Space Based Radar: Theory & Applications

Space Based Radar Theory & Applications S. Unnikrishna Pillai Polytechnic University, New York Ke Yong Li C & P Tech...

Author: S Pillai | Ke Yong Li | Braham Himed

325 downloads 750 Views 4MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!

Report copyright / DMCA form

DOWNLOAD PDF

Space Based Radar Theory & Applications S. Unnikrishna Pillai

Polytechnic University, New York

Ke Yong Li

C & P Technologies, Inc., New Jersey

Braham Himed

Air Force Research Lab, New York

New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto

Copyright © 2008 by The McGraw-Hill Companies. All rights reserved. Manufactured in the United States of America. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher. 0-07-159592-9 The material in this eBook also appears in the print version of this title: 0-07-149756-0. All trademarks are trademarks of their respective owners. Rather than put a trademark symbol after every occurrence of a trademarked name, we use names in an editorial fashion only, and to the benefit of the trademark owner, with no intention of infringement of the trademark. Where such designations appear in this book, they have been printed with initial caps. McGraw-Hill eBooks are available at special quantity discounts to use as premiums and sales promotions, or for use in corporate training programs. For more information, please contact George Hoare, Special Sales, at [email protected] or (212) 904-4069. TERMS OF USE This is a copyrighted work and The McGraw-Hill Companies, Inc. (“McGraw-Hill”) and its licensors reserve all rights in and to the work. Use of this work is subject to these terms. Except as permitted under the Copyright Act of 1976 and the right to store and retrieve one copy of the work, you may not decompile, disassemble, reverse engineer, reproduce, modify, create derivative works based upon, transmit, distribute, disseminate, sell, publish or sublicense the work or any part of it without McGraw-Hill’s prior consent. You may use the work for your own noncommercial and personal use; any other use of the work is strictly prohibited. Your right to use the work may be terminated if you fail to comply with these terms. THE WORK IS PROVIDED “AS IS.” McGRAW-HILL AND ITS LICENSORS MAKE NO GUARANTEES OR WARRANTIES AS TO THE ACCURACY, ADEQUACY OR COMPLETENESS OF OR RESULTS TO BE OBTAINED FROM USING THE WORK, INCLUDING ANY INFORMATION THAT CAN BE ACCESSED THROUGH THE WORK VIA HYPERLINK OR OTHERWISE, AND EXPRESSLY DISCLAIM ANY WARRANTY, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. McGraw-Hill and its licensors do not warrant or guarantee that the functions contained in the work will meet your requirements or that its operation will be uninterrupted or error free. Neither McGraw-Hill nor its licensors shall be liable to you or anyone else for any inaccuracy, error or omission, regardless of cause, in the work or for any damages resulting therefrom. McGraw-Hill has no responsibility for the content of any information accessed through the work. Under no circumstances shall McGraw-Hill and/or its licensors be liable for any indirect, incidental, special, punitive, consequential or similar damages that result from the use of or inability to use the work, even if any of them has been advised of the possibility of such damages. This limitation of liability shall apply to any claim or cause whatsoever whether such claim or cause arises in contract, tort or otherwise. DOI: 10.1036/0071497560

Professional

Want to learn more? We hope you enjoy this McGraw-Hill eBook! If you’d like more information about this book, its author, or related books and websites, please click here.

About the Authors S. Unnikrishna Pillai is a Professor of Electrical and Computer Engineering at Polytechnic University in Brooklyn, New York. His research interests include radar signal processing, blind identiﬁcation, spectrum estimation, and waveform diversity. Dr. Pillai is the author of Array Signal Processing and co-author of Spectrum Estimation and System Identiﬁcation and Prof. Papoulis’ Probability, Random Variables and Stochastic Processes (fourth edition). Ke Yong Li is a senior engineer at C & P Technologies, Inc. in Closter, New Jersey. His areas of research include Space-Time Adaptive Processing (STAP), waveform diversity, and radar signal processing. Braham Himed is a senior research engineer at the U.S. Air Force Research Laboratory, Sensors Directorate, in Rome, New York. Dr. Himed’s research interests include radar signal processing, detection, estimation, multichannel adaptive processing, time series analysis, and array processing.

Copyright © 2008 by The McGraw-Hill Companies. Click here for terms of use.

About the Technical Reviewers Dr. Peter Zulch received his Bachelors, Masters, and Doctorate from Clarkson University in 1988, 1991, 1994 respectively. From 1994 to April 2007 he has been employed by the Air Force Research Laboratory, Sensors Directorate, Rome, NY. From April 2007 to the present, he has been employed by Air Force Research Laboratory Information Directorate, also in Rome NY. His interests include multidimensional adaptive signal processing with applications to Airborne Early Warning Radar, space based radar, multi-static and distributed radar, and adaptive waveform diversity for radar. Dr. Zulch is a senior member of the IEEE. Dr. James Ward is Assistant Head of the ISR Systems and Technology Division at MIT Lincoln Laboratory. His areas of technical expertise include signal processing for radar and sonar systems, adaptive array signal processing, detection and estimation theory, and sensor systems analysis. Dr. Ward has given tutorials on spacetime adaptive processing for radar at several IEEE radar and phased array conferences. He has been an organizer and regular lecturer for Lincoln Laboratory short courses on radar systems. He is a past recipient of the MIT Lincoln Laboratory Technical Excellence Award, and the IEEE Aerospace and Electronic Systems Society Fred Nathanson Young Radar Engineer Award. Dr. Ward earned a Bachelor’s degree from the University of Dayton, and both M.S. and Ph.D. degrees in electrical engineering from The Ohio State University. Dr. Ward is a Fellow of the IEEE.

Copyright © 2008 by The McGraw-Hill Companies. Click here for terms of use.

For more information about this title, click here

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

1

Introduction

..........................................

1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The Radar Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Notations and Matrix Identities . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . 1.3.2 Hermitian Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Singular Value Decomposition (SVD) . . . . . . . . . . . 1.3.4 Schur, Kronecker, and Khatri-Rao Products . . . . . 1.3.5 Matrix Inversion Lemmas . . . . . . . . . . . . . . . . . . . . . . Appendix 1-A: Line Spectra and Singular Covariance Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

The Conics

..........................................

What Is a Conic? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Parabola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Hyperbola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Solar System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 2-A: Spherical Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1

3

Two Body Orbital Motion and Kepler’s Laws

.......

Orbital Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 The Motion of the Center of Mass . . . . . . . . . . . . . . . 3.1.2 Equations of Relative Motion . . . . . . . . . . . . . . . . . . . 3.2 Kepler’s Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Synchronous and Polar Orbits . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Satellite Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 3-A: Kepler’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 3-B: Euler’s Equation and the Identiﬁcation of Comets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 3-C: Lambert’s Equation for Elliptic Orbits . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1

1 3 7 9 10 12 16 17 25 26 28

31 31 33 39 40 44 46 50

51 51 52 54 57 60 61 67 71 74 76

v

vi

Space Based Radar 4

Space Based Radar—Kinematics

....................

Radar-Earth Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maximum Range on Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mainbeam Footprint Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Packing of Mainbeam Footprints . . . . . . . . . . . . . . . . . . . . . . Range Foldover Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Mainbeam Foldover . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Total Range Foldover . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Doppler Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Crab Angle and Crab Magnitude: Modeling Earth’s Rotation for SBR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.1 Range Foldover and Crab Phenomenon . . . . . . . . Appendix 4-A: Ground Range from Latitude and Longitude Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 4-B: Nonsphericity of Earth and the Grazing Angle Correction Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 4-C: Doppler Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 4-D: Oblate Spheroidal Earth and Crab Angle Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 4.2 4.3 4.4 4.5

5

Space-Time Adaptive Processing 5.1

5.2 5.3

5.4

5.5 5.6

5.7

...................

Spatial Array Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Why Use an Array? . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Maximization of Output SNR . . . . . . . . . . . . . . . . . . Space-Time Adaptive Processing . . . . . . . . . . . . . . . . . . . . . Side-Looking Airborne Radar . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Minimum Detectable Velocity (MDV) . . . . . . . . . . 5.3.2 Sample Matrix Inversion (SMI) . . . . . . . . . . . . . . . . 5.3.3 Sample Matrix with Diagonal Loading (SMIDL) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Eigen-Structure Based STAP . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Brennan’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Eigencanceler Methods . . . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Hung-Turner Projection (HTP) . . . . . . . . . . . . . . . . . Subaperture Smoothing Methods . . . . . . . . . . . . . . . . . . . . . 5.5.1 Subarray Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . Subaperture Smoothing Methods for STAP . . . . . . . . . . . 5.6.1 Subarray Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.2 Subpulse Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.3 Subarry-Subpluse Method . . . . . . . . . . . . . . . . . . . . . Array Tapering and Covariance Matrix Tapering . . . . . 5.7.1 Diagonal Loading as Tapering . . . . . . . . . . . . . . . . .

77 77 81 83 86 90 90 94 97 101 118 121 123 130 134 137

139 139 140 148 153 155 162 162 165 165 166 167 171 173 177 183 183 184 184 188 192

Contents Convex Projection Techniques . . . . . . . . . . . . . . . . . . . . . . . . 5.8.1 Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.2 Toeplitz Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.3 Positive-Deﬁnite Property . . . . . . . . . . . . . . . . . . . . . 5.8.4 Methods of Alternating Projections . . . . . . . . . . . . 5.8.5 Relaxed Projection Operators . . . . . . . . . . . . . . . . . . 5.9 Factor Time-Space Approach . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 Joint-Domain Localized Approach . . . . . . . . . . . . . . . . . . Appendix 5-A: Uniform Array Sidelobe Levels . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8

6

STAP for SBR

......................................

215

SBR Data Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Mainbeam and Sidelobe Clutter . . . . . . . . . . . . . . . 6.1.2 Ideal Clutter Spectrum . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Minimum Detectable Velocity (MDV) . . . . . . . . . . . . . . . . 6.3 MDV with Earth’s Rotation and Range Foldover . . . . . 6.4 Range Foldover Minimization Using Orthogonal Pulsing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Scatter Return Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Terrain Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 ICM Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 MDV with Terrain Modeling and Wind Effect . . . . . . . . 6.6.1 Effect of Wind on Doppler . . . . . . . . . . . . . . . . . . . . . 6.6.2 General Theory of Wind Damping Effect on Doppler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Joint Effect of Terrain, Wind, Range Foldover, and Earth’s Rotation on Performance . . . . . . . . . . . . . . . . . 6.8 STAP Algorithms for SBR . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 6-A: Matrix Inversion Identity . . . . . . . . . . . . . . . . . . Appendix 6-B: Output SINR Derivation . . . . . . . . . . . . . . . . . . . . Appendix 6-C: Spectral Factorization . . . . . . . . . . . . . . . . . . . . . . Appendix 6-D: Rational System Representation . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

280 283 296 297 298 303 307

Performance Analysis Using Cramer-Rao Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

309

6.1

7

194 195 196 197 198 200 201 205 208 213

7.1 Cramer-Rao Bounds for Multiparameter Case . . . . . . . . 7.2 Cramer-Rao Bounds for Target Doppler and Power in Airborne and SBR Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

216 218 223 232 234 246 255 256 261 268 270 275

309 320 331 338

vii

viii

Space Based Radar 8

Waveform Diversity

................................

Matched Filter Receivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Matched Filter Receivers in White Noise . . . . . . . 8.1.2 Matched Filter Receivers in Colored Noise . . . . . 8.2 Chirp and Pulse Compression . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Joint Transmitter–Receiver Design in Noise . . . . . . . . . . . 8.4 Joint Time Bandwidth Optimization . . . . . . . . . . . . . . . . . . Appendix 8-A: Transform of a Chirp Signal . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1

9

Advanced Topics

...................................

An Inﬁnitesimal Body Around Two Finite Bodies . . . . . 9.1.1 Particular Solutions of the Three-Body Problem 9.1.2 Stability of the Particular Solutions . . . . . . . . . . . . 9.1.3 Stability of Linear Solutions . . . . . . . . . . . . . . . . . . . 9.1.4 Stability of Equilateral Solutions . . . . . . . . . . . . . . . Appendix 9-A: Hill Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1

Index

...................................................

341 344 346 353 358 364 376 385 391

393 394 401 405 409 412 417 420

421

Preface

T

his book is primarily intended for seniors/graduate students as well as professional engineers who have a basic understanding of radar fundamentals. The material is organized so that it can readily be covered in a classroom over a semester. Some basic background on probability theory and stochastic processes will be helpful in understanding the statistical aspects addressed in this book. To help with the signal processing portion, the book has introductory chapters on sensor array processing and Space-Time Adaptive Processing (STAP). Additional material such as lecture notes and homework problems and their solutions are available for the instructor at the course website (visit: www.mhprofessional.com for more details). All celestial bodies and artiﬁcial satellites move around in space subject to Newton’s inverse square law of attraction. In the case of two bodies, this leads to various conic sections such as circles, ellipses, parabolas, and hyperbolas for their orbits. In Chapters 2 and 3 background material on various conic sections and their relationship to Newton’s inverse square law in the form of Kepler’s laws are presented. Chapter 4 presents the space based radar kinematics, including radar-Earth geometry, grazing angle, range and mainbeam footprint size on Earth, range foldover phenomenon, Doppler shift due to Earth’s rotational effects, and the resulting crab angle derivation. Related topics such as the effect of Earth’s non-sphericity on grazing angle, as well as ground range as a function of the local latitude/longitude are treated in the appendices. Chapter 5 gives an introduction to array signal processing and STAP and some of the basic methods for receiver processing are discussed in some detail. The list presented there is by no means exhaustive, and is used only for illustrating the advantages of STAP. Chapter 6 gives detailed accounts of space based radar (SBR) clutter modeling and target detection performance evaluation. Various factors that affect the clutter data such as the range foldover effect,

ix Copyright © 2008 by The McGraw-Hill Companies. Click here for terms of use.

x

Space Based Radar crab phenomenon due to Earth’s rotation, wind and terrain effects are discussed here. The effect of these factors on various STAP algorithms for clutter suppression and target detection are quantiﬁed and illustrated for different situations. To help with the terrain model, maps generated from NASA’s Terra Satellite are used in this book. They categorize each 1 km2 patch of actual Earth into one of 16 land cover types—forest, urban, lakes, etc. The tabulated mean radar cross section (RCS) values of these patches are then used to simulate individual random scatter returns. For even higher ﬁdelity, effects such as grazing angle dependency on RCS are included. Wind simulation is accomplished by modeling the wind as a low order stable ARMA process that depends on the carrier frequency, wind speed, and radar pulse repetition frequency (PRF). An interesting analysis of the effect of wind on algorithm performance is also presented. Chapter 7 presents performance analysis in terms of Cramer-Rao bounds for multiple parameters in a multi-sensor, multi-pulse environment for the airborne case as well as the SBR case. The presentation deals with two unknown parameters; namely target azimuth and power level. Chapter 8 gives an introduction to the joint transmitter receiver waveform diversity. For a given transmit waveform, target (channel) response and noise scene, it is well known that the classical matched ﬁlter gives the optimum receiver characterization. The problem of transmitter optimization and the potential advantages to be gained by transmitting a speciﬁc waveform in a particular context—speciﬁc target (channel) and interference/noise scene—are addressed in this chapter. Chapter 9 deals with additional topics of interest such as locating suitable places in space to park future space stations, as well as near-Earth asteroid-tracking SBRs. In this context, a special three body problem, where an inﬁnitesimal body moves under the inﬂuence of two rotating ﬁnite bodies, is reviewed along with their stable solutions. These stable solutions are illustrated in well-known situations such as the Trojan asteroids near Jupiter and the Gegenschein patch of light in the sky on the earth’s side away from the Sun. Some of these stable solutions such as the Sun-Earth or the Earth-Moon systems may be of interest to man-made missions as well. In the long run, our ultimate survival may depend on the ability of such vigilant deep space radars to detect and track the near-Earth heavenly objects such as asteroids and comets and ultimately deﬂect those that are on a collision course with Earth. The authors would like to take this opportunity to thank Drs. Peter Zulch, AFRL (Rome, NY) and James Ward of Lincoln Labs (Lexington, MA) for their review of the manuscript. Their feedback has been very useful in improving the quality of the book. Peter has been an

Preface enthusiastic supporter of all our efforts from the very beginning, and technical discussions with him on various topics presented here are gratefully acknowledged, in addition to his efforts in helping us with the multiple level public release approval process at AFRL. To that extent, efforts by Mr. Paul Gilgallon, AFRL, and Mr. William Baldygo, Chief, AFRL, Radar Signal Processing branch are also acknowledged here. Drs. Mark Davis, Joseph Guerci, Michael Wicks, S. Radhakrishnan Pillai, and Stephen Mangiat also deserve special thanks for their useful feedback and comments. The authors would like to extend their appreciation also to Mr. Gerard Genello, Yuhong Zhang, Abdelhak Hajjari, and Mr. Lawrence Adzima for their support and encouragement. Finally, the visionaries at AFRL also deserve special credit for their far-sightedness in creating broad based technical programs and for their end-to-end execution. The team at McGraw-Hill—Ms. Wendy Rinaldi, Editorial Director for Engineering; Ms. Mandy Canales, Acquisitions Coordinator; and Ms. Harleen Chopra, Project Manager—deserves special credit for their highly efﬁcient coordinated work and guidance throughout the period of the production process. Ms. Rinaldi with her efﬁcient management style has made this whole process seem effortless for us, and we wish to thank her for the same. Finally, the ﬁrst author wishes to take this opportunity to express his deep gratitude to his mentor Prof. Dante Youla of Polytechnic University, Brooklyn, New York, who has been a true inspiration to the author in many ways from his day one at Polytechnic. To express his appreciation, a quote from John Bunyan is appropriate in this context: . . . You have been so hearty in counseling of us that we shall never forget your favor towards us. . . S. Unnikrishna Pillai Ke Yong Li Braham Himed

xi

This page intentionally left blank

List of Abbreviations AMTI AR ARMA CMT CNR CR DFT EC ECSASPFB

Air Moving Target Indication Auto Regressive Auto Regressive Moving Average Covariance Matrix Tapering Clutter to Noise Ratio Cramer-Rao Discrete Fourier Transform Eigen Canceller Eigen Canceller with Subarray-Subpulse and Forward-Backward smoothing (EC-SASP-FB) EFA Extended Factored Time-Space Approach FTS Factored Time-Space Approach GMTI Ground Moving Target Indication HTP Hung-Turner Projection HTPSASPFB Hung-Turner Projection with Subarray-Subpulse and Forward-Backward smoothing (HTP-SASP-FB) i.i.d. Independent and Identically Distributed JDL Joint Domain Localized approach MDV Minimum Detectable Velocity MF Matched Filter ML Maximum Likelihood PRF Pulse Repetition Frequency PRI Pulse Repetition Interval RCS Radar Cross Section SAR Synthetic Aperture Radar SBR Space Based Radar SINR Signal to Interference plus Noise Ratio SMI Sample Matrix Inversion SMIDL Sample Matrix Inversion with Diagonal Loading SMIDLSASPFB Sample Matrix Inversion with Diagonal Loading, Subarray-Subpulse and Forward-Backward smoothing (SMIDL-SASP-FB) SMIPROJ Sample Matrix Inversion with Convex Projection SMISASPFB Sample Matrix Inversion with Subarray-Subpulse and Forward-Backward smoothing (SMI-SASP-FB) SNR Signal to Noise Ratio STAP Space-Time Adaptive Processing SVD Singular Value Decomposition UAV Unmanned Aerial Vehicle

xiii Copyright © 2008 by The McGraw-Hill Companies. Click here for terms of use.

How shall I comprehend this thing thou sayest “From the beginning it was I who taught”? –Srimat. Bhagavad Gita, Ch. 4 Edwin Arnold translation

CHAPTER

1

Introduction Man made satellites and heavenly objects such as planets and comets move around in space mainly subject to a central force of attraction. In the case of planets the Sun exerts the central force, and in the case of satellites, the Earth plays that role. The central force of attraction is Newton’s inverse square law of gravitation (Principia, 1687), and in the case of two bodies an interesting feature of this force is that the resulting orbits are planar—circular, ellipses, parabolas, or hyperbolas—that all come under the general term conics. Most of the satellites move around the Earth in nearly circular orbits. According to Kepler (1571–1630), the Earth and other planets in the Solar system move around the Sun in elliptical orbits with Sun at one focus, as does the Moon around the Earth, although the orbits of the Earth and the Moon are nearly circular. Parabolic orbits are useful for shifting a spacecraft from one orbit to another, and to escape altogether from the central force as in the case of interplanetary voyages. To realize this goal in a hurry, hyperbolic orbits are more efﬁcient. Space based radar (SBR) once launched into an orbit, moves around the Earth while the Earth continues to rotate around its own axis. By virtue of its location, an SBR can cover a very large area on Earth for intelligence, surveillance, and monitoring of ground moving targets. By adjusting the SBR speed and orbital parameters it is thus possible to periodically scan various parts of the Earth and collect data. Such an SBR based surveillance system can be controlled remotely and may require very little human intervention. The system has a rapid response time, and provides accurate information. As a result, targets of interest can be identiﬁed and tracked in greater detail or images can be made with very high resolution [1]. Depending on the speciﬁc application such as ground/air moving target indication (GMTI/AMTI), or imaging using synthetic aperture radar (SAR), the objectives of the SBR mission can vary. In general, near continuous global coverage and near real-time tasking are the requirements while some of the technical challenges include affordability,

1 Copyright © 2008 by The McGraw-Hill Companies. Click here for terms of use.

2

Space Based Radar constellation size and cost, as well as the interference cancellation capabilities of the receiver processor. Wide area surveillance systems are very important for monitoring Earth resources, mitigation of natural disasters such as ﬂoods, seismic activity monitoring, and border and homeland security. These systems provide visibility over as wide an area as possible, with revisit times commensurate with the mobility and characteristics of the targets of interest. The target signatures may include fast airborne vehicles and very slow or stationary structures hidden by foliage. Historically, manned airborne surveillance platforms such as the Joint Surveillance Target Attack Radar System (Joint STARS) have been ﬁelded for air vehicle detection, surface vehicle imaging, and moving target detection. Recently, however, there has been a growing need in augmenting these assets with space based capabilities. SBR systems have been under consideration for several years, and have only been made commercially available for SAR modes. This is due to the limited availability of space based component technologies and the high cost of manufacturing large space systems. Several recent technology programs have promised signiﬁcant advances in affordable antenna array radar design. However, the design of these radars will require aperture sizes and average power signiﬁcantly larger than previously considered. Such systems would be capable of providing both wide-area surveillance and tracking of airborne and ground moving targets. This capability is particularly attractive because it provides deep coverage into areas typically denied airborne access, greater ease and ﬂexibility for deploying the sensor platform on station and meeting coverage tasking, greater area coverage rate performance, and steep look-down capability for foliage penetration (FOPEN) operation [2]. Several factors must be considered in the SBR system design. These include size of area of interest, revisit rate to cover the search volume, mode scheduling—what other modes need to be scheduled that will affect the area coverage rate-obstruction of targets due to terrain blockage, foliage or other interference effects, and minimum detectable velocity (MDV). The major drivers on area coverage are the altitude of the radar platform and the ﬁeld of view of the radar sensor. Figure 1.1 shows the large ground range coverage obtained using an SBR platform, located at 500 km altitude versus that obtained with an airborne platform located at a height of 10 km [2]. Space based surveillance requires a nominal area coverage rate of several hundred km2/s with revisit rates of one to two minutes, while tracking requires somewhat shorter revisit times. Several studies have shown a constellation of satellites may be necessary to meet these requirements. Even though it may seem that the altitude of a satellite can be freely chosen, the two Van Allen radiation belts limit

Chapter 1:

Introduction

Platform Altitude (km)

500

10 0

200

400 600 800 1,000 Ground Range (km)

1,200

FIGURE 1.1 Ground coverage for airborne vs. SBR platforms.

practical orbit selection. The two Van Allen radiation belts are centered on the Earth’s geomagnetic axis, at altitudes ranging from 1,500 km to 5,000 km and from 13,000 km to 20,000 km. To minimize the radiation damage to electronic components, the satellites would have to be placed in orbits outside of these belts. Therefore, either a mediumearth orbit (MEO) at altitudes of 5,000 km to 13,000 km or a low-earth orbit (LEO) at an altitude less than 1,500 km is desirable. Today, both LEO and MEO constellations are being considered for SBR operations. This book, however, focuses on LEO modes of operation [3]. Another major difference between airborne and space based scenarios arises from the fact that non-geosynchronous spaceborne sensors view a rotating Earth. For airborne geometries, both the sensor platform and the surveillance scene are essentially rotating together, resulting in an equivalent stationary, non-rotating Earth. But for SBR operations, there is a range-dependent shift of the clutter rangeDoppler response, which makes clutter suppression a difﬁcult task, and this in turn limits performance of adaptive techniques such as space-time adaptive processing (STAP) algorithms.

1.1 Overview SBR systems can be categorized into three major areas: Earth observing radars, planetary radars, and defense radars. Seasat (1978), RadarSat-1 (1995) and RadarSat-2 (2007), Shuttle Imaging Radar (SIRA/B/C) (1981–1994), European Remote Sensing Satellite (ERS-1 and 2) are some of the Earth observing radars. The Jet Propulsion Laboratory (JPL) Earth science projects include Seasat (1978) for ocean

3

4

Space Based Radar science, SIR-A/B/C for land imaging, the Shuttle Radar Topography Mission (SRTM) in 2002 for global and topography, the TOPography EXperiment (TOPEX) in 1992, Jason-1 (2001) and Jason-2 (2008), the NASA SCATterometry (NSCAT) in 1996–1997 for ocean topography and wide data set collection [4], [5], [6]. Seasat was the ﬁrst Earth-orbiting satellite designed for remote sensing of the Earth’s oceans and had onboard the ﬁrst spaceborne SAR technology. It was designed to demonstrate the feasibility of global satellite monitoring of oceanographic phenomena and to help determine the requirements for an operational ocean remote sensing satellite system. The speciﬁc objectives were to collect data on sea-surface winds, sea-surface temperatures, wave heights, internal waves, atmospheric water, sea ice features, and ocean topography [7]. Seasat, managed by NASA’s JPL, was launched in June, 1978 into a nearly circular 800-km orbit with an orbital inclination of 108◦ . Seasat carried ﬁve major instruments designed to return the maximum information from ocean surfaces: (i) Radar altimeter to measure spacecraft height above the ocean surface; (ii) Microwave scatterometer to measure wind speed and direction; (iii) Scanning multi-channel microwave radiometer to measure sea surface temperature; (iv) Visible and infrared radiometer to identify cloud, land, and water features; and (v) SAR L-band, HH Polarization, ﬁxed look angle to monitor the global surface wave ﬁeld, and polar sea ice conditions. Seasat operated for 105 days until October 10, 1978, when a short circuit in the satellite’s electrical system ended the mission. RadarSat-1 is Canada’s ﬁrst commercial Earth observation satellite [8]. It was launched on November 4, 1995 from Vandenberg Air Force Base (AFB) in California, into a sun-synchronous (dawn-dusk) orbit above the Earth with an altitude of 798 km and inclination of 98.6◦ . Developed under the management of the Canadian Space Agency (CSA) in cooperation with Canadian provincial governments and the private sector, it provides images of the Earth for both scientiﬁc and commercial applications. RadarSat-1’s images are useful in many ﬁelds, including agriculture, cartography, hydrology, forestry, oceanography, geology, ice and ocean monitoring, arctic surveillance, and detecting ocean oil slicks. It uses a SAR sensor to image the Earth at a single microwave frequency in the C band. Unlike optical satellites that sense reﬂected sunlight, SAR systems transmit microwave energy toward the surface and record the reﬂections. Thus, RadarSat-1 can image the Earth, day or night, in any atmospheric condition, such as cloud cover, rain, snow, dust, or haze. Each of RadarSat-1’s seven beam modes offer a different image resolution. It also has the unique ability to direct its beam at different angles. With an orbital period of 100.7 minutes, RadarSat-1 circles the Earth 14 times a day. The orbit path repeats every 24 days, and hence the

Chapter 1:

Introduction

satellite is in exactly the same location and it takes the same image (same beam mode and beam position) every 24 days. This is useful for interferometry and detecting changes at any particular location that took place during the 24 days. Using different beam positions, a location of interest can also be scanned every few days. It is a side-looking satellite, where the microwave beam transmits and receives on the right side of the satellite, relative to its orbital path. As it descends in its orbit from the north pole, it faces west, and when it ascends from the south pole, it faces east. Locations can therefore be imaged from opposite sides. Combined with the different beam modes and positions, this provides users with many possible perspectives from which to image a location. NASA provided the Delta II rocket to launch it in exchange for access to its data. Estimates are that the project, excluding launch, cost about $620 million (Canadian). The Canadian federal government contributed about $500 million, the four participating provinces (Quebec, Ontario, Saskatchewan, and British Columbia) about $57 million, and the private sector about $63 million. MacDonald, Dettwiler and Associates (MDA) has announced the launch of RadarSat-2 for December 2007. RadarSat-2 is scheduled to be launched on a Soyuz vehicle from Russia’s Baikonur Cosmodrome in Kazakhstan. RadarSat-2 has been designed with signiﬁcant and powerful technical advancements which include high-resolution imaging, polarization ﬂexibility, left and right-looking imaging options, superior data storage, and more precise measurements of spacecraft position and attitude. Some of the planetary radars are Magellan (1990–1994) for Venus probe, Cassini (2004) for Titan probe, Marsis (2003) and Mars reconnaissance orbiter (2006) for Mars probe. The objectives of the CassiniHuygens radar are: (i) Determine the three-dimensional (3D) structure and dynamic behavior of the rings; (ii) Determine the composition of the satellite surfaces and the geological history of each object; (iii) Determine the nature and origin of the dark material on lapetus’s leading hemisphere; (iv) Measure the 3D structure and dynamic behavior of the magnetosphere; (v) Study the dynamic behavior of Saturn’s atmosphere at cloud level; (vi) Study the time variability of Titan’s clouds and hazes and (vii) Characterize Titan’s surface on a regional scale. Cassini-Huygens is a joint NASA, European Space Agency (ESA), the Italian Space Agency (ASI) unmanned space mission intended to study Saturn and its Moons. The spacecraft consists of two main elements: (i) The Cassini orbiter, named after the Italian-French astronomer Giovanni Domenico Cassini; (ii) The Huygens probe, named after the Dutch astronomer Christiaan Huygens. It was launched on October 15, 1997 and entered Saturn’s orbit on July 1, 2004. On December 25, 2004, the probe separated from the orbiter, with deployment conﬁrmed by JPL. The probe reached Saturn’s Moon Titan on

5

6

Space Based Radar January 14, 2005, where it made an atmospheric descent to the surface and relayed scientiﬁc information. It is the ﬁrst spacecraft to orbit Saturn and the fourth spacecraft to visit Saturn. NASA has several spacecraft in orbit around Mars. The latest in that group is the reconnaissance orbiter and the others are Mars Global Surveyor, Mars Odyssey and the ESA’s Mars Express. On the Mars surface, the NASA robotic rovers Spirit and Opportunity continue to perform their geologic missions. The latest orbiter has an advanced camera to photograph the surface and a ground perpetrating radar to probe underground for ice and possible evidence of water. Military satellites include the Lacrosse Series (1, 2, 3, 4, 5) high resolution radar imaging satellites, and various reconnaissance satellites for intelligence purposes, that include high resolution photography (IMINT), signal intelligence (SIGINT), communications, detection of missile launches and nuclear tests [9]. StarLite represented a new light-weight satellite concept advanced by the Defense Advanced Research Project Agency (DARPA) in early 1997. Its study reported the feasibility of developing, deploying and operating a constellation of relatively inexpensive radar satellites designed to provide useful information to the warﬁghter that could be directly tasked by the warﬁghter and downlinked to theater for processing and exploitation. The concept was modiﬁed to incorporate a low cost approach to space based High Range Resolution Ground Moving Target Indication (HRR-GMTI) as well as synthetic aperture radar (SAR) imaging to augment the airborne capabilities, including the Unmanned Aerial Vehicle (UAV), U-2, and JSTARS battleﬁeld HRRGMTI. TechSat 21 represents new innovations in microsatellite cluster program using three satellites that can be deployed in different conﬁgurations and at various separations [10]. The Innovative Space Based Radar Antenna Technology (ISAT) deploys extremely large antennas that are electronically scanned for coherent beamforming so that continuous tracking of surface targets (GMTI) from an MEO is possible [11], [12]. Other countries that are major players in the space radar research and development include Russia with their Cosmos, Almaz, Yantar, Zenit programs, Germany (SAR-Lupe 1-5), France (Hellos 1B, 2A), United Kingdom (Zircon), China, and India (RISAT 2007). Monostatic versus bistatic or multi-static mode is another way to categorize the SBRs. In the monostatic case, the transmitter and receiver are collocated on the same platform, whereas in the bistatic situation they are on different platforms [13]. For example, the transmitter maybe on a high-value high-altitude platform, whereas the receiver maybe on a lower orbit platform and/or on an Unmanned Aerial Vehicle (UAV). The bistatic/multistatic situation affords extra

Chapter 1:

Introduction

degrees of freedom; however, clutter characterization and compensation is much more complex compared to the monostatic case [14], [15]. The purpose of this book is to provide a detailed understanding of the basic SBR principles, and study the analysis and synthesis aspect to its data collection. Several related topics are reviewed in this context here. A review of the conics followed by the dynamics of orbital motion that result in Kepler’s laws are reviewed ﬁrst in this book for an in-depth understanding of the fundamentals. This is followed by a systematic study of SBR—its geometry, mainbeam footprint size, range ambiguities on footprints, SBR clutter modeling, and clutter spectrum. A study of STAP algorithms is taken up next for GMTI and AMTI applications. Other topics covered in this book include an introduction to waveform diversity as well as a detailed analysis of a special case of the three body problem, where an inﬁnitesimal body moves under the inﬂuence of two ﬁnite bodies that revolve around their common center of mass. This is an important conﬁguration for parking space stations as well as deep space based platforms such as asteroid tracking space radars in the Sun-Earth or Earth-Moon frame for long-term surveillance.

1.2 The Radar Equation The fundamental relation between the power characteristics of a radar, target, and the receiver is given by the radar equation [16] that takes the two-way scattering into account. Suppose the radar transmits power PT using an antenna with gain G T then the power spreads over a sphere of radius equal to the transmitter-target distance RT . This gives the power per unit solid angle in the direction of the target to be P1 = PT G T /4π RT2 . If As represents the effective area of the scatter and η its reﬂectivity gain power, then σ = As η

(1.1)

represents the radar cross section (RCS) and P2 = P1 σ represents the power reradiated by the scatter in the direction of the receiver. With RR representing the target-receiver distance, the power/unit angle in the direction of the receiver equals P3 = P2 /4π R2R , which after discounting for the transmitter to target and target to receiver propagation factors FT and F R reduces to P4 = P3 FT2 F R2 . If AR represents the receiver antenna aperture area, we get PR = P4 AR to be the received power. In terms of receiver antenna gain pattern G R , we have

7

8

Space Based Radar AR = λ2 G R /4π so that we obtain the receiver power to be [17] PR = P4 AR =

PT G T G R λ2 σ FT2 F R2 , (4π) 3 RT2 R2R

(1.2)

where λ is the operating wavelength. The receiver average thermal noise power is given by No = kTo Bn ,

(1.3)

where k is the Boltzmann’s constant, To the average noise temperature and Bn the noise bandwidth. Thus the signal-to-noise ratio (SNR) at the receiver is given by SNR =

PT G T G R λ2 σ FT2 F R2 PR = No (4π ) 3 kTo Bn RT2 R2R

(1.4)

and it represents the general radar equation connecting receiver output SNR and range. In the monostatic case where then transmitter and receiver are collocated, we have G T = G R = G, FT = F R = F , RT = RR = R, and the radar equation reduces to SNR =

PT G 2 F 2 λ2 σ KM = 4 (4π) 3 kTo Bn R4 R

(1.5)

and in the bistatic case, (1.4) reduces to SNR =

KB , RT2 R2R

(1.6)

where K M and K B are monostatic and bistatic constants, that depend on radar parameters, target type and geometry. Equations (1.5)–(1.6) can be used to make some interesting observations. For example, in the monostatic constant RCS case, ground points that are at constant range (iso-range) are circles. Similarly from (1.5) points of constant SNR (iso-SNR) correspond to constant range and hence they too are associated with circular paths on the ground. However, in the bistatic case, from (1.6) iso-SNR plots correspond to RT RR = constant. For a ﬁxed transmitter-receiver geometry, these plots represent the Ovals of Cassini, a more complicated situation. Similarly if we deﬁne the transmitter-target-receiver distance RT + RR to represent the bistatic range, then the iso-range plots correspond to ellipses with its two foci at the transmitter and receiver. The constant transmitter-target-receiver distance represents the major axis of this ellipse. As a result, in the bistatic case, iso-range and iso-SNR plots do not coincide with each other and the clutter modeling and associated target detection or imaging becomes a much more complex issue [18], [19].

Chapter 1:

Introduction

1.3 Notations and Matrix Identities Notation for scalars, vectors, and matrices used throughout this book are as follows: Scalars: Regular lower case or upper case Roman and Greek letters (e.g., a , A, α, or λ); Vectors: Lower case bold letters with or without underline (e.g., a, a, or β), or lower case letters with underline (e.g., a , α, or β); Matrices: Upper case bold Roman letters and upper case Greek (e.g., R, U, or ). ¯ AT , A∗ , tr(A), det (A) = |A|, A−1 represent the comFurther, A, plex conjugate, transpose, complex conjugate transpose, trace, determinant, and inverse of the matrix A respectively. The identity matrix is represented by I. A diagonal matrix D with diagonal entries d1 , d2 , . . . , dm is denoted by



d1



0

0

0

0 

d2

0

0 0

0

..

0   .  0 

0

0

D = diag [d1 , d2 , . . . , dm ] =  

.

(1.7)

dm

Let a ij or Aij represent the (i, j)th entry of the matrix A. For any two square matrices A and B of same size we have (AB) ∗ = B∗ A∗ tr (AB) = tr (BA) =

i

(1.8) a i j b ji

det(AB) = |AB| = |A||B| (AB)

−1

−1

(1.9)

j

−1

=B A

(1.10) (1.11)

provided their inverses exist. A principal minor of A is the determinant of a submatrix of A formed with the same numbered rows and columns. If the rows and columns involved in forming a principal minor are consecutive, then the determinant is said to be a leading principal minor. Thus an n × n matrix A has n leading principal minors given by

a 11 a 12 a 13 a 11 a 12 , a 21 a 22 a 23 , . . . , det(A). a 11 , a 21 a 22 a 31 a 32 a 33

(1.12)

9

10

Space Based Radar A square matrix A is said to be nonsingular (singular) if its determinant is nonzero (zero).

1.3.1 Eigenvalues and Eigenvectors For an n × n matrix A, the eigenvalues λ satisfy the equation Ae = λe,

e = 0,

(1.13)

where the vector e is an eigenvector corresponding to the eigenvalue λ. All eigenvalues of A are given by the roots of the characteristic polynomial det(λI − A) = λn + c 1 λn−1 + · · · + c n−1 λ + c n = (λ − λ1 ) m1 (λ − λ2 ) m2 · · · (λ − λr ) mr ,

(1.14)

where λ1 , λ2 , . . . , λr are distinct and m1 , m2 , . . . , mr are the algebraic multiplicities of the eigenvalues. Clearly m1 + m2 + · · · + mr = n. The matrix A has distinct eigenvalues if all mi = 1 and r = n in (1.14). If A has n distinct eigenvalues λ1 , λ2 , . . . , λn , then it has n linearly independent eigenvectors e 1 , e 2 , . . . , e n each pair satisfying (1.13) [20]. Proof If not, suppose these eigenvectors are linearly dependent. Then we must have a 1e 1 + a 2e 2 + · · · + a ne n =

n

a i e i = 0,

(1.15)

i=1

where (at least some) a i are nonzero constants. Applying (A − λn I) to (1.15) we get (A − λn I)

n

ai ei =

i=1

n

a i ( Ae i − λn e i ) =

i=1

=

n−1

n

a i (λi − λn )e i

i=1

(λi − λn )a i e i =

i=1

n−1

b i e i = 0,

(1.16)

i=1

where b i = (λi − λn )a i . From (1.15) and (1.16) we get n−1

bi e i = 0

(1.17)

i=1

and continuing this process, we get e 1 = 0, and hence e 2 = 0, . . . , a contradiction. Hence the eigenvectors associated with distinct

Chapter 1:

Introduction

eigenvalues are independent. Hence we have Ae i = λi e i ,

i =1→n

that gives

A [ e 1,



e 2,

T

···

e n ] = [ e 1,

   e 2, · · · e n ]  

  T

(1.18)

λ1

0

0 .. .

λ2 .. .

0

0

··· .. . ..

.

0 0 .. .

    ,   

· · · λn

(1.19) where T is nonsingular. Equation (1.19) has the form AT = T

(1.20)

A = TT−1 .

(1.21)

which can be written as

Equation (1.21) represents a similarity transformation1 . Thus A is similar to a diagonal matrix if it has distinct eigenvalues. More generally, A is similar to a diagonal matrix if and only if A has n linearly independent eigenvectors. Let v1 , v2 , . . . , vn represent the row vectors of the matrix T−1 . Thus





v1

 v2     T−1 =   ..  = V.  . 

(1.22)

vn In that case (1.21) can be rewritten as A = TV =

n

λi e i vi

(1.23)

n 1 ev. λi i i

(1.24)

i=1

and similarly A−1 = T−1 V =

i=1

1 Two n × n matrices A and B are said to be similar if there exists a nonsingular matrix T such that A = TBT−1 .

11

12

Space Based Radar In the case of repeated eigenvalues, the number of linearly independent eigenvectors (geometric multiplicity) αi associated with an eigenvalue λi is upper bounded by its algebraic multiplicity mi in (1.14). Thus αi ≤ mi .

(1.25)

The number of linearly independent rows (or columns) of a matrix A represents its rank ρ(A). In case of a square matrix, its rank coincides with the total number of nonzero eigenvalues (including repetitions). For any two matrices A and B of dimensions m × n and n × r , the rank of their product satisﬁes Sylvester’s inequality [21] given by ρ(A) + ρ(B) − n ≤ ρ(AB) ≤ min(ρ(A), ρ(B)).

(1.26)

A matrix of full rank is said to be nonsingular.

1.3.2 Hermitian Matrices A square matrix A of size n × n is said to be Hermitian if A = A∗ , i.e., a ij = a ji∗ , i, j = 1 → n. The matrix A is said to be nonnegative deﬁnite if for any n × 1 vector x, the quantity x∗ Ax ≥ 0. When strict inequality holds, i.e., x∗ Ax > 0 for x = 0, the matrix A is said to be positive deﬁnite. A square matrix U is said to be unitary if UU∗ = U∗ U = I. Two n × n matrices A and B are said to be unitarily similar if there exists a unitary matrix U such that A = UBU∗ . A classical result due to Schur states that every n × n matrix A is unitarily similar to a lower triangular matrix. Thus A = ULU∗ ,

(1.27)

where L is lower triangular. In particular, it follows that if A is Hermitian, then L is diagonal, and if A is positive deﬁnite, L is diagonal and positive deﬁnite and hence all its principal diagonal entries L ii > 0. Hence diagonalization of a Hermitian matrix is always possible using a unitary similarity transformation. Thus L = D2 where D is a real diagonal matrix with Dii = ±

L ii

(1.28)

and A = ULU∗ = UD2 U∗ = UDU∗ UDU∗ = C2 ,

(1.29)

where C = UDU∗ . Notice that C is Hermitian and it represents the square root of A. Clearly, from (1.28), C is not unique [22]. How√ ever, it can be made unique by choosing all Dii = L ii > 0. In that

Chapter 1:

Introduction

case, C is positive deﬁnite. Thus for every Hermitian positive deﬁnite matrix, there exists a unique Hermitian square root that is also positive deﬁnite. For Hermitian matrices, the necessary and sufﬁcient condition for it to be nonnegative deﬁnite (positive deﬁnite) can be given in terms of the signs of its principal minors.2 To be speciﬁc, a Hermitian matrix is nonnegative deﬁnite (positive deﬁnite) if and only if all its principal minors are nonnegative (positive). More interestingly, a Hermitian matrix is positive deﬁnite if and only if all its leading principal minors are positive. A matrix is said to be Toeplitz if its entries along every diagonal are the same. Thus if T is Toeplitz, then Tij = ti− j . The inverse of an invertible Toeplitz matrix is not Toeplitz unless it is a 2 × 2 or lower (upper) triangular. Alternatively, any two lower (upper) triangular Toeplitz matrices commute and their product is again [lower (upper) triangular and] Toeplitz! However, lower (or upper) triangular block Toeplitz matrices do not commute. If an n × n Toeplitz matrix is also Hermitian, then it has only at most n independent entries, namely, those along the ﬁrst (last) row (column). Thus the (n + 1) × (n + 1) matrix



ro

r1

 r∗ ro  1   .. Tn =  . ···   r∗  n−1 · · · ∗ rn∗ rn−1

· · · rn−1 r1

···

..

···

.

r1∗ ···

ro r1∗

rn



rn−1    ..  .   r1

(1.30)

 

ro

is Hermitian Toeplitz and moreover it has the recursive form



rn

    Tn =    

Tn−1 rn∗

∗ rn−1

· · · r1∗



rn−1   ..   . .



(1.31)

r1  

r0

Let λi denote an eigenvalue of A. Then, there exists an eigenvector ui = 0, such that Au i = λ i ui . The eigenvectors are in general complex and for positive deﬁnite matrices, they can be made unique 2 A principal minor is the determinant of a submatrix formed with the same numbered rows and columns.

13

14

Space Based Radar by normalization together with a constraint of the form uii ≥ 0. The eigenvalues of a Hermitian matrix are real. For ui∗ Aui = λi ui∗ ui = λi ,

(1.32)

ui∗ Aui = (Aui ) ∗ ui = λi∗ ui∗ ui = λi∗ .

(1.33)

with ui∗ ui = 1. Also

Thus λi∗ = λi or λi is real. If A is also positive deﬁnite then ui∗ Aui > 0 and hence it follows that its eigenvalues are all real and positive. Moreover, eigenvectors associated with distinct eigenvalues of a Hermitian matrix are orthogonal. This follows by noticing that, if λi , λ j and ui , u j are any two such pairs, then ui∗ Au j = ui∗ λ j u j = λ j ui∗ u j .

(1.34)

However, we also have ui∗ Au j = (Aui ) ∗ u j = (λi ui ) ∗ u j = λi ui∗ u j .

(1.35)

Thus λ j ui∗ u j = λi ui∗ u j or equivalently (λi −λ j )ui∗ u j = 0, or ui∗ u j = 0 provided λi = λ j . As a result, if A = A∗ has n distinct eigenvalues, then it can be diagonalized by a unitary similarity transformation as in (1.21) (where T is unitary, T−1 = T∗ ). Interestingly, the diagonalization of any Hermitian matrix is always possible irrespective of whether its eigenvalues λ1 , λ2 , . . . , λn are distinct or not. To prove this, the following result is helpful: Lemma If an n × n matrix A maps a subspace Z into itself, then A has an eigenvector in that subspace Z. Proof Let the subspace Z be of dimension m ≥ 1 and b 1 , b 2 , . . . , b m be any basis for Z. Then b i ⊂ Z and since A maps Z into itself we have Ab i ⊂ Z. Hence Ab i =

m

c ij b i , j = 1 → m.

(1.36)

j=1

Let B = [b 1 , b 2 , . . . , b m ].

(1.37)

AB = BC,

(1.38)

Using (1.36) we get

where C = (c ij ) is the m × m matrix deﬁned in (1.36).

Chapter 1:

Introduction

Let µ represent a nonzero eigenvalue of C; then there exists a vector y such that Cy = µy.

(1.39)

ABy = B(Cy) = B(µy) = µBy.

(1.40)

z = By.

(1.41)

Hence using (1.38)

Deﬁne

Then z is a linear combination of the basis vectors b 1 , b 2 , . . . , b m and hence z ⊂ Z and from (1.40) Az = µz ,

(1.42)

i.e., z is an eigenvector of A belonging to Z. This proves the Lemma. Referring back to the n × n Hermitian matrix A, let u1 , u2 , . . . , um , m < n represent a set of m orthogonal unit eigenvectors, and S the subspace orthogonal to them. S maps into itself since x ⊂ S gives ui∗ x = 0, i = 1 → m and ui∗ (Ax) = (x ∗ Aui ) ∗ = (x ∗ λi ui ) ∗ = λi ui∗ x = 0,

(1.43)

i.e., x ⊂ S gives ui ⊥x, i = 1 → m

implies

ui ⊥Ax, Axi ⊂ S, i = 1 → m.

(1.44)

By the above Lemma, there exists a unit eigenvector y of A that belongs to S, and y = um+1 ⊥{u1 , u2 , . . . , um }. This process has generated m + 1 orthonormal eigenvectors. Continuing this process, a full set of n orthonormal eigenvectors u1 , u2 , . . . , un with corresponding eigenvalues λ1 , λ2 , . . . , λn can be formed. Note that λi need not be all distinct. As a result, if an eigenvalue has multiplicity L, it is always possible to choose a new set of L orthonormal vectors from the above L-dimensional subspace to act as an eigenvector set for that eigenvalue. Thus, for an n × n Hermitian matrix A, if λ1 , λ2 , . . . , λn and u1 , u2 , . . . , un represent its eigenvalues and an orthonormal set of eigenvectors, then Aui = ui λi , i = 1, 2, . . . , n, or in compact form AU = U

(1.45)

where U = [u1 , u2 , . . . , un ],

= diag[λ1 , λ2 , . . . , λn ].

(1.46)

15

16

Space Based Radar Clearly, UU∗ = U∗ U = I, i.e., U is a unitary matrix, and consequently AU = U gives (see (1.27)) A = UU∗ .

(1.47)

Thus, any hermitian matrix can be diagonalized by a unitary matrix whose columns represent a complete set of its normalized eigenvectors. Moreover |A| = |U||||U∗ | = λ1 , λ2 , . . . , λn and tr (A) = λ1 + λ2 + · · · + λ n .

1.3.3 Singular Value Decomposition (SVD) Any m × n matrix A of rank r can be expressed as A = UDV∗ ,

(1.48)

where U and V are unitary matrices of sizes m × m and n × n respectively, and D is a diagonal matrix with r positive diagonal entries. Proof The eigendecomposition of the m × m nonnegative deﬁnite Hermitian matrix AA∗ gives r positive eigenvalues λ21 , λ22 , . . . , λr2 with corresponding eigenvectors u1 , u2 , . . . , ur . Thus AA∗ ui = λi2 ui ,

i = 1 → r.

(1.49)

Deﬁne vi =

1 ∗ A ui , λi

i = 1 → r.

Then vi∗ v j

=

ui∗ AA∗ u j λi λ j

=

λ2j ui∗ u j λi λ j

λj ∗ = u u = λi i j

(1.50)

1,

i = j,

0,

i = j,

i, j = 1 → r.

(1.51) Expand ui , i = 1 → r to form a complete orthonormal basis ui , i = 1 → m. Similarly expand v j , j = 1 → r to form a complete orthonormal basis v j , j = 1 → n. Let U = [u1 , u2 , . . . , um ], V = [v1 , v2 , . . . , vn ].

(1.52)

Clearly U is m × m and V is n × n and UU∗ = Im , VV∗ = In .

(1.53)

Further AA∗ ui = 0,

i > r.

(1.54)

Chapter 1:

Introduction

Thus A = UU∗ A = =

r i=1

m i=1

ui ui∗ A =

ui (λi vi ) ∗ =

n i=1

m

ui ( Aui ) ∗

i=1

λi ui vi∗ = UDV

(1.55)

where D is m × n given by

     D=    

λ1

0

0 .. . .. .

λ2

0

0

··· ··· ..

0

λr

0  ..   .  ≥ 0.  ..  . 

···

0

.

···



(1.56)

Or alternatively Avi =

λ2 1 AA∗ ui = i ui = λi ui , λi λi Avi vi∗ = λi ui vi∗ ,

(1.57) (1.58)

and hence A

n

vi vi∗ =

i=1

In

n

λi ui vi∗ =

i=1

r

λi ui vi∗

(1.59)

i=1

or A=

r

λi ui vi∗ = UDV

(1.60)

i=1

as before.

1.3.4 Schur, Kronecker, and Khatri-Rao Products Schur Product For any two matrices A and B of same size, the Schur (or the SchurHadamard) product A ◦ B represents their element-wise multiplication. Thus if C=A◦B

(1.61)

then

c ij = a ij b ij .

(1.62)

17

18

Space Based Radar We have A ◦ B = B ◦ A,

(1.63)

A ◦ (B ◦ C) = (A ◦ B) ◦ C,

(1.64)

(A ◦ B) ∗ = A∗ ◦ B∗ ,

(1.65)

and the rank of A ◦ B satisﬁes ρ(A ◦ B) ≤ ρ(A)ρ(B).

(1.66)

Let A and B be m × m Hermitian nonnegative deﬁnite matrices with eigenvalues {λi (A)} and {λi (B)} respectively. Suppose λ1 (A) ≥ λ2 (A) ≥ · · · λm (A) ≥ 0

(1.67)

λ1 (B) ≥ λ2 (B) ≥ · · · λm (B) ≥ 0.

(1.68)

and

Thus λ1 and λm represent the largest and smallest eigenvalues respectively. Also let b 1 and b m represent the largest and smallest entries respectively among the diagonal entries of B. Then b 1 ≥ b m > 0, and we have [23] b m λm (A) ≤ λi (A ◦ B) ≤ b 1 λ1 (A)

(1.69)

λm (A)λm (B) ≤ λi (A ◦ B) ≤ λ1 (A)λ1 (B).

(1.70)

and also

It follows from (1.69) that if either A or B is positive deﬁnite, then their Schur product is also positive deﬁnite.

Kronecker Product For any two matrices A and B of arbitrary sizes m × n and p × q respectively, the Kronecker product A⊗B is given by the concatenated block matrix C whose (i, j)th block-entry is given by Cij = a ij B,

i = 1 → m, j = 1 → n.

(1.71)

Notice that the (i, j)th block of C is a scaled version of the matrix B (scaled by the entry a ij ). Hence



a 12 B

···

a 1n B



 a B a B ··· a B  22 2n   21   ··· ··· ··· ···  a m1 B a m2 B · · · a mn B

C=A⊗B=

is of size mp × nq .

a 11 B

(1.72)

Chapter 1:

Introduction

In the case of arbitrary matrices A, B, and D, we obtain [24] (A ⊗ B) ⊗ D = A ⊗ (B ⊗ D),

(1.73)

A ⊗ (B + D) = A ⊗ B + A ⊗ D.

(1.74)

and

For conformable pairs of matrices (i.e., if A is m1 × n1 , B is m2 × n2 , then C is n1 × n3 and D is n2 × n4 ) we obtain the mixed product rule (A ⊗ B)(C ⊗ D) = AC ⊗ BD

(1.75)

that follows by noticing that the (i, j)th block of both sides of (1.75) is given by a ik c kj BD. Other useful relations in this context include k

(A ⊗ B) ∗ = A∗ ⊗ B∗ , (A ⊗ B)

−1

=A

−1

(1.76)

−1

⊗B ,

(1.77)

and ρ(AB) = ρ(A)ρ(B).

(1.78)

Interestingly, we can rewrite the Kronecker product (1.72) using the column vectors of A and B as follows. Let {a i } and {b i } represent the columns of A and B respectively. Thus A = (a 1 , a 2 , . . . , a n ), B = (b 1 , b 2 , . . . , b q ).

(1.79)

Then from (1.72) it follows that C = A ⊗ B = (a 1 ⊗ B, a 2 ⊗ B, . . . , a i ⊗ B, . . . , a n ⊗ B) = (a 1 ⊗ b 1 , a 1 ⊗ b 2 , . . . , a 1 ⊗ b q , a 2 ⊗ b 1 ,

(1.80)

a 2 ⊗ b 2 , . . . , a 2 ⊗ b q , . . . , a n ⊗ b q ). Thus the columns of the Kronecker product A ⊗ B are {a i ⊗ b j } for all i, j combinations arranged in lexicographic order.

Khatri-Rao Product When A and Bhave equal number of columns (n = q ), the special subset a i ⊗ b i , i = 1, 2, . . . , n of (1.80) arranged in lexicographic order deﬁnes the Khatri-Rao product A B. Thus [25]

A B = a 1 ⊗ b1, a 2 ⊗ b2, . . . , a n ⊗ bn

(1.81)

19

20

Space Based Radar and it represents a subset of the columns of the Kronecker product A ⊗ B. Hence it follows that A B = (A ⊗ B) En

(1.82)

where En is an n2 × n matrix given by [26]

En = e 1 , e n+2 , e 2n+3 , . . . , e n2 ,

(1.83)

where e k is an n2 × 1 column vector with a unity in the kth location and zeros elsewhere, i.e.,



T

e k = 0, 0, . . . , 1 0, . . . , 0 .

(1.84)

k

In a similar manner, we can express the Schur product in (1.61) in terms of the Khatri-Rao product. To see this, consider any two column vectors a i , b i of same length m. Then from (1.81), their Khatri-Rao product and the Kronecker product are the same, and it is given by



a 1,i b i



 a 2,i b i     a i bi = a i ⊗ bi =   .. .  .  a m,i b i

(1.85)

From (1.61) and (1.62), the Schur product of a i and b i is given by



a 1,i b 1,i

 a 2,i b 2,i  .. . 

a i ◦ bi =  

   ,  

(1.86)

a m,i b m,i and it is clearly a subset of the rows of (1.85). Proceeding as in (1.82)– (1.84), we can rewrite (1.86) in terms of (1.85) as T a i ◦ b i = Em (a i b i )

(1.87)

with Em deﬁned as in (1.83) with n replaced by m. Thus for any two matrices A and B of size m × n, we obtain T A ◦ B = Em (A B).

(1.88)

Together with (1.82), we obtain the useful identify T A ◦ B = Em (A ⊗ B)En .

(1.89)

Chapter 1:

Introduction

We can use (1.82) and (1.89) to establish an important result: For any two matrices A and B of size m × n and p × n, we get (A B) ∗ (A B) = EnT (A ⊗ B) ∗ (A ⊗ B)En = EnT (A∗ ⊗ B∗ )(A ⊗ B)En = EnT (A∗ A ⊗ B∗ B)En = A∗ A ◦ B∗ B.

(1.90)

Thus with Rx = A∗ A, R y = B∗ B we can write Rx ◦ R y = C∗ C,

(1.91)

where C = A B. More generally, suppose the matrix products AB and CD are of common size m × n, where A is of size m × p and C is of size m × q , and consider their Schur product AB ◦ CD. Using (1.89) together with (1.75) we get

T T AB ◦ CD = Em (AB ⊗ CD)En = Em (A ⊗ C) {(B ⊗ D)En }

= {[(AT ⊗ CT )Em ]T }(B D) T

(1.92)

T T

= (A C ) (B D), where the last two steps follow from (1.82). Notice that (1.90) is a special case of (1.92). In particular for vectors B = x and D = y, (1.92) simpliﬁes to (n = 1) Ax ◦ Cy = (AT CT ) T (x ⊗ y).

(1.93)

Similarly, for conformal matrices A, B, C, D, the useful relation [25] AB CD = (AB ⊗ CD) En = (A ⊗ C) {(B ⊗ D) En } = (A ⊗ C) (B D)

(1.94)

is analogous to (1.75). In particular for vectors x and y, (1.94) reads

Ax By = (A ⊗ B) x ⊗ y .

(1.95)

Let A and B be two square matrices with eigenvalues λ1 , λ2 , . . . , λm and µ1 , µ2 , . . . , µn respectively. Also, consider the situation where A and B have a full set of eigenvectors {ui } and {v j } respectively. In that case, we have Aui = λi ui ,

i = 1 → m,

(1.96)

Bvi = µi vi ,

i =1→n

(1.97)

21

22

Space Based Radar which in compact form reads AU = UΛ1

(1.98)

BV = VΛ2

(1.99)

where U = ( u1

u2

...

um )

V = ( v1

v2

...

vn )

   Λ1 =       Λ2 =   

(1.100) (1.101)

λ1

0

···

0



0 .. .

λ2 .. .

···

0 .. .

    

0

0

· · · λm

µ1

0

···

0



0 .. .

µ2 .. .

···

0 .. .

  .  

0

0

..

.

..

.

(1.102)

(1.103)

· · · µn

From (1.98) and (1.99) A = UΛ1 U−1

(1.104)

−1

B = VΛ2 V

(1.105)

and hence repeated application of (1.75) gives

A ⊗ B = U Λ1 U−1 ⊗ V Λ2 V−1 = (U ⊗ V) Λ1 U−1 ⊗ Λ2 V−1

= (U ⊗ V)(Λ1 ⊗ Λ2 )(U−1 ⊗ V−1 ) = (U ⊗ V)(Λ1 ⊗ Λ2 )(U ⊗ V) −1 .

(1.106)

Thus the columns of U ⊗ V represent the eigenvectors of A ⊗ B and the product set λi µ j , i = 1 → m, j = 1 → n represents the eigenvalues of A ⊗ B. It follows from (1.106) that if both A and B are positive deﬁnite, then so is A ⊗ B. A matrix function can be vectorized in many ways. Neudecher’s vector function of a matrix is obtained by stacking its column vectors in lexicographic order. Thus for the matrix A as deﬁned in (1.79), we

Chapter 1:

Introduction

get [27], [28]



a1



 a2     vec(A) =   .. .  .  am

(1.107)

Using (1.107), Neudecker has shown that for any conformal matrices A, X, and B vec(AXB) = (BT ⊗ A)vec(X).

(1.108)

If A is a diagonal matrix with diagonal entries a ii , i = 1 → m, then (1.107) with its predominant zero entries is an inefﬁcient representation of A, and in that case, a more efﬁcient representation of A is given by the m × 1 vector

   vecd(A) =   

a 11 a 22 .. .

   .  

(1.109)

a mm As a result, if X is a diagonal matrix in (1.108), then using the KhatriRao product it can be more efﬁciently represented as vec(AXB) = (BT A)vecd(X),

(1.110)

where vecd(X) is as deﬁned in (1.109). Given the matrices A, B, C, D, (1.108) can be used to solve for the unknown matrix X in the linear matrix equation AXB + CXD = Q

(1.111)

(BT ⊗ A + DT ⊗ C)vec(X) = vec(Q).

(1.112)

by rewriting it as

Notice that (1.112) is of the form Ay = b. If X is a priori known to be diagonal in (1.111), then using (1.110), we obtain the more compact representation (BT A + DT C)vecd(X) = vec(Q)

(1.113)

that can be used to solve for the unknown diagonal matrix X. For example, the discrete form of the Lyapunov equation X − AXA∗ = −Q

(1.114)

23

24

Space Based Radar where Q is positive deﬁnite, gives the solution (use (1.108)) ¯ ⊗ A)vec(X) = −vec(Q) (I ⊗ I − A

(1.115)

vec(X) = −B−1 vec(Q)

(1.116)

¯ ⊗ A. B=I⊗I− A

(1.117)

or

where

It is well known that the matrix solution X in (1.114) is positivedeﬁnite if and only if A is a stable matrix, i.e., |λi (A)| < 1, i = 1, 2, . . . , n. Although (1.115) and (1.116) do not explicitly exhibit the positive-deﬁnite nature of X, nevertheless it can be simpliﬁed to demonstrate that form. Toward this, consider the eigen decomposition A = TΛV =

n

λi e i vi

(1.118)

i=1

in (1.23). Then by direct computation

B( e¯ j ⊗ e i ) = 1 − λi λ∗j ( e¯ j ⊗ e i )

(1.119)

B(T ⊗ T) = (T ⊗ T)Λ B ,

(1.120)

so that

where Λ B is diagonal with entries 1 − λi λ∗j . This gives B = (T ⊗ T)Λ B (T ⊗ T) −1 = (T ⊗ T)Λ B (T

−1

⊗ T−1 )

= (T ⊗ T)Λ B (V ⊗ V),

(1.121)

where V is as deﬁned in (1.22). Hence as in (1.24), we get B−1 =

n n n n ( e¯ j ⊗ e i )( v¯ j ⊗ vi ) ( e¯ j v¯ j ) ⊗ (e i vi ) = . (1.122) ∗ 1 − λi λ j 1 − λi λ∗j i=1 j=1

i=1 j=1

Using (1.122) in (1.116) we get vec(X) = −

n n ( e¯ j v¯ j ) ⊗ (e i vi ) i=1 j=1

1 − λi λ∗j

vec(Q)

(1.123)

or X=−

n n (e i vi )Q(e j v j ) ∗ i=1 j=1

1 − λi λ∗j

,

(1.124)

Chapter 1:

Introduction

where we have used (1.108). Observe that (1.124) is of the form Y∗ RY where the block matrices Y = ( Y1

Y2

. . . Yn ) T ,

Yi = e i vi , Ri, j = −

Q . 1 − λi λ∗j

(1.125)

Clearly X is Hermitian positive deﬁnite if and only if R is positive deﬁnite, and it can be shown that such is the case if and only if |λi | < 1,

i = 1, 2, . . . , n.

(1.126)

1.3.5 Matrix Inversion Lemmas Given a square matrix A and commensurate vectors a and b we have [29] (A + a b∗ ) −1 = A−1 −

A−1 a b∗ A−1 1 + b∗ A−1 a

.

(1.127)

For square matrices A and B that are also invertible, (A + B) −1 = A−1 − A−1 (A−1 + B−1 ) −1 A−1

(1.128)

from which it follows that (A + BQB∗ ) −1 = A−1 − A−1 B(B∗ A−1 B + Q−1 ) −1 B∗ A−1 ,

(1.129)

provided A, B, Q are invertible. Let R = I + P1 α 1 α ∗1 + P2 α 2 α ∗2 .

(1.130)

Then R−1 = I − γ (γ1 α 1 α ∗1 + γ2 α 2 α ∗2 − ρs γ1 γ2 α 1 α ∗2 − ρs∗ γ1 γ2 α 2 α ∗1 ) (1.131) where ρs = α ∗1 α 2 ,

(1.132)

P1 γ1 = , 1 + P1 α ∗1 α 1

(1.133)

γ2 =

P2 , 1 + P2 α ∗2 α 2

(1.134)

γ =

1 . 1 − |ρs |2 γ1 γ2

(1.135)

and

25

26

Space Based Radar Also

−1 −1 A B A + FEG −FE = C D −EG E

(1.136)

where E = (D − CA−1 B) −1 , −1

F = A B,

(1.137) (1.138)

and G = CA−1 .

(1.139)

Also if |A| = 0, then

A 0 A B I −A−1 B = C D 0 I C D − CA−1 B

(1.140)

from which follows the determinantal identity

A B = |A| |D − CA−1 B|. C D

(1.141)

In a similar manner, we also obtain (|D| = 0)

A B −1 C D = |D| |A − BD C|.

(1.142)

Identities (1.127), (1.128)–(1.136), and (1.140) can be veriﬁed by direct multiplication.

Appendix 1-A: Line Spectra and Singular Covariance Matrices Consider the Hermitian Toeplitz matrix



ro

r1

 r∗ ro  1  ..  ··· Tn =  . r ∗  n−1 · · · ∗ rn∗ rn−1

· · · rn−1 r1 .. . r1∗ ···

··· ··· ro r1∗

rn





 rn−1    ..   . =   r1  ro

Tn−1    ∗ rn∗ rn−1 · · · r1∗

rn



rn−1  ..   . ,  r1   ro (1A.1)

Chapter 1:

Introduction

where rk is the covariance between the random variables x(i T) and x ((i + k)T) generated from a wide sense stationary process x(t) sampled every T seconds. Then rk = E{x((i + k)T)x∗ (i T)}

(1A.2)

x = [x(i T), x((i + 1)T), . . . , x((i + n)T)]T ,

(1A.3)

and with from (1.30) we have Tn = E{x x∗ } ≥ 0.

(1A.4)

Thus the covariance matrices are always nonnegative deﬁnite. Let n = det(Tn ) ≥ 0

(1A.5)

represent the determinant of Tn . Then every n is nonnegative. Consider a stationary stochastic process {x(i T)} with covariance matrices Tn as in (1.30). For some no , suppose Tno −1 is positive deﬁnite. Then no −1 > 0 and moreover all the leading principal minors of Tno −1 are positive. Thus no −1 > 0 ⇒ o > 0,

1 > 0,

...,

no −2 > 0.

(1A.6)

We can use this observation to establish an interesting result: From (1A.5), at the next stage no can be either positive or zero. Suppose no = 0. Then we must have no +1 = 0,

no +2 = 0,

...

(1A.7)

If not, assume no +1 > 0. In that case from (1A.6), in particular no > 0, a contradiction. Hence we must have no +1 = 0, and following the same argument we have k = 0, k > no . Since Tno only involves only one unknown rno compared to Tno −1 (see (1.31)), the conditions no −1 > 0 and no = 0 completely determines rno in terms of ro , r1 , . . . , rno −1 . The remaining higher order covariances can be similarly determined as well in terms of ro , r1 , . . . , rno −1 . To summarize, at any stage if the covariance matrix of a stochastic process becomes singular, then all higher order covariance matrices are also singular. In the case of stationary processes, the higher order covariances are completely determined in terms of the lower order ones. Thus no −1 > 0,

no = 0 ⇒ no +1 = 0,

no +2 = 0,

···

(1A.8)

In particular, if the covariances satisfy rk =

no

Pi e − jkωi , k = 0 → ∞,

(1A.9)

i=1

where Pi > 0, then substituting these into (1.30), we get Tn = An PA∗n

(1A.10)

27

28

Space Based Radar S(w)

FIGURE 1.2 Line spectra.

P2 Pi P1 Pn w1

wi

w2

wn

w

where the (n + 1) × no matrix



1

 e − jω1   − j2ω1 e An =   .  ..  e − jnω1

1

···

e − jω2

···

e − j2ω2 .. .

···

e − jnω2

· · · e − jnωno

1



e − jωno  



e − j2ωno , ..  . 

···

(1A.11)



and the no × no matrix



P1



0

0

0

0 

P2

0

0 0

0

..

0   .  0 

0

0

P= 

.

(1A.12)

Pno

Since rank(Tn ) = no for n ≥ no , we get no −1 > 0,

n = 0.

(1A.13)

Hence the autocorrelations in (1A.9) satisfy (1A.8) and they correspond to line spectra given by (see Figure 1.2) S(ω) =

+∞ k=−∞

rk e

− jkω

=

no

Pi δ(ω − ωi ).

(1A.14)

i=1

In summary, only pure sinusoidal processes possess the determinantal property in (1A.8).

References [1] G.L. Guttrich, W.E. Sievers, and N.M. Tomljanovich, “Wide Area Surveillance Concepts Based On Geosynchronous Illumination and Bistatic Unmanned Airborne Vehicles or Satellite Reception,” Proc. 1997 IEEE National Radar Conference, pp. 126–131, Syracuse, NY, 13–15 May 1997.

Chapter 1:

Introduction

[2] M.E. Davis, B. Himed, and D. Zasada, “Design of Large Space Based Radar for Multimodee Surveillance,” Proc. 2003 IEEE Radar Conference, Huntsville, AL, 5–8 May 2003. [3] Y. Zhang, A. Hajjari, L. Adzima, and B. Himed, “Application of BeamDomain STAP Technologies to Space-Based Radars,” Proc. 36th IEEE Southeastern Symposium on System Theory (SSST) Conference, Atlanta, GA, 14–16 March 2004. [4] L.L. Fu and B. Holt, “ Seasat Views Oceans and Sea Ice with Synthetic Aperture Radar,” Technical Final Report, JPL Publication 81–120, Pasadena, CA, February 1982. [5] T. Freeman, “JPL Imaging Radar: A Tutorial,” also available at http:// southport. jpl.nasa.gov/, July 1994. [6] http://www.jpl.nasa.gov/missions/past missions.cfm [7] R. Friedl, “Exploring Our Home Planet: JPL’s Earth Science and Technology,” Proc. 9th Annual NASA/JPL Space Science Symposium for Small Business, Washington DC, June 2006. [8] Information on RADARSAT-1 and -2 available at http://www.space.gc .ca/asc/eng/default.asp [9] “The Space-Based Radar Plan,” Air Force Magazine, August 2002. [10] J. Hazen, et al., “Stacked Reconﬁgurable Antennas for Space-Based Radar Applications,” IEEE Antennas and Propagation Society International Symposium, and USNC/URSI National Radio Science Meeting, Vol. 1, pp. 158–161, Boston, MA, July 08–13, 2001. [11] S. Chien, et al., “The Techsat-21 Autonomous Space Science Agent,” Proc. AAMAS 2002 conference, Bologna, Italy, June 15–19, 2001. [12] M. Martin, et al., “Techsat 21 qnd Revolutionizing Space Missions Using Microsatellites,” Proc. of 15th AIAA/USU conference on small satellites, Logan, VT, August 2001. [13] R. Klemm, “Comparison Between Monostatic and Bistatic Antenna Conﬁgurations for STAP,” IEEE Transactions on Aerospace and Electronic Systems, Vol. 36, No. 2, April 2000. [14] H.D. Grifﬁths, “Bistatic and Multistatic Radar,” IEE Military Radar Seminar, Shrivenham, September 7, 2004. [15] W.L. Melvin, B. Himed, and M. E. Davis, “Doubly Adaptive Bistatic Clutter Filtering,” 2003 IEEE Radar Conference, Huntsville, AL, May 5–8, 2003. [16] M. Skolnik, Radar Handbook, Second Edition, McGraw Hill, New York, NY, 1990. [17] M. Skolnik, Radar Handbook, First Edition, McGraw Hill, New York, NY, 1961. [18] M.C. Jackson, “The Geometry of Bistatic Radar Systems,” IEE Proc., Vol. 133, Part F., No. 7, pp. 604–612, December 1986. [19] R.E. Kell, “On the Derivation of Bistatic RCS from Monostatic Measurements,” Proc. IEEE, Vol. 53, pp. 983–988, 1965. [20] G.H. Golub and C.F.V. Loan, Matrix Computations, Third Edition, The Johns Hopkins University Press, Baltimore, MD, 1996. [21] F.R. Gantmacher, The Theory of Matrices, Chelsea, New York, NY, 1977. [22] P. Lancaster and M. Tismenetsky, The Theory of Matrices, Second Edition, Academic Press, New York, NY, 1985. [23] C.R. Rao and M.B. Rao, Matrix Algebra and its Applications to Statistics and Econometrics, World Scientiﬁc, Singapore, 1998. [24] R. Bellman, Introduction to Matrix Analysis, Chap. 20, McGraw-Hill, New York, NY, 1960. [25] C.G. Khatri, and C.R. Rao, “Solutions to Some Functional Equations and Their Applications to Characterization of Probability Distributions,” Sankhya: The Indian J. Stat., Series A, 30, pp. 167–180, 1968. [26] H. Lev-ari, “Efﬁcient Solution of Linear Matrix Equations with Application to Multistatic Antenna Array Processing,” Communications in Information and Systems, Vol. 5, No. 1, pp. 123–130, 2005.

29

30

Space Based Radar [27] H. Neudecker, “Some Theorems on Matrix Differentiation with Special Reference to Kronecker Matrix Products,” J. Amer. Stat. Assoc., Vol. 64, pp. 953–963, 1969. [28] J.W. Brewer, “Kronecker Products and Matrix Calculus in System Theory,” IEEE Trans. On Circuits and Systems, Vol. Cas-25, No. 9, pp. 772–781, September 1978. [29] R. Horn and C. Johnson, Topics in Matrix Analysis, Cambridge University Press, New York, NY, 1991.

CHAPTER

2

The Conics All heavenly objects such as planets, comets, and man made satellites move around in space subject to a central force of attraction—viz., the inverse square law of gravitation. An interesting aspect of this force is that the resulting orbits are planar—circles, ellipses, parabolas, and hyperbolas that all come under the general term conics.

2.1 What Is a Conic? A conic represents any plane section of a cone (see Figure 2.1)—circle, ellipse, parabola, and hyperbola—and they can be represented by a general quadratic (with suitable change of axes) in two variables such as [1] ax2 + by2 + c = 0.

(2.1)

A more useful deﬁnition is that a conic is the locus of a point Psuch that the ratio of its distance from a ﬁxed point F (the focus) to its distance from a ﬁxed line L (directrix) is a constant. Thus in Figure 2.2, FP = e, PL

(2.2)

where e is a constant known as the eccentricity of the conic. Let s represent the distance FC from the ﬁxed point to the ﬁxed line. Then since FP = r and PL = s − r cos θ, we have r =e (2.3) s − r cos θ or p r= (2.4) 1 + e cos θ with

p = se

(2.5)

31 Copyright © 2008 by The McGraw-Hill Companies. Click here for terms of use.

32

Space Based Radar FIGURE 2.1 Sections of a cone. Circle

Ellipse

Parabola

Hyperbola

representing the parameter or the semilatus rectum of the conic. From (2.4), by setting θ = π/2 the chord DE through F that is perpendicular to FC has length equal to 2 p. By setting θ = 0, the chord FA has length q = p/(1 + e).

(2.6)

Equation (2.4) is the polar form of a general conic. If e = 0 in (2.4), then r = p represents the equation of a circle. If e < 1, we have r bounded and it represents a closed ﬁgure, the ellipse. Notice that the focus lies on the x-axis and the directrix is parallel to the y-axis.

y-axis D

x-axis

O

P

p

r

F

q

s – r cosq

A s

Focus

L

C

E

Directirx

FIGURE 2.2 A general conic section.

Chapter 2:

The Conics

The conic represents (I) A circle if e = 0 (II) An ellipse if e < 1 (III) A parabola if e = 1 (IV) A hyperbola if e > 1 To study the ellipse, we start with another equivalent deﬁnition and show that it leads to a special case of (2.4) with e < 1.

2.1.1 Ellipse The sum of distances from two ﬁxed points (foci) to any point on the ellipse is a constant (see Figure 2.3). Thus with F1 P = r1 and F2 P = r2 we have r1 + r2 = 2a .

(2.7)

Consider a Cartesian coordinate system centered at the origin of the ellipse with x-axis along the line joining the foci F1 and F2 . If 2c represents the distance between the foci and (x, y) any point P on the ellipse, then (2.7) gives

or

(x + c) 2 + y2 = 4a 2 − 4a

(x − c) 2 + y2 = 2a

(2.8)

(x − c) 2 + y2 + (x − c) 2 + y2 ,

(2.9)

(x + c) 2 + y2 +

(x − c) 2 + y2 = −

P

xc − a 2 xc =a− , a a

(x,y) r1

r2 B

A

O

F2

F1

2c

2a

FIGURE 2.3 Ellipse.

(2.10)

2b

33

34

Space Based Radar

x 2 + c 2 − 2xc + y2 = a 2 − 2xc +

x2

x2 c 2 , a2

a 2 − c2 + y2 = a 2 − c 2 , a2

(2.11)

(2.12)

x2 y2 + = 1. a2 a 2 − c2

(2.13)

b2 = a 2 − c2

(2.14)

Deﬁne

so that the equation of the ellipse becomes (for the Cartesian coordinate system based at O) x2 y2 + = 1. a2 b2

(2.15)

a and b are known as the semimajor and semiminor axes of the ellipse. Since b < a , deﬁne the eccentricity e of an ellipse through the relation b 2 = a 2 (1 − e 2 ).

(2.16)

c = ae

(2.17)

This gives e < 1,

and hence eccentricity represents the position of the focus as a function of the semimajor axis a. Let (r, θ ) represent the polar coordinates of a point P with respect to a focus F1 as in Figure 2.4 and (x, y) its Cartesian coordinates with respect to the center O. Then x = c + r cos θ = ae + r cos θ,

(2.18)

y = r sin θ.

(2.19)

Substituting these into (2.15) we get (ae + r cos θ ) 2 r 2 sin2 θ + = 1, a2 a 2 (1 − e 2 )

(2.20)

(ae + r cos θ ) 2 (1 − e 2 ) + r 2 sin2 θ = a 2 (1 − e 2 ),

(2.21)

Chapter 2:

The Conics

T2 P

(x,y)

r O

F2

q F1

c

T1

FIGURE 2.4 Polar coordinates centered at a focus of an ellipse.

(a 2 e 2 + r 2 cos2 θ + 2a e r cos θ )(1 − e 2 ) + r 2 sin2 θ = a 2 (1 − e 2 ), (2.22) r2

−

e 2 r 2 cos2 θ

+ 2a (1 −

e 2) e r

cos θ +

a 2 e 2 (1

−

e 2)

=

a 2 (1

− e 2 ), (2.23)

a 2 (1 − e 2 ) 2 − 2a (1 − e 2 ) e r cos θ + e 2 r 2 cos2 θ = r 2

(2.24)

which gives r = ± (e r cos θ − a (1 − e 2 )).

(2.25)

When e = 0, r = ± a . But r is always positive, hence r = −(er cos θ − a (1 − e 2 )),

(2.26)

or r=

a (1 − e 2 ) . 1 + e cos θ

(2.27)

Equation (2.27) represents the polar form of an ellipse since e < 1. From (2.4) and (2.27) we have the parameter p = a (1 − e 2 )

(2.28)

for an ellipse. In Figure 2.4, the focal radii F1 P and F2 P make equal angles with the tangent at P, i.e., F1 P T1 = F2 P T2 . To prove this, reﬂect F1 on the tangent T1 T2 at P (Figure 2.4) to produce F1 and join F2 F1 to intersect the tangent at P1 (see Figure 2.5). Then F2 P1 F1 is the shortest path from F2 through a point on the tangent to F1 . For any other point P2 on the tangent we have

35

36

Space Based Radar F 1′

FIGURE 2.5 Reﬂection of F1 on tangent T1 T2 .

T2

P1

F2

P2

T1

F1

F2 P2 F1 > F2 P1 F1 and hence F2 P2 F1 = F2 P2 F1 > F2 P1 F1 = F2 P1 F1 .

(2.29)

Thus P1 on T1 T2 satisﬁes the property that for any other point P2 on T1 T2 F1 P1 F2 < F1 P2 F2 ,

(2.30)

i.e., the shortest path from F1 to F2 meeting the tangent is through P1 . Yet, such a shortest path must be through the point of contact P on the tangent (Figure 2.4) since every other point on the tangent lies outside the ellipse which is a closed convex surface. Hence P1 in Figure 2.5 coincides with P in Figure 2.4, and as a result in Figure 2.5

F1 P1 T1 = F1 P1 T1 = F2 P1 T2

(2.31)

and this proves the claim. Notice that we have made use of the closed convexity property of the ellipse in proving (2.31). From Figure 2.5 we also obtain F1 P1 = P1 F1 . But from (2.7) F2 P1 + F1 P1 = 2a ,

(2.32)

F2 P1 + P1 F1 = F2 F1 = 2a ,

(2.33)

so that

i.e., F1 (the reﬂection of F1 about the tangent at P) lies on the circumference of a circle with center at the other focus F2 and radius equal to 2a (see Figure 2.6). Yet another useful interpretation of an ellipse is also possible: In Figure 2.7, let P be any point on the ellipse with center-based Cartesian coordinates (x, y). Draw an eccentric (auxiliary) circle of radius a with center at O. Draw a perpendicular line to the major axis O A through P to cut the circle at Q and the major axis at R. Let QOR = E and it is known as the eccentric angle of P.

Chapter 2:

The Conics

F 2′ 2a

F 1′

2a

P

P

F2

2a

F1

F2

2a

F1

(a)

(b)

FIGURE 2.6 Reﬂections of F1 and F2 about tangents on the ellipse.

Then x = OR = OQ cos E = a cos E

(2.34)

and substituting this into (2.15) we get y = b sin E = PR.

(2.35)

QR = a sin E.

(2.36)

Also

FIGURE 2.7 Alternate deﬁnition of an ellipse using an eccentric (major) circle.

Q (0,b)

a P

O

r q

E

B

PR = b QR a

(x,y)

R

F1

A (a,0)

37

38

Space Based Radar Hence for any point P on the ellipse, we obtain PR b sin E b = = , QR a sin E a

a constant less than 1.

(2.37)

From (2.37), an ellipse can be viewed as the locus of points P (on the line QR) such that PR/QR is a constant less than 1, where Q is any point on the circle of radius a and QR is perpendicular to the diameter AOB. P will trace out an ellipse with semi-axes a and b (
(2.38)

r cos θ = a (cos E − e)

(2.39)

PR = r sin θ = b sin E.

(2.40)

so that

and

Squaring and adding (2.39) and (2.40), and using (2.16) we get r = a (1 − e cos E).

(2.41)

r (1 − cos θ ) = a (1 + e)(1 − cos E)

(2.42)

r (1 + cos θ) = a (1 − e)(1 + cos E)

(2.43)

From (2.39) and (2.41)

and

and hence from (2.42) and (2.43) we obtain the desired relation connecting the true anomaly and the eccentric anomaly to be [1], [2], [3]

tan(θ/2) =

1+e tan( E/2). 1−e

(2.44)

Another (minor) eccentric circle can be drawn with center at O and radius b as shown in Figure 2.8. Let P be any point on the ellipse with coordinates (x, y). Draw a perpendicular line to the minor axis OC through P to cut the circle at Q and the minor axis at R. Let QOA = E . Then y = OR = OQ sin E = b sin E

(2.45)

Chapter 2: FIGURE 2.8 Minor eccentric circle.

The Conics

PR QR

C (0,b) R B

Q b

=

P(x,y )

E′

O

a b

F1

A (a,0)

D

and substituting this into (2.15) we get x = a cos E = PR.

(2.46)

QR = b cos E .

(2.47)

Also

Hence for any point on the ellipse we get (see also (2.37)) PR a a cos E = > 1. = QR b cos E b

(2.48)

Hence similar to (2.37), an ellipse can be viewed as the locus of points P on the line QR such that PR/QR is a constant greater than one. In summary, for any ellipse there are two eccentric circles—one inscribing and the other circumscribing the ellipse—and any one of them can be used to generate the ellipse.

2.1.2 Parabola For a parabola the eccentricity e = 1 in (2.4) and hence often it can be thought of as the limiting case of an ellipse with e → 1. Since the semilatus rectum p = F1 D is ﬁnite for a parabola, if we let e → 1 in an ellipse, then from (2.28) a → ∞ for a parabola. From (2.4) with e = 1 we obtain p r = sec2 (θ/2) = q sec2 (θ/2) (2.49) 2 to be the polar equation of a parabola (with focus F1 as the origin). Here AF1 = q = p/2 as in (2.6). The Cartesian equation of a parabola centered at A is y2 = 2px,

(2.50)

39

40

Space Based Radar y - axis D P

A

q

q

r θ

F1

E

FIGURE 2.9 Parabola.

since D has coordinates ( p/2, p). Parabola has only one (ﬁnite) focus F1 at ( p/2, 0) since F1 A = p/2. The second focus is at inﬁnity since the distance F1 F2 = 2ae → ∞ as e → 1 for an ellipse. Since F1 P and F2 P make equal angles with its tangent at P, it follows that the tangent at P makes equal angles with F1 P and the line parallel to the x-axis through P. Hence a ray of light traveling parallel to the x-axis gets focused at F1 after reﬂection by the parabola as shown in Figure 2.9.

2.1.3 Hyperbola For a hyperbola the eccentricity e > 1 and hence r is not bounded in (2.4). Proceeding as in (2.7)–(2.16), we have a 2 (1 − e 2 ) to be negative, and hence if we deﬁne b 2 = a 2 (e 2 − 1)

(2.51)

x2 y2 − =1 a2 b2

(2.52)

then

represents the equation of a hyperbola. Hyperbola can be deﬁned as the locus of points P such that the difference of distances from the two foci is a constant. Thus in Figure 2.10 F1 P − F2 P = 2a

(2.53)

and proceeding as in (2.8)–(2.15) we can derive (2.52). This also gives the semilatus rectum p = F1 D to be p = a (e 2 − 1).

(2.54)

For large x and y, (2.52) is nearly the same as

x y + a b

x y − a b

=0

(2.55)

Chapter 2:

The Conics

FIGURE 2.10 Hyperbola. D P p B F2

A O

F1

E

which represents the equation of two lines that pass through the origin with slope ±b/a . Hence the larger (x, y) becomes in (2.52), the more nearly the hyperbola resembles these two lines, known as the asymptotes as shown in Figure 2.10. From (2.52) there is no point on the hyperbola for which −a < x < a , and hence the hyperbola has two branches. When r → ∞ in (2.4), we have θmax = cos−1 (−1/e)

(2.56)

and this equals the gradient of the asymptotes given by tan−1 (−b/a ). Hence tan−1 (−b/a ) = cos−1 (−1/e)

(2.57)

from which we obtain sin(cos−1 (1/e)) b = tan(cos−1 (1/e)) = a 1/e

=

1 − cos2 (cos−1 (1/e)) = e 2 − 1, 1/e

(2.58)

same as (2.51). From (2.52) the Cartesian coordinates (centered at O) of any point (x, y) on the hyperbola can be parameterized as x = a cosh ψ,

y = b sinh ψ,

(2.59)

and hence the name “hyperbola.” In Figure 2.10, OA = a , and hence the tangent at A cuts the asymptotes at (a , ±b). The distance F1 A = p/(1 + e) = a (e − 1). The orbits of all planets are ellipses with the Sun at one focus, the other focus being empty. Earth moves around the Sun in an elliptical orbit with eccentricity e = 0.016726. The Moon moves around the

41

42

Space Based Radar Earth in an elliptical orbit (the Earth is at one focus) with eccentricity e = 0.05490. Both orbits are nearly circular. In fact all planetary orbits are nearly circular except for Mercury and Pluto which have slightly prolonged eccentric elliptical orbits with eccentricities 0.250 and 0.205 respectively. The orbits of all planets are more or less in the same plane called the ecliptic (plane of the Earth’s orbit around the Sun) except that of Pluto which has a 17◦ inclination with respect to the ecliptic. The ecliptic itself is inclined only at 7◦ from the plane of the Sun’s equator. All planets in the Solar system revolve in the counterclockwise direction around the Sun (as seen from above the North Pole of the Sun). All planets except Venus, Uranus, and Pluto also rotate in that same sense. Comets are a collection of rocks and ice that travel around the Sun in highly eccentric orbits surrounded by a haze of gases and dust that for some reason did not get included into the planets during the formation of the Solar system. Comets are thought to be stored in a region outside the orbit of Pluto in the outer regions of the Solar system called the Oort cloud. From time to time, due to perturbations from passing stars, some of them enter the inner part of the Solar system traveling in parabolic orbits, and many return to the Oort cloud after a close encounter with the Sun. There are many periodic comets in the Solar system moving in elliptic orbits including Halley’s comet. Chance encounters close to Jupiter are thought to be the main cause that perturbs their orbits from parabolic to highly eccentric ellipses. Comets have a nucleus that is relatively solid and stable. Made of ice and gas it is surrounded by a hydrogen cloud extending to millions of kilometers in diameter. The nucleus is followed by a dust tail up to ten million kilometers long composed of particles escaping from the nucleus that get “heated up” as the comet passes around the Sun. Many comets have pronouncedly elongated elliptical orbits, often being nearly parabolic with periods on the order of 10–3,000 years with the Sun at their focus. Of the 1,000 comets whose orbits have been computed, fewer than 100 have periods less than 100 years. Many comet orbits extend considerably farther out into space beyond Pluto (trans-plutonian space), that is still dominated by the gravitational ﬁeld of the Sun. The Sun in fact is able to hold bodies as far as two to three light years away from its center. The average period of the famous Halley’s Comet is about 77 years and this gives the major axis of its elliptical orbit to be about 2.4933 light hours or 18.07 A.U. After a ﬁnite number of passes (500–1,000) around the Sun, a comet loses most of its ice and gas leaving a rocky object similar to an asteroid. Meteor showers occur when the Earth passes through the dust tail of a comet. There are millions of asteroids in the Solar system. Most of them are in stable elliptical orbits in the asteroid belt between Mars and Jupiter. However, some of them have Earth crossing orbits as well. As the Earth moves around the Sun, these objects can occasionally strike the

Chapter 2:

The Conics

Earth and other planets causing destruction. The degree of destruction depends on the size and speed of the striking object. For example, a 50-meter object can destroy a city; an object of size 1–2 km striking at 45,000 km/h can practically wipe out life on Earth and “sterilize” the planet. About 2,000 such objects circulate around the Sun. Less catastrophic attacks are more often. For example, the Tunguska explosion over Siberia in 1908 was caused by a 60-meter object that exploded over the atmosphere ﬂattening thousands of kilometers of forest. It is easy to estimate the probability of Earth impact due to such an object. Recall that Earth travels around the Sun in approximately a circle of radius Ro = 150 million km. Most of the objects of interest that loop around the Sun cut through this “Ro radius sphere” that has a surface area of 4π Ro2 (total exposed area). Earth’s surface area 4πre2 represents the desired area. Thus probability of Earth impact is given by (Laplace’s deﬁnition) pH

Earth s area 4πre2 = = = Total area 4π Ro2

re Ro

2

=

6,378 150 × 106

2

= 1.8 × 10−9 ≈ 1 in a billion (per object).

(2.60)

Table 2.1 shows the number of various sizes of objects round the Sun and an estimate of probability of impact on Earth due to each class.

NEO Size

N: Number of Objects Around the Sun/Year

10 km

∼30

2 × 10−8 ∼ once every 50 Million years∗

1 km

2,000

3.6 × 10−6 ∼ once every 200,000 years

100 m

300,000

5.4 × 10−5 ∼ once every 20,000 years

50 m

∼1 − 10 Million

2 × 10−2 − 5 × 10−3 ∼ once every 50–100 years♣

Prob. of Hit: p =N× pH

∗ (i)

Last hit at Chicxulub, Yucatan ∼65 Million years ago (∼20 km size). (ii) Earth crossing asteroid 4179 Toutatis (4.5 × 2.4 × 1.9 km size) passed Earth at a distance of 1 million miles on September 2004. (Loops around the Sun every 4 years).

♣ (i)

Last hit at Tunguska in 1908. (ii) Asteroids 1989FC (around 800 meters) and JA1 (100 meters) missed the Earth by about six hours on March 1989 and May 1996 respectively. (iii) Asteroid 2004 MN4 (400 meters): p = 0.003. Close encounter around April 2029?

TABLE 2.1 Number of various sizes of near Earth objects (NEO) around the Sun and the estimated probability of impact on Earth.

43

44

Space Based Radar

2.2 The Solar System A long time ago, a cloud of interstellar gas and dust began to collapse under its own gravity, possibly prompted by shock wave disturbances from nearby exploding supernovas. As the cloud collapsed it heated up vaporizing gas and the center began to compress forming a star in about 100,000 years. The rest of the gas ﬂowed toward the center in a rotating manner while adding to the mass of the star—the Sun. Some of that gas formed a disc around the star that cooled off and began to condense into metal, rock, and other particles. These particles collided forming larger particles and asteroids. With nontrivial gravity eventually they pulled in more particles and after about 10–100 million years planets were formed around the Sun. The Sun is at the center of the Solar system with about 99.2% of the total mass of the Solar system in it (Ms 1.99 × 1030 kg) and Jupiter contains most of the rest. The Sun is giant with its radius about 109 times the radius of the Earth. In terms of sphere packing, about 1,300,000 Earths could ﬁt easily inside the Sun. Jupiter and Saturn are next with respective radii about 11 and 9 times the radius of the Earth. In terms of actual size, the radius of the Sun is much larger than that of the Moon’s orbit around the Earth (696,265 km versus 384,400 km). There are nine planets in the Solar system (including Pluto)—Mercury, Venus, Earth, and Mars, forming the inner Solar system and Jupiter, Saturn, Uranus, Neptune, and Pluto forming the outer Solar system. If we take the Earth to be at one Astronomical unit (A.U. = 1.49 × 108 km) away from the Sun, then planets lie in the range of 0.38–39.5 A.U. In addition there are about 1,500 known asteroids, 31 satellites such as the Moon, and a large number of comets and meteors in the Solar system. The Sun is about 70% hydrogen, 28% helium, and everything else amounts to less than 2% by mass. This composition changes slowly as the Sun converts hydrogen to helium through nuclear fusion at its core. This process also generates gamma rays that eventually travel toward the Sun’s surface. As the gamma rays progress toward the Sun’s surface, energy is continuously absorbed and reemitted at lower and lower temperatures within the Sun’s body, resulting in visible light at the surface. It took about 4.5-billion light years for the Sun to get to its current state. By now the Sun has used up about half of the hydrogen supply in its core. The Sun will continue to shine peacefully for another 5 billion years with its intensity roughly doubling during that time. In the end, the Sun will exhaust all of its hydrogen fuel and it will collapse triggering a nuclear fusion reaction of the elements at its core. This in turn will push the surrounding hydrogen outward to

Chapter 2:

The Conics

FIGURE 2.11 Darkness to light to darkness.

(a) In the beginning, there was nothing; total darkness everywhere.

(b) Then there was bright light. Earth

Sun

(c) Bright light turned out to be good at some places.

(d) At the end, back to darkness.

form a red giant star whose orbit will reach beyond that of Jupiter, and all that will ultimately cause the total destruction of the Earth. Future is indeed bleak, but hopefully, not the immediate one1 (see Figure 2.11). 1 The “immediate” cause for concern is the (partial/total) destruction of life on Earth that can be caused by an impact of an Earth-crossing asteroid or a comet of size 1–2 km or more. Such an event (roughly once every 200,000 years) can practically “sterilize” the planet. Direct impact of even smaller size asteroids can cause tremendous destruction. Hopefully the SBR technology will be in place on time to detect, track, and if necessary to deﬂect these Earth-crossing objects.

45

46

Space Based Radar A

FIGURE 2.12 Ordinary triangle.

A

c

B

b

B

C a

C

Appendix 2-A: Spherical Triangles In an ordinary planar triangle ABC (Figure 2.12), the following relation holds among the angles A, B, C and opposite sides a , b, c: a b c = = . sin A sin B sin C

(2A.1)

To prove (2A.1), drop a perpendicular from A to the opposite side a and if p represents the length of this perpendicular, then sin B =

p p , sin C = c b

(2A.2)

from which (2A.1) follows. However, in the case of a spherical triangle (2A.1) is no longer true.

Spherical Triangle If three points are chosen on the surface of a sphere, then three unique great circles can be drawn through them by selecting two of these points at a time. These three great circles intersect generating three arcs on the surface of the sphere that form the boundaries of a spherical triangle (see Figure 2.13). The planes of these great circles form a trihedral at the center of the sphere as in Figure 2.13. The tangents to the arcs AB and AC at A along these planes are perpendicular to the radius OA and the interior angle deﬁned by these tangents gives the spherical angle A. In a spherical triangle located on a unit sphere, the angles A, B, C so deﬁned and the opposite side (arcs) a , b, c satisfy the following relation [4]:

The Law of Sines sin b sin c sin a = = . sin A sin B sin C

(2A.3)

Chapter 2:

The Conics

FIGURE 2.13 Spherical triangle. A A

b

c O

C B

a

Proof We shall ﬁrst prove (2A.3) for a right spherical triangle that has one angle equal to 90◦ as shown in Figure 2.14. Draw a plane through A perpendicular to the line OB that cuts the spherical pyramid in the triangle ADE. Since OD is perpendicular to DA and DE, the angle ADE equals the spherical angle B. From triangle AOE, by ordinary trigonometry we have AE = sin b.

(2A.4)

From triangle AOD, we have AD = sin c.

(2A.5)

A A b

1 C b O c

90°

90° a 1

90°

B

C

E

90°

B

D B

FIGURE 2.14 Right spherical triangle.

90°

a

47

48

Space Based Radar Hence from triangle ADE, we get sin B =

sin b AE = AD sin c

(2A.6)

sin b . sin B

(2A.7)

or sin c =

Next, consider a general spherical triangle ABC in Figure 2.15, where p is the arc of a great circle through A that is perpendicular to side a. Then ABD and ACD each has one right angle and applying (2A.7) to the spherical triangles ABD and ACD, we obtain sin c =

sin p , sin B

(2A.8)

sin b =

sin p . sin C

(2A.9)

and

This gives sin c sin B = sin b sin C

(2A.10)

sin b sin c = . sin B sin C

(2A.11)

or

Similarly, considering another great circle arc from B that is perpendicular to the opposite side b we get sin a sin c = . sin A sin C

(2A.12)

A

FIGURE 2.15 General spherical triangle.

A b

c p

B

C

B 90° a

D

ρ

C

Chapter 2:

The Conics

Combining (2A.11) and (2A.12), we get (2A.3) and that completes the proof for the law of sines. Finally, if the sphere is of radius r, then a great arc of length a extends an angle a /r at the origin, and hence (2A.3) holds good in that case with a , b, c replaced by α = a /r , β = b/r , and γ = c/r , i.e., sin α sin β sin γ = = , sin A sin B sin C

(2A.13)

where α, β, γ are respectively the angles generated by the great arcs a , b, c at the origin. From (2A.3) and (2A.13), it follows that the order of magnitude of the sides of a spherical triangle is the same as the order of magnitude of the respective opposite angles, i.e., a
↔

A< B < C .

(2A.14)

The Law of Cosines For spherical triangles, the law of cosines reads as follows: cos a = cos b cos c + sin b sin c cos A.

(2A.15)

Proof Referring back to the right triangles in Figure 2.14, we get OD = cos c,

OE = cos b,

AD = sin c

(2A.16)

so that from triangle ODE we obtain DE = OD tan a = cos c tan a

(2A.17)

DE = OE sin a = cos b sin a .

(2A.18)

and

Equating (2A.17) and (2A.18) we get cos c = cos a cos b .

(2A.19)

Further, from the right triangle ADE in Figure 2.14, we get cos B =

DE cos c tan a = AD sin c

(2A.20)

or tan a = tan c cos B.

(2A.21)

Equations (2A.19)–(2A.21) are valid for right spherical triangles To prove (2A.15) in the general case, refer to the general spherical triangle in Figure 2.15, and let the great arc DC = ρ. Then arc

49

50

Space Based Radar BD = a − ρ, and applying (2A.19) to the right spherical triangles ABD and ACD, we get cos c = cos p cos(a − ρ)

(2A.22)

cos b = cos p cos ρ

(2A.23)

and

or cos c cos(a − ρ) = = cos a + sin a tan ρ. cos b cos ρ

(2A.24)

But for the right spherical triangle ACD, (2A.21) gives tan ρ = tan b cos C

(2A.25)

and using this in (2A.24), we get cos c = cos b (cos a + sin a tan b cos C) = cos a cos b + sin a sin b cos C

(2A.26)

the desired cosine relation for a spherical triangle.

References [1] J.M.A. Danby, Fundamentals of Celestial Mechanics, The Macmillan Co, New York, NY, 1964. [2] F.R. Moulton, An Introduction to Celestial Mechanics, The Macmillan Co, New York, NY, 1964. [3] W.M. Smart, Text Book on Spherical Astronomy, Cambridge University Press, Cambridge, 1965. [4] L.M. Kellas, W.F. Kern, J.R. Bland, Spherical Trigonometry, McGrawHill, New York, NY, 1942.

CHAPTER

3

Two Body Orbital Motion and Kepler’s Laws The problem of two body motion was ﬁrst solved by Newton around 1685 and the solution is given in his Principia. From studying Kepler’s laws, the gravity at the Earth’s surface,1 and the motion of the Moon around the Earth, Newton was led to the universal law of gravitation which states that “every two bodies of matter in the universe attract each other with a force that acts in the line joining them, and whose intensity varies as the product of their masses and inversely as the square of their distance.” The law of gravitation involves considerably more than planetary motion; however, Newton had nothing more than Kepler’s laws to build it on. Nevertheless, by a master stroke of genius Newton stated the law of gravitation in immense generality applying it to the whole universe and it has stood in its entirety without change for the last 300 years. It is however noteworthy to observe that the question of whether the law of gravitation is truly universal has been proved only to hold in the Solar system and in the motion of double stars.

3.1 Orbital Mechanics Orbital motion of planets and satellites can be derived from Newton’s law of gravitation [1], [2], [3]. Let (x1 , y1 , z1 ) and (x2 , y2 , z2 ) denote the coordinates of the Sun and the planet with respect to an inertial reference frame centered at O as in Figure 3.1. Further, let M and m denote the masses of the Sun and the planet. By Newton’s second law, the planet P is attracted to the Sun with force G Mm/r 2 where r is the distance SP. The component of this force 1 Among other things, the interesting anecdote of an apple falling on Newton’s head.

51 Copyright © 2008 by The McGraw-Hill Companies. Click here for terms of use.

52

Space Based Radar Z

FIGURE 3.1 The Sun and a planet.

r

P (x2,y2,z2)

qx

S

(x1,y1,z1) O

Y

X

in the positive x-direction equals −

GMm x2 − x1 GMm cos θx = − 2 . r2 r r

(3.1)

2

Thus if d x22 denotes the acceleration of the planet P parallel to the dt x-axis, we have m

d 2 x2 dt2

=−

GMm (x2 − x1 ). r3

(3.2)

Similarly for the Sun there is an equal and opposite force resulting in M

d 2 x1 dt

2

=

GMm (x2 − x1 ). r3

(3.3)

From (3.2) and (3.3) we get d 2 (x2 − x1 ) dt

2

=−

G( M + m) (x2 − x1 ). r3

(3.4)

Similar equations can be obtained for y and z directions as well. What happens to the center of mass of this two-body system under the inverse square law?

3.1.1 The Motion of the Center of Mass To investigate the motion of the center of mass under the inverse square law, consider the center of mass of the two-body system in Figure 3.1 is given by Mx1 + mx2 M+m My1 + my2 y¯ = M+m Mz1 + mz2 z¯ = M+m

x¯ =

(3.5)

Chapter 3:

Two Body Orbital Motion and Kepler’s Laws

and it lies on the line joining M and m. To determine its motion under inverse square law, we can make use of (3.2) and (3.3). From there we get d 2 x1

M

dt2

+m

d 2 x2 dt2

= 0.

(3.6)

Similarly for the y and z coordinates we get M

d 2 y1

M

dt

2

d 2 z1 dt

2

+m +m

d 2 y2 dt2 d 2 z2 dt2

=0 = 0.

(3.7)

Consecutive integration of these equations yield d x1 d x2 +m = α1 dt dt dy2 dy1 M +m = α2 dt dt dz1 dz2 M +m = α3 dt dt

M

(3.8)

and Mx1 + mx2 = α1 t + β1 My1 + my2 = α2 t + β2

(3.9)

Mz1 + mz2 = α3 t + β3 . Using (3.1)–(3.3) in (3.6) we get ( M + m) x¯ (t) = α1 t + β1 ( M + m) y¯ (t) = α2 t + β2

(3.10)

( M + m) z¯ (t) = α3 t + β3 or ( M + m) x¯ − β1 ( M + m) y¯ − β2 ( M + m) z¯ − β3 = = , (3.11) α1 α2 α3 i.e., the coordinates of the center of mass move in a straight line. Taking the derivatives in (3.10), squaring and adding we also obtain ¯ V(t) =

2 2 2 α 2 + α22 + α32 d x¯ d y¯ d z¯ + + = 1 = c, dt dt dt ( M + m) 2

(3.12)

i.e., under the inverse square law, the center of mass of a two-body system moves in a straight line with constant speed.

53

54

Space Based Radar What is really important is to determine the relative motion of one body with respect to the other under the inverse square law (e.g., motion of the Earth with respect to the Sun).

3.1.2 Equations of Relative Motion In this context, let x = x2 − x1 , y = y2 − y1 , z = z2 − z1

(3.13)

represent the coordinates of the planet P referred to the rectangular axes passing through the Sun (heliocentric coordinate system). Then with µ = G( M + m)

(3.14)

from (3.4) we obtain d2x 2

dt

= −µ

x , r3

(3.15)

= −µ

y , r3

(3.16)

= −µ

z . r3

(3.17)

and in a similar manner we get d2 y dt

2

d2z dt

2

Equations (3.15)–(3.17) represent the motion of the planet with reference to the Sun. From (3.15)–(3.16) we get x or

d2 y dt

2

−y

d2x

=0

dt2

(3.18)

dy dx d x −y dt dt dt

=0

(3.19)

which gives x

dy dx −y = A. dt dt

(3.20)

Similarly from (3.16)–(3.17) we get y

dy dz −z = B, dt dt

(3.21)

and from (3.15) and (3.17) we have z

dz dx −x = C. dt dt

(3.22)

Chapter 3:

Two Body Orbital Motion and Kepler’s Laws

From (3.20)–(3.22) we also get Az + Bx + Cy = 0

(3.23)

which is the equation of a plane passing through the origin of the (x, y, z) coordinate system, i.e., the heliocentric coordinate system. Equation (3.23) shows that the motion of the planet under the inverse square law in (3.1) takes place in a plane with the Sun at the origin. Hence we can refer to the motion of any planet by considering two axes (x and y) passing through the Sun. Thus let (3.15) and (3.16) represent the equations of motion for the planet. To analyze this two-dimensional (2D) motion further, let (r, θ) represent the heliocentric polar coordinates of the planet with reference to the Sun as the origin as in Figure 3.2. Then with x and y as in (3.13) we have x = r cos θ,

y = r sin θ.

(3.24)

dx dr dθ = cos θ − r sin θ dt dt dt

(3.25)

dy dr dθ = sin θ + r cos θ . dt dt dt

(3.26)

Hence

and

Also from (3.25)

2 dθ d 2θ − r sin θ dt dt2 dt2 2 2 d θ dr dθ d 2r dθ = − r cos θ − r + 2 sin θ (3.27) dt dt 2 dt dt dt2

d2x

d 2r

dr dθ = 2 cos θ − 2 sin θ − r cos θ dt dt dt

y

FIGURE 3.2 The Sun centered polar coordinate system.

P

(r,q )

r q S

x

55

56

Space Based Radar and from (3.26)

2 dθ d 2θ + r cos θ dt dt2 dt2 2 d 2r dθ d 2θ dr dθ = − r sin θ + r +2 cos θ. (3.28) 2 2 dt dt dt dt dt

d2 y

d 2r

dr dθ = 2 sin θ + 2 cos θ − r sin θ dt dt dt

Let αr and αθ represent the components of the acceleration of the planet along the radial direction SP and at right angles to SP respectively. Then d2x

αr =

d2 y

cos θ +

dt2

dt2

sin θ,

(3.29)

and αθ = −

d2x 2

dt

sin θ +

d2 y dt2

cos θ.

(3.30)

Substituting (3.27) and (3.28) into (3.29) we get

2 dθ αr = 2 − r . dt dt d 2r

(3.31)

Similarly using (3.15)–(3.16) and (3.24) in (3.29) gives αr = −

µ µ (x cos θ + y sin θ ) = − 2 . r3 r

(3.32)

From (3.31)–(3.32) we get αr =

d 2r dt2

−r

2 µ dθ = − 2. dt r

(3.33)

With (3.27) and (3.28) in (3.30) we get

d 2θ

dθ dr dθ 1 d αθ = r 2 + 2 r2 . = dt dt r dt dt dt

(3.34)

However (3.15)–(3.16) in (3.30) gives µ µ (x sin θ − y cos θ) = 3 (r cos θ sin θ − r sin θ cos θ) = 0 r3 r (3.35) and hence from (3.34)–(3.35) we get αθ =

d dθ r2 dt dt

=0

(3.36)

Chapter 3:

Two Body Orbital Motion and Kepler’s Laws Q

FIGURE 3.3 Area swept by the planet P around the Sun S.

rdq P

r dA

dq

S

or r2

dθ = h(constant). dt

(3.37)

But 12 r 2 dθ = d A, the area swept by the planet through an angular motion dθ centered at the Sun (see Figure 3.3). Hence from (3.37), dA 1 dθ = r2 = h/2 dt 2 dt

(3.38)

A(t) = ht/2.

(3.39)

or

From (3.39) for any t2 = t1 + with > 0, we get A(t1 + ) − A(t1 ) =

h , 2

(3.40)

i.e., the radius vector SP of a planet sweeps equal areas in equal intervals of time. Thus depending on the size of the radius vector, the planet has to move faster or slower to satisfy this requirement. This remarkable result is the mathematical expression of Kepler’s second law.

3.2 Kepler’s Laws As we have seen, Equations (3.37)–(3.39) are the mathematical expression of Kepler’s second law that states that “the radius vector SP of a planet sweeps equal areas in equal time intervals.” To obtain a parametric expression for planetary motion, we need to eliminate t in (3.33) and (3.38). We deﬁne u=

1 r

(3.41)

57

58

Space Based Radar so that from (3.37) dθ = hu2 dt

(3.42)

dr 1 du 1 du dθ du =− 2 =− 2 = −h dt u dt u dθ dt dθ

(3.43)

and

and d 2r d = dt 2 dθ

dr dθ d 2u = −h 2 u2 2 . dt dt dθ

(3.44)

Also

2 dθ r = h 2 u3 . dt

(3.45)

Substituting (3.44) and (3.45) into (3.33) we obtain − h 2 u2

d 2u − h 2 u3 = −µu2 , dθ 2

(3.46)

d 2u µ +u= 2 2 dθ h

(3.47)

or

which represents the path of the planet around the Sun in polar coordinates. 2 The general solution of ddθu2 + u = 0 is easily seen to be c 1 cos θ + c 2 sin θ and hence the particular solution to (3.47) is given by u=

µ µ + c 1 cos θ + c 2 sin θ = 2 + c o cos(θ − θo ), h2 h

(3.48)

or r=

h 2 /µ 1 + e cos(θ − θo )

(3.49)

e = h 2 c o /µ.

(3.50)

with

On comparing with (2.4), Equation (3.49) represents a special case of the general equation of a conic section given by r=

p 1 + e cos θ

(3.51)

Chapter 3:

Two Body Orbital Motion and Kepler’s Laws

where p=

h2 µ

(3.52)

represents the parameter of the conic and e its eccentricity. From (3.51) if r is bounded—as in the case of planetary motions around the Sun—then e < 1 in (3.51) and it represents an ellipse.2 Since r in (3.49) represents a closed orbit, we have e < 1 in (3.49) and it represents an ellipse. Hence all closed planetary orbits around the Sun are ellipses with the Sun at one focus (the other focus is empty), and this is Kepler’s ﬁrst law. On comparing (3.49)–(3.52) and (2.28), for an ellipse we have h2 = p = a (1 − e 2 ) µ

(3.53)

h 2 = µa (1 − e 2 ).

(3.54)

and hence we have

From (3.39) if T represents the orbital period of a planet then since πa b represents the total area of an ellipse, we have A = π ab = hT/2

(3.55)

or h 2 = 4π 2

a 2b2 4π 2 a 4 (1 − e 2 ) = . 2 T T2

(3.56)

From (3.54)–(3.56), every planet obeys a3 µ G( M + m) = = . T2 4π 2 4π 2

(3.57)

Hence if a i , a j and Ti , Tj represent the semimajor axes of two planetary orbits and their corresponding orbital periods, then

3 2 ai M + mi Ti = , aj M + m j Tj

(3.58)

and this represents the correct form of Kepler’s third law. If mi , m j are negligible compared to the mass of the Sun M in (3.58), then

2 3 Ti ai Tj aj 2 Notice

(3.59)

that (3.49)–(3.51) can generate parabolic and hyperbolic orbits as well.

59

60

Space Based Radar represents the classical form of Kepler’s third law. From (3.57)–(3.59), the orbital period of a planet or a satellite depends only upon the size of the major axis of the ellipse and not on its eccentricity or mass. From (3.15)–(3.16) and (3.49)–(3.51), for closed orbits, the inverse square law leads to ellipses. The converse is also true. Thus assume that orbital motion is governed by an ellipse. Then from (3.51)–(3.54) u=

1 + e cos θ a (1 − e 2 )

(3.60)

and hence d 2u e cos θ 1 =− = −u 2 2 dθ a (1 − e ) a (1 − e 2 )

(3.61)

1 d 2u +u= 2 dθ a (1 − e 2 )

(3.62)

or

which agrees with (3.47), an equation derived from the inverse square law, thus proving the claim that for closed orbits inverse square law and elliptical orbits are equivalent. Notice that from the single law of gravitation, Kepler’s three laws can be derived.

3.3 Synchronous and Polar Orbits Positioning of synchronous satellites gives an interesting application of Kepler’s third law. For a satellite to be geosynchronous, its period T needs to be 24 h. From Kepler’s third law this gives a = 42, 241 km (use (3.57) with G = 6.673 × 10−11 m3 kg−1 s−2 , Me = 5.9742 × 1024 kg). Thus if a space craft is projected into a circular orbit as in Figure 3.4(a) that is 35,863 km (22,300 miles) above the Earth’s surface in the plane of the equator traveling in the same direction as the Earth, then both the satellite and the Earth will complete one revolution in every 24 h and they both will rotate about the same axis. As a result, the satellite will remain over the same spot over the equator at all times. Notice that only an orbit in the Earth’s equatorial plane has this “stationary” property. A geosynchronous location is ideal for a communication satellite. However, to scan the Earth, the satellite (SBR) needs to provide maximum Earth coverage. A satellite placed in a polar orbit (which is at right angles to the equator and passes through the Earth’s poles) on the other hand provides maximum Earth coverage for a single satellite. For example, a polar satellite at 500 km above the Earth’s surface has a

Chapter 3:

Two Body Orbital Motion and Kepler’s Laws Polar satellite paths B

Geosynchronous Polar orbit orbit B

North America

Equator

Equator

A

Pacific Ocean South America

A

0

(a) Geosynchronous and polar orbits

4447 km

Asia

Europe

Atlantic Ocean 0°

45°

90°

Africa Australia Indian Ocean

Antarctica

(b) Mercator’s projection chart

FIGURE 3.4 Geosynchronous and polar orbits. A and B refer to geosynchronous and polar satellites respectively.

period of 1.57 h, and while it completes a circle around the Earth that is ﬁxed with respect to the stars, the Earth turns through 22.5◦ or 1/16 of a revolution about its axis (see the Mercator’s projected Earth map in Figure 3.4(b)). Thus every time the spacecraft crosses the equator the Earth has moved 2,500 km eastward giving an “automatic” scan of the surface below to the onboard radar. Essentially the radar is able to scan the Earth in both latitude and longitude by virtue of the Earth’s rotation.

3.4 Satellite Velocity The objective here is to determine the planetary velocity as a function of the instantaneous radius vector. Toward this, let VP denote the planet velocity at P and it is directed along the local tangent at P. From (3.25)–(3.26), VP has a component dr dt along the radius vector at right angles to the radius vector as in Figure 3.5 [1]. and r dθ dt This gives

VP2

2 2 2 dr du 2 dθ 2 2 = +r =h +u dt dt dθ

(3.63)

where we have used (3.43) and (3.45). From the general solution (3.48)– (3.49) u=

µ (1 + e cos(θ − θo )) h2

(3.64)

61

62

Space Based Radar dr dt P

VP

r

dq dt

r

O

F2

F1

FIGURE 3.5 Planet velocity.

and hence VP2 =

µ2 µ2 (1 + 2e cos(θ − θo ) + e 2 ) = 2µu − 2 (1 − e 2 ) 2 h h

=µ

2 1 − e2 , − r p

(3.65)

where we have used (3.52). Equation (3.65) gives the planet velocity as a function of the radius vector r and the semilatus rectum p of the conic. Since e < 1, e = 1, e > 1 correspond to an ellipse, parabola, and hyperbola in that order, substituting the appropriate value of p, the respective orbital velocities are given by [1], [3]

 2 1   µ , −   r a   µ    , r 2 VP = 2µ   ,   r    2 1   , + µ r a

elliptical orbit circular orbit , parabolic orbit

µ = G( M + m).

(3.66)

hyperbolic orbit

The velocity in the case of elliptical orbit in (3.66) can be given an interesting interpretation by relating it to the free fall motion under the inverse square law. Toward this, consider a stationary body of mass m at a distance so from the focus F1 that falls freely under the inverse square law toward a mass M + m at the focus (see Figure 3.6).

Chapter 3:

Two Body Orbital Motion and Kepler’s Laws P1

FIGURE 3.6 Free fall of a mass from the circle to the ellipse.

2a

r B

F2

2a

F1

P A

The acceleration of the mass m at distance s equals d 2s µ G( M + m) = − 2. =− dt s2 s

(3.67)

Multiplying (3.67) by 2 ds dt and integrating we obtain

2 2µ 1 ds 1 = , + c = 2µ − dx s s so

(3.68)

since the body was at rest at s = so . Hence if Vp denotes the velocity at a distance r from F1 we get Vp2 = µ

2 2 2 1 =µ , − − r so r a

(3.69)

for so = 2a . From (3.69) and (3.66) we conclude that the speed of a body at any point in an elliptical orbit is the same as that of a stationary body under free fall from the circumference of a circle along its radius to the surface of the ellipse; here the radius of the circle equals to the major axis of the ellipse and the center of the circle is located at F1 as in Figure 3.6. Clearly the planet at its perihelion A moves the fastest and at its aphelion B moves the slowest in its orbit (see also Figure 2.6). The parabolic speed in (3.66) represents the escape velocity. From (3.66), suppose a satellite of mass m is projected with speed VP from a point P. Let r represent the distance between √ P and another mass M. Then VP less than, equal to, or greater than 2µ/r gives rise to elliptic, parabolic, or hyperbolic orbits around the other mass M. Notice that the major axis and the period of the resulting orbit depend only on r and VP and not on the direction of projection. From (3.66), an elliptical orbit VP is greatest when r is the smallest (r = a (1 − e)), which

63

64

Space Based Radar corresponds to the perihelion. Similarly the velocity is minimum when the planet is in aphelion3 (r = a (1 + e)). The direction of projection, however, is very important in determining the shape of the orbit [4]. Together with the speed with which the satellite is ﬁnally launched into its orbit, it determines the type and √ character of the satellite orbit. If the speed is Vc = µ/r at the launch height r and if the velocity vector is truly horizontal to the ﬂight path (i.e., perpendicular to the radius vector at P), the orbit will be circular. If either condition is not satisﬁed the orbit will be elliptical. Figure 3.7 shows three orbits all launched from P that √ is at a height a from the center of the Earth with speed equal to= µ/a . The circular orbit corresponds to a horizontal launch. The two elliptical paths in Figure 3.7 correspond to launch directions that are high (ellipse E 1 ) and low (ellipse E 2 ) compared to the horizontal. Note that both ellipses have semimajor axis equal to a , same as the radius of the circular orbit, and if both satellites are launched at the same small angle α to the horizontal at P one upward and one downward the resulting orbits will have the same eccentricity and will be symmetrically located with respect to the projection point. Their perigee 3 If we assume the orbits of planets around the Sun are approximately circular and those of the comets parabolic, then from (3.66), the orbits of comets around the Sun cross the planets’ orbits with velocities 1.414 times those of the respective planets.

Halley’s Comet: Using (3.59) and (3.66), we can give a more accurate ﬁgure in the case of Halley’s comet whose observed period of 77 years gives its semi-major axis of its elliptic orbit (reference to Earth’s orbit) to be (use (3.59))

#a $ H

#T $2/3 H

=

# 77 $2/3

= 18.09 aE TE 1 or a H = 18.09a E . Using the elliptical orbit formula in (3.66), this gives the speed of the Halley’s comet at a distance r = a E from the Sun to be (i.e., at Earth’s orbit) # $ 35 µ 35 µ 2 1 VP2 = = = − V2 aE 1 18 18 a E 18 E

=

µ 35 or VP = 18 VE , since VE = a E . Assuming that Halley’s comet intersects with the Earth’s orbit in a perpendicular direction, the relative speed between the Earth and Halley’s comet equals

V=

12

35 + VE = 18

53 VE = 1.715VE = 1.715 × 29.772 = 51.08 km/sec 18 8

2π ×1.4952×10 km E since VE = 2πa TE = 365.24×24×3600 sec = 29.772 km/sec. Kepler himself never applied his remarkable laws to comets, and it remained for Halley (1705) to prove that one comet at least, and hence many comets, moved in highly eccentric periodic elliptical orbits around the Sun. Halley noticed that several observations that lead to nearly identical orbital parameters such as perihelion were always apart by a 75–77 years interval, and he concluded that a single comet has made repeated returns to the perihelion at about 0.5871 A.U. Halley was able to trace the comet as far back as 1305 (Halley’s comet).

Chapter 3:

Two Body Orbital Motion and Kepler’s Laws

FIGURE 3.7 Effect of direction of projection for circular orbit velocity.

Vc P α Vc Vc 2a

E2

a

Elliptical

2a

E1

2a Earth

C Circular

(point closest to Earth) will be less than the projected height a . To allow for launch velocity vector errors the launch height must be above a minimum safety distance. The eccentricity e of the elliptical orbits in Figure 3.7 is a function of the launch angle α (off the horizontal at P). In that case Vp cos α represents the component of the velocity that is at right angles to the radius vector a at P. Hence (see Figure 3.5)

µa (1 − e 2 ) , (3.70) a √ where we have used (3.37) and (3.54). But Vp = µ/a = Vc and hence we get cos α = 1 − e 2 (3.71) dθ h Vp cos α = a = = dt a

or e = sin α.

(3.72)

From Figure 2.3 the perigee of an ellipse equals F1 A = a (1 − e) = a (1 − sin α)

(3.73)

and hence for a satellite at launch height H above the ground that is launched at an angle α as in Figure 3.7, the perigee is given by ( Re + H)(1 − sin α). This gives the minimum distance that the satellite clears above the ground to be ( Re + H)(1 − sin α) − Re ≥ h min

(3.74)

65

66

Space Based Radar

P

Satellite Vc Hyperbola r

E1 Earth Elliptical

Parabola

C

Circular E2 Elliptical

E3

Elliptical

FIGURE 3.8 Satellite orbits for various horizontal projection velocities.

or α ≤ sin−1

H − h min , Re + H

(3.75)

where h min is chosen to be about 325 km to minimize the atmospheric drag and prevent burn out. From (3.75), this gives α ≤ 1.5◦ for a satellite orbital height of 500 km and it shows that velocity deviation in either up or down directions from the horizontal are equally undesirable and the margin of error is very small. Figure 3.8 shows the various orbits that result for horizontal launches when the launch speed VP is different from Vc , the speed required for circular orbit [4]. If VP < Vc , the orbit is a subcircular ellipse E 1√with its perigee less than the projected height r . If Vc < VP < 2µ/r , the orbit will be elliptical as in E 2 and E 3 but the height of apogee (point farthest away from the Earth) will be greater than the projected height.4 4 Finally consider the case where the launch speed is different from V and the c launch direction is different from the horizontal at an angle α away from the horizontal (up or down) as in Figure 3.7. Thus let Vp = Vc (1 + ε) with ε < 0.414 so that it leads to an elliptical orbit with semimajor axis given by (use (3.66)-a)

a = r/(1 − 2ε − ε 2 ) > r. The eccentricity e of this ellipse however depends on the launch angle, and proceeding as in (3.70) we get

1 − e2 = 1 − ε 2 (2 + ε) 2 cos α < cos α which shows that to maintain the same eccentricity the elliptical orbits have greater tolerance for launch angle errors compared to circular orbits.

Chapter 3:

Two Body Orbital Motion and Kepler’s Laws

√ If VP = 2µ/r , the elliptical orbits√become a parabola and the satellite is lost as it never returns. If VP > 2µ/r the path is a hyperbola, and once again the satellite never returns, and consequently hyperbolic orbits are important for spaceships that are attempting to leave the Earth for voyages into space. Earth itself moves around the Sun in nearly a circular orbit (orbit C in Figure 3.8 with the center representing the Sun) with speed of 29.772 km/s that is directed at right angles to the line connecting the Sun and the Earth. If the Earth’s velocity is slowed down to 15 km/s, then the Earth will follow the subcircular elliptic path E 1 in Figure 3.8. As the Earth falls toward the Sun its speed increases so that at perigee its centrifugal force will exceed the gravitational pull of the Sun (even though that has also increased), and the Earth will pull away from the Sun, slowing down as it goes until it arrives at P with the same velocity in the same direction. If we speed up the Earth in the 30–40 km/s range, the Earth will move away from the Sun along ellipses E 2 and E 3 slowing down near the apogee where the centrifugal force is still insufﬁcient to overcome the weak gravitational pull of the Sun, and consequently the Earth will fall back toward the Sun regaining speed until it reaches P with the same velocity. Increasing Earth’s velocity to 41.84 km/s will make the Earth move in the parabolic orbit in Figure 3.8 away from the Sun forever since the gravitational attraction of the Sun will be insufﬁcient to slow down the Earth and cause it to return.

Appendix 3-A: Kepler’s Equation Kepler’s second law that states “equal areas are swept in equal intervals of time by the radius vector of a planet centered at a focus (Sun)” can be used to calculate the position of a planet in its elliptical orbit at any time. Let τ represent the time at perihelion at A in Figure 3.9 and P the position of the planet at time t. Then from (3.39) Area AF1 P = A(t) = h(t − τ )/2

(3A.1)

and together with (3.55) that states Total Area = πab = hT/2,

(3A.2)

we obtain Area AF1 P =

1 πab(t − τ ) = nab(t − τ ) T 2

(3A.3)

67

68

Space Based Radar

Q a P r B

A

q

E F2

O

F1

R

Perihelion at t

FIGURE 3.9 Area swept by a planet.

where 2π T represents the mean angular motion of the planet. From Figure 3.9, n=

Area AF1 P = Area F1 PR + Area RPA.

(3A.4)

(3A.5)

But the area of the triangle F1 PR is given by 1 1 ( F1 R)(PR) = r cos θ r sin θ 2 2 (3A.6) 1 1 = a (cos E − e)b sin E = ab sin E(cos E − e), 2 2

Area F1 P R =

where we have used (2.39) and (2.40). We can make use of (2.37) to simplify the elliptical segment area RPA as well. Since PR b = , QR a

(3A.7)

by considering thin rectangular slices of width xi along AR that are parallel to QR, and adding them up we obtain

b xi Area RPA b i = , = Area RQA a xi a i

(3A.8)

Chapter 3:

Two Body Orbital Motion and Kepler’s Laws

or b (Area OQA − Area OQR) a 1 b 1 2 a E − a cos E a sin E = a 2 2

Area RPA =

=

1 ab( E − cos E sin E). 2

(3A.9)

Substituting (3A.6) and (3A.9) into (3A.5) we get Area AF1 P =

1 a b( E − e sin E) 2

(3A.10)

and ﬁnally substituting this into (3A.3) we get n(t − τ ) = E − e sin E,

(3A.11)

Kepler’s equation. With N = n(t − τ )

(3A.12)

Kepler’s equation is usually written as N = E − e sin E.

(3A.13)

Knowing t and T we have N, and the transcendental equation (3A.13) must be solved to obtain the eccentric anomaly E. It is easy to show that Kepler’s equation has one, and only one, real solution for every N and every e such that 0 ≤ e ≤ 1. To see this, let φ( E) = E − e sin E − N,

(3A.14)

and suppose mπ ≤ N < (m + 1)π , then φ(mπ) = mπ − N < 0

(3A.15)

φ((m + 1)π) = (m + 1)π − N > 0

(3A.16)

and

so that there are an odd number of solutions for φ( E) inside the interval (mπ, (m + 1)π ). But dφ( E) = 1 − e cos E > 0 dt

(3A.17)

and hence φ( E) monotonically increases and takes the value zero only once inside the interval (mπ, (m + 1)π), thus proving the uniqueness claim.

69

70

Space Based Radar

a

a 0 + e sina

a3 a2 a1 a0

a0

a1 a2 a3

a

FIGURE 3.10 Graphical procedure for solving Keplar’s equation.

To ﬁnd this unique solution for E, write N = mπ + α0 , 0 < α0 < π. Then from the above discussion E = mπ + α, 0 < α < π. Substituting these values into the Kepler’s equation E = N + e sin E,

(3A.18)

α = α0 + e sin α.

(3A.19)

we obtain

Equation (3A.19) represents the intersection of two convex functions α and α0 + e sin α and to ﬁnd the solution the method of alternating projections can be employed using the iteration αk+1 = α0 + e sin αk .

(3A.20)

Starting with α0 , we get α1 = α0 + e sin α0 > α0 , similarly α2 > α1 and the graphical procedure is shown in Figure 3.10. From the method of alternating projections, the above iteration converges to the true value α; i.e., αk → α and the desired E = mπ + α. Knowing E, the location of the planet can be determined using (2.41) and (2.44), provided we know the semimajor axis a and the eccentricity e of the planet’s orbit.5 5 The solution of Kepler’s equation was ﬁrst discovered by Kepler himself followed by Newton in his Principia. For the next 200 years, every prominent mathematician gave his/her attention to the solution of Kepler’s equation and a very large number of analytic and graphical solutions have been discovered on this topic.

Chapter 3:

Two Body Orbital Motion and Kepler’s Laws

Appendix 3-B: Euler’s Equation and the Identiﬁcation of Comets Comets move reasonably rapidly through the sky and it is possible to make intermittent observations about their positions. Let (r1 , θ1 ) and (r2 , θ2 ) correspond to two comet sightings (radii and true anomalies) at instants t1 and t2 respectively (see Figure 3.11). An important problem in this context is to determine whether these observations correspond to the same comet or different comets. An equation derived by Euler connecting the two radii and their common chord s for parabolic orbits can be used to answer this question. For parabolic orbits, their eccentricity equals unity, so that from (2.49) r = q sec2 (θ/2).

(3B.1)

As we have seen, the central inverse square law of attraction leads to (3.37), and using (3B.1) for r , it simpliﬁes to q 2 (sec2 (θ/2) + sec2 (θ/2) tan2 (θ/2))dθ = h =

2µq dt,

(3B.2)

since from (3.52) h 2 = µp = 2µq , or √ (sec2 (θ/2) + sec2 (θ/2) tan2 (θ/2))dθ =

2µ dt. q 3/2

(3B.3)

The integral of this expression is tan(θ/2) +

1 1 tan3 (θ/2) = 3/2 µ/2(t − τ ), 3 q

(3B.4)

where τ is the time of perihelion passage at A in Figure 3.11. Let the radii at instants t1 and t2 be r1 and r2 with respective true anomalies θ1 and θ2 . Further let s represent the chord length joining the extremities of r1 and r2 as in Figure 3.11. From (3B.4) and with √ κ = µ/2, we have κ 1 (t1 − τ ) = tan(θ1 /2) + tan3 (θ1 /3), q 3/2 3

(3B.5)

κ 1 (t2 − τ ) = tan(θ2 /2) + tan3 (θ2 /3). q 3/2 3

(3B.6)

71

72

Space Based Radar Comet P2 s

P1 r1 q

q1

A

r2 q2 F1

FIGURE 3.11 Identiﬁcationof comets using two observations.

q

Thus κ (t − t1 ) 3/2 2 = tan

θ2 θ1 θ2 θ1 1 tan3 − tan + − tan3 2 2 3 2 2

= tan

θ2 θ1 − tan 2 2

%

1+

θ2 θ1 1 θ2 θ1 tan2 + tan tan + tan2 3 2 2 2 2

&

' 2 ( θ2 θ1 θ1 θ1 θ2 1 θ2 = tan 1 + tan tan tan . − tan + − tan 2 2 2 2 3 2 2 (3B.7) The equation for the chord P1 P2 in Figure 3.11 is s 2 = r12 + r22 − 2r1 r2 cos(θ2 − θ1 ) = (r1 + r2 ) 2 − 4r1 r2 cos2

θ2 − θ1 2 (3B.8)

which gives

√ θ2 − θ1 2 r1 r2 cos = ± (r1 + r2 + s)(r1 + r2 − s), 2

(3B.9)

with the plus sign corresponding to θ2 − θ1 < π and the negative sign corresponding to θ2 − θ1 > π . From (3B.1) we have r1 = q sec2

θ1 , 2

r2 = q sec2

θ2 2

(3B.10)

Chapter 3:

Two Body Orbital Motion and Kepler’s Laws

so that

r1 + r2 = q 2 + tan2

θ1 θ2 + tan2 2 2

(3B.11)

and √ θ1 θ2 q r1 r2 = q sec sec . = 2 2 cos θ21 cos θ22

(3B.12)

Substituting (3B.12) into (3B.9) we get √ θ1 (r1 + r2 + s)(r1 + r2 − s) θ2 1 + tan tan =± . 2 2 2q

(3B.13)

Finally by direct expansion6

√ 2 √ r1 + r2 + s ∓ r1 + r2 − s = r1 + r2 ∓ (r1 + r2 + s)(r1 + r2 − s) 2 2 θ1 θ2 = q tan , (3B.14) − tan 2 2 where we have used (3B.11) and (3B.13). From (3B.14) we obtain

√ √ θ1 θ2 r + r + s ∓ r + r − s = 2q tan . − tan 1 2 1 2 2 2 a1

(3B.15)

a2

Finally substituting (3B.15) and (3B.13) into (3B.7) we get

κ(t2 − t1 ) (a 1 ∓ a 2 ) a1a2 1 (a 1 ∓ a 2 ) 2 √ = ± + q 3/2 2q 3 2q 2q

(3B.16)

or √ 6 2κ(t2 − t1 ) = (a 1 ∓ a 2 )(±3a 1 a 2 + (a 1 ∓ a 2 ) 2 )

= (a 1 ∓ a 2 ) a 12 + a 22 ± a 1 a 2 = a 13 ∓ a 23

(3B.17)

or √ 6 µ(t2 − t1 ) = (r1 + r2 + s) 3/2 ∓ (r1 + r2 − s) 3/2 ,

(3B.18)

the Euler’s equation. For small arcs, the minus sign in (3B.18) should be used. When θ2 − θ1 = π, the second term in (3B.18) is zero and for larger difference the plus sign should be used. 6 If θ − θ < π, then a plus sign corresponds to (3B.13) and in that case minus 2 1 sign in (3B.14), and vice-versa.

73

74

Space Based Radar Euler’s equation is remarkable since it does not involve q and it only involves the observations (r1 , θ1 ) and (r2 , θ2 ) at instants t1 and t2 . Clearly if the given observations belong to the same comet, then they must satisfy (3B.18) and hence Euler’s equation is useful to test the hypothesis of whether two given observations correspond to the same comet or not.7 There is a corresponding equation for elliptic orbits due to Lambert (see Appendix 3C).

Appendix 3-C: Lambert’s Equation for Elliptic Orbits Let E 1 and E 2 be the eccentric anomalies corresponding to two positions P1 and P2 of a planet (or a comet) in an elliptic orbit with respective radial distances r1 and r2 as in Figure 3.12. Let t1 and t2 denote the associated time instants. Suppose E 2 > E 1 and deﬁne G=

E2 + E1 , 2

E2 − E1 . 2

(3C.1)

r2 = a (1 − e cos E 2 )

(3C.2)

g=

Then from (2.41) r1 = a (1 − e cos E 1 ), so that r1 + r2 = a (2 − e(cos E 1 + cos E 2 )) = 2a (1 − e cos G cos g).

(3C.3)

Using (2.34) and (2.35), the chord length s in Figure 3.12 equals s 2 = (x2 − x1 ) 2 + ( y2 − y1 ) 2 = a 2 (cos E 2 − cos E 1 ) 2 + b 2 (sin E 2 − sin E 1 ) 2 2

2

2

2

2

(3C.4)

2

= 4a (sin G sin g + (1 − e ) cos G sin g). 7 A more interesting situation is when a comet on its routine visit around the Sun undergoes a chance encounter with an outer planet such as Jupiter that results in altered orbital parameters. How is the comet to be recognized with any certainty in its subsequent visits?, i.e., how does one test the hypothesis that two comets appearing several years apart are the same comet considering that random planetary perturbations might have altered their orbital parameters? A condition derived by Tisserand in connection with the three body problem gives an invariant relation that is unchanged by such perturbations [2].

Chapter 3:

Two Body Orbital Motion and Kepler’s Laws Planet P2 s

r2 E2

a

E1

O

F2

P1

r1

F1

FIGURE 3.12 Identiﬁcation of planets/comets using two observations.

Let e cos G = cos c.

(3C.5)

Then (3C.4) simpliﬁes to s 2 = 4a 2 sin2 g(1 − cos2 c),

(3C.6)

s = 2a sin g sin c.

(3C.7)

r1 + r2 = 2a (1 − cos g cos c).

(3C.8)

α =c+g

(3C.9)

β = c − g.

(3C.10)

or

From (3C.3), we also have

Let

and

Then from (3C.7) and (3C.8) we get r1 + r2 + s = 2a (1 − cos(g + c)) = 4a sin2 (α/2)

(3C.11)

r1 + r2 − s = 2a (1 − cos(g − c)) = 4a sin2 (β/2).

(3C.12)

and

75

76

Space Based Radar Finally from Kepler’s equation in (3A.12) and (3A.13) we get n(t2 − t1 ) = ( E 2 − E 1 ) − e(sin E 2 − sin E 1 ) = (α − β) − 2e cos G sin g = (α − β) − 2 cos c sin g

(3C.13)

n(t2 − t1 ) = (α − β) − (sin α − sin β),

(3C.14)

or

Lambert’s equation for elliptic motion. As remarked earlier, t1 and t2 in (3C.14) correspond to the time instants at which the planet passes through P1 and P2 in Figure 3.12. Notice that α and β are given in term of r1 + r2 , s and a in (3C.11) and (3C.12).

References [1] J.M.A. Danby, Fundamentals of Celestial Mechanics, The Macmillan Co, New York, NY, 1964. [2] F.R. Moulton, An Introduction to Celestial Mechanics. The Macmillan Co, New York, NY, 1964. [3] W.M. Smart, Text Book on Spherical Astronomy, Cambridge University Press, Cambridge, UK, 1965. [4] S. Herrick, “Earth Satellites and Repeat Orbit and Perturbation Theory,” Chapter 5, Space Technology, H. Seifert, ed. John Wiley & Sons, New York, NY, 1959.

CHAPTER

4

Space Based Radar----Kinematics A space based radar (SBR) located at some height above the Earth’s surface points its mainbeam at a point of interest on the ground where it generates a certain grazing angle. Surrounding the mainbeam, the antenna projects sidelobes all of which contribute toward the data collected for that speciﬁc point of interest. The SBR continues to collect data in this fashion as it travels along its orbit around the Earth. The objective is to detect slowly moving ground/air targets by suppressing the clutter returns and perform other tasks such as target identiﬁcation and imaging. In this context, the SBR-Earth geometrical relationships are ﬁrst derived for an ideal spherical Earth, with various correction factors due to an ellipsoidal (oblate spheroid) Earth model considered in the appendices. Another important issue addressed here is the shift in Doppler due to Earth’s rotational effects and a detailed analysis of this and related phenomena are carried out in this chapter.

4.1 Radar-Earth Geometry An SBR located at an orbital height H above its nadir point1 has its mainbeam focused to a point of interest on the ground located at range R. Figure 3.4 and Figure 4.1 (a) show an SBR on a polar orbit moving in the North-South direction. In general, the SBR can be in an orbit that is inclined at an angle to the equator. The inclination of the SBR orbit is usually given at the point where it crosses the equator from which its local inclination at other latitudes can be determined. The range is measured from the nadir point B (that is directly below 1 Nadir point is on the Earth’s surface directly below the satellite and it is obtained by joining a line between the SBR and the center of the Earth (see Figure 4.1 (b)).

77 Copyright © 2008 by The McGraw-Hill Companies. Click here for terms of use.

78

Space Based Radar SBR SBR

Region of interest

Vp qAZ

A

Slant range H

B

North America

Rs

qEL R

B

Atlantic

Nadir point

Pacific Ocean South America

0

Re

Range

qe

Grazing angle Point of interest on ground

y p 2 Re

D Ground

4447 km Reference map (c) 2000 ESRI

(a)

C Center of the Earth (b)

FIGURE 4.1 Space Based Radar. (a) An SBR on a polar orbit and the region of interest. (b) The parameters of an SBR pointing its mainbeam to a ground point D.

the satellite) to the antenna mainbeam footprint along a great circle on Earth (see Figure 4.1 (b)). Alternatively, instead of specifying the range R to the point of interest, the latitude-longitude pairs (α1 , β1 ) and (α2 , β2 ) of the satellite footprint B and the point of interest D maybe given (see Appendix 4-A for a detailed derivation of range from these parameters). For example, a polar orbit satellite at 506 km above the Earth’s surface has a period of 1.57 h. While it completes a circle around the Earth that is ﬁxed with respect to the stars, the Earth turns through 22.5◦ or 1/16 of a revolution about its axis (see Figure 3.4 and Figure 4.1 (a)). Thus every time the space craft crosses the equator the Earth moves 2500 km eastward giving an “automatic” scan of the surface below to the onboard radar. As shown in Figure 4.1 (a), the radar is able to scan the Earth in both latitude and longitude by virtue of the Earth’s rotation. In Figure 4.1 (b), the SBR is located at A, and B represents the nadir point. The point of interest D is located at range R from B along the great circle that goes through B and D with C representing the center of the Earth. The main parameters of an SBR setup are as follows [1]: R : Actual ground range from the nadir point to the point of interest along a great circle on the surface of the Earth. H : SBR orbit height above the nadir point.

Chapter 4:

S p a c e B a s e d R a d a r-----K i n e m a t i c s

Rs : Radar slant range from the satellite to the antenna footprint center at range R. ψ : Grazing angle at the antenna footprint at range R (i.e., the angle at which the surface is illuminated by the radar beam). Re : Spherical Earth’s radius (3,440 miles or 6,373 km). θ E L : Mainbeam elevation from the vertical line associated with range R. θAZ : Azimuth point angle measured between the plane of the array (generally also the SBR velocity vector) and the elevation plane AB D. φ E L : 3 dB mainbeam width in the elevation plane. φAZ : 3 dB mainbeam width in the azimuth plane. VP : Satellite velocity vector. θe : Core angle between the nadir point and the grazing point measured at the Earth’s center. (α1 , β1 ) : Latitude and longitude of the SBR nadir point B. (α2 , β2 ) : Latitude and longitude of the range point D. ηi : Inclination of the SBR orbit at the equator (with respect to the equator). From Figure 4.1 (b), the core angle subtended at the center of Earth by the range arc BD is given by θe = R/Re

(4.1)

and from triangle ACD we get Rs2 = Re2 + ( Re + H) 2 − 2Re ( Re + H) cos θe .

(4.2)

Thus the slant range Rs equals Rs =

)

Re2 + ( Re + H) 2 − 2Re ( Re + H) cos( R/Re ).

(4.3)

Similarly, the grazing angle ψ is also a function of range. To see this, referring back to the triangle ACD we have sin(π/2 + ψ) sin θEL sin θe = , = ( Re + H) Rs Re

(4.4)

sin θe sin θEL cos ψ = = . ( Re + H) Rs Re

(4.5)

or

79

Space Based Radar

5,500

Slant Range Rs vs Range Rs,max

5,000 H = 2,000 km

4,500 Slant Range Rs (km)

80

4,000 R s,max 3,500 3,000

H = 1,000 km Rs,max

2,500 H = 506 km

2,000 1,500 1,000 500

Rmax 0

1,000

2,000

Rmax

3,000

Rmax 4,000

5,000

Range R (km)

FIGURE 4.2 Slant range vs. range.

Thus we obtain the grazing angle at range R to be2 ψ = cos−1

Re + H sin(R/Re ) , Rs

(4.6)

and the corresponding elevation angle is given by θEL = sin

−1

Re sin(R/Re ) . Rs

(4.7)

Notice that both the grazing angle ψ and the elevation angle θEL are range dependent. From (4.5) we also have θEL = sin−1

1 cos ψ . 1 + H/Re

(4.8)

Similarly from triangle ACD we obtain an alternate formula θEL = π/2 − θe − ψ = π/2 − ψ − R/Re

(4.9)

2 The above derivation assumes Earth to be a perfect sphere. However, Earth is more like an ellipsoid and this introduces an error in the grazing angle that must be adjusted through a correction factor. Appendix 4-B gives a detailed derivation of this correction factor given the latitude-longitude pairs of the SBR footprint and the point of interest on the ground.

Chapter 4:

S p a c e B a s e d R a d a r-----K i n e m a t i c s Elevation Angle qEL vs Range

H = 506 km H = 1,000 km H = 2,000 km

80 70

Elevation Angle q EL (degree)

Grazing Angle y (degree)

Grazing Angle vs Range 90

60 50 40 30 20 10 0

Rmax 0

1,000

2,000

Rmax

3,000

70 60 50

q EL,max

H = 506 km

q EL,max

H = 1,000 km

q EL,max

H = 2,000 km

40 30 20 10

Rmax

0

4,000

Rmax 0

1,000

2,000

3,000

Rmax

Rmax

4,000

Range R (km)

Range R (km)

(a) Grazing angle vs range

(b) Elevation angle vs range

FIGURE 4.3 Grazing angle and elevation angle vs. range.

for the angle of elevation as well. The slant range, grazing angle, and elevation angle as functions of the range are shown in Figures 4.2– 4.3. Interestingly as these ﬁgures show, for an SBR located at a given height, the curvature of the Earth sets limits on the maximum available range, slant range, and elevation angle. This phenomenon is discussed next.

4.2 Maximum Range on Earth The curvature of Earth limits the maximum range achievable by a satellite located at height H as shown in Figure 4.4. At maximum range, the slant range becomes tangential to the Earth so that the grazing angle ψ = 0 and from Figure 4.4 we have θEL = π/2 − θmax . Thus from (4.4), for a spherical Earth cos θmax =

Re 1 = Re + H 1 + H/Re

or θmax = cos−1

1 1 + H/Re

(4.10)

.

(4.11)

The maximum range on Earth for an SBR located at height H is given by Rmax = Re θmax = Re cos

−1

1 1 + H/Re

.

(4.12)

81

Space Based Radar SBR A

FIGURE 4.4 Maximum range on ground.

qEL Rs,max

H

Rmax

B

D

p/2 Re Re

q max C

Similarly maximum slant range at the same height is given by

*

Rs,max = ( Re + H) sin θmax = ( Re + H) sin cos−1 =

)

( Re + H) 2 − Re2 =

1 1 + H/Re

H(2Re + H),

+

(4.13)

and the maximum elevation angle equals θEL,max

π π = − θmax = − cos−1 2 2

1 1 + H/Re

.

(4.14)

For low Earth orbit (LEO) satellite located at 506 km above the ground, the maximum range is 2,460 km and θEL,max = 67.9◦ (see Figure 4.5). Rmax vs Height

qEL,max vs Height

8,000

80

7,000

70

qEL,max (degree)

Rmax (km)

82

6,000 5,000 4,000 3,000

50 40 30

2,000 1,000

60

1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 9,000 10,000

Height (km)

20

1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 9,000 10,000

Height H (km)

FIGURE 4.5 Maximum range and elevation angle vs. satellite height.

Chapter 4:

S p a c e B a s e d R a d a r-----K i n e m a t i c s

4.3 Mainbeam Footprint Size The mainbeam of the radar generates a footprint on the ground whose size depends upon the actual range R. Let φEL represent the 3 dB mainbeam width of the antenna pattern in the elevation plane. Further let RT and RH denote the ranges of the “toe” and “heel” of the mainbeam footprint whose center is at range R as shown in Figure 4.6. Let ψT and ψ H represent grazing angles at the “toe” and “heel” of the mainbeam footprint. Thus from triangle ACE in Figure 4.7 that corresponds to the footprint “toe”, we have sin(π/2 + ψT ) sin(θEL + φEL /2) , = Re + H Re

(4.15)

where θEL represents the elevation at range R. This gives the grazing angle at the “toe” to be ψT = cos−1

*

H Re

1+

sin θEL +

φEL 2

+ ,

(4.16)

A fEL (Beamwidth) qEL

Mainbeam

H R RH

B

L “Toe”

“Heel”

D

E RT

Mainbeam footprint qH

C

FIGURE 4.6 Mainbeam footprint at range R. Distances RT and RH correspond to ranges at the “toe” and “heel” of the footprint. Range R represents the curved distance BD to the center of the footprint.

83

84

Space Based Radar A

fEL 2

qEL +

H

Mainbeam yT

B RT

p/2

E

Re Re qT

C

FIGURE 4.7 Range calculation at the “toe” of the mainbeam footprint.

and similarly the grazing angle at the “heel” is given by ψ H = cos−1

*

1+

H Re

sin θEL −

φEL 2

+

.

(4.17)

Also from Figure 4.7 the core angle at the center of Earth for the “toe” equals θT =

π φEL − θEL − − ψT 2 2

and the range to the mainbeam “toe” equals

RT = Re θT = Re

(4.18)

π φEL − θEL − − ψT . 2 2

(4.19)

Similarly, the range to the “heel” of the mainbeam equals

RH = Re θ H = Re

π φEL − θEL + − ψH . 2 2

(4.20)

This gives the length of the footprint of the mainbeam at range R to be L = RT − RH = Re (ψ H − ψT − φEL ).

(4.21)

S p a c e B a s e d R a d a r-----K i n e m a t i c s

450 400 350 300 250

H = 1,000 km H = 506 km

200 H = 2,000 km

150 100 50 0 500

Width of Mainbeam Footprint (km)

Length of Mainbeam Footprint (km)

Chapter 4:

1,000 1,500 2,000 2,500 3,000 3,500

80 70 H = 2,000 km

60 50 40

H = 1,000 km

30 20 10 500

Range (km)

H = 506 km

1,000 1,500 2,000 2,500 3,000 3,500 Range (km)

FIGURE 4.8 Length and width of mainbeam footprint vs. range. Mainbeam 3 dB beamwidths in both elevation and azimuth directions are assumed to be 1◦ .

Let φAZ represent the 3 dB beamwidth in the azimuth direction, then the horizontal mainbeam beamwidth equals W = Rs φAZ .

(4.22)

As a result π L W/4 represents the area of the elliptically shaped mainbeam on the ground. As Figure 4.8 shows, both the length and width of the footprint are functions of the range and SBR height. In summary, when the antenna mainbeam is focused along θEL , returns from the illuminated region of the corresponding mainbeam footprint will contribute toward clutter from that range [2] (see Figure 4.9). For example, an SBR located at a height of 506 km generates a mainbeam size of 55 km × 20 km at a range of 1,000 km. The mainbeam footprint size L = RT − RH and W = Rs φAZ are functions of the range R and height H, and they need to be recomputed for different ranges. Notice that the region of coverage on the Earth’s surface is annular in shape since the region around the nadir point is inaccessible (nadir hole) and the farthest range is limited by the Earth’s curvature as in Figure 4.10.

FIGURE 4.9 Mainbeam footprint on the ground.

Illuminated regions

W = RsfAZ L = RT − RH

85

86

Space Based Radar SBR

Nadir point

qELmax

Nadir hole

Rmin

Annular region of coverage

Rmax

Equator

FIGURE 4.10 Annular region of coverage and mainbeam footprints.

4.4 Packing of Mainbeam Footprints Next, we examine the interesting question of determining midfootprint range points R1 , R2 , · · · along the conical strips in the annular region in Figure 4.10 such that the corresponding mainbeam footprints are non-overlapping. Toward this, Figure 4.11 shows a conical section corresponding to a ﬁxed azimuth angle from the annular region. Let L i represent the

yTi –1

L2 qAZ

yi Wi ...

... RTi–1

Rmin R1

R2

R3

R

FIGURE 4.11 Mainbeam footprint vs. range.

RHi Ri

Rmax

Chapter 4:

S p a c e B a s e d R a d a r-----K i n e m a t i c s

mainbeam footprint size at range Ri . Thus L i = RTi − RHi .

(4.23)

To have non-overlapping mainbeam footprints as in Figure 4.11, the range points R1 , R2 , · · · at the centers of the footprints must be selected such that the ith bin satisﬁes RHi = RTi−1 ,

i = 2, 3, · · ·

(4.24)

and ψ Hi = ψTi−1 ,

(4.25)

i.e., the heel of the ith bin coincides with the toe of the (i − 1)th bin. This can be achieved by incrementing the antenna mainbeam in the elevation plane by the 3 dB elevation beamwidth φEL . Since,

RHi = Re

φEL π − θELi + − ψ Hi 2 2

(4.26)

where θEL i represents the elevation angle at Ri , we get θELi =

RT π φEL + − ψTi−1 − i−1 . 2 2 Re

This gives the grazing angle at range Ri to be ψi = cos−1

*

and from (4.9) Ri = Re

1+

#π 2

H Re

(4.27)

+

sin θELi

(4.28)

$

− ψi − θELi .

(4.29)

As a result the grazing angle and range at the toe of the ith bin equals ψTi = cos

−1

*

H 1+ Re

sin θELi

φEL + 2

+

(4.30)

and the corresponding range at the toe is given by

RTi = Re

π φEL − θELi − − ψTi . 2 2

(4.31)

From (4.23), this gives L i = RTi − RHi = Re (ψ Hi − ψTi − φEL ).

(4.32)

Starting with the initial point, RH1 = Rmin , and ψ H1 = ψmin , we obtain RT1 = Re ( π2 − θELmin − φEL − ψT1 ). These initial values are used

87

88

Space Based Radar in (4.24)–(4.25) and the procedure is repeated to generate the range values R1 , R2 , . . . that guarantee full overage of the ground using a minimum number of mainbeams. Table 4.1 shows the range points Ri and elevation angles corresponding to the centers of the non-overlapping mainbeam footprints

Beam #i

θ E Li (◦ )

Ri (km)

ψi (◦ )

RHi = RTi−1 (km)

Li (km)

R (km)

Na ,i

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

390.6 405.7 421.3 437.5 454.3 471.8 489.9 508.8 528.5 549.1 570.8 593.5 617.4 642.7 669.5 697.9 728.2 760.7 795.5 833.2 874.1 918.8 967.9 1,022.6 1,084.0 1,153.9 1,234.9 1,331.1 1,449.6

49.5 48.4 47.2 46.1 44.9 43.8 42.6 41.4 40.2 39.1 37.9 36.7 35.4 34.2 33.0 31.7 30.5 29.2 27.8 26.5 25.1 23.7 22.3 20.8 19.3 17.6 15.9 14.0 12.0

383.2 398.1 413.4 429.4 445.8 463.0 480.7 499.2 518.5 538.7 559.8 582.0 605.3 629.9 655.9 683.5 712.8 744.2 777.8 814.0 853.2 895.9 942.7 994.5 1,052.3 1,117.7 1,192.8 1,280.7 1,386.9

14.9 15.4 15.9 16.5 17.1 17.8 18.5 19.3 20.2 21.1 22.2 23.3 24.6 26.0 27.6 29.3 31.3 33.6 36.2 39.2 42.7 46.8 51.8 57.8 65.4 75.0 88.0 106.2 134.3

107.4 105.6 103.8 102.2 100.6 99.0 97.5 96.1 94.7 93.4 92.2 91.0 89.8 88.7 87.6 86.6 85.6 84.6 83.7 82.9 82.0 81.2 80.4 79.7 79.0 78.3 77.7 77.1 76.5

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2

TABLE 4.1 Ground range locations and elevation angles for non-overlapping antenna mainbeams corresponding to H = 506 km, Rmin = RH1 = 383 km, and Rmax = RT29 = 1,521 km.

Chapter 4:

S p a c e B a s e d R a d a r-----K i n e m a t i c s

for a LEO satellite located at a height of 506 km. In this case the grazing angle is allowed to vary between 50◦ and 10◦ , and it corresponds to ground range of 383–1521 km. In a pulsed radar situation, consecutive pulse returns from different range points overlap generating range ambiguity. With Tr representing the pulse repetition interval, the range ambiguity is given by R cTr /2, where c represents the velocity of light (see also (4.35) and (4.37)). A 2 KHz pulse repetition frequency (PRF) is used here to compute the range ambiguity R . Notice that the antenna mainbeam footprint sizes on the ground vary from 15–135 km and the number of range ambiguities in this case varies from 1 to 2 within the mainbeam footprint. If a 10 kHz PRF is used instead, the number of range ambiguities in a footprint would vary from 1 to 9 within the mainbeam. The footprints corresponding to ranges R1 , R2 , . . . in Figure 4.11 are non-overlapping in the range direction and cover the entire region of interest. However, their widths Wi = Rsi φAE will increase as Ri increases. This situation is shown in Figure 4.12 where the footprints are bounded in a conical surface along the range direction. The end widths of the conical sections are W1 and Wend = Rs J φAZ respectively. After ﬁnishing a range sweep along a particular θAZ , the azimuth angle can be incremented by the beamwidth φAZ and the procedure is repeated to cover the entire annular region of interest in Figure 4.12.

Wend

qAZ …

…

…

…

R1

Annular region of coverage

R2

RsfAZ RJ

…

Rmax

Nadir hole

FIGURE 4.12 Region of coverage for SBR.

89

90

Space Based Radar

4.5 Range Foldover Phenomenon To detect targets, radar transmits pulses periodically. Range foldover occurs when clutter returns from previously transmitted pulses, returning from farther range bins, get combined with returns from the point of interest (see Figure 4.13). Depending on the size of the mainbeam footprint, the two-dimensional (2D) antenna array pattern and the radar pulse repetition frequency, range foldover can occur both from within the mainbeam as well as from the entire 2D region. The effect of mainbeam foldover is discussed ﬁrst, followed by its extension to the entire 2D region.

4.5.1 Mainbeam Foldover Let τ represent the radar output pulse length and Tr the pulse repetition interval. Pulses travel along the slant range and interact with the ground through the mainbeam as well as the sidelobes of the antenna array as shown in Figure 4.14. Each pulse travels along the slant range and hence the slant range that can be recovered unambiguously is of size cτ2 . Thus slant range resolution is given by δ SR =

SBR A

(4.33)

itt sm an Tr lses pu ed

…

FIGURE 4.13 Common return wavefront showing all range ambiguity returns corresponding to a point of interest at range R.

cτ . 2

e

ng ra d ar s rw ver Fo ldo fo

s er d D ar ldov w ck e fo a B ng ra R R- 2 -1

Returns due to later pulses

R1

rn nt tu efro e R av w

R2

Range point of interest at R

R3

Returns due to earlier pulses

Chapter 4:

S p a c e B a s e d R a d a r-----K i n e m a t i c s

Vp

A

qAZ t

qEL

H

Tr Range ambiguities

y

B

Mainbeam Pulse impression

D Re

Sidelobes Re

C

FIGURE 4.14 Mainbeam range ambiguities.

Translating to the ground plane, since the pulse wave front is perpendicular to the slant range direction, we get the range resolution on ground to be (see Figure 4.15) δR =

cτ cτ = sec ψ. 2 cos ψ 2

(4.34)

Thus δ R represents the ground-plane spatial resolution that can be realized by the SBR. Two objects that are separated by a distance less than δ R will be indistinguishable by the radar. Notice that only the output pulse length contributes to the range resolution and it can be orders of magnitude smaller than the actual pulse length because of pulse compression effects. For example, using chirp waveforms it is possible to realize 1:100 or higher order compression. From (4.34) for short range regions where the grazing angle ψ is closer to π/2, the range resolution is very poor, and for long range the resolution approaches its limiting value δ SR as ψ → 0. Radar transmits pulses every Tr seconds and for high PRF situations, following (4.34), the distance R between range ambiguities on the ground (distance between

FIGURE 4.15 Ground range resolution.

Pulse wavefront ct /2 y dR

91

92

Space Based Radar A

FIGURE 4.16 Distance between range ambiguities.

Rs

Rs

H

y B

R

cTr/2

D

∆R E

Re

Re qe

Re ∆qe

C

consecutive pulse shadows) is given by R =

cTr sec ψ. 2

(4.35)

Equation (4.35) assumes a high PRF situation where the grazing angles at various range ambiguities are assumed to be equal. The general situation that takes the change in grazing angle into account is shown in Figure 4.16. If Rs represents the slant range in (4.3) at the end of one pulse (say at D), then Rs + cTr /2 is the new slant range at the end of the next pulse at E. Let R1 = R + R represent the new range corresponding to the second pulse shadow on the ground at E. From triangle ACE in Figure 4.16

2

( Rs + cTr /2) =

Re2 + ( Re

or R = Re cos

−1

2

+ H) − 2Re ( Re + H) cos

Re2 + ( Re + H) 2 − ( Rs + cTr /2) 2 2Re ( Re + H)

R + R Re

(4.36)

− R.

(4.37)

From Figure 4.17, interestingly R is a decreasing function of R, and when R is relatively small, the distance between the pulse shadows on the ground is large and it decreases as R increases (Figure 4.18 (a)). However, for large values of range, R remains constant at its limiting value cTr /2. This also follows from (4.35) since for large R the grazing angle approaches zero. As a result, as the range increases this has the additional effect of packing several range ambiguities within

Chapter 4:

S p a c e B a s e d R a d a r-----K i n e m a t i c s

∆R vs Range

∆R vs Range 220

650 H = 506 km H = 1,000 km H = 2,000 km

600

180 ∆R (km)

∆R (km)

550

H = 506 km H = 1,000 km H = 2,000 km

200

500 450

160 140 120

400

100

350

80

300 500

1,000 1,500 2,000 2,500 3,000 3,500 Range (km)

60 500

(a) PRF = 500 Hz

1,000 1,500 2,000 2,500 3,000 3,500 Range (km)

(b) PRF = 2 kHz

FIGURE 4.17 R vs. range for PRF = 2 kHz and 500 Hz.

the mainbeam footprint. The situation in the mainbeam is shown in Figure 4.18(b). If we assume R to be constant within a mainbeam footprint, then the number of range ambiguities within the mainbeam is given by Na =

RT − RH . R

(4.38)

Otherwise, starting at the heel of the mainbeam located at range RH , sequentially R1 , R2 , · · · are calculated within the mainbeam footprint till the toe is reached. Thus the number of range ambiguities Na within a mainbeam satisﬁes the equation (see Figure 4.18) R1 + R2 + · · · + RNa = RT − RH .

(4.39)

Figure 4.19 shows Na versus range for different heights. Notice that Na is also a function of range. To have no range ambiguities within the

ct secy 2

∆R1

Mainbeam footprint

∆R2

ct /2

cTr/2

Range ambiguities

RSfAZ

RSfAZ

RT – RH

RT – RH

(a) Short range

(b) Long range (y ~ 0)

FIGURE 4.18 Mainbeam footprint and range ambiguities.

93

Space Based Radar Number of Range Ambiguities vs Range

Number of Range Ambiguities vs Range 3 2.5

7

H = 506 km H = 1,000 km H = 2,000 km

6

H = 506 km H = 1,000 km H = 2,000 km

5 Na

2 Na

94

1.5

4 3

1

2

0.5 0 500

1 1,000 1,500 2,000 2,500 3,000 3,500 Range (km)

0 500

1,000 1,500 2,000 2,500 3,000 3,500 Range (km)

(a) PRF = 500 Hz

(b) PRF = 2 kHz

FIGURE 4.19 Number of range ambiguities Na within a mainbeam vs. range (for PRF = 2 kHz and 500 Hz).

mainbeam, the PRF must be selected so that Na = 1. Often a higher PRF is used resulting in Na > 1. From Figure 4.18, we also obtain the total area illuminated on the Earth’s surface by the mainbeam to be Ac = Rs φAZ Na

cτ sec ψ 2

(4.40)

and returns from this illuminated region corresponding to the Na range ambiguities contribute to the mainbeam clutter at this particular range [3], [4].

4.5.2 Total Range Foldover The total number of range ambiguities can be much more than that given in (4.38) if the sidelobe pattern of the antenna beam in the elevation direction is signiﬁcant compared to the mainbeam. In that case, the pulse returns from sidelobe regions also must be considered. This situation is shown in Figure 4.20. In Figure 4.20, the point of interest (D) is within the mainbeam, and the return of the radar pulse from there represents the main clutter. However, because of the 2D antenna pattern, previous pulse returns returning from adjacent “range ambiguity points”—both forward and backward—that have been appropriately scaled by the array gain pattern get added to the mainbeam return causing additional range foldover. To compute the immediate forward and backward range ambiguity points (E and F respectively), the geometry in Figure 4.21 can be used. Notice that the forward range ambiguity increments R+ are the same as R in (4.36) and (4.37). From there, if R1 represents the ﬁrst

Chapter 4:

S p a c e B a s e d R a d a r-----K i n e m a t i c s

(q EL,m , qAZ,j) …

Mainbeam

qAZ

…

D: Point of interest

Azimuth sidelobes

∆R−

…

∆R+ Elevation sidelobes

Range ambiguities Range of interest

FIGURE 4.20 “Range foldover” phenomenon.

forward range ambiguity point then

R1 = R + R+ = Re cos

−1

Re2 + ( Re + H) 2 − ( Rs + cTr /2) 2 2Re ( Re + H)

(4.41)

and this procedure can be repeated till Rmax is reached to determine all forward range ambiguity points R1 , R2 , · · · for a given R. Similarly, the backward range ambiguity increments R− are given by the new relation (see Figure 4.21)

2

( Rs −cTr /2) =

Re2 +( Re + H) 2 −2Re ( Re + H) cos

R − R− . Re

(4.42)

95

96

Space Based Radar A

Rs 2 T r/ –c

H Backward range ambiguity point

Rs

Rs

y B F

R

D ∆R +

Re Re

∆qe -

Forward range ambiguity point

cTr/2

∆R -

E

Point of interest

∆qe + qe

C

FIGURE 4.21 Forward and backward range ambiguity points.

This gives R− = R − Re cos

−1

Re2 + ( Re + H) 2 − ( Rs − cTr /2) 2 2Re ( Re + H)

(4.43)

or the ﬁrst backward range ambiguity point R−1 = R − R− = Re cos

−1

Re2 + ( Re + H) 2 − ( Rs − cTr /2) 2 2Re ( Re + H)

. (4.44)

Once again, this procedure is repeated till the nadir hole is reached. In general, for range R, the kth forward and backward range ambiguity points are given by R±k = Re cos−1

Re2 + ( Re + H) 2 − ( Rs ± kcTr /2) 2 2Re ( Re + H)

, k = 1, 2, . . . (4.45)

Chapter 4:

S p a c e B a s e d R a d a r-----K i n e m a t i c s

40 35 H = 1,000 km

30 H = 1,500 km Na

25 20

H = 506 km

15 10 5 500

1,000 PRF (Hz)

1,500

2,000

FIGURE 4.22 Number of range ambiguities as function of SBR height and PRF.

where R+k = Rk . Notice that the actual range R enters (4.45) through Rs given in (4.3). Figure 4.14 shows the return wavefront from all range ambiguities corresponding to a point of interest D at range R. Let Na refer to the total number of range ambiguities (both forward and backward) corresponding to a range bin of interest. The clutter returns from forward and backward range ambiguities get scaled by the array gain corresponding to those locations and get added to the returns from the point of interest. Figure 4.22 shows the total number of range ambiguities in the 2D region as a function of SBR height and PRF. On comparing this ﬁgure with Figure 4.19, we notice that the total number of range ambiguities at 500 Hz PRF jumps from 2 in the mainbeam to 7 in the elevation direction.

4.6 Doppler Shift Interestingly, the Doppler shift in the case of an SBR is contributed by two moving components—the motion of the SBR and Earth’s rotation around its own axis. To determine the overall Doppler component, consider an SBR at height H above the Earth on a great circular orbit that is inclined at an angle ηi (with respect to the equator). By virtue

97

98

Space Based Radar SBR

Vp qAZ

A

Slant range H

Rs

qEL

Grazing angle R

B

y p 2

Range

Nadir point

Point of interest on ground D Ground

Re

Re qe

C Center of the Earth

FIGURE 4.23 The parameters of an SBR pointing its mainbeam to a ground point D.

of Earth’s gravity the SBR is moving with velocity3 Vp =

G Me /( Re + H)

(4.46)

in a circular orbit and this contributes to a relative velocity of Vp cos θAZ sin θEL

(4.47)

along the line of sight for a point of interest D on the ground that is at an azimuth angle θAZ with respect to the ﬂight path and an elevation angle θEL with respect to the nadir line as shown in Figure 4.23. To derive (4.47), from Figure 4.23 we notice that Vp cos θAZ represents the relative velocity of the SBR along the array azimuth direction, and Vp cos θAZ sin θEL represents the relative velocity along the slant range direction AD. 3 In a circular orbit the gravitational pull due to the inverse square law mVp2 r

GmMe r2

GmMe /r 2

G Me must equal the centripetal force Thus = gives Vp = r where Vp represents the satellite orbit speed, m its mass, Me mass of Earth, G universal gravitational constant (= 6.673×10−11 m3 kg−1 s−2 ) and r = Re + H. For a satellite located at height H = 506 km, this gives Vp = 7.61 km/s and it corresponds to an orbital period of 1.579 h. Notice that (3.66) gives the more accurate formula given by VP = G( Me + m)/r .

mVp2 /r .

Chapter 4:

S p a c e B a s e d R a d a r-----K i n e m a t i c s

If Tr represents the radar pulse repetition rate and λ the operating wavelength, then the Doppler frequency ωd contributed by (4.47) equals (see (4C.10) in Appendix 4-C) ωd =

2Vp Tr sin θEL cos θAZ λ/2

(4.48)

and (4.48) accounts for the Doppler frequency of the stationary ground return due to the SBR motion. Given R, θAZ , and H from (4.1)–(4.5), we obtain sin θEL =

Re sin ( R/Re ) Rs

=

Re sin ( R/Re )

,

(4.49)

Re sin ( R/Re ) cos θAZ 2Vp Tr . 2 λ/2 Re + ( Re + H) 2 − 2Re ( Re + H) cos( R/Re )

(4.50)

Re2

+ ( Re + H) 2 − 2Re ( Re + H) cos( R/Re )

and hence the Doppler frequency in (4.48) has the form ωd =

From (4.50), the Doppler frequency is clearly range dependent.4 For short ranges, (4.50) reduces to ωd ( R) ≈

2Vp Tr R cos θAZ . λ/2 H

(4.51)

Figure 4.24 shows the Doppler dependency on range as a function of the azimuth angle. Clearly, the Doppler is an increasing function of range R as the azimuth angle moves away from 90◦ . From (4.51), the Doppler increment ωd = ωd ( R2 )−ωd ( R1 ) due to the range difference R = R2 − R1 increases as azimuth angle moves away from 90◦ , since ωd ∝ R cos θAZ . This is seen in the iso-Doppler plots shown in Figure 4.25 where Doppler magnitude is minimum (zero) for θAZ = 90◦ and maximum at θAZ = 0◦ , 180◦ . If the Earth’s rotation is included as we shall see in Section 4.7, the Doppler difference due to range generates an undesirable “Doppler ﬁlling” effect when data samples from different range bins are used to estimate the covariance matrix.

4 A more accurate treatment for the Doppler frequency that takes into account the Earth’s rotation is given in the next section.

99

Space Based Radar R3 = 1,750 km

200

120 qAZ = 60°

100 R2 = 1,000 km

0

Doppler

Doppler

100

R1 = 250 km

80 qAZ = 75°

60 40

−100

qAZ = 85°

20 −200

qAZ = 90°

0

−1

−0.5

0 cos qAZ

0.5

1

0

500

1,000 1,500 Range (km)

(a)

2,000

(b)

FIGURE 4.24 Doppler dependency on range vs. azimuth angle.

In practice the effect of difference in Doppler is even more severe because of the clutter ridge slope parameter (Doppler foldover parameter) βo =

2Vp Tr λ/2

(4.52)

that appears in (4.50), which for Vp = 7.61 km/s, Tr = 0.002 s (PRF = 500 Hz), and operating frequency 1.25 GHz gives βo = 253.57. As a result ωd can be much larger than unity at various azimuth angles. Since the Doppler term ωd appears in complex sinusoidal form (Appendix 4-C), ωd > 1 causes the Doppler to foldover as shown in Figure 4.26. To demonstrate the effect of βo on Doppler spread, Figure 4.26 shows the Doppler-azimuth pattern in Figure 4.24 for βo = 6. Notice that Doppler foldover corresponding to various range bins occurs at different azimuth angles. For operating conditions corresponding to

180

−1

-200

160 140

-200

−0.5

120

cos (qAZ)

Azimuth (deg)

100

100 0

80

0

0

60 0.5

40 20 0

200 0

500

1,000

1,500

Range (km)

2,000

1

200 0

500

1,000

1,500

Range (km)

FIGURE 4.25 Iso-Doppler plots in the range-azimuth domain.

2,000

Chapter 4:

S p a c e B a s e d R a d a r-----K i n e m a t i c s R1 = 1,000 km

1

R2 = 500 km

0.8 0.6

Doppler

0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1

−0.8

−0.6

−0.4

−0.2

0 0.2 Azimuth

0.4

0.6

0.8

1

FIGURE 4.26 Doppler spread due to range and clutter ridge slope βo = 6.

βo = 253.57, it follows that this effect is more severe for any azimuth angle other than the immediate neighborhood of the broad-side.

4.7 Crab Angle and Crab Magnitude: Modeling Earth’s Rotation for SBR [5] As we have seen in Section 4.5, the range foldover phenomenon— clutter returns that correspond to previous/later radar pulses— contributes to the SBR clutter. Another important phenomenon that affects the clutter data is the effect of Earth’s motion around its own axis. At various locations on Earth this contributes differently to Doppler, and the effect is modeled here. For any point on Earth at range R that is at an elevation angle θEL and azimuth angle θAZ with respect to an SBR at height H, the Doppler shift due to the SBR motion equals [1] ωd =

2Vp Tr sin θEL cos θAZ , λ/2

(4.53)

as derived in Section 4.6. Let ηi denote the inclination of the SBR orbit with respect to the equator (see Figures 4.27–4.28). As the SBR moves around the Earth, the Earth itself is rotating around its own axis on a 23.9345 hour basis in a west-to-east direction.

101

102

Space Based Radar Vp qAZ

SBR A

North pole

qEL

b Z

g

Ve cos a2 (eastward) Latitude a1

D

y

R

Point of contact on Earth (a 2, b 2)

B

a SBR path

a2

a1 C

hi

Longitude b 2

b 2 – b1 b1

Equator

Longitude b 1

FIGURE 4.27 Doppler contributions from SBR velocity and Earth rotation. North pole Z Z b1 a2

SBR path

D

Latitudes g a1

Y2

p/2

b

B1

Ve cos a 2

R qAZ

d

Region of interest

Y1

B

hi

SBR inclination at equator

Equator

B0

b1

b2 Longitudes

FIGURE 4.28 Effect of Earth rotation on Doppler frequency.

Chapter 4:

S p a c e B a s e d R a d a r-----K i n e m a t i c s

This contributes an eastward motion with equatorial velocity of Ve =

2π Re = 0.4651 km/s. 23.9345 × 3600

(4.54)

Let (α1 , β1 ) refer to the latitude and longitude of the SBR nadir point B and (α2 , β2 ) those of the point of interest D as shown in Figure 4.27 and Figure 4.28. As a result, the point of interest D on the Earth at latitude α2 rotates eastward with velocity Ve cos α2 and this will contribute to the Doppler in (4.48) as well. To compute this new component in Doppler shift, from Figure 4.28 the angle BDY2 between the ground range vector R and the Earth velocity vector at D equals π/2 + β so that [6] Vo = Ve cos α2 cos(π/2 + β) = −Ve cos α2 sin β

(4.55)

represents the Earth’s relative velocity at D along the ground range direction toward B. Since the grazing angle ψ represents the slant range angle with respect to the ground range at D (see Figure 4.27), we have Vo cos ψ = −Ve cos α2 sin β cos ψ

(4.56)

represents the relative velocity contribution between the SBR and the point of interest D due to the Earth’s rotation toward the SBR. Combining (4.47) and (4.56) as in (4.48), we obtain the modiﬁed Doppler frequency that also accounts for the Earth’s rotation to be ωd =

2Tr Vp sin θEL cos θAZ − Ve cos α2 sin β cos ψ . λ/2

(4.57)

From triangle ACD in Figure 4.23, we have sin (π/2 + ψ) sin θEL = Re + H Re

(4.58)

so that

cos ψ =

1+

H Re

sin θEL

(4.59)

and hence (4.57) becomes

ωd = =

2Vp Tr Ve sin θEL cos θAZ − λ/2 Vp

1+

H Re

2Vp Tr sin θEL (cos θAZ − cos α1 sin γ ) , λ/2

cos α2 sin β (4.60)

103

104

Space Based Radar where we deﬁne Ve = Vp

H 1+ Re

.

(4.61)

In (4.59) we have also used the identity cos α2 sin β = cos α1 sin γ

(4.62)

that follows from (4A.10) and (4A.11). In Figures 4.27–4.28, ZBD = γ represents the azimuth angle of the ground range segment R along the great circle relative to the north measured at B. To simplify (4.60) further, from Figure 4.28 we have

ZBD = γ ,

ZBBo = δ,

B1 BD = θAZ

(4.63)

and hence γ = ZBB1 + B1 BD = π − δ + θAZ .

(4.64)

This gives sin γ = sin(δ − θAZ ) = sin δ cos θAZ − cos δ sin θAZ .

(4.65)

From the spherical triangle ZBo B formed by the north pole and the SBR locations at the equator and at (α1 , β1 ), we get (see Figure 4.29)

ZBo B =

π − ηi , 2

Bo ZB = β1 ,

ZBBo = δ.

Z (North pole)

FIGURE 4.29 Spherical triangle ZBo B formed by the north pole, SBR locations at equator (Bo ) and (α1 , β1 ) at B.

b1 p a1 2 p 2

B1

qo p - hi 2 hi Bo

d B

SBR path

(4.66)

Chapter 4:

S p a c e B a s e d R a d a r-----K i n e m a t i c s

Hence using (2A.3), we obtain sin δ sin(π/2 − ηi ) cos ηi = = sin π/2 sin(π/2 − α1 ) cos α1

(4.67)

or5 sin δ =

cos ηi . cos α1

(4.68)

Since δ > π/2, we have cos δ < 0, which gives

cos δ = −

cos2 α1 − cos2 ηi . cos α1

(4.69)

Substituting (4.68)–(4.69) into (4.65) we get

cos ηi cos θAZ + sin θAZ cos2 α1 − cos2 ηi sin γ = cos α1

(4.70)

and ﬁnally with (4.70) in (4.60) we obtain the modiﬁed Doppler frequency to be ωd =

2Vp Tr sin θEL (1 − cos ηi ) cos θAZ λ/2 −

=

cos2 α1 − cos2 ηi sin θAZ

2Vp Tr ρc sin θEL cos(θAZ + φc ) λ/2

where φc = tan and ρc =

−1

cos2 α1 − cos2 ηi 1 − cos ηi

1 + 2 cos2 α1 − 2 cos ηi .

(4.71)

(4.72)

(4.73)

In (4.71)–(4.73), φc represents the crab angle and ρc represents the crab magnitude. In summary, the effect of Earth’s rotation on the 5 In (4.68), α ≤ η since the maximum latitude of the SBR never exceeds its ini 1 clination at the equator. Equation (4.64) together with (4.68) compute the azimuth angle corresponding to the point of interest (α2 , β2 ) on Earth from the SBR parameters (α1 , β1 ) and ηi in terms of γ . Here γ can be expressed in terms of (α1 , β1 ) and (α2 , β2 ) as in (4A.7) and (4A.10). Thus, given the current location of the SBR and the point of interest in terms of (α1 , β1 ), (α2 , β2 ), and ηi , the instantaneous azimuth angle θAZ , elevation angle θEL , and grazing angle ψ can be determined. The latter two require the SBR height as well.

105

106

Space Based Radar Doppler velocity is to introduce two distortions—a crab angle and crab magnitude into the SBR azimuth angle and modify it accordingly [7], [8]. Interestingly both these quantities depend only on the SBR orbit inclination and its latitude, and not on the latitude or longitude of the clutter patch of interest. Equation (4.71) corresponds to the case where the region of interest D is to the east of the SBR path as shown in Figure 4.28. If on the other hand, the region of interest is to the west of the SBR path as in Figure 4.30, then Vo = Ve cos α2 cos(β − π/2) = Ve cos α2 sin β

(4.74)

represents the Earth’s velocity component along DB (refer to (4.55)).

North pole Z

Z b1

Region of interest a2

D

SBR path b

B1

Y2

Latitudes p –d

b – p/2

g

R

a1

d

hi

qAZ

Y1

B

SBR inclination at equator

Equator

Bo

b1

b2 Longitudes

FIGURE 4.30 Effect of Earth rotation on Doppler frequency when point of interest is on the west of the SBR path.

Chapter 4:

S p a c e B a s e d R a d a r-----K i n e m a t i c s

This gives the modiﬁed Doppler to be (see (4.57)) ωd = =

2Tr (Vp sin θEL cos θAZ + Ve cos α2 sin β cos ψ) λ/2 2Tr sin θEL (cos θAZ + cos α1 sin γ ). λ/2

(4.75)

However, in this case from Figure 4.30, we have θAZ = γ + π − δ. Hence γ = θAZ + δ − π

(4.76)

so that sin γ = − sin(δ + θAZ ) = − sin δ cos θAZ − cos δ sin θAZ − cos ηi cos θAZ + sin θAZ = cos α1

cos2 α1 − cos2 ηi

,

(4.77)

where we have used (4.68) and (4.69). Substituting (4.77) into (4.75) and simplifying we get ωd =

2Vp Tr sin θEL ((1 − cos ηi ) cos θAZ λ/2

+ =

cos2 α1 − cos2 ηi sin θAZ )

2Vp Tr ρc sin θEL cos(θAZ − φc ), λ/2

(4.78)

where φc and ρc are as deﬁned in (4.72) and (4.73). Combining (4.71) and (4.78) we obtain the modiﬁed Doppler to be [5] ωd =

2Vp Tr ρc sin θEL cos(θAZ ± φc ). λ/2

(4.79)

In (4.79), the plus sign is to be used when the region of interest is to the east of the SBR path and the minus sign is to be used when the point of interest is to the west of the SBR path. Figure 4.31 shows the crab angle and crab magnitude variation as a function of SBR height for a satellite located at the equator on three different orbits. For example, on a polar orbit (Figure 4.31 (a)) at a low altitude of 506 km, the crab angle is about 3.77◦ and the crab magnitude is about 1.002. However, at a medium Earth orbit of 5,000 km, the crab angle increases to 7.98◦ and the crab magnitude becomes 1.01. From Figure 4.31 (a), both these parameters are increasing functions of the SBR height for a satellite on a polar orbit. Figure 4.31 (b) shows the satellite on an equatorial orbit. In this case, the crab angle is zero at all altitudes, however, the crab magnitude

107

Space Based Radar

50

1.8

40

Crab Magnitude

Crab Angle (deg)

1.7

30 20

1.6 1.5 1.4 1.3 1.2

10

1.1 0

0

1

2 H (km)

3

1

4 4 ×10

0

1

(i) fc

2 H (km)

3

4 4 ×10

(ii) rc

(a) SBR on a polar orbit (a1 = 0°, hi = 90°) 1

0.5

Crab Magnitude

Crab Angle (deg)

1

0 −0.5 −1

0.8

Geosynchronous orbit

0.6 0.4 0.2

0

1

2 H (km)

3

0

4 4 ×10

0

1

(i) fc

2 H (km)

3

H0 4 4 ×10

(ii) rc

(b) SBR on an equatorial orbit (a1 = 0°, hi = 0°) 80

1

60

Crab Magnitude

Crab Angle (deg)

108

40

20

0

0.8 0.6 0.4 0.2

0

1

2 H (km) (i) fc

3

0

4 ×10

4

0

1

2 H (km)

3

4 4

×10

(ii) rc

(c) SBR on an inclined orbit (a1 = 0°, hi = 45°)

FIGURE 4.31 Crab angle and crab magnitude as functions of SBR height on three different orbits.

Chapter 4:

S p a c e B a s e d R a d a r-----K i n e m a t i c s

monotonically decreases till it becomes zero at Ho = 35,868 km, which corresponds to the geosynchronous situation. This also follows from (4.73) since in that case with α1 = 0, ηi = 0 we get ρc = 1 − = 1 −

Ve Vp

1+

H Re

and ρc = 0 gives the SBR height Ho to be √ G Me T Vp Re + Ho = Re = √ Re Ve 2π Re Re + Ho or √ 2/3 T G Me Re + Ho = = 42,241 km 2π

(4.80)

(4.81)

(4.82)

which gives Ho = 42,241 − 6,373 = 35,868 km.

(4.83)

From (4.79) ρc = 0 gives the overall Doppler frequency to be zero, in agreement with the geosynchronous nature of the SBR orbit. The importance of crab magnitude ρc in the modiﬁed Doppler frequency given in (4.79) is clearly evident from the above argument, and it must be taken into account when its contribution is signiﬁcant. For example, referring back to Figure 4.31 (a), the effect of ρc maybe negligible on a low Earth orbit, whereas in a high orbit such as 10,000 km and above, the effect of ρc is signiﬁcant. Figure 4.31 (c) shows the crab parameters variation for an SBR orbit inclinated at 45◦ . In this case, the crab angle increases whereas the crab magnitude decreases up to about 3,000 km and then increases as the SBR altitude increases. In what follows for illustrative purposes, the SBR height is set at 506 km. Figure 4.32 shows the crab angle and crab magnitude for an SBR located on a polar orbit (ηi = π/2) as a function of its latitude α1 . To determine the peak values of the crab angle and the crab magnitude we can equate dφc = 0, dα1

(4.84)

dρc = 0. dα1

(4.85)

and

After some simpliﬁcation, Equation (4.84) and (4.85) together with (4.72) and (4.73) yield sin 2α1 = 0.

(4.86)

109

Space Based Radar rc 1.0025

fc 4 3.5

1.002

3 2.5 deg

110

1.0015

2 1.001

1.5 1

1.0005

0.5 0

0

10

20

30

40

50 a1

60

70

80

1

90

0

10

20

30

40 50 a1

60

70

80

90

(b) Crab magnitude

(a) Crab angle

FIGURE 4.32 Crab angle and crab magnitude as a function of α1 for an SBR on a polar orbit (SBR height = 506 km).

Interestingly α1 = 0 and α1 = ±π/2 are the only solutions to (4.86), where α1 = 0 corresponds to the maximum and α1 = ±π/2 corresponds to the minimum. Hence irrespective of the SBR inclination and height, the crab angle peaks when the SBR is above the equator. In particular, for an SBR on a polar orbit, the crab angle peaks globally when it is above the equator and its minimum (zero) occurs when it is above the poles. Figure 4.33 shows the crab angle and crab magnitude as functions of the SBR latitude and orbit inclination. From Figure 4.33, for an SBR located at an altitude of 506 km, the crab angle varies between ±3.77◦ and it has maximum effect for an SBR on a polar orbit located at the equator. The crab magnitude on the other hand has maximum effect

rc 1 0.99 0.98 0.97 0.96 0.95 0.94

fc 3 2 1 0

80

60 hi

40

20

0 0

20

(a) Crab angle

40 a1

60

80

80

60

40 a1 20

0 0

20

40

60 hi

(b) Crab magnitude

FIGURE 4.33 Crab angle and crab magnitude as a function of SBR latitude and orbit inclination (SBR height = 506 km).

80

Chapter 4:

S p a c e B a s e d R a d a r-----K i n e m a t i c s 1.01

4 hi = 60° hi = 30°

2

hi = 90°

1 Crab Magnitude

Crab Angle (deg)

3

1 0

hi = 0°

−1 −2

0.98 hi= 60°

0.97 0.96 0.95

hi = 30°

0.94

−3 −4 −100

hi = 90°

0.99

−50

0 Latitude

50

100

0.93 −100

(a) Crab angle

hi = 0° −50

0 Latitude

50

100

(b) Crab magnitude

FIGURE 4.34 Crab angle and crab magnitude vs. SBR latitude for different inclination angles (SBR height = 506 km).

when the SBR is on an equatorial orbit (ηi = 0, α1 = 0). In that case ρc = 1 −

(4.87)

which leads to a maximum of 6% error. Thus, the crab angle has maximum effect on a polar orbit and crab magnitude has maximum effect on an equatorial orbit. Figure 4.34 shows the crab angle and crab magnitude as functions of SBR latitude for different inclination angles. Once again about 3.77◦ error can be expected for the crab angle in the worst case. On comparing (4.48) and (4.79), we notice that the effect of the Earth’s rotation on the Doppler is to modify the cos θAZ term in (4.48) as ρc cos (θAZ ± φc ). To determine the associated error, deﬁne u = ρc cos (θAZ + φc ) − cos θAZ

(4.88)

uo = cos (θAZ + φc ) − cos θAZ ,

(4.89)

and

where uo represents only the effect of crab angle on the Doppler (ρc ≡ 1). Notice that both u and uo are functions of the azimuth angle θAZ , orbit inclination ηi , and SBR latitude αi . Figure 4.35 (a)–(c) show the error uo introduced only by the crab angle as a function of the SBR latitude and azimuth angle for orbit inclinations ηi = 90◦ , 45◦ , and 0◦ . Similarly Figure 4.35 (d)–(f) show the overall error u due to Earth’s rotation as a function of α1 and θAZ . From these ﬁgures for any orbit inclination, the maximum error occurs when the SBR is directly above the equator (α1 = 0). However, as Table 4.2 shows, the azimuth angle at which these errors peak are different for u and uo . From Table 4.2, the maximum crab error occurs

111

Space Based Radar

0.02

0

0

−0.02

−0.02

u

u0

0.02

−0.04

−0.04

−0.06

−0.06

−0.08 150

100 qAZ

50 0 0

20

40

60

−0.08 150

80

a1

100 qAZ

50 0 0

(a)

20

40

60

80

a1

(d) hi = 90° (Polar orbit)

0.05 u

u0

0.1 0 −0.01 −0.02 −0.03 −0.04 −0.05

0 −0.05 −0.1

150

100 qAZ

50 0 0

10

(b)

20

30

150

40

100 50 qAZ

a1

0 0

10

20 a

30

40

1

(e)

hi = 45° (Inclined orbit)

1

0.08 0.06 0.04

0.5

0.02 0

u

u0

112

0 −0.02 −0.04

−0.5

−0.06 −1

0

50

100 qAZ

150

−0.08

0

(c)

50

100 qAZ

150

(f) hi = 0° (Equatorial orbit)

FIGURE 4.35 Error due to crab angle and crab magnitude as functions of SBR latitude and the azimuth angle.

when the SBR is vertically above the equator (α1 = 0) and the point of interest is directly ahead on the equator in which case the error peaks at 6.6% irrespective of the SBR orbit inclination. This phenomenon seems to be true when the SBR is at other latitudes as well. Table 4.3 shows the maximum crab error and the corresponding SBR azimuth for various orbit inclinations and SBR latitudes. From there the crab error peaks when the azimuth for the point of interest

Chapter 4:

S p a c e B a s e d R a d a r-----K i n e m a t i c s

Crab Angle and Crab Angle Only Crab Magnitude SBR Orbit Max Error SBR Azimuth Max Error SBR Azimuth Inclination η, (u0 ) θ AZ (u) θAZ 90◦ 0.066 86.11◦ 0.066 90◦ 60◦ 0.059 88.38◦ 0.066 60◦ ◦ ◦ 45 0.049 88.56 0.066 45◦ ◦ ◦ 30 0.035 88.92 0.066 30◦ ◦ ◦ 15 0.018 89.46 0.066 15◦ 0◦ 0 0◦ 0.066 0◦ TABLE 4.2 Maximum error in (4.88) and (4.89) for various orbit inclinations ηi (global maximum occurs for α1 = 0◦ )

is such that the mainbeam is directly ahead along the local latitude. In that case, as seen from Figure 4.36 and Table 4.3, we have θo +θAZ = π/2 for all inclinations. Once again the maximum error depends only on the SBR latitude and is independent of the orbit inclination. Table 4.4 shows the minimum crab error and the corresponding SBR azimuth for various orbit inclinations and SBR latitudes. From there the crab angle is zero when the azimuth for the point of interest is perpendicular to the local latitude (i.e., along the local longitude). These results are summarized in Figure 4.36 which shows three points of interest D1 , D2 , and D3 for an SBR located at B. From the above discussion, the beam B D1 that looks directly ahead along the local latitude generates the maximum crab error. The beam B D2 along the local longitude has minimum (zero) crab error, and the beam B D3 that is off the bore-side of the array has nonzero crab error and provides a tradeoff between crab error and array azimuth ambiguity. However, the overall effect of crab error on the estimated clutter covariance matrix and the corresponding adaptive weight vector is much more complicated because of the Doppler-Azimuth dependency on range and the range foldover phenomenon. The effect of crab angle on Doppler as a function of azimuth angle for various range values is shown in Figures 4.37–Figure 4.38. As (4.71) shows, for an SBR altitude of 506 km, the effect of Earth’s rotation is to shift the azimuth angle appearing in the Doppler by approximately φc = 3.77◦ as shown in Figure 4.37 and simultaneously modify the Doppler magnitude as well. As a result, even for θAZ = 90◦ , the Doppler peak values occur away from ωd = 0 depending on the range. This shift in Doppler with and without crab effect is illustrated in Figure 4.38 for various azimuth angles. The actual Doppler frequencies of the incoming clutter will depend upon the Doppler foldover parameter in (4.52) for the array conﬁguration.

113

114

TABLE 4.3 Maximum error in (4.88) for various orbit inclinations ηi and local latitude α1

Space Based Radar

α1 0◦ 15◦ 30◦ 45◦ 60◦ 75◦ 90◦

η i = 90◦ η i = 45◦ η i = 30◦ Angle Between Angle Between Angle Between Max Err Azimuth Local Longitude Max Err Azimuth Local Longitude Max Err Azimuth Local Longitude u θ AZ and SBR Path θ 0 u θ AZ and SBR Path θ 0 u θ AZ and SBR Path θ 0 0.066 90◦ 0◦ 0.066 45◦ 45◦ 0.066 30◦ 60◦ 0.064 90◦ 0◦ 0.064 42.94◦ 47.06◦ 0.064 26.28◦ 63.72◦ 0.057 90◦ 0◦ 0.057 35.27◦ 54.73◦ 0.057 0◦ 90◦ ◦ ◦ ◦ ◦ 0.047 90 0 0.047 0 90 – – – 0.033 90◦ 0◦ – – – – – – 0.017 90◦ 0◦ – – – – – – 0 – 0◦ – – – – – –

Chapter 4:

S p a c e B a s e d R a d a r-----K i n e m a t i c s North pole

Z

Z

SBR path

Y2

a2 D3

Latitude q0 a1

Large crab effect, good azimuth

qAZ

Y1

B D1 Small crab effect, poor azimuth

hi

Equator D2 Longitude b2

b1

FIGURE 4.36 Effect of crab angle/magnitude and azimuth for three points of interest on an SBR located at latitude α1 . fc

Without crab effect

R2

With crab effect

Ro

100 Doppler

{

{

{

200

R1 0 R1 < Ro < R2 −100 −200

qAZ

1

−1

−0.5

0

qAZ 0.5 2

qAZ

3

1

Azimuth (cos qAZ)

FIGURE 4.37 Effect of crab angle on azimuth-Doppler proﬁle as a function of range (SBR height = 500 km).

115

116

TABLE 4.4 Minimum error in (4.88) for various orbit inclinations ηi and local latitude α1

Space Based Radar

α1 0◦ 15◦ 30◦ 45◦ 60◦ 75◦ 80◦

η i = 90◦ η i = 45◦ η i = 30◦ Angle Between Angle Between Angle Between Max Err Azimuth Local Longitude Min Err Azimuth Local Longitude Min Err Azimuth Local Longitude u θ AZ and SBR Path θ 0 u θ AZ and SBR Path θ 0 u θ AZ and SBR Path θ 0 0 0◦ , 180◦ 0◦ 0 135◦ 45◦ 0 120◦ 60◦ 0 0◦ , 180◦ 0◦ 0 132.94◦ 47.06◦ 0 116.28◦ 63.72◦ 0 0◦ , 180◦ 0◦ 0 125.28◦ 54.73◦ 0 90◦ 90◦ ◦ ◦ ◦ ◦ ◦ ◦ 0 0 , 180 0 0 90 90 – – – 0 0◦ , 180◦ 0◦ – – – – – – 0 0◦ , 180◦ 0◦ – – – – – – 0 – 0◦ – – – – – –

Chapter 4:

S p a c e B a s e d R a d a r-----K i n e m a t i c s

Without crab effect (

60

).

qAZ = 75°

50

With crab effect (

).

Doppler

40 qAZ = 80°

30 20 10 0

qAZ = 90°

−10 −20

0

500

1,000

1,500

2,000

Range (km)

FIGURE 4.38 Effect of crab angle on range-Doppler proﬁle (SBR height = 500 km).

180

180

160

160

140

140 Azimuth (deg)

Azimuth (deg)

Figures 4.39–4.41 show the iso-Doppler plots for an SBR on a polar orbit with and without Earth’s rotation taken into account. Figure 4.39 shows the iso-Doppler proﬁle in the range-azimuth domain. Figures 4.40—4.41 show the iso-Doppler proﬁle in the latitudelongitude domain for an SBR on a polar orbit located at the equator and latitude 30◦ respectively.

120 100 80 60

120 100 80 60

40

40

20

20

0

0

500

1,000

1,500

Range (km) W/ Earth’s rotation

(a) a1 = 0°

0

0

500

1,000

1,500

Range (km) W/O Earth’s rotation

(b) a1 = 30°

FIGURE 4.39 Iso-Doppler proﬁle in range-azimuth domain for an SBR on a polar orbit (ηi = 90◦ ) with and without Earth’s rotation. (a) SBR at equator, (b) SBR at latitude 30◦ (SBR height = 506 km).

117

118

Space Based Radar 1

1

0.8 0.6

0.5 Latitude a 2

Latitude a 2

0.4 0.2 0 −0.2 −0.4

0

−0.5

−0.6 −0.8 −1

0

5 10 Longitude b 2−b 1

−1

15

0

5

5

Latitude a 2

Latitude a2

15

(b)

(a)

0

−5

5 10 Longitude b 2−b 1

0

5 10 Longitude b 2−b 1 W/ Earth’s rotation

(c)

15

W/O Earth’s rotation

0

−5

0

5 10 Longitude b 2−b 1 W/ Earth’s rotation Iso-azimuth

15

W/O Earth’s rotation Iso-range

(d)

FIGURE 4.40 Iso-Doppler proﬁle in the latitude-longitude domain for an SBR on a polar orbit at the equator with and without Earth’s rotation. (c) and (d) show the extended latitude region. (b) and (d) contain the iso-azimuth and iso-range proﬁles as well (SBR height = 506 km).

Figures 4.42–4.43 show the iso-Doppler plots for an SBR on a 60◦ inclination orbit with and without Earth’s rotation taken into account. Figure 4.42 shows the iso-Doppler proﬁle in range-azimuth domain. Figure 4.43 shows the iso-Doppler proﬁle in latitude-longitude domain with details as indicated there. From these ﬁgures, there is signiﬁcant difference in Doppler when Earth’s rotation in taken into account, and its effect on target detection performance is taken up in Chapters 6 and 7.

4.7.1 Range Foldover and Crab Phenomenon When crab phenomenon and range foldover are present simultaneously, multiple Doppler frequencies are generated from the associated range foldover points as in Figure 4.20. These Doppler frequencies

Chapter 4:

S p a c e B a s e d R a d a r-----K i n e m a t i c s 35

31 30.8 30.6 Latitude a 2

Latitude a 2

30.4 30.2 30 29.8

30

29.6 29.4 29.2 29

0

5

10 Longitude b 2−b 1

25

15

W/ Earth’s rotation

0

5 10 Longitude b 2−b 1

15

W/O Earth’s rotation

(a)

(b)

FIGURE 4.41 Iso-Doppler proﬁle in the latitude-longitude domain for an SBR on a polar orbit at 30◦ latitude with and without Earth’s rotation, (b) the extended latitude region (SBR height = 506 km).

180

180

160

160

140

140

120

Azimuth (deg)

Azimuth (deg)

now correspond to the solid curves in Figure 4.44 that contain the crab effect. For every range point of interest, these two phenomena together generate a sequence of Doppler frequencies and they are plotted in Figure 4.44 with and without the crab effect for an SBR located at height 506 km above ground. Notice that at PRF = 500 Hz, there are seven foldover Doppler frequencies (see also Figure 4.22).

100 80 60 40

100 80 60 40

20 0

120

20 0

500

1,000

1,500

Range (km) W/ Earth’s rotation

(a) a1 = 0°

0

0

500

1,000

1,500

Range (km) W/O Earth’s rotation

(b) a1 = 30°

FIGURE 4.42 Iso-Doppler in range-azimuth domain for an SBR on a 60◦ inclination orbit with and without Earth’s rotation. (a) SBR at equator, (b) SBR at latitude 30◦ (SBR height = 506 km).

119

Space Based Radar 5

1

4 3 2 Latitude a 2

Latitude a 2

0.5

0

1 0

−1 −2

−0.5

−3 −4

−1

0

5 10 Longitude b 2−b 1

15

W/ Earth’s rotation

−5

0

5 10 Longitude b 2−b 1

15

W/O Earth’s rotation

(a)

(b)

FIGURE 4.43 Iso-Doppler proﬁles in the latitude-longitude domain for an SBR on a 60◦ inclination orbit at the equator with and without Earth’s rotation. (b) shows the extended region. (SBR height = 506 km).

The clutter corresponding to these range bins will be associated with these modiﬁed Doppler frequencies. The effect of crab phenomenon on clutter Doppler spread and limitations imposed on processing gains are discussed in detail in Chapter 6.

Foldover returns

120

Without crab effect 100

With crab effect

80 Doppler

120

60 40 20 0

Main return 500

1,000 1,500 Range (km)

2,000

FIGURE 4.44 Effect of range foldover and crab angle on range-Doppler proﬁle for an SBR located at height 506 km above ground. PRF = 500 Hz, θAZ = 60◦ .

Chapter 4: SBR

S p a c e B a s e d R a d a r-----K i n e m a t i c s

North

A

y (Grazing angle) Z

Point of contact on Earth

g

D

R

Latitude a1 B

Longitude b 2

Longitude b1 a a2

a1

Y

C X F

E

b1

Equator

Vernal equinox

Æ ECB = a 1, Æ XCE = b 1, Æ FCD = a 2, Æ XCF = b 2 Æ BCD = a, R = Rea

FIGURE 4.45 Ground range between nadir point and point of interest from their geo-coordinates.

Appendix 4-A: Ground Range from Latitude and Longitude Coordinates Let (α1 , β1 ) and (α2 , β2 ) represent the latitude and longitude of the nadir point B and the point of interest D on Earth respectively as shown in Figure 4.45. To determine the range R from point B to point D, and the core angle α between the radii CB and CD, we proceed as follows: We have

EC B = α1 ,

F C D = α2 ,

XC E = β1 ,

XC F = β2

(4A.1)

so that C E = Re cos α1 ,

E B = Re sin α1 ,

C F = Re cos α2 ,

F D = Re sin α2 .

(4A.2)

121

122

Space Based Radar North North pole

Z

SBR A

b 2- b1 R

b

Point of interest on ground at (a 2, b 2)

D

g

H

G

B a Nadir point (a 1, b 1)

a1 b1

Y

b 2 - b 1 cos a 2 b2

E

Vernal equinox

Re

C

Re X

sin a 2

a2

sin a 1

F cos a1 Equator

FIGURE 4.46 Nadir point—point of interest plane.

Further EC F Figure 4.46),

= β2 − β1 , so that from triangle ECF (see

E F 2 = Re2 (cos2 α1 + cos2 α2 − 2 cos α1 cos α2 cos(β2 − β1 ))

(4A.3)

Since BC D = α from triangle BCD we also obtain B D2 = 2Re2 (1 − cos α).

(4A.4)

Let BG be parallel to EF so that BG = EF and FG = EB as in Figure 4.46. Hence DG = FD − EB = Re (sin α2 − sin α1 ).

(4A.5)

Substituting these values into the right triangle BGD we obtain BD2 = BG2 + DG2

(4A.6)

which simpliﬁes to cos α = sin α1 sin α2 + cos α1 cos α2 cos(β2 − β1 )

(4A.7)

Chapter 4:

S p a c e B a s e d R a d a r-----K i n e m a t i c s

and hence R = Re α = Re cos−1 (sin α1 sin α2 + cos α1 cos α2 cos(β2 − β1 )) . (4A.8) From Equation (2A.13) in Appendix 2-A, using the spherical triangular law applied to the spherical triangle ZBD in Figure 4.46, we obtain sin γ sin β sin(β2 − β1 ) = = (4A.9) sin(π/2 − α2 ) sin(π/2 − α1 ) sin α which gives sin γ = cos α2

sin(β2 − β1 ) sin α

(4A.10)

sin β = cos α1

sin(β2 − β1 ) . sin α

(4A.11)

and

Notice that γ and β represent the azimuth scan of the SBR nadir point B and the point of interest D relative to the North pole.

Appendix 4-B: Nonsphericity of Earth and the Grazing Angle Correction Factor The Earth is usually assumed to be a spheroid with radius Re = 6,373 km. However, Earth is not an exact sphere and it is more like an ellipsoid (oblate spheroid) with major and minor axes given by (Hayford, 1909) a = 6,378.4 km, b = 6,356.9 km.

(4B.1)

The eccentricity of an ellipse is deﬁned through the relation b 2 = a 2 (1 − e 2 )

(4B.2)

which gives Earth’s eccentricity e to be 0.08199. Consider an ellipsoidal Earth with major and minor axis given by a , b, and an ideal Earth of radius Re = a circumscribing the ellipsoid as shown in Figure 4.47. Let A represent the location of the SBR with its nadir point B at latitude α1 , C the center of the ideal Earth, D the point of interest on Earth with latitude α2 , and E its counterpart on the ideal Earth. Let ψe and ψ represent the grazing angles at D and E. Clearly ψe represents the grazing angle of interest based on the actual ellipsoidal model, and

123

124

Space Based Radar

Spherical Earth

A

North pole

SBR

H Rs2

Elliptical Earth

ye

r1 r

a a1 a2 C

1

y

B

a

Rs

G

E D

u f

X F

b

FIGURE 4.47 Nonspherical Earth and grazing angle correction.

ψ that based on the ideal spherical model given by (4.6). The objective is to determine the correction factor of the ideal grazing angle ψ so as to obtain the actual grazing angle ψe at D. Let ρ1 and ρ represent the radial distances CB and CD to the nadir point and point of interest respectively. Further, let6 α = α1 − α 2

(4B.3)

represent the angle at the center of the Earth extended by the desired range R. From ACE, the slant range AE is given by Rs1 =

a 2 + ( H + ρ1 ) 2 − 2a ( H + ρ1 ) cos α

(4B.4)

and sin α cos ψ = H + ρ1 Rs1

(4B.5)

6 We will ﬁrst assume that points B and D have the same longitudes. The additional adjustment when their longitudes β1 and β2 are different is given in (4B.35)– (4B.41).

Chapter 4:

S p a c e B a s e d R a d a r-----K i n e m a t i c s

so that the ideal grazing angle equals ψ = cos

−1

H + ρ1 sin α . Rs1

(4B.6)

Similarly from ACD, the slant range AD equals Rs2 =

ρ 2 + ( H + ρ1 ) 2 − 2ρ( H + ρ1 ) cos α

(4B.7)

and with CDG = υ, where DG represents the perpendicular to the tangent at D, we have cos(ψe − υ) sin α = H + ρ1 Rs2

(4B.8)

so that the actual grazing angle is given by ψe = υ + cos−1

H + ρ1 sin α . Rs2

(4B.9)

From (4B.4)–(4B.8), to obtain the actual grazing angle ψe in (4B.9) we need to determine ρ, ρ1 , and υ in terms of α1 and α2 . As a ﬁrst approximation, if we assume ρ = ρ1 = Re

(4B.10)

so that α = R/Re , then Rs1 = Rs2 = Rs =

)

Re2 + ( Re + H) 2 − 2Re ( Re + H) cos( R/Re ) (4B.11)

as in (4.3), and from (4B.6) and (4B.9) we get ψe = ψ + υ

(4B.12)

and hence υ represents the correction factor to be applied to (4.6) to accommodate the nonsphericity of Earth. To determine υ, we proceed as follows.

Determination of the Correction Factor υ The equation of an ellipse is given by x2 y2 + = 1, a2 b2

(4B.13)

125

126

Space Based Radar where (x, y) represents the coordinates of the point of interest D. Differentiating (4B.13) the slope of the target point at D is given by x b2 dy = − 2. dx ya

(4B.14)

But DG is perpendicular to the tangent at D, and in terms of DG X = φ (the celestial latitude), the slope of the above tangent equals tan(π/2 + φ). Hence

dy x b2 = tan(π/2 + φ) = −cotφ = − 2 dx ya

(4B.15)

or tan φ =

y a2 . x b2

(4B.16)

Substituting (4B.16) into (4B.13) we get 2 x2 2b + x tan2 φ = 1 a2 a4

(4B.17)

or x2 =

a 4 cos2 φ a 2 cos2 φ + b 2 sin2 φ

=

a 2 cos2 φ 1 − e 2 sin2 φ

(4B.18)

and y2 =

a 2 (1 − e 2 ) 2 sin2 φ 1 − e 2 sin2 φ

.

(4B.19)

Let ρ denote the radius CD. Then from Figure 4.47 and (4B.18)– (4B.19), we obtain x = ρ cos α2 =

a cos φ 1 − e 2 sin2 φ

a (1 − e 2 ) sin φ y = ρ sin α2 = 1 − e 2 sin2 φ

(4B.20)

(4B.21)

which gives ρ2 = a 2

1 − (2e 2 − e 4 ) sin2 φ 1 − e 2 sin2 φ

= a2

1 + (1 − e 2 ) 2 tan2 φ . 1 + (1 − e 2 ) tan2 φ

(4B.22)

Using the relation tan α2 =

y b2 = 2 tan φ = (1 − e 2 ) tan φ x a

(4B.23)

Chapter 4:

S p a c e B a s e d R a d a r-----K i n e m a t i c s

we can rewrite ρ in (4B.22) in terms of α2 as ρ2 =

a 2 (1 − e 2 ) sin2 α2 + (1 − e 2 ) cos2 α2

or

, ρ=a

=

a 2 (1 − e 2 ) 1 − e 2 cos2 α2

1 − e2 . 1 − e 2 cos2 α2

(4B.24)

(4B.25)

Here ρ represents the Earth’s local radius at D, and it is given in terms of the equatorial radius a , Earth’s eccentricity e and the local latitude α2 at D. Notice that the local radius at latitude α2 is smaller than that at the equator. Since ρ1 corresponds to latitude α1 , we also have

,

ρ1 = a

1 − e2 . 1 − e 2 cos2 α1

(4B.26)

Further from (4B.20)–(4B.21), we have a e 2 sin φ cos φ x sin φ − y cos φ = ρ sin(φ − α2 ) = . 1 − e 2 sin2 φ

(4B.27)

But from CDG υ = φ − α2 .

(4B.28)

a e 2 sin(2φ) . ρ sin υ = 2 1 − e 2 sin2 φ

(4B.29)

Hence

Similarly

)

x cos φ + y sin φ = ρ cos(φ − α2 ) = ρ cos υ = a

1 − e 2 sin2 φ. (4B.30)

Thus tan υ =

e 2 sin(2φ) 2(1 − e 2 sin2 φ)

=

e 2 sin(2α2 ) . 2(1 − e 2 cos2 α2 )

(4B.31)

Equations (4B.25)–(4B.31) can be used in (4B.4)–(4B.9) to determine the actual grazing angle ψ. Notice that it depends on α1 , α2 , and e. However, if we use the approximation in (4B.12) we get ψe = ψ + tan−1

e 2 sin(2α2 ) 2(1 − e 2 cos2 α2 )

(4B.32)

127

Space Based Radar where ψ = cos

−1

Re + H sin( R/Re ) Rs

(4B.33)

represents the grazing angle corresponding to a spherical Earth as in (4.6). Thus the correction factor to the grazing angle at latitude α2 equals −1

υ = tan

e 2 sin(2α2 ) 2(1 − e 2 cos2 α2 )

(4B.34)

and it is plotted in Figure 4.48. Notice that the maximum correction occurs at 45◦ latitude and the grazing angle at that latitude computed using the spherical Earth must be incremented by 0.1932◦ . Finally suppose (α1 , β1 ) and (α2 , β2 ) represent the latitudelongitude pair for the nadir point B and the point of interest D respectively and assume that β1 = β2 as shown in Figure 4.49. In this case consider the great circle that goes through B and D and let it intersect the equator at X at latitude αo = 0 and longitude βo (see Figure 4.49). The angles BCD and DCX are in the plane of this great circle and they need to be re-evaluated. Correction Factor vs a 2

0.2 0.18 0.16 Correction Factor (deg)

128

0.14 0.12 0.1 0.08 0.06 0.04 0.02 0

0

10

20

30

40

50

60

70

80

Latitude a 2 (deg)

FIGURE 4.48 Correction factor for grazing angle at different latitudes.

90

Chapter 4:

S p a c e B a s e d R a d a r-----K i n e m a t i c s

SBR

North

A

y Z

Point of contact on Earth D

R B Great circle through B and D

a

X a2

a1

(0, bo)

C F

E

b1

Vernal equinox

Equator

Longitude b 2

FIGURE 4.49 Great circle through nadir point and ground point of interest.

From (4A.7)

BC D = α = cos−1 (sin α1 sin α2 + cos α1 cos α2 cos(β1 − β2 )) . (4B.35)

But

BCD = BCX − DCX

(4B.36)

and by repeated use of (4B.35) we obtain (consider the points (α1 , β1 ) and (0, βo ))

BCX = cos−1 (cos α1 cos(β1 − βo ))

(4B.37)

and (consider the points (α2 , β2 ) and (0, βo ))

DCX = cos−1 (cos α2 cos(β2 − βo )) .

(4B.38)

Using (4B.37) and (4B.38) in (4B.36) we get cos−1 (cos α1 cos(β1 − βo )) − cos−1 (cos α2 cos(β2 − βo )) = cos−1 (sin α1 sin α2 + cos α1 cos α2 cos(β2 − β1 )),

(4B.39)

129

130

Space Based Radar and this equation is ﬁrst used to determine the longitude of interest βo . Knowing βo , the corrected versions of α1 and α2 are given by α1 = cos−1 (cos α1 cos(β1 − βo ))

(4B.40)

α2 = cos−1 (cos α2 cos(β2 − βo )) .

(4B.41)

and

These values and the projected versions of α1 and α2 onto the plane containing the great circle that goes through B and D. From (4B.39), notice that α1 − α2 = α. These projected values must be used in place of α1 and α2 in (4B.4)–(4B.9), (4B.25)–(4B.26), and (4B.34) to determine the correction factor for the actual grazing angle ψe .

Appendix 4-C: Doppler Effect In radar, to detect moving targets (and to estimate their velocities), pulses are transmitted periodically since a single pulse return contains unambiguous information about the instantaneous range only.

What Is Doppler Effect? Suppose V represents the relative velocity between the transmit source s(t) and the target along their line of sight, and let D1 be the range of the target from the source at time t = to , when the ﬁrst pulse is transmitted toward the target for a ﬁnite duration τo (see Figure 4.50). Assume the source to be stationary and the target moves toward the source with velocity V. Let τ1 represent the time it takes for the transmit signal to arrive at the moving target. During that time interval, the waveform travels a distance of cτ1 and the target travels a distance Vτ1 toward each other, so that from Figure 4.50 (a) (c + V)τ1 = D1

(4C.1)

or τ1 =

D1 c+V

(4C.2)

where c represents the velocity of light. The ﬁrst pulse arrives back at the stationary transmitter after another τ1 seconds at t = to + 2τ1 and the corresponding received signal equals x1 (t) = s(to + 2τ1 ).

(4C.3)

Chapter 4:

S p a c e B a s e d R a d a r-----K i n e m a t i c s

First pulse

Target

s(to)

D1

V

t = to ct1

t = to + t1

Vt1

V

x1(t ) 2Vt1

V

t = to + 2t1

(a) Propagation of the first pulse

Second pulse s(to + T ) D2 = D1- VT

t = to + T

ct 2

t = to + T + t2

V

Target V Vt2

x2(t ) t = to + T + 2t2

V

2Vt 2

(b) Propagation of the second pulse

FIGURE 4.50 Propagation of two consecutive pulses.

Notice that the delay τ1 contains combined information about the range D1 and target velocity V, both of which are unknown. Let T represent the radar pulse repetition interval. Thus a second pulse is transmitted at t = to + T. At that instant, the distance to target equals D2 = D1 − VT.

(4C.4)

As a result, from (4C.2), the time τ2 it takes for the second pulse to reach the target equals τ2 =

D2 VT D1 VT = − = τ1 − . c+V c+V c+V c+V

(4C.5)

131

132

Space Based Radar Consequently, the second pulse arrives at the transmitter at t = to + T + 2τ2 . This gives the corresponding received signal to be

x2 (t) = s(to + T + 2τ2 ) = s

2VT to + 2τ1 + T − c+V

.

(4C.6)

If the carrier signal s(t) is sufﬁciently narrowband of the form s(t) = e j (ωo t+φ) ,

(4C.7)

then (4C.6) simpliﬁes to x2 (t) = s(to + 2τ1 + T)e − j2ωo c+V . VT

(4C.8)

But ωo = 2πc/λ, so that the exponent in (4C.8) simpliﬁes to 2ωo VT =π c+V

1 1 + V/c

2VT 2VT ≈π = π ωd , λ/2 λ/2

(4C.9)

where we have used c V, which is the case in radar (but not in sonar since sound travels much slower than light). In (4C.9), we have deﬁned ωd =

2VT . λ/2

(4C.10)

Substituting (4C.9) into (4C.8), we get x2 (t) = s(to + 2τ1 + T)e − jπ ωd .

(4C.11)

On comparing (4C.3) and (4C.11), the envelope of the second pulse return is a shifted version of the ﬁrst pulse return (shifted by a known amount T) and hence knowing the ﬁrst pulse return, the envelope of the second return is also known. In particular, if these envelopes are constant over the pulse duration, then the ﬁrst and second returns have the form x1 (t) = A,

(4C.12)

x2 (t) = Ae − jπ ωd .

(4C.13)

and

In (4C.13), the factor ωd appearing in the phase shift factor represents the Doppler frequency. Knowing this additional phase shift in (4C.13), the target velocity can be estimated from (4C.10). If this procedure is repeated with M consecutive identical pulses of constant envelopes that are separated by a common interval T, then from (4C.12)–(4C.13) their outputs x1 (t), x2 (t), · · · xM (t) can be

Chapter 4:

S p a c e B a s e d R a d a r-----K i n e m a t i c s

conveniently expressed in vector form as



x1 (t)





1



 x (t)   e − jπ ωd   2     .  = A . ..  .     .    . xM (t) e − j ( M−1)π ωd

(4C.14)

The vector on the right side of (4C.14) is referred to as the temporal steering vector, since phase shifting and adding the received output xi (t) generates

y(ωd ) =

M

xi (t)e jπ(i−1)ωd = MA.

(4C.15)

i=1

Thus the target echoes can be coherently ampliﬁed in magnitude, provided the Doppler factor ωd used in (4C.15) coincides with the true target Doppler ωdo . Observe that y(ωd ) in (4C.15) represents a ﬁnite impulse response (FIR) Doppler ﬁlter (Figure 4.51), and its output peaks as the Doppler variable ωd in (4C.15) coincides with the true target Doppler frequency.

0 −5

y (wd) in dB

−10 −15 −20 −25 −30 −1

−0.5

FIGURE 4.51 FIR Doppler ﬁlter.

0 wd

wd0 0.5

1

133

134

Space Based Radar

Appendix 4-D: Oblate Spheroidal Earth and Crab Angle Correction The Earth is not an exact sphere and it is more like an oblate spheroid which introduces a correction factor υ to the grazing angle ψ that is based on the spheroidal model (see Appendix 4-B). From (4B.32) and (4B.34), if ψe represents the actual grazing angle, then ψe = ψ + υ

(4D.1)

where

cos ψ =

1+

H Re

sin θEL

(4D.2)

and υ = tan−1

e 2 sin(2α2 ) 2(1 − e 2 cos2 α2 )

.

(4D.3)

To account for the Earth’s oblate spheroidal shape, the actual grazing angle ψe must be substituted in (4.56)–(4.57) for the crab angle. This gives the total Doppler in (4.57) to be ωde =

2Tr (Vp cos θAZ sin θEL − Ve cos α2 sin β cos ψe ). λ/2

(4D.4)

From (4D.1) we have cos ψe = cos(ψ + υ) = cos ψ cos υ − sin ψ sin υ cos ψ − υ sin ψ (4D.5) and using (4D.5) in (4D.4) we get ω de =

2Tr (Vp cos θAZ sin θEL − Ve cos α2 sin β cos ψ) λ/2 +

2Tr υVe cos α2 sin β sin ψ λ/2

= ωd +

2Tr υVe cos α1 sin γ sin ψ, λ/2

(4D.6)

where ωd is the (crab angle) corrected Doppler due to the spheroidal Earth as in (4.57)–(4.71), and the second term in (4D.6) represents the error ξ due to the oblate spheroidal Earth.

Chapter 4:

S p a c e B a s e d R a d a r-----K i n e m a t i c s

To express this error as a correction factor to the crab angle, we can make use of (4.70). This gives the error term in (4D.6) to be ξ = =

2Ve Tr υ sin ψ(cos ηi cos θAZ + sin θAZ cos2 α1 − cos2 ηi ) λ/2 2Vp Tr ε cos α1 cos (θAZ − u) υ sin ψ λ/2

where

, u = tan

(4D.7)

−1

cos2 α1 −1 cos2 ηi

(4D.8)

ε=

Ve . Vp

(4D.9)

and

Substituting (4D.7) into (4D.6) and using (4.71) and (4.59), the actual Doppler simpliﬁes to ω de =

2Vp Tr (ρc sin θEL cos (θAZ + φc ) + ευ cos α1 cos (θAZ − u) sin ψ) λ/2

* sin ψ ρc + ευ cos α1 cos(φc + u) cos (θAZ + φc) sin θEL + sin ψ sin (θAZ + φc ) + ευ cos α1 sin(φc + u) sin θEL

2Vp Tr = sin θEL λ/2

=

2Vp Tr sin θEL {(ρc + υ cos α1 cos(φc + u) tan ψ) cos (θAZ + φc ) λ/2 + υ cos α1 sin(φc + u) tan ψ sin (θAZ + φc )}

=

2Vp Tr ρ sin θEL cos(θAZ + φc − φe ) λ/2

where φe = tan

−1

υ cos α1 sin(φc + u) tan ψ ρc + υ cos α1 cos(φc + u) tan ψ

(4D.10)

(4D.11)

and ρ=

(ρc + υ cos α1 cos(φc + u) tan ψ) 2 + (υ cos α1 sin(φc + u) tan ψ) 2 .

(4D.12)

135

Space Based Radar

4

×10−4

r 1.0022

2 fe (deg)

0 1.0022 −2 −4 −20

−10

0

10

20

1.0022 −20

−10

0

a 2 (deg)

a 2 (deg)

(a) fe

(b) r

10

20

FIGURE 4.52 φe and ρ as function of α2 for an SBR with polar orbit at equator.

Here, is as deﬁned in (4.61). From (4D.10) and (4D.11), the quantity φe represents the correction to the crab angle φc due to the oblate spheroidal shape of the Earth. Figure 4.52 shows φe and ρ as functions of α2 for an SBR on a polar orbit at the equator at an altitude of 506 km. The point of interest is assumed to be at 15◦ longitude and α2 is varied by moving the point of interest along that longitude. From there, the correction factor φe is between −0.0004◦ and 0.0004◦ and the correction factor ρ stays at 1.0022 as α2 changes. Interestingly, ρ is the same as ρc in (4.73) that corresponds to the spherical Earth. Figure 4.53 shows ρ

1.0022

r, rc

136

1.0021

1.002 −20

−10

0 a 1 (deg)

10

20

FIGURE 4.53 ρ and ρc as function of α1 for an SBR with polar orbit.

Chapter 4:

S p a c e B a s e d R a d a r-----K i n e m a t i c s

and ρc as functions of the SBR latitude. From there, ρ (solid line) and ρc (dotted line) coincide with each other indicating that they are the same. In summary, the correction factor to the crab angle due to the oblate spheroidal shape of the Earth is negligible.

References [1] Leopold J. Cantaﬁo, Space-Based Radar Handbook, Artech House, Boston, 1989. [2] Troy L. Hacker, “Performance Analysis of a Space-Based GMTI Radar System Using Separated Spacecraft Interferometry,” MS Thesis, Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Lexington, MA, May 2000. [3] Mark E. Davis, Braham Himed, and David Zasada, “Design of Large Space Radar for Multimode Surveillance,” IEEE Radar Conference, Huntsville, AL, pp. 1–6, May 2003. [4] S.M. Kogon, D.J. Rabideau, and R.M. Barnes, “Clutter Mitigation Techniques for Space-Based Radar,” IEEE International Conference on Radar, Vol. 4, pp. 2323– 2326, March 1999. [5] S.U. Pillai, B. Himed, Y.K. Li, “Effect for Earth’s Rotation and Range Foldover on Space-Based Radar Performance”, Proc. IEEE Transactions on Aerospace and Electronic Systems, Vol. 42, no. 3, July 2003. [6] G.A. Andrews and K. Gerlach, “SBR Clutter and Interference,” Ch. 11, SpaceBased Radar Handbook, Ed. Leopold J. Cantaﬁo, Artech House, Boston, 1989. [7] J.R. Guerci, Space-Time Adaptive Processing for Radar, Artech House, Boston, MA, 2003. [8] S. Hensley and E. Chapin, “Write Up of Scott’s Notes on Earth’s Rotation— Version 2.0,” JPL, September 2003.

137

This page intentionally left blank

CHAPTER

5

Space-Time Adaptive Processing Space-Time Adaptive Processing (STAP) refers to spatially and temporally distributed data collection and processing. For spatial processing, an array of sensors that are distributed in space (in one or two dimensions) can be used; for temporal processing, a sequence of pulses that are transmitted consecutively can be used (Appendix 4-C); by combining them appropriately, such as in the case of phased arrays, spatio-temporal processing can be achieved. Why use spatio-temporal processing? To see the advantages in processing gain that can be realized using spatio-temporal processing, the simpler situation—spatial (array) processing is considered ﬁrst in this chapter. This is followed by a discussion on an optimal weight vector for combining the array outputs so as to maximize the output signal to interference plus noise ratio (SINR) in the spatial domain as well as the spatio-temporal domain. The STAP issues are considered next for a side-looking airborne radar. Angle-Doppler clutter power spectral properties, various techniques to illustrate clutter cancellation are discussed next including the eigen-structure based methods and their variations.

5.1 Spatial Array Processing Array processing refers to transmitting/receiving using a set of sensors rather than a single sensor. The use of multiple sensors calls for additional processing both prior to transmitting and/or after receiving signals.

139 Copyright © 2008 by The McGraw-Hill Companies. Click here for terms of use.

140

Space Based Radar

5.1.1 Why Use an Array? Consider a transmit signal that gets reﬂected from a target. Let s(t) represent the return signal—at a single receiver. Why use a set of sensors instead of a signal sensor at the receiver? The advantage in using an array can be best illustrated in a narrowband situation. Thus if s(t) represents a narrowband signal, then s(t) has the form s(t) = a e j (ωo t+φ)

(5.1)

and let this represent the signal at the ﬁrst (reference) sensor in Figure 5.1. Thus x1 (t) = s(t) = a e j (ωo t+φ).

(5.2)

Let θ represent the angle that the normal to the wavefront makes with the line of the array. If the second sensor is d˜ apart from the ﬁrst sensor, then d˜ cos θ represents the additional distance to be covered by the wavefront to reach the second sensor. Hence the wavefront arrives at the second sensor after d˜ cos θ τ= (5.3) c seconds. Hence the second sensor output with reference to the ﬁrst sensor is given by x(t) = s(t − τ ) = s(t)e − jωo

d˜ cos θ c

d˜

= s(t) e − jπ λ/2 cos θ = s(t) e − jπd cos θ

(5.4)

d˜ λ/2

(5.5)

where d=

s(to) s(t )

Planar wavefront

t

∼ d cos q s(t – t)

q

Reference sensor x1(t )

FIGURE 5.1 Spatial processing.

……

……

∼ d

……

…… x2(t )

xi (t )

xN (t )

Chapter 5:

Space-Time Adaptive Processing

represents the interelement distance normalized to half-wavelength. In general, let di represent the ith sensor location normalized to λ/2 with respect to the reference sensor. Then following the above argument, the ith sensor output equals x(t) = s(t)e − jπdi cos θ,

i = 1, 2, . . . , N.

(5.6)

In vector form, these outputs can be expressed as





  1  x (t)   2   e − jπd2 cos θ      ..  ..     .    .    x(t) =  = s(t)  − jπ d cos θ    = s(t)a (θ) i  xi (t)  e  .  .    ..  .     .  − jπ d cos θ N e xN (t) x1 (t)

where



1

 e − jπd2 cos θ  ..   .  a (θ) =  − jπd cos θ i e ..   .

(5.7)

         

(5.8)

e − jπ d N cos θ represents the spatial steering vector associated with the direction of arrival θ. For a uniform array with interelement spacing equal to λ/2, we have di = (i − 1),

i = 1, 2, . . . , N

(5.9)

and the spatial steering vector takes the form



1

 e − jπ cos θ   ..   . a (θ ) =   e − jπ(i−1) cos θ   ..   .

      .     

(5.10)

e − jπ( N−1) cos θ In general, the sensor measurements are buried in noise and let n1 (t), n2 (t), · · · refer to the input noise signals at the sensors. Thus the

141

142

Space Based Radar ith sensor output equals xi (t) = s(t)e − jπ di cos θ + ni (t),

i = 1, 2, . . . , N

(5.11)

In vector form, (5.11) becomes x(t) = s(t)a (θ ) + n(t)

(5.12)

n(t) = [n1 (t), n2 (t), . . . , n N (t)]T

(5.13)

where

represents the input noise vector. We are in a position to demonstrate the advantage in using multiple sensors instead of a single sensor. If we only have one sensor, say the ﬁrst one, then the signal to noise ratio (SNR) at its input equals (SNR) i =

E{|s(t)|2 } P = 2 E{|n1 (t)|2 } σn

(5.14)

where σn2 = E{|ni (t)|2 }

(5.15)

refers to the input noise power that is common to all sensors. To exhibit the processing gain that can be achieved using a set of sensors, consider the simple phase shifting scheme where each output in (5.11) is phase shifted so as to combine their signal parts coherently. From (5.11), it follows that the phase shift for the ith sensor must be by the amount e jπ di cos θ as shown in Figure 5.2. Combining the phase shifted outputs, we obtain the ﬁnal output y(t) to be y(t) =

N

xi (t)e jπdi cos θ .

(5.16)

i=1

x1(t)

x2(t )

...

xi (t )

... e jp d1cosq

e jp d2cosq

xN (t )

... e jpdicosq

Σ y(t)

FIGURE 5.2 Phase shifted array.

...

e jp dN cosq

Chapter 5:

Space-Time Adaptive Processing

From (5.11), we have y(t) = Ns(t) +

N

ni (t)e jπdi cos θ = Ns(t) + no (t)

(5.17)

i=1

where no (t) =

N

ni (t)e jπdi cos θ .

(5.18)

i=1

From (5.17) and (5.18), the useful signal part has been combined coherently in amplitude; whereas the noise part gets added only incoherently. From (5.16), the ﬁnal output SNR is given by (SNR) o =

E{|Ns(t)|2 } N2 E{|s(t)|2 } = 2 ∗ jπ(di −dk ) cos θ E{|no (t)| } E i k ni (t)nk (t)e

= N i

=N

N2 P E{|ni (t)|2 }

=

N2 P Nσn2

P = N · (SNR) i . σn2

(5.19)

In (5.19), we have assumed the various sensor noise outputs to be uncorrelated, i.e., E{ni (t)n∗j (t)} = 0 for i = g. The advantage of using an array of sensors is obvious. From (5.19), in the case of uncorrelated noise the simple phase shifting and sum operation improves the array output SNR for the desired signal by a factor equals to the number of sensors! From (5.17)–(5.19), the desired signal gets added coherently in amplitude, whereas the noise gets added in power only; the above operation of phase shifting and adding is also known as “beamforming” since the array has been “steered” along the desired direction (i.e., along θ, the direction of the desired signal) to create maximum gain along that direction.

Uniform Array An interesting question in the context of beamforming is the nature of the gain pattern along other angles while the array is steered along a speciﬁc direction, i.e., the nature of the sidelobe patterns of the array (see Figure 5.3). From (5.16), in the case of omnidirectional sensor, the normalized array pattern is given by (xi (t) ≡ 1)

2 N 1 jπdi cos θ e G(θ ) = , N i=1

(5.20)

143

Space Based Radar Look direction

Main lobe

Sidelobes

FIGURE 5.3 Array sidelobe patterns.

which for a uniform array (di = i − 1) reduces to (see Figure 5.4)

G(θ) =

Nπ cos θ 2 2 θ . N sin π cos 2 sin

(5.21)

From (5.21), the width of the mainbeam is proportional to 2 N→∞ −→ 0. N 0 −13.46 dB

−10

G(q ) in dB

144

−20 −30 −40 −50 −1

−0.5

0 cos (q )

0.5

FIGURE 5.4 Array pattern for a 15-element uniform array with half-wavelength interelement spacing.

1

(5.22)

Chapter 5:

Space-Time Adaptive Processing

Thus the array mainbeam width can be decreased and made narrower by increasing the number of sensor elements. However, this is not true for the sidelobe levels. The sidelobe levels of a uniformly placed array get settled around −13.46 dB, and cannot be improved further by increasing the number of sensor elements in the array. To see this, deﬁne ω=

π cos θ , 2

(5.23)

so that the gain pattern in (5.21) is given by

G(ω) =

sin( Nω) N sin(ω)

2 .

(5.24)

Clearly the ﬁrst null in (5.24) and Figure 5.4 occurs at ωo = π/N,

(5.25)

and the second null occurs at ωo = 2π/N, so that the ﬁrst dominant sidelobe level occurs approximately at the center of these two nulls at ω1 = ωo +

π 3π = . 2N 2N

(5.26)

The corresponding gain at ω = ω1 is given by

G(ω1 = 3π/N) =

sin(3π/2) N sin(3π/2N)

= -

# N

= #

2

1 [N sin(3π/2N)]2

1 3π 2N

−

(3π/2N) 3 3!

−

9π 2 16N2

+ ···

$.2

1 3π 2

=

+ ···

$2 ≈

2 3π

2 (5.27)

or 10 log G(ω1 ) 10 log(2/3π ) 2 = −13.46 dB.

(5.28)

Thus for a uniform array, the maximum sidelobe level stays around −13.46 dB irrespective of the number of sensor elements present in the array.1 However, the sidelobe levels in the array gain pattern in Figure 5.4 can be further suppressed by introducing shading factors at the sensor outputs [1]. 1 See Appendix 5-A for a detailed treatment on the sidelobe levels of a uniform array.

145

146

Space Based Radar In general, direction-dependent weights wi (θ) can be used in (5.16) to combine the array output as in Figure 5.5. In that case, the output is given by y(t) =

N

wi∗ xi (t).

(5.29)

i=1

Equation (5.29) can be expressed more conveniently in matrix form as y(t) = w ∗ x(t) where



w1 (θ )

(5.30)



 w (θ )   2   w=  .. .  .  w N (θ )

(5.31)

Notice that in the case of beamformer w = a (θ ),

(5.32)

i.e., the weight vector is simply the complex conjugate of the steering vector. In general, x(t) is more complex than a single signal buried in uncorrelated noise (e.g., signal plus clutter or multiple signals arriving simultaneously). Let R = E{x(t)x ∗ (t)} > 0

(5.33)

represent the N × N array output covariance matrix. Notice that the (i, j) element of R equals

Ri, j = E xi (t)x ∗j (t)

(5.34)

and it represents the cross covariance between the outputs of the ith and jth sensor elements. R is in general a positive-deﬁnite hermitian (R = R∗ > 0) matrix. For example, for a single source situation as in FIGURE 5.5 Array with directiondependent weights.

x1(t ) w∗1

x2(t ) w∗2

…

xi (t ) …

…

…

wi∗

Σ y(t )

xN (t ) wN∗

Chapter 5:

Space-Time Adaptive Processing

(5.12), with θ = θo and p = E{|s(t)|2 }, we get R = E{|s(t)|2 }a (θo )a ∗ (θo ) + σn2 I = Pa (θo )a ∗ (θo ) + σn2 I,

(5.35)

where we have assumed the signal and noise parts to be uncorrelated. Further, the noise itself is assumed to be sensor to sensor uncorrelated and of equal variance σn2 (or more generally, noise elements are independent and identically distributed (i.i.d.) random variables). If the noise term is taken to be broad enough to include interferences and clutter, then the element to element uncorrelated assumption may no longer be true for noise and in that case (5.35) generalizes to R = Pa (θo )a ∗ (θo ) + Rn

(5.36)

Rn = E{n(t)n∗ (t)} > 0

(5.37)

where using (5.13),

represents the array input interference plus noise covariance matrix. Although the signal part of (5.35) and (5.36) has rank one, the noise covariance matrix is invariably full rank and hence R is also generally of full rank. Using (5.30) and (5.33), the array output power for look direction θ equals P(θ) = E{|y(t)|2 } = E{|w ∗ (θ )x(t)|2 } = E{w ∗ (θ )x(t)x ∗ (t)w(θ )} = w ∗ (θ ) E{x(t)x ∗ (t)}w(θ) = w ∗ (θ)Rw(θ) > 0.

(5.38)

The importance of the array output matrix R must be clear from (5.38); it plays a central role in shaping the output power for every look direction. In the case of beamformer, w = a (θ), and the output power equals PB (θ ) = a ∗ (θ )R a (θ ).

(5.39)

In the case of a single source scene with the source located along θo , using (5.35) in (5.39), we obtain the beamformer output to be ∗

2

PB (θ) = P|a (θ)a (θo )| +

Nσn2

2

=N P

sin( Nω) N sin ω

2 + Nσn2 ,

(5.40)

where ω = π(cos θ − cos θo )/2. Figure 5.6 shows the beamformer output for two situations: a single source case with θo = 90◦ as in Figure 5.6 (a) and a two source case with θ1 = 70◦ and θ2 = 75◦ as shown in Figure 5.6 (b).

147

Space Based Radar 2

2

0

0

−2

−2 PB(q ) in dB

PB(q ) in dB

148

−4 −6 −8

−10

−4 −6 −8

−10

−12

−12 0

50

100 q in deg

150

0

(a) q 0 = 90°

50

100 q in deg

150

(b) q 1 = 70°, q 2 = 75°

FIGURE 5.6 Beamformer.

As Figure 5.6 (b) shows, the ﬁnite size of the mainbeam width can cause merging of peeks when multiple signals are present simultaneously along close, but different directions, and it becomes necessary to use more complicated weight vectors that suppress the unwanted signals for better resolution when the array is focused along a speciﬁc direction. One approach in this context is to choose the weight vector so as to maximize the array output SNR.

5.1.2 Maximization of Output SNR Consider a desired source s(t) located along the direction θ in presence of other possible sources, interference, and noise. The array output vector can be written as x(t) = s(t)a (θ ) + n(t)

(5.41)

where n(t) contains all other undesired signals including noise. Let Rn = E{n(t)n∗ (t)} > 0

(5.42)

represent the covariance matrix of the unwanted “noise” part as in (5.37). The objective is to determine the optimum weight vector w(θ ) that maximizes the array output SNR along the desired direction θ . Clearly with the signal power along θ being ﬁxed, the maximization of SNR occurs at the expense of minimizing the noise components, thereby reducing the unwanted component in the output signal. The array output signal equals y(t) = w ∗ (θ )x(t) = s(t)w ∗ a (θ ) + w ∗ n(t).

(5.43)

Chapter 5:

Space-Time Adaptive Processing

This gives the output SNR to be SNRo = =

E{|s(t)|2 }|w ∗ a (θ )|2 E{|w ∗ n(t)|2 } P|w ∗ a (θ)|2 P|w ∗ a (θ )|2 = . w ∗ E{n(t)n∗ (t)}w w ∗ Rn w

(5.44)

To maximize (5.44), we can use Schwarz’s inequality that states for any two vectors u and v, |u∗ v|2 ≤ (u∗ u)(v∗ v),

(5.45)

with equality in (5.45) if and only if v = ku∗ .

(5.46)

To use (5.45) in (5.44), we rewrite the numerator term in (5.44) as2

−1/2 w ∗ a (θ) = w ∗ R1/2 a (θ ) = Rn1/2 w n Rn

∗

R−1/2 a (θ ) = u∗ v n

(5.47)

a (θ ) so that the numerator in (5.44) with u = Rn1/2 w and v = R−1/2 n becomes |w ∗ a (θ )|2 = |u∗ v|2 ≤ (u∗ u)(v∗ v).

(5.48)

But in this case

u∗ u = R1/2 n w

∗

Rn1/2 w = w ∗ Rn w

(5.49)

and

a (θ ) v∗ v = R−1/2 n

∗

R−1/2 a (θ ) = a ∗ (θ)R−1 n n a (θ).

(5.50)

Substituting these into (5.48), we get

|w ∗ a (θ )|2 ≤ (w ∗ Rn w) a ∗ (θ )R−1 n a (θ)

(5.51)

and hence (5.44) becomes SNRo ≤ Pa ∗ (θ)R−1 n a (θ) = SNRmax .

(5.52)

From (5.46), clearly the maximum SNR in (5.52) can be realized if Rn1/2 w = kRn−1/2 a (θ ).

(5.53)

2 If R is an N × N Hermitian positive-deﬁnite matrix, then R = UU∗ = R∗ , n n n where U is unitary (U U∗ = I ) and is diagonal with positive real entries λ1 , 1/2 −1/2 ∗ ∗ λ2 , . . . , λ N . In that case, Rn = U1/2 U and Rn = U−1/2 U where ±1/2 is ±1/2 ±1/2 ±1/2 again diagonal with entries λ1 , λ2 , . . . , λ N , etc.

149

150

Space Based Radar

x (t )

R−1/2 n

y(t ) = s(t )b(q ) + wn (t )

b(q ) Beamformer

z (t ) = b ∗(q )y(t ) = w ∗x(t)

FIGURE 5.7 Matched ﬁlteras whitening followed by beamformer.

or3 w = R−1 n a (θ ).

(5.54)

Equation (5.54) represents the optimum desired weight vector that maximizes the output SNR for the look direction θ . Notice that in the case of a single source present in i.i.d. noise, (5.54) reduces to the ordinary beamformer in (5.32); however, if the noise is not i.i.d. (colored noise), the optimum weight vector is not the beamformer even in a single source scene and is given by (5.54). Interestingly, the optimum weight vector in (5.54) that maximizes the output SNR can be given another interpretation as well. Suppose the data x(t) in (5.41) is ﬁrst passed through a whitening ﬁlter so that the colored noise n(t) becomes white noise w n (t). From (5.42), Rn−1/2 whitens the noise, since y(t) = Rn−1/2 x(t) = s(t)R−1/2 a (θ ) + R−1/2 n(t) n n = s(t)b(θ) + w n (t), and

(5.55)

Rwn = E w n (t)w ∗n (t) = Rn−1/2 E{n(t)n∗ (t)}R−1/2 n = R−1/2 Rn R−1/2 = I. n n

(5.56)

Thus the noise component w n (t) in (5.55) indeed represents white noise, and the signal component b(θ) is given by

a (θ ). b(θ ) = R−1/2 n

(5.57)

Equation (5.55) represents a signal s(t) with steering vector b(θ ) buried in white noise, and from (5.54) the optimum matched ﬁlter is the ordinary beamformer b(θ). These two operations are shown in Figure 5.7. From Figure 5.7, the ﬁnal output z(t) = b ∗ (θ ) y(t) = a ∗ (θ )Rn−1/2 y(t)

−1 = a ∗ (θ )R−1 n x(t) = Rn a (θ)

∗

x(t) = w ∗ x(t),

(5.58)

3 The constant k in (5.46) can be chosen to be unity since it does not alter the output SNR.

Chapter 5:

Space-Time Adaptive Processing

where w is as given in (5.54). In general, w = R−1 n a (θo )

(5.59)

represents the optimum weight vector that maximizes the output SNR for a target located along θo . Thus the optimum weight vector is equivalent to whitening followed by matched ﬁltering operation. The whitening operation in Figure 5.7 shows that the adaptive processor cancels out any dominant interference signal present in Rn by generating nulls along their respective directions of arrival, and the matched ﬁltering operation boosts the output along the desired direction a (θo ). For example, let the interference/noise signal scene consist of three spatially discrete uncorrelated signals of equal power along directions θ1 , θ2 , θ3 , and white noise. This gives Rc = a (θ1 )a ∗ (θ1 ) + a (θ2 )a ∗ (θ2 ) + a (θ3 )a ∗ (θ3 ) + σn2 I

(5.60)

and the maximum SNR output SNRmax = a ∗ (θ)R−1 n a (θ)

(5.61)

in (5.52) is plotted in Figure 5.8. Clearly the three interferences are mitigated as indicated by the nulls in Figure 5.8.

Interferences

0

SINRmax (dB)

−5 −10 −15 −20 −25 0

50

100

150

qAZ (deg)

FIGURE 5.8 Three uncorrelated arrivals along θ1 = 55◦, θ2 = 75◦, and θ3 = 105◦ . Interference to noise ratio = 20 dB. A 12-sensor array is used here.

151

Space Based Radar The adaptive processor output

2

P(θ) = |w ∗ a (θ)|2 = a ∗ (θo )R−1 n a (θ)

(5.62)

is shown in Figure 5.9 using (5.59) with a desired target along θo = 85◦ in (5.59). Notice that the adaptive processor nulls out the three interference sources and boosts the target output along θo . To implement (5.54), Rn needs to be known. Generally, the ideal covariance matrices are unknown, and the next best thing (in terms of maximum likelihood (ML) estimation) is to use its ML estimate of Rn using data simples x 1 , x 2 , . . . , x K , where xi refers to a particular observation of the array output “noise” vector. If the “noise” represents jointly Gaussian random variables, then the ML estimate for their covariance matrix Rn is given by K ˆn = 1 x k x ∗k R K

(5.63)

k=1

ˆ n , we need K > N, and that is based on K samples. Clearly, to invert R ˆ when Rn is invertible, its inverse may be used in (5.54) as an estimate for R−1 n . In that case (5.54) is referred to as Sample Matrix Inversion (SMI) method. Thus, the weight vector for the SMI method equals −1

ˆ n a (θ). w SMI = R

(5.64)

Interferences 10 Target direction

0 −10 SINR (dB)

152

−20 −30 −40 −50 −60 0

50

qo 100 qAZ (deg)

150

FIGURE 5.9 Adaptive processor output with desired target at θo = 85◦ .

Chapter 5:

Space-Time Adaptive Processing

If the array size is large, then to implement SMI more and more data need to be collected, all of which must have the same ideal covariance matrix, i.e., the data must be at least wide sense stationary. This may be a difﬁcult condition to meet in practice especially for large arrays (as in STAP), and alternative approaches to weight vector estimation are desired.

5.2 Space-Time Adaptive Processing Space-Time-Adaptive Processing (STAP) refers to joint spatiotemporal processing where spatial diversity is realized utilizing a set of sensors conﬁgured in one or two dimensions and temporal processing is achieved using returns from a sequence of periodically transmitted pulses. The primary reason for repeated temporal pulse transmission is to detect target Doppler (i.e., moving targets), and in the presence of an array of sensors, each transmitted pulse produces a data vector. Stacking these vectors corresponding to different pulses, we can generate a space-time data vector. The situation corresponding to N sensors and M pulses is shown in Figure 5.10. From Figure 5.10 (b), the space-time data vector is of size MN × 1 and corresponds to a single sample snapshot for the STAP scheme. The spatial part of the output at time t is given by the column vector (5.12) and the temporal returns due to two consecutive pulses at the same reference sensor are given by (4C.11)–(4C.13). From Figure 5.10 and (5.12), the N array output vector due to ﬁrst pulse equals x 1 (t) = s(t)a (θ ) + n1 (t),

(5.65)

where s(t) refers to the return from the desired target located along θ. From (4C.13), the second pulse output vector equals x 2 (t) = s(t)a (θ )e − jπ ωd + n2 (t),

1

2

N ……

x(t ) = x1

[x 1]

x(t + T ) = x 2 x=

[x 2] …

…

FIGURE 5.10 Space-time data corresponding to N sensors and M pulses in two conﬁgurations.

(5.66)

x(t + (M − 1)T ) = xM

[x M]

(a)

(b) STAP vector (MN × 1)

153

154

Space Based Radar where the identical nature of the transmit waveform from pulse to pulse is utilized. Here ωd represents the Doppler frequency from the desired target of interest. In general, the return due to the ith pulse equals xi (t) = s(t)a (θ)e − jπ(i−1)ωd + ni (t),

i = 1, 2, . . . , M

(5.67)

and stacking these M column vectors as in Figure 5.10 (b), we obtain







a (θ ) x1  a (θ)e − jπ ωd  x2    .   ..  .   .  .  x=  = s(t)   a (θ)e − jπ(i−1)ωd  xi    .   ..  .   . . xM a (θ )e − jπ( M−1)ωd where





     + n(t)    

(5.68)



n1 (t)  n2 (t)   n(t) =   ..  . n M (t)

(5.69)

represents the external noise vector. Following (4C.14), (5.8), we deﬁne the temporal steering vector (M × 1)



1



 e − jπ ωd      ..   .  b(ωd ) =  − jπ(i−1)ωd  . e     ..   . − jπ( M−1)ω d e

(5.70)

Using (5.70) in (5.68), the spatio-temporal data vector at time t equals x = s(t)b(ωd ) ⊗ a (θ) + n(t),

(5.71)

where ⊗ represents the Kronecker product in (5.71). Deﬁne s(θ, ωd ) = b(ωd ) ⊗ a (θ)

(5.72)

to be the space-time steering vector associated with arrival angle θ and Doppler frequency ωd . Then xk = s(tk )s(θ, ωd ) + n(tk )

(5.73)

Chapter 5:

Space-Time Adaptive Processing

represents the space-time data vector at time t = tk and such K observations corresponding to time instants tk , k = 1, 2, . . . , K represent the space-time data cube. Equation (5.73) has the same form as (5.12) and hence all the processing techniques described earlier are applicable here as well. As the following application shows, the speciﬁc form of the Doppler will depend on the nature of the problem.

5.3 Side-Looking Airborne Radar In an airborne radar, the platform is moving with velocity V, and periodic pulses are transmitted toward the ground to perform ground moving target indication (GMTI). In this case, the ground is stationary, however, the returns do generate a Doppler frequency because of the relative velocity of the ground with respect to the moving platform. For a one-dimensional side-looking radar with N uniformly placed sensors that are spaced d˜ apart (with d representing the normalized

No bins

R

qk V ~ d

Airborne radar

FIGURE 5.11 Airborne side-looking radar.

155

156

Space Based Radar interelement distance with respect to half-wavelength (Figure 5.11)), the spatial steering vector has the form (see (5.8))



1



 e − jπd cos θ      ..   .  a (θ) =  − jπ(i−1)d cos θ   e     ..   . − jπ( N−1)d cos θ e

(5.74)

corresponding to the look direction θ from the line of the array. Here a stationary ground patch in the direction θ has a relative velocity V cos θ with respect to the moving platform and hence from (4C.10), it generates a Doppler frequency ωd =

2VT cos θ, λ/2

(5.75)

where T represents the pulse repetition interval. In this case the temporal steering vector b(ωd ) is given by (5.70). On comparing the exponents in the spatial and temporal steering vector, we obtain ωd = =

2VT 2VT cos θ = d cos θ λ/2 dλ/2 2VT d cos θ = β(d cos θ) d˜

(5.76)

VT 2VT = ˜ d˜ d/2

(5.77)

where β=

refers to the Doppler foldover factor.4 From (5.77), the Doppler foldover factor represents the number of half-interelement spacing traveled by the platform during one pulse repetition interval T. From (5.76), the Doppler frequency ωd for a side-looking radar is proportional to the spatial angular parameter d cos θ appearing in (5.74), the proportionality term given by the Doppler foldover factor β. If β = 1, the clutter ﬁlls the angle-Doppler space exactly once; if β > 1, the clutter folds over (aliasing) into the Doppler space creating clutter Doppler ambiguity. The space-time steering vector corresponding to ground clutter returns for a side-looking radar is given by (5.72) with a (θ ) and b(ωd ) 4β

is also known as the Brennan factor.

Chapter 5:

Space-Time Adaptive Processing PB(q, w d) in dB

1

0

0.8 −5

0.6 0.4

−10

wd

0.2 0

−15

−0.2 −0.4

−20

−0.6 −0.8 −1 −1

−25 −0.5

0

0.5

1

cos (q )

FIGURE 5.12 Angle-Doppler proﬁle.

given by (5.74), (5.70) and ωd as in (5.75)–(5.76). Using these values and deﬁning φ = π d cos θ,

(5.78)

the space-time steering vector corresponding to ground clutter patches takes the explicit form

% . s(θ, ωd ) = 1, e − jφ , · · · e − j ( N−1)φ , .. e − jβφ , · · · e − j(β+( N−1))φ , . &T .. − j(( M−1)β+( N−1))φ . (5.79) ··· . ···e . The peculiar single variable dependency in (5.79) is noteworthy, and it gives special structure to the clutter spectra when viewed in the angle-Doppler domain (see Figure 5.12), that can be exploited for better clutter suppression through STAP. In a simpliﬁed model, the ground clutter data vector corresponding to a particular range has the form xc =

No k=1

c k s(θk , ωdk ),

(5.80)

157

158

Space Based Radar where the summation is carried out over all ground patches at the same range and s(θk , ωdk ) is as given in (5.79) with θ replaced by θk (see Figure 5.11). Here c k represents the scatter strength from the kth patch for the particular range of interest. If we assume the clutter returns are uncorrelated from patch to patch, the clutter covariance matrix has the form

Rc = E xc x∗c =

No

Pk sk s∗k

(5.81)

k=1

where Pk = E{|c k |2 }

(5.82)

represents the kth clutter patch power and

sk = s(θk , ωdk ).

(5.83)

In addition to clutter, there is always noise and if we include the noise term into (5.80), the total covariance matrix (in the case of i.i.d. noise) has the form Rc =

No

Pk sk s∗k + σn2 I.

(5.84)

k=1

Here σn2 represents the independent and identically distributed common noise variance at the array input. Thus,

CNR =

k Pk σn2

(5.85)

deﬁnes the clutter power to noise power ratio (CNR). To compute the angle-Doppler power distribution, we can employ a two-Dimensional (2D) beamformer with w B = s(θ, ωd ).

(5.86)

Thus

2 PB (θ, ωd ) = E w∗B x = s∗ (θ, ωd )Rc s(θ, ωd )

(5.87)

represents the angle-Doppler power distribution. Figure 5.12 shows the beamformer output using the ideal clutter covariance matrix Rc in (5.87). The concentration of clutter power along the angle-Doppler diagonal ridge in Figure 5.12 is not surprising since from (5.75), the Doppler is proportional to cos θ. The slope of the ridge is determined by the clutter foldover factor β appearing in (5.77). Figure 5.13 shows the clutter power distribution for β < 1, β = 1, and β > 1. Notice that

Chapter 5:

Space-Time Adaptive Processing PB(q, wd) in dB

PB(q, wd) in dB

1

0

0.8 −5

0.6 0.4 wd

0

−15

−0.2

−20

−0.4 −0.6

−25

−0.8 −1−1

wd

−10

0.2

−0.5

0

−5 −10 −15

−0.2 −0.4

−20

−0.6 −0.8 −1 −1

1

0.5

0

1 0.8 0.6 0.4 0.2 0

−25 −0.5

cos (q )

0 cos (q )

(a) b = 0.5

(b) b = 1

0.5

1

PB(q, wd) in dB 1

0

0.8 −5

0.6 0.4

−10

wd

0.2 0

−15

−0.2 −0.4

−20

−0.6 −0.8

−25

−1 −1

−0.5

0 cos (q )

0.5

1

(c) b = 2

FIGURE 5.13 Clutter power distribution for different β.

β > 1 generates clutter foldover and weak targets present at these clutter ridges are impossible to identify. Generally any target present is buried in clutter, and to identify these targets the dominant clutter needs to be suppressed. We can make use of the maximum output SINR strategy discussed in Section 5.1.2 to determine the desired weight vector in this case. Let H1 and Ho represent the two hypotheses corresponding to the presence or absence of the target. Thus, the data vector under these two hypotheses has the form

*

x=

s(θt , ωdt ) + xc + n,

H1

xc + n,

H0 ,

(5.88)

where xc is as in (5.80). Here θt and ωdt refer to the true target parameters. The corresponding covariance matrix in this case is given by R = E{x x∗ } =

*

Pt s(θt , ωdt )s∗ (θt , ωdt ) + Rc ,

H1

Rc ,

H0 ,

(5.89)

159

Space Based Radar PB(q, wd) in dB 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1

PB(q, wd) in dB 0 −5 −10 −15 −20 −25

−0.5

0 0.5 cos (q )

Target 0

1 0.8 0.6 0.4 0.2 wd

wd

160

1

(a) PB(q, wd)

0 −0.2 −0.4 −0.6 −0.8 −1 −1

−20 −40 −60 −80 −100 −120 −0.5

0 cos (q )

0.5

1

(b) PMSNR (q, wd)

FIGURE 5.14 Target in clutter, N = 14, M = 16, CNR = 40 dB, and SNR = 0 dB.

where Rc represents the total clutter plus noise covariance matrix as in (5.84). From (5.54), it follows that the desired space-time adaptive vector that maximizes the output signal to interference plus noise ratio (SINR) is given by wopt = R−1 c s(θt , ωdt ). The 2D adaptive pattern

2

(5.90)

2

P(θ, ωd ) = w∗opt s(θ, ωd ) = s∗ (θt , ωdt )R−1 c s(θ, ωd )

(5.91)

can be used to evaluate the performance of the adaptive weight vector in (5.90) for suppressing the clutter and enhancing target detection. Figure 5.14 corresponds to a 14-element array, 16-pulse conﬁguration with a target buried in clutter and noise with the target located at θt = 90◦ and ωdt = 0.3. The clutter power to noise power ratio (C NR) is 40 dB and SNR = 0 dB. The ideal covariance matrix R corresponds to hypothesis H1 in (5.89). The ideal beamformer output power PB (θ, ωd ) in (5.87) and the P(θ, ωd ) in (5.91) are shown in Figure 5.14 (a) and Figure 5.14 (b) respectively. Notice that the target is undetectable using the beamformer, whereas the optimum weight vector detects the target by suppressing the clutter. Notice that the dominant clutter ridge in PB (θ, ωd ) along the angle-Doppler line is nulled out in P(θ, ωd ) generating a valley thereby exposing the target. Once again, to evaluate the performance of these methods with measured data, the ideal clutter covariance matrix is replaced by its ML estimate; thus K 1 ˆc = 1 R xk x∗k = Y K Y K K K k=1

(5.92)

Chapter 5:

Space-Time Adaptive Processing

Training cells

Range bin of interest

FIGURE 5.15 Range bin of interest and training cells.

where Y K = [ x1 , x2 , . . . , x K ].

(5.93)

Here K training cells around the range bin of interest5 (see Figure 5.15) are used to estimate Rc , under the assumption that they all correspond to the same ideal covariance matrix. The sample clutter power beamformer is given by ˆ c s(θ, ωd ) Pˆ B (θ, ωd ) = s∗ (θ, ωd ) R

(5.94)

and this is illustrated in Figure 5.16 along with the ideal case.

−5 −10 −15 −20 −25 −0.5

−30 0 cos (q )

(a) PB(q, wd)

0.5

1

1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1

0 −5 −10 −15

wd

0

wd

1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1

−20 −25 −30 −0.5

0 cos (q )

0.5

1

^ (b) PB(q, wd)

FIGURE 5.16 Ideal vs. Estimated clutter power using 10 data samples.

5 The range bin of interest by hypothesis may contain target, and hence it is not ˆ c in (5.92). used in estimating R

161

162

Space Based Radar

5.3.1 Minimum Detectable Velocity (MDV) To study the clutter suppression capabilities of an algorithm, one may assume a hypothetical target at θ with unknown Doppler ωd , which gives the output SINR to be [2] SINR =

|w∗ s(θ, ωd )|2 ≤ s∗ (θ, ωd )R−1 c s(θ, ωd ) = SINRmax . w∗ Rc w

(5.95)

Clearly, any weight vector other than wopt = R−1 c s(θ, ωd )

(5.96)

in (5.95) performs inferior to (5.96) resulting in output SINR that is lower compared to the matched ﬁlter output SINRmax = s∗ (θ, ωd )R−1 c s(θ, ωd ).

(5.97)

The array parameters used in this chapter are shown in Table 5.1. Observe that some of the parameters selected here agree with those of the Mountain top program [3]. Figure 5.17 shows the ideal matched ﬁlter output in (5.97) as a function of the target velocity for θ = 90◦ . From there, for unambiguous detection up to 5 dB loss, the target velocity has to exceed 7 m/s (25.2 km/h) and this gives the minimum detectable velocity (MDV) bound for a side-looking array conﬁguration.

5.3.2 Sample Matrix Inversion (SMI) To implement the optimal weight vector as in (5.96) using the training (secondary) data samples, the inverse of the sample matrix estimate ˆ −1 R c is required. Obviously, for this estimate to be meaningful, the number of samples K ≥ MN, which can violate the underlying stationarity assumption in (5.92).

Number of sensors

(N)

14

Number of pulses

(M)

16

Platform velocity (m/s)

(Vp )

100

Operating frequency (MHz)

( fc ) ˜ (d)

435 0.33

(d)

0.957

Interelement spacing (m) Normalized interelement spacing with respect to half-wavelength Pulse repetition frequency (Hz) TABLE 5.1 Array parameters

(PRF)

625

Chapter 5:

Space-Time Adaptive Processing

0 −10

SINR max

−20 −30 −40 MF

−50

−100

7 0 Velocity (m/s)

−50

50

100

FIGURE 5.17 MDV performance using ideal matched ﬁlter (MF) output.

ˆ c is invertible, the SMI can be employed using (5.92). Using When R this in (5.90) we obtain ˆ −1 w SMI = R c s(θt , ωdt )

(5.98)

and the corresponding estimate for (5.91) is given by

2 2 ˆ −1 PSMI (θ, ωd ) = w∗SMI s(θ, ωd ) = s∗ (θt , ωdt ) R c s(θ, ωd ) .

(5.99)

Figure 5.18 shows this sample estimator output in the angleDoppler domain. The array parameters are given in Table 5.1. The target is at range bin 200 with arrival angle of 90◦ and Doppler = 0.3. 1

0 −10 −20

0

−30 −40

−0.5

−1 −1

−50 −0.5

0 cos (q )

0.5

(a) Top view

1

0 Power (dB)

Doppler

0.5

−5 −10 −15 −20 −25 1 Doppler

0 −1 −1

−0.5

(b) Side view

FIGURE 5.18 Angle-Doppler performance for SMI using (5.99).

0 cos (q )

0.5

1

163

Space Based Radar Ideal

SINR vs Velocity

−10

SMI

−20 SINR (dB)

164

−30 −40 −50 −60 −100

MF SMI −50

0 V (m/s)

50

100

FIGURE 5.19 MDV performance for SMI.

The CNR is 40 dB and target-to-noise ratio is 0 dB. Here, 400 samples are used to estimate the clutter covariance matrix. The MDV perforˆ −1 mance of this algorithm obtained by replacing R−1 c by Rc in (5.97) is shown in Figure 5.19. Figure 5.20 shows the 2D adaptive pattern in (5.99) in the rangeDoppler domain for a ﬁxed arrival angle θ. The arrival angle is chosen to be the target arrival angle θt . To compute the adaptive pattern for range bin i, the corresponding clutter covariance matrix Rc is estimated using the identify K ˆi = 1 xk x∗k R K

(5.100)

k=i

and the desired weight vector equals ˆ i−1 s(θt , ωdt ). wi = R

(5.101)

This gives the adaptive pattern to be

2

P(i, ωd ) = wi∗ s(θ, ωd ) .

(5.102)

Chapter 5:

Space-Time Adaptive Processing

Power (dB)

0 −5 −10 −15 −20 −25

1 210

0 Doppler

−1 180

190

220

200 Range

FIGURE 5.20 Range-Doppler for SMI.

5.3.3 Sample Matrix with Diagonal Loading (SMIDL) The number of data samples K that can be used in (5.92) or (5.100) may be small because of non-stationarity issues. Heterogeneous terrain, large discretes such as lakes or paved roads, multiple targets can also contribute to non-stationarity. Thus, if K < MN, then the sample ˆ c is singular and the SMI method cannot be implemented matrix R directly. One way to circumvent this issue is to diagonally load the sample covariance matrix by an appropriate amount ε as ˆε = R ˆ c + εI, ε > 0 R

(5.103)

ˆ ε being invertible, it can be used in ˆ ε is invertible. Notice that R so that R ˆ (5.98) and (5.99) in place of Rc . Figure 5.21 shows the adaptive pattern for SMIDL using a diagonal loading factor of ε = 10−7 .

5.4 Eigen-Structure Based STAP The clutter covariance matrix corresponding to a uniformly spaced array that receives signals in a constant PRF (pulse repetition frequency) mode exhibits additional structural properties, provided the clutter data is uncorrelated. To see this, we start with the clutter covariance matrix in (5.81) that corresponds to an uncorrelated clutter scene. Thus Rc =

No k=1

Pk sk s∗k

(5.104)

165

Space Based Radar

0 Power (dB)

166

−5 −10 −15 −20 −25

1

220 0 Doppler

210

−1 180

190

200 Range

FIGURE 5.21 Adaptive range-Doppler pattern for SMIDL using 80 samples.

where No represents the number of clutter bins used. Note that the number of data samples No can be very large compared to MN, the size of the matrix. As a result, it is tempting to conclude that Rc is always a full rank matrix. However, this is often not the case because of the special structure of the space-time steering vector in (5.79) corresponding to the clutter patches [4]. From (5.79), if β is an integer less than N, then clearly, some of the entries in s(θ, ωd ) repeat. For example, N = 3, M = 2, and β = 2 in (5.79) gives

%

. sk = 1, e − jφk , e − j2φk .. e − j2φk , e − j3φk , e − j4φk .

&T (5.105)

which has two identical rows, and hence sk s∗k has two identical rows No Pk sk s∗k and two identical columns. As a result, the 6 × 6 matrix k=1 also has two identical rows and two identical columns irrespective of the actual entries and the large number of terms used in the summation, thus making it rank deﬁcient.

5.4.1 Brennan’s Rule Returning to the general case in (5.104), the space-time steering vector is given by (5.79). Clearly the ﬁrst segment in (5.79) has N independent entries, and thereafter each segment generates only β new entries provided β is an integer less than N. To see this, notice that the integer exponent in the last entry of the ﬁrst segment corresponding to the

Chapter 5:

Space-Time Adaptive Processing

ﬁrst pulse is N − 1, whereas the second segment has exponent entries β, β +1, . . . , N−1, ( N−β) +β, . . . , ( N−2) +β, ( N−1) +β.

(5.106)

Clearly, only the last β entries in (5.106) are new and there are M− 1 such segments which make the total number of distinct entries in any sk to be ( M − 1)β + N. Thus each sk s∗k and hence the ideal covariance matrix Rc can have at most ( M− 1)β + N independent rows/columns. Thus rank(Rc ) = min{( M − 1)β + N, No , MN}.

(5.107)

Usually, the number of scatters No is much more than MN, and hence (5.107) reduces to rank(Rc ) = min{( M − 1)β + N, MN} = ( M − 1)β + N,

if β ≤ N − 1. (5.108) The signiﬁcance of the clutter foldover factor β is clear. In particular, the rank for Rc can be as low as M + N − 1 when β = 1. We will refer to [5] rc = ( M − 1)β + N

(5.109)

as the Brennan’s rule for computing the rank of a clutter covariance matrix when the clutter foldover factor is an integer. Thus, the rank of a clutter covariance matrix is approximately given by (5.109), where β in (5.77) represents the number of half-interelement spacing traveled by the platform within one pulse interval. Brennan’s rule is an effective indicator of both the severity of the clutter scene, and the number of degrees of freedom required to produce an effective clutter cancellation. The eigen decomposition of Rc gives further insight into this phenomenon.

5.4.2 Eigencanceler Methods From (5.108) and (5.109), we have Rc in (5.104) is nonnegative-deﬁnite hermitian matrix with rank rc . Hence its eigen decomposition gives Rc = UU∗

(5.110)

U = [u1 , u2 , . . . , urc , . . . , u MN ]

(5.111)

where

167

168

Space Based Radar represents a unitary matrix and

      =    

µ1

 µ2

..

   .   µrc ,  0  ..  .  0

(5.112)

where µi > 0, i = 1, 2, · · · , rc represent the nonzero eigenvalues of Rc [6]. Using (5.111)–(5.112) in (5.110), we get Rc =

rc

µi ui ui∗ .

(5.113)

i=1

From (5.113), it is clear that the clutter covariance matrix has only rc degrees of freedom. If we combine the noise covariance matrix σn2 I along with the clutter covariance matrix to deﬁne the total clutter plus noise covariance matrix as in (5.84), we have

Rc = UU∗ + σn2 I = U + σn2 I U∗ =

rc

λi ui ui∗ +

MN

σn2 ui ui∗

(5.114)

i=rc +1

i=1

where

λi = µi + σn2 , i = 1, 2, · · · rc .

(5.115)

The ideal eigenvalues when plotted exhibits the “brick wall” shape shown in Figure 5.22. In term of (5.114), the maximum output SINR in (5.97) can be rewritten as SINRmax =

rc |u∗ s(θ, ωd )|2 i

i=1

µi + σn2

+

MN |ui∗ s(θ, ωd )|2 . σn2

(5.116)

i=rc +1

We can use the above eigen-decomposition of Rc to explain the sharp null found in the MDV performance of the ideal matched ﬁlter output in Figure 5.17.

Chapter 5:

Space-Time Adaptive Processing

70 60

l in dB

50 40 30 20 Brick wall

Flat bed

10 0 0

rc

50

100

150

200

250

Index

FIGURE 5.22 Eigenvalues of the ideal covariance matrix.

Sharp Null in MDV Performance The SINR output in Figure 5.17 shows the ideal matched ﬁlter output in (5.97) for a ﬁxed look angle θ = θ1 while spanning the Doppler frequency ωd . In that case s(θ1 , ωd1 ) = s1 is the only steering vector generated in (5.97) that coincides with the clutter scatter set {s1 , s2 , . . . , s No } in (5.104). From (5.109) the rank rc of the clutter covariance matrix Rc is much less than MN and hence from (5.104)–(5.114), the clutter steering vector set {s1 , s2 , . . . , s No } spans the same subspace as {u1 , u2 , . . . , urc }. But the clutter subspace and noise subspace are orthogonal since they correspond to distinct eigenvalues, i.e., {u1 , u2 , . . . , urc } ⊥ {urc +1 , urc +2 , . . . , u MN }.

(5.117)

As a result, the clutter steering vector set is also orthogonal to the noise subspace, i.e., {s1 , s2 , . . . , s No } ⊥ {urc +1 , urc +2 , . . . , u MN }, and hence s∗1 R−1 c s1

rc ∗ 2 ui s1 = ≤ s∗ (θ1 , ωd )R−1 c s(θ1 , ωd ). µi + σ 2

(5.118)

(5.119)

i=1

In (5.119), the inequality follows for all other steering vectors s(θ1 , ωd ) generated by spanning the Doppler frequencies as in Figure 5.17, since these steering vectors are not orthogonal to the

169

170

Space Based Radar noise subspace eigenvectors as in (5.118). Consequently, all other steering vectors s(θ1, ωd ) = s1 contribute to the noise subspace terms in (5.114) as well as in (5.116) and generates a higher SINRmax output. Hence, a unique null appears at ωd = ωd1 in Figure 5.17 and it corresponds to the clutter ridge.

Eigencanceler [4], [6] Equation (5.114) can be used to derive an alternate expression for the optimum weight vector in (5.90). From (5.114), we obtain

rc MN 1 1 ui ui∗ + ui ui∗ λi σn2 i=1 i=rc +1 MN / r c 2 σ ui ui∗ − 1 − n ui ui∗ λi i=rc +1 i=1 / rc σn2 ∗ I− 1− ui ui . (5.120) λi

2 R−1 c = U + σn I

=

1 σn2

=

1 σn2

−1

U∗ =

i=1

Thus w opt in (5.90) simpliﬁes to wopt = R−1 c s(θt , ωdt ) = s(θt , ωdt ) −

rc

1−

i=1

= s(θt , ωdt ) −

rc

σn2 λi

(ui∗ s(θt , ωdt ))ui

b i ui

(5.121)

i=1

where

bi =

σ2 1− n λi

(ui∗ s(θt , ωdt )).

(5.122)

Equation (5.121) shows that the optimum weight vector is the quiescent steering vector st = s(θt , ωdt ) corresponding to the target that has been freed from the clutter subspace components. The clutter subspace components of the quiescent steering vector is subtracted out from the quiescent steering vector corresponding to the target to obtain the optimum weight vector. Direct implementation of (5.121) represents the eigencanceler (EC) approach. In the ideal case, there is no difference in performance when either formula (5.90) or (5.121) is used for clutter suppression. However, this is not the case when data samples are used to estimate the total clutter plus noise covariance matrix Rc . If λˆ i , uˆ i , σˆ n2 , and

Chapter 5:

Space-Time Adaptive Processing

80

l in dB

60

40

20

0

−20

Ideal

Estimated 100 150 Index

50

0

200

250

FIGURE 5.23 Estimated eigenvalue plot.

rˆc represent the estimate of λi , ui , σn2 , and rc in (5.121) respectively, then wec = st −

rˆc i=1

σˆ 2 1− n λˆ i

(ui∗ st ) uˆ i = st −

rˆc

bˆ i uˆ i

(5.123)

i=1

represents the EC weight vector. Figure 5.23 shows the estimated eigenvalue plot using 400 data samples. Notice that the brick wall/ﬂatbed ideal nature in Figure 5.22 is no longer present here. Figure 5.24 shows the EC adaptive pattern given by

2

Pec (θ, ωd ) = w∗ec s(θ, ωd ) .

(5.124)

and the 2D adaptive pattern in the range-Doppler domain for the target arrival angle is shown in Figure 5.25. When the numbers of data sample K in (5.92) for estimating Rc is less than MN, as we have seen, diagonal loading can be used as in (5.103) with interesting results. The limiting case of this method as ε → ∞ deserves special attention.

5.4.3 Hung-Turner Projection (HTP) From (5.103), the optimum weight vector in the diagonal loaded case equals wε = R−1 ε s(θt , ωdt )

(5.125)

171

Space Based Radar 0

1

−20 −30

0

−40

−0.5

Power (dB)

−10 0.5 Doppler

−1 −1

−0.5

0 cos (q )

0.5

1

0 −5 −10 −15 −20 −25 1

−50

0 Doppler

−60

(a) Top view

0 −0.5 −1 −1 cos (q )

0.5

1

(b) Side view

FIGURE 5.24 Eigencanceler (EC) angle-Doppler pattern with 80 samples.

where Rε =

1 Y K Y∗K + εI. K

(5.126)

To study the limiting behavior of (5.125), it is beset to invert Rε in (5.126) using the matrix inversion identity (see (1.129))

−1

(P−1 + MQ−1 M∗ ) −1 = P − PM M∗ PM + Q

M∗ P.

(5.127)

This gives R−1 ε =

−1 ∗ 1 I − Y K Y∗K Y K + εK I YK ε

(5.128)

0 Power (dB)

172

−5 −10 −15 −20 −25

1 0 Doppler

−1 180

190

200

210

220

Range

FIGURE 5.25 Eigencanceler (EC) range-Doppler pattern with 80 samples.

Chapter 5:

Space-Time Adaptive Processing 0

1

−10 −20

Power (dB)

Doppler

0.5

−30

0

−40 −0.5 −1 −1

0 cos (q )

0.5

1

−10 −15 −20 −25

−50 −0.5

0 −5

1 0 Doppler

−60

−1 −1

−0.5

0.5 0 cos (q )

1

(b) Side view

(a) Top view

FIGURE 5.26 HTP angle-Doppler pattern with 80 samples.

so that ∗ −1 ∗ whtp = lim εR−1 ε st = {I − Y K (Y K Y K + εK I) Y K }st ε→∞

= st − Y K (Y∗K Y K ) −1 Y∗K st .

(5.129)

From (5.123) and (5.129), the projection operator φ K = I − Y K (Y∗K Y K + εK I) −1 Y∗K

(5.130)

approximates the EC approach, without an explicit eigen-decompoˆ c . From (5.129), the optimum weight vector is obtained by sition of R subtracting out K “clutter subspace” components from the quiescent steering vector st . Clearly, for this method to be effective, K rc . If K > rc , then additional K − rc eigen-subspace components are subtracted out in (5.129). Hence, using a large number of samples in (5.129) can cause degradation in performance and the method is suitable when the number of samples available are small due to stationarity and logistic issues. Figures 5.26–5.27 show the estimated angle-Doppler pattern using

2

Phtp (θ, ωd ) = w∗htp s(θ, ωd ) .

(5.131)

and the adaptive pattern in the range-Doppler domain.

5.5 Subaperture Smoothing Methods Subaperture smoothing methods refer to subdividing the outputs of a large array into (overlapping or non-overlapping) smaller subarrays, and averaging their outputs either in the data domain or in the covariance domain in some appropriate manner.

173

Space Based Radar

0 Power (dB)

174

−5 −10 −15 −20 −25

1 0 Doppler

−1 180

190

200

210

220

Range

FIGURE 5.27 HTP range-Doppler pattern with 80 samples.

In traditional array processing, maximally overlapping subarrays are formed and their output covariance matrices are averaged to generate a smoothed covariance matrix that has improved decorrelation properties compared to the original array. This procedure has direct applications in implementing direction ﬁnding algorithms in coherent sources scenes. In such cases, to obtain performance similar to those in uncorrelated or correlated source scenes, the subarray smoothing procedure can be implemented [7]. To illustrate this, consider an N element uniformly spaced linear array receiving signals from its ﬁeld of view from K sources located along directions θ1 , θ2 , . . ., θ K . Following (5.80), we have x(n) =

K

c k (n)a (θk ) + w(n),

(5.132)

k=1

where a (θk ) refers to the direction vector associated with the kth source as in (5.8), and c k (n) = c k (nT) refers to the kth source signal at t = nT, and w(n) represents the noise vector as in (5.13). Deﬁne c(n) = [c 1 (n), c 2 (n), . . . , c k (n)]T

(5.133)

A = [a (θ1 ), a (θ2 ), . . . , a (θ K )]

(5.134)

and

to represent the sources and the direction vectors of interest. Thus x(n) = Ac(n) + w(n)

(5.135)

Chapter 5:

Space-Time Adaptive Processing

and hence the N × N array output covariance matrix has the form R = E{x(n)x ∗ (n)} = ARs A∗ + Rw

(5.136)

Rs = E{c(n)c ∗ (n)},

(5.137)

Rw = E{w(n)w ∗ (n)}

(5.138)

where

and

represent the K × K source covariance matrix and the N × N noise covariance matrix respectively. If all the K sources are uncorrelated, then Rs is diagonal with the kth diagonal entry representing the kth power level, i.e., Pi = E{|c i (n)|2 } and



P1  0

  0 0

Rs = 

0 P2 0 0

0 0 .. . 0

(5.139)



0 0 

  0  PK

(5.140)

and it represents a full rank matrix. In general, the K sources can be correlated, in which case the nondiagonal entries of Rs are nonzero. The degree of source correlation determines the rank of Rs . One extreme case is represented by (5.140) where all sources are completely uncorrelated with each other. The other extreme case is where all sources are fully correlated as in a multipath scene shown in Figure 5.28. In c2(t) ck (t )

c1(t) ......

qk

q2

q1

...... x1(t )

x2(t )

FIGURE 5.28 Multipath scene.

......

......

...... xi (t )

xN (t)

175

176

Space Based Radar this case c k (n) = αk c 1 (n)

(5.141)

c(n) = [α1 , α2 , . . . , α K ]T c 1 (n) = αc 1 (n)

(5.142)

and hence

which gives the source covariance matrix in (5.137) to be Rs = P1 α α ∗ .

(5.143)

In this extreme case, Rs has rank one. In general6 , 1 ≤ rank(Rs ) ≤ K

(5.144)

and a variety of source correlation scenes can be expected in practice. Interestingly, the adaptive processor in (5.54) doesn’t perform well either. Recall that the optimum adaptive processor in (5.54) nulls out the undesired interferences by generating nulls along their arrival angles (see (5.60) and Figure 5.8). In a coherent interference scene with uncorrelated noise of equal variance σn2 , the interference plus noise covariance matrix has the form R = ARs A∗ + σn2 I = P1 s s ∗ + σn2 I

(5.145)

where from (5.136) and (5.143) s = Aα =

K

αk a (θk ).

(5.146)

k=1

Hence the optimum processor R−1 a (θ ) is only able to generate a null along the vector s. But s in (5.145) and (5.146) is a linear combination of all steering vectors, and it doesn’t correspond to an actual steering vector. Consequently, the adaptive processor is unable to generate nulls along the true coherent arrival directions θ1 , θ2 , . . . , θ K . This is shown in Figure 5.29 for a three-source scenario that is the same as in Figure 5.8, where the two sources along 55◦ and 105◦ are coherent and the third one along θ2 = 75◦ is uncorrelated with the rest. Notice that the processor nulls out the uncorrelated interference along 75◦ , but the two coherent interferences are not nulled out. In a fully coherent K sources scene such as (5.143), the signal portion of the array output covariance matrix R is also reduced to rank one, thereby making it ineffective in determining the associated directions of arrival. In the case of a uniform array, the rank structure of R in 6 The

rank structure of Rs forces the same structure on R as well.

Chapter 5:

Space-Time Adaptive Processing

0

SINR in dB

−5 −10 −15 −20 −25

0

50

100 Angle (deg)

150

FIGURE 5.29 SINR output for two coherent sources with one uncorrelated source.

(5.136) can be improved in a fully coherent source scene by employing subarray smoothing techniques [7].

5.5.1 Subarray Smoothing The uniform linear array with N sensors is subdivided into overlapping subarrays with Ns sensors, with sensors {1, 2, . . . , Ns } forming the ﬁrst subarray, sensors {2, 3, . . . , Ns + 1} forming the second subarray, etc., up to the last subarray formed by sensors {N − Ns + 1, N − Ns + 2, . . . , N} (see Figure 5.30). In (5.135), the ﬁrst subarray output has the form (suppressing the time index) x 1 = As c + n1 , Lth subarray

1st subarray

1

(5.147)

2nd subarray

2

Ns

FIGURE 5.30 Forward subarrays.

Ns + 1

N−1

N

177

178

Space Based Radar where As is Ns × K (see (5.134)). In the second subarray, the direction vectors are scaled by γk = e − jπ cos θk , k = 1, 2, . . . , K respectively so that with

 

γ1

B= 



γ2

..

 , 

.

(5.148)

γK we get the second subarray output vector to be x 2 = As Bc + n2

(5.149)

and in general the ith subarray output vector is given by xi = As Bi−1 c + ni , i = 1, 2, . . . , L ,

(5.150)

where ni represents the noise input vector for the ith subarray. With uncorrelated noise of equal variance σ 2 , this gives the corresponding array output covariance matrices to be ∗

Ri = E{xi xi∗ } = As Bi−1 Rs B(i−1) A∗s + σ 2 I, i = 1, 2, . . . , L .

(5.151)

Although each Ri is of rank one in the fully coherent case, that is not the case for their average R f given by L 1 Ri = As DA∗s + σ 2 I L

Rf =

(5.152)

i=1

where D=

L 1 i−1 ∗ B Rs B(i−1) . L

(5.153)

i=1

In the fully coherent case, D simpliﬁes to D=

L 1 i−1 ∗ (i−1)∗ 1 B αα B = E E∗ L L

(5.154)

i=1

where E = [α, Bα, B2 α, . . . , B L−1 α]

 

= 

α1

 α2

..

. αK



γ1L−1



γ1

···

γ2 .. .

··· .. .

γ2L−1    ..  = E1 V. . 

1 γK

···

γ KL−1

1

 1    .. .

(5.155)

Chapter 5:

Space-Time Adaptive Processing

0

SINR in dB

−5 −10 −15 −20 −25 −30

0

50

100 Angle (deg)

150

FIGURE 5.31 Optimum SINR using subarray smoothing scheme. Two coherent interferences along 55◦ and 105◦ , and one uncorrelated signal along 75◦ . Spatial smoothing is used to decorrelate the coherent sources.

Clearly, E and D are of full rank and hence the forward smoothed covariance matrix R f in (5.152) is also full rank, provided L ≥ K . From (5.152), R f maintains the same structure as any Ri ; however, its rank has increased from one to K by the above smoothing operation. Interestingly, we can use this procedure to null out coherent interference signals by generating the optimum adaptive processor in (5.54) using the smoothed covariance matrix R f rather than using R. Figure 5.31 shows the adaptive processor output corresponding to the coherent scheme in Figure 5.29 where subarray smoothing has been employed on the original covariance matrix using two subarrays (L = 2). Clearly, the two coherent interference signals are resolved and consequently the adaptive processor is able to null them out. Starting from the far end of the array at sensor N, the reversed and complex conjugated data has the same structure as well. Toward this, deﬁne L backward subarrays by grouping elements {N, N−1, . . . , N− Ns + 1} to form the ﬁrst subarray, {N − 1, N − 2, . . . , N − Ns } to form the second subarray, etc., as shown in Figure 5.32. Let yl denote the complex conjugate of the lth backward subarray element outputs, i.e.,

0

∗ ∗ ∗ yl = xN−l+1 , xN−l , . . . , xL−l+1

1T

= As (B N − l c) + n˜ l

= As Bl−1 (B N−1 c) + n˜ l = As Bl−1 c˜ + n˜ l ,

l = 1, 2, . . . , L ,

(5.156)

179

180

Space Based Radar

1

N − Ns

2

N − Ns + 1

N−1

2nd backward subarray Lth backward subarray

N

1st backward subarray

FIGURE 5.32 Backward subarray.

where we have used (5.150). Here the top bar refers to ordinary complex conjugation operation. This gives the corresponding backward covariance matrix to be ∗

E{yl yl∗ } = As Bl−1 Rs˜ Bl−1 A∗s + σ 2 I,

l = 1, 2, . . . , L ,

(5.157)

where the effective source covariance matrix equals ∗ ∗ ¯ s B( N−1) . Rs˜ = B( N−1) E{˜c c˜ ∗ }B( N−1) = B( N−1) R

(5.158)

In the case of a coherent scene, ∗

Rs˜ = B( N−1) α˜ α T B N−1 = β β ∗

(5.159)

where ∗

β = B( N−1) α,

(5.160)

i.e., βk = γk−( N−1) αk = 0,

k = 1, 2, . . . , K

(5.161)

and the covariance matrices in (5.157) are also of rank one. From (5.157), the backward-smoothed covariance matrix is given by L 1 Rb = E{yi yi∗ } = As F A∗s + σ 2 I L

(5.162)

l=1

where F=

L 1 l−1 ∗ B Rs˜ B(l−1) . L i=1

(5.163)

Chapter 5:

Space-Time Adaptive Processing Lth forward subarray

1st forward subarray 2nd forward subarray

1

N − Ns N − Ns + 1

Ns Ns + 1

2

N−1

N

st 2nd backward subarray 1 backward subarray

Lth backward subarray

FIGURE 5.33 Forward/backward subarray.

In the fully coherent case using (5.159), we obtain F = G G∗ ,

(5.164)

with G = [β, Bβ, B2 β, · · · B L−1 β]

  = 

β1

β2

..

. βK

  1  1    .. .

γ1L−1



γ1

···

γ2 .. .

··· .. .

γ2L−1    ..  = E2 V . 

1 γK

···

γ KL−1

(5.165)

as in (5.155), and hence F and Rb are full rank provided L ≥ K . Thus in a K source scene, to fully decorrelate the sources using either the forward smoothing scheme or the backward smoothing scheme, K subarrays are required which gives the total number of sensors to be N = Ns + L ≥ Ns + K ⇒ L ≥ K .

(5.166)

Interestingly, the number of additional sensors required can be reduced by combining the forward and backward smoothing methods. Thus deﬁne the forward/backward smoothed covariance (see Figure 5.33) to be [7] ˜ = R f + Rb . R 2

(5.167)

181

182

Space Based Radar Using (5.152)–(5.164), we obtain ˜ = 1 As (D + F)A∗s + σ 2 I = As PA∗s + σ 2 I R 2

(5.168)

where 1 1 (D + F) = (E E∗ + G G∗ ) 2 2L % ∗& 1 1 E = [E G] Q Q∗ . = G∗ 2L 2L

P=

(5.169)

But from (5.155) and (5.165) E2 V ] = E1 [ V

Q = [ E G ] = [ E1 V

εV ] = E1 H

(5.170)

where ε is a diagonal matrix with ε = E−1 1 E2 ,

εk = βk /αk = 0.

(5.171)

Clearly, H is of size K × 2L and if all the K sources in H are linearly ˜ are of rank K provided independent, then H and hence Q, P, and R 2L ≥ K

or

L ≥ K /2.

(5.172)

In this case, compared to (5.166) the number of required subarrays has been reduced from K to K /2 and hence the total number of sensors satisfy N = Ns + L ≥ Ns + K /2,

(5.173)

thus resulting in a saving in the number of sensors. Finally, referring back to (5.150)–(5.151), if all the sources are uncorrelated, then each forward subarray covariance matrix equals ∗

Ri = As Bi−1 Rs B(i−1) A∗s + σ 2 I = ARs A∗ + σ 2 I = R

(5.174)

and same is true for backward subarrays as well. In this context, we can visualize these subarray data vectors to represent a sample data vector corresponding to the unknown covariance matrix R. Thus starting with a single data vector of size N × 1, 2L data vectors (N forward, N backward) of reduced size Ns × 1 (Ns = N − L + 1) can be generated. This scheme is particularly attractive in sparse sample situations such as in a nonstationary clutter scene where only a few measured

Chapter 5:

Space-Time Adaptive Processing

data vectors maintain stationarity. This is explored next for STAP applications.

5.6 Subaperture Smoothing Methods for STAP In STAP, data is available both in spatial and temporal domains and hence subapertures can be generated in both domains using subarrays and subpulses to simulate additional data vectors.

5.6.1 Subarray Method In this case, Ns elements are grouped together in an overlapping manner to generate L = N − Ns + 1 subarray data vectors in the spatial domain. Let x(ti ) refer to the N element output vector at t = ti , and yl (ti ), l = 1, 2, . . . , L, the subarray vectors of size Ns × 1. Then we have x(ti) = [x1(ti), x2(ti), …, xNs(ti), …, …, xN(ti)]T y1(ti)

(5.175)

…

y2(ti) …

yL(ti)

Thus y l (ti ) = [xl (ti ), xl+1 (ti ), . . . , xl+Ns −1 (ti )]T ,

(5.176)

and

 ul,i

  =  



y l (ti )

  ,  

y l (ti + T) .. .

i = 1, 2, . . . , K ,

y l (ti + ( M − 1)T)

l = 1, 2, . . . , L

(5.177)

represents the MNs × 1 data vector formed by M such pulses; their common covariance matrix can be expressed as Ry =

Pk ss (k)s∗s (k) + σ 2 I

(5.178)

k

where ss (k) = b(ωdk ) ⊗ a Ns (θk )

(5.179)

183

Space Based Radar with a Ns (θk ) representing the top Ns × 1 subvector of the spatial steering vector a (θk ) in (5.8). R y is independent of i and l implying that every data vector in (5.177) has the same covariance matrix. Further, their complex conjugated and transposed (backward) data vector also have the same covariance matrix. Thus, for every sample, 2L = 2( N−Ns +1) new data samples of size Ns M × 1 can be generated in this manner.

5.6.2 Subpulse Method Here, Ms pulses are grouped together in an overlapping manner to form subpulse vectors. Let x(ti ) refer to the M pulses output vector at t = ti and x(ti ), the array vectors of size N × 1. We have y1(i)

x(ti) …

x(ti + T) y2(i)

xi = x(ti + (Ms − 1)T)

…

(5.180)

…

… …

184

yJ(i)

x(ti + (M − 1)T )

Then

 y

 

j,i

= 



x(ti + ( j − 1)T) x(ti + j T) .. .

    , i = 1, 2, . . . , K ,  

j = 1, 2, . . . , J

x(ti + ( j + Ms − 2)T) (5.181) represents the Ms N × 1 data vector from N sensor output. Once again, these subpulse data vectors and their backward forms of size Ms N × 1 have the same covariance matrix when the clutter components are uncorrelated. Thus, every original data vector of size MN×1 generates 2J = 2( M − Ms + 1) reduced data vectors of size Ms N × 1.

5.6.3 Subarry-Subpluse-Method To jointly exploit the spatio-temporal characteristics, notice that each subarray vector in (5.177) contains M pulses and the subpulse operation in (5.181) can be performed on each one of them to generate J = ( M − Ms + 1) subpulses. Conversely, each subpulse in (5.181) consists of N sensors and the subarray operation in (5.177) can be performed to generate

Chapter 5:

Space-Time Adaptive Processing

L = N − Ns + 1 subarrays. Either method generates J L = ( N − Ns + 1)( M − Ms + 1) data samples and together with their backward data, we obtain 2J L = 2( N − Ns + 1)( M − Ms + 1) reduced data samples of size Ms Ns × 1 for each original sample of size MN × 1. Thus Ns = N − 1, Ms = M − 1 generates two subarrays and two subpulses with a total of 8 data samples for every original sample vector. To complete the task, we can perform the subpulse operation on the subarray vector in (5.177). We have z(i)l,1

yl(ti) …

yl(ti + T) z(i)l,2

ul,i = yl(ti + (Ms − 1)T)

…

… …

…

(5.182) z(i)l,J

yl(ti + (M − 1)T)

Deﬁne

  

zl,(i)j =  





y l (ti + ( j − 1)T)

  , l = 1, 2, . . . , L ,  

y l (ti + j T) .. .

j = 1, 2, . . . , J ,

y l (ti + ( j + Ms − 2)T) (5.183) where y l is given by (5.176); (5.183) represents Ms Ns × 1 data vectors from the subarrays and subpulses. The common covariance matrix of the data vectors in (5.183) and their backward data form can be shown to be Rs,t =

Pk ss,t (k)s∗s,t (k)

(5.184)

k

where the reduced spatio-temporal vector ss,t (k) = b Ms (ωdk ) ⊗ a Ns (θk ) with b Ms (ωdk ) representing the top Ms × 1 subvector of b(ωdk ).

(5.185)

185

186

Space Based Radar Thus, for every MN ×1 original data, the forward reduced size data vectors are given by

% Zf =

z(i) 1,1

(i) z1,2

···

z(i) 1, J

.. (i) (i) (i) .. z2,1 z2,2 · · · z2, J

.. (i) .. · · · z L , J

& (5.186)

and the corresponding backward data vectors are given by Zb = J o Z¯ f

(5.187)

where J o equals



0

0

0

1



0 0 1 0   . 0 1 0 0 1 0 0 0

Jo = 

(5.188)

This gives





Z = Zf

.. .  .. Zb .

(5.189)

to be the 2L J samples associated with each original data set. Finally, the estimated covariance matrix corresponding to Z is given by ˆ s,t = R

1 Z Z∗ . 2L J

(5.190)

Interestingly, to estimate the subaperture smoothed covariance maˆ s,t , it is not necessary to employ (5.190) when the original data trix R has been spread out both in space and time as in (5.186) that uses (5.177) and (5.181). Instead, these operations can be performed efﬁˆ c in (5.81) and its complex ciently on the sample covariance matrix R conjugate by identifying the appropriate subblocks. This procedure is illustrated in Figure 5.34 for M = 3, N = 2, Ms = 2, and Ns = 2, which gives L = J = 2. Figure 5.35 shows the estimated adaptive pattern of ordinary HTP and HTP with forward/backward subarray subpulse (HTPSASPFB) smoothing method using ten data samples for a 14-element array with 16 pulses with C NR = 40 dB with Ns = 13 and Ms = 15. Injected target is at range bin 200 at θt = 90◦ and ωdt = 0.2. Clearly, ordinary HTP is unable to detect the target with ten samples, whereas HTPSASPFB has in effect 80 samples and it is able to detect the target unambiguously.

Chapter 5:

Space-Time Adaptive Processing

(2 × 2) 2

1 1 Rc =

3

4

(3 × 3)

(3 × 3)

4

3

(3 × 3)

(3 × 3)

2

(3 × 3)

(3 × 3)

=

(3 × 3)

(3 × 3)

(3 × 3)

(3 × 3)

1

2 1

3

2 (2 × 2)

4 4

3

(a) Rc of size 9 × 9 + 1

2

3

4

(2 × 2) (2 × 2) (2 × 2) (2 × 2)

(b) Subaperture Rsp of size 4 × 4

FIGURE 5.34 Subaperture in covariance matrix domain. Region 1 in (a) gets averaged and goes over to region 1 in (b), etc.

0 −5

Power (dB)

Power (dB)

Figure 5.36 shows the estimated adaptive pattern for ordinary EC and EC with forward/backward subarray subpulse (ECSASPFB) smoothing with Ns = 13 and Ms = 15 using ten data samples. In summary, when the clutter components are uncorrelated, subaperture smoothing methods simulate a large number of samples at the expense of a reduction in the array aperture.

−10 −15 −20

0 −5 −10 −15 −20

1

1 0 Doppler

−1 180

(a) HTP

190

210 200 Range

220

0 Doppler

−1 180

190

210 200 Range

220

(b) HTPSASPFB

FIGURE 5.35 Adaptive pattern for HTP and HTP with forward/backward subarray subpulse (HTPSASPFB) smoothing.

187

Space Based Radar

0 −5

Power (dB)

Power (dB)

188

−10 −15 −20

0 −5 −10 −15 −20

1 0 Doppler

200 190 −1 180 Range

210

1

220

0 Doppler

(a) EC

−1 180

210 200 Range

190

220

(b) ECSASPFB

FIGURE 5.36 Adaptive pattern for EC and EC with forward/backward subarray subpulse (ECSASPFB) smoothing.

5.7 Array Tapering and Covariance Matrix Tapering Traditionally, array tapering is used to decrease the sidelobes by weighting the array outputs by individual tapering coefﬁcients as shown in Figure 5.37 (b). Thus if α = [α1 , α2 , . . . , α N ]T

(5.191)

represents a set of tapering coefﬁcients to be applied to the array output vector x(ti ), we have y(ti ) = [α1 x1 (ti ), α2 x2 (ti ), . . . , α N xN (ti )]T (5.192) = α ◦ x(ti ),

x1(t )

...... ......

x2(t )

a1

a2

y1(t )

y2(t )

......

xN (t ) aN

...... x1(t )

x2(t )

...

xN (t )

(a) Traditional array

FIGURE 5.37 Tapering an array.

(b) Tapered array

yN (t )

Chapter 5:

Space-Time Adaptive Processing

where ◦ represents the Schur product (element-wise multiplication) as in (1.61)–(1.62). Similarly, if β = [β1 , β2 , . . . , β M ]T

(5.193)

represents the temporal tapering to be applied to the various pulse outputs x(ti ), x(ti − T), · · · x(ti − ( M − 1)T), we have



β1 x(ti )

 







x(ti )

   β2 x(ti − T) x(ti − T)    = β ⊗α ◦  .. ..        . . β M x(ti − ( M − 1)T) x(ti − ( M − 1)T)

yi =  

= c ◦ xi ,

(5.194)

where xi is the original data vector and c=β ⊗α

(5.195)

represents the spatio-temporal tapering vector. Clearly, the tapered covariance matrix R y is given by Ry = E{yi yi∗ } = E{xi xi∗ } ◦ c c∗ = Rx ◦ c c∗ ,

(5.196)

and similarly, the corresponding sample estimates are given by K K 1 ∗ ˆy = 1 ˆ x ◦ c c∗ R yi yi∗ = xi xi ◦ c c∗ = R K K i=1

(5.197)

i=1

where ˆ x = 1 X X∗ R K

(5.198)

X = [ x1 x2 . . . x K ]

(5.199)

and

represents the original data vector matrix. It easily follows that if we apply different tapering vector c1 , c2 , . . . , c p to the data set X, we have the enhanced data set Z = [ Z1 Z2 · · · Z p ]

(5.200)

where Zi = [ci ◦ x1 , ci ◦ x2 , . . . , ci ◦ x K ] ,

i = 1, 2, . . . , p.

(5.201)

Notice that X and Zi are of size MN× K , whereas the enhanced data set if of size MN × pK , implying that the number of data samples has increased by a factor of p. However, unlike X, the samples in Z are not

189

190

Space Based Radar independent, and hence statistically the improvement will not be by a factor of p. The estimated covariance matrix corresponding to Z in (5.200) is given by p p ˆ z = 1 ZZ∗ = 1 ˆx ◦ 1 ˆx ◦T R Zi Zi∗ = R ci ci∗ = R pK pK p i=1

(5.202)

i=1

where T=

p 1 c j c∗j ≥ 0 p

(5.203)

j=1

represents the equivalent covariance matrix tapering (CMT). Observe that T is a non-negative Hermitian matrix of rank p (provided all the ˆ z corresponds tapering vectors are linearly independent) and since R ˆ ˆz to pK effective number of samples, compared to Rx , the rank of R must have improved. This also follows from the matrix rank identity in (1.66) where equality is possible. ˆ z in (5.202) can In particular, by adjusting p so that pK > MN, R be made to be full rank by a judicious choice of the tapering vector ˆ z is a better compensated estimate compared to R ˆ x in c1 , c2 , . . . , c p . R terms of their condition numbers. This also follows from the eigenvalue spread condition (see also (1.70)) λmin (A)λmin (B) < λ (A ◦ B) ≤ λmax (A)λmax (B),

(5.204)

where A and B are nonnegative-deﬁnite matrices. Conversely, any nonnegative-deﬁnite covariance matrix tapering T ˆ x can be equivalently represented as an effective increment applied to R in the available data samples. To see this, since ˆ x ◦ T, ˆz = R R

(5.205)

and if T represents a nonnegative-deﬁnite tapering matrix, then its eigenvalue decomposition gives T=

p

λk ek e∗k =

k=1

where ck =

p 1 ∗ ck ck p

(5.206)

k=1

pλk ek .

(5.207)

ˆ z in (5.205) corresponds It now follows from (5.200) and (5.203) that R to K p samples as in (5.200). Conceptually, to incorporate a given non-negative matrix T in (5.205), one may equivalently construct the new data set Z as in (5.200) and (5.201) with ci ’s obtained as in (5.206)

Chapter 5:

Space-Time Adaptive Processing

and (5.207). However, this approach will not be computationally efﬁcient. Other methods such as forward/backward smoothing, subarraysubpulse methods can be applied to this data set to further enhance the data set. Since subarray-subpulse smoothing can be efﬁciently ˆ x , tapering together carried out using diagonal block matrices of R with subaperture smoothing can be efﬁciently incorporated in the ˆ z. covariance domain using R To illustrate this, consider subaperture smoothing applied to data X in (5.199) by employing two sets of sub-data sets X1 and X2 , where X=

X1

=

X2

.

(5.208)

In an uncorrelated scene, X1 and X2 have the same covariance matrix and consequently, the extended data set 0 . 1 Xs = X1 ..X2 (5.209) can be used to estimate the smoothed covariance matrix ˆ x,s = 1 Xs X∗s = 1 X1 X∗1 + X2 X∗2 R (5.210) 2K 2K ˆ1 that has twice the number of samples compared to X. Further, if R ˆ 2 represent the corresponding upper and lower blocks of R ˆ x in and R (5.198), i.e., ^ = R x

^ R 1

=

^ R2

(5.211)

then we also have ˆ ˆ ˆ x,s = R1 + R2 . R (5.212) 2 Equation (5.212) turns out to be useful to analyze the joint implementation of CMT and subaperture smoothing methods. To see this, observe that for a given nonnegative-deﬁnite T as in (5.203), the CMT representation in (5.202) corresponds to the enhanced data set Z in (5.200). Subarray smoothing on Z as in (5.208) generates the data sets Z1 and Z2 given by Z=

Z1

=

Z2

.

(5.213)

The corresponding estimated smoothed covariance matrix equals ˆ z,s = R

1 Z1 Z∗1 + Z2 Z∗2 . 2 pK

(5.214)

191

192

Space Based Radar Notice that Z1 and Z2 have the same structure as in (5.200) and (2) (1) (2) (5.201) with c j replaced by c(1) j and c j , where c j , c j represent the “top” and the “bottom” portions of c j respectively, i.e., c(1)j

cj =

= c(2)j

(5.215)

·

This gives 1 ˆ (i) (i)∗ 1ˆ 1 cj cj = R Zi Zi∗ = Ri ◦ i ◦ Ti , 2 pK 2p 2 p

i = 1, 2, . . .

(5.216)

j=1

ˆ i as in (5.211) and with R

Ti =

p 1 (i) (i)∗ cj cj , p

i = 1, 2, . . .

(5.217)

j=1

On comparing (5.217) and (5.203), it follows that T1 and T2 correspond to the upper-left and lower-right principal block matrices of T, i.e., T=

T1

=

T2

·

(5.218)

Substituting (5.216) into (5.214), we obtain ˆ ˆ ˆ z,s = R1 ◦ T1 + R2 ◦ T2 . R 2

(5.219)

ˆ z,s can From (5.219), the tapered and smoothed covariance matrix R ˆ x and T by applying tapering on the apbe obtained directly from R ˆ x and summing them. Clearly, the procedure propriate subblocks of R can be extended to spatial and temporal domains, and additional proˆ z,s in (5.219) as cessing such as forward-backward can be applied to R well. Observe that (5.219) avoids working directly on the extended data set Z, Z1 , and Z2 ; instead, all additional preprocessing such as ˆ x and CMT, smoothing, etc., are carried out on the original estimate R its partitioned blocks only.

5.7.1 Diagonal Loading as Tapering The diagonal loading scheme (SMIDL) discussed in Section 5.3.3 can be shown to be a special case of data-dependent tapering. To see this, observe that diagonal loading has the form ˆ DL = R ˆ x + εI = R ˆ ◦ Tx R

(5.220)

Chapter 5: where

 r1,1 +ε r1,1

  1   1 Tx =    ..  .  1  ε    =  

r1,1

0 .. . 0

Space-Time Adaptive Processing

1

1

···

1

r2,2 +ε r2,2

1

···

1

r3,3 +ε r3,3

···

1

1 .. .

.. .

1

..

1

.

1





1 r N, N +ε r N, N

        

  1 1 ··· 1  0  1 1 ··· 1 r2,2    +  = Dx + ee∗ ..    1 1 · · · 1 . 0  ε 1 1 ··· 1 0 r N, N 0 ··· 0 ε

(5.221)

where e = [ 1, 1, . . . , 1 ]T .

(5.222)

Eigen-decomposition of Tx gives Tx = Dx + ee∗ =

MN

ck c∗k ,

(5.223)

l=1

since Dx is a data-dependent full rank diagonal matrix, and hence the equivalent tapering vectors ck in (5.223) are also data-dependent. From the earlier discussion, it follows that the effect of diagonal loading as in (5.220) is to generate an equivalent data set as in (5.200) and (5.201) with p = MN, where the tapering vectors are data-dependent (adaptive). The overall effect is to generate a full rank matrix with excellent performance as has been well documented in [5]. Interestingly, when viewed abstractly, other preprocessing schemes such as subarray subpulses, relaxed projection method discussed in Section 5.8 can be viewed as data-dependent covariance matrix taperˆ obtained ing as well. This follows since the ﬁnal covariance matrix R ˆ x can always be expressed as from the original estimate R ˆ =R ˆ x ◦ T, R

(5.224)

where T is an appropriate data-dependent CMT. However, it is impossible to know a-priori such a T before the actual processing. Figure 5.38 shows the comparison between a conventional STAP method (SMIDL) with that of a covariance matrix tapering approach (SMIDLCMT). In this example, a 14-element array with 16 pulses receives data with C NR = 40 dB. Injected target with SNR = 0 dB is

193

Space Based Radar

0

Power (dB)

Power (dB)

194

−5 −10 −15 −20

0 −5 −10 −15 −20

1 0 Doppler

210

−1 180

190

1

220

0 Doppler

200 Range

(a) SMIDL

−1 180

190

210 200 Range

220

(b) SMIDLCMT

FIGURE 5.38 Adaptive pattern responses of (a) SMIDL and (b) SMIDLCMT methods using 30 snapshot data samples.

located at range 220 at an angle of 90◦ and with a Doppler of 0.3. The tapering matrix T used here is given by [8] T = T f¯M ⊗ Tθ¯ N ,

(5.225)

where T f¯M is of the size M × M and given by

T f¯M (i, j) = sinc

|i − j| f π

and Tθ¯ N is of size N × N and given by

Tθ¯ N (i, j) = sinc

|i − j| θ π

(5.226)

.

(5.227)

The parameters f and θ are set at f = θ = 0.01 to generate the tapering matrix in (5.226) and (5.227).

5.8 Convex Projection Techniques Algorithms based on ideal clutter covariance matrix Rc always outperform the same algorithms based on their estimated counterpart ˆ c . The ideal covariance matrix generally has well-deﬁned structural R properties—such as block Toeplitz nature in an uncorrelated scene, positive-deﬁniteness, etc. The corresponding estimate on the other hand need not possess any of these properties; for example, low number of data samples can make the estimate ill-conditioned. The ideal covariance matrix might assume stationary behavior for the underlying data and in reality the data collected may be nonstationary leading ˆ c one approach is to to poor performance. In such situations, given R obtain a better estimate prior to processing that shares some of the

Chapter 5:

Space-Time Adaptive Processing

FIGURE 5.39 Convex set.

f2 f1

f

ˆ c , ﬁnd a matrix B that is properties of Rc . In other words, given R “closer” to it and possesses the desired properties of Rc :

2 2 2 2 2R ˆ c − B 2 ≤ 2R ˆ c − Rc 2 .

(5.228)

Obviously, the above minimization needs to be performed in the absence of Rc . One approach in this context is to examine whether the structural properties satisﬁed by Rc exhibit any convexity constraints. Members of a convex set satisfy the convexity constraint that states if C is a convex set and if f 1 ∈ C and f 2 ∈ C, then (see Figure 5.39) f = α f 1 + (1 − α) f 2 ∈ C,

0 ≤ α ≤ 1.

(5.229)

5.8.1 Convex Sets Convex sets have the remarkable property that for any point x outside C, there exists a “unique” nearest neighbor xo that belongs to C; i.e., for any x, there exist a unique member xo ∈ C such that x − xo ≤ x − x1 , x1 ∈ C.

(5.230)

This unique member xo is known as the projection of x onto C. Thus in Figure 5.40 xo = P x,

(5.231)

where P is the projection operator associated with the closed convex set C. Interestingly, many useful notions in signal processing form convex sets. For example, the set of all signals that are band limited to Bo form a convex set C B , since f 1 (t) and f 2 (t) are band limited to Bo

FIGURE 5.40 “Unique” neighbor property.

C Px x

xo

x1

195

196

Space Based Radar implies α f 1 (t) + (1 − α) f 2 (t) is also band limited to Bo . As a result, given any signal f (t), it is easy to show that its projection onto C B is given by band limiting its Fourier transform F (ω) to Bo (setting F (ω) = 0, |ω| > Bo ). Thus

*

PB f (t) ↔

F (ω), |ω| ≤ Bo . 0, |ω| > Bo

(5.232)

Similarly, the set of all non-negative signals form a convex set C+ . Given any f (t) its projection onto C+ is given by setting its negative part to zero, i.e.,

*

P+ f (t) =

f (t), 0,

f (t) ≥ 0 . f (t) < 0

(5.233)

Another example is the set of all signals with common phase function ϕ(ω). All such signals form a convex set since if f 1 (t) ↔ F1 (ω) = A1 (ω)e jϕ(ω) and f 2 (t) ↔ F2 (ω) = A2 (ω)e jϕ(ω) , then f (t) = α f 1 (t) + (1 − α) f 2 (t) ↔ α F1 (ω) + (1 − α) F2 (ω) = {α A1 (ω) + (1 − α) A2 (ω)}e jϕ(ω) = A(ω)e jϕ(ω)

(5.234)

has the same phase function ϕ(ω).

5.8.2 Toeplitz Property Toeplitz matrices of the same size form convex sets C T since T = αT1 + (1 − α)T2 is also Toeplitz, provided T1 and T2 are Toeplitz matrices. Given an arbitrary matrix R of size n × n, let T denote its nearest Toeplitz neighbor with to , t ± 1 , t ± 2 , t ± 3 , representing the diagonal entries of the Toeplitz matrix. Then T − R2 needs to be minimized. This gives = T − R2 =

n

|to − ri, j |2 +

n

|to − ri,i |2 +

i=1

|t j−i − ri, j |2

i= j

i=1

=

n

n−|k|

|tk − ri,i+k |2 .

(5.235)

∂ =2 (to − ri,i ) = 0 ∂to

(5.236)

i=1

Hence n

i=1

and

∂ =2 (tk − ri,i+k ) = 0, ∂tk n−|k|

i=1

k = ±1, ±2 · · ·

(5.237)

Chapter 5:

Space-Time Adaptive Processing

gives 1 ri,i n n

to =

(5.238)

i=1

and tk =

n−|k| 1 ri,i+k , n − |k|

k = ±1, ±2 · · ·

(5.239)

i=1

and the Toeplitz matrix so formed is the “nearest neighbor” Toeplitz solution to the given matrix R . Thus the projection operator in this case amounts to averaging the respective diagonal entries of R to obtain the corresponding Toeplitz entry. Notice that if R is Hermitian, then so will be its projection T. Suppose R is Hermitian positive-deﬁnite, then its projection onto the Toeplitz domain gives a Toeplitz matrix T that is Hermitian. However, T need not be nonnegative-deﬁnite. It can have negative eigenvalues. An interesting question in this context is given a Hermitian matrix A, how to obtain the closest nonnegative-deﬁnite matrix B?

5.8.3 Positive-Deﬁnite Property Interestingly, the nonnegative-deﬁnite (positive-deﬁnite) property also forms a closed convex set C p since R1 ≥ 0, R2 ≥ 0 gives R = αR1 + (1 − α)R2 ≥ 0,

0 ≤ α ≤ 1.

(5.240)

All n × n hermitian matrices A admit the eigen decomposition Aui = λi ui ,

i = 1, 2, . . . , n,

(5.241)

where λi is real and ui ⊥ u j if λi = λ j . Thus A = UU∗ . Moreover, B − A = B − UU∗ = U(U∗ BU − )U∗ = U(C − )U∗

(5.242)

where C = U∗ B U.

(5.243)

So that when B − A2 = tr (B − A)(B − A) ∗ = tr [U(C − )U∗ U(C − ) ∗ U∗ ] = tr [(C − )(C − ) ∗ U∗ U] = tr (C − )(C − ) ∗ = C − 2 =

i

|Ci,i − λi |2 +

i= j

|Ci, j |2 ,

(5.244)

197

198

Space Based Radar since UU∗ = U∗ U = I. Clearly, (5.244) is minimized by setting Ci, j = 0, i = j and7

* Ci,i =

λi , λi > 0 . 0 λi ≤ 0

(5.245)

(5.246)

Recall that the constraint B nonnegative-deﬁniteness implies C is nonnegative-deﬁnite as well. From (5.245) and (5.246), C is diagonal with non-negative entries. Let + deﬁne the diagonal matrix deﬁned in (5.246). Then C = + ,

(5.247)

where + is the same as with its negative entries replaced by zero. From (5.243) B = U+ U∗

(5.248)

∗

represents the projection of A = UU onto the non-negative matrix domain. The projection operator simply resets the negative eigenvalues of A to zero as in (5.246) and recompute the matrix, and is quite simple to implement. If the ideal covariance matrix R is positive-deﬁnite and block Toeplitz, then it belongs to both the convex sets C T (block Toeplitz) and C P , i.e., to their intersection convex set C = CT ∩ C P

(5.249)

and in that case, the problem is to ﬁnd a “nearest neighbor” to C ˆ that is neither block Toeplitz nor positivestarting from an estimate R deﬁnite. In this context, the method of alternating projections can be employed.

5.8.4 Methods of Alternating Projections Consider the closed convex sets C1 , C2 , . . . , Cn and their associated projection operators P1 , P2 , . . . , Pn . If the desired function belongs to each Ci , then it belongs to their intersection Co = C1 ∩ C2 ∩ · · · Cn .

(5.250)

Starting from an outside point x (usually an approximant), the problem is to determine its “nearest neighbor” xo within the derived convex set Co ; if not at least a point within Co . If the projection operator 7 If the original covariance matrix is known to have lowest eigenvalues equal to σ 2 , then this quantity can be used in (5.246) in place of zeros when λi < σ 2 .

Chapter 5:

Space-Time Adaptive Processing

Po corresponding to the new convex set Co is known, then that nearest neighbor to x equals xo = Po x as in (5.231). However, Po is usually unknown since Co is deﬁned only through (5.250), and only P1 , P2 , . . . , Pn are known. Thus the problem reduces to ﬁnding a neighbor in Co starting at x given P1 , P2 , . . . , Pn [9], [10], [11]. The method of alternating projections originally proposed by John Neumann [12] has been generalized to address this problem [13]. Deﬁne P = Pn Pn−1 · · · P2 P1

(5.251)

and consider the iteration xi = P xi−1 ,

i = 1, 2, · · ·

(5.252)

Starting with xo = x. Then it has been shown that the iteration in (5.252) converges weakly to a point in the interest Co in (5.250) [10], [11], [13]. This is illustrated in Figure 5.41 for two convex sets. If we let C1 = ˆ in Figure 5.41, then starting with an estimated C T , C2 = C P and x = R ˆ the iteration covariance matrix R, Rn = P2 P1 Rn−1 ,

n = 1, 2, · · ·

(5.253)

ˆ gives the best Toeplitz approximation that is also positivewith Ro = R deﬁnite. This procedure can be extended in STAP to obtain a better ˆ by preserving the block Toeplitz and positive deﬁnite estimate for R property. The iterations in (5.252) and (5.253) can be accelerated in their convergence by making use of relaxed projection operators as shown in Figure 5.42.

P1x

C1

x

P1x

C1

x P2P1x

Co xn

C2 Co (a)

FIGURE 5.41 Method of iterative projections.

(b)

C2

199

Space Based Radar FIGURE 5.42 Relaxed projection method.

C1

y1 = Q1x P2 y1

P1x

Q2 y1

x

Co

C2

5.8.5 Relaxed Projection Operators The projection operators Pk can be “relaxed” to generate Qk = I + λk ( Pk − I )

(5.254)

that are nonexpansive8 operators for 0 ≤ λk ≤ 2, with I representing the identity operator. The quantity λk in (5.254) represents the relaxation parameter and it can be chosen to form a sequence of adaptive coefﬁcients for faster convergence as in Figure 5.42. Thus, relaxed projection operators can be substituted in (5.252) and (5.253) and by controlling λk s, the convergence rate can be controlled as well. Either the projection operators or the relaxed projection operators corresponding to the block Toeplitz convex set and positive-deﬁnite convex set can be used in STAP for better clutter covariance matrix estimation since in an uncorrelated clutter scene the ideal covariance matrix possesses these two properties.

0

Power (dB)

Power (dB)

200

−5 −10 −15 −20 1

0 −5 −10 −15 −20

0 Doppler

200

190 −1 180 Range

(a) SMI

210

220

1 0 Doppler

200 −1 180 190 Range

210

220

(b) SMIPROJ

FIGURE 5.43 Adaptive pattern responses of (a) SMI and (b) SMIPROJ methods using 30 snapshot data samples. 8 If T is a nonexpansive operator, then T x − T y ≤ x − y. If T x − T y < x − y then T represents a contraction (strict inequality).

Chapter 5:

Space-Time Adaptive Processing

The new covariance matrix so obtained can be used in conjunction with any other standard STAP algorithm such as SMIDL, EC, etc. Figure 5.43 shows the performance of the projection-based STAP algorithm for the SMI case.

5.9 Factored Time-Space Approach The factored time-space (FTS) is also known as post-Doppler dimension-reducing beamforming [5], [14], [15]. It is applied adaptively after Doppler ﬁltering has been ﬁrst applied to each channel. The simplest form of the FTS is the single-bin method, which transforms an MN dimensional space-time ﬁltering problem into N dimensional spatial-only adaptive beamforming as shown in Figure 5.44 [14]. In FTS, the kth range cell MN × 1 space-time data vector xk is reconﬁgured to an M × N signal matrix [14]



xk (1, 1)

 x (2, 1)  k Xk =  ..   . xk ( M, 1)



xk (1, 2)

···

xk (2, 2) .. .

··· .. .

xk (2, N)    = [x 1,k , x 2,k , . . . , x N,k ] ..   .

xk ( M, 2)

···

xk ( M, N)

xk (1, N)

(5.255) where

x n,k = [xk (1, n), xk (2, n), . . . , xk ( M, n)]T

(5.256)

is the M×1 vector containing all pulses for channel n. Doppler processing is then performed on each column of Xk giving the transformed

FIGURE 5.44 Factored time-space approach (FTS).

FFT ......

FFT ......

FFT ......

Adaptive processor Adaptive spatial nulling Y1

201

202

Space Based Radar signal matrix to be



yk (1, 1)

  yk (2, 1)  Yk =  ..  .  yk ( M, 1)



yk (1, 2)

···

yk (1, N)

yk (2, 2)

···

yk (2, N) 

.. .

.. .

.. .

yk ( M, 2)

···

yk ( M, N)

   = [y1,k , y2,k , . . . , y N,k ],   (5.257)

where vector yn,k is the DFT of the vector x n,k . The desired Doppler ﬁlter corresponding to a given row of Yk is selected and the N spatial samples are adaptively combined. The output of the adaptively ﬁltered mth Doppler ﬁlter for the kth range cell is given by −1

˜ k,m y˜ zk (m) = a ∗ R , k,m

(5.258)

where a is the N × 1 spatial steering vector, y˜ k,m is the transpose of the mth row of Y given by y˜ k,m = [yk (m, 1), yk (m, 2), . . . , yk (m, N)]T

(5.259)

˜ k,m is the N × N spatial covariance matrix and R ˜ k,m = 1 R K

k+K /2

i=k−K /2

y˜ k,m y˜ ∗k,m ,

i = k.

(5.260)

˜ k,m . Notice only 2N i.i.d. samples are needed for estimation of R The 1D Doppler ﬁltering can be represented as [14]

    y˜ k =     

y˜ k,1





I

wo∗

1

I

···

wo∗

M−1

I



∗ 1 ∗ M−1    I w1 I ··· w1 I        xk ..  . . . . = . . . .   .  . . . .    ∗ 1 ∗ M−1  y˜ k, M I I w M−1 I · · · w M−1 y˜ k,2 

f ∗t,o ⊗ I





o



x k,1



 ∗      f t,1 ⊗ I   1   x k,2        xk =  .   .  = ..    .  .  .    .  .       ∗ f t, M−1 ⊗ I M−1 x k, M

(5.261)

Chapter 5:

Space-Time Adaptive Processing

where m is the N × NM transformation matrix m = f ∗m ⊗ I

(5.262)

and I is the N × N identity matrix. The M × 1 vector f m is the mth Doppler DFT steering vector given by



f t,m



1



1



 1   1   wm   τt,m       = b(ωd ) ◦  .  = .  .   .   .   .      M−1 wmM−1 τt,m

(5.263)

where τt,m = e − j2πm/M .

(5.264)

The mth Doppler bin y˜ k,m is then processed as follows:

˜ −1 zk (m) = w ˜ ∗m y˜ k,m = R k,m s˜ m

∗

= s˜ m

∗

−1 m R∗m y˜ k,m

y˜ k,m (5.265)

where s˜ m = m s,

(5.266)

y˜ k,m = m xk .

(5.267)

and

Here s in the space-time steering vector as deﬁned in (5.72). On comparing (5.265) with the regular SMI approach zk = s∗ R−1 xk ,

(5.268)

the effective steering vector and data vector are given by (5.266) and (5.267) respectively. The effective covariance matrix equals ˜ k,m = m R∗m . R

(5.269)

203

Space Based Radar Equation (5.265) shows the single-bin FTS approach; however, its performance is usually poor. A better strategy is to use two or more Doppler bines [15]. In general, let Ms be the number of Doppler bins, then the transformation matrix is given by [14]





m−( Ms −1)/2   ..   .  

   ˜ m =       

         

m−1 m

m+1 .. .

(5.270)

m+( Ms −1)/2 and the effective steering vector, data vector, and covariance matrix are as given by (5.266), (5.267), and (5.269) respectively. This method is also known as the extended factored time-space (EFA) approach. Figure 5.45 shows the adaptive angle-Doppler pattern of the FTS using one Doppler bin. Fifty snapshot data samples are used here. However, as pointed out earlier, the performance of a single Doppler bin FTS is poor [15]. Figure 5.46 shows the adaptive angle-Doppler pattern of EFA using two Doppler bins and the number of snapshot data samples used here is also 50. As seen from Figure 5.46, the target is clearly visible. The improvement over a single bin approach is signiﬁcant even when only two bins are used.

1

0

0.5

−20 −30

0

−40 −50

−0.5

−60 −1 −1

−0.5

0

0.5

cos (q )

(a) Top view

1

Power (dB)

−10

Doppler

204

0 −5 −10 −15 −20 −25

1 0 Doppler

−1 −1

−0.5

0

0.5

1

cos (q )

(b) Side view

FIGURE 5.45 Adaptive angle-Doppler pattern of FTS with one Doppler bin. Fifty snapshot data samples are used here.

Chapter 5: 1

0 −10 −20

0

−30

−0.5 −1 −1

−40

0 Power (dB)

0.5 Doppler

Space-Time Adaptive Processing

−5 −10 −15 −20 −25 1

−50 −0.5

0 cos (q )

0.5

0 Doppler

1

(a) Top view

−1 −1

−0.5

0 cos (q )

0.5

1

(b) Side view

FIGURE 5.46 Adaptive angle-Doppler pattern of EFA with two Doppler bins. Fifty snapshot data samples are used here.

5.10 Joint-Domain Localized Approach Joint-Domain Localized (JDL) processing that adapts over a local processing region consists of adjacent angle and Doppler bins [16]. The MN × 1 space-time data vector is ﬁrst transformed to angle-Doppler domain via the two-dimensional discrete Fourier transform (2D-DFT). To achieve this, the kth range cell MN × 1 spacetime data vector xk is reconﬁgured to a M × N signal matrix as in (5.255). Thus



xk (1, 1)

 x (2, 1)  k Xk =  ..   . xk ( M, 1)



xk (1, 2)

···

xk (2, 2) .. .

··· .. .

xk (2, N)   , ..   .

xk ( M, 2)

···

xk ( M, N)

xk (1, N)

(5.271)

where xk (m, n) represents the data from the mth pulse and the nth sensor. 2D-FFT is then performed on the space-time domain data matrix Xk to create the angle-Doppler domain data matrix



zk (1, 1)

 z (2, 1)  k Zk =  ..   . zk ( M, 1)



zk (1, 2)

···

zk (2, 2) .. .

··· .. .

zk (2, N)   , ..   .

zk ( M, 2)

···

zk ( M, N)

zk (1, N)

(5.272)

where zk (m, n) represents the mth Doppler bin and nth angle bin data. Then a set of these adjacent angle and Doppler bins are grouped and

205

Space Based Radar processed adaptively for clutter cancellation. Let the number of anglebins be Ns and the number of Doppler-bins be Ms , then the total number of angle-Doppler bins is given by L = Ms Ns . Notice that the numbers of angle bins Ns and Doppler bins Ms (usually only 3–5 bins are needed) are usually much smaller than the number of sensors N and pulses M [16]. As a result, L is much less than MN and the number of independent data samples required for clutter covariance estimation and computation requirement are reduced. Figure 5.47 shows an example of the angle-Doppler grouping with Ms = 3 and Ns = 3. The JDL processing is similar to the FTS approach discussed before. To see this, let the mth Doppler bin and nth angle bin data to be [14], [16] zk (m, n) = f∗m,n xk

(5.273)

where fm,n is the MN × 1 transformation vector given by fm,n = f t,m ⊗ f s,n .

(5.274)

In (5.274), the M × 1 vector f t,m is the mth Doppler (temporal) DFT steering vector as given in (5.263). Similarly, the N × 1 vector f s,n is the nth angle (spatial) DFT steering vector given by

 f s,n

1



 τ1   s,n   = a (θ) ◦   ..   .  N−1 τs,n

(5.275)

Clutter ridge

Doppler bins

206

Angle bins

FIGURE 5.47 Angle-Doppler bins grouping in JDL processing.

Chapter 5:

Space-Time Adaptive Processing

where τs,n = e − j2π n/N .

(5.276)

Thus when Ms Doppler bins and Ns angle bins are grouped, the MN × Ms Ns Doppler-angle transformation matrix at mth Doppler bin and nth angle bin is given by



 T

T fm−( Ms −1)/2,n−( Ns −1)/2   ..   .   T   fm,n−( Ns −1)/2     .  ..      T  fm+( Ms −1)/2,n−( Ns −1)/2 

Fms ,ns

      . .  .. =        T  fm−( Ms −1)/2,n+( Ns −1)/2    ..    .   T   fm,n+( Ns −1)/2    ..     . T fm+( Ms −1)/2,n+( Ns −1)/2

(5.277)

This gives the Ms Ns × 1 Doppler-angle data vector to be zk,ms ,ns = F∗ms ,ns xk

(5.278)

and it is processed as follows:

˜ ms ,ns pk,ms ,ns = w∗ms ,ns zk,ms ,ns = K−1 k,ms ,ns s

= s˜ ∗ms ,ns F∗ms ,ns R Fms ,ns

−1

∗

zk,ms ,ns

zk,ms ,ns (5.279)

where the effective steering vector and effective clutter covariance matrix are given by s˜ ms ,ns = F∗ms ,ns s,

(5.280)

Kk,ms ,ns = F∗ms ,ns R Fms ,ns

(5.281)

and

respectively. Figures 5.48–5.49 show the adaptive angle-Doppler pattern and range-Doppler pattern of the JDL approach using ﬁve angle bins and three Doppler bins. Fifty snapshot data samples are used here and the target is clearly detected.

207

208

Space Based Radar 1

0 −10 −20 −30

0

−40 −0.5 −1 −1

0 Power (dB)

Doppler

0.5

−50

−5 −10 −15 −20 −25

1

−60 −0.5

0 cos (q )

0.5

0 Doppler

1

(a) Top view

−1 −1

0 −0.5 cos (q )

0.5

1

(b) Side view

FIGURE 5.48 Adaptive angle-Doppler pattern of JDL with ﬁveangle and three Doppler bins. Fifty snapshot data samples are used here.

1

0

Doppler

0.5

−20 −30

0

−40 −0.5 −1 180

−50 −60 190

200 Range

210

(a) Top view

220

Power (dB)

−10

0 −5 −10 −15 −20 −25 1 0 Doppler

−1 180

200 190 Range

210

220

(b) Side view

FIGURE 5.49 Adaptive range-Doppler pattern of JDL with ﬁveangle and three Doppler bins. Fifty snapshot data samples are used here.

A variety of STAP techniques are discussed above for adaptive clutter cancellation and target detection. The list of methods discussed here is not exhaustive, and is used here for illustration purposes.

Appendix 5-A: Uniform Array Sidelobe Levels For an N element uniform array with interelement spacing equal to half-wavelength, the array gain pattern is given by

2 2 N 1 θ sin N π cos 2 jπ(k−1) cos θ , G(θ) = e = θ N N sin π cos 2 k=1

(5A.1)

Chapter 5:

Space-Time Adaptive Processing

where θ represents the look angle off the line of the array. Deﬁne ω=

π cos θ 2

(5A.2)

so that the above gain pattern reduces to (see also (5.24))

G(ω) =

sin( Nω) N sin ω

2 .

(5A.3)

Clearly G(ω) is periodic with period π and G(0) = 1,

G(ω) < 1,

|ω| < π/2.

(5A.4)

Figure 5.50 shows the array gain pattern for a 10-element uniform array with half-wavelength spacing. From (5A.3), the mainlobe width 2ωo is dictated by the ﬁrst zero in the numerator and hence π sin( Nωo ) = 0 ⇒ Nωo = π ⇒ ωo = (5A.5) N indicating that the mainlobe width tends to zero as the number of sensors increases. Beyond the mainlobe, there are ( N − 1) sidelobes with distinct peaks occurring at frequencies ω1 , ω2 , . . .. To determine an approximate location of these peaks and their actual values, we can proceed as follows: To start with, from (5A.5), the ﬁrst zero of G(ω) occurs at π/N and similarly the second zero occurs at 2π/N so that the ﬁrst sidelobe

0 −13.46 dB

G(w) in dB

−10 G(w 1) −20 −30 −40 −50 −p/2

0 w

w1

w2

−p/2

FIGURE 5.50 Array gain pattern for a 10-element uniform array with half-wavelength spacing.

209

210

Space Based Radar peak located in between these zeros occurs at approximately at their midpoint 3π/2N. Substituting this frequency into (5A.3) we get the ﬁrst sidelobe peak value to be (approximately)

G(ω = 3π/2N) =

→

sin(3π/2) N sin(3π/2N) 2 3π

2 = 3

# N

1 3π 2N

−

(3π/2N) 3 3!

+ ···

$42

2 = −13.46 dB.

(5A.6)

Thus for a uniform array, the maximum sidelobe level stays around 13.46 dB below the mainlobe peak and it is not possible to reduce it any further by increasing the number of sensors (unlike the mainlobe width that tends to be zero as N → ∞). It is possible to reﬁne the above argument to obtain a more accurate location of the sidelobe peaks and their values. To start with, the ﬁrst sidelobe level peaks at frequency ω1 with a peak value of G(ω1 ). To obtain these peak values and their locations, we can proceed as follows: At a peak, we have

dG(ω) =0 dω ω=ωk so that from (5A.3) dG(ω) = 2 dω

sin Nω N sin ω

(5A.7)

N sin ω cos( Nω) − sin( Nω) cos ω =0 ( N sin ω) 2 ω=ωk (5A.8)

gives9 N sin ωk cos( Nωk ) = sin( Nωk ) cos ωk

(5A.9)

N tan ωk = tan( Nωk ).

(5A.10)

or

Thus, the sidelobe peak locations ωk satisfy the equation (5A.10) and from Figure 5.51, their solutions are of the form ωk =

(2k + 1)π − k , 2N

(5A.11)

where k > 0. 9 The other solution sin( Nω) = 0 corresponds to the locations of the zeros of the gain pattern.

Chapter 5:

Space-Time Adaptive Processing

10 10

tan Nw

0

p 2N

N tan w w1

p N

p 2

3p 2N

N tan w ∆1

w tan Nw

−10

3p 2N

0

w1

(a) Sidelobe locations

(b) First sidelobe location

FIGURE 5.51 Sidelobe locations and roots of the equation (5A.10) for N = 7.

To obtain an expression for k , we can substitute (5A.11) in (5A.10). This gives

N tan

(2k + 1)π − k 2N

= tan =

(2k + 1)π − N k 2

= cot( N k )

1 1 . ≈ tan( N k ) N tan k

(5A.12)

Expanding the left side of (5A.12) we get

#

N tan (2k+1)π − tan k 2N 1 + tan

(2k+1)π 2N

$

tan k

1 . N tan k

(5A.13)

Let x = tan k

(5A.14)

so that (5A.13) reduces to the quadratic equation N2 x 2 − ( N2 − 1) tan

(2k + 1)π x+1=0 2N

(5A.15)

whose roots are given by ( N2 x1,2 =

− 1) tan

(2k+1)π 2N

±

#

( N2 − 1) tan (2k+1)π 2N

$2

− 4N2

2N2 (5A.16)

211

212

Space Based Radar First Sidelobe Peak Locations and Peak Values Computed Using (5A.18)–(5A.20) Actual N

1 (degree) ω1 (degree) G(ω1 ) in dB ω1 (degree) G(ω1 ) in dB

10

1.1842

25.8158

−12.9662

25.8329

−12.9662

20

0.6267

12.8733

−13.1883

12.8835

−13.1882

50

0.2545

5.1455

−13.2498

5.1498

−13.2498

100

0.1275

2.5725

−13.2586

2.5746

−13.2585

200

0.0638

1.2862

−13.2608

1.2873

−13.2607

500

0.0255

0.5145

−13.2614

0.5149

−13.2613

1000 0.0128

0.2572

−13.2615

0.2575

−13.2614

TABLE 5.2 Computed first sidelobe location and value vs. actual sidelobe location and value

and the smallest among these roots gives the desired solution. Hence we obtain ( N2 tan k =

− 1) tan

(2k+1)π 2N

#

−

( N2 − 1) tan (2k+1)π 2N

$2

− 4N2

2N2 (5A.17)

which for k = 1 gives



1 = tan−1 

3π ( N2 − 1) tan 2N −

)

3π ( N2 − 1) tan 2N

2

− 4N2

2N2

 .

(5A.18) Thus the ﬁrst sidelobe peaks at ω1 =

3π − 1 2N



3π ( N2 − 1) tan 2N − 3π = − tan−1  2N

)

3π ( N2 − 1) tan 2N

2N2

2

− 4N2

 

(5A.19) and for any N, the peak sidelobe value is given by

G(ω1 ) = G

3π − 1 2N

=

cos( N 1 ) N sin

3π

2N

− 1

2

.

(5A.20)

Chapter 5:

Space-Time Adaptive Processing

Table 5.2 shows the computed peak values and their location using (5A.18)–(5A.20) and their actual values. To determine the limiting value of the peak sidelobe level, from (5A.18) we get (for large N)

)

2

3π 1 (3π/2) 2 − 4 − 4N 2N $ 3π # 2 = 1 − 1 − 4(2/3π ) 2 (5A.21) 4N 3π N

tan 1 1 =

3π N2 2N −

3π N2 2N

− 4N2

2N2

=

so that 3π 2 − 2N 3π N which when substituted into (5A.3) gives ω1 =

G(ω1 ) =

cos(2/3π) N sin (3π/2N − 2/3π N)

(5A.22)

2

→

cos(2/3π ) 3π/2 − 2/3π

= (0.217228) 2 = −13.26 dB.

2

(5A.23)

Also, from Table 5.2, for large N G(ω1 ) → −13.26 dB

(5A.24)

indicating that for a uniform array although the mainlobe width decreases as N → ∞, the sidelobe levels stay at −13.26 dB. In general, the kth sidelobe peak location is given by (2k + 1)π 2 ωk = − →0 (5A.25) 2N (2k + 1)π N and the corresponding peak level equals



#

cos

G(ωk ) →  (2k+1)π 2

2 (2k+1)π

−

$ 2

2 (2k+1)π

 .

(5A.26)

From (5A.26), the second sidelobe level saturates at −17.83 dB and the third sidelobe level saturates at −20.79 dB below the mainlobe peak, etc. In summary, the peak sidelobe level of a uniform array saturates at about 13.26 dB below the mainlobe peak and to lower them further additional weighting factor must be introduced at the array output.

References [1] B. Friedlander, “The MDVR Beamformer for Circular Arrays,” Proc. 34th Asilomar Conf. Signals, Systems, Computers, Vol. 1, pp. 25–29, November 2000. [2] H. Wang, L. Cai, “On Adaptive Spatial-Temporal Processing for Airborne Surveillance Radar Systems,” IEEE Transaction on Aerospace and Electronic Systems, Vol. 30, No. 3, pp. 660–670, July 1994.

213

214

Space Based Radar [3] G.W. Titi, D.F. Marshall, “The ARPA/NAVY Mountain top Program: Adaptive Signal Processing for Airborne Early Washing Radar,” 1996 IEEE International Conference on Acoustics, Speech and Signal Processing, Atlanta, Georgia, May 7–10, 1996. Mountain Top Radar Site Parameters, http://spib .rice.edu/spib/mtn top.html [4] A.M. Haimovich, “The Eigencanceler: Adaptive Radar by Eigenanalysis Methods,” IEEE Transaction on Aerospace and Electronic Systems, Vol. 32, No. 2, pp. 532–542, April 1996. [5] J. Ward, Space-Time Adaptive Processing for Airborne Radar, MIT Technical Report 1015, MIT Lincoln Laboratory, Lexington, MA, December 1994. [6] A.M. Haimovich, “Eigenanalysis Based Space-Time Adaptive Radar,” IEEE Transaction on Aerospace and Electronic Systems, Vol. 33, No. 4, pp. 1170–1179, October 1997. [7] S.U. Pillai, Array Signal Processing, Springer-Verlag, New York, 1989. [8] J.R. Guerci, “Theory and Application of Covariance Matrix Tapers for Robust Adaptive Beamforming,” IEEE Trans. on Signal Processing, pp. 977–985, April 1999. [9] D.C. Youla, “Generalized Image Restoration by the Method of Alternating Projections,” IEEE Transactions on Circuits and Systems, Vol. 25, pp. 694–702, September 1978. [10] L.G. Gubin, B.T. Polyak, E.V. Raik, “The Method of Projections for Finding the Common Point of Convex Sets,” U.S.S.R. Computational Mathematics and Mathematical Physics, Vol. 7, No. 6, pp. 1–24, 1967. [11] Z. Opial, “Weak Convergence of the Sequence of Successive Approximations for Nonexpansive Mappings,” Bull. Amer. Soc. Vol. 73, pp. 591–597, 1967. [12] J.V. Neumann, “Functional Operators, Vol. II,” Annals of Mathematics Studies, No. 22, Theorem 13.7, p. 55, Princeton, NJ, 1950. [13] D.C. Youla, “Mathematical Theory of Image Restoration by the Method of Convex Projections,” Chapter 2, Image Recovery Theory and Application, Henry Stark, ed., Academic Press, Inc., New York, NY, 1987. [14] J.R. Guerci, Space-Time Adaptive Processing for Radar, Artech House, Boston, MA, 2003. [15] R.C. DiPietro, ”Extended Factored Space-Time Processing for Airborne Radar,” Proc. 26th ASILOMAR Conf., Paciﬁc Grove, CA, pp. 425–430, October 1992. [16] H. Wang and L.J. Cai, “On Adaptive Spatial-Temporal Processing for Airborne Surveillance Radar Systems,” IEEE Transaction on Aerospace and Electronic Systems, Vol. 30, No. 3, July 1994.

CHAPTER

6

STAP for SBR This chapter deals with clutter data modeling from an SBR platform by taking into consideration the various phenomena that affect the data modeling, and examining how they affect the target detection performance. These phenomena are imperfections present in any realistic data scene compared to the ideal conditions. They include the array pattern due to the ﬁnite antenna size, effect of Earth’s rotation on the Doppler shift (Section 4.6–4.7), range foldover generated by the presence of multiple pulse returns on the received data (Section 4.5), and the dependence of the scatter power proﬁle on the local terrain type. In addition, factors such as the effect of wind and altitude information can be added to the model for higher ﬁdelity clutter modeling. In real life, all these items affect the SBR data and hence any realistic data modeling must consider and account for these items. An important aspect in this context will be to understand the impact on performance of each such phenomenon separately, as well as when taken together in various combinations. Performance measures to quantify clutter nulling effects are reviewed in this context, and performance evaluations are carried out both in the matched ﬁlter case and the estimated case using the various STAP algorithms discussed in Chapter 5. Transmit beams scale the transmit pulses according to the transmit array pattern, and they get reﬂected from various range foldover points depending upon the local terrain reﬂectivity and respecting the radar equations (1.2)–(1.5). The returns once again get weighted by the receiver array pattern, and generate the receiver output. The following aspects must be taken into consideration for clutter modeling: Scatter Power Proﬁle: Reﬂectivity of the terrain as a function of elevation and azimuth. Local terrain dependence can be accommodated using a Knowledge Aided Sensor Signal Processing and Expert Reasoning (KASSPER) like approach [1]. Array Gain Pattern: Both transmitter and receiver array conﬁgurations with weights can be used to control the sidelobe pattern.

215 Copyright © 2008 by The McGraw-Hill Companies. Click here for terms of use.

216

Space Based Radar Range–Foldover Return (Elevation): Returns due to earlier/later transmitted pulses that arrive from range points other than the point of interest must be taken into account for data generation. Azimuth Return: Scatter returns from isorange spanning all azimuth angles. Earth’s Rotation: Modiﬁes the Doppler due to SBR motion resulting in a crab angle effect, and should be accounted in data modeling. Performance measures to quantify clutter nulling effect are reviewed, and performance evaluations are carried out both in the ideal case and estimated case using various STAP algorithms discussed in Chapter 5. Performance degradation is severe when both range foldover and Earth’s rotation are present in the data. However, when either one is present, the degradation in performance can be corrected. Orthogonal transmit pulsing schemes are used to minimize range foldover effect, and this results in improved overall performance.

6.1 SBR Data Modeling Consider an SBR array with N sensors and M pulses. If the incoming wavefront makes an azimuth angle θ AZ and elevation angle θEL with reference to the array (Figure 4.1 (b)), then for two sensors that are d˜ apart, the propagation delay τ at the second sensor (with respect to the ﬁrst sensor) is τ=

d˜ sin θ EL cos θ AZ , cυ

(6.1)

where c υ represents the velocity of light. Hence the second sensor output x2 (t) is given by (see (5.3)–(5.4)) d˜

x2 (t) = x1 (t − τ ) = x1 (t)e − jωo τ = x1 (t)e − jπ λ/2 sin θ E L cos θ AZ ,

(6.2)

where x1 (t) refers to the ﬁrst sensor output. If the interelement distance d˜ is normalized with respect to λ/2, then the normalized interelement spacing is given by d=

d˜ . λ/2

(6.3)

Let c = sin θ EL cos θ AZ

(6.4)

represent the “cone angle” associated with the spatial point (θEL , θ AZ ) for the SBR array. In that case, from (6.2), the output vector (spatial)

Chapter 6:

STAP for SBR

for a uniform N-sensor array becomes



x1 (t)

 x2 (t)  ..  .

x(t) =  

    = a (c)x1 (t).  

(6.5)

xN (t) Here a (c) represents the spatial steering vector given by



1



 e − jπdc     − j2πdc   e  a (c) =  . ..   .     e − j ( N−1)π dc

(6.6)

Similarly, if M pulses are transmitted with a certain pulse repetition frequency (PRF), then the returns due to different pulses have a similar relation as in (6.5) and (6.6). To be speciﬁc [2], [3], with Vp denoting the SBR platform velocity and T the pulse repetition period, we have ωd in (4.53) or (4.71) represents the clutter Doppler for that speciﬁc range bin. The return vector due to the various pulses at the ﬁrst sensor is given by (see also (5.67)–(5.70)) y1 (t) = b(ωd )x1 (t)

(6.7)

where b(ωd ) represents the temporal steering vector given by

     b (ωd ) =    

1 e − jπ ωd e − j2π ωd .. . e − j ( M−1)π ωd

     .   

(6.8)

Combining the effects of both the spatial array and temporal pulses, the concatenated data vector due to the N sensors and M pulses has the form



y1 (t)

 y (t)  2 .  ..

x(t) =  

y M (t)



   = s(c, ωd )x1 (t)  

(6.9)

217

218

Space Based Radar where s(c, ωd ) = b(ωd ) ⊗ a (c)

(6.10)

represents the spatio-temporal steering vector with ⊗ representing the Kronecker product. From (4.34) the ground resolution is represented by δ R and hence the range cells must be separated by at least δ R . Let rk , k = 1, 2, . . . represent the actual range bin locations along the azimuth and elevation angles θ AZ and θEL,k , respectively. The clutter corresponding to any range bin has contributions from the mainbeam antenna footprint located along (θEL,k , θ AZ ) as well as from the sidelobe footprints located along (θEL,k , θ AZ, j ), where θ AZ, j covers various azimuth angles along the ﬁeld of view of the SBR antenna that is pointed along (θEL,k , θ AZ ).

6.1.1 Mainbeam and Sidelobe Clutter From Figures 4.10–4.12, if rk falls within the ith footprint then there are Na i distinct range ambiguities within that footprint and they all contribute to the clutter for that range. Here Na i represents the number of range ambiguities within the ith mainbeam footprint as in (4.38)– (4.39). Thus if RHi < rk < RTi ,

(6.11)

then the Na i range ambiguities located at RHi ≤ rk ± m R ≤ RTi ,

m = 0, 1, . . .

(6.12)

contribute to the clutter return for range rk (see Figure 6.1). To include the clutter contribution from the sidelobes, the azimuth angle can be varied to cover the entire ﬁeld of view of the SBR antenna. In that case, the returns from the elevation direction will correspond to

FIGURE 6.1 Range ambiguities that contribute to the mainbeam clutter from range rk .

Range ambiguities that contribute to clutter from range rk ∆R dR

rk rk +1 Range ambiguities that contribute to clutter from range rk+1

Chapter 6:

STAP for SBR

(q EL,m’ qAZ,j ) …

Mainbeam

Iso-cone clutter discretes

…

qAZ D: Point of interest, cone angle C 0

SBR A Nadir hole

…

Azimuth sidelobes

Elevation sidelobes

Range of interest

Range ambiguities

FIGURE 6.2 Mainbeam and sidelobe clutter for SBR antenna pointing along azimuth direction θAZ and range rk (elevation angle θEL,k ).

all range foldover points rk ± m R , m = 0, 1, . . . , Na (see Figure 6.2). Let θ AZ j = θ AZ + j θ, j = 0, 1, 2, . . . , No represent the azimuth angles associated with the ﬁeld of view, and θELm , m = 0, 1, 2, . . . , Na the elevation angles corresponding to the total number of range ambiguities in the ﬁeld of view. Further let c m, j = sin θELm cos θ AZ j

(6.13)

represent the cone angle and A(θ m, j ) the overall 2D array amplitude pattern associated with the spatial point of interest where

θ m, j = θELm , θ AZ j .

(6.14)

219

220

Space Based Radar In general, the overall array pattern A(θ m,n ) is a product of the transmit array pattern AT (θ m,n ) and the receiver pattern AR (θ m,n ), and hence A(θ m,n ) = AT (θ m,n ) AR (θ m,n ).

(6.15)

When omnidirectional sensors are used, the receiver pattern arises when subarrays are formed at the receiver for data collection. In that case, sensor elements in the subarray are combined using beamformer and the array pattern so generated by the subarray plays the role of AR (θ m,n ) in (6.15). Each of these subarray outputs generates the entries of the spatial data vector in (6.5). With {a i,k } representing the appropriate array element weights (transmit array or receiver subarray with omnidirectional sensors), we have N E L −1 N AZ −1

Ax (θ m,n ) =

i=0

a i,k e − jπ [i d E L (cos θ E L m −cos θ E L )+kd AZ (cos θ AZn −cos θ AZ )] ,

k=0

x = T or R,

(6.16)

where (θEL , θ AZ ) represents the point of interest on the ground. For a separable array, we have a i,k = αi βk ,

(6.17)

where {αi } and {βk } represent the weights in the azimuth and elevation direction, respectively. Let um, j represent the random clutter scatter returns from location θ m, j for the kth range bin. From the radar equation (1.2), the transmit waveform as well as the scatter returns are attenuated by the slant range projected by the range bin of interest, and let αm, j represent the scatter returns received by the array. Then αm, j =

um, j , Rs2j

(6.18)

where Rs j represents the slant range from the SBR to the ground location θ m, j as in (4.3). Then the total clutter return from range rk is given by (see Figure 6.2) xk =

Na No

αm, j A(θ m, j )s(c m, j , ωdm, j ) + n,

(6.19)

j=0 m=0

with the inner summation representing the various range foldovers (m = 0 represents the actual range of interest rk ), and the outer summation spanning over all azimuth angles including the sidelobes in the ﬁeld of view.

Chapter 6:

STAP for SBR

This gives the ensemble average clutter covariance matrix associated with range rk to be

Rk = E xk x∗k =

Na No

Pm, j G(θ m, j )sm, j s∗m, j + σn2 I.

(6.20)

j=0 m=0

Here

Pm, j = E |αm, j |2 ,

(6.21)

represents the scatter power from θ m, j and

2

G(θ m, j ) = A(θ m, j )

(6.22)

represents the array gain pattern and sm, j = s(c m, j , ωdm, j ).

(6.23)

To obtain realistic scatter returns um, j , “site-speciﬁc” information can be incorporated to determine the terrain types. In this approach, a land cover map with speciﬁc grazing angle dependent mean radar cross section (RCS) values can be used to generate the clutter scatter power levels according to speciﬁc Weibull distribution. A further reﬁnement can be realized by introducing wind induced internal clutter motion (ICM). A detailed account of this approach along with performance analysis is given in Section 6.5. In (6.19) and (6.23), ωdm, j represents the Doppler associated with the range bin θ m, j and is given by (see (4.53), (4.71)) [4]

ωdm, j =

no Earth s rotation,

βo c m, j ,

ω˜ dm, j = βo ρc sin θ ELm cos(θ AZ j + φc ), with Earth s rotation. (6.24)

Using (5.76), we obtain βo =

2VT . λ/2

(6.25)

From (5.76) and (5.77), it also follows that β=

βo 2VT = ˜ d d

(6.26)

represents the Brennan factor. From (5.5), d˜ and d represent the actual interelement spacing as well as the normalized interelement spacing

221

Space Based Radar # of sensors # of pulses Radius of Earth (km) Altitude of SBR (km) SBR velocity (m/s) Normalized interelement spacing with respect to half wavelength Pulse repetition frequency (Hz)

(N) (M) (Re ) (H) (Vp ) (d)

32 16 6,373 506 7,160 13.4

(PRF)

500–2,000

TABLE 6.1 SBR parameters

with respect to half wavelength, respectively. With φ = dc representing the array factor exponent in (6.6), in the absence of Earth’s rotation, ωd = βφ as in (6.24). For example, for the SBR conﬁguration in Table 6.1, we obtain the Brennan factor β = 19.47. Figures 6.3–6.4 show Doppler as a function of the cone angle with and without Earth’s rotation present. Note that in the absence of Earth’s rotation, there is a one-to-one linear correspondence between Doppler and cone angle. As a result, points that project the same cone angle in the range-azimuth domain generate the same Doppler. Thus in the absence of Earth’s rotation, all points on the isocone contour in Figure 6.2 project the same Doppler frequency. It follows in that case, all points in the ﬁeld of view align along a single line in the Dopplercone angle domain. However, this is no longer true when Earth’s rotation is present, since different range points on a given cone angle contour generate different Doppler frequencies as shown in Figures 6.3–6.4. Figure 6.3 shows Doppler frequencies corresponding to R = 500 km and its 150 15

50

No Doppler spread without Earth’s rotation

Doppler (kHz)

100 Doppler (kHz)

222

0 Doppler spread due to Earth’s rotation

−50

−100 −150 −1

10

Doppler spread/splitting due to Earth’s rotation No Doppler spread without Earth’s rotation

5 0 −5

−10 −15

−0.5

0 Cone Angle (a)

0.5

1

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Cone Angle (b)

FIGURE 6.3 Doppler spread due to Earth’s rotation and range foldover for range 500 km. (a) All cone angles (b) “Zoomed in” view around zero cone angle.

Chapter 6:

STAP for SBR

No Doppler spread without Earth’s rotation

Doppler (kHz)

0

−5

Doppler spread/splitting due to Earth’s rotation

−10

−15 −0.04

−0.02

0 Cone Angle

0.02

0.04

FIGURE 6.4 Doppler spread due to Earth’s rotation for all points of interest in the ﬁeldof view.

seven foldovers. Since the seven range foldovers intersect any isocone contour at seven distinct points (see Figure 6.2), from (6.24) they generate seven distinct Doppler frequencies for the same cone angle. Figure 6.3(a) shows the Doppler spread generated by the seven foldovers for all cone angles, and Figure 6.3(b) shows a magniﬁed view around zero cone angle. Figure 6.4 shows the band of Doppler frequencies corresponding to all range points of interest as a function of the cone angle, and in the limit they generate a continuous band of Doppler frequencies. In summary, when Earth’s rotation is present, every point on Earth projects a different Doppler that is conﬁned within a band of frequencies. Figure 6.5 shows the Doppler frequency variation along the isocone contour (c o = 0.3) shown in Figure 6.2 without and with Earth’s rotation as a function of range. From (6.24), without Earth’s rotation, as the dashed curve shows, Doppler along an isocone contour is a constant; whereas with Earth’s rotation present, as the solid curve shows, each range point on the isocone projects a different Doppler.

6.1.2 Ideal Clutter Spectrum To appreciate the effective clutter suppression capabilities of the matched ﬁlter receiver, it is instructive to examine the true clutter spectrum. The clutter spectrum corresponding to the beamformer output is given by [2], [5], [6]

PB ( Ro , ωd ) = E |x∗ s(c, ωd )|2 = s∗ (c, ωd )Rs(c, ωd )

(6.27)

223

Space Based Radar 80 Without Earth’s rotation 75 Doppler

224

70

65 With Earth’s rotation 60

0

500

1,000 1,500 Range (km)

2,000

2,500

FIGURE 6.5 Doppler frequency along an isocone contour with co = 0.3.

where Ro is the range of interest. To show the range dependency of the Doppler, the range-Doppler clutter power proﬁle is plotted in Figure 6.6 for a ﬁxed azimuth angle θ AZ = 89.5◦ in the absence of both Earth’s rotation and range foldover by varying the range parameter k and the Doppler ωd in (6.27). As Table 6.1 shows, a 32-element array with 16 pulses, and normalized interelement spacing d = 13.4 with PRF = 500 Hz is used in the simulations. From Figure 6.6, the Doppler is an increasing function of the range and this is in agreement with (4.53). This should not be confused with Figure 6.5 that shows the Doppler along the iso-cone contour (Figure 6.2). Figures 6.7–6.8 show the angle-Doppler proﬁle of the clutter spectrum using the parameters listed in Table 6.2. Here the clutter power spectrum PB,k (θ AZ , ωd ) = s∗ (c k , ωd )Rk s(c k , ωd )

(6.28)

and the elevation angle θEL,k correspond to the kth range bin and it remains ﬁxed in (6.28). The azimuth angle θ AZ and the Doppler parameter ωd in (6.28) are varied here to obtain Figures 6.7–6.8. Figure 6.7 shows the ideal angle-Doppler dependency of the clutter spectrum for range rk equal to 500 km. Notice that for a particular azimuth angle, the clutter covers almost the entire Doppler region. As a result the clutter-free region in the Doppler domain is at a premium. Obviously to detect a target, the target Doppler must fall in the clutter-free region. Figure 6.8 shows the “zoomed in” version of the

Chapter 6:

STAP for SBR

Range-Doppler Pattern

0

1,100

−1 1,000

−2 −3

Range (km)

900

−4 −5

800

−6 700

−7 −8

600

−9 −10

500

−11 400 −1

−0.5

0 Doppler

0.5

1

FIGURE 6.6 Clutter spectrum using 2D beamformer without range foldover and Earth’s rotation for θAZ = 89.5◦ .

angle-Doppler proﬁle for two different ranges. As expected, the clutter Doppler in proportional to cos θ AZ . From Figure 6.8, the “slow” range dependence of the clutter Doppler can be observed from the relative position of the clutter ridge as a function of range. As range increases (Figure 6.8 (a)–(b)) the slope of the clutter ridge increases and this is consistent with (4.53)–(4.50) since in that case θEL increases with range R. Table 6.2 and Figure 6.9 list another set of SBR parameters corresponding to a hypothetical array studied in the literature [7], [8], [9]. A planar array consisting of 384 elements in azimuth and 12 elements in elevation is considered here. The azimuth and elevation element spacings are taken to be 0.54λ and 0.695λ, respectively. It is often impractical to place an analog to digital converter behind each element. As a result, in the receive mode 32 column subarrays of 12 by 12 elements separated by 6.48λ are considered. Although each subarray is phased to point to the “target,” from a spatial sampling basis, the clutter will be weighted by the corresponding antenna pattern, and will exhibit grating lobes based on the spacing between the subarrays. All sensors in the azimuth and elevation directions transmit at the same time generating the transmit pattern in Figure 6.10. However,

225

226

( H) (Vp ) ( M) PRF

Transmit Number of sensors in azimuth direction Interelement spaceing in azimuth direction normalized to half wavelength Number of sensors in elevation direction Interelement spaceing in elevation direction normalized to half wavelength

TABLE 6.2 Realistic SBR array parameters

Space Based Radar

Altitude of SBR SBR velocity Number of pulses Pulse Repetition Frequency

506 km 7160 m/s 16 500 Hz Receive

( NT x−AZ )

384

(dT x−AZ )

1.08

( NT x−E L )

12

(dT x−E L )

1.39

Number of sensors in azimuth direction/subarray Interelement spaceing in azimuth direction normalized to half wavelength Number of sensors in elevation direction/subarray Interelement spaceing in elevation direction normalized to half wavelength Number of subarrays in azimuth direction Number of subarrays in elevation direction

( NRx−AZ )

12

(d Rx−AZ )

1.08

( NRx−E L )

12

(d Rx−E L )

1.39

32 1

Chapter 6:

STAP for SBR

Angle-Doppler Pattern 1

0

0.8 −5

0.6

Doppler

0.4 −10

0.2 0

−15

−0.2 −0.4

−20

−0.6 −0.8 −1 −1

−25 −0.5

0.5

0

1

Azimuth (cosq )

FIGURE 6.7 Ideal angle-Doppler proﬁleof the clutter spectrum for R = 500 km.

on receive, 12 sensors in the azimuth and 12 sensors in the elevation direction are grouped together to form a subarray block of size 12 × 12. This procedure generates 32 subarray blocks (1D) of size 12 × 12. Figure 6.9 shows the conﬁguration of the SBR receiver array and

Angle-Doppler Pattern

−0.6 −0.8 −1 −0.05

0 −5 −10 −15 −20 −25 0 Azimuth (cosq )

(a) R = 500 km

0.05

Doppler

Doppler

Angle-Doppler Pattern 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4

1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −0.05

0 −5 −10 −15 −20 −25 0 Azimuth (cosq )

0.05

(b) R = 1,400 km

FIGURE 6.8 Zoomed in ideal angle-Doppler proﬁlefor different ranges (PRF = 500 Hz).

227

228

Space Based Radar

(a)

Vp

qAZ qEL

1st

2nd

block

block

…

12 by 12 sensor block

32nd …

…

12 sensors

block

384 sensors

Toward the center of the Earth (b)

FIGURE 6.9 Array conﬁguration.

Table 6.2 shows the rest of the array parameters. All simulations that follow use these parameters unless otherwise speciﬁed. Figure 6.10 shows the transmit patterns in the azimuth and elevation directions and the overall transmit pattern for the array pointing at 90◦ azimuth and 90◦ elevation. Figure 6.11 shows the corresponding receiver pattern also with uniformly weights. Figure 6.12 shows the clutter spectrum generated using the set of parameters shown in Table 6.2 with θ AZ = 90◦ for range 500 km; Figure 6.12(a) shows the spectrum in the full azimuth domain and Figure 6.12(b) shows detailed version of the spectrum around 90◦ azimuth. Finally Figure 6.13 shows the clutter spectrum for a ﬁxed azimuth angle θ AZ = 90◦ in the range-Doppler domain, corresponding to the four situations with and without Earth’s rotation and range foldover. Note that when the Earth’s rotation is enabled, the Doppler dependency on range becomes highly nonlinear, making target detection much more difﬁcult.

Chapter 6: 0 −10 −20 −30 −40 −50 −60 −70 −80 −90 −100 −110

STAP for SBR

0 −10

Tx in dB

Tx in dB

−20 −30 −40 −50 −60 0

20

40

60 80 100 120 140 160 180

0

20

40

60 80 100 120 140 160 180

Azimuth Angle

Elevation Angle

(a) Transmit pattern in azimuth direction

(b) Transmit pattern in elevation direction

Tx Pattern in dB

0 −50 −100 −150 150 100 Azimuth Angle

50 0

0

50

100

150

Elevation Angle

(c) Overall transmit pattern

FIGURE 6.10 Transmit pattern—uniform weights.

Transmit Array Weights One approach to reduce the transmit sidelobe levels in Figure 6.10 is to use array weights {a i,k } as shown in (6.16). Taylor weights that are suitably scaled can be used to lower sidelobes, at the expense of the mainlobe width, and sidelobe levels can be reduced as shown in Figure 6.14. Figure 6.14 shows the azimuth transmit pattern with and without applying Taylor weights in the azimuth direction only. From there, sidelobe levels of the transmit pattern in the azimuth direction can be further reduced by 20 dB when Taylor weights are used. In this case, elevation direction has uniform weight so that the transmit pattern is same as that in Figure 6.10 (b). Similarly the receiver pattern is also uniformly weighted for the array conﬁguration described here.

229

0

0

−10

−10

−20

−20 Rx in dB

Rx in dB

Space Based Radar

−30 −40 −50

−70

−30 −40 −50

−60 0

20

40

−60 0

60 80 100 120 140 160 180 Azimuth Angle

20

Rx Pattern in dB

40

60 80 100 120 140 160 180 Elevation Angle

(b) Receiver pattern in elevation direction

(a) Receiver pattern in azimuth direction

0 −20 −40 −60 −80 −100 −120 150 100 Azimuth Angle 50

0

50

0

100

150

Elevation Angle

(c) Receiver transmit pattern

FIGURE 6.11 Receiver pattern—uniform weights.

1

1

0 −5 −10 −15 −20

0

−25 −30 −35

−0.5

0 −5 −10

0.5 Doppler

0.5 Doppler

230

−15 −20

0

−25 −30 −35

−0.5

−40

−40 −1 −1

−45 −0.5

0 Azimuth

(a)

0.5

1

−1 −0.05

−45 0

0.05

Azimuth

(b)

FIGURE 6.12 Ideal clutter spectrum without Earth’s rotation and range foldover in angle-Doppler domain for range = 500 km. (a) All azimuth angles (b) Around 90◦ azimuth angle.

Chapter 6:

Range (km)

2,000 1,800 1,600 1,400 1,200 1,000 800 −1

−0.5 0 0.5 Normalized Doppler

2,400 2,000 1,800 1,600 1,400 1,200 1,000 800 −1

1

(a) w/o range foldover, w/o Earth’s rotation

Range (km)

2,000 1,800 1,600 1,400 1,200 1,000 0

0.5

1

2,400

0 −2 −4 −6 −8 −10 −12 −14 −16 −18

2,200 2,000 Range (km)

0 −2 −4 −6 −8 −10 −12 −14 −16 −18 −20

2,400

−0.5

−0.5 0 0.5 Normalized Doppler

(b) w/ range foldover, w/o Earth’s rotation

2,200

800 −1

0 −2 −4 −6 −8 −10 −12 −14 −16 −18 −20

2,200 Range (km)

0 −2 −4 −6 −8 −10 −12 −14 −16 −18 −20

2,400 2,200

STAP for SBR

1,800 1,600 1,400 1,200 1,000 800 −1

1

−0.5

0

0.5

Normalized Doppler

Normalized Doppler

(c) w/o range foldover, w/ Earth’s rotation

(d) w/range foldover, w/ Earth’s rotation

1

FIGURE 6.13 Clutter spectrum with/without range foldover and with/without Earth’s rotation, azimuth angle = 90◦ .

Tx Array Gain in AZ 0

−20

−20

−40

−40

Array Gain in dB

Array Gain in dB

Tx Array Gain in AZ 0

−60 −80 −100 −120

−80 −100 −120 −140

−140 −160

−60

0

20

40

60

80 100 120 140 160 180

−160

0

20

40

60

80 100 120 140 160 180

Azimuth Angle (deg)

Azimuth Angle (deg)

(a) Uniform weights

(b) Taylor weights

FIGURE 6.14 Azimuth transmit pattern with and without Taylor weights.

231

232

Space Based Radar

6.2 Minimum Detectable Velocity (MDV) In general the total received signal consists of returns from target (if any), mainbeam clutter, sidelobe clutter, and noise. Let xk represent the data vector at an N element array corresponding to M pulses received from range rk . This gives

* xk =

qt + ck + nk H1 ck + nk

Ho

,

k = 1, 2, . . .

(6.29)

Here qt = αt s(c t , ωdt ) corresponds to the target (if any) present at range rk (see Figure 6.15). Notice that both xk and qt are MN × 1 data vectors. The optimum adaptive weight vector corresponding to (6.29) is given by wk = R−1 k s(c t , ωdt )

(6.30)

and the adaptive matched ﬁlter power output is given by

2

Popt,k (θ AZ , ωd ) = w∗k s(c k , ωd ) .

(6.31)

In actual practice, the covariance matrix in (6.30) corresponding to range rk is unknown and it can be estimated from the actual data from neighboring range bins, using the expression ˆk = R

xk+ j x∗k+ j .

(6.32)

j

In (6.32) the number of range bins over which the summation is ˆ k . The carried out is chosen so as to maintain stationary behavior for R

FIGURE 6.15 Monostatic airborne target detection using SBR STAP.

Rx

Tx

H R H1

Mainbeam footprint

Chapter 6:

STAP for SBR

estimated adaptive weight vector corresponding to (6.30) is given by the sample matrix inversion (SMI) method as ˆ −1 s(c t , ωdt ) ˆk =R w k

(6.33)

and the corresponding estimated power output is given by ˆ ∗k s(c k , ωd )|2 . PSMI,k (θ AZ , ωd ) = |w

(6.34)

Another useful metric for evaluating the performance of a particular STAP algorithm is the signal power to interference plus noise ratio (SINR) deﬁned in (5.95) SINR =

|w∗ s|2 , w∗ Rw

(6.35)

where w is the associated adaptive weight vector and R is the ideal clutter plus noise covariance matrix deﬁned in (6.20). In the case of SMI, for example, (6.35) can be written as (use (6.33)) SINR =

ˆ −1 s|2 |s∗ R . ˆ −1 s ˆ −1 RR s∗ R

(6.36)

Clearly the performance of (6.36) is bounded by the ideal matched ˆ = R in (6.36). This gives ﬁlter output SINRideal obtained by letting R SINRideal = s∗ (c, ωd )R−1 s(c, ωd ).

(6.37)

Figure 6.16 shows the ideal matched ﬁlter output SINRideal in (6.37) for PRF = 500 Hz as a function of target velocity V with and without using Taylor weights on transmit. From there, when uniform weights are used on transmit, the target velocity has to exceed 28 m/s for unambiguous detection up to 5-dB loss (or an appropriate user deﬁned threshold). When Taylor weights are used on transmit, the target velocity has to exceed 15 m/s for unambiguous detection and this gives the Minimum Detectable Velocity (MDV) bound for this SBR conﬁguration [9], [10]. See also Section 5.3.1 for a discussion on MDV. MDV deﬁned here can be used to evaluate the performance of any algorithm. In what follows we examine the ideal performance in terms of clutter suppression for an SBR by considering the two phenomena— Earth’s rotation and range foldover—that invariably affect the SBR data collection process.

233

Space Based Radar 0

0

−10

−10

−20

−20

SINR in dB

SINR in dB

234

−30 −40

−30 −40

MDV

MDV −50 −60

−50 −30

−20

−10

0 10 V (m/s)

20

30

(a) Uniform weights

−60

−30

−20

−10

0 10 V (m/s)

20

30

(b) Taylor weights

FIGURE 6.16 Matched ﬁlteroutput and Minimum Detectable Velocity (MDV) (PRF = 500 Hz).

6.3 MDV with Earth’s Rotation and Range Foldover The two phenomena—Earth’s rotation (crab angle and crab magnitude; Section 4.6–4.7) and range foldover (Section 4.5)—contribute to the degradation in performance for clutter suppression and target detection. To quantify their effects in target detection performance, we consider the following four possible scenarios: A. Without Earth’s rotation and without range foldover B. Without Earth’s rotation and with range foldover C. With Earth’s rotation and without range foldover

(6.38)

D. With Earth’s rotation and with range foldover These four situations are ﬁrst analyzed using the ideal clutter coˆc variance matrix Rc and then using the estimated covariance matrix R (Section 6.8) for various STAP algorithms. Recall that range foldover occurs when pulse returns from previously (later) transmitted pulses that are returning from farther (near) range points contribute to the current pulse return. For example, in Table 6.3 the range bin of interest is located at R = 500 km. However, as Table 6.3 shows the pulse returns from that range are also complemented by those arriving from ranges R1 = 858.7 km, R2 = 1180.5 km, R3 = 1489.8 km, R4 = 1793.7 km, R5 = 2095 km, and R6 = 2395.2 km (see Figure 4.13). The corresponding elevation angles and array gains for these locations are shown in Table 6.3 as well. In addition, the normalized Doppler frequencies associated with these ranges are also tabulated in Table 6.3 for θ AZ = 90◦ and 60◦ . Notice that the

500 43.541 0 0.4723 −0.6463

TABLE 6.3 Range foldover information for range = 500 km

858.73 56.635 −26.163 0.0238 −0.2164

1180.5 62.346 −43.881 −0.8226 −0.5364

1489.8 65.289 −35.159 0.7980 0.0086

1793.7 66.852 −42.879 0.6129 −0.7490

2095 67.618 −52.267 0.5263 −0.1680

2395.2 67.88 −57.565 0.4974 0.0257

Chapter 6:

Ranges that contribute to clutter return (km) Corresponding elevation angle (degree) Array gain in elevation direction (dB) Normalized Azimuth Angle = 90◦ clutter Doppler Azimuth Angle = 60◦

STAP for SBR

235

Space Based Radar 0

0

−10

−10 A, B, C SINR in dB

−20 SINR in dB

236

−30 −40

D

−20

A, B, C, D

−30 −40 −50

−50

−60

−60 −40 −30 −20 −10

0

10

20

30

40

V(m/s) No Crab, No Foldover

(a) Azimuth angle = 90°

No Crab, Foldover

−40 −30 −20 −10 0 10 V(m/s) Crab, No Foldover

20

30

40

Crab, Foldover

(b) Azimuth angle = 60°

FIGURE 6.17 Matched ﬁlteroutput SINR in (6.37) vs. velocity with/without Earth’s motion and with/without range foldover for range = 500 km. (A) No Earth’s rotation, no foldover, (B) No Earth’s rotation, foldover, (C) Earth’s rotation, no foldover, and (D) Earth’s rotation, foldover.

most signiﬁcant range foldover occurs at −26 dB that corresponds to ωd = 0.0238 (velocity = 1.04 m/s) for θ AZ = 90◦ and its effect can be seen in Figure 6.17 (a) as widening the notch when both Earth’s rotation (crab) and range foldover are present. In another example, Table 6.4 corresponds to range of interest 1,200 km with corresponding matched ﬁlter performance details as shown in Figure 6.18. From Table 6.4, the most signiﬁcant clutter foldover return occurs at −4 dB that corresponds to ωd = 0.7823 (velocity = 26.4 m/s) for θ AZ = 90◦ and ωd = 0.1144 (velocity = 3.9 m/s) for θ AZ = 60◦ . Their effect is visible in Figure 6.18 in terms of wider notches and lower SINR levels when both Earth’s rotation and range foldover are present. From Figures 6.17–6.18, when Earth’s rotation and range foldover are present together, the SINR performance substantially degrades compared to the three other case (A, B, C) in (6.38). To illustrate the limitations imposed on processing gains due to Earth’s rotational phenomenon and range foldover, Figure 6.19 shows the matched ﬁlter output in terms of SINR loss deﬁned in (6.37) without (ideal) and with range foldover and Earth’s rotation present when Taylor weights are used on transmit. The resulting range dependent Doppler shift in Figure 6.19 (c) can be adjusted to reﬂect the normalized performance as shown in Figure 6.19 (a). However, as seen in Figure 6.19 (d), the performance is signiﬁcantly degraded when both range foldover and Earth’s rotation are jointly present and rangedependent Doppler compensation is not possible.

524.87 44.819 −57.035 0.2046 −0.8496

TABLE 6.4 Range foldover information for range = 1,200 km

879.42 57.125 −16.458 −0.0544 −0.3083

1200 62.589 0 −0.8555 −0.3162

1508.9 65.419 −4.076 0.7823 0.1144

1812.5 66.919 −10.46 0.6053 −0.6974

2113.7 67.647 −15.283 0.5232 −0.1465

2413.9 67.883 −17.208 0.4971 0.0285

Chapter 6:

Ranges that contribute to clutter return (km) Corresponding elevation angle (degree) Array gain in elevation direction (dB) Normalized Azimuth Angle = 90◦ clutter Doppler Azimuth Angle = 60◦

STAP for SBR

237

0

0

−10

−10

A, B, C

A, B, C

−20

−20

SINR in dB

SINR in dB

Space Based Radar

−30 −40

−30 −40 D

−50

−50

D

−60 −30

−60 −20

−10

0 10 V(m/s)

20

No Crab, No Foldover

−30

30 No Crab, Foldover

−20

−10

Crab, No Foldover

(a) Azimuth angle = 90°

0 10 V(m/s)

20

30

Crab, Foldover

(b) Azimuth angle = 60°

2,400 2,200 2,000 1,800 1,600 1,400 1,200 1,000 800 −1

0 −10 −20 −30 −40

Range (km)

Range (km)

FIGURE 6.18 Matched ﬁlteroutput SINR in (6.37) vs. Velocity with/without Earth’s rotation and with/without range foldover for range = 1,200 km. (A) No Earth’s rotation, no foldover, (B) No Earth’s rotation, foldover, (C) Earth’s rotation, no foldover, and (D) Earth’s rotation, foldover.

−50 −0.5 0 0.5 Normalized Doppler

1

−60

(a) w/o range foldover, w/o Earth’s rotation 2,400 2,200 2,000 1,800 1,600 1,400 1,200 1,000 800 −1

0 −10 −20 −30 −40 −50 −0.5 0 0.5 Normalized Doppler

1

(c) w/o range foldover, w/Earth’s rotation

2,400 2,200 2,000 1,800 1,600 1,400 1,200 1,000 800 −1

0 −10 −20 −30 −40 −50 −0.5 0 0.5 Normalized Doppler

1

−60

(b) w/range foldover, w/o Earth’s rotation

−60

Range (km)

Range (km)

238

2,400 2,200 2,000 1,800 1,600 1,400 1,200 1,000 800 −1

0 −10 −20 −30 −40 −50 −0.5 0 0.5 Normalized Doppler

1

−60

(d) w/range foldover, w/Earth’s rotation

FIGURE 6.19 STAP matched ﬁlteroutput with and without range foldover and Earth’s rotation for an SBR located at height 506 km above ground with PRF = 500 Hz and θAZ = 90◦ . Taylor weights (azimuth direction) are used on transmit.

Chapter 6:

STAP for SBR

0 −10 A SINR in dB

−20 −30 −40 D

−50 −60 −30

−20

−10

0 V(m/s)

10

20

30

FIGURE 6.20 Matched ﬁlteroutput with and without (ideal) range foldover and Earth’s rotation for range 2,400 km and azimuth angle of 90◦ . (A) No Earth’s rotation, no foldover, and (D) With Earth’s rotation, and foldover present.

This effect can be seen in Figure 6.20 as well, which corresponds to the SINR loss in (6.37) as range equals to 2,400 km. The performance in terms of clutter nulling is seen to be signiﬁcantly inferior when both of these phenomena are present. This is evidently the case in practice, which in turn presents signiﬁcant challenges for target detection and tracking [8], [11]. To understand why the clutter notch widens and degrades in the presence of Earth’s rotation, as shown in Figure 6.20, it is necessary to review (6.24), (6.20)–(6.21), and (6.37) simultaneously. Figure 6.20 is plotted for a ﬁxed ( Ro , θ AZo ) which ﬁxes the cone angle c in (6.4), while varying ωd in (6.37). As a result, the cone angle c o = sin θ E L o cos θ AZo

(6.39)

remains ﬁxed. However, referring to (6.20)–(6.23) and (6.37), there are many other points ( Rk , θ AZk ) or equivalent locations at (θ E L k , θ AZk ) that satisfy the identity c o = sin θ E L k cos θ AZk

(6.40)

for the c o in (6.39) and the dotted line in Figure 6.21 shows the isocone contour plot in (6.40). These isocone contour can also be seen in Figure 6.22 that shows the azimuth-range proﬁle as seen from the SBR.

239

Space Based Radar 60

1

k

0

80

58

qAZ

56

wd

k

k

54

wd

0

70

1

wd

1

50

Projected doppler w/ Earth’s rotation Projected doppler w/o Earth’s rotation Iso-cone

50

75°

60

52

40 30

46

w~d

w~d ≠ w~d ≠ w~d 44 k

0

qAZ

Iso-cone

48

1

qAZ

Azimuth

wd = wd = wd

Doppler

240

k

20

w~d

0

0

42

R0 0

500 Rk

1,000

w~d

1

R1 1,500 Range

10 2,000

2,500

FIGURE 6.21 Projected Doppler with and without Earth’s rotation. Rt = 1,000 km, θAZ = 75◦ .

The points on the isocone contour give the best match for the spatial part of the steering vector in (6.10), and hence they contribute the maximum toward the null in Figure 6.20. However, the Doppler projected by these points satisfy (4.48) or (4.79) depending on whether Earth’s rotation is absent or present (see also (6.24)). Thus, when Earth’s rotation is absent, the projected Doppler ωdk at these selected points agree exactly with ωdo , the Doppler associated with ( Ro , θ AZo ), giving a perfect match, together with the temporal part of the steering vector in (6.10) (see dashed line in Figure 6.21). This results in a single null in the output SINR plot. However, in the presence of Earth’s rotation, the projected Doppler values ω˜ dkm = βo ρc sin θ E L m cos(θ AZk + φc ) at these selected points are different from the Doppler associated with ( Ro , θ AZo ), as seen from the solid curve in Figure 6.21. Hence, the adaptive processor tries to null out the “secondary sources” at ω˜ d1 , ω˜ d2 , . . . , All points on an isoc0 = 0 cone line project a single scatter

Iso-cone (Iso-Doppler)

c0 = 0 Iso-Doppler

Points on an isocone line project different scatters

Iso-cone −ck

ck

Isorange

(a) Earth’s rotation absent

−ck

ck

(b) Earth’s rotation present

FIGURE 6.22 Clutter with and without Earth’s rotation.

Isorange

Chapter 6:

STAP for SBR

and taken together, the corresponding SINR plot results in a wider clutter null.

Clutter Notch Width To quantitatively evaluate the width of the clutter notch in the presence of Earth’s rotation, consider a point of interest at range Ro with azimuth angle θ AZo , and let R1 represent its dominant foldover range. Let θ E L o and θ E L 1 represent the elevation angles at these range points. Thus, c o = sin θ E L o cos θ AZo = sin θ E L 1 cos θ AZ1

(6.41)

represents the cone angle of interest, and θ AZ1 represents the azimuth angle at range R1 that projects the same cone angle c o . Then, with ωdo and ωd1 representing the Doppler at points ( Ro , θ AZo ) and ( R1 , θ AZ1 ), we have in the absence of Earth’s rotation (Figure 6.23 (a)) ωdo = ω d1 = β o c o

(6.42)

and with Earth’s rotation present, we have (Figure 6.23 (b)) ω˜ do = βo ρc sin θ E L o cos(θ AZo + φc ),

(6.43)

ω˜ d1 = βo ρc sin θ E L 1 cos(θ AZ1 + φc ).

(6.44)

SINR

SINR

s*0 R−1s0

wd0 = wd1 = wdk

s~*1 R −1s~1

s~0* R −1s~0 w~dk wd

(a) Without Earth’s rotation

~

w d0

w~d1

wd

(b) With Earth’s rotation present

FIGURE 6.23 Widening of the clutter notch with Earth’s rotation (a) without Earth’s rotation, (b) with Earth’s rotation.

241

242

Space Based Radar From (6.41) and (6.42) and in the absence of Earth’s rotation, both points ( Ro , θ AZo ) and ( R1 , θ AZ1 ) project the same steering vector so = s(c o , ωdo ) = s(c o , βo c o ).

(6.45)

More generally, it follows that in the absence of Earth’s rotation all points on the isocone contour in Figure 6.22 (a) project the same space-time steering vector. However, in the presence of Earth’s rotation this is no longer true, and as Figure 6.22 (b) shows, the isocone plots and iso-Doppler plots do not coincide. As a result, every point on the isocone contour projects a different Doppler and hence they generate distinct steering vectors. Thus for the situation in (6.43) and (6.44), we obtain the distinct steering vectors s˜ o = s(c o , ω˜ do ),

s˜ 1 = s(c o , ω˜ d1 ).

(6.46)

This results in the total covariance matrix in (6.20), due to ranges Ro and R1 , to be given by

* R=

Qo + so s∗o

without Earth’s rotation

Q1 + s˜ o s˜ ∗o + g˜s1 s˜ ∗1

with Earth’s rotation,

(6.47)

where g < 1 represents the normalized array gain at (θ E L 1 , θ AZ1 ). Using the matrix identity (see Appendix 6-A for a proof) (Q + ss∗ ) −1 = Q−1 −

Q−1 ss∗ Q−1 1 + s∗ Q−1 s

,

(6.48)

and with the speciﬁc eigenstructure of R in (6.47), it is easy to show that in the absence of Earth’s rotation (see (6.64) for a proof) s∗ R−1 s ≥ s∗o R−1 so =

s∗o Q−1 o so

1 + s∗o Q−1 o so

(6.49)

for any s corresponding to ωd = ωdo indicating the existence of a unique null at ωdo as shown in Figure 6.23 (a). However, in the presence of Earth’s rotation (see paragraph after (6.64)) s∗ R−1 s > s˜ ∗o R−1 s˜ o ,

s∗ R−1 s > s˜ ∗1 R−1 s1

(6.50)

are satisﬁed locally, indicating multiple local nulls as shown in Figure 6.23 (b). Here, repeated use of (6.48) in (6.47) gives (when Earth’s rotation is present, see Appendix 6-B) s˜ ∗o R−1 s˜ o =

A + ( AB − |C|2 )g 1 + A + ( B + AB − |C|2 )g

(6.51)

Chapter 6:

STAP for SBR

and s˜ ∗1 R−1 s˜ 1 =

B + ( AB − |C|2 ) 1 + A + ( B + AB − |C|2 )g

(6.52)

where ˜ o , B = s˜ ∗1 Q−1 ˜ 1 , and C = s˜ ∗o Q−1 ˜1. A = s˜ ∗o Q−1 1 s 1 s 1 s

(6.53)

It is reasonable to assume that A B. However, since the normalized gain g < 1, from (6.51) and (6.52) we obtain s˜ ∗o R−1 s˜ o < s˜ ∗1 R−1 s˜ 1 .

(6.54)

The situation is shown in Figure 6.23 (b), that exhibits two dips corresponding to ω˜ do and ω˜ d1 with the dominant one at ω˜ do , and hence a wider overall null. The sharp null shown in Figure 6.23 (a) and its absence in Figure 6.23 (b) can be explained by expressing the covariance matrix R in terms of cone angles. From (6.42) and in the absence of Earth’s rotation, all range foldover points project the same Doppler for the same cone angle. Hence, using (6.20) we get R=

No

Pi si si∗ + σn2 I

(6.55)

i=0

where si = s(c i , βo c i ),

i = 1, 2, . . .

(6.56)

as in (6.45), and from (6.20), we have Pi =

Na

Pi,m G(θ i,m ).

(6.57)

m=0

The summation in (6.57) is along the isocone curve c = c i in Figure 6.23 (a) and it represents the total power reﬂected along the cone angle c i . Notice that the summation in (6.55) covers the entire cone angle set (the ﬁeld of view of all scattering points). However, in the presence of Earth’s rotation and for the same cone angle c i , different range foldovers generate different Doppler frequencies ω˜ di,o , ω˜ di,1 , . . ., that are obtained as in (6.43) and (6.44), with c o replaced by c i in (6.41). Deﬁne s˜ 2i = s(c i , ω˜ di,o ),

s˜ 2i+1 = s(c i , ω˜ di,1 ),

i = 0, 1, 2, . . .

(6.58)

243

Space Based Radar for the two range foldovers, and (6.55) becomes R=

No

P˜ 2i s˜ 2i s˜ ∗2i + P˜ 2i+1 s˜ 2i+1 s˜ ∗2i+1 + σn2 I.

(6.59)

i=0

Note that using the particular structure (linear dependency between Doppler and cone angle) in (6.56), together with Brennan’s rule in (5.108), the rank of R in (6.55) is given by (β = βo /d) r B = min{MN, N + β( M − 1), No }

(6.60)

which is much smaller than the total number of scatters No . However, no such relation exists in (6.59) and hence, in general the clutter subspace has full rank in presence of Earth’s rotation. This allows the following eigendecomposition for R in (6.55)–(6.59)

R=

 r MN B   (λk + σn2 )uk u∗k + σn2 ui ui∗ ,   i=r B +1

k=1

MN    ( λ˜ k + σn2 ) u˜ k u˜ ∗k , 

without Earth’s rotation, with Earth’s rotation,

k=1

(6.61)

where λk , uk , and λ˜ k , u˜ k represent the clutter subspace eigenvalues and eigenvectors without and with Earth’s rotation, respectively. Figure 6.24 shows the eigenvalue spread of the clutter covariance 60 With Earth’s rotation

50 Eigenvalues in dB

244

40 30 20 Without Earth’s rotation

10 0 −10

0

100

200

300 Index

400

500

600

FIGURE 6.24 Clutter subspace rank with and without Earth’s rotation. A 32-sensor array with 16 pulses is used with Brannan factor β = 10.

Chapter 6:

STAP for SBR

matrix R with and without Earth’s rotation. Note that in the absence of Earth’s rotation and using Brennan’s rule, the clutter subspace is of smaller dimension compared to the whole space. However, the clutter subspace occupies the whole space in the presence of Earth’s rotation. Thus, irrespective of range foldover effect, the clutter subspace rank is determined by the presence or absence of Earth’s rotation. In particular, when Earth’s rotation is absent, from (6.55) and (6.61), MN the noise subspace eigenvectors {ui }i=r are orthogonal to the clutter B +1 scatter steering vector set {so , s1 , s2 , . . .} in (6.55) and (6.56); i.e., {so , s1 , s2 , . . . , s No }⊥{ur B +1 , ur B +2 , . . . , u MN }.

(6.62)

Using (6.61), the output SINR in (6.37) simpliﬁes to

 r B ∗ 2 MN uk s ∗ 2  2  ui s ,  + σ  n 2  λ + σ k n i=r B +1 s∗ R−1 s = k=1 MN ∗ 2  ˜ u s  k   ,  ˜ k + σn2 λ k=1

without Earth’s rotation

with Earth’s rotation. (6.63)

This is plotted in Figure 6.20 and Figure 6.23, by ﬁxing the steering vector cone angle to that corresponding to the desired location (c = c o ) and spanning over all Doppler frequencies. As a result, in the absence of Earth’s rotation, so in (6.45) is the only steering vector generated in this manner that corresponds to the clutter scatter set in (6.62) and hence, from (6.62) and (6.63) we get, s∗o R−1 so

r B ∗ 2 uk s o = ≤ s∗ R−1 s. λk + σn2

(6.64)

k=1

In (6.64), the inequality follows for all other steering vectors s = so in (6.62) since they are not orthogonal to the noise subspace eigenvectors, and hence contribute to the noise subspace terms as well. Therefore, a unique null appears at ωdo as shown in Figure 6.23 (a) when Earth’s rotation is absent. A similar reasoning shows that in the presence of Earth’s rotation, with a ﬁxed cone angle c o for s in (6.63), s˜ o and s˜ 1 given in (6.46) are the only two steering vectors generated that correspond to the clutter scatter set in (6.58), and they span the same subspace as that of the clutter subspace eigenvectors {u˜ 1 , u˜ 2 , . . .}. Hence, at both ω˜ do and ω˜ d1 , the output SINR tends to be lower than at other Doppler frequencies as in (6.50), thus generating a wider null as shown in Figure 6.23 (b). This is especially true when array gain factors are included in (6.59) as in (6.20). In this case, the clutter subspace need not span the whole space even in the presence of Earth’s rotation, and using a similar

245

246

Space Based Radar reasoning as in (6.64), sharper nulls can be expected at both ω˜ do and ω˜ d1 thus, resulting in a wider clutter notch.

6.4 Range Foldover Minimization Using Orthogonal Pulsing When range foldover and Earth’s rotation are present simultaneously, distinct Doppler frequencies are generated from the associated range foldover points (see Figure 6.25). These range foldover points will project different gains depending on their position with respect to the array gain pattern (Figure 6.25 (b)). For every range point of interest, the two phenomena together generate a sequence of Doppler frequencies. For example, at PRF = 500 Hz, there are seven range foldover Doppler frequencies for an SBR located at a height of 506 km above Earth’s surface. The clutter corresponding to these range bins will be associated with these modiﬁed Doppler frequencies. As Figures 6.17–6.20 show, having both range foldover and Earth’s rotation present at the same time results in unacceptable performance degradation. Interestingly, the performance can be resorted to an acceptable level if only either effect is present in the data. As remarked earlier, the linear relationship between the spatial and Doppler frequencies is lost when a crab angle, induced by the Earth’s

(qEL,m’ qAZ,j)

it sm an Tr lses pu

Iso-cone …

SBR

ge

...

n ra d ar rs rw ve Fo ldo fo

Mainbeam

rn nt tu fro Re ave w

qAZ

…

D: Point of interest, cone angle c0

…

Azimuth sidelobes

d ar w s ck ver Ba ldo fo

Returns due Range point of interest to later pulses

Elevation sidelobes

Returns due to earlier pulses

Range ambiguities Range of interest

(a) Side view

FIGURE 6.25 Range foldover phenomenon.

(b) Top view from SBR

Chapter 6:

STAP for SBR

rotation around its own axis, appears in the formulation of the Doppler frequency. As we have seen, this in turn, signiﬁcantly degrades the performance of adaptive processing algorithms. Several solutions have been proposed to mitigate the impact of the crab angle. One such solution involves inclining the array with an angle commensurate (in the negative direction) with the induced crab angle. This solution appears to be highly sensitive to errors in such compensation [8]. As a result, a signal processing solution involving waveform diversity is generally preferred over the mechanical correction solution. Waveform diversity, in general, refers to the use of various waveforms in a transmitter/receiver design, with the objective of improving the overall system performance, including detection, estimation, and identiﬁcation of targets embedded in clutter, jamming, and noise. Depending upon the platform structure (monostatic, bistatic, multistatic) and the use of single vs. multiple apertures, waveform diversity can spatially augment the dimensionality of the processing space, which then permits the transmission of different waveforms, with the goal of achieving a ﬁner separation of the target from the interference. Waveform diversity also allows for distinct waveforms of different durations over different spectral bands to be used in both time and frequency domains. The extended multidimensionality allows for improved target detection and interference cancellation since the target and interference signals are well localized in this extended space. In presence of additional Doppler component induced by Earth’s rotation around its axis and the presence of range ambiguities, waveform diversity can effectively enhance the performance of adaptive processing techniques. In the present context, waveform diversity can be used on the sequence of transmitted radar pulses to realize the above goal by minimizing the effects of range foldover returns. As a result, the data contains mainly the effect of Earth’s rotation only and the performance can be restored. Recall that in ordinary practice, a set of identical pulses are transmitted as in Figure 6.26 (a). To suppress the returns due to range foldover, for example, individual pulses f 1 (t), f 2 (t), . . . , as shown in Figure 6.26 (b) can be made orthogonal to each other so that

5

To

f i (t) f j (t)dt = δi, j ,

i, j = 1, 2, . . . , Na ,

(6.65)

o

with To representing the common pulse length and Na corresponding to the maximum number of distinct range foldovers present in the data. Here δi, j is the standard Kronecker delta product. Then, with appropriate matched ﬁltering as shown in Figure 6.27, the range ambiguous returns can be minimized from the main return corresponding

247

248

Space Based Radar

f (t)

f (t)

Rectangular orthogonal pulses (Na pulses)

Conventional radar pulses f1 T0

Tr

f2

f3

Tr

...

... t

t

(a)

(b)

FIGURE 6.26 Radar pulse stream with and without waveform diversity.

to the range of interest. The decision instant T satisﬁes T ≥ To to maintain (6.65). In this case, performance will be closer to that shown in Figure 6.19 (c). Note that for range foldover elimination, waveform diversity needs to be implemented only over Na pulses as shown in Figure 6.26 (b). For an SBR located at a height of 506 km and an operating PRF = 500 Hz, we get Na = 7. To understand how waveform diversity helps to minimize the range foldover effect, in Figure 6.28 assume that we have Na = 3 different range foldover points and the range R1 is the main return. Let the corresponding clutter scatter returns for these three different range foldovers be c 1 , c 2 , and c 3 . Also assume that M = 6. At t = 0, the array sends out pulse f 1 (t) and it arrives at R1 at t = T. At this time instant as shown in Figure 6.28, the earlier waveforms f 3 (t) and f 2 (t) returning from locations R2 and R3 are also present at t = kTr

t = kT Data

f1(t - T )

x

k

T Data

f1(t −T ) t = (k +1)Tr f2(t −T ) t = (k + 2)Tr

Σ

f3(t −T )

xk

f4(t −T ) t = (k + M − 1)Tr (a) Conventional matched filter

(b) Bank of matched filters

FIGURE 6.27 Matched ﬁlterswith and without waveform diversity.

Chapter 6: Tx

f3

f2

f1

STAP for SBR

c1 f1 + c2 f3 + c3 f2 f3

at t = ∆T f2

f1 c3 f2 c1 f1 R1

c2 f3 R2 R3

FIGURE 6.28 Wavefront present at R1 for t = T.

R1 . Thus, the wavefront at R1 at t = T equals c 1 f 1 (t) + c 2 f 3 (t) + c 3 f 2 (t).

(6.66)

At t = 2 T, the ﬁrst pulse wavefront returns to the array and the array uses the output of the receiver h 1 (t) matched to the pulse f 1 (t). Thus the array output received data equals x 1 = c 1 ρ11 a R1 e − j0ωd1 + c 2 ρ31 a R2 e − j0ωd2 + c 3 ρ21 a R3 e − j0ωd3

(6.67)

where

5Tr ρi j =

f i (t)h ∗j (t)dt

(6.68)

0

and a Ri corresponds to the spatial steering vector for range Ri and ωdi corresponds to the Doppler shift for range Ri . Similarly (see Figure 6.29), at t = Tr +2 T, the second pulse returns to the receiver and the output is given by the receiver h 2 (t) that is matched to f 2 (t). Thus, the received data for second pulse is x 2 = c 1 ρ22 a R1 e − jωd1 + c 2 ρ12 a R2 e − jωd2 + c 3 ρ32 a R3 e − jωd3 .

(6.69)

In a similar manner, the received data for the third, fourth, ﬁfth, and sixth pulses are given by x 3 = c 1 ρ33 a R1 e − j2ωd1 + c 2 ρ23 a R2 e − j1ωd2 + c 3 ρ13 a R3 e − j1ωd3 ,

(6.70)

x 4 = c 1 ρ11 a R1 e − j3ωd1 + c 2 ρ31 a R2 e − j3ωd2 + c 3 ρ21 a R3 e − j3ωd3 ,

(6.71)

249

250

Space Based Radar Tx

f3

f2

f1

c1 f2 + c2 f1 + c3 f3 f3

at t = Tr + ∆T f2

f1 c3 f3 c2 f1

c1 f2 R1

R2 R3

FIGURE 6.29 Wavefront present at R1 for t = Tr + T.

x 5 = c 1 ρ22 a R1 e − j4ωd1 + c 2 ρ12 a R2 e − j4ωd2 + c 3 ρ32 a R3 e − j4ωd3 ,

(6.72)

x 6 = c 1 ρ33 a R1 e − j5ωd1 + c 2 ρ23 a R2 e − j5ωd2 + c 3 ρ13 a R3 e − j5ωd3 ,

(6.73)

respectively. If we stack up the data for all the pulses, we get











x1 ρ11  x2   ρ22       x=  ..  = c 1  ..  b(ωd1 )  ⊗ a R1 . . xM ρ33













ρ31 ρ21 ρ  12    ρ32    .  b(ωd2 )  ⊗ a R + c 3  .  b(ωd3 )  ⊗ a R +c 2  2 3  .    .   . . ρ23 ρ13 (6.74) Note that the correlation coefﬁcient ρii = 1 so that the space-time steering vector corresponding to R1 will not be changed. For truly orthogonal waveforms, ρi j = 0, i = j and hence the inner summation in (6.19) and (6.20) that represent the range foldovers returns is eliminated. However, for approximately orthogonal waveforms, ρi j = 0, i = j. In that case, for the range foldover points R2 and R3 , the space-time steering vector will change since ρi j are not the same when i = j. This amounts to an amplitude modulation for the temporal part of the steering vector and results in inferior performance if the waveforms have different correlations (see also Section 6.6). However, if all ρi j can be made equal for i = j, the steering vector will not change for the range foldover data. In that case with

Chapter 6:

STAP for SBR

ρi j = ρ, i = j, we have



 





x1 1 x  2  1    = c 1  .  b(ωd1 )  ⊗ a R x= . 1  .   .   . . 1 xM       ρ ρ  ρ    ρ    .  b(ωd2 )  ⊗ a R + c 3  .  b(ωd3 )  ⊗ a R +c 2  2 3  .    .   . . ρ ρ = c 1 b(ωd1 ) ⊗ a R1 + ρc 2 b(ωd2 ) ⊗ a R2 + ρc 3 b(ωd3 ) ⊗ a R3 = c 1 s(θ 1 , ωd1 ) + ρc 2 s(θ 2 , ωd2 ) + ρc 3 s(θ 3 , ωd3 ). (6.75) In summary, the effect of waveform diversity is to scale down the scatter return power from range foldover return points so that the impact of the inner summation terms in (6.20) is minimized except for the dominant ﬁrst return term. Figures 6.30–6.31 show the improvement in SINR obtained by using eight rectangular orthogonal pulses as shown in Figure 6.26 (b). Note that the performance is restored since the eight waveforms are able to successfully eliminate the seven range foldover ambiguities present there. Figure 6.33 shows the SINR improvement using a more practical set of waveforms—four up/down chirp waveforms all with equal 0

0 −10

−10 (i), (iii)

−20

SINR in dB

SINR in dB

(i), (iii) (ii)

−30 −40

Ideal Conventional 8-Ortho.

−50 −60 −40

−20

0

20

V (m/s)

(a) Range = 500 km

40

−20 −30 (ii)

−40

Ideal Conventional 8-Ortho.

−50 −60

−30

−20

−10

0

10

20

30

V (m/s)

(b) Range = 1,200 km

FIGURE 6.30 SINR performance improvement with and without using eight rectangular waveforms. (i) Ideal (no Earth’s rotation, no range foldover), (ii) Earth’s rotation and range foldover with conventional pulsing, (iii) Earth’s rotation and range foldover with eight-orthogonal pulsing.

251

Space Based Radar 2,400 2,200 2,000

0

2,400

0

−10

2,200

−10

2,000

−20

1,800 1,600

−30

1,400

−40

1,200

Range (km)

Range (km)

252

800 −1

−0.5

0

0.5

−30

1,400

−40 −50

1,000

−60

1

1,600 1,200

−50

1,000

−20

1,800

800 −1

−0.5

0

0.5

−60

1

Normalized Doppler

Normalized Doppler

(a) Conventional

(b) Eight rectangular waveforms

FIGURE 6.31 Matched ﬁlteroutput SINR with range foldover and Earth’s rotation for two different pulsing schemes (a) Conventional pulsing (same as Figure 6.20 (d)), (b) Eight rectangular orthogonal waveforms.

bandwidth and whose instantaneous frequencies are as shown in Figure 6.32. Quadrature phase shifting of these waveforms will generate an additional set of four waveforms, resulting in a pool of eight waveforms. Note that these eight waveforms are only approximately orthogonal. In this case, although the performance has improved over the conventional pulsing scheme considerably, the remaining degradation compared to the ideal case can be attributed to the approximate orthogonal nature of these waveforms. Figure 6.34 (b) shows the improvement in SINR as a function of range and Doppler obtained by using these chirp waveforms. For comparison purposes, Figure 6.34(a) shows the performance using conventional pulsing when both range foldover and Earth’s rotation are present. Note that using waveform diversity at transmit, the performance in Figure 6.34(b) is restored to that shown in Figure 6.19 (c), where only Earth’s rotation is present. Although the correlation between the chirp waveforms shown in Figure 6.32 is about −20 dB, their correlation is not uniformly low. The off-diagonal entries of their correlation matrix R1 are unequal as

B0

w 2(t )

w 1(t )

B0

B0

w4(t ) B0

t

t

T

T

T (a)

w 3(t )

(b)

t

t

(c)

FIGURE 6.32 Up/down chirp waveforms in frequency domain.

T (d)

0

0

−10

−10 (i), (iii)

−20

SINR in dB

SINR in dB

Chapter 6:

(ii) −30 −40 Ideal Conventional 8-Chirp

−50 −60

−20

−40

0

20

STAP for SBR

(i)

−20

(iii)

−30 (ii) −40 Ideal Conventional 8-Chirp

−50 −60

40

−30

−20

−10

10

0

30

20

V (m/s)

V (m/s)

(a) Range = 500 km

(b) Range = 1,200 km

FIGURE 6.33 SINR performance improvement with and without orthogonal pulsing using eight up/down chirp waveforms for range = 1,200 km. (i) Ideal (no Earth’s rotation, No range foldover), (ii) Earth’s rotation and range foldover with conventional pulsing, (iii) Earth’s rotation and range foldover with eight up/down chirp pulsing.

shown below.



1

0.0121

0.0007

   1 0.0055 0.0027 0.0196 0   . 0.0055 1 0.0167 0.0049 0.0001    0.0027 0.0167 1 0.0034 0.0043   0.0196 0.0049 0.0034 1 0 

 0.0164   0.0121   0.0007  R1 =   0.0062   0.0057   0.0006

0.0054

0.0179

0.0079

0.0203

0.0036

0.0087

0.0052

0.0030

0

1

0.0073

0.0073

1

0.0049

0.0140

0.0062

0.0057



0.0164

0.0006

0.0052

0.0030 

0.0049

0.0054

0.0079

0.0036

0.0140

0.0179

0.0203

0.0087

0

0

0.0001

0.0043

0

1 (6.76)

2,400

0

2,400

0

2,200

−10

2,200

−10

−20

1,800 1,600

−30

1,400

−40

1,200

−50

1,000 800 −1

−0.5

0

0.5

1

−60

2,000 Range (km)

Range (km)

2,000

−20

1,800 1,600

−30

1,400

−40

1,200

−50

1,000 800 −1

−0.5

−60 0

0.5

1

Normalized Doppler

Normalized Doppler

(a) Conventional

(b) Eight chirp waveforms

FIGURE 6.34 Matched ﬁlteroutput SINR with range foldover and Earth’s rotation for two different pulsing schemes (a) Conventional pulsing (same as Figure 6.19 (d)), (b) Eight-chirp waveforms that are approximately orthogonal.

253

Space Based Radar As a result, the residual leakages at the matched ﬁlter output in Figure 6.27 (b) are different and the steering vector model for STAP needs to be modiﬁed to accommodate them. Of course, equal correlations among the waveforms will retain the standard STAP steering vector, and hence an interesting problem is to design a set of waveforms whose correlations are approximately equal. The hybrid-chirp waveforms f k (t) cos(ωo t+bt 2 ), f k (t) sin(ωo t+bt 2 ), k = 1, . . . , 4, with { f k (t)}4k=1 represents an orthogonal set satisfying this “equal correlation” property. With R2 representing their correlation matrix, we have



1

0.0060

0.0064

 0.0060   0.0064   0.0059  R2 =   0.0062  0.0063    0.0064

0.0063

0.0064

0.0062

0.0064

0.0064

0.0062

0.0066

0.0070

0.0064

0.0059

0.0062

0.0063

0.0064

1

0.0065

0.0056

0.0063

0.0062

0.0064

0.0065

1

0.0055

0.0064

0.0064

0.0062

0.0056

0.0055

1

0.0063

0.0070

0.0064

0.0070

1

0.0061

0.0065

0.0070

0.0061

1

0.0066

0.0064

0.0065

0.0066

1

0.0062

0.0059

0.0057

0.0056

0.0063



0.0070 



0.0064  

0.0062   . 0.0059  

  0.0056 

0.0057  1

(6.77)

Notice that the off-diagonal entries are approximately equal in this case. Figures 6.35–6.36 show the performance improvement for various range levels. On comparing Figures 6.35–6.36 and Figures 6.33– 6.34, we observe that additional improvements can be realized by using hybrid-chirp waveforms.

0

0

−10

−10 (i), (iii)

−20

(ii) SINR in dB

SINR in dB

254

−30 −40 Ideal Conventional 8-Hybrid

−50 −60

−40

−20

0

20

V (m/s)

(a) Range = 500 km

40

−20

(i)

(iii)

−30

(ii)

−40 Ideal Conventional 8-Hybrid

−50 −60

−30

−20

−10

10 0 V (m/s)

20

30

(b) Range = 1,200 km

FIGURE 6.35 SINR performance improvement with and without orthogonal pulsing using eight hybrid-chirp waveforms for different ranges. (i) Ideal (no Earth’s rotation, no range foldover), (ii) Earth’s rotation and range foldover with conventional pulsing, (iii) Earth’s rotation and range foldover with eight hybrid-chirp pulsing (coincides with (i) for range = 500 km).

Chapter 6: 2,400

0

2,400

0

2,200

−10

2,200

−10

−20

1,800 1,600

−30

1,400

−40

1,200

−50

1,000 800 −1

−0.5

0

0.5

1

−60

2,000 Range (km)

2,000 Range (km)

STAP for SBR

−20

1,800 1,600

−30

1,400

−40

1,200

−50

1,000 800 −1

−0.5

0

0.5

1

Normalized Doppler

Normalized Doppler

(a) Conventional

(b) Eight hybrid-chirp waveforms

−60

FIGURE 6.36 Matched ﬁlteroutput SINR with range foldover and Earth’s rotation for two different pulsing schemes (a) Conventional pulsing (same as Figure 6.19 (d)), (b) Eight hybrid-chirp waveforms.

Thus, the waveforms only modify the scatter return for the range foldover point. Hybrid-chirp waveforms give better performance compared to up/down chirp waveforms because the off-diagonal entries of the correlation matrix for the hybrid-chirp waveforms are almost the same whereas the off-diagonal entries of the correlation matrix for the up/down chirp waveforms are more dispersed (see (6.76) and (6.77)).

6.5 Scatter Return Modeling To obtain realistic scatter return amplitudes for clutter data in (6.19), “site-speciﬁc” information can be incorporated into the clutter data modeling problem. In this approach, a land cover map using NASA’s terra satellite based images with 1 km2 resolution categorizes the Earth into 16 land types (desert, lake, forest, etc.) with speciﬁc grazing angle dependent mean radar cross section (RCS) values. Making use of Weibull-type modeling for each type of terrain, random scatter returns can be generated for the entire ﬁeld of view. A further reﬁnement can be realized by introducing wind induced internal clutter motion (ICM) [12]. Terrain classiﬁcation using NASA’s terra satellite image map with a 1 km2 resolution that shows various land cover types and their area of coverage using a 16 type land/water classiﬁcation scheme is considered here [13]. Knowing the type of terrain from the NASA map, speciﬁc terrain statistical data can be used to determine the mean RCS and the actual backscatter amplitude return um, j in (6.18) and (6.19) for

255

256

Space Based Radar that speciﬁc location [14]. At low grazing angles Weibull distributions and at large grazing angles Rayleigh distributions have been found to effectively model the radar backscattering [12], [14]. The parameters required to model these random variables can be tabulated from the mean RCS data available for various terrain types. ICM due to wind is also studied here. The effect of wind is to amplitude modulate the various pulse returns. By modeling the effect of wind as a covariance tapering matrix that modiﬁes the clutter covariance matrix, the performance degradation is studied with site-speciﬁc examples [15]. To illustrate the effect of wind on detection performance, Billingsley’s windblown clutter model is used to generate the “wind” random variables that modulate the various pulse returns. Although Billingsley’s wind model gives the autocorrelation functions in an airborne context, that model is used in the SBR case here to evaluate the ﬁrst order effects of wind on target detection. Parametric modeling of the wind spectrum allows to draw some interesting conclusions in this case.

6.5.1 Terrain Modeling As we have seen earlier, SBR array gain pattern modulates the transmit waveform and depending upon the terrain mean RCS, backscattered returns are generated from the points of interest on the ground. With PT representing the SBR transmitter power, for the clutter patch located at slant range Rs j , the input power density is given by 4πPRT 2 . sj

If a m, j represents the planar scattering area of the clutter patch, then Pi = 4πPRT 2 a m, j represents the input power incident on the patch. With sj

o σm, j representing the normalized mean RCS for the (m, j)th clutter o patch, in this case Pi σm, j represents the scatter power off the clutter patch of interest. This gives the average power received at the receiver sensors located at a distance Rs j from the (m, j)th clutter patch to be

Pr(m, j) =

o Pi σm, j

4π Rs2j

a r = κo

o σm, j

Rs4j

,

(6.78)

where a r represents the effective receiver sensor area. The random backscattered return from the (m, j)th cell will have statistical features that are characteristic to the local terrain type, and this must be taken into account in any meaningful simulation. Figure 6.37 shows a typical map generated from NASA’s Terra Satellite [13]. The map has 1 km patches of the actual Earth categorized into 16 land cover types—forest, urban, croplands, lakes, etc. The o mean RCS value σm, j in (6.78) for each terrain type for moderate to large grazing angles is listed in Table 6.5 along with their shape parameter β that is useful for parametric modeling later (see (6.83)–(6.85)).

Chapter 6:

STAP for SBR

FIGURE 6.37 Land cover map (NASA’s terra satellite). SBR

(m, j )th patch

For instance, urban has higher reﬂectivity, desert has lower reﬂectivity, and water has signiﬁcantly lower reﬂectivity. Equation (6.78) represents the average backscatter return at the sensor from the (m, j)th patch, and to determine the corresponding random return, let um, j represent the random return from the (m, j)th bin such that

o E |um, j |2 = σm, j,

(6.79)

so that the received power (6.78) at the reference sensor can be expressed as Pr(m, j)

   u 2  m, j = κo E 2 = κo E |αm, j |2 ,  Rs j 

(6.80)

with αm, j is as deﬁned in (6.18). Assuming that the clutter patches are of equal size, the constant κo in (6.78)–(6.80) can be normalized. In that case from (6.80), αm, j represents the random scatter return signal from the (m, j)th patch that is received at the reference receiver sensor. As a result, after taking into account the array factor and the space-time steering vector, the received data vector has the form in (6.19). Observe that for data modeling, the random clutter return um, j has been made site-speciﬁc using (6.79) as discussed below.

257

258

Space Based Radar

Terrain Type Urban Deciduous Broadleaf Evergreen Broadleaf Evergreen Needleleaf Deciduous Needleleaf Mixed Forest Vegetation Mosaic Cropland Grassland Savanna Woody Savanna Closed Shrubland Open Shrubland Wetland Snow Barren Water

Low Grazing Angles (0–6◦ ) σ ◦ (dB) Shape Parameter β −20 .5 −21 .77 −23 .77 −26 .77 −25 .77 −24 .77 −28 .42 −30 .42 −34 .77 −38 .77 −36 .67 −37 .67 −39 .67 −40 .77 −50 .91 −45 .91 −60 .33

High Grazing Angles (45◦ ) σ ◦ dB (β = 1) −6 −10 −12 −15 −14 −13 −17 −19 −19 −23 −21 −22 −24 −25 −32 −27 −36

(see also [13])

TABLE 6.5 Mean RCS for various terrain types

o In general, the mean RCS value σ o = σm, j in (6.79) depends on the grazing angle, and various models have been proposed to accommodate the grazing angle factor. In the simplest constant gamma model [12], we have

σ o (ψ) = γ sin ψ,

(6.81)

where γ is a terrain constant. This model is found to be useful for 10◦ – 60◦ grazing angles, barring high grazing angle values that correspond to near nadir points. At high grazing angles, the return power increases signiﬁcantly and to accommodate this an additional term can be introduced to the constant gamma model in (6.81). With an extra constant term added to determine the plateau region, the near RCS equation takes the form [16] π E σ o (ψ) = A + B sin ψ + Ce −D( 2 − ψ) .

(6.82)

Notice that ﬁve parameters (A, B, C, D, E) are required to represent this model and hence it is known as the ﬁve parameter model [16]. These parameters in turn are determined by the terrain type. Figure 6.38

Chapter 6: 20 10

Urban Deciduous Broadleaf Evergreen Broadleaf Evergreen Needleleaf Deciduous Needleleaf Mixed Forest Vegetation Mosaic Cropland Grassland Savanna Woody Savanna Close Shrubland Open Shrubland Wetland Snow Barren Water

Urban

RCS (dB) s o(y)

0 −10 −20 −30 −40 Water

−50 −60

0

10

20

30

40

50

60

STAP for SBR

70

80

90

Grazing Angle

FIGURE 6.38 Five parameter RCS model.

shows σ o (ψ) using the ﬁve parameter model for a variety of terrains. Observe that the midregion value (ψ 10◦ → 70◦ ) agrees with those in Table 6.5 for high grazing angles. By ﬁtting various statistical models to experimental data, it has been well documented that Weibull distributions can effectively model the amplitude or power levels of the backscattered signal, especially at low grazing angles [14]. Recall that Weibull random variables have the following probability density function [17]

f X (x) =

αx β−1 e −αx

β

/β

0

x≥0 otherwise.

(6.83)

Let X = |um, j |2

(6.84)

represent the random backscattered power associated with the backscatter return um, j . From (6.79) its mean value is given by the normalized RCS. The random backscatter power return X in (6.84) is generally modeled as a Weibull random variable with parameters α and β [15]. This gives E{X} = µ X =

β1 β o (1 + 1/β) = σm, j, α

(6.85)

where the last equality follows from (6.79) and (6.84). If we let β = 2 in (6.83), we obtain the Rayleigh distribution. In general, knowing

259

Space Based Radar 0.8

Urban

Mountain

Weibull -shape Parameter b

0 −10 −20 Mean RCS

260

−30

Farm

Rural

−40

Rural Forest Farm Desert Mountain Urban

−50 Desert

−60 −70

0

1

2

3

4

5

6

Desert

0.75 0.7

Rural

0.65 Mountain

0.6 0.55

Urban

0.5 0.45

Farm

0.4 0.35

0

1

Rural Forest Farm Desert Mountain Urban

Grazing Angle (deg)

2 3 4 Grazing Angle (deg)

5

(a) Mean RCS

(b) Weibull parameter b

6

FIGURE 6.39 (a) Mean RCS and (b) Weibull shape parameter β for low-angle clutter over different terrains [12]. See also Table 6.5.

o the shape parameter β and the mean terrain RCS σm, j from Table 6.5, the other Weibull parameter α in (6.83) can be computed from (6.85). o Figure 6.39 shows the mean RCS σm, j and the Weibull shape parameter β for low values of grazing angle [12]. In general Figure 6.38 together with Figure 6.39 give an accurate mean RCS for various terrain types that takes the grazing angle dependency into consideration. Consequently, knowing the locations of the point of interest on Earth and that of the SBR, the terrain types, the mean RCS, and the Weibull random variable parameters for the backscattered power return can be computed for the entire ﬁeld of view using this approach. Interestingly, the return amplitude (magnitude) random variable √ |um, j | = X that is useful for simulation is√also Weibull with parameters 2α and 2β. This follows since Y = X, and with f X (x) in (6.83) representing the probability density function of X, we obtain the probability density function of Y to be1

1 2β f Y ( y) = dy f X ( y2 ) = 2yf X ( y2 ) = 2αy2β−1 e −α y /β

dx β α1 yβ1 −1 e −α1 y 1 /β1 , y ≥ 0 = 0, otherwise

(6.86)

with α1 = 2α, β1 = 2β. This procedure is adapted here to simulate the backscatter amplitude return random variables |um, j | for all points in the ﬁeld of view of the point of interest. Using a uniform random 1 More generally, if X ∼ w(α, β), then Y = Xa is also Weibull with parameters α/a and β/a .

Chapter 6:

STAP for SBR

phase for um, j , scatter returns can be then faithfully simulated. This approach is used in our simulations to generate clutter data as in (6.19) and to study the effect of nonuniform terrain on target detection.

6.5.2 ICM Modeling A realistic clutter model should include other secondary effects in addition to accounting for terrain type variations. For example, forests and lakes are constantly modulated by wind and they affect the pulse returns by suitably amplitude modulating the temporal returns thus affecting the Doppler. For a uniform pulse sequence with PRF = 1/Tr , the temporal steering vector b (ωd ) corresponding to M pulses is given by (6.8). The wind modulated temporal steering vector has the form [2] b˜ (ωd ) = b (ωd ) ◦ w

(6.87)

w = [w1 , w2 , . . . w M ]T ,

(6.88)

where

with w1 , w2 , . . . representing the “wind” random variables and ◦ representing the Schur Product as in (1.61) and (1.62). In other words, the wind random variables amplitude modulate the pulse returns. Let

∗ rw (kTr ) = E wi wi+k

(6.89)

represent the autocorrelation coefﬁcients of the “wind” random variables in (6.88) with Tr representing the pulse repetition interval. For the airborne case, Billingsley has modeled these windblown autocorrelations using the real symmetric function [12] rw (τ ) =

µ + r (τ ) = c o + r (τ ) 1+µ

(6.90)

where co =

µ , 1+µ

(6.91)

and the time-dependent term r (τ ) in (6.90) is given by r (τ ) =

1 (cλ) 2 . 1 + µ (cλ) 2 + (4πτ ) 2

(6.92)

In (6.90)–(6.92), µ represents the DC to AC ratio deﬁned by µ = 489.8 vw−1.55 f o−1.21

(6.93)

261

Space Based Radar 1 10

10 mph 20 mph 30 mph 40 mph 50 mph 60 mph 70 mph 80 mph

0.9 20

0.8 0.7 rw(kTr)

262

0.6

40

0.5 0.4 0.3 0.2

80 0

50

100

150

200

250

300

350

400

450

500

k

FIGURE 6.40 Windblown clutter autocorrelation function with PRF = 500 Hz and a carrier frequency of 1.25 GHz.

with vw representing the wind speed in mph and f o the carrier frequency in GHz and [12] c −1 = 0.1048(log10 (vw ) + 0.4147).

(6.94)

Figure 6.40 shows the windblown autocorrelations rw (kTr ) in (6.89) and (6.90) for PRF = 500 Hz and carrier frequency of f o = 1.25 GHz at various wind speeds. To simulate the windblown random variables in (6.87) and (6.88) that satisfy the autocorrelations in (6.89) and (6.90), only the timedependent portion r (τ ) in (6.92) needs to be modeled, and a variety of techniques can be used for this purpose [18]. A direct approach in this case is to sample the autocorrelations at the desired PRF and thereby deﬁne rk = r (kTr ), k = 0, 1, 2, . . .

(6.95)

in (6.92) and compute the discrete-time spectrum S(ω) =

+∞

rk e − jkω

(6.96)

k=−∞

and identify the underlying real minimum phase system H(z) given by [18] S(ω) = |H(e jω )|2

(6.97)

Chapter 6:

STAP for SBR

where H(z) =

∞

h k z−k .

(6.98)

k=0

Identifying the minimum phase system H(z) involves a spectral factorization, and knowing the system H(z) in (6.98) the desired “wind random variables” in (6.87) and (6.88) can be generated using the relation wk =

h k−i vi +

√

co ,

(6.99)

i

where {vi } represents a white noise sequence of unit spectral density and c o = µ/(1 + µ) as in (6.91), which compensates for the constant term in (6.90). We can use the classic Bauer-type factorization [19] to identify the underlying minimum phase system in (6.98) (see Appendix 6-C). Interestingly in this particular case the system so generated being real can be further modeled using a set of damped real sinusoids. This gives the system response to be h k = h(kTr )

(6.100)

where h(t) =

no

c i e −(αi + jωi )t =

i=−no

no

|c k | e −αk t cos(ωk t − φk ),

t≥0

k=0

(6.101) corresponds to a real ARMA minimum phase system model H(z) =

B(z) b 0 + b 1 z−1 + · · · + b m−1 z−(m−1) = −1 −m a0 + a1z + · · · + amz A(z)

(6.102)

in (6.98), where the model order m = 2no + 1. Using (6.102), the iteration in (6.99) can be expressed as wk = −

m i=1

a i wk−i +

m−1

b i vk−i +

√ c0 ,

(6.103)

i=0

where {a i } and {b i } represent the real coefﬁcients of A(z) and B(z) in (6.102). To illustrate this approach, Table 6.6 gives the damped sinusoidal components and Table 6.7 gives the corresponding ARMA system coefﬁcients for three typical wind speeds (10, 40, and 80 mph) at carrier frequency 1.25 GHz and PRF = 500 Hz [20]. To obtain a similar

263

264

Space Based Radar

|c i |

Wind (mph) 10

40

80

0.068 0.079 0.0223 0.3225 0.3552 0.0417 0.026 0.412 0.812 0.095

Damped Sinusoid Parameters ωi αi f i = 2π 6.5 17.21 4.0500 14.3500 40.0000 61.9000 4.37 15.98 43.34 65.39

0 3.49 0 0 3.68 11.25 0 0 4.33 13.07

φi

0 −146 0 0 −120.62 73.48 0 0 −126 60

TABLE 6.6 Wind modeling using damped sinusoidal components for Billingsley model at carrier frequency = 1.25 GHz, PRF = 500 Hz

parametric form at other frequency and PRF sets, a separate spectral factorization must be carried out. From Tables 6.6–6.7, for example, the effect of wind (10–80 mph) at 1.25 GHz is equivalent to two damped sinusoids at frequencies

Wind (mph)

Order 3

10 6 40

6 80

a0 → am (Top to Bottom)

b 0 → b m−1 (Top to Bottom)

1.000000000 −2.919100657 2.842025760 −0.922887282 1.000000000 −5.557239739 12.879352421 −15.935236822 11.102506392 −4.130502782 0.641120586 1.000000000 −5.522320350 12.723036421 −15.655963091 10.853636468 −4.019910201 0.621520855

0.002425544 −0.005076445 0.002797401 0.006565227 −0.030211407 0.055984419 −0.052176135 0.024462356 −0.004624028 0.0074985190 −0.0341922920 0.0629394360 −0.0583378060 0.0272206780 −0.005127624

TABLE 6.7 ARMA system coefﬁcientsfor Billingsley model at carrier frequency = 1.25 GHz, PRF = 500 Hz. (see Figure 6.42)

Chapter 6:

STAP for SBR

1.1 1 Exact Recomputed Simulated

0.9 0.8 0.7 0.6 0.5 0

0.5

1 t (sec)

1.5

2

FIGURE 6.41 Exact (6.95), recomputed (6.104), and simulated (6.106) autocorrelations for wind speed of 40 mph.

of about 3–5 Hz and 10–14 Hz as well as one to two exponentially decaying components. Interestingly, this trend seems to be true for a much larger set of frequencies (100 MHz–10 GHz). Figure 6.41 shows the exact autocorrelations in (6.90)–(6.95), those recomputed from the system model in (6.100) using the identity r˜w (kTr ) =

µ + r˜k 1+µ

(6.104)

with r˜k =

h i+k h i∗

(6.105)

i=0

and the unbiased estimated autocorrelations rˆw generated from the “wind random variables” in (6.99), using the relation 1 ∗ wi wi+k . n n

rˆw =

(6.106)

i=0

We have used these random variables {wi } in (6.87)–(6.88) to simulate the effect of wind on the SBR clutter data. Figures 6.42–6.43 show the original spectrum given by the Billingsley model and its various rational approximations for two different situations [20]. Figure 6.42

265

Space Based Radar

0

Spectrum in dB

−20

Rational Model (Rank 6)

−40 −60

Discrete-Time Spectrum

−80 −100

Billingsley Spectrum

−120 −50

−25

0

25

50

Freq. (Hz)

FIGURE 6.42 Wind spectrum and its rational approximation—Billingsley model, discrete-time spectrum obtained from the exact autocorrelations sampled at PRF, and spectrum corresponding to a sixth order rational approximation. Wind speed = 40 mph, carrier frequency = 1.25 GHz, PRF = 500 Hz.

0 −20 Spectrum in dB

266

−40 Rational model (Rank 6)

−60 −80 −100

Billingsley spectrum −150

−100

−50

0 50 Freq. (Hz)

100

150

FIGURE 6.43 Wind spectrum and its rational approximations—Billingsley model and spectrum corresponding to a sixth order rational approximation. Breezy wind conditions (6 mph), carrier frequency = 10 GHz, PRF = 690.8 Hz.

Chapter 6:

STAP for SBR

corresponds to a wind speed of 40 mph, carrier frequency of 1.25 GHz, and a PRF of 500 Hz. The original spectrum corresponding to the Billingsley model in (6.90)–(6.92) given by

SB (ω) =

1 cλ −cλ|ω|/2 µ , δ(ω) + e 1+µ 1+µ 4

(6.107)

the spectrum in (6.96) associated with the discrete-time autocorrelations in (6.95), and the spectrum given by (6.97)–(6.98) corresponding to the sixth order rational model in Table 6.6 and Table 6.7 are shown there. In this case the spectrum associated with the sixth order rational approximation is able to faithfully reproduce the true wind spectrum in (6.107) up to about −60 dB. Figure 6.43 corresponds to another set of parameter values (wind speed = 6 mph, carrier frequency = 10 GHz, and PRF = 690.8 Hz). The spectrum associated with the rational system of order six with ﬁlter coefﬁcients as given in Table 6.8 is also shown in Figure 6.43 [21]. In this case, the proposed sixth order system, consisting of two damped sinusoids and two exponentially decaying terms, is able to faithfully reproduce the wind spectrum up to about −80 dB. From (6.107), the slope in Figures 6.42–6.43 is given by −cλ/2 and hence it has different values in those two cases. Consequently the rational approximations are valid over different frequency bands (20 Hz vs. 125 Hz). This can also be explained using the location of the two sets of complex poles of the rational system in each case. In Figure 6.42 they are located around frequencies 3.7 Hz, 11 Hz, whereas Figure 6.43 corresponds to 16 Hz and 50 Hz and hence the pole near 50 Hz is able to stretch the spectral match to about 125 Hz in Figure 6.43.

a0 → a6 1.000000000 −4.448829511 8.262476857 −8.228964421 4.654376421 −1.423964559 0.184975558

b0 → b5 0.006086379 −0.014867925 0.020668209 −0.015636555 0.005389510 −0.001382418

TABLE 6.8 ARMA(6,5) system coefﬁcientsfor Billingsley model at wind speed = 6 mph, carrier frequency = 10 GHz, PRF = 690.8 Hz (see Figure 6.43).

267

268

Space Based Radar

6.6 MDV with Terrain Modeling and Wind Effect Site-speciﬁc terrain modeling and the wind phenomenon (ICM) studied in Section 6.5 also affect the clutter suppression performance. To quantify their effects on target detection, the ideal SINR in (6.37) vs. target velocity can be studied as in Figure 6.16. When terrain modeling is used, the scatter power return Pm, j in the ideal covariance matrix is given by

Pm, j = E |αm, j |

2

E |um, j |2 = Rs4j

=

o σm, j

Rs4j

(6.108)

where we have used (6.21) and (6.79)–(6.80). From (6.87), the effect of wind is to modulate the pulse returns and as a result the wind modulated clutter data vector x˜ k has the same form as in (6.19) with the space-time steering vector si, j replaced by s˜ i, j where ˜ dm, j ) ⊗ a (θ m, j ) s˜ m, j = b(ω

(6.109)

˜ d ) as in (6.87). This gives the wind modulated clutter covariwith b(ω ˜ to be ance matrix R

˜ = E x˜ k x˜ ∗k = E xk x∗k ◦ E{yy∗ } = R ◦ E{yy∗ } R

(6.110)

where y = w ⊗ 1N

(6.111)

Here w represents the temporal wind random variables in (6.88) and the “all one” N × 1 vector 1 N represents the spatial component. Let T y = E{yy∗ }

(6.112)

T = E{w w ∗ }

(6.113)

˜ = R ◦ T y = R ◦ {T ⊗ 1 N×N }. R

(6.114)

and

so that T y = T ⊗ 1 N×N and

Here, 1 N×N represents an “all one” M × M matrix. Notice that T represents the wind autocorrelation matrix with Ti, j = rw ((i − j)Tr )

(6.115)

−10 Single −15 Terrain −20 −25 −30 Multi-Terrain −35 −40 −45 −50 −55 −40 −30 −20 −10

SINR

SINR

Chapter 6:

Single Terrain Multiple Terrains 0

10

20

30

40

−5 −10 −15 −20 −25 −30 −35 −40 −45 −50 −55

STAP for SBR Multi-Terrain

Single Terrain

Single Terrain Multiple Terrain −40 −30 −20 −10

0

10

20

30

Velocity (m/s)

Velocity

(a) Location 1 (Delmarva)

(b) Location 2 (Ontario)

40

FIGURE 6.44 The Effect of Terrain Modeling: Performance at Rt = 500 km for two different points of interest: (a) Location 1 is at latitude of 39◦ , longitude of −76◦ (Delmarva region, USA) and (b) Location 2 is at latitude of 50◦ , longitude −92◦ (Ontario, Canada). CNR = 40 dB.

as in (6.89)–(6.94). T is Toeplitz in nature indicating the wide sense stationary nature of the random process associated with the wind phenomenon. Figure 6.44 shows the effect of terrain modeling (uniform vs. multiple terrains) on the SINR performance computed using the ideal covariance matrix R in (6.20) for two different locations on Earth shown in Figure 6.45. Figure 6.44 (a) corresponds to the Delmarva region considered in [15] with latitude 39◦ , longitude −76◦ , and Figure 6.44 (b) corresponds to the Ontario region with latitude 50◦ and longitude −92◦ . In both cases the CNR is set to 40 dB. Note that the multiterrain performance in Figure 6.44(a) is worse than the single terrain performance and the opposite is true in Figure 6.44(b). In Figure 6.44 (a), the mainlobe covers a terrain with signiﬁcantly lower RCS compared to the nearest sidelobes. Therefore, the sidelobe clutter is much more difﬁcult to suppress. In the Delmarva region considered here, land juts out into the Chesapeake Bay, and the degradation is due to the mainbeam covering a section of water, leading to smaller returns in the mainbeam. The relatively high power of the sidelobes (caused by neighboring coastline/islands) leads to severe performance degradation. Conversely, in Figure 6.44 (b) the sidelobes are weaker as they cover areas with low reﬂectivity such as lakes, leading to superior nulling capabilities for the processor in the multiterrain case [20]. Thus for a ﬁxed CNR, incorporating site-speciﬁc terrain effect into the clutter model does not always lead to inferior performance compared to a uniform terrain model. Site-speciﬁc terrain performance depends on the exact combination and location of these different terrains with respect to the mainlobe and their RCS characteristics.

269

270

Space Based Radar 120°W 60°N

90°W

Ontario

60°W

2

45°N

A 30°N

B

1

New York City Delmarva

Southwest

FIGURE 6.45 Two locations for clutter simulation: (a) Location 1 is at latitude of 39◦ , longitude of −76◦ (Delmarva region) and (b) Location 2 is at latitude of 50◦ , longitude −92◦ (Ontario).

Depending on the location, performance can either improve or worsen compared to a uniform terrain model. Figures 6.46–6.47 show the effect of wind on the SINR performance. Figure 6.46 shows the wind effect on SINR for the single terrain model with wind speeds varying from 0 to 80 mph. Figure 6.47 shows the combined terrain/wind effect for a single terrain as well as multiterrain (Delmarva region shown in Figure 6.45) with wind speeds again varying from 0 to 80 mph. In both cases, wind deteriorates the SINR performance by increasing the width of the clutter notch, thus making target detection much more difﬁcult [20].

6.6.1 Effect of Wind on Doppler To understand why the “wind effect” deteriorates the SINR performance by widening the clutter notch, we can make use of analysis similar to the clutter notch width in (6.47)–(6.64). Let so in (6.45) represent the space-time steering vector associated with the point of interest (i o , jo ). In that case, (6.49) and (6.50) hold true for steering vectors with ωd = ωdo , indicating the existence of a unique null at ωdo as shown in Figure 6.23 (a). However, the situation is different when wind effects are included. From (6.87) the effect of wind is to modulate the pulse returns, and as a result, the wind modulated clutter data vector has the same form as in (6.19) with the space-time steering vector sm, j replaced by s˜ m, j ˜ given by (6.109). This gives the wind modulated covariance matrix R

Chapter 6:

STAP for SBR

No wind −10 −15 −20 Windy (80 mph)

SINR

−25 −30

0 mph 10 mph 20 mph 30 mph 40 mph 50 mph 60 mph 70 mph 80 mph

−35 −40 −45 −50 −55 −40

−30

−20

−10

0

10

20

30

40

Velocity (m/s)

FIGURE 6.46 The effect of wind on single terrain SINR: Wind speed = 0–80 mph.

No wind

Single terrain

−10 −15 −20

SINR

−25

Multi terrain

−30

Windy (80 mph)

−35 −40 −45 −50 −55 −40

−30

−20

−10

0

10

20

30

40

Velocity (m/s)

FIGURE 6.47 Combined effect: Wind speeds of 0–80 mph applied to a single terrain and multiple terrains (Delmarva region).

271

272

Space Based Radar to be as in (6.110)–(6.114) with T representing the wind autocorrelation matrix. The parametric nature of the underlying system in (6.98) can be used to make further progress. In fact, from (6.100) the effect of wind is equivalent to two damped sinusoids together with one to three decaying exponentials. Substituting (6.100) into (6.104) or by directly ﬁtting a Prony-type model into the autocorrelations in (6.92), it is possible to show that the Billingsley autocorrelation model in (6.92) itself can be expressed as a sum of two damped sinusoids together with one or two damped exponentials (see Appendix 6-D for a proof). As a ﬁrst approximation, if we assume the damping to be negligible (αi = 0 in (6.100)) then the wind autocorrelations in (6.92) satisfy rw (kTr ) = c o + 2

2

q i cos(ωi k) = c o +

i=1

2

q i (e − jωi k +e + jωi k ) (6.116)

i=1

with ωi representing the frequencies given in Table 6.6, q i > 0, k ≥ 0, and c o denoting the positive constant term in (6.90). With (6.116) in (6.113)–(6.115) we get T = c 0 1 M×M +

2 q i b i b i∗ + b −i b ∗−i

(6.117)

i=1

where b i = b(ωi ) = [1, e − jπ ωi , e − j2π ωi , . . . e − j ( M−1)π ωi ]T

(6.118)

b −i = b(−ωi ) = [1, e jπ ωi , e j2π ωi , . . . e j ( M−1)π ωi ]T

(6.119)

and

represent temporal steering vectors and 1 M×M represents an “all one” M× M matrix as in (6.114). Substituting (6.116)–(6.117) into (6.114) and making use of (6.46) and (6.47) for R we obtain ˜ =Q ˜ o + c o so s∗o + R

2 q i si si∗ + s−i s∗−i .

(6.120)

i=1

Here ˜ o = Qo ◦ T y Q

(6.121)

and si = so ◦ (b i ⊗ 1 N ) = {b(ωdo ) ◦ b(ωi )} ⊗ a (θ o ) = b(ωdo + ωi ) ⊗ a (θ o ),

i = 1, 2

(6.122)

Chapter 6:

STAP for SBR

and s−i = so ◦ (b −i ⊗ 1 N ) = b(ωdo − ωi ) ⊗ a (θ o ),

i = 1, 2.

(6.123)

From (6.120) the effect of the simple model in (6.116) for the wind autocorrelations is to modify the original clutter component at Doppler frequency ωdo to four additional components with nearby Doppler frequencies ωd±1 = ωdo ± ω1 and ωd±2 = ωdo ± ω2 . Furthermore, from (6.120), these new components are uncorrelated with the original component at ωdo and the rest. Interestingly, for the location of interest θ io , jo , from (6.117) and (6.120), c o + 2(q 1 + q 2 ) = ro represents the total backscattered power factor, and the terrain RCS σioo , jo gets scaled by this factor. However, ro = 1 and hence the RCS is not modiﬁed due to the wind effect. From (6.48) and (6.49) in the absence of wind, the clutter ridge passes through the Doppler ωdo associated with the point of interest. However, this is no longer true when the modiﬁed Doppler frequencies ωd±1 and ωd±2 are also present. In that case the effect of the adaptive processor is to generate local nulls at all these ﬁve frequencies. This results in an overall wider null and, consequently, the clutter nulling effect with wind present will be inferior to that in the absence of wind. To see this explicitly let q 1 > q 2 in (6.120). This allows us to rewrite (6.120) as

˜ = q 1 Q1 + s1 s∗1 + gs2 s∗2 R

(6.124)

where 1 Q1 = q1

˜ o + c o so s∗o + Q

2

q i s−i s∗−i

,

g = q 2 /q 1 < 1.

(6.125)

i=1

Equation (6.124) is similar to the situation where Earth’s rotation is present in (6.47), and following the same analysis we obtain ˜ −1 s1 = s∗1 R

A + ( AB − |C|2 )g 1 q 1 1 + A + ( B + AB − |C|2 )g

(6.126)

˜ −1 s2 = s∗2 R

1 B + ( AB − |C|2 ) q 1 1 + A + ( B + AB − |C|2 )g

(6.127)

and

where A = s∗1 Q−1 1 s1 ,

B = s∗2 Q−1 1 s2 ,

C = s∗1 Q−1 1 s2 .

(6.128)

273

274

Space Based Radar with A B. However, since the normalized gain g < 1 from (6.126), and (6.127), we obtain ˜ −1 s1 < s∗2 R ˜ −1 s2 , s∗1 R

(6.129)

indicating that the dominant component in (6.120) introduces a deeper notch (lower SINR). To compare the widths of the clutter notches without and with wind present, it is instructive to compare the SINR expressions in (6.49) and (6.126) at an off-notch Doppler frequency such as ωd1 = ωdo +ω1 . From (6.120) this corresponds to the space-time steering vector s1 at θ = θ o , ωd1 = ωdo + ω1 and substituting this into (6.49)–(6.50) we get s∗1 R−1 s1 > s∗o R−1 so .

(6.130)

Thus without the effect of wind, SINR at the modiﬁed Doppler ωd1 is at a higher value compared to that at the notch frequency ωdo (see Figure 6.48). Equation (6.126) gives the corresponding SINR in the presence of wind. To simplify this expression assume that the second frequency component in (41) is much weaker compared to the ﬁrst one (q 1 = 1, q 2 0). In that case g 0 in (6.126) and we get ˜ −1 s1 = s∗1 R

1 A 1 s∗1 Q−1 1 s1 . = q 1 1 + A q 1 1 + s∗1 Q−1 1 s1

(6.131)

Because of the local null at ωd1 due to the adaptive processor ˜ −1 s1 < s∗1 R−1 s1 s∗1 R

SINR

(6.132)

(Without Wind)

s∗1 R−1s1

(With Wind)

~

s∗3 R−1s3 = s1∗ R−1s1

wd

wd

o

1

wd

3

FIGURE 6.48 Clutter notch widening due to wind.

wd

Chapter 6:

STAP for SBR

i.e., the wind induced SINR at ωd1 projects a smaller value than the one without the wind effect. This is shown in Figure 6.48. As a result the wind induced SINR is potentially wider than the one without the wind effect. The real nature of the covariance in (6.90) guarantees conjugate symmetry for the frequencies in (6.116). Thus the shifted frequencies occur at ωdo ± ω1 and ωdo ± ω2 forcing symmetry in the SINR output. Although this symmetry is evident in Figure 6.47 (a)–(b), the suggested secondary level sharp nulls are not present there. To understand the absence of these speciﬁc nulls due to wind, it is necessary to examine the wind model in further detail (Section 6.6.2). The model in (6.114)–(6.116) assumes the wind autocorrelations to be the sum of two pure sinusoids plus a constant term. From Table 6.6 and Figure 6.41, the wind autocorrelations are more accurately represented by the sum of damped sinusoids together with damped exponentials and a constant. Next, we analyze this model to understand the effect of wind damping factor on Doppler.

6.6.2 General Theory of Wind Damping Effect on Doppler If we assume the wind autocorrelations to be represented as a sum of m damped sinusoids together with damped exponentials and a constant, the model in (6.116) can be rewritten as rw (kT) = c o +

m

q i e −αi k e − jωi k ,

(6.133)

i=1

where the damped exponentials can be represented with ωi = 0. Substituting (6.133) into (6.115), the temporal tapering matrix T in (6.113) takes the form T = c o 1 M×M +

m

q i Hi ◦ b i b i∗

(6.134)

i=1

where Hi is a real symmetric positive-deﬁnite Toeplitz matrix given by



1

  ρi   2 Hi =  ρi  .  .  . ρiM−1

ρi

ρi2

1

ρi

ρi .. .

1 .. .

ρiM−2

ρiM−3



···

ρiM−1

···

ρiM−2 

  · · · ρiM−3  >0 ..  ..  . .  ···

1

(6.135)

275

276

Space Based Radar where ρi = e −αi > 0

(6.136)

and b i s are as in (6.118). Notice that ωi = 0 gives b i = 1 M . In general, Hi has full rank and let Hi =

M

T µi,k e i,k e i,k

(6.137)

k=1

represent its eigen-decomposition. Hi is real positive deﬁnite and symmetric implies that µi,k , k = 1, 2, . . . , M are positive and e i,k are real. Substituting (6.137) into (6.134), it simpliﬁes to T = c o 1 M×M +

M m

q i µi,k (e i,k ◦ b i )(e i,k ◦ b i ) ∗ .

(6.138)

i=1 k=1

With (6.138) in (6.114), we obtain the wind modiﬁed clutter covariance matrix to be

˜ = R ◦ (T ⊗ 1 N×N ) = Qo + so s∗o ◦ (T ⊗ 1 N×N ) R ˜ o + c o so s∗o + =Q

m M

∗ µ ˜ i,k s˜ i,k s˜ i,k

i=1 k=1

˜ o + c o so s∗o + =Q

m

Ui ,

(6.139)

i=1

where the last summation represents m uncorrelated covariance matrices with each one corresponding to one damped sinusoidal term in (6.133). Thus (6.139) suggests that the returns due to the sinusoidal terms in (6.133) are uncorrelated with each other. In (6.139) ˜ o = Qo ◦ (T ⊗ 1 N×N ) Q Ui =

M k=1

∗ , µ ˜ i,k s˜ i,k s˜ i,k

µ ˜ i,k = q i µi,k > 0,

(6.140) (6.141)

and s˜ i,k = so ◦ {(e i,k ◦ b i ) ⊗ 1 N }.

(6.142)

Observe that (6.141) itself represents a bundle of M uncorrelated returns, each of them associated with the same damped sinusoidal term in (6.133). Together with (6.139), it now follows that each damped sinusoidal term present in the wind generates a bundle that

Chapter 6:

STAP for SBR

is uncorrelated with the remaining sinusoidal returns, with each such bundle consisting of M uncorrelated returns. To analyze the returns in (6.141) further, notice that the vector s˜ i,k in (6.142) can be rewritten as (use (6.138)) s˜ i,k = so ◦ {(e i,k ◦ b i ) ⊗ 1 N } = {b(ωdo ) ⊗ a (θ o )} ◦ {(e i,k ◦ b i ) ⊗ 1 N } = {e i,k ◦ b(ωdo ) ◦ b(ωi )} ⊗ {a (θ o ) ◦ 1 N } = {e i,k ◦ b(ωdo ) ◦ b(ωi )} ⊗ a (θ o ) = {e i,k ◦ b(ωdo + ωi )} ⊗ a (θ o ) = {e i,k ◦ b(ωdi )} ⊗ a (θ o ) = b˜ i,k (ωdi ) ⊗ a (θ o )

(6.143)

where b˜ i,k (ωdi ) = e i,k ◦ b(ωdi ),

k = 1, 2, . . . M

(6.144)

represents amplitude modulated temporal steering vectors corresponding to the shifted Doppler frequency ωdi = ωdo + ωi ,

i = 1 → m.

(6.145)

As a result, s˜ i,k in (6.139)–(6.143) represents a temporally modulated space-time steering vector associated with the Doppler frequency ωdi and spatial location θ o . On comparing (6.120) and (6.139), the effect of each sinusoidal wind frequency ωi is to shift the original Doppler ωdo to ωdo + ωi , and the effect of damping is to generate M new amplitude modulated temporal steering vectors as in (6.144) for each such frequency. From (6.141), these amplitude modulated steering vectors correspond to M uncorrelated returns all originating from the same spatial location θ o . In summary, the effect of each damped sinusoidal component present in the wind autocorrelation is to generate an additional Doppler frequency as in (6.145), and then generate M new uncorrelated scattered returns each corresponding to distinct amplitude modulated temporal steering vectors as in (6.144). Finally, to see the effect of amplitude modulation in (6.144), let f i,k represent the DFT vector associated with each real eigenvector e i,k . The entries in e i,k correspond to a sampling period of Tr . Hence its 1 DFT coefﬁcients f i,k (n) are sampled at = MT apart in the frequency r domain, we get



e i,k =

M/2 n=−M/2

  

f i,k (n) 

1

e − j2π n/M .. . e − j2π( M−1)n/M



M/2   = 

f i,k (n)b

2n M

,

n=−M/2

(6.146)

277

278

Space Based Radar where b is deﬁned as in (6.118). Finally substituting (6.146) into (6.144) we get M/2

b˜ i,k (ωdi ) =

f i,k (n)b

2n M

◦ b(ωdi )

n=−M/2 M/2

=

f i,k (n)b ωdi +

2n M

(6.147)

n=−M/2

and using this in (6.143) we obtain s˜ i,k =

M/2

f i,k (n)b ωdi +

2n M

⊗ a θo

n=−M/2

=

M/2

f i,k (n)si,k θ o , ωdi +

2n M

.

(6.148)

n=−M/2

Equation (6.148) represents a scaled sum of space-time steering vectors associated with the Doppler frequencies ωdi + 2n M , n = −M/2 → M/2, all of them originating from the same spatial location θ o , and as such they represent a coherent set of return signals at these incrementally shifted Doppler frequencies. Together with (6.148), Equation (6.139) can be given the following interpretation: The effect of wind can be represented as a constant term together with a ﬁnite number of damped sinusoids. For each spatial location θ o on Earth with inherent clutter Doppler ωdo , each sinusoidal wind component with frequency ωi , i = 1 → m generates an additional clutter Doppler frequency ωdi = ωdo + ωi and the return bundle for each such frequency is uncorrelated with the returns from other frequencies. The effect of damping is two fold: Each such Doppler frequency ωdi generates a bundle of M uncorrelated returns as in (6.141). Further from (6.148), each such uncorrelated return in that bundle contains at most M coherent returns at new Doppler frequencies ωdi ±

2n , M

n = 0, 1, . . . , M/2.

(6.149)

This situation is illustrated in Figure 6.49, where the wind induced frequencies ω1 and ω2 generate two additional Doppler returns at frequencies ωd1 = ωdo + ω1 and ωd2 = ωdo + ω2 that are uncorrelated with each other and also with the return at the original Doppler frequency ωdo . The effect of wind damping factor is to generate a bundle of M uncorrelated returns for each new Doppler frequency ωd1 and

Chapter 6:

o

A

Uncorrelated returns wd

o

B

C1

A

wd

C2

Coh

eren

t

Uncorrelated wd ± 2n 2 M

B

STAP for SBR

C3 D1

wd

2

D2

wd ± 2n 1 M

Uncorrelated returns

Bundle for wd 1

D3 O

O

(a) Without wind

wd

1

(b) With wind

FIGURE 6.49 Scatter returns without and with wind effect. (a) Without wind, a single Doppler return frequency is generated, (b) With wind, the return consist of m uncorrelated bunches, with each bunch consisting of M uncorrelated returns. Further, each such return contains M coherent signals (m = 2, M = 3 shown here).

ωd2 . Further each return in that bundle consists of a coherent sum of M returns. A sum of coherent returns does not correspond to a steering vector and consequently, they cannot be nulled out by the standard adaptive weight vector w = R−1 s. This is illustrated in Figure 6.50 where the

Coherent returns

SINR in dB

0

−10

−20

Uncorrelated returns 0

50

100

150

Angle (deg)

FIGURE 6.50 SINR loss in an uncorrelated scene vs. a coherent scene. Dashed curve represents three uncorrelated returns of equal power arriving from 75◦ , 95◦ , and 112◦ at a seven-sensor element array. The solid curve represents two identical coherent returns from 75◦ and 112◦ that are uncorrelated with the return from 95◦ . SNR = 15 dB.

279

280

Space Based Radar dotted line corresponds to three uncorrelated sources in white noise, and the solid line corresponds to two coherent sources and an uncorrelated source in white noise. In the uncorrelated case, the array output covariance matrix R1 equals R1 = s1 s∗1 + s2 s∗2 + s3 s∗3 + σ 2 I

(6.150)

and in the second case, the sources corresponding to steering vectors s1 and s3 are coherent. Hence R2 = aa∗ + s2 s∗2 + σ 2 I

(6.151)

a = s1 + s3 .

(6.152)

where

Clearly, the adaptive processor w1 = R−1 1 s

(6.153)

is able to null out the three uncorrelated sources as shown in the SINR output as the steering vector s spans through the ﬁeld of view. However, the adaptive processor w2 = R−1 2 s

(6.154)

in the coherent case is only able to null out the uncorrelated component s2 present in R2 . It follows that the wind induced coherent components cannot be nulled out by the STAP processor and as a result, speciﬁc nulls do not show up at these frequencies ωd1 , ωd2 , . . .. Thus the wind induced SINR performance will be potentially wider compared to the no-wind situation as evident in Figures 6.46–6.47.

6.7 Joint Effect of Terrain, Wind, Range Foldover, and Earth’s Rotation on Performance Finally, to illustrate the combined effect of terrain, wind, range foldover, and Earth’s rotation on clutter nulling performance for the optimum adaptive processor, two locations A and B in Texas shown in Figure 6.45 are selected (Location A: Latitude = 37.21◦ , Longitude = −106.4◦ , and Location B: Latitude = 37.89◦ , Longitude = −105.3◦ ), and the SINR performance is analyzed at a range of 1,200 km. Figures 6.51–6.52 show the performance levels for location A with conventional waveform and hybrid-chirp waveform. In these two

0

0

−10

−10

−20

SINR in dB

SINR in dB

Chapter 6:

Ideal

−30 −40

−20 −30 −40 −50

−50 −60

−30

−20

−10

−60 20

0 10 V (m/s)

30

−30

−20

−10

0

0

−10

−10

−20

−20

SINR in dB

SINR in dB

0 10 V (m/s)

20

30

(b)

(a)

−30 −40

Ideal

−30 −40 −50

−50 −60

STAP for SBR

−30

−20

−10

−60 0 10 V (m/s)

20

30

−30

−20

−10

0 10 V (m/s)

20

30

(d)

(c)

FIGURE 6.51 Location A in Texas, SBR range 1,200 km, using conventional waveform. (a) Terrain only, (b) Terrain and wind, (c) Terrain and range foldover, (d) Terrain, wind, range foldover, Earth’s rotation present.

Ideal

0

SINR in dB

−10 −20 −30 −40 −50 −60

−30

−20

−10

0

10

20

30

V (m/s)

FIGURE 6.52 Location A in Texas. SBR using hybrid-chirp waveforms. Terrain, wind, range foldover, and Earth’s rotation are present.

281

Space Based Radar 0

Ideal

−10

−10

−20

−20

SINR in dB

SINR in dB

0

−30

−30

−40

−40

−50

−50 −30

−20

−10

0 10 V (m/s)

20

−30

30

−20

−10

0

−10

−10

−20

−20

SINR in dB

0

−30

−40

−50

−50 −20

−10

0 10 V (m/s)

(c)

20

30

20

30

Ideal

−30

−40

−30

0 10 V (m/s)

(b)

(a)

SINR in dB

282

−30

−20

−10

0 10 V (m/s)

20

30

(d)

FIGURE 6.53 Location B in Texas. SBR range 1,200 km, using conventional waveform. (a) Terrain only; (b) Terrain and wind; (c) Terrain and range foldover; (d) Terrain, wind, range foldover, Earth’s rotation present.

ﬁgures, the dashed lines show the performance of each case under consideration, and the solid lines show the ideal performance (uniform terrain, no wind, no Earth’s rotation, and no range foldover) for comparison purposes. Notice that when all these effects are present, conventional waveform output has extremely poor performance, whereas waveform diversity improves the performance to a considerable level. Similar conclusions follow from Figures 6.53–6.54 that correspond to location B in Texas. In summary, when terrain effects, ICM due to wind, range foldover effect, and Earth’s rotational effects are present in the clutter data; conventional waveforms are unable to detect targets. However, signiﬁcant performance improvement can be realized using waveform diversity.

Chapter 6: 0

STAP for SBR

Ideal

SINR in dB

−10 −20 −30 −40 −50 −30

−20

−10

0

10

20

30

V (m/s)

FIGURE 6.54 Location B in Texas. SBR using hybrid-chirp waveforms. Terrain, wind, range foldover, and Earth’s rotation are present.

6.8 STAP Algorithms for SBR In Section 4.7, we have modeled the effect of Earth’s rotation on SBR clutter data in terms of crab angle and crab magnitude that affects the clutter Doppler. The matched ﬁlter (MF) performance in terms of SINR loss is analyzed in Section 6.3. From there, the performance is signiﬁcantly degraded when both range foldover and Earth’s rotation are jointly present. To understand the performance of the STAP algorithms described in Chapter 5, simulated clutter data is used to estimate the clutter covariance matrix for the four cases (A, B, C, D) listed in (6.38). Figure 6.55 shows the SINR output for ground range of 500 km. The ideal (no Earth’s rotation, no range foldover) MF output, as well as the MF output using (6.37), the estimated SINR output using the traditional SMI algorithm for each of the four cases above are shown here. Recall that traditional SMI algorithm requires at least MN = 32 × 16 = 512 samples in the present setup for satisfactory performance. As a result, 800 data samples are used for clutter covariance matrix estimation. From Figure 6.55, the MF performance is the same as the ideal MF performance when either Earth’s rotation or range foldover is absent (cases A, B, and C). However, the performance is signiﬁcantly degraded when both Earth’s rotation and range foldover are present together (case D). This is in agreement with the results from Section 6.3. In addition, the estimated performance using SMI with 800 data

283

0

0

−10

−10

−20

−20

SINR in dB

SINR in dB

Space Based Radar

−30 −40

Ideal MF SMI

−50 −60

−40

−20

0 V (m/s)

20

−30 −40

Ideal MF SMI

−50 −60

40

−40

(a) Without Earth’s rotation, without range foldover 0

0

−10

−10

−20

−20

−30 −40

Ideal MF SMI

−50 −60

−40

−20

0 V (m/s)

20

40

(c) With Earth’s rotation, without range foldover

−20

0 V (m/s)

20

40

(b) Without Earth’s rotation, with range foldover

SINR in dB

SINR in dB

284

−30 −40 Ideal MF SMI

−50 −60

−40

−20

0 V (m/s)

20

40

(d) With Earth’s rotation, with range foldover

FIGURE 6.55 SINR vs. velocity on clutter data without/with Earth’s rotation and range foldover for ground range of 500 km. SMI with 800 data samples is shown here.

samples is close to the MF performance for cases A, B, and C. However, for case D, since the MF performance is quite poor to start with, other methods need to be introduced to improve the MF performance. Waveform diversity introduced in Section 6.4 can minimize the range foldover return and result in improved performance. The performance evaluation with waveform diversity in the data case is carried out later in this section. When Earth’s rotation is present (case C), eventhough the MF performance is close to the ideal performance, the estimated output is inferior to the MF output. This is due to the Doppler spread caused by the Earth’s rotation. The impact of Doppler spread and method to compensate it is later studied. To analyze the clutter nulling performance of different STAP algorithms, the “clean” data corresponding to range 500 km for case A (without Earth’s rotation and range foldover) is used.

Chapter 6:

STAP for SBR

From Figure 6.55, SMI with 800 data samples gives good performance. However, the number of samples required is quite large and results in higher computation requirement. Furthermore, large data samples are not always available in a nonstationary environment. In this context other STAP algorithms introduced in Section 5.3.3 and 5.4 can be used. Figure 6.56 shows the estimated performance with 200 samples using SMIDL, EC, and HTP methods. From there, these STAP algorithms perform close to the MF output using only 200 samples. The number of samples required can be further reduced using the forward/backward, subarray, subpulse smoothing methods introduced in Section 5.5 and 5.6. Figure 6.57 shows the performance for SMIDL, SMIDLSASPFB, EC, ECSASPFB with 40 samples. From there, the traditional STAP methods (SMIDL, EC) give poor performance using a small number of samples. However, when the forward/backward, subarray, subpulse smoothing methods are used, the resulting performance is improved. Observe that the number of

0

SINR in dB

−10 −20 −30 −40 −50 −60

MF SMIDL −40

−20

0 V (m/s)

20

40

0

0

−10

−10

−20

−20

SINR in dB

SINR in dB

(a) SMIDL with 200 samples

−30 −40 −50 −60 −40

MF EC −20

0

20

40

−30 −40 −50 −60

MF HTP −40

−20

V (m/s)

0 V (m/s)

20

(b) EC with 200 samples

(c) HTP with 200 samples

40

FIGURE 6.56 SINR vs. velocity on clutter data without Earth’s rotation and range foldover for ground range of 500 km. Two hundred data samples are used here. Different STAP algorithm (a) SMIDL, (b) EC, and (c) HTP are shown here.

285

0

0

−10

−10

−20

−20

SINR in dB

SINR in dB

Space Based Radar

−30 −40 −50 −60

MF SMIDL −40

−20

0 V (m/s)

20

40

−30 −40 −50

0 −10

−20

−20 SINR in dB

0

−30 −40

−60

MF EC −40

−20

0 V (m/s)

20

(c) EC with 40 samples

−20

0 V (m/s)

20

40

(b) SMIDLSASPFB with 40 samples

−10

−50

MF SMIDLSASPFB

−60 −40

(a) SMIDL with 40 samples

SINR in dB

286

40

−30 −40 −50 −60

MF ECSASPFB −40

−20

0 V (m/s)

20

40

(d) ECSASPFB with 40 samples

FIGURE 6.57 SINR vs. velocity on clutter data without Earth’s rotation and range foldover for ground range of 500 km. 40 data samples are used here. Different STAP algorithms (a) SMIDL, (b) SMIDLSASPFB, (c) EC, and (d) ECSASPFB are shown here.

samples used here is very small compared to the full degree of freedom (512 in this case). Other reduced rank/dimension STAP algorithms such as factored time-space (FTS), and Joint-Domain Localized (JDL) discussed in Section 5.9 and 5.10 also give excellent performance results. Figure 6.58 shows the performance for the FTS method using one Doppler bin and Figure 6.59 shows the performance of extended factored time-space (EFA) method using two and three Doppler bins. The improvement over a single bin approach is signiﬁcant even when only two bins are used. However, as the number of Doppler bins is increased, the number of samples required for clutter covariance matrix estimation also increases. For the two bin approach, the size of the matrix is 2N × 2N and the number of data samples required is 4N which turns out to be 128 for the current SBR setup. Interestingly, methods such as SMIDLSASPFB give similar performance even with smaller number of samples (around 40 samples).

Chapter 6:

FTS, K = 200, N2 = 1 0

−10

−10

−20

−20

SINR in dB

SINR in dB

FTS, K = 60, N2 = 1 0

−30 −40

−30 −40 −50

−50 −60

STAP for SBR

−40

−20

0

20

−60

40

−40

−20

0

20

40

V (m/s)

V (m/s)

(a) One bin with 60 samples

(b) One bin with 200 samples

FIGURE 6.58 Performance of FTS with one Doppler bin. Range = 500 km. Clutter without Earth’s rotation and range foldover.

Figure 6.60 shows the performance for JDL using three angle bins and various Doppler bins and Figure 6.61 shows the performance for JDL using ﬁve angle bins and various Doppler bins. As Figure 6.60 shows, the performance is poor when number of angle bins used is three irrespective of how many Doppler bins are used. As Figure 6.61 (b)–(c) shows, JDL method using ﬁve angle bins and two Doppler gives acceptable performance with as few as 40 samples.

Computational Effect of SMI, EFA, and JDL In FTS, dimension reduction is achieved by solving a different spatialonly adaptive beamforming problem for each Doppler bin. Thus only inversion of an N × N matrix (Ms N × Ms N for EFA) is required in EFA, K = 200, N2 = 3 0

−10

−10

−20

−20

SINR in dB

SINR in dB

EFA, K = 200, N2 = 2 0

−30 −40 −50 −60

−30 −40 −50

−40

−20

0

20

40

−60

−40

−20

0

20

40

V (m/s)

V (m/s)

(a) Two bins with 200 samples

(b) Three bins with 200 samples

FIGURE 6.59 Performance of EFA with two and three Doppler bin. Range = 500 km. Clutter without Earth’s rotation and range foldover.

287

Space Based Radar JDL, K = 50, N2 = 3, M2 = 7

0

0

−10

−10

−20

−20

SINR in dB

SINR in dB

JDL, K = 40, N2 = 3, M2 = 3

−30 −40

−60

−30 −40 −50

−50 −40

−20

0

20

−60

40

−40

−20

V (m/s)

0

20

40

V (m/s)

(b) Three angle bins and three doppler bins with 40 samples

(b) Three angle bins and seven doppler bins with 50 samples

FIGURE 6.60 Performance of JDL using three angle bins and various Doppler bins. Range = 500 km. Clutter without Earth’s rotation and range foldover.

contrast to NM × NM in the case of full degree of freedom STAP problem. This results in computational savings for one look angle and one Doppler of interest. From Section 5.3 and 5.4, there is only one matrix inversion needed for different look angles and Doppler in the case of SMI, or SMIDL, etc. However, the effective covariance matrices for FTS and EFA depends on the temporal steering vector b(ωd ) as seen from (5.269). Hence multiple matrix inversions are needed when different Doppler values are searched.

JDL, K = 20, N2 = 5, M2 = 1

JDL, K = 40, N2 = 5, M2 = 2

0

0

−10

−10

−20

−20

SINR in dB

SINR in dB

288

−30 −40 −50 −60

−30 −40 −50

−40

−20

0

20

40

−60

−40

−20

0

20

40

V (m/s)

V (m/s)

(a) Five angle bins and one Doppler bin with 20 samples

(b) Five angle bins and two Doppler bins with 40 samples

FIGURE 6.61 Performance of JDL with ﬁveangle bins and various Doppler bins. Range = 500 km. Clutter without Earth’s rotation and range foldover.

Chapter 6:

STAP for SBR

As mentioned in Section 5.10, for JDL the size of the effective covariance matrix is only Ms Ns × Ms Ns and it is also smaller than the full size case of MN × MN. As Figure 6.61 (d) shows, in this case only ﬁve angle bins and two Doppler bins are required for good performance for our SBR simulation setup. The resulting covariance matrix to be inverted is 10 × 10 instead of 512 × 512. Thus there is a huge computational saving for one look angle and Doppler of interest. The computational requirement for JDL is even smaller compared to the FTS method in this case. However, from Section 5.3–5.8, for methods such as SMI, SMIDL, etc., the clutter covariance matrix does not dependent on the angle and Doppler of interest and only one matrix inversion is required for different angles and Dopplers; on the other hand, in the case of JDL, the effective clutter covariance matrix depends both on the angle and Doppler of interest. The number of matrix inversion needed equals the product of number of angles and number of Doppler of interest.

Effect of Earth’s Rotation on Clutter Nulling Performance The crab angle φc appearing in (4.71) generates additional undesirable effects on the estimated clutter covariance matrix. As Figure 6.55 (c) shows, when Earth’s rotation is present and a large number of samples is used to estimate the clutter covariance matrix, the resulting performance is very poor. This is due to the Doppler spread caused by the Earth’s rotation. In addition, the performance degradation at range 500 km is worse than at far ranges. To see this, from Figure 6.21 in Section 6.3, when the data samples from range 500 km are used for estimating the clutter covariance matrix, the Doppler spread there is larger than the Doppler spread at range 1,200 km. Thus, the performance degradation at closer range is worse than the performance at far range when Earth’s rotation is present in the data. This Doppler spreading effect is illustrated in Figure 6.62 which ˆ k s at range equal to 500 km shows the estimated clutter spectrum s∗ R in the Doppler–azimuth domain for θ AZ = 90◦ . Once again when the crab effect is absent (Figure 6.62 (a)), the clutter Doppler peaks at ωd = 0 for θ AZ = 90◦ . Moreover, the clutter Doppler spread around ωd = 0 is ﬁnite, irrespective of the number of samples used to estimate the clutter covariance matrix. However, as Figure 6.62 (b) shows these two conclusions are not true when crab effect is present in the clutter data. From there, in addition to the shift in the Doppler peak, as the ˆ increases, the clutter Doppler number of samples used in estimating R begins to spread wider around its peak value (along the y-axis). To illustrate this, the clutter spectrum using 4,000 samples is shown in Figure 6.62 (b). From there, the clutter covers the entire useful Doppler

289

Space Based Radar

Doppler

0.5 0 −0.5 −1 −0.05

0

0.05

0 −5 −10 −15 −20 −25 −30 −35 −40 −45 −50

1 0.5 Doppler

1

0 −0.5 −1 −0.05

0

0.05

0 −5 −10 −15 −20 Doppler −25 spreads in a −30 finite region −35 −40 −45 −50

cos qAZ

cos qAZ

4,000 Samples. Range: 450–550 km

100 Samples. Range: 499–501 km

(a) Clutter data without crab effect

0.5 0 −0.5 −1 −0.05

0 cos qAZ

0.05

100 Samples. Range: 499–501 km

0 −5 −10 −15 −20 −25 −30 −35 −40 −45 −50

1

0 −5 −10 −15 −20 −25 −30 −35 −40

0.5 Doppler

1

Doppler

290

0 −0.5 −1 −0.05

0 cos qAZ

Doppler spread covers the entire region

0.05

4,000 Samples. Range: 450–550 km

(b) Clutter data with crab effect

FIGURE 6.62 Clutter power in the angle-Doppler domain using estimated clutter covariance matrix as a function of number of samples for range = 500 km (θAZ = 90◦ ).

region making target detection difﬁcult for that look direction when a large number of samples are used to estimate the clutter covariance matrix. This effect (spreading of clutter Doppler as the number of samples increases) is also validated in Figure 6.63 using the MDV analysis. The performance of the STAP algorithms using various number of samples are shown there. In Figure 6.63 (a), where the crab effect is absent, the performance of the STAP algorithms improves as the number of samples is increased. However, in Figure 6.63 (b) where the crab effect is present, increasing the number of samples degrades the performance. For example, performance using 800 samples is inferior to that using 200 samples. Recall that traditional STAP algorithms require at least MN = 32 × 16 = 512 samples for reasonable performance. However, in the presence of Earth’s rotation, the above analysis shows that using over 200 samples will lead to inferior performance. Interestingly, the forward/backward, subarray, subpulse smoothing methods introduced in Section 5.5 and 5.6 only require a small set of data samples compared

0

0

−10

−10

−20

−20

SINR in dB

SINR in dB

Chapter 6:

−30 −40 −50 −60

−20

0 V (m/s)

20

−30 −40 −50

MF SMIDL −40

STAP for SBR

−60

40

MF SMIDL −40

−20

200 Samples

0 V (m/s)

20

40

800 Samples

0

0

−10

−10

−20

−20

SINR in dB

SINR in dB

(a) Clutter data without crab effect

−30 −40 −50 −60

MF SMIDL −40

−20

0 V (m/s)

20

40

−30 −40 −50 −60

MF SMIDL −40

−20

200 Samples

0 V (m/s)

20

40

800 Samples

(b) Clutter data with crab effect

FIGURE 6.63 SINR without/with crab effect. Range = 900 km, θAZ = 90◦ . SMIDL is used for processing.

to the traditional algorithms and hence they are also suitable to address the clutter data containing Earth’s rotation. Figure 6.64 shows the SINR performance without and with Earth’s rotation for 500 km range. Performance using SMIDLSASPFB with 40 data samples are shown here. From there, the performance with crab effect is improved compared with Figure 6.63(b). However, the performance is still inferior compared to Figure 6.64(a) and additional processing must be done to reduce the remaining performance degradation.

Doppler Warping Doppler warping can be used on the secondary data set to reduce the Doppler spread caused by Earth’s rotation when range foldover is absent. When Doppler warping is used, the Doppler frequency of the secondary data with the same cone angle as the primary data

291

0

0

−10

−10

−20

−20

SINR in dB

SINR in dB

Space Based Radar

−30 −40 −50 −60

−20

0

20

−40 −50

MF SMIDLSASPFB

−40

−30

−60

40

MF SMIDLSASPFB

−40

−20

0

20

V (m/s)

V (m/s)

(a) Without crab effect

(b) With crab effect

40

FIGURE 6.64 SINR without/with crab effect. Range = 500 km, θAZ = 90◦ , number of samples = 40. SMIDLSASPFB is used here.

are realigned to those of the primary data [22]. This is shown in Figure 6.65. In this case, ω˜ do and ω˜ dk are the Doppler frequencies of the primary data and secondary data corresponding to the cone angle of interest respectively. Deﬁne the differential Doppler between the primary data and the secondary data to be ωd = ω˜ do − ω˜ dk .

(6.155)

~ w dk

−4 −4.5 Doppler (KHz)

292

Secondary data Doppler Primary data Doppler Secondary data Doppler

−5 −5.5 −6 −6.5 −7 −5

0 Cone Angle

FIGURE 6.65 Doppler warping.

~ w do

5 ×10−3

Chapter 6:

STAP for SBR

The Doppler on the secondary data can be realigned by performing the following operation: xˆ k = xk ◦ s(0, ωd ).

(6.156)

Here, xk represents the secondary data and s(0, ωd ) is the spacetime steering vector given by (6.10). The role of s(0, ωd ) is to refocus the Doppler on the secondary data by applying the differential Doppler to the data. Thus, the Doppler on the adjusted secondary data xˆ k is same as that in the primary data and the Doppler spread is minimized. The adjusted data xˆ k is then used for clutter covariance matrix estimation. The Doppler spread can be eliminated when only Earth’s rotation is present because there is only one Doppler frequency present in the data for a ﬁxed cone angle. However, when range foldover is also present, there are multiple Doppler frequencies present in the data for a ﬁxed cone angle and all of them cannot be corrected. Figure 6.66 shows the SINR output without and with Doppler warping. From there, when Doppler warping is applied to the secondary data, the performance is restored close to the ideal case performance (see Figure 6.56(b) and Figure 6.57). Thus, Doppler warping can reduce the Doppler spread resulting in improved performance.

STAP Performance with Waveform Diversity When both Earth’s rotation and range foldover are present at the same time, the performance degrades signiﬁcantly and it cannot be corrected completely (Section 6.3). This can also be seen in Figure 6.55(d) where the estimated performance is poor. Waveform diversity introduced in Section 6.4 can eliminate/ minimize the range foldover return resulting in improved matched ﬁlter performance. However, even when waveform diversity is used, crab effect still exists in the clutter data. Performance will be poor (Figure 6.63(b)) if the crab effect is not taken into account when the secondary data is used to estimate the clutter covariance matrix. Doppler warping introduced earlier can be used for this purpose. Figure 6.67 shows the SINR output for range 500 km. Clutter data is generated using orthogonal waveforms with Earth’s rotation and range foldover present. Figure 6.67(a) shows the performance of SMIDLSASPFB with Doppler warping and Figure 6.67(b) shows performance of HTPSASPFB with Doppler warping. From Figure 6.67, the MF output is restored to the ideal case performance with orthogonal waveform diversity. In addition, with Doppler warping, various STAP algorithms give close to MF performance using 40 data samples. About 25-dB improvement can be achieved

293

0

0

−10

−10

−20

−20

SINR in dB

SINR in dB

Space Based Radar

−30 −40 −50 −60

−20

0 V (m/s)

20

−40 −50

MF SMIDL

−40

−30

−60

40

MF SMIDLSASPFB

−20

−40

SMIDL with 200 samples

0 V (m/s)

20

40

SMIDLSASPFB with 40 samples

(a) Without Doppler warping 0

0

−10

−10

−20

−20

SINR in dB

SINR in dB

294

−30 −40 −50 −60

MF SMIDL

−40

−20

0 V (m/s)

20

SMIDL with 200 samples

40

−30 −40 −50 −60

MF SMIDLSASPFB

−40

−20

0 V (m/s)

20

40

SMIDLSASPFB with 40 samples

(b) With Doppler warping

FIGURE 6.66 SINR without and with Doppler warping. Clutter data with Earth’s rotation. Range = 500 km.

with waveform diversity and Doppler warping compared to that in Figure 6.55 (d). Figure 6.68 shows the performance when hybrid-chirp waveforms are used for range 1,200 km. Performance of ECSASPFB with Doppler warping using 40 samples is shown here. From there, the MF performance is restored close to the ideal case output by waveform diversity and the estimated performance achieves more than 35 dB improvement compared to the performance in Figure 6.35 (b). In summary, high resolution STAP algorithms together with forward/backward, subarray, and subpulse smoothing methods reduce the number of secondary data required for clutter covariance matrix estimation and improves SINR performance. When Earth’s rotation is present in the clutter data, Doppler spread occurs. In that case, Doppler warping can be used to realign the clutter Doppler frequency.

0

0

−10

−10

−20

−20

SINR in dB

SINR in dB

Chapter 6:

−30 −40 Ideal MF SMIDLSASPFB

−50 −60

−40

−20

0

20

40

STAP for SBR

−30 −40 Ideal MF HTPSASPFB

−50 −60

−40

−20

0

V (m/s)

V (m/s)

(a) SMIDLSASPFB

(b) HTPSASPFB

20

40

FIGURE 6.67 SINR performance with waveform diversity. Clutter data with Earth’s rotation and range foldover. Orthogonal waveforms are used to generate clutter data. Range = 500 km. (a) SMIDLSASPFB and (b) HTPSASPFB with Doppler warping using 40 samples are shown.

Waveform diversity can be used to eliminate or minimize range foldover returns. High resolution STAP algorithms, Doppler warping and waveform diversity together result in acceptable SINR performance for the current SBR setup. Selection of a particular STAP algorithm may be irrelevant compared to addressing all these other issues—Earth’s rotational effect,

0

SINR in dB

−10 −20 −30 −40 Ideal MF ECSASPFB

−50 −60

−30

−20

−10

0 10 V (m/s)

20

30

FIGURE 6.68 SINR performance with waveform diversity. Clutter data with Earth’s rotation and range foldover. Hybrid-chirp waveforms are used to generate clutter data. Range = 1, 200 km. ECSASPFB with Doppler warping using 40 samples is shown.

295

296

Space Based Radar site speciﬁc clutter modeling, effect of wind, etc., and then taking the necessary steps to correct/minimize these undesired effects prior to actual processing.

Appendix 6-A: Matrix Inversion Identity Consider the identity Q + ss∗ = (I + ss∗ Q−1 )Q

(6A.1)

so that its inverse is given by (Q + ss∗ ) −1 = Q−1 (I + ss∗ Q−1 ) −1 .

(6A.2)

(I + ss∗ Q−1 ) −1 = I − A,

(6A.3)

(I − A)(I + ss∗ Q−1 ) = I

(6A.4)

A + Ass∗ Q−1 = ss∗ Q−1 .

(6A.5)

As + Ass∗ Q−1 s = ss∗ Q−1 s

(6A.6)

As(1 + s∗ Q−1 s) = ss∗ Q−1 s,

(6A.7)

In (6A.2), let

or

which gives

Thus

or

and hence we obtain the identity As = ∗

ss∗ Q−1 1 + s∗ Q−1 s

(6A.8)

s

−1

ss Q from which A = 1+s is clearly a solution in (6A.3). Substituting ∗ Q−1 s this value of A in (6A.2)–(6A.3) we get the desired result

(Q + ss∗ ) −1 = Q−1 −

Q−1 ss∗ Q−1 1 + s∗ Q−1 s

.

(6A.9)

Chapter 6:

STAP for SBR

Appendix 6-B: Output SINR Derivation In the presence of Earth’s rotation, the clutter covariance matrix has the form (refer to (6.47)) R = s˜ o s˜ ∗o + g˜s1 s˜ ∗1 + Q1 = s˜ o s˜ ∗o + Q

(6B.1)

so that using (6A.9) we get Q−1 s˜ o s˜ ∗o Q−1

R−1 = Q−1 −

(6B.2)

1 + s˜ ∗o Q−1 s˜ o

where Q = g˜s1 s˜ ∗1 + Q1 .

(6B.3)

Hence Q−1 = Q−1 1 −

˜ 1 s˜ ∗1 Q−1 gQ−1 1 s 1 ˜1 1 + g˜s∗1 Q−1 1 s

.

(6B.4)

Thus

s˜ ∗o Q−1 s˜ o

=

˜o s˜ ∗o Q−1 1 s

−

2

˜1 g s˜ ∗o Q−1 1 s

˜1 1 + g˜s∗1 Q−1 1 s

.

(6B.5)

Deﬁne ˜o , A = s˜ ∗o Q−1 1 s

˜1, B = s˜ ∗1 Q−1 1 s

˜1, C = s˜ ∗o Q−1 1 s

(6B.6)

so that (6B.5) becomes s˜ ∗o Q−1 s˜ o = A −

g|C|2 A + g( AB − |C|2 ) = . 1 + gB 1 + gB

(6B.7)

From (6B.2), we also obtain s˜ ∗o R−1 s˜ o

∗ −1 2 s˜ o Q s˜ o

=

s˜ ∗o Q−1 s˜ o

=

A + g( AB − |C|2 ) , 1 + A + g( B + AB − |C|2 )

−

1 + s˜ ∗o Q−1 s˜ o

=

s˜ ∗o Q−1 s˜ o

1 + s˜ ∗o Q−1 s˜ o (6B.8)

where we have made use of (6B.7). Similarly we also obtain s˜ ∗1 R−1 s˜ 1

=

s˜ ∗1 Q−1 s˜ 1

−

∗ −1 2 s˜ o Q s˜ 1 1 + s˜ ∗o Q−1 s˜ o

.

(6B.9)

297

298

Space Based Radar But using (6B.4) we get

s˜ ∗1 Q−1 s˜ 1

=

˜1 s˜ ∗1 Q−1 1 s

2

˜1 g s˜ ∗1 Q−1 1 s

−

1+

˜1 g˜s∗1 Q−1 1 s

=

˜1 s˜ ∗1 Q−1 1 s

˜1 1 + g˜s∗1 Q−1 1 s

=

B . 1 + gB (6B.10)

Also ˜1 − s˜ ∗o Q−1 s˜ 1 = s˜ ∗o Q−1 1 s

˜ 1 s˜ ∗1 Q−1 ˜1 g˜s∗o Q−1 1 s 1 s

=C−

˜1 1 + g˜s∗1 Q−1 1 s

gC B C = . 1 + gB 1 + gB (6B.11)

Substituting (6B.7), (6B.10), and (6B.11) into (6B.9) we get s˜ ∗1 R−1 s˜ 1 =

B |C|2 /(1 + g B) 2 − 1 + gB (1 + A + g( B + AB − |C|2 ))/(1 + g B)

=

B(1 + A + g B + g AB − g|C|2 ) − |C|2 (1 + g B)(1 + A + g( B + AB − |C|2 ))

=

(1 + A)(1 + g B) B − (1 + g B)|C|2 (1 + g B)(1 + A + g( B + AB − |C|2 ))

=

B + ( AB − |C|2 ) . 1 + A + g( B + AB − |C|2 )

(6B.12)

Appendix 6-C: Spectral Factorization For a stable system with transfer function H(z) =

∞

h k z−k ,

|z| > 1,

(6C.1)

rk e − jkω ≥ 0

(6C.2)

k = 0, 1, 2, . . .

(6C.3)

k=0

deﬁne its power spectrum S(ω) as

+∞

2

S(ω) = H(e jω ) =

k=−∞

so that its autocorrelations satisfy rk =

∞ i=0

h k+i h i∗ = rk∗ ,

Chapter 6:

STAP for SBR

∞ |h k |2 = ro < ∞ implies h i → 0 as Notice that from (6C.3), i=0 i → ∞. Every stable system transfer function H(z) can be expressed in terms of a unique minimum phase factor Ho (z) as

H(z) = Ho (z) A(z),

(6C.4)

where A(z) represents a regular (stable) (|A(e jω )| = 1). For example, the rational stable system H(z) =

all-pass

z+2 3z + 1

function

(6C.5)

corresponds to the minimum phase factor2 Ho (z) =

2z + 1 , 3z + 1

(6C.6)

where the stable all-pass factor is given by A(z) = (z + 2)/(2z + 1). Notice that in (6C.4) both H(z) and its minimum phase factor Ho (z) have the same spectrum and hence they both generate the same set of autocorrelations. Thus from (6C.2), given the power spectrum S(ω), the corresponding stable system transfer function H(z) is unique only upto an all-pass factor A(z) since

2

2

2

S(ω) = H(e jω ) = Ho (e jω ) A(e jω ) = Ho (e jω )

(6C.7)

and hence the system and the corresponding impulse response {h k }∞ k=0 in (6C.1) represent a minimum phase system uniquely. Equation (6C.3) represents the relation between the impulse response of a system and its autocorrelations. Given the impulse response, (6C.3) can be used to compute its autocorrelations. However, to obtain the impulse response sequence from the autocorrelations involves a spectral factorization. As (6C.7) shows, spectral factorization does not lead to a unique factor. However, the minimum phase factor associated with the spectral factorization is unique, and all other factors can be obtained by cascading it with a regular all-pass function. Interestingly, the unique minimum phase factor can be obtained from the autocorrelation sequence using an iterative algorithm. 2 Poles and zeros of a minimum phase system with series representation as in (6C.1) are inside the unit circle.

299

300

Space Based Radar To make further progress, deﬁne the two inﬁnite dimensional matrices



ro  r1∗   .. T=  .∗  rn  .. . and



ho

0   0  H=  ...   0  .. .

r1 ro .. .

∗ rn−1 .. .

· · · rn · · · rn−1 .. .. . . · · · ro .. .. . .

h1

h2

···

hn

ho

h1

···

h n−1

0 .. .

ho .. .

···

h n−2 .. .

0 .. .

0 .. .

··· .. .

..

.

ho .. .



··· ··· 



···  ···  .. . ···

(6C.8)



· · · 



· · ·  . · · · 

(6C.9)

 

· · · ..

.

Notice that T is a Hermitian Toeplitz matrix whose ﬁrst row is given by (ro , r1 , r2 , . . . rn , . . .) and H is an upper triangular Toeplitz matrix whose ﬁrst row equals (h o , h 1 , h 2 , . . . h n , . . .). In terms of T and H, the inﬁnite set of equations in (6C.3) can be expressed compactly as T = HH∗ > 0.

(6C.10)

Let Tn and Hn represent the top-left (n + 1) × (n + 1) block matrices of T and H respectively. Then



ro

  ro  ∗ rn−1  r   1 ..   =  .. .   . ro rn∗

· · · rn

r1

r ∗ ro · · · 1 Tn =  .. . .  .. . . . ∗ ∗ rn rn−1 · · ·

r1 · · · rn

Tn−1

        Tn−1 =    rn∗ · · · r1∗



rn ..  . > 0



r1 

ro (6C.11)

and



ho

0   0 Hn =  .  ..  0

h1

h2

···

ho

h1

···

0 .. .

ho .. .

···

0

0

···

..

.

hn



h n−1  



h n−2  = ..  .  

ho h1 · · · hn 0

Hn−1

=

ho

b

0 Hn−1

.

ho (6C.12)

Chapter 6:

STAP for SBR

Using (6C.12), we can rewrite H in (6C.9) as

H=

Hn Xn

(6C.13)

0 Yn

where the semi-inﬁnite matrix

   Xn =   

···



h n+1

h n+2

h n+3

hn .. .

h n+1 .. .

h +2 .. .

···  ,  ···

ho

h1

h2

···

(6C.14)

and Yn can be deﬁned accordingly from (6C.9). Substituting (6C.13) in (6C.10), and making use of (6C.11), we get Tn = Hn H∗n + Xn Xn∗ .

(6C.15)

Equation (6C.15) represents an exact identity where the key observation is that the role of Xn becomes less and less signiﬁcant as n → ∞. This is true for any ﬁnite power system (ro < ∞), since in that case, h i → 0 as i → ∞. The Bauer-type factorization exploits this fact and shows that the iterative matrix factorization procedure above leads to a minimum phase system impulse response sequence [19]. This result is based on Jensen’s inequality [23], which in the general matrix case states that for an m × m transfer function F(z) =

∞

Fk z−k ,

det Fo = 0,

(6C.16)

k=0

we have 1 2π

5π

ln det F(e jθ ) dθ ≥ ln |det Fo | > −∞,

(6C.17)

−π

with equality in the ﬁrst part of (6C.17) impling that F(z) in (6C.16) is minimum phase, i.e., det F(z) is free of zeros in |z| > 1. In fact from (6C.15), as n → ∞, we obtain Tn → Hn H∗n ,

(6C.18)

301

302

Space Based Radar where the ﬁrst row of Hn gives the desired impulse response sequence. Following [19], let

      Ln =     



L (n) 01

L (n) 01

L (n) 02

···

L (n) 0n

0

L (n) 11

L (n) 12

···

 L (n) 1n 

0

0

L (n) 22

···

.. .

.. .

.. .

..

0

0

0

···

.

 

 L (n) 2n   ..  .  

(6C.19)

L (n) nn

represent the unique upper triangular factor with positive diagonal entries of the positive deﬁnite Toeplitz matrix Tn in (6C.11). Thus Tn = Ln L∗n .

(6C.20)

Then (6C.15)–(6C.18) imply that lim L i(n) j = h j−i

n→∞

(6C.21)

and in particular the entries of the ﬁrst row of Ln gives lim L(n) 0j = h j,

n→∞

j = 0, 1, 2, . . .

(6C.22)

i.e., as n → ∞, the ﬁrst row of Ln gives the desired impulse response sequence. Interestingly, the upper triangular factor in (6C.19) can be computed iteratively. Let

Ln = Then



c b . 0A

(6C.23)



ro r1 · · · rn  r∗  d bA∗  1  ∗ Tn =  .  = Ln Ln = Ab ∗ AA∗  .. Tn−1  rn∗

(6C.24)

where d = c 2 + b ∗ b.

(6C.25)

AA∗ = Tn−1 = Ln−1 L∗n−1

(6C.26)

But from (6C.24)

Chapter 6:

STAP for SBR

and hence we get A = Ln−1 .

(6C.27)

Using the identity [r1 , r2 , . . . , rn ] = bA∗ in (6C.24) we get

b = [r1 , r2 , . . . , rn ] L∗n−1

−1

,

(6C.28)

and

ro − b ∗ b.

c=+

(6C.29)

Using (6C.27)–(6C.29) in (6C.23), we obtain the iterative algorithm

Ln =

c b 0 Ln−1

n ≥ 1.

,

(6C.30)

Thus at every stage, only the ﬁrst row of the upper triangular matrix needs to be computed using (6C.28) and (6C.29). From (6C.21) and (6C.22) for sufﬁciently large n, the ﬁrst row of Ln tends to be desired minimum phase impulse response. Finally to establish that the sequence {h j }∞ j=0 in (6C.21) leads to a minimum phase factor, from Jensen’s inequality in (6C.17), it is necessary to show that 1 2π

5π ln S(ω)dω = ln |h o |2 .

(6C.31)

−π

This key identity is established in [19].

Appendix 6-D: Rational System Representation Let H(z) =

b o + b 1 z−1 + · · · + b m z−m B(z) = 1 + a 1 z−1 + · · · + a n z−n A(z)

(6D.1)

represent an ARMA(n, m) rational system, whose poles z1 , z2 , . . . , zn are given by the zeros of its denominator polynomial. Thus A(z) = a o + a 1 z−1 + · · · + a n z−n =

n 9 k=1

(1 − zk z−1 ),

zk = e −(αk + jωk ) . (6D.2)

303

304

Space Based Radar Partial fraction expansion of (6D.1) gives H(z) =

n k=1

=

∞

i ck = ck zk z−1 −1 1 − zk z n

∞

n

i=0

k=1

k=1

c k zki

i=0

z−i =

∞

h i z−i

(6D.3)

i=0

where

hi =

n

c k zki ,

i ≥0

(6D.4)

k=1

represents the impulse response of the system. Equation (6D.4) represents a sum of complex sinusoids since with zk = e −(αk + jωk ) ,

k = 1, 2, . . .

(6D.5)

we obtain hi =

n

c k e −(αk + jωk )i ,

k = 0, 1, 2, . . .

(6D.6)

k=1

If (6D.1) represents a system with real coefﬁcients, then the poles and residues in (6D.4) occur in complex conjugate pairs so that hi = 2

n/2

|c k | e −αk i cos(ωk i + φi ).

(6D.7)

k=1

For a stable system representation as in (6D.1) and (6D.2), the poles are within the unit circle. Thus αk > 0 in (6D.5) and (6D.6) and hence (6D.7) represents a sum of exponentially damped sinusoids at frequencies ω1 , ω2 , . . . . It is easy to show that in the rational case, the h i s in (6D.4) are linearly dependent beyond a certain stage. This follows by equating (6D.1) and the right side of (6D.3). Thus

∞ −1 −n −i ao + a1z + · · · + anz hi z = b o + b 1 z−1 + · · · + b m z−m . i=0

(6D.8)

Equating coefﬁcients of like terms on both sides, we get a o h o = bo , a o h 1 + a 1 h o = b1 , .. . a o h m + a 1 h m−1 + · · · + a m h o = b m ,

(6D.9)

Chapter 6:

STAP for SBR

and, in general, a o h k + a 1 h k−1 + · · · + a n h k−n = 0,

k > m.

(6D.10)

Notice that (6D.10) represents an inﬁnite set of equations in the unknowns a o , a 1 , . . . , a n . With a o = 1 (see (6D.1)), Equation (6D.10) gives h k = − (a 1 h k−1 + · · · + a n h k−n ) ,

k>m

(6D.11)

showing the linear dependency of the response coefﬁcients beyond a certain stage. From (6D.10) for k = n, n + 1, . . . , 2n − 1 we get

       

ho

h1

h1

h2

.. .

.. .

h n−1

hn

···

h n−1



an





hn



     h n+1          ..    ..  = −  .. . ··· .  .   .      a1 h 2n−1 · · · h 2n−2 ···

h n   a n−1 

(6D.12)

Equation (6D.12) uses the ﬁrst 2n impulse response coefﬁcients 2n {h i }i=0 of the system, and it can be used to determine the denominator n n coefﬁcients {a i }i=0 and hence the poles {zi }i=1 of the system. Knowing the {a i }, the numerator coefﬁcients {b i } in (6D.1) can be computed using (6D.9). More generally, deﬁne the Hankel matrix



ho

h1

h2

  h1 h2 h3   . .. ..  Hk =  .. . .  h  k−1 h k h k+1  h k h k+1 h k+2

···

hk



    ..    ··· . =     · · · h 2k−1   hk · · · h 2k · · · h k+1 



hk h k+1 

Hk−1

h k+1



..   . 

···

h 2k (6D.13)

that uses the ﬁrst 2k + 1 impulse response coefﬁcients of the system. Then from (6D.12), the matrix Hn−1 is full rank, and moreover using (6D.11) for k > n, the last row/column of Hk in (6D.13) is linearly dependent on its previous n rows/columns. Hence rank Hn = rank Hn−1 = n

(6D.14)

and in general for an ARMA (n, m) system rank Hk−1 = n,

k ≥ n,

(6D.15)

305

306

Space Based Radar an old result due to Kronecker (1881). Equation (6D.15) represents the classic necessary and sufﬁcient condition for an inﬁnite sequence {h k }∞ k=0 to represent the impulse response of a rational system. Equation (6D.15) can also be used to determine the pole locations. From (6D.10) with k = n, n + 1, . . . , 2n, we get



an



   a n−1     .  .  Hn  .   = 0.    a1    ao

(6D.16)

Notice that Hn is of size (n+1) ×(n+1) and has rank n (see (6D.14)). Hence a = [a n , a n−1 , . . . , a 1 , a o ]T

(6D.17)

formed from the denominator coefﬁcients represents the unique eigenvector associated with the zero eigenvalues of Hn . The power spectrum associated with any system transfer function H(z) is given by

2

S(ω) = H(e jω ) =

∞

rk e − jkω ≥ 0,

(6D.18)

k=−∞ ∗ , k ≥ 0 represents its autocorrelation sequence. Substiwhere rk = r−k ∞ h i z−i into (6D.18), we obtain tuting H(z) = i=0

S(ω) =

∞

h m e − jmω

∞

m=0

=

∞ ∞

k=−∞

h i∗ e jiω =

i=0

h i+k h i∗

∞ ∞

h m h i∗ e − j(m−i)ω

m=0 i=0

e − jkω =

∞

rk e − jkω

(6D.19)

k=−∞

i=0

which gives (see (6C.3)) rk =

∞

h k+i h i∗ ,

k = 0 → ∞.

(6D.20)

i=0

For a rational system as in (6D.1), its impulse response is given by (6D.4). To show that, a similar relation holds among its

Chapter 6:

STAP for SBR

autocorrelations, we can use (6D.4) in (6D.20). This gives rk =

n ∞ i=0

=

n m=1

i+k c m zm

n

m=1 k c m zm

cl∗ (zl∗ ) i =

l=1 n l=1

cl∗ = 1 − zm zl∗

n n

c m cl∗

m=1 l=1 n m=1

k c m dm zm =

qm

n

∞

i k zm zl∗ zm

i=0 k q m zm

m=1

dm

=

n

q m e −(αm + jωm )k ,

k=0→∞

(6D.21)

m=1

where q m = c m dm = c m

n l=1

cl∗ . 1 − zm zl∗

(6D.22)

Thus the autocorrelations of a rational system also has a representation similar to (6D.4) as the sum of damped exponentials. On comparing (6D.6)–(6D.15) with (6D.21), it follows that Hankel matrices generated using the autocorrelations also satisfy similar rank conditions [18].

References [1] Knowledge Aided Sensor Signal Processing and Expert Reasoning (KASSPER), Third Annual Workshop, Clearwater, Florida, February 22–24, 2005. [2] J.R. Guerci, Space-Time Adaptive Processing for Radar, Artech House, Boston, MA, 2003. [3] R. Klemm, Principles of Space-Time Adaptive Processing, IEE Publishing, London, UK, 2002. [4] S.U. Pillai, B. Himed, K.Y. Li, “Effect of Earth’s Rotation and Range Foldover on Space-Based Performance”, Proc. IEEE Transactions on Aerospace and Electronic Systems, Vol. 42, No. 3, July 2006. [5] S.U. Pillai, Array Signal Processing, Springer-Verlag, New York, NY, 1989. [6] K.Y. Li et al., “STAP for Space Based Radar,” Air Force Research Laboratory Final Technical Report, AFRL-SN-RS-TR-2004-170, AFRL, Rome, NY, June 2004. [7] M.E. Davis, B. Himed, D. Zasada, “Design of Large Space Radar for Multimode Surveillance,” IEEE Radar Conference, Huntsville, AL, pp. 1–6, May 2003. [8] P. Zulch, et al., “The Earth Rotation Effect on a LEO L-Band GMTI SBR and Mitigation Strategies,” IEEE Radar Conference, Philadelphia, PA, April 2004. [9] M.E. Davis, B. Himed, “L Band Wide Area Surveillance Radar Design Alternatives,” Proc. of International Radar 2003–Adelade Australia, September, 2003. [10] J. Maher, et al., “High Fidelity Modeling of Space-Based Radar,” Proc. 2003 IEEE Radar Conference, pp. 185–191, Huntsville, AL, May 5–8, 2003. [11] S.U. Pillai, B. Himed, K.Y. Li, “Waveform Diversity for Space Based Radar,” Proceedings of Waveform Diversity and Design, Edinburgh, SL, November 8–10, 2004.

307

308

Space Based Radar [12] J.B. Billingsley, Low-Angle Radar Land Clutter, William Andrew Publishing Norwich, NY, 2002. [13] NASA TERRA Satellite. http://terra.nasa.gov [14] M. Long, Radar Reﬂectivity of Land and Sea, Artech House, Boston, MA, 2001. [15] P. Techau, B. Jameson, J.R. Guerci, “Effects of Internal Clutter Motion on STAP in a Heterogeneous Environment” IEEE Radar Conference, Atlanta, GA, 2001. [16] P. Zulch, “Five Parameter Clutter Model,” Private Communications, Air Force Research Lab, Rome, NY, 2005. [17] A. Papoulis, S.U. Pillai, Probability, Random Variables and Stochastic Processes, McGraw-Hill Higher Education, New York, NY, 2002. [18] S.U. Pillai, Spectrum Estimation and System Identiﬁcation, Spring-Verlag, New York, NY, 1993. [19] D.C. Youla, N. Kazanjian, “Bauer-Type Factorization of Positive Matrices and the Theory of Matrix Polynomials Orthogonal on the Unit Circle,” IEEE Transactions on Circuits and Systems, Vol. 25, Issue. 2, Feburary 1978. [20] S. Mangiat, K.Y. Li, S.U. Pillai, B. Himed, “Effect of Terrain Modeling and Internal Clutter Motion on Space-Based Radar Performance”, 2006 IEEE Radar Conference, Verona, NY, April 24–27, 2006. [21] P. Mountcastle, “New Implementation of the Billingsley Model for GMTI Data Cube Generation,” IEEE Radar Conference, Philadelphia, PA, 2004. [22] G.K. Borsari, “Mitigating Effects on STAP Processing Caused by an Inclined Array,” Proceedings of IEEE National Radar Conference, Dallas, TX, May 1998. [23] R.P. Boas, Entire Functions, Academic Press, New York, NY, 1954.

CHAPTER

7

Performance Analysis Using Cramer-Rao Bounds A theorem independently developed by Rao (1945) and Cramer (1946) states that the variance of any unbiased estimator for a nonrandom parameter can be always lower bounded under some very general conditions. Known as the Cramer-Rao bound it is related to the Fisher information content about the unknown parameter contained in the joint density function of the data set (observations). When the parameter set contains more than one unknown, the bounds become coupled and can be expressed in matrix form.

7.1 Cramer-Rao Bounds for Multiparameter Case Let θ = [θ1 , θ2 , . . . , θm ]T represents the unknown set of nonrandom parameters contained in the data set x under the probability density function (p.d.f.) f (x, θ ), and T(x) = [T1 (x), T2 (x), . . . , Tm (x)]T

(7.1)

an unbiased estimator vector for the unknown set of parameters θ . Then E{T(x)} = θ

(7.2)

and the covariance matrix for T(x) is given by cov{T(x)} = E{(T(x) − θ)(T(x) − θ ) ∗ }.

(7.3)

309 Copyright © 2008 by The McGraw-Hill Companies. Click here for terms of use.

310

Space Based Radar The cov{T(x)} is of size m × m and it represents a positive-deﬁnite matrix. The Cramer-Rao bound in this case has the form1 cov{T(x)} ≥ J−1 (θ )

(7.4)

where J(θ ) represents the m × m Fisher information matrix associated with the parameter set θ under the p.d.f. f (x, θ ). The entries of the Fisher information matrix are given by

*

Ji j = E

∂ log f (x; θ ) ∂ log f (x; θ ) ∂θi ∂θ j

* = −E

∂ 2 log f (x; θ) ∂θi ∂θ j

+

+

,

i, j = 1 → m

(7.5)

provided the following regularity conditions are satisﬁed [1], [2]: ∂ ∂θi and ∂ ∂θi

5

5

f (x, θ) d x =

5

∂ f (x, θ) d x = 0, ∂θi

5 T(x) f (x, θ) d x =

T(x)

∂ f (x, θ) d x. ∂θi

(7.6)

(7.7)

Here the integrals represent n-fold integration.

Proof. Using the unbiased property, for any parameter θi we have 5+∞ E{Ti (x) − θi } = (Ti (x) − θi ) f (x; θ) d x = 0.

(7.8)

−∞

Differentiate the later part of (7.8) with respect to θi on both sides to obtain

5+∞ 5+∞ ∂ f (x; θ) (Ti (x) − θi ) dx − f (x; θ) d x = 0, ∂θi

−∞

(7.9)

−∞

where we have made use of the regularity conditions given in (7.6) and (7.7). Thus

5+∞ ∂ f (x; θ) (Ti (x) − θi ) d x = 1. ∂θi

(7.10)

−∞

1 In (7.4), the notation A ≥ B is used to indicate that the matrix A − B is a nonnegative-deﬁnite matrix. Strict inequality would imply positive-deﬁniteness.

Chapter 7:

Performance Analysis Using Cramer-Rao Bounds

But 1 ∂ f (x; θ) ∂ log f (x; θ ) = , ∂θi f (x; θ) ∂θi

(7.11)

∂ f (x; θ) ∂ log f (x; θ ) = f (x; θ) ∂θi ∂θi

(7.12)

so that2

and (7.10) becomes

5+∞ ∂ log f (x; θ ) (Ti (x) − θi ) f (x; θ) d x = 1, ∂θi

i = 1 → m.

(7.13)

−∞

From (7.13), we obtain the useful identity

* E

(Ti (x) − θi )

∂ log f (x; θ ) ∂θi

+ = 1,

i = 1 → m.

(7.14)

Also using (7.7) we obtain

* E

∂ log f (x; θ ) (Ti (x) − θi ) ∂θ j

+

5+∞ ∂ f (x; θ ) = (Ti (x) − θi ) dx ∂θ j −∞

5+∞ 5+∞ ∂ ∂ Ti (x) f (x; θ ) d x − θi f (x; θ) d x = ∂θ j ∂θ j −∞

∂θi = 0, = ∂θ j

2 From

−∞

i = j = 1 → m.

(7.11), we get the identity E f (x; θ ) d x = 0.

∂ log f (x;θ ) ∂θi

(7.15)

=

: +∞ −∞

∂ f ( x;θ ) ∂θi

dx =

∂ ∂θi

: +∞ −∞

311

312

Space Based Radar To exploit the key identities obtained in (7.14) and (7.15), deﬁne the 2m × 1 vector





T1 (x) − θ1

 T2 (x) − θ2   ..  .    Tm (x) − θm   z=  ∂ log f (x;θ )  ∂θ1   ∂ log f (x;θ )  ∂θ2   ..  .  ∂ log f ( x;θ )

        y1    = =    y2       

  . 

(7.16)

∂θm

Then using (7.2) and (7.11), we get E{z} = 0 and hence cov{z} = E{zz∗ } = E

(7.17)

y1 y∗1

y1 y∗2

y2 y∗1

y2 y∗2

.

(7.18)

But E{y1 y∗1 } = cov{T(x)}

(7.19)

and from (7.14) and (7.15) E{y1 y∗2 } = Im

(7.20)

the identity matrix. Also from (7.5)

 E{y2 y∗2 } =

J11

J12

J  21 J22 J= ..  ..  . . Jm1 Jm2

···

J1m

··· .

J2m .. .

···

Jmm

..

     

(7.21)

represents the Fisher information matrix with entries Ji j = J ji as in (7.5). Thus

cov{z} =

cov(T(x))

I

I

J

≥ 0.

(7.22)

Chapter 7:

Performance Analysis Using Cramer-Rao Bounds

To make further progress, deﬁne the matrix

S=

I

0

−D−1 C

I

(7.23)

and obtain the matrix identity S∗

A

B

C

D

S=

A − BD−1 C B − C∗ D∗−1 D 0

D

.

(7.24)

With A = cov{T(x)}, B = C = I, and D = J in (7.24), it represents a positive-deﬁned matrix that reduces to

A − BD−1 C

B − C∗ D∗−1 D

0

D

=

cov{T(x)} − J−1

0

0

J

≥0 (7.25)

which gives cov{T(x)} − J−1 ≥ 0

(7.26)

cov{T(x)} ≥ J−1 (θ),

(7.27)

or

the desired result in (7.4). To obtain the second form in (7.5), integrate (7.12) and use the regularity condition once again to obtain2

5

5

∂ log f (x, θ ) ∂ f (x, θ) dx = ∂θi ∂θi

5

f (x, θ) d x = 0.

(7.28)

Differentiate this expression with respect to θ j to get ∂ f (x, θ) ∂ log f (x, θ ) dx + ∂θ j ∂θi

5

f (x, θ)

∂ 2 log f (x, θ) d x = 0, ∂θi ∂θ j

(7.29)

or using (7.12) again, we get the desired identity

5

Ji j =

f (x, θ)

5 =−

∂ log f (x, θ ) ∂ log f (x, θ) dx ∂θi ∂θ j

f (x, θ)

∂ 2 log f (x, θ) d x, ∂θi ∂θ j

(7.30)

the desired result. In particular, from (7.27)

var{Ti (x)} ≥ J ii = (J−1 ) ii ,

i = 1 → m.

(7.31)

Thus in the multiparameter case, the Cramer-Rao bound is given by the diagonal entries of the inverse of the Fisher information matrix.

313

314

Space Based Radar When the unknown set consists of a single parameter θi , the corresponding CR bound in (7.27), (7.31) becomes var{Ti (x)} ≥

1 = Jii

E

1 ∂ log f (x; θi ) ∂θi

2 / =

* E

−1 +. ∂ 2 log f (x; θi ) ∂θi2

(7.32) In (7.31) and (7.32) if the variance of an estimator agrees with the corresponding bound, then that estimator is said to be efﬁcient. Interestingly (7.27)–(7.32) can be used to evaluate the degradation factor for the CR bound caused by the presence of additional unknown parameters in the scene. For example, when θ1 is the only unknown, the CR bound is given by 1/J11 , whereas it is given by J11 in presence of other parameters θ2 , · · · θ3 , · · · θm . With

J=

J11

b∗

b

G

(7.33)

in (7.21), from (7.24) we have J11 =

1 ∗

−1

J11 − b G b

=

1 1 . ∗ J11 1 − b G−1 b/J11

(7.34)

Thus 1/(1 − b∗ G−1 b/J11 ) > 1 represents the effect of the remaining unknown parameters on the bound for θ1 . As a result, with one additional parameter, we obtain the increment on the bound for θ1 to be J11 −

J212 1 1 = ≥ 0. J11 J11 J11 J22 − J212

(7.35)

Referring back to (7.21) and (7.33), if the Fisher information matrix turns out to be diagonal, then the estimators are independent, and their bounds are not degraded by the presence of other parameters. Conversely, the freedom present in setting up an experiment can be used sometimes to realize diagonal Fisher information matrices. To illustrate how the performance degrades when an additional unknown parameter is introduced, next we consider some examples. Example 7.1 Let x1 , x2 , · · · , xn represent i.i.d. gamma distributed random variables with parameters α and β. Then their joint probability density function is given by [3] β nα f (x; α, β) = n (α)

n 9

xiα−1

e

−β

n i=1

xi

.

(7.36)

i=1

Determine the Cramer-Rao bounds for α and β when either or both parameters are unknown.

Chapter 7:

Performance Analysis Using Cramer-Rao Bounds

Solution From (7.36), we have

L = log f (x; α, β) = nα log β − n log (α) + (α − 1)

n

log xi − β

i=1

n

xi . (7.37)

i=1

The parameters α and β may be unknown one at a time or together. In either case, we have n (α) ∂L log xi , = n log β − + ∂α (α)

(7.38)

nα ∂L xi . = − ∂β β

(7.39)

n

i=1

and n

i=1

This gives ∂2 L = −n ∂α 2

(α) − (α)

# (α) $2 (α)

= nϕ(α),

∂2 L n = , ∂α ∂β β

(7.40)

(7.41)

nα ∂2 L =− 2 , ∂β 2 β

(7.42)

and hence from (7.5)

* J11 = −E

* J22 = −E

* J12 = −E

∂2 L ∂α 2 ∂2 L ∂β 2

+ = nϕ(α) ,

(7.43)

nα , β2

(7.44)

+ =

∂2 L ∂α ∂β

+

n . β2

=−

(7.45)

Hence if either α or β is the only unknown, then their respective CR bounds are given by 2 σCR (α) =

1 1 = J11 nϕ(α)

(7.46)

1 β2 = . J22 nα

(7.47)

and 2 σCR (β) =

However, if both α and β are unknown simultaneously, we have

J=

J11

J12

J21

J22

=n

so that J

−1

1 = n(αϕ(α) − 1)

− β1

ϕ(α)

(7.48)

α β2

− β1 α

β

β

β 2 ϕ(α)

.

(7.49)

315

316

Space Based Radar This gives the corresponding bounds to be 2 (α) = J11 = σCR

α 1 > n(αϕ(α) − 1) nϕ(α)

(7.50)

β 2 ϕ(α) β2 > . n(αϕ(α) − 1) nα

(7.51)

and 2 (β) = J22 = σCR

Clearly when both parameters are unknown, the bounds are inferior compared to the case where only either parameter is unknown.

As the next example shows, the presence of an extra parameter does not always lead to inferior bounds, however, at times it can lead to inferior estimators. Example 7.2 Let x1 , x2 , · · · , xn be i.i.d. Gaussian data samples with common mean µ and common variance σ 2 . Find the Cramer-Rao bounds for µ and σ 2 when either or both parameters are unknown. Solution In this case, the joint density function of the observation is given by f (x; µ, σ 2 ) =

1 − e (2π σ 2 ) n/2

n

i=1

(xi −µ) 2 2σ 2

(7.52)

which gives

(xi − µ) 2 n . log(2π σ 2 ) − 2 2σ 2 n

L = log f (x; µ, σ 2 ) = −

(7.53)

i=1

Hence

(xi − µ) ∂L = ∂µ σ2

(7.54)

(xi − µ) 2 ∂L n =− 2 + . ∂σ 2 2σ 2(σ 2 ) 2

(7.55)

n

i=1

and n

i=1

As a result, we obtain ∂2 L = ∂µ2

n −1

σ2

i=1

=−

n , σ2

(xi − µ) ∂2 L = − , ∂µ ∂σ 2 (σ 2 ) 2

(7.56)

n

(7.57)

i=1

and

(xi − µ) 2 ∂2 L n = − . 2 2 2 2 ∂(σ ) 2(σ ) (σ 2 ) 3 n

i=1

(7.58)

Chapter 7:

Performance Analysis Using Cramer-Rao Bounds

Hence

* J11 = −E

* J12 = −E

* J22 = −E

∂2 L ∂µ2

+ =

∂2 L ∂µ ∂σ 2 ∂2 L ∂(σ 2 ) 2

+

n , σ2 =

(7.59)

n E{xi − µ}

σ4

i=1

+

= 0,

E{(xi − µ) 2 } n + 2σ 4 σ6 n

=−

i=1

n nσ 2 n =− 4 + 6 = . 2σ σ 2σ 4 Hence

(7.60)

(7.61)

n

0 = . (7.62) J= n J21 J22 0 2σ 4 The diagonal nature of the Fisher information matrix in (7.62) indicates that in the case of i.i.d. Gaussian data samples, the bounds on the variance of any unbiased estimator for parameters µ and σ 2 are the same whether only one parameter is unknown or both parameters are unknown simultaneously. In particular, these bounds for µ and σ 2 are given by J11

J12

2 (µ) = σCR

σ2

1 σ2 = J11 n

(7.63)

1 2σ 4 = . J22 n

(7.64)

and 2 (σ 2 ) = σCR

Thus in the i.i.d. Gaussian data case, whether the mean µ is known or unknown, it does not inﬂuence the CR bound on the variance parameter σ 2 and vice versa.

It is interesting to note that the bounds may be insensitive to the status of the other parameter; however, the actual unbiased estimators in these cases may not exhibit similar behavior. We shall illustrate this for the i.i.d. Gaussian data case under discussion here. In that case, for the unknown parameter µ, the sample mean 1 xi = x¯ n

(7.65)

E{µ} ˆ = E{¯x} = µ

(7.66)

n

µ ˆ =

i=1

is an efﬁcient estimator since

and var{µ} ˆ = E{( x¯ − µ) 2 } =

n 1 nσ 2 σ2 E{(xi − µ) 2 } = 2 = 2 n n n i=1

(7.67)

317

318

Space Based Radar that agrees with (7.63). Thus the sample mean estimator is always efﬁcient whether σ 2 is known or unknown. However, the situation with the parameter σ 2 is quite different. If µ is known, then 1 (xi − µ) 2 n n

σˆ 2 =

(7.68)

i=1

is an efﬁcient estimator for σ 2 since 1 nσ 2 E{(xi − µ) 2 } = = σ2 n n

(7.69)

2σ 4 n

(7.70)

n

E{σˆ 2 } =

i=1

and3 var{σˆ 2 } =

that agrees with the bound in (7.64). Notice that in this case the bound is proportional to the square of the unknown parameter of interest. However if µ is unknown, the estimator in (7.68) is not useful. In that case, it can be shown that [1] 1 (xi − x¯ ) 2 n n

θˆ2 =

(7.71)

i=1

is the best unbiased estimator for σ 2 in terms of attaining the lowest possible variance. Moreover, after some algebra, its variance is given by [1], [2], [3] var{θˆ2 } =

2σ 4 . n−1

(7.72)

From (7.70) and (7.72), the variance of the unknown parameter σ 2 is inferior (and not efﬁcient) when the other parameter µ is unknown. Thus the quality of the best unbiased estimator for one unknown parameter can depend on whether the other parameter is known or unknown. Next we address the general multichannel Gaussian case. 3 We

have yi = xi − µ ∼ N(0, σ 2 ). Hence E{yi2 } = σ 2 , E{yi4 } = σ 4 . In term of yi ,

we have σˆ 2 =

1 n

n

i=1

yi2 and hence var{σˆ 2 } =

E{yi4 }−E{yi2 } n

=

3σ 4 −σ 4 n

=

2σ 4 n .

Chapter 7:

Performance Analysis Using Cramer-Rao Bounds

Gaussian Data When n independent and identically distributed observations x = [x1 , x2 , . . . , xn ]

(7.73)

are jointly Gaussian with covariance matrix R, the Fisher information matrix J in (7.4) and (7.5) can be further simpliﬁed. In that case [4] f (x; θ) =

1 e− |πR|n

n

xi∗ R−1 xi =

i=1

1 −n tr (R−1 S) e |π R|n

(7.74)

where 1 ∗ xi xi S= n n

(7.75)

i=1

represents an unbiased estimate for R. Thus log f (x; θ) = −n(log |π R| − tr {R−1 S}) so that

*

∂ log f (x; θ ) ∂|R| = −n |R−1 | + tr ∂θi ∂θi

* *

= −n tr R−1

*

−1 ∂R

= −n tr R since

∂R ∂θi

*

∂θi

+

∂R−1 S ∂θi

*

− tr R−1

+

−1

++

+ =

++

∂R −1 R S ∂θi

(I − R S)

∂R ∂|R| |R | = tr R−1 ∂θi ∂θi −1

*

(7.76)

(7.77)

1 ∂λk , λk ∂θi

(7.78)

k

and ∂R−1 ∂R −1 = −R−1 R ∂θi ∂θi

(7.79)

m where {λi }i=1 represent the eigenvalues of R. Differentiating (7.77) again with respect to θ j we get

*

∂ 2 log f (x; θ ) ∂R −1 ∂R = −n tr −R−1 R (I − R−1 S) ∂θi ∂θ j ∂θ j ∂θi + R−1

∂ 2R ∂R (I−R−1 S)+ R−1 ∂θi ∂θ j ∂θi

+ ∂R −1 R−1 R S . ∂θ j (7.80)

319

320

Space Based Radar Taking expectation on both sides we obtain

* Ji j = −E

∂ 2 log f (x; θ ) ∂θi ∂θ j

+

*

−1 ∂R

= n tr R

∂θi

−1

R

∂R ∂θ j

+ (7.81)

since E{S} = R. Equation (7.81) is also known as the Slepian-Bang’s formula [5]. We can use (7.81) to determine the Fisher information matrix by substituting the appropriate R in each case.

7.2 Cramer-Rao Bounds for Target Doppler and Power in Airborne and SBR Cases Consider a target located at (θ EL t , θ AZt ) with power level Pt and Doppler parameter ωdt . In general, all these four target parameters are unknown. In practice, however, the data from a speciﬁc location is analyzed for the presence or absence of a target. Hence we will assume the target location (θ EL t , θ AZt ) and its associated cone angle c t = sin θ EL t cos θ AZt

(7.82)

are known, and the power level Pt and the Doppler ωdt are unknown parameters. The simpler situation is to assume that only one parameter is unknown at a time, and in the general case both parameters are unknown simultaneously. Both in the airborne and the space based radar (SBR) case, the target is buried in clutter and noise; although the clutter model depends upon the type of platform being used. For example in the SBR case, as we have seen in Chapter 6, the clutter covariance matrix Rc takes different forms depending on whether or not Earth’s rotation and range foldover are modeled into the formulation. Naturally with increasing clutter complexity, one expects poor performance and a quantitative measure of what can be expected in terms of target parameter estimation accuracy is of crucial importance both in the airborne as well as the SBR case. The theory developed here is platform independent and can be used to evaluate the performance of either platforms by substituting the clutter covariance matrix and steering vector models that are appropriate for the case under consideration. In all these situations, a baseline can be established by considering the total interference to be consisting only of independent and identically distributed sensor noise with common input variance σn2 . Note that the total interference covariance matrix in that case becomes Rc = σn2 I.

(7.83)

Chapter 7:

Performance Analysis Using Cramer-Rao Bounds

The performance in all other situations can be compared with this reference case to determine the degradation due to each factor present in that speciﬁc situation. In that context, the array conﬁguration as well as the array reference points—both spatial and temporal—can inﬂuence the estimation accuracy and the bounds in question may be used to design optimum array conﬁgurations as well as new protocols for determining the best data reference points. The target at location (θ EL t , θ AZt ) that projects an unknown Doppler ωdt at the array can be represented using the steering vector st = b(ωdt ) ⊗ a (c t )

(7.84)

where c t is as in (7.82) and a and b represent the spatial and temporal portions of the steering vector (see (6.6)–(6.8)). In that case the covariance matrix associated with the received data vector x (containing target) is given by R = E{xx∗ } = Pt st s∗t + Rc .

(7.85)

Here Pt represents the target power and Rc represents the total interference (clutter plus noise) covariance matrix at the array output. For example, in the airborne case in the absence of range ambiguities (see (5.84)) Rc =

Pi G(θi )s(θi )s∗ (θi ) + σn2 I

(7.86)

i

and in the SBR case, from (6.20) Rc =

Na No

Pm, j G(θ m, j )sm, j s∗m, j + σn2 I.

(7.87)

j=1 m=1

Here the inner summation is over the Na range foldovers at R1 , R2 , · · · , RNa , and the outer summation is over No azimuth angles of interest including sidelobes. Further, Pm, j and G(θ m, j ) correspond to the clutter power and array gain, respectively, and sm, j = s(c m, j , ωdm, j )

(7.88)

represents the steering vector for the (m, j)th clutter patch. In (7.88), c m, j represents the cone angle for the (m, j)th patch given by c m, j = sin θ EL m cos θ AZ j

(7.89)

and

* βo c m, j , no Earth’s rotation, ωdm, j = ω˜ dm, j = βo ρc sin θ EL m cos(θ AZ j ± φc ), with Earth’s rotation, (7.90) depending on whether Earth’s rotation is absent or present in (7.87).

321

322

Space Based Radar Let θ1 = ωdt

θ2 = Pt

and

(7.91)

represent the two unknown target parameters. In this case from (7.21) and (7.81), the Fisher information matrix is of size 2 × 2 where the (i, j)th entry is given by

Ji j = n tr R

−1 ∂R

∂θi

R

−1

∂R ∂θ j

,

(7.92)

where R is as in (7.85). To make further progress, deﬁne = R−1/2 RR−1/2 c c

(7.93)

where Rc−1/2 represents any square root of the positive-deﬁnite matrix Rc . Then using (7.85), takes the convenient form = I + Pt α α ∗

(7.94)

α = R−1/2 st . c

(7.95)

R = Rc1/2 Rc1/2

(7.96)

where

From (7.93) we have

and substituting this into (7.92) we get

Ji j = n tr

∂ 1/2 −1/2 −1 −1/2 1/2 ∂ 1/2 R−1/2 −1 Rc−1/2 Rc1/2 R Rc Rc Rc R c ∂θi c ∂θ j c

= n tr −1

∂ −1 ∂ . ∂θi ∂θ j

(7.97)

We shall use the later form in (7.97) together with (7.94) to compute Ji j . From (7.94) −1 = I −

Pt α α ∗ Pt α α ∗ = I − 1 + Pt α ∗ α 1+γ

(7.98)

where we deﬁne γ = Pt α ∗ α = Pt s∗t R−1 c st =

Pt σ2

(7.99)

and σ2 =

1 s∗t R−1 c st

(7.100)

represents the “effective interference pulse noise power” at the target site. Notice that γ represents the effective SINR at the target site since

Chapter 7:

Performance Analysis Using Cramer-Rao Bounds

in the noise only case we have Rc = σn2 I and from (7.100) 2 s∗t R−1 c st = MN/σn

and4 γ = MN Pt /σn2 . From (7.94) we also obtain

∂ = Pt ∂ωdt where

∂α ∗ ∂α ∗ α +α ∂ωdt ∂ωdt

(7.101)

= Pt

∂α ∗ ∂α ∗ ∂st ∗ ∂s∗ α +α = Rc−1/2 st + st t ∂ωdt ∂ωdt ∂ωdt ∂ωdt

=

(7.102)

R−1/2 . c

(7.103)

With st as in (7.84) we get





0

e − jπ ωdt



 ∂st ∂b(ωdt ) = ⊗ a (c t ) = − jπ   ∂ωdt ∂ωdt 

   ⊗ a (c t ).  

2e − j2π ωdt .. . ( M − 1)e − j ( M−1)π ωdt

(7.104) Using (7.98) and (7.102) we have

−1

∂ = ∂ωdt

α α∗ I − Pt 1+γ

and hence from (7.97) J11 = Jωdt ,ωdt = n tr

* =

nPt2 tr

Pt = Pt

∂ −1 ∂ωdt

α α∗ − Pt 1+γ

(7.105)

2 /

α α ∗ 2 α α ∗ α α ∗ α α ∗ − Pt − Pt + Pt2 1+γ 1+γ (1 + γ ) 2

+

2

= nPt2 tr {2 } − 2Pt

α ∗ 2 α (α ∗ α) 2 + Pt2 1+γ (1 + γ ) 2

.

(7.106)

Similarly ∂ = α α∗ ∂ Pt

(7.107)

4 In the white noise case, let σ 2 represents the noise at each sensor element input. n Then Pt /σn2 represents the input SNR at each sensor input. Thus γ = Pt /σ 2 = MNPt /σn2 = MN(SNR), and the SINR at the target site has been “boosted up” by a factor of MN by the beamformer action of the array.

323

324

Space Based Radar so that −1

∂ α α∗ α α∗ γ = α α ∗ − Pt = 1− ∂ Pt 1+γ 1+γ

and hence

J22 = n tr

* = n tr

−1

∂ ∂ Pt

α α∗ =

α α∗ 1+γ

2

(7.108)

2 /

α α∗ α α∗ 1+γ 1+γ

+

γ n(α ∗ α) 2 = =n 2 (1 + γ ) 1+γ

1 . Pt2

(7.109)

+

Finally from (7.97)

* J12 = n tr

−1

*

=

*

nPt 1+γ

=n

α∗ α ∗ α α − Pt α α 1+γ ∗

1−

γ 1+γ

= n tr Pt

α α ∗ α α ∗ nPt tr α α ∗ − Pt 1+γ 1+γ

nPt = 1+γ =

+

∂ −1 ∂ ∂ωdt ∂ Pt

α ∗ α =

+

α α∗ − Pt 1+γ

α α∗ 1+γ

nPt α ∗ α (1 + γ ) 2

α ∗ α γ γ =n α ∗ o α (1 + γ ) 2 α ∗ α (1 + γ ) 2

(7.110)

where we have deﬁned = ∗ −1 . α∗ α st Rc st

o =

(7.111)

Now using (7.95) and (7.103) we have ∗

α α =

s∗t R−1 c

= s∗t R−1 c st

∂st ∗ ∂s∗ st + st t ∂ωdt ∂ωdt

s∗t R−1 c

R−1 c st

∂st ∂s∗t −1 + R st ∂ωdt ∂ωdt c

(7.112)

and hence using (7.111) α ∗ o α =

∂st α ∗ α ∂s∗t −1 = s∗t R−1 + R st = 2Re c ∗ α α ∂ωdt ∂ωdt c

*

s∗t R−1 c

∂st ∂ωdt

+ .

(7.113)

Chapter 7:

Performance Analysis Using Cramer-Rao Bounds

Deﬁne ρ = s∗t R−1 c

∂st = a + jb ∂ωdt

(7.114)

to represent the correlation between the target steering vector and its Doppler derivative weighted with respect to the inverse of the clutter covariance matrix. Then from (7.113) α ∗ o α = 2a

(7.115)

represents twice its real part and using this in (7.110) we get

J12 = n

γ 1+γ

2

2a . γ

(7.116)

To simplify (7.106) further, using (7.111) we can rewrite it as

J11 = nPt2 (α ∗ α) 2 tr 2o − 2Pt

=n

γ 1+γ

2#

α ∗ 2o α (α ∗ o α) 2 + Pt2 1+γ (1 + γ ) 2

$

(1 + γ ) 2 tr 2o − 2Pt (1 + γ )α ∗ 2o α + Pt2 (α ∗ o α) 2 . (7.117)

Now using (7.95) and (7.103) we get ∗

2

α α=

s∗t R−1 c

∂st ∗ ∂s∗ st + st t ∂ωdt ∂ωdt

= s∗t R−1 c st

R−1 c

∂st ∗ ∂s∗ st + st t ∂ωdt ∂ωdt

ρ 2 + (ρ ∗ ) 2 + |ρ|2 + s∗t R−1 c st

R−1 c st

∂s∗t −1 ∂st R ∂ωdt c ∂ωdt

,

(7.118) with ρ is as deﬁned in (7.114). Let

µ=

∂s∗t −1 ∂st R ∂ωdt c ∂ωdt

s∗t R−1 c st

(7.119)

represents the weighted (Mahalanobis) distance of the gradient steering vector normalized with respect to the “effective interference plus noise power” σ 2 deﬁned in (7.100). In that case using (7.114) and (7.119) in (7.118), we get α ∗ 2o α =

α ∗ 2 α α ∗ 2 α 2 2 2 = 2 = σ (3a − b + µ). ∗ 2 −1 ∗ (α α) st Rc st

(7.120)

325

326

Space Based Radar Similarly

* 2

tr { } = tr

=2

Rc−1/2

∂st ∗ ∂s∗ st + st t ∂ωdt ∂ωdt

∂s∗t −1 ∂st R ∂ωdt c ∂ωdt

R−1 c

∂st ∗ ∂s∗ st + st t Rc−1/2 ∂ωdt ∂ωdt

+

2 ∗ 2 2 2 s∗t R−1 c st + ρ + (ρ ) = 2(µ + a − b ),

(7.121) so that

tr 2o =

tr {2 } = 2σ 4 (µ + a 2 − b 2 ). (α ∗ α) 2

(7.122)

Finally substituting (7.115), (7.120), and (7.122) into (7.117) we get

J11 = n

γ 1+γ

2

0 (1 + γ ) 2 2σ 4 (µ + a 2 − b 2 )

− 2Pt (1 + γ )σ 2 (3a 2 − b 2 + µ) + Pt2 4a 2

= 2n

1

2

γ 1+γ

σ 4 [(1 + γ )µ + a 2 (1 − γ ) − b 2 (1 + γ )].

(7.123)

From (7.32), if either the target Doppler ωdt or target power Pt is the only unknown, then 1/J11 or 1/J22 acts as the respective lower bound for their variance and (7.123) or (7.109) can be used to compute them. However, when both the target parameters are unknown, the 2 × 2 Fisher information matrix must be used to compute the corresponding bounds. In that case [6]

J = 2n

γ 1+γ

2

   

σ 4 [(1 + γ )µ + a 2 (1 − γ ) − b 2 (1 + γ )] a γ

a γ 1 2Pt2

   

(7.124) so that J

−1

1 = 2n o



1+γ γ

1  2Pt2 ×  a − γ

2

−

a γ



  (7.125)  σ 4 [(1 + γ )µ + a 2 (1 − γ ) − b 2 (1 + γ )]

Chapter 7:

Performance Analysis Using Cramer-Rao Bounds

where o = =

σ4 a2 2 2 [(1 + γ )µ + a (1 − γ ) − b (1 + γ )] − γ2 2Pt2 1+γ 1+γ (µ − (a 2 + b 2 )) = (µ − |ρ|2 ). 2γ 2 2γ 2

Hence J−1



1 2 1 1+γ  2P  t = 2  a n µ − |ρ| − γ

(7.126)



a − γ σ 4 [(1 + γ )µ + a 2 (1 − γ ) − b 2 (1 + γ )]

 . 

(7.127) Equation (7.127) can be used to determine the Cramer-Rao bounds for target Doppler and power estimates when both these quantities are simultaneously unknown. From (7.127) the variance of the target Doppler estimate is lower bounded by [6] 2 σCR (ωdt ) = J 11 =

1 1+γ 1 1 1 + 1/γ = 2 2 2 n µ − |ρ| 2Pt 2nσ Pt (µ − |ρ|2 )

(7.128)

and the variance of the target power estimate is lower bounded by 2 σCR ( Pt ) = J 22 =

1 + 1/γ Pt σ 2 [(1 + γ )µ + a 2 (1 − γ ) − b 2 (1 + γ )] . n µ − |ρ|2 (7.129)

Equations (7.128) and (7.129) represent the Cramer-Rao lower bounds for target Doppler and power estimates when both are unknown. Recall that µ represents the normalized Mahalanobis distance of the Doppler gradient of the target steering vector as in (7.119), and ρ represents the “weighted cross correlation” between the target steering vector and its Doppler derivative as deﬁned in (7.114). Using Cauchy-Schwarz’ inequality we have |ρ|2 ≤ µ

(7.130)

so that (7.128) and (7.129) represent meaningful bounds. From (7.128) and (7.129) it also follows that when both target parameters are unknown, array design freedom may be utilized to minimize the weighted correlation factor ρ so that the bounds are as low as possible. From (7.123), J11 represents the bound on target Doppler estimate when the power level is also unknown; whereas from (7.123), 1/J11 represents the same bound when the power level is also known.

327

328

Space Based Radar Hence their difference η1 = J11 −

1 ≥0 J11

(7.131)

can be used as a measure of the degradation in target Doppler estimation due to not knowing the target power. Similarly η2 = J22 −

1 ≥0 J22

(7.132)

represents the degradation factor for target power when target Doppler is unknown. To make meaningful comparisons using these bounds, the noise only case (clutter is absent) can be used as a reference frame.

Noise Only Case In this case, the total interference matrix is diagonal so that Rc = σn2 I

(7.133)

and substituting this into (7.86), (7.114) and (7.119), we get σ2 =

1 s∗t R−1 c st

=

σn2 , MN

γ = Pt /σ 2 = MN ρ=

(7.134)

Pt = MN(SNR), σn2

1 ∗ ∂st s σn2 t ∂ωdt

(7.135) (7.136)

and

2

2

2 MN ∂s∗t ∂st 1 2 ∂st 2 2. µ= 4 = 42 σn ∂ωdt ∂ωdt σn 2 ∂ωdt 2

(7.137)

In (7.135), SNR represents the input signal to noise ratio at each sensor input4 . Using these in (7.128) we obtain the target Doppler bound in the noise only case to be 2 σCR (ωdt ) =

(1 + γ )/γ 2 MN 1 + 1/γ ( MN) 2 = 2 ∂ s 22 ∗ ∂ s 2 . 2 2 2nσn Pt (µ − |ρ| ) 2n MN2 ∂ωdt 2 − st ∂ωdt t

t

(7.138) We can readily compute (7.138) for various speciﬁc array conﬁgurations.

Chapter 7:

Performance Analysis Using Cramer-Rao Bounds

Uniform Linear Array In this case the Doppler derivative of the target steering vector is given in (7.104) and from there5

2 2 ∂st 2 2 ∂ωd

t

22 ∗ M−1 2 2 = (a ∗ (c t )a (c t )) ∂b (ωdt ) ∂b(ωdt ) = Nπ 2 k2 2 ∂ωd ∂ωd t

= Nπ 2

(M − 3

= MNπ 2

1) 3

+

t

(M − 2

M 1 M2 − + 3 2 6

1) 2

+

M−1 6

k=1

.

(7.139)

Similarly the cross correlation in (7.136) is given by s∗t

∂b(ωdt ) ∂st = (a ∗ (c t )a (c t )) b ∗ (ωdt ) ∂ωdt ∂ωdt = − j Nπ

M−1 k=1

k=−

jπ MN( M − 1) . 2

(7.140)

From (7.136) and (7.140), ρ is purely imaginary so that a in (7.114) is zero. As a result, the Fisher information matrix in (7.124) is diagonal, implying that in the noise only case, the two target parameter estimators are independent. In that case, substituting (7.139) and (7.140) into (7.138) we obtain the bound on the target Doppler estimator to be 2 σCR (ωdt ) =

1 6 1+γ . π 2 n γ 2 M2 − 1

(7.141)

From (7.135) and (7.141), higher values of input SNR improve the target Doppler bound. To simplify the target power bound in (7.129), from (7.140) notice that ρ in (7.136) is purely imaginary so that using (7.114) we get a = 0,

ρ = jb.

(7.142)

Substituting these values into (7.129) we get the desired bound to be 2 σCR ( Pt )

(1 + γ ) 2 (1 + γ ) 2 Pt σn2 1 = Pt σ 2 = = nγ nγ MN n

1+γ γ

2 Pt2 (7.143)

that agrees with (7.109). Next, we examine a centro-symmetric array. 5 The

identity

n k=1

k2 =

n3 3

+

n2 2

+

n 6

can be easily veriﬁed by induction.

329

330

Space Based Radar

Centro-Symmetric Uniform Linear Array For a centro-symmetric uniform linear array, the reference pulse is chosen to be the center one, so that the temporal vector [7] b(ωd ) = [ e jπ(

M−1 2 )ωd

, . . . , e jπ ωd , 1, e − jπ ωd , . . . , e − jπ(

M−1 2 )ωd

]T (7.144)

and using this as in (7.104) we get M−1

s∗t

2 ∂st = − jπ k = 0. ∂ωdt M−1

k=−

(7.145)

2

Hence ρ = 0, however, in this case

2 2 ∂st 2 2 ∂ωd

t

M−1 22 2 2 2 = 2Nπ 2 k2 2

k=1

3 2 1 M − 1 M − 1 M − 1 1 1 = 2Nπ 2 + + 3 2 2 2 6 2 = MNπ 2

M2 − 1 12

(7.146)

and once again using (7.138) we get 2 σCR (ωdt ) =

1 6 1+γ . π 2 n γ 2 M2 − 1

(7.147)

In (7.141)–(7.147), the Doppler parameter is normalized to (−1, +1). For a renormalized Doppler parameter (with respect to (−π, +π)), (7.145) reduces to 2 σCR (ωdt ) =

1 61+γ 6 n γ 2 M2 − 1 nγ ( M2 − 1)

(7.148)

that agrees with [7]. In this case, ρ = 0, and hence from (7.129) we obtain the bound for the power estimate to be 2 σCR ( Pt )

(1 + γ ) 2 Pt σn2 1 = = nγ MN n

1+γ γ

2 Pt2 .

(7.149)

which is same as (7.143). Notice that the noise only case bounds in (7.138)–(7.147) are the same for an airborne platform as well as an SBR platform, and hence they can be used as a reference baseline for comparison with clutter situations. Observe that the centro-symmetric array does not result in any improvement in the noise only case. Figure 7.1 shows the various

Chapter 7:

Performance Analysis Using Cramer-Rao Bounds CR Bounds

White Noise Only

Clutter and Noise

Ro SBR

Airborne or SBR

Airborne

Single or Multiple Parameters

Single Multiple Parameter Parameters

Centrosymmetric Array (C)

Regular Linear Array (L)

C

L

C

L

Single Parameter

Multiple Parameters

R1 R2 R3 R4 ………

R1 R2 R3 R4 …………

C

L

C

L

FIGURE 7.1 Various possible options for Cramer-Rao.

possible options when clutter is present. In the case of SBR, the presence/absence of range foldover and/or Earth’s rotation introduces four different situations with associated clutter covariance matrices R1 , R2 , R3 , and R4 as in (7.86)–(7.90). Here (see also (6.38)) R0 : White noise only R1 : No range foldover; no Earth’s rotation R2 : Range foldover present; no Earth’s rotation

(7.150)

R3 : No range foldover; Earth’s rotation present R4 : Range foldover present; Earth’s rotation present Thus in the SBR case, σR2 4 (ωd ) will refer to the Cramer-Rao bound for the target Doppler in presence of both Earth’s rotation and rangefoldover effects for multiparameter case using a regular linear array.

7.3 Simulation Results In what follows the Cramer-Rao bounds are computed for target Doppler and power, and compared with their variance estimates obtained from simulation results corresponding to various airborne and SBR situations shown in Figure 7.1. In the airborne case, a 14-sensor array with 16 pulses is used, and the parameter set for the SBR case is as shown in Table 6.2. In all these cases the array look angle is set at boreside (θ AZ = 90◦ ). Figures 7.2 and 7.3 show the CR bounds for target Doppler and power and their variance estimates for the airborne case. Figure 7.2 corresponds to the noise only case and Figure 7.3 corresponds to the clutter and noise case with CNR = 40 dB.

331

Space Based Radar SNR = −10 dB SNR = 0 dB SNR = 20 dB

0

0 −2

−4

CR bound

−6 −8 −10 −1

Var. in dB

Var. Est.

Var. in dB

−2

−0.5

0

0.5

wd = 0.1 wd = 0.25 wd = 0.55

Var. Est.

−4 −6

CR bound

−8 −10 −10

1

−5

0

5

10

15

20

Target Doppler

SNR in dB

(a) CR bound and variance estimates for target Doppler as a function of Doppler

(b) CR bound and variance estimates for target Doppler as a function of SNR

FIGURE 7.2 Airborne case (noise only): CR bounds for target Doppler and their variance estimates as function of (a) target Doppler and (b) SNR.

Signiﬁcant degradation in target Doppler estimation is to be expected around the clutter dominant neighborhood, which is diagonal in the angle-Doppler domain. For a side-looking array with look angle along boreside (θ AZ = 90◦ ), clutter is dominant at ωd = 0, and the CR bound and the variance estimates have worst performance around that region as is evident in Figure 7.3. From Figures 7.2 and 7.3, the CR bound and the variance estimates for target Doppler decreases as SNR increases. Figures 7.4 and 7.6 correspond to the SBR situations Ro → R4 shown in Figure 7.1 for Range = 500 km. Figures 7.7 and 7.8 refer to target

SNR = −10 dB SNR = 0 dB SNR = 20 dB

CR bound

−6 −8

−10 −1

−0.5

0

0.5

1

Var. Est.

−4

−2 −4 −6 −8 −10 −10

−5

0

5

10

15

20

Target Doppler

SNR in dB

(a) CR bound and variance estimates for target Doppler as a function of Doppler

(b) CR bound and variance estimates for target Doppler as a function of SNR

CR bound

Var. Est.

−2

wd = 0.1 wd = 0.25 wd = 0.55

0

Var. in dB

0

Var. in dB

332

FIGURE 7.3 Airborne case (clutter and noise): CR bounds for target Doppler and their variance estimates as a function of (a) target Doppler and (b) SNR.

Chapter 7:

Performance Analysis Using Cramer-Rao Bounds SNR = 0 dB

SNR = 0 dB

−3 R4

−5 −6 −7

CR Bound for Doppler in dB

CR Bound for Doppler in dB

−2 −4

−1

R0 R1 R2 R3 R4

−1

R1 → R3

−8

Ro

−9 −1

−0.5

0 0.5 Target Doppler

−3 −4

R4

−5 −6

R1 → R3

−7 −8

Ro

−9 −1

1

R0 R1 R2 R3 R4

−2

−0.5

0 0.5 Target Doppler

1

(b) Target power is known

(a) Target power is unknown

FIGURE 7.4 SBR case: CR bounds for target Doppler as function of Doppler. Five cases are shown here (Ro –R4 ). Range = 500 km, CNR = 40 dB. (a) Target power is unknown, (b) Target power is known.

Doppler CR bounds and its estimates in the noise only case (Ro ) as functions of target Doppler and SNR. Similar situations for cases R1 and R4 in (7.150) are shown in Figures 7.9–7.11 for ranges 500 km and 1,200 km. In the SBR case performance degradation occurs in the clutter dominant region also (see Figures 7.4, 7.6, 7.9). In addition the CR bounds for Doppler improve as SNR increases (Figure 7.5). However, as crab effect and/or range foldover are included into the modeling (cases

Power is Unknown, Doppler = 0.1 R4

−2 −4

R1 → R3

−6 −8 −10 −12 −10

Power is Unknown, Doppler = 0.6 −3

R0 R1 R2 R3 R4

CR Bound for Doppler in dB

CR Bound for Doppler in dB

0

Ro wd = 0.1 −5

0

5 10 SNR (dB)

15

(a) Target Doppler = 0.1

20

−4

R0 R1 R2 R3 R4

R4

−5 −6

R1 → R3

−7 −8 −9 −10 −11 −10

Ro wd = 0.6 −5

0

5 10 SNR (dB)

15

20

(b) Target Doppler = 0.6

FIGURE 7.5 SBR case: CR bounds for target Doppler as function of SNR for three different target Doppler. Five cases are shown here (Ro –R4 ). Range = 500 km, CNR = 40 dB, target power is unknown. (a) Target Doppler = 0.1, (b) Target Doppler = 0.6.

333

Space Based Radar SNR = 0 dB R0 R1 R2 R3 R4

8 7 6 5

R1 → R3

R4

4 3 Ro

2 −1

−0.5

Doppler = 0.6

7 CR Bound for Doppler in dB

CR Bound for Doppler in dB

9

0 0.5 Target Doppler

1

wd = 0.6

6 5 4

R4

3

R0 R1 R2 R3 R4

2 Ro → R3

1 0 −10

(a) CR bound for target power as a function of Doppler

−5

0

5 10 SNR (dB)

15

20

(b) CR bound for target power as a function of SNR

FIGURE 7.6 SBR case: CR bounds for target power as a function of Doppler and SNR. Five cases are shown here (Ro –R4 ). Range = 500 km, CNR = 40 dB, target Doppler is unknown.

R1 → R4 ), the performance degradation becomes even more predominant. Notice that the CR bounds and the variance estimates have very similar performance for cases R1 , R2 , and R3 (see Figures 7.4–7.5, 7.9) whereas performance corresponding to case R4 is much worse (Figure 7.10). This is in agreement with previous conclusions (see Section 6.3; Figures 6.17–6.20) showing that when either crab effect or range foldover is present, it is possible to compensate for that effect and obtain accurate estimates, whereas when both effects are present, their effect cannot be compensated. SNR = −10 dB SNR = 0 dB SNR = 20 dB

−3

1 Var. Est.

−4 −5 −6

CR bound

−7 −8 −9 −10 −11 −1

−0.5

0

0.5

1

Target Doppler

(a) CR bound and variance estimates for target Doppler as a function of Doppler

Mean of Estimated Doppler

−2

Var. in dB

334

SNR = −10 dB SNR = 0 dB SNR = 20 dB

0.5 0 −0.5 −1 −1

−0.5

0

0.5

1

Target Doppler

(b) Mean of estimated target Doppler as a function of Doppler

FIGURE 7.7 SBR case: Noise only case (Ro ). Range = 500 km. (a) CR bounds for target Doppler and its estimates as a function of Doppler and (b) Mean of estimated target Doppler as a function of Doppler.

Chapter 7:

Performance Analysis Using Cramer-Rao Bounds Var. Est.

−4 Mean of Estimated Doppler

0.7

−5 Var. in dB

−6

wd = 0.1 wd = 0.25 wd = 0.55

−7 −8 −9

CR bound

−10 −11 −10

−5

0

5

10

15

0.6 0.5

wd = 0.1 wd = 0.25 wd = 0.55

0.4 0.3 0.2 0.1 0 −10

20

−5

0

5

10

15

20

SNR in dB

SNR in dB

(a) CR bound and variance estimates for target Doppler as a function of SNR

(b) Mean of estimated target Doppler as a function of SNR

FIGURE 7.8 SBR case: Noise only case (Ro ). Range = 500 km. CR bounds for target Doppler and its estimates as function of SNR and (b) Mean of estimated target Doppler as a function of SNR.

To demonstrate the effect of clutter on parameter estimation, the mean value of the estimated target Doppler is plotted as a function of Doppler and SNR in Figures 7.7(b)–7.10(b). The estimates are unbiased in the noise only case (Ro ), whereas when either the range foldover effect or Earth’s rotation is present, the estimates are not very accurate in the clutter dominant region (ωd ≈ 0). The degree of bias depends on a variety of parameters and gets worse when both effects are present (Figure 7.10(b)). SNR = −10 dB SNR = 0 dB SNR = 20 dB

−4 −6 −8 −10 −1

−0.5

0

0.5

1

Mean of Estimated Doppler

−2 CR bound Var. Est.

Var. in dB

0

1

SNR = −10 dB SNR = 0 dB SNR = 20 dB

0.5 0 −0.5 −1 −1

−0.5

0

0.5

1

Target Doppler

Target Doppler

(a) CR bound and variance estimates for target Doppler as a function of Doppler

(b) Mean of estimated target Doppler as a function of Doppler

FIGURE 7.9 SBR case: Clutter and noise case (R1 ), w/o range foldover, w/o Earth’s rotation. Range = 500 km. CR bounds for target Doppler and its estimate as a function of Doppler and (b) Mean of estimated target Doppler as a function of Doppler.

335

Space Based Radar SNR = −10 dB SNR = 0 dB SNR = 20 dB

1

Var. Est.

Var. in dB

0 −2

CR bound

−4 −6 −8 −1

−0.5

0 0.5 Target Doppler

Mean of Estimated Doppler

2

SNR = −10 dB SNR = 0 dB SNR = 20 dB

0.5

0 −0.5 −1 −1

1

(a) CR bound and variance estimates for target Doppler as a function of Doppler

−0.5

0 0.5 Target Doppler

1

(b) Mean of estimated target Doppler as a function of Doppler

FIGURE 7.10 SBR case: Clutter and noise case (R4 ), w/ range foldover, w/ Earth’s rotation. Range = 500 km. CR bounds for target Doppler and its estimate as a function of Doppler and (b) Mean of estimated target Doppler as a function of Doppler.

The performance degradation becomes further pronounced as a function of range. This is exhibited in Figure 7.11 that corresponds to range 1,200 km. From there, it is clear that when both crab effect and range foldover effect are present in the data at far ranges, target detection is difﬁcult, and it is necessary to introduce waveform diversity into transmit design to minimize the effect of clutter and other interference.

SNR = −10 dB SNR = 0 dB SNR = 20 dB

0

1 0

−6 −8 −0.5

0

0.5

1

−1

Var. Est.

Var. in dB

CR bounds Var. Est.

−4

−10 −1

SNR = −10 dB SNR = 0 dB SNR = 20 dB

2

−2 Var. in dB

336

−2 −3

CR bounds

−4 −5 −6 −1

−0.5

0

0.5

Target Doppler

Target Doppler

(a) Without range foldover, without Earth’s rotation

(b) With range foldover, with Earth’s rotation

1

FIGURE 7.11 SBR case: CR bounds for target Doppler and its estimate as a function of Doppler. Range = 1,200 km. (a) Case R1 , w/o range foldover, w/o Earth’s rotation; (b) Case R4 , w/ range foldover, w/ Earth’s rotation.

Chapter 7:

Performance Analysis Using Cramer-Rao Bounds

Figures 7.12 and 7.13 show the effect of waveform diversity on the CR bounds for ranges 500 km and 1,200 km. Notice that the bound associated with any waveform diversity method is smaller than that associated with the conventional waveform when Earth’s rotation and range foldover effects are present (case R4 ). This is generally true for both Doppler and power estimates indicating that introducing waveform diversity improves performance. As Figure 7.13 shows, at higher range (R = 1,200 km), different waveforms give different performance, but all of them are superior to the conventional waveform for case R4 . In general, the

Doppler = 0.25

SNR = 0 dB R0 R1 R4, Conv. R4, Orth. R4, Chirp R4, Hybrid

−2 −3 −4

R4: Conv

−2 CR Bound for Doppler in dB

CR Bound for Doppler in dB

−1

−5 −6 R , R : Orth, 1 4 −7 Chirp, Hybrid −8 −9 −1

Ro −0.5

0 0.5 Target Doppler

R4: Conv −4 −6 −8 −10 −12 −10

1

(a) CR bounds as a function of target Doppler

R1, R4: Orth, Chirp, Hybrid

R0 R1 R4, Conv. R4, Orth. R4, Chirp R4, Hybrid

Ro wd = 0.25 −5

0

5 10 SNR (dB)

15

20

(b) CR bounds as a function of SNR

Range = 500 km Doppler = 0.25

SNR = 0 dB R0 R1 R4, Conv. R4, Orth. R4, Chirp R4, Hybrid

0 −2 −4 −6 −8 −1

−0.5

0 0.5 Target Doppler

R0 R1 R4, Conv. R4, Orth. R4, Chirp R4, Hybrid

0 CR Bound for Doppler in dB

CR Bound for Doppler in dB

2

1

−2 −4 −6 −8 −10 −12 −10

wd = 0.25 −5

0

5 10 SNR (dB)

15

20

(b) CR bounds as a function of SNR (a) CR bounds as a function of target Doppler Range = 1,200 km

FIGURE 7.12 CR bound for target Doppler with waveform diversity for range = 500 km and 1,200 km.

337

338

Space Based Radar R1 R4, Conv. R4, Orth. R4, Chirp R4, Hybrid

−1 −2 −3 −4

0

R4: Conv

−1 −2

R4: Conv

−3 −4

−5 R4 with waveform diversity

−6 −7 −8 −1

R1 R4, Conv. R4, Orth. R4, Chirp R4, Hybrid

1

−6 −7

R1 −0.5

R4 with waveform diversity

−5

0

0.5

(a) Range = 500 km

1

−8 −1

R1 −0.5

0

0.5

1

(b) Range = 1,200 km

FIGURE 7.13 CR bound for target Doppler without and with waveform diversity for case R4 . SNR = 0 dB. (a) Range = 500 km; (b) Range = 1,200 km.

performance in the case of R4 with waveform diversity is similar to that of case R1 . In summary, for target detection using SBR, when both range foldover and Earth’s rotation effects are jointly present, the performance bounds are inferior to those associated with when only either one of the effect is present. Waveform diversity can be used to minimize the effect of range foldover and consequently leads to superior performance. These conclusions are also supported by the Cramer-Rao bounds associated with various scenarios presented here. As a result, waveform diversity should be considered to improve target detection especially for large ground range values [6]. Interestingly, the Cramer-Rao bounds for target Doppler and power have asymmetric behavior with respect to SNR, i.e., the Doppler bound decreases as SNR increases whereas the corresponding bound for the power increases with SNR. This is in agreement with the general Gaussian data case with unknown variance as in (7.64) and (7.72) [1]. The asymmetric behavior suggests an optimum SNR for the joint parameter estimation problem.

References [1] C. Radhakrisna Rao, Linear Statistical Inference and Its Applications (2nd edition), John Wiley and Sons, New York, NY, 2001. [2] Rohatgi, Vigay K., A.K. Saleh, An Introduction to Probability and Statistics, John Wiley and Sons, New York, NY, 2001. [3] A. Papoulis, S. Unnikrishna Pillai, Probability, Random Variables and Stochastic Processes, McGraw-Hill Higher Education, New York, NY, 2002. [4] S.U. Pillai, Array Signal Processing, Springer-Verlag, New York, NY, 1989. [5] P. Stoica, R.L. Moses, Spectral Analysis of Signals, Prentice Hall, New Jersey, NJ, 2005.

Chapter 7:

Performance Analysis Using Cramer-Rao Bounds

[6] S.U. Pillai, K.Y. Li, B. Himed, “Cramer-Rao Bounds for Target Parameters in Space Based Radar Applications,” To appear in Proc. IEEE Transactions on Aerospace and Electronic Systems, Vol. 44, No. 4, October 2008. [7] J. Ward, “Cramer-Rao Bounds for Target Angle and Doppler Estimation with Space-Time Adaptive Processing Radar,” 29th Asilomar Conference on Signals, Systems and Computers (2-Volume Set), Paciﬁc Grove, CA, October 30– November 2, 1995.

339

This page intentionally left blank

CHAPTER

8

Waveform Diversity Waveform diversity refers to the use of various waveforms (signals) in both transmitter (Tx ) and receiver ( Rx ) design for improving the overall performance such as detection and/or identiﬁcation of targets in interference and noise. Waveform diversity can be exploited spatially using a multiple set of sensors for both transmission and/or reception, and in the time-frequency domain using distinct waveforms of different durations over different spectral bands. In addition, other features such as polarization, energy distribution of various transmit signals can be used for further diversity. Spatial diversity can be realized from a single platform or multiple platforms for both transmission and reception. For example, in monostatic mode the same set of sensors are used for transmission and reception, whereas in the bistatic mode, different sensors are used for these two functions (Figure 8.1). Thus in the bistatic case, a dedicated sensor is used for transmission and a multiple set of sensor outputs can be used as the receiver. In both cases, the platform involved may be stationary or in motion. In a typical airborne or Space Based Radar (SBR) situation, the platform is moving and both transmitter and receiver may be time sharing the same set of antennas in different conﬁgurations giving rise to a monostatic situation. If we envision a set of unmanned aerial vehicles (UAVs) into this situation used in the receiver mode, then we have a multisensor bistatic situation at hand. The objective in such situations also may comprise of multiple tasks such as Ground Moving Target Indication (GMTI), Airborne Moving Target Indication (AMTI), and Synthetic Aperture Radar (SAR). In the case of GMTI, the task is to improve the overall system design so that slowly moving targets can be detected and identiﬁed in the presence of clutter, jamming signals, and noise. Once the transmitter and receiver array conﬁgurations and their relative positions are known,

341 Copyright © 2008 by The McGraw-Hill Companies. Click here for terms of use.

342

Space Based Radar Temporal Tx diversity

f2(t)

f1(t)

F1(w)

t

w0 Spectral diversity

Polarization diversity

SBR or Airborne Radar (Tx−Rx 1)

w

Rx-3 (UAV)

AMTI

Spatial diversity

Beam #1 Multi tasks

Tx-Rx 2 (UAV)

Spatial Beam #4 diversity GMTI

Rx-4

f ′2(t)

f ′1(t) t

Temporal Tx diversity

FIGURE 8.1 Multichannel waveform diversity.

the problem is to select various transmitter and receiver waveforms for target return enhancement and simultaneous suppression of interference and noise. Target scene may consist of single or multiple targets which may be stationary or moving under various conditions. The system goal is to detect and identify all of them, and estimate the relevant parameters such as their range, Doppler, elevation, and azimuth angles, etc. Usually target return is buried in competing interference which consists of stochastic return signals from the environment and noise. These random returns are transmit signal dependent and hence they directly compete with the target returns usually dominating them. One task of waveform diversity is to distinguish the interference spectral characteristics and design waveforms accordingly to minimize the competing signal-dependent interferences. In the spectral domain, it is easy to state the waveform design goal. The transmit waveform should put out energy at those spectral regions where target characteristics are dominant, and minimize transmission at those regions where interference and noise are dominant. However, this is to be attained using ﬁnite duration waveforms in some optimal manner, and the problem becomes more challenging when target, interference, and noise have overlapping spectral regions. The problem considered here is equally applicable in the communication scene as well, where the channel response plays the role of the

Chapter 8: Interference

Noise

Waveform Diversity

MTI Rx Rx −1

f1 (t)

Moving Target

.. .

f2 (t)

.. .

Rx −2

t = to

∑

Ry −1 SAR

fN (t)

.. .

Output

Rx −m1

Ry −2

t = to

∑

Ry −m2

wdo

MTI qo

SAR

Output SAR Rx

Multiple transmitter waveforms fi (t)

Noise

Multi receiver set for coherent processing

FIGURE 8.2 Transmitter–receiver design.

target. In that case, the transmitted signal passes through the channel to generate an output signal that also gets modiﬁed by interference and noise. Once again, the goal is to enhance the signal part of the channel output and minimize the interference and noise components through proper joint transmitter–receiver design. Clearly, the target or channel characteristics must play a role in the transmitter as well as the receiver design—to enhance the corresponding ﬁnal desired output. At the same time, the design must deal with suppressing the interference and noise output components. In this context, the receiver output signal to interference plus noise ratio (SINR) may be used as an optimization goal. This is illustrated in Figure 8.2 where the receiver outputs are combined to make a decision at t = to . Thus SINR =

Receiver output signal (target) power at t = to . Average interference pulse noise output power

(8.1)

To maximize the output SINR, the following design components must be simultaneously taken into account: Transmitter: Set of ﬁnite duration waveforms f i j (t) ↔ Fi j (ω) (i: spatial and j: temporal) with an energy constraint on the whole set or subsets of waveforms. Known transmitter output ﬁlters Pi j (ω) can be used to control the transmitter bandwidth Bo . The spatial set of waveforms can be realized by partitioning the transmit array, and the temporal set by utilizing a pulse train of waveforms.

343

344

Space Based Radar Target (channel): Impulse response q (t) ↔ Q(ω). Simplest model is the point (impulse) target (Q(ω) ≡ 1, ﬂat spectrum). In general, Q(ω) is arbitrary. In the multi-input–multi-output (multichannel) case, Q(ω) represents the target transform matrix whose (i, j)th entry Qi j (ω) is the transfer function between the ith transmitter and the jth receiver. Noise: Stationary stochastic process wn (t) with power spectrum G n (ω) = σn2 that corresponds to the white (ﬂat) noise case. In general, G n (ω) is an arbitrary spectrum. (High interference (clutter dominant) situation may be represented as G n (ω) ≈ 0.) In the multichannel case, G n (ω) is represented by a positive-deﬁnite spectral matrix. Interference (clutter): Interference can be modeled as a stationary stochastic process wc (t) with power spectrum G c (ω) = σc2 that corresponds to ﬂat interference. In general G c (ω) is an arbitrary spectrum with speciﬁc bandwidth. (Noise dominant situation may be represented as G c (ω) ≈ 0.) In the multichannel case, G c (ω) represents a positive-deﬁnite spectral matrix. Receiver ﬁlter: A multiple set of waveforms h i (t) ↔ Hi (ω) that may be causal or noncausal. Decision instant to : The overall goal is to maximize the receiver output SINR at the decision instant. The simplest situation is to consider a single transmitter and a single receiver with a target (channel) and no interference signal. To start with assume the transmitter signal and the target to be known. Hence the goal is to design the optimum receiver. The receiver in this case is presented with the target output (ﬁxed signal) that is corrupted by noise. The simplest noise situation is white noise (equal strength at all frequencies). Thus the simplest waveform design problem is to design a receiver to maximize the receiver output signal to noise ratio (SNR) for a given incoming signal that is buried in white noise. The solution is the well-known “matched ﬁlter” receiver.

8.1 Matched Filter Receivers Consider the problem of designing a receiver to detect an incoming signal s(t) that is buried in noise. This situation occurs—in radar, sonar, and communication scenarios. In a communication scene, a transmitted signal f (t) goes through a channel with impulse response q (t). The channel output signal s(t) is further corrupted by noise n(t), so

Chapter 8:

Waveform Diversity

f (t) f (t)

q(t)

Q(w)

s(t)

⊗

r (t)

q(t)

s(t)

n(t) (a) Communication scene

(b) Radar/sonar scene

FIGURE 8.3 Noisy channel output/radar return.

that the observed signal r (t) as shown in Figure 8.3 (a) is given by r (t) = s(t) + n(t)

(8.2)

where

5+∞

s(t) = f (t) ∗ q (t) =

f (τ )q (t − τ )dτ .

(8.3)

−∞

In radar and sonar, the same model represents the target return in noise, where s(t) represents the target return signal (if any), and q (t) the target impulse response (Figure 8.3 (b)). In all these cases, the received signal r (t) is passed through a receiver with transfer function H(ω) to minimize the effect of noise as shown in Figure 8.4. The problem is how to design a good receiver? Toward this, let y(t) represent the receiver output. Then y(t) = sˆ (t) + w(t),

(8.4)

where sˆ (t) represents the signal part of the output and w(t) the output noise. Clearly sˆ (t) = s(t) ∗ h(t)

(8.5)

w(t) = n(t) ∗ h(t)

(8.6)

and

r(t) = s(t) + n(t)

FIGURE 8.4 Receiver.

H(w)

y(t) = s(t) + w(t)

345

346

Space Based Radar where h(t) ↔ H(ω)

(8.7)

represents the unknown receiver characteristics. Obviously the receiver transfer function H(ω) should be selected so as to maximize the output signal and minimize the output noise effect.

8.1.1 Matched Filter Receivers in White Noise Consider the detection problem where the receiver output must be used to decide in favor of the target being present or absent in the data. Toward this the output SNR of the received signal evaluated at some desired time instant t = to can be used to select the optimum receiver so as to maximize the receiver output SNR at t = to . Note that the output SNR at t = to is given by the instantaneous signal power at t = to divided by the average noise power.1 Thus (see (8.1))

SNR|t=to =

|ˆs (to )|2 . E{|w(t)|2 }

(8.8)

From (8.5) ˆ sˆ (t) ↔ S(ω) = S(ω) H(ω)

(8.9)

5+∞ jωt ˆ S(ω)e dω,

(8.10)

so that 1 2π

sˆ (t) =

−∞

and using (8.10), we obtain 1 sˆ (to ) = 2π

5+∞ S(ω) H(ω)e jωto dω.

(8.11)

−∞

Let G n (ω) and G w (ω) represent the noise power spectral densities of n(t) and w(t) respectively. Hence 1 E{|w(t)| } = 2π 2

5+∞ G w (ω)dω.

(8.12)

−∞

1 The deﬁnition in (8.8) uses the instantaneous output signal power vs. average output noise power. The numerator here is not an average value of the signal power as in the case of the denominator.

Chapter 8: FIGURE 8.5 White noise.

Waveform Diversity

Gw (w) s2 w

But from (8.6) G w (ω) = G n (ω) |H(ω)|2

(8.13)

and hence 1 E{|w(t)| } = 2π 2

5+∞ G n (ω) |H(ω)|2 dω.

(8.14)

−∞

If we assume G n (ω) to be white noise as shown in Figure 8.5, we have G n (ω) = σ 2 ,

(8.15)

and using this in (8.14), we get σ2 E{|w(t)| } = 2π 2

5+∞ |H(ω)|2 dω.

(8.16)

−∞

Substituting (8.11) and (8.16) into (8.8), we get the output SNR at t = to to be [1]

SNRo =

+∞ 2 : S(ω) H(ω)e jωto dω −∞

2πσ 2

+∞ : −∞

.

(8.17)

|H(ω)|2 dω

Obviously, the unknown receiver H(ω) in (8.17) should be chosen so as to maximize the output SNR at the observation time t = to . We can use Schwarz’ inequality to simplify (8.17). Note that Schwarz’ inequality states that

5 2 5 5 A(ω) B(ω)dω ≤ |A(ω)|2 dω |B(ω)|2 dω

(8.18)

347

348

Space Based Radar and equality in (8.18) is achieved if and only if B(ω) = µA∗ (ω),

(8.19)

where µ is a constant. To make use of (8.18) in the present problem, let A(ω) = S(ω)e jωto

(8.20)

B(ω) = H(ω).

(8.21)

and

We can choose the constant multiplier µ in (8.19) to be unity here, since that gets canceled out from the numerator and the denominator of (8.17). Using (8.20) and (8.21) in (8.17), from (8.18) we obtain

+∞ 2 5 5+∞ 5+∞ 2 S(ω) H(ω)e jωto dω ≤ |S(ω)| |H(ω)|2 dω. dω −∞

−∞

(8.22)

−∞

With (8.22) in (8.17), we get 1 SNRo ≤ 2πσ 2

5+∞ |S(ω)|2 dω.

(8.23)

−∞

Thus the maximum value of the output SNR is given by 1 2π

SNRmax =

+∞ : −∞

+∞ :

|S(ω)|2 dω σ2

=

−∞

|s(t)|2 dt σ2

=

E , σ2

(8.24)

a quantity independent of H(ω), and from (8.21) this maximum value is achieved if and only if H(ω) = A∗ (ω) = (S(ω)e jωto ) ∗ = S∗ (ω)e − jωto ,

(8.25)

or, in the time domain we obtain h(t) = s ∗ (to − t).

(8.26)

If s(t) is real, the matched ﬁlter solution reduces to the classic form h(t) = s(to − t)

(8.27)

which is a time reversed and shifted version of s(t). Thus, under additive white noise, the optimum receiver that maximizes the output SNR

Chapter 8:

150

−5

r(t)

Waveform Diversity

−2

100

0

50

2

0

5 0

0.1

0.2 0.3 t (sec)

0.4

−50

0.5

0

h(t) = s(to − t)

r(t) = s(t) + n(t )

0.2

0.4 0.6 t (sec)

0.8

1

r(t) = s(t) + w(t)

FIGURE 8.6 Optimum receiver (matched ﬁlter).

at t = to is given by (8.25) and (8.26). Notice that h(t) depends only on the receiver input signal waveform s(t), or the receiver is matched to its input signal s(t). To summarize, if s(t) + n(t) is received and the noise n(t) is white, then the optimum receiver is given by the classical matched ﬁlter solution as in Figure 8.6. In Figure 8.6, the time index to represents the time instant at which the output SNR is maximized and hence that instant (t = to ) must be used to make the decision. Note that the optimum ﬁlter in (8.26) need not represent a causal solution. However, if s(t) is a ﬁnite duration signal, then by suitably selecting to , the receiver h(t) can be made causal. From (8.3), we have S(ω) = F (ω) Q(ω)

(8.28)

and hence the optimum receiver H(ω) = S∗ (ω)e − jωto = F ∗ (ω) Q∗ (ω)e − jωto .

(8.29)

From (8.29), the optimum ﬁlter is matched to the input signal and channel characteristics. From (8.24), the maximum value of output SNR is also given by +∞ :

SNRmax =

−∞

+∞ : −∞

1 2π

=

σ2 1 2π

=

|s(t)|2 dt

+∞ : −∞

|S(ω)|2 dω σ2

|F (ω) Q(ω)|2 dω σ2

.

(8.30)

349

350

Space Based Radar The same conclusions can also be reached by making use of a time domain analysis. To see this, referring back to (8.8) and (8.5)–(8.6), we have

+∞ 2 : h(τ )s(t − τ )dτ −∞ SNRt = 2 / +∞ : E h(τ )n(t − τ )dτ −∞

=

+∞ 2 : h(τ )s(t − τ )dτ +∞ : +∞ : −∞ −∞

−∞

.

(8.31)

h(τ1 )h ∗ (τ2 ) E {n(t − τ1 )n∗ (t − τ2 )} dτ1 dτ2

If we assume n(t) to be a stationary white noise, then

E n(t − τ1 )n∗ (t − τ2 ) = Rnn (τ1 − τ2 ) = σ 2 δ(τ1 − τ2 ),

(8.32)

and using this, (8.31) becomes

SNRt=to =

+∞ 2 : h(τ )s(to − τ )dτ +∞ : +∞ : −∞ −∞

=

−∞

h(τ1 )h ∗ (τ2 )σ 2 δ(τ1 − τ2 )dτ1 dτ2

+∞ 2 : h(τ )s(to − τ )dτ −∞

σ2

+∞ : −∞

.

(8.33)

|h(τ )| dτ 2

Once again, using Schwartz’ inequality, the numerator in the above expression gives

+∞ 2 5 5+∞ 5+∞ 2 h(τ )s(to − τ )dτ ≤ |h(τ )| dτ |s(to − τ )|2 dτ −∞

−∞

(8.34)

−∞

and (8.33) reduces to SNRmax |t=to

1 ≤ 2 σ

5+∞ 5+∞ 1 E 2 |s(to − τ )| dτ = 2 |s(t)|2 dt = 2 σ σ

−∞

−∞

(8.35)

Chapter 8:

Waveform Diversity

f(t)

FIGURE 8.7 Triangular pulse. 1

t T

with equality if and only h(t) = s ∗ (to − t),

(8.36)

which agrees with (8.26). Example 8.1 A triangular pulse f (t) as shown in Figure 8.7 is transmitted through an ideal channel with transfer function Q(ω) = e − jωtc .

(8.37)

If white noise corrupts the input signal, ﬁnd the optimum receiver. Solution Since r (t) = s(t) + n(t)

(8.38)

and n(t) is white noise, the optimum receiver is the matched ﬁlter given by (8.25) and (8.26), i.e., h(t) = s(to − t).

(8.39)

S(ω) = F (ω) Q(ω) = F (ω)e − jωtc

(8.40)

s(t) = f (t − tc )

(8.41)

Here

so that

and this is illustrated in Figure 8.8. In particular, if to = tc , then h(t) = s(tc − t) and it is shown in Figure 8.9 (a). On the other hand, if to = tc + T, then we get h(t) as shown in Figure 8.9 (b).

h(t) = s(to − t )

s(−t)

s(t ) 1

1 t tc

(a) s(t)

T + tc

−(T + tc) −tc

1 t

−(T + tc)

(b) s(−t)

FIGURE 8.8 Optimum receiver for decision instant t = to .

to (c) h(t)

t −tc + to

351

352

Space Based Radar h(t) = s(tc − t )

h(t) = s(tc + T − t ) = f (t)

1

1 t

−T

t T

(a) Optimum receiver for decision instant to = tc .

(b) Optimum receiver for decision instant to = tc + T .

FIGURE 8.9 Optimum receiver for decision instant to = tc .

In Figure 8.9 (a), h(t) is noncausal, whereas in Figure 8.9 (b) it represents a causal ﬁlter. From (8.24) and (8.30), the maximum output SNR at t = to is given by

:

+∞

SNRmax =

:

T/2

|s(t)|2 dt

−∞

σ2

2 =

(2t/T) 2 dt

0

=

σ2

8

# T $3

3T 2 σ 2

2

=

T . 3σ 2 (8.42)

To summarize, if r (t) = s(t) + n(t), where s(t) has the form shown in Figure 8.10 (a), then if n(t) is white noise, the optimum receiver impulse response is given by

h(t) = s(to − t),

(8.43)

and this is illustrated in Figure 8.10 (a)–(c) for some arbitrary to .

s(−t)

s(t)

t T (a)

s(−t)

t −T

t −T

(b)

FIGURE 8.10 Optimum receiver for some arbitrary to .

−(T − to) (c)

to

Chapter 8:

Waveform Diversity

Matched Filter as a Correlator Receiver Suppose the received data r (t) is available in the interval (0, t). In that case using (8.26), the matched ﬁlter output is given by

5t rˆ (t) = r (t) ∗h(t) =

5t r (τ )h(t − τ )dτ =

0

r (τ )s ∗ (to − t + τ )dτ ,

(8.44)

0

since h(t − τ ) = s ∗ (to − (t − τ )) = s ∗ (to − t + τ ). The matched ﬁlter maximizes its output SNR at t = to , and hence the output rˆ (t) must be used at that instant to make further decision. Thus from (8.44), we have

5to rˆ (to ) =

∗

5to

r (τ )s (τ )dτ = 0

r (t)s ∗ (t)dt.

(8.45)

0

Notice that (8.45) can be interpreted as a correlator receiver and it can be represented as in Figure 8.11. From Figure 8.11, the received waveform is correlated with s(t) and integrated until t = to to generate the output rˆ (to ). Since the correlator receiver in (8.45) follows from the matched ﬁlter receiver in (8.43), we conclude that, in the case of additive white noise, both matched ﬁlter receiver as well as the correlator receiver are equivalent realizations.

8.1.2 Matched Filter Receivers in Colored Noise What if the receiver noise is of nonwhite nature? How does one design the optimum causal ﬁlter in the colored noise case? Interestingly, by increasing to alone the optimum solution cannot be made causal in this case. To see this, let G n (ω) represent the input noise spectrum so

r(t)

r(t) to

t

r(t)

⊗

∫(⋅)dt

0

s(t)

FIGURE 8.11 Correlator receiver.

t = to r(to)

t to

353

354

Space Based Radar that the receiver output noise power is given by 1 E{w (t)} = 2π

5+∞ G n (ω) |H(ω)|2 dω.

2

(8.46)

−∞

Every spectrum G(ω) that is non-negative and satisfying the PaleyWiener condition [1]

5+∞ −∞

|ln G(ω)| dω < ∞ 1 + ω2

(8.47)

can be factorized in terms of its Wiener factor L(s) as G(ω) = L( jω)L ∗ ( jω),

(8.48)

where L(s) is analytic together with its inverse L −1 (s) in the right half plane of the complex s = σ + jω plane (minimum phase system). Let L n (s) represent the Wiener factor associated with G n (ω) so that G n (ω) = |L n ( jω)|2

(8.49)

and

SNR|t=to =

+∞ 2 1 : jωt o H(ω)S(ω)e dω 2π 1 2π

−∞ +∞ : −∞

=

+∞ 2 1 : jωt o H(ω)S(ω)e dω 2π

G n (ω) |H(ω)| dω

−∞ +∞ :

1 2π

2

−∞

.

|L n (ω) H(ω)| dω 2

(8.50) To accommodate the new denominator, let us rewrite the term H(ω)S(ω) in the numerator as

{L n ( jω) H(ω)} L −1 n ( jω)S(ω) .

(8.51)

Substituting this into (8.50) and a straightforward application of Schwarz’ inequality gives 1 ρ(to ) ≤ 2π

5+∞ 5+∞ −1 |S(ω)|2 L n ( jω)S(ω) 2 dω = 1 dω, 2π G n (ω)

−∞

(8.52)

−∞

with equality if and only if

L n ( jω) H(ω) = L −1 n ( jω)

∗

S∗ (ω)e − jωto

(8.53)

Chapter 8:

Waveform Diversity

or H(ω) =

S∗ (ω)e − jωto . G n (ω)

(8.54)

Thus in the time domain if linv (t) ↔ L −1 n ( jω)

(8.55)

then the optimum receiver impulse response is given by ∗ (−t) ∗ s(to − t). h(t) = linv (t) ∗ linv

(8.56)

Clearly (8.56) represents a noncausal waveform, and a simple ﬁnite shift by to alone (however large) may not be enough to maintain the causal nature of the receiver.

Optimum Causal Matched Filter in Colored Noise To obtain the optimum causal receiver in this context, it is necessary to proceed differently. Toward this, let [2] υ(t) ↔ H(ω)L n ( jω)

(8.57)

and let g(t) represent the inverse transform of L −1 n ( jω)S(ω) −1 g(t) ↔ L −1 n ( jω)S(ω) = L n ( jω) Q(ω) F (ω)

(8.58)

so that

g ∗ (−t) ↔ L −1 n ( jω)S(ω)

∗

(8.59)

and

g ∗ (to − t) ↔ L −1 n ( jω)

∗

S∗ (ω)e − jωto .

(8.60)

Since L n (s) and L −1 n (s) are analytic in Re s > 0, we have υ(t) and g(t) are causal waveforms and by Parseval’s theorem

5+∞ 5+∞ υ(t)g(to − t)dt = υ(t){g ∗ (to − t)}∗ dt −∞

−∞

1 = 2π 1 = 2π

5+∞ jωto {H(ω)L n ( jω)} L −1 dω n ( jω)S(ω)e −∞

5+∞

H(ω)S(ω)e jωto dω = Q −∞

(8.61)

355

356

Space Based Radar same as the numerator factor in (8.50). But from (8.61)

5+∞ 5+∞ Q= υ(t)g(to − t)dt = υ(t)g(to − t)dt, −∞

(8.62)

0

because of the causal nature of υ(t). However, by making use of the unit step function u(t), we can rewrite the later integral in (8.62) as follows Q=

5+∞ 5+∞ υ(t)g(to − t)dt = υ(t)g(to − t)u(t)dt.

(8.63)

−∞

0

Let K (ω) represent the transform of the causal ﬁlter g ∗ (to − t)u(t). Thus g ∗ (to − t)u(t) ↔ K (ω),

(8.64)

and once again Parseval’s theorem applied to (8.63) gives

5+∞ ∗ {υ(t)} g ∗ (to − t)u(t) dt Q= −∞

1 = 2π

5+∞ {H(ω)L n ( jω)} {K ∗ (ω)}dω.

(8.65)

−∞

Using this in (8.50), we get

ρ(to ) = SNR|t=to =

+∞ 2 1 : ∗ H(ω)L n ( jω) K (ω)dω 2π −∞ 1 2π

1 ≤ 2π

+∞ :

−∞

|L n (ω) H(ω)|2 dω

5+∞ |K (ω)|2 dω,

(8.66)

−∞

with equality if and only if H(ω)L n ( jω) = K (ω)

(8.67)

∗ H(ω) = L −1 n ( jω) K (ω) ↔ linv (t) ∗ g (to − t)u(t).

(8.68)

or equivalently

Chapter 8:

Waveform Diversity

Note that the impulse response h(t) of this matched ﬁlter is necessarily causal by design. Thus h c (t) = linv (t) ∗ g ∗ (to − t)u(t).

(8.69)

In addition, using (8.64) in (8.66) we obtain 1 ρmax (to ) = 2π

5+∞ 5to 2 |K (ω)| dω = |g(t)|2 dt. −∞

(8.70)

0

Equations (8.67)–(8.70) represent the optimum causal solution, where K (ω) and g(t) are as deﬁned in (8.64) and (8.58) respectively. Notice that the causal solution in (8.68) is neither a simple shifted version of the noncausal solution in (8.56), nor can it be obtained directly from (8.56). Using (8.54)–(8.55) and (8.60), we can rewrite (8.56) as h nc (t) = linv (t) ∗ g ∗ (to − t)

(8.71)

and on comparing (8.71) with (8.68), we notice that the causal solution in (8.68) and (8.69) drops the noncausal portion of g ∗ (to − t) prior to its convolution with linv (t) as in Figure 8.12. We also have h nc (t) = h c (t) + linv (t) ∗ g ∗ (to − t)u(−t).

(8.72)

Since linv (t) ∗ g ∗ (to − t)u(−t) has a causal part, it will be impossible to obtain the optimum causal ﬁlter h c (t) from the noncausal one at the ﬁnal state. The causal matched ﬁlter receiver in (8.68) that corresponds to the colored noise case can be given an interesting interpretation as shown in Figure 8.13. From Figure 8.13, the input noise n(t) is ﬁrst “whitened” using the ﬁlter L −1 ( jω) to generate white noise w(t). During this transformation, the input signal s(t) becomes g(t) that is given by (8.58). Since g(t) is contaminated by white noise, from (8.26), g ∗ (to − t) represents the optimum noncausal matched ﬁlter receiver for the input g(t) +w(t). g(to − t)

g(to − t)u(t)

t (a)

linv (t)

t (b)

FIGURE 8.12 Causal solution of the matched ﬁlter.

t (c)

357

358

Space Based Radar

r(t) = s(t) + n(t)

L−1(

g(t) + w(t ) jw)

g (to − t)u(t ) K(w)

y(t)

t = to r (to)

FIGURE 8.13 Matched ﬁlter in colored noise.

However from (8.68) and (8.69), the causal matched ﬁlter receiver for the input g(t) +w(t) is given by g ∗ (to −t)u(t) ↔ K (ω). Hence taken together the concatenated overall receiver has the form in (8.68), where the input is ﬁrst whitened followed by matched ﬁltering to generate the desired output. Finally, in the white noise case G n (ω) = σ 2 so that from (8.58), g(t) = s(t)/σ and substituting this into (8.70), we obtain SNRmax

1 = ρ(to ) = 2 σ

5to |s(t)|2 dt.

(8.73)

0

As a result, the causal matched ﬁlter in the white noise case is given by (compared with the noncausal receiver in (8.26)) h c (t) = s ∗ (to − t)u(t).

(8.74)

From (8.70) and (8.73), in the causal case the SNR at the matched ﬁlter output at t = to is proportional to the energy in the input signal up to that instant. This relation is unlike (8.35), where the SNR is proportional to the total energy. Thus in the case of causal receiver, the maximum SNR output is a monotonically nondecreasing function of the observation instant to , and as we shall see later, this property has certain interesting implications. Since the matched ﬁlter generates outputs with large peaks, they contain the potential for generating time-compressed waveforms that may lead to accurate target range estimation. Next, we investigate this possibility and show that the chirp waveform indeed possesses the desired time-compression property.

8.2 Chirp and Pulse Compression [3] A receiver matched to the incoming signal produces large peaks at the output and hence it has the potential for time compression (see Figure 8.14). For example, the matched ﬁlter receiver corresponding to the incoming signal s(t) is given by s ∗ (to −t), where to represents the output observation instant and for ﬁnite duration input signals, to is

Chapter 8:

Waveform Diversity y(t )

s(t)

h(t ) = s(to − t )

y(t)

t

t

to

t = to

FIGURE 8.14 Matched ﬁlter and time compression.

usually chosen to be the signal duration T itself, so that the matched ﬁlter represents a causal ﬁlter (see Figure 8.9 (b)). In that case, from (8.26) the matched ﬁlter output y(t) equals

5t y(t) =

s ∗ (τ )s(to − t + τ )dτ

(8.75)

0

which peaks at t = to and the peak value is given by

5to |s(t)|2 dt.

y(to ) =

(8.76)

0

It follows that if the input waveform is delayed by To due to target echo, the matched ﬁlter output also is delayed by the same amount, and hence the method can be used for range estimation as well, provided the output is time compressed enough to give the desired range resolution. To analyze this concept further, consider a ﬁnite energy signal s(t) with corresponding transform S(ω). Then s(t) is square integrable and hence

5+∞ 5+∞ 1 2 |s(t)| |S(ω)|2 dω < ∞. Es = dt = 2π −∞

(8.77)

−∞

Assume that |s(t)| M , the maximum value of s(t) is ﬁnite. Then we may deﬁne the effective duration of s(t) to be [3] Ts =

Es , |s(t)|2M

(8.78)

and similarly, its effective bandwidth equals Bs =

Es |S(ω)|2M

(8.79)

359

360

Space Based Radar s(t)

so(t)

|s(t)|M

|s(t)|M Ts

t

t Ts

FIGURE 8.15 An arbitrary signal and its equal energy equivalent rectangular pulse.

where |S(ω)| M represents the maximum value of its transform S(ω). From (8.78), a rectangular pulse with amplitude |s(t)| M and width Ts has the same energy as s(t) as shown in Figure 8.15 and Figure 8.16. A ﬁlter matched to s(t) has transform (Figure 8.14 or (8.25) with to = 0) H(ω) = S∗ (ω)

(8.80)

and hence its output signal transform is given by H(ω)S(ω) = |S(ω)|2 . This gives the ﬁlter output signal to be 1 y(t) = 2π

5+∞ |S(ω)|2 e jωt dω,

(8.81)

−∞

whose peak value is obtained at t = 0. Hence, its peak value is given by

|y(t)| M = y(0) =

1 2π

5+∞ |S(ω)|2 dω = E s .

(8.82)

−∞

s(w )

so(w )

|s(w)|M

|s(w )|M

w

-Bs

Bs

FIGURE 8.16 An arbitrary spectrum and its equal energy equivalent rectangular spectrum.

w

Chapter 8:

Waveform Diversity

The output signal energy E y equals

5+∞ 5+∞ 1 2 |y(t)| dt = |S(ω)|4 dω Ey = 2π −∞

−∞

≤ |S(ω)|2M

1 2π

5+∞ |S(ω)|2 dω = |S(ω)|2M E s .

(8.83)

−∞

Hence Ey =

|S(ω)|2M Es , η

(8.84)

where η is an unknown constant that is greater than unity. Using (8.78), (8.82)–(8.84), the effective duration of the output signal y(t) is given by (see Figure 8.17) Ty =

|S(ω)|2M E s |S(ω)|2M 1 Ey 1 = = = , 2 2 η Es η Es ηBs |y(t)| M

(8.85)

where Bs represents the effective bandwidth of s(t) as in (8.79). To determine the pulse compression realized by the matched ﬁlter output compared to its input, we can examine the input to output effective pulse length ratio Ts /Ty . From (8.85), [3] Ts Tinput = = Ts ηBs = ηTs Bs ≥ Ts Bs Toutput Ty

(8.86)

since η > 1. Thus large values of time compression can be realized at the matched ﬁlter output by any input signal with large timebandwidth product Ts Bs . In that case, the matched ﬁlter output will appear as a narrow pulse whose peak corresponds to the input delay. As Appendix 8-A shows, the chirped pulse signal (both modulated as well as centered at baseband) has the capacity to realize large time-bandwidth products. For example, from (8A.35)–(8A.37) the

s(t)

y(t)

Ts

t

h(t) = s(to − t)

ty

y(t)

t to

FIGURE 8.17 Effective duration of the matched ﬁlter output y(t).

361

362

Space Based Radar transform of the causal chirped signal

* s(t) =

cos(ωo t + βt 2 ),

0 < t < T,

0,

other wise

(8.87)

is given by [1]

* 1√

|S(ω)| ≈

2

ωo < |ω| < ωo + 2βT,

π/β,

0,

other wise

(8.88)

provided βT 1. From (8.88), the bandwidth Bs of a chirped signal equals 2βT and the transform is ﬂat over that region. Notice that the bandwidth can be increased by increasing the duration of the chirp signal. Finally, substituting these values into (8.86) we get Ts = Tη2βT = 2ηβT 2 ≥ 1 Ty

(8.89)

since βT 1. From (8.83)–(8.84) and (8.88), |S(ω)|2M

η= 1 2π

+∞ : −∞

Es ≈

|S(ω)| dω 4

(π/β) 2 (π/β) 2

= 1.

(8.90)

From (8.89) by increasing the pulse duration of the chirp signal, it is possible to realize output pulse compression to any degree. This is illustrated in Figure 8.18. This remarkable property of the chirp signal is exploited in radar for accurate target range estimation. The above analysis assumes the channel to be impulsive so that the transmit signal s(t) is returned with an unknown delay τo as s(t−τo ). In that case in the presence of additive noise, the matched ﬁlter achieves excellent pulse compression when s(t) is a chirp signal. However if the target (channel) interaction is signiﬁcant enough to change the shape of the return signal, chirp waveform may not be the optimum

3 2 1 0 −1 −2 −3

150 100

h(t) = s(to − t)

y(t)

50 0

0

0.1 0.2 0.3 0.4 t (sec)

0.5

−50

0

0.2 0.4 0.6 0.8 t (sec)

FIGURE 8.18 Chirp signal and its matched ﬁlter output y(t).

1

Chapter 8:

Waveform Diversity

transmit signal. Clearly in that case the designer must pay attention to the target (channel) characteristics as well. Before addressing this general design problem, we show that if the target/channel effect is simply to modulate the chirp signal with a baseband waveform a (t) so that b(t) = a (t)e − j (ωo t+βt ) , 2

0
(8.91)

is received at the receiver input, then remarkably enough the original matched ﬁlter matched to the chirp signal still retains the desirable pulse compression property. The target interaction can result in either modulating the transmit chirp signal by a time-varying gain function a (t) as in (8.91), or it can result in convolving the chirp by a target impulse response function a o (t). Interestingly, the output b o (t) has the same form as in (8.91), since

b o (t) = a o (t) ∗ e − jβt = 2

5T

a o (τ )e − jβ(t−τ ) dτ 2

0

= e − jβt

2

5T

a o (τ )e − jβτ (τ −2t) dτ = Ao (t)e − jβt . 2

(8.92)

0

Thus the target interaction either in the form of convolution or multiplicative gain can be modeled as in (8.91) and as we show below in both cases it is still possible to attain pulse compression property.

Modulated Chirp Signal Suppose a suitable baseband signal a (t) is used to modulate the chirp signal as in (8A.30) to generate the output in (8.91). Clearly, a (t) and b(t) have the same duration but the bandwidth of b(t) is larger than that of a (t). Also since |a (t)| = |b(t)|

(8.93)

the energy contents of both signals are the same. Consider the receiver h o (t) that is matched to only the chirp signal (use (8.26) with to = 0), i.e., h o (t) = s ∗ (−t) = e j (−ωo t+βt

2

)

(8.94)

363

364

Space Based Radar FIGURE 8.19 Modulated chirp signal and matched ﬁlter output.

a(t )e- j(wot+b t

2)

ho(t ) = e j(-wot + b t

2)

y(t )

so that the matched ﬁlter output y(t) equals (see Figure 8.19)

5+∞ 5+∞ 2 2 b(τ )h o (t − τ )dτ = a (τ )e − j (ωo τ +βτ ) e j (−ωo (t−τ )+β(t−τ ) ) dτ y(t) = −∞

= e j (−ωo t+βt

−∞ 2

5+∞

a (τ )e − j2βtτ dτ = e j (−ωo t+βt ) A(2βt).

)

2

(8.95)

−∞

Thus the output pulse width Ty depends only on the width of A(ω) and β and it is given by Ty =

width of A(ω) . 2β

(8.96)

As a result by selecting the chirp modulation parameter β large enough, once again the output can be made as narrow as possible, even when an unknown carrier modulates the transmit chirp signal. This property is very desirable when the channel has time-dependent ampliﬁcation gains. Equation (8.91) can be given another interpretation as well. From a signal design view point, the freedom present in selecting a (t) ↔ A(ω) can be used for enhancing the target features and suppressing the interferences.

8.3 Joint Transmitter–Receiver Design in Noise Additive White Noise Case Consider the problem of transmitting a signal f (t) over a channel with impulse response q (t). The channel output s(t) gets corrupted by additive white noise n(t), and the received signal is given by (see Figure 8.3) r (t) = s(t) + n(t).

(8.97)

If maximization of the output SNR at the decision instant t = to is chosen to be the criterion for receiver design, then as we have seen

Chapter 8:

Waveform Diversity

in the previous section, for a given s(t) the matched ﬁlter is the optimum solution, and the optimum causal receiver h(t) is given by (8.74). Thus h(t) = s ∗ (to − t)u(t).

(8.98)

In that case, since s(t) = f (t) ∗ q (t), the maximum SNR given by (8.35) or (8.73) is a function of f (t) and it makes sense to select f (t) optimally so as to further maximize the output SNR at t = to . In the causal case, since f (t) and q (t) are causal, s(t) is also causal, and from the relation

5+∞ s(t) = f (t) ∗ q (t) =

5t f (τ )q (t − τ )dτ =

−∞

f (τ )q (t − τ )dτ , (8.99) 0

where s(to ) depends only on the segment of q (t) up to t = to . Since the maximum SNR also depends only on s(t) up to t = to , the signal design problem reduces to designing the optimum pulse f (t), for 0 < t < to that maximizes (8.73). Notice that the SNR in (8.73) can be obviously increased by simply scaling f (t), and hence it makes sense to restrict the energy in f (t) to a prescribed constant for this optimization problem. Thus the problem reduces to ﬁnding an f (t) for 0 < t < to , such that

5to | f (t)|2 dt = 1,

(8.100)

0

and

5to |s(t)|2 dt

ρ(to ) =

(8.101)

0

is maximized where s(t) is given by (8.99). This formulation is meaningful in many communication problems as well where f (t), for example, can represent the baseband carrier used to modulate the information bearing symbols. Usually a rectangular pulse is used as the baseband carrier. That formulation assumes that we use the same rectangular pulse irrespective of the actual channel characteristics. But the rectangular pulse may not be the best pulse shape for all channels. Thus, given some additional information about the channel, the problem is to ﬁnd a meaningful way to incorporate that information in selecting the best baseband pulse. Notice that since information symbols arrive at every To seconds in this model, at the

365

Space Based Radar FIGURE 8.20 Zeros of a minimum phase signal transform.

jw Ze ro s

366

s = s + jw s

output, decisions must be made also at every To seconds, and hence to could be chosen equal to To in this case.

Minimum Phase Character of the Optimum Solution [2] Interestingly, it can be shown that the input causal pulse f (t) that maximizes ρ(to ) in (8.101) must be a minimum phase signal. A signal f (t) is said to be minimum phase, if its Laplace transform F (s) is free of zeros and poles in the open right half plane (Res > 0) as in Figure 8.20. To establish the minimum phase condition, assume that the ﬁnite pulse f (t) is not minimum phase. Then

5to F (s) =

f (t)e −st dt

(8.102)

0

has at least one zero in Re s > 0, say s = so . Rewrite F (s) as

F (s) = Fo (s)

s − so s + so∗

= Fo (s) A(s),

σo = Re s > 0,

(8.103)

where A(s) is a regular all-pass function.2 Let f o (t) represent the inverse Laplace transform corresponding to Fo (s). By extracting all an-pass, the remaining portion Fo (s) is guaranteed to have the same energy as the original pulse f (t). However, for f o (t) to act as a possible candidate in this optimization problem, it should also be a ﬁnite pulse of duration to . To examine this, notice that

Fo (s) = F (s)

s + so∗ s − so

= F (s) + 2σo

F (s) s − so

(8.104)

2 A regular all-pass function has all its poles in the open left-half plane, and represents a causal system.

Chapter 8:

Waveform Diversity

so that, its inverse transform in given by

5∞ f o (t) = f (t) + 2σo f (t) ∗ e

=

so t

f (τ )e so (t−τ ) u(t − τ )dτ

u(t) = f (t) + 2σo 0

 :t  so t  f (t) + 2σ e f (τ )e −so τ dτ,  o 

t < to

 :to    f (t) + 2σo e so t f (τ )e −so τ dτ,

t ≥ to

0

0

=

 t   f (t) + 2σo e so t : f (τ )e −so τ dτ,  

t < to

0

2σo

e so t F (s

o)

= 0,

.

(8.105)

t ≥ to

Thus f o (t) is also of ﬁnite duration to , and it possesses the same energy as f (t), establishing the fact that extraction of an all-pass factor from a ﬁnite pulse transform still retains its ﬁnite pulse shape characteristics. Toward establishing the minimum phase character of the optimum pulse f (t) that maximizes (8.101), let s(t) and so (t) represent the channel output due to inputs f (t) and f o (t) respectively (see Figure 8.21). Recall that f o (t) is also a ﬁnite pulse with the same energy, but its transform contains one less zero in the right-half plane compared to that of f (t). Thus s(t) = f (t) ∗ q (t);

so (t) = f o (t) ∗ q (t).

(8.106)

We will show that the running energy up to t = to in so (t) is greater than that in s(t); i.e.,

5to

5to 2

|s(t)|2 dt,

|s(t)| dt ≥ 0

(8.107)

0

indicating that extraction of a right-half plane zero, or more precisely a regular all-pass function, from the input excitation pulse will result in greater running energy for the system output. Clearly, by continuing this process of regular all-pass extraction, maximum running energy at the output will be realized by a pulse whose transform is free of

f(t)

q(t)

s(t ), fo(t)

FIGURE 8.21 Transmit waveform f(t) and fo (t).

q(t)

so(t),

367

368

Space Based Radar so(t)

t

a(t) ↔ A(s)

so(t)

s(t)

FIGURE 8.22 Minimum phase input signal.

zeros in the right-half plane; i.e., among all pulses with the same input energy, the minimum phase pulse generates maximum running energy at the output of any system. We can make use of (8.103) to complete the minimum phase proof. Together with (8.106), this gives (see Figure 8.22) s(t) = a (t) ∗ so (t),

(8.108)

i.e., s(t) is the output of the all-pass ﬁlter a (t) ↔ A(s) due to the input so (t). By making use of the regular all-pass nature of A(s) and Parseval’s relation, we have

5∞

5∞ |so (t)| dt =

|s(t)|2 dt.

2

0

(8.109)

0

Let x(t) represent the output of the same all-pass ﬁlter A(s) due to the truncated input s1 (t) as shown in Figure 8.23. Thus

* s1 (t) =

s1(t)

so (t),

0 < t < to

0,

other wise

.

(8.110)

so(t)

to

t s1(t)

FIGURE 8.23 Truncated input signal.

A(s)

x(t)

Chapter 8:

Waveform Diversity

Then once again using Parseval’s relation, by equating the total input and output energy, we get

5∞

5to |s1 (t)|2 dt =

0

5∞ |so (t)|2 dt =

0

|x(t)|2 dt.

(8.111)

0

But, because of the causal nature of A(s), we have

5t x(t) =

s0 (τ )a (t − τ )dτ = s(t),

0 < t < to ,

(8.112)

0

(i.e., the outputs in Figures 8.22 and 8.23 agree up to t < to ) and substituting this into the later half of (8.111) we get

5to

5to |so (t)| dt =

0

5∞ |s(t)| dt +

2

|x(t)|2 dt,

2

(8.113)

to

0

or

5to

5to |so (t)| dt ≥

|s(t)|2 dt,

2

0

(8.114)

0

establishing our original claim. Thus to maximize the running energy of any system output, the input pulse f (t) must be necessarily minimum phase.

Optimum Solution Returning to the original optimization problem in (8.101), to make further progress, we can substitute (8.99) into (8.101). This gives

2 5to 5∞ ρ(to ) = f (τ )q (t − τ )dτ dt 0

0

  5to 5∞ 5∞ =  f ∗ (τ1 )q ∗ (t − τ1 ) f (τ2 )q (t − τ2 )dτ1 dτ2  dt 0

5∞ = 0

0

0

  5∞ 5to f ∗ (τ1 )dτ1  q ∗ (t − τ1 )q (t − τ2 )dt  f (τ2 )dτ2 . 0

0

K (τ1 , τ2 )

(8.115)

369

370

Space Based Radar Deﬁne

5to

K (τ1 , τ2 ) =

q ∗ (t − τ1 )q (t − τ2 )dt

(8.116)

0

to represent the kernel associated with the channel impulse response q (t). Note that only the channel response up to t = to takes part in deﬁning the above kernel. Further, q (t) causal implies that if |τ1 | > to ,

K (τ1 , τ2 ) = 0,

or |τ2 | > to .

(8.117)

Using (8.116) in (8.115), it simpliﬁes to

5∞ ρ(to ) =

5∞

∗

K (τ1 , τ2 ) f (τ2 )dτ2

f (τ1 )dτ1 0

0

5to

5to

=

f ∗ (τ1 )dτ1

0

(8.118) K (τ1 , τ2 ) f (τ2 )dτ2 ,

0

where the later limits in the above integral follow form (8.117). The above integral deﬁnes an integral operator T( f ) given by

5to T( f ) =

K (τ1 , τ2 ) f (τ2 )dτ2 = ψ(τ1 ),

(8.119)

0

and using this, (8.118) can be expressed also as an inner product given by

5to ρ(to ) =

f ∗ (τ1 )ψ(τ1 )dτ1 .

(8.120)

0

However, using Schwartz’ inequality, the above expression yields

 t 1/2  t 1/2 5o 5to 5o ρ(to ) ≤  | f (τ1 )|2 dτ1 |ψ(τ1 )|2 dτ1  =  |ψ(t)|2 dt  0

0

0

(8.121) and (8.120) is maximized if and only if ψ(τ1 ) is chosen to be proportional to f (τ1 ) in 0 < τ1 < to , i.e., for maximum SNR f (τ1 ) must be a

Chapter 8:

Waveform Diversity

solution of the integral equation3

5to ψ(τ1 ) =

K (τ1 , τ2 ) f (τ2 )dτ2 = λ f (τ1 ),

0 < τ1 < to .

(8.122)

0

Substituting (8.122) into (8.120), we get

5to | f (τ1 )|2 dτ1 = λ ≤ λ1 ,

ρ(to ) = λ

(8.123)

0

where λ1 is the largest (positive) eigenvalue of the above integral equation. Notice that the right-hand side of (8.123) is independent of f (t) and hence to maximize ρ(to ), λ in (8.123) must be chosen as the largest eigenvalue of the above integral equation. In other words, ρ(to ) is maximized by selecting f (t) to correspond to the eigenfunction associated with the largest eigenvalue λ of the integral equation (8.122), and the maximum possible SNR is given by ρ(to ) = λmax (to ),

(8.124)

where λmax (to ) is the largest value of λ that satisﬁes (8.122). From the previous argument, the eigenfunction f (t) corresponding to the largest eigenvalue of the kernel in (8.122) also represents a minimum phase function.

Optimum Input and the Matched Filter Receiver It is easy to show that in the white noise case, the optimum solution for f (t) that satisﬁes the integral equation (8.122) is identical to the matched ﬁlter receiver solution. In fact, more generally if f (t) satisﬁes (8.122) for some eigenvalue λ, then h(t) = f (t),

0 < t < to ,

(8.125)

where h(t) is the optimum causal receiver deﬁned in (8.98). To see this, from (8.98) and (8.99), we get h(t) = s ∗ (to − t)u(t) =

5to

q ∗ (to − t − τ ) f ∗ (τ )dτ ,

(8.126)

0

3 Every eigenvalue that results as a solution of the integral equation in (8.122) will be positive, because of the symmetric positive-deﬁnite nature of the kernel ∗ K (τ1 , τ2 ) in (8.116). Note: that : K (x, y) ∗is said to be symmetric if K (x, y) = K ( y, x), and positive-deﬁnite if K (x, y) f (x) f ( y)d xd y > 0 for any f (x).

371

372

Space Based Radar and consider the integral

5to K (τ1 , τ2 )h(τ2 )dτ2 ,

(8.127)

0

obtained by replacing f (τ2 ) in (8.122) with h(τ2 ). Substituting (8.126) into (8.127), and making use of (8.116), we obtain

5to

5to K (τ1 , τ2 )

0

q ∗ (to − τ2 − τ ) f ∗ (τ )dτ2 dτ

0

5to

5to

=

5to

∗

q (t − τ1 )q (t − τ2 ) 0

5to =

0

q ∗ (to − τ2 − τ ) f ∗ (τ )dτ2 dτ dt

0

q ∗ (t − τ1 )

5to

5to

0

0

0

q (t − τ2 )q ∗ (to − τ2 − τ ) f ∗ (τ )dτ2 dτ dt. (8.128)

Let to − τ2 = u, so τ2 = to − u, and the above integral becomes

5to

5to 5to

∗

q (t − τ1 ) 0

0

q (t − to + u)q ∗ (u − τ ) f ∗ (τ )du dτ dt.

(8.129)

0

But by deﬁnition

5to

q (u − (to − t)) q ∗ (u − τ )du = K ∗ (to − t, τ ).

(8.130)

0

Thus (8.129) becomes

5to

 t ∗ 5o q ∗ (t − τ1 )  K (to − t, τ ) f (τ )dτ  dt

0

0

5to = 0

q ∗ (t − τ1 )(λ f ∗ (to − t))dt,

(8.131)

Chapter 8:

Waveform Diversity

where we have made use of (8.122). Finally, let to − t = v, then t = to − v, and hence

5to λ

∗

∗

5to

q (t − τ1 ) f (to − t)dt = λ 0

q ∗ (to − τ1 − v) f ∗ (v)dv = λh(τ1 ),

0

(8.132)

where we have made use of (8.126). Thus from (8.127)–(8.132) we have shown that

5to K (τ1 , τ2 )h(τ2 )dτ2 = λh(τ1 ),

0 < τ1 < to ,

(8.133)

0

where f (t) in (8.126)–(8.131) is the actual eigenfunction in (8.122) corresponding to any eigenvalue λ there. Obviously, (8.133) is true even when f (t) corresponds to the largest eigenvalue λmax , and in that case h o (t) is the desired matched ﬁlter. Figures 8.24 and 8.26 show the optimal input waveforms for various channel characteristics with details as indicated there. In Figure 8.24, the channel is assumed to be the usual rectangular pulse with width To , and the optimum transmit waveform f (t) for to = To obtained by solving (8.122) and the corresponding matched ﬁlter responses are shown in Figure 8.24 (b)–(d). Figure 8.24 (e) shows the optimum input pulse shape f (t), for different values of to .4 Similarly, Figures 8.25 and 8.26 exhibit the desired transmit pulse f (t) as well as the corresponding matched ﬁlters for various channel characteristics as shown there. This leads us to an interesting question: for a given channel, is there a best value for to at which decision should be made? If we use the output SNR ρ(to ) as the optimality criterion, this question can be answered by examining ρ(to ) vs. to . It is easy to see that ρ(to ) is in fact a monotonically nondecreasing function of to , i.e., ρ(t1 ) ≥ ρ(to ),

if

t1 > to .

(8.134)

To prove this, let us assume the contrary, and suppose that for some t1 > to , the optimum eigenvector f 1 (t) associated with t1 leads to a smaller SNR compared to the optimum eigenvector f o (t) associated with t = to . In that case, since so (t) = f o (t) ∗ q (t), s1 (t) = f 1 (t) ∗ q (t), 4 For the decision instant t , since f (t) exists only in the interval 0 < t < t , o o it should be easy to identify the corresponding input pulse in Figure 8.24 (e) for various value of to .

373

Space Based Radar q(t)

f(t)

t

To

to = To

(a) Channel impulse response

t

(b) Optimum pulse f(t) for to = To

s(t) = q(t) * f(t)

h(t) = s(to - t)u(t)

to

to + To

t

t

to (d) Matched filter for to = To

(c) Channel output f(t)

r (to) 25 20 dB

374

15 10

To

2To

3To

4To

t

To

2To

3To

4To

to

(f) Output SNR for different values of to

(e) Optimum pulse f(t) for different values of to

FIGURE 8.24 Optimum transmit pulse and matched ﬁlter for a rectangular channel response.

and if we have

5to

5t1 |so (t)|2 dt > ρ(t1 ) =

ρ(to ) = 0

|s1 (t)|2 dt,

(8.135)

0

by redeﬁning

* f 1 (t) =

f o (t),

0 < t ≤ to ,

0,

to < t ≤ t1 ,

(8.136)

Chapter 8: q(t)

Waveform Diversity

f(t)

To

t

to = To

t

(a) Target impulse response

(b) Optimum pulse f(t) for to = To

h(t)

r(to)

dB

25 20 15

To

10

t

(c) Matched filter for to = To

To

2To

3To

4To

to

(d) Output SNR for different values of to

FIGURE 8.25 Optimum transmit pulse and matched ﬁlter for an exponentially decaying target response. q(t)

f(t)

To

t

to = T o

t

(b) Optimum pulse f(t) for to = To

(a) Target impulse response

r (to)

dB

h(t)

To (c) Matched filter for to = To

t

20 15 10 5 0 −5 To

2To

3To

4To

to

(d) Output SNR for different values of to

FIGURE 8.26 Optimum transmit pulse and matched ﬁlter for an arbitrary target.

375

376

Space Based Radar the inequality in (8.134) can be satisﬁed with equality. Thus (8.134) is always true, indicating that under the joint optimality condition, delaying the decision instant to cannot make it worse in terms of discriminating the signal from the noise. Figures 8.24 (f) and 8.25 (d)–8.26 (d) show the monotonic nondecreasing nature of ρ(to ) as a function of to . These ﬁgures indicate that ρ(to ) → ρ(∞), the maximum possible value of the output SNR obtained by letting to → ∞, and the ρ(t0 ) curve can be used to determine the best value for t0 . To summarize, for a given pulse f (t), the channel output results in a speciﬁc waveform s(t) and maximization of the output SNR at t = to leads to the matched ﬁlter receiver. Since the output SNR depends on f (t), further improvement in the output SNR can be obtained by exploiting the freedom present in selecting f (t) at the transmitter. Thus for a given channel, by combining the matched ﬁlter design together with the input pulse design problem, we have been able to achieve joint optimal transmitter–receiver design. The optimum input pulse will always be minimum phase, and moreover in the white noise case the matched ﬁlter is identical to the optimum input pulse. The key idea in this section is to make use of the kernel of the channel characteristics to deﬁne an optimal input waveform, and as we show in the next section, this approach has been used to solve another interesting problem in communication theory.

8.4 Joint Time Bandwidth Optimization Suppose a pulse e(t) ↔ E(ω) of duration to needs to be transmitted through a channel with impulse response q (t) ↔ Q(ω). Let r (t) ↔ R(ω) denote the channel output. Then

5to r (t) = e(t) ∗ q (t) =

e(τ )q (t − τ )dτ ,

(8.137)

0

or, in terms of their transforms R(ω) = E(ω) Q(ω).

(8.138)

Suppose −Bo < ω < Bo represents the desired spectral band at the receiver as shown in Figure 8.27(b). We would like to choose the input signal e(t) such that the output signal r (t) has maximum energy concentrated in the above band. This formulation makes sense in communication problems for minimizing cochannel interference, since the above design criterion makes sure that most of the energy of the transmitted signal will be concentrated in the desired band (−Bo , Bo ), and hence very little energy will spill over to the adjacent bands. In a different context, we may be

Chapter 8:

Waveform Diversity

|E(w)|2 Desired band

e(t) Bo

-Bo

t e(t)

q(t)

Q(w)

w

r(t)

to (a)

(b)

FIGURE 8.27 Joint time-bandwidth optimization.

interested in ﬁnding the best time-limited signal that also has maximum energy in a given band. Since time-limited signals cannot be band limited, the next best thing that one could hope for in terms of bandlimiting a time-limited signal is to maximize the energy within a given bandwidth [2]. Returning to the original problem, the objective is to select the best pulse e(t) such that the output energy 1 E= 2π

5Bo

1 |R(ω)| dω = 2π

5Bo |Q(ω)|2 |E(ω)|2 dω

2

−Bo

(8.139)

−Bo

in the spectral band (−Bo , Bo ) is maximized. Since simple scaling of e(t) can increase the value of the above E, it is necessary to normalize e(t) by requiring it to possess unit (or, constant) energy. Thus our problem reduces to maximizing (8.139) by selecting a suitable timelimited signal e(t), 0 < t < to under the energy constraint

5to |e(t)|2 dt = 1.

(8.140)

0

To solve this problem, once again we can make use of the concept of the channel kernel discussed in the previous section. However, it is necessary to bring in the desired bandwidth information into this kernel in some suitable manner. To express this problem in the time domain, we can make use of Parseval’s theorem. If we deﬁne

(ω) =

*

|Q(ω)|2 E(ω),

−Bo < ω < Bo

0,

otherwise,

(8.141)

377

378

Space Based Radar q(t)

|Q(w)|2

t

w

−Bo

Bo

FIGURE 8.28 Inverse Fourier transform of the channel restricted to (−Bo < ω < Bo ).

then using Parseval’s theorem in (8.139) we obtain 1 E= 2π

5+∞ 5+∞ 5to ∗ ∗ (ω) E (ω)dω = ψ(t)e (t)dt = ψ(t)e ∗ (t)dt, −∞

−∞

0

(8.142) where ψ(t) represents the inverse transform of (ω) deﬁned above. To simplify ψ(t), deﬁne K (τ ) to represent the inverse Fourier transform of the channel transfer function gain |Q(ω)|2 restricted to the frequency band (−Bo < ω < Bo ) as in (8.141). Thus (Figure 8.28) 1 K (τ ) = 2π

5Bo

|Q(ω)|2 e jωτ dω = K ∗ (−τ ).

(8.143)

−Bo

The function K (τ ) acts as the kernel associated with the channel transfer function gain |Q(ω)|2 over the desired bandwidth (−Bo , Bo ). Note that K (τ ) represents a stationary kernel (autocorrelation function), since it is a function of only one argument. Using (8.141)–(8.143) and ψ(t) ↔ (ω) we get

5+∞ 5to ψ(t) = K (t) ∗ e(t) = K (t − τ )e(τ )dτ = K (t − τ )e(τ )dτ , −∞

(8.144)

0

since e(τ ) exists only in the interval 0 < τ < to . Note that ψ(t) in (8.144) deﬁnes an integral operator T(e) through the above kernel, and using this, the positive quantity E in (8.142) can be represented as an inner product. Thus

5to E= 0

ψ(t)e ∗ (t)dt,

(8.145)

Chapter 8:

Waveform Diversity

and once again using Schwartz’ inequality, the above expression gives

 t 1/2  t 1/2 5o 5to 5o E ≤  |ψ(t)|2 dt |e(t)|2 dt  =  |ψ(t)|2 dt  , 0

0

(8.146)

0

and it is maximized if and only if we choose ψ(t) = λe(t),

0 ≤ t ≤ to ,

(8.147)

i.e., if and only if e(t) satisﬁes the integral equation

5to K (t − τ )e(τ )dτ = λe(τ ),

0 ≤ t ≤ to .

(8.148)

0

In this case

5to |e(t)|2 dt = λ.

E =λ

(8.149)

0

Since the right side of the above equation is independent of e(t), to maximize E, the eigenvalue λ in (8.148) must be chosen as the largest possible value. Thus the optimal input waveform e(t) that maximizes E is the solution of the integral equation in (8.148) corresponding to the largest eigenvalue of λ, where the positive-deﬁnite kernel K (t − τ ) is as deﬁned in (8.143). From (8.146),

E max

; <5to < < = = |ψmax (t)|2 dt = λmax

(8.150)

0

represents the maximum value for the energy, where ψmax (t) is as deﬁned in (8.144) with e(t) representing the eigenfunction associated with the largest eigenvalue in (8.148). Since this situation is identical to that in (8.101), using the arguments developed in that section it follows that to maximize E max , the input waveform e(t) must be necessarily minimum phase. Thus the above optimum solution given by the eigenfunction associated with the largest eigenvalue of (8.148) must be necessarily minimum phase. In addition, it is easy to show that if e(t) satisﬁes the integral equation (8.148), then so does

f (t) = e ∗ (to − t),

0 ≤ t ≤ to .

(8.151)

379

380

Space Based Radar This follows easily, since

5to

5to K (t − τ ) f (τ )dτ =

0

K (t − τ )e ∗ (to − τ )dτ

0

 t ∗ 5o =  K (to − t − x)e(x)d x  0 ∗

= λe (to − t) = λ f (t),

0 < t < to ,

(8.152)

where we have used the substitution to − τ = x, and made use of the symmetric nature of the kernel given in (8.143). Thus if λ represents a distinct eigenvalue, then these two solutions must equal. This gives e(t) = e ∗ (to − t),

0 < t < to ,

(8.153)

i.e., e(t) represents a symmetric solution in the interval (0, t0 ) in the sense that it agrees with its matched ﬁlter with respect to to . Alternatively, in terms of their transforms E(s) = e −to s E ∗ (−s ∗ ).

(8.154)

Thus E(s) must possess zeros symmetrically in both half planes. However, because of its minimum phase character, the eigenfunction corresponding to the largest eigenvalue has no zeros in the strict right half plane. Consequently it is also free of zeros in the strict left half plane. Thus all of its zeros must lie on the jω − axis. To summarize, the best input pulse e(t) of ﬁnite duration to that also maximizes the channel output energy over the band (−Bo , Bo ) is given by the eigenfunction associated with the largest eigenvalue λmax (to ) of the integral equation in (8.148). It represents a minimum phase symmetric function with all the zeros of its Laplace transform E(s) distributed along the jω − axis. The maximum possible value of the energy in (−Bo , Bo ) is given by λmax (to ). Note that the channel characteristics comes into play in the kernel through |Q(ω)|2 along with the desired frequency band (−Bo , Bo ). Once again, it is easy to show that λmax (to ) is a monotone nondecreasing function of to . To see this, using (8.146), we have

E max

; <5to < < = λmax (to ) = = |ψ(t)|2 dt, 0

(8.155)

Chapter 8:

Waveform Diversity

where ψ(t) is given by (8.144). With t1 > to , let e 1 (t) and e(t) represent the solution of (8.148) for t1 and to respectively. Thus

; <5t1 < < λmax (t1 ) = = |ψ1 (t)|2 dt

(8.156)

0

: t1

where ψ1 (t) = 0 K (t − τ )e 1 (τ )dτ , and if λmax (t1 ) < λmax (to ), then it is easy to satisfy this inequality with an equality by redeﬁning e 1 (t) as

* e 1 (t) =

e(t),

0 < t < to

0,

otherwise.

(8.157)

Thus if t1 > t0 , then λmax (t1 ) ≥ λmax (to ), which establishes the monotone nondecreasing nature of λmax (to ). To illustrate the scheme presented here, Figures 8.29 and 8.30 show the optimum waveform e(t) for a typical low-pass channel gain function |Q(ω)|2 given by |Q(ω)|2 =

1 1 + ω2

(8.158)

for different values of to and Bo . In this case, from (8.143) we have 1 K (τ ) = π

5Bo

cos ωτ dω, 1 + ω2

(8.159)

0

1.5

e(t )

1

0.5

0

0.5

1

1.5

2

2.5

t

FIGURE 8.29 Optimum e(t) for to = 0.5, 1, 1.5, 2, and 2.5 with Bo ﬁxed to 0.5 normalized bandwidth.

381

Space Based Radar 1.1

Bo

1 e(t )

382

0.9

0

0.2

0.4

0.6

0.8

1

t

FIGURE 8.30 Optimum e(t) for Bo = 0.2, 0.4, 0.6, 0.8, and 1 with to ﬁxed equal to 1.

and together with (8.148) it can be used to solve for the desired e(t). Figure 8.29 shows the optimum e(t) for various values of to with Bo ﬁxed equal to the normalized bandwidth of 0.5. Similarly, Figure 8.30 shows the corresponding e(t) for different values of Bo with to ﬁxed equal to unity. Notice the symmetric nature of these waveforms. As noted earlier, in all these cases the transforms of these waveforms have all their zeros on the jω − axis. Figure 8.31 shows the square magnitude of the transforms of the optimum waveforms in Figure 8.30 as a function of the normalized frequency and for comparison purpose, they are plotted along with |Q(ω)|2 .

Prolate Spheroidal Functions [3] As a special case, suppose we need to determine the best time-limited signal that is also maximally band-limited. In other words, this is equivalent to ﬁnding the best time-limited signal of unit energy, that has maximum energy within a given bandwidth Bo as in Figure 8.32. Thus, the problem is to maximize

1 E= 2π

5Bo |E(ω)|2 dω −Bo

(8.160)

Chapter 8: 0

Waveform Diversity

|Q(w)|2

−20 dB

|E(w)|2 for Bo = 0.2, 0.4, 0.6, 0.8, and 1 −40

−60 −1

−0.5

0 w

1

0.5

FIGURE 8.31 Transforms of optimum waveform for Bo = 0.2, 0.4, 0.6, 0.8, and 1 with to = 1 vs. |Q(ω)|2 .

subject to

5to

1 |e(t)| dt = 2π 2

0

5+∞ |E(ω)|2 dω = 1.

(8.161)

−∞

Notice that E in (8.160) represents the energy compaction ratio inside the frequency band (−Bo , Bo ) for e(t), and it has the same form as (8.139), with |Q(ω)|2 as shown in Figure 8.33. From (8.143), this gives the corresponding kernel to be 1 K (τ ) = 2π

5Bo e jωτ dτ = −Bo

1 sin Bo τ , π τ

(8.162)

|E(w)|2

e(t)

t to

−Bo

w Bo

FIGURE 8.32 Optimum pulse of unit energy that has maximum energy within a given bandwidth.

383

384

Space Based Radar FIGURE 8.33 Flat spectrum for |Q(ω)|2 .

|Q(w)|2 1

–Bo

w

Bo

and the desired signal e(t) is given by the eigenfunction associated with the largest eigenvalue of the integral equation [6]

5to 0

1 sin Bo (t1 − t2 ) e(t2 )dt2 = λe(t1 ), π t1 − t2

0 < t1 < t0 .

(8.163)

By a simple change of variables (t = Bo t1 , τ = Bo t2 ), we can rewrite the above equation as 1 π

5to Bo

sin(t − τ ) e(τ )dτ = λe(τ ), t−τ

0 < t < t0 Bo ,

(8.164)

0

which shows that the time-bandwidth product η = to Bo is the only independent parameter in this problem. Solutions of the integral equation 1 π

5η

sin(t − τ ) e(τ )dτ = λ(η)e(τ ), t−τ

0
(8.165)

0

for various values of the time-bandwidth product η represent the well known prolate–spheroidal functions [5]. These functions are plotted in

FIGURE 8.34 Energy compaction ratio for prolate–spheroidal functions.

Chapter 8:

Waveform Diversity

l max (h)

FIGURE 8.35 λmax (η) as a function of η with to ﬁxed equal to 1.

0

dB

–1 –2 –3 –4 0

2

4

6

8

10

h

Figure 8.34 for various values of η. As pointed out earlier, the Laplace transform E(s) will have all of its zeros on the jω − axis in this case. Figure 8.35 shows λmax (η) as a function of η with to ﬁxed equal to unity. As η → ∞, λmax (η) → 1, since this is equivalent to ﬁxing t = to and letting Bo → ∞. In that case, E in (8.160) tends to unity as well.

Appendix 8-A: Transform of a Chirp Signal Consider the chirped rectangular symmetric pulse (noncausal)

f o (t) =

2

e jβt ,

−T/2 < t < T/2,

0,

otherwise,

(8A.1)

whose transform is given by

5T/2 Fo (ω) =

e

jβt 2 − jωt

e

dt = e

− jω2 /4β

−T/2

Let

√ β t−

ω 2β

5T/2

e

ω jβ t− 2β

2 dt.

(8A.2)

−T/2

= τ . Then

1 2 Fo (ω) = √ e − jω /4β β

√ β (βT−ω)/2 5

2

e jτ dτ , √

−(βT+ω)/2 β

(8A.3)

385

386

Space Based Radar or



√ (βT−ω)/2 β 5

1 2  Fo (ω) = √ e − jω /4β  β

√ (βT+ω)/2 β 5

2

e jτ dτ +

 2



e jτ dτ . (8A.4)

0

0

Deﬁne the Fresnel integral

K (ω) =

2 π

5ω

2

e jt dt

(8A.5)

0

so that

K (∞) =

2 π

5∞

1 2

2

e jt dt = 0

1 = √ jπ = 2π

2 π

5∞

2

e jt dt −∞

1 j = √ e jπ/4 , 2 2

(8A.6)

and hence

5∞

5∞ cos t 2 dt =

0

sin t 2 dt =

1 π/2. 2

(8A.7)

0

In terms of (8A.5), (8A.4) becomes Fo (ω) = e

− jω2 /4β

π 2β

* + βT + ω βT − ω √ √ K +K 2 β 2 β

(8A.8)

and it represents the transform of the pulsed symmetric√ chirp signal in (8A.1). Notice that near the baseband region, (i.e., for βT 1)

K

βT + ω √ 2 β

√ ≈K

Hence for large values of

βT 2

≈ K (∞) =

j /2.

(8A.9)

√ βT, in the baseband region (see (8A.6))

Fo (ω) ≈ or

π 2β

|Fo (ω)| ≈

π , β

j + 2

j 2

|ω| < βT

(8A.10)

(8A.11)

Chapter 8:

|Fo(w)|

FIGURE 8.36 Transform of the chirp signal 2 fo (t) = e jβ t , −T/2 < t < T/2.

p|b

−b T

K

βT − ω √ 2 β

= −K

ω − βT √ 2 β

w

bT

Bs = 2b T

and for ω > βT (use K (−ω) = −K (ω))

Waveform Diversity

≈ −K (ω)

(8A.12)

and hence Fo (ω) = 0, Hence |Fo (ω)| =

ω > βT.

*√ π /β, 0,

(8A.13)

|ω| < βT,

(8A.14)

|ω| > βT.

As Figure 8.36 shows, the chirp signal in (8A.1) has constant magnitude in the baseband region, and it is essentially zero outside that region. >√ Notice that even in the passband the magnitude goes down as 1 β for the chirp. Next consider the modulated chirped pulse signal 2

f (t) = e j (ωo t+βt ) ,

−

T T
(8A.15)

Then F (ω) = Fo (ω − ωo ).

(8A.16)

Hence we get F (ω) = e

− j (ω−ωo ) 2 /4β

%

π K 2β

ω − ω1 √ 2 β

+K

ω2 − ω √ 2 β

& (8A.17)

where ω1 = ωo − βT,

ω2 = ωo + βT.

(8A.18)

387

388

Space Based Radar |F(w)|

FIGURE 8.37 Transform of the chirp signal 2 f(t) = ej(ωo t+β t ) , T T −2
p|b

w 1 = wo − b T

w 2 = wo + b T

wo

w

Bs = 2b T

Once again for large values of

√

βT compared to ω, we have

√ π βT π j π |F (ω)| ≈ 2 K ≈ 2 , = 2β 2 2β 2 β

(8A.19)

and for |ω − ωo | βT, we have

F (ω) ≈

π {K (ω) + K (−ω)} ≈ 0, 2β

for ωo − βT > ω > ωo + βT. (8A.20)

Hence (see Figure 8.37) *√ π /β, |F (ω)| = 0,

ωo − βT < ω < ωo + βT, otherwise.

(8A.21)

To obtain the transform of a causal chirp, deﬁne using (8A.15) f 1 (t) = f (t − T/2) = e j{ωo (t−T/2)+β(t−T/2) 2 2 = e − j (2ωo T−βT )/4 e j (ωo −βT)t+βt ,

2

}

0 < t < T.

(8A.22)

Notice that f 1 (t) has a new carrier at o = ωo − βT

(8A.23)

>

rather than at ωo . Hence with φo = (2ωo T − βT 2 ) 4, we have f 1 (t) = e − jφo e j (o t+βt ) , 2

0
(8A.24)

has a transform equal to F (ω)e − jωT/2 . Hence F1 (ω) = F (ω)e − jωT/2 .

(8A.25)

Chapter 8:

Waveform Diversity

|F2(w)|

FIGURE 8.38 Transform of the chirp signal 2 f (t) = ej(o t+β t ) ,

p|b

2

0 < t < T.

Ωo + b T

Ωo

Ωo + 2b T

w

Let (see Figure 8.38) 2

f 2 (t) = e j (o t+βt ) ,

0 < t < T.

(8A.26)

From (8A.24) we have f 2 (t) = e jφo f 1 (t),

0 < t < T,

(8A.27)

and hence F2 (ω) = e jφo F1 (ω) = e jφo e − jωT/2 F (ω) 2 2 = e j ( 2ωo T−βT )/4 e − jωT/2 e − j (ω−ωo ) /4β G(ω)

=e =e =e

−j −j −j

(ω−ωo )2 4β

ω−ωo √ 2 β

+

+ √

βT 2 4

(ω−ωo )T 2

βT 2 2 G(ω)

ω−(ωo −βT) 2 √ 2 β

+

G(ω) = e

−j

G(ω)

ω−o 2 √ 2 β

G(ω) (8A.28)

where from (8A.17)

G(ω) =

π 2β

* + ω − o o + 2βT − ω √ √ K +K , 2 β 2 β

(8A.29)

since ωo = o + βT. Notice that the transform in (8A.28) and (8A.29) is nonzero in the interval o < ω < o + 2βT for βT 1 and zero elsewhere. As Figure 8.38 shows, when a carrier modulated chirp as in (8A.26) is transmitted in 0 < t < T (causal waveform), its nonzero transform region is in (o , o + 2βT) where o represents the carrier frequency. Thus, in standard form (see Figure 8.39)

f 3 (t) =

2

e j (ωo t+βt ) ,

0 < t < T,

0,

other wise

(8A.30)

389

390

Space Based Radar FIGURE 8.39 Transform of the chirp signal 2 f (t) = ej(ωo t+β t ) ,

|F3(w)| p|b

3

0 < t < T.

wo + b T wo + 2b T

wo

w

Bs = 2b T

has the transform (see (8.29) and (8.30)) F3 (ω) = e

−j

ω−ωo 2 √ 2 β

%

π K 2β

ω − ωo √ 2 β

+K

ωo + 2βT − ω √ 2 β

& .

(8A.31) To down convert F3 (ω) to baseband, one must translate it by the center frequency ωo + βT, and not by the carrier frequency ωo alone. Equation (8A.31) may be also rewritten as F3 (ω) = e

−j

ω−ωo 2 √ 2 β

1 β

5x

2

e jt dt

(8A.32)

√ x− βT

where x=

ω − ωo √ . 2 β

(8A.33)

Finally since cos(ωo t + βt 2 ) =

f 3 (t) + f 3∗ (t) , 2

(8A.34)

from (8A.30)–(8A.34) we have the transform pair (see Figure 8.40) f 4 (t) = cos(ωo t + βt 2 ),

0 < t < T ↔ F4 (ω) =

F3 (ω) + F3∗ (ω) . 2 (8A.35)

From (8A.32) F3∗ (−ω) = e

−j

ω+ωo 2 √

2 β

1 β

√ y+ 5 βT

y

e − jt dt 2

(8A.36)

Chapter 8:

Waveform Diversity |F4(w)| p|b| 2

cos (wot + b t 2 )

t

wo + b T

−(wo + b T ) −(wo + 2b T )

−wo

wo + 2b T

wo

w

Bs = 2b T

FIGURE 8.40 Transform of a real and causal chirp signal f4 (t) = cos(ωo t + βt2 ), 0 < t < T.

where y =

ω+ω √ o. 2 β

Using (8A.30)–(8A.34) we get

1 |F4 (ω)| =

2 0,

π/β,

ωo < |ω| < ωo + 2βT,

(8A.37)

otherwise.

References [1] A. Papoulis, Signal Analysis, McGraw-Hill, New York, NY, 1977. [2] S.U. Pillai, et al., “Optimum Trans Receiver Design in the Presence of SignalDependent Interference and Channel Noise,” IEEE Transactions on Information Theory, Vol. 46, No. 2, pp. 577–584, March 2000. [3] D.C. Youla, “Chirp and Pulse Compression,” Private Communications, Polytechnic University, Melville, NY, 2004. [4] J.H.H. Chalk, “The Optimum Pulse Shape for Pulse Communication,” Proc. Inst. Elec. Eng. London, UK, Vol. 87, pp. 88–92, 1950. [5] D. Slepian, H.O. Pollak, “Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty – I,” Bell Syst. Tech. J. 40, pp. 43–63, January 1961. [6] H.L. Van Trees, Detection, Estimation, and Modulation Theory, Part I, Wiley, New York, NY, 1968.

391

This page intentionally left blank

CHAPTER

9

Advanced Topics The motion of an inﬁnitesimal body when it is attracted by two ﬁnite bodies that revolve around their common center of mass is an important conﬁguration for deep space-based platforms such as satellites, space stations, and SBRs. Applications of such a conﬁguration include determining locations to park space stations and Space Based Radars in the Sun-Earth, or the Earth-Moon frame for long-time surveillance over a considerable period. Obviously stability of these artiﬁcial objects is an important goal and in that context preferential stable orbits are of great importance. This situation falls under the problem of three bodies, where a number of important results have been established rigorously under speciﬁc initial conditions on the positions and velocities of the three bodies. For example, the particular solutions of the motion of three ﬁnite bodies such that the ratios of their mutual distances are constants were ﬁrst given by Lagrange in a prize memoir in 1772. The theorem due to Lagrange states that it is possible to start three ﬁnite bodies in such a manner that their orbits will be similar ellipses, all described in the same time [1]. The general three-body problem is insolvable, i.e., given the positions and velocities of three mass points at a certain time that attract each other according to Newton’s second law, it is impossible to predict the future progress of their motion. Thus in general, for such a system it is no longer possible to state with certainty that no member will drift away to inﬁnity (stability of the system), or that collisions between two or more mass points will not occur. However, a special case is of considerable interest—the motion of an inﬁnitesimal body around two ﬁnite bodies, and more deﬁnite statements can be made regarding their motion.

393 Copyright © 2008 by The McGraw-Hill Companies. Click here for terms of use.

394

Space Based Radar

9.1 An Inﬁnitesimal Body Around Two Finite Bodies An inﬁnitesimal body is attracted by two ﬁnite bodies, but it does not attract the ﬁnite bodies. Consider two ﬁnite bodies (such at the Sun and Earth pair, Sun and Jupiter pair, or Earth and Moon pair) revolving in circles1 around their common center of mass with the inﬁnitesimal body subject to their combined attraction. Assume that the unit of mass is chosen so that the sum of the masses of the two ﬁnite bodies is unity, and let µ and 1 − µ represent the two masses with µ ≤ 1/2. Similarly, let the unit of distance be such that the constant distance between the ﬁnite bodies is unity, and the unit of time is adjusted so that the gravitational constant G is unity. Assume the center of mass of the ﬁnite bodies represent the origin of the coordinate system, with the α-β plane representing the plane of the motion of these ﬁnite bodies. Let the coordinates of the ﬁnite bodies 1 − µ and µ be (α1 , β1 , 0) and (α2 , β2 , 0) respectively, and let (α, β, γ ) represent the coordinate of the inﬁnitesimal body. Then following (3.2) and (3.3), the differential equation of motion for the inﬁnitesimal body can be written as d 2α α − α1 α − α2 = −(1 − µ) −µ , 3 dt 2 r1 r23

(9.1)

β − β1 β − β2 d 2β = −(1 − µ) −µ , dt 2 r13 r23

(9.2)

d 2γ γ γ = −(1 − µ) 3 − µ 3 2 dt r1 r2

(9.3)

and

where

(α − α1 ) 2 + (β − β1 ) 2 + γ 2 , r2 = (α − α2 ) 2 + (β − β2 ) 2 + γ 2 .

r1 =

(9.4) (9.5)

From (3.57), the mean angular motion of the ﬁnite bodies is given by √ 2π G( M + m) n= = 1, (9.6) = T a 3/2 since G ≡ 1, M + m = 1, and a = 1. Thus the ﬁnite bodies are rotating in the α-β plane with uniform angular velocity of unity and 1 For more realistic elliptical orbits treatment, refer to F. R. Moulton, “Periodic Orbits”.

Chapter 9:

Advanced Topics Infinitesimal body (x, y )

FIGURE 9.1 Inﬁnitesimal body around two ﬁnite bodies.

y r1 r2

x1 = −m

x2 = 1 − m

m1 = 1 − m

m2 = m

x

let (x, y, z) represent the coordinates of the inﬁnitesimal body in this rotating system with the new axes having the same origin as the old ones. Then the coordinates of the new system are given by (see Figure 9.1) α = x cos t − y sin t,

(9.7)

β = x sin t + y cos t,

(9.8)

γ =z

(9.9)

and αi = xi cos t − yi sin t, βi = xi sin t + yi cos t,

i = 1, 2,

(9.10)

where as Figure 9.1 shows, (xi , yi ), i = 1, 2 refer to the coordinates of the ﬁnite bodies in the rotating system. From (9.7) dα dx dy = cos t − x sin t − sin t − y cos t dt dt dt

(9.11)

so that d 2α d2x dx = cos t − 2 sin t − x cos t dt 2 dt 2 dt −

d2 y dy sin t − 2 cos t + y sin t. 2 dt dt

(9.12)

395

396

Space Based Radar Substituting (9.7), (9.10), and (9.12) into (9.1) we get

d2x dy −2 − x cos t − dt 2 dt

d2 y dx +2 − y sin t dt 2 dt

*

x − x1 x − x2 = − (1 − µ) +µ 3 r1 r23

*

+ (1 − µ)

y − y1 y − y2 +µ 3 r1 r23

+ cos t

+ sin t

(9.13)

where r1 =

(x − x1 ) 2 + ( y − y1 ) 2 + z2

(9.14)

r2 =

(x − x2 ) 2 + ( y − y2 ) 2 + z2 .

(9.15)

and

Similarly using the derivatives of (9.8) and (9.10)–(9.12) in (9.2) we get

dy d2x −2 − x sin t + dt 2 dt

*

x − x1 x − x2 = − (1 − µ) +µ r13 r23

*

− (1 − µ)

dx d2 y +2 − y cos t dt 2 dt

y − y1 y − y2 +µ 3 r1 r23

+ sin t

+ cos t.

(9.16)

Multiply (9.13) and (9.16) by cos t and sin t respectively and add; similarly multiply them by − sin t and cos t respectively and add to obtain d2x dy x − x2 x − x1 −2 −µ − x = −(1 − µ) 3 2 dt dt r1 r23

(9.17)

d2 y dx y − y2 y − y1 +2 −µ . − y = −(1 − µ) 3 2 dt dt r1 r23

(9.18)

and

Finally from (9.9), (9.10), and (9.3) we also obtain d2z z z = −(1 − µ) 3 − µ 3 . dt 2 r1 r2

(9.19)

Chapter 9:

Advanced Topics

If the x-axis is chosen to continuously pass through the centers of the ﬁnite bodies, we have y1 = 0,

y2 = 0,

(9.20)

and we get the simpliﬁed equations of motion d2x dy x − x2 x − x1 −2 −µ , = x − (1 − µ) dt 2 dt r13 r23

(9.21)

d2 y dx y y +2 = y − (1 − µ) 3 − µ 3 , 2 dt dt r1 r2

(9.22)

and d2z z z = −(1 − µ) 3 − µ 3 2 dt r1 r2 where r1 =

(x − x1 ) 2 + y2 ,

r2 =

(x − x2 ) 2 + y2 .

(9.23)

(9.24)

From (9.21)–(9.23), in general the motion of the inﬁnitesimal body is a sixth-order problem, and it becomes fourth order if the body moves in the plane of the motion of the ﬁnite bodies. The only general integral solution to (9.21)–(9.23) was ﬁrst given by Jacobi (1843) by deﬁniting U(x, y, z) =

1 2 1−µ µ + . (x + y2 ) + 2 r1 r2

(9.25)

In terms of U(x, y, z), (9.21)–(9.23) becomes dy ∂U d2x −2 = , 2 dt dt ∂x

(9.26)

dx ∂U d2 y = , +2 2 dt dt ∂y

(9.27)

d2z ∂U = . dt 2 ∂z

(9.28)

dz Following Jacobi, multiply (9.26)–(9.28) by 2 ddtx , 2 dy dt , and 2 dt respectively, and sum those equations to obtain

d dt

dx dt

2

+

dy dt

2

+

dz dt

2 / =2

∂U d x ∂U dy ∂U dz + + ∂ x dt ∂ y dt ∂z dt

=2

dU(x, y, z) dt

(9.29)

397

398

Space Based Radar and upon integrating, we get the desired general integral to be

2 2 2 dx dy dz 2(1 − µ) 2µ + + = 2U − C = x 2 + y2 + + − C. dt dt dt r1 r2 (9.30) But

dx dt

2

+

dy dt

2

+

dz dt

2 = V2

(9.31)

represents the square of the velocity of the inﬁnitesimal body, so that the integral solution due to Jacobi reduces to [1] V 2 = 2U − C = x 2 + y2 +

2(1 − µ) 2µ + − C, r1 r2

(9.32)

and it represents a relation between the velocity and coordinates of the inﬁnitesimal body in the rotating frame. Once the initial conditions determine the constant of integration C, equation (9.32) determines the velocity of the inﬁnitesimal body for all locations, or for a given velocity the corresponding locations of the inﬁnitesimal body are given by (9.32). Interestingly, V = 0 in (9.32) gives the surface of zero velocity, one side of which represents locations where the body velocity is real and consequently the region where the inﬁnitesimal body can move around, and the other side represents prohibited region for the body since the velocity is imaginary (V 2 < 0) there. From (9.32), the equation of surface of zero velocity equals x 2 + y2 +

2(1 − µ) 2µ + =C r1 r2

(9.33)

with r1 and r2 as in (9.14)–(9.15), and (9.20). As z → ∞, (9.33) reduces to x 2 + y2 = C

(9.34) √ which represents a cylinder with radius C and whose axis coincides with the z-axis. Hence all surfaces represented by (9.33) are contained within the above cylinder. These surface are symmetric with respect to the x-y and x-z planes since only the square of y and z appear in (9.33). In the x-y plane, we have z = 0, and the surface curves in the x-y plane are given by x 2 + y2 +

2(1 − µ) (x − x1

)2

+

y2

2µ + = C. (x − x2 ) 2 + y2

(9.35)

Chapter 9:

Advanced Topics

FIGURE 9.2 Equation of surface for large values of x and y.

For large values of x and y, the third and fourth terms on the left side of (9.35) are unimportant, and the equation reduces to x 2 + y2 = C − ε1 (x, y),

(9.36)

where ε1 is small. This is the equation of a circle with radius √ C − ε1 (x, y) and hence this represents an approximately circular oval within the asymptotic cylinder deﬁned in (9.34). For larger values of C, corresponding large values of x and y satisfy (9.36), and hence ε1 gets smaller, and the curves nearly approach the asymptotic circle (see Figure 9.2). For very small values of x and y in (9.35), on the other hand, the ﬁrst two terms are negligible and the equation reduces to 1−µ µ C + = − ε2 (x, y). 2 2 2 2 2 (x − x1 ) + y (x − x2 ) + y

(9.37)

For large values of C, (9.37) represents ovals around the bodies 1−µ and µ (since for x x1 , y 0, the ﬁrst term in (9.37) is dominant which reads (x − x1 ) 2 + y2 C , etc.), and for small values of C, these ovals merge forming dumbbell-shaped ﬁgures as in Figures 9.3 and 9.4. Finally, for even smaller values of C the dumbbell shape enlarges to become an oval that covers both the bodies. These surfaces for various values of C are shown in Figure 9.4. Interestingly, for large values of C, since the ovals around the ﬁnite bodies are closed and if the velocity is positive inside, it will be positive everywhere inside those ovals. As a result, an inﬁnitesimal body

399

Space Based Radar 100

0.4

y

400

0

50

−0.4 10 −0.5

0 x

0.5

FIGURE 9.3 Equation of surface for very small values of x and y.

trapped inside one of these ovals in Figure 9.3 will always remain there since it could not cross a surface of zero velocity. Motion of the Moon in the Sun-Earth frame corresponds to this situation and it was in this manner that Hill proved that the Moon cannot recede indeﬁnitely from Earth and its distance from Earth has a ﬁnite upper limit.

FIGURE 9.4 Equation of surface for small values of x and y.

Chapter 9:

Advanced Topics

9.1.1 Particular Solutions of the Three-Body Problem If the inﬁnitesimal body is placed on a surface of zero velocity, depending on the value of the acceleration terms in (9.21)–(9.23), the body will move inward into the permissible region unless the acceleration terms themselves are zeros. In that case, the inﬁnitesimal body with zero velocity will remain forever relatively at rest, unless distorted by some external force. To examine these particular solutions, where the inﬁnitesimal body in a three body system stays relatively at rest forever, we can equate both the velocity and acceleration terms to zero in (9.21)–(9.23). From (9.26)–(9.28), this is also equivalent to setting the derivative of the Jacobi function in (9.25) to zero, i.e., ∂U = 0, ∂x

∂U = 0, ∂y

∂U = 0. ∂x

(9.38)

This gives (use (9.25)) x − x2 x − x1 −µ = 0, r13 r23 y y y − (1 − µ) 3 − µ 3 = 0, r1 r2 z z (1 − µ) 3 + µ 3 = 0. r1 r2

x − (1 − µ)

(9.39) (9.40) (9.41)

Clearly, (9.41) is satisﬁed only for z = 0 which shows that all these particular solutions lie in the x-y plane. They are special cases of the Lagrangian solutions known as the Lagrange libration points, where the particle stays at rest forever. To determine these particular solutions, we examine (9.40). From (9.40), y = 0 is a solution and together with z = 0 when substituted into (9.39) gives ψ(x) = x − (1 − µ)

x − x1 x − x2 −µ = 0. 3 |x − x1 | |x − x2 |3

(9.42)

As Figure 9.5 shows, (9.42) is positive for x = +∞, negative at x = x2 + ε, ε > 0, positive at x = x2 − ε; it is negative at x = x1 + ε, positive for x = x1 − ε, and negative at x = −∞. As a result, ψ(x) crosses the x-axis at these distinct points—once between x1 and x2 at L 1 , once between x2 and +∞ at L 2 , once between −∞ and x1 at L 3 . These three Lagrange points are all on the x-axis, and the inﬁnitesimal body under proper initial conditions will remain stationary at these locations.

401

402

Space Based Radar y (x)

L3

x1 m1 = 1 − m

L1

x2

L2

x

m2 = m

FIGURE 9.5 Three co linear Lagrange points on the x-axis.

Since x1 = −µ, x2 = 1 − µ, the three real Lagrange solutions L 1 , L 2 , and L 3 of (9.42) can be expressed in terms of µ. To obtain explicit expressions for them, consider the solution L 1 located between x1 and x2 , and let ρ represent the actual distance of L 1 from the second mass m2 located at x2 . Thus x2 − x = ρ, or x − x2 = −ρ,

r1 = x − x1 = ρ,

x =1−µ−ρ

(9.43)

and substituting these into (9.42) and simplifying we obtain f (ρ) = ρ 5 − (3 − µ)ρ 4 + (3 − 2µ)ρ 3 − µρ 2 + 2µρ − µ = 0.

(9.44)

Equation (9.44) has ﬁve sign variations among its coefﬁcients, and hence applying Descartes’ theorem2 it has at least one real positive root. Further, for µ = 0, the above equation reduces to ρ 3 (ρ 2 − 3ρ + 3) = 0

(9.45)

which has three roots at ρ = 0 and two complex roots. Hence for sufﬁciently small values of µ, the three roots of (9.44) are expressible as powers of µ1/3 , one of which is real and positive and the other two are complex conjugate pairs. Thus the only real and positive root of 2 Descartes’ Theorem: If ν represents the sign variations among the coefﬁcients of a polynomial with real coefﬁcients, and p the number of its positive real roots, then ν = p + k, where k is an even number.

Chapter 9:

Advanced Topics

(9.44) has the form ρ = a µ1/3 + bµ2/3 + · · ·

(9.46)

Substituting this into (9.44) and equating the coefﬁcients of the lowest powers of µ in the expansion to zero, we obtain (3a 3 − 1)µ = 0,

(−3a 3 + 2 + 9a b)a µ4/3 = 0

(9.47)

which gives

1/3 1 a= ; 3

1 1 =− b=− 9a 3

2/3 1 . 3

(9.48)

Hence (9.46) yields [1] ρ

# µ $1/3 3

−

1 # µ $2/3 + ··· 3 3

(9.49)

to be the approximate distance of L 1 from the second mass located at x2 . Using a similar approach, it is easy to show that the second solution L 2 is located approximately at a distance of [1] ρ

# µ $1/3 3

+

1 # µ $2/3 + ··· 3 3

(9.50)

from the second mass at x2 in the other direction (toward inﬁnity). Similarly, the distance of the L 3 solution from the ﬁrst mass at x1 can be shown to be [1] ρ 1−

7 µ + ···. 12

(9.51)

For the Sun-Earth frame, since the unit of distance (Sun to Earth) equals a = 149, 597, 890 km, and µ = 1/333, 000, we get the location of the Lagrange points L 1 and L 2 to be (ﬁrst-order approximation) ρa

# µ $1/3 3

1, 496, 477 km 1.5 million km

(9.52)

from Earth on either side along the Sun-Earth line. Clearly, the Moon located at around 384,403 km from Earth is well within the closed ovals around the Earth and hence as Hill has shown the Moon cannot recede beyond a certain limiting distance from Earth [2] (see Figures 9.3 and 9.4). For the Sun-Earth conﬁguration, these linear solutions are of special signiﬁcance to NASA and several missions have been planned around them. For example, the Solar Heliospheric Observatory (SOHO) and the Advanced Composition Explorer (ACE) are currently positioned at the Sun-Earth L 1 point that is ideally suited to make observations

403

404

Space Based Radar

L2 L3

Sun

L1

Earth

Moon

Visible gegenschein

Earth’s orbit

FIGURE 9.6 The Gegenschein phenomenon.

about the Sun since these satellites are never shadowed by either the Earth or the Moon. The Sun-Earth L 2 point is a good location for Space Based Radars since an object at L 2 will maintain the same orientation with respect to the Sun-Earth system and hence, calibration and shielding are much simpler in this case. The Wilkinson Microwave Anisotropy probe is already at the Sun-Earth L 2 location, and NASA plans to launch the future Herschel space observatory and the next generation James Webb space telescope to the Sun-Earth L 2 location. The Earth-Moon L 2 location can be used for a communication satellite that covers the far side of Moon. Interestingly, the linear solution L 2 in the Sun-Earth system has been pointed out to explain the hazy patch of light observed at the side of Earth away from Sun, around the ecliptic, above the horizon known as the Gegenschein (German for “counter-glow”), that was independently discovered by astronomers Brorsen (1855), Backhouse (1868), and Barnard (1875). It is a faint glow of light stretching about 10◦ on the opposite side of the Sun, and hence it rises when the Sun sets reaching its peak at midnight (Figure 9.6). According to a suggestion made by astronomer Gylden, meteors under the right initial conditions that are passing near the Sun-Earth L 2 point might get trapped around that point for a while, and if a very large number of such meteors are trapped in this manner around L 2 , their collective glow originating from back-scattered sunlight would appear from Earth as a hazy patch of light with its center approximately at the anti-Sun location (very near to L 2 ) similar to the observed Gegenschein phenomenon. No certain answers can be given to this explanation, although the

Chapter 9:

Advanced Topics

low brightness glow area at the anti-Sun location above the horizon is unmistakable on a dark night. It is quite possible that there is such a collection at the Sun-Earth L 3 point as well. However, its veriﬁcation is difﬁcult from Earth since L 3 is behind the Sun on the other side away from the Earth. To determine the location of other Lagrange points in the x-y plane that are not along the x-axis, we return to (9.40) and assume y = 0. In that case, (9.40) reduces to 1−

1−µ µ − 3 = 0. 3 r1 r2

(9.53)

Upon multiplying (9.53) with (x − x2 ) and (x − x1 ) respectively and subtracting them from (9.39) we obtain x2 − (1 − µ)

x2 − x1 =0 r13

(9.54)

and x1 − µ

x1 − x2 = 0. r13

(9.55)

Since x1 = −µ, x2 = 1 − µ, x2 − x1 = 1, from (9.54)–(9.55) we get 1−

1 = 0 ⇒ r1 = 1 r13

(9.56)

1−

1 = 0 ⇒ r2 = 1. r23

(9.57)

and

From (9.56) and (9.57), the only other real solutions to (9.39)–(9.41) are r1 = 1, r2 = 1, and these two Lagrange points L 4 and L 5 form equilateral triangles with the ﬁnite bodies (see Figure 9.7). Once again, under the right initial conditions, the inﬁnitesimal body at these locations will remain trapped there forever, provided they represent stable solutions.

9.1.2 Stability of the Particular Solutions There are three particular solutions of the motion of the inﬁnitesimal body along the x-axis (L 1 , L 2 , and L 3 ) and two off the x-axis in the x-y plane (L 4 and L 5 ) where it can remain relatively at rest in the absence of external forces (Figure 9.7). Known as the Lagrange libration points, these solutions are potential places to build Space Based Radars and space stations.

405

Space Based Radar y (x ) L4

=1

=1

r2

r1

406

L3

x1

L1

m1 = 1 − m

x2

L2

m2 = m

x

L5

FIGURE 9.7 The two Lagrange points L4 and L5 that form equilateral triangles with the ﬁnite bodies.

An important question in this context is the stability of these particular solutions that correspond to (9.21)–(9.23) with the velocity and acceleration terms there equal to zero. Hence, with x − x1 x − x2 −µ , r13 r23 y y g(x, y, z) = y − (1 − µ) 3 − µ 3 , r1 r2

f (x, y, z) = x − (1 − µ)

(9.58) (9.59)

and h(x, y, z) = −(1 − µ)

z z −µ 3, 3 r1 r2

(9.60)

if (xo , yo , zo ) represents a particular solution then from (9.39)–(9.41), we have f (xo , yo , zo ) = g(xo , yo , zo ) = h(xo , yo , zo ) = 0.

(9.61)

The important question in this context is the stability of the particular solutions L 1 through L 5 mentioned earlier. If the inﬁnitesimal body is located at a stable solution, then a slight displacement from the exact solution along with a small velocity and acceleration will only cause the small body to oscillate around the stable point for a considerable amount of time; however, if the solution

Chapter 9:

Advanced Topics

is unstable, the small body will eventually depart from the exact solution forever and drift away. To examine the nature of stability of these particular solutions, the small body is given a small displacement and a small velocity so that its coordinates and velocities are [1] x = xo + u,

y = yo + v,

z = zo + w

(9.62)

and du dx = , dt dt

dy dv = , dt dt

d2x d 2u = , dt 2 dt 2

dz dw = ; dt dt

d2 y d 2v = , dt 2 dt 2

d2z d 2w = . dt 2 dt 2

(9.63)

Thus (9.21)–(9.23) can be rewritten as d 2u dv −2 = f (xo + u, yo + v, zo + w), dt 2 dt

(9.64)

d 2v du +2 = g(xo + u, yo + v, zo + w), dt 2 dt

(9.65)

d 2w = h(xo + u, yo + v, zo + w). dt 2

(9.66)

and

Using Taylor’s formula f (xo + u, yo + v, zo + w)

∂ f ∂ f ∂ f u+ v+ w + ···, = f (xo , yo , zo ) + ∂x ∂y ∂z xo, yo ,zo

g(xo + u, yo + v, zo + w)

xo, yo ,zo

xo, yo ,zo

∂g ∂g ∂g = g(xo , yo , zo ) + u+ v+ w + ···, ∂x ∂y ∂z xo, yo ,zo

(9.67)

xo, yo ,zo

xo, yo ,zo

(9.68)

and h(xo + u, yo + v, zo + w) = h(xo , yo , zo ) +

∂h ∂h ∂h u+ v+ w + ··· ∂x ∂y ∂z xo, yo ,zo

xo, yo ,zo

(9.69)

xo, yo ,zo

If the disturbances u, v, w are very small at least initially, the second and higher order terms may be neglected, in which case (9.67)–(9.69) represent a linearized model. Together with (9.61), (9.64)–(9.66) now

407

408

Space Based Radar reduce to dv d 2u ∂f ∂f ∂f −2 = u+ v+ w, dt 2 dt ∂x ∂y ∂z

(9.70)

d 2v du ∂g ∂g ∂g +2 = u+ v+ w, 2 dt dt ∂x ∂y ∂z

(9.71)

and ∂h d 2w ∂h ∂h = u+ v+ w, (9.72) 2 dt ∂x ∂y ∂z where the partial derivatives are evaluated at x = xo , y = yo , z = zo as in (9.67)–(9.69). With f (x, y, z), g(x, y, z), and h(x, y, z) as deﬁned in (9.58)–(9.60) and r1 , r2 as in (9.14) and (9.15) with y1 = 0, y2 = 0, we obtain their partial derivatives to be

r 3 − (x − x1 ) 32 r1 2(x − x1 ) ∂f = 1 − (1 − µ) 1 ∂x r16

3 2 r2 2(x − x2 ) , r26 (x − x1 ) − 32 2y (x − x2 ) − 32 2y ∂f −µ , = −(1 − µ) ∂y r15 r25 −µ

r23 − (x − x2 )

and

(9.74)

(x − x2 ) − 32 2z (x − x1 ) − 32 2z ∂f − µ . = −(1 − µ) ∂z r15 r25 Similarly, we obtain

(9.73)

(9.75)

y − 32 2(x − x2 ) y − 32 2(x − x1 ) ∂g − µ , (9.76) = −(1 − µ) ∂x r15 r25

r23 − y 32 r2 2y r 3 − y 32 r1 2y ∂g − µ , = 1 − (1 − µ) 1 ∂y r16 r26

(9.77)

y − 32 2z y − 32 2z ∂g − µ , = −(1 − µ) ∂z r15 r25

(9.78)

z − 32 2(x − x2 ) z − 32 2(x − x1 ) ∂h − µ , = −(1 − µ) ∂x r15 r25

(9.79)

y − 32 2y z − 32 2y ∂h − µ , = −(1 − µ) ∂y r15 r25

(9.80)

r23 − z 32 r2 2z r 3 − z 32 r1 2z ∂h − µ . = −(1 − µ) 1 ∂z r16 r26

(9.81)

Chapter 9:

Advanced Topics

These partial derivatives (9.73)–(9.81) can be evaluated for each particular solution and substituted into (9.70)–(9.72) to determine the nature of their stability.

9.1.3 Stability of Linear Solutions In the case of the linear solutions, L 1 , L 2 , and L 3 in Figures 9.5–9.7, let (xo , yo , zo ) represent a particular solution. Then yo = zo = 0

(9.82)

and xo > 0 for L 1 and L 2 and xo < 0 for L 3 . Also r1 = |xo − x1 | ,

r2 = |xo − x2 |

so that from (9.73)–(9.75)

∂ f = 1− ∂x

xo, yo ,zo

1−µ µ + 3 3 r1 r2

= 1+2

+3

1−µ µ + 3 r13 r2

1−µ µ + 3 3 r1 r2

(9.83)

= 1 + 2α

(9.84)

where we deﬁne [3] α= and

µ 1−µ + 3 >0 r13 r2

∂ f = 0, ∂y xo, yo ,zo

(9.85)

∂ f = 0. ∂z xo, yo ,zo

Similarly, from (9.76)–(9.81)

∂g = 0, ∂x xo, yo ,zo

∂g = 0, ∂z

∂h = 0, ∂x xo, yo ,zo

and

∂h =− ∂z xo, yo ,zo

(9.87)

xo, yo ,zo

µ 1−µ ∂g =1− + 3 = 1 − α, ∂y r13 r2

xo, yo ,zo

(9.86)

(9.88)

∂h = 0, ∂y

(9.89)

xo, yo ,zo

1−µ µ + 3 3 r1 r2

= −α.

(9.90)

409

410

Space Based Radar Substituting (9.84)–(9.90) in (9.70)–(9.72) we obtain d 2 u(t) dv(t) −2 = (1 + 2α)u(t), 2 dt dt

(9.91)

d 2 v(t) du(t) +2 = (1 − α)v(t), 2 dt dt

(9.92)

and d 2 w(t) = −αw(t). (9.93) dt 2 Equations (9.91)–(9.92) are coupled, whereas (9.93) is independent of (9.91) and (9.92). Hence (9.93) gives the stable solution √ √ (9.94) w(t) = a cos( α t) + b sin( α t) >√ that is periodic with period 2π α. From (9.83) and (9.85), the period is different for the three linear solutions. To examine the solutions of the coupled set of linear equations (9.91) and (9.92), consider the Laplace transforms

5∞

u(t)e −st dt

(9.95)

v(t)e −st dt.

(9.96)

U(s) = 0

and

5∞ V(s) = 0

Taking Laplace transforms of (9.91) and (9.92), we obtain the coupled set of equations

%

s 2 − (1 + 2α)

% =

A(s)

su(0) + u (0) − 2v(0) sv(0) + v (0) + 2u(0)

whose solution is given by

%

U(s) V(s)

&

= A−1 (s) = =

&%

%

1 det A(s) 1 det A(s)

U(s)

&

V(s) s 2 − (1 − α)

2s

−2s

a 1 (s)

&

%

=

a 1 (s)

%

(9.97)

a 2 (s)

&

a 2 (s)

%

&

s 2 − (1 − α)

2s

−2s

s 2 − (1 + 2α)

x1 (s) x2 (s)

&

,

&%

a 1 (s)

&

a 2 (s) (9.98)

Chapter 9:

Advanced Topics

where x1 (s) and x2 (s) are two polynomials of degree three at most. Hence x1 (s) , a (s) x2 (s) V(s) = a (s)

U(s) =

(9.99) (9.100)

where the denominator a (s) = det A(s) = (s 2 − (1 − α))(s 2 − (1 + 2α)) + 4s 2 = s 4 + (2 − α)s 2 + (1 + 2α)(1 − α)

(9.101)

is a biquadratic polynomial of degree four. Hence if s1 , s2 , s3 , and s4 represent the four roots of the determinantal polynomial in (9.101), then from (9.99)–(9.100) their partial fraction expansion followed by inverse Laplace transform gives u(t) =

4

a i e −si t ,

(9.102)

b i e −si t ,

(9.103)

i=1

and v(t) =

4 i=1

where a i , b i are constants that depend on the initial disturbances in (9.97). The locations of these roots si , i = 1, 2, 3, 4 determine the nature of the stability of the above solutions. Let λ = s2

(9.104)

in (9.101) so that it reduces to λ2 + (2 − α)λ + (1 + 2α)(1 − α) = (λ − λ1 )(λ − λ2 ) and in terms of its roots λ1 and λ2 we have s1 =

λ1 ,

s2 = −

λ1 ,

s3 =

λ2 ,

s4 = −

λ2 .

(9.105)

(9.106)

Clearly for (9.102) and (9.103) to represent stable solutions, both λ1 and λ2 must be negative. In that case (9.106) represents purely imaginary solutions and (9.103) and (9.104) represent stable periodic solutions. If λ1 and λ2 are both negative, then the product λ1 λ2 > 0 and from (9.105) this leads to the condition (1 + 2α)(1 − α) must be positive, or we must have 1−α >0 since α in (9.85) is positive.

(9.107)

411

412

Space Based Radar To determine whether (9.107) is satisﬁed at the desired solutions L 1 , L 2 , and L 3 , from (9.42) with (xo , 0, 0) representing any one such solution, we obtain (use (9.85)) ψ(xo ) = xo − (1 − µ)

xo − x1 xo − x2 −µ 3 |xo − x1 | |xo − x2 |3

= xo (1 − α) − µ(1 − µ)

1 1 − 3 r13 r2

since x1 = −µ, x2 = 1 − µ. Hence we obtain 1−α =

µ(1 − µ) xo

= 0,

(9.108)

1 1 − 3 . r13 r2

(9.109)

For L 1 and L 2 , we have xo > 0 and r1 > r2 so that 1 − α < 0. For L 3 , xo < 0, and r1 < r2 so that 1 − α < 0, and hence all three straight line solutions L 1 , L 2 , and L 3 are unstable! As a result, inﬁnitesimal bodies at these solutions under small perturbations eventually will drift away to great distances.

9.1.4 Stability of Equilateral Solutions In the case of the equilateral solutions L 4 and L 5 , we have r1 = r2 = 1

(9.110)

and from Figure 9.7 and 9.8 for L 4 and L 5 √ 1 3 xo = − µ, yo = ± , zo = 0, 2 2 so that 1 1 1 xo − x1 = , xo − x2 = − µ − (1 − µ) = − . 2 2 2 Substituting (9.110)–(9.112) into (9.73)–(9.75), we obtain

(9.111)

(9.112)

∂ f 1 3 1 3 1 1 = 1 − (1 − µ) 1 − · · 2 · −µ 1− − · ·2· − ∂x 2 2 2 2 2 2

xo, yo ,zo

= 1−

(1 − µ) µ 3 − = , 4 4 4

(9.113) √ √ 3 3 ∂ f 1 3 1 3 = −(1 − µ) − · 2 · − · − · 2 · · − µ ∂y 2 2 2 2 2 2 xo, yo ,zo

=

√ 3 3 (1 − 2µ), 4

∂ f = 0. ∂z xo, yo ,zo

(9.114) (9.115)

Chapter 9:

Advanced Topics

y

FIGURE 9.8 Stability of Lagrange solutions L4 and L5 .

L4

(xo,yo,zo)

√3/2 1/2 x1 = −m

x2 = 1 − m

0

x

L5

Similarly (9.76)–(9.78) give √ √ ∂g 3 3 1 3 1 3 = −(1 − µ) − · 2 · − µ − · 2 · − · · ∂x 2 2 2 2 2 2 xo, yo ,zo

√ 3 3 = (1 − 2µ), 4

(9.116) √ √ √ √ 3 3 3 3 3 3 ∂g = 1 − (1 − µ) 1 − −µ 1− · ·2· · ·2· ∂y 2 2 2 2 2 2 xo, yo ,zo

9 = 1 − (1 − µ + µ) 1 − 4

=

9 4

∂g = 0, ∂z

(9.117) (9.118)

xo, yo ,zo

and ﬁnally (9.79)–(9.81) gives

∂h = 0, ∂x xo, yo ,zo

∂h = 0, ∂y xo, yo ,zo

∂h = −(1 − µ) − µ = −1. ∂z

(9.119)

xo, yo ,zo

Using (9.113)–(9.115) in (9.70) we obtain √ d 2 u(t) dv(t) 3 3 3 −2 = u(t) + (1 − 2µ)v(t). dt 2 dt 4 4

(9.120)

413

414

Space Based Radar Similarly (9.116)–(9.118) in (9.71) gives √ d 2 v(t) du(t) 3 3 9 +2 = (1 − 2µ)u(t) + v(t) dt 2 dt 4 4 and (9.119) in (9.72) gives

(9.121)

d 2 w(t) = −w(t). (9.122) dt 2 As before equations (9.120) and (9.121) are coupled, whereas (9.122) is independent of (9.120) and (9.121), and it gives the stable solution w(t) = a cos t + b sin t

(9.123)

that is periodic with period 2π, which is the same as the revolution of the two ﬁnite bodies. To examine the solution of the coupled set of linear equations (9.120) and (9.121), as before taking Laplace transforms of (9.120) and (9.121), we obtain the set of equations √   3 3 3 ' ( 2 s − 2s + − (1 − 2µ)   U(s) 4 4   √   V(s) 3 3 9 2 2s − (1 − 2µ) s − 4 4

' =

B(s) su(0) + u (0) − 2v(0) sv(0) + v (0) + 2u(0)

(

'

=

b 1 (s) b 2 (s)

( (9.124)

whose solution is given by

%

U(s) V(s)

&

=B

−1

%

(s)

b 1 (s)

&

b 2 (s)

=

√  9 3 3 & % − 2s + (1 − 2µ)   b 1 (s) 1 4 4   √  b 2 (s) det B(s)  3 3 3 s2 − (1 − 2µ) − 2s − 4 4



s2

=

1 det B(s)

%

c 1 (s) c 2 (s)

&

,

(9.125)

where c 1 (s) and c 2 (s) are two polynomials of degree three at most. Hence c 1 (s) U(s) = , (9.126) b(s) c 2 (s) V(s) = (9.127) b(s)

Chapter 9:

Advanced Topics

where the denominator

√ 2 9 3 3 3 2 2 2 b(s) = det B(s) = s − s − + (2s) − (1 − 2µ) 4 4 4 = s4 + s2 +

27 µ(1 − µ) 4

(9.128)

is a biquadratic polynomial of degree four. As before if s1 , s2 , s3 , s4 represent the four roots of the determinantal polynomial in (9.128), then from (9.126) and (9.127), after partial fraction expansion their inverse transform gives u(t) =

4

c i e −si t

(9.129)

di e −si t

(9.130)

i=1

and v(t) =

4 i=1

where c i , di are constants that depend on the initial disturbances in (9.124). The locations of the roots s1 , s2 , s3 , s4 clearly determine the nature of the stability of the above disturbances (solutions). Equation (9.128) represents a biquadratic and let λ = s2

(9.131)

in (9.128), so that it reduces to λ2 + λ +

27 µ(1 − µ) = (λ − λ1 )(λ − λ2 ) 4

(9.132)

whose roots are given by λ1,2 = From (9.131), clearly s1 =

λ1 ,

s2 = −

−1 ±

λ1 ,

√ 1 − 27µ(1 − µ) . 2

s3 =

λ2 ,

(9.133)

s4 = −

λ2

(9.134)

represent the four roots in (9.129)–(9.130), and hence for (9.129) and (9.130) to represent stable solutions both λ1 and λ2 in (9.133) must be negative. In that case, si , i = 1, 2, 3, 4 represent purely imaginary solutions and (9.129) and (9.130) represent stable periodic disturbances. From (9.133), the condition for periodic stability is that the discriminant there be positive, i.e., 1 − 27µ(1 − µ) > 0,

(9.135)

415

416

Space Based Radar or µ2 − µ + But µ2 − µ +

1 27

1 > 0. 27

(9.136)

= 0 gives the solution

µ1,2

1 1 = ± 2 2

and since µ < 12 , we obtain 1 µ2 = − 2

4 1 1− = ± 27 2

23 108

23 = 0.03852 108

(9.137)

(9.138)

as the only solution to the above quadratic. Thus (9.136) is satisﬁed if µ < µ2 = 0.03852.

(9.139)

For any such µ, from (9.131)–(9.134), the corresponding normalized periodic frequencies are given by

ω1 =

1+

√ 1 − 27µ(1 − µ) , 2

ω2 =

1−

√ 1 − 27µ(1 − µ) 2 (9.140)

so that the respective orbital periods equal 1 1 , and T2 = . (9.141) ω1 ω2 In summary, the Lagrange libration points L 4 and L 5 represent (linear) stable solutions, and an inﬁnitesimal body situated at those locations under small disturbances will continue to evolve around those points in stable orbits that are a combination of the periodic orbits with both periods given by (9.141), provided the mass of one of the ﬁnite bodies is less than 0.03852 of the mass of their sum. Interestingly, this condition is satisﬁed by the Sun-Jupiter, the SunEarth combination as well as Earth-Moon combination. The mass of the Jupiter being only 1/1,000 of that of the Sun, condition (9.138) is actually satisﬁed by the equilateral solution on the Jupiter orbit. Three Trojan asteroids Achilles (1904), Agamemon, and Hector have been found at the L 4 and L 5 locations on the Jupiter’s orbit around the Sun. There are several thousands of asteroids at these Trojan locations as well. With µ = 1/1,000, their normalized (Jupiter years) orbital periods are T1 =

T1 = 1.0034,

and

T2 = 12.1363

(9.142)

that correspond to 11.9 years and 144 years respectively. Although a similar asteroid Eureka was found in 1990 in the orbit of Mars, no

Chapter 9:

Advanced Topics

such Trojan objects have been found either in the Sun-Earth system or in Earth-Moon system. Interestingly, large concentrations of dust glowing fainter than the Gegenschein has been reported in the L 4 and L 5 points in Earth-Moon system by Kordylewski (1960), and there is still controversy as to its existence due to the extreme faintness. For Earth-Moon system, µ = 0.0123 which is less than µ2 = 0.03852 so that the L 4 and L 5 solutions in Earth-Moon system are also stable. With µ = 0.0123 in (9.141), we obtain the normalized orbital periods to be T1 = 1.04835

and

T2 = 3.33099.

(9.143)

Thus space stations and Space Based Radars located at these L 4 and L 5 points will ﬂoat around those points along paths that are combinations of periodic solutions with periods 1 and 3.33 sidereal months respectively. Interestingly, the L 4 and L 5 locations both in the SunEarth system as well as Earth-Moon system are possible candidates for future permanent space station colonies. It should be possible to have any number of inﬁnitesimal bodies revolving around the same point without interfering with each other.

Appendix 9-A: Hill Sphere The Hill sphere refers to the gravitational sphere of inﬂuence of one astronomical body such as Earth in presence of perturbations from another heavier body (Sun) around which its orbits. In the restricted three-body problem containing a heavier mass and a lighter mass, the Hill sphere refers to the region around the lighter mass located at x2 within which the total gravitational inﬂuence on an inﬁnitesimal mass will be directed toward the lighter mass. Let r H represent the radius of the Hill sphere. Then according to Hill [2]

rH = a

µ 3(1 − µ)

1/3 ,

(9A.1)

where a represents the distance between the two dominant masses and µ the ratio of the lighter mass to the heavier mass. To derive (9A.1), we can rewrite the Jacobi function in (9.25) in terms of r1 and r2 as

1 2 U = (1 − µ) r12 + 2 r1

1 2 + µ r22 + 2 r2

−

µ(1 − µ) . 2

(9A.2)

Here r1 and r2 refer to the distance of the inﬁnitesimal body from the heavier and lighter masses respectively. Equation (9A.2) follows

417

418

Space Based Radar

r1

L3

r2

L1 A

1−m

(x,y)

m

L2

x

rH

FIGURE 9.9 Hill Sphere.

by noticing that r12 = (x − x1 ) 2 + y2 = x 2 + y2 + 2µx + µ2 , r22

= (x − x2

)2

+

y2

=

x2

+

y2

− 2(1 − µ)x + (1 −

(9A.3) µ) 2 ,

(9A.4)

so that (1 − µ)r12 + µr22 = x 2 + y2 + µ(1 − µ).

(9A.5)

Since the inﬂuence of the second mass is smallest along the x-axis, it acts as the limiting factor for the size of the Hill sphere. Thus, when the inﬁnitesimal body is at location A as in Figure 9.9, we have in particular r1 + r2 = 1 and the equilibrium condition ∂U ∂U = . ∂r1 ∂r2

(9A.6)

From (9A.2) ∂U = (1 − µ)(r1 − r1−2 ) ∂r1 ∂U = µ(r2 − r2−2 ) ∂r2 so that (9A.6) gives [4]

(9A.7) (9A.8)

r22 1 − r13 r22 3r2 − 3r22 + r23 r1 − r1−2 µ = = = 1−µ 1 − r23 r12 1 − r23 (1 − r2 ) 2 r2 − r2−2

=

3r23

3

1 1 − r2 + r22 1 + r23 + · · · 3

× 1 + 2r2 − r22 + 2r2 − r22

2

+ ···

4

Chapter 9:

Advanced Topics

1 = 3r23 1 − r2 + r22 1 + r23 + · · · 1 + 2r2 + 3r22 − 4r23 + · · · 3

3r23

=

4 1 + r2 + r22 + · · · . 3

(9A.9)

Deﬁne µ = 3α 3 . 1−µ

(9A.10)

From (9A.9) we get

α = r2

4 1 + r2 + r22 + · · · 3

1/3

/ 11 2 4 2 4 2 3 3 −1 = r2 r2 + r2 + · · · + r2 + r2 + · · · + · · · 3 2 3 1 1 = r2 1 + r2 + r22 + · · · , (9A.11) 3 3 1 1+ 3

or r2 =

α 1+

1 3 r2

+ 13 r22 + · · ·

1 1 1 1 1 − r2 1 + r2 + · · · + r22 1 + r2 + · · · 3 3 9 3

=α

*

1 1 = α 1 − α 1 − α + ··· 3 3

1 1 + α2 1 + α + · · · 9 3

1 1 − α + ··· 3

2

+ ···

/

+ ···

1 1 2 = α 1 − α − α2 + · · · + α2 + · · · 3 9 9 =α−

/

2

α2 α3 − + ··· 3 9

(9A.12)

where (use (9A.10))

α=

µ 3(1 − µ)

1/3 .

(9A.13)

419

420

Space Based Radar But in this limiting case r2 = r H . Thus to a ﬁrst-order approximation from (9A.12) and (9A.13) the normalized Hill radius equals

r2 = r H = α =

µ 3(1 − µ)

1/3 .

(9A.14)

For the Sun-Earth system (1 − µ 1) so that the Hill radius r H in (9A.14) is the same as the distance to the Lagrange solutions L 1 and L 2 from Earth. Beyond the Hill sphere the inﬁnitesimal body will be more and more inﬂuenced by the larger mass and would eventually end up orbiting the larger mass.

References [1] F.R. Moulton, An Introduction to Celestial Mechanics, The Macmillan Co, New York, NY, 1964. [2] G.W. Hill, “Researches in the Lunar Theory”, The Collected Mathematical Works, Memoir No. 32, Vol. I, pp. 284–335, Carnegie Institution of Washington, June, 1905. [3] J.M.A. Danby, Fundamentals of Celestial Mechanics, The Macmillan Co, New York, NY, 1964. [4] H.C. Plummer, An Introductory Treatise on Dynamical Astronomy, Dover Publications, New York, NY, 1960.

Index 2D beamformer, 224, 225 2D (two-dimensional) motion, 55 2D-DFT (two-dimensional discrete Fourier transform), 205

A adaptive beamforming, 201, 287 adaptive clutter cancellation, 205, 208 adaptive processor output, 152, 176, 179, 201 additive white noise case, 364–366 Airborne Moving Target Indicator (AMTI), 341–342 airborne systems Cramer-Rao bounds, 320–331 ground coverage, 2–3 noise only case bounds, 330 side-looking, 155–165 target Doppler/power, 320–338 vs. space-based systems, 3 all-pass function, 299, 366–368 alternating projections, 198–201 AMTI (Airborne Moving Target Indicator), 341–342 angle bins, 206–208 angle-Doppler dependency, 224 angle-Doppler domain, 230, 290 angle-Doppler pattern, 204 angle-Doppler performance, 163, 164 angle-Doppler proﬁle, 157, 225, 227 angle-Doppler space, 157–163

ARMA systems Billingsley model, 264, 267 rational system representation, 303–307 wind speeds, 263–264, 267 array processing, 139–153 array tapering, 187–194 arrays. See also subarrays beamforming and, 143–150 centro-symmetric uniform linear array, 330 direction-dependent weights, 146 gain patterns, 215, 220 mainbeam width, 144–145 noise and, 146–152 N-sensor, 216–218 overall amplitude patterns, 219–220 phase shifted, 142–143 sensors, 140–148 separable, 220 sidelobe patterns, 143–145 SMI method, 152–153 SNR and, 142–143 subdividing into subarrays, 173–187 uniform. See uniform arrays uses for, 140–148 weight vector, 139, 148–153 weights, 229–231 asteroids. See also comets; meteors impact on Earth, 43, 45 number of, 42, 44

421 Copyright © 2008 by The McGraw-Hill Companies. Click here for terms of use.

422

Space Based Radar asteroids. (Cont.) overview, 42–43 tracking, 7 Trojan, 416–417 asymptotes, 41, 399 attraction, central force of, 1 azimuth angles, 100, 111–113, 216–219 azimuth domain, 230, 290 azimuth return, 216 azimuth transmit pattern, 229, 231

B backscatter amplitude return, 256 backward subarrays, 179–182 baseband carrier, 365 baseband region, 386 baseband signal, 363 beamforming. See also mainbeam 2D, 158 adaptive, 201, 287 arrays and, 143–150 clutter power, 161 coherent, 6 gain pattern and, 143–144 output power, 160 overview, 143 uniform arrays and, 143–148 Billingsley’s wind model, 256, 261, 263–267, 272 bistatic constants, 8 bistatic mode, 341 bistatic SBRs, 6–7 Boltzmann’s constant, 8 Brennan’s rule Earth rotation and, 221–222, 244–245 overview, 166–167

C Canadian Space Agency (CSA), 4 Cartesian coordinate system ellipses, 33–36 hyperbolas, 41 parabolas, 39 Cassini orbiter, 5

Cassini radar, 5 Cassini-Huygens radar, 5 causal ﬁlters, 352, 357 causal receivers, 355, 365, 371 causal waveforms, 355–356, 389 center of mass, 52–54 centripetal force, 98 centro-symmetric uniform linear array, 330 channel kernel, 370–371, 376–380, 383 chirp compression, 358–364 chirp signal, 364, 385–391 chirp waveforms, 91, 251–255, 281–283, 294 chirped rectangle symmetric pulse, 385–387 circles, 32 circular orbit velocity, 65 closed orbits, 59, 60 clutter. See also interference corresponding to range bins, 218 Doppler frequency and, 156–162 Earth rotation and, 234–246 mainbeam, 218–223 range ambiguities and, 218–219 sidelobe, 218–223 waveform diversity, 344 clutter covariance matrix Brennan rank, 167–168 Eigen-structure based STAP, 165–173 ground patches and, 158 clutter data Earth rotation, 234–246 range foldover, 234–246 scatter return modeling, 255–267 clutter Doppler ambiguity, 156 clutter foldover, 158–159, 167, 236 clutter modeling, 215 clutter notch width Earth rotation and, 239, 241–246 overview, 241 wind effect on, 270–275 clutter nulling effects, 215, 216, 273, 280, 289–292

Index clutter nulling performance Earth rotation and, 280–283, 289–292 effect of combined inﬂuences on, 280–283 effect of terrain on, 280–283 range foldover, 280–283 STAP algorithms, 283–296 wind effect on, 280–283 clutter power beamformer, 161 clutter power distribution, 158–160 clutter power to noise power ratio (CNR), 160 clutter ridge slope, 100, 101 clutter scene Brennan’s rule, 166–167 nonstationary, 182 clutter spectrum, 223–231 clutter steering vector, 169 clutter subspace, 169, 170, 173, 244–245 clutter suppression minimum detectable velocity and, 162, 232–234 range-Doppler response and, 3 terrain modeling, 268–280 wind effect, 268–280 CMT (covariance matrix tapering), 188–194 colored noise, 353–358 comets. See also asteroids; meteors altered orbital parameters, 74 Halley’s comet, 64 identiﬁcation of, 71–74 number of, 44 orbits of, 42, 64 overview, 42 communication scene, 345 compression chirp, 358–364 pulse, 91, 358–364 conics, 31–50 circle, 32 eccentricity of, 31, 34 ellipse, 32, 33–39 gravitation and, 1

hyperbola, 32, 40–43 overview, 31–33 parabola, 32, 39–40 sections of, 32 solar system, 44–46 spherical triangles, 46–50 covariance matrix tapering, 188–194 convex projection techniques, 194–201 convex sets, 195–196 correlator receivers, 353 cosines, law of, 49–50 covariance matrix smoothed, 174, 179–180 subaperture in, 186–187 tapered, 187–189, 192 crab angle, 101–120 Doppler frequencies and, 117, 119, 246–247 Earth rotation and, 216 oblate Earth, 133–136 range foldover and, 118–120 crab angle correction, 134–137 crab error, 111–113 crab magnitude, 101–115 crab phenomonen, 117–120 Cramer-Rao bounds, 309–338 airborne platform, 320–331 Fisher information matrix, 310–320, 326, 329 Gaussian data, 319–320 multiparameter case, 309–320 SBR platform, 320–331 simulation results, 331–338 target Doppler/power, 320–338 CSA (Canadian Space Agency), 4

D DARPA (Defense Advanced Research Project Agency), 6 decision instant, 344, 351–352, 373, 376 Defense Advanced Research Project Agency (DARPA), 6 DFT (Doppler temporal) steering vector, 169, 202–206

423

424

Space Based Radar diagonal loading schemes, 165, 192–193 Doppler bins, 204–208 Doppler (temporal) DFT steering vector, 169, 202–206 Doppler effect, 130–133 “Doppler ﬁlling” effect, 99 Doppler ﬁlters, 133, 202 Doppler frequencies airborne systems, 320–338 along isocone contour, 224 angle-Doppler dependency, 224 angle-Doppler domain, 230, 290 angle-Doppler pattern, 204 angle-Doppler performance, 163, 164 angle-Doppler proﬁle, 225, 227 angle-Doppler space, 157–163 clutter and, 156–162 crab angle and, 117, 119, 246–247 Cramer-Rao bounds and, 320–331 detecting with temporal pulse transmission, 153–154 effect of crab angle on, 113, 115, 117 effect of Earth rotation on, 102–107 effect of wind on, 270–275 foldover factor, 117–120, 156 iso-Doppler plots, 99–100, 117–120 range bins, 221 range dependency of, 99–101, 224 range foldover points, 118–120 target Doppler/power, 320–338 wind dampening effect on, 275–280 Doppler shift, 97–101 Doppler spread, 100–102, 222–223, 294–295 Doppler warping, 292–294 Doppler-azimuth pattern, 100 Doppler-azimuth proﬁle, 224–229

E Earth destruction of, 44–45 eccentricity of, 123

grazing angle correction factor, 123–130 impact of asteroids on, 43, 45 maximum range on, 81–82 nonsphericity of, 123–130 oblate spheroidal shape of, 134–137 place in solar system, 44 size of, 44 Earth observing radars, 3–5 Earth rotation Brennan’s rule, 221–222, 244–245 clutter data, 234–246 clutter notch width, 239 clutter nulling performance, 280–284, 289–292 crab angle effect, 216 Doppler frequency and, 102–107, 216 Doppler spread, 100–102, 222–223, 294–295 MDV, 234–246 range foldover due to, 222–223 SINR performance, 236, 291 Earth-Moon system, 393, 404, 416, 417 EC (eigencanceler) methods, 167–171 EC with forward/backward subarray subpulse (ECSASPFB) smoothing, 187–188 eccentricity conics, 31, 34 Earth, 123 ecliptic orbits, 42 ECSASPFB (EC with forward/backward subarray subpulse) smoothing, 187–188 EFA (extended factored time-space) approach, 204–205, 286 eigencanceler (EC) methods, 167–171 eigen-structure based methods, 165–173

Index eigenvalues matched ﬁlter receiver, 371–374, 379–380, 384 matrices, 10–12 eigenvectors, 10–12, 202–206 elevation angle, 81, 87, 88 ellipses conics, 32, 33–39 eccentric, 42 inverse square law, 60 iso-range plots and, 8 semimajor axis, 64 Sun, 59, 67 elliptic orbits, 42, 74–76 elliptical orbits, 41–43, 62–67 energy compaction ratio, 383, 384 equations Euler’s, 71–74 Kepler’s, 67–70 Lambert’s, 74–76 Lyapunov, 23–24 radar, 7–8 relative motion, 54–57 equatorial orbit, 107–112 escape velocity, 63 Euler’s equation, 71–74 extended factored time-space (EFA) approach, 204–205, 286

F factor time-space (FTS) approach, 201–205 ﬁlters causal, 352, 357 Doppler, 133, 202 matched ﬁlter output, 162–163 matched ﬁlter receivers, 344–358, 371–376 noncausal, 352 receiver, 344, 346–352 ﬁnite bodies, 393–420 ﬁnite duration waveforms, 343 ﬁnite impulse response (FIR) Doppler ﬁlter, 133 FIR (ﬁnite impulse response) Doppler ﬁlter, 133

Fisher information matrix, 310–320, 326, 329 Five parameter RCS model, 258–259 foldover. See also range foldover clutter, 158–159, 167, 236 Doppler, 117–120, 156 mainbeam, 90–94 foliage penetration (FOPEN) operation, 2 footprint, mainbeam, 82–89 FOPEN (foliage penetration) operation, 2 free fall motion, 62–63 FTS (factor time-space) approach, 201–205

G gamma rays, 44 Gaussian data, 319–320 Gaussian data samples, 316–317 Gegenschein phenomenon, 404–405, 417 geosynchronous satellites, 60–61 GMTI (ground moving target indication), 155, 341–342 gravitation, law of, 1, 31, 51, 52–55 gravitational pull, 98 grazing angle correction factor for, 123–130 MEO/LEO satellites, 88–89 radar-Earth geometry, 77–81 scatter return modeling, 255–260 vs. range, 81 ground moving target indication (GMTI), 155, 341–342 ground range from latitude/longitude coordinates, 120–123 between nadir point and point of interest, 121–122 ground range resolution, 91

H Halley’s comet, 64 heliocentric coordinate system, 54–55

425

426

Space Based Radar helium, 44 Hermitian matrices, 12–16, 146, 149, 197 Hermitian Toeplitz matrix, 26–27 High Range Resolution Ground Moving Target Indication (HRR-GMTI), 6 high resolution photography (IMINT), 6 Hill sphere, 417–420 HRR-GMTI (High Range Resolution Ground Moving Target Indication), 6 HRR-GMTI coverage, 6 HRR-GMTI surveillance, 6 HTP (Hung-Turner projection), 171–173 HTP angle-Doppler pattern, 173–174 HTP with forward/backward subarray subpulse (HTPSASPFB) smoothing method, 186–187 HTPSASPFB (HTP with forward/backward subarray subpulse) smoothing method, 186–187 Hung-Turner projection (HTP), 171–173 Huygens probe, 5–6 hybrid-chirp waveforms, 254–255, 280–283, 294–295 hydrogen, 44 hyperbola conics, 32, 40–43 planet velocity and, 62 satellite orbits, 66 hyperbolic orbits, 1

I ICM (internal clutter motion) effect of wind on, 221, 255–257, 261–267 scatter return modeling, 261–267 white noise and, 263 ICM modeling, 261–267

IMINT (high resolution photography), 6 inﬁnitesimal bodies, 393–420 Innovative Space Based Radar Antenna Technology (ISAT), 6 interelement spacing, 221–222 interference, 342. See also clutter internal clutter motion. See ICM Inverse Fourier transform, 378 inverse square law of gravitation, 1, 31, 51, 52–55 ISAT (Innovative Space Based Radar Antenna Technology), 6 isocone contour, 223, 224, 239–242 iso-Doppler plots, 99–100, 117–120 iteration projections, 199

J Jacobi function, 397–398, 401, 417 JDL (Joint-Domain Localized) approach, 205–208 Joint-Domain Localized (JDL) approach, 205–208 Joint STARS (Joint Surveillance Target Attack Radar System), 2 Joint Surveillance Target Attack Radar System (Joint STARS), 2 joint time-bandwidth optimization, 376–385 joint transmitter-receiver design, 364–376 Jupiter, 44

K KASSPER (Knowledge Aided Sensor Signal Processing and Expert Reasoning), 215 Kepler’s equation, 67–70 Kepler’s laws, 57–60 kernel, channel, 370–371, 376–380, 383 Khatri-Rao product, 19–25 kinematics, 77–137

Index Knowledge Aided Sensor Signal Processing and Expert Reasoning (KASSPER), 215 Kronecker product, 18–19

L Lacrosse Series, 6 Lagrange libration points, 401, 405, 416 Lagrange theorem, 393, 401–409 Lambert’s equation, 74–76 Laplace transforms, 366, 380, 385, 410–411 launch height, 65 launch speed, 66 launch velocity vector errors, 65 law of cosines, 49–50 law of gravitation, 1, 31, 51, 52–55 law of sines, 46–49 lemmas, 14–16, 25–26 LEO (low-earth orbit), 3 LEO/MEO satellites, 88–89 line spectra matrices, 26–28 low-earth orbit (LEO), 3 Lyapunov equation, 23–24

M M pulses, 216–218 Mac Donald, Dettwiler and Associates (MDA), 4 Magellan radar, 5 Mahalanobis distance, 325, 327 mainbeam footprints, 82–89 mainbeams. See also beamforming arrays, 144–145 clutter, 218–223 foldover, 90–94 non-overlapping, 88 range ambiguities, 89–97 Mars, 44 Mars Express, 6 Mars Global Surveyor, 6 Mars Odyssey, 6 Mars reconnaissance orbiter, 5 Marsis radar, 5 matched ﬁlter (MF) receivers, 344–358

chirp signal, 361–364 eigenvalues, 371–374, 379–380, 384 optimum input and, 371–376 overview, 344–345 performance, 283–284 matrices clutter covariance matrix, 158, 165–173 covariance. See covariance matrix described, 9 eigenvalues, 10–12 eigenvectors, 10–12 Fisher information matrix, 310–320, 326, 329 Hermitian, 12–16, 146, 149, 197 line spectra, 26–28 Sample Matrix Inversion (SMI), 162–165 singular covariance, 26–28 Toeplitz, 13, 196–197 wind autocorrelation matrix, 268–269 matrix inversion identity, 296 matrix inversion lemmas, 25–26 MDA (Mac Donald, Dettwiler and Associates), 4 MDV (minimum detectable velocity) angle-Doppler performance for, 164 clutter suppression and, 161–162, 232–234 Earth’s rotation, 234–246 overview, 161–162 performance, 169–170 range foldover, 234–246 signal to interference plus noise ratio, 233 terrain modeling, 268–280 wind effect, 268–280 medium-earth orbit (MEO), 3 MEO (medium-earth orbit), 3 MEO/LEO satellites, 88–89 Mercator’s projection chart, 61 Mercury, 44

427

428

Space Based Radar meteor showers, 42 meteors. See also asteroids; comets glow from, 404 number of, 44 trapped, 404 MF receivers. See matched ﬁlter (MF) receivers military satellites, 6 minimum detectable velocity. See MDV minimum phase pulse, 368 minimum phase signal, 366–369 modulated chirp pulse signal, 387–389 modulated chirp signal, 363–364 monostatic constants, 8 monostatic mode, 341 monostatic SBRs, 6–7 Moon Earth-Moon system, 393, 404, 416, 417 elliptical orbits and, 41 Gegenschein phenomonen, 404 orbit, 1, 44 receding of, 400, 403 Saturn’s moons, 5 motion free fall, 62–63 internal clutter motion. See ICM mean angular, 67–68 relative, 54–57 two-dimensional, 55 motion of center of mass, 52–54 multichannel waveform diversity, 341–344 multiparameter case, 309–320 multipath scenes, 175–176 multi-static SBRs, 6–7

N nadir hole, 86, 96 nadir point great circle through, 128–130 inaccessible, 85 latitude/longitude of, 120–121 overview, 77–79

NASA terra satellite image map, 255–261 Neptune, 44 noise. See also SNR angle-Doppler power distribution, 158–160 arrays and, 146–152 colored, 353–358 joint transmitter-receiver design, 364–376 matched ﬁlter receivers, 344–358 waveform diversity, 344 white. See white noise noise bandwidth, 8 noise only case, 328 noise power spectral densities, 346–347 noise subspace eigenvectors, 170, 245 noise temperature, 8 noncausal ﬁlters, 352 noncausal waveforms, 355 nonnegative-deﬁnite property, 197–198 north pole, 104 N-sensor array, 216–218

O oblate spheroidal Earth, 133–136 omnidirectional sensors, 220 Oort cloud, 42 Opportunity rover, 6 optimization problem, 365–371 optimum transmit pulse, 374–376 optimum waveforms, 382–383 orbital mechanics, 51–57 orbits closed, 59, 60 comets, 42, 64 elliptic, 41, 74–76 elliptical, 41–43, 62–67 equatorial, 107–112 grativation and, 1 hyperbola, 66 hyperbolic, 1 inclination of, 77 low-earth, 3

Index maximum error, 111–114 medium-earth, 3 minimum error, 116 Moon, 1, 44 parabola, 66 parabolic, 1 planets, 51–57 polar, 60–61, 77–78, 107–109 sun and, 51–57 synchronous, 60–61 two-body orbital motion, 51–57 orthogonal pulsing minimizing range foldover with, 246–255 SINR performance and, 253 orthogonal transmit pulsing schemes, 216 output SINR, 297–298 output SNR, 143, 148–153 Ovals of Cassini, 8

P Paley-Wiener condition, 354 parabola conics, 32, 39–40 satellite orbits, 66 parabolic orbits, 1 parabolic speed, 63 Parseval’s relation, 368–369 Parseval’s theorem, 355–356, 378 phase shifted arrays, 142–143 planar wavefront, 140 planetary radars, 5–6 planetary velocity, 61–67 planets. See also speciﬁc planets elliptical orbits and, 41–42, 58–60 location of, 70 mean angular motion of, 67–68 orbital motion of, 51–57 relative motion and, 54–57 polar coordinates, 34, 35 polar orbits, 60–61, 77–78, 107–109 positive-deﬁnite property, 197–198 PRF (pulse repetition frequency), 162, 165 projection operators, 198–201

prolate spheroidal functions, 382–385 pulse compression, 358–364 pulse repetition frequency (PRF), 162, 165 pulse repetition interval, 89, 90 pulse wave front, 91 pulsed radar, 89 pulses ﬁnite impulse response (FIR), 133 M, 216–218 minimum phase, 368 optimum transmit, 374–376 rectangular, 360, 365, 373 temporal, 217–218 vectors corresponding to, 153–154

Q quadrature phase shifting, 252 quiescent steering vectors, 170

R radar. See also SBR (space based radar) airborne. See airborne systems Cassini, 5 Cassini-Huygens, 5 Earth observing, 3–5 ISAT, 6 Magellan, 5 mainbeam of, 83–89 Marsis, 5 planetary, 5–6 pulsed, 89 side-looking airborne, 155–165 synthetic aperture. See SAR radar cross section (RCS) described, 7 Five parameter RCS model, 258–259 grazing angle dependent, 221 scatter return modeling, 255–261 radar equation, 7–8 radar pulse repetition rate, 99 radar-Earth geometry, 77–81

429

430

Space Based Radar RadarSat-1 satellite, 4–5 RadarSat-2 satellite, 5 radar/sonar scene, 345 random returns, 342 range Doppler dependency on, 99–100 mainbeam footprints along, 86–89 maximum range on Earth, 81–82 vs. slant range, 80–81 range ambiguities clutter and, 218–219 mainbeam footprints, 89–97 range bin of interest, 161 range bins clutter corresponding to, 218 Doppler frequencies, 221 range foldover, 90–97. See also foldover clutter data, 234–246 clutter nulling performance, 280–283 crab phenomenon and, 118–120 Doppler frequencies, 118–120 due to Earth rotation, 222–223 minimizing with orthogonal pulsing, 246–255 minimum detectable velocity, 234–246 performance degradation and, 216 scatter return power, 251 SINR peformance, 236 total range foldover, 94–97 waveform diversity and, 248–251 range-azimuth domain, 100, 117–119, 222 range–foldover return, 215 Rao bounds. See Cramer-Rao bounds rational system representation, 303–307 Rayleigh distributions, 256, 259 RCS (radar cross section) described, 7

Five parameter RCS model, 258–259 grazing angle dependent, 221 scatter return modeling, 255–261 receiver subarrays, 220 receivers causal, 355, 365, 371 correlator, 353 ﬁlters, 344, 346–352 joint transmitter-receiver design, 364–376 matched ﬁlter. See matched ﬁlter (MF) receivers rectangular channel response, 373–374 rectangular pulse, 360, 365, 373 reference sensor, 140–141 relative motion, 54–57 relaxed projection operators, 200–201 return signal, 140

S Sample Matrix Inversion. See SMI Sample Matrix Inversion with Diagonal Loading (SMIDL), 165, 192–194 SAR (synthetic aperture radar) imaging support, 6, 341–343 SAR sensor, 4 SAR systems, 1, 4 satellite image map, 255–261 satellite velocity, 61–67 satellites geosynchronous, 60–61 MEO/LEO, 88–89 military, 6 number of, 44 orbits. See orbits polar, 60–61 RadarSat-1, 4–5 RadarSat-2, 5 Sesat, 4 synchronous, 60–61 velocity, 61–67 Saturn, 44

Index SBR (space based radar). See also radar array parameters, 226 challenges, 1–2 Cramer-Rao bounds, 320–331 data modeling, 216–231 described, 1 Doppler shift, 97–101 ground coverage, 2–3 modeling Earth’s rotation for, 101–120 orbits. See orbits parameters, 222 region of coverage, 89 requirements, 1 STAP algorithms for, 283–296 target Doppler/power, 320–331 SBR systems design considerations, 2 introduction of, 2 main parameters of, 78–81 overview, 3–7 types of, 3–7 vs. airborne systems, 3 scalars, 9 scatter power, 220 scatter power proﬁle, 215 scatter return modeling clutter data, 255–267 grazing angle, 255–260 ICM modeling, 261–267 Rayleigh distributions, 256, 259 RCS (radar cross section), 255–261 terrain modeling, 256–261 Weibull distributions, 255–256, 259–260 windblown autocorrelations, 256, 261–277 scatter return power, 251 scatter returns, 220 Schur product, 17–18, 261 Schwartz’ inequality, 347–351, 370, 379 sensors array of, 140–148 input noise and, 142

N-sensors, 216–218 omnidirectional, 220 reference, 140–141 SAR, 4 SNR, 142 Sesat satellite, 4 sidelobes array gain patterns, 215 arrays, 143–145 including clutter contribution from, 218–219 saturation levels, 208–213 side-looking airborne radar, 155–165 SIGINT (signal intelligence), 6 signal intelligence (SIGINT), 6 signal to interference plus noise ratio. See SINR signal to noise ratio. See SNR sines, law of, 46–49 singular covariance matrices, 26–28 singular value decomposition (SVD), 16–17 SINR (signal to interference plus noise ratio) clutter data, 160–164, 168–170 MDV and, 233 optimum, 179 output SINR derivation, 297–298 receiver output signal, 343 SINR loss, 236, 279, 283 SINR performance Earth’s rotation, 236, 291 orthogonal pulsing, 253 range foldover, 236 terrain modeling, 269–270 waveform diversity, 251–253, 295 slant range, 79–80, 220 SMI (Sample Matrix Inversion), 162–165 angle-Doppler performance for, 163 arrays and, 152–153 computational effect of, 289 overview, 162–163

431

432

Space Based Radar SMIDL (Sample Matrix Inversion with Diagonal Loading), 165, 192–194 SMIDLCMT method, 193–194 SMIPROJ methods, 201 SNR (signal to noise ratio). See also noise additive white noise case, 364–365, 370–376 arrays and, 142–143, 148–153 described, 8 matched ﬁlter receivers, 345–353 multiple sensors and, 142 output, 143, 148–153 solar system, 44–46 space based radar. See SBR space-time adaptive processing. See STAP space-time steering vectors, 153–157, 166, 169, 270 space-time vectors, 153–154 spatial array processing, 139–153 spatial arrays, 216–218 spatial diversity, 153–154 spatial-steering vectors, 140–148, 169, 204–206, 216–218 spatio-temporal processing, 139, 153–154 spectral factorization, 298–303 spherical triangles, 46–50 Spirit rover, 6 STAP (space-time adaptive processing) algorithms, 3, 283–296 array tapering, 188–194 covariance matrix tapering, 188–194 convex projection techniques, 194–201 eigen-structure based methods, 165–173 FTS approach, 201–205 JDL approach, 205–208 overview, 139, 153–155 performance, 293–296 side-looking airborne radar, 155–165

spatial array processing, 139–153 subaperture smoothing methods, 173–187 waveform diversity, 293–296 STAP steering vectors, 254 stationary stochastic process, 344 steering vectors. See also vectors clutter, 169 distinct, 242 Doppler temporal, 169 quiescent, 170 space-time, 153–157, 166, 250, 270 STAP, 254 stochastic process, 344 subaperture smoothing methods, 173–187 subarray vectors, 183–184 subarrays. See also arrays backward, 179–182 receiver, 220 subarray-subpulse method, 184–187 subpulse smoothing method, 184 Sun ellipses, 59, 67 elliptical orbits and, 41 gravitational force of, 42 orbital motion of, 51–57 overview, 44–45 relative motion and, 54–57 size of objects around, 43 SVD (singular value decomposition), 16–17 synchronous orbits, 60–61 synchronous satellites, 60–61 synthetic aperture radar. See SAR

T tapered covariance matrix, 187–189, 192 target Doppler, 320–338 target impulse response, 374 target power, 320–338 target return, 342

Index Taylor weights, 229–231 TechSat 21, 6 temporal processing, 153–154 temporal pulse processing, 139, 153–154 temporal pulses, 217–218 temporal steering vector, 156 terra satellite image map, 255–261 terrain clutter nulling performance, 280–283 NASA terra satellite image map, 255–261 reﬂectivity of, 215 terrain classiﬁcation, 255–261 terrain modeling MDV and, 268–280 scatter return modeling, 256–261 SINR performance, 269–270 three-body problem, 393, 401–405 time compression, 359, 361 time-bandwidth products, 361, 384 Toeplitz matrices, 13, 196–197 Toeplitz property, 196–197 total range foldover, 94–97 training cells, 161 transmit array weights, 229–231 transmit beams, 215 transmit signal, 140 transmit waveform, 154, 220, 342, 367, 373 transmitter output ﬁlters, 343 transmitter-receiver design, 343, 364–376 transmitters joint transmitter-receiver design, 364–376 waveform diversity, 341–344 triangles, spherical, 46–50 Trojan asteroids, 416–417 truncated input signal, 368 two-body orbital motion, 51–57 two-dimensional discrete Fourier transform (2D-DFT), 205 two-dimensional (2D) motion, 55

U UAVs (unmanned aerial vehicles), 341 uniform arrays beamforming and, 143–148 centro-symmetric, 330 linear, 329–330 sidelobe saturation levels, 208–213 spatial steering vector, 141 subarray smoothing and, 176–177 uniform linear array, 329 universal gravitational constant, 98 unmanned aerial vehicles (UAVs), 341 Uranus, 44

V Van Allen radiation belts, 2–3 vectors corresponding to pulses, 153–154 described, 9 eigenvectors, 10–12, 202–206 errors, 65 input noise, 142 launch velocity, 65 space-time, 153–154 steering. See steering vectors subarray, 183–184 weight, 139, 148–153 velocity escape, 63 launch, 65 minimum detectable, 161–162 planetary, 61–67 satellite, 61–67 Venus, 44

W waveform diversity, 341–391 bistatic mode, 341 chirp signal, 364, 385–391 clutter, 344

433

434

Space Based Radar decision instant, 344, 351–352, 373, 376 interference, 344 joint time-bandwidth optimization, 376–385 joint transmitter-receiver design, 364–376 matched ﬁlter receivers, 344–358, 371–376 monostatic mode, 341 multichannel, 341–344 noise, 344 overview, 247, 341–344 prolate spheroidal functions, 382–385 range foldover and, 248–251 SINR performance, 251–253, 295 STAP performance, 293–296 target (channel), 344 target Doppler, 335–338 transmitters, 341–344 waveforms causal, 355–356, 389 chirp, 91, 251–255, 281–283, 294 noncausal, 355 optimum, 382–383 transmit, 154, 220, 342, 367, 373 wavefront, planar, 140 Weibull distributions, 255–256, 259–260 Weibull-type modeling, 255 weight vector, 139, 148–153

white noise additive white noise case, 364–366, 370–376 ICM modeling and, 263 matched receiver ﬁlters in, 346–352, 358 output SNR, 150, 151 sensor element input, 323 uncorrelated sources in, 280 wide area surveillance systems, 2 Wiener factor, 354 wind Billingsley’s wind model, 256, 261, 263–267, 272 clutter nulling performance, 280–283 dampening effect, 270–275 effect on clutter notch width, 270–275 effect on Doppler frequency, 270–280 effect on ICM, 221, 255–257, 261–267 MDV and, 268–280 random variables, 256, 261–262, 263 Schur product, 261 wind autocorrelation matrix, 268–269 wind modeling, 256, 264, 275 wind spectrum, 256, 266 windblown autocorrelations, 256, 261–277