase at the antenna is a function of distance along the line antenna, as indicated in Figure 6.7. v
occp
2
UlrO
_ c OVIU Zt-E
c e o C0O.P
3 *S8 8
RE-
'1
a
OS
DET
ESSES
!fl£Ojiii
CKM
muz
Sid
246
+ 1 2
ARRAY OF LENGTH I WITH AN INFINITE NUMBER OF ARRAY ELEMENTS; I.E. A CONTINUOUS LINE APERTURE RADIATION FROM + i, RADIANS OFF BORESIGHT
BORESIGHT FOR RADIATION FROM + + 6R = X SIN + SO THAT PHASE VS. x BECOMES 2 it
V(x)
oR
Figure 6.7 Real-aperture line antenna.
Consider the signal from a point target at angle +
(6.2)
By expanding the exponential in terms of its quadrature components, we have Ztf) = j cos[^x)](Lr + j y j smMxm 7
K
n
(6.3)
The imaginary term, being odd, integrates to zero, leaving only the real component, which integrates to 1
I. The response of (6.4) is based on Ihe assumption of a uniformly weighted aperture, which for small 4> his (he familiar (tin 4>V4 shape. A typical antenna will be aperture weighted lo reduce sidelobes with the penalty of slightly increased beamwidth.
247
sinf j / sin Hi) = —
6 4
r-
<->
(l' s i n *)
If normalized to unity at
,
sin ! -rl sin
rz(0)f L Z ( 0 ) J
V _
I-*, • J
( 6 5 )
At the half-power points, we have
stn'i sin2^/
sin
^
(l / s i n *J which is a transcendental expression. The graphical solution for the argument is ^ / s i n # = ±1.39 rad
(6.7)
At the half-power points of the response, we have from (6.7) sin 4>\
= ±0.44 j
iM
(6.8)
For the small beamwidth where R t> Rkll, the off-boresight angle corresponding to the half-power points is
(6.9)
Thus, the one-way half-power beamwidth is (fcdB
= 0.88y
(6.10)
248
The one-way beamwidth defined at the HIT (Rayleigh) points of (6.S) is similarly determined as' A* =7
(6.11)
From (6.1) with (6.10) and (6.11) the cross-range half-power resolution and cross-range IIIT Rayleigh resolution for small-beamwidth antennas are, respectively, Ar | , „ « 0.88/cy c
(6.12)
and L\r - A? j c
(6.13)
The half-power resolution based on the two-way power gain pattern can be showj to be ArJ, ^two-way) •= 0.64/fy
(6.14)
We can see that improved resolution occurs for short wavelengths and large aperture dimensions in the cross-range dimension. Also, the cross-range resolution is a direct function of range. A side-looking real-aperture radar used by the U.S. Coast Guard for environmental resource monitoring flies in the Falcon HU-2SA aircraft shown in Figure 6.8. This is AIREYE, an advanced ocean surveillance system developed by Aerojet. The radar is the Motorola side-looking airborne radar (SLAR) [71. The imagery shown in Figure 6.9 was recorded over the Santa Barbara Channel near Santa Barbara, California. The radar can detect oil seepage and spills, which would appear as darker streaks on the imagery. Oil spills appear as "holes" in the radar image, because the oil dampens the surface-wave' motion on the water, making it less reflective. Other ocean wave phenomena" can be detected, as well as ice and icebergs. Excellent mapping is possible by using real-aperture radars. However, our discussion now turns to the means for improving resolution by synthetically generating a very long aperture. 63 SAR THEORY (UNFOCUSED APERTURE) Figure 6.10 illustrates the geometry for generating a side-looking synthetic aperture. The radar platform is traveling a straight-line path above the earth while the radar illuminates
249
F1(wt 6.8 A1REYE ocean surveillance aircraft carrying a real-aperture tide-looking radar (nole the long realaperture antenna hanging below the aircraft). (From (7). Reprinted with permission.)
Igaurt 6.9 Santa Barbara Channel as seen with a side-looking radar. (From (7). Reprinted with permission.)
to the right with the real aperture. A moving side-looking synthetic aperture much than the real aperture is generated continuously. Figure 6.10 illustrates the synthetic (tenure of length f£ generated by the airborne side-looking radar as it flies along the •dicated ground track from -v,772 to +v.7"/2, where v is the platform's ground-track •docity and T is the integration time. Available integration time is the dwell time during Brain vger
r
250
Figure 6.10 Geometry Tor generating a side-looking synthetic aperture.
which each point target remains illuminated by the side-looking radar antenna. A large real beamwidth increases dwell time and thus the maximum possible synthetic aperture X, which is also referred to as integration length. The beamwidth in Figure 6.10 is assumed to be wide enough to produce illumination over the length f£ at range R from the radar. (In a typical system, range R is much larger than integration length X so that the beamwidth of the illuminating antenna would be much smaller than that illustrated in Figure 6.10.) Point targets are shown at both boresight and cross-range displacement y. Radar range R + SR to the point target at y = 0 first decreases then increases as the radar travels from -v Ttl to +v TI2 with minimum range R occurring at t = 0 and where SR - 0. The resolution of the synthetic aperture of Figure 6.10 will now be determined by developing the resulting power gain response to a point target at cross-range distance y from boresight. Similarly to the above real-aperture analysis, the resolution will be taken as the cross-range distance between the half-power and 21 tr points of the power gain response to a point target. The SAR response is the integration of the signal received r
f
25/
from a point target at y during time -7/2 to 4-772. The signal phase seen at each position along x is the two-way phase relative to transmission phase, such as would be obtained by quadrature mixing with the radar's stable oscillator. (This is in contrast to the oneway phase of the signal at each position along the real aperture discussed above.) Scatterers in Figure 6.10 will be assumed to be uniformly illuminated during integration time 7. This is approximated for integration performed over a small segment of the available integration length, which is approximately equal to R times the real-aperture beamwidth.
63.1 Small Integration-Length SAR First consider the far-field case in which the processed synthetic aperture length X is small enough that targets on boresight produce echo signals that arrive in phase over the whole real aperture. The associated geometry in Figure 6.11 can be seen to be similar to that of Figure 6.7 for the real-aperture line antenna, where the source was at infinite range. The two-way phase advance in Figure 6.11 for the target at cross-range distance y from boresight is ip(x) = (AiHAyx sin
+2724-
Fltwt 6.11 Geometry for small synthetic aperture.
252
Z(y) = J ci*"'df m
(6.15)
where ^(JC) = (4n(A)vjy(R. The integral (6.15) after performing steps similar to those leading to (6.4) becomes .
f 2 n y j y \
Z(y) = -i—^-T
(6.16)
RX The resolution at the half-power points is obtained by following the procedures in (6.4) through (6.7) to obtain the argument for which the normalized power response [Z(y)]V (Z(0)] = 1/2. Half-power points occur for J
2
^ f
= ±1.39
(6.17)
for which
RX .
y U = ±0.22-^
(6.18)
so that resolution in terms of aperture length i£ is Ar |,« = 0.44/?^, SR
(6 19)
where f£ = v,T. Rayleigh resolution, which is resolution defined at the 21 IT amplitude points, obtained in a similar manner becomes Ar, = i*A SR
253
2
(R + SR)
(6.2!)
The maximum allowable one-way phase error for real apertures is commonly taken to be W8 rad [8]. Range deviation for two-way phase deviation of nfB rad for a synthetic aperture corresponds to SR = A/32. Solving (6.21) for maximum synthetic aperture length with SR < A/32, assuming R> SR.v/e obtain (6.22)
0.5 ^RA
For example, the maximum integration length determined by (6.22) for a 3 GHz radar operating at R = 50 nmi (96 km) is % = 49m. Integration time for a platform velocity of 100 m/s is about 0.5 sec. Half-power resolution from (6.19) is 86m. m
6.3.2 Optimum Unfocused SAR Integration Length The integration-length criteria (6.22) sets a limit on the length X of an unfocused synthetic aperture beyond which resolution will be degraded significantly from the ideal resolution of (6.19) and (6.20). The optimum unfocused synthetic aperture extent and associated resolution will now be determined from the power gain response obtained when deviation SR from constant range is significant and remains uncorrected. The two-way phase advance at wavelength A of the echo signal from a point target at position y = 0, with SR from Figure 6.10, is Air HU 0) = - d 7 c = T
Anvji^ 4 i r - A 2R
(6.23)
T
This is a quadratic-phase term resulting from straight-line platform motion past the scene to be mapped. We will now examine what is referred to as unfocused SAR, which results when this term is not corrected and thus remains in the expression for power gain response of the synthetic aperture. The two-way phase advance of the echo signal from a point target at position y is
H>, y) =
(6.24)
where x = vy. Note that fait, y) in (6.24) includes the ^,(r, 0) term of (6.23). A target at azimuth position y having a minimum radar-to-target range R produces the normalized azimuth signal expressed in complex form as
254
j*y..,» _
e
c o s
^
v )
+
j
s j n
(6.25)
v )
= cos [-fait. y)J - j sin[- &(/, y)J It will again be assumed that the PRF is high enough so that the signal can be treated as though it were continuous. The summed response of the azimuth signal, assuming uniform illumination during integration time T, is expressed as a function of point target location y as z(
=
» lZ
c
(v
*p[" 2>S '' •
y ) 1
]
d t
( 6 2 6 )
By expanding the exponential of (6.26) in terms of its quadrature components, we have
Z ( y )
" L
C O S
(
[RA ^ " » f 2
~i L
Sin
(
[/7A V "
-
( 6 2 7 )
The power response is then given by
'
J
|Z( y)| = { C
C
0
S
[M
(
V
~
y ) J
d
] '}
( 6 2 8 )
These integrals can be written in terms of the Fresnel sine and cosine integrals:
C(z) = £ c o s ^ j d i
(6.29)
S(z) = Isin f o l-s* ( f ' ))dS
(6.30)
and
, =
S i n
2
where s is a variable of integration. The power gain response of an unfocused synthetic aperture, when normalized to peak gain at y = 0, can be expressed in terms of the Fresnel integrals of (6.29) and (6.30) as
255
|Z(y)P |Z(0)|
(6.3!)
l
[C(t, -fl
+ C(r,+
flr + [S(r, -Q
4[C\ ) V
+
+ S(y + QY
S\v)]
where •n = -f=
and (=
- 7 =
(6.32)
The power gain response |Z(0)P of (6.31) to a target at y = 0 is maximized when C\r,) + S\tj) is maximum. Examination of the values of Fresnel integrals plotted in Figure 6.12 for C(z) and S(z) indicates that the peak of the quantity C'(rj) + S\rf) occurs at ,*1.2. Thus, the peak power response to a point target from (6.32) occurs for
71=1.2 =
Flgort 6.12 Plots of the C(z) and S(z) Fresnel integrals.
^f^U
(6.33)
256
The optimum unfocused aperture length corresponding to rj = 1.2 is (6.34]
X = v,T~ [.2yJk~A
which, for a given range R, can be seen to be slightly over twice the length limit of (6.22) for maximum unfocused integration length based on less than 71/8 rad of phase deviation at ±l£/2. Longer integration time does not contribute to the integrated response at y = 0 because of the uncorrected quadratic-phase term fa't, 0) of (6.23), which is part of the total phase advance ip (i, y) associated with the two-way echo delay. The half-power resolution corresponding to this length is found by solving for (he value of (that satisfies |Z(y)|V|Z(0)| = 0.5 at r) = 1.2. The result is {= 0.5, so that with (6.32). \J }
J
Ar = 2|y\ = {yJRA •» 0 . 5 ^ r
(6.35)
We can see here that the resolution of an unfocused synthetic aperture degrades as the square root of range and wavelength. We will show in the following discussion that the Fresnel terms disappear when the quadratic-phase term tp,(t, 0) of (6.23) is canceled, which is attempted in most SAR designs. However, the above analysis is sometimes used to assess the reduced resolution due to the presence of residual uncorrected quadratic phase. 6.4 SAR THEORY (FOCUSED APERTURE) The quadratic-phase term, produced by the straight-line platform motion, can be subtracted from the total phase advance to generate what is called a focused aperture. Then, from (6.23) and (6.24), the corrected two-way phase term for the side-looking SAR of Figure 6.10 is
Ht. y) - Ht. 0) = -
~
(- 4 r )
A\
R
6
3
<-«
2R)
where x = vj. The corrected response to a point target at a cross-range distance y from boresight then becomes
257
The response Z( y) after simplification can be expressed as z
(»=L«p(jT-^) ' d
( 6 3 8 )
The expression for Z(y) from (6.38) for the focused aperture can be seen to be identical to that for the small-aperture approximation (6.IS) of the uncorrected response for an unfocused aperture. Resolution of a focused synthetic aperture measured at the half-power points is therefore Ar |,„ = r
0.44/4
(6.39)
and the Rayleigh resolution is Ar, = ^ R±
(6.40)
which are the resolutions given, respectively, by (6.19) and (6.20) for an unfocused aperture of maximum length given by (6.22).
y 6.4.1 Focusing in Terms of Matched Filtering SAR focusing in the azimuth dimension is analogous to the chirp-pulse matched-filter processing described in Chapter 3. The azimuth response to a point target, for typical SAR parameters, approximates a linear FM signal of duration T. Azimuth chirp rate in hertz per second applying (3.2) with either (6.23) or (6.24) is expressed as (6.41) where foil) is the instantaneous Doppler frequency of the azimuth response. Azimuth chirp bandwidth for integration time T is (6.42) The half-power duration of the matched-filter compressed response from (4.37) with (6.42) is 0.886 ^ RX '• = ~ T ~ = 0.886—— A 2v\T n
oa
(6.43)
258
The resolution in the cross range for SAR analogous to c/2 times compressed-pulse duration for resolution in the slant range is Ar = V , = 0.44 R~ r
(6.44)
which with = v T is the resolution given by (6.39) for focused SAR. Quadratic range shift is produced in the ideal situation of Figure 6.10, where the radar platform moves in a straight line over a fixed surface. Deviation from straight-line motion produces additional range shift analogous to chirp-pulse nonlinearity. Cross-track motion of the radar platform past the surface can be produced by wind with aircraft platforms and by earth (or planet) rotation beneath satellite platforms. Minimizing and/ or correcting for these and other sources of phase error become a major part of any SAR design or processing effort. Toleration of uncorrected phase error can be analyzed using the paired-echo theory of Chapter 3. f
6.4.2 SAR Resolution for Nonuniform Illumination
s
Up to this point, we assumed SAR response functions based on reflectivity obtained for uniform illumination of point targets on the earth's surface. This assumption in (6.38) predicts somewhat optimistic resolution for focused SAR processing over the entire available integration length, which is not uniformly illuminated by practical antennas. A more accurate SAR response function which includes the weighting produced by the illuminating antenna is expressed as
I
I
where |Z(v,///f)| /|Z(0)| is the normalized two-way illumination function produced by the real beam. With x = v l, (6.45) becomes p
z(
j
>K4wM ^*)^
(646)
for narrow illumination beamwidth \Z(x/R)f = \Z(4>)f, which is the power gain pattern of the illuminating antenna. In practical systems, integration is likely to be performed over the cross-range extent -v,772 to +w,772, corresponding to illumination over just the mainlobe of the real beam. Integration length ££ = v,T is then approximately equal to R
259
2
width of the response function (6.46) for a selected integration time T with |Z(d>)|'/lZ(0)| obtained from known or measured beam patterns. SAR resolution will now be determined for the normalized power gain pattern (6.3) obtained for the uniformly weighted line antenna of length /. We will assume small real beamwidth so that the SAR response is determined primarily by illumination within small deviations
(6.47)
If we let u = 2x/(AR), we can write (6.47) as
(6.48)
Rewriting (6.48) in the form of the Fourier transform of the product of two (identical) functions and employing the inverse of the convolution theorem expressed in the form of (4.54), we obtain
(6.49) 2
2
where rect(2y/) = 1 from -1/4 to +1/4 and zero elsewhere is the inverse Fourier transform of each of the (sin x)lx forms and the symbol * refers to convolution. Figure 6.13 illustrates
260
Figure 6.13 Normalized response of focused side-looking SAR for integration over the entire beam produced by a uniformly illuminated line antenna.
the triangular response Z(y)/Z(0) produced by the convolution of the two identical rectangular functions after normalization by Z(0) = 2A/V(v,i ). Resolution from Figure 6.13 is J
Ar, = ^ (measured at the half-amplitude points)
\6.50)
Ar = 0.29/ (measured at the half-power points)
(6.51)
and r
6.4.3 Equivalent Rectangular Beamwidth It is convenient to estimate SAR resolution based on SAR integration length determined by an equivalent rectangular real beamwidth defined in terms of the antenna gain response
|Z(0)P as
*' = J ^ D Z ( 4 W
^6.52)
The equivalent rectangular beamwidth of the one-way power gain response (6.5) for the unweighted line antenna using the definition (6.52), in radians, is
(6.53)
261
which from (6.11) is also the beamwidth defined at the 21 n points. SAR resolution can be expressed directly in terms of the illuminating antenna length / by substituting ££ = Rifi, = RXIl in the expression (6.20) for SAR Rayleigh resolution with uniform illumination. The result is 1/2, which happens to be identical to the half-amplitude resolution given by (6.50) for the nonuniform illumination produced by the unweighted line antenna for which All is the effective beamwidth. We conclude that 112 is a goodfirstapproximation to the SAR cross-range resolution. A more accurate estimate is the width of the response function (6.45) with |Z(y,/)| obtained from the actual power gain pattern of the illuminating antenna. Figure 6.14 compares the response to a point target for (a) a uniformly illuminated synthetic aperture, and (b) a synthetic aperture formed by the nonuniform beam pattern of an unweighted antenna of length /. For a side-looking focused SAR, we can see that the resolution is limited by the . real-aperture size. A small real aperture along the cross-range dimension results in better 2
Illumination function
Illumination
Response to a point target
(a)
se
i. 2se
nix
y
Figure (.14 Response for uniform and nonuniform illumination functions over which coherent integration is performed: (a) uniform gain segment or small real beam: (b) unweighted narrow-beam line antenna of length I.
262
SAR resolution, in contrast to real-aperture mapping, where a large azimuth aperture dimension produces better resolution. In summary, it is possible to increase the cross-range resolution of surface-mapping radars over that of real-aperture mapping radars by coherently integrating target echo signals as the radar platform passes by the area to be mapped. Maximum possible resolution occurs for focused SAR when quadratic-phase and phase errors caused by deviation from straight-line motion are corrected before integration. The SAR technique is essential for spaceborne radar mapping of the earth's surface, where useful resolution is not likely to be achieved with practical real apertures. The following equations were derived above for resolution associated with three types of apertures in increasing order of resolution. 1. Real aperture (6.14): &r » 0.64 « y r
(6.54)
2. Optimum unfocused SAR (6.35): Ar = 0.5 yfRA f
(6.55)
3. Focused SAR (6.50): Ar f
(6.56)
Resolution as a function of range for the above three types of apertures is plotted in Figure 6.15 for a 3m real aperture at A = 3m. 6.S SAR THEORY FROM DOPPLER POINT OF VIEW The focused aperture SAR concept can also be explained from the point of view of differential Doppler signals produced by scatterers separated in azimuth. j Consider the airborne side-looking SAR of Figure 6.16 at the instant that the aircraft is directly beamed on boresight to the center of two point targets, both at range R, which are located in the cross range at -y and +y, respectively, from boresight. Focusing can be thought of as correcting for the range deviation SR of the straight-line flight path from the constant radius dashed curve. The instantaneous velocity of the radar past the two targets will produce an echo signal containing a pair of instantaneous Doppler offset frequencies -2<wy/A and +2<wy/A for wavelength A, where to is the instantaneous angular velocity of the aircraft relative to the centroid of the two targets. The Doppler frequency separation is
263
1
10
100
1000
RANGE R (nmi) Figure 6.1S Resolution versus range for three generic types of mapping radars at 0.03m wavelength.
Sf = ja,y
(6.57)
D
For a Doppler frequency resolution A/ , the cross-range resolution becomes D
Ar = 2|>i = ~M r
D
(6.58)
264
Figure 6.16 Source of SAR cross-range Doppler.
6.5.1 Uniform Illumination Doppler frequency resolution A/ is also the Doppler frequency bandwidth of the spectrum of the echo signal from a point target. Figure 6.17(a) illustrates the envelop of a point target's echo signal and signal spectrum for uniform illumination over a selected small integration angle if/and for the corresponding integration time T. The two-way echo signal can be represented as the impulse response D
Z(r) = rect'^j = r c c t ^ T
(6.59) T
= 1 for r = - - < r < - and zero elsewhere The Doppler frequency spectrum of the point target response is FT[rect(//7")]. From (2.52), the Doppler frequency resolution A/„ associated with coherent integration of an
265
Illumination
Point target echo signal vs. angle
Normalized spectrum
SAR Impulse response
Doppler h(f ) Q
- 1 As/- _ J L _
A,
c~
2
0)
W
* D "
2
ror
(a)
i V i
.j
sin' A
(
^c = H
8
/
o =
T
M
(b)
2, L Figure 6.17 SAR resolution determined from Doppler frequency spectrum: (a) uniform gain segment of small real beam; (b) beam pattern of unweighted narrow-beam antenna of length I.
echo signal having a rectangular envelope is the bandwidth L\f = 1/7 at the 21 ir points of the (sin x)lx form of the Doppler frequency spectrum of the rectangular-shaped response. Therefore, from (6.58), the cross-range resolution obtained by coherent integration over a small uniform illumination segment of a real beam is D
A
1
A
(6.60)
For coherent integration over an integration angle if/, regardless of w, we have A
1
A
(6.61)
45.2 Nonuniform Illumination Figure 6.17(b) illustrates the envelope of the echo signal and its spectrum for the nonuniform illumination over the entire two-way response (6.4) of an unweighted line antenna
266
of length /. As for the above analysis carried out from the aperture viewpoint, we will assume that the SAR impulse response is determined primarily by illumination within a small angular deviation from boresight, so that in (6.4) sin tf> =» tf>. The two-way echo signal for
(6.62)
which can be shown to have the triangular spectrum illustrated in Figure 6.17(b). The, Doppler frequency resolution at the half-amplitude bandwidth is lu>/A. The cross-range resolution from (6.S8) at the half-amplitude points is . A r
Ito
A X
-2^ T
i =
( 6 6 3 )
2
which checks with (6.50) determined from the aperture viewpoint. The expressions given above for cross-range SAR resolution were based on Ihe response produced by coherent integration during the real-beam dwell time of the reflected signals from point targets. From the Doppler viewpoint, coherent integration in the forjn of the Fourier transform of the reflected signal produced fine Doppler resolution, which was shown to be related directly to cross-range resolution. In practice, the coherent integration process may take several forms. A common approach (discussed later in this chapter) is to correlate the azimuthal signal data collected along known range-versusazimuth trajectories with a suitable azimuthal reference to achieve azimuth compression. 6.6 CHIRP-PULSE SAR 6.6.1 Resolution
\
Fine range resolution produced by conventional side-looking SAR, spotlight SAR, or Doppler beam sharpening is often obtained via some type of pulse-compression method, chirp-pulse compression being the most common. Later, we will discuss a stepped-frequency SAR concept, wherein range resolution is achieved synthetically as described in Chapter 5. The resolution Ar, in range for pulse-compression SAR is c/2/3, where j3 is the frequency-weighted bandwidth, which equals A for uniform weighting over the chirppulse frequency excursion A. Regardless of how slant-range resolution is achieved, the cross-range resolution for side-looking SAR produced by integration ever small real beams is approximately
267
A r , - - J
[
(664)
where iff, is the equivalent rectangular beamwidth of (6.53). The cross-range resolution possible with spotlight SAR for integration over a small angle tfi, in radians, is Ar = i^
(6.65)
r
6.6.2 Data Collection Up to this point it was assumed that echo signal data samples were so closely spaced that continuous integration could be assumed in the calculation of resolution in the slant range and the cross range. Requirements for data collection of discrete samples will now be defined for SAR using chirp-pulse-compression radar waveforms. The term data collection, as before, does not preclude real-time processing, but is intended to clarify separate requirements for signal sampling and signal processing. Figure 6.18(a) illustrates the process of collecting SAR data obtained with a chirppulse-compression radar. As the platform containing the radar travels above (at a small down-look angle) and alongside of the area to be mapped, chirp pulses are transmitted at some PRF, assumed here to be/onstant. The time between pulses is made sufficiently long to prevent ambiguous range responses, at least over the effective illuminated range extent wherein echo signals may appear above the noise. A slightly different patch of the earth's surface is illuminated by each transmitted pulse. Each time a pulse is transmitted, the echo signal is sampled at some range-sample spacing or continuously recorded over some portion of the illuminated range extent, called range swath. The collected data comprise a set of reflectivity measurements in two dimensions. The dimensions of the data format can be referred to in several ways: slant range versus cross range, range delay versus time history, fast time versus slow time, and range versus azimuth. A data record (Fig. 6.18(b)) will extend in the slant range over the range swath and continuously in the cross range along the flight path over which data are collected. Data sampled in azimuth at a given range sample position is called an azimuth data line, and data sampled from the echo signal for a single pulse is called a range data line. Typically, a data record will consist of unresolved dispersed responses from a continuum of scatterers. The data set before processing does not resemble a map of the terrain.. Rather, echoes from individual point targets are dispersed in both range and azimuth, as illustrated by the data collection element in Figure 6.18(b). Range and azimuth compression, to be described later, produce the desired maps. It will be convenient to refer to the approximatelyrectangulardata collection element illustrated in Figure 6.18(c) for side-looking SAR. This element is sampled by approximately TJ, x i t samples. This is the area bounded in the cross-range extent by two slantN
r
268
III CO -I o.
zer. I-a
DCO llllllllllllllllllllll llllllllllllllllllllll iiiiiiiliii>M!!:;tiii m«s Ullllllll Miirmii iiiiikiii llllllllllllllllllllll
| c
I 3 111 >
-I 3
z
(9
oz
8 V :
1
ouicr.il
i
u
a
I CO Ul
oS2£ui uiq -o -J|-cos ocjuiOk. zzp uix -ii-
8
1
ECO
5
269
(C)
Figure 6.18 (continued)
range segments, each of length cT /2, which is the slant-range integration length for chirp-pulse duration T,. The slant-range extent is bounded by two cross-range arc segments, each of approximate length R tfr, which is the cross-range integration length for integration angle if/. For side-looking SAR, Rifi is the synthetic aperture length. When defined in this way, the data collection element contains the unprocessed two-dimensional dispersed response to a point target. The data collection element approaches a true rectangle for small if when R *> cT,/2. A total of % complex echo samples are collected in each data collection element during range integration time T, for each of N transmitted pulses occurring within the azimuthal integration angle if/. A total of •n = N samples is collected along the length Rip of each resulting range cell. t
c
270
Data collected from the slant-range and cross-range space indicated in Figure 6.18 are processed to achieve range and azimuth compression. Compression in range for each point-target response is from cT,/2 to Ar„ where Ar, is the slant-range resolution. The compression in azimuth is from Rift to Ar , where Ar is the cross-range resolution. Echo signals produced from each linear FM pulse of chirp bandwidth A to meet the Nyquist sampling criteria must be sampled by at least A complex samples per second. This corresponds to a complex sample spacing of 1/A in range delay and to a range resolution of Ar, = c/(2A). In other words, the dispersed range-delay signal produced by each chirp pulse is required to be sampled at slant-range spacing equal to or less than the slant-range resolution Ar, associated with the transmitted chirp bandwidth. Sampling requirements in the cross range are similar. At the nearest approach of a side-looking SAR in straight-line motion past a surface point target, the range rate and therefore the Doppler shift will vary approximately linearly with time (history), passing through zero frequency at boresight. During the target dwell time for small real beamwidth, the Doppler shift therefore approximates linear FM. Azimuth compression, then, can be thought of as compression of the FM Doppler signal produced during the integration length Rip. Therefore, the azimuth echo signal in each range cell isrequiredto be sampled at a cross-range spacing equal to or less than the cross-range resolution Ar as jciated with the Doppler FM seen across the real beamwidth during its dwell time at R? Unambiguous data sampling of the two-dimensional dispersed response occurs when r
r
r
.
vAr, * Y
(6.66)
r)Ar > Rip
(6.67)
and r
with one complex sample per resolution cell. 6.6J Slant-Range Sampling Criteria e minimum number ofrequiredsamples following each transmitted pulse for unambiguous slant-range sampling is obtained from (6.66) as
(»,.)... =
^
=
r
,
A
(6.68)
2. Low-level Doppler signals from sidelobes of the illuminating antenna produce Doppler frequencies outside the FM bandwidth seen across the main beam. Samples collected at cross-range spacing equal to the crossrange resolution A r produced by the Doppler FM across lite effective real beamwidth, therefore, do not strictly meet the Nyquist criteria. r
271
\
where A is the chirp-pulse bandwidth. In practice, the design of pulse-compression radars that use data sampling techniques is often limited in resolution by the maximum available A/D conversion rates. For example, if the A/D conversion rate is 100 megasamples per sec, this translates to about l.Sm slantrange resolution for unambiguous sampling, according to (6.66); that is.
Ar,=
2V, I
(6.69)
2 (y,fT,)
3 x 10*
I
100 x 10
6
= 1.5m
High-resolution SAR systems, to avoid sampling at very high rates, have tended in the past to rely on analog means for recording echo data on film. This is followed by optical processing. Stepped-frequency SAR, to be described later in this chapter, is a concept that avoids the requirement for high A/D converter rates to achieve high resolution with sampled data. High-speed A/D conversion can also be avoided by means of stretch waveforms, mentioned in Chapter 4. The rate at which analog data can be sampled, converted into digital quantities, and stored is increasing rapidly at this writing. Trends in high-speed sampling and A/D conversion are indicated in Figure 3.17. 6.6.4 Cross-Range (Azimuth) Sampling Criteria The minimum number of samples required for unambiguous azimuth sampling of the azimuthal integration length Rip at each range position is the integration length divided by the cross-range resolution associated with the integration length of the entire real beam. From (6.67) and (6.64), we obtain Rip
Rip
(»7c)». = £7 = j - J samples
(6.70)
Integration for side-looking SAR may be carried out over the entire real beam. An estimate of the minimum number of samples can then be made under the assumption that the effective integration beamwidth is given by the equivalent rectangular beamwidth tp, of (6.53). For ib =
ip* 5
(^)m» = —-p- samples
(6.71)
272
Equation (6.70) applies for the general case in which coherent integration may be performed on data collected over an integration angle ip that may be less than or greater than the real beamwidth. Equation (6.71) applies when resolution is achieved by coherent integration over the effective beamwidth of the real antenna for which Ar = AI2
-
r
6.6.5 PRF Requirements From Doppler Point of View Radar PRF must meet the Nyquist criteria for sampling of the Doppler signal produced by the effective rotation, as seen by the radar, of the cross-range extent of the earth's J surface from which echo signals may arrive. From Figure 6.19, Doppler bandwidth produced by scatterers at range R, extending over cross-range length Rip, is f t-fo,=j^R>P
(6.72)
D
Cross-range length Rip, expressed in terms of Doppler bandwidth, becomes RA
r?t*=^r(/ -/ ) w
(6.73)
D1
A cross-range ambiguity length (Rip)**, associated with the Nyquist PRF fm - /DI, can be defined from (6.73) as
1/T7equal to 2
RA
= sry
(674)
An illuminated cross-range extent greater than (Rip)** will be undersampled. Required PRF, for the reasons stated above, must be defined in terms of real antenna beamwidth.
273
Figure 6.19 Doppler spread associated with cross-range ambiguity length.
Nyquist sampling in a side-looking SAR mode requires that (Rip)** £ Rif>„ so that we have, from (6.74) and (6.53). as in Section 6.6.4,
7,
(675
>
Multiple coherent looks taken across Ihe real azimuth beamwidth produce the crossrange resolution given by (6.61), where ^ = toT is the single-look integration angle. This resolution reduced from that available from the entire real beam is traded off for reduced speckle noise when superimposed multiple looks are summed noncoherently. However,
274
the received Doppler spectrum is produced by rotation past the cross-range extent illuminated by the entire real beam. Adequate sampling of this band of Doppler frequencies requires the PRF given by (6.75). 6.6.6 PRF Requirements From the Point or View of Grating Lobes Equation (6.75) can also be obtained from the viewpoint of SAR grating lobes. From Figure 6.20, the two-way pulse-to-pulse phase advance seen at the SAR's real antenna for an off-boresight scatterer, displaced <j> rad from boresight, is (4viA)v Ti sin
2
designers as a grating
lobe.
Grating lobes occur for (6.76) where n is an integer. The first grating-lobe angle at n = 1 of (6.76) is
ECHO RADIATION V FROM ON-BORESK3H1 SCATTERER
FOR RADIATION FROM + | CR - d tin + • v , T tin | t
Figure 6.20 SAR grating-lobe geometry.
275
(6.77)
Grating-lobe angles that are large compared to real antenna beamwidth result in negligible responses. This is the situation for most aircraft SAR applications. A minimum PRF criterion can be established as the PRF that produces a grating lobe at the first null of the real-beam pattern. A higher PRF will then produce grating lobes that fall outside the real main beam. The first null of a uniformly weighted antenna can be seen from (6.4) to occur for sin d> = A/1, leading to the criterion from (6.76) that (6.78) Note that this is identical to (6.7S), which was determined by requiring that PRF be greater than the Nyquist sampling in the cross range. It may be desirable to avoid grating-lobe responses more safely by setting the grating-lobe angle farther from the main beam (for example, at the second null). There is no cross-track beam traveling past the target in an ideal spotlight SAR mode. An unambiguous cross-range length for-small integration angle, however, can still be defined by (6.74) through substitution of v,|cos
6.6.7 Square Resolution In a spotlight SAR mode, it may be desirable to maintain square resolution, defined as i r = Ar, = Ar . Slant-range resolution expressed in terms of chirp bandwidth A is t
(6.79)
276
Cross-range resolution expressed in terms of angular rotation rate and integration time from (6.60) is
r
l
* -' 2-Zr
( 6 8 0 )
For squint angle
RA
Ar = ^ , ,, ' 27V,|cos vH c
(6.81)
where integration angle t/> is |7V, cos qtyR. The chirp bandwidth that results in square resolution, based on expressions (6.79) and (6.81), becomes A = ^ | c o s
(6.82)
Theoretically, square-resolution zooming could be done dynamically by incuasing chirp bandwidth, according to (6.82), to improve slant-range resolution as dwell time is increased to improve cross-range resolution and vice versa. Square resolution for side-looking SAR, if desired, requires that the slant-range resolution c/(2A) equal the cross-range resolution A/(2/) for integration over a beam segment ifi. Square resolution then requires that A = apt A. If integration over the entire beam is considered, tp= tp, = All from (6.53), so that A = ell. Variable resolution may become practical for pulse-compression radar designs using the DDS chirp generator described in Chapter 4. The stepped-frequency approach described below could alsc provide this capability in future systems.
J
6.6.8 Design Tables and Block Diagrams Table 6.1 lists the expressions derived above for chirp-pulse SAR. A generic block diagram is illustrated in Figure 6.21. A stable master oscillator supplies a timing reference, radar center frequency/. IF reference at/;, and difference frequency/-/, as shown. A motion sensor operates at the antenna. Such motion data can be converted into phase-correction signals to the transmitter or phase-correction data to be used in a motion-compensation computer. Additional processing commonly carried out, but not indicated in Figure 6.21, may be for weighting, equalization, and correction for earth curvature, cell migration, and quadratic-phase distortion. The processor itself may be optical, which carries out optical processing of data records on Film, or a digital processor, which processes data collected in digital form. The trend is toward digital processing. Both approaches will be discussed further in this chapter.
277
Tabic 6.1 Summary of Chirp-Pulse-Compression SAR Equations Parameters
Slant-range resolution
Symbol
Side-Looking Expression
Spotlight Expression
Ar,
c 24
c 2A
Unambiguous slant-rang extent
cr, 2 cT, 2 A A 1
Slant-range integration length Cross-range resolution
Ar,
Cross-range ambiguity length Available cross-range integration length (synthetic aperture) Minimum number of complex samples per slant-range integration length Minimum number of complex samples per cross range integration length Unambiguous PRF Required bandwidth for Ar, = Ar, Maximum integration length. angle for unforcused SAR
cT, 2 cT, 2 A
A
2e>r lip 2
ltsr 2t*
AM 2v,T,
AM 2T,v,|cos «H Rip
X = R$ * 71 ("7,)«.
TA
TA
2 A H , 2R>tf J • A
(TJ,)*.
2v,», 2v, A " / c>_ c A
J_ T, 0
"7
=
ifc..
0.5"V"M
IR'frh A 2<»,v^cos 2v,|cos
_ M ~ ' \ A" fl
Resolution in Table 6.1, except for Ar = 1/2, refers to the Rayleigh criteria, which assumes uniform illumination. The resolution given as 112 refers to resolution denned at the half-amplitude points or as Xl(2tp,). Equations apply for focused SAR and for unfocused jSAR for which if < (Rtf)**. A small integration angle is assumed. c
r
t 7 STEPPED-FREQUENCY SAR It is possible to generate SAR maps by using the stepped-frequency waveform discussed in Chapter S to replace pulse compression as a means to obtain range resolution. Although as yet untested for SAR, these waveforms for ISAR (to be described in Chapter 7) have been under investigation for several years. Cross-range resolution can be obtained as in
278
LSE RAT
XDUI UO.Z • •1 o
111
;
O
SAMPLER
oc
o
UADRATURI MIXER
UJ
IRP. inr*
oc 111 O Z Ui OC
O UJ
J
w
0Ctf) CUJUi-J JQ.3
o.
2
a.
279
pulse-compression SAR by coherently integrating range-resolved responses obtained during the real-beam dwell time. Slant-range resolution with the pulse-to-pulse steppedfrequency waveform, in contrast with that for the chirp-pulse waveform, is obtained synthetically from the pulse-to-pulse frequency-domain reflectivity data. Stepped-frequency SAR maps can be generated from data collected in the frequencyversus-viewing-angle domain by using two transforms: (1) an IDFT of the complex samples of reflectivity data obtained during each stepped-frequency burst to acquire the complex range profile for each burst, and (2) a D F T of samples of the Doppler timehistory response in each synthetically generated range cell to obtain Doppler resolution. Stepped-frequency SAR, in principle, can perform high-resolution mapping with frequency-agile, narrowband radars (including magnetron radars, using concepts to be discussed in Chapter 9). In addition, zooming capability is possible by interrupting the side-looking mode to adjust the stepped-frequency bandwidth to match the resolution associated with target dwell time.
6.7.1 Resolution Synthetic slant-range resolution produced by each burst of n pulses stepped in A / Hz frequency steps from (5.21) for uniform frequency weighting is
A r . - | ^
(6.83)
Cross-range (azimuth) resolution is obtained, as in the case of chirp-pulse SAR, by coherent processing over the integration angle 0 in each (synthetic) range cell. Crossrange resolution for both the side-looking and spotlight SAR modes is expressed in exactly the same way as for pulse-compression SAR.
6.7.2 Slant-Range Sampling Criteria
0
Unambiguous sampling of the extended range swath is achieved by collecting samples at discrete coarse-range positions as described in Section S.S. The frequency step size for stepped-frequency waveforms is set to produce the unambiguous range-delay extent I/A/ equal to the duration of the filtered baseband response to a point target, which for an ideal matched filter, matched to a transmitted pulse of duration T, is 2T,. The minimum frequency step size for unambiguous sampling in the matched-filter case therefore becomes
A/= ^
(6.84)
A slant-range ambiguity window is defined for each coarse-range sample position as
280
w, = °- x ±-
f
(6.85)
However, the maximum unambiguous slant-range extent obtained by superposition of samples collected from multiple coarse-range bins as described in Chapter 5 is not limited to the ambiguity window of (6.85). Instead, the total maximum unambiguous slant-range extent is
L\R,
86
= ^
(°- >
which is the range associated with the time between pulses, as for chirp-pulse SAR. 6.7.3 Cross-Range (Azimuth) Sampling Criteria and PRF The required number of samples in each resolved range cell per azimuthal integration length Rip is given by (6.70). Azimuth sampling occurs at the burst rate U(nTi). The minimum number of bursts required for unambiguous sampling during integration angle ip in radians at range R from (6.70) with y} -N bursts becomes c
2RM
NZ.—^
bursts
(6.87)
where
bursts
(6.88)
Each burst of n pulses corresponds to a single pulse of a pulse-compression waveform. The effective PRF is thereby reduced by n. Equation (6.74) for unambiguous cross-range length in terms of PRF when applied to stepped-frequency waveforms becomes
Equation (6.75) for required PRF becomes
281
6.7.4 Spotlight Zooming A target area observed in thefixed-beamside-looking mode may be selected for magnification by initiating a spotlight zooming mode wherein the target area is angle tracked over a viewing angle larger than the real beamwidth, as in the top-down view in Figure 6.22, to improve cross-range resolution while bandwidth is increased to improve slant-range
Figure 6.22 Spotlighted ire* produced by N burst of n pulses sampled at range delay 2R/c over integration angle *V.
282
resolution. The bandwidth nA/ is continuously adjusted according to (6.82) with nA/ substituted for chirp bandwidth A. Squint angles for spotlight SAR are not restricted to near zero as suggested in Figure 6.22, but cross-range resolution degrades rapidly as the squint angle approaches ±v/2 (straight ahead), as indicated by (6.81). 6.73 Design Tables, Waveforms, and Block Diagram Table 6.2 lists expressions derived above for focused stepped-frequency SAR. Expressions for slant-range resolution, slant-range and cross-range integration lengths, and number of samples per integration length for stepped-frequency SAR are identical to those for pulsecompression radar with chirp bandwidth A = nL\f. The unambiguous PRF for either sidelooking or spotlight modes is higher than that for chirp-pulse SAR by a factor of n, where\ Table 6.2 Summary of Stepped-Frequency SAR Equations Parameters
Slant-range resolution Umambiguous slant-range extent
Symbol
Ar,
Side-Looking Expression
Spotlight Expression
c 2nHf cT, 2
c 2nA/ ch 2
cT,
cT,
'
Slant-range integration length Cross-range resolution
Ar,
A
A
Cross-range ambiguity length
<*<»».
RA
Minimum number of bursts per cross-range integration length
N
Required bandwidth for \r, = Ar, Maximum integration length. angle tor unfocused SAR
R
**
cT, AT,
cT, Ar,
l
n
Unambiguous PRF
ZnT,vJcos <M
2 = **
Pulses per bunt
2*M A
1 T, nA/
*
A
2u>r 2
RA
2v/iT, Available cross-range integration length (synthetic aperture)
A
1 2
2o>r 2f
2Rtf. '
2»yfc
2v> 1
A
at
c
A
"7
«
A
A
O.S-JRA
_ 2*vJcos*| 2v,|cos* 2n • - : • n ' . X t c$ c7VJcos ft A XX A
is
n V\ /—
283
n is the number of pulses per burst. Compare (6.90) and (6.75). Pulse-tb-pulse frequency separation, however, may allow operation in the otherwise range-ambiguous region of PRF. Equations in Table 6.2, as for Table 6.1 for chirp-pulse SAR, apply for focused and unfocused SAR, small integration angle, and Rayleigh resolution. Waveforms for unambiguous and ambiguous range are illustrated in Figures 6.23 and 6.24, respectively. In the side-looking mode of stepped-frequency SAR, each burst produces k sets of n complex echo samples, spread throughout the desired swath-delay interval. Figure 6.23 illustrates sampling when the PRF corresponds to the unambiguous range so that PRF < c/(2A/?,), where A/?, is the illuminated range extent over which significant echo power is received. Figure 6.24 illustrates sampling when PRF > i7(2A/? ). In this figure, echo foldover is avoided by frequency separation between pulses. In this way, the PRF might be made sufficiently high to avoid synthetic aperture grating lobes while also avoiding range ambiguity. Receiver blanking would likely be required during each transmitted pulse. Further study is needed to characterize degradation of system performance by relative motion of the target and radar platform during the burst time, and to develop appropriate motion-compensation algorithms. In addition, further study is needed in the areas of memory and computation speed requirements for mapping operations. Figure 6.25 is a generic block diagram of a stepped-frequency system. Similarity to the block diagram in Figure 6.21 for pulse-compression SAR is apparent. The key difference is the means for achieving the fine resolution in the slant range. A controlledfrequency synthesizer is used in stepped-frequency SAR to generate the waveforms for synthetic range-profile processing, instead of a chirp generator and the pulse-compression scheme as for pulse-compression processing. It may be possible to avoid platform motion compensation by using a variation of the technique to be described in Chapter 7 for ISAR data motion compensation with ;
SWATH DELAY INTERVAL
ECHO
COARSE RANGE-DELAY CELL EXTENT
. £
TIME
ONE BURST OF n PULSES-
Figure 6.23 Stepped-frequency SAR sampling (unambiguous range).
284
ECHO
SWATH INTERVAL
1
I
Miiliiiii^ t t . 0
t
DELAY
tk-1
-COARSE RANGE-DELAY CELL EXTENT
~32^JIIIIIII
tk-1
2T,t„t,...
-3T,
-I
tk-1
ilWlftnnmr t,t,...
START OF NEW BURST
I
T,
Figure 6.24 Stepped-frequency SAR sampling (ambiguous range).
stepped-frequency waveforms. This is suggested by Che dotted lines associated with motion compensation in Figure 6.2S. The hopped-frequency alternative to stepped-frequency waveforms, discussed in Chapter 5, may be able to provide advantages, such as improved electronic countercountermeasures (ECCM) performance and ability to unambiguously sample Doppler
QUAORATU MIXER
oc
SAMPLER
285
UjCC_J
? OC < H I T J D-UJOT
286
frequencies at up to the PRF instead of up to only PRF/n with stepped-frequency waveforms. 6.8 RANGE CURVATURE AND RANGE WALK 6.8.1 Side-Looking SAR •Sange curvature refers to the curved response in range that occurs for side-looking SAR as the radar platform first approaches, then recedes from, each scatterer entering the real antenna beam. This change in range, for integration lengths associated with a given crossrange resolution, becomes more pronounced at long ranges. For airborne SAR, range is relatively small, so the curvature may produce much less than one range-cell shift as the beam travels across scatterers. Multiple range cells of curvature, however, may occur for spaceborne radars. In Figure 6.10, the range shift of a target versus time history can be seen as
S
R
=
i
j
!
t
l R ^
(
6
9
,
)
where y is the target displacement from boresight. The associated two-way phase advance from (6.24) is the quadratic function
Both optical and digital processing of SAR data from spaceborne radars may require that the range curvature be removed before carrying out azimuth compression. Optical processing removes the curvature with conical lenses (or equivalent tilted cylindrical lenses). Digital processing to achieve azimuth compression of range-compressed data requires algorithms that achieve integration of range-compressed data along known curved paths of range versus azimuth. Correction of the quadratic-phase response ^j(f) produced by range curvature in SAR can be considered as a focusing procedure. As is the case in optics, SAR focusing is a two-dimensional process. The collected SAR data are focused in both the slant range and the cross range to form an image. Ordinary optical telescopes are normally focused simultaneously in the slant-range and cross-range dimensions by adjusting one focal length. SAR data focusing, in contrast, is usually carried out separately, first in the slant-range dimension (range compression), then in the cross-range dimension (azimuth compression). Independent focusing in azimuth and range is adequate when range curvature results in range migration of less than one range ceil. A two-dimensional focusing process is required when range curvature exceeds one range cell.
287
Range walk is produced when scatterers enter and leave the azlmuthal integration extent at different slant ranges. This occurs with spaceborne side-looking SAR because of the earth's cross-track rotation beneath the satellite. Range curvature and range walk are illustrated together in Figure 6.26. Range walk can be removed before azimuth focusing by corrections based on known orbit or flight-path parameters. Azimuth focusing can then be achieved within some range focusing depth to be defined below. Tracks of range walk and range curvature have the same shape for every scatterer for limited range excursion, as shown in Figure 6.27. To achieve azimuth focusing, integration over the coherent integration angle if> must be carried out along the rangedelay path associated with range curvature and range walk for each processed twodimensional resolution cell in the final image. Shortcut methods are possible when curvature is limited. The amount of range curvature and the focusing depth for a given SAR design can be determined by reference to Figure 6.28. The slant range to a scatterer at the edge of the antenna beam is
(6.93)
The integration time T used here will first refer to time of travel past a scatterer from the leading to trailing edge of the effective angular extent if, of the entire real beam. This may include multiple-look integration time if multiple sequential looks are noncoherently added for speckle reduction. For small angular extent ip„ we have Rh
(6.94)
With this substitution into (6.93), we obtain
(6.95)
The range shift from the real-beam center to either edge, obtained by subtracting R from both sides, is
(6.96)
By means of a binomial series expansion for small tfr„ we obtain
288
289
From the real-beam center to either edge, the number of cells of slant-range cell migration for cell size Ar, is AY' = ? - 5 A ^ Ar, 8 Ar,
(6.98)
290
Figure 6.28 Range curvature geometry.
Coherent integration angle fa is substituted for fa, when (6.96) and (6.97) are written for coherent integration of a single look over beam segment fa < fa,. The SAR azimuth response to a point target, because of the curved range response, is quadratic in phase. This results in a chirped Doppler echo signal, centered with zero frequency at boresight (for side-looking radars). Assume now that by some means the range-compressed azimuth response to a point target is obtained along the curved path associated with M' range cells of migration from some range R of the nearest approach The range walk is assumed to be insignificant or corrected. A filter matched to the. curved azimuth response of the point target at this range will then be mismatched at all othet ranges. At ranges of nearest approach greater than or less than R, the azimuth response exhibits quadratic-phase deviation from the response, at R. This is exactly analogous to quadratic-phase distortion, discussed in Chapter 4. A limit of fa((o,) = TT rad of phase deviation from the matched condition at the frequency band edges was suggested in Chapter 4 as a value of phase deviation that results in acceptable output pulse distortion and sidelobe levels in the weighted response from a pulse-compression filter. Some references assume irfl rad. A SAR range focusing depth (A/?) can be established based on a maximum quadraticphase error
f
29/
LEADING EDGE OF BEAM
TRAILING EDGE OF BEAM
/
2
DEPTH t
„,
1
FOCUS (AR,)
Figure 6.29 Range focusing depth.
(L\R),=
(6.99)
6^.2 Range Curvature for Spotlight SAR Figure 6.22 above is a top-down view of mapping a spotlight area as the radar platform flies a level, straight-line path. Range curvature encountered when integration is to be carried out over an integration angle / centered at zero squint angle is given by (6.97),
292
with ^substituted for iff, assuming Rifi < R. Squint angle and range to the spotlighted area, however, vary continuously for typical spotlight operation. The expressions above for range curvature for side-looking SAR are therefore not directly applicable to spotlight SAR. The issues associated with correcting for range curvature seen in spotlight SAR data are better understood from the viewpoint of data collection in the polar coordinates of range and angle, as discussed for ISAR in Chapter 7.
6.9 SPECKLE NOISE The quality of SAR imagery, because it is produced by coherent processing, is degraded by speckle noise. Speckle noise arises from random variations in earth-surface roughness. The mechanism can be understood by considering a single SAR image picture element (called pixel) and its corresponding resolved, uneven patch of earth surface. Two-way distances traveled by energy reflected from various unresolved surface areas within the resolved uneven patch are likely to differ by multiple wavelengths for most earth features at commonly used SAR frequencies. Pixel intensity is product 1 by the detection of coherently added reflections from all surface areas of a resolved patch. An adjacent patch differing only in detailed roughness produces a different pixel intensity. The result is unordered pixel-to-pixel variation, unrelated to the macroscopic features of the terrain being mapped. The variance of pixel intensity about some mean intensity is called speckle noise. It should be noted that speckle noise consists of spatial, not temporal, variation ol pixel intensity. Speckle noise has been referred to as multiplicative noise, in contrast to kT,B thermal noise, which is additive. The ratio of signal to thermal noise can be increased by increasing the radar transmitter power or reducing the receiving system noise temperature and losses. These methods leave the ratio of signal to speckle noise unaffected. Rather, it is reduced by noncoherent integration of two or more independent coherent looks at the same pan of the earth's surface. The most common way to achieve this is by noncoherently summing the superimposed processed SAR scenes viewed from different portions of the real beam. Figure 6.30 shows how data from different portions of the real side-looking bean can be separated into independent looks of the same scene. Figure 6.30(b) shows the continuous spectrum of the rectangular echo pulse of Figure 6.30(a) from a single scatterer. Figure 6.30(c) illustrates the line spectrum that occurs in a given range cell for a continuous train of echo pulses received at a Fixed PRI from a target at a fixed range. The expanded view of Figure 6.30(d) is the spectrum of the baseband response with overlapped Doppla spectra observed when the PRF is approximately equal to the Doppler bandwidth sect when the real beam illuminates a continuum of scatterers as it passes over the earth'! surface. Doppler spectra centered at each PRF line have the shape of the real-beam patten with peak responses at Doppler frequencies 0, 1/T , 2/T respectively. Reduce* responses in each lobe are seen for Doppler shifts corresponding to the forward am rearward beam edges, respectively. Figure 6.30(e) shows the sum of the overlapped spectra 2
2
293
Single pulse (a)
1h Single-pulse echo spectrum (b)
1h Echo spectrum lor PRI = Tg (c)
f
{
Overlapping Doppler spread corresponding to beam shape
2
T
2
Echo spectrum at baseband, expanded about the carrier
T
2
4 looks per beam Repetitive spectral pattern (e) J_ f 2
_2_ T,
Figure 6JO SAR Doppler spectrum.
_3_ T 2
294
The four Doppler regions centered at zero Doppler correspond to four looks. Each look contains data collected from a different azimuth region of the beam pattern. Azimuth input data can be separated into independent looks by Doppler frequency filtering in a manner to be illustrated below for the processing of SEASAT data. Individual looks can also be separated in the time domain by separate correlation with the reference function for each look. An azimuth data line, when correlated with the azimuth reference for a selected look, produces azimuth-compressed responses from data in that look and reduced responsesvfrom data in adjacent looks. Look-to-look independence is provided by the slightly diffefenTaspect angles from which image pixels are generated. Reduced independence that could occur for smaller dimensions of the resolved terrain patch and/or operation at increased wavelength tends to be offset by corresponding larger angular separation between look centers associated with the increased integration angle needed to obtain finer cross-range resolution and/or the same resolution at a lower transmitted center frequency. The advantages of reduced speckle noise provided by the multiple-look processing described above are traded for the coarser azimuth resolution that re ults from the smaller integration angles of each look. Two methods for obtaining the advantages of multiplelook processing for speckle reduction, without this penalty, are polarization agility and wavelength agility. Three independent looks are obtained with a polarization-agile radar able to simultaneously (or near simultaneously) collect HH, HV, and VV transmit/receive polarization data, respectively, where H refers to horizontal polarization and V to vertical' polarization (HV and VH do not represent independent looks). A disadvantage is radar system complexity. Multiple looks are also obtained with a frequency-agile radar able to collect data at separate frequency bands pulse to pulse, one band for each look. A disadvantage is reduced range resolution for a given available bandwidth. An analysis of speckle can be made on the basis of the probability density of pixel intensity as detected from quadrature components of processed coherent data from each look. Although often referred to as a detector, a quadratic detector is also a muter. As with any mixer, operation in the square-law region produces a signal output amplitude that is linearly related to signal input amplitude. A random echo signal will produce bipolar / and Q outputs of random amplitude. The mean amplitude is assumed to be zero in each channel. The process of generating image pixels from the processed quadrature outputs is also referred to as a detection process. Here the process is square-law detection, not mixing, because pixel intensity is made to be proportional to the echo power from the corresponding resolved feature on the earth's surface. For a single look, the pixel intensity at a single pixel location is expressed as l(\)=x\+x\
(6.100)
where JC, and JC are the amplitudes of the / and Q outputs, respectively, from the quadrature mixer. We will assume that pixel amplitude components x, and x have zero-mean Gaussian density, given by 2
t
295
1
p(» = —Use-* "''
(6.101)
2
where a is the variance of x. Intensity produced by n, multiple looks is given by /(«,) = x] + x\ + ... +xl
(6.102)
where n = In,. If the random variables x, are normal and independent with the same variance a , the probability density of pixel intensity for single and multiple looks can be represented by the chi-squared density (9) given by 2
1
where y = x} + x] +...
+ xl
(6.103)
and
r(i'+l) = /! if» = 1.2
n, where 0! = 1
The symbol y in the chi-squared density corresponds here to the intensity obtained by summing the squares of the amplitudes x , ..., x.. Look-1 pixel intensity is x\ + jj. Look-2 intensity is x\ + JC}. Look-3 intensity is x\ + JtJ, and so forth. The probability densities of pixel intensity for single-look and multiple-look processing can be respectively expressed as 2
p[/(D] = jL
exp[-/(l)/2o-M
(6.104)
for single-look processing (n = 2), and p\Kn,)] = j^j-j
1
[/(«,)]"-' expHinWo ]
(6.105)
X The tymbul n here does not refer to the number of radar pulses as elsewhere in the text, but rather to Ihe •mill, i of Gaussian-density random variables forming the chi-squared distributions.
296
for multiple-look processing, where 7(1) and I(n,) are single-look and multiple-look pixel intensities, respectively, and n, = n/2 is the number of looks. The variance of the random < variable x, is given by 2
(6.106)
where E(x) refers to the"expected value of x,. Because we assumed zero-mean density for the / and Q amplitude components (balanced quadrature processing), we have Efa) = 0 so that 1
a = E(x})
(6.107)
Mean pixel intensity resulting from single-look processing is 7(l) = £(jtJ + xl)
(6.108)
= E(x}) + E(x\)
The expected values of x] and x\ produced by identical / and Q processing are themselves identical, so that .
7(l) = 2cr
1
(6.109)
where cr is the standard deviation of the / and Q components of pixel intensity. By substituting 2& = 7(1) into (6.104) and (6.105), we obtain, respectively, 1
PVO)}
= jj^ exp(-/(l)/7(l)]
(6.U0)
and
"
[ /
^
]
= rife) TTTit^f]"''
- P I - ' ^ ^ ^
6
M
< - >
The single-look (6.110) and multiple-look (6.111) densities can be shown to have standard deviations given by l
(6.112)
o-UM) = V"^0)
(6-U3)
and
297
The signal-to-speckle-noise ratio for a single-look is the ratio of mean to standard deviation 7(l)/
, {n
n.7(l) )=
N - ^T(Tr^'
r
( 6 n 4 )
Actual single-look and multiple-look speckle as seen with the SEASAT system described below are shown in Figures 6.31 and 6.32. 6.10 DESIGN EXAMPLES Performance will now be evaluated for two types of SAR designs to illustrate the use of the various expressions developed above. The first design to be evaluated will be a spaceborne SAR with the approximate parameters of the JPL SEASAT design [6,10], which was put into orbit in June 1978. SEASAT was in operation for a total of 105 days.
Figure 6,31 Four-look SEASAT SAR map of Ihe Sonora Sand Dune Field in Baja California. (Courtesy of Dr. D. N. Held, formerly of (he NASA Jet Propulsion Laboratory (JPL).)
298
Figure (.32 Speckle pattern as seen in a single look at the area outlined in Figure 6.31. (Courtesy of Dr. D. N. Held, formerly of the NASA Jet Propulsion Laboratory.)
During that time, about SO hours of SAR data were collected for the NASA. The second SAR design to be evaluated will be a hypothetical airborne SAR. The performance of these (wo types of SAR systems will be evaluated on the basis of their basic radar design parameters independently of the type of processing that may have been or would be implemented. Processing architectures are to be discussed later in this chapter. Expressions from Tables 6.1 and 6.2 will be used to determine most performance parameters. We should bear in mind that parameters for an actual design would be derived iteratively by using similar expressions to achieve desired performance. 6.10.1 SEASAT The SEASAT system was the first spaceborne SAR put in orbit by NASA. Other spaceborne systems have followed, orbiting the earth and other planets in our solar system. Some of the highest resolution mapping, however, was performed by SEASAT. It is i well-documented radar and presents an ideal system to illustrate the principles discussed in this chapter. The actual SEASAT design operated at a center frequency of 1,275 MHz. The antenna was a 10.7m-by-2.16m array producing a 1-deg real beamwidth in azimuth and a 6-deg elevation beamwidth. This real beam illuminated an earth surface of about
299
15 by 100 km in area. The SEASAT concept is shown in Figure 6.33. Several PRFs were actually used, but a nominal value is given in the list of SEASAT parameters in Table 6.3. The transmitted pulse was a chirp pulse. Data was down-linked to three U.S. stations (Fairbanks, Goldstone, and Merritt Island) and two foreign stations. The ratio of pixel signal to thermal noise is determined by solving the radar equation for the nominal SEASAT parameters of Table 6.3. The SNR for each look, produced by n coherently integrated echo pulses per look, is
Figure 6J3
SEASAT SAR system.
300
Table 6 J SEASAT Design Parameters (Approximate) Design
Parameter
Center frequency Bandwidth Pulse duration Peak power PRF
Symbol
Value
1.275 MHz (A = 0.235m) 19 MHz 34 us I.OOOW 1,500 Hz (nominal)*
J
P T, P,
Radar system noise temperature Satellite altitude Antenna gain Incident angle Antenna beamwidth in azimuth Platform velocity (ground track) System loss Number of looks Single-look coherent integration angle Range to center of swath Pulses per look
r, r.
650K 800 km 35 dB (3.162) 67 deg (at beam center) I deg (17.4 x Ifr' rad) 6.6km/s 2(3dB) 4 0.30 der (5.24 x 10° rad) 854 km^ 1,024
c «4
<*>. *, L
* R n
'Actual SEASAT PRF selections were 1.463, 1.537, and 1,645. A PRF o r 1,500. however, will be used for illustration.
Radar cross section or of resolved surface features depends on the terrain to be mapped. Values for the average land clutter return parameter, y = o~°/sin 0,, for three types of terrain at 0 = IS- to 70-deg incidence at 1.25 GHz was obtained from Nathanson [II, Table 7.13]. Values for the sea clutter reflection coefficient
t
m
t
d
(6.116) for SEASAT parameters listed in Table 6.3. Results obtained by using (6.116) for the three terrain conditions and sea state 3 with tr values from Figure 6.4 are listed in Table 6.5. Other performance factors were calculated from expressions listed in Table 6.1 and expressions (6.98) and (6.99). The SNR required to detect the presence of an earth feature can be estimated by assuming a fluctuation model for look-to-look signal power. On the basis of Swerling case 2 statistics (fast fluctuation look to look) for P = 0.5 and P = 10"*, an SNR of D
fA
301
Tabic 6.4 Land and Sea Clutter Return
Terrain
y.'
-32 -15 -11
6.31 x lO" 3.16 x lO"' 7.94 x lO"
Desert and roads Open woods Cities
4
1
•1-3)
Sea State ' -
\
0 3
o\nf) (For 25m x 25m Resolution Cell at ft = 67 deg)
dB (y.) (for ft = IS lo 70 deg)'
r
at ft = 60 deg -32 -20
•£)
at ft = 60 deg 6.31 x lO"' 1 x 10-'
0.36 18.2 45.7
oW) at ft = 60 deg 0.39 6.25
'Source: F. E. Nathenson, Radar Design Principles, New York: McGraw-Hill, 1969, pp. 238 and 273. 'Sea clutter returns listed are for horizontal polarization. '*„ = - — — = mean clutter return with o* in square meters of RCS per square meter of surface area resolved, sin tjj ff * (25 x 25)o* = (25 x 25) y . sin ft. Incident power density is proportional to sin ft so dial the quantity y. - /sin ft tends to remain constant over wide variations (15 to 70 deg) of incident angle.
about +7 dB is required after the noncoherent integration of four looks. The SNR produced by actual earth surface features will vary above and below those predicted in Table 6.5, but pixel SNR appears to be adequate to observe most terrain features, except desert and very low sea states. The SEASAT design, to achieve the needed SNRs with its modest transmitter power, was required to look down steeply with a 67-deg incidence angle in order to increase the backscatter coefficient and to shorten the range. Table 6.5 shows that at the center of its range swath the SEASAT design results in a resolution capability of about 25m by 25m on the earth's surface. The PRF is sufficiently high to provide a cross-range ambiguity length of 23 km. This is adequate for unambiguous sampling of Ihe Doppler spread produced by the 18.7-km illuminated cross-range extent at the range-swath center associated with the effective beamwidth of Art = 0.022 rad (1.25 deg) in azimuth. At the same time, the PRF is sufficiently low to provide a maximum unambiguous illuminated ground-range extent of 256 km, which is quite adequate to sample the approximately 100 km of illuminated ground extent associated with the 6-deg elevation beamwidth. Synthetic aperture size is 4.5 km for each coherent look. Regarding range curvature, six cells of range migration at the range center are produced in four looks. The resulting depth of focus is 17 km, based on v/2 rad residual two-way phase variation across the real-beam response. Additional issues concerning orbit position uncertainties and range walk are not considered in this example. A realistic analysis of space radar design would also involve precise determination of orbital mechanics. 6.10.2 Airborne SAR The second example for which performance will be evaluated is hypothetical. It is an illustration of the stepped-frequency SAR concept described above. Let us assume an
302
Table 63 SEASAT Performance Calculations
Performance Parameter
Value for SEASAT Design Parameter
Expression
Cross-range resolutuion (four-look) processing Slant-range resolution
1A 2* . I c ' 2 B Ar, sin 23* A/?, cT, 1 sin 23* 2 sin 23* '
A r
A
Ground-range resolution
r
22.4m*
=
7.89m
=
20.2m»
p
Maximum unambiguous illuminated range extent (along the ground assuming a flat earth) Cross-range ambiguity length (at center of swath) Single-look integration length at center of swath Single-look integration time Average pixel SNR (per look)
Range migration during four looks
_ *
tpjn
256 km 23 km
A
4.5 km
X = Rip
7"=nr,
'desert S woods S cities (.sea (SS - 3) _ 1 R^tf ^ 8 Ar,
w
0.68 sec
-4.4 dB +12.6 dB +16.7 dB +8.0 dB 6 cells
=
Range focusing depth per look for •JC*) = w/2
S(btf
17 km
T h e actual SEASAT resolution cell size after processing is reported to be 25m x 25m [10].
antenna of approximately 2m in length by 0.35m in height, which would provide about 35-dB gain with an effective azimuth beamwidth ip, = All of 0.015 rad (0.86 deg). Only the side-looking mode will be evaluated. If a target area of interest were discovered in the relatively low-resolution side-looking mode, the radar operator could slew the antenna to zoom in on this area. At the same time, bandwidth would be adjusted upward to product a slant-range resolution equal to the increased cross-range resolut' ^ resulting from the increased target dwell time that occurs as the antenna beam spotlights the target area oi interest. Table 6.6 lists design parameters for the radar. Frequency step size was selected it accordance with the criteria (6.84) that unambiguous integration length equals or exceed! the range extent associated with the matched-filter response to the transmitted pulse. Calculations are for range R = 85 km. The number n of pulses per burst is the total bandwidth nA/ divided by the stef size A/. Pulse repetition rate 1/T was selected in accordance with the criteria (6.90) fa effective real beamwidth if/,. The number N of bursts per azimuth single-look integratiox 2
303
Table 6.6 Hypothetical Airborne SAR Parameters Design parameter Center frequency Pulse duration Peak power Radar system noise temperature Aircraft altitude Incident angle Antenna gain Antenna effective beamwidth in azimuth Platform velocity System loss Number of looks Single-look integration angle Bandwidth Frequency-step size Number of steps per burst PRF
Symbol
Value
J T, P, T. hi
10 GHz (X = 0.03m) 1.25 fa 2kW I.000K 15 km 10 deg (at 85 km) 35 dB (3.162) 0.86 deg (0.015 rad) 150 m/s 2 (3 dB) Up to 12 1.25 x 10-'rad 12.8 MHz 400 kHz 32 5.783
«*.
G
"f L n.
*
n\f Af n 1 T N nN t
Bunts per azimuth integration length at R = 85 km Number of pulses per look at R = 85 km
128 4.096
length Ri/i was selected in accordance with (6.87). Parameters A/, n, 1/7*2, and N above were selected so that the number of pulses nN per integration length became 2" * 4,096 according to the expression
,i/V = — : J r
(6.117)
evaluated at R = 85 km. Azimuth resolution obtained by integration over the entire effective illumination beamwidth from (6.64) is lm. Each of the four looks of Table 6.6 with integration angle i>= 1.25 x 10"' rad provides a cross-range resolution of 12m, as indicated in Table 6.8. This remains the resolution following noncoherent look-to-look integration to reduce speckle. Resolution with four looks could be improved to about 3m by increasing integration angle for each look to 4 x 10~ rad to roughly fill Ihe available illumination beamwidth of 15 X 10*' rad, or the ratio of signal to speckle noise could be improved by adding more of the 1.25 x 10" rad looks to fill the beam. Pixel SNR for the hypothetical airborne SAR design parameters of Table 6.6 is determined by using (6.115) with Nn substituted for n. The radar cross section a, as for the SEASAT example, depends on the terrain to be mapped. Clutter cross section from }
J
304
Table 6.7 Land and Sea Clutter Returns (for 8 = 10 deg) a
Terrain Desert Open woods Cities Sea Slate
For 12m x 12m resolution cell
aim')
-Ki"
2.51 x I0"» 5.01 x 10"' 31.6 x 10-'
-26 -23
0.36 0.72 4.55
0 3
1.25 x 10-' 6.31 x I0-
0.0018 0.091
4
'Source: F. E. Nalhenson, Radar Design Principles, New York: McGraw-Hill, 1969, pp. 236 and 263.
'Median backscatter for vertical polarization.
Table 6.8 Hypothetical Airborne Radar Performance Performance
Expression
Parameter
Cross-range resolution (per look)
A
A
1
r
=
*'
Unambiguous illuminated range extent
RA
^ - = 2v7n v
Single-look integration lime
12m
C
2nTf
,_ , (
Single-look integration length at R = 85 km
12m
A
2 *
Slant-range resolution (~ ground-range resolution)
Cross-range ambiguity length
1
Value
1.536m
X = R+
107m
T=nNT,
0.71 sec
desert 5, woods N cities .sea (SS - 3)
Average pixel SNR (per look)
26 km
Range migration during four looks
+11 dB +14 dB +22 dB +5dB 0.02 cells
8 Ar, 38 km
Range focusing depth per look for
Nathanson [11, Tables 7.6 and 7.11] is tabulated here in Table 6.7. The SNR for four noncoherently added looks at 85 km is S
P,G'AVr,/Vn = 32.2o-
305
J
The calculated performance parameters are listed in Table 6.8. The hypothetical stepped-frequency SAR system operates at much shorter ranges than the SEASAT design. Thus, range curvature is greatly reduced. Also, the shorter range permits greatly reduced sampling rates for the cross-range Doppler response. In the SEASAT design, the PRF of 1,500 pulses per second was shown to provide 23 km of unambiguous cross-range sampling. This was adequate at midswath range with some margin to sample the Doppler band of frequencies produced by the 18.7-km cross-range extent illuminated by the real antenna. The shorter range of the airborne SAR design allows the use of a stepped-frequency waveform of 32 pulses per burst to generate range profiles. The resulting unambiguous cross-range length of 1,536m at the far range of the swath (85 km) is adequate for sampling the Doppler frequency spread produced by the azimuth extent Rip, = 1,275m at that range associated with the 0.015-rad effective beamwidth of the real antenna. The high PRF, however, results in an unambiguous range of only 26 km. To achieve the specified 85-km range would require sampling as shown in Figure 6.24. Range migration, because it is less than one cell, is not seen as an issue. The stepped-frequency design, because of multiple pulses required for each coarse-range cell, tends to require high PRF as resolution increases or when longer ranges are necessary. Implementation of the pulse-to-pulse hopped-frequency waveform, discussed Chapter 5, can, in principle, allow unambiguous sampling of Doppler frequencies at up to the PRF, which could allow longer unambiguous range. 6.11 SAR PROCESSING SAR processing began in the mid-1950s using optical techniques. Radar data recorded on film rolls on board the SAR aircraft were processed into maps on optical benches on the ground by using special lenses and coherent light sources. Optical processing of this type is often considered to be the conventional method for SAR mapping and optical techniques are still employed when the application requires extreme resolution. The trend in SAR mapping, however, is now clearly toward digital processing. Although quite complex, SAR digital processing offers the advantages of accuracy and flexibility. Digital processing techniques have advanced dramatically since early SAR development, while conventional optical techniques have not experienced significant improvement. 6.11.1 Input Data for Chirp-Pulse SAR o
The following discussion of SAR processing methods will be restricted to conventional chirp-pulse SAR systems with antenna beams looking 90 deg from the direction of platform motion. Processing of stepped-frequency SAR data will not be discussed here, but will be treated in connection with ISAR in Chapter 7. After first defining the form of received signals from chirp pulses, we will discuss both optical and digital processing methods. Finally, methods for processing SEASAT digitized data will be discussed as an example.
306
The signal received from surface features by a chirp-pulse SAR is dispersed in both range delay and azimuth time-history. Range and azimuth compression of this twodimensional signal produce a /focused image (map) of the surface. The form of the signal will be defined under tlnvassumptions of uniform illumination, rectangular chirp waveforms, and straight-line SAR platform motion. The chirp waveform, as discussed in Chapter 4, was expressed by (4.15) in complex form as l
,
c
sM = ^ect^e' *> •' '•'
7,
(6.119)
where / is the center frequency, T, is the uncompressed pulse width, and K is the chirp rate with zero delay set to be at the center of the chirp pulse. A single point target in the real beam at range delay r will produce a normalized response expressed as s(t -T) = rcctl^^^ »¥»-a*m-m
(6.120)
t
The response, after mixing with a continuous reference signal at /, is at baseband and may be expressed as ,
*'(f- T) = K c l ^
i
y ^
l
-
K
U
- ^
^
(6.121)
Now consider the geometry of Figure 6.34 compared to that of Figure 6.10. The range delay rto a point target, displaced y in the cross range from boresight for minimum range distance R at / = 0 and a platform velocity v is r
r-=
a + ^-VirH
(6122)
where t = 0 is at boresight and R > v t. Range delay expressed in (6.122) is the delay to the point target during the azimuthal integration time T, which is the duration of time that the real antenna beam illuminates the single point target. De'^/y will be assumed ;o remain.constant during the chirp pulse. Substitution for range delay from (6.122) into the first term of the exponential of (6.121) will now be carried out to illustrate the twodimensional quadratic-phase response to each scatterer. Also required is a second amplitude term, which, assuming uniform illumination during integration time T, is r
rect^y^
(6.123)
307
Figure 6-34 SAR range delay to a single scatterer.
where T is the time-history (azimuth) delay ylv . The baseband response to a single point target, with J- cl\, becomes t
"^(^T^)
s\t - r) =
xexpJ - 4^-^ l
J
T r
--^-^-
(6
+
124)
-jJ
As we can see from (6.124), the phase of the baseband response to a point target is the sum of two quadratic-phase functions and a constant range-delay phase. The frequency of die response for the complex representation varies above and below zero. Zero frequency corresponds to the target positioned at range delay r and azimuth position y . The first exponential term of (6.124) contains a slowly varying quadratic-phase function corresponding to the Doppler frequency change that occurs as the slant-range distance to the point target first decreases, and then increases as the SAR platform passes
308
by. A second point target, offset in the crossaange from the first, would produce a phase response corresponding to an offset Doppler response. The second exponential term contains a rapidly varying quadratic-phase function corresponding to the chirp frequency deviation from the center frequency of the delayed pulse. A second point target, offset in the slant range from the first, would produce a phase response corresponding to an offset frequency deviation. The two-dimensional signal from a single point target is illustrated in Figure 6.35 for an azimuthal integration length that is sufficiently short to make range curvature appear negligible. (Curved responses are illustrated later.) The received echo signal from a point AMPLITUDE
AZIMUTH RESPONSE AT t, = t
PHASE
y/v.
•T-
t, =
t, AMPLITUDE
yN,
PHASE
RANGE/AZIMUTH PHASE RESPONSE RANGE RESPONSE AT t, => y/v,,
Figure 6.35 Two-dimensional quadratic-phase response to a point target observed with a side-looking SAR assuming small range curvature.
309
target will extend over the uncompressed pulse duration T, and will be centered in range delay at the target's range-delay position r. The signal at the target's range delay will extend in azimuth over integration time 7", corresponding to the synthetic aperture size, and will be centered at the target's azimuth delay. It is assumed that the delay r is essentially constant during each echo pulse, but varies according to (6.122) during the target's dwell time T. Symbols /, and h in Figure 6.35, sometimes called fast time and slow time, refer to range delay and time history, respectively. The third exponential term of (6.123) is a phase term dependent on the closest approach in range of the radar platform to the point target, a term ideally made a constant by flying the platform in a straight line. The response of a single point target in terms of data collection space will extend over the integration lengths in range and azimuth. The area cT,/2 x v,7"is the data collection element (for small curvature) that contains the dispersed response to a point target.
6.11.2 Optical Processing Conventional optical SAR processing is carried out on film rolls that contain the twodimensional phase history of the response of target scatterers, which were produced as the SAR platform traveled above and alongside the range swath to be mapped. Both range and azimuth compression can be performed optically. Film rolls are exposed on an optical film scanner, illustrated in Figure 6.36. The input to the scanner is the coherent signal heterodyned down to bipolar video. Light intensity from a CRT in the scanner is modulated by the bipolar video signal. This corresponds to the baseband signal produced by reflection from multiple scatterers on the illuminated earth surface. The film roll is exposed as it moves past the CRT in a direction perpendicular to the range sweep of the intensitymodulated light spot. A bias voltage may be used to produce the desired film exposure. After recording the SAR phase history, the input film roll is brought to an optical bench, where it is focused to form the output film roll, which is the SAR strip map. Data recording of the response from a single point target is illustrated in Figure 6.37. An actual film record would contain the phase histories of the numerous scatterers on the surface to be mapped. Phase histories of individual scatterers are likely to overlap one another, but will ideaily focus to individual points. Recorded phase history on film is similar to the Fresnel zone plates used in optics. The quadratic nature of the phase response makes "it possible to diffract collimated coherent light passing through the film to produce focused images. Phase history recorded on a SAR film roll focuses incident coherent light at different focal lengths in azimuth and elevation. The situation can be thought of as astigmatism, which can be optically corrected by the use of cylindrical lenses. Separate focal lengths occur because recorded phase is the sum of separate phase components in each dimension. From (6.124), the slant-range component of a recorded phase of the echo from a point target is 4irK(t - r)V4. This dimension of phase is recorded in the range dimension at a
310 T
Figure 636 SAR optical film scanner.
x
sweep velocity of v,. Also from (6.124), the azimuth component of a phase of the same point target is -4ir\v,t - y)V(2XR). This component of phase is recorded in the azimuth dimension at the film transport velocity of v.. The resulting two-dimensional phase history in range and azimuth focuses collimated light passing through the film at different focal lengths in range and azimuth. The focal length associated with SAR phase history recorded on film can be compared to other, more familiar optical focusing mechanisms. Figure 6.38 illustrates three equivalent focusing mechanisms (each shown for one dimension). Light in each mechanism propagates in a manner so as to encounter quadratically distributed delay in the cross-axis dimension labeled x. The quadratic-phase function in'Figure 6.38(a,b) results from qua- . dratic variation of delay along the x-dimension. Figure 6.38(c) illustrates one dimension of quadratically distributed phase history recorded on SAR film. F^rr each case, the oneway phase function for light at wavelength X, is )=
^ -xA
(6,25)
where 9 is the optical focal length. This result for the reflector in Figure 6.38(b) is directly analogous to that found from Figure 6.10 and the accompanying discussion regarding the quadratic-phase response produced by the SAR platform moving past a point target on
311
RANGE DELAY, ti
AZIMUTH TIME HISTORY, t,
Figure 6.37 Optical film record of the phase history of a single point target (side-looking SAR with insignificant range curvature).
the earth's surface. With the proper optics, focusing results when collimated coherent light is passed through the zone plate formed by the film record of the quadratically distributed responses to individual scatterers. For sweep velocity v, and film transport velocity v„ the ^-dimension of the recorded signal is x, = vft, - T) in range and jr = v,(r - ylv ) in azimuth. Therefore, uV(jr) of (6.125), written in terms of I, and /j, becomes 2
2
r
In v|((, - r)
A, in the range dimension, and
29,
a
(6.126)
312
LENS
(»)
REFLECTOR
(b)
(c)
Figure 6.38 Equivalent focusing mechanisms.
in the azimuth dimension, where /, and fj refer to time associated with range delay and time history, respectively. Optical focal lengths 9 , and JF, can be expressed in terms of radar parameters by setting the magnitude of the two optical phase ct ^ponents given in (6.126) and (6.127) equal to their corresponding RF phase components from (6.124) as follows.
2 7 r v
in range, and
'(>.-r)»
„(», -
T)
1
(6.128)
313
^""AT
2*.
~T~2R
"T—*—
( 6 1 2 9 )
in azimuth. By solving for the two focal lengths and recalling from Chapter 4 that chirp slope K = A/7",, we obtain
" A,K ~ A,A
(6.130)
for the optical focal length in the range dimension, and
9
(6.131)
for the optical focal length in the azimuth dimension. The two focal lengths are illustrated in Figure 6.39. Azimuth focal length varies linearly with range across the width of the film because of increased radius of range curvature of input data at increasing range. SAR optical processing corrects for the astigmatism by using cylindrical lenses. In addition, conical or tilted cylindrical lenses correct for the linear variation of azimuth focal length with range. Figure 6.40 illustrates a simplified configuration. The data film on the left has a vertical range focal plane followed by the tilted azimuth focal plane. A cylindrical lens is oriented so that its input focal plane coincides with the tilted azimuth focal plane of the data film to coliimate rays in the azimuth dimension. A second cylindrical lens, further to the right, is placed so that its input focal plane coincides with the vertical range focal plane of the data film to coliimate rays in the range dimension. With both dimensions collimated, targets will be focused at infinity to the right. A spherical lens focuses targets on the SAR image plane. Actual optical processors are far more complicated in practice. Usually, the SAR image is made continuously. Both the SAR data film and output SAR image film are driven, and a slit in the range dimension produces continuous exposure of the SAR image.
6.113 Digital Processing The generation of SAR images is a two-dimensional process, regardless of the processing technique employed. Optical SAR processors process the range-azimuth analog data simultaneously in time. Digital SAR processors often resort to a series of two one-
314
Figure 639 Range and azimuth focal lines of point-target phase history. (Modification of Fig. 23. p. 1191 • (10). Reprinted with permission.).
dimensional processes to produce the two-dimensional result from digitized input data. The advantages of increased accuracy and flexibility in digital processing are obtained at the expense of considerable complexity. It is beyond the scope of this section to cover the field of SAR digital processing. Rather, a two-dimensional | prrelation method of processing that is applicable to chirp-pulse-compression SAR wnl be discussed in a t attempt to report some of the important issues. Two-dimensional correlation achieves pulse compression
in the slant range (range compression) and azimuth compression m
the cross range (azimuth compression). The idealized response to a single point target viewed with a chirp radar wat expressed in (6.124). This equation contains similar quadratic terms in both range-delay and time-history dimensions. Lenses are able to perform the two-dimensional compressioa in optical SAR processors. The lenses were shown to possess quadratic-phase functions, which collimated the light through the data film so that individual target responses could be focused into points on the image film. This process has also been described as two-
315
RANGE
A
Z
|
M
u
T
H
AZIMUTH FOCAL PLANE
RANGE COLLIMATOR
SARIMAGE RECONSTRUCTION PLANE o
p
T
|
c
A
L
/AXIS
COLLIMATED COHERENT LIGHT RANGE FOCAL PLANE
INFINITE CONJUGATE IMAGING LENS
TILTED AZIMUTH COLLIMATOR
Figure 6.40 Simple optical SAR processor. (From (101, Fig. 24. p. 1191. Reprinted with permission.)
dimensional optical convolution [3]. Digital processors for pulse-compression SAR, in an analogous process, may convolve the digitized two-dimensional data with a digitized two-dimensional, matched-filter impulse response function instead of lenses. The response function, in general, is made up of the impulse response h(t ) of the chirp signal for range compression and a similar function /t('i) for azimuth compression. As before, t, refers to range delay (fast time) and tj refers to time history (slow time). SAR processing, however, is often described in terms of correlation rather than convolution. Instead of referring to the impulse response of the matched Filter in range delay or in azimuth time history, the concept of range and azimuth reference functions B used. The equivalent reference functions in range and azimuth are the time inverses of the complex conjugates of the respective matched-Filter impulse responses. Correlation of the range-delayed signal with a range reference is the equivalent of convolution of the same signal with the impulse response of the matched filter to the transmitted waveform. A similar equivalence holds in the azimuth dimension. The reference function for range correlation is the point-target response in range. The reference function for azimuth correlation is the point-target response in azimuth. A two-dimensional reference function is the dispersed response in range and azimuth. Azimuth and range compression of two-dimensional signal data will now be described for Ihe ideal case in which the two dimensions of the reference function can be defined independently. This idealization is valid for the processing of a data block for which range and azimuth extent is sufficiently small that range curvature and range walk can be neglected. Then a single azimuth reference produces azimuth focusing at all ranges in ne block. Figure 6.41 illustrates a block of digitized two-dimensional data that includes the idealized response from a single point target at delay rand azimuth position y . Each t
316
Hrl RANGE-DELAY EXTENT OF INPUT DATA BLOCK
POINTTARGET UtRESPONSEl
TIME HISTORY -EXTENT OF INPUTDATA BLOCK
RANGE DELAY, t, (FROM TRANSMIT)
AZIMUTH TIME HISTORY, t, (FROM BORESIGHT)
Figure 6.41 SAR dan Mock for chirp waveform showing response lo a point target centered at /, = r, fi« -y/v (small range curvature). f
resolved element contains a complex data sample. T h e response tn two separate point targets is illustrated in Figure 6.42(a). Two-dimensional correlation with the two-dimensional reference produces an image block containing the two targets in focus as indicated in Figure 6.42(c). Columns of range data lines are first correlated against the range reference. The correlated result for each range data line is a set of range-compressed data lines. Range-correlated results are shows in Figure 6.42(b). Rows of azimuth data lines are then correlated against the azimuth reference to obtain two-dimensional correlated results, shown in Figure 6.38(c). The two one-dimensional processes produce the required two-dimensional image of Figure 6.42(c) without distortion because the same range reference was assumed valid for all range columns and the same azimuth reference was assumed valid for all azimuth rows.
317
RANGE DELAY t,
AZIMUTH TIME HISTORY t ,
DISPERSED RESPONSE F R O M TARGET 1
DISPERSED RESPONSE FROM TARGET 2
n-1
(a)
RANGE REFERENCE 1
(b) 02
N-1
1 3 AZIMUTH REFERENCE
(e)
Figure 6.42 Processing of SAR input data containing two point targets (small range curvature): (a) input data • block; (b) range-correlated data; (c) image frame.
318
6.11.4 Nonindependent References Independent range and azimuth references were employed in Figures 6.41 and 6.42. This was possible because of the stated assumption of sufficiently small range curvature, range swath, azimuth integration angle, and range walk. Consider the case in which range compression produces such closely spaced azimuth lines that azimuth responses are not contained along individual lines. This occurs when range migration M' of (6.98) exceeds unity. Azimuth compression for each image pixel must then be carried out along curved paths in range to achieve full resolution capability. The azimuth reference is also range-dependent. Figure 6.43 reprr ^nts the phase history of a chirp-pulse response to two point targets at the same azimuth position, but separated in range at opposite edges of a SAR range swath. The phase history for both targets remains quadratic in both range and azimuth (as viewed along their curved range responses), but we can see that the azimuth reference needed for azimutfi focusing at near range differs from that at far range. The curve is longer but less pronounced for the response to the target at far range. Therefore, an azimuth reference for range R matches a larger FM Doppler slope than that for /? . The range reference, because it is determined only by radar waveform, is independent of azimuth position. Finally, range walk caused by cross-track earth motion beneath a satellite SAR, unless corrected, produces responses that walk through range cells. Range curvature and range walk result in responses from individual scatterers that travel through range cells and require a range-dependent reference. In principle, image formation is still possible by using two-dimensional processing from known geometry. For example, after range compression, the reflectivity for a selected two-dimensional resolution cell could be established by processing range-compressed data obtained along the range-azimuth path on which a scatterer would travel to produce a response in the selected resolution cell. The process would be repeated for each cell. This approach is avoided in practical processors because of its complexity. An example of a shortcut method for carrying out two-dimensional processing is that for the SEASAT digital SAR processor, described below. Another method is polar reformatting, which is described for ISAR in Chapter 7. t
2
6.11.5 Fast Correlation We discussed the convolution of sampled and digitized target signal data produced by a chirp radar using FFT processing in Chapter 4. The method was called fast convolution. Fast convolution of digitized data was accomplished with a digital version of the matchedfilter impulse response to the chirp-pulse waveform. The same process could have been described in terms of fast correlation with a reference function equal to the time inverse of the conjugate of the digitized point-target response in range. SAR data sets can be processed by using a two-dimensional fast-correlation method. Such a method, because
Figure 6.43 Response tt> two point Urgets at the same aiimuth position but separated in range (chirp-pulsecompression SAR).
320
of the use of the FFT algorithm, is usually faster than direct correlation, just as fast convolution is faster than direct convolution. Fast convolution is based on the discrete form of the convolution theorem, which for input signal s,(t) convolved with impulse response h(t), was expressed in ( 4 . 5 4 ) as FTta(f) * /.(f)] = SAf) x / / ( / )
(6.132)
where / / ( / ) is the Fourier transform of h(t) and SAf) is the Fourier transform of 5,(r). An equivalent expression can be written in terms of correlation. From the definitions of convolution and correlation, the following equivalence can be written: FTUAD * HO] = F T U W ® h'(-t)]
(6133)
where <8> denotes cross correlation. The matched-filter transfer function H(f) for the transmitted waveform s\(i) is S",{f), and the time inverse of the complex conjugate /i'(-f) of the matched-filter impulse response h(t) is 5,(0, where s,(t) is the point-target response, which becomes the reference function. Equation (6.132) for the convolution theorem, therefore, can be rewritten as the correlation theorem, expressed as FT[5,<0 ® *,(/)] = SAf) xS'Af)
(6.134)
where SAf) = FT[f,{f)l and SAf) = FT[J,(')J. Thus, the correlation of a signal with a reference function is obtained by multiplying their respective Fourier transforms, then using the inverse Fourier transform to translate back to the time domain. Of interest for fast correlation is the discrete version of the correlation theorem, which is expressed in shorthand notation as D F T I J , ( / A / ) ® 5,(/Af)l = SAiAf) • S ; ( I A / ) , for /, / = 0 , 1, 2
n-1
(6.135)
where sAIAt) and Ji(/Ar) are both periodic with the same period nA/ for sampling interval Af, and 5,(iA/°) and S\(i&.f) are also periodic, with period nA/. Statec,' in words, the DFT. of the correlation of two periodic discrete functions is equal to tne product of their individual discrete Fourier transforms. Fast correlation is carried out digitally by using the FFT process. Fast correlation of two-dimensional SAR data with independent range and azimuth references is illustrated in Figure 6.44. Here, rows of range response data sAh) are transformed into rows of frequency response SAf)- (Note that rows and columns are the reverse of those in Figure 6 . 4 2 to facilitate arrangement of the block diagram.) Rows of frequency response are then multiplied, element by element, by the frequency-domain form of the range reference to form rows of frequency-response products. Row-by-row inverse Fourier transforms of these products produce cross-correlated responses in the range domain versus time history f- The data have now undergone range compression. Range-compressed rows are stored in a corner-turn memory, from which cross-range 2
x
321
8,(t )vs.t,
wo
t
t,
/
I
FFT
f,
J
_sm
_
|
i,
I
t,
H-FFT-CZ3 ROW-BY-ROW RANGE REFERENCE RANGE CORRE•?(t») LATION FOR RANGE f, COMPRESSION 1
t. INPUT DATA BLOCK
1 FFT' i
J
^ FFT—,
CORNER TURN MEMORY: READ IN ROWS, READ OUT COLUMNS
, COLUMN-BY- I COLUMN I AZIMUTH I CORRELATION! FOR AZIMUTH I COMPRESSION
I AZIMUTH REFERENCE
-FFT^L.
OUTPUT IMAGE (BEFORE DETECTION) (N x n PIXELS) Figure 6.44 Fast condition with one-dimensional references.
322
(azimuth) data columns are read out column by column. Azimuth correlation is then carried out in exactly the same way as range correlation, except that it is done in the time-history dimension instead of the range-delay dimension. The final result is the output image frame. The above fast-correlation process is illustrated in terms of a processor block diagram in Figure 6.45. The same process could also be described in terms of fast convolution. Fast convolution or correlation requires special care because of the aperiodic nature of the data and reference. This was discussed in Chapter 4 in connection with digital pulse compression of one-dimensional range data. We showed that it is possible to generate a valid convolution of a finite length of sampled range data by establishing a common periodic length for both the signal and impulse responses, which is sufficiently large so that the convolution result of one period does not overlap that of the succeeding period. We saw that this is possible by adding zeros to the discrete impulse response function so as to satisfy (4.56). Digital SAR processing, however, may involve subimage processing from relatively large input data blocks. Correlation in the range or azimuth dimension using the FFT for the resulting long data lines can be more conveniently carried out by techniques referred to as overlap-save or overlap-add [12]. In each of these processes, the input data line is divided into subsections that are overlapped by the extent of the reference function. Subsections will then correspond to periodic lengths, which, when correlated with a reference function of the same period, produce a periodic response equivalent to direct aperiodic correlation valid over the subsection. The overlap-save technique saves the cross-correlation result for the part of the period that excludes the end effect at the front of each period. The end-effect result is invalid in that it does not represent the true aperiodic result. Except for thefirstsection, the lost end-effect part of the cross-correlation result is restored when we make a composite reconstruction of the individual sections. The overlap-add process is similar, but data samples within the overlap portion of each data section are replaced with zeros. Composite reconstruction of the cross-correlation result of all sections then provides valid representation of the true aperiodic result with no invalid end effect. Subimages can be generated by initially correlating the first section of each input range data line with the range reference. The azimuth section length is the number of correlated range data line sections chosen to be read into the comer-turn memory. Azimuth data line sections read out of the corner-turn memory are correlated with the azimuth reference. The result is one subimage. The next subimage in the range dimension is developed by repeating the process in the next section of range data lines. The next subimage in the azimuth dimension is developed by processing the series of range data line sections corresponding to the next azimuth section, and so forth. The process will now be described for SEASAT SAR data as an example. 6.11.6 SEASAT Processing Example SEASAT data were collected during the 100 days of SEASAT's life by down-linking signals to tracking stations over an analog data line. Down-linked echo signals were
II
WE
ill
EU I
A;
NO I
t u.
1
OOiuia REI
UTH ENCE
7 iC u.
RANGE REFERENCE
R
il FORM,
MULTILOOK PROCESSING
DETE CTION
c Ul
i
at <
o V E
s I e-
a.
s
E o V
u.
xl
I
c
324
digitized and recorded on a high-speed recorder. Data processing has since been carried out both optically and digitally. The general type of digital process carried out by the Jet Propulsion Laboratory (JPL)'s interim digital SAR processor (IDP) on SEASAT data is discussed here to illustrate the principles discussed above. Details of the IDP have been described in reports by the JPL for NASA and in the open literature by Elachi et al. (10) andWu(13]. A SEASAT range data line was obtained from each pulse of the radar extending over a sampling range-delay window of 288 /JS. An A/D sampling rate of 45 x 10 real samples per second provided the equivalent of one complex sample for each 6.7m of the slant range. Slant-range resolution associated with the 19-MHz waveform bandwidth is 7.89m. SEASAT azimuth (cross-range) resolution associated with its synthetic aperture length, based on its real-aperture size of 10.7m in azimuth, from (6.64), is about 5.35m. The nominal SEASAT PRF of 1,500 for the 6.6 km/s of beam travel over the ground produces one complex sample in each range cell for each 4.4m of cross-range (azimuth) travel. Both range and azimuth sampling are thus shown to occur at a rate slightly higher than the Nyquist rate of one complex sample per resolution cell. The processor generates 100- by 100-km SAR maps from about 15 sec of time history of the 2%%-fts range data blocks. Four coherent looks are noncoherently integrated, which degrades the available resolution from the real beam by a factor of four. Four-look-processed resolution is nominally 25m by 25m. The processing follows the general form illustrated in Figures 6.44 and 6.45. The 100- by 100-km coverage is made up of multiple subimages generated by range and azimuth sectioning of the data using the overlap-save procedure in each dimension. Section lengths are 2,048 complex data samples in range by 2,048 complex data samples in azimuth, as indicated in Figures 6.46(a) and 6.48(a), respectively. 6
4
The range reference function for range correlation is about 34 /us in duration. It is represented by 768 complex digital values, corresponding approximately to SEASAT's 34-/*s pulse duration multiplied by one-half the actual sampling rate of 45 x 10* real samples per second. Data integration for four coherent looks in azimuth extends slightly beyond the I-deg half-power points of the real beamwidth in azimuth. Antenna beam dwell time, corresponding to the four 0.3-deg looks at the beam's center in elevation at range center, is approximately
>
(854 x 10 )(2T»/360)(4)(0.3) 6,600
(6.136)
= 2.71 sec 4. Input data of the actual processor were sampled at an offset frequency from baseband. Equivalent complei data will be assumed for this example.
325
a
<•>
-SECT. 1
JTL
J
r—SECT. 2
H-SECT. 3—1 -SECT. 4 -
M
204S EACH SECTION, 768 OVERLAP
0>)
768
.—
1280
- DISCARD
(e)
768
NOTE: 768,1280 ft 2048 REFER TO NUMBER OF COMPLEX SAMPLES OR COMPUTED VALUES
1260 ^DISCARD ?
768
1280 DISCARD
768
1280
UNRECOVEREO PORTION (END EFFECT) 6 3
(0
-»«( 1280.
12801280,
1280
ETC
,1
RANGE SUBIMAQE LINES
Figure 6.46 SEASAT range correlation of data line containing dispersed response to eight (hypothetical) point targets: (a) range data line; (b) range reference; (c) section I correlation; (d) section 2 correlation; (e) section 3 correlation; (0 composite correlation (one image line).
326
As with the range reference function, the azimuth reference function is represented by complex digital values of the same time spacing as that of the data. Data spacing in azimuth is the reciprocal of the 1,500 complex azimuth samples per second (one sample per PRI) produced in each range cell, which is 4,065 samples per azimuth line, generated during the 2.71 sec of beam dwell time at beam center. Actually, 4,096 complex values are used to represent four coherent-integration looks. At the 4.4m sample spacing in azimuth, this covers an azimuth extent equal to the four-look cross-range integration length of about 18 km. Image quality depends on the accuracy with which the azimuth reference function represents the phase history of point targets in the real beam. Azimuth phase history can be represented by a quadratic function analogous to the quadratic function that represents the range reference. The azimuth reference function can be defined if the Doppler frequency at the azimuth center of illumination (Doppler centroid) and Doppler frequency slope (hertz per second) are known. This corresponds to the requirement that the instantaneous frequency at the center of the FM chirp pulse and FM slope (hertz per second) be known. Uncertainties in SAR platform attitude and Doppler echo spectrum produced by the earth's rotation below the satellite can require special preprocessing programs to generate the azimuth reference function. However, no such variations occur in the range reference, because the FM chirp generator in the radar determines the reference independently of platform attitude and orbit considerations. Clutter lock and autofocusing are methods used to estimate the Doppler centroid and Doppler frequency slope, respectively, based on the SAR data [6,10]. The clutterlock method sets the Doppler centroid of the reference to that of the received spectral response from the illuminated surface area. Autofocusing sets the Doppler frequency slope to produce minimum azimuth blur in the processed image, as determined by spatial frequency analysis, or by adjusting for minimum azimuth registration error between looks. Range correlation is performed first. The sectioned range data are illustrated in Figure 6.46(a). The reference function in range is shown in Figure 6.46(b). The range reference in the speclral domain can be separately generated in a preprocessing program. Fast correlation is performed on each of the 2,048-element sections of the input data, with the results as indicated in Figure 6.46(c-e). The composite result for one range data line is shown in Figure 6.46(f). The overlap-save process results in 2,048 - 768 = 1,280 complex values saved from each section. Range walk in the IDP is corrected to the first order by sliding the range lines as needed to align their starting samples. Finer correction is carried out by selecting from one of several range reference functions that vary by a fraction of a range cell in delay. This provides range-walk interpolation to within a fraction of a range cell. Correlated range data is stored in a corner-turn memory, then read out in the azimuth dimension. The data read out are transformed line by line (or column by column, as in Figure 6.44) into azimuth spectral data. Range curvature in the IDP processor is compensated for in the azimuth spectral domain by using a process that is efficient in terms of processor time, covering the known range curvature [13]. This process is illustrated in
327
Figure 6.47. The range curvature of a particular point target is shown plotted in the azimuth spectral data domain as range delay versus Doppler frequency. Because quadraticphase history is assumed, the range delay of a point target versus Doppler frequency, and its range delay versus azimuth time history are represented by the same curve, except for a constant factor. To correlate the near-range azimuth spectral line of data in Figure 6.47, the spectral form of the azimuthal reference requires vector multiplication by the spectral data that appear along the curve path. The product comprises the composite spectral line in the lower part of the figure. Another composite spectral line is obtained from spectral data that appear along the same curve when it is shifted outward in range to the next azimuth spectral line. The process of shifting to the next azimuth data line is repeated until all the curved data in the spectral domain have been converted into composite lines free of curvature. The piecewise-linear approximation of the curved delay provides advantages in terms of memory storage requirements and flexibility in reference updating [13]. RANGE
AZIMUTH (SPECTRAL DOMAIN)
CURVED RANGE DELAY OF
COMPOSITE AZIMUTH SPECTRAL LINE
Figure 6.47 Range curvature compensation in the azimuth spectral domain. (Based on Fig. 3 from C. Wu et al.. " S E A S A T Synthetic-Aperture Radar Data Reduction Using Parallel Programmable Array Processors," IEEE Trans. Geoscience and Remote Sensing, Vol.GE-20, No. 3. July 1982. Reprinted with permission.)
328
At this point, azimuth correlation can be thought of as being performed on curvaturefree azimuth time-history data. The fast correlation process proceeds for each composite line by inverse-transforming spectral regions for each of four looks. Figure 6.48 illustrates correlation for one look. Figure 6.48(a) illustrates a single azimuth line of curvature-free data from which an image line is to be generated. Individual looks at a given point target occur at separate portions of the total Doppler spectral response to the target. The first look at the leading edge of the beam contains only positive Doppler shift because range decreases during the first look. The last look contains only negative Doppler shift because range increases. The spectrum of the reference functions for each look likewise occupies a separate portion of the spectrum of a hypothetical reference for the total beam response, as is illustrated in Figure 6.48(b). Fast correlation for each look uses the overlap-save process to correlate azimuth data sections of2,048 elements, each with its 1,024-element, single-look reference function. The result is 2,048 - 1,024 = 1,024 azimuth values saved per subimage data line. Only 2S6 of the 1,024 values ultimately must be saved, however. This can be understood by recalling that for the SEASAT velocity and PRF, the total synthetic aperture length of 18 km is sampled with 4,096 complex samples spaced 4.4m apart. Pixels produced by one look will represent the equivalent of about 22m in resolution. In other words, azimuth data is oversampled by about a factor of four for integration during each look. Azimuth data are originally sampled at or above the Nyquist rate for the total aperture because the actual phase history is that of the Doppler spectrum produced by the total aperture. Therefore, 2,048 complex values per section are retained up to the point where the inverse FFT is performed in the azimuth compression process. The inverse FFT then must be performed on only 512 of the 2,048 spectral values for unambiguous representation of the reduced single-look resolution. Of the 512 resulting time-domain values, only 256 are saved, as indicated in Figure 6.48 (c-e), which are detected (converted to magnitude only) to form image pixels. Figure 6.48(f) indicates that saved azimuth subimage lines register side by side to form a contiguous azimuth image line, as in the range domain. A contiguous set of subimages of 1,280-by-256 azimuth-elevation pixels are generated. The corresponding subimages from four looks are overlapped and noncoherently summed. Subimages are assembled to form an image of 5,800-by-5,144 azimuth/elevation pixels. This is called a SEASAT-A SAR frame and covers an area of 100 by 100 km with about 25m by 25m resolution. A four-look SEASAT image of the San Diego, California, area, obtained from data collected during revolution 107, is shown in Figure 6.49.
6.12 DOPPLER BEAM SHARPENING The third type of SAR mentioned at the beginning of this chapter was called Doppler beam sharpening (DBS). It is discussed separately here from side-looking and spotlight SAR because the theory takes a somewhat different form. To date, DBS radar has been
329
(I) AZIMUTH OATA LINE
(b) AZIMUTH REFERENCE
EEL
^ 1 " II I L -SECT. —SECT. 2—J H—SECT. -—SECT. -"—SECT. 4 — —
1st LOOK
T I
2nd I 3rd I 4th , LOOK | LOOK | LOOK I 1
Ul
1
-4V/ 2048 EACH SECTION. \ 1024 OVERLAP
1st LOOK
_L
1024 SPECTRUM OF AZIMUTH REFERENCE \ 256 *
(c) SECTION 1 LOOK1
i
li .
.
25S (d) SECTION 2 LOOK1 CORRELATION
NOTE: 256. 512,1024, & 2048 REFER TO NUMBER OF COMPLEX SAMPLES OR COMPUTED VALUES
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(I) SECTION 3 LOOK 1 CORRELATION
-£ARD'
/////, UNRECOVERED PORTION (END EFFECT)
256
(I) COMPOSITE CORRELATION (LOOK1)
AZIMUTH SUBIMAGE LINES
Figure 6.4U SEASAT azimuth correlation of data line containing dispersed response to seven (hypothetical) point targets: (a) azimuth data line; (b) azimuth reference; (c) section I look I correlation: (d) section 2 look I correlation: (e) section 3 look I correlation: ( 0 composite correlation (look I).
no
\ •
ft
7-
..•*
•••
•
»;* .... .'•if*'/'
.>•..<•;•>;•;.•;
^
v
: . i ? •<,«.,»
i
; • 11
1
/ " i M» ...
Figure 6.49 SEASAT image of San Diego area.
used almost exclusively in the form of an air-to-ground mode on surface-strike aircraft to locate and target enemy land-based installations. Typically, DBS is one of several modes available from the nose-mounted radar on modern multipurpose combat aircraft. The radars for these aircraft usually sector-scan up to ±irfl nA from straight ahead. Maximum beam sharpening occurs for scan angles toward ±irfl. The beam sharpening
331
ratio reduces to unity as the beam scans past 0 rad, straight ahead. In recent years, DBS has been considered for a few nonmilitary applications.
6.12.1 DBS Radaf Resolution Cross-range resolution with focused DBS, as for focused side-looking and spotlight SAR, is determined by wavelength and coherent integration angle if/, where, as before, coherent integration angle is the view angle change over which signal data from the target is processed coherently. The coherent integration angle for a constant effective rotation rate to, about the target during integration time T is to,T. The effective instantaneous rotation rate for DBS, as for side-looking and spotlight SAR, is the instantaneous component of platform velocity (v,),„ normal to the line of sight to the target divided by target range. Integration time 7* for DBS, unlike that for either side-looking or spotlight SAR, however, is determined primarily by antenna scan rate. Figure 6.S0 illustrates DBS low-altitude surveillance for a straight and level flight. The antenna rotates at rotation rate to,. The instantaneous effective rotation rate, at angle
...... (6.137)
R
where v, is the platform velocity. Available coherent integration time with effective beamwidth if/, of a uniformly illuminated antenna scanned at constant angular scan rate to, > to is r
T ^
to,
=^ to,
(6.138)
where if/, = AH from (6.53) for cross-range aperture dimension /. The cross-range resolution from (6.61) in terms of effective rotation rate
Ar
=
' =^ 4 r
<
6,39)
where it is assumed that to, remains constant during time T. From (6.139) with (6.137) and (6.138). Rto,l Ar, = ~
:—7
2v, sin
(6.140)
332
Figure C.50 Doppler beam sharpening radar.
6.12.2 DBS Ratio The term beam sharpening ratio, often used to denote DBS performance, is defined as the ratio of the illuminating antenna beamwidth to the synthetic angular resolution. Synthetic angular resolution from (6.140) is Ar,
co. I
R
2v, sin di
tft, = = r = w
J
The beam sharpening ratio from (6.141) with (6.138) and (6.53) is then
(6.141) '
v
333
sin
(6.142)
6.12.3 DBS Radar for Commercial Navigation Radars have been used for many years on ships and small craft for collision avoidance. A typical commercial navigation radar may operate at A = 0.03m with a scanning 2m antenna and provide range-azimuth search over 360 deg. Azimuth resolution for this size aperture is approximately A/f = 0.015 rad, corresponding to about 60m cross-range resolution at 2 nmi, which is roughly also the typical range resolution. This resolution provides generally adequate all-weather and night-time vision for navigation at sea or on large bodies of water, where there is a need to keep track of low-density surface vessel traffic, surface obstacles, and shoreline features. More recently, there has been interest in extending the concept of radar navigation and collision avoidance to something approaching true vision under otherwise low-visibility conditions. Applications are for ship, aircraft, and land-vehicle operation in relatively close-in, complex surface environments where high-density traffic may exist. Examples are (1) ship and small craft operation in close quarters such as docking, (2) aircraft landing, and (3) ground-vehicle operation. Resolution limits of conventional range-azimuth search radar does not provide adequate vision for safe vehicle operation in these complex surface environments. Available space for antennas even at millimeter wavelengths tends to limit azimuth resolution to less than what is acceptable. Commercial navigation applications of DBS, unlike the military air-to-ground targeting application, are likely to require constant resolution over the scan sector with a minimum straight-ahead blind sector. Constant azimuth resolution can be obtained by varying scan rate at, to produce a constant DBS ratio. The scan rate from (6.142) versus scan angle for constant DBS ratio / ? is M S
sin
(6.143)
Mo* The scan rate in (6.143) approaches zero at
334
meters of the auxiliary antenna would probably need to roughly equal that of the main antenna to obtain the needed SNR. The aperture length transverse to the vehicle motion, however, would exceed that of the main antenna by the equivalent of the constant DBS ratio for the scan sector outside the DBS blind zone. The limited scan sector and reduced height of this auxiliary antenna may make this an acceptable solution for some applications. DBS radar, like side-looking and spotlight SAR, is basically a terrain mapping radar. Moving targets will be seen displaced in the cross range unless their radial component of motion is corrected. 6.12.4 Short-Range DBS Commercial applications of DBS may involve terrain mapping at short ranges for which the assumption for (6.138) that scan rate is much higher than the effective target rotation rate is no longer valid. In principle, dwell time, as for spotlight SAR, could be made to increase without limit by continuously adjusting the scan angle to cancel the effective rotation* produced by platform motion. Actually, for typical scan rates, the increase in resolution tends to occur for impractically short ranges. Enhanced near-range beam sharpening may be possible for some applications for which close-in mapping is needed. The improvement occurs only in one scan direction. In the other direction, resolution is degraded. PROBLEMS Problem 6.1 What is the approximate cross-range (azimuth) resolution at a 50-km range for a lS-GHz radar with a 1 m-diameter side-looking antenna operating in the following ground-mapping modes: (a) real aperture, (b) optimized unfocused SAR, and (c) focused SAR? Problem 6.2 A real-aperture radar in a low-altitude aircraft is desired to produce radar maps with 25m by 25m resolution at a range of 25 nmi (46.3 km). Aerodynamic considerations limit antenna length to 10m. What are the required radar parameters of bandwidth and radar frequency? Assume the system transfer function of Figure 6.6. Problem 6.3 A real-aperture mapping radar employs 0.25-/iS pulses (no pulse compression). What is the matched-filter output SNR level against a 1-m point target at 10 nmi (18.52 km)? 1
335
Parameters are as follows: P, = 100 kW, G = 33 dB. A = 0.03 m. L - 6 dB, and T, = 500K. Problem 6.4 (a) An unfocused SAR operating at A = 0.03m is in an aircraft traveling at 200 m/s. What is the data integration time that produces peak responses at a range of 40 km? (b) What is the corresponding cross-range (linear azimuth) resolution? Problem 6.5 Show that the effective rectangular beamwidth if), that produces the same integrated response as a uniformly weighted antenna is 1.14 times its one-way half-power beamwidth 4>i (IB-
Problem 6.6 Real-beam antenna gain for narrow-beam lossless antennas can be defined to be G -
W
^ 4>,6,
where if>, and 0, are the equivalent rectangular beamwidths in azimuth and elevation, respectively. Show from Problem 6.5 that for uniformly illuminated apertures, „
4-7r
where F, = 0.88. All beamwidths are in terms of radians. Problem 6.7 A side-looking airborne radar with a lm-diameter antenna is to operate in a focused SAR mode to generate surface maps at low grazing angles. Echo signals can be detected over a range extent from 10 to 110 nmi (18.5 to 185.2 km), (a) What is the maximum PRF required to avoid range ambiguity? (b) At this PRF, what is the maximum platform speed to avoid Doppler ambiguity? Problem 6.8 An airborne spotlight SAR operates at a center frequency of 5.4 GHz. Platform speed is 780 kn (400 m/s). (a) What is the Finest possible cross-range resolution for 10 sec of
336
spotlight time at 100 nmi (185.2 km)? (b) What is the required minimum PRF to avoid azimuth foldover for a real beamwidth of if, = 1.43 deg (0.025 rad)? (c) What is the maximum illuminated slant-range extent at the minimum PRF before range-ambiguous responses occur? Problem 6.9 What bandwidth will produce square resolution for a focused side-looking SAR operating at a center wavelength of 0.03m. assuming integration is carried out over the entire radar antenna beamwidth of 1.5 deg (0.0262 rad)? Problem 6.10 An airborne side-looking SAR flying at 300 kn (154 m/s) and operating at 9.4 GHz sees a railway train at 25 nmi (46.3 km). How far in the cross range does the train appear to be off the track if it is moving with a radial component of 5 kn (2.57 m/s) relative to the radar? Problem 6.11 Use (6.64) to show that range curvature expressed in (6.98) can also be expressed independently of beam integration angle as
32Ar,(Ar )
J
r
Also, show that allowable range curvature from (6.97) corresponds to the maximum integration length SE^, given by (6.22) for allowable curvature SR = A/32. Problem 6.12 Show that (6.99) for the range depth of focus can be expressed in terms of allowable quadratic-phase error 4Xa>,) as
(A*VProblem 6.13 Range curvature is corrected at some range R . Show by using (6.97) that the range extent over which there is less than one range cell of range migration is given by
337
where if/ is the angle over which either single or multiple looks are to be taken. Problem 6.14 (a) What is the cross-range resolution of the SEASAT design of Table 6.3 associated with data from four looks that are processed coherently as a single look for increased resolution? (b) What is the new range depth of focus? Assume uniform illumination over the four looks. Problem 6.15 Use the expression in the text for chi-squared density to show that the probability densities of pixel intensity for single-look and multiple-look processing are given by (6.104) and (6.105), respectively. Assume that the outputs of the / and Q channels have the zeromean Gaussian probability density p(x) given in the text. Problem 6.16 What improvement factor (decibels) in signal-to-speckle-noise ratio is achieved compared with that for one look in the SEASAT design by noncoherently adding four coherent looks based on zero-mean, Gaussian-distributed quadrature inputs to the pixel intensity detector for each look? Problem 6.17 (a) Show that 12 single looks are possible for the hypothetical radar of Table 6.6 within the effective beamwidth of the real beam, (b) What is the corresponding improvement in pixel signal-to-speckle-noise ratio over that for four looks (assuming zero-mean Gaussian quadrature pairs are detected to produce pixel intensity values)? (c) What is the increased range migration from beam center to either beam edge at the far edge of the range swath (85 km) across the increased total multiple-look angle? Problem 6.18 What is the minimum PRF that could be used in the SEASAT radar for unambiguous sampling of the Doppler bandwidth produced across the antenna's 3-dB beamwidth?
338
Problem 6.19 (a) What are range and azimuth sizes on data collection film of the dispersed response recorded from a point target during a single SEASAT look if CRT sweep velocity is 100 m/s and film transport velocity is 0.1 m/s? (b) What are the optical focal lengths in range and azimuth for optical processing with a source at 0.6 pan (0.6 x I0~'m)? Assume a range of 854 km. Problem 6.20 (a) How many sections of 2,048 range data samples are required after overlap to carry out range processing of the total 2&$-pts range swath of SEASAT echo data produced by each 34-/xs pulse? (b) What range extent of data in meters is discarded at the beginning of the range profile for each section? Problem 6.21 (a) How many sections of 2,048 complex azimuth data samples are required after overlap to carry out 100 km of azimuth processing of SEASAT data? (b) What azimuth extent of data in meters is discarded at the beginning of each section of the 100 x 100 km of processed image data? Problem 6.22 A 35-GHz DBS radar is used for road-scene mapping from a surface vehicle traveling at 30 m/s. The radar antenna has an azimuth dimension of 0.1 m and scans a forward azimuth scan sector of ±ir/4 rad at wrad/s. (a) What is the cross-range resolution at 1 km at ±77/ 4 rad from straight ahead? (b) At what angle from straight ahead does the cross-range resolution degrade to that of the real beamwidth? (c) At what range does the instantaneous effective rotation rate produced by platform motion approach the antenna scan rate at the ±JT/4 edges of the scan sector? REFERENCES (I) Sherwin, C. W., J. P. Ruina, and R. D. Rawcliffe, "Some Early Developments in Synthetic Aperture Radar Systems," IRE Trans. Military Electronics, Vol. MIL-6. No. 2, April 1962, pp. 11 l - l 15. (21 Kovaly, J. J., Synthetic Aperture Radar, Dedham, MA: Artech House, 1976. (A collection of 33 reprints covering the development, theory, performance, effect of errors, motion compensation, processing, and application of SAR.) (3] Cutrona. L. J., "Synthetic Aperture Radar," Ch. 21 of Radar Handbook 2nd Ed., M. I. Skolnik, ed„ New York: McGraw-Hill. 1990, pp. 21-1 to 21-23.
339
(4) Hirger. R. O.. Synthetic Aperture Radar Systems: Theory and Design, New York: Academic Press. 1970. [5] Hovancssian, S.A., Introduction to Synthetic Array and Imaging Radars, Dedham, MA: Artech House, 1980. (6] Curlander, J. C. and R. N. McDonough, Synthetic Aperture Radar: Systems and Signal Processing, New York: John Wiley & Sons. 1991. (7) Senlz. J. D., Jr., and J. D. Wiley, "Radar's Crowing Role in Ice, Pollution Surveillance." Sea Technology, Aug. 1985, pp. 27-29. (8) Balanis, C. A., Antenna Theory Analysis and Design, New York: John Wiley & Sons, 1982. p. 114. [9] Papoulis, A., Probability, Random Variables and Stochastic Processes, New York: McGraw-Hill, 1965, pp. 250-253. [10] Elachi, C , et a!., "Spaceborne Synthetic-Aperture Imaging Radars: Applications, Techniques and Technology," Proc. IEEE, Vol. 70. No. 10. Oct. 1982, pp. 1174-1209. ( I I ) Nathanson. F. E., Radar Design Principles, New York: McGraw-Hill, 1969, pp. 236, 238. 263. and 273. 112) Brigham, E. O., 77ir Fast Fourier Transform, Englewood Cliffs. NJ: Prentice-Hall. 1974. pp. 209-217. (13) Wu, C . " A Digital Fast Correlation Approach to Produce SEASAT SAR Imagery," IEEE 1980 Int. Radar Conf. Record. 28-30 April 1980. pp. 153-160.
Chapter 7 Inverse Synthetic Aperture Radar 7.1 COMPARISON OF SAR AND ISAR Inverse synthetic aperture radar (ISAR) is a version of SAR that can be used operationally to image targets such as ships, aircraft, and space objects. The technique also has application to instrumentation radar for evaluating radar cross section ot targets and target models. Basic theory with respect to instrumentation-range measurements is covered in a book by Mensa [1] and in journal articles by Chen and Andrews [2] and Walker [3]. Imaging of aircraft targets is discussed by Chen and Andrews [4], Haywood and Evans [5], and Steinberg [6]. Imaging of planets and space objects is discussed by Ausherman et al. (7]. Theory and design principles from the viewpoint of operational applications will be discussed in this chapter with emphasis on ship and aircraft imaging. A SAR map is generated from reflectivity data collected as the radar platform moves past the target area to be mapped, whereas ISAR target imagery is generated from reflectivity data collected as the target rotates while remaining in the radar beam. In Chapter 6 it was shown to be possible to achieve a form of SAR called spotlight SAR, illustrated in Figure 7.1. Spotlight SAR is obtained as the radar antenna constantly tracks a particular target area of interest. Here, the cross-range resolution is determined not by the size of the illuminating antenna as is the case for side-looking SAR, but by target dwell time. ISAR can be explained in terms of SAR by referring to the spotlight form. After correcting for deviation from straight-line motion, a spotlight SAR can be thought of as if the radar were flying an angular segment ^ of a circle around the target area, as in Figure 7.2. We can see from this figure that although the radar moves about the target, the same data would be collected if the radar were stationary and the target area were rotated. This is precisely what occurs with ISAR. The aspect (viewing angle) rotation of the target relative to the radar is used to generate the target map, which is the target image. The ISAR process will now be discussed in more detail. The discussion will be limited to ISAR imagery of targets having dimensions that are small compared with both 341
342
Figure 7.1 Spotlight SAR.
target range to the radar and cross-range extent of the radar antenna beam. It will also be assumed initially that the images are obtained from observations in the form of target reflectivity data collected over small segments of the viewing angle. This corresponds to applications for long-range imaging of relative small targets such as aircraft at, say, Xband (8.5 to 10.68 GHz) and higher. 7.2 ISAR THEORY FROM APERTURE VIEWPOINT The theory of ISAR will first be explained from the viewpoint of cross-range resolution produced by an equivalent unfocused SAR operating at far-field range. Figure 7.3(a) illustrates data collection from a ship target by a stationary radar as the target rotates in azimuth through an angle segment tfi over which collected data is to be coherently integrated. The SAR equivalent is the moving radar of Figure 7.3(b) collecting the same data while flying the dashed-line segment of the circle around an identical but nonrotating target. The SAR aperture length 2! in Figure 7.3(b) corresponds to integration angle ifi in Figure 7.3(a). Figure 7.3(b) is approximated by the unfocused far-field SAR illustrated in Figure 7.3(c) where deviation in range between the curved and straight-line paths of the moving radar is much less than a wavelength. For this assumption, the two-way phase advance of the target scatterer at azimuth position
,„ (7.1)
343
Figure 7 J Spotlight SAR—circular flight paths.
as for the SAR geometry in Figure 6.11. The term sin
i(j(
(72)
The integrated response over integration angle ifi of Figure 7.3, for a IV signal, is
Z(y) = £V"»M^ which integrates to
(7.3)
344
Figure 7 J SAR/ISAR equivalence: (a) ISAR; (b) SAR equivalent; (c) approximation for 2 < R and y < X.
345
sim
(7.4)
Z(y) =
The cross-range resolution defined in Section 6.3.1 as the half-power points of the normalized power response [Z(y)] /[Z(0)] becomes 2
1
A r U = 0.44A/
(7.5)
The Rayleigh resolution, which is defined at the 21 rr amplitude points, becomes (7.6)
7.2.1 Maximum Unfocused Integration Angle The maximum synthetic aperture length, before uncorrected phase deviation produces defocusing in the cross range, was defined for SAR in terms of allowable two-way phase deviation of return seen from a scatterer at the real-beam extremes from that seen from the same scatterer at beam center. The maximum integration angle for ISAR before uncorrected phase produces defocusing occurs in the cross range can be defined in terms of allowable two-way phase deviation of return seen from a scatterer at the integration angle extremes from that seen from the same scatterer at the center of the integration angle. Maximum phase deviation occurs for return from the near- and far-range edges of an ISAR target at long range. Two scatterers of a rotating ISAR target are illustrated in Figure 7.4. A third scatterer at the center of rotation remains in focus as the target rotates. Scatterer I near the range edge of the target toward the radar travels along a circular arc at radius r from the center of target rotation as the target rotates from
(7.7) Solving for ip with r > Sr, we obtain
346
-\»/2. . Scatter 2 ^"-•^^^^
Center of rotatk rotation
Radar-
Scatter 1 Target 91 - 0
Figure 7.4 Unfocused ISAR.
v*=f— j
(7.8)
Assume, as we did for SAR, a maximum allowable two-way phase deviation of iri% rad as the criteria for focus. The corresponding allowable range deviation Sr is A/32. The maximum integration angle before defocusing occurs, from (7.8), then becomes
The maximum target size in terms of radius r, from (7.9), before defocusing begins to occur is expressed in terms of focused resolution Ar from (7.6), as r
(7.10)
Processing to correct for range curvature is required to obtain focused imagery for data collected over larger integration angles than indicated by (7.9) or from targets with larger radii than indicated by (7.10). 7.2.2 Optimum Unfocused ISAR Integration Angle The optimum unfocused integration angle and associated cross-range resolution for ISAR are analogous to the respective SAR parameters. Figure 7.4 for ISAR is analogous to
347
Figure 6.10 and the associated analysis for SAR, where scatterer I in Figure 7-4 corresponds to the SAR scatterer on boresight in Figure 6.10. The phase advance with time t for the echo signal from scatterer 1 located at rotation angle d» = 0 when / = 0 is
,
W
.
q
.
-
£
»
.
_
£
<
=
(7.11)
£
for small rotation angle d>= tot with constant angular rotation rate to. The two-way phase advance for scatterer 2, which is located at r, -d> when I = 0 is
W
»»,(/,-
(7.12)
2 r
Equations (7.11) and (7.12) for ISAR correspond to (6.23) and (6.24) for SAR. In a manner similar to the analysis for the optimum unfocused SAR aperture, we obtain the optimum unfocused integration angle ft
1.2VrA
"
""
(7.13)
The resolution at this angle is &r = O.SyJr~A
(7.14)
c
For example, for wavelength A = 0.1m and target radius r = 100m, the optimum unfocused integration angle is 0.1
•A* = l-*y-Joo =
0
0
3
8
rad
2
<-
2 d e
6)
(715)
The cross-range resolution for this integration angle is Ar, -
0.5V(100)(0.1) =
1.58m
(7.16)
7.2.3 ISAR Theory (Focused Aperture) A focused aperture for ISAR is achieved, in principle, by subtracting the two-way phase advance tf>,(t, 0) of (7.11) of the echo signal from scatterer 1 of Figure 7.4, from the two-
348
way phase advance i/fj(t, -
«».-«*)-
W-°> = - t [ - 7
: +
£]
717
< >
where for constant rotation rate, x = tort and y = r<£. The integrated response for ISAR is given by (6.38) for SAR with R = r and v = wr. The half-power resolution for ISAR from (6.39) with i£ = v,7" = torT becomes f
ArJ,* = 0 . 4 4 r ^ = 0 . 4 4 ^
(7.18)
The Rayleigh resolution from (6.40) becomes *r = \ ± r
(7.19)
Equations (7.18) and (7.19) for focused ISAR with a>T= ^correspond to (7.5) and (7.6) for a nonfocused, small-integration-angle ISAR. The uniform illumination assumption for SAR used to obtain resolution (6.39) and (6.40), while useful for analysis, does not accurately represent illumination by practical physical SAR apertures, but the uniform illumination assumption does apply to most practical ISAR situations where the target azimuth extent is small compared to the illumination beamwidth. Thus, (7.18) and (7.19) more closely represent observed resolution for ISAR than (6.39) and (6.40) do for SAR. A fundamental difference should be noted between procedures required for focusing side-looking SAR data and those for ISAR data. Range focusing in both cases is performed based on the range-independent point-target response determined by the radar waveform. Azimuth focusing for side-looking SAR can be performed by correlation to the rangedependent point-target reference response determined by the SAR geometry. However, the azimuth point-target response for ISAR systems used to image ships and aircraft in operational environments is determined by the angular rotation part of the target geometry, which is generally not known a priori with sufficient accuracy for useful target imaging. Fortunately, azimuth focusing is not required for many smaller targets that meet the criteria of (7.10). We will see later how rotational motion occurring during data collection from larger targets can be determined from the data by seeking the rotational motion solution that results in the sharpest focus. In addition, some ISAR waveforms such as steppedfrequency waveforms require that collected data be corrected for target translational motion, which is also not generally known a priori with sufficient accuracy to focus in range. Translational motion solutions for these ISAR waveforms can also be generated from the data.
349
13 RANGE-DOPPLER IMAGING
Resolution and sampling requirements for ISAR are probably most easily understood in terms of range-Doppler imaging. Assume the target model of Figure 7.5, consisting of a three-dimensional rigid set of scatterers from which wideband echo data is collected during target rotation about a fixed rotation axis in the far field. The target is assumed to be uniformly illuminated, and processing to obtain an image is assumed to be performed on data collected during target rotation through a small integration angle segmentfathat meets the criteria of (7.9). This is approximately equivalent to assuming that the integration angle is small enough that the slant-range and Doppler frequency of scatterers at the target extremes shift less than the corresponding processed slant-range and Doppler resolution. The processed image consists of estimates of the magnitude and position of scatterers in the slant range and cross range. The slant range is the radar LOS range dimension.
SCATTERER
10
•—RADAR
RANGE-PROFILE SAMPLE NUMBER l
0^VV"' V
h T 1» N
l f t
"
(I OR Q AMPLITUDE)
* «nT«1 RADIAN > INTEGRATION TIME * NUMBER OF RANGE SAMPLES = NUMBER OF PROFILES PER INTEGRATION TIME
Figure 7 J Range-Doppler sampling of a routing target.
350
The cross range is the dimension lying normal to the plane contained by the radar LOS and target rotation axis. The range-Doppler model will be used to develop expressions for slant-range and cross-range resolution, sampling requirements, and the target image plane. Basic principles of processing will be described. This will be followed in succeeding sections by analysis of the defocusing effects of target translational motion and processing over integration angles that exceed the criteria of (7.9). Methods for translational motion correction (TMC) and rotational motion correction (RMC) will be described for the practical situation in which target motion is not known a priori. After this, a generalized target model will be developed that includes both target translational and rotational motion. Automatic focusing methods using this model will be described for applications where the focusing criteria of (7.9) and (7.10) for maximum integration angle and target size are exceeded. Range-Doppler imaging is further discussed by Ausherman et al. [7], Chen and Andrews [2], and Walker [3].
7.3.1 Basic ISAR Theory for Small Integration Angle Figure 7.S suggests a series of range profiles produced by an HRR radar as it observes a rotating target. Range sample increments correspond to target dimensions in meters, and profile-to-profile increments correspond to time in seconds. The signal along one range profile is illustrated, and the response to a resolved scatterer at one range position is shown in time history. The response to the scatterer can be seen to produce a few cycles of Doppler shift during integration time T while the target rotates through if rad. Not shown in the figure are Doppler responses produced in other range cells corresponding to other scatterers on the target. Data for one image are sampled at baseband with rj, inphase and quadrature-phase (/ and Q) samples per range profile for each of N range profiles obtained during time T. Waveforms used to obtain the range profiles and sampling criteria will be discussed later. Doppler frequency shift produced by a given slant-range resolved scatterer for small if is proportional to the target angular rotation rate as well as to the cross-range distance between the scatterer and the center of the target rotation. One or more Doppler spectral lines can exist for each slant-range cell, one for each Doppler-resolved scatterer. The magnitude of a spectral line is proportional to the reflectivity of the resolved sotterer. The target's reflectivity, therefore, can be mapped in both the slant range and cross range with the cross-range scale factor dependent on the target angular rotation rate. Target track and other data can be used in some applications to estimate rotation rate and orientation of the rotation axis relative to the radar LOS. The orientation of the target's rotation relative to the radar establishes the orientation of the image plane. The image is bounded by slant-range and cross-range windows, the significance of which will be discussed later.
351
13.2 Cross-Range Resolution The basic relationship between target rotational motion, scatterer position, and the resulting Doppler frequency shift can be seen by referring to Figure 7.6. Neither the radar nor the target has any translational motion in this example. Radar LOS is in the plane of the paper. The target rotates at a constant angular rotation rate to in radians per second about a fixed axis perpendicular to the plane of the paper. A single scatterer at a cross-range distance r can be seen with instantaneous velocity tor, toward the radar. The instantaneous Doppler frequency shift is c
(7.20) where / is the carrier or center frequency of the radar, A is the wavelength, and c is the propagation velocity. We will initially assume that f is constant during the viewing-angle change that occurs during a small integration time T. Later, we will show that defocusing is produced by variation of the Doppler frequency, which increases as processing is performed over larger viewing angles. If two scatterers in the same slant-range cell are separated in the cross range by a distance Sr„ then the separation between the frequencies of the received signals, from (7.20), is D
-rtoSr,
Sf
D
(7.21)
so that (7.22)
Sr.
INSTANTANEOUS SCATTERER VELOCITY TOWARD THE TARGET
LOS RADAR
Figure 7.6 Radial velocity produced by a scatterer on a rotating target.
352
Then, for a radar that has a Doppler frequency resolution of Lfo, we have a crossrange resolution given by
The cross-range resolution Ar can be seen to be dependent on the resolvable difference in the Doppler frequencies from two scatterers in the same slant-range cell. The Doppler resolution in turn can be related to the available coherent integration time T of a constant-level signal. The relationship, from (2.S2) of Chapter 2, in terms of the Rayleigh resolution, is A/ = \IT. The coherent integration time will also be called the image frame time. The cross-range resolution obtained by coherent integration of the echo signal received during the viewing-angle change if/ that occurs during integration time T is thus obtained from (7.23) as c
D
Ar
--5^-53f-B
(724)
Where coherent angle ifr= a>T for uniform target rotation. Equation (7.24) is the same as (7.6) and (7.19) obtained from the aperture viewpoint of ISAR, and the same as (6.60) and (6.61) for SAR. Typically, a DFT process, in the form of an FFT, is used to convert the set of timehistory samples collected in each range cell during the time segment T into a discrete Doppler spectrum. Resolution from (7.24) assumes uniform weighting during integration. The precise relationship between Doppler frequency resolution and integration time depends on the type of transform and the window function used to weight the segment of time-history response. Figure 7.7(a) illustrates a series of echo samples available in the same range cell of N range profiles. The DFT of the data is illustrated in Figure 7.7(b). 7.3 J Slant-Range Resolution As for SAR, the slant-range resolution for ISAR is obtained by using wideband waveforms. Regardless of the type of waveform, the achievable range resolution is approximately cl (20), where 0 is the waveform bandwidth. In principle, any of the waveforms discussed in Chapters 4 and 5 would be suitable. Only two are discussed here: chirp pulse and stepped frequency; these are the same two waveforms discussed for SAR in Chapter 6. While chirp-pulse and stretch waveforms are the most common for SAR, stepped-frequency waveforms have been found to be useful for ISAR when the application requires extreme resolution. Rayleigh resolution for chirp waveforms is (7.25)
353
SAMPLES OF RESPONSE
TIME HISTORY
DISCRETE FOURIER TRANSFORM
I
Figure 7.7 Sampled lime history and associated Doppler spectrum in one range cell (illustrated for the case where one range profile is generated from the received response from each chirp pulse).
where A is the chirp bandwidth. Synthetic processing of stepped-frequency waveforms, as discussed in Chapter 5, requires the conversion of echo data, collected in the frequency domain, into synthetic range profiles. This is typically carried out by using a DFT process, as illustrated in Figure 7.8. The resolution for n steps of A/Hz each, from (5.21), is (7.26) Synthetic ISAR involves two dimensions of the Fourier transform: (1) frequencydomain reflectivity into range-delay reflectivity for each burst to resolve targets in range.
354
Figure 7.8 Echo spectrum and associated synthetic range (delay) profile for a single burst.
followed by (2) time-domain reflectivity in each range cell into Doppler frequency-domain reflectivity for each range cell to resolve targets in the cross range. The above twodimensional transformation process, in the most fundamental sense, transforms reflectivity data obtained in frequency and viewing-angle space into object-space reflectivity estimates.
7.3.4 Slant-Range Sampling Targets to be imaged using ISAR are usually isolated moving targets, in contrast to the large fixed surface areas to be mapped with SAR. For ISAR, therefore, we assume that some type of angle and range tracking is used to keep a selected target immersed in the radar antenna beam during data collection. Samples from each of N range profiles, regardless of waveform, will be assumed as the input data shown in Figure 7.5 for one image. The n, samples of the range profile produced by real processing of the received response from a transmitted chirp pulse are collected directly in the time domain. When stepped-frequency bursts are transmitted, sampling can be said to occur in the frequency domain. The synthetic range profile obtained from each burst by the DFT process is effectively~sampled by n pulses of each burst so that r\, = n. A slant-range window will now be defined for each type of waveform.
355
A sampled target range profile received from the transmission of a single chirp pulse is illustrated in Figure 7.9. The unambiguously sampled slant-range extent, called the slant-range
window, is given by cL\t
c
w. = V.-J- = l . ^
(727)
for T}, complex samples spaced by Ar sec in range delay. Samples are obtained, as described in Chapter 4, by using some form of range tracking that starts the first sample just before the target echo arrives from each pulse. Additional samples are collected during a total delay interval corresponding to the slant-range window given by (7.27). To meet Nyquist's criterion, the complex / and Q sampling rate during this interval must equal or exceed A complex samples per second. At least one complex sample of the baseband response for each pulse of a pulseto-pulse stepped-frequency burst is required for unambiguous sampling of targets of rangedelay extent less than the duration of the baseband response. In other words, we require at least one complex sample of the target signal produced at each frequency. As discussed in Chapter 3, a target's reflectivity for unambiguous sampling in the frequency domain requires complex sample pairs spaced by A/ £ l/(5r), where St is the range-delay extent over which the target reflects incident waves. As stated in terms of an unambiguous range window, also called range ambiguity window, we write
=
T
=
2A?
(
7
2
8
)
The synthetic range window is effectively sampled by the t), = n samples per burst collected over bandwidth 0 = nA/. Targets that exceed the slant-range window defined by (7.27) for sampling of real profiles produced by chirp-pulse radars will be imaged over only that portion of the rangedelay extent of the target where samples were taken. Targets that exceed the range ambiguity window defined by (7.28) for stepped-frequency pulse sequences will produce images that are folded over within the range window.
Figure 7.9 Sampled range (delay) profile.
356
The slant-range integration length for chirp-pulse waveforms of pulse duration T, is cT,f2, as for SAR. The sampling window is unaffected by the integration length. When stepped-frequency sequences are transmitted, the attempt is typically to sample at the delay position corresponding to the center of the narrowband / and Q video response. The slant-range integration length, when the receiver bandwidth before sampling is perfectly matched to the transmitted pulse duration T,, is approximately cT,, which is the approximate effective length associated with the duration 2T, of the triangular matched-niter output pulse. This integration length, for one sample per frequency step, needs to equal or exceed the target length in order to image the entire target. To approach uniform weighting over the target's range extent requires that integration length exceed target length by a factor of two or more. The integration length for a given transmitted pulse duration can be increased before sampling by reducing receiving system bandwidth to less than that for the matched-filter case. The integration length approaches cT, when samples of extended targets taken at multiple coarse-range sample positions are superimposed as described in Chapter 5. 7.3.5 Cross-Range Sampling Cross-range sampling refers to sampling along time history in each resolved range cell. Samples are separated in time by the radar PRI 7j for chirp-pulse waveforms and by nT for stepped-frequency waveforms. Analogously to SAR, a cross-range ambiguity window for ISAR can be defined as the largest cross-range target extent that can be unambiguously sampled for a given PRF, viewing-angle rotation rate, and wavelength. For ISAR, however, the target is usually immersed in the illuminating antenna beamwidth so that the ambiguity window refers to the target size in the cross range, rather than to the instantaneous illuminated cross-range extent on the earth's surface, as is the case for SAR. From (7.20), which expresses the Doppler frequency produced by a single scatterer on a uniformly rotating target, we can show that the Doppler frequency bandwidth produced by scatterers extending over a cross-range window w, is liowJK. The PRF required for unambiguous sampling of reflectivity data produced from a chirp-pulse radar when viewing a target of cross-range extent w„ therefore, is t
1
2a>w,
(7.29)
assuming complex samples are collected, one sample in each range cell for each transmitted pulse. Synthetic processing of stepped-frequency bursts of n pulses per burst requires a PRF of 1
2n cow.
(7.30)
357
Later in this chapter, we will estimate PRF requirements for ISAR images of ships and aircraft. Regardless of waveform, the number of range profiles needed for unambiguous sampling of a target of cross-range extent w„ based on the requirement that NITtfo, is (7.31) The unambiguous cross-range length w = A/Ar is the maximum cross-range extent of a target that can be examined unambiguously with N stepped-frequency bursts of n pulses for synthetic processing or with N = n pulses for real processing, one profile for each transmitted chirp pulse. For a small integration angle o>T = ifr, the unambiguous crossrange window A/Ar, with Ar from (7.24) or from (7.31) is expressed as f
r
c
Wavelength A, when referring to chirp-pulse or stepped-frequency waveforms, is the wavelength at the center frequency. Narrow fractional bandwidth is assumed in both cases. 73.6 Square Resolution Square resolution with ISAR, as for SAR, is defined as equal resolution in the slant range and cross range. Required bandwidth caiTIk to obtain square resolution is obtained by solving for bandwidth A or nA/ for chirp-pulse or stepped-frequency waveforms, respectively, for which the cross-range resolution given by (7.24) equals the slant-range resolution given by (7.25) and (7.26). A summary of basic ISAR equations written in terms of chirp and stepped-frequency waveforms is given in Table 7.1. 7.4 SOURCES OF TARGET ASPECT ROTATION So far, we have considered only the viewing-angle rotation produced by target rotational motion. Target aspect change is also produced by the tangential translation of the target relative to the radar. Radial translation (motion along the radar LOS) produces no viewingangle change, but tangential translation (motion normal to the LOS), like target rotation, produces a viewing-angle rotation that results in a Doppler gradient associated with target scatterers distributed in the cross range. Figure 7.10 illustrates how a differential Doppler shift is produced between two scatterers of a radar target that has a tangential velocity component relative to the radar. The scale in this drawing has been exaggerated to clarify the relationship between radial velocities v«, and v»».
358
Table 7.1 Summary of Equations for ISAR Waveform Symbol
Chirp-Pulse
Stepped-Frequency
Slant-range resolution* Slant-range window
Ar, w.
Slam-range integration length Cross-range resolution*' Cross-range ambiguity window*
c/(2nt\f) c 2A/ cT,/2
Ar
c/(2A) c "•2A cT,/2 A/(2t» A/A A IhiT loiTi uT r,A
AM _ ,» 2o)7" 2n
(it)*.
w,/(A/v)
w /{&r.)
(1/r.U A, nA/
2«w,M cwTM
InanvJA caiTIA
Parameter
*
1
Cross-range integration angle Minimum number of complex samples per slant-range integration length Minimum number of range profiles per cross-range integration angle Unambiguous PRF* Required bandwidth for Ar, = Ar, 1
r
r
y*r
Maximum integration angle before azimuth defocusing occurs I.2>/JV
Optimum nonfocused integration angle
l2s[A/r
•Rayleigh. 'Focused or ^ £ '•: 'Assumes constant target angular rotation rate. 'Assumes target reflectivity collected from one range-sample gate.
A tangential velocity component v>of a target relative to a radar at range R produces a component of target angular rotation rale as seen by the radar of (7.33) The tangential vector component of the total angular rotation vector of the target as seen by the radar is _ <"T =
v, x
ft
—S~
(7.34)
where ft is the unit vector along the radar LOS and v, is the target-velocity vector relative to the radar. Equation (7.34) reduces to (7.33) when the target's relative velocity vector v, is perpendicular to the radar LOS unit vector ft.
359
RADAR
Figure 7.10 Cross-range Doppler produced by large! translation. (Note: The exaggerated target size highlights the slight differences that occur between radial velocity components of target scatterers when the target translates relative to the radar and has a nonzero tangential component of translation velocity. For the target velocity v as shown, it can be teen that v „ is greater than v„.)
The aspect rotation produced by both target translation and target angular motion is indicated in Figure 7.11. The rotation produced by target angular motion is made up of three components of rotation: pitch, roll, and yaw (or turn). Translation is likely to be the major contributor to a useful Doppler gradient when viewing aircraft targets from ground-based radar at frequencies up to about the X-band region (8.S0 to 10.68 GHz). For ship targets observed at ranges beyond about 10 nmi, target pitch, roll, and yaw predominate to produce the image. 7.5 TARGET IMAGE PROJECTION PLANE The image produced from echo signal data collected from a long-range target during the time interval that it rotates through some small viewing angle can be thought of as a set of reflectivity estimates plotted in a rectangular slant-range-versus-Doppler-frequency coordinate system, which is called the image projection plane. This plane contains the radar LOS and is normal to the effective rotation vector. The effective rotation vector is the projection of the actual target rotation vector on a plane normal to the radar LOS. Lines of constant slant range lie in the projection plane normal to the LOS. Lines of constant cross range lie in the projection plane parallel to the LOS. 7.5.1 Image Plane for SAR and ISAR The effective rotation vector of a typical ISAR target such as a ship at sea varies continuously as the target pitches, rolls, and yaws. A continuous series of range-Doppler ISAR
360
ROTATION VECTOR PRODUCED BY TARGET TRANSLATION
VECTOR* •VECTOR COMBINATION OF TURNING, PITCH, ROLL, AND YAW COMPONENTS Flgurc7.lt Components oY target rotation.
images or the target will therefore appear on a constantly changing image projection plane. This is in contrast to the situation for SAR. The effective rotation vector produced by the constant straight-line platform motion of a side-looking SAR past a flat surface illuminated by a narrow real beam approaches a fixed vector that tilts from vertical toward the target area by an amount equal to the grazing angle. For mapping at relatively long ranges from a low elevation platform, the image projection plane lies nearly parallel to the surface so that we see the SAR map as though we were viewing from directly above, as shown in Figure 6.49. 7.5.2 Vector Relationships for ISAR Instantaneous Doppler frequency shift foil), produced at radar frequency / by a single scatterer on a target that is moving with radial velocity v, toward the radar, is a vector function of v, and the following time-variable vector quantities: target rotation vector a>i(f), scatterer velocity vector v(f) produced by the target's rotation vector, and target LOS unit vector
R(f).
The LOS unit vector for radar range that is large relative to target cross-range extent can be chosen to point toward the radar from some point within the target along its rotation
361
vector wj, as shown in Figure 7.12(a). The Doppler frequency shift produced by the scatterer can then be expressed as
ML5M %,
mm
.)
f+
(7 35
By dropping the time notation and combining terms, we obtain = ^ ( v M ft + v,)
(7.36)
If r is the position vector of the scatterer measured from the intersection of 3 then the Doppler frequency becomes /„ = | , ( 3 i x r) * ft + v.)
l
and ft,
(7.37)
Later in this chapter, we will see that, as part of the ISAR image processing, the target's radial velocity component v, can be removed, leaving only the Doppler shift produced by target rotation. Then fo = ^ ( ( " i x r) x
ft]
(7.38)
Equation (7.38), written to express the Doppler shift produced by a scatterer at crossrange displacement r , becomes r
/ = 7[(«*xfcr,)xi] D
(7.39)
For angletf>between
(740)
The quantity an sin
fo = ~
(7.41)
362
Figure 7.12 lm.ge-pl.ne geometry: (.) target geometry viewed in . coordin.te system moving at velocity K (b) target geometry viewed in the R. 3 i plane normal to the cross-nutge vector fir,: (c) ISAR image plane.
363
as given in (7.20) for center frequency/equal t o / i n (7.41). The image projection plane in Figure 7.12(a) lies in the x, z plane. The image projection plane in a more general x, y, z coordinate system is shown in Figure 7.12(c). If the z-axis is vertical, the effective rotation axis is tilted upward at an angle 4>, from the horizon as viewed by a radar along the x-axis. The target's actual rotation vector aii, and hence its effective rotation vector « , is not directly known from the target echo data. Therefore, the cross-range scale factor and image plane are said to be ambiguous. The effect of the target's rotation vector on the resulting ISAR image plane can be explained in a qualitative manner by examining Figure 7.13, where the image plane is illustrated for pitch, roll, and yaw motions of a ship target (no translation). In each case, the actual rotation vector is equal to the effective rotation
b)
"55c: ^ ) ^ 5 T ~ S b ^ 5 " r\ r
1
II n
n
S c)
THE RADAR LIES ANYWHERE IN THE PLANE OF THE PAPER DIRECTED AT THE TARGET
Figure 7.13 ISAR image views produced by target (a) pitch, (b) roll, and (c) yaw motions.
364
vector and is normal to the plane of the paper. Radar range is assumed to be large relative to target size. The image planes of the three views of Figure 7.13 are those that result when the radar LOS unit vector ft lies anywhere in the plane of the paper. Figure 7.14 illustrates the image plane of a rolling ship that is resolved in range and Doppler by an airborne radar. The ship and radar are both in the plane of the paper.
7.6 ISAR DATA COLLECTION AND PROCESSING FOR CHIRP-PULSE RADAR HRR target range-profde data for chirp-pulse ISAR processing is typically obtained by sampling the range-delay extent of the matched-filter responses produced by chirp-pulse target illumination. Both the amplitude and phase of the sampled range profiles must be retained to achieve the coherent ISAR processing described above. With analog pulse compression, the target's range profile is produced at the output of the pulse-compression filter, usually at an intermediate frequency. At this point, quadrature detection typically occurs by mixing to baseband. Then sampling and digitizing follow. The steps leading to data collection for the generation of one ^-element range profile are illustrated in Figure 7.15. Each range profile contains i), complex / and Q sample pairs. Sequentially generated range profiles, one profile per pulse, are typically tracked in range to obtain a time history of range-sampled profiles that are in alignment profile to profile. This is essential to ensure that the r) = N cross-range samples (for an A/pulse sequence) in each range cell correspond to the same respective range positions along the target. The resulting set of /V aligned range profiles, illustrated in Figure 7.5, is processed to form ISAR images. c
Figure 7.14 Image plane for ship target with roll motion (example of Fig. 7.13(b)).
365
FROM MIXER
\
IF
/
PULSECOMPRESSION FILTER
l| REFERENCE •
n
IJ u
ONE RANGE PROFILE AT IF
QUADRATURE DETECTION
A
ft
ANALOGQ
ANALOG I SAMPLING AND A/D CONVERSION
J_l_L.
11 < • SAMPLED I
SAMPLED Q TO ISAR PROCESSOR (N RANGE PROFILES, ONE FROM EACH PULSE)
Figure 7.15 Range-profile data collection for chirp-pulse ISAR.
As before, the term data collection will be used here to refer to the signal sampling process as distinct from signal processing. The term does not preclude real-time imaging. Consider first a fixed target observed with a fixed radar. The range-delay trigger for each pulse can start the sampling and high-speed A/D conversion sequence just before the
366
arrival of the range-profile baseband signal. This sets the beginning of the range window. Because the target is at a fixed range position, the ith range sample from each of the successive target responses, one from each transmitted pulse, will correspond to the same range position along the target. (Not shown in Figure 7.15 are low-pass filters at the / and Q outputs of the quadrature detector that pass the video chirp bandwidth but reject the sum signal and harmonics.) Figure 7.16 illustrates the image processing sequence for sampled range-profile data represented as a row-column matrix for % range profiles (rows) of r,, complex echo samples (columns). This matrix can be thought of as the time history of the target's reflectivity sampled for each of N echo pulses at discrete range-delay positions within the sampled range window as the target viewing-angle changes due to its rotation relative to the radar. Complex samples in the time-history matrix are represented by amplitude and phase (A/
tJ
c
Time history of target's range profile Range Range Range ceno c e l l cell n,-1
• • • (^•)n,.i.
0
• • • (A'4)o n .i l
IFTI
Symbol vV4i
°t/
T
u
Definition
PD.0
Amplitude and phase In flh range eel of the synthetic range profile from the Mh chirp pulse Magnitude in ithrangeeel and fix Doppler eel of range-Doppler Image Image frame time
WJlrttlc-i
e
iFTt
I FTt
•• •
Oo.i
• • •
tange Dopple r image
(T=NT) 2
RangeFigure 7.16 image processing of chirp-pulse data.
367
Range alignment for a moving target requires range tracking, wherein the delay trigger is continuously adjusted in delay to track the target. In practice, manual range tracking of moving targets cannot be done with sufficient precision for ISAR processing. Servo loops have been developed to set the range-trigger delay to track a prominent target scatterer. The resulting samples are then ideally equivalent to samples of the same target at a fixed range.
7.7 ISAR DATA COLLECTION AND PROCESSING FOR STEPPEDFREQUENCY RADAR The concept of stepped-frequency ISAR imaging was originally developed to image air and surface targets in real time at full radar range from fixed or moving radar platforms using narrowband radar and signal processing methods. This form of ISAR has also proved to be useful for extremely high resolution imaging of scale models of real targets. The concept uses the waveforms discussed in Chapter 5, consisting of contiguous sequences, also called bursts, of either stepped- or hopped-frequency monotone pulses. Elements of the application to ISAR include (I) the use of these waveforms to obtain a time history of the target's reflectivity sampled in the frequency domain, (2) correction of spectral samples for radar system phase and amplitude ripple, (3) correction of spectral samples for target translation motion, (4) conversion of the time history of the corrected samples of target frequency-domain reflectivity, burst to burst, into a time history of the target's synthetic range profile, and (5) conversion of the target's range-profile history into a slant-range-versus-cross-range image. When the technique is referred to as stepped-frequency imaging, it refers to the processing of sequences of pulses within each burst, with the frequency of each successive pulse increased by a constant frequency step. The total bandwidth required for transmission of all the pulses is much greater than the instantaneous bandwidth of a single pulse. Uniformly increasing pulse-to-pulse frequency stepping is not essential. Hence, pseudorandom, pulse-to-pulse hopped-frequency radars designed for improved electronic countercountermeasures (ECCM) capability can also produce target images. Pulse-to-pulse frequency excursion is harmonically related to the fundamental step size A/. The process of generating images from target data collected with stepped-frequency waveforms is illustrated in Figures 7.17 and 7.18. In Figure 7.17, the image processing sequence is illustrated, beginning with a rowcolumn matrix for N bursts (rows) of n complex echo samples (columns). Corrections for radar phase and amplitude ripple and target translational motion are assumed to have been made. This matrix can be thought of as the time history of the target's reflectivity sampled at discrete frequencies in the RF domain as the target viewing-angle changes due to its rotation relative to the radar. Complex echo samples in the time-history matrix are represented by amplitude and phase (A/
368
Time history of target's frequency-domain „. „. signature Step ' Step Step 0 1 n-1 Burst 0
<<«/t)o.o
Time history of target's range profile Range Range Range call 0 cell 1 cell n-1
-FT-1-*
Burst 1
Time Nstory
Burst AM ("•)O.N.I
<"•),.* (A'*)',,* °>J T
• • • ("•1'O.AM
Frequency — » Symbol
• • •W»i.o
(A'tJ'o.o
i.ol
Jit 1
Definition Amplitude and phase of f and Q echo samples from Kh step Mh burst
Do
Amplitude and phase In Ahrangecell of the synthetic range profile from the Mh burst
• • •
Magnitude In Ah range cell and Jth Doppler cell of range-Doppler image Image frame time (TVirVTj)
J. I
1
• • • Do-i.o
Po.i
OO.AM
itange
t)opple r Image 1
1
D
M.AM
Range - • -
Figure 7.17 Image processing of stepped-frequency data. Din*
of each of the N bursts of n frequency samples is taken, row by row, to generate N new rows of synthetic range profiles, each with n synthetic range cells. The result can be represented as a second row-column matrix, called the comer turn or time history of, the target's range-profile signature. Complex values in each synthetic range cell of this matrix are represented by the amplitude and phase (A/
369
10 M • • •
'n-1 T R A N S M I T T E D WAVEFORM S H O W I N G A SINGLE BURST. (BURSTS C A N BE » , C O N T I G U O U S O R SEPARATED IN TIME)
II fl II11 fl...
1
•TP I I I I I I I I I I (A/y)l.k ' O j l * *j •
t
ll I'
(A/*)i.K
I
1
' '
'n-1
l Mill
ii IT
till
I I 'I
2
* "i I* , , 'ni . . .''Ill . ,». I I 11 I I _ \TT-T< Tj | ' 1
,
r
I 1
„,
1
FREQUENCY-DOMAIN SIGNATURE OF TARGET (I OR O CHANNEL SHOWN) DISCRETE RANGE-PROFILE SIGNATURE OF TARGET OBTAINED BY TAKING IDFT op FREQUENCY-DOMAIN SIGNATURE (I OR Q CHANNEL SHOWN)
ui CD
3
3 Z
2 (N-1)
TIME HISTORY OF A SERIES OF RANGE PROFILES, ONE FROM EACH OF A SERIES OF N BURSTS, OBSERVED DURING FRAME TIME T FOR CONSTANT PRI - T (I OR Q CHANNEL SHOWN)
2
RESULTING TIME HISTORY OF ONE RANGE CELL OBSERVED DURING TIME T.
Figure 7.18 Waveforms and signals associated with synthetic ISAR.
Above, range-Doppler descriptions of chirp-pulse and stepped-frequency ISAR assumed that (1) real and synthetic range profiles, respectively, were produced by wideband echo data collected from, or corrected to equivalent data collected from the target rotating about a rotation axis at a fixed range from the target, and (2) processing was performed on data collected over a small target rotation angle.
370
The generation of focused ISAR images, regardless of the processing method, requires coherent integration of reflectivity data collected during rotation from each resolution element Ar, x Ar, determined by radar bandwidth, wavelength, and integration angle. This requirement is met when range-Doppler processing is performed on data collected from a target rotated through a small integration angle about a fixed-range rotation axis, because for this case (1) there is no translation of target scatterers in range, and (2) the integration angle is small enough that the circular arcs through which scatterers travel do not exceed Ar, or Ar in length. In practical operational situations, images are usually required from data collected from targets moving with unknown radial translational motion, and images may be required from data collected during a relatively large integration angle generated at an unknown, nonuniform target angular rotation rate. Individual scatterer reflectivity seen in real or synthetic range-profile time history will therefore tend to shift profile to profile in range, and scatterer reflectivity seen in the processed image may be spread along circular arcs that pass through slant-range and cross-range resolution cells when processing is performed over an integration angle that exceeds the criteria of (7.9). Effects of target motion and data correction procedures to produce focused images will be described in the following three sections. c
7.8 RANGE OFFSET AND RANGE WALK Target translational motion can produce both profile-to-profile range walk and a constant range offset. Range walk results directly from a change in target range from profile to profile. Range offset results from range-Doppler coupling inherent to both chirp-pulse and stepped-frequency waveforms. (With waveforms such as short pulses, hopped-frequency sequences, or phase-coded waveforms, there is no range offset.) A moving target observed, with an analog or discrete linear FM waveform, such as a chirp-pulse or a pulse-to-pulse stepped-frequency sequence, can be seen offset in range delay from that of a nonmoving target at the same range. Range offset is a fixed offset for constant radial velocity. Range walk, however, accumulates from profile to profile so that range-profile alignment is destroyed, even for constant-velocity targets. TMC of some form, therefore, is required to restore the slant-range alignment between adjacent slant-range profiles. Range walk can be circumvented in real ISAR processing by continuously sampling each new profile at equivalent positions along the range profile. One way is to track a single prominent scatterer. For synthetic ISAR, however, sampling occurs before there is a slant-range profile to track, and hence sample position does not determine the slantrange position of the resulting synthetic slant-range profile. Correction for target translational motion, therefore, is required after sampling as part of the processing, as will be described below. First, a single expression will be developed for chirp-pulse waveforms to express range offset and range walk produced by target radial velocity. Then an equivalent expression will be developed for stepped-frequency waveforms.
371
7.8.1 Range Walk and Range Offset for Chirp-Pulse Waveforms Range walk is exhibited by both short-pulse and pulse-compression radars when the target moves in a radial direction relative to the radar. This may be observed when the rangesampling position remains fixed. Then pulse-to-pulse delay shift for PRI T, and constant inbound target velocity v, is -2v,TJc. The accumulated number of range-delay cells shifted, called range walk or range-cell migration, due to target translation during the acquisition of it range profiles (one per pulse) is
where At ~ 1/A is the range-delay resolution of the radar with chirp bandwidth A. Range walk, as expressed in (7.42) for real-profile processing, considers only the delay shift associated with target translation between radar pulses. Range offset is an additional but nonaccumulating range-delay shift that occurs with chirp radar for targets of constant speed. Range-delay shift Sr from the zero-Doppler response for a linear chirp pulse with frequency slope K Hz/s is coupled to Doppler frequency in hertz according to the expression /„ = -KST
(7.43)
Delay shift associated with the positive Doppler shift of a target at constant velocity v, toward (he radar, therefore, is given by
S
r
=
_ -
=
(7.44)
where A is the chirp bandwidth for chirp-pulse duration 7", and / i s the center frequency. The number of range cells shifted for delay resolution 1/A is the range offset expressed as
M, = - ^'fv,
(7.45)
The accumulated number of range cells shifted during the acquisition of k = //echo pulses for ISAR imaging with fixed range sampling, from (7.42) and (7.45), is
m
The first term, range offset, is the number of range cells of fixed delay shift produced by a constant Doppler frequency shift. The second term, range-cell migration or range walk, is the number of range cells of accumulated delay shift due to pulse-to-pulse target translation for the N pulses associated with the ISAR image. Range walk for chirp-pulse processing is usually avoided by employing range tracking, as we discussed above and in Chapter 4. Range offset does not destroy alignment between profiles unless target velocity changes significantly during data collection for the processed integration angle. Target motion with chirp waveforms, in addition to producing range shift and range walk, also produces distortion of the compressed response for the usual case in which the pulse-compression filter is matched to the transmitted pulse at zero Doppler. Distortion, however, is small until the target Doppler shift becomes larger than l/T,. This is not likely for ship and aircraft targets viewed by most pulsed microwave radar systems.
7.8.2 Range Walk and Range Offset for Stepped-Frequency Waveforms The range shift and range-profile distortion described above for real range-profile processing of chirp-pulse data becomes more complicated for synthetic range-profile processing of stepped-frequency data. The effective transmitted pulse duration for synthetic processing to produce one range profile from an n-pulse burst is nT in length as compared with 7) for real processing, where 7* is the PRI and 7", is the chirp-pulse duration. Thus, even slow targets may produce enough range walk and range offset to cause severe rangeprofile distortion with stepped-frequency waveforms. Synthetic range-profile distortion due to target velocity was discussed in Chapter S. Only the resulting range offset and profile-to-profile range walk will be discussed here. The processing of stepped-frequency data produces synthetic target range profiles, which appear within an unambiguous range window determined by frequency step size. The delay of a point target seen within the range window is the group delay determined by the phase-versus-frequency function i/Kf, v,) of the stepped-frequency radar system as affected by target radial motion. For a point target moving with constant velocity v„ we have t
2
( 7 4 7 )
The part of the total delay r associated with target velocity is the delay shift ST. The nonshifted component of the delay for the target at range R is 2Rlc. Delay shift due to velocity is therefore given by 1 d, ,
5 T =
„
2R
-2lrd7 ^ ' -T l
V
) ]
(7.48)
m
The phase function ip'f, v,) of (7.47) and (7.48) can be estimated by assuming a linear slope of phase versus frequency during each burst (i.e., no dispersion in the propagation media or radar hardware over bandwidth nA/). Then, for the ith burst, we have (7.49) The phase of the sampled echo data from one burst of a stepped-frequency waveform, from (5.3), is expressed in terms of sampling time S as t
ft = -217/7(5,) = - 2 « - / J y - ^ 5 J
(7.50)
f
where, for receiving system transfer delay r„ from (5.8), S, = iTi + r, + —
(7.51)
c
When multiple bursts are to be considered, as for ISAR, we have 2R Su, = (i + nk)Tt + T, +
(7.52)
—
The phase advance of echo data sampled according to (7.52) from k bursts is (7.53) Equation (7.49) with (7.53) for / = / + (A/and * *> 1 becomes 0
2/r /.T, + A/I^r + ™ + n*r,j r
# / . V,) = 27T/
-t]
A/
(7.54)
The total delay, from (7.47) with (7.54), is
2v,
T= - •
[f„T + i
L\f(j
r
+ — +
A/
nkT ^ 1
2/? + — c*
(7.55)
374
The delay shift due to target velocity, from (7.48), is
*
T =
"c^[
/ o 7
"
l
A /
+
( ' T T
+
+ nJt7
' )] 1
( 7 5 6 )
An approaching target (v, positive), therefore, has a response shifted to less delay. This was illustrated in Figures S.8 and 3.9 of Chapter 5. The number of cells M of range shift for radar delay resolution Ar = l/(*iA/) is M = ^=nAfSr
(7.57)
By substituting ST from (7.56), we obtain M =
"7 [
/tTl +
T +
V[[ ' T
+ nkT2
v
(758)
)] '
Because we are interested in the range shift for at least one synthetic range profile: 1R
nJtTi *> T, + —
(7.59)
c
Then (7.58) is approximated as
2-7" M- -
+ nk£kf)v,
(7.60)
When an entire image generated from N bursts is considered, k - N - 1. From (7.60), when N is large, the accumulated number of range cells shifted due to target velocity during the entire image frame time is M"-
—
(/o + nNHf)v, = -I — — U +
k
(761)
Equation (7.61) is plotted in Figure 7.19 for parameters used in an experimental imaging radar at the NOSC in San Diego. The first term of (7.61) is analogous to the Doppler-produced range offset term of (7.46) for chirp-pulse processing. The stepped-frequency burst length nT corresponds to a chirp pulse of duration T,, and the starting frequency f corresponds (approximately, for small-percentage bandwidth) to the chirp center frequency/. The second term of (7.61) is identical to the range walk predicted in (7.46) for Tj = nT and chirp bandwidth A = }
t
2
375
100 M/vt -
2
^2
c
(f
0
+ nNAf) 1852/3600
OFFSET lo n Af
= 3 0 GHz = 256 = 1.0 MHz
0.1 1
10 100 NUMBER OF BURSTS
1000
Figure 7.19 Range migration produced by target velocity for stepped-frequency ISAR.
nA/. For these substitutions, the stepped-frequency waveform required for one ISAR image is approximated in (7.46) by N contiguous chirp pulses of length nT , PRI = nT , and chirp bandwidth nA/. Uncorrected range walk produces distortion in the ISAR image. Responses from scatterers walk out of the synthetic range-cell columns of Figure 7.17 in which the IDFT is being performed to resolve scatterers in the cross range. The result will be less integration of the scatterer's response in time history, which in turn results in degraded cross-range resolution. In addition, scatterer responses walk into adjacent range-cell columns. This produces range spread in scatterer responses seen in the processed image. We can see from Figure 7.19 that range walk of about 10 range cells per knot can occur in a 256 x 256 element image for the given parameters. Unless corrected, the image distortion for even very slow targets would be unacceptable. By examining Figure 7.19 with Figures 5.8 and 5.9 of Chapter 5, we can see that significant range walk in ISAR image data begins at a much smaller target velocity than that for the distortion produced by related effects within a target's individual synthetic range profile. For example, from Figure 7.19, range walk during the integration of the data from a 256 x 256-element data set approaches one range cell for an uncorrected target velocity of about 0.2 m/s (0.1 kn), whereas distortion of an individual range profile from Figure 5.8 appears to begin for an uncorrected velocity of about 30 m/s. We account for this through the increase by a factor of N in imaging integration time as compared with that for range-profile generation for a given number of pulses per burst. 2
2
376
7.9 TRANSLATIONAL MOTION CORRECTION FOR SYNTHETIC ISAR Range walk and range-profile distortion occurring with synthetic ISAR processing of data collected from moving targets cannot be avoided by precise range tracking before sampling as would be the case for short-pulse or pulse-compression radar systems. Instead, TMC must be carried out on the sampled data. Range tracking before sampling, however, need only be sufficiently precise to obtain samples near the peak of the narrowband response, where the SNR is highest. Assume that the target range during data collection of N bursts of n pulses per burst is represented as R for frequency step i and burst k. Two-way phase error produced by translational motion is then u
A. = -4irf,R /c
(7.62)
it
Both the range walk of Figure 7.19 and the range-profile distortion illustrated in Figures 5.8 and 5.9 will tend to be canceled by complex multiplication of the complex sampled data G = A exp(Mi) by the exponent ( defined as a
u
u
(u = exp[j47r/* /c] u
(7.63)
where A? is the estimated target translational motion, which generally includes velocity, acceleration, and higher terms. For some applications of synthetic ISAR, the target's translational motion may be known precisely from auxiliary data. One example is data collection at a fixed range from rotating target models on a turntable. Another example is space-object identification for which precise auxiliary target range data may be available. When images are to be generated from tactical targets, such as ships and aircraft, however, sufficiently precise target translational motion data may not be available. In early tests at the NOSC against ship and aircraft targets, an estimate of initial target range and average velocity was obtained from the delay positions of the sampling pulses generated by the range tracker. The velocity estimate was obtained by calculating the average range rate from the series of Nn range-sample positions used to collect the data for each image frame. Application of the correction factor fa of (7.63) with estimates of range and range rate obtained in this manner was found to be marginal at best, even for slow targets like ships. It was found that the target's effective instantaneous velocity varied sufficiently during the image frame time to cause unacceptable image distortion. A process was then sought that would produce a burst-to-burst correction factor. Both the Hughes Aircraft Company and Syracuse Research Corporation, under U.S. Navy contracts, developed methods for TMC that were tested on NOSC data. In the phase method developed by Hughes for the NOSC, the phase change in each range cell is first determined relative to that for the same range cell from the previous profile. The average phase change for the entire new profile is then calculated, and the result after profile-tou
377 T
profile smoothing is used to correct the phase in the frequency domain, t h e sequence of operations is indicated in Figure 7.20. Instantaneous target velocity and the range history for a ship target are shown in Figures 7.21 and 7.22, respectively. Burst-to-burst, two-way phase changes larger than 2»rrad are ambiguously related to target range motion. Therefore, the above process for wavelength A will produce distortion when 2\v,\nT is equal to or greater than A in any range cell. Maximum tolerable uncorrected velocity error before phase correction based on this criterion is plotted in Figure 7.23. To meet this criterion for faster targets, cross correlation can be employed 2
l&Q DATA
MAGNITUDE PHASE
RANGE DFT
PHASE COMPENSATION
FORM AVERAGE PHASE DIFFERENCE PROFILE TO PROFILE
POLAR-TORECTANGULAR CONVERSION
LOW-PASS FILTERING TO REDUCE NOISE
TWO-DIMENSIONAL DFT
INTEGRATE TO ESTIMATE RANGE HISTORY
RETAIN ONLY MAGNITUDES
LINEAR INTERPOLATION FOR CROSSRANGE SCALING FOR DISPLAY
i IMAGE
Figure 7.20 Synthetic ISAR image generation process.
379
BURST INTERVAL m"2 (seconds) Figure 7.23 Maximum tolerable velocity error before phase correction.
to prealign range profiles. The concept is to set up a rough alignment profile to profile, before attempting the phase compensation. Here, the synthetic range profile for each burst is cross-correlated with the previous range profile to measure the range shift produced by target radial motion. The measured range shift can be removed to set up a rough alignment of the time history of synthetic range profiles. The motion estimate obtained from the cross-correlation process and various smoothing techniques is then applied for correction of the original frequency-domain data before phase alignment. The entire rangevelocity correction process is further fine-tuned in several other ways to produce the best images. The phase method can be improved by weighting the contribution in each range cell to average phase change of the new profile by the magnitude in that cell. In this way, the value calculated for average phase change for the new profile is biased by that part of the range window containing the target and by positions along the range profile where the response is highest. Despite this and other improvements, the phase method, as well as other, earlier TMC methods, performed with only limited success against ship and air targets. A large fraction of the collected data was typically not adequately corrected for translational motion until iterative techniques to be described below were employed that sought motion solutions yielding the best focused imagery based on entropy or other measures of focus. Two additional noniterative TMC techniques will be briefly mentioned. A Doppler analysis method obtains estimates of instantaneous velocity during the data collection time history of the baseband signal at one or more of the n stepped frequencies of each burst. The TMC exponent fa of (7.63) is determined from relative range obtained by
380
time integration of the instantaneous target velocity. Results can be averaged for two or more of the n frequencies. Another TMC method is based on the selection and then tracking of a prominent peak seen on the uncorrected or roughly corrected series of synthetic range profiles from which the image is to be generated. Magnitude tracking is performed first, followed by phase tracking of the selected prominent peak. The exponent of (7.63) is calculated from the relative range travel determined for the prominent peak. Target TMC procedures developed at NOSC for ship and aircraft imaging are discussed further by Tran [8] and Bocker et al. [9,10]. In addition, target TMC is discussed by Lush and Hudson [11], Ausherman et al. [7], Chen and Andrews [4], Haywood and Evans [S], and Steinberg [6].
7.10 DISTORTION PRODUCED BY TARGET ROTATION Image distortion is also produced by target rotation. Target rotation produces the target Doppler gradient that makes cross-range resolution possible in the first place. However, because the motion of the target's individual backscattering centers about the target's rotation axis is circular, their velocity components toward the radar are not constant, and they follow curved paths in a rectangular range/cross-range data collection space. Conversion of time-aligned profiles into ISAR images using IDFT, range cell by range cell, therefore results in a distorted image. Deviation in Doppler and range, although small during the 1 to 3 deg of target aspect rotation typically required for imaging of ships or aircraft, can severely reduce image quality. Distortion becomes more severe as resolved pixel size is reduced, target size increases, and wavelength increases. The phenomenon is analogous to the effects of quadratic-phase error and cell migration produced by range curvature associated with straight-line SAR platform motion past the target area. Correction for the quadratic-phase error is commonly referred to as focusing. Correction for cell migration requires two-dimensional processing to integrate the target responses through their curved tracks in data collection space. Later in this chapter, a polar-refoitnatting technique will be discussed that produces distortion-free images by reformatting data collected in the polar domain of frequency versus viewing angle into a rectangular domain of frequency versus frequency. First we will discuss the distortion-producing mechanisms and their effect on image quality. Analysis up to this point has assumed that the target rotation angle required to form the image was small enough that the Doppler frequency produced by individual scatterers and change in scatterer range during the image frame time were less than the processed Doppler and range resolution, respectively. Target rotation angles that are large enough to produce significant change in Doppler frequency and range during the image frame time will be examined next. Target geometry is shown in Figure 7.24, which illustrates a single scatterer at radius r from the target's rotation axis. Figure 7.25 is a view seen in the plane containing the radar LOS and is normal to the target rotation axis.
381
r cos Figure 7.25 Single scatterer of a routing target.
0
382
The physical rotation of the target produces time-dependent echo delay to the scatterer. Since the resulting distortion of the image is not dependent on the method used to generate range profiles of the target, distortion will be analyzed in terms of real rather than synthetic range profiles for simplicity. Range delay to the single scatterer of Figure 7.25 is rit) = -[R - v,t - r cos(tot - 0)]
(7.64)
c
where to is the target's effective aspect angular rotation rate. Following perfect radialvelocity correction, (7.64) becomes 1R
1r
c
c
tit) = — - — cos(w/ - 0)
(7.65)
Phase associated with rit) relative to the transmitted pulse is
(7.66)
where 7 is the center frequency of the radar. From (7.65), we have = - 2 i r / ^ - j cos(«* - 0)J
(7.67)
The Doppler frequency shift produced by the scatterer is
m
= 2lr^r
°
m
With ifr(r) from (7.67), we have foil) = - / — sin(«/ c
ff)
(7.69)
The idealized analysis at the beginning of this chapter assumed 6 = ml. at / = 0, small aspect change toT during image frame time T, and constant aspect rotation rate to. For these conditions, the Doppler frequency shift produced by a scatterer at cross-range distance r reduces to c
383 +
where r is the component of the radius r seen in the cross-range dimension normal to both the target axis of rotation and the radar LOS. This is the same as expression (7.20) derived at the beginning of this chapter. Our discussion now turns to distortion, which is produced when the actual Doppler phase slope Si/X.t)/dt begins to deviate from a constant during the image frame time 7". r
7.10.1 Quadratic-Phase Distortion The effect of nonlinear phase slope on ISAR image quality can be analyzed in terms of quadratic-phase distortion and distortion produced by cell migration. First, let us consider quadratic-phase distortion. The delay to each scatterer versus time history observed during a small but significant segment of aspect rotation is a small segment of the sinusoidal delay term of (7.65). Small segments of either sinusoidal delay versus time history or phase versus time history are approximately quadratic. Quadratic phase is evident by expanding (7.67) about the argument tot - 6 for 0 = 0, where the maximum phase nonlinearity occurs. For small tot, the phase response of a scatterer at radius r normal to the effective rotation axis becomes
The phase function (7.71), expressed in terms of wavelength and the scatterer's velocity, becomes AirR
4nr
AirvH*
where v = tor is the cross-range velocity of the scatterer at 0=0. The first two terms of (7.72) express the total phase advance to the scatterer at / = 0. The third term is the quadratic-phase term, which is identical to that for SAR, expressed in (6.23), when the scatterer's cross-range velocity v is substituted for the SAR platform's cross-range velocity v,, and the scatterer's radius r from the center of rotation is substituted for the SAR range R. One effect of uncorrected quadratic phase in ISAR, as in the case of SAR, is to reduce the image cross-range resolution. Removal of the quadratic phase, or correction for it, can be thought of as a focusing process, as in the case of SAR. 7.10.2 Cell Migration Produced by Target Rotation Cell migration occurs with ISAR when the data integration angle toT is large enough to cause scatterers toward the edge of the target to shift by at least one resolution cell. Cell
384
migration for ISAR will be shown to occur in both the range and Doppler frequency dimensions. The slant-range between the radar and the scatterer of Figure 7.2S, after targetvelocity correction, is R - r cos(wt - 6). The change in the slant range of the scatterer during target rotation is maximum at the cross-range extremes when 0= itfl. The number M' of slant-range resolution cells for resolution Ar, by which the scatterer at r will migrate during integration time 7* when rotation begins at itfl is M' = ahs-r^- ([R - r cos(>f - «/2)]U - [R - r cos(tot - n/2)]Ur) t\r
(7.73)
t
. r sin toT
For small rotation angles, the number of cells is approximated as ht - abs^-^u>r - ^
+ . . .J
(7.74)
The change in the cross range of the scatterer of Figure 7.2S during target rotation is maximum at the slant-range extremes when 0=0. The number X? of cross-range Doppler resolution cells through which the scatterer at r will migrate during integration time T for rotation angles beginning at t = 0 is A?=abs-ir(/ U-/ Ur) 0
D
(7.75)
With the Doppler resolution given by [IT, we have
J?=abs(r/ U-77 Ur) D
D
(7.76)
With f from (7.69) for rotation beginning at 0=0, we obtain 0
X ? = a b s ^ ^ s i n toT
(7.77)
For small rotation angles, the number of cells is approximated as A7 = a b s ^ [
W
r-
From the expression (7.24) for cross-range resolution.
(7.78)
385
2/
(7.79)
so that the number of cells of cross-range migration from (7.78) becomes
K
L
M = abs-^ \wT- -^
+...\
r
(7.80)
From (7.74) and (7.80) with square resolution in which Ar = Ar„ we have M' = M. Therefore, for square resolution, the number of cells of slant-range or cross-range migration for small toT becomes f
M = 4-uT Ar
, (7.81)
for Ar = Ar, = Ar,. We would like to express (7.81) in terms of radar center frequency or wavelength and target size. From the first two expressions of (7.24) for cross-range resolution, we write (7.81) as
* ' £ » ' : * ' Ar 2/Ar
Ar 2 Ar
(7
.
82)
Consider a target extending from the target rotation axis to a radius r resolved into n = N = rILr resolution cells in the slant range and cross range. The maximum number of cells of migration from (7.82) is then expressed in terms of the target radius measured in range or cross-range resolution cells as Af»f£ =f £ Ar 2 / Ar 2
(7.83)
Results obtained by using (7.83) for three radar frequency bands are plotted in Figure 7.26. The quantity n here refers to the number of cells offset from the target's center of rotation. Cross-range cell migration caused by target rotation blurs the image in the crossrange dimension at all slant-range positions within the target, except at the corrected slantrange position where radial velocity, and thus cross-range migration, reduces to zero. Slantrange cell migration due to target rotation blurs the image in the slant-range dimension at all cross-range positions within the target, except at the corrected cross-range position where Doppler frequency, and thus slant-range migration, reduces to zero. Only one point is focused in image space. It lies on the radar LOS where Doppler shift is corrected to zero at a corrected range, which may be set to zero. This position is equivalent to the
386
Figure 7.26 Maximum slant-range and cross-range migration produced by target motion.
target center of rotation (centroid). Blurring increases with radius as measured from the centroid. 7.10.3 Blur Radius A practical limit for target size beyond which the unfocused ISAR image becomes blurred at its edges is called the blur radius. This is defined as the target radius that results in a maximum of one cell of slant-range or cross-range migration during the required integration angle to achieve a given cross-range resolution. In Figure 7.27, migration approaches one cell when rift = Ar. For Ar = Ar = Ar, (square resolution), the required rotation angle, from (7.24), is r
(7.84) The target blur radius for one cell of migration is therefore given by Ar r
~
2(Ar) A
:
(7.85)
387
CROSS RANGE
Figure 7.27 Blur radius.
Note that the criteria of (7.85) based on cell migration is similar to the criteria of (7.10) based on 77/8-rad phase deviation at the edges of the integration angle. 7.11 ROTATIONAL MOTION CORRECTION USING POLAR REFORMATTING Data samples of a target's reflectivity versus frequency that are taken as the target rotates are said to be collected in polar format. Although such data may be collected at uniformly - spaced frequency steps and rotation-angle positions, the samples are nonuniformly spaced in a rectangular format called frequency space. The result is that the two-dimensional Fourier transform to target-space reflectivity results in unfocused images for the larger rotation angles needed for high resolution. The notion of a blur radius as discussed above is another perspective on the same effect. The process of resampling from polar to rectangular format produces uniformly spaced data in two-dimensional frequency space. Two-dimensional Fourier-transform processing then produces focused images. Polar reformatting to produce uniformly spaced two-dimensional data in frequency space requires that the target rotational motion is either known a priori or obtained iteratively based on a focus measure of the processed image. The process will be referred to as target KMC, which follows TMC processing. Two additional benefits may be obtained when the polar-reformatting process is performed iteratively on data collected from targets
388
that are significantly larger than the blur radius defined by (7.85): the range/cross-range scale factor is established, and focused imagery is possible with data collected from targets during nonuniform target rotation, as is the case near the extremes of ship pitch, roll, and yaw. Details of the polar-reformatting concept are described below for stepped-frequency data. Applying the concept to chirp-pulse and other coded-pulse or short-pulse data would require that the collected range-profile histories first be transformed to frequency-domain histories. Before application to ISAR, the polar-to-rectangular-reformatting process was applied to spotlight SAR [7]. Optical processing of range-Doppler data collected on polarformat film [3] was an early method of ISAR imaging of turntable targets.
7.11.1 Frequency-Space Aperture Radar target imagery can be explained in terms of Fourier-transforming of frequencyspace reflectivity data into target-space reflectivity estimates. The term frequency-space aperture is sometimes used to refer to the extent of the two-dimensional frequency space over which data are collected. The transformation from frequency space to target space may not be readily apparent from the range-Doppler viewpoint of ISAR imagery discussed up to this point. First, consider data collected by using a pulse-compression radar. Target images obtained by processing the time history of complex range-profile data were thought of as range-Doppler images. Range-profile data are in target space, but data collected in each range cell over a limited range of polar viewing angles can be thought of as one dimension offrequency-space data to be defined below. Target-space reflectivity estimates in Ihe cross-range dimension in each range cell, as discussed up until now, were obtained from such data by performing a Fourier transform. Data that was processed into the ISAR image, therefore, existed in both target space and frequency space. The notion cf transforming from frequency space to target space becomes clearer in the synthetic ISAR process, whereby the input data can be shown to exist entirely in frequency and viewingangle coordinates, which for small viewing angles approximate a rectangular frequencyspace format. We will now formalize the concept of frequency space by defining the quantities:
cos $
(7.86)
v..sin 0
(7.87)
and
389
where /, and f in meters"' are the spatial frequency components associated with the measurement of target reflectivity at frequency / and target rotational angle 0 projected along the horizontal and vertical axes, respectively, of a rectangular frequency-space coordinate system, such as that shown in Figure 7.28. Consider first a target rotating at a constant rate about a z-axis normal to the paper. Data samples are collected at some signal frequency / at uniformly spaced polar-angle positions. Positions of the data samples, in frequency, appear in Figure 7.28 as dots along a constant radius. When the radar is shifted to a new frequency, another series of samples is generated at a new radius. Data represented by the dots collected at uniformly spaced polar angles and frequencies are polar-formatted data. Stepped-frequency data can be interpreted from Figure 7.28 as data collected along radial lines that are uniformly spaced at discrete polar-angle positions. This assumes that reflectivity data is collected over the radar's stepped-frequency bandwidth at each polar-angle position. The effect of constant target rotation rate will be discussed later. To understand the problem associated with polar-formatted data, we should review the nature of the DFT. Any set of complex quantities can be converted by the DFT into a second set of complex quantities. However, this transform or the equivalent FFT, when applied to actual systems, assumes that samples are collected at uniformly spaced intervals in space, time, or frequency. For example, the frequency spectrum of a waveform can be accurately represented by the DFT of a set of samples of the waveform that are spaced uniformly in time. Conversely, a waveform can be accurately represented by the IDFT of a set samples of the waveform's spectrum that are uniformly spaced in frequency.
'y
I
* DATA COLLECTED IN POLAR FORMAT O RESAMPLED DATA IN RECTANGULAR FORMAT
o
6
o
o .o * o o o
o © • o o. #
«~ix Figure 7.28 Coordinate system for frequency-space data collection.
390
7.11.2 Polar-Reformatting Process
The problem posed by polar-formatted data, represented by the dots in Figure 7.28, is that these data samples are not uniformly spaced in spatial frequency. If, somehow, the frequency-space data could be collected in the rectangular format illustrated by the matrix of small circles, then the requirements for uniformly spaced data sampling would be met in both dimensions of data collection space, and the transformed data would thus produce a focused image. This, however, would require nonuniformly spaced frequency samples at known nonuniformly spaced polar angles, where the nonuniformities in both dimensions have been carefully controlled to produce the rectangular spatial-frequency format. A more convenient process is to reformat data collected at uniform frequency and polar-angle spacing into a rectangular format. The circles in Figure 7.28 would then represent reformatted samples of the target's reflectivity in two dimensions of spatial frequency: reflectivity versus cycles per meter along f, and f Data samples of this reformatted data set can be transformed into focused target-space imagery in terms of reflectivity estimates versus x and y in meters. Discrete values of the complex reflectivity estimates in resolved two-dimensional cells in x, y space would be video-detected to form image pixels. The need to reformat target data collected in the polar format increases for a given desired resolution as radar frequency decreases. In Figure 7.29, a target's sampled reflectivity data appear along discrete polar-angle positions in a single plane of rotation. The radar lies in the plane of rotation and along the/, spatial-frequency axis. The resulting data format is illustrated for two radar bands. Figure 7.29(a,b) illustrates how the set of sampled data points deviate from a rectangular format: At the high band, a relatively small angle of rotation may be adequate to produce the needed cross-range resolution. Differences between the polar and rectangular formats are then small. Producing the same resolution at the lower radar band requires that data be collected over a greater angle of target rotation. Data points acquired in the polar format over the larger rotation angle can be seen to deviate significantly from uniform spacing in spatial frequency. The blur radius, discussed above, has been exceeded when deviation across the data collection space exceeds the sampling interval corresponding to one resolution cell in image space. DFT processing of the polar-formatted data to produce an image would then result in blurring at the edges of the image. Polar reformatting is a process that can be carried out on spotlight SAR or ISAR data collected by using frequency-sampled data for waveforms such as the stretch or stepped frequency. Reformatting of stepped-frequency ISAR data will be discussed here to illustrate the process. The stepped-frequency waveform, because it has been used to provide extremely high resolution, often results in significant scatterer migration. The dots in Figure 7.30 illustrate input data samples collected burst by burst, while the target rotates through a range of viewing angles at a constant rotation rate. (We assume that data at this point has been corrected for target translation motion.) The first step after r
391
Figure 7.29 Discrete frequency data at two radar bands that produce the same image resolution: (a) tower radar frequency; (b) higher radar frequency.
V -
VIEW ANGLE —»,
Figure 7JO Frequency-stepped input data and resampled output data (first step).
TMC is to resample the curved data from each frequency burst onto radii at discrete, uniformly spaced polar angles. The resampled data are indicated by the small circles. The next step, shown in Figure 7.31, is to resample for uniform spacing in the frequency-space dimension along the central viewing angle. Finally, the data along the
392
INPUT
•
o
®
o
®
o
®
O
O
OUTPUT O O
O
6 • O
£ g C
-VIEW ANGLE -
Figure 7.31 Resampled data along the cental viewing angle (second step).
discrete polar angles are resampled in the direction normal to the central viewing angle, as shown in Figure 7.32. At this point, the data can be converted by the DFT independently in both dimensions, then detected to form a focused ISAR image. Reformatting with one dimension of frequency space aligned with the central viewing angle, as discussed above, is not required. The resampled format can have any orientation relative to the polarformatted data, but the apparent viewing angle seen in the imaged target will change with orientation. The target rotation rate is likely to be known and is often quite uniform when ISAR images are to be generated from data, such as data collected from space objects or turntable models. By contrast, when images of ship or air targets are to be generated, the target's rotational motion is generally not known and is not likely to be uniform. The reformatting process then needs to be carried out iteratively by seeking estimates for the target's angular
VIEW ANGLE
Figure 7.32 Resampled data in the direction normal to the central angle (final reformatting step).
393 i
I
,
,
rotation rate and acceleration that produce the best focus. Nearly uniform rotational motion can be assumed for limited rotation angles, typically up to several degrees for many applications, without serious image degradation. An example of a stepped-frequency image before and after polar reformatting is shown in Figure 7.33. SAR maps of the earth's surface often appear to be of photographic quality. In contrast, ISAR images of individual targets, although often obtained at much higher resolution, sometimes appear to be of lower quality. The difference is due to the extended
Figure 7 J 3 Focused and unfocused stepped-frequency ISAR image.
394
surface areas being mapped by SAR as compared with individual targets imaged by ISAR that contain comparatively fewer scatterers. SAR images of nonmoving ships and aircraft are roughly similar to ISAR images of the same targets in motion obtained at the same resolution. Considerable room for improvement in ISAR imagery remains, but photographic-quality imagery does not appear to be likely. Unfocused stepped-frequency ISAR images of two commercial aircraft (circa 197S) are shown in Figure 7.34(a). Simulated ship imagery observed with the Texas Instruments AN/APS-137 pulse-compression airborne radar appears in Figure 7.34(b). 7.12 AUTOMATIC ISAR FOCUSING METHODS Correction for defocusing effects of target translation and rotational motion can be performed automatically by iteratively improving target motion estimates based on measurements of sharpness of focus of the image obtained from corrected data. One measure of sharpness is the average information content in terms of entropy of the processed image. Another measure is the bandwidth observed by Fourier-transforming image pixel data (a)
'o n Al
3.0 GHz 256 1.08 MHz
T2 = 185 |ts N s 64
j
•
• •<
•u -A
Figure 7.34 (a) ISAR Images of two commercial aircraft, (b) simulated ship images. (Courtesy of Texas Instruments.)
395
(b) Figure 734 (continued)
appearing along one or more straight lines through the image space; for example, along one or more cross-range or slant-range slices. A burst derivative measure by Bocker et al. [9] and Bocker and Jones [10] has also been evaluated. The burst derivative method estimates focusing performance in the spatial-frequency domain, thus avoiding the requirement with the entropy method to generate an image in target space with each iteration. Use of the entropy measure will be discussed here. First, an expression will be developed based in part on the Bocker and Jones analysis for the data set collected with a target geometry that includes both translational and rotational motion. 7.12.1 ISAR Geometry The ISAR geometry in Figure 7.35 depicts an ISAR target with translational and rotational motion relative to an observing radar. Angle b\t) is the instantaneous rotational position
396
Figure 7J5 ISAR geometry.
of the target in the u, v Cartesian coordinate system, where the u-axis lies along the radar LOS. The effective rotation vector is normal to the plane of the x, y coordinate system at the point marked center of rotation and in the plane normal to the page containing the radar LOS and actual rotation vector (not shown). The instantaneous range to the center of target rotation is R(t). The instantaneous range to a target reflection point at target coordinates jr. y is r'(t). The radar waveform is the stepped-frequency waveform described in Chapter 5.
7.12.2 Sampled Data From an ISAR Target The generation of a target's image from data collected in frequency space can be thought of as the process of estimating the target's two-dimensional reflectivity function p(x, y). Thus, we proceed to first develop the expression for collected data in terms of target reflectivity. Consider uniform illumination over the target's two-dimensional cross-range extent by one monotone transmitted pulse, of RF amplitude B, represented in complex form as 1
x(f) = Be** , =0
0 £ r £ T,
(7.88)
otherwise
where / i s the frequency and 7", is the pulse duration. The infinitesimal reflectivity of the target differential area dxdy at target coordinates x, y for target reflectivity density p(x, y) in meters per square meter can be represented at frequency / as p(x. y) =
BV^-^dJcdy
(7.89)
397
where B" is the magnitude of the reflectivity, and distance r'(t) to the target reflection point x, y is represented by r' for short. The expression for the received echo signal for target dimensions that are small relative to cT,H becomes y(1) = 4 £.£./>(* yycW^dxdy,
IRIc
+ IRIc
(7.90)
where A includes the amplitude associated with the transmitted signal, antenna gain, propagation attenuation, receiver, processing gain, and radar system loss. It will be assumed that A remains constant during the time required to collect the reflectivity data set that is to be processed into an ISAR image. The baseband echo pulse produced by mixing the echo signal of (7.90) with the reference signal t*' ' is 1
m(t) = A jljlpU y^'^dxdy,
2R/c £ t <. 7, + IRIc
•
(7.91)
where the constant phase difference existing between the reference and transmitted signal is assumed to be zero. It can be shown from Figure 7.3S that, for target dimensions that are small relative to target range R, the range to an individual reflection point at x, y is r' = R + u *R + x cos 0 - y sin 0
(7.92)
for 6X0 written as 0 for short. With (7.92), the baseband signal (7.91) can be written in terms of target coordinates x, y and rotation angle 0 as m(i) = Ae*'"" [_f„p(x,
y^-^l-'^dxAy,
IRIc Zt£T,+
IRIc
(7.93)
where f = (2//c) cos 0 and f, = (2flc) sin 0 are the spatial-frequency quantities of (7.86) and (7.87) defined at frequency/and target rotation angle 0. These quantities are assumed to remain constant during each pulse, but vary continuously and unpredictably during the data collection period for a processed image. The instantaneous target range and rotation angle in terms of the time history of target motion during data collection can be expressed, respectively, as s
R(t) = R + v t + ^flo' a
1
(7.94)
t
and 6\0 = 6" + «o' + ~«o' 0
3
(7.95)
398
where Ra, v , and ao represent initial values of target range, velocity, and acceleration, respectively, and 4. wo. and an represent initial values of target rotation angle, angular rotation rate, and angular acceleration, respectively. The data record obtained from N bursts of n transmitted pulses stepped from frequency f to f,. is sampled at time t = S for frequency step i of burst k as defined in (7.52). Thus, we express target translational and rotational positions (7.94) and (7.95), respectively, in terms of sample times as 0
0
R
t
it
= R + v S + jaoSj,
(7.96)
0 = b\ + wo£ + ^aoSl
(7.97)
it
0
0
it
and
U
u
The sampled output of the baseband signal (7.93) at frequency step i of burst k becomes ,
,u
G = A e * ^ * £ £ / K * )e^ W- >->* dJtdy a
(7.98)
v
wheref.'i.k) = 2(//c) cos 0^ and f,ii.k) - 2(//c) sin 0 . The quantity ! represents the integral term of (7.98), which in turn represents the reflectivity data obtained when the effect of target translation is removed by multiplying G by ( = exp[j4ir//?f.i/c] of (7.63) for perfect range estimates R . The frequency of Ihe ith pulse of every burst is given by lk
u
u
ik
Ll
/ = / , + /A/.
/ = 1. 2. 3 . . . . , n - 1
(7.99)
where A/ is the frequency step size. The objective of ISAR processing, including motion compensation, is to obtain an estimate of the target's reflectivity density function p(x, y), which represents the data collected from a translating and rotating target with this reflectivity density function. The purpose of TMC is to correct for the target's translation motion R . After perfect TMC, the corrected data, ignoring the constant A, can be represented by / , which from (7.98) can be seen to be the two-dimensional Fourier transform of the target's reflectivity density function at frequency step i of burst k. The inverse Fourier transform of /*,». which represents spatial-frequency data for a nonmoving target, becomes the estimate of the target-space reflectivity function. Once the data are corrected for translational motion, the RMC process can begin followed by transformation into the ISAR image. Both TMC and u
u
399
RMC can proceed open-loop as described above in Sections 7.9 and 7.1 l However, we now proceed to describe a closed-loop method, which iteratively seeks the translational and rotational motion solution that provides the sharpest image focus based on minimized entropy. Ideally, this occurs for the true (ground-truth) target motion relative to the radar. r
7.12.3 Minimum-Entropy TMC Minimum-entropy TMC begins by multiplication of the data represented in (7.98) by the exponential of (7.63), in terms of translation positions R obtained from a first set of estimates f and <J for initial values of target velocity and acceleration, respectively. The result, after multiplication by is the corrected data set expressed as a
t
0
i
ff
G' = Ac- " *«-'«*l Lt
(7.100)
u
where R represents true target positions. At this point, the two-dimensional DFT (2DDFT) process, described in Figure 7.17, is performed on the data set represented by (7.100) to obtain a first-iteration range-Doppler image in terms of reflectivity estimates D,, at discrete target slant-range and cross-range positions /, j along x, y, respectively, of Figure 7.35. The 2D-DFT process of Figure 7.17, which in actual computer implementation can be substituted by a two-dimensional FFT, is defined as u
A./= I X W i i e ^ H ll>0
(7.101)
I
for 0 <> 1£ n - 1 and 0 <j £ N - 1, and (Ald>)' of Figure 7.17 represented as W , where Wii is expressed as the IDFT: u
u
•-I
«/.i = Xc:.e*-"'«
(7.102)
for (Al
u
u
k
»-l AM D
l o
E(v, a) = X X ' / 6 Du
(7103)
where D,j of (7.101) represents image pixel magnitudes at discrete image space positions /,;'. The process is repeated with different sets of initial range, velocity, and acceleration
400
estimates until a translational motion solution is found that produces minimum entropy. Minimum entropy tends to occur for the sharpest focus when the translational motion closely approximates ground-truth motion. The initial range value Rg of (7.96) does not affect focus but positions the image in range. Search algorithms exist that obtain minimumentropy values in less than about 10 iterations. Figure 7.36(a-d), from Rocker et al. [9], illustrates velocity and acceleration slices through the entropy surface for a minimum-entropy TMC process applied to a simulated target for which translational and rotational motion parameters are assigned. Figure 7.36(c,d) shows the slices of Figure 7.36(a,b), respectively, expanded near the minimumentropy regions. Only the correction for translational motion is illustrated. The simulated target appears in Figure 7.37(a) in terms of its two-dimensional reflectivity density function p{x, y), which for the simulation consists of a set of point scatterers of equal reflectivity positioned in an image window scaled to 64m by 64m. Reflectivity versus frequency step and burst number were calculated for N = 64 bursts of n = 64 frequencies stepped in A/ = 2.2-MHz steps at lS0-/xs PRI intervals starting for each burst at / = 3 MHz. Assigned target translational and rotational groundtruth motion solutions (7.96) and (7.97), respectively, were based on translational motion 0
213.0
216.0 (c)
219.0
-16.7
-I3J
-10.7
(d)
Figure 736 Velocity and acceleration slices through the entropy surface. (From [9], Fig. 5, p. 309.)
401
(b)
(d)
(0
Figure 7.37 Motion-compensated ISAR images using the minimum-entropy measure. (From [9], Fig. 6, p. 309.)
J
3
coefficients R = 22.567.2m. v„ = 216.02 m/s, a* = 13.69 m/s , and j = -0.82 m/s (jerk coefficient); and rotational motion coefficients b\, = 133.94 deg, a\, = 3.77 deg/s, and at, = -0.03 deg/s . Figure 7.37(b) is the gray-scale plot of the image generated by the 2D-FFT of the uncorrected reflectivity data set of 64 by 64 complex values. Figure 7.37(c) is the grayscale plot of the image generated by the 2D-FFT of the reflectivity data set corrected for translational motion using the minimum-entropy velocity value of Figure 7.36(a,c). Figure 7.37(d) is the image obtained with the minimum-entropy acceleration value of Figure 7.36(b,d). Figure 7.37(e,f) is the result obtained when ground-truth, jerk, and slant-range are included, respectively, in the motion correction. It is convenient to begin the search for R that minimizes entropy of the image by searching for the initial value v = v, of velocity in (7.96) that minimizes entropy with initial acceleration a set equal to zero. Then the process continues by searching for the value of initial acceleration a* that produces minimum entropy for v = v,. Minor adjustments of v, using a, may then be required to obtain the final minimum-entropy values in the u, v entropy surface. Rotational motion for the above example produced an integration angle of \ji = (3.77) x (0.614) - (1/2) x (0.03) x (0.614) = 2.31 deg (0.040 rad) during integration time T= A/nA/ = 0.614 sec. The maximum integration angle i/t^ before defocusing occurs. a
0
2
Lk
0
a
0
2
402
based on (7.9) with A = O.lm and 2r = 64m, is 0.027 rad. The maximum allowable integration angle before defocusing begins to occur can be seen to be only slightly exceeded for this example. We see an image that appears well focused without correction for rotational motion. 7.12.4 Minimum-Entropy RMC Collected data from a translating and rotating target is represented after ideal TMC by the two-dimensional integral term / of (7.98), which represents the set of polar data at frequency step i of burst k that would be collected from the same target rotating about a fixed axis at zero range. Correction for rotational motion proceeds by seeking the rotation angle solution 0 of (7.97) that results, after reformatting, in a rectangular data set from which a minimum-entropy image can be generated. The first RMC iteration requires estimates a\ and So for initial values of angular rotation rate and acceleration, respectively, for the rotation angle solution 6\» representing angle positions at which frequency step i of burst k were sampled. TMC data / is then arranged in a polar format determined by (7.97) for estimates of initial values of wo, 4> equal to u\. So, respectively. The rectangular data set obtained by reformatting the resulting polar data set is processed as in Figure 7.17 to obtain the first-iteration image. Image entropy is calculated from (7.103) and the process is repeated with a suitable search algorithm to obtain the minimum-entropy rotation motion solution with its associated image. Dramatic improvements in sharpness of focus are realized for integration angles for which integration angle if/ greatly exceeds if™ of (7.9). l t
ik
tJ
7.13 MULTIPLE-LOOK ISAR PROCESSING The speckle noise present in SAR maps also appears in ISAR target imagery. Speckle reduction by noncoherent summation of single-look pixel intensity 7(1) at each pixel location of images obtained at each of n, independent looks at the target will theoretically increase the ratio of signal to speckle noise from unity to yfn, according to (6.114). Pixel intensity associated with pixel magnitude D is" D\ H. As for SAR, independent ISAR looks can be obtained simultaneously, or nearly simultaneously, by frequency or polarization diversity. Improved imagery results from either. Unlike for SAR, however, independent ISAR looks obtained from nonsimultaneous data collected at different positions of the illuminating beam pattern via Doppler frequency or time separation are not possible, because with ISAR, there is no intentional beam travel past the target. Speckle reduction can, however, be obtained by noncoherent summation of superimposed ISAR imagery generated from data sets collected in time sequence while the target is being tracked in range and angle. The best results are obtained by noncoherent summation of images generated from a contiguous moving-window data set. Performing the motion solutions for automatic RMC makes it possible to present, in a fixed two-dimensional orientation, the series of images generated from data collected it
k
403
as the target aspect varies to present independent looks and reveal additional target features. The series of images in the same orientation can be noncoherently summed to reduce speckle. Automatic focusing, as discussed above, is not required to achieve this capability if target motion is accurately known, which is often the case when imaging data collected on instrumentation ranges or when imaging space-object target data. Automatic focusing is useful, however, for moving targets, such as aircraft and ships, for which the target angular motion is generally not known. Figure 7.38 illustrates the generation of images obtained from data collected over a significant range of viewing angles during a series of m data looks at an aircraft target. Data from looks 1 through m collected from the aircraft during its flight path of Figure 7.38(a) are arranged, using final motion solution results, into the polar data formats in Figure 7.38(b). Focused images generated from the series of rectangular-formatted data sets of fixed orientation shown in Figure 7.38(b) will then also appear in the same orientation, as suggested for looks 1 and m in Figure 7.38(c). Noncoherent look-to-look summation of the series of images will tend to not only reduce speckle, but include features revealed at different viewing angles. The method suggested in Figure 7.38 requires that looks are selected that extend over enough target viewing angle to be able to require an accurate motion solution for focusing. For small viewing angles, the entropy or other focus measures will not produce precise estimates of motion solution coefficients 6\, OJO, and a& of (7.97). 7.14 ALTERNATIVE ISAR PROCESSING METHODS ISAR imaging was discussed above in terms of a focusing process achieved by means of convolution and Fourier-transform processing of motion-compensated target reflectivity data. Processing of two types of sampled target reflectivity data sets was discussed: ( I ) "complex reflectivity data from pulse-compression radar systems collected at discrete range delay and viewing-angle positions for a selected target range window, and (2) complex reflectivity data from stepped-frequency radar systems collected at discrete frequency and viewing-angle positions for a selected stepped-frequency bandwidth. These two types of data sets, typically associated with ISAR imaging of ship and air targets, are illustrated in Figure 7.39. Figure 7.39(a) represents sampled data collected from the output of a pulse-compression radar. The image is generated by the series of % one-dimensional DFTs performed on the N samples, one per pulse, obtained from the convolved response at sample times t to l . respectively, during target rotation angle if/. 0
% lt
Figure 7.39(b) represents sampled data collected at each frequency of a steppedfrequency radar. The image is generated by the equivalent of a single 2D-DFT process performed on the burst-to-burst echo data set collected at frequency f to /„_, of each of N bursts transmitted during target rotation angle if/. The convolution and Fourier-transform processing techniques have the advantage that no a priori information about the target's reflectivity function is required. In addition. 0
404
Figure 7.38 Multiple-look ISAR: (») urget ground truth; (b) minimum-entropy polar data and reformatted rectangular data; (c) images generated from reformatted data.
405
(a)
(b)
Figure 7.39 Data collection methods: (a) pulse-compression data collection: (b) stepped-frequency data collection.
for chirp radar systems, analog and discrete convolution for matched-filter processing is convenient for range focusing and one-dimensional FFT processing is convenient for azimuth focusing for the typical operational situation where range-Doppler processing is performed over limited viewing-angle segments for which a rectangular format can be assumed. The two-dimensional FFT process is convenient for processing data sets obtained with stepped-frequency radar. A third advantage of convolution and Fourier-transform processing is the inherently high performance obtained with data at a low SNR. Despite these advantages, promising results have been achieved with the limited viewing-angle data sets of Figure 7.39 using other techniques. A brief mention of some of these techniques follows.
7.14.1 Deductive Methods Several procedures have been investigated for obtaining target shape information from analysis of range-only, Doppler-only, and range-Doppler reflectivity data in terms of various interference effects and other data characteristics that infer dimensions and locations of target features, including target edges. SNR requirements tend to be high, and
406
stringent requirements for radar system phase and amplitude ripple and dynamic range exist for some approaches. 7.14.2 Tomography ISAR can be thought of as a form of tomography, which refers to a class of imaging techniques now commonly used for medical diagnostics and nondestructive testing. Signal sources for these purposes include x-radiation, ultrasound, nuclear radiation, and imbedded radioactive material. Imaging may be performed from both reflection and transmission characteristics of the target object. An ISAR instrumentation range that collects and processes data from targets on a turntable at a single frequency over 360 deg of target viewing angle comes nearest to resembling tomography as performed with the above sources. Even here, however, a major difference is the coherent nature of radar tomography compared to the intensityonly nature of most other types of tomography. Operational ISAR imaging of aircraft and ship targets from limited viewing-angle data sets using wideband radar waveforms is only distantly related to its tomography counterparts that perform medical and other types of diagnostics from noncoherent data collected over 360 deg of target viewing angle under controlled conditions. Figure 7.40 illustrates the relationship between data collected for ISAR and x-ray tomography. Figure 7.40(a) represents one tomographic projection produced at viewing-
(a) Figure 7.40 Tomography and ISAR.
(b)
407
A.
angle $ by moving the narrow-beam x-ray source and sensor together along the ordinate u. The projection along u is the relative power transmission versus u through the object. Power transmission at every point along u is the line integral of object density g'x. y) for the path from x-ray source to sensor. Figure 7.40(b) represents the corresponding HRR profile of signal power versus u for a radar target at the same viewing angle. Note the orthogonal relationship between the LOS for the x-ray and radar system. Tomographic applications to spotlight SAR are discussed by Munson et al. [12]. Mensa et al. [13], Gerlach [14], and Mensa [I] discuss tomographic processing of ISAR data collected from instrumentation ranges. 7.14.3 System Identification Imaging A method for using the data set of Figure 7.39(b) for imaging based on matrix calculus has been developed by Haywood and Evans [5]. The approach is to analyze the data to obtain the maximum likelihood estimate of the image array obtained for a model of the entire process from transmitting to sampling of the response from the translating, rotating target. The approach has the advantage that target motion correction is part of the process. The concept has been successfully tested with data collected from ships and aircraft. 7.14.4 Super Resolution The Rayleigh resolution in the slant range and cross range given by c/(2B) and A/(2i//), respectively, results from coherent processing, as described throughout this book, using convolution or Fourier transformation. Methods for obtaining finer resolution than the Rayleigh resolution are sometimes referred to as super-resolution techniques. Tomographic techniques described above can provide super resolution from radar target data collected during up to 360 deg of target rotation from instrumentation-range turntables. However, resolution tends to degrade to the Rayleigh resolution for application to operational situations where data is collected over the limited viewing angles represented in Figure 7.39. Spectral estimation is a method by which super resolution can be achieved from the ISAR data sets represented in Figure 7.39. Consider, for example, applications for improved ISAR cross-range resolution. The Doppler spectral estimation for the limited time history of data in each resolved range cell of Figure 7.39(a) can improve the processed Doppler resolution and thus the cross-range resolution over that for convolution and FFT processing. Autoregressive digital spectral estimation techniques described by Marple [IS] have been successfully employed to improve the cross-range resolution obtained for ship and aircraft targets imaged from short sequences of chirp-pulse-compression data sets of the form of Figure 7.39(a). Linear predictive coding has been shown by Nandagopal et al. [16] to provide the same type of benefit with stepped-frequency ship target data of the
408
form of Figure 7.39(b). The advantage in each case is a reduced target dwell time requirement. This becomes particularly important for ISAR systems operating against ships and aircraft at lower microwave frequencies. The dwell time needed at S-band (2.30 to 2.50 GHz and 2.70 to 3.70 GHz) and below to obtain data for coherent processing over the integration angle required for useful resolution tends to be excessive for operational applications. For example, from (7.24), to obtain lm resolution at 3 GHz with coherent processing requires data collected over about 3 deg of target rotation, which for a typical ship target requires a dwell time of 1 to 3 sec. A short sequence of range-profile data collected during, say, 0.5 sec of dwell time and processed using autoregressive spectral estimation methods could provide the same cross-range resolution. As with any spectral estimation method, a priori knowledge of the unsampled signal characteristics is required, and artifacts tend to occur in the processed signal. For ISAR processing, a priori knowledge can be assumed in the form of the approximate number of prominent scatterers.
7.14.5 Polarimetrlc ISAR Additional information about target features can be obtained from ISAR data collected at orthogonal polarizations. Stokes parameters calculated for each predetected pixel of the processed image can be used to assign pixel color based on the location of the Stokes vector on a color-coded Poincard sphere. Results obtained by Patel et al. [17] indicate that a predominate color tends to persist regardless of target viewing angle. 7.14.6 Maximum Entropy A maximum-entropy reconstruction method developed by Van Roekeghem and Heidbreder [18] has been tested with aircraft model data of the form of Figure 7.39(b). Advantages claimed are improved resolution and dynamic range and elimination of sidelobes due to windowing associated with conventional techniques. Images are said to be free of artifacts at levels below sidelobes associated with conventional image processing. 7.15 PREDICTED CROSS-RANGE RESOLUTION OF SHIP TARGETS It is possible to predict image quality based on ship class and sea state when the crossrange Doppler gradient is generated from small segments of a ship's pitch, roll, and yaw angular motion. Pitch, roll, and yaw periods for oceangoing ships are on the order of 5 to 30 sec. Blurring of the target image occurs with range-Doppler processing methods when integration time becomes a significant fraction of the ship's motion period. Blurring results from significant change in rotation rate and direction that occurs within the integration angle. Experiments have shown that, without focusing, the optimum integration time for ship targets is about 2 to 3 sec at S-band (2.30 to 2.50 GHz and 2.70 to 3.70 GHz)
409
and 0.025 to 0.5 sec at X-band (8.50 to 10.68 GHz). Tangential motion of the radar relative to the target is an insignificant source of cross-range Doppler for ship imaging, except for cases of calm seas and very short ranges. Table 7.2 lists computed values of pitch, yaw, and roll amplitudes and periods for two types of ships. Worst-case heading and speed for sea-state-5 wave modeling was used, worst-case heading and speed being those producing the largest ship motion. It is estimated that typical or average heading and speed would reduce computed rotation amplitude values by about one-half. The periods would remain relatively constant. Ship rotational motion can be approximated by a sinusoidal function, as shown in Figure 7.41. Here, the instantaneous angular position of the ship in degrees from vertical is represented as
Table 7.2 Computed Worst-Case Ship Motion for Two Ship Types' in Sea-Slate 5 Double Amplitude q (deg)
Ship Type Destroyer
Average Period 0 (sec)
3.4 pitch 3.8 yaw 38.4 roll 0.9 pitch 1.33 yaw 5.0 roll
Carrier
6.7 14.2 12.2 11.2 330 26.4
'Unpublished Navy hull design data.
NORMALIZED ANGULAR AMPLITUDE. Y (degrees)
Y, Y NORMALIZED ANGULAR . VELOCITY. Y (degrees/a)
I
/
PERIOD, ll(s)
-
Figure 7.41 Sinusoidal representation of ship pitch, roll, or yaw motion.
DOUBLE AMPLITUDE, q (degrees)
410
(7.104)
where q is the double-amplitude excursion in degrees and fl is the period of motion in seconds. For this sinusoidal representation, the angular velocity of ship motion in degrees per second becomes
=
2wq - - c
0
/„ t\ [ 2 „ - }
&
The average magnitude of angular velocity is M
2 i*
2a
J
l^ = n L ^
=
( 7 I 0 6 )
TI
Table 7.3 lists predicted values of the cross-range resolution at a center frequency of 3 GHz, based on computed ship motions in Table 7.2. Equations used to obtain the results in Table 7.3 are as follows:
'^
lm =
if
d c g / s > q
a n d n
f r o m T a b , c 1 2
( 7 - 1 0 7 )
2-JJ*7*
=-^IfU
rad
«
w h e r e
T
=
1
0
8 4 5 0
<
Table 7 J Predicted Cross-Range Resolution Produced by l-sec Integration for Two Ship Types in Sea-State 5 Using an S-Band Radar lj- 3 GHz)
Ship Type Destroyer
Carrier
Average Magnitude of Angular Velocity l ? U (deg/,) 1.01 pitch 0.54 yaw 6.30 roll 0.16 pitch 0.08 yaw 0.38 roll
Average Angular Excursion in l-sec Frame Time (rod) 0.0177 0.0093 0.1100 0.0028 0.0014 0.0066
Average CrossRange Resolution 4 r A „ (m) 2.82 5.35 0.46 17.83 35.54 7.56
7 1 0 8
)
411
Ar U = ^ x - ^ - = ^ m , f o r / = 3 . 0 G H z
(7.109)
f
Yiit
Vive
Results in Table 7.3 must be considered optimistic from an imaging standpoint. Sea-state 3 has wave heights about half those of sea-state 5 ; for typical rather than worstcase speed and headings, the above resolution values would degrade by about a factor of four. Actual motion during target observation would be the vector sum of the target aspect rotation produced by pitch, yaw, and roll and relative radar-to-target tangential velocity. 7.16 SAMPLE DESIGN CALCULATIONS FOR ISAR 7.16.1 Air Targets Table 7.4 defines a set of stepped-frequency ISAR parameters that are roughly consistent with use against air targets. Table 7.3 lists calculated values of interest that result when the radar of Table 7.4 is used against aircraft having no significant pitch, roll, yaw, or turning motion. It is assumed that the principal source of target aspect rotation is that produced by the target's 400-kn tangential velocity component relative to the radar. Equations used to generate Table 7.5, including conversion of knots to meters per second and nautical miles to meters, are, from (7.33), w
4 376o r x
( r a d ; s )
( 7 1 , 0 )
from (7.24) and (7.33), Tabic 7.4 Hypothetical Radar Design for the Imaging of Air Targets Equation
Value
/.-. -/.
c
ISO MHz
Pulses per burst
n
w. A7
60 MHz
Bursts per integration time
N
Parameter Slant-range and cross-range window Slant-range and cross-range resolution Center frequency Bandwidth
Symbol
Specified 60m
Ar,. Ar,
lm
7
3 GHz
60 MHz Ar,
Frequency-step size
/.-.
n- 1
2.5 MHz
412
Table 7.5 Sample Calculations for the Radar of Table 7.4 (For an Air Target With Translational Motion Only) Relative Tangential Velocity v (knots)
Radar Range R (nmi)
400 400 400 400
100 so 25 12.5
r
Effective Required Dwell Angular Time for Rotation Rate Ar, «= lm T(s) <•><*') I I I x 10-' 2.22 x 10"' 4.44 x 10"' 8.88 x 10-'
45 22.5 11.3
2.86 2.86 2.86 2.86
5.7
T=-^—x
from
Doppler Target Rotation Angle Bandwidth •Kdcg) Po(Hz)
3,600 (sec)
1.33 2.66 5.32 10.64
Uin PRF 1/T, (Hz)
80 160 320 640
(7.111)
(7.110) and (7.111) >
= w r =
X
X
( d
f ^ 3700 ^
7
<-
Il2)
from A> = liowjlc with (7.33)
= 2
x
^ 7r 3^oo
( H z )
<
7M3)
and
= -~-(Hz)
,
(7.114)
Actual flight paths of air targets may deviate sufficiently from straight-line motion to permit imaging with less dwell time than indicated in Table 7.5. In general, however, al frequencies below X-band (8.50 to 10.68 GHz), several seconds of dwell time seem to be needed to achieve useful resolution for air targets. Pitch, roll, and yaw motions above X-band may be able to generate Doppler gradients that are adequate for useful imaging in a fraction of a second. For example, at K,-band (33.4 to 36.0 GHz), the integration angle required to produce the same cross-range resolution (lm) is only about 0.3 deg as compared with nearly 3 deg at 3 GHz. Above K.-band, a small fraction of a degree of aircraft pitch, roll, and yaw could produce lm resolution. Super-resolution techniques could further reduce the required target aspect-angle change.
413
7.16.2 Ship Targets The situation changes for imaging ship targets. A ship's tangential motion relative to the radar is small, unless the radar is airborne at close range. Also, pitch, roll, and yaw angular motion of ships during straight-line travel is larger than that for aircraft targets. Thus, the major source of Doppler gradient for ISAR ship imaging is target pitch, roll, and yaw motion. When tangential motion of the radar platform generates a significant part of the viewing-angle change, the image can be thought of as being generated by both ISAR and SAR processes. Table 7.6 defines a set of synthetic ISAR parameters that are roughly consistent with use against ship targets. It is assumed that the ship's pitch, roll, and yaw motion is adequate to generate cross-range resolution as fine as 2m. Recall that a synthetically generated ISAR image is made up of Nn pixels, 1 pixel on the average for each transmitted pulse. The number of pulses n per burst and number of bursts N per image frame for unambiguous sampling must equal or exceed the number of image resolution cells in the slant range and cross range, respectively. The minimum pulse repetition frequency is determined by the number N of frequency bursts, each of n pulses, that are transmitted and received during the data collection time T over which data is processed to create an image frame. Image frame time will be assumed to be less than about 3 sec in order to stay well within ship pitch, roll, or yaw periods. The PRF listed in Table 7.6 could be inconveniently high from a radar design standpoint. The PRF could be reduced without reducing resolution by sizing the window for a better fit to the target shape. For example, if the largest expected ship size were Table 7.6 Hypothetical Radar Design for the Imaging of Ship Targets Parameter
Symbol
Specified
Slant-range and cross-range window Slant-range and cross-range resolution Center Frequency Bandwidth
w„ w,
400m
Ar„ Ar,
2m
J /..i - J o
3 GHz
Pulses per burst
n
Bursts per integration time
N
Frequency-step size
A/
Minimum PRF (for T i 3 sec)
l/T,
Equation
Value
c 2Ar, w. Ar,
75 MHz
H»,
Ar^ /.-, -fo n- 1 nNIT
200 200 0.38 MHz 13 3 kHz
414
400m by 80m, the PRF for bow, stern, port, or starboard aspects could be reduced from 13.3 kHz to 1 nM 200 x 40 „„,„ f = ~f = 3 = 2.7kHz i
(7.115)
7.17 CHIRP-PULSE COMPARED TO STEPPED-FREQUENCY ISAR ISAR imaging of target models, aircraft, ships, and space objects has been achieved to date mostly by using three basic waveforms: (1) chirp pulse, (2) stretch, and (3) pulse-topulse stepped frequency. Chirp-pulse and stepped-frequency waveforms, although related, provide the most contrasting performance. 7.17.1 Chirp-Pulse ISAR A chirp-pulse radar produces a target range-profile signature for each pulse. This profile is obtained in digital form by sampling the output of an analog pulse-compression filter or by digital pulse compression of the sampled baseband signal as described in Chapter 4. To avoid ambiguity in either case, samples should be spaced in time by no more than the range-delay resolution associated with the chirp bandwidth. Resolution, therefore, can be limited by A/D conversion-rate performance. On the other hand, because an entire target range profile is obtained from each pulse, profile-to-profile sampling at each rangeresolved position of the target occurs at the radar's PRF. The bandwidth of the Doppler spectrum of the presampled response in each range cell, even with large targets, such as ships at maximum pitch, roll, and yaw rates, is therefore low enough at microwave frequencies to be adequately sampled at radar PRFs as low as a few hundred pulses per second. For example, consider a 10-GHz chirp radar with a PRF of 1,000 Hz that samples and digitizes each pulse at a rate of 10* complex samples per second, which will be assumed to be the maximum rate of the A/D converter. Doppler frequencies extending over the range as low as -500 Hz to as high as +500 Hz are unambiguously sampled at this PRF. In the worst-case sea-state-5 conditions, the predicted roll rate for the destroyer of Table 7.3 is 6.3 deg/s. If its exposed height h above the surface is 30m, the maximum Doppler bandwidth to be sampled following velocity correction to zero at the center of rotation is A, = - 6 * 7
3 x 10* = 220 Hz
X
( |) 63X
X (30) x (10 x 10')
(7.116)
415
This is well within the unambiguous range of Doppler frequencies provided by the assumed PRF of 1,000 Hz. The same radar, however, must limit chirp bandwidth to less than 100 MHz to avoid undersampling in range delay. The useful slant-range resolution is therefore limited to c/(2B) = 1.5m. 7.17.2 Stepped-Frequency ISAR In contrast, consider a stepped-frequency ISAR, which produces a range-profile signature from each burst of n pulses. Samples can be thought of as being collected in the illumination-frequency domain. The spacing of the samples in time is the radar's PRI. A/D conversion is therefore performed at the radar's PRF. This does not present a problem, even at extremely high radar PRFs. The range resolution, therefore, is limited only by the radar stepped-frequency bandwidth. On the other hand, because sampling in the Doppler frequency domain effectively occurs at the burst rate, the required PRF is n times as high as that for an equivalent chirp radar operating against the same target. For the destroyer example above, the required PRF in a stepped-frequency mode of 110 steps would be 110 x 220 Hz = 24 kHz. Operation at this PRF at useful ranges means that the echo pulse at each frequency arrives multiple PRIs following the transmitted pulse at that frequency. The frequency synthesizer should then be programmed to set up transmission and local oscillator references that allow measurement of echo signal phase relative to the phase of the corresponding transmission signal at each frequency. In other words, a transmission phase reference must be recalled at each frequency after several other pulses are transmitted. This is conveniently achieved using a direct digital synthesizer. However, multiple pulses in the air tend to reduce the clear range region of reception between pulse transmissions. The PRF requirements for stepped-frequency ISAR can be determined by recognizing that the cross-range window u> is equal to the number of bursts N multiplied by the cross-range resolution Ar provided by target rotation through the u>T rad of viewing angles that occur for target rotation rate at during imaging time T. The relationship, from (7.32), is c
r
w = NAr = N^~ c
c
(7.117)
By recalling that data collection time to collect N bursts of n stepped-frequency pulses is T = NnT for a PRI of T , we can express (7.117) as 2
2
w
=
< i*h
when N = n (7.118) can be written independently of N and n as
< > 71,8
416
1/2
(7.119) 1/2 3
2a/"" ( r ) 2
The minimum PRF corresponding to this window is
(7.120)
For example, sampling a cross-range window w, equal to the 30m height of the destroyer during Ihe 6.3-deg/s roll rate listed for sea-state 5 requires that the PRF of the above 10GHz radar for N = n and O.S-sec integration time exceed the value
(7.121)
= 24,181 Hz
With N = n, the number of pulses n per burst and number of bursts N per integration time T = nNTi becomes (7.122) Assume the above typical integration time at 10 GHz of T = 0.S sec and the PRI from (7.121) of r = 1/(24,181) sec. From (7.122), we then obtain n = 110 pulses per burst and N = 110 bursts per integration time. The burst rate, for T = 0.5 sec, is therefore 220 bursts per second, which is just adequate to sample the 220-Hz Doppler bandwidth shown by (7.116) to be produced by the rolling destroyer. The expression (7.32) for a cross-range window with (7.122) and cross-range resolution Ar from (7.24) is expressed in terms of resolution Ar, image frame time T, and radar PRI T as 2
c
2
(7.123) for n = N and where Ar = Ar, = Ar,.
417
7.17.3 Summary Chirp and stepped-frequency waveforms contrast in their technical requirements. Chirp radar design for high resolution requires samples closely spaced in range delay, which translates to a requirement for high-speed, wide-dynamic-range sampling and A/D conversion. Stepped-frequency radar systems designed for high resolution require samples closely spaced in frequency, which translates to a requirement for a frequency synthesizer capable of being programmed to switch rapidly from frequency to frequency while maintaining phase coherence. Figure 7.42 illustrates the general range of operation for the two types of waveforms. The use of hopped-frequency waveforms described in Chapter 5 allows unambiguous Doppler sampling with PRF greater than Doppler bandwidth B instead of greater than n times B for stepped-frequency waveforms. 0
D
7.18 RADAR TARGET IMAGING RANGE In this book, we have described wideband radar waveforms and systems that are capable of resolving individual targets into one dimension of slant range (HRR) and the two orthogonal dimensions of slant range and cross range (ISAR), as illustrated in Figure 7.43. Ideally, the resolution performance of HRR, SAR, and ISAR is independent of SNR. The degradation of image quality with range is therefore fundamentally determined by reduced SNR of resolved image picture elements (pixels), rather than by resolution. The radar range at which targets can be imaged, therefore, may be defined in terms of the SNR of image pixels. The term imaging as used here will refer to both onedimensional and two-dimensional target resolution. Because the image process resolves the target into target-space resolution elements, the radar cross section to be considered in the range equation is that of the individually resolved target-space elements illustrated in Figure 7.43. This is in contrast to the situation for target detection, in which radar detection range is ordinarily based on the radar cross section presented by the contributions of all scatterers of the target using narrowband waveforms. This effect tends to be offset by the image processing gain G produced by pulse-to-pulse coherent integration inherent in the imaging process, which is performed over integration time typically much longer than the target dwell time associated with detection. f
7.18.1 Image Processing Gain The average per-pulse SNR at the output of the matched filter, for free-space propagation conditions, by a radar operating in a narrowband, low-resolution detection mode, from (2.47) of Chapter 2, is S
P.G'AT,
_
(7.124)
O
IS
HP I
0.Z ;3ui
UJ o
i s m Ui
so o
o O
!!!!
111 (ft
3 0.
lift '. ,, i'
o
3
jtfl Oi
II
' I'll"
in
(u) N o i i m o s a u
t
I
419
1
,
A 1 TARGET , / I 1 SPACE
0
1
2 \
v
\
I
1
1
i
i n-1
J
m TARGET ELEMENTS
(a)
SLANT-RANGE WINDOW
0
1
SLANT-RANGE WINDOW 2
~i—i—i—i—i—i—i—i
(b)
n-1
r
ui O z < rr
nN pixels TOTAL
oDC O
m TARGET ELEMENTS
N-1
Figure 7.43 Occupation of image space by the target: (a) range-profile image; (b) ISAR image.
where 7", is the transmitted pulse duration and ~a is the target cross section seen with a narrowband radar. Now consider a radar operating in a wideband imaging mode. The cross section of resolved target elements, from (2.21) of Chapter 2, is approximated by (7.125) where tr is the average narrowband cross section of targets resolved into m resolution elements. The average output SNR of image pixels can be expressed as
420
(S\
P,GWT
t
a
where G, is the image processing gain produced by the coherent pulse-to-pulse integration associated with image processing. Equations (7.124) and (7.126) apply to either chirp-pulse or stepped-frequency waveforms. Chirp-pulse radar receiving systems are said to provide processing gain because the SNR referred to the receiving system input is increased at the output by pulse compression. For the purpose of this discussion, however, we wish to focus on the processing gain provided by the pulse-to-pulse coherent integration inherent in image processing. By taking T, of (7.124) to be the pulse duration and P, to be transmitted pulse power, correct results for output SNR will be obtained for a matched filter, matched to the transmitted pulse regardless of any coding within the pulse. Then image processing gain greater than 1.0 for both chirp-pulse and stepped-frequency waveforms can be attributed to the pulse-to-pulse integration provided by image processing. The minimum image processing gain (G,)«i. occurs when the number of echo pulses integrated is just adequate to unambiguously sample the slant-range and cross-range sampling windows of the image. As examples, we have (G,)^. = nN for an /i/V-pulse synthetic ISAR image frame of nN pixels obtained using a stepped-frequency waveform, ( G , ) „ = N for an A/pulse ISAR image frame of nN pixels obtained using chirp-pulse waveforms, (G,)«i. = n for an n-pulse synthetic range profile using stepped-frequency waveforms, and (G,) = I for the range profile produced by matched-filter-processing the return from one chirp pulse. Actual image processing gain will exceed the minimum gain (G,)„, when the number of radar pulses that have been coherently integrated to generate the image exceeds the minimum required for unambiguous imaging or, in other words, when the target image space is oversampled. For example, if a single range profile is generated by coherently adding the responses from 10 chirp pulses, the summed range profile is oversampled by a factor of 10 and (G,) = 10.
7.18.2 Fraction or Visible Target Elements So far, we have discussed only average SNRs. The actual situation is complicated by variations in reflectivity of different scatterers of the target. The image of a target, viewed at a range where the SNR is high, will contain visible pixels produced by scattered of both high and low reflectivity. As range increases, the responses from small scatterers will become invisible in the noise, but responses from larger scatterers will remain above some visibility threshold. The term visibility as used here need not be restricted to the sensor capability or human observation. Automatic radar image recognizers would also use some visibility threshold criterion to recognize those pixels containing responses produced by scatterers.
421
The fraction of visible target resolution elements will be defined as the measure of radar target image visibility in noise. This fraction can be related to the output SNR produced by the target's ensemble of scatterers as seen in a narrowband mode. (In a typical application, a target detected during operation in the radar's narrowband surveillance mode is selected for imaging. The surveillance mode is then interrupted to illuminate the selected target long enough to obtain recognition.) Because for many applications the target to be imaged and recognized must first be detected, it is convenient to express the fraction of visible target resolution elements in terms of the SNR required for narrowband detection. This can be done by defining image quality in terms of the single-pulse SNR for the radar in a narrowband mode, such as detection, at the same range as for imaging. The derivation follows. The target's average cross section
'A^WkT.L Pfl»AT,
S N
( 7 , 2 7 )
The quantity S/N may be considered to be the SNR available for detection of the target of average cross section a. Now consider the same radar in an imaging mode. All radar parameters will be assumed to remain the same except for the substitution of pixel SNR for S/N and the inclusion of pulse-to-pulse image processing gain G A chirp-pulse radar mode obtains one range profile from each pulse. A stepped-frequency mode obtains one range profile from n pulses. The pulse duration is 7", in each case. The visibility threshold cross section of a resolved target element, above which the corresponding image pixel will be visible, is r
(7,28) y
~
P.GWT,
G,(NI
where (S/N), is the pixel SNR that by some criterion is determined to be required for pixel visibility. The threshold cross section y will be evaluated by assuming that imaging is carried out at the range where narrowband echo pulse SNR produced by the target's average cross section a is the S/N of (7.127). The quantities T„ G, L, P„ A, R, and T are assumed to remain unchanged from the detection to imaging modes. From the ratio of (7.128) and (7.127), we then have {
a
(S/N),
Resolved target elements with cross section above y will be considered visible. The image quality can be defined in terms of the fraction of visible target elements by using
422
(7.129) together with an assumed reference distribution function for the radar cross section of resolved scatterers. The target element cross section will be assumed to be distributed according to the power form of the Rayleigh distribution. The probability density for element cross section o~„ based on this distribution is given by *«••> = 3 e x p ( = ^ = I? explfa,
\cr,J
a
\
a
/
(7.130)
where a, = aim is the average element cross section. The probability that a, of (7.130) will equal or exceed the threshold value y is P, = P\a, 2 y)
= f r exp[(-mcr,fo)]dcr, *y tr
=
(7.131)
cxp[-(mfa)y]
Image visibility, defined as the fraction of resolved target image elements that are visible, can be expressed by substituting y from (7.129) into (7.131). The result is P. = expj-
m
G,
m\ S/N
J
(7.132)
Equation (7.132) provides a way of evaluating target image quality in terms of singlepulse signal-to-pulse ratio S/N produced by the radar operating in a narrowband mode, such as detection, at the same range as is used for imaging. To make use of this expression, we must assign values of processed pixel SNR per pulse required for pixel visibility and matched-filter SNR per pulse for narrowband operation, both at the same range. Figure 7.44 shows (7.131) plotted for three values of mlG . p
7.18.3 Calculation of Image Visibility For illustrative purposes, consider a slowly fluctuating target to be imaged by using a stepped-frequency waveform of N = 50 bursts of n = 40 pulses per burst. Frorrt. Figure 2.8(a) of Chapter 2, the required single-pulse SNR in the detection mode for a Swerling case 1 target is about +18 dB. Assume that image pixels become visible when the processed pixel SNR is 6 dB below that for single-pulse detection. Then (S/N), in decibels is 12 dB. For these assumptions, (S/N)J(S/N) = 0.25. The ratio m/G, is the number of image pixels occupied by the target image divided by the number of pulses integrated to generate the image. Assume synthetic ISAR processing to create one n x /V-element image. Then,
423
m >ui ui(9 ? 2
RATIO OF REQUIRED S/N FOR IMAGING TO THAT REQUIRED FOR SINGLE-PULSE DETECTION, ( S / N ) , ASH*) Figure 7.44 Image visibility (fraction of visible pixels).
G, = (G,)^, = «/V = 2,000
(7.133)
Next, assume that the target occupies one-half of the available n x /V-element image space. For these assumptions,
From Figure 7.44 or (7.132), the fraction of target pixels containing visible responses is 0.88. The 2,000-pixel image frame generated by the N= 50 bursts of n = 40 pulses would, from (7.134), contain 1,000 target resolution elements, of which 880 would be visible at the radar's single-pulse detection range. Under the same assumptions, if a second target to be imaged occupies only onequarter of the image space, mlG = 0.25. From (7.132), a fraction of about 0.94 of the r
424
imaged target pixels would then contain visible responses. The second target would occupy 500 of the 2,000 available image pixels, of which 470 would be visible at the radar's single-pulse detection range. 7.18.4 Radar Range Equation for Imaging A radar range equation can be defined for target imaging in terms of S/N for single-pulse narrowband operation against a target of average radar cross section a, minimum acceptable pixel visibility fraction P, with required pixel SNR (S/N), for pixel visibility, processing gain G,, and number of imaged target resolution elements m. From (7.132),
With narrowband S/N in (7.135) expressed in terms of radar parameters from (7.124), we obtain m
i
(Air) R*kT,L/S\
(7l36)
^ - ^ - f ^ PfiWT, T J W G
Solving (7.136) for free-space imaging range,
R
r
P W T , *
x
Hf
,
(7 37)
The ratio of single-pulse detection range to imaging range from (7.124) and (7.137) is R (detection) \m R (imaging) " J C ,
(SM). 1 f S/rV * [-Ln(P,)]J
*''"
B |
For the above example where (S/N)J(S/N) = 0.25 and m/G, = 0.5 for which P, = 0.88, we confirm with (7.138) that /((detection) = /{(imaging), as assumed. Image visibility improves as the target range decreases below the maximum single-pulse detection range. 7.19 SPATIAL FREQUENCY BANDWIDTH AND RESOLUTION LIMITS Fundamental limits to ISAR resolution can be understood in terms of spatial bandwidth of ISAR data defined in a rectangular spatial-frequency format. Figure 7.45 represents spatial-frequency data obtained from a fixed and a rotating target. The expressions for
Figure 7.45 Spatial bandwidth produced by RF bandwidth and target rotation: (a) target space with target viewed at rotation angle 8; (b) frequency space for target data collected over RFs/, —»/, at target rotation angle fr, (c) target space with target viewed al rotation angles 0, and 8,; (d) frequency space for target data collected al rotation angles 0, -» 8 with constant RF /. t
1
spatial bandwidth in meters' along the/, a n d / coordinates of Figure 7.45(b), for data collected from frequencies/, to/ from the fixed target of Figure 7.45(a), is obtained from (7.86) to (7.87). respectively, as 2
0 = — cos 0 - — c o s 6 X
and
c
c
(7.139)
426
B, = — smO-^-%m0 c c
(7.140)
1
The expression for spatial bandwidth in meters' along the f, and / , coordinates of Figure 7.45(d) for data collected at fixed frequency / from the rotating target of Figure 7.45(c) is obtained, respectively, as B, = - cos ft. - ^ c o s c c
ft
(7.141)
B. = ^ s i n Bi - ^ s i n c c
ft.
(7.142)
and
The Rayleigh resolution in meters along target-space coordinates x and y , obtained by two-dimensional inverse Fourier transformation of the spatial frequency data in rectangular format, can be shown to be limited to the reciprocals of spatial bandwidth B, and B respectively. The limits of resolution in target space can therefore be seen to be determined by the spatial bandwidth produced by the available RF bandwidth and target rotation angle. For example, consider the resolution limits in the slant range and cross range associated with a given RF bandwidth and target rotation angle. Orient the rectangularformatted data in Figure 7.45(b) with 0 = 0 so that /, is initially along the radar LOS. The reciprocal of B, at this initial position is then the slant-range Rayleigh resolution limit, which for B, evaluated for RF bandwidth P=fi-f\ from (7.139), becomes v
Ar, = ^
(7.143) r
The cross-range resolution limit produced by target rotation angle ip from 0= 0 is the reciprocal of spatial bandwidth B, along/, in Figure 7.45(d). With spatial bandwidth B, from (7.142) for ft. = ^and 0, = 0, the expression for the cross-range Rayleigh resolution evaluated at the highest frequency / becomes
A
r
r = ^
•
2/ sin ip 2 sin ip
0^^Sirf2
(7.144)
The limit-Cor resolution produced by target rotation through angle ^can be obtained from (7.142) by letting 0, = -uV2 and b\ = +uV2. Then rotation from ft. to ft produces the highest possible spatial frequency bandwidth expressed as
427
^ji^iyt^m,
(
7
,
4
5
)
The expression for the Rayleigh resolution limit along y in Figure 7.45(c) then becomes
&
r
X
= A
, , x > 4 sin(vV2) n
0
<,tr
(7.146)
Resolution obtained by processing data over large integration angles would be limited for actual targets by the extent of angular segments over which target reflection points remain illuminated and at the same RCS. We conclude that, for target images produced by Fourier-transform or equivalent convolution processing of ISAR data, the limit of slant-range resolution is c/(2/3) for RF bandwidth f3 and the limit of cross-range resolution is A/(2 sin ip) for rotation angles up to ir/2. The resolution limit without reference to the slant range or cross range is A/[4 sin(yV2)] for rotation angles up to ir rad. Resolution limits imply focused ISAR processing. As an example, consider the resolution limits associated with data collected from a radar operating from 750 to 1,250 MHz during target rotation through 60 deg. Slantrange resolution from (7.143) is c A r
'
=
2^
=
3 x 10" m/s (2X500x 10* Hz)
=
„ °-
„ , „ 3 0
m
(
7
U
7
)
8
)
The cross-range resolution from (7.144) is 3 x 10* m/s (2X1.25 x 10'Hz)(sin60») =
c A r
'
=
=
0
J
4
m
(
7
I
4
Based on wavelength at the maximum frequency 1,250 MHz, the resolution associated with target rotation from (7.146) is determined as c
" 4/sin(vW2)
=
3 x 10* m/s (4)(1.25 x 10' Hz)(sin 30°) ~
0
l
2
m
(
7 U 9 )
Super-resolution methods produce resolution exceeding that for the above limits based on spatial-frequency bandwidth. PROBLEMS Problem 7.1 A 300m ship target in a smooth sea is making a 1-deg/s turn, (a) What is the Doppler bandwidth of the echo signal seen with a 3.5-GHz shipboard radar viewing the target
m
from broadside at long range assuming no other motion? (b) What is the focused crossrange resolution in meters for a target dwell time of 5 sec? (c) What is the optimum resolution possible without focusing? (d) What is the maximum-length data record in terms of data collection time that can be processed before defocusing occurs? (e) What resolution is obtained from the data record of (d)? Problem 7.2 The ship target in Problem 7.1 is viewed from broadside with the same radar during 2.0 deg of roll in a rough sea. What is the best possible cross-range resolution in meters? Problem 7.3 An aircraft is flying straight with a tangential component of velocity of 200 kn (103 mi s) at a range of SO nmi (92.6 km) relative to a ground-based radar operating at a wavelength of 0.03m. What is the best possible cross-range resolution in meters for a five-second target dwell time? Problem 7.4 A chirp radar transmits 0.S-/ts RF pulses with dispersion D= 180. Center wavelength is 0.03m. What is the best possible slant-range and cross-range resolution in meters against a ship target viewed bow-on during 0.5 deg of pitch motion? Problem 7.5 What is the best possible slant-range and cross-range resolution in meters if the radar of Problem 4 transmitted monotone pulses stepped in frequency pulse to pulse, in repeating bursts of 180 pulses per burst, spaced by 2 MHz? Problem 7.6 Two-dimensional ISAR images of aircraft are to be produced by a chirp radar operating at 0.03m wavelength. Resolution capability is to be as fine as 0.5m in both dimensions, and target sizes of up to 75m in each dimension will be viewed, (a) What is the minimum PRF required to image unambiguously targets that have up to 2-deg/s aspect rotation rate7 (b) What is theTninimum required A/D converter speed in terms of complex sample pairs per second?
429
Problem 7.7 The radar of Problem 7.6 is to be operated in a stepped-frequency mode. Compute the new requirements for (a) PRF and (b) sampling rate. Problem 7.8 A short-pulse radar is to be used to generate ISAR images containing 128 x 128 pixels. (a) What is the minimum number of transmitted pulses required to form a single image? (b) How many complex samples per pulse are required? Problem 7.9 What are the answers to (a) and (b) of Problem 7.8 if stepped-frequency waveforms are to be employed instead of short pulses? Problem 7.10 Two-dimensional ISAR images are to be obtained of a ship target viewed from a shipboard radar. The ship target has the following dimensions: 200m long, 20m wide, and 30m high out of the water. Assume that by some method the image was calibrated in true slant range and cross range. What would be the image dimensions in meters for the following conditions: (a) bow view and pitch.motion only? (b) broadside view and roll motion only? (c) broadside view and yaw motion only? (d) bow view and yaw motion only? (e) bow view and roll motion only? Problem 7.11 A stepped-frequency radar has the following parameters: lowest frequency = 3 GHz, frequency-step size = 1 MHz, pulses per burst = 256, PRF = 5 kHz, pulse-width = 3 LIS, and bursts per image frame = 256. A target is rotating at 1 deg/s in a plane containing the radar LOS. What, in meters, is (a) the cross-range resolution? (b) the slant-range resolution? (c) the cross-range window? (d) the slant-range window? Problem 7.12 A chirp-pulse radar is to be used to generate ISAR images of a 75m-long air target moving at 200 m/s directly toward the radar. (Cross-range resolution is produced by target pitch and yaw motion.) Sampling is carried out on the pulse-compressed signature using a fixed
430
sampling window through which the target passes (fixed-range sampling gate). A total of 100 complex samples is collected from each pulse at the rate of ISO x 10 samples per second (one complex sample per meter). The slant-range resolution is lm and the PRF is 400 pulses per second, (a) How many complete range-profile signatures of the target will be sampled? (b) What is the accumulated number of cells of range walk? (c) How many ISAR image pixels are produced per image following velocity correction? 6
Problem 7.13 What is the target range-measurement error caused by range-Doppler coupling in terms of number of resolution cells for Problem 7.12 if the chirp-pulse width is 2 /is and the center frequency is 9.S GHz? Problem 7.14 ISAR images of the target of Problem 7.12 are to be generated using a stepped-frequency waveform. The PRF is 10 pulses per second, there are 100 pulses per burst, and processed resolution is the same as for the Problem 7.12 chirp waveform. A single quadrature pair of samples is collected from each pulse as the target passes a fixed sampling gate, (a) How many synthetic range profiles will be obtained if the 3-dB width of the sampled pulse is 0.67 /is? (b) What is the accumulated number of range cells of range walk? Neglect samples obtained outside the pulse 3-dB edges. 4
Problem 7.15 A 200m ship in a 2-deg/s turn is viewed at long range by a 1.3-GHz radar from a 0-deg elevation angle, (a) What is the bandwidth of Doppler frequencies observed when the average azimuth aspect angle to the radar is 45 deg from bow-on? (b) Following radial motion correction that corrects to zero Doppler at ship center, what is the change in Doppler frequency during 5 sec of observation time produced by a scatterer on the bow? (c) What is the resulting cross-range shift in terms of number of resolution cells? Problem 7.16 In Problem 7.15, what is the maximum possible number of range cells shifted during the 5 sec of target dwell time assuming that range resolution is 10m? Problem "7^7 A radar is to be designed to generate ISAR images of 256 slant-range cells by 256 crossrange cells with as fine as 1.5m resolution. What is the minimum radar center frequency
431
that prevents cell migration due to target rotation from exceeding one cell? Assume that velocity correction centers the image in the slant range and cross range. Problem 7.18 1
A target at long range is viewed at 5.4 x 10' Hz. (a) What is the spatial frequency / , (m" ) at the instant the/, coordinate axis is aligned with the radar LOS? (b) What is the spatial frequency f,(m~') at that instant along an axis at right angles to f?. (c) If the target then rotates 10 deg, what are the new values along / and /,? Problem 7.19 A target is viewed over a bandwidth of 300 MHz at a center frequency of 3 GHz. (a) What is the spatial frequency bandwidth (m* ) along the / , axis when it is aligned with the radar LOS? (b) What viewing-angle change in degrees is required to produce the same spatial-frequency bandwidth along an f axis that is in the plane of rotation and perpendicular to the / , axis? 1
f
Problem 7.20 Show that the result for Problem 7.19(b) is also obtained based on expressions Ar, = cl (23) and Ar = Xl(2fa) by requiring Ar, = Ar . c
f
Problem 7.21 A stepped-frequency radar is to be designed to provide an ISAR imaging mode with slantrange and cross-range resolution as fine as 0.7m during an integration time of 5 sec. The radar PRF is 5,000 pulses per second. What is the largest size target in meters that can be handled unambiguously if aspect is unknown? Assume equal slant-range and crossrange windows. Problem 7.22 Air targets are to be imaged while they are flying at up to 600 m/s at ranges as near as 30 km from a 10-GHz ground-based radar. Target length can reach 40m. (a) What is the maximum received Doppler bandwidth? (b) What is the minimum required PRF for a pulse-compression waveform? (c) What is the minimum required PRF for a steppedfrequency waveform of 128 pulses per burst?
432
Problem 7.23 A target is to be viewed with a radar capable of 0.5m slant-range resolution. A slantrange (delay) window of 0.5 LIS is to be established. What is the minimum processing gain (number of pulses coherently integrated) required to divide up unambiguously the target window in slant-range resolution cells using (a) chirp-pulse waveforms? (b) pulseto-pulse stepped-frequency waveforms? Problem 7.24 A radar is to be designed for ISAR imaging. A 120m slant-range by 75m cross-range image space is to be established with a resolution of 0.5m in both dimensions. What is the minimum number of pulses required to be integrated to divide up unambiguously the image space using (a) chirp-pulse waveforms? (b) pulse-to-pulse stepped-frequency waveforms? Problem 7.25 A space-based radar in its surface-search surveillance mode obtains a single-pulse SNR of 18 dB from a 75m-by-10m ship target. A chirp-pulse-compression mode is then selected for ship-target imaging, which involves chirping the transmit pulse used in the surveillance mode. An ISAR image from a 100-pulse look is generated with 1.5m-by-1.5m resolution. Based on (7.132), how many visible target pixels can be expected for a pixel visibility threshold of 8 dB? Problem 7.26 (a) What is the limit of slant-range resolution produced by data collected from an idealized impulse radar for which the spectrum is uniform from 0 to / Hz? (b) What is the limit of cross-range resolution for data collected during itfl rad of target rotation? (c) What is the limit of resolution obtained for data collection over it rad of target rotation? Express above answers in terms of wavelength. REFERENCES ( I ) MeTBa. D. L „ High Resolution Radar Cross-Section Imaging, Norwood, MA: Artech House, 1991.
(2] Chen, C , and H. C. Andrews, "Multifrcquency Imaging of Radar Turntable Data," IEEE Trans. Aerospace and Electronics Systems, Vol. AES-16. No. I, Jan. 1980, pp. 15-22. (3) Walker, J. L.. "Range-Doppler Imaging of Rotating Objects," IEEE Trans. Aerospace and Electronics Systems. Vol. AES-16, No. I, Jan. 1980. pp. 23-52.
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[4] Chen, C , and H. C. Andrews, "Target-Motion-Induced Radar Imaging," IEEE Trans. Aerospace and Electronics Systems. Vol. AES-16. No. I, Jan. 1980. pp. 2-14. |5) Haywood, B.. and R. J. Evans. "Discrete 2-D System Identification for ISAR Imaging," I EE International Radar Con/.. Brighton. U.K.. 12. 13 October 1992. pp. 411-414. (61 Steinberg, B. D.. "Microwave imaging of Aircraft." Proc. IEEE, Vol. 76. No. 12. December 1988. pp. 1578-1592. (71 Ausherman, D. A., et al. "Developments in Radar Imaging," IEEE Trans. Aerospace and Electronics Systems, Vol. AES-20. No. 4, July 1984. pp. 363-400. [8) Tran, S. T , "Computer Processing of High Resolution Radar Data at Naval Ocean Systems Center," IEEE International Conf. Systems Engineering. 9-11 August 1990. pp. 482-485. (9) Bocker, R. P., T. B. Henderson, S. A. Jones, and B. R. Frieden, " A New Inverse Synthetic Aperture Radar Algorithm for Translational Motion Compensation.'' SPIE Conf.. Vol. 1569. July 1991. pp. 298-310. [10) Bocker, R. P., and S. A. Jones. "ISAR Motion Compensation Using the Burst Derivative Measure as a Focal Quality Indicator," International J. Imaging Systems & Technology (1JIST), Vol. 4, 1992. pp. 285-297. ( I l l Lush, D. C . and D. A. Hudson. "Resolution Analysis of Large Time-Bandwidth Radars for Non-Uniform Target Motion." I EE International Radar Conf. Brighton, U.K.. 12. 13 October 1992. pp. 407-410. [121 Munson. D. C . Jr., J. D. O'Brien, and W. K. Jenkins, " A Tomographic Formulation of Spotlight-Mode Synthetic Aperture Radar," Proc. IEEE, Vol. 71. No. 8. August 1983, pp. 917-925. [13] Mensa, D. L., S. Halevy, and G. Wade. "Coherent Doppler Tomography for Microwave Imaging." Proc. IEEE, Vol. 71. No. 2. February 1983. pp. 254-261. (14) Gerlach, D., "Radar Cross Section (RCS) Measurements Using a Tomographic Inverse Synthetic Aperture Radar (ISAR) Algorithm." Radarcon 90, Adelaide, Australia, 18-20 April 1990, pp. 445-451. [15] Marple, S. L „ Jr.. Ch. 8 in Digital Spectral Analysis With Applications, Englewood Cliffs. NJ: PrenticeHall. 1987. (16) Nandagopal, D., D. Longstaff, G. Nash, D. J. Heilbronn, B. Haywood, and N. Martin. "Application of Selective Linear Predictive Coding (SLPC) in Enhancing Cross-Range Resolution of Inverse Synthetic Aperture Radar (ISAR)," SPIE Conf, Vol. 1154. 1989. pp. 77-88. (17) Patel, I. R., A. Jain, and D. R. Wehner, "Ship and Aircraft Identification From Dual Polarized HRR and ISAR." 1988 Polarimelric Technology Workshop, to be published in the IEEE Trans. Antennas and Propagation. [181 Van Roekeghem. F., and G. Heidbreder, "Maximum Entropy Reconstruction of Radar Images of Rotating Targets," International Symp. Signal Processing and Applications, Brisbane, Australia, August 1987. pp. 486-490.
Chapter 8 Three-Dimensional Imaging With Monopulse Radar 8.1 SHORTCOMINGS OF ISAR Target imaging in an ISAR mode is usually possible at the full detection range of the same radar in its surveillance mode. Once detected, a target selected for imaging can be imaged during, typically, a fraction of a second to several seconds of target dwell time in a track mode. Despite the advantage provided by the ISAR process to provide imaging at the full radar detection range, there remain the following inherent shortcomings. 1. The cross-range dimension scale is a direct function of the target's aspect angularrotation rate. Distorted images result for small-inlegration-angle data unless the rotation rate can be determined from auxiliary data. 2. The ISAR image plane does not reveal the true aspect of the target, which is unknown because the radar cannot directly determine the direction of the target's rotation vector that produces the cross-range Doppler gradient. 3. Target dwell time required to produce a given cross-range resolution is dependent on the target's aspect rotation rate relative to the radar. Therefore, a long range, nonmaneuvering air target may require tens of seconds to image. The three problems with ISAR mentioned above result from its inherent dependence on the target changing its viewing angle to the radar. The direction of the target's aspect rotation vector is not determined in the ISAR approach, except from auxiliary data, and the magnitude may be too small to produce adequate cross-range resolution during an acceptable dwell time. Despite this limitation, the ISAR technique remains attractive for surveillance problems that require target identification. Consideration has been given to methods by which the above shortcomings associated with ISAR imaging can be resolved in part. A slantrange to cross-range scale factor can be estimated for small-integration-angle data from 435
436
a priori knowledge of target shapes and expected pitch, roll, and yaw rates. The target aspect rotation rate, and thus the slant-range to cross-range scale factor, can be estimated from target track data for those ISAR images produced by tangential motion of the target relative to the radar, as in the case of straight-line flying aircraft targets. When data is collected over large viewing angles, scale factor can be determined as part of the focusing process, as described in the previous chapter. The image plane and true aspect angle of imaged targets can be inferred in some cases from the target image itself or from its behavior in time during several image frame times. Finally, the relatively long target dwell time requirement may be lessened by employing super-resolution processing techniques, which obtain cross-range resolution Ar, corresponding to less than the dwell time T = A/(2o>Ar ) associated with the Rayleigh resolution. The target dwell time is also reduced for those applications in which it is practical to operate at shorter wavelengths. Three-dimensional (3-D) monopulse radar imaging, described in this chapter, entirely avoids all three of the problems cited above by generating images from monopulse sum and difference signals independently of the target's aspect motion. This concept as of this writing remains in the experimental phase. A major disadvantage with respect to ISAR appears to be limited range performance. The basic principle is the extraction of crossrange scatterer position from normalized monopulse error signals along an HRR profile of the target, as indicated in Figure 8.1. Figure 8.2 illustrates the general process. c
8.2 MONOPULSE THREE-DIMENSIONAL IMAGING CONCEPT Wideband monopulse radar processing makes it possible to measure the position of an isolated point target in two orthogonal dimensions of the cross range. When carried out at each resolved slant-range cell of a complex target, the result is a 3-D image of the target. Orthogonal cross-range dimensions of resolved scatterers are obtained from differential error signals produced in the azimuth and elevation channels of a monopulse radar. Early work at the NRL [1] demonstrated cross-range signatures of an aircraft obtained by using short-pulse waveforms. A stepped-frequency waveform with potential for imaging at longer ranges will be discussed in this chapter. Figure 8.3 is a generic block diagram of a 3-D monopulse radar using steppedfrequency waveforms. Amplitude-comparison monopulse processing is assumed. The bandwidth corresponds to that required for the desired slant-range resolution. Following the three-channel quadrature detection, the signals are digitized for processing into 3-D images. Two alternative types of stepped-frequency processing are indicated in Figures 8.4 and 8.S. In Figure 8.4, single bursts, each of n frequencies, are processed to form either one dimensional (1-D) profiles or 3-D images. Stepped-frequency sum-channel signals and error signals from the two difference channels are first corrected for target motion. Each burst of n echo signals from the three channels is then converted by the use of discrete versions of the Fourier transform into synthetic range profiles: sum signals into slantrange profiles and error (or difference) signals into profiles of error signal versus range.
437
Figure 8.1 Generation of cross-range error signals with HRR monopulse radar.
Error signals in both channels are then normalized, range cell by range cell, by the sum signal to produce bipolar cross-range position data. A 3-D display format is illustrated in Figure 8.6. Cross-range positions of resolved scatterers in each slant-range cell are displayed in azimuth and elevation. A single display point is shown. The 3-D image can be displayed isometrically on a conventional twodimensional (2-D) display using conventional processing techniques. Generic block diagrams for a pulse-compression version of a 3-D imaging radar and the required processing would be somewhat simpler. The processing in Figure 8.4 will result in false cross-range estimates of scatterer position where two or more scatterers remain unresolved in a slant-range cell. The resulting error signal in such a slant-range cell will be that produced by the effective phase center
439 1
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ELEVATION CROSS RANGE 4 i
ONE ELEVATION RANGE/CROSS-RANGE - ADDRESS; CONTAINS CROSS-RANGE POSITION ' AND INTENSITY
AZIMUTH CROSS RANGE Figure 8.* One frame of a 3-D image display (from one burst of n frequency steps).
of both (or all) scatterers in the cell, and generally will vary widely about the position of whichever scatterer has the strongest instantaneous signal, as a result of interference between scatterers with varying phase angles, and will,generally not correspond to the position of any one scatterer. This problem can be overcome in part, while also achieving improved range performance, by the use of coherent processing of multiple steppedfrequency bursts, as shown in Figure 8.S. Here, an additional Fourier transform is carried out, similar to that used for stepped-frequency ISAR processing, to separate scatterers in Doppler as well as in the slant range and cross range. Multiple-pulse processing would achieve the same result with a pulse-compression version. The Doppler frequency, as in ISAR, is produced by target aspect rotation relative to the radar. Processing and display is suggested-in Figure 8.7. Multiple Doppler cells are shown to exist for each slant-range cell. Therefore, if multiple scatterers exist in a single range cell, they may be separable' in terms of Doppler frequency, and therefore displayed separately. The capability of the multiple-burst processing of Figure 8.5 to resolve scatterers is a function of the target aspect motion that occurs during the data collection time. Thus,
443
ONE RANGE/CROSS-RANGE DOPPLER CELL CONTAINS CROSS-RANGE OISTANCE ANO BACKSCATTER INTENSITY
DIFFERENTIALOOPPLER INDEX
* • SLANT-RANGE INDEX
' AZIMUTH CROSS RANGE
DIFFERENTIAL DOPPLER INDEX
Figure 8.7 One frame of a 3-D image display (from N bursts of n frequency steps).
(he advantage of reduced image frame time associated with monopulse processing as compared with ISAR processing is traded off. Tables 8.1 and 8.2 define terms used in Figures 8.4 and 8.5. respectively. Concept details for the stepped-frequency type of 3-D image processing illustrated in Figures 8.3 through 8.5 are given later in this chapter. 8.3 RANGE PERFORMANCE High-resolution monopulse processing makes it possible to measure cross-range positions of target scatterers that are resolved in the slant-range dimension. The position of resolved scatterers is measured in two orthogonal cross-range dimensions. Measurements are rela-
444
Table 8.1 Symbol! Used in Figure 8.4* Symbol
1(0 2'(0 6.M
KM) tUD
&M0
/-»' 1(0
a«(0
A*(0 r<(0 |X(0l
Definition Sum signal echo al frequency step /. Velocity-corrected turn signal at frequency step i. Azimuth error signal at frequency step i. Velocity-corrected azimuth error signal at frequency step i. Elevation error signal at frequency step i. Velocity-corrected elevation error signal at frequency step i. Transform from frequency domain to synthetic slant-range domain. Sum signal at Oh range position. Azimuth error signal at Ah range position. Elevation error signal at Ah range position. Azimuth cross-range (bipolar) amplitude in Ah range position. Elevation cross-range (bipolar) amplitude in Ah range position. Sum signal (magnitude) in Ith range position.
•All signals are complex digital values unless otherwise noted.
Table 8 J Symbols Used in Figure 8.S* Symbol
x«. *) I'tt « A«(i.*)
AMU)
/-»'
!-»/, Xtt*) A-tt*) 1(1/) AJif) IULJ) rMJ) rMJ)
Definition Sum signal echo at frequency step i, burst k. Velocity-corrected sum signal at step i, burst k. Azimuth error signal at step I, burst k. Velocity-corrected azimuth error signal at step i. burst k. Elevation error signal al step i. burst k. Velocity-corrected elevation error signal at step /. burst k. Transform from frequency domain to synthetic slant-range domain, burst by burst Transform from time-history domain at each synthetic range position to Dopplerfrequency domain. Sum signal al Mi range position of burst k. Azimuth error signal at Ah range position of burst k. Elevation error signal at Ah range position of burst k. Sum signal al Ith range position and Ah Doppler frequency. Azimuth error signal al Ah range position and jlh Doppler frequency. Elevation error signal at Ah range position and Ah Doppler frequency. Azimuth cross-range bipolar amplitude at Ah range position and .Ah Doppler frequency. Elevation cross-range bipolar amplitude in Ah range position and Ah Doppler frequency. Sum signal (magnitude) in Ah range position and All Doppler frequency.
•All signals are complex digital values unless otherwise noted.
445
live to scatterers in adjacent resolved slant-range cells. True resolution is not actually achieved in the cross-range dimension; that is, multiple scatterers in a slant-range cell (or range-Doppler cell) cannot be resolved in the cross range. However, the word "resolution" in quotes will be used for convenience to indicate ability to measure range-cell-to-rangecell differences, or spreads, in the cross-range scatterer position. Slant-range resolution in conventional radar and both slant-range and cross-range resolution in ISAR are independent of target range. For 3-D monopulse imaging, however, we will see that cross-range "resolution" of the target decreases with target range. In Chapter 7, ISAR image performance was estimated in terms of the fraction of resolved target elements that exceeded a radar cross-section threshold value. This threshold value was related to the ratio of the pixel signal to noise considered adequate for visibility. Pixel SNR was based on the radar range equation so that we could calculate image quality in terms of single-pulse SNR. This method is applicable when resolution itself is independent of target range so that the number of visible image elements constitutes the measure of image quality. "Resolution" in the cross range for monopulse 3-D imaging, however, is a function of target range. As target range increases, the difference-channel signals, which are measures of cross-range angle positions of resolved target reflection points, become smaller. The ability to measure changes in scatterer position will degrade as the corresponding changes in signal voltage approach the level of receiver noise. It is, therefore, more appropriate to estimate image quality directly in terms of cross-range "resolution." Therefore, expressions will now be developed for 3-D target imaging range in terms of cross-range "resolution" as defined above. The SNR referred to the input of the sum channel of a monopulse radar, produced by a resolved target scatterer at range R of radar cross section
kT 0.FL o
where: St Vi N G
= sum-channel received power out of the comparator; •
= sum-channel received voltage (rms);
= thermal noise power; = antenna gain; A = wavelength; P, = transmitted-pulse power; L = radar system and propagation losses; k = Boltzmann's constant; To = standard noise temperature (290K); receiving system noise bandwidth; A = receiver input impedance; = z„ F = radar system noise figure.
(8.1)
446
For a circular monopulse antenna of half-power beamwidth aSa, an error slope K. may be defined as the ratio of normalized difference-channel error voltage VJVi out of the comparator to normalized offset angle 4>J4h . Thus, m
„_.yjy
.
s
(8 2)
where V is the rms voltage out of one of the difference-channel comparators. The error slope defined in this manner is quite independent of antenna beamwidth. Practical values for K„ based on Barton [2] and others, vary from about 1.2 to 2.0. For K = l.S, the rms error voltage, from (8.2), is 4
M
V -1.5-^-V 4
(8.3)
x
The angular offset in radians from boresight produced by a target scatterer at cross-range displacement y at range R is & =£
(8.4)
By using an approximation given by Jasik [3] applied to parabolic reflectors, we obtain
The rms error voltage out of the comparator from (8.3) with (8.4) and (8.S) then becomes v/ G \ V
1J
w
'- 5(i32)*
(86)
The difference in rms error voltage dV produced by a shift dy in the scatterer crossrange position is A
The difference-channel input signal power associated with a cross-range shift dy is then
447
c _«WA) -
J
04
= 2.25 The SNR referred to the input of the difference channel, produced by a scatterer's small but finite cross-range shift Sr, from (8.1) and (8.8), is
(8.9)
It is convenient to obtain a range expression written in terms of antenna diameter rather than gain. For an antenna aperture A, the gain from (2.4) is
G = -JT
(
8 | 0
>
. If we assume 50% antenna aperture efficiency and an antenna diameter of D„, then the effective area of the antenna is
The antenna gain of (8.9) with (8.11) then becomes „ G
4TT l/wDh =
X
ir'Dl
^ 2-(—)"W
8,2
< >
We will assume that targets to be imaged are made up of scatterers that have physical dimensions equal to their cross-range separation Sr. The estimation of the radar target range at which scatterers can be separated will be based on the radar cross section of two ideal types of passive reflectors: spherical and flat-plate. In other words, the target is assumed to be made up entirely of scatterers, which are either spherical reflecting surfaces or flat plates facing the radar.
448
Spheres have the smallest radar cross section for their size of any of the simple geometric shapes. Ideal nonspherical reflectors produce relatively high specular backscattering, which increases with the ratio of their physical dimensions to wavelength. The effect is not expected to occur for most real targets, however, because their normal curved shapes would not present flat regions to the radar over more than a few wavelengths in extent. We therefore conclude that the assumption of spherical scatterers, while appearing conservative, probably roughly represents the expected population of scatterers for real targets. The validity of this conclusion remains to be tested experimentally. The radar cross section of a conducting sphere with diameter Sr, for Sr > A, is [4]
•J-*?* For specular return from a flat plate of size SrxSr,
(8-13) the radar cross section is [4]
< /=—\T-
<- >
R
8
I4
The SNR, referred to the input of the difference channels and produced by a cross-range shift fir of a spherical scatterer at range R, is expressed from (8.9) in terms of the antenna gain from (8.12) and the radar cross section of a conducting sphere from (8.13) as
(a-
j '(f)"
m^h^ kToB.FL
(8.15)
Equation (8.15) was derived for a single scatterer that changes its cross-range position by S. The same expression will be assumed to represent the SNR:
((S.),-W/V
(8.16)
associated with any two range-resolved scatterers that are separated by Sr. The resulting range to obtain "resolution" Sr for the assumption of spherical scatterers is
where (S/N)^ is the SNR referred to the input of each of the difference channels.
449
At this point, we have said nothing about the transmitted waveform. Range is of interest in terms of the SNR at the output of a receiving system matched to the transmitted waveform. Output SNR from either matched-filter difference channel is T 3.(S/N)^ for a transmitted pulse of duration T,. The radar range expressed in (8.17) for spherical scatterers, when written in terms of output signal-to-noise ratio S/N of a matched-filter difference channel, becomes lr
The range equation (8.18) differs significantly from the conventional radar range equation for detection (2.47) in Chapter 2. First, the term taken to the one-sixth power in (8.18) contains parameters that formed a term taken to the one-fourth power in (2.47). The key parameters are radar transmitted-pulse power, transmitted pulse width, and required output SNR. To double 3-D imaging range requires an 18-dB increase in radar transmitted-pulse power, compared with 12 dB needed to double detection range. The comparison also applies for changes in transmitted pulse width and required output SNR. Thus, range performance is expected to degrade 6 dB per octave faster for 3-D imaging than for detection. Nonetheless, we can see that 3-D imaging performance in terms of range is directly proportional to antenna diameter while being nearly proportional to radar frequency and the desired cross-range "resolution" Sr.
8.3.1 Range Performance With Short Pulses and Chirp Pulses Equation (8.18) expresses range at which the output SNR produced in the monopulse difference channel by cross-range scatterer separation Sr is equal to S/N for a single transmitted pulse of duration 7*,. Before applying (8.18) to the stepped-frequency processing outlined in Figures 8.3, 8.4, and 8.5, consider range performance using short pulses and chirp pulses. Range performance by using (8.18) increases with the pulse energy P,T,. We obtain increased range resolution for uncoded short pulses that are peakpower limited at the expense of range performance because the pulse duration must be reduced in order to increase resolution. The problem is overcome with chirp-pulse waveforms (and other coded waveforms) because range resolution is determined by waveform bandwidth, not pulse duration. 8.3.2 Range Performance With Stepped-Frequency Waveforms Stepped-frequency waveforms have an advantage over both short-pulse and chirp-pulse waveforms in terms of range performance for monopulse imaging, because pulse-to-pulse coherent integration is inherent in the processing. The SNR is increased by a factor of n
450
following coherent processing associated with the frequency-to-time transformation of each stepped-frequency burst of n echo signals. Pulse-to-pulse coherent processing, in principle, could also be carried out with chirp-pulse waveforms, but A/D conversion becomes more difficult for wideband signals, as discussed in previous chapters. Burst-to-burst coherent integration with stepped-frequency waveforms provides additional processing gain after application of target TMC algorithms, which were discussed previously in connection with 2-D ISAR imaging. Motion-correction inputs for 3D monopulse processing could be derived by motion solutions obtained for the relatively high SNR range profiles produced from sum-signal data using techniques described for ISAR. The resulting phase corrections would be applied to the error signal from each step of each echo burst in the two relatively low signal level difference channels, as shown in Figure 8.5. Burst-to-burst integration, carried out by the second (discrete) Fourier transform in the difference channels, then produces an SNR improvement factor equal to the number N of integrated bursts. The SNR referred to the input of the sum channel, expressed in (8.1) for each pulse, exceeds that of the difference channel, as given in (8.9), by (SvN)x
(8.19)
The SNR referred to the input of the sum channel associated with an S/N at the output of the difference channel for N bursts of n steps is S/N
(8.20)
As an example, consider the hypothetical spaceborne radar parameters of Table 8.3, for which 128 bursts of 128 steps are integrated coherently. The input SNR in the sumsignal channel for a required SNR of 10 dB in the output cross-range data is calculated from (8.20) and parameters from Table 8.3. Results are indicated in Figure 8.8. Except for very low "resolution" performance (extreme radar target range), there appears to be an ample SNR of the sum signal for target detection, tracking, and application of motioncorrection algorithms required for burst-to-burst integration. Furthermore, the sum SNR expressed in (8.20) is that for a single scatterer for which a difference-channel signal output is obtained. A typical target consists of multiple spherical reflecting points, which will produce a sum SNR that can be expected to be higher than that expressed in (8.20) by a factor of roughly Wcr, for average target cross section a and conducting sphere cross section a,.
s1
S q
o
^ § 2 -? S 8 S
„
m
» q K
o F^l OO O
O
« i + — N
o N
o
_
,
2 2 •» ''l
^ 8 8 CJei ^
X
n m
s
-< t£ -J e < % !*.
-o 3 1
8 -e S £ -3 o S c e
6v
s i .c « 5 y
s
J
«
> -a
5 £2
«J Si
H
a.
co
09
to O £
0.
5j
i
A B
*
o
452
CROSS-RANGE "RESOLUTION" (It) Figure 8.8 Sum-signal SNR versus cross-range "resolution" for hypothetical spaceborne radar operating at I difference SNR of 10 dB.
Equation (8.18), for a single-pulse imaging range, can be expressed for coherent processing of N bursts of n pulses of velocity-corrected stepped-frequent data as
'•=°-m'°(e8^)" 8 3 3 Range Performance Assuming Flat-Plate Scatterers Equations (8.18) and (8.21) apply for the assumption that the target consists of spherical scatterers. The range R for specular return from flat-plate scatterers is f
(8.22) Then, with a, and
/4«(c5r)V*
453
As an example of the improvement in range performance for the assumption of flat-plate scatterers over that for the assumption of spherical scatterers, the calculated ranges from (8.21) and (8.23) for the tracking radar (parameters listed in Table 8.3) are as follows.
Similar results are obtained by using parameters for other radars listed in Table 8.3. 8.3.4 Range Performance Calculation Examples Range versus cross-range "resolution" was calculated from (8.21) for the parameters of each of the four radars of Table 8.3. These radars are representative of three generic types of military radars and a hypothetical spaceborne radar. Plots are shown in Figure 8.9. Stepped-frequency waveforms were assumed in each case. Stepped-frequency monopulse radars could generate 3-D images while operating in their tracking modes. A shipboard or ground-based radar would be required to acquire the target and then maintain target tracking. The target tracking function would be performed with a separate channel of the same stepped-frequency data being processed into imagery. An airborne radar or missile-seeker radar with a track-while-scan (TWS) mode would'probably need to switch to a track-only mode on a single target of interest. The spaceborne radar would identify assigned space or air targets from a space platform. Targets of interest would be acquired and then tracked for up to several seconds to obtain the image. 8.4 CONCEPT DETAILS FOR STEPPED-FREQUENCY APPROACH Figure 8.10 illustrates the stepped-frequency waveform described in Chapter 5 for synthetic processing. Frequency is changed from pulse to pulse in uniform frequency steps over a burst of frequencies / to/,.,. The frequency of the reference signal to the first mixer of Figure 8.3 is also changed from pulse to pulse to produce a fixed narrowband IF signal in each of the three channels. The quadrature mixer then mixes the three IF signals with a coherent reference at the same IF to produce baseband outputs. These outputs are sampled and digitized, and then processed, as in Figures 8.4 and 8.5, into outputs suitable for 1-D. 2-D, or 3-D display. The digital processing indicated in Figures 8.4 and 8.5 includes target motion correction, DFTs, and phase comparisons to produce sum-anddifference-channel range profiles. The square of the output magnitude of the sum signal 0
454
CROSS-RANGE " R E S O L U T I O N " (ft) Figure 8.9 Monopulse radar target range versus cross-range "resolution" (assuming spherical scatterers).
in a given synthetic range cell, assuming system linearity, is proportional to the radar cross section of the resolved scatterer in that cell. The output magnitude of the difference signal in a given synthetic range cell, after normalization by the sum-signal output in that cell, is proportional to the scatterer cross-range distance in that range cell. Normalization is carried out in the digital phase comparator described below. We will discuss the detailed operation of the stepped-frequency 3-D image process for a four-feed-hom monopulse radar from the viewpoint of an image produced by echo signals from a single point target, illuminated by a single burst of n pulses, stepped in frequency from f to/.-,. The target signal levels from either the azimuth or elevation pair of an amplitude-sensitive monopulse antenna feed are equal when the target is on boresight. Offset antenna patterns produced by two of the four feeds of a monopulse antenna ate illustrated in Figure 8.11. Difference patterns as seen at baseband are illustrated in Figure 8.12. Slightly off boresight, the RF signals as seen at the output of feeds 1 and 2 will be unbalanced, as illustrated in Figure 8.13(a). The cross-range distance of the target from a
455
Figure 8.11 Idealized antenna patterns from each of a pair of feeds feeding a single reflector.
456
AZ
•
Figure 8.12 Monopulse antenna difference patterns seen at baseband from two pairs of feeds.
boresight is proportional to the amount of imbalance. The direction, left or right, is indicated by the difference-channel error signal phase, 0 or w, relative to the sum signal, as indicated in Figure 8.13(b). The synthesizer in the block diagram of Figure 8.3 generates the series of steppedfrequency bursts of pulses of Figure 8.10, a stepped-frequency reference signal, and a fixed IF reference signal. All three signals are assumed to be generated by multiplying up from a fixed master oscillator, and are therefore all coherently related. The resulting RF sum signal and two difference signals (X, A , and Ad of Fig. 8.3) from the monopulse comparator at each frequency step are the RF signal sum of all four feed antennas and the difference RF signal from each of the two orthogonal pairs, respectively, at each frequency step. u
457
RIGHT OF BORESIGHT 1
LEFT OF BORESIGHT
2
Figure 8.13 Monopulse waveforms at one frequency for echo from point large! to the right and left of boresight (single cycle of echo from one pulse is shown): (a) outputs from feeds I and 2; (b) sum- and difference-channel comparator outputs.
The phase of each of the two difference-channel signals, as stated above, will be either 0 or IT rad, relative to the phase of the sum signal. The three channels of echo signal are mixed to a suitable IF, and then mixed again to baseband. (Although how at baseband, these signals are still symbolized as X , A„, and A in Fig. 8.3.) The signals are sampled and then converted to three channels of complex digital data, which are symbolized by £(/), A (i), and A,I(I), where / denotes the frequency-step index. This completes the RF processing and data sampling. Figure 8.4 indicated the required digital processing. The first digital process is target TMC, the methods for which were discussed in Chapter 7. Motion-corrected data are designated X (i), AL(0. and AJi(i'). The quadrature mixer output in the sum channel and in either of the difference channels, respectively, at frequency step i for a point target to the right or left of boresight at time-dependent delay tit) is expressed as d
u
c
S(i) = |X(i)|e A(/) = ±|A(i)|e-*•*•*«>
( 8
'
2 4 )
with the plus sign applying to the point target when right of boresight, and the minus applying to the point target when left of boresight (for the azimuth channel). For velocitycorrected signals, we have
458
2R
2vl
2R
(8.25)
The magnitudes |I(i)| and |A(i)| in (8.24) refer to the sum and difference signal magnitudes, respectively. The sum-channel and the azimuth difference-channel responses for one of the frequency steps from a burst of n stepped frequency pulses is illustrated in Figure 8:14. The symbols A(i), and A(i). indicate the difference response that would occur at that frequency for the target at each side of boresight, left or right, respectively. Next, the resulting range profiles 2.(1), A„(/), and A^/) for each of the sum and difference channels are generated in the same way as for single-channel burst data. By using the 1DFT, from (5.11) of Chapter 5, the sum and difference range profiles of the point target at range R, respectively, become
(8.26a)
(8.26b)
| !
A
- f » 0 (target right of boresight) " (target left of boresight) £
T A ~ T X
b
Q
Figure 8.14 Sampled turn and difference signals shown for one frequency of an n-slep burst (azimuth channel).
459
where I is the synthetic range increment and T = IRIc. This result is illustrated in Figure 8.15 for outputs occurring in one difference channel when the target being tracked produces a single large response at / and one smaller response. The two complex difference signals A(/) at each range position / are divided by the complex sum signal 2(/) at that range position. The result is a bipolar amplitude proportional to target distance from boresight, independent of target size. An explanation of this final process follows in terms of the phase comparison between sum and difference rangeprofile responses to a single point target. The output r(7) of the digital phase comparator for either the azimuth or elevation channel is obtained by dividing (8.26b) and (8.26a); hence. 0
(8.27)
For a point target at range delay r, a peak response occurs for some range element /o in the n-element synthetic sum and difference range profiles. The peak occurs for the same argument in each exponential of (8.27). For the assumption of a point target, the echo amplitudes are identical at each frequency, so that |2 (i)l = | I | and |A (i)| = |A|. The phase comparator output expressed in (8.27) at / then becomes c
c
0
Figure 8.15 Synthetic sum and azimuth difference-channel signals (shown for a target producing two responses).
460
r('o) = ±Tvi
(8.28)
As we can see from (8.28), Ihe magnitude of the difference response |A| is normalized by the magnitude of the sum response In terms of the received rms voltages V and Vx from the comparator, (8.28) then becomes 4
r(/ ) = ± V A i 0
(8.29)
where, as before, positive and negative polarities, respectively, result when the resolved scatterer is to the right and left of boresight. Equation (8.28) can be expressed directly in terms of error slope and normalized offset angle. From (8.2), we have (8.30) Thus, we can see that the output of the digital phase comparator produces a signal r(/ ), which for a given scatterer is proportional to scatterer offset angle from boresight rf> , independent of scatterer size. 0
4
8.5 SUMMARY The monopulse 3-D imaging concept provides a possible means for generating target images dimensioned in two orthogonal cross-range dimensions versus slant range. HRR processing, as for pulse-compression or stepped-frequency processing, can be used to resolve targets in the slant range. Monopulse processing may then be used to measure the cross-range position of resolved scatterers in two orthogonal components of the cross range. The use of stepped-frequency waveforms provides convenient pulse-to-pulse coherent integration. This may make it possible to estimate the cross-range positions of target scatterers from the relatively weak angle-sensitive (error) signals in the difference channels of a monopulse receiver at useful target ranges. The relationship of range resolution and range-window size to radar pulse width, step size, and other parameters for stepped-frequency 3-D monopulse imaging radar is identical to that for 2-D steppedfrequency ISAR imaging radar. In synthetic 3-D monopulse processing, as for synthetic ISAR processing, successive stepped-frequency echo bursts of data are converted by the use of the DFT to obtain synthetic range profiles. Three channels are required for 3-D monopulse processing: a sum channel and two difference channels. Early results of 2-D monopulse imagery obtained from a short-pulse radar developed at the NRL (1] are shown in Figure 8.16. Preliminary results using a 3.2-GHz steppedfrequency wavelhrm, shown in Figures 8.17 and 8.18 for a small craft at sea, were obtained at the NOSC in 1980. Advantages and disadvantages of 3-D monopulse imaging are summarized below.
461
WV-2 SUPER CONSTELLATION
Q _l 0. S RANGE VIDEO
R " 1
'-yl
ANGLE VIDEO
x (3 rr
/
/ £. _
CORRESPONDING ZERO ANGLE ERROR VOLTAGE TIME
Figure 8.16 2-D slanl-range and cross-range data obtained from an aircraft in flight with an HRR monopulse radar using a short-pulse waveform. (From (11. p. 753. Reprinted with permission.)
8.5.1 Advantages • • • • • •
Target aspect change is not required. Unambiguous cross-range scale factors are achieved. A third dimension of target information is generated. Three-dimensional capability can augment ISAR imaging for a monopulse radar. Short image-frame time is possible (at high SNRs). Compatibility with the normal monopulse tracking function is possible.
462
464
8.5.2 Issues • • • •
Good image quality is yet to be demonstrated as of this writing. Range limitations are likely. Target acquisition may be a problem. Required antenna characteristics may be severe (tolerances, bandwidth, and motion stability). • Multiple unresolved scatterers within individual range cells may distort the image. • Millimeter-wave frequencies may be essential.
8.53 Potential Applications Three potential future applications are: • Airborne intercept (Al) radar for target confirmation; • Shipboard fire control for radar target confirmation; • Space and airborne radar for ship and aircraft identification. Target imaging using monopulse processing is not as yet known to have progressed beyond an experimental stage. Eventual applicability to military or nonmilitary users is speculative. PROBLEMS Problem 8.1 (a) At what range in nautical miles will the tracking radar of Figure 8.9 operating in a 250-MHz chirp-pulse mode obtain 5m of cross-range "resolution" based on one pulse? (b) What would be the range if 256 chirp pulses were coherently summed to form the error signal? Assume all other parameters remain as in Table 8.3. Problem 8.2 1
Two point targets, each having a radar cross section of 0.1 m , are separated by 5m. What is the output SNR produced by the difference channel of the missile seeker of Table 8.3' at 2-km range by processing one stepped-frequency burst if the point-target separation is cross-range to the radar LOS? Problem S3 1
A ship target having a radar cross section of 2,000 m is 90m in length by 18m in width. The target is to be imaged from a broadside aspect using the monopulse 3-D imaging
465
method, (a) Based on (2.21) of Chapter 2, what is the average range-resolved radar cross section for a 3m slant-range resolution cell? (b) The target is observed over an azimuth viewing-angle change sufficient to produce 3m (ISAR) cross-range resolution. What is the corresponding radar cross section in the resulting 3m-by-3m ISAR resolution cell using the same equation? (c) What is the cross section of a range-resolved spherical reflecting surface of 3m diameter assuming that wavelength is much less than 3m? (d) What is the SNR out of the difference channel produced by a change of 3m in the crossrange position of resolved scatterers in (a), (b), and (c) above, using the missile-seeker parameters of Table 8.3 at 3 nmi (5,556m) from the target? Assume one burst of 128 pulses for (a) and 20 bursts of 128 pulses for (b) and (c). Problem 8.4 Monopulse processing is one method of increasing radar-target angular location accuracy over that associated with the half-power beamwidth of the radar antenna. Sometimes the term beam-splitting is used to refer to splitting the radar main-beam response to a target into fine increments of angular position. A beam-splitting ratio can be defined as the ratio of half-power beamwidth to angular accuracy following beam-splitting, (a) What is the beam-splitting ratio obtained by the airborne intercept radar of Table 8.3 at 10 nmi (18.5 km) when cross-range "resolution" is lm? (b) What is the beam-splitting ratio associated with an absolute tracking accuracy of ±1 mrad for the same radar at the same range in a conventional narrowband angle-tracking mode? REFERENCES (1) Howard, D. D., "High Range-Resolution Monopulse Tracking Radar," IEEE Trans. Aerospace and Electronic Systems, Vol. AES-I I, No. 5. Sept. 1975, pp. 749-755. [2] Barton, D. K., Radar System Analysis, Dedham, MA: Artech House, 1979, p. 275. (3) Jasik. H., Antenna Engineering Handbook, New York: McGraw-Hill, 1961, p. 12. (4) Jasik, H., Antenna Engineering Handbook. New York: McGraw-Hill. 1961. pp. 13-10.
Chapter 9 Target Imaging With Noncoherent Radar Systems 9.1 COHERENCY REQUIREMENTS FOR TARGET SIGNATURE PROCESSING Pulse compression, synthetic range-profile generation, and imaging, as discussed up to this point, have implied the use of some type of coherent power-amplifier transmitter. In our discussion of pulse compression, we assumed intrapulse coherence. Intrapulse coherence here means that the waveform phase is preserved through the transmitter. In this way, signals for each echo pulse can be compressed on the basis of known phase characteristics of the transmitted pulse. In our discussion of synthetic HRR, interpulse coherence was also implied in that a stable radar master oscillator (RMO) existed, from which RF transmission and LO signals were generated, and to which were referenced the PRF and echo sample delay triggers. A radar with interpulse coherence defined in this manner can measure the RF phase difference between the transmitted and echo pulses. We had assumed that echo phase was conveniently measured relative to the RF signal input to the transmitter, thus taking advantage of phase coherence through the transmitter. Thus, up to this point, we have assumed that intrapulse and interpulse coherence, when required, were maintained through the radar transmitter by using RF power amplifiers to amplify the transmitted signal waveform. In this chapter, the possibility of maintaining the required coherence with noncoherent transmitters will be examined: first, briefly, for pulse compression requiring intrapulse coherence; and then in more detail with respect to pulse-to-pulse coherent integration for synthetic range-profile generation and ISAR image processing requiring transmission-to-reception interpulse coherence. Pulse-compression systems require intrapulse coherence. This is difficult to achieve without a coherent power-amplifier transmitter. However, a power oscillator, such as a magnetron transmitter, in principle, could be designed to generate a frequency-coded pulse, such as a chirp pulse, the echo from which would be compressed by a matched 467
468
filter in the radar receiver. This would require some means for intrapulse control of a magnetron-oscillator output frequency of sufficient accuracy for useful pulse compression. Achievement of this capability in practice is not known to exist. Synthetic range-profile processing requires coherence from transmission to reception because the technique depends on frequency-stepped measurements, pulse by pulse, of the target echo's relative magnitude and range-delay phase. The received echo amplitude and phase can be measured relative to the transmitted pulse at each frequency by quadrature detection at baseband. The sampled set of complex values for each burst, representing echo amplitude and phase at each frequency step, is the target's reflectivity sampled in the frequency domain. Synthetic range profiles are generated, one from each burst, from these samples. We can see that for synthetic range-profile generation, it is transmissionto-reception coherence that is required, not intrapulse coherence. The set of complex samples representing amplitude and phase in each resulting synthetic range cell, taken in time history, is the discrete form of the time-domain response of target scatterers in that range cell. The processed two-dimensional distribution of Doppler spectrum magnitudes versus synthetic range cell and Doppler frequency is the (unfocused) range-Doppler ISAR image. A range-Doppler image is generated, as described in Chapter 7, by coherent profile-to-profile processing, which requires burst-to-burst coherence. As viewed another way, synthetic processing fundamentally depends on measurements of the pulse-to-pulse changes in the phase and amplitude of target reflectivity with time and frequency. Therefore, a radar using a pulsed power oscillator as a transmitter, while possessing no means with which to retain coherence through the transmitter, can perform coherent interpulse processing if the phase of the received signal is measured relative to the transmitted output pulse, rather than to the input pulse waveform, as is done for convenience in the case of radars using power-amplifier transmitters. Thus, it is conceptually possible to carry out synthetic generation of target range profiles, images, and SAR maps with a noncoherent radar if its power-oscillator transmitter can be made to be frequency-agile over adequate bandwidth. We will therefore see that an important consideration is the required precision of step size. The most familiar example of a noncoherent radar system is probably that of a pulsed magnetron radar operating at microwave frequencies. The magnetron for these radars oscillates at microwave frequencies upon application of a high-voltage video pulse, and continues to oscillate for the duration of the video pulse, thus producing a high-power RF transmitted pulse. The starting phase and oscillation frequency vary randomly from pulse to pulse. Magnetrons can be made to preserve phase coherence by injection locking, in which a lower power RF pulse is injected just before and during the high-voltage video pulse. This technique, however, has a disadvantage for high-power radars in that a second, moderate-power coherent amplifier is required as the driver. Injection locking, therefore, is not commonly found in long-range surveillance and target-tracking radars. However, for most military applications, there remains the possibility of carrying out target-signature processing, including ISAR imaging and SAR mapping, with noncoherent but frequencyagile pulsed magnetron radars by measuring echo phase relative to transmitter output.
469
This concept will be discussed in more detail in this chapter, starting with background information on frequency agility and coherent-on-receive methods for nonimaging radar surveillance functions. 9.2 FREQUENCY-AGILE AND COHERENT-ON-RECEIVE RADARS Frequency agility generally refers to the ability to change the radar transmission frequency from pulse to pulse by an amount equal to or greater than the pulse bandwidth. Typically, the pulse-to-pulse shift is 2 to 5 MHz, with total excursions of up to 500 MHz. The advantage, from the viewpoint of surveillance, is improved ECCM or detection performance, as will be discussed in Chapter 10. Frequency agility in coherent radar systems is achieved by generating a series of transmission and LO signal frequencies arranged such that a fixed IF exists, regardless of the transmission frequency. In this way, the radar's receiver bandwidth remains relatively narrow with a relatively wide frequencyagile transmitter bandwidth. Magnetrons, however, are oscillators. Thus, frequency agility for magnetron radars is achieved by rapid tuning to change the oscillation frequency. This has been achieved in several ways. The most common method is to dither the magnetron frequency by an electronically driven tuning plunger, which is part of the magnetron resonance structure (see Fig. 3.30 of Chapter 3). Magnetron radars, because they are noncoherent, require the receiver to acquire the transmitter frequency at each pulse. To do so, a new LO frequency must be generated within a few microseconds following each transmitted pulse. The LO frequency for each pulse would ideally be the frequency that resulted in IF echo signals centered in frequency at the band center of the IF amplifier. The minimum requirement is that the resulting IF signals for each pulse should fall somewhere within the radar's IF bandwidth. This is achieved in frequency-agile magnetron search radars by applying a frequency readout voltage from the dither tuning system, at the time of each newly transmitted pulse, for coarse setting of a VCO. A block diagram is shown in Figure 9.1. The coarse frequency setting is followed by automatic frequency control (AFC) to acquire the frequency of the transmitter. The VCO setting is then held during the interpulse period to serve as the LO until the transmission of the next pulse. The term tracking local oscillator (TLO) has been used to describe the circuitry that acquires and holds the frequency of each transmitted pulse. We should note that the TLO tracks the carrier frequency of the transmitted pulse, not the phase. A dithered frequency-agile magnetron radar with TLO tracking, therefore, does not allow measurements of target signal phase. Furthermore, the resulting transmitted pulses, while spread in frequency, are not stepped with sufficient precision in frequency for synthetic HRR processing. Requirements for HRR, SAR, and ISAR are thus not met by conventional frequency-agile radar techniques using dithered magnetrons and a TLO. (Frequency-agile radar techniques are discussed in more depth by Barton [1].) The desired coherence between transmitted and received pulses, while not achieved for conventional frequency-agile magnetron radars, has been possible for many years with
470
Z3
a.
PS SO
471
fixed-frequency magnetron radars. The technique is called coherent-on-rec\eive. This refers to the method of achieving phase coherence between transmitted and received pulses with noncoherent radar systems. The technique was developed during World War II [2] to provide moving target indication (MTI) with magnetron radars. Suitable power amplifiers were not available during that lime at the microwave frequencies required for many surveillance applications. The coherent-on-receive radar illustrated in Figure 9.2 works as follows: a coherent (local) oscillator (COHO) is phase-locked pulse to pulse to the difference in frequency between the coupled transmitted pulse and a stable local oscillator (STALO). The COHO continues to oscillate at this frequency during the interpulse interval associated with each transmitted pulse. The COHO is then relocked to the next magnetron pulse. Received signals for each transmitted pulse are mixed with the STALO to produce an IF signal, which is amplified and then mixed with the COHO in a quadrature mixer to baseband. The coherent-on-receive capability can be understood by noting that both the COHO signal for each transmitted pulse and the resulting received signal (from a fixed target) at IF are down-conversions from the same two signal sources, the magnetron and STALO. The instantaneous target signal phase at IF relative to that of the COHO is ideally determined only by the target's instantaneous reflectivity at the transmitted pulse frequency and its instantaneous delay. Deviation of the target signal frequency at IF from the COHO frequency is ideally determined only by the target's radial velocity relative to the radar. The MTI is provided by a pulse-to-pulse cancellation loop, a simple form of which is illustrated in Figure 9.2. Historically, then, magnetron radars have existed for a considerable time that are either frequency-agile or coherent-on-receive, but not both. Both features are required for synthetic HRR and image processing. At this point, our discussion will proceed to the techniques that are expected to be required to achieve a stepped-frequency, coherent-onreceive capability when using the currently available pulsed magnetron-oscillator designs. Then we will assess the required magnetron frequency control accuracy as a function of target size and desired signature fidelity.
9.3 STEPPED-FREQUENCY MAGNETRON IMAGING RADAR The concept for imaging to be discussed here is based on a technique to measure the relative phase and amplitude of the series of target responses produced by transmitting precisely controlled frequency-stepped bursts of magnetron pulses. The pulses from the magnetron are controlled so that their frequencies approximate / = / + iA/, i = 0, 1,2, . . . . n - 1, for the ith pulse of each burst, where A/ is a fixed step size. Two possible methods to achieve magnetron frequency control are (1) by controlling the trigger time of frequency-dithered magnetrons, and (2) by rapid tuning of electronically controlled magnetrons. The feedback technique for control used in each method would employ a frequency discriminator to produce a frequency-sensing voltage for each frequency step. A precise clock, generated from the base frequency of the stepped-frequency source. 0
472
473
serves as a reference to the range tracker, which establishes delay positions for sampling transmitted and received pulse pairs. Transmitted and received pulses are sampled at a baseband frequency produced by heterodyning down from their respective carriers using reference LO signals obtained from a stepped-frequency source. Target signal phase at each frequency step is obtained as the difference in phase between each sampled transmitted pulse and the corresponding sampled received pulse. Resulting time histories of complex signal data are processed to generate target range profiles, SAR maps, and ISAR images. Three-dimensional stepped-frequency monopulse images of targets could be generated as well. The concept is a means to achieve coherent processing of stepped-frequency data collected from noncoherent magnetron radars for which the starting time, phase, and frequency of the transmitted pulse are not known precisely. An earlier approach for obtaining high range resolution from magnetron radars is described by Ruttenberg and Chanzit [3]. In the Ruttenberg-Chanzit approach, a frequencyagile, coherent-on-receive method is used whereby echo signals are summed in a recirculating delay line of delay equal to the radar's PRI. The summed output is a high-resolution response.
9.4 RESPONSE TO A SINGLE FIXED-POINT TARGET The block diagram of a conceptual coherent-on-receive, stepped-frequency radar appears in Figure 9.3. Waveforms are given in Figures 9.4 and 9.5. The frequency of the magnetron /«, for each pulse is controlled to approximate the desired frequency / for each step. Frequency control will be briefly discussed later in this chapter. The phase of the echo pulse is measured at each frequency step from i = 0 to / = n - I of each burst as the phase of the sampled quadrature-detected transmitted pulse subtracted from that of the sampled quadrature-detected echo pulse, the latter at delay 2R/c for a target at range R, where c is the propagation velocity. Quadrature-detected outputs for nonmoving targets are not at zero frequency, as for a coherent transmitter, except for the unlikely situation where/., = / . . A fixed-range target is assumed in the following analysis, whereas velocity correction would be required for moving targets. Transmission and echo phase are obtained from the two in-phase and quadrature sample pairs produced by two sample gates, which occur at S, and S sec, respectively, from the main trigger pulse. The sample gate at S, within the transmitted pulse duration occurs T, sec after the leading edge of the transmitted pulse. The sample gate at 5 occurs within the received echo pulse duration T sec after the leading edge of the echo pulse arrives at the radar receiver. The quantity is the delay from a start pulse to the leading edge of the transmitted pulse, produced by a magnetron at frequency step i. Subtraction of the transmitted phase from the received phase is carried out digitally by a digital mixer. The echo signal phase and transmission phase of the ith pulse pair for a point target at fixed range R, as seen by the quadrature detector at sample points S and S respectively, are 2
2
2
u
t
-UJF-O
475
TRIGGER PULSE i + 1
TRIGGER PULSE I PULSE REPETITION INTERVAL-
o . D
1) TRANSMIT
TRANSMIT AND ECHO PULSES AT RF
ECHO
.Villi SYNTHESIZER OUTPUT
—U
-a
TRANSMIT AND ECHO SIGNALS AT IF
\J
TRANSMIT AND ECHO SIGNALS OUT OF QUADRATURE DETECTOR
^
Figure 9.4 Waveforms and signals of sleppcd-frequency. coherent-on-receive radar.
Figure 9.5 Quadrature-detected transmitted pulse and echo pulse for a single fixed target a; range R.
476
(
— +T )\
+ 2rrf \S -[ m
m
1
(9.1)
and + lirf [S> - T J
(dS), = -lufA
(9.2)
m
The phase difference at frequency step / from (9.1) and (9.2) is fa = (•*!)> - (
J
(9.3)
Sample positions S, and S can be expressed, respectively, as 2
1R
S = + r t
T
+T
m
(9.4)
t
and S, = r , + 7,
(9.5)
m
i
so that Si-S^^
+ Tt-K
(9.6)
where T, - 7" is the sampling delay offset from transmitting to receiving. The measured echo phase relative to transmitted phase of the ith pulse from (9.3) with (9.6) becomes 5
fa = -2njlf^- + T,-T^
+ 2irUT< - T ) t
(9.7)
After rearrangement. 2R fa =
+
2TKT - W , - / ) 4
(9.8)
where / is the synthesizer frequency and /„ is the transmitted frequency, respectively, at each frequency step i.
477
9.5 RESPONSE FOR A RANGE-EXTENDED TARGET The output of the digital mixer for a nonmoving point target, from (5.10) of Chapter 5, can be expressed as the complex echo function G = A,c>*
(9.9)
(
for fa of (9.8) and where A, is the amplitude at frequency step i. With a radar having flat amplitude response observing a point target, A, = A for all i. A range-extended target composed of multiple reflection points will alter the resulting echo phase and amplitude so that the output from the digital mixer is G, = Be'* x Voie*'"
(9.10)
1
where yfo-fi'' is the target-dependent component of the echo transfer function (2.18) of the target at frequency step i. The quantity B, which includes the yfirrR quantity in (2.18), is a constant at a given target range. Thus, (9.10) for a range-extended target becomes d; = B^afi**'**
(9.11)
with phase ip, + 0, dependent on both the target's range and its instantaneous reflectivity at frequency step /'. By substituting /, from (9.8), we obtain G, = Byfit exp j [ - 2 7 r / y + 2it(T - mf A
-/) +
m
flj
(9.12)
where Byfa, is the relative amplitude at frequency step i. The target quantities yfa, and phase 0, are independent of range R. Figure 9.6 illustrates the generation of the transmission and echo phase quantities. Figure 9.7 tracks phase relationships throughout the entire process. 9.6 SYNTHETIC RANGE-PROFILE DISTORTION Distortion of synthetic range profiles, as we saw in Chapter 5, is produced by random deviation (error) from precise synthesized f r e q u e n c i e s i ' A / . Additional distortion is produced in a coherent-on-receive system by random fluctuations in magnetron frequency f . Phase error produced by magnetron frequency deviation f - / of (9.8) can be seen to be the deviation multiplied by 2ir(T - T ), where T, - T, < IRIc. In contrast, phase error produced by synthesized frequency deviation from/ is the deviation times 2ir(2Rlc) associated with the total range delay. Fortunately, for the practical situation where T m
m
t
}
t
T,
TARGET TRANSFER FUNCTION
R/c DELAY
R/c DELAY
TARGET TRANSFER FUNCTION
T,
2"lm|(l-»m|)
2R c DELAY
2*im|(l-'m|) + « i
2(if , |l-|?^ m
ACTUAL
R«
R«
+t ,)J m
EQUIVALENT
+ Hi
Figure 9.4 Functional block diagram and phase relationships for a stepped-frequency, coherent-on-receive radar.
T •* 2Rlc, tolerance to random pulse-to-pulse frequency fluctuations of the magnetron is much higher than to random pulse-to-pulse frequency fluctuations of the synthesized frequencies. Distortion due to random frequency fluctuation will now be analyzed. Synthetic range profile values H, at discrete range positions / of a single point target at range R are obtained from the series of n complex echo signals G, of (9.12) for frequency steps / = 0, I, 2 , . . . . n - 1. Using the IDFT of (5.11) given by }
(9.13) where, from (9.8) and (9.9) with A, = 1 for all i, we have 2R G, = exp jjj-2irj -2ir/,— + 2ir(T - W . 4
(9.14)
The synthesizer frequency / is stepped A/ in frequency pulse to pulse, so that /Wo+«"*./
f9.1.«
9.6.1 Analysis for Ideal System The expression (9.14) for sampled data from a point target collected from a steppedfrequency system, free of either magnetron frequency fluctuation or sampling offset, becomes
480
2
+'A/) ^j
G, = cxp j|-2ir(/o
(9.16)
where the exponent does not contain the random phase error term +2ir T t
i
- 7") x (/„ 3
B of (9.14). By substituting (9.16) into (9.13), we can show, as in Chapter 5, that the magnitude of the synthetic range profile of a point target is given by (5.19); sin try .
(9.17)
If
sin—y n where y== ^
+
/
(9.18)
This is the synthetic range profile of a single point target when frequency and sampling errors are zero. The effect of random frequency fluctuation of the frequency synthesizer on a target's range profile was found in Chapter 5 to be small at radar ranges of common interest for random frequency fluctuation associated with commonly available synthesizers. Next we will analyze distortion that occurs with the stepped-frequency, coherenton-receive radar of Figures 9.3, 9.6, and 9.7. Two additional sources of distortion are produced in this radar by the magnetron's frequency fluctuation: random phase error associated with target range extent and random phase error associated with range-sample delay offset. We will first examine distortion for the case of a single point target. 9.6.2 Random Phase Error for Point Targets To examine the effect of phase error, produced by frequency fluctuation, on the synthetic range profile of a point target, (9.14) is rewritten in the form of (S.41) as C = expjJ-27r/y-Kr,J (
(9.19)
where vx, is random cumulative phase error produced by frequency error x,. For random error (x,), in the synthesizer frequency/, the resulting random cumulative phase error will be -IvilRIc) x (or,),. Thus, the phase constant vassociated with the synthesizer is defined to be v, =
2irx2R/c.
For random error (JC,). in magnetron frequency/j,, the resulting random cumulative phase error from (9.8) will be 2ir(T, - T ) x (*,•)„ for (x,) > (JC,),. Thus, v associated with y
m
481
sampling offset in range delay is defined to be v, = range-delay offset.
-IITSS,
where
SS =\Tt
- Tj is the
9.6.3 Random Phase Error for Extended Targets The third source of phase error is associated with the target itself. This is produced by random changes in the target transfer function term yfa^ exp(+jfl) of (9.12) produced by frequency error of the magnetron pulses. Changes in the transfer function are caused by deviation of the interscatterer delay phase from that which occurs when the frequencies are separated by precisely A/. The effect becomes more severe for targets of large range extent. However, the transfer function is not changed at all if the target contains only one scatterer. Echo phase error associated with target range-delay extent will be expressed similarly to that for total range delay. Consider first the delay phase that would be measured for a point target. This phase, from (9.8), is measured relative to the synthesizer signal at frequency / , independently of the magnetron signal frequency if samples S, and S of Figure 9.5 are set exactly at the centers of the transmitted and received pulses, respectively. The only phase error is then -27r x IRIc x (x,), produced by random frequency error of the synthesizer. Now consider the delay phase from the sample position to a scatterer near either edge of the target echo pulse in Figure 9.8. Note that this delay phase is measured with reference to the magnetron's transmitted pulse at frequency f^. For random error (;Cj)„ of the magnetron frequency, the resulting random phase error will be 2ir x d/ c x (Xj) - Thus, v associated with target range extent d is defined to be v = -2tr x d/c. 2
m
4
SAMPLE POSITION S , SET TO CENTER OF ECHO PULSE
d c
RANGE DELAY
TARGET RANGE-DELAY EXTENT
^
Figure 9.8 Echo pulse from a target of range extent d.
9.6.4 Three Types of Random Phase Error (Summary) Random error in the synthesizer frequency will produce phase errors associated with target range delay IRIc. Random error in the magnetron frequency produces a similar type of phase error, but one that is associated only with the range extent of the target itself. A related type of phase error produced by magnetron random frequency error is that associated with the relative sampling delay offset SS = T - 7 between transmission and echo pulse sample positions. These three types of phase error are listed in Table 9.1 along with their respective characteristic functions, which are discussed below. The quantities rr, and
3
m
m
9.6.5 Effect on Peaks and Nulls of the Profile As we saw in Chapter 5, the synthetic range profile observed when random frequency error is present is the same as for the error-free case, except that the resulting cumulative Tabic *.l Summary of Random Echo Phase Error and Resulting Characteristic Functions
Type of Random Echo Phase Error
Phase Error
Random transmit-lo-receive frequency error (jtj, of
-2wx
Characteristic Function, C
f
2R — x(4
synthesizer output signal
2R where v, = 2tt x —
related to target range R. Random pulse-to-pulse frequency error (x,)
m
of
2trx£xU).
target range exlent d.
frequency error ( * , ) . of magnetron pulses related to relative sampling delay offset SS between transmitted and echo pulse.
,
n
where v, = - 2 j t c
magnetron pulses related lo
Random pulse-to-pulse
v
e '" - '
2»x«xU).
e
^^n,
where v, = -2nSS
phase errors reduce the peak value of the range profile and increase the noise floor. If 'we assume a normal probability distribution of magnetron frequency error, the expected value E[H,(x,)] of the peaks is reduced by the factor C,, which is the characteristic function listed in Table 9.1. Positions of the peaks and nulls remain undisturbed. Equations (5.42), (5.43), and (5.45), when expressed in terms of Q of Table 9.1, apply for the determination of reduced peak signal, phase-noise floor, and ratio of signal to phase noise, respectively. 9.6.6 Tolerance to Frequency Error Tolerable values for the standard deviation a of the frequency error will be calculated for illustrative purposes on the basis of vcr= 1 (4.3-dB loss) for each type of phase noise. Tolerable values for the standard deviation
|KrO-.|
=
2TTx
d c
X
a <. 1.0
(9.20)
m
By solving for the tolerable standard deviation, we obtain ^ c 47.7 x 10* ^ i l T d ^ — — (
k
H
z
)
(
9
2
,
)
Tolerable deviation becomes smaller as the target range extent becomes larger. For a 300m target, (9.21) indicates a maximum tolerable frequency deviation of 159 kHz. Tolerable deviation and minimum unambiguous frequency step size A/ as a function of slant-range extent d is shown in Table 9.2. Tolerable magnetron frequency deviation, for the criteria of (9.20), can be seen to be approximately one-third of the minimum frequency step size. Tabic 9 3. Tolerable Standard Deviation of Magnetron Frequency for Several Values of Target Range Extent d{m)
cr. (kHz)
60 200 300 600
796 239 159 80
C\f=±j(kHz) 2,500 750 500 250
0.32 0.32 0.32 0.32
I. A practical design is likely to require a value for mcorresponding to a loss of less than 4.3 dB.
484
Tolerance of magnetron random frequency error related to sampling delay offset SS is computed from the requirement \v,a \ = 2nSScr <, m
m
1.0
(9.22)
so that
Tolerable magnetron frequency deviation for several values of sampling delay offset are shown in Table 9.3. Sampling time offset will not likely exceed the transmitted pulse width. Assume, for example, a pulse width of 1 /is. Tolerable magnetron deviation cr„ then, is 139 kHz. 9.7 MAGNETRON FREQUENCY CONTROL Pulse-to-pulse frequency stability of fixed-frequency magnetrons is typically quoted as about 15 kHz rms for coaxial magnetrons and 100 kHz for conventional magnetrons (a. = 15 kHz and 100 kHz, respectively). These values presumably apply for single-frequency operation and modulation pulses having no pulse-to-pulse ripple. If so, the numbers represent inherent frequency deviation expected of magnetrons, independent of modulator and frequency-control characteristics. These deviations are well below the 159-kHz rms tolerable deviation derived above for 300m targets or l-/ts delay offset. For steppedfrequency operation, in Figure 9.3, random error in magnetron frequency as a function of frequency readout voltage u would introduce an additional source of magnetron frequency uncertainty. Frequency accuracy commonly quoted for frequency-agile magnetrons is usually several megahertz, but such quoted values are likely to, be absolute values, which include drift, voltage measurement error, and deviations associated with applications that do not require greater precision. By using constant calibration referenced to precise synthesizerTable « J Tolerable Standard Deviation of Magnetron Frequency for Several Values of Sampling Delay Offset
SS (us)
a. (kHi)
0.1 0.5 1.0 5.0
1,390 318 159 32
485
generated signals, we would expect that magnetron frequency deviation could be held to values near the inherent deviation of fixed-frequency magnetrons. 9.8 INTRAPULSE FM In the technique described above, the magnetron frequency is changed during each pulse, but constant frequency during the pulse was assumed for calculation of the tolerable frequency error. Intrapulse frequency modulation would produce no additional phase error and thus no additional distortion of the range profile if the FM were identical for each frequency step. Were the FM to undergo significant random step-to-step change, restrictions would need to be placed on the step-to-step sampling timing error SS = T - 7y No known data exist on the random characteristics of intrapulse FM slope. The effect, however, is not considered to be significant with respect to performance for range-profile processing or imaging performance. t
9.9 EFFECT OF FREQUENCY ERROR ON CROSS-RANGE DISTORTION Cross-range Doppler frequencies are generated by target aspect motion relative to the radar. The cross-range Doppler profile is extracted in each synthetic slant-range cell for ISAR imaging by use of a second transform carried out in each slant-range cell of the series of synthetic range profiles. Phase error produced by long-term frequency fluctuation would probably be indistinguishable from error due to target slant-range motion and could be corrected as part of the TMC process. However, remaining uncorrected is burst-toburst random phase error at any frequency step. Because linear processes are involved, the two series of transforms that produce an ISAR image can be interchanged without affecting the resulting image. Thus, after velocity correction, Doppler spectra could be obtained first, followed by the range profiles. If thought of in this way, the source of cross-range profile distortion is random burst-to-burst phase error, rather than random pulse-to-pulse phase error as discussed above for slant-range profile distortion. Unless the random pulse-to-pulse frequency error differs significantly from random burst-to-burst error, the cross-range distortion may be expected to be equivalent to range distortion in the ISAR image. PROBLEMS Problem 9.1 A RMO of a coherent radar system provides RF pulses to the input of a traveling-wavetube amplifier used as a pulse-radar transmitter. The RMO also provides the reference signal to a quadrature detector to mix the target signal to baseband from a suitable
486
intermediate frequency. The RMO has an rms frequency shift of one part in 10' during the echo delay associated with a target at 250 nmi (463 km). What is the rms phase error of the quadrature-detected signal produced by the frequency shift if the radar operates at 3.2 GHz? Problem 9.2 Target echo pulses from the single-frequency, coherent-on-receive radar of Figure 9.2 are sampled at the output of the quadrature detector. The MTI canceler is not used. Pulseto-pulse random frequency error in the COHO output is 100 Hz rms. A target detection is based on the output of the DFT of each burst of n samples (one complex sample per pulse), (a) What is the rms phase error of received pulses caused by the COHO frequency error for targets at a range of 150 nmi (277.8 km)? (b) What is the resulting loss in the ratio of input signal to phase noise? Problem 9.3 The standard deviation of the pulse-to-pulse frequency of the magnetron in Figure 9.3 from the desired uniform frequency steps produced by the frequency synthesizer is 0.5 MHz. (a) If inphase and quadrature-phase samples are taken at the centers, respectively, of each transmitted and echo pulse at baseband, without sampling-delay error, what is the attenuation for scatterers seen in the resulting synthetic range profile ±50m from the target center? (b) What is the attenuation for scatterers seen at the center of the target? Assume that the center of the echo pulse corresponds to the target center. REFERENCES ( l | Barton, D. K., ed., "Frequency Agility and Diversity," Radars, Vol. 6, Dedham, MA: Artech House, 1977. [2] Barton. D. K., Radar System Analysis, Dedham, MA: Artech House, 1979, pp. 191—195. (3) Rutlenberg, K.. and L. Chanzit, "High Range Resolution by Means of Pulse-to-Pulse Frequency Shifting," IEEE EASCON Record, 1968. pp. 47-51.
Chapter 10 Applications for Surveillance
The main emphasis up to this point has been the use of wideband, high-resolution radar for mapping and imaging. We now turn to existing and potential future benefits for improved surveillance performance. Surveillance, here, will refer to the traditional role of detection and tracking of multiple surface or air targets within a designated surveillance region. Surveillance functions to be discussed are as follows: 1. 2. 3. 4. 5.
Electronic counter-countermeasures; Low-flyer detection; Low-probability-of-intercept radar; Reduced target fluctuation loss; Small-target detection in clutter.
The importance of radar bandwidth and range-Doppler resolution to achieve these benefits has been well known by the radar community for many years. More recently, the ability to carry out coherent and noncoherent processing at up to 500 MHz of bandwidth and beyond has created new opportunities and renewed interest. In the 1980s, new challenges in terms of military surveillance requirements and expected countermeasure environments emerged, which necessitated a re-examination of fundamental approaches to military radar designs. Jamming, while always a serious threat, has become more formidable. The capability for nearly undetectable radar surveillance, called low-probability-of-intercept radar (LPIR), after years of controversy, has finally become recognized as both necessary and feasible for some surveillance applications. Extremely low flying missiles, difficult to detect with radar, are now the trend for some important air-to-surface attack roles, such as antiship attack. Aircraft and missiles with a greatly reduced radar cross section exist, and the requirement for reduced signature has come to be strongly emphasized for most new designs. Furthermore, new long-range, high-speed airborne threats require extended surveillance coverage for early warning, while retaining high target-revisit rates for close-in surveillance. 487
488
The role of increased radar resolution and bandwidth, although long recognized as crucial for some of these surveillance tasks, has now become more fully appreciated. To meet growing challenges, it is possible that future radar systems will operate over very wide bandwidths with multiple transmitting and receiving beams, to provide more surveillance information via increased target dwell time without sacrificing target-revisit rate. In contrast, conventional radar designs today commonly use single-beam scanning, whether the antennas are electronically steered or mechanically scanned, and waveforms are commonly narrowband, although radar operating bandwidth is often quite wide. Performance for conventional designs in the past had been improved by increasing transmitter power, usable antenna gain, and clutter cancellation, or by reducing system noise factor, antenna sidelobes, and system loss. Unfortunately, the potential for improvement made possible by these means has approached its useful limits. On the other hand, there is growing awareness of the potential for improvement by the exploitation of new technology advances in the areas of high resolution, wide bandwidth, and the associated increased information content. This chapter will illustrate the role of radar bandwidth and resolution for improving surveillance performance to meet some existing and likely future surveillance challenges.
10.1 ELECTRONIC COUNTER-COUNTERMEASURES The term electronic countermeasures (ECM) commonly refers to a broad range of electronic methods used to deny an enemy the full use of its electromagnetic assets. The use of some type of RF jamming signal is the best known ECM technique. Electronic countercountermeasures (ECCM) usually refers to methods employed to offset the effect of ECMs so that radar surveillance can continue to be performed in expected ECM environments. Antiradar jammers typically attempt to reduce radar sensitivity by radiating energy into the radar receiver via the radar antenna's main beam, sidelobes, or both. The radar designer has at his or her disposal a number of methods to provide ECCM, including high radar power, beam agility, sidelobe cancellation and sidelobe blanking, special waveforms, and
power management. Our discussion here will deal primarily with the improved ECCM performance provided by increased radar bandwidth and resolution. Radar designers and users are well aware of the advantage of a wide-bandwidth capability to provide improved resistance to jamming and other countermeasures. An enemy jammer is least effective, all else being equal, when the radar is able to transmit rapidly and randomly over a wide bandwidth of frequencies while maintaining a very narrow receiving-system bandwidth. On a one-on-one basis (one jammer against one radar), this forces an enemy to spread its available jammer power over the radar's wide transmitting bandwidth, thereby reducing the jammer signal power seen at the outpurof the narrowband radar receiving system. We will consider this case by looking at an example. An ECCM performance factor P can be defined as follows: c
489
P W) E
= 10 l ° g . o ( ^ r )
001)
where P, is the radar's transmitter power, B, is the transmitter's frequency-agile bandwidth, and B, is the receiving system's noise bandwidth. Assume a hypothetical long-range search radar with parameters as follows: P, = 100 kW, B, = 200 MHz, and 0, = 10 kHz (matched to monotone pulses of \00-/is duration). Then from (10.1), 1
n . , . (lOOx 1 0 ) x ( 2 0 0 x ltf) P (dB) = 10 log ) I M k
n
c
l0
| f )
| f |
= 93 Likely parameters of a hypothetical long-range, frequency-agile mode for target imaging with the same radar are P, = 100 kW, B, = 200 MHz, and B. = 1 Hz. Then, P (dB) = .n 10 ilog ( I D
£
0
0
*
, 0 J
)
X
(
2 0
°
X
, O 6
)
l0
1
0
(10.3)
= 133 A radar in an imaging mode, because it must dwell on the order of a second on each target or scene to be imaged, cannot perform useful surveillance of the large coverage regions thought of as typical for search-radar applications. On the other hand, the wide transmitted bandwidth required for slant-range resolution, together with the very narrow receiving-system bandwidth associated with coherent processing during the long dwell time needed to obtain useful cross-range resolution, provides inherent (but not always implemented) jammer tolerance. As measured by the performance factor, expressed by (10.1), the above radar, operating in a frequency-agile imaging mode, can be seen to have a 40-dB advantage in ECCM over that in its conventional search mode using the same transmitting bandwidth and transmitter power. The advantage in this case occurs because of the narrow processing bandwidth of the imaging mode. ECCM evaluation for operation in a surveillance mode would involve other factors, including available target dwell time, antenna sidelobe levels, and expected jamming strategy. A more complete but still idealized analysis now follows to illustrate the role of transmitted bandwidth in ECCM performance of surveillance radars for which the receiving-system bandwidth is restricted to that of a matched filter to the transmitted pulse. The analysis will be carried out in terms of an increase in effective system noise factor produced by jamming signals entering the radar receiver from a noise jammer. The analysis will be further idealized by assuming that received wideband jamming signals are indistinguishable from radar system kT,B, noise.
490
Wideband noise jamming is called barrage noise jamming. The performance of this type of counlermeasure is commonly stated in terms of effective radiated power density (ERPD) (e.g., watts per hertz, watts per megahertz, kilowatts per megahertz). The ERPD takes into account the jammer power, jammer antenna gain, and jammer radiating bandwidth. (In many practical situations, the jammer ERPD is determined by the frequency coverage needed to jam over several radar operating bands, in which case the one-onone assumption above would not apply.) Radar ECCM performance in terms of radar bandwidth and jammer ERPD will now be analyzed by using the concept of effective system noise figure. The free-space radar range equation, expressed by (2.9) in Chapter 2, can be rewritten as J
I
}
4
(/ ,G AVy|(4ir) /? L)
(10.4)
where F is the radar system noise figure (without jammer) and T is the standard noise temperature (290K assumed). The product FT„ is the radar system noise temperature T,. The SNR referred to the input to the radar receiver in a noise-jamming environment can be expressed as B
(10.5)
where S Ni p, kT F
9
= = = = =
the received echo signal power; the received jammer power, the radar's noise bandwidth; Boltzmann's constant times standard noise temperature; the radar system noise figure with no jamming signal present. 1
The jammer power received via the radar antenna sidelobes at jammer-to-radar range R, is
(10.6)
where I. Mainlobe jamming is not considered here.
491
PJGJIB, =
Pj Gj L, G,
= = = =
jammer's ERPD (ERP per hertz) for jammer bandwidth matched to radar frequency-agile bandwidth /?,; jammer transmitter power; jammer antenna gain; radar receiving-system loss; radar antenna sidelobe gain.
For simplicity, any jammer losses are assumed to be included in the terms P, and G,, and all radar system loss is assumed to arise from radar receiving-system loss. The SNR at the receiver input, from (I0.S) and (10.6), then, is (10.7)
where Fai is the effective system noise figure of the radar system, which, using (10.6) and (10.7), is defined as F
f
+
** (' iw)
_
R
PJGJG,/
B,
A \'
I
\47rRj)kToL,
For sidelobe jamming, the radar antenna sidelobe gain is _ ( GIL, ' ~ yGI'Lb)
(in the sidelobes) (in the sidelobes after cancellation)
(10.9)
where G is the mainlobe gain of the radar antenna, L, is the radar antenna's mainlobeto-sidelobe ratio, and L, is the radar's sidelobe cancellation ratio (for radars employing sidelobe cancellation ). The input SNR (10.4), written in terms of effective system noise figure, becomes 1
2. Sidelobe cancellation refers to a well-known method for cancellation of the received jamming signal from the radar's main antenna with the same jammer signal received in an auxiliary quasiomnidirectional antenna. The signal in the auxiliary channel is adaptively adjusted in amplitude and phase to effect cancellation of the jammer signal in the main receiving-system channel, ideally leaving only the echo signal present in the main receiver.
492
(P^MV)/[(4«
Q
The radar waveform will be assumed to consist of monotone pulses of duration T,, which change in frequency from pulse to pulse. The output SNR, from (2.45) of Chapter 2, is
(10.11)
Free-space detection range R, from (10.10), in terms of required output signal-to-noise, from (10.11) for detection in jamming, then becomes
f'
c2Avr
'
r
(,o..2)
The ECCM performance of a hypothetical 3-D (range, azimuth, elevation) air-search radar will now be evaluated in a hypothetical jamming environment as a function of the radar's frequency-agile bandwidth. Radar parameters are listed in Table 10.1. The scanning strategy and coverage are not stated for this example, but 55 and 3 pulses per dwell are assumed for the high and low PRF modes, respectively. Effective system noise figure versus bandwidth is evaluated using (10.8) and plotted for the high-PRF mode in Figure 10.1, based on effective radiated power (ERP) of 1 MW of a noise barrage jammer located 100 nmi from the radar. The ERPD available from the jammer is assumed to be its ERP divided by the radar's agile bandwidth 0. Radar range versus effective system noise figure with jamming present is evaluated using (10.12) and plotted in Figure 10.2. Finally, radar range versus operating bandwidth of the frequency-agile radar with jamming present is evaluated in Figure 10.3, based on results in Figures 1,0.1 and 10.2. We should note that the effective system noise figure in Figure 10.1 and the radar range with jamming in Figure 10.3, although plotted in terms of frequency-agile radar bandwidth, are actually functions of the jammer's frequency spread. A radar with narrow frequency-agile bandwidth would enjoy the same benefits as an otherwise equivalent wideband radar, unless the jammer increased its ERPD by reducing its bandwidth. The frequency spectrum of actual jamming signals will not, in general, be matched to a particular radar's transmitter bandwidth. Furthermore, actual jamming signals that appear within the radar's receiver bandwidth will affect the radar detection range somewhat differently than kT,/3. noise in the same bandwidth. Equation (10.12), however, is submitted as being useful in two ways: (1) to illustrate the advantage of wide-bandwidth radars against barrage noise jammers, and (2) for ECCM performance comparison of alternative frequency-agile radar and noise-jammer designs.
493
Tabic 10.1 Hypothetical 3-D Air-Search Radar Parameters Parameter
Symbol
Peak power PRF
inr,
Hits' (pulses) per dwell Antenna gain (radar) Average wavelength Radar cross section Radar receiving system loss Boltzmann's constant Standard noise temperature Output SNR required for detection
n G A a U k T, &W
P.
Value 50 kW 5,000 (high) or 300 (low) pulses per second 55 (high-PRF) or 3 (low-PRF) 10' (40 dB) 0.1m 1.0 m 2.0 (3 dB) 1.38 x l f r J/K 290K 1.12 (0.5 dB) for 55 hits. 7.9 (9 dB) for 3 hits, for P„ = 0.5. P = lO" , Swerling case I 20 us for 55 hits per dwell; 333 us for 3 hits per dwell 2 (3 dB) Variable (Hz) 1
u
4
fK
1
Radar pulse width assuming 10% duty factor
T,
Radar system noise figure (no jamming) Transmitter agile bandwidth
fi,
F
"Number of pulses transmitted during the dwell time of the radar beam on the target. 'Radar performance might be improved over that for SC-I if wide bandwidth were obtained by pulse-topulse frequency agility. See Section 10.4.
Smart-jammer designs have been envisioned that can measure radar frequency from each pulse of a frequency-agile radar in sufficient time to jam the radar's receiver before reception of an echo pulse. For situations where this or other smart-jammer techniques are feasible and practical, the above analysis does not apply. A wide instantaneous bandwidth radar (as opposed to the above wide frequency-agile bandwidth radar), however, in addition to countering nonsmart noise jammers could, in principle, also defeat a smart jammer. A pulse-compression radar, for example, might be designed to transmit individual pulses containing energy spread over the radar's entire operating bandwidth. In principle, (10.12) may also predict the range performance in a nonsmart, wideband, barrage-noisejammer environment of a wideband pulse-compression radar. For example, a chirp-pulsecompression radar, with chirp bandwidth A = B, in (10.8) and other parameters identical to those for the frequency-agile radar of Table 10.1, will produce identical performance. In practice, however, pulse-compression radars may have serious disadvantages in an ECCM environment. The receiving system's matched-filter bandwidth for a pulsecompression radar must cover that of the transmitted pulse from the antenna through detection. The jammer signal from a narrowband jammer of moderate power operating anywhere within the radar's bandwidth may exceed the saturation level at some point in the radar's receiver, thereby desensitizing the receiver to small echo signals. The same jammer operating against a frequency-agile radar will have only a small probability of
494
0
400 800 R A D A R F R E Q U E N C Y - A G I L E B A N D W I D T H p (MHz)
1200
t
Figure 10.1 Effective noise figure for radar of Table 10.1 versus radar transmitter frequency-agile bandwidth (assuming that the jammer spreads its available power uniformly over Ihe radar frequency-agile bandwidth).
entering the radar's narrowband receiver. Furthermore, the narrowband receiver's IF amplifiers, which follow the preamplifier in a frequency-agile system, will be less susceptible to saturation because of their inherently wider dynamic range. Dynamic range can be viewed here as the ratio of maximum output power before saturation to the thermal noise power floor. The thermal noise power floor is proportional to the receiver noise bandwidth so that dynamic range is reduced as bandwidth increases. In any actual radar design, a major issue is automatic gain control (AGC) operation. An AGC system can be designed to protect the receiver against saturation over a large variety of jammer and signal conditions.
495
Figure 10.2 Radar range with noise jamming versus effective noise figure (for radar of Table 10.1).
A common type of the antijamming feature related to AGC that is commonly used in military radars in various forms is hard limiting. Hard limiting is applied near the input stage of the radar receiving system to suppress large jamming interference signals so as to prevent receiver saturation. Hard limiting reduces false alarms at the expense of desensitization target signals that overlap strong jamming signals in time. 10.2 LOW-FLYER DETECTION The ability to detect low-flying antiship missiles is of crucial concern to modern naval fleets. Radar detection of low-flying air targets, in general, is a radar problem that dates back to the early days of air-surveillance radar. Two fundamental problems are involved.
496
250
^ L j L c = -70 dB 200 -
¥ tr a
^ ^ - ^ ^ * ^ L , L
= -60 dB
C
z
§150 Z UJ
O
1100 <
^ ^ - - - - <
<
<
T
S
L
C
= -45 dB P j G j = 10«W R j = 100 nmi
tr 50
_ ... . .,1 200
i
i
i
400 600 800 FREQUENCY-AGILE BANDWIDTH p (MHz)
1000
1200
t
Figure 10 J Radar range with noise jammer for radar or Table 10.1 (assuming that die jammer spreads its available power uniformly over the radar's frequency-agile bandwidth).
which are associated with RF propagation phenomena. First, propagation over the earth's horizon is normally weak. Exceptions are for radar operation in the 3- to 30-MHz region, where surface-wave propagation is enhanced and skywave propagation can extend radar detection ranges to beyond 2,000 nmi, and for propagation-ducting conditions, which can provide over-the-horizon (OTH) microwave radar range performance up to abouMhe equivalent of that for free-space propagation. A second propagation-related problem associated with low-flyer detection is the cancellation of the direct signal in each direction by the indirect signal reflected from the
497
earth's surface. This problem will now be addressed as it affects detection by shipboard radars of extremely low-flying targets over the sea surface. We will show why operation at higher microwave frequencies (above 10 GHz) appears to be better than at lower microwave frequencies for detection of these targets at ranges out to the earth's horizon. We will then show, by using two examples, how high-resolution radar techniques provide the means to discriminate targets from sea clutter at higher microwave frequencies. It should be recognized, however, that the problem of shipboard surveillance against small low-flying missiles is a complex one that involves other important issues besides those associated with the propagation phenomena addressed here to illustrate potential advantages of high-resolution radar for surveillance. The radar detection range of sea-skimming missiles or aircraft, at the microwave frequencies commonly used for shipboard air surveillance, may be less than the range to the horizon (about 15 nmi for a 100-ft radar antenna height). For most operational radars, this is far less than the free-space detection range to even small missile targets. The problem of low-flyer detection can be illustrated by using Figure 10.4. For the lowgrazing angles associated with surveillance against low-flying targets, the magnitude of the reflection coefficient at microwave frequencies for reflection from the sea surface approaches unity, regardless of RF polarization [1]. As target height decreases for a given radar height, the difference in length between reflected and direct paths decreases. For low-flying targets, at lower microwave frequencies, the path-length difference may be the equivalent of much less than a wavelength. In this situation, near-perfect cancellation will occur in both the transmitting and receiving directions. The phase of the reflection coefficient at microwave frequencies is about 180 deg at near-grazing angles for both horizontal and vertical polarization (2]. From the viewpoint of the radar, there arc two targets: the real target in the direct path to the radar and a mirror image of the target below the sea surface, associated with the indirect path reflected from the sea surface. As target height increases, for a given fixed target range, some height will be reached at which the difference between the direct and sea-reflected paths will approach one-half wavelength. At this point, constructive interference occurs. At fixed frequencies in the DIRECT PATH
Figure 10.4 Sea-surface reflection.
TARGET
498
higher microwave frequency bands, the familiar multiple-lobing phenomenon becomes evident at these relatively short ranges. This results when the phase difference between direct and sea-reflected paths goes through multiple 2irrad of phase change as the radar or target moves in range or elevation. Figure 10.5 illustrates, for a radar antenna height of 100 ft, the relative field strength at the target versus target height at a radar range of 15 nmi, and relative field strength at the target versus range for a constant target height of 20 ft, both for several microwave radar frequencies. The range to the horizon from the 100-ft antenna is about 15 nmi. Relative field strength, also called the propagation factor, was defined in (2.32) of Chapter 2 as Resultant field at the target Field at the target due to direct path only
(10.13)
1
Incident power density at the target is modified by the factor \F \ from that for free space. The propagation factor, together with other propagation effects, determines the range attenuation g(R), defined by (2.27) of Chapter 2. The received power at the radar is I
100-
T A R G E T
Horizontal Polarization Radar Height 100 ft Range IS nmi
H E I G H T ft
(a)
~i
r
-20 -10 PROPAGATION FACTOR dB
Figure 10£ Propagation factor plots for low-flying targets illuminated by shipboard radars at various frequencies (a) for a target range of IS nmi; (b) for a target height of 20 ft (generated from PC software supplied as part of EREPS: Engineer's Refractive Effects Prediction System Software and User's Manual, developed by the NOSC Tropospheric Branch, Ocean and Atmospheric Sciences Division published by Artech House, Norwood, MA).
499
Figure 10.5 (continued)
modified by the factor \FJ? from that for free space because of the two-way propagation effects caused by the presence of the reflecting sea surface. We can infer from Figure 10.5 that for target heights below about 50 ft, the range-averaged multipath propagation is improved at ranges near the optical horizon by operating shipboard radars at the two higher microwave frequencies. Multiple peaks and nulls of the propagation factor can be seen to appear for targets below 100 ft at frequencies above 9.5 GHz. Peaks and nulls can also be observed at the higher frequencies for low-elevation targets, viewed at constant elevation but changing range. Low-flyer detection is effectively improved at the higher frequencies by the existence of these lobes because targets moving through multiple lobes in range will present multiple opportunities for detection. In contrast, a 1-GHz system will experience severe propagation loss at all ranges out to the horizon. Thus, for targets below 50 ft, the detection range is clearly improved by operating at or above X-band (8.50 to 10.68 GHz). Operation at K.-band (13.4 to 14.0 GHz and 15.7 to 17.7 GHz) is particularly useful because of the additional advantage of favorable ducting characteristics, which have been found to prevail near the ocean surface at these frequencies. Unfortunately, the detection of small targets flying over ocean surfaces requires clutter discrimination. Clutter discrimination historically has been carried out by using narrowband MTI techniques, which are difficult to implement above X-band. Two well-
500
known techniques are high-PRF pulsed-Doppler and low-PRF clutter cancellation? Pulsed Doppler methods are commonly used at frequencies from S-band (2.30 to 2.S0 GHz and 2.70 to 3.70 GHz) to X-band. Clutter cancellation methods are commonly used below L-band (1.2IS to 1.400 GHz). Hence, we will show that wideband, high-resolution methods may be capable of providing the required clutter discrimination at frequencies above X-band, where multipath cancellation is more favorable. First, we will discuss some basic principles of conventional narrowband clutter discrimination. 10.2.1 Clutter Discrimination With Narrowband Radars Narrowband clutter-discrimination radars usually depend primarily on coherent processing of echo signals to separate moving targets from fixed targets. Conceptually, the simplest type of radar that discriminates moving targets from clutter is probably the CW radar. The output response to a constant-velocity air target can become a steady signal separated in frequency from the clutter signal by homodyne mixing the received signal with the transmitter reference signal. Moving targets produce a Doppler frequency given by the expression (2.50) as f„A,
(10.14)
for/>/i>, where v, is the target radial velocity, c is the propagation velocity, and / i s the radar transmitter frequency. At low transmitter power levels, a single antenna can serve for both transmitting and receiving. Highway police use this type of radar to measure automobile speed relative to the radar. In most military applications, the required power precludes the use of one antenna for both transmitting and receiving because reflected transmitted energy from the antenna itself or nearby objects for higher power transmission would desensitize or even destroy the receiver. For this reason, a modification of the technique, to be discussed below, called pulsed-Doppler radar, was developed to allow the use of a single antenna. Coherent MTI radar systems using clutter cancellation measure the residual response resulting from the phasor subtraction of range-delay responses produced by consecutive pairs of transmitted pulses. Subtraction (also called phase detection) is achieved as shown in Figure 10.6. The canceled response at any instant is the phasor difference between (1) the response versus range from the previous pulse delayed by the PRI, and (2) the response 3. The terms high PRF, low PRF, and medium PRF commonly refer to pulse waveforms or to radars using these waveforms according to their respective range- and Doppler-ambiguily characteristics for maximum expected range delay and Doppler frequencies of return signals. Return signals for high-PRF radars are likely to be ambiguous in range and unambiguous in Doppler, whereas return signals for low-PRF radars are 'likely to be ambiguous in Doppler and unambiguous in range. Return signals for medium PRF radars are likely to be ambiguous in both range and Doppler. Selection of average PRFs in each case depends on wavelength and expected maximum target range and velocity. Ambiguities are resolved by PRF shifting and other techniques.
SOI
TRANSMIT PULSE
TARGET ECHO (CLOSING RANGE)
R A N G E DELAY 2ND R E C E I V E D PULSE
1ST R E C E I V E D P U L S E OUTPUT
D E L A Y * 1/PRF i INPUT
1 DELAY SINGLE-LOOP CALCELER
CANCELED OUTPUT VELOCITY
Figure 10.6 Coherent MTI canceler.
versus range from the most recent undelayed pulse. The output of the MTI canceler, ideally, will be zero at range delays for which no moving targets exist. Responses from moving targets will tend not to cancel because there will be a shift in the phase of successive target echo responses. Blind speeds occur for targets moving so fast toward or away from the radar that the responses from successive transmitted pulses are shifted in phase by 2ir rad or multiples of 2n rad. Low-PRF MTI radars typically perform the phase detection with digitized baseband signals or at some convenient IF, where quartz delay lines can be used to provide the delay between pulses. Multiple-loop cancelers coherently process three, four, or five pulses to increase the spectral width over which clutter is canceled. Regardless of design details, however, blind speeds will occur for targets at velocities that produce 0, 2ir, Air,... rad of phase difference between successive received RF pulses. Below about 1 GHz, it is possible for many applications to operate at sufficiently low PRF to minimize the rangeambiguous return arriving at delays greater than the radar PRI, while at the same time avoiding blind-speed problems. This is possible at these relatively large wavelengths because the change in phase between received pulses from air targets tends to remain less than a wavelength even at the relatively low PRF associated with unambiguous longrange air surveillance. Pulsed-Doppler processing at high or medium PRF is commonly employed for clutter discrimination in airborne systems, where the radar frequency is typically at Sband (2.30 to 2.50 GHz and 2.70 to 3.70 GHz) or X-band (8.50 to 10.68 GHz). Here, Doppler filtering is often used to separate moving targets from the ground clutter. Note that the ground clutter seen by airborne radars will be spread in Doppler frequency. Whereas the low-PRF MTI system tends to become ambiguous in velocity, the high-PRF pulsed-Doppler system is usually ambiguous in range.
502
Detections with pulsed-Doppler radar are normally made from output responses that are obtained in the Doppler-frequency domain using various types of Doppler filtering techniques. Target range can be measured after detection by observing the Doppler shift of target responses produced by controlled frequency ramping of the transmitted signal. The high-PRF radar can be thought of as a CW radar that avoids the need for two antennas. While often referred to as being unambiguous in velocity, a single-sideband (SSB) pulsedDoppler radar will produce ambiguous results when a target's Doppler shift exceeds onehalf of the radar PRF. To prevent this, the PRF is made at least twice as high as the highest range of Doppler frequencies expected from targets of interest. The resulting transmitter switching rates, however, tend to become impractically high above X-band frequencies. 10.2.2 Clutter Discrimination Using HRR Techniques At frequencies above X-band, problems of multiple blind speeds with low-PRF MTI radar and transmitter switching rates with pulsed-Doppler radar may be avoided by using HRR methods. One conceptual approach is to employ high-resolution noncoherent MTI, also called short-pulse area MTI (NRL Report 8162, A Short-Pulse Area MTI, by Ben Cantrell, 22 September 1977). In this approach, subtraction of successive radar video returns from an HRR radar is performed to retain the moving-target signals and to reject the stationary clutter signals (see Figs. 10.7 and 10.8). Use of HRR waveforms for MTI reduces clutter cell size and avoids multiple blind speeds. Note that the resolution of responses from separate two-way propagation paths would require extreme resolution for the above lowflyer geometries, so it is not considered here. As an example of noncoherent MTI cancellation, assume that a radar sends out pulses of l-ns width and 500-fis separation. A moving target with a radial speed of 300 m/s will then produce an interpulse range-delay shift of (21c) x 300 m/s x 500 x 10"* = 1 ns (i.e., a shift equal to the pulse width). This shift would result in an uncanceled response at the target's range-delay position, indicating the presence of the moving target. Fiber-optic delay lines appear capable of providing, at sufficiently low loss, the required DELAY =
1 PRF
ENVELOPE-) INPUT DETECTED HRR ECHO CANCELEDPULSE OUTPUT FOR MOVING TARGET) Figure 10.7 Noncoherent MTI canceler.
503
TARGET POSITION DURING PULSE; 1
ECHO FROM PULSE 1
ECHO FROM PULSE 2
RANGE DELAY Figure 10Jt Low-flyer detection with HRR (target moving toward radar).
delay and bandwidth [3]. Transducers at the input and output of the canceler convert to and from light and wideband video. The noncoherent MTI canceler of Figure 10.7 can also be implemented with high-speed A/D conversion followed by high-speed digital delay. As we will see later, to achieve a given level of signal-to-clutter performance, the noncoherent MTI canceler need not cancel as completely as the coherent canceler. A second approach is to employ an HRR track-before-detect (TBD) concept. Here, the reduced range-cell size resulting from the HRR processing First increases the targetto-clutter ratio by decreasing the size of the clutter patch, as indicated in Figure 10.9. (The clutter patch can be defined as the resolved area on the sea surface corresponding to resolution in range and azimuth.) Then, following a suitable threshold circuit, the threshold crossings are converted into relatively narrowband responses, illustrated in Figure 10.10. The narrowband responses are tracked by a suitable tracking algorithm. Input data for TBD processing are illustrated in Figure 10.11 for threshold crossings over a limited range extent for five antenna scans. Only those threshold crossings that behave like a target track are declared to be targets. At least one system of this type has been tested experimentally, using pulse compression to achieve the required high resolution. It is likely that a combination of noncoherent cancellation followed by TBD may be
504
RADAR CLUTTER PATCH y , IS RADAR ANTENNA AZIMUTH BEAMWIDTH (RECTANGULAR EQUIVALENT) R IS RANGE TO CLUTTER PATCH Ar.lS RADAR RESOLUTION IN SLANT RANGE AR IS RANGE INCREMENT TO BE CONSIDERED FOR FALSE ALARM (Ar,= AR FOR NARROWBAND RADAR CONSIDERED IN THE TEXT) Ry.Ar, IS THE CLUTTER-PATCH AREA FOR HIGH-RESOLUTION RADAR a'
= - i — (CLUTTER COEFFICIENT, m* PER m») Ry.Ar,
Figure 10.9 Clutter patch on the sea surface (view looking down at the sea).
optimum. Only two possible means have been suggested above for low-flyer detection with HRR radar. Other high-resolution techniques are currently under investigation. 10.23 Wideband Versus Narrowband Radar for Clutter Discrimination At this point, we will show how sea clutter, which when seen at low resolution can be thought of as diffuse scattering, is resolved into individual sources of clutter as resolution cell size is reduced. Then we will show tradeoffs for high-resolution MTI compared to low-resolution MTI for the simplifying assumption of'Rayleigh distributed clutter in each case. This will be followed by an illustrative example. Figure 10.12 illustrates some relationships between radar sea clutter as characterized for narrowband radar and in terms of amplitude as a function of range and time history for HRR radar. In the 3-D sketch on the right side of Figure 10.12, a series of successive HRR profiles are indicated with the clutter sources moving away from a stationary radar. On the left-hand side of Figure 10.12, measurements of sea-clutter return are illustrated in terms of amplitude distribution, power spectrum and correlation in time, radar frequency, and range. Characterizations on the left are useful for representing sea-clutter return when individual clutter sources are unresolved, as in the case of narrowband radar. Examples of actual sea-clutter backscatter observed with HRR radar are shown in Figures 10.13 and 10.14. The coordinate system is that used to represent the sequential
506
if*
So
St 2
UI
O
<
5o
Is at UI
UI
3 o
t3 o
3a 40
3+ o
st 3 o
ui
t3
I 3
s
t
& E
507
S E Q U E N T I A L HRR PROFILES O F SEA CLUTTER
NARROWBAND MEASUREMENTS
RELATED T O DISTRIBUTION O F C L U T T E R SOURCE AMPLITUDES
AMPLITUDE DISTRIBUTION
AMPLITUDE
RELATED DISTRIBUTION OF C L U T T E R SOURCE VELOCITIES or/St
POWER SPECTRUM
• DOPPLER
t
/RANGE WINDOW
_/ / REAL TIME MINUS S s
•DOPPLER
^ / R E L A T I V E R A N G E " E A L TIME RELATED T O CLUTTERSOURCE LIFETIME
TIME -
CORRELATION { FREQUEr
RANGE
J
RELATED T O C L U T T E R - S O U R C E •* R A N G E SEPARATIONS RELATED T O .CLUTTER-SOURCE. PATTERN C H A N G E WITH R A N G E
Figure 10.12 Characierizalion of narrowband sea clutter versus high-resolution sea clutter.
HRR profiles on the right side of Figure 10.12. Figure 10.13 shows a series of measured, successive HRR profiles. The radar was fixed and the clutter sources were observed in the open sea. Isolated sources of sea clutter, probably associated with a sea wave moving toward the radar, are shown in Figure 10.14. Their lifetime seems to be about three to five seconds. Range extent appears to be about 10 to 30 ft. It was found that the velocity was proportional to the wind velocity relative to the radar. The clutter patch associated with matched-filter processing of narrowband pulses is large. The clutter echo signal sampled at any instant at a given range and azimuth position is the instantaneous phasor sum of the echo signals from all scatterers in the patch centered at that position. At high sea states, the clutter return, even at the shallow grazing angles associated with shipboard radars, can be much higher than echo signals from some reduced-RCS low-altitude air targets. For this reason, cancelers have been
508
Figure 10.13 HRR sea clutter (open sea).
developed for narrowband radars to provide high cancellation ratios. For HRR, on the other hand, the range dimension of the resolved clutter patch is small. The backscatter sources are resolved for the most part. Highly resolved sea backscatter, therefore, is much reduced from that of low-resolution sea backscatter, even though the clutter coefficient (square meters per square meter of illuminated sea-surface area) is roughly the same.
509
RANGE WINDOW DISPLAY TIME
250 ft 4.9 s
Figure 10.14 HRR sea duller (two typical views).
The reduced clutter return means that required cancellation ratios for HRR radar are expected to be correspondingly lower than those for narrowband radar. HRR radar, on the other hand, has many more range cells, each of which can produce a false alarm. Therefore, detection thresholds will be set for a much lower false-alarm probability. The result of these effects will now be analyzed and then illustrated in an example. The probability P A that canceled clutter for cancellation ratio R will exceed the radar cross-section threshold o> is expressed as F
r
-OH-
P(a
c
£ R
T
(10.15)
From Figure 10.9, the clutter patch size at range R, beamwidth >/>„ and resolution L\r, is Rt//,Ar,. The resulting average sea-clutter cross section is (10.16) where a" is the clutter coefficient. Sea-clutter fluctuation can be described by a probability density function, p(tr ), defined such that pi
r
c
r
r
Pier, £ y) = f p(
e
t
(10.17)
The exponential Rayleigh probability density function has been used to represent the statistics of sea-clutter return. Other density functions used to represent sea-clutter
sw backscatter statistics are the Ricean, log normal, and Weibull densities [4], but here we will assume the more simple exponential Rayleigh distribution, which as applied to sea clutter is expressed as
p(cr) = i- expHTi/oi)
(10.18)
f
where a is the average sea-clutter cross section of the illuminated sea-surface patch. For a detection threshold oy with clutter cancellation ratio R„ the probability of false alarm, from (10.15), (10.17), and (10.18), becomes c
P
fA
= P[u > y] = f - exp(-o- /7J- )do- = exp(-y/aj r
c
c
(10.19)
c
where y=/? o>with y S O
(10.20)
c
We can see that for y = 0, P = 1, indicating that there is unity probability that
t
c
y/Zr, = -ln(P J
(10.21)
f
The detection threshold, in terms of target RCS, from (10.20) with (10.21) becomes
(10.22)
F A
For clutter patch area
Rfa,t\r,
0
and clutter coefficient tr , (10.22) with (10.16) becomes
a r =
_?M^
ln(
P
(,0.23)
FA)
The required cancellation ratio R from (10.23) for detection of target size tr thus is r
T
0~T
The ratio of signal to mean clutter power before cancellation is
SI I
(10.25)
and after cancellation. RcO-T
R<
T
R^A^o*
(10.26)
Differences between narrowband radar and HRR radar for detection of targets in clutter will now be illustrated using the above expressions. The two types of radars are compared in Table 10.2 at the same false-alarm rate (FAR). Characteristics for these two hypothetical radars were chosen to be identical, except for range resolution. Azimuthal scanning over 360 deg is assumed for each radar. A maximum false-alarm rate is assigned only for a 1-nmi surveillance ring to simplify calculations. A different clutter patch area would have to be considered for each range increment if a single FAR were assigned for the entire extent of range coverage. The HRR radar can be seen to require almost four orders (40 dB) of magnitude lower probability of false alarm than that of the narrowband radar, but this disadvantage is offset by requiring almost 40 dB less clutter cancellation. For this reason, a noncoherent canceler (cancellation at video), although inherently less efficient than a coherent canceler (cancellation at baseband or IF), may be appropriate for HRR and could solve the problem of carrying out MTI at frequencies above 10 GHz, where low-flyer detection is easier to achieve. A more complete analysis would use a better sea-clutter model than the Rayleigh probability distribution, which is known to be optimistic for sea echo as seen with HRR radar. Experimental confirmation of any model, however, would be difficult, if not impractical, for the extremely low P values involved with HRR radar. A more complete analysis would also recognize slight differences in er° values for HRR as compared with low-resolution radar and the deviation from diffuse scattering that occurs as resolution cell size decreases. The latter effect is indicated in Figures 10.13 and 10.14. Values chosen for the above analysis, a" = 10~\ were for vertical polarization in medium seas at neargrazing angles [5]. More complete analysis indicates that noncoherent HRR-MTI radar used for low-flyer detection may require about 10- to 20-dB sea-clutter cancellation, instead of 2 dB as calculated in Table 10.2. Methods for exploiting the isolation of sources of sea clutter seen at HRR are being investigated. One important additional consideration is involved for the above noncoherent type of clutter cancellation. The resolution must be sufficiently high to produce a detectable uncanceled residual for the slowest targets of interest. Returns from successive transmitted pulses will tend to cancel for slowly moving targets as well as clutter. The problem becomes more severe as the radar PRF is increased. We will try to gain some insight into this consideration by estimating the required resolution for a target moving at the relatively FA
512
Table 10.2 Calculation Sheet for Wideband (HRR) Versus Narrowband Clutter Cancellation Parameter
Symbol
Radar range to be considered
R
Incremental range Tor which false alarms are to be considered (centered at 10 nmi) Scanning rate Azimuth beamwidth (equivalent rectangular) Beam dwells per scan Target size Radar center frequency Range resolution
SR
10 nmi (18.52 km) 1 nmi (1.852m)
Narrowband Radar
10 nmi (18.52 km) 1 nmi (1,852m)
1 2ffscan/s 1 deg (0.0175 rad) 360 1 m' 10 GHz 0.2m
IT*
10"' m'/rn'
1 2rrscan/s 1 deg (0.0175 rad) 360 1 m» 10 GHz 1 nmi (1.852m) (12-/is pulse) 10-' mVm'
R+Ar, (VC),
65 m 12 dB
6.0 x I0 m' -28 dB
N.
1.20 x 10™ h r
Ifc
o>
;
Ar,
Clutter coefficient (near grazing, medium seas) Clutter patch area at 10 nmi Signal-to-clutter ratio before cancellation. Eq. (10.25) Alarm opportunities per hour in range ring „ « „ «R R - Y to A" + y
Wideband (HRR) Radar
1
5
1
1.30 x ir/hr'
\N. = w,x — x — x3600J Required probability of false alarm corresponding to FAR = 1.0 per hour in range ring SR' Required cancellation for er = 1 m . Eq. (10.24) Signal-lo-cluller ratio after cancellation. Eq. (10.26) !
r
8.3 x 10"
7.7 x I0-'
R,
2dB
39 dB
(ttC),
14 dB
I I dO
'From P,» = 1W„ which deviates slightly from P (See Barton (9|. p. 19.)
1 FA
based on the Marcum definition of false alarm lime.
slow radial velocity of 150 m/s toward the radar. For a PRI of 500 pis, the target will travel (150 m/s) x (500 x 10"* sec) = 0.075m between pulses. For example, uncanceled video responses from a 0.15m resolution radar would then overlap to produce roughly a 50% degradation in the amplitude of the MTI response. This is illustrated using the idealized triangular pulses of Figure 10.15. Handling slower targets without further degradation would require lower PRF, higher resolution, or both. Targets at radial speeds of 150 m/s or higher produce less than the 50% amplitude degradation for the above example. We should note that R in (10.24) is the degraded cancellation ratio. {
513
PULSE 2 ——I
I-—0.075m
V
PULSE 1 AND PULSE 2 BEFORE CANCELLATION
-t
PULSE 2 MINUS PULSE 1 AFTER CANCELLATION
t
Figure 10.15 Cancellation of two HRR video pulses.
The arguments given above concerning the use of HRR for sea-clutter discrimination also apply, but in a more complex way, to the TBD technique for HRR clutter discrimination. The HRR-TBD concept depends on reduced clutter patch size to limit the number of threshold crossings to be sorted out from real targets while maintaining threshold values sufficiently low to detect small targets. 10.3 LOW-PROBABILITY-OF-INTERCEPT RADAR A severe problem with military radar is the vulnerability of its transmitted signal to exploitation by an enemy. These radars normally emit high-power radiation. Therefore, a typical radar transmitted signal can be easily intercepted and homed in on by relatively simple receivers at ranges well beyond that of the radar. Problems faced by the tactical radar user are (1) enemy detection of the high-value radar platforms, (2) antiradiationmissile (ARM) attack, and (3) enemy electronic warfare countermeasures. We will examine the advantage of high-resolution radar for reducing vulnerability to interception of the radar transmitted signal. The term electronic support measures (ESM) refers to methods and equipment for interception and analysis of emissions from radars. The development of radars that can carry out useful surveillance, guidance, and tracking functions while remaining immune to enemy detection has been pursued, at least sporadically, since the 1960s. This development has been one of the most controversial in radar. It has been claimed by some that it is impossible ever to realize practical LPIR, because the ESM receiver detects the radar over a one-way path, while the radar detects
514
targets over a two-way path. Some proponents argue, on the other hand, that the radar "knows" the exact nature of the waveform it is transmitting and could successfully exploit this advantage. The actual situation, however, is now generally recognized as being much more complicated than could be inferred by either argument. It is not the intent of this section to resolve the issue, but rather we will assess the role played by radar bandwidth and resolution. Four generic methods for achieving low probability of intercept (LPI) are suggested in Figure 10.16. Spread spectrum, with echo signal integration carried out over extended time, is probably the most commonly recognized LPI method. Here, the radar waveform is spread over a wide band of frequencies, while at the same time there is a relatively long integration time associated with processing the target signals. Two example waveforms are illustrated in Figure 10.17. Both the radar transmitter and the receiver LO in the CW waveform of Figure 10.17(a) may be shifted in frequency to generate a constant IF. Such a waveform was described in Chapter 4. Various forms of high-duty-cycle waveforms, such as that indicated in Figure 10.17(b), may also be employed to spread the transmitted energy in time and frequency. A second LPI method, not necessarily involving wideband waveforms at all, is simply to operate at a frequency for which atmospheric absorption is relatively high. The lowest of these frequencies is the water-vapor absorption line at 22.234 GHz. This technique is illustrated in Figure 10.18. For a uniform radar-to-target and radar-to-interceptor propagation loss environment, the path loss as seen by the radar will just equal that seen by the interceptor when its range to the radar is twice the radar's range to the target. For larger interception ranges, the path loss advantage belongs to the radar. For practical situations, the propagation environments are nonuniform. Absorption decreases rapidly with elevation so that the radar antenna beam must be shaped in elevation to prevent radiation at high elevation angles, where atmospheric loss would not offer protection. Propagation loss at the water-vapor absorption line also changes with humidity. Operation at this frequency results in problems of reduced radar performance in rain, although this can be countered to some degree by shifting the frequency out of the absorption peak, thereby allowing the rain to provide part of the required path loss. Analysis and experiments have suggested that it may be possible to obtain a radar range of about 15 to 25 nmi against some low-elevation targets, while maintaining operationally useful signal-interception resistance. A third method for achieving LPI is to operate with multiple-simultaneous antenna beams to increase target dwell time without compromising target revisit time. An example of this method will be given later. A fourth method to obtain LPI, called signal matching, is mentioned for completeness. The concept is to operate the radar with waveforms, beam patterns, and antenna rotation rates that can be confused with radars that are not considered threatening by an enemy. As in the case of the absorption method, wideband waveforms are not necessarily involved here.
SIS
f ^ S T O ^ L MATCHINQ^
Figure 10.16 LPIR techniques.
103,1 Basic LPIR Expressions The spread-spectrum, extended-time-integration method will be discussed further to illustrate the role of wide-bandwidth, high-duty-cycle waveforms to achieve LPI. The combined advantages of high-resolution and multiple-beam processing will be considered next. First, however, we will outline some basic principles involved in analyzing LPIR performance.
516
»n-1
«n-1 »0
I fl
7 o
(a)
J (b) Figure 10.17 Spread-spectrum, emended-lime-integration waveforms for LPIR: (a) CW waveform; (b) highduty-cycle waveform.
The radar transmitter power P, required to produce an echo signal power 5 at the receiving-system input from a target of cross section rrat free-space range R is expressed from (2.7) as i
,
S(.4n) R L P,=
(10.27)
2
G)\ cr
where L is the total radar system loss (Li. 1), including two-way propagation loss and C, is the gain of the radar antenna. The maximum free-space range R at which the radar's transmitted signal can be intercepted with an ESM receiver of sensitivity 5, through a receiver antenna of gain C, is expressed as t
(10.28)
J
V(47T) 5,i S,L,
where L, is the total ESM receiver system loss (L2. 1), including one-way propagation loss. By solving (10.28) for the radar transmitter power given (10.27), with echo signal power S equal to radar receiver sensitivity S the maximum signal-interception range for a radar that is just able to detect a target of size tr at up to radar range R, becomes n
„
D
,/47r
L
G,
SA"
(10.29)
5/7
(b) Figure 10.18 Exploitation of RF absorption to achieve LPI: (a) LPIR operation in a uniform absorption-loss environment; (b) atmospheric absorption loss versus frequency.
Note that wavelength and radar power do not appear in (10.29). Radar signal-interception range is reduced for the following: large target cross section, high ESM receiver loss, low radar loss, low ESM receiver antenna gain, high radar antenna gain, poor ESM receiver sensitivity, and good radar receiver sensitivity. Equation (10.29), when applied to search radar, contains interrelated terms. Radar receiving-system sensitivity depends on the dwell time of the radar antenna beam on the target. The dwell time, in tum, for azimuth rotation scanning depends on antenna rotation rate and azimuth beamwidth. The dwell time for a phased-array radar depends on the required surveillance volume and average revisit time, as well as on the scan strategy selected for a particular mission. The azimuth and elevation beamwidths relate to antenna gain. Despite the interrelated nature of terms in the expression (10.29) for interception range, it is possible to use it to investigate the effect of radar bandwidth and other parameters on LPI performance. The best radar receiving-system sensitivity occurs when the receiving system is matched to the target echo signal that occurs during the available beam dwell time on target. Radar receiving-system sensitivity, from (2.46) of Chapter 2, ideally, can approach
518
5 »*7-i|
(10.30)
r
where t is the equivalent rectangular target dwell time and S/N is the signal-to-noise power ratio required at the output of the radar receiving system to declare a target, based on target echo energy integrated during the dwell time. The role of the transmitted radar waveform bandwidth for improved radar performance will first be discussed, for purposes of illustration, for two hypothetical singlebeam search radar examples. The additional advantages of including multiple-beam processing will be illustrated afterward. 4
103.2 Examples Single-Beam LPIR Example I As afirstexample, assume a radar having a single antenna, specifically, a rotating antenna operating at IS rpm (4-sec data rate) with an equivalent rectangular beamwidth of 2 deg in azimuth and an antenna gain of 40 dB. The transmitted bandwidth will be assumed to be 2 GHz. The periodic, discrete frequency-coded CW waveform in Figure 10.17(a) will be assumed to have the capability of lossless coherent signal integration over target dwell time, which is 1
2
A
3 6 0
(10.31)
= 0.022 sec The radar receiving-system sensitivity from (10.30) for an assumed 15-dB output SNR requirement for signal-dwell detection and 900K. system noise temperature is S, = (1.38 x 10-") x (900) x ( - j - ^ x 31.6
V
0022
/
(10.32)
= 1.78 x 10-"W (-167 dBW) Next, consider the ESM receiver. An operational ESM receiving system, for practical reasons, is not likely to have the sensitivity of the radar receiving system. Further, it is not usually practical to optimize for detecting any particular type of radar. The ESM receiver can often be represented adequately by a wideband predetection amplifier, covering a specified radar frequency band of interest, followed by a relatively narrowband postdetection video amplifier. The ESM receiver's sensitivity is then dependent on its selected predetection and postdetection bandwidths. This simple model of an ESM receiver.
519
shown in Figure 10.19, is convenient for comparative analysis of the interception susceptibility of different radar designs. Actual ESM systems vary considerably, as indicated in Table 10.3. Assume that interception is possible with an output signal-to-noise ratio S/N. ESM receiving-system sensitivity is then approximated by the expression
S,-kTJ^j
(10.33)
where 0, is an involved function of predetection and postdetection bandwidths, signal level, video detection characteristics, and type of signal. The quantity T, is the system noise temperature of the receiving system. An approximate expression for 0, of the receiver in Figure 10.19 with square-law detection is [6] 0, - -flaji,
for a > 8
(10.34)
for which a is the predetection bandwidth and B is the postdetection bandwidth. Typically, postdetection bandwidth is set to correspond to expected pulse widths of pulsed radars likely to be encountered. For purposes of illustration, assume a predetection bandwidth a matched to the CW radar's transmission bandwidth of 2 GHz and a postdetection bandwidth 0 of 1 MHz. With these values for a and 0 in (10.34), 01 = 63 MHz. Assume further that T, = 5.000K and S/N = 20 (13 dB). The receiver's sensitivity according to (10.33) for these parameters becomes ANTENNA
)
PREDETECTION a
DETECTION
POSTDETECTION P
Figure 10.19 ESM receiver model.
Table 10J Generic ESM Receiver Systems Receiver Types Crystal video Wideband predetection (Fig. 10.19) Frequency-tuned superheterodyne Instantaneous frequency measurement (IFM)
Antenna Types High-gain scanning antenna Omnidirectional (in azimuth)
520
IJ
6
S, = (1.38 x l(r ) x (5000) x (63 x 10 ) x 20 (10.35)
= 0.87 x 10-'° W (-101 dBW) 10
A value of S, = 10" W (-100 dBW) will be assumed here. Typical antenna gain for an ESM receiver may be about 10 dB to obtain nearly omnidirectional coverage in azimuth. Table 10.4 summarizes radar and ESM parameters for this first LPIR example. Radar LPI performance will be examined for 100-nmi and 20-nmi free-space radar ranges against a 1-m target. With the parameters of Table 10.4, using (10.29), the free-space range within which radiation from the main beam of the radar could be intercepted is 1
1
= (,85,200)^-x —
7
8
x
l0
~" x12. ^ x 10 1 x 104 X
10
(10.36)
= 1,026 km (554 nmi) If the radar power were reduced so that it would just detect the same sized target at 20 nmi, the range at which radiation from the main beam could be intercepted is 22 nmi. Single-Beam
LPIR Example 2
As a more complex example, consider the LPI potential of the hypothetical long-range, 3-D air-search radar described earlier in our radar ECCM performance example. The LPI potential of this radar will be calculated from its parameters given in Table 10.1, except for peak power, which will be a dependent variable to be calculated. For this second LPIR example, assume the same ESM receiver parameters as in Example 1. The ESM receiver bandwidth extends beyond the operating frequency range of the radar. The performance for the 3-D air-search radar example is calculated in Table 10.5 and the results appear in Table 10.6. Detection of radiation from the radar's main beam in its short-range surveillance mode can be seen to occur when the ESM receiver platform approaches to within a freespace range of 56 nmi from the radar. For long-range area defense (100 nmi) against 10Tabic 10.4 Hypothetical ESM Receiver and Radar Parameters ESM Receiver G,= IO(IOdB) S, = 10-" W (-100 dBW) £,= I O ( I O d B )
Radar 67, = 10* (40 dB) S, = 1.78 x 10-" W (-167 dBW) L * 4 (6 dB)
521
Table 10 S LPIR Performance Calculation Sheet Parameter Hits per dwell Required receiver output SNR
Predetection bandwidth Postdetection bandwidth Radar pulse width (10% duty factor)
Symbol
ESM Receiver
Radar
n
NA 20(l3dB)
a
2 GHz 1 MHz NA
3 (LPRF) or 55 (HPRF) 1.12 (0.5 dB) for 55 hits: 7.9 (9.0 dB) for 3 hits: Swerling case 1' >,, = 0.5./>„= KT*) NA
fi T,
Receiver bandwidth Receiving system noise temperature Bollzmann's constant ESM system sensitivity calculated from kT&SM) Radar system sensitivity, calculated fromtrxl/T.XWV)'
fi
Target radar cross section System loss Antenna gain
a
T, k s, S,
y]2a0 = 63 MHz 3.000K 1.38 x Ifr" J/K 0.87 x I0-* W (-100 dBW) NA
NA IO(IOdB) lO(IOdB)
G,.G,
20 MS for 55 hits per dwell: 333 us for 3 hits per dwell NA 600K 1.38 x Ifr" J/K NA 4.64 x I0- W. n = 55 (-153 dBW); 1.96 x 10-" W. » = 3 (-157 dBW) 1. 10 m' 2 (3 dB) 10* (40 dB) 14
*Radar performance might be improved over that for SC-I if wide bandwidth were obtained by pulse-topulse frequency agility. See Section 10.4. 'Equation (2.46) of Chapter 2.
Table 10.6 Calculation of Free-Space Interception Range R, for Two Values of Free-Space Detection Range R by Using (10.29) With the Results of Table 10.5
1 m> 10 m"
R
R, for Low PRF In = 3)
R, for High PRF (n = 55)
20 nmi 100 nmi
56 nmi 441 nmi
86 nmi 678 nmi
r 1
m air targets, main-beam interception from Table 10.4 can occur at free-space ranges of up to 441 nmi. Interception at greater ranges can be seen to occur for the higher radar PRF. This is because the radar then requires more peak power to detect targets of the same size at the same range. The radar peak and average power required to achieve these results when operating in the low-PRF mode is obtained from (10.27) with S = S . By using the parameters of Table 10.5, (10.27) is evaluated in Table 10.7 for A = 0.1m. r
522
Table 10.7 LPIR Transmitter Power Calculations by Using (10.27) With the Results of Table 10.5 (Low PRF) Transmitter Power R
Peak (P.)
Average (for 10% duty)
20 nmi 100 nmi
1 m» 10 m'
I.5W 92.0W
0.15W 9.2 W
Multiple-Beam
LPIR Example
A potentially powerful, though complex, approach to achieving LPI performance is to transmit and receive through multiple simultaneous antenna beams with high-resolution waveforms. In this approach, the multiple-beam processing allows increasing dwell time without compromising surveillance revisit time. High-resolution processing tends to convert fluctuating targets into highly resolved, steady responses, which provide detection at a reduced SNR. Coherent processing over the increased target dwell time provides more received signal energy for a given level of radiated RF power density at the target than provided with single-beam scanning at the same revisit time for a given aperture size. The advantage of multiple-beam processing is illustrated in Figure 10.20(a) for a hypothetical 10 beam azimuth-scanning antenna compared to a single-beam system. In this idealized illustration, the beams are instantaneously shifted from one dwell to the next. For a revisit time of 4 sec, the 10-beam system is thus capable of achieving 10 times the target dwell as the single-beam antenna. Implementation of multiple-simultaneous-beam surveillance will refer here to transmission and reception through n» separate RF connections to a single aperture. These antennas are also called focal-plane arrays in contrast to conventional aperture-plane arrays, which produce a single steerable beam. An example of a simple multiple-beam antenna is a parabolic reflector type of antenna with several closely spaced feeds stacked vertically near the focal point. A separate transmitting or receiving beam then exists for each feed. Azimuth scanning with multiple beams stacked in elevation is illustrated in Figure 10.20(b). When operated as a radar, each beam in this example scans its own solid angle of coverage so that during each scan the radar surveys n» times as much solid-angle coverage as the same aperture fed by a single feed. A single transmitter could supply RF power to the entire array of feeds, but an independent receiver, processor, and detector function would be required for each independent beam. Discussion of practical approaches for achieving high-resolution multiple-beam radar surveillance is beyond the scope of this book. Rather, the advantages and potential performance will be assessed for idealized processing of high-resolution range data collected in each beam during the extended dwell time provided by multiple-beam scanning.
523
DWELL 2
RADAR Figure 10.20(a) Single-beam compared lo multiple-beam scanning.
Contiguous m-cell segments of high-resolution video range data extending over the surveillance range for each beam are converted into coarse-range cells. The response in each coarse-range cell is the summation of the high-resolution video responses in each m-cell segment, where m is selected to correspond to the range-delay extent of expected targets. Target detection is based on the amplitude of the summed response in each coarse-range cell. High-resolution processing for each beam will be assumed to convert each fluctuating target into a group of equal steady point-target responses that are distributed in range delay within the range-delay extent of the target. These responses are square-law-detected before summing to form a nonfluctuating response to the target in a given coarse-range cell. The average cross section of the unresolved target is assumed to be approximately
524
/ Radar
Figure 10.20(b) Azimuth scanning with multiple beams slacked in elevation.
equal to the sum of the cross sections of the resolved scattering elements [7]. Therefore, the received signal power from the average resolved, nonfluctuating-target scattering element can be treated as equal to the signal power produced by the target's average cross section divided by the number n of significant scatterers. This idealization allows LPI performance to be expressed in terms of a nonfluctuating but conventionally specified target RCS, rather than in terms of the RCS of resolved scatterers, which is more difficult to specify. In practice, some target fluctuations would remain. All reflection sources of actual targets would likely not be resolved. Those that would be might not be resolved into point targets, but rather into reflection sources that produce range-extended echoes. Even resolved point targets would produce range-extended responses because of time sidelobes associated with the radar. As the target changed aspect to the radar, interference among echoes of unresolved reflection sources and the range-extended responses of resolved and unresolved sources would produce fluctuating responses to be summed noncoherently. The resulting summed response would then also fluctuate, but mildly it is hoped, as compared with the single-frequency response. Finally, deviations from the assumption of equal-size scatterers will affect predicted LPIR performance.
525
Resolved responses from individual scatterers of the target, regardless of radar waveform, will be assumed for this analysis to be steady signals that can be coherently integrated during the entire target dwell time tj before square-law detection into video signals. The receiving system can then be assumed to be matched to nonfluctuating signals of duration t . Target detection is based on the video sum of the n video signals produced by the n significant resolved scatterers within an m-element coarse-range cell. The resulting SNR can then be assumed to be that produced by a nonfluctuating target of cross section 7r/n, where a is the average cross section of the unresolved target. The sensitivity of the receiving system when written as the minimum signal power, averaged over dwell time tj, required for detection is expressed from (10.30) as 4
d
Sr=kT,yJ-
(10.37)
where S/N is the output SNR, for the video response to the average resolved scatterer, required for detection of the summed video responses. The receiving-system sensitivity of (10.37), when written in terms of pulse duration T\ and PRI T , is expressed as 2
< i¥i
s
=kT
(,038)
The quantity TJTt is the reciprocal of the radar's duty cycle. Average dwell time on target during n beam dwells of the array of beams is d
r, = -
(10.39)
where T, is the surveillance revisit time. If the half-power beamwidths in elevation and azimuth are 8, andtf> . respectively, the number of beam dwells required to scan a solid-angle surveillance coverage of ft sr, assuming 4ir > 9 x dS , is given by m
3d B
J l S
1
^nimr
)dB
ft ( 1 0 4 0 )
for n simultaneous beams arranged so that adjacent beams and beam dwell positions cro^s at the half-power points in azimuth and elevation. The average dwell time on target from (10.39) with (10.40) then becomes b
(.^y'
(10.4D
4. Note that the receiver is matched to the signal from individual resolved target scatterers, not to the target's high-resolution signature, which is usually unknown.
526
so that the sensitivity in terms of peak echo power from (10.38) with (10.41) is expressed as ns/N
T
2
'Mr03dB
h
Antenna gain and half-power beamwidth are related by the expression G, = j ^ - F \
(10.43)
where F, is typically quoted as having values between about 0.65 and 0.9. By substituting for ^3dBv*3dB from (10.43) into (10.42), the radar receiving-system sensitivity becomes
and the interception range, from (10.29) with (10.44), becomes
x
^ F ^
x
/ v
x
r ; j
( 1 0
-
4 5 )
The interception range for the radar and ESM receiver combination specified in Table 10.5 was calculated by using (10.45) for an elevation coverage of 16 deg (0.28 rad) and a surveillance revisit time of 6 sec. The SNR required for detection is based on treating the n resolved video signals as n pulses integrated in Figure 2.8(b). Results are plotted in Figure 10.21 assuming n = 10 prominent scatterers of average RCS = 7Hn. The bandwidth, while not an explicit parameter of interception range as calculated by (10.45), was assumed adequate to resolve targets into nonfluctuating scatterers,and to force the enemy to operate with a wide (2-GHz) predetection bandwidth. Potential improvement over that of the single-beam version of the Table 10.5 radar system can be dramatically shown by comparing results in Figure 10.21 with those in Table 10.6. Small changes in the estimate of the number of prominent scatterers, since it is coupled with required SNR of each corresponding video response, and small deviation from the assumption of equal scatterers can both be shown to produce only slight changes in the calculated results. An actual high-resolution multiple-beam system, however, will fall short of providing the equivalent of lossless coherent integration (true matched-filter processing to individual target reflection points), and we would not achieve the idealized result of eliminating target fluctuation. An estimate of the associated losses can be included in the total radar loss term L.
527
528
10.3.3 Some Final Remarks Regarding LPIR Factors that would enter into a more complete evaluation of radar LPI performance are as follows. 1. 2. 3. 4.
Atmospheric propagation loss and ducting; Radar horizon relative to target and ESM receiver; Potential for highly sensitive, special-purpose, multiple-channel ESM receivers; ESM receiver platform type (surface or air).
An ESM receiver could use a high-gain scanning antenna to search for radar sidelobes, but then the probability per scan of intercepting radar main-beam radiation for practical scanning rates would be very low. To achieve the same interception ranges against radar antenna sidelobes as for omnidirectional mainlobe interception, the gain of the scanning antenna, all other factors being equal, would have to be increased with respect to that of the omnidirectional (azimuthal) antenna assumed above. In fact, the gain increase required is equal to the radar antenna's peak-to-sidelobe ratio. For example, if that ratio is 33 dB, the interceptor antenna of Table 10.5 with a gain of 10 dB will have to be increased in size to produce a gain of 43 dB. In the example of Table 10.5, this happens to be greater than the radar antenna gain, and for a wavelength of 0.1m would require something on the order of a 20-ft-diameter aperture. This aperture size is not likely to be practical for most applications. Therefore, we may conclude that mainlobe interception is probably the worst case that the radar will encounter in LPI operation. We should note that radar bandwidth and resolution are but two factors that determine LPI capability. Other important factors have been shown to be the number of simultaneous beams and the required surveillance revisit time. A more complete discussion of radar signal interception is given by Wiley [8]. 10.4 REDUCTION IN TARGET FLUCTUATION LOSS FOR SURVEILLANCE RADAR In the above discussion of LPI radar, we evaluated means for processing high-resolution data to reduce the increased transmitted power required to overcome the loss in receiver sensitivity caused by target fluctuation. We will now refer back to the definition of target fluctuation in terms of fluctuation models and define fluctuation loss. Then we will analyze the application of wide radar bandwidth and high-resolution processing to reduce fluctuation loss in order to improve radar range performance of surveillance radars apart from consideration of LPI performance. 10.4.1 Sources of Fluctuation Loss As discussed in Chapter 2, a fluctuation loss is associated with the detection of most targets of interest. The source of this loss is the target's fluctuating reflectivity as seen
529
by the radar as the target is viewed at continuously changing target aspects. A target's narrowband reflectivity observed at typical surveillance frequencies, antenna scanning rates, and target dwell periods is relatively steady during a single beam's dwell, but varies significantly from scan to scan. Figures 2.9 and 2.10 of Chapter 2 illustrate that the RCS of aircraft targets can be expected to vary significantly, even for small aspect changes on the order of a degree or less. Therefore, a target's pitch, roll, yaw, turn, and tangential translations relative to the radar produce large scan-to-scan variations in target reflectivity. The effect is even greater for ship targets, which are more complex and normally experience more pitch, roll, and yaw motion than aircraft. One common mathematical description of target fluctuation is referred to as the exponential form of the Rayleigh probability density, which we used earlier in this chapter to represent sea-clutter reflectivity statistics. The probability density of the instantaneous cross section for a target of average cross section cr, when represented by this form, is expressed according to (10.18) as P(
(10.46)
exp(-or/tr)
The single-pulse probability of detection computed for this representation and that for a Swerling case 3 representation, given by (10.47) are compared [9] in Figure 10.22 with respect to target fluctuation loss. Fluctuation loss Lf is defined as the ratio by which the required average SNR produced by a fluctuating target must exceed that produced by a steady target to achieve a given probability of detection. The fluctuation loss plotted in Figure 10.22 applies for a wide range of falsealarm probabilities centered about P = 0.33 [9]. Note that loss exists only for values for detection probability above P = 0.33. Below this value, fluctuations of the target echo enhance rather than reduce detection relative to a steady target. Only high values of P will be considered in our discussion of fluctuation loss reduction to follow. Fluctuation loss also occurs when target reflectivity is sampled (at a single frequency) with multiple pulses per beam dwell. Multiple-pulse fluctuation loss was illustrated in Figure 2.8 of Chapter 2 for the Swerling case 1 model of a fluctuating target with P . Here, we can see that loss is about 5.2 dB for P - 0.8 and 1.5 dB for P = 0.5, regardless of the number of pulses integrated before detection. 0
D
D
FA
D
D
10.4.2 Frequency-Agility Method A search-radar design that exploits narrowband but pulse-to-pulse frequency-agile waveforms to improve ECCM or LPI performance may also inherently provide improved
530
-8
0
+8
FLUCTUATION LOSS L
+16 (
(dB)
Figure 10.22 Fluctuation loss versus detection probability. (From D. K. Barton, Radar System Analysis, Dedham. MA: Artech House, 1979, p. 24. Reprinted with permission.)
target-detection performance relative to an otherwise equivalent design operating at a single frequency. The result is improved detection for high-probability-of-detection requirements when multiple pulses transmitted during a dwell period are sufficiently spaced in frequency to produce independent measures of target reflectivity. In other words, there is a reduction in the loss of detection performance associated with target fluctuation. Reduction in detection loss by this means becomes significant for slowly fluctuating targets, as represented by the Swerling case 1 model, and when the single-scan detection probability criterion is greater than about 50%. Target fluctuation as seen by a scanning search radar is illustrated in Figure 10.23. The target's narrowband echo signal, due to aspect motion relative to the radar, changes rapidly relative to a scan period (typically, 2 to 10 sec), while very little change occurs during a beam dwell (typically 5 to 50 ms). A single-frequency radar then sees a steady target during a beam dwell. In other words, target echo pulses received during a beam dwell tend to be of equal amplitude, rather than independent samples of the slowing fluctuating echo response. The average echo signal level in Figure 10.23 necessary to obtain a reasonably high probability of detection on each dwell must be quite high relative to noise to prevent excessive false alarms. However, if the individual pulses available during a dwell could be converted into independent measures of target reflectivity data,
531
NOISE L E V E L , N 1 SCAN PERIOD
1
1
1
1
TIME
_
Figure 10.23 Single-frequency target echo and receiver noise characteristics.
then a lower average signal level relative to noise could result in the same probability of detection and false alarm. The result would be increased range performance. Independent samples of a slowly fluctuating target can be obtained during the short interval of a typical dwell by changing the transmitted frequency from pulse to pulse during the dwell. The frequency change between pulses alters the phase relationships among interfering echo signals reflected by the multiple unresolved reflection points of the target. The set of received pulses obtained during a single dwell, therefore, will vary in amplitude from pulse to pulse, as though we were observing a fast fluctuating target. Thus, a slowly fluctuating target tends to be converted into a fast fluctuating target by employing frequency agility. In terms of the Swerling models, a slowly fluctuating target represented by the Swerling case 1 fluctuation model may approach that of a Swerling case 2 target, as illustrated in Figure 10.24. In this figure, the number of received pulses per dwell that are integrated before detection is taken as 30 and 300. For single-scan detection probabilities below about 33%, Swerling case 1 target characteristics can be seen to require a lower SNR. Above 33% detection probability, Swerling case 2 characteristics can be seen to have the advantage. For a single-scan detection probability of P = 0.9, the input SNR per pulse required for detection is reduced by about 8 dB due to pulseto-pulse decorrelation. The relationships illustrated are only weakly related to the number of pulses integrated or the probability of false alarm P . An approximate analysis will now be carried out by viewing frequency agility as a means for conversion from Swerling case 1 to Swerling case 2 statistics. D
FA
532
99.8
SWERLING 1 TARGET • SWERLING 2 TARGET -
n = NUMBER PULSES INTEGRATED (VIDEO INTEGRATION)
-6
-4
J
I
I
I
I
L
-2
0
2
4
6
8
10
J
I
I
1
L
12
14
16
18
20
22
24
SIGNAL-TO-NOISE RATIO (dB)
Figure 10.24 Single-scan detection probability versus SNR for Swerling case I and 2 targets. (From unpublished material supplied by Tom Lund of Teledyne Ryan, San Diego, California.)
It has been shown [10] that decorrelation occurs when the frequency change between pulses is greater than a critical difference frequency L\f , defined as c
(10.48)
533
where c is the propagation velocity and / is the range extent of the target's ensemble of scattering elements. For example, assume a minimum target dimension in the range direction of 20m. The critical difference frequency L\f necessary to obtain pulse-to-pulse independence from (10.48) is 7.5 MHz. Barton [10] discusses an approximate relationship between the number of integrated independent frequency samples n, and the reduction in fluctuation loss L,. Frequencyagility gain G(n ), obtained by video integration of the n, independent samples, is defined by Barton in terms of reduction in single-frequency fluctuation loss L,( 1) to be c
t
G(A) = [L,0 )]'"""'
(10.49)
10 log G(n ) = ^1 - j^j 10 log Lf(\)
(10.50)
which in decibel form becomes
t
The quantity 7^(1) from Figure 10.24 for P = 0.9 is about 8 dB. The number of available independent samples is limited by the total frequency-agile bandwidth B, and critical bandwidth A/ according to the expression D
r
n,
(10.51)
Available target dwell time may further limit the number n, of independent samples. Plots of frequency-agility gain (reduction in SNR) required for detection compared to that required for a Swerling case 1 target, as a function of the number of independent frequency samples integrated, are given in Figure 10.25. The results plotted in Figure 10.25 are obtained from (10.50) with Lf(\) obtained from Figure 10.24 for several values of P . Also shown in the figure is the Swerling case 2 limit for each plot. The greatest gain is obtained from the first few independent samples. Improvement achieved for more than about six samples is quite small. In the above example, where / = 20m, six independent samples correspond to six pulses during the dwell spaced 7.5 MHz apart for a total frequency-agile bandwidth, from (10.51), of D
B, > (n - 1)A/ = (6 - 1)7.5 = 37.5 MHz t
C
(10.52)
Frequency agility for detection improvement is a well-known technique that is used with a number of existing search radar systems. Frequency-agile magnetrons provide an inexpensive means to achieve wideband frequency agility. Video pulse integration can, for simple radars, be provided by luminosity addition on the plan-position-indicator (PPI) display and by the human observer.
534
Figure 10.25 Frequency-agility gain versus number of independent samples. (From unpublished material supplied by Tom Lund of Teledyne Ryan Electronics, San Diego, California.)
535
10.4.3 High-Resolution Method Frequency agility, as described above, reduces fluctuation loss by video integration of multiple target responses produced by transmitting pulses that are dispersed in frequency pulse to pulse during the beam dwell. A second method will now be described for reducing detection loss by using high-resolution processing to resolve targets into individual responses to scatterers before detection. If extended targets could be resolved into pointtarget scatterers, then the fluctuation for each scatterer would be reduced to zero. As described above for multiple-beam LPIR, nonfluctuating video responses from resolved target scatterers are noncoherently summed before detection. The required SNR, as before, is based on the number n of significant resolved scatterers of the target. Then, as for the LPIR case, the radar range equation can be conveniently expressed in terms of the required S/N produced by a nonfluctuating cross section a/n. The dwell time for a single-beam scanning radar with elevation and azimuth beamwidths 6 and d> , respectively, is given approximately by (10.41) with n = 1 as Jit
iiB
h
where T is the target revisit time and il is the solid angle of surveillance coverage. The transmitted power to be used in the radar equation to compute single-dwell echo power is the transmitted power averaged over dwell time regardless of whether we assume pulsed or CW transmission. Residual fluctuation loss associated with unresolved scatterers will be considered as part of the total radar system loss L along with other processing losses. The radar range equation (2.47) of Chapter 2 can then be written in terms of detection of target scatterers of size a/n as r
(10.54) where dwell time t is substituted for signal duration 7",. With t from (10.53), the radar detection range becomes d
t
(10.55) This expression can be simplified by relating antenna gain and beamwidth using the substitution (10.56)
536
where, as before, F, is an antenna factor that typically varies between about 0.65 and 0.9. The expression (10.55) for radar range based on detection of the summed predetected video responses from the target's resolved scatterers is written in terms of antenna gain from (10.56) as
Waveform bandwidth does not appear explicitly in (10.57), but the assumption of resolved scatterers implies a bandwidth on the order of 500 MHz. Assumptions regarding the summation of predetected resolved responses and aspectaveraged RCS, as for the LPIR analysis, were chosen here to result in a closed-form expression. Potential detection improvement for the idealized concept is illustrated by comparing the respective performance of the radar specified in Table 10.8 for highresolution and low-resolution processing. Performance for high-resolution processing will be predicted by (10.57). Performance for low-resolution processing will be predicted by the more common form of the radar equation of (2.47), given by Table 10.8 Surveillance Radar Parameters Parameter Azimuth beamwidth Elevation beamwidth Average power Peak power Center frequency Antenna gain Solid angle of surveillance System noise temperature Total system loss PRF Average azimuthal rotation rate Target revisit time Pulse width Duty cycle Probability of detection (each dwell) Probability of false alarm (each dwell) Antenna factor Number of prominent target scatterers (assumed equal size)
Symbol
Value
Pb
0.017 rad (I deg) 0.28 rad (16 deg) 3,600 W 100 kW 1 GHz (A = 0.3m) 2.000 (33 dB) 2m%„ = 1.76 sr 500K 4 (6 dB) 360 2«r-rad azimuth scans each 4 sec 4 sec 100 MS 0.036 0.9
PfA
l(r»
n
0.88 10
P, f G n
T, L 1/Ti
T, T, 7-,/r,
537
l
R =
P,G \*T a (4TT)\ kT,(S/N)L x
(10.58)
:i
where P, is the peak power. In each case there are about four pulses per dwell. For the high-resolution mode, sampled high-resolution range data from the four transmitted pulses per dwell are assumed to be first coherently summed, and then predetected to form the set of n HRR video responses extending over the desired range coverage. For lowresolution processing, the four narrowband output pulses per dwell are first predetected to form a set of low-resolution video responses, which are summed before making a detection decision. The SNR in (10.57) for high-resolution processing will be assumed to be that required to meet the P and P criterion in Table 10.8 for each of the n steady video responses that are summed for detection. The SNR in (10.58) for low-resolution processing will be assumed to be that required to meet the P and P criterion in Table 10.8 for a fluctuating target (Swerling case 1) with the integration of four predetected video pulses. Detection parameters for each type of processing are summarized in Table 10.9 and the results are plotted in Figure 10.26 for a target of 10 resolved scatterers. A range improvement of nearly 2:1 with high resolution is predicted for this example. This improvement results from (1) the assumption of resolved steady responses for high-resolution processing compared to a fluctuating target for low-resolution processing, and (2) the advantage of coherent processing during the entire beam dwell for the high-resolution case over fourlook video integration for the low-resolution case. Pulse-to-pulse coherent integration may not be feasible for practical applications. This would slightly reduce advantage for highresolution radars for which multiple pulses per dwell occur. D
rk
D
fK
10.5 DETECTION OF SMALL, SLOWLY MOVING TARGETS IN CLUTTER Target returns immersed in sea, land, weather, or chaff clutter are typically detected using velocity discrimination. If a target has a higher radial velocity relative to the radar than that of the clutter sources, the target can often be detected in the presence of the clutter by MTI processing. However, if there is a requirement to detect small, slowly moving or Table 10.9 Low-Resolution and High-Resolution Detection Parameters for the Design at Table 10.8
Parameter Number of video responses integrated Assumed target fluctuation model Required SNR
Expression
Low-Resolution Design
High-Resolution Design
n
4 pulses
S/N
Swerling case I 55 (17.4 dB)
10 HRR video responses Steady target 4 ( 6 dB)
538
539
small stationary targets in clutter, then MTI cannot provide discrimination against clutter return on the basis of velocity. High-resolution methods have proved to be useful for this application. By reducing range-cell size down toward the range extent of the target, the average clutter cross section becomes smaller relative to that of the target. A threshold criterion can then be selected to produce false alarms on only high clutter spikes. A typical design will declare a detection based on a selected number of threshold crossings during a given number of antenna scans. For example, a target may be declared for three crossings in five scans. PROBLEMS Problem 10.1 The radar of Table 10.1 operates over a 150-MHz pulse-to-pulse frequency-agile bandwidth. A jammer having an effective radiated power of lO'W toward the radar is operating at a 50-nmi (92.6-km) range to the radar with a uniform noise bandwidth of 250 MHz. (a) What is the effective system noise figure of the radar? (b) What is the radar range against a 1-m target? Assume that the radar antenna sidelobe levels are expected to be -40 dB relative to the peak antenna gain. Assume the high-PRF mode and no sidelobe cancellation. 2
Problem 10.2 The duty cycle of the radar of Table 10.1 in a bum-through mode can be increased briefly to 25% in critical coverage areas when jamming is severe. Assume -35-dB antenna sidelobe levels and 25-dB sidelobe cancellation. What is the burn-through range performance against a 1-m target for the standoff noise jammer considered in Figure 10.1 if jammer power is spread uniformly over the radar's 200-MHz transmitter bandwidth? Assume the high-PRF mode. 2
Problem 10.3 Self-protection noise jammers on board attacking aircraft attempt to deny range information to radars on the defending target. Show that range information is first obtained by the defending search radar at a closing range of
P,GaT,L,0, 4njjPjGjL
540
where S/N is the signal-to-noise (jamming) ratio per pulse required for ranging. Assume that receiving-system noise is predominately due to noise jamming. (The terms L and L, represent total radar system loss and radar receiving-system loss, respectively. Other parameters correspond to those in Table 10.1.) Problem 10.4 The radar of Table 10.1 is performing shipboard air surveillance in its high-PRF mode. A standoff jammer is not present, but the ship is under attack from an aircraft (cr= 5 m ) equipped with self-protection jamming equipment. The jammer transmits 50W over a frequency band covering the radar's 200-MHz frequency-agile bandwidth through a 12dB gain antenna directed at the ship. From Problem 10.3, at what closing range does the radar first obtain range information on the attacking aircraft? Assume range information is obtained at the same SNR as for target detection. Also assume that all radar losses arise from receiving-system losses so that L = L,. J
Problem 10.5 A high-PRF radar is to be designed to operate at 9.4 GHz for airborne intercept of other aircraft. What is the minimum required PRF to avoid ambiguous velocity responses for closing speeds of up to 2,500 m/s? Neglect clutter. Problem 10.6 A low-PRF MTI radar is to be designed to operate at 425 MHz for airborne early warning against air targets, (a) What is the PRF that results in the first blind speed for a target at a velocity of 500 m/s? (b) What is the maximum unambiguous range at this PRF? Assume MTI is achieved using a single canceler. Problem 10.7 The Rayleigh density of x for x S 0 is defined as f(x) =
-s'W
where a is a constant. Show that when the amplitude of the response from sea clutter is said to be Rayleigh distributed, the sea clutter cross section itself has the probability density given by (10.18). This is sometimes referred to as the log form or exponential
541
form of the Rayleigh density. (Hint: Let the instantaneous sea-clutter cross section a = x and let the average sea-clutter cross section cr = 2a , which is the expected value of c
2
2
0
Problem 10.8 3
If the clutter coefficient a" of sea clutter is 10~ for a low-flyer geometry, what is the probability for any pulse during the dwell that clutter return from a range of 10 km will exceed that of a target of a 1 -m cross section observed with a shipboard surface-search radar having 0.5m resolution? Assume a fan-beam antenna of 1-deg azimuth beamwidth and assume a Rayleigh probability density for the clutter statistics. 2
Problem 10.9 Assuming sea-clutter decorrelation time of 0.05 sec, how much data collection time is required to confirm a clutter-statistics model at threshold value y for which P[
5
Problem 10.10 Low-flyer detection is to be carried out with a noncoherent MTI radar design by using the wideband delay-line canceler described in the text. The compressed pulse width is 2 ns. Radar PRF is 1,000 pulses per second. What is the minimum detectable target radial velocity based on the criterion that the pulse-to-pulse delay difference must be greater than one-half the compressed pulse width? Problem 10.11 Show that the ratio of radar received echo power from a target to power received by an ESM receiver in the target at range R is given by (4tiiA )(G /Gi)g(R)cr, where g(R) is the range attenuation factor defined in (2.27), G, is the radar antenna gain, G, is the gain of the ESM receiver antenna, and a is the target's radar cross section. 2
r
Problem 10.12 A shipboard LPI radar is to be developed at 22.234-GHz center frequency with the objective of providing 15-nmi detection range against a 0.1-m target while remaining 2
542
immune to signal interception beyond 50 nmi by an ESM receiver of -70-dBm sensitivity and 10-dB antenna gain (when system loss is included). This requirement is to apply for both target and ESM receiver at 0-deg elevation and for standard atmospheric conditions and no multipath. What is the minimum required radar antenna gain if the receiving system is to be matched to a target dwell time of 10 ms? Required radar SNR per dwell is 16 dB, system noise temperature is 1,500K, and radar system loss is 6 dB. (The twoway propagation loss at 0-deg elevation under standard conditions at 22.2 GHz is 9 dB and 32 dB for 15 nmi and 50 nmi, respectively, at sea level.) Problem 10.13 An ESM receiver operates with an antenna having 0-dB gain. Total system loss is 6 dB. Predetection bandwidth is 4 GHz and postdetection bandwidth is 0.25 x 10 MHz. The system noise temperature is 2.500K. At what free-space range are -45-dB sidelobes of the radar in Table 10.1 detectable, assuming that an SNR of 16 dB per pulse is adequate for signal-interception confirmation? Assume negligible atmospheric loss. 6
Problem 10.14 The radar evaluated in Table 10.5 is modified for improved LPI performance by operating with a low-power, frequency-coded periodic waveform (100% duty) to permit lossless coherent integration of the target echo signal for each dwell. What is the new free-space signal-interception range with the same ESM receiver if radar transmitter power is adjusted to just detect a 1-m Swerling case I target at 20 nmi with P = 0.5 and P = 10" for each dwell? Assume that the target is steady during each dwell (i.e., equivalent to one pulse integrated per dwell). Assume that the antenna has a 1-deg azimuth beamwidth and scans at 15 rpm. 2
6
D
FA
Problem 10.15 2
A multiple-beam shipboard radar is to operate at just enough power to detect 1-m antiship missiles at a free-space range of 20 nmi. Radar and ESM receiving-system parameters are as defined in Figure 10.21 except that the target consists of a single prominent scatterer instead of 10 equal resolved scatterers. What is the percentage decrease in intercept range from that predicted in Figure 10.21 for the 10-scatterer target? Use Figure 2.8 for P = 0.5, P = lO". D
fA
Problem 10.16 A high-resolution radar resolves a target into 30 steady responses of roughly equal amplitude scatterers. Determine the percentage decrease in intercept range to a radar having a
543
receiving system somehow able to perform detections based on the matched-filter response to the target's steady HRR signature itself compared to that for a radar having a receiving system that performs detections based on the sum of the video forms of the matchedfilter responses from each of the 30 scatterers. Assume equal target range and dwell time on target and P = 0.5, P = 10~ . Use Figure 2.8. 6
D
TK
Problem 10.17 A radar with a PRF of 400 pulses per second scans at 30 rpm while azimuth searching for targets on the sea surface. The effective rectangular beamwidth is 2 deg. What is the maximum possible frequency-agility gain with P = 0.8 and P = 0.7 x 10" for each dwell over that for a single-frequency radar? Use Figure 10.25. ffi
Problem 10.18 What minimum frequency-agile bandwidth is required for the radar of Problem 10.17 for optimum detection of ship targets of 50m expected range extent? Problem 10.19 A fast-scanning (180 rpm) airborne anti-submarine warfare (ASW) radar is designed to detect small, slowly moving targets in sea clutter at low grazing angles. The range resolution is 0.3m, the effective rectangular beamwidth is 2 deg and tr° is 10" . (a) What is the signal-to-clutter ratio per scan against a 5-m target at 40-km range assuming one pulse per dwell? (b) How many scan-to-scan noncoherent integrations are required and what is the required surveillance time to produce a probability of detection of 0.8 and a probability of false alarm of 10 based on the signal-to-clutter ratio per pulse for Swerling case 1 target statistics? This corresponds to assuming that the target slowly fluctuates from scan to scan and that the sea clutter is noiselike and decorrelated from scan to scan. 3
2
-6
REFERENCES [I] Long, M. W.. Radar Reflectivity of Land and Sea, 2nd edition, Dedham, MA: Artech House, 1983, Fig. 4.4, p. 102. (2] Long, M. W.. Radar Reflectivity of Land and Sea, 2nd edition, Dedham. MA: Artech House. 1983, Fig. 4.5, p. 103. [3] Chang, C. T., et al., "Noncoherent Radar Moving Target Indicator Using Fiber Optic Delay Lines," IEEE Trans. Circuits and Systems. Vol. CAS-26, No. 12, Dec. 1979, pp. 1132-1135. HI Long, M. W., Radar Reflectivity of Land and Sea, 2nd edition, Dedham, MA: Artech House, 1983. p. 42. [5] Skolnik, M. I., ed., Radar Handbook, New York: McGraw-Hill. 1970, Fig. 3, pp. 26-27. [6] Boyd, J. A., et al.. Electronic Countermeasures, Ann Arbor, MI: Institute of Science and Technology, University of Michigan, pp. 9-41, 9-42, 1961.
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[7] Weinstock, W., Ch. 5 in Modem Radar, R. S. Berkowitz, ed.. New York: John Wiley and Sons, 1965, p. 567. [8] Wiley, R. G„ Electronic Intelligence: The Analysis of Radar Signals, Dedham, MA: Artech House, 1982. [9] Barton. D. K.. Radar System Analysis, Dedham, MA: Artech House, 1979. p. 24. (10) Barton, D. K., "Simple Procedures for Radar Detection Calculations," IEEE Trans. Aerospace and Electronic Systems. Vol. AES-5, No. 5, Sept. 1969, pp. 837-846.
Appendix Using the High-Resolution Radar Software Tutorial
The High-Resolution Radar Software Tutorial is designed to help the reader by demonstrating the waveforms, signal processing, and imaging methods for high-resolution radar described in the book. The tutorial demonstrates each concept by showing an example of the waveform or processing method related to that concept. As much as possible, the tutorial uses realistic parameters from actual radar for the examples. However, the signals and targets used in the tutorials are much simpler than would appear in a real operational radar. So the tutorial will demonstrate radar imaging using a target with just a few scatterers, but an operational radar imaging a ship would detect hundreds of significant scatterers.The tutorial is organized by topics from the book. Each topic of the tutorial is self-contained and can be viewed in any order. However, because the tutorial is based on the material in the book, the appropriate chapter of the book should be read first before viewing the tutorial. Each tutorial topic displays pictures of waveforms and radar images along with text describing the significant aspect of that picture. The tutorial also allows the user to interact with the program and modify the radar parameters for each topic to see how a change in the parameter will change the displayed waveform or radar image. The tutorial assumes that the user understands the following basics of using Microsoft Windows: how to start Windows, how to use the mouse, how to start a program, and how to select commands from the menu bar. This tutorial does not require the user to understand any computer programming. The rest of this appendix discusses how to install the tutorial software, how to use the tutorial software, and a list of the topics covered in the tutorial software. INSTALLING THE TUTORIAL SOFTWARE The High-Resolution Radar Tutorial Software can be obtained by contacting Artech House, Inc. This software requires that the user have the following software and hardware: 545
546
• • • • •
Microsoft Windows Version 3.1; Personal computer with 80286 processor or better; 2 MB of RAM memory; 2 MB of free hard disk space; Video graphics card and monitor (VGA or better resolution recommended).
The tutorial software comes with an automatic installation program and must be installed on the hard disk (it cannot be run from the floppy disk). Use the following steps to run the automatic installation program. 1. 2. 3. 4.
Turn your computer on and start Windows. Insert the Install Disk in your computer's floppy disk drive. From the Program Manager's File menu, choose Run. In the Run window, click in the Command Line box. Then type a: install if you are installing from drive A, or type b: install if you are installing from drive B. 5. Click the OK button to continue. The installation window appears. 6. The installation software puts the tutorial software into the directory C: \HRRTUTOR by default. If the tutorial software needs to be placed in another directory, edit the installation path. 7. Click the INSTALL button to start the installation.
USING THE TUTORIAL SOFTWARE This section of the appendix describes the basic steps for using the tutorial software. There is additional documentation in the "Read Me" file. Additional information also exists on the online help in the tutorial. Starting the Tutorial Software The installation program created a Program Manager group called HRR Tutorial. Open this program group by double clicking on it. In this program group, there are two icons. One icon is used to read the release notes and the other icon is used to start the tutorial software. The release notes contain recent information about the tutorial software that is not included in this appendix. To read the release notes, double click on the icon called Read Me. This will start the Windows Notepad editor and display the release notes for the tutorial software.To start the tutorial software, double click on the icon called HRR Tutorial. This will start the tutorial software and show the topic selection window. The tutorial program uses a standard Windows menu bar and toolbar. All of the operations of the tutorial can be selected by either using the pull-down menus from the
547
menu bar or by using the buttons on the toolbar. Generally, selecting an operation from the toolbar is the most convenient method. Selecting a Tutorial Topic The topic selection window appears when the tutorial program is first started and when the user wants to select a new topic. This window contains a list of all the tutorial topics, a description of the selected topic, an OK button, and a Cancel button. The user selects a tutorial topic by clicking with the mouse on the title for that topic. The selected topic will be highlighted and a short description of it will appear in the description window. A complete list of the available tutorial topics is given in section A.3 of this appendix. To start the selected topic, click on the OK button. Viewing a Tutorial Topic Each tutorial topic is divided into pages. Each page of the topic is displayed with a picture on top of the window and some text that explains the picture underneath. There are scroll bars that can be used to view the explanation text if it is too big to fit into the bottom half of the window. The first page of every topic contains an introduction to that topic and the block diagram for the system described by that tutorial topic. The block diagram shows the physical representation of the system that will be used and also defines the signals that will be described. At any time during the tutorial, the user can select the Block Diagram button on the toolbar to see the opening page again. The rest of the pages for each topic contain a picture of either a waveform plot or a generated radar image. The user can move to the next page or the previous page by selecting the Next Page or Previous Page buttons on the toolbar. At any time, the user can go to a new topic by selecting the Topic button on the toolbar or may exit the program by selecting the Exit button on the toolbar. Experimenting With Changes to System Parameters The tutorial allows the user to modify the system parameters and see the resulting changes in a waveform or in a processed radar image. This allows the user to understand how the parameters affect the resulting waveform or image. Each tutorial topic is a simple example of a radar system arranged to demonstrate a certain topic in the book. In a radar system there are many parameters, such as operating frequency, transmitter power, pulse width, transmitted waveform, and target range. However, in a particular tutorial topic there may be only one or two parameters of interest, such as pulse width and transmitted waveform. For this topic, the user would only be
548
able to modify pulse width and transmitted waveform within a preset range of valid values. The user selects the Experiment button on the toolbar to enter the experiment mode. In the experiment mode, the descriptive text for that page is replaced by a description of each of the parameters that can be modified for this topic along with an edit field to change the value of the parameter. The user presses the Plot button on the toolbar to see the new waveform plotted on the same axis along with the original waveform. The user presses the Return button on the toolbar to leave the experiment mode and return to the tutorial topic page. LIST OF TUTORIAL TOPICS This section contains a list of each tutorial topic by chapter, along with a short description of the material covered in the topics. It is recommended that the chapter in the book be read first before viewing the tutorial topics for that chapter. Tutorial Topics for Chapter 1, Introduction 1.1 Narrowband Approximation. Demonstrates the limitations of the narrowband approximation for representation of high-resolution signals. The signal spectrum and deconvolved signal envelope are shown for several values of modulation fractional bandwidth. Tutorial Topics for Chapter 2, Application of the Radar Range Equation to HighResolution Radar 2.1 Target Resolution. Demonstrates the fundamentals of radar range and velocity resolution using a model of two point targets. The concept of resolution is demonstrated and the effect produced by interference between backscattered radiation from two closely spaced targets is shown. 2.2 Target Fluctuation. Demonstrates the effect of radar resolution on target fluctuation. The unresolved response from a simulated target consisting of multiple scatterers is shown to fluctuate with target view angle and transmitted frequency. Reduced fluctuation is observed as resolution is increased to resolve scatterers. 2.3 North's Theorem. Shows how the coherently integrated response of the received signal plus added noise for a point target has a peak signal-to-noise ratio that is related to received pulse energy and is independent of pulse duration. Tutorial Topics for Chapter 3, High-Resolution Radar Design 3.1 Signal Distortion. Demonstrates distortion that occurs in signals that are passed through systems or system components for which the transfer function exhibits deviation from
549
flat amplitude and linear phase over the signal spectrum. The convolved response of wide fractional bandwidth signals with the impulse response of linear systems is shown for system transfer functions containing assigned amplitude ripple, phase ripple, and other sources of delay dispersion over their passbands. Delay dispersion, response widening, and time sidelobes are shown. 3.2 Sampling and Digitizing. Illustrates the basic principles of quadrature detection, I and Q sampling, and digitizing for a simple pulse radar and target model. 3.3 Matched-Filter Processing and Correlation. Shows that the signal-to-noise ratio of a processed point-target signal, produced either by correlation with the transmitted waveform or by matched-filter processing, depends on signal energy independent of waveform. 3.4 Effect of System Phase Noise. Shows the degradation of coherently processed target data that occurs when the data are contaminated by transmitter phase noise. First, the tutorial demonstrates how cumulative phase noise results from frequency noise on the transmitted signal. Then a simple radar-target model is used to show the effect of cumulative phase noise on signal amplitude and noise floor. Tutorial Topics for Chapter 4, High-Range-Resolution Waveforms and Processing 4.1 Effect of Waveform on Resolution. Shows the effect of waveform bandwidth on the resolution of multiple-scatterer targets for matched filtering of received target signals using discrete phase-coded pulse waveforms, frequency-coded pulse waveforms, and chirp-pulse waveforms. 4.2 Effect of Frequency Weighting. Shows the effect of weighting on processed resolution and sidelobe level. 4.3 Effect of Waveform on Doppler Tolerance. Shows the advantage of chirp waveforms over phase-coded and pseudorandom frequency-coded pulse waveforms in terms of Doppler tolerance. Tutorial Topics for Chapter 5, Synthetic High-Range-Resolution Radar 5.1 Synthetic Range-Profile Generation. Shows the process of generating a target's synthetic range profile from data collected by transmitting one stepped-frequency burst of pulses. Radar parameters are selected. A simulated target is generated at a fixed radar range. Complex data collected from the simulated target, illuminated by the pulse-topulse stepped-frequency waveform, are shown in phasor form. One-dimensional DFT processing is demonstrated and the resulting target range profile is shown. 5.2 Synthetic-Range-Profde Distortion Produced by Target Translational Motion. Shows distortion, produced by target radial motion, in processed target synthetic range profiles.
550
A target is modeled and assigned a target motion scenario relative to the radar. Synthetic range profiles are generated from the resulting collected data. Distortion due to phase error and range migration is shown. Translational motion correction is applied to the collected data based on known target motion, and motion-corrected data is processed. Synthetic range profiles obtained with corrected data are shown. 5.3 Distortion Produced by Transmitter Frequency Instability. Shows how distortion is produced in synthetically generated range profiles by cumulative phase noise resulting from transmitter frequency instability. Radar parameters, including transmitter frequency stability, will be assigned. A simple target is modeled. Transmitter random frequency fluctuation is added to the transmitted carrier frequency. The resulting pulse-to-pulse cumulative phase error will be added to the collected data and processed range profiles will be shown. 5.4 Processing of Range-Extended Target Data. This tutorial performs synthetic processing of stepped-frequency data collected from range-extended target regions. A fixed-target ground scene is modeled. Radar parameters are assigned. Processing and summation of data collected from multiple coarse-range gates are demonstrated to show how continuous extended-range profiles are generated. 5.5 Processing of Hopped-Frequency Data. Demonstrates processing of target data collected from pulse-to-pulse pseudorandomly hopped-frequency sequences. Radar parameters and hopped-frequency sequences are assigned. A moving target is simulated. Collected data is shown and processing is performed to obtain synthetic range profiles. Comparisons to results for linearly stepped sequences are shown.
Tutorial Topics for Chapter 6, Synthetic Aperture Radar
6.1 SAR Processing (Unfocused). Demonstrates the basic principles of side-looking SAR data collection and processing. A limited surface target region to be mapped is modeled. Radar parameters and SAR geometry including radar platform motion are assigned. Data collection and the resulting data set over the limited target azimuth region are shown. Unfocused SAR processing is performed to generate a strip map covering the limited range-azimuth target region. Resolution limits are shown. 6.2 SAR Processing (Focused). Demonstrates focused SAR processing of data collected for the above radar and target scene. Chirp-pulse waveforms are assumed. Dispersed responses from point targets are shown at near and far range. Range and azimuth references are determined, respectively, from radar parameters and SAR geometry. Range curvature, depth of focus, and speckle noise are shown. Multiple-look processing to reduce speckle noise and PRF limitations is also shown.
551
Tutorial Topics for Chapter 7, Inverse Synthetic Aperture Radar 7.1 ISAR Concept. Demonstrates the basic principle of ISAR imaging. A simple target is modeled. Target rotation is assigned at a fixed target range. The target is illuminated by short monotone pulses. A data collection set is obtained over a small target rotation angle. Range-Doppler processing, required to obtain ISAR images, is performed. Resolution limits and cross-range ambiguity are shown. 7.2 Stepped-Frequency ISAR. Demonstrates the basic principle of range-Doppler processing of ISAR data collected with pulse-to-pulse stepped-frequency waveforms. Radar parameters are assigned. Simple target models are generated and target rotational motion is assigned about a rotation axis at a fixed range from the radar. Sampled complex data from the illuminated targets are collected in time history. Two-dimensional range-Doppler processing, required to obtain ISAR images, is performed. Target imagery is shown for several integration angles to illustrate resolution limits. Slant-range and cross-range ambiguity windows are also shown. 7.3 Effects of Target Motion. Demonstrates stepped-frequency data collection and processing for a target with translational and rotational motion relative to the radar. Simple targets are modeled and radar and target geometries are generated. Radar parameters are assigned. Images are generated for several target-motion scenarios. Image distortion, produced by uncorrected translational and rotational motion, is shown. 7.4 Target Motion Correction. Demonstrates corrections required for target translational and rotational motion to produce focused ISAR imagery. Data collected in tutorial 7.3 is corrected using ground truth data. Data is first corrected for translational motion. Then polar-to-rectangular reformatting is performed from known rotational data. Focused images and the entropy measure of focusing are shown.
List of Symbols
(See also Tables 8.1 and 8.2) A Antenna aperture; amplitude. si Physical area. A, Amplitude at ith frequency step or ith pulse. Aj Amplitude of y'th pulse. (A/
1
f
;
553
554
E(x) F 8F 9% F\(f ) Fl F„ F F, F F G G Gj G (G )„ G, m
e
t
p
T
t
r
r
Gj (G ) Q Q, "' G(f) G(f) G(n ) y
c
u
e
G (G ) „ G, p
r m
G, G, G(
>
H/d). ••
H
n
Expected value of x. Radar system noise figure. Optical focal length. Optical focal lengths of SAR data film in the azimuth and range dimensions, respectively. Two-sided frequency noise-power spectral density. White frequency noise-power spectral density. Effective system noise figure. Correction factor for relating antenna beamwidth to antenna gain. Noise figure. Propagation factor. Transmission line factor. Gain; antenna gain. Interceptor-receiver antenna gain. Jammer antenna gain. Transmission line transfer function. Transmission line transfer function in the matched condition. Sampled complex output at ith frequency step of a pulse-to-pulse hopped-up stepped-frequency sequence. Sampled complex output at jth transmitted pulse of a pulse-to-pulse hopped-frequency sequence. Gj after translation motion correction (TMC). Sampled complex output at ith frequency step of ifcth burst before and after TMC, respectively. Gain function of frequency. Gain at center frequency. Frequency-agility gain produced by noncoherent integration of n, pulse-to-pulse frequency-independent pulses. Pulse-to-pulse processing gain. Minimum pulse-to-pulse processing gain. Radar antenna gain (transmitter-receiver) or radar receiving antenna gain for separate transmitting, receiving antennas. Antenna sidelobe gain. Transmitting antenna gain. Antenna gain function of azimuth angle. Transfer function. Transfer function at discrete frequency iA/. Synthetic range profile evaluated at /th range increment. Synthetic range profiles obtained from data collected at range gates 2
1. - • • •
Synthetic range profile evaluated at /th range increment offcthburst.
555
Hfa) H(w) I /i, /
Jo(b\), J\(bi),
Hi obtained with transmitted frequency error x,. Transfer function in terms of angular frequency a>. Current; inphase component out of quadrature detector, Input current and output current, respectively, of a long transmission line. G for range-motion-corrected data. Pixel intensity produced by one SAR look. Pixel intensity produced by n, independent SAR looks, noncoherently added. Joules. Bessel functions of the first kind.
Ubi), • • • K K K L i£, SE^ %(f„) L\ L, L/(\) Li L L, M M' M M, M N N. Nj Wo P p p p p Pj />, (P)
Chirp rate (Hz/s). Kelvin. Normalized error slope of a monopulse antenna. Loss. Synthetic aperture length, maximum iC. One-sided phase-noise power spectral density. Antenna sidelobe cancellation ratio. Target fluctuation loss. Single-frequency target fluctuation loss. ESM receiving-system loss. Radar receiving-system loss. Antenna sidelobe-to-mainlobe ratio. Number of cells of slant-range or cross-range migration. Number of cells of slant-range migration. Number of cells of cross-range migration. Number of cells of range shift caused by Doppler shift. Number of cells of range walk. Noise power; a positive integer. Alarm opportunities per hour. Received power from a noise jammer. Single-sideband noise power per hertz. Delay slope (sec/Hz). Probability of detection. ECCM performance factor. Fraction of visible pixel elements of imaged target. Probability of false alarm. Jammer transmitter power. Radar transmitter power. Radar average transmitter power.
2
I 1(1) l(n ) it
u
f
J
m
r
w
D
s
t
FA
m
556
P(x > y) Q
R R R R(T)
R't) R(0 "DBS
Ru *u *r R, Rj
R, R„Ri S S
s, Sr
s. Si Si
s« s t
S, S(f) S,(f) SJLf)
SJLf) SAf)
Probability that any variable x is greater than some threshold y. Quadrative-phase component out of quadrature detector. Radar range; minimum range to a scatterer in side-looking SAR; range to target rotation axis in ISAR. Estimated target range. Unit vector directed to the radar along the radar LOS. Autocorrelation at shift r. Instantaneous target range. Instantaneous target-range unit vector. Doppler beam sharpening ratio. Clutter cancellation ratio. Target range at ith pulse of the kth burst of pulses. Estimated Monopulse imaging range based on flat-plate reflectors. Radar signal interception range. Jammer range from radar. Monopulse imaging range based on spherical reflectors. Initial target range. Range to near and far edges, respectively, of range data collection interval. Signal power. Peak signal power. Signal power sensitivity of ESM receiving system. Signal power sensitivity of radar receiving system. Signal power out of the difference channel of a monopulse comparator. Signal power out of the sum channel of a monopulse comparator. Sampling time of frequency step i. Sampling time of frequency step i of burst k. Sampling delay of transmitted magnetron radar pulse. Sampling delay of received pulse of a magnetron radar. Spectrum of signal s(t). Spectrum of input signal s,{t). Spectrum of output signal s,(t). Spectrum of real signal s£t). Spectrum of waveform s^'t). Spectrum of point target response s,(t>) in range delay and s,(h) in azimuthal delay, respectively. Value, at frequency step /, of discrete spectrum of range-delay input signal s,{ri) (Fig. 6.44).
557
MA/) S(z) SW
(SW),
S/N(n ) t
(S/C)u (S/Ch
WJ r r. r. To
r, r, r, 7", 7",
r. V V(t)
v,
v,
Value, at frequency step /', of discrete angular-frequency spectrum of input signal s,{t). Angular-frequency spectrum of output signal s,{t). Value, at iA/, of discrete spectrum of input signal s,{lAt)Value, at i'A/, of discrete spectrum of waveform S|(/Ar)Fresnel sine integral. Output signal-to-noise ratio. Signal-to-noise ratio referred to receiving-system input. Peak value of ratio of signal-to-average-noise out of a matched filter. Pixel signal-to-noise ratio. Pixel signal-to-noise ratio required for a visible response. Signal-to-noise ratio out of the difference channel of a monopulse comparator. Signal-to-noise ratio out of the sum channel of a monopulse comparator. Ratio of signal-to-speckle-noise following noncoherent integration of n, looks. Ratio of signal-to-clutter before, after cancellation. Common symbol for *iv(/JCommon symbol for FUfJIntegration time; sampling spacing; signal duration. Antenna noise temperature. Effective noise temperature of an amplifier. Standard noise temperature (290K). Quadratic-phase error constant. Surveillance revisit time; scan period. System noise temperature. Pulse width; duration of a waveform frequency segment. Pulse repetition interval (PRI). Delay from leading edge to sample position of magnetron transmitted pulse; sample spacing. Delay from leading edge to sample position of received signal from magnetron transmitted pulse. Radiation intensity. Voltage. Instantaneous voltage. Source voltage at input to a transmission line. RMS voltage out of difference channel of a monopulse comparator. RMS voltage out of sum channel of a monopulse comparator. Input voltage to a transmission line; voltage of first of two inputs to a mixer.
558
V
2
Y Y I Y\„ Zo Z|, Z2 Z( y) Z(t) Z(
p
a a. Oo, d ao, a,, a , . . . a(t) a (t) a(f ) b bo, b , b , ... b(t) b(-t) c d d e / / / / fa 0
2
N
m
0
k
f ,fo7 DI
/, /LO f /, / /„ f c
m
0
Output voltage from a transmission line; voltage of second of two inputs to a mixer. Instantaneous angular position in ship pitch, roll, or yaw. Angular rotation rate of ship pitch, roll, or yaw. Average magnitude angular rotation rate Y. Characteristic impedance; load impedance. Source, load impedances of a transmission line. Response to reflection points at azimuth position y from boresight. Response at time t. Antenna response at angle > from boresight. Antenna illumination signal at cross-range distance x at range R. Antenna illumination signal at cross-range distance x = iy at range R. Dimension of geometric shape; acceleration; a constant. Target acceleration correction that minimizes image entropy. True, estimated initial target acceleration. Fourier series coefficients of amplitude term of a transfer function. Instantaneous amplitude; amplitude modulation. Instantaneous amplitude of noise modulation. One-sided phase-noise voltage in 1 -Hz band at /„. A constant. Fourier series coefficients of phase term of a transfer function. Unit-value bit increment at time it, < / < ( / + 1)/,. Time reverse of b(t). Propagation velocity; a constant. A constant; distance; dimension; target range extent; gain imbalance. Relative radar range to Jtth scatterer of a target, Exponential, 2.718. Frequency. Peak frequency modulation. ' A Carrier frequency; average frequency; center frequency. Frequency of a spurious signal. Doppler frequency. Doppler frequencies associated with scatterers 1 and 2, respectively. Intermediate frequency. Local oscillator frequency. Cutoff frequency of waveguide. Band-edge frequency. The ith frequency. Frequency of input signal to a frequency synthesizer. Magnetron frequency; modulation frequency or offset frequency.
559
/«, /, /, /, fo, f ^ i /i,/
/(0, T)
Magnetron frequency at frequency step /'. Sampling rate. The x-component of spatial frequency. The y-component of spatial frequency. First and last of n frequencies. Frequencies associated with fast-lime and slow-time data, respectively, for SAR; first and last frequency at which data is collected, Instantaneous frequency. Instantaneous frequency of FM-noise signal. Instantaneous Doppler frequency. Instantaneous local oscillator frequency. Instantaneous magnetron frequency. Average frequency deviation during the interval from t to t + T. Samples of average frequency deviation during the interval from t to t + T starting at discrete sample time iT. First of two samples of average frequency deviation during interval
7(7", T)
Second of two samples of average frequency deviation during interval
f,(i, k) f(i, k) g(R) h h A, hi h{t) h(t - T) h{t - T) Kt\), Kh)
f at frequency step i of burst k. f at frequency step i of burst k. One-way range attenuation factor. Echo transfer function of a target; exposed target height. Echo transfer function of the ktb scatterer of a target. Radar height; antenna height. Target height. Impulse response. Impulse response delayed by r. Time reverse of h(t - T). Impulse response in range delay (fast time) and azimuth delay (slow time), respectively. Value of impulse response at discrete time (Ar. Positive integer, Unit vector along x-axis. Denotes imaginary component of complex quantity. Positive integer. Jerk coefficient. Unit vector along y-axis. Boltzmann's constant (1.38 x 10" J/K); a positive integer. Unit vector along z-axis. Length; positive integer; target range extent.
2
/(r) /„(') /o(f) f (t) f (t) Jit, r) /(iT, r) w
m
T.
T.
t
h(lL\t) / i j j ;'o j k k /
J}
560
k m m
D
m, m, m, m(t) D
m,(0. m,'(0
n
n. n n, r
P pU) p(Xj),
p{Xj)
p(y) P(
P(cr ) c
P(
c
r /(/) c
r r(D r{L) r,, r s
2
s.
Discrete range-delay position of peak response. Positive integer. Number of Doppler-resolved cells of a target. Sampled output from /-channel of quadrature detector at frequencj step /. Number of range-resolved cells of a target. Number of range-Doppler-resolved cells of a target. Instantaneous complex baseband echo signal. Instantaneous /- and 2-channel output signal, respectively, of a quad rature mixer at frequency step i. Instantaneous /- and g-channel output signal, respectively, of a quad rature mixer at frequency step i of burst k. Positive integer. Number of beams of a multiple-beam antenna. Number of beam dwells. Number of noncoherently summed pulses, SAR looks, or ISAR look Delay-trigger reset count. Main-trigger reset count. Positive integer. Probability density of x. Probability density of random frequency error at frequency step step j , respectively. Chi-square density of y. Probability density of radar cross section cr. Probability density of sea clutter radar cross section a . Probability density of element cross section cr . Probability density of pixel intensity for one SAR look. Probability density of pixel intensity for the sum of n, SAR look Double amplitude excursion in ship pitch, roll, or yaw. Scatterer distance from,target rotation axis; blur radius. Position vector of a scatterer. Cross-range distance. Cross-range distance vector. Instantaneous range to reflection point at target position coordina c
c
x, y. Short for r'(t) Normalized monopulse output signal at range index /. Normalized monopulse output signal at range index of peak respon VSWR at input and output, respectively, of a long transmission li Variable of integration. Echo power density.
561
s, ^(0 Sj(t ) s(t - T) s'(t - T ) Si(t) Sj(-t) t
Sj(t) Sj(t - T) I S I ( T - t) • s„(t) ; s,(lht) s {t, f ) s&t) $Ht) e
D
S](-t) s (r) t
s (t - T) t
Power density incident on a radar target. Complex representation of a waveform or signal. Input range-delay response. Signal delayed by r. Baseband signal delayed by r. Input signal. Time-reversed input signal. Input signal at delay r. Input signal delayed by T. Time reverse of input signal delayed by r. Output signal. Input signal at discrete time l&t. Output signal at Doppler frequency f . Representation of real waveform or signal. Hilbert transform of «X0Waveform; point target response. Time reverse of s,(t). Waveform at delay T. D
Waveform delayed by T.
sjfo, f „ - i fi, f i u v
Waveform or point-target response in range delay (fast time) and azimuthal delay {slow time), respectively. Waveform at discrete time l&t. Time variable. Target dwell time. Sample spacing. Bit length of discrete coded waveform; (short) pulse width; compressed response. Sample times at beginning, end of range window. Fast time (also referred to as range delay), slow time (also referred to as time history or azimuthal delay). Readout voltage; coordinate in u, v system; variable. Velocity; coordinate in u, v system.
V|, v If v v„ v. v s(v ) r ni>. V„ \>, i,
Velocities of targets 1 and 2, respectively. Velocity vector. SAR film transport velocity. Target velocity correction that minimizes image entropy. Radar platform velocity. Effective platform velocity for squinted SAR. Range sweep velocity of SAR C R T scanner. ^ True, estimated target radial velocity. Target velocity vector.
*i('2)
Ji(/Af)
2
2
p
f
e(
562
v(f) V/ji, v„ v V|, v
2
T
2
w w, x x x c
u
2
x
t
x(t) *,(/) (JC,)„
(x,), y
y(r) y,(f) z z,(0 Zi, z a ato, &o 0 0 0, 0, 0, y y y S Sf SI SR 2
D
m
p
D
Sr
Instantaneous velocity vector. Radial velocity of target scatterers 1 and 2 , respectively. Target tangential velocity. Scatterer velocity at beam edges 1 and 2 , respectively, for sidelooking SAR. Cross-range window. Slant-range window. Random variable; distance; abscissa; variable; inphase signal. Range and azimuth cross-axis dimensions of SAR film; /, Q amplitudes for one SAR look. Random amplitudes of / or Q outputs; frequency error of ith pulse (or average frequency error during delay rfor ith pulse); ith random sample of variable x. Instantaneous inphase signal; complex transmitted signal. The ith transmitted waveform at the ith frequency step. Magnetron frequency error of ith frequency step. Synthesizer frequency error of ith frequency step. Cross-range distance from boresight, sum of squares of n random variables in chi-square density; ordinate; quadratic-phase signal; variable; integer. Instantaneous quadratic-phase signal; complex received signal. The received signal at the ith frequency step, The z-axis; variable. The reference signal at the ith frequency step. Respective arguments of sine and cosine Fresnel integrals. Variable; predetection bandwidth. True, estimated initial target angular acceleration. Bandwidth; postdetection bandwidth; phase constant; variable. Doppler bandwidth. Bandwidth of ESM receiving,system. Noise bandwidth. Transmitted bandwidth. Threshold; integer. Land-clutter return. Propagation constant. Phase imbalance. Doppler frequency separation; change in Doppler frequency. Change in current. Range increment; deviation from SAR minimum range to a fixed point on the earth surface. Slant-range or cross-range shift.
563
Sr
e
St SS
SV ST S
{
Ca V V, Vc E
ft ft ft.. 6X0 ftjdB
A A, V
Cross-range position of scatterers; monopulse cross-range "resolution." Delay interval; range-delay extent. Range-delay sample offset. Change in voltage. Delay shift, separation. Difference between matched and mismatched insertion phase of long transmission line. Maximum Sip. Complex exponential for target range motion correction; part of argument of Fresnel integral (unfocused SAR). Target range translation motion correction factor for /th pulse of k\h burst. Part of argument of Fresnel integral (unfocused SAR). Number of slant-range samples. Minimum required number of slant-range samples. Number of cross-range samples. Rotation angle; view angle; polar angle; azimuth or elevation angle; phase angle; short for COincident angle. Initial target rotation angle. Phase at frequency step i. Target rotation angle at frequency step i of burst k. Instantaneous: phase; phase modulation; angle. Half-power beamwidth in elevation. Wavelength. Optical wavelength. /- and g-channel bias and yj/i] + fi$, respectively. Phase error associated with random frequency error x . Phase constant (rad/Hz) for frequency error associated with target range extent for a magnetron radar. Phase constant (rad/Hz) for frequency error produced by sampling offset in a magnetron radar. Phase delay constant (rad/Hz) for frequency error produced by the radar's frequency synthesizer. 3.1416. Target two-dimensional reflectivity function. Radar cross section; standard deviation. Clutter coefficient. Average radar cross section. Variance. t
t>4
v. v, IT
ftx,
tr tr a tr
0
1
y)
564
2
OH
t
T
e
c
t
2
cr , (
D
2
a [n, T,r] £^[2, T, T] a [Hix,)] r r. T 2
M
r(r) T (
Tj(ai,)
r,(/), r,(w) T (J~) P
4> 4> 4>, d\ 4> fa y
Variance of H (short for a [H/'x )]). Target detection threshold in terms of RCS. Clutter cross section; standard deviation of cumulative phase noise. Average clutter cross section. Radar cross section of resolved target element. Average radar cross section of resolved target element. Radar cross section of a flat plate. Radar cross section of kth scatterer of a target. Standard deviation of frequency error of a magnetron. Radar cross section of a conducting sphere; standard deviation of frequency error of a frequency synthesizer. Fractional Allan variance of frequency deviation. Radar cross section of Doppler resolution cell, Radar cross section of range resolution cell. Radar cross section of range-Doppler resolution cell. Sample variance. Standard deviation of SAR single-look pixel intensity. Standard deviation of SAR pixel intensity following summation of n, looks. Allan variance for n samples of frequency, sample spacing T, and averaging time (or time interval) r. Allan variance for two samples of frequency. Variance of H associated with random frequency error x . Delay; averaging time (or time interval), Receiving-system transfer delay. Delay from start pulse to leading edge of transmitted pulse of a magnetron. Time-dependent delay. Delay error at ±
i
t
565
03 dB. 04 dB 0(/)
4>(t + r) MO *Um 0 4>{o>)
2
\xir,f )\ D
* (ft
A.
TK»l). "JOl)
•KX) m
Hu 0) y)
Ht. CO
m
-4>)
Half-power, 4-dB beamwidth, respectively, in azimuth; half-power, 4-dB beamwidth, respectively, of circular antenna. Insertion phase as a function of frequency. Instantaneous phase. Instantaneous phase at delay r. Instantaneous cumulative phase noise at delay T. Instantaneous phase noise. Instantaneous phase noise at delay r. Instantaneous phase noise in 1-Hz band at offset frequency/,. Phase as a function of angular frequency. Quadratic-phase deviation at band edges. Ambiguity function. Antenna beam segment over which coherent integration occurs (integration angle); phase. Maximum integration angle for SAR and ISAR before defocusing occurs. Optimum unfocused integration angle. Average excursion in rotation angle of target during ISAR image frame time. Synthetic angular resolution for Doppler beam sharpening. Equivalent rectangular beamwidth. Phase of sampled data at ith frequency step or of ith pulse; phase at code-bit position i; input phase. Phase of sampled data at frequency step i of burst k. Phase of sampled data of jth pulse of a hopped-frequency sequence. Tilt angle of effective target rotation axis. Phase of input signals 1 and 2, respectively. Phase as a function of frequency and target velocity. Instantaneous phase; pre-envelope or analytic signal. Range-delay phase, azimuthal delay phase. Phase of incident wave as a function of position x along an aperture. Instantaneous phase at frequency step i. Two-way phase advance versus time of the echo signal from a point target at boresight. Two-way phase advance versus time of the echo signal from a point target displaced y from boresight. Two-way phase advance versus time of the echo signal from a point target displaced -
566
To co. to to, to p
T
To
T
to , 6J toi 0
0
To-i 7oi(t) Hi + 1)
V(n,) A A/ A/ c
A/ A/?, (A/?),^ (A/?,); Ar Ar Ar |j Ar |„, Ar, Ar,| A/ Av, A(i') A (/') 0
c
c
dB
c
3dB
c
A(/) X 2(0 2 (i) 2(f) c
Effective target rotation vector. Band-edge angular frequency (rad/s). Effective rotation rate (rad/s) of DBS platform. Antenna scanning rate (rad/s). Magnitude of target rotation rate (rad/s) produced by target tangential motion. Vector component of target rotation rate produced by target tangential motion. True, estimated initial target rotation rate (rad/s). Magnitude of vector sum of components of target rotation rate (rad/s). Vector sum of components of target rotation rate. Instantaneous atj. Gamma function of i + 1. Gamma function of nil. Gamma function of the number n, of SAR looks. Chirp-pulse frequency excursion; monopulse difference signal. Frequency step; frequency spacing. Minimum frequency shift to produce an independent sample of reflectivity data. Doppler frequency resolution. Illuminated range extent. Maximum unambiguous illuminated range extent. SAR range depth of focus. Slant-range or cross-section resolution. Cross-range resolution (Rayleigh). Cross-range resolution (half-power). Average cross-range resolution. Slant-range resolution (Rayleigh); sampling resolution. Cross-range resolution (half-power). Time resolution; sample time spacing; aperture fill time. Target velocity resolution. Monopulse difference signal at frequency step /'. Target-motion-corrected monopulse difference signal at frequency step i. Monopulse difference signal at synthetic range index /. Sum-channel signal out of a monopulse comparator. Monopulse sum signal at frequency step i. Target-motion-corrected monopulse sum signal at frequency step i. Monopulse sum signal at synthetic range index /.
567
T
*(/) ft
ft. r X® Y X*Y oo
Azimuth delay. Azimuth scan sector. Phase of 77,. Two-sided phase-noise power spectral density. Phase of sampled data collected at frequency step i from a moving target. Spectrum of analytic function
List of Acronyms
1-D 2-D 2D-DFT 3-D A/D AFC AGC Al AM ARM ASW BCD CFA COHO CRPL CRT CW D/A DBS DDS DFT ECCM ECM EOS ERP ERPD ESM FAR
one-dimensional two-dimensional two-dimensional DFT three-dimensional analog-to-digital automatic frequency control automatic gain control airborne intercept amplitude modulation antiradiation missile antisubmarine warfare binary-coded-decimal crossed-field amplifier coherent (local) oscillator Central Radio Propagation Laboratory cathode ray tube continuous-wave digital-to-analog Doppler beam sharpening direct digital synthesizer discrete Fourier transform electronic counter-countermeasures electronic countermeasures end of sequence effective radiated power effective radiated power density electronic support measures false-alarm rate 569
570
FFT FTML HF HRR IDFT IDP IF IFM IREPS ISAR ITU JPL LO LOS LPI LPIR MTI NEL NELC NOAA NOSC NRL OTH PM PPI PRF PRI radar RAM RCS RF RMC RMO rms ROM SAR SAW SF SLAR SNR SPECAN
fast Fourier transform folded-tape meander line high frequency high range resolution inverse discrete Fourier transform interim digital SAR processor intermediate frequency instantaneous frequency measurement Integrated Refractive Effects Prediction System inverse synthetic aperture radar International Telecommunication Union Jet Propulsion Laboratory local oscillator line of sight low probability of intercept low-probability-of-intercept radar moving target indication Naval Electronics Laboratory Naval Electronics Laboratory Center National Oceanic and Atmospheric Administration Naval Ocean Systems Center Naval Research Laboratory over-the-horizon phase modulation plan position indicator pulse repetition frequency pulse repetition interval radio detection and ranging random access memory radar cross section radio frequency rotational motion correction radar master oscillator root mean square read-only memory Synthetic aperture radar surface acoustic wave stepped frequency side-looking airborne radar signal-to-noise ratio spectral analysis
SSB STALO TBD TE TEM TLO TM TMC TWS TWT VCO VSWR
single-sideband stable local oscillator track-before-detect transverse-electric transverse electric and magnetic tracking local oscillator transverse-magnetic translational motion correction track-while-scan traveling-wave tube voltage-controlled oscillator voltage standing-wave ratio
Solutions Answers to Even-Numbered Problems Chapter 1 1.2: 1.33 fis; 1.4(a): one sample; 1.4(b): 1,000 samples; 1.6(a): 50 MHz; 1.6(b): 20 ns. Chapter 2 2.2: 146W; 2.4(a): 1.93 deg; 2.4(b): 624m; 2.8: 149 km; 2.10: 50 pulses; 2.12: s(t) = B expj27r/(r-2R/c); 2.14(a):-177 dBW; 2.14(b):-107 dBW; 2.18: 50.3 mi/h; 2.20:600m. Chapter 3 3.2: -1,257 rad, 66.7 ns; 3.4(a): 10.0000200 GHz at t = 0, 10.0000233 GHz at t = 10 sec; 3.4(b): 20 kHz at t = 0, 23.33 kHz at t - 10 sec; 3.6: 10 ns; 3.8: -20 dB at -0.2 /JS and -20 dB at +0.2 /us; 3.10(a): 5; 3.10(b): 30 dB; 3.12(a): zero; 3.12(b): zero; 3.12(c): 6.4 deg; 3.18: 0 to 50 MHz; 3.22: 3 dB; 3.24: 1.10 to 1.60 GHz; 3.26: 2 m\ 2.4 m ; 3.28: 24 dB; 3.30: +0.3 dB, -29 dB; 3.32: 0.027 Hz; 3.34: 1000; 3.38: 6.6 ns. 2
Chapter 4 4.4: 0.664 sec; 4.6(a): 32; 4.6(b): 23.4m; 4.6(c): 6.4 MHz; 4.12(a): 15 MHz; 4.12(b): 0.5 MHz; 4.12(c): 30; 4.12(d): 60 /is; 4.14: 1.0 MHz; 4.16: 5.0 MHz; 4.18: 0.086 dB; 4.20(a): 100 MHz; 4.20(b): 0.50 dB; 4.20(c): 10 ns; 4.20(d): 20 dB; 4.24: 15; 4.26: 116m; 4.28: -28.5m. 573
574
Chapter 5 5.2(c): 24dB; 5.4: 0.22 m/s; 5.6: 6.0 km, 48.7 m/s; 5.8: 4,363 Hz. Chapter 6 6.2: /3= 6 M H z , / = 36 GHz; 6.4(a): 0.208 sec; 6.4(b): 17.3m; 6.8(a): 1.29m; 6.8(b): 360 Hz; 6.8(c): 417 km; 6.10: 772m; 6.14(a): 5.61m; 6.14(b): 1,070m; 6.16: +3 dB; 6.18: 977 Hz; 6.20(a): 5; 6.20(b): 5.12 km; 6.22(a): 7.4m; 6.22(b): 3.5 deg; 6.22(c): 6.75m. Chapter 7 7.2: 1.23m; 7.4: Ar, = 0.417m, Ar = 1.72m; 7.6(a): 175 Hz; 7.6(b): 300 megasamples/ sec; 7.8(a): 128; 7.8(b): 128; 7.10(a): 200 x 30; 7.10(b): 20 x 30; 7.10(c): 20 x 200; 7.10(d): 200 x 20; 7.10(e): 200 x 0; 7.12(a): 50; 7.12(b): - 2 5 ; 7.12(c): 5,000; 7.14(a): 50; 7.14(b): -100; 7.16: 1.23; 7.18(a): 36 nr ; 7.18(b): 0 nr'; 7.18(c): 35.5 nr ; 7.18(d): 6.25 nr ; 7.22(a): 53 Hz; 7.22(b): 53 Hz; 7.22(c): 6,827 Hz; 7.24(a): 150; 7.24(b): 36,000; 7.26(a): A/2; 7.26(b): A/2; 7.26(c): A/4. c
1
1
1
Chapter 8 8.2: 10.0 dB; 8.4(a): 537; 8.4(b): 29. Chapter 9 9.2(a): 1.16 rad (rms); 9.2(b): 5.8 dB. Chapter 10 5
10.2: 319 km; 10.4: 23.8 km; 10.6(a): 1,417; 10.6(b): 106 km; 10.8: 1.05 x 10" ; 10.10: 150 m/s; 10.12: 48 dB; 10.14: 27 km; 10.16: 37%; 10.18: 9 MHz.
About the Author
Donald Wehner is a private consultant in radar and electromagnetic systems. He served as the head of the radar branch of the Naval Ocean System Center from 1971 to 1991. He earned his MS in physics from Union University. His most recent research has revolved around the development of consumer applications of radar and the development of highresolution RCS measurement techniques. He is a senior member of the IEEE.
575
Index
3-D image display digital processing to generate, 440 from echoes, 441 frame of, 442,443 3-D monopulse imaging, 435-64 advantages, 461 concept, 436-43 cross-range resolution for, 445 illustrated, 438 HRR image, 461 illustrated small craft, 462-63 issues, 464 potential applications, 464 range performance, 443-53 assuming flat-plate scatterers, 452-53 calculation examples, 453 with short/chirp pulses, 449 with stepped-frequency waveforms, 449-52 stepped-frequency approach, 453-60 summary of, 460-64 target range vs. cross-range resolution, 454 using stepped-frequency waveform, 439 See also 3-D image display Active chirp generation, 159, 164-67 illustrated, 167 Add-and-divide frequency synthesizer, 114,115-17 Airborne intercept radar, parameters, 451 Airborne SAR, 298,301-5 hypothetical parameters, 303 hypothetical performance, 304 land and sea clutter returns, 304 See also Synthetic aperture radar (SAR) Aircraft AIREYE, 249 RCS, 31-32 AIREYE, 248
illustrated, 249 Air targets ISAR design calculations, 411-12 ship targets, 413-14 Allan variance, 110 cumulative phase noise from, 113 defined, 110 frequency stability and, 110-12 illustrated, 111 in range-profile distortion, 227-28 Ambiguity function, 74-75 of chirp pulse, 76 common form of, 74 defined, 74 magnitude of, 74 of monotone pulse, 75 Amplitude-comparison monopulse, 436 Amplitude equalization, 67 Amplitude ripple, 18 Amplitude weighting, 165 Analog-to-digital (A/D) converters, 93 line of constant difficulty, 94 Analytic signal, 7 phasor representation of, 8 transform of, 9 Angular resolution, 49 Antenna boresight, 49 Antenna gain, 18 wavelength and, 19 Antenna noise, 38 Antennas directivity of, 18 effective aperture of, 15 wavelength and, 19 wideband, 124-25 _^ Antiradiation-missile (ARM), 513 577
578
Antiship missile detection, 495 Aperture, 4 beamwidth, 124 data collection, 4-5 effective, 14-15 efficiency factor, 18 fill time, 124 focused, 256 frequency-space, 4 optical, 5 plane array, 18 synthetic, 4 side-looking, 248-51 Aperture-plane arrays, 522 A-scope display, 182 Atmospheric absorption, 514 Autofocusing, 326 Automatic frequency control (AFC), 469 Automatic gain control (AGC), 494-95 hard limiting, 495 Azimuth compression in cross range, 314 Azimuth correlation, 328 SEASAT, 329 Azimuth reference function, 31S definition of, 326 image quality and, 326 range dependence of, 318 Azimuth scanning, 218 Backscattering centers, 28 Backscatter sources, 22-25 characteristics, 24 illustrated, 23 major, 22-23 Backward-wave CFAs, 123 Bandwidth. See Radar bandwidth Barker Codes, 138-39 Barrage noise jammer, 490 Beam sharpening ratio, 332-33 See also Doppler beam sharpening (DBS) Beam solid angle, 49 Binary-coded-decimal (BCD) frequency synthesizer, 114, 117 comb frequency selection, 117 illustrated, 116 Binary phase coding, 136-42 3-bit code and waveform, 138 digital matched filtering, 139 quadrature processing, 140 defined, 136 eight-bit phase-coded waveform, 139,141 signal discrete convolution, 142
Bistatic radar, 22 Blur radius, 386-87 defined, 386 illustrated, 387 for one cell of migration, 386 polar reformatting and, 390 Bulk acoustic wave devices, 163-64 Burst derivative, 395 Carrier frequency, 5 Cell migration cross-range, 385 illustrated, 386 slant-range, 384 illustrated, 386 target rotation, 383-86 Characteristic function, 99 Chirp, 149 Chirp generation active, 159,164-67 coherent, 169 DDS, 170-71 passive, 160,164-67 Chirp pulse, 5 3-D monopulse imaging range performance with, 449 delay error, 173 form weighting, 173-74 quadratic-phase error in, 173 rectangular, 158-59 waveform, 150 illustrated, 6, 151 Chirp-pulse compression, 149-74 analysis based on phase equalization, 153-57 DDS chirp generation, 170-71 duration, 156 hardware implementation, 161-67 network characteristics, 152 quadratic-phase distortion, 171-74 • rectangular pulse shape effect, 157-60 time jitter, 168-70 weighting and, 160-61 Chirp-pulse imaging, 418 See also Imaging Chirp-pulse ISAR, 364-67,414-15 image processing, 366 range-profile data collection, 365 range walk/range offset for, 371-75 stepped-frequency ISAR vs., 414-17 See also Inverse synthetic aperture radar (ISAR) Chirp-pulse SAR, 266-77 block diagrams, 276-77,278 cross-range sampling criteria, 271-72
579
data collection. 267-70 illustrated, 268-69 design tables, 276-77 equation summary, 277 input data, 305-9 PRF requirements, 272-75 from Doppler point of view, 272-74 from grating lobes point of view, 274-75 resolution, 266-67 slant-range sampling criteria, 270-71 square resolution, 275-76 See also Synthetic aperture radar (SAR) Chirp radars, 134 Chirp waveform Doppler-invariant, 181 Doppler-tolerant, 181 Clutter cancellation, 500 calculation sheet, 512 noncoherent, 511 ratio, 510 Clutter coefficient, 509 Clutter discrimination, 499-500 with narrowband radars, 500-502 using HRR techniques, 502-4 wideband vs. narrowband radar for, 504-13 Clutter lock, 326 Coherent (local) oscillator (COHO), 471 Coherent-on-receive mode, 200 Coherent-on-receive radar, 469-71 illustrated, 472 stepped-frequency, 474 block diagram, 478 phase relationships, 478,479 waveforms, 475 Coherent radars, 121 Complex range profile, 206 Complex samples, 91,93 Compression analog pulse, 177 chirp-pulse, 149-74 digital-pulse, 174-80 range, 177 Continuous-wave (CW) radar, 17-18 Comer-turn memory, 320 Correlation, 315 azimuth, 328 SEASAT, 329 fast, 318-22 range, 324 performance of, 326 SEASAT, 325 Creeping-wave reflection, 22-23
Cross-correlation, 74 for prealigning range profiles, 377-79 Crossed-field amplifiers (CFAs), 123 backward-wave, 123 forward-wave, 123 Cross-range antenna, 243 power gain pattern, 244 Cross-range distortion, frequency error effect on, 485 Cross-range integration length, 269 Cross-range resolution, 5, 239 for 3-D monopulse imaging, 445 ISAR and, 342-45 of ship targets, 408-11 ISAR integration angle, 347 ISAR limits, 426.427 range-Doppler imaging, 351-52 of real-aperture antenna, 244 of real-aperture radar, 243 scale factor, 435-36 See also Resolution Cross-range sampling criteria chirp-pulse SAR, 271-72 frequency-stepped SAR, 280 range-Doppler imaging, 356-57 See also Sampling Cross-range scale factor, 350, 363 CRPL exponential reference atmosphere, 36 Cumulative phase noise, 97, 106-8 from Allan variance, 113 estimates of, 100 limit for integration, 109 signal loss, 101 signal-to-noise floor, 101 variance, 107-8 for white-noise approximation, 110 Data collection, 267 aperture, 4-5 chirp-pulse ISAR, 364-67 illustrated, 365 chirp-pulse SAR, 267-70 illustrated, 268-69 frequency-space, 389 pulse-compression, 405 stepped-frequency, 405 Data lines, 316 Data sampling. See Sampling DDS chirp generator, 170 Delay equalizer, 153 Delay-trigger counter, 229, 231 Deramp FFT processing, 149
580
Detection, 78-88 high-resolution, parameters, 537 linear, 87-88 low-flyer, 495-513 low-resolution, parameters, 537 mixer diode current vs. voltage, 79 phase, 500 pulsed-Doppler, 502 quadrature, 39, 81-87 radar, range, 497 of small, slowly moving targets in clutter, 537-39 square-law, 87-88 target, 78 threshold, 510 video, 78 Digital mixer, 473 Digital phase comparator, 454 output, 459,460 Digital processing, SAR, 313-17 advantages, 314 input data, 317 range correlation reference function, 315 Digital pulse compression, 174-80 fast-convolution example, 178 Digital-to-analog (D/A) converter, 118, 119 Direct digital synthesizer (DDS), 118-19 chirp generation, 170-71 illustrated, 118 Direct frequency synthesizers, 114 Directivity, 18 Discrete convolution, 179 minimum acceptable period lengths for, 180 Discrete Fourier transform (DFT), 47, 179 Discrete frequency-coded waveform, 143 signal convolution with, 146, 147 Discrete frequency coding, 142-43 continuous, 142-49 Dispersive delay, 153 surface acoustic wave, 166 Distortion cross-range, frequency error effect on, 485 quadratic phase, 171-74 synthetic range-profile, 477-84 from target radial motion, 180-81 target rotation, 380-87 cell migration and, 383-86 quadratic phase, 383 from transmission through waveguides, 120 Doppler beam sharpening (DBS), 240-41, 328-34 illustrated, 242 radar for commercial navigation, 333-34
illustrated, 332 resolution, 331 ratio. 332-33 scan rate vs. scan ratio, 333 short-range, 334 Doppler filtering, 47 Doppler frequency resolution, 263 at half-amplitude bandwidth, 266 Doppler frequency separation, 262-63 Doppler-invariant, 181 Doppler resolution, 46-47 coherent integration and, 47 illustrated, 47 See also Resolution Doppler spectrum, SAR, 293 Doppler-tolerant, 181 Down counters, 229 Duplexer, 17 Dynamic range, 494 Early-late gate, 231 Echo amplitude, 28 Echo power density, 14 Echos amplitude deviation vs., 66 paired, 57, 63 amplitude of, 66 phase deviation vs., 66 Echo signals, for two closely spaced points, 46 Effective aperture, 14 antenna, 15 Effective instantaneous velocity, 376 Effective radiated power density (ERPD), 490 Effective radiated power (ERP), 492 Effective trihedral size, 25 Eight-bit phase-coded waveform, 139 illustrated, 141 Electronic counter-countermeasures (ECCM), 284.367,488-95 defined, 488 evaluation for operation, 489 hypothetical 3-D, performance, 492 parameters, 493 methods of providing, 488 performance factor, 488-89 pulse-compression radars and, 493 radar, performance, 490 See also Electronic support measures (ESM) Electronic countermeasures (ECM), 488 Electronic support measures (ESM), 513 receiver, 518 high-gain scanning antenna and, 528 hypothetical, 520
581
illustrated, 519 interception range and, 526 receiver systems, 518 generic,519 End-of-sequence (EOS) burst input, 231 Entropy measure, 395 Envelope delay, 59 Equalization niters, 67 FTML, 163 illustrated, 164 rectangular-shaped pulse and, 157 transversal, 67-68 waveguide, 121 Equi-Doppler lines, 241-42 Equirange lines, 241 Equivalent rectangular beamwidth, 260-62 False-alarm rate (FAR), 511 Far field, 22 Fast correlation, 318-22 with one-dimensional references, 321 overlap-save process, 322, 328 processor, 323 special care of, 322 of two-dimensional SAR data, 320 See also Correlation Fast Fourier transform (FFT), 47,179 Fast time, 309 Flat-plate scatterers 3-D monopulse range performance with, 452-53 specular return, 452 Fluctuation loss defined, 528 detection probability vs., 530 mathematical description of, 529 reduction of, 538-37 frequency-agility method, 529-34 high-resolution method, 535-37 from scanning search radar, 530, 531 single-frequency, 533 sources of, 528-29 FM ramps, 149 FM slope adjustment, 121 Focal-plane arrays, 522 Focused aperture, 256 Focused SAR, 240 equivalent rectangular beamwidth, 260-62 matched filtering and, 257-58 resolution, 262 theory, 256-62 See also Synthetic aperture radar (SAR) Folded-tape meander line (FTML), 163 filter illustration, 164
Forbidden zones, 88 between zero and 10 GHz, 90 Forward-wave CF As, 123 Four-frequency periodic waveform, 144 illustrated, 145 Fractional bandwidth, 9 Free-space propagation, 34-36 echo power, 36 Free-space radar equations, 34-36,42 Frequency accumulator, 119 Frequency accuracy, 484 Frequency-agile radars, 469-71 dithered, 470 Frequency agility, 469 for detection improvement, 533 fluctuation loss method, 529-34 gain, 533 independent samples vs., 534 Frequency dispersion, 120 Frequency-domain sampling, 3,93-95 frequency steps required for, 95 illustrated, 95 pulse-to-pulse, 93 Frequency-domain target signatures, 197-200 Frequency error effect on cross-range distortion, 485 magnetron, 483 random, 484 tolerance lo, 483-84 Frequency fluctuation, 96 effect of. 97-102 illustrated, 98 of transmitted signal, 102 Frequency instability. See Frequency fluctuation Frequency-modulation bandwidth, 10 Frequency noise modulation, 103-6 Frequency space, 387 Frequency-space aperture, 4 Frequency-space data collection, 389 Frequency stability measures of, 102 power spectral density of phase noise and, 102-3 pulse-to-pulse, 484 in terms of Allan variance, 110-12 transmitter requirements, 96-114 of wideband transmitted signal, 102 Frequency synthesizers, 114-19 add-and-divide, 115-17 binary-coded-decimal, 117 direct, 114 "~ direct digital (DDS), 118-19 indirect, 114
582
Frequency synthesizers (com.) summary of, 119 Frequency translation, 78 Gaussian-taper filter, 161, 162 Grating lobes, 274 PRF requirements and, 274 SAR geometry for, 274 Group delay, 59-62 illustrated, 61 Hard limiting, 495 High-range-resolution (HRR), 5, 18, 181-86 clutter discrimination using, 502-4 early designs of, 163 low-flyer detection with, 503 monopulse radar, 437 illustrated image, 461 radar types, 134 range tracker, 184 using motor-driven phase shifter, 185 reflectivity, 198 sea clutter, 508, 509 signatures, 187, 188, 189 synthetic, 143, 197-235 track-before-detect (TBD), 503, 505,506 waveforms, 133-86 list of, 135 See also HRR target responses High-resolution method, 535-37 High-resolution radar bandwidth, 10 design, 57-125 radar detection range for, 49-51 RCS for, 28-34 system loss, 34 See also High-range resolution (HRR); Synthetic HRR radar High-Resolution Radar Software Tutorial, 545-51 installing, 545-46 starting, 546-47 system parameter experimentation, 547-48 topic list, 548-51 topic selection in, 547 using, 546-48 viewing topic in, 547 Hopped-frequency bandwidth, 10 Hopped-frequency data collection, 219 Hopped-frequency sequences, 214 parameter selection, 224 for surveillance, 222 Hopped-frequency waveforms, 213-20 azimuth scanning with, 218
defined, 214 processing, 215-17 pulse-to-pulse, 213 HRR. target responses, 181-86 display of, 182-83 preprocessing, 184-86 range profiles, 183 recording, 183-84 IF signal, 80 Image frame time, 352 Image processing gain, 417-20 defined, 417 Image projection plane, 359-64 defined, 359 for ISAR, 359-60 for SAR, 360 vector relationships, 360-64 Image visibility calculation of, 422-24 illustrated, 423 Imaging with noncoherent radar, 467-85 radar range equation for, 424 radar target, range, 417-24 range-Doppler, 349-57 three-dimensional, 435-64 Impulse waveforms, 134 Indirect frequency synthesizers, 114 Inphase output, 83 Insertion delay, 153 Insertion phase, 59 long-line, 71 matched, 70 mismatched, 70 Instantaneous bandwidth, 10 Instantaneous delay, 59 Instantaneous frequency, 58-59, 153 defined, 58 Integration length, 250 cross-range, 269 slant-range, 269 Interpulse, 59 coherence, 467 Intrapulse, 59 coherence, 467 Intrapulse FM, 485 Inverse discrete Fourier transform (IDFT), 200 Inverse synthetic aperture radar (ISAR), 1,341-427 aircraft, images, 394 automatic, focusing methods, 394-402 chirp-pulse, 364-67,414-15 image processing, 366
range-profile data collection for, 365 stepped-frequency vs., 414-17 cross-range resolution, 342-45 of ship targets, 408-11 defined, 341 design calculations, 411-14 air targets, 411-12 ship targets, 413-14 equation summary, 358 focused, images, 370 geometry, 395-96 illustrated, 396 half-power resolution for, 348 high-resolution, 32 image-plane geometry, 362 image projection plane, 359-64 for ship target, 364 vector relationships, 360-64 image views from target motion, 363 integrated response for, 348 maximum unfocused integration angle and, 345-46 multiple-look, processing, 402-3 illustrated, 404 polarimetric, 408 polar reformatting and, 390 processing alternative, 403-8 automatic, 394-402 deductive methods of, 405-6 maximum entropy, 408 multiple-look, 402-3,404 objective, 398 super resolution, 407-8 system identification imaging, 407 tomography, 406-7 range-Doppler imaging, 349-57 cross-range resolution, 351-52 cross-range sampling, 356-57 slant-range resolution, 352-54 slant-range sampling, 354-56 small integration angle and, 350 square resolution, 357 range offset/range walk, 370-75 for chirp-pulse waveforms, 371-75 uncorrected, 375 Rayleigh resolution, 345, 348 resolution limits, 424-27 cross-range, 426,427 Rayleigh, 426,427 target rotation, 426-27 SAR equivalence, 344
SAR vs., 341-42 ship, images, 395 of ships/planes, 114 shortcomings, 435-36 side-looking data resolution, 348 spatial frequency bandwidth, 424-27 stepped-frequency, 367-70,415-16 chirp-pulse vs., 414-17 data collection, 367-70 focused, 393 image processing, 368 range migration, 375 unfocused, 393 waveforms, 348 synthetic, 353-54 image generation process, 377 range walk and, 370 translational motion correction for, 376-80 waveforms and signals, 369 target aspect rotation sources, 357-59 target imagery, 341 target imaging range, 417-24 equation, 424 image processing gain, 417-20 image visibility calculation, 422-24 visible target elements, 420-22 target sampled data, 396-99 in terms of SAR, 241 theory, 342-48 focused aperture, 347-48 for small integration angle, 350 unfocused, 346 See also Synthetic aperture radar (SAR) Klystron power amplifiers, 121 Linear detection, 87-88 Linear FM, 59 Linear FM chirp, 57-59 radial motion distortion and, 180-81 Line of sight (LOS), 5 Local-oscillator frequency forbidden, 88-89 between zero and 10GHz, 90 selection of, 88-89 Local oscillator (LO) signal, 80 Long-line effect, 68-72 defined, 68 Long-range tracking radar, parameters, 451 Low-flyer detection, 495-513 with HRR, 503 Low-pass filters (LPFs), 221
584
Low probability of intercept (LPI), 514 performance factors, 328 RF absorption to achieve, 517 Low-probability-of-intercept radar (LPIR), 487, 513-28 atmospheric absorption method, 514 expressions, 515-18 multiple-beam, 522-27 single-beam vs., 523, 524 multiple-simultaneous antenna beams method, 514, 522-27 performance calculation sheet, 521 signal matching method, 514 single-beam (example 1), 518-20 single beam (example 2), 520-22 spread spectrum method, 514 extended-time integration, 516 techniques, 514 illustrated, 515 transmitter-power calculations, 522 Low-resolution radar, RCS for, 26-28 Magnetron frequency control, 484-85 Magnetron frequency error, 483 random, 484 tolerable, 483 Magnetrons, 121 deviation tolerance of, 484 fixed-frequency, 485 frequency-agile, 122-23. 484 tunable, 122 Matched-fdter, 72-74 complex, output, 155 defined, 72 focusing and, 257-58 half-power duration of, 257 impulse response, 143-44 function, 72 input signal, 78 processing, 17 digital, 138 receiving system, 43 response for periodic discrete frequency-coded waveforms, 148 response function, 75-78 illustrated, 77 response to point target, 148 SNR. 41-44 of target echo signal, 74 transfer function, 73, 76 Maximum entropy, 408 Minimum entropy RMC. 402
TMC, 399-402 ISAR images using, 401 velocity and acceleration slices, 400 Missile seeker, parameters, 451 Mixers, 80-81 baseband signal, 83 inphase outputs, 83 output phase, 205-6 products, 80, 81 Monopulse 3-D imaging, 435-64 advantages, 461 concept, 436-43 cross-range resolution for, 445 illustrated, 438 HRR image, 461 small craft, 462-63 issues, 464 potential applications, 464 range performance, 443-53 assuming flat-plate scatterers, 452-53 calculation examples, 453 with short pulses and chirp pulses, 449 with stepped-frequency waveforms, 449-52 stepped-frequency approach, 453-60 summary of, 460-64 target range vs. cross-range resolution, 454 using stepped-frequency waveform, 439 See also Imaging Monopulse antennas illustrated patterns, 455 at baseband of two pairs of feeds, 456 Monopulse waveforms, 457 Monostatic cross section, 22 Monostatic radar, 13 Moving target indication (MTI), 471 short-pulsed area, 502 MTI canceler. 501, 502 coherent, 501 noncoherent, 502 Multiple-frequency echo responses, 143 Multiple lobing, 498 Multiple-look ISAR processing, 402-3 illustrated, 404 Multiple-simultaneous antenna beams, 514,522-27 implementation of, 522 Multiplicative noise. See Speckle noise Noise bandwidth, 17,40 Noise figure, 40 Noise jammer, 489 nonsmart, 493 radar range with, 496
585
Noise power, 16 calculation of, 39 expression of, 38 Noise temperature, 38 effective, 40 standard, 39 Noncoherent radars, 121 coherent-on-receive radar, 469-71,472 frequency-agile radar, 469-71 frequency error effect, 485 intrapulse FM, 485 magnetron frequency control, 484-85 power-amplifier transmitters, 468 pulsed magnetron radar, 468 range-extended target response, 477 stepped-frequency magnetron imaging radar, 471-73 synthetic range-profile distortion, 477-84 target imaging with, 467-85 Non-free-space propagation, 37 Nonuniform illumination Doppler frequency and, 265-66 illustrated response for, 261 SAR resolution for, 258-60 Nyquist criteria, 4 , 9 1 , 355 PRF and, 272 sampling rate, 179 One-sided power spectral density of phase noise, 102-3, 105 Optical aperture, 5 Optical film, 309 focusing method, 312 illustrated, 311 Optical film scanner, 309 illustrated, 310 Optical processing, SAR, 309-13 astigmatism and, 313 illustrated, 315 phase history, 309-10 range and azimuth focal lines, 314 See also Synthetic aperture radar (SAR) Optical region, 23 Optimum codes, 138-39 Output image frame, 322 Overlap-save (overlap-add), 322, 328 Over-0»e-horizon (OTH) microwave performance, 496 surveillance, 10 Paired echos, 57,63 amplitude of, 66 Passive chirp generation, 160,164-67
illustrated, 167 of transmitted pulse, 165 video pulse spectrum for, 168 Phase accumulator, 119 Phase delay, 59 Phase detection, 500 Phase equalization, 67 chirp-pulse analysis, 153-57 transfer function, 157 Phase equalizer, 153 Phase fluctuation. See Phase noise Phase method, 376 cross-correlation and; 377-79 improvements, 379 weighting and, 379 Phase-modulated carrier, 106 phasor diagram of, 107 Phase modulation index, 103 Phase noise, 57 cumulative, 97,106-8 from Allan variance, 113 estimates of, 100 limit for integration, 109 signal loss, 101 signal-to-noise floor, 101 variance, 107-8 for white-noise approximation, 110 power spectral density, 103 power spectrum and, 104-5 specifying, 108-9 from white noise, 108 pulse-to-pulse, 97 from pushing transmitter transfer phase, 228 pushing of, 99 random sources, 96 Phase pulling, 112 Phase pushing, 112 Phase-reversal modulation, 136 illustrated, 137 See also Binary phase coding Phase ripple, 18,66-67, 123 Phase-steered arrays, 124 Pixel intensity, 292 Plan-position-indicator (PPI) display, 533 Polar format, 387 Polar-formatted data, 389 Polarimetric ISAR, 408 Polar processing, 149 Polar reformatting, 318 blur radius and, 390 discrete frequency data, 391 frequency-stepped input data, 391
586
Polar reformatting (com) on ISAR data, 390 on spotlight SAR data, 390 process of, 390-94 resampled data, 392 rotational motion correction with, 387-94 of stepped-frequency ISAR data, 390-94 Power gain pattern for cross-range antenna, 244 one-way, 247 two-way, 248 Power spectral density one-sided, 102-3, 105 of phase noise, 102-3 power spectrum and, 104-5 specifying, 108-9 from white noise, 108 Predetection process, 28 Pre-envelope signal, 7 Processing, ISAR, 405-7 Processing gain, 17 Propagation factor, 498 path attenuation, 37 plots, 498,499 Pulse compression, 18 filter, dispersive, 57 network illustration, 151 radar, 149 in slant range, 314 Pulsed-Doppler, 500 detections, 502 processing, 501 Pulsed-Doppler radar, 500 SSB, 502 Pulsed magnetron radar, 468 Pulse power, 15 Pulse repetition frequency (PRF), 75 requirements, 272 SAR, from Doppler point of view, 272-74 SAR, from grating lobe point of view, 274-75 stepped-frequency SAR, 280 SEASAT, 299, 324 Pulse repetition interval (PRI), 83 Pulse-to-pulse frequency-agile bandwidth, 10 Pulse-to-pulse phase noise, 97 from pushing transmitter transfer phase, 228 Pulse width, 13 Quadratic phase distortion, 171-74 from 30 mm waveguide length, 174 defined, 171
illustrated, 171 magnitude, 173 production of, 171 target rotation, 383 undesired, 172 Quadratic-phase error, 121 for focused SAR, 240 Quadratic range shift, 258 Quadrature detection, 39,81-87 defined, 81 errors, 83-87 gain/phase imbalance, 84, 86 illustrated, 82, 84 output phase, 97 practical design, 85 uses, 81 waveforms, 84 Quadrature mixing, 81-83 illustrated, 82 Radar, 13 Radar bandwidth, 2 categories, 10 defined, 3 high-resolution, 10 increased, 2 advantages of, 3-4 Radar cross section (RCS), 14,19-34 of aircraft, 31,32 of conducting sphere, 448 definitions, 19-22 shorthand, 20 Doppler-resolved element sums, 34 element types, 33-34 for high-resolution radar, 28-34 of ideal geometric shapes, 24 for low-resolution radar, 26-28 narrowband, 27 of passive reflectors, 447 ship length vs., 33 sphere, 448 of spherical conductor, 21 Radar detection range free-space calculations, 50 for high-resolution radar, 49-51 Radar Handbook,/! 36. 123 Radar interception range, 526 radar beams vs., 527 Radar master oscillator (RMO), 467 Radar ovens, 121 Radar range equation derivation of, 13-17 free-space, 34-36,42
587
for imaging, 424 key elements, 14,16 Radar resolution. See Resolution Radar target image, 3-4 Radial translation, 3S7 Radial velocity, 351 from scatterer on rotating target, 351 Radiation density, 15 Radiation intensity, 19 Random frequency error degradation from, 231-35 maximum tolerable, 228 normal distribution of, 233 range-profile distortion from, 226-28 synthetic range profile variance for, 234 Random phase error, 480 for extended targets, 481 for point targets, 480-81 types of, 482 Range ambiguity window, 355 Range attenuation, 34-37 free-space propagation, 34-36 non-free-space, 37 Range compression, 174 Range correlation, 324 performance of, 326 SEASAT, 325 See also Correlation Range curvature, 286 compensation in azimuth spectral domain, 327 geometry, 290 illustrated, 288 individual scatterer tracks, 289 in SAR processing, 318 for spotlight SAR, 291-92 tracks, 287 Range-Doppler coordinate, 242 Range-Doppler coupling, 181 Range-Doppler imaging, 349-57 cross-range resolution, 351-52 cross-range sampling, 356-57 illustrated, 349 rotational motion correction (RMC) and, 350 slant-range resolution, 352-54,354-56 small integration angle and, 350 square resolution, 357 translational motion correction (TMC) and, 350 See also Imaging Range-extended targets, 220-26 isolated targets, 221
surveillance applications, 221-24 surveillance example, 224-26 Range focusing depth, 287,290-91 illustrated, 291 Range offset constant, 370 ISAR, 370-75 for chirp-pulse waveforms, 371-75 Range profile complex, 206 envelope-detected, 198-99 of isolated target, 221 synthetic, 367 complex form of, 207 of fishing boat, 213 generation, 199-209 with hopped-frequency processing, 217 of moving small craft, 214 of point target, 211-12 processing, 198 for random frequency error, 234 response to fixed point target, 208 Range-profile distortion from random frequency error, 226-27 reduction in, 226-27 Range profile signature, comer turn of, 368 Range reference function, 315 Range resolution, 5,28,44-46 Rayleigh criterion for, 44 See also Range-velocity resolution Range-sample delay offset, 480 Range swath, 267 Range tracker digital, 229.230 for HRR radar, 184 using motor-driven phase shifter, 185 Range tracking, stepped-frequency radar, 229-31 Range-velocity resolution, 48-49 Range walk, 287 illustrated, 288 individual scatterer tracks, 289 in ISAR processing, 370-75 for chirp-pulse waveforms, 371-75 circumventing, 370 uncorrected, 375 profile-to-profile, 370 in SAR processing, 318 tracks, 287 Rayleigh resolution, 257 criterion for, 44 ISAR, 345, 348 limits, 426,427
588
Real aperture, 239 resolution, 262 Real-aperture line antenna, 244 illustrated, 246 Real-aperture radar antenna, 243 cross-range resolution of, 243 illustrated, 244 transfer function, 245 Real-aperture radar mapping, 242-48 in side-looking mode, 243 Real-time imaging, 365 Receiving-system noise temperature, 17 Receiving-system sensitivity, 16, 37-41 from output SNR, 42 preamplification noise specification, 40-41 Reflectivity, 198 HRR, 198 time-domain measurement of, 197 Resolution, 44-49 angular, 49 chirp-pulse SAR, 266-67 cross-range, 5, 239 ISAR, 342-45 ISAR integration angle, 347 ISAR limits, 426,427 range-Doppler imaging, 351-52 of real-aperture antenna, 244 of real-aperture radar, 243 scale factor, 435-36 DBS, 331 Doppler, 46-47 Doppler frequency, 263, 266 focused SAR, 262 half-power, 256 high-range (HRR), 5,18 ISAR limits, 424-27 cross-range, 426, 427 Rayleigh, 426,427 target rotation, 426-27 optical, 44 optimum unfocused SAR, 262 for pulses with rectangular spectrum, 45 range, 5,28,44-46 range-velocity, 48-49 Rayleigh, 257 criterion for, 44 ISAR, 345, 348 ISAR limits, 426,427 real aperture, 262 sampling, 208 SAR
cross-range, 266 from Doppler frequency spectrum, 265 for nonuniform resolution, 258-60 slant-range, 5,239 range-Doppler imaging, 352-54 scale factor, 435 square, 275-76 cell migration and, 385 range-Doppler imaging, 357 zooming, 276 stepped-frequency SAR, 279 super, 407-8 for synthetic aperture, 250 target-space, elements, 417 See also High-resolution radar; Low-resolution radar Response function, matched-filter, 75-78 Ringing filter, 167 Rotational motion correction (RMC), 350 automatic, 402 minimum-entropy, 402 target, 387 Samples, complex, 91,93 Sample variance, 110-11 Sampling cross-range, 271-72 frequency-domain, 3,93-95 line of constant difficulty, 94 rates, 91 resolution, 208 slant-range, 270-71 target reflectivity, 95 time-domain, 3,89-93 Scatterers. See Backscatter sources Scatterer velocity vector, 360 Scattering cross section, 22 Sea clutter, 504 backscatter examples, 507 fluctuation, 509 HRR, 508, 509 narrowband vs. high-resolution, 507 SEASAT, 297,298-301 azimuth correlation, 329 design parameters, 300 four-look, SAR map, 297 JPL, design, 297 land and sea clutter return, 301 performance calculations, 302 pixel signal to thermal noise ratio, 299 PRFs, 299, 324 processing example, 322-28 range correlation, 325
589
range data line, 324 system illustration, 299 Sea-surface reflection, 497 Semiconductor diodes, 78-79 output voltage, 79 Serial sampling system, 184 Short monotone pulse 3-D monotone imaging range performance with, 449 radial motion distortion and, 180-81 rectangular, 182 Short-pulsed area MTI, S02 Short-pulse waveforms, 134—36 with ringing filter, 136 search-radar application, 136 Short-term pulse stability, 18 Sidelobe jamming, 491 Side-looking airborne radar (SLAR), 248 Side-looking SAR, 239-40 focused, 260 fundamental concepts, 241-42 illustrated, 240 range curvature, 286-90 range focusing depth, 287, 290-91 range walk, 287-90 real-aperture mapping, 242-48 See also Synthetic aperture radar (SAR) Side-looking synthetic aperture, 248-51 geometry for generating, 250 illustrated example, 249 resolution for, 250 Signal distortion, 62 in components, 62-68 in wideband systems, 62-68 Signal matching, 514 Signal-to-noise ratio (SNR) average, 42 degradation, 161 illustrated, 162 matched-filter, 41-44 Signatures discrete frequency, 198 frequency-domain, 197-200 HRR of C-45 aircraft, 188 of diving diesel submarine, 189 ofT-28, 187 range-profile, 368 Single-beam LPIR, 518-22 Single-look integration angle, 273 Single-sideband (SSB) pulsed-Doppler radar, 502 Skywave propagation, 496
Slant-range integration length, 269,356 Slant-range resolution, 5, 239 range-Doppler imaging, 352-54 scale factor, 435 See also Resolution Slant-range sampling criteria chirp-pulse SAR, 270-71 frequency-stepped SAR, 279-80 range-Doppler imaging, 354-56 See also Sampling Slant-range window, 355 Slow time, 309 Small integration-length SAR, 251-53 Smart-jammer, 493 Spaceborne radar, parameters, 451 Spatial bandwidth, ISAR, 424-27 Speckle noise defined, 292 illustrated, 298 pixel intensity and, 292 SAR, 292-97 signal-to-speckle-noise ratio, 297 Spectral analysis (SPECAN) processing, 149 Spectral purity, 103 Spikes, 183 Spotlight SAR, 240 circular flight paths, 343 illustrated, 241,342 polar reformatting and, 390 range curvature for, 291-92 See also Synthetic aperture radar (SAR) Spotlight zooming, 281-82 illustrated, 282 Spread spectrum, 514 extended-time integration waveforms, 515 Spurious responses, 96 Square-law detection, 87-88 Square-law region, 80 Square resolution, 275-76 cell migration and, 385 range-Doppler imaging, 357 zooming, 276 See also Resolution Stable local oscillator (STALO). 471 Steady-state transfer function, 62^ Stepped-frequency bandwidth, 10 Stepped-frequency imaging, 418 radars, 67 Stepped-frequency ISAR, 367-70,415-16 chirp-pulse ISAR vs., 414-17 data collection, 376-70
590
Stepped-frequency ISAR (com.) focused, 393 image processing of, 368 polar reformatting and, 390-94 range migration, 375 unfocused, 393 See also Inverse synthetic aperture radar (ISAR) Stepped-frequency magnetron imaging radar, 471-73 block diagram, 478 illustrated, 474 phase relationships, 478,479 waveforms, 475 Stepped-frequency pulses, 202 illustrated, 203 Stepped-frequency radar, 200 functional block diagram, 201 practical design, 202 pulses, 203 resolution cells, 216 synthetic cells, 216 transmitted and reference waveforms, 204 waveforms, 202 echo pulses and, 204 Stepped-frequency SAR, 277-86 A/D conversion and, 271 block diagram, 285 cross-range sampling criteria, 280 design tables, 282-86 PRF requirements, 280 resolution, 279 sampling (ambiguous range), 284 sampling (unambiguous range), 283 slant-range sampling criteria, 279-80 spotlight zooming, 281-82 waveforms, 283-86 See also Synthetic aperture radar (SAR) Stepped-frequency sequences, 214 for surveillance, 222 Stepped-frequency waveforms, 199,200 3-D monopulse imaging using, 439 illustrated, 202,455 ISAR, 348 synthetic processing of, 353 transmitted and reference, 204 Step transform processing, 149 Stretch waveforms, 149 illustrated, 150 processing of, 150 Subimage processing, 322 Sum and difference signals sampled, 458
synthetic, 459 Superheterodyne (superhef) receiver, 80 Super resolution, 407-8 processing, 436 Surface acoustic wave (SAW) devices, 163-64 dispersive delay line, 166 Surface-wave propagation, 496 Surveillance applications, 487-539 extended, regions, 222 hopped-frequency sequences for, 222 multiple-simultaneous-beam, 522 range-extended target application, 221-24 example, 224-26 stepped-frequency sequences for, 222 Surveillance radar parameters, 536 target fluctuation loss reduction for, 528-37 Swerling models, S31 single-scan detection probability vs. SNR for, 532 Synthetic aperture, 4 defined, 239 Synthetic aperture radar (SAR), 1, 239-334 airborne, 298, 301-5 chirp-pulse, 59,266-77 block diagrams, 276-77,278 cross-range sampling criteria, 271-72 data collection, 267-70 design tables, 276-77 equation summary, 277 input data for, 305-9 PRF requirements, 272-75 resolution, 266-67 slant-range sampling criteria, 270-71 square resolution, 275-76 cross-range, resolution, 266 cross-range Doppler, 264 DBS, 240-41,328-34 illustrated, 242 defined, 239 design examples, 297-305 digital processing, 313-17 advantages, 314 input data, 317 range correlation reference function, 315 Doppler spectrum, 293 focused, 240 equivalent rectangular beamwidth and, 260-62 resolution, 262 theory, 256-62
591
high-resolution, 32 image projection plane for, 359-60 ISAR, in terms of, 241 ISAR equivalence, 344 ISAR vs., 341-42 map, 341 mapping, 305 optical processing, 309-13 astigmatism and, 313 illustrated, 315 plane history, 309-10 range and azimuth focal lines, 314 processing, 305-28 digital, 313-17 fast correlation, 318-22 nonindependent references, 318 optical, 309-13 SEASAT example, 322-28 range delay to single scatter, 307 resolution for nonuniform illumination, 258-60 SEASAT, 297,298-301 side-looking, 239-40 focused, 260 fundamental concepts, 241-42 illustrated, 240 range curvature, 286-90 range focusing depth, 287, 290-91 range walk, 287-90 real-aperture mapping, 242-48 small integration-length, 251-53 speckle noise, 292-97 spotlight, 240 circular flight paths, 343 illustrated, 241,342 polar reformatting and, 390 range curvature for, 291-92 stepped-frequency, 277-86 A/D conversion and, 271 cross-range sampling criteria, 280 design tables, 282-86 equation summary, 286 polar reformatting and, 390-94 PRF requirements, 280 resolution, 279 sampling (ambiguous range), 284 sampling (unambiguous range), 283 slant-range sampling criteria, 279-80 spotlight zooming, 281-82 waveforms, 283-86 theory from Doppler point of view, 262-66 types of, 239 unfocused, 240
integration length, 253-56 optimum, resolution, 262 theory, 248-56 uses, 239 See also Inverse synthetic aperture radar (ISAR) Synthetic HRR radar, 143, 197-235 frequency-domain target signatures, 197-200 hopped-frequency sequences, 214—20 range-extended targets, 220-26 range-profile generation, 200-209 target velocity effects, 209-14 See also High-range resolution (HRR) Synthetic ISAR, 353-54 image generation process, 377 range walk and, 370 translational motion correction for, 376-80 waveforms and signals, 369 See also Inverse synthetic aperture radar (ISAR) Synthetic processing, 148 of stepped-frequency waveform, 353 Synthetic range profile, 367 complex form of, 207 of fishing boat, 213 generation, 199-200 concept, 200-209 summary, 200 with hopped-frequency processing, 217 of moving ship, 215 of moving small craft, 214 of point target, 211-12 for random frequency error, 234 response to fixed target, 208 Synthetic range profile distortion, 477-84 frequency error tolerance, 483-84 ideal system analysis, 478-80 peaks and nulls effect, 482-83 random phase error for extended targets, 481 random phase error for point targets, 480-81 random phase error types, 482 System identification imaging, 407 System loss, 34 System noise, 39 radar, 41 Tangential translation, 357 Target aspect rotation, 357-59 illustrated, 360 from target angular morion, 359 from target translation, 359 Target detection, 78 Target image, 241 See also Imaging Target LOS unit vector, 360
592
Target range extent, 480 Target recognition, 3-4, 197-200 Target rotation center of, 396 distortion from, 380-87 blur radius, 386-87 cell migration and, 383-86 quadratic-phase, 383 geometry, 381 motion correction with polar reformatting, 387-94 single scatterer, 381 Target rotation vector, 360, 363 Target signature processing, 467-69 Target velocity, 209-14 distortion, 211-12 Thermal noise power floor, 494 Thumbtack ambiguity response, 149 Time-bandwidth product, 43-44, 153 Time-dependent frequency, 58 Time-domain sampling, 3, 89-93 at baseband, 92 Time jitter, 168-70 Tomography, 406-7 applications, 407 ISAR and. 406 Track-before-detect (TBD), 503 input data, 506 threshold-circuit operation, 505 Tracking local oscillator (TLO), 469 Transfer function amplitude and phase components of, 63 matched-filter, 73, 76 phase equalization, 157 of real-aperture mapping radar, 245 transmission line, 69 Translational motion correction (TMC), 350 Doppler analysis method, 379-80 minimum-entropy, 399-402 ISAR images using, 401 velocity and acceleration slices, 400 phase method, 377-79 prominent peak method, 380 for synthetic ISAR, 376-80 Transmission lines equation, 69 illustrated, 68 theory, 68-69 transfer function, 69 for wideband radar, 119-21 Transmitted average power, 15 Transmitted peak power, 15
Transmitter phase ripple, 100 Transmitter power, 17-18 of phased array radar, 18 Transmit-to-receive leakage, 143 Transverse electric and magnetic (TEM) propagation, 119 Transverse-electric (TE) mode, 120 Transverse-magnetic (TM) mode, 120 Traveling-wave tubes (TWTs), 121 Two-dimensional optical convolution, 314-15 Ultrawideband waveforms, 134 Unambiguous range window, 355 Unfocused ISAR, 346 optimum, integration angle, 346-47 See also Inverse synthetic aperture radar (ISAR) Unfocused SAR, 240 integration length, 253-56 resolution, 262 theory, 248-56 See also Synthetic aperture radar (SAR) Uniform illumination, 261 Doppler frequency and, 264-65 Up-down counters, 229,231 Video detection, 78 Video pulse spectrum, 168 Visibility threshold, 420 Visible target resolution elements, 420 fraction of, 420-22 Voltage-controlled oscillator (VCO), 159 rale switch, 231 as second stabilizer, 183 voltage drive, 183 Voltage standing-wave ratio (VSWR), 70 input, 72 output, 72 Waveforms binary phase-coded, 139-42 chirp-pulse, 6 illustrated, 151 range walk/range offset for, 371-75 discrete frequency-coded, 143 signal convolution with, 146,147 dispersive, 57 eight-bit phase-coded, 139 illustrated, 141 exponential function, 7 four-frequency periodic, 144-45 frequency-stepped SAR, 283-86 hopped-frequency, 213-20 azimuth scanning with, 218 defined, 214
processing, 215-17 pulse-to-pulse, 213 HRR, 133-86 list of, 135 impulse, 134 monopulse, 457 quadrature detection, 84 real/complex representation of, 8 selection of, 134 short-pulse, 134-36 with ringing filter, 136 search-radar application, 136 spread-spectrum, 516 stepped-frequency, 199,200 3-D monopulse imaging using, 439 illustrated, 202,455 ISAR. 348 synthetic processing of, 353 transmitted and reference, 204 stretch, 149 illustrated, 150 processing of, 150 ultrawideband, 134 Waveguides equalization filters, 121 transmission distortion through, 120 Wavelength, 19 Weighting filters, 161 amplitude, 165 illustrated, 161 performance, 166 pulse widening due to, 163 Weighting functions, 48 White noise, 39 Wideband antennas, 124-25 Wideband microwave power tubes, 121-23 Wideband radar, 10 transmission lines for, 119-21 Wideband solid-state microwave transmitters, 123-24 Wide bandwidth, 10