Space Based Radar Theory & Applications

Space Based RadarTheory & Applications

S. Unnikrishna PillaiPolytechnic University, New York

Ke Yong LiC & P Technologies, Inc., New Jersey

Braham HimedAir Force Research Lab, New York

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About the AuthorsS. Unnikrishna Pillai is a Professor of Electrical andComputer Engineering at Polytechnic University inBrooklyn, New York. His research interests includeradar signal processing, blind identification, spectrumestimation, and waveform diversity. Dr. Pillai is theauthor of Array Signal Processing and co-author ofSpectrum Estimation and System Identification and Prof.Papoulis’ Probability, Random Variables and StochasticProcesses (fourth edition).

Ke Yong Li is a senior engineer at C & P Technologies,Inc. in Closter, New Jersey. His areas of research includeSpace-Time Adaptive Processing (STAP), waveformdiversity, 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 interestsinclude radar signal processing, detection, estimation,multichannel adaptive processing, time series analysis,and array processing.

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About the Technical ReviewersDr. Peter Zulch received his Bachelors, Masters, andDoctorate from Clarkson University in 1988, 1991,1994 respectively. From 1994 to April 2007 he hasbeen employed by the Air Force Research Laboratory,Sensors Directorate, Rome, NY. From April 2007 to thepresent, he has been employed by Air Force ResearchLaboratory Information Directorate, also in RomeNY. His interests include multidimensional adaptivesignal processing with applications to Airborne EarlyWarning Radar, space based radar, multi-static and dis-tributed radar, and adaptive waveform diversity forradar. Dr. Zulch is a senior member of the IEEE.

Dr. James Ward is Assistant Head of the ISR Systemsand Technology Division at MIT Lincoln Laboratory.His areas of technical expertise include signal process-ing for radar and sonar systems, adaptive array signalprocessing, detection and estimation theory, and sensorsystems analysis. Dr. Ward has given tutorials on space-time adaptive processing for radar at several IEEE radarand phased array conferences. He has been an orga-nizer and regular lecturer for Lincoln Laboratory shortcourses on radar systems. He is a past recipient of theMIT Lincoln Laboratory Technical Excellence Award,and the IEEE Aerospace and Electronic Systems SocietyFred Nathanson Young Radar Engineer Award. Dr.Ward earned a Bachelor’s degree from the Universityof Dayton, and both M.S. and Ph.D. degrees in electricalengineering from The Ohio State University. Dr. Wardis a Fellow of the IEEE.

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Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 The Radar Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Notations and Matrix Identities . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3.1 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . 101.3.2 Hermitian Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.3.3 Singular Value Decomposition (SVD) . . . . . . . . . . . 161.3.4 Schur, Kronecker, and Khatri-Rao Products . . . . . 171.3.5 Matrix Inversion Lemmas . . . . . . . . . . . . . . . . . . . . . . 25

Appendix 1-A: Line Spectra and Singular CovarianceMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2 The Conics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.1 What Is a Conic? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.1.1 Ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.1.2 Parabola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.1.3 Hyperbola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.2 The Solar System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44Appendix 2-A: Spherical Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . 46References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3 Two Body Orbital Motion and Kepler’s Laws . . . . . . . 513.1 Orbital Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.1.1 The Motion of the Center of Mass . . . . . . . . . . . . . . . 523.1.2 Equations of Relative Motion . . . . . . . . . . . . . . . . . . . 54

3.2 Kepler’s Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.3 Synchronous and Polar Orbits . . . . . . . . . . . . . . . . . . . . . . . . . 603.4 Satellite Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Appendix 3-A: Kepler’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 67Appendix 3-B: Euler’s Equation and the Identification

of Comets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Appendix 3-C: Lambert’s Equation for Elliptic Orbits . . . . . . . 74References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

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4 Space Based Radar—Kinematics . . . . . . . . . . . . . . . . . . . . 774.1 Radar-Earth Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.2 Maximum Range on Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.3 Mainbeam Footprint Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.4 Packing of Mainbeam Footprints . . . . . . . . . . . . . . . . . . . . . . 864.5 Range Foldover Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.5.1 Mainbeam Foldover . . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.5.2 Total Range Foldover . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.6 Doppler Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974.7 Crab Angle and Crab Magnitude: Modeling

Earth’s Rotation for SBR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014.7.1 Range Foldover and Crab Phenomenon . . . . . . . . 118

Appendix 4-A: Ground Range from Latitude and LongitudeCoordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Appendix 4-B: Nonsphericity of Earth and the GrazingAngle Correction Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

Appendix 4-C: Doppler Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130Appendix 4-D: Oblate Spheroidal Earth and Crab

Angle Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

5 Space-Time Adaptive Processing . . . . . . . . . . . . . . . . . . . 1395.1 Spatial Array Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

5.1.1 Why Use an Array? . . . . . . . . . . . . . . . . . . . . . . . . . . . 1405.1.2 Maximization of Output SNR . . . . . . . . . . . . . . . . . . 148

5.2 Space-Time Adaptive Processing . . . . . . . . . . . . . . . . . . . . . 1535.3 Side-Looking Airborne Radar . . . . . . . . . . . . . . . . . . . . . . . . 155

5.3.1 Minimum Detectable Velocity (MDV) . . . . . . . . . . 1625.3.2 Sample Matrix Inversion (SMI) . . . . . . . . . . . . . . . . 1625.3.3 Sample Matrix with Diagonal

Loading (SMIDL) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1655.4 Eigen-Structure Based STAP . . . . . . . . . . . . . . . . . . . . . . . . . . 165

5.4.1 Brennan’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1665.4.2 Eigencanceler Methods . . . . . . . . . . . . . . . . . . . . . . . . 1675.4.3 Hung-Turner Projection (HTP) . . . . . . . . . . . . . . . . . 171

5.5 Subaperture Smoothing Methods . . . . . . . . . . . . . . . . . . . . . 1735.5.1 Subarray Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

5.6 Subaperture Smoothing Methods for STAP . . . . . . . . . . . 1835.6.1 Subarray Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1835.6.2 Subpulse Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1845.6.3 Subarry-Subpluse Method . . . . . . . . . . . . . . . . . . . . . 184

5.7 Array Tapering and Covariance Matrix Tapering . . . . . 1885.7.1 Diagonal Loading as Tapering . . . . . . . . . . . . . . . . . 192

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5.8 Convex Projection Techniques . . . . . . . . . . . . . . . . . . . . . . . . 1945.8.1 Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1955.8.2 Toeplitz Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1965.8.3 Positive-Definite Property . . . . . . . . . . . . . . . . . . . . . 1975.8.4 Methods of Alternating Projections . . . . . . . . . . . . 1985.8.5 Relaxed Projection Operators . . . . . . . . . . . . . . . . . . 200

5.9 Factor Time-Space Approach . . . . . . . . . . . . . . . . . . . . . . . . . 2015.10 Joint-Domain Localized Approach . . . . . . . . . . . . . . . . . . 205Appendix 5-A: Uniform Array Sidelobe Levels . . . . . . . . . . . . . 208References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

6 STAP for SBR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2156.1 SBR Data Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

6.1.1 Mainbeam and Sidelobe Clutter . . . . . . . . . . . . . . . 2186.1.2 Ideal Clutter Spectrum . . . . . . . . . . . . . . . . . . . . . . . . 223

6.2 Minimum Detectable Velocity (MDV) . . . . . . . . . . . . . . . . 2326.3 MDV with Earth’s Rotation and Range Foldover . . . . . 2346.4 Range Foldover Minimization Using Orthogonal

Pulsing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2466.5 Scatter Return Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

6.5.1 Terrain Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2566.5.2 ICM Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

6.6 MDV with Terrain Modeling and Wind Effect . . . . . . . . 2686.6.1 Effect of Wind on Doppler . . . . . . . . . . . . . . . . . . . . . 2706.6.2 General Theory of Wind Damping Effect

on Doppler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2756.7 Joint Effect of Terrain, Wind, Range Foldover,

and Earth’s Rotation on Performance . . . . . . . . . . . . . . . . . 2806.8 STAP Algorithms for SBR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283Appendix 6-A: Matrix Inversion Identity . . . . . . . . . . . . . . . . . . 296Appendix 6-B: Output SINR Derivation . . . . . . . . . . . . . . . . . . . . 297Appendix 6-C: Spectral Factorization . . . . . . . . . . . . . . . . . . . . . . 298Appendix 6-D: Rational System Representation . . . . . . . . . . . . 303References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

7 Performance Analysis Using Cramer-RaoBounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3097.1 Cramer-Rao Bounds for Multiparameter Case . . . . . . . . 3097.2 Cramer-Rao Bounds for Target Doppler and Power

in Airborne and SBR Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . 3207.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338

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8 Waveform Diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3418.1 Matched Filter Receivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344

8.1.1 Matched Filter Receivers in White Noise . . . . . . . 3468.1.2 Matched Filter Receivers in Colored Noise . . . . . 353

8.2 Chirp and Pulse Compression . . . . . . . . . . . . . . . . . . . . . . . . 3588.3 Joint Transmitter–Receiver Design in Noise . . . . . . . . . . . 3648.4 Joint Time Bandwidth Optimization . . . . . . . . . . . . . . . . . . 376Appendix 8-A: Transform of a Chirp Signal . . . . . . . . . . . . . . . . 385References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391

9 Advanced Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3939.1 An Infinitesimal Body Around Two Finite Bodies . . . . . 394

9.1.1 Particular Solutions of the Three-Body Problem 4019.1.2 Stability of the Particular Solutions . . . . . . . . . . . . 4059.1.3 Stability of Linear Solutions . . . . . . . . . . . . . . . . . . . 4099.1.4 Stability of Equilateral Solutions . . . . . . . . . . . . . . . 412

Appendix 9-A: Hill Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421

Preface

This book is primarily intended for seniors/graduate studentsas well as professional engineers who have a basic understand-ing of radar fundamentals. The material is organized so that

it can readily be covered in a classroom over a semester. Some ba-sic background on probability theory and stochastic processes will behelpful in understanding the statistical aspects addressed in this book.To help with the signal processing portion, the book has introductorychapters on sensor array processing and Space-Time Adaptive Pro-cessing (STAP). Additional material such as lecture notes and home-work problems and their solutions are available for the instructor atthe course website (visit: www.mhprofessional.com for more details).

All celestial bodies and artificial satellites move around in spacesubject to Newton’s inverse square law of attraction. In the case oftwo bodies, this leads to various conic sections such as circles, el-lipses, parabolas, and hyperbolas for their orbits. In Chapters 2 and3 background material on various conic sections and their relation-ship to Newton’s inverse square law in the form of Kepler’s laws arepresented.

Chapter 4 presents the space based radar kinematics, includingradar-Earth geometry, grazing angle, range and mainbeam footprintsize on Earth, range foldover phenomenon, Doppler shift due toEarth’s rotational effects, and the resulting crab angle derivation.Related topics such as the effect of Earth’s non-sphericity on graz-ing angle, as well as ground range as a function of the local lati-tude/longitude are treated in the appendices.

Chapter 5 gives an introduction to array signal processing and STAPand some of the basic methods for receiver processing are discussedin some detail. The list presented there is by no means exhaustive, andis used only for illustrating the advantages of STAP.

Chapter 6 gives detailed accounts of space based radar (SBR) clut-ter modeling and target detection performance evaluation. Variousfactors that affect the clutter data such as the range foldover effect,

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crab phenomenon due to Earth’s rotation, wind and terrain effects arediscussed here. The effect of these factors on various STAP algorithmsfor clutter suppression and target detection are quantified and illus-trated for different situations. To help with the terrain model, mapsgenerated from NASA’s Terra Satellite are used in this book. Theycategorize each 1 km2 patch of actual Earth into one of 16 land covertypes—forest, urban, lakes, etc. The tabulated mean radar cross sec-tion (RCS) values of these patches are then used to simulate individ-ual random scatter returns. For even higher fidelity, effects such asgrazing angle dependency on RCS are included. Wind simulation isaccomplished by modeling the wind as a low order stable ARMAprocess that depends on the carrier frequency, wind speed, and radarpulse repetition frequency (PRF). An interesting analysis of the effectof wind on algorithm performance is also presented.

Chapter 7 presents performance analysis in terms of Cramer-Raobounds for multiple parameters in a multi-sensor, multi-pulse envi-ronment for the airborne case as well as the SBR case. The presenta-tion deals with two unknown parameters; namely target azimuth andpower level.

Chapter 8 gives an introduction to the joint transmitter receiverwaveform diversity. For a given transmit waveform, target (channel)response and noise scene, it is well known that the classical matchedfilter gives the optimum receiver characterization. The problem oftransmitter optimization and the potential advantages to be gainedby transmitting a specific waveform in a particular context—specifictarget (channel) and interference/noise scene—are addressed in thischapter.

Chapter 9 deals with additional topics of interest such as locat-ing suitable places in space to park future space stations, as well asnear-Earth asteroid-tracking SBRs. In this context, a special three bodyproblem, where an infinitesimal body moves under the influence oftwo rotating finite bodies, is reviewed along with their stable solu-tions. These stable solutions are illustrated in well-known situationssuch as the Trojan asteroids near Jupiter and the Gegenschein patchof light in the sky on the earth’s side away from the Sun. Some ofthese stable solutions such as the Sun-Earth or the Earth-Moon sys-tems may be of interest to man-made missions as well. In the longrun, our ultimate survival may depend on the ability of such vigilantdeep space radars to detect and track the near-Earth heavenly objectssuch as asteroids and comets and ultimately deflect those that are ona collision course with Earth.

The authors would like to take this opportunity to thank Drs. PeterZulch, AFRL (Rome, NY) and James Ward of Lincoln Labs (Lexington,MA) for their review of the manuscript. Their feedback has beenvery useful in improving the quality of the book. Peter has been an

P r e f a c e xi

enthusiastic supporter of all our efforts from the very beginning,and technical discussions with him on various topics presentedhere are gratefully acknowledged, in addition to his efforts inhelping us with the multiple level public release approval pro-cess at AFRL. To that extent, efforts by Mr. Paul Gilgallon, AFRL,and Mr. William Baldygo, Chief, AFRL, Radar Signal Processingbranch are also acknowledged here. Drs. Mark Davis, Joseph Guerci,Michael Wicks, S. Radhakrishnan Pillai, and Stephen Mangiat alsodeserve 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. LawrenceAdzima for their support and encouragement. Finally, the vision-aries at AFRL also deserve special credit for their far-sightednessin creating broad based technical programs and for their end-to-endexecution.

The team at McGraw-Hill—Ms. Wendy Rinaldi, Editorial Directorfor Engineering; Ms. Mandy Canales, Acquisitions Coordinator; andMs. Harleen Chopra, Project Manager—deserves special credit fortheir highly efficient coordinated work and guidance throughout theperiod of the production process. Ms. Rinaldi with her efficient man-agement style has made this whole process seem effortless for us, andwe wish to thank her for the same.

Finally, the first author wishes to take this opportunity to expresshis deep gratitude to his mentor Prof. Dante Youla of PolytechnicUniversity, Brooklyn, New York, who has been a true inspiration to theauthor in many ways from his day one at Polytechnic. To express hisappreciation, a quote from John Bunyan is appropriate in this context:

. . . You have been so hearty in counseling of usthat we shall never forget your favor towards us. . .

S. Unnikrishna PillaiKe Yong Li

Braham Himed

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List of Abbreviations

AMTI Air Moving Target IndicationAR Auto RegressiveARMA Auto Regressive Moving AverageCMT Covariance Matrix TaperingCNR Clutter to Noise RatioCR Cramer-RaoDFT Discrete Fourier TransformEC Eigen CancellerECSASPFB Eigen Canceller with Subarray-Subpulse and

Forward-Backward smoothing (EC-SASP-FB)EFA Extended Factored Time-Space ApproachFTS Factored Time-Space ApproachGMTI Ground Moving Target IndicationHTP Hung-Turner ProjectionHTPSASPFB Hung-Turner Projection with Subarray-Subpulse

and Forward-Backward smoothing (HTP-SASP-FB)i.i.d . Independent and Identically DistributedJDL Joint Domain Localized approachMDV Minimum Detectable VelocityMF Matched FilterML Maximum LikelihoodPRF Pulse Repetition FrequencyPRI Pulse Repetition IntervalRCS Radar Cross SectionSAR Synthetic Aperture RadarSBR Space Based RadarSINR Signal to Interference plus Noise RatioSMI Sample Matrix InversionSMIDL Sample Matrix Inversion with Diagonal LoadingSMIDLSASPFB Sample Matrix Inversion with Diagonal Loading,

Subarray-Subpulse and Forward-Backwardsmoothing (SMIDL-SASP-FB)

SMIPROJ Sample Matrix Inversion with Convex ProjectionSMISASPFB Sample Matrix Inversion with Subarray-Subpulse

and Forward-Backward smoothing (SMI-SASP-FB)SNR Signal to Noise RatioSTAP Space-Time Adaptive ProcessingSVD Singular Value DecompositionUAV Unmanned Aerial Vehicle

xiiiCopyright © 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. 4Edwin Arnold translation

C H A P T E R 1Introduction

Man made satellites and heavenly objects such as planets and cometsmove 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 caseof satellites, the Earth plays that role.

The central force of attraction is Newton’s inverse square law ofgravitation (Principia, 1687), and in the case of two bodies an interest-ing feature of this force is that the resulting orbits are planar—circular,ellipses, parabolas, or hyperbolas—that all come under the generalterm conics. Most of the satellites move around the Earth in nearlycircular orbits. According to Kepler (1571–1630), the Earth and otherplanets in the Solar system move around the Sun in elliptical orbitswith Sun at one focus, as does the Moon around the Earth, although theorbits of the Earth and the Moon are nearly circular. Parabolic orbits areuseful for shifting a spacecraft from one orbit to another, and to escapealtogether from the central force as in the case of interplanetary voy-ages. To realize this goal in a hurry, hyperbolic orbits are more efficient.

Space based radar (SBR) once launched into an orbit, moves aroundthe Earth while the Earth continues to rotate around its own axis. Byvirtue of its location, an SBR can cover a very large area on Earth forintelligence, surveillance, and monitoring of ground moving targets.By adjusting the SBR speed and orbital parameters it is thus possibleto periodically scan various parts of the Earth and collect data. Suchan SBR based surveillance system can be controlled remotely andmay require very little human intervention. The system has a rapidresponse time, and provides accurate information. As a result, targetsof interest can be identified and tracked in greater detail or imagescan be made with very high resolution [1].

Depending on the specific application such as ground/air movingtarget indication (GMTI/AMTI), or imaging using synthetic apertureradar (SAR), the objectives of the SBR mission can vary. In general, nearcontinuous global coverage and near real-time tasking are the require-ments while some of the technical challenges include affordability,

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

2 S p a c e B a s e d R a d a r

constellation size and cost, as well as the interference cancellationcapabilities of the receiver processor.

Wide area surveillance systems are very important for monitoringEarth resources, mitigation of natural disasters such as floods, seis-mic activity monitoring, and border and homeland security. Thesesystems provide visibility over as wide an area as possible, with re-visit times commensurate with the mobility and characteristics of thetargets of interest. The target signatures may include fast airborne ve-hicles and very slow or stationary structures hidden by foliage. His-torically, manned airborne surveillance platforms such as the JointSurveillance Target Attack Radar System (Joint STARS) have beenfielded for air vehicle detection, surface vehicle imaging, and movingtarget detection. Recently, however, there has been a growing need inaugmenting these assets with space based capabilities.

SBR systems have been under consideration for several years, andhave only been made commercially available for SAR modes. This isdue to the limited availability of space based component technolo-gies and the high cost of manufacturing large space systems. Severalrecent technology programs have promised significant advances inaffordable antenna array radar design. However, the design of theseradars will require aperture sizes and average power significantlylarger than previously considered. Such systems would be capable ofproviding both wide-area surveillance and tracking of airborne andground moving targets. This capability is particularly attractive be-cause it provides deep coverage into areas typically denied airborneaccess, greater ease and flexibility for deploying the sensor platformon station and meeting coverage tasking, greater area coverage rateperformance, and steep look-down capability for foliage penetration(FOPEN) operation [2].

Several factors must be considered in the SBR system design. Theseinclude size of area of interest, revisit rate to cover the search volume,mode scheduling—what other modes need to be scheduled that willaffect the area coverage rate-obstruction of targets due to terrain block-age, foliage or other interference effects, and minimum detectable ve-locity (MDV). The major drivers on area coverage are the altitude ofthe radar platform and the field of view of the radar sensor. Figure 1.1shows the large ground range coverage obtained using an SBR plat-form, located at 500 km altitude versus that obtained with an airborneplatform located at a height of 10 km [2].

Space based surveillance requires a nominal area coverage rateof several hundred km2/s with revisit rates of one to two minutes,while tracking requires somewhat shorter revisit times. Several stud-ies have shown a constellation of satellites may be necessary to meetthese requirements. Even though it may seem that the altitude of asatellite can be freely chosen, the two Van Allen radiation belts limit

C h a p t e r 1 : I n t r o d u c t i o n 3

10

200 400 600 800 1,000 1,200

500

Ground Range (km)

Plat

form

Alt

itud

e (k

m)

0

FIGURE 1.1 Ground coverage for airborne vs. SBR platforms.

practical orbit selection. The two Van Allen radiation belts are cen-tered on the Earth’s geomagnetic axis, at altitudes ranging from 1,500km to 5,000 km and from 13,000 km to 20,000 km. To minimize the ra-diation damage to electronic components, the satellites would have tobe placed in orbits outside of these belts. Therefore, either a medium-earth orbit (MEO) at altitudes of 5,000 km to 13,000 km or a low-earthorbit (LEO) at an altitude less than 1,500 km is desirable. Today, bothLEO 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 sce-narios arises from the fact that non-geosynchronous spaceborne sen-sors view a rotating Earth. For airborne geometries, both the sensorplatform and the surveillance scene are essentially rotating together,resulting in an equivalent stationary, non-rotating Earth. But for SBRoperations, there is a range-dependent shift of the clutter range-Doppler response, which makes clutter suppression a difficult task,and this in turn limits performance of adaptive techniques such asspace-time adaptive processing (STAP) algorithms.

1.1 OverviewSBR systems can be categorized into three major areas: Earth obser-ving radars, planetary radars, and defense radars. Seasat (1978),RadarSat-1 (1995) and RadarSat-2 (2007), Shuttle Imaging Radar (SIR-A/B/C) (1981–1994), European Remote Sensing Satellite (ERS-1 and2) are some of the Earth observing radars. The Jet Propulsion Lab-oratory (JPL) Earth science projects include Seasat (1978) for ocean


science, SIR-A/B/C for land imaging, the Shuttle Radar TopographyMission (SRTM) in 2002 for global and topography, the TOPographyEXperiment (TOPEX) in 1992, Jason-1 (2001) and Jason-2 (2008), theNASA SCATterometry (NSCAT) in 1996–1997 for ocean topographyand wide data set collection [4], [5], [6].

Seasat was the first Earth-orbiting satellite designed for remotesensing of the Earth’s oceans and had onboard the first spaceborneSAR technology. It was designed to demonstrate the feasibility ofglobal satellite monitoring of oceanographic phenomena and to helpdetermine the requirements for an operational ocean remote sens-ing satellite system. The specific objectives were to collect data onsea-surface winds, sea-surface temperatures, wave heights, internalwaves, atmospheric water, sea ice features, and ocean topography [7].Seasat, managed by NASA’s JPL, was launched in June, 1978 intoa nearly circular 800-km orbit with an orbital inclination of 108.Seasat carried five major instruments designed to return the maxi-mum information from ocean surfaces: (i) Radar altimeter to measurespacecraft height above the ocean surface; (ii) Microwave scatterome-ter to measure wind speed and direction; (iii) Scanning multi-channelmicrowave radiometer to measure sea surface temperature; (iv) Visi-ble and infrared radiometer to identify cloud, land, and water features;and (v) SAR L-band, HH Polarization, fixed look angle to monitor theglobal surface wave field, and polar sea ice conditions. Seasat oper-ated for 105 days until October 10, 1978, when a short circuit in thesatellite’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 ForceBase (AFB) in California, into a sun-synchronous (dawn-dusk) orbitabove 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 andthe private sector, it provides images of the Earth for both scien-tific and commercial applications. RadarSat-1’s images are useful inmany 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 Earthat a single microwave frequency in the C band. Unlike optical satel-lites that sense reflected sunlight, SAR systems transmit microwaveenergy toward the surface and record the reflections. Thus, RadarSat-1can image the Earth, day or night, in any atmospheric condition, suchas cloud cover, rain, snow, dust, or haze. Each of RadarSat-1’s sevenbeam modes offer a different image resolution. It also has the uniqueability to direct its beam at different angles.

With an orbital period of 100.7 minutes, RadarSat-1 circles the Earth14 times a day. The orbit path repeats every 24 days, and hence the


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 forinterferometry and detecting changes at any particular location thattook place during the 24 days. Using different beam positions, a loca-tion of interest can also be scanned every few days. It is a side-lookingsatellite, where the microwave beam transmits and receives on theright side of the satellite, relative to its orbital path. As it descends inits orbit from the north pole, it faces west, and when it ascends from thesouth pole, it faces east. Locations can therefore be imaged from oppo-site sides. Combined with the different beam modes and positions, thisprovides users with many possible perspectives from which to imagea location. NASA provided the Delta II rocket to launch it in exchangefor access to its data. Estimates are that the project, excluding launch,cost about $620 million (Canadian). The Canadian federal govern-ment contributed about $500 million, the four participating provinces(Quebec, Ontario, Saskatchewan, and British Columbia) about $57million, and the private sector about $63 million.

MacDonald, Dettwiler and Associates (MDA) has announced thelaunch of RadarSat-2 for December 2007. RadarSat-2 is scheduled tobe launched on a Soyuz vehicle from Russia’s Baikonur Cosmodromein Kazakhstan. RadarSat-2 has been designed with significant andpowerful technical advancements which include high-resolutionimaging, polarization flexibility, left and right-looking imaging op-tions, superior data storage, and more precise measurements of space-craft position and attitude.

Some of the planetary radars are Magellan (1990–1994) for Venusprobe, Cassini (2004) for Titan probe, Marsis (2003) and Mars recon-naissance orbiter (2006) for Mars probe. The objectives of the Cassini-Huygens radar are: (i) Determine the three-dimensional (3D) structureand dynamic behavior of the rings; (ii) Determine the composition ofthe satellite surfaces and the geological history of each object; (iii) De-termine the nature and origin of the dark material on lapetus’s lead-ing hemisphere; (iv) Measure the 3D structure and dynamic behaviorof the magnetosphere; (v) Study the dynamic behavior of Saturn’satmosphere at cloud level; (vi) Study the time variability of Titan’sclouds and hazes and (vii) Characterize Titan’s surface on a regionalscale. Cassini-Huygens is a joint NASA, European Space Agency(ESA), the Italian Space Agency (ASI) unmanned space mission in-tended to study Saturn and its Moons. The spacecraft consists of twomain elements: (i) The Cassini orbiter, named after the Italian-Frenchastronomer Giovanni Domenico Cassini; (ii) The Huygens probe,named after the Dutch astronomer Christiaan Huygens. It was launch-ed on October 15, 1997 and entered Saturn’s orbit on July 1, 2004. OnDecember 25, 2004, the probe separated from the orbiter, with deploy-ment confirmed by JPL. The probe reached Saturn’s Moon Titan on


January 14, 2005, where it made an atmospheric descent to the surfaceand relayed scientific information. It is the first spacecraft to orbit Sat-urn and the fourth spacecraft to visit Saturn. NASA has several space-craft in orbit around Mars. The latest in that group is the reconnais-sance orbiter and the others are Mars Global Surveyor, Mars Odysseyand the ESA’s Mars Express. On the Mars surface, the NASA roboticrovers Spirit and Opportunity continue to perform their geologic mis-sions. The latest orbiter has an advanced camera to photograph thesurface and a ground perpetrating radar to probe underground for iceand possible evidence of water.

Military satellites include the Lacrosse Series (1, 2, 3, 4, 5) highresolution radar imaging satellites, and various reconnaissance satel-lites for intelligence purposes, that include high resolution photog-raphy (IMINT), signal intelligence (SIGINT), communications, detec-tion of missile launches and nuclear tests [9]. StarLite representeda new light-weight satellite concept advanced by the Defense Ad-vanced Research Project Agency (DARPA) in early 1997. Its studyreported the feasibility of developing, deploying and operating aconstellation of relatively inexpensive radar satellites designed toprovide useful information to the warfighter that could be directlytasked by the warfighter and downlinked to theater for processingand exploitation. The concept was modified to incorporate a low costapproach to space based High Range Resolution Ground MovingTarget Indication (HRR-GMTI) as well as synthetic aperture radar(SAR) imaging to augment the airborne capabilities, including theUnmanned Aerial Vehicle (UAV), U-2, and JSTARS battlefield HRR-GMTI. TechSat 21 represents new innovations in microsatellite clus-ter program using three satellites that can be deployed in differentconfigurations and at various separations [10]. The Innovative SpaceBased Radar Antenna Technology (ISAT) deploys extremely large an-tennas that are electronically scanned for coherent beamforming sothat continuous tracking of surface targets (GMTI) from an MEO ispossible [11], [12].

Other countries that are major players in the space radar researchand 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 tocategorize the SBRs. In the monostatic case, the transmitter and re-ceiver are collocated on the same platform, whereas in the bistaticsituation they are on different platforms [13]. For example, the trans-mitter maybe on a high-value high-altitude platform, whereas thereceiver maybe on a lower orbit platform and/or on an UnmannedAerial Vehicle (UAV). The bistatic/multistatic situation affords extra


degrees of freedom; however, clutter characterization and compensa-tion is much more complex compared to the monostatic case [14], [15].

The purpose of this book is to provide a detailed understandingof the basic SBR principles, and study the analysis and synthesis as-pect to its data collection. Several related topics are reviewed in thiscontext here.

A review of the conics followed by the dynamics of orbital mo-tion that result in Kepler’s laws are reviewed first in this book foran in-depth understanding of the fundamentals. This is followed bya systematic study of SBR—its geometry, mainbeam footprint size,range ambiguities on footprints, SBR clutter modeling, and clutterspectrum. A study of STAP algorithms is taken up next for GMTI andAMTI applications.

Other topics covered in this book include an introduction towaveform diversity as well as a detailed analysis of a special caseof the three body problem, where an infinitesimal body moves underthe influence of two finite bodies that revolve around their commoncenter of mass. This is an important configuration for parking spacestations as well as deep space based platforms such as asteroid track-ing space radars in the Sun-Earth or Earth-Moon frame for long-termsurveillance.

1.2 The Radar EquationThe fundamental relation between the power characteristics of a radar,target, and the receiver is given by the radar equation [16] that takes thetwo-way scattering into account. Suppose the radar transmits powerPT using an antenna with gain GT then the power spreads over asphere of radius equal to the transmitter-target distance RT . This givesthe power per unit solid angle in the direction of the target to beP1 = PT GT/4π R2

T . If As represents the effective area of the scatter andη its reflectivity gain power, then

σ = Asη (1.1)

represents the radar cross section (RCS) and P2 = P1σ representsthe power reradiated by the scatter in the direction of the receiver.With RR representing the target-receiver distance, the power/unit an-gle in the direction of the receiver equals P3 = P2/4π R2

R, which afterdiscounting for the transmitter to target and target to receiver prop-agation factors FT and FR reduces to P4 = P3 F 2

T F 2R. If AR represents

the receiver antenna aperture area, we get PR = P4 AR to be the re-ceived power. In terms of receiver antenna gain pattern G R, we have


AR = λ2G R/4π so that we obtain the receiver power to be [17]

PR = P4 AR = PT GT G Rλ2σ F 2T F 2

R

(4π )3 R2T R2

R, (1.2)

where λ is the operating wavelength. The receiver average thermalnoise power is given by

No = kTo Bn, (1.3)

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

SNR = PR

No= PT GT G Rλ2σ F 2

T F 2R

(4π )3kTo Bn R2T R2

R(1.4)

and it represents the general radar equation connecting receiver out-put SNR and range. In the monostatic case where then transmitterand receiver are collocated, we have GT = G R = G, FT = FR = F ,RT = RR = R, and the radar equation reduces to

SNR = PT G2 F 2λ2σ

(4π )3kTo Bn R4 = KM

R4 (1.5)

and in the bistatic case, (1.4) reduces to

SNR = K B

R2T R2

R, (1.6)

where KM and K B are monostatic and bistatic constants, that dependon radar parameters, target type and geometry.

Equations (1.5)–(1.6) can be used to make some interesting obser-vations. For example, in the monostatic constant RCS case, groundpoints that are at constant range (iso-range) are circles. Similarly from(1.5) points of constant SNR (iso-SNR) correspond to constant rangeand hence they too are associated with circular paths on the ground.However, in the bistatic case, from (1.6) iso-SNR plots correspondto RT RR = constant. For a fixed transmitter-receiver geometry, theseplots represent the Ovals of Cassini, a more complicated situation.Similarly if we define the transmitter-target-receiver distance RT + RRto represent the bistatic range, then the iso-range plots correspond toellipses with its two foci at the transmitter and receiver. The constanttransmitter-target-receiver distance represents the major axis of thisellipse. As a result, in the bistatic case, iso-range and iso-SNR plotsdo not coincide with each other and the clutter modeling and asso-ciated target detection or imaging becomes a much more complexissue [18], [19].


1.3 Notations and Matrix IdentitiesNotation for scalars, vectors, and matrices used throughout this bookare 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 ).

Further, A, AT , A∗, tr(A), det (A) = |A|, A−1 represent the com-plex conjugate, transpose, complex conjugate transpose, trace, deter-minant, and inverse of the matrix A respectively. The identity ma-trix is represented by I. A diagonal matrix D with diagonal entriesd1, d2, . . . , dm is denoted by

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

d1 0 0 0

0 d2 0 0

0 0. . . 0

0 0 0 dm

. (1.7)

Let aij or Aij represent the (i, j)th entry of the matrix A. For any twosquare matrices A and B of same size we have

(AB)∗ = B∗A∗ (1.8)

tr (AB) = tr (BA) =∑

i

∑j

ai j bji (1.9)

det(AB) = |AB| = |A||B| (1.10)

(AB)−1 = B−1A−1 (1.11)

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

a11,

∣∣∣∣ a11 a12

a21 a22

∣∣∣∣ ,∣∣∣∣∣∣∣

a11 a12 a13

a21 a22 a23

a31 a32 a33

∣∣∣∣∣∣∣ , . . . , det(A). (1.12)


A square matrix A is said to be nonsingular (singular) if its deter-minant is nonzero (zero).

1.3.1 Eigenvalues and EigenvectorsFor 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 eigenvalueλ.

All eigenvalues of A are given by the roots of the characteristicpolynomial

det(λI − A) = λn + c1λn−1 + · · · + cn−1λ + cn

= (λ − λ1)m1 (λ − λ2)m2 · · · (λ − λr )mr , (1.14)

where λ1, λ2, . . . , λr are distinct and m1, m2, . . . , mr are the algebraicmultiplicities of the eigenvalues. Clearly m1 + m2 + · · · + mr = n.

The matrix A has distinct eigenvalues if all mi = 1 and r = nin (1.14). If A has n distinct eigenvalues λ1, λ2, . . . , λn, then it has nlinearly independent eigenvectors e1, e2, . . . , en each pair satisfying(1.13) [20].

Proof If not, suppose these eigenvectors are linearly dependent. Thenwe must have

a1e1 + a2e2 + · · · + anen =n∑

i=1

ai ei = 0, (1.15)

where (at least some) ai are nonzero constants. Applying (A − λnI) to(1.15) we get

(A − λnI)n∑

i=1

ai ei =n∑

i=1

ai ( Aei − λnei ) =n∑

i=1

ai (λi − λn)ei

=n−1∑i=1

(λi − λn)ai ei =n−1∑i=1

bi ei = 0, (1.16)

where bi = (λi − λn)ai . From (1.15) and (1.16) we get

n−1∑i=1

bi ei = 0 (1.17)

and continuing this process, we get e1 = 0, and hence e2 = 0, . . . ,a contradiction. Hence the eigenvectors associated with distinct


eigenvalues are independent. Hence we have

Aei = λi ei , i = 1 → n (1.18)

that gives

A [ e1, e2, · · · en ]︸︷︷︸T

= [ e1, e2, · · · en ]︸︷︷︸T

λ1 0 · · · 0

0 λ2... 0

......

. . ....

0 0 · · · λn

︸︷︷︸

,

(1.19)

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

AT = T (1.20)

which can be written as

A = TT−1. (1.21)

Equation (1.21) represents a similarity transformation1. Thus A issimilar to a diagonal matrix if it has distinct eigenvalues. More gen-erally, A is similar to a diagonal matrix if and only if A has n linearlyindependent eigenvectors. Let v1, v2, . . . , vn represent the row vectorsof the matrix T−1. Thus

T−1 =

v1

v2

...

vn

= V. (1.22)

In that case (1.21) can be rewritten as

A = TV =n∑

i=1

λi ei vi (1.23)

and similarly

A−1 = T−1V =n∑

i=1

1λi

ei vi . (1.24)

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


In the case of repeated eigenvalues, the number of linearly inde-pendent eigenvectors (geometric multiplicity) αi associated with aneigenvalue 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 matrixA represents its rank ρ(A). In case of a square matrix, its rank coincideswith the total number of nonzero eigenvalues (including repetitions).For any two matrices A and B of dimensions m ×n and n× r , the rankof 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 MatricesA square matrix A of size n × n is said to be Hermitian if A = A∗,i.e., aij = a∗

ji , i, j = 1 → n. The matrix A is said to be nonnegativedefinite if for any n × 1 vector x, the quantity x∗Ax ≥ 0. When strictinequality holds, i.e., x∗Ax > 0 for x = 0, the matrix A is said to bepositive 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 ifthere exists a unitary matrix U such that A = UBU∗. A classical resultdue to Schur states that every n × n matrix A is unitarily similar to alower triangular matrix. Thus

A = ULU∗, (1.27)

where L is lower triangular. In particular, it follows that if A is Hermi-tian, then L is diagonal, and if A is positive definite, L is diagonal andpositive definite and hence all its principal diagonal entries Lii > 0.Hence diagonalization of a Hermitian matrix is always possible usinga unitary similarity transformation. Thus L = D2 where D is a realdiagonal matrix with

Dii = ±√

Lii (1.28)

and

A = ULU∗ = UD2U∗ = UDU∗UDU∗ = C2, (1.29)

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

Lii > 0. In that


case, C is positive definite. Thus for every Hermitian positive definitematrix, there exists a unique Hermitian square root that is also positivedefinite.

For Hermitian matrices, the necessary and sufficient condition forit to be nonnegative definite (positive definite) can be given in termsof the signs of its principal minors.2 To be specific, a Hermitian matrixis nonnegative definite (positive definite) if and only if all its principalminors are nonnegative (positive). More interestingly, a Hermitianmatrix is positive definite if and only if all its leading principal minorsare positive.

A matrix is said to be Toeplitz if its entries along every diagonalare the same. Thus if T is Toeplitz, then Tij = ti− j . The inverse of aninvertible Toeplitz matrix is not Toeplitz unless it is a 2 × 2 or lower(upper) triangular. Alternatively, any two lower (upper) triangularToeplitz matrices commute and their product is again [lower (upper)triangular and] Toeplitz! However, lower (or upper) triangular blockToeplitz matrices do not commute. If an n × n Toeplitz matrix is alsoHermitian, then it has only at most n independent entries, namely,those along the first (last) row (column).

Thus the (n + 1) × (n + 1) matrix

Tn =

ro r1 · · · rn−1 rn

r∗1 ro r1 · · · rn−1

... · · · . . . · · · ...

r∗n−1 · · · r∗

1 ro r1

r∗n r∗

n−1 · · · r∗1 ro

(1.30)

is Hermitian Toeplitz and moreover it has the recursive form

Tn =

rn

rn−1

Tn−1...

r1

r∗n r∗

n−1 · · · r∗1 r0

. (1.31)

Let λi denote an eigenvalue of A. Then, there exists an eigenvec-tor ui = 0, such that Aui = λ i ui . The eigenvectors are in generalcomplex and for positive definite matrices, they can be made unique

2A principal minor is the determinant of a submatrix formed with the samenumbered rows and columns.


by normalization together with a constraint of the form uii ≥ 0. Theeigenvalues of a Hermitian matrix are real. For

u∗i Aui = λi u∗

i ui = λi , (1.32)

with u∗i ui = 1. Also

u∗i Aui = (Aui )

∗ui = λ∗i u∗

i ui = λ∗i . (1.33)

Thus λ∗i = λi or λi is real. If A is also positive definite then u∗

i Aui > 0and hence it follows that its eigenvalues are all real and positive. More-over, eigenvectors associated with distinct eigenvalues of a Hermitianmatrix are orthogonal. This follows by noticing that, if λi , λ j and ui ,u j are any two such pairs, then

u∗i Au j = u∗

i λ j u j = λ j u∗i u j . (1.34)

However, we also have

u∗i Au j = (Aui )

∗u j = (λi ui )∗u j = λi u∗

i u j . (1.35)

Thus λ j u∗i u j = λi u∗

i u j or equivalently (λi −λ j )u∗i u j = 0, or u∗

i u j = 0provided λi = λ j . As a result, if A = A∗ has n distinct eigenvalues,then it can be diagonalized by a unitary similarity transformation asin (1.21) (where T is unitary, T−1 = T∗).

Interestingly, the diagonalization of any Hermitian matrix is alwayspossible irrespective of whether its eigenvalues λ1, λ2, . . . , λn are dis-tinct or not.

To prove this, the following result is helpful:

Lemma If an n × n matrix A maps a subspace Z into itself, then Ahas an eigenvector in that subspace Z.

Proof Let the subspace Z be of dimension m ≥ 1 and b1, b2, . . . , bmbe any basis for Z. Then bi ⊂ Z and since A maps Z into itself wehave Abi ⊂ Z. Hence

Abi =m∑

j=1

cijbi , j = 1 → m. (1.36)

Let

B = [b1, b2, . . . , bm]. (1.37)

Using (1.36) we get

AB = BC, (1.38)

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


Let µ represent a nonzero eigenvalue of C; then there exists a vectory such that

Cy = µy. (1.39)

Hence using (1.38)

ABy = B(Cy) = B(µy) = µBy. (1.40)

Define

z = By. (1.41)

Then z is a linear combination of the basis vectors b1, b2, . . . , bm andhence 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 thesubspace orthogonal to them. S maps into itself since x ⊂ S givesu∗

i x = 0, i = 1 → m and

u∗i (Ax) = (x∗Aui )

∗ = (x∗λi ui )∗ = λi u∗

i 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 thatbelongs to S, and y = um+1⊥u1, u2, . . . , um. This process has gener-ated m + 1 orthonormal eigenvectors. Continuing this process, a fullset of n orthonormal eigenvectors u1, u2, . . . , un with correspondingeigenvalues λ1, λ2, . . . , λn can be formed. Note that λi need not be alldistinct.

As a result, if an eigenvalue has multiplicity L , it is always pos-sible to choose a new set of L orthonormal vectors from the aboveL-dimensional subspace to act as an eigenvector set for that eigen-value. Thus, for an n × n Hermitian matrix A, if λ1, λ2, . . . , λn andu1, u2, . . . , un represent its eigenvalues and an orthonormal set ofeigenvectors, then Aui = uiλi , i = 1, 2, . . . , n, or in compact form

AU = U (1.45)

where

U = [u1, u2, . . . , un], = diag[λ1, λ2, . . . , λn]. (1.46)


Clearly, UU∗ = U∗U = I, i.e., U is a unitary matrix, and conse-quently AU = U gives (see (1.27))

A = UU∗. (1.47)

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

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 respec-tively, and D is a diagonal matrix with r positive diagonal entries.

Proof The eigendecomposition of the m × m nonnegative definiteHermitian matrix AA∗ gives r positive eigenvalues λ2

1, λ22, . . . , λ2

r withcorresponding eigenvectors u1, u2, . . . , ur . Thus

AA∗ui = λ2i ui , i = 1 → r. (1.49)

Define

vi = 1λi

A∗ui , i = 1 → r. (1.50)

Then

v∗i v j = u∗

i AA∗u j

λiλ j= λ2

j u∗i u j

λiλ j= λ j

λiu∗

i u j =

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 completeorthonormal 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)


Thus

A = UU∗A =m∑

i=1ui u

∗i A =

m∑i=1

ui ( Aui )∗

=r∑

i=1ui (λi vi )

∗ =n∑

i=1λi ui v

∗i = UDV

(1.55)

where D is m × n given by

D =

λ1 0 · · · · · · 0

0 λ2 0...

. . ....

... λr...

0 0 · · · · · · 0

≥ 0. (1.56)

Or alternatively

Avi = 1λi

AA∗ui = λ2i

λiui = λi ui , (1.57)

Avi v∗i = λi ui v

∗i , (1.58)

and hence

An∑

i=1

vi v∗i︸︷︷︸

In

=n∑

i=1

λi ui v∗i =

r∑i=1

λi ui v∗i (1.59)

or

A =r∑

i=1

λi ui v∗i = UDV (1.60)

as before.

1.3.4 Schur, Kronecker, andKhatri-Rao Products

Schur ProductFor any two matrices A and B of same size, the Schur (or the Schur-Hadamard) product A B represents their element-wise multiplica-tion. Thus if

C = A B (1.61)

then

cij= aijbij. (1.62)


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 witheigenvalues λi (A) and λi (B) respectively. Suppose

λ1(A) ≥ λ2(A) ≥ · · · λm(A) ≥ 0 (1.67)

and

λ1(B) ≥ λ2(B) ≥ · · · λm(B) ≥ 0. (1.68)

Thus λ1 and λm represent the largest and smallest eigenvalues re-spectively. Also let b1 and bm represent the largest and smallest entriesrespectively among the diagonal entries of B. Then b1 ≥ bm > 0, andwe have [23]

bmλm(A) ≤ λi (A B) ≤ b1λ1(A) (1.69)

and also

λm(A)λm(B) ≤ λi (A B) ≤ λ1(A)λ1(B). (1.70)

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

Kronecker ProductFor any two matrices A and B of arbitrary sizes m × n and p × qrespectively, the Kronecker product A⊗B is given by the concatenatedblock matrix C whose (i, j)th block-entry is given by

Cij = aijB, 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 aij). Hence

C = A ⊗ B =

a11B a12B · · · a1nB

a21B a22B · · · a2nB

· · · · · · · · · · · ·am1B am2B · · · amnB

(1.72)

is of size mp × nq .


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

(A ⊗ B) ⊗ D = A ⊗ (B ⊗ D), (1.73)

and

A ⊗ (B + D) = A ⊗ B + A ⊗ D. (1.74)

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) isgiven by

∑k

aikckjBD. Other useful relations in this context include

(A ⊗ B)∗ = A∗ ⊗ B∗, (1.76)

(A ⊗ B)−1 = A−1 ⊗ B−1, (1.77)

and

ρ(AB) = ρ(A)ρ(B). (1.78)

Interestingly, we can rewrite the Kronecker product (1.72) using thecolumn vectors of A and B as follows. Let a i and bi represent thecolumns of A and B respectively. Thus

A = (a1, a2, . . . , an), B = (b1, b2, . . . , b q ). (1.79)

Then from (1.72) it follows that

C = A ⊗ B = (a1 ⊗ B, a2 ⊗ B, . . . , a i ⊗ B, . . . , an ⊗ B)

= (a1 ⊗ b1, a1 ⊗ b2, . . . , a1 ⊗ b q , a2 ⊗ b1,

a2 ⊗ b2, . . . , a2 ⊗ b q , . . . , an ⊗ b q ).(1.80)

Thus the columns of the Kronecker product A ⊗ B are a i ⊗ b j forall i , j combinations arranged in lexicographic order.

Khatri-Rao ProductWhen A and B have equal number of columns (n = q ), the specialsubset

a i ⊗ bi

, i = 1, 2, . . . , n of (1.80) arranged in lexicographic

order defines the Khatri-Rao product A B. Thus [25]

A B = (a1 ⊗ b1, a2 ⊗ b2, . . . , an ⊗ bn

)(1.81)


and it represents a subset of the columns of the Kronecker productA ⊗ B. Hence it follows that

A B = (A ⊗ B) En (1.82)

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

En = (e1, en+2, e2n+3, . . . , en2

), (1.83)

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

ek =0, 0, . . . , 1︸︷︷︸

k

0, . . . , 0

T

. (1.84)

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

a i bi = a i ⊗ bi =

a1,i bi

a2,i bi

...

am,i bi

. (1.85)

From (1.61) and (1.62), the Schur product of ai and bi is given by

a i bi =

a1,i b1,i

a2,i b2,i

...

am,i bm,i

, (1.86)

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

a i bi = ETm(a i bi ) (1.87)

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

A B = ETm(A B). (1.88)

Together with (1.82), we obtain the useful identify

A B = ETm(A ⊗ B)En. (1.89)


We can use (1.82) and (1.89) to establish an important result: Forany two matrices A and B of size m × n and p × n, we get

(A B)∗(A B) = ETn (A ⊗ B)∗(A ⊗ B)En

= ETn (A∗ ⊗ B∗)(A ⊗ B)En = ET

n (A∗A ⊗ B∗B)En

= A∗A B∗B. (1.90)

Thus with Rx = A∗A, Ry = B∗B we can write

Rx Ry = 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

AB CD = ETm(AB ⊗ CD)En = ET

m(A ⊗ C)(B ⊗ D)En

= [(AT ⊗ CT )Em]T (B D) (1.92)

= (AT CT )T (B D),

where the last two steps follow from (1.82). Notice that (1.90) is aspecial 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, . . . , λmand µ1, µ2, . . . , µn respectively. Also, consider the situation where Aand B have a full set of eigenvectors ui and v j respectively. In thatcase, we have

Aui = λi ui , i = 1 → m, (1.96)

Bvi = µi vi , i = 1 → n (1.97)


which in compact form reads

AU = UΛ1 (1.98)

BV = VΛ2 (1.99)

where

U = ( u1 u2 . . . um ) (1.100)

V = ( v1 v2 . . . vn ) (1.101)

Λ1 =

λ1 0 · · · 0

0 λ2 · · · 0...

.... . .

...

0 0 · · · λm

(1.102)

Λ2 =

µ1 0 · · · 0

0 µ2 · · · 0...

.... . .

...

0 0 · · · µn

. (1.103)

From (1.98) and (1.99)

A = UΛ1U−1 (1.104)

B = VΛ2V−1 (1.105)

and hence repeated application of (1.75) gives

A ⊗ B = U(Λ1U−1)⊗ V

(Λ2V−1) = (U ⊗ V)

(Λ1U−1 ⊗ Λ2V−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 ⊗ Band the product set λiµ j , i = 1 → m, j = 1 → n represents theeigenvalues of A ⊗ B. It follows from (1.106) that if both A and B arepositive definite, then so is A ⊗ B.

A matrix function can be vectorized in many ways. Neudecher’svector function of a matrix is obtained by stacking its column vectorsin lexicographic order. Thus for the matrix A as defined in (1.79), we


get [27], [28]

vec(A) =

a1

a2

...

am

. (1.107)

Using (1.107), Neudecker has shown that for any conformal matri-ces A, X, and B

vec(AXB) = (BT ⊗ A)vec(X). (1.108)

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

vecd(A) =

a11

a22

...

amm

. (1.109)

As a result, if X is a diagonal matrix in (1.108), then using the Khatri-Rao 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)

by rewriting it as

(BT ⊗ A + DT ⊗ C)vec(X) = vec(Q). (1.112)

Notice that (1.112) is of the form Ay = b. If X is a priori known tobe diagonal in (1.111), then using (1.110), we obtain the more compactrepresentation

(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)


where Q is positive definite, gives the solution (use (1.108))

(I ⊗ I − A ⊗ A)vec(X) = −vec(Q) (1.115)

or

vec(X) = −B−1vec(Q) (1.116)

where

B = I ⊗ I − A ⊗ A. (1.117)

It is well known that the matrix solution X in (1.114) is positive-definite 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 thepositive-definite nature of X, nevertheless it can be simplified todemonstrate that form. Toward this, consider the eigen decomposition

A = TΛV =n∑

i=1

λi ei vi (1.118)

in (1.23). Then by direct computation

B(e j ⊗ ei ) = (1 − λiλ∗j

)(e j ⊗ ei ) (1.119)

so that

B(T ⊗ T) = (T ⊗ T)ΛB, (1.120)

where ΛB is diagonal with entries 1 − λiλ∗j . This gives

B = (T ⊗ T)ΛB(T ⊗ T)−1 = (T ⊗ T)ΛB(T−1 ⊗ T−1)

= (T ⊗ T)ΛB(V ⊗ V), (1.121)

where V is as defined in (1.22). Hence as in (1.24), we get

B−1 =n∑

i=1

n∑j=1

(e j ⊗ ei )(v j ⊗ vi )

1 − λiλ∗j

=n∑

i=1

n∑j=1

(e j v j ) ⊗ (ei vi )

1 − λiλ∗j

. (1.122)

Using (1.122) in (1.116) we get

vec(X) = −n∑

i=1

n∑j=1

(e j v j ) ⊗ (ei vi )

1 − λiλ∗j

vec(Q) (1.123)

or

X = −n∑

i=1

n∑j=1

(ei vi )Q(e j v j )∗

1 − λiλ∗j

, (1.124)


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

Y = ( Y1 Y2 . . . Yn )T , Yi = ei vi , Ri, j = − Q1 − λiλ

∗j. (1.125)

Clearly X is Hermitian positive definite if and only if R is positivedefinite, 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 LemmasGiven a square matrix A and commensurate vectors a and b wehave [29]

(A + a b∗)−1 = A−1 − A−1a b∗A−1

1 + b∗A−1a. (1.127)

For square matrices A and B that are also invertible,

(A + B)−1 = A−1 − A−1(A−1 + B−1)−1A−1 (1.128)

from which it follows that

(A + BQB∗)−1 = A−1 − A−1B(B∗A−1B + Q−1)−1B∗A−1, (1.129)

provided A, B, Q are invertible. Let

R = I + P1α1α∗1 + P2α2α

∗2. (1.130)

Then

R−1 = I − γ (γ1α1α∗1 + γ2α2α

∗2 − ρsγ1γ2α1α

∗2 − ρ∗

s γ1γ2α2α∗1) (1.131)

where

ρs = α∗1α2, (1.132)

γ1 = P1

1 + P1α∗1α1

, (1.133)

γ2 = P2

1 + P2α∗2α2

, (1.134)

and

γ = 11 − |ρs |2γ1γ2

. (1.135)


Also (A B

C D

)−1

=(

A−1 + FEG −FE

−EG E

)(1.136)

where

E = (D − CA−1B)−1, (1.137)

F = A−1B, (1.138)

and

G = CA−1. (1.139)

Also if |A| = 0, then(A B

C D

)(I −A−1B

0 I

)=(

A 0

C D − CA−1B

)(1.140)

from which follows the determinantal identity∣∣∣∣∣A B

C D

∣∣∣∣∣ = |A| |D − CA−1B|. (1.141)

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

∣∣∣∣ = |D| |A − BD−1C|. (1.142)

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

Appendix 1-A: Line Spectra and SingularCovariance Matrices

Consider the Hermitian Toeplitz matrix

Tn =

ro r1 · · · rn−1 rn

r∗1 ro r1 · · · rn−1... · · · . . . · · · ...

r∗n−1 · · · r∗

1 ro r1

r∗n r∗

n−1 · · · r∗1 ro

=

rnrn−1

Tn−1...

r1

r∗n r∗

n−1 · · · r∗1 ro

,

(1A.1)


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

rk = Ex((i + k)T)x∗(iT) (1A.2)

and with

x = [x(iT), x((i + 1)T), . . . , x((i + n)T)]T , (1A.3)

from (1.30) we have

Tn = Ex 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(iT) 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−1are 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. Supposeno = 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 particularno > 0, a contradiction. Hence we must have no+1 = 0, and fol-lowing the same argument we have k = 0, k > no . Since Tno onlyinvolves only one unknown rno compared to Tno−1 (see (1.31)), theconditions no−1 > 0 and no = 0 completely determines rno in termsof ro , r1, . . . , rno−1. The remaining higher order covariances can be sim-ilarly determined as well in terms of ro , r1, . . . , rno−1.

To summarize, at any stage if the covariance matrix of a stochasticprocess becomes singular, then all higher order covariance matricesare also singular. In the case of stationary processes, the higher ordercovariances are completely determined in terms of the lower orderones. Thus

no−1 > 0, no = 0 ⇒ no+1 = 0, no+2 = 0, · · · (1A.8)

In particular, if the covariances satisfy

rk =no∑

i=1

Pi e− jkωi , k = 0 → ∞, (1A.9)

where Pi > 0, then substituting these into (1.30), we get

Tn = AnPA∗n (1A.10)


wi wnw

w1 w2

S (w)

Pn

P2

P1

Pi

FIGURE 1.2 Linespectra.

where the (n + 1) × no matrix

An =

1 1 · · · 1

e− jω1 e− jω2 · · · e− jωno

e− j2ω1 e− j2ω2 · · · e− j2ωno

...... · · · ...

e− jnω1 e− jnω2 · · · e− jnωno

, (1A.11)

and the no × no matrix

P =

P1 0 0 0

0 P2 0 0

0 0. . . 0

0 0 0 Pno

. (1A.12)

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 corres-pond to line spectra given by (see Figure 1.2)

S(ω) =+∞∑

k=−∞rke− jkω =

no∑i=1

Piδ(ω − ωi ). (1A.14)

In summary, only pure sinusoidal processes possess the determi-nantal 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 Air-borne Vehicles or Satellite Reception,” Proc. 1997 IEEE National Radar Confer-ence, pp. 126–131, Syracuse, NY, 13–15 May 1997.


[2] M.E. Davis, B. Himed, and D. Zasada, “Design of Large Space Based Radar forMultimodee Surveillance,” Proc. 2003 IEEE Radar Conference, Huntsville, AL,5–8 May 2003.

[3] Y. Zhang, A. Hajjari, L. Adzima, and B. Himed, “Application of Beam-Domain STAP Technologies to Space-Based Radars,” Proc. 36th IEEE Southeast-ern Symposium on System Theory (SSST) Conference, Atlanta, GA, 14–16 March2004.

[4] L.L. Fu and B. Holt, “ Seasat Views Oceans and Sea Ice with Synthetic ApertureRadar,” Technical Final Report, JPL Publication 81–120, Pasadena, CA, February1982.

[5] T. Freeman, “JPL Imaging Radar: A Tutorial,” also available at http://southport. jpl.nasa.gov/, July 1994.

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[12] M. Martin, et al., “Techsat 21 qnd Revolutionizing Space Missions UsingMicrosatellites,” Proc. of 15th AIAA/USU conference on small satellites, Logan,VT, August 2001.

[13] R. Klemm, “Comparison Between Monostatic and Bistatic Antenna Configu-rations for STAP,” IEEE Transactions on Aerospace and Electronic Systems, Vol. 36,No. 2, April 2000.

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[15] W.L. Melvin, B. Himed, and M. E. Davis, “Doubly Adaptive Bistatic ClutterFiltering,” 2003 IEEE Radar Conference, Huntsville, AL, May 5–8, 2003.

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Part F., No. 7, pp. 604–612, December 1986.[19] R.E. Kell, “On the Derivation of Bistatic RCS from Monostatic Measurements,”

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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, Aca-

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Applications to Characterization of Probability Distributions,” Sankhya: TheIndian J. Stat., Series A, 30, pp. 167–180, 1968.

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[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.

C H A P T E R 2The Conics

All heavenly objects such as planets, comets, and man made satellitesmove around in space subject to a central force of attraction—viz., theinverse square law of gravitation. An interesting aspect of this forceis that the resulting orbits are planar—circles, ellipses, parabolas, andhyperbolas 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 ageneral quadratic (with suitable change of axes) in two variables suchas [1]

ax2 + by2 + c = 0. (2.1)

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

FPPL

= e, (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 haver

s − r cos θ= e (2.3)

or

r = p1 + e cos θ

(2.4)

with

p = se (2.5)



Circle

Ellipse Parabola

Hyperbola

FIGURE 2.1Sections of a cone.

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 perpendicularto FC has length equal to 2p. By setting θ = 0, the chord FA haslength

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 tothe y-axis.

LP

p

F

Focus

s – r cosq

sC

y-axis

x-axis O

Directirx

r

q

D

E

A

FIGURE 2.2 A general conic section.

C h a p t e r 2 : T h e C o n i c s 33

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 andshow that it leads to a special case of (2.4) with e < 1.

2.1.1 EllipseThe sum of distances from two fixed points (foci) to any point on theellipse is a constant (see Figure 2.3). Thus with F1 P = r1 and F2 P = r2we have

r1 + r2 = 2a . (2.7)

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

(x + c)2 + y2 +√

(x − c)2 + y2 = 2a (2.8)

or

(x + c)2 + y2 = 4a2 − 4a√

(x − c)2 + y2 + (x − c)2 + y2, (2.9)

√(x − c)2 + y2 = −xc − a2

a= a − xc

a, (2.10)

r1r2

F2 F1

2a

2b

P

OB

(x,y)

2c

A

FIGURE 2.3 Ellipse.


x2 + c2 − 2xc + y2 = a2 − 2xc + x2c2

a2 , (2.11)

x2 a2 − c2

a2 + y2 = a2 − c2, (2.12)

x2

a2 + y2

a2 − c2 = 1. (2.13)

Define

b2 = a2 − c2 (2.14)

so that the equation of the ellipse becomes (for the Cartesian coordi-nate system based at O)

x2

a2 + y2

b2 = 1. (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

b2 = a2(1 − e2). (2.16)

This gives e < 1,

c = ae (2.17)

and hence eccentricity represents the position of the focus as a functionof the semimajor axis a.

Let (r, θ ) represent the polar coordinates of a point P with respectto a focus F1 as in Figure 2.4 and (x, y) its Cartesian coordinates withrespect 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

a2 + r2 sin2 θ

a2(1 − e2)= 1, (2.20)

(ae + r cos θ )2(1 − e2) + r2 sin2 θ = a2(1 − e2), (2.21)


r qF2 F1c

(x,y)P

OT1

T2

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

(a2e2 + r2 cos2 θ + 2a e r cos θ )(1 − e2) + r2 sin2 θ = a2(1 − e2),

(2.22)

r2 − e2r2 cos2 θ + 2a (1 − e2) e r cos θ + a2e2(1 − e2) = a2(1 − e2),

(2.23)

a2(1 − e2)2 − 2a (1 − e2) e r cos θ + e2r2 cos2 θ = r2 (2.24)

which gives

r = ± (e r cos θ − a (1 − e2)). (2.25)

When e = 0, r = ± a . But r is always positive, hence

r = −(er cos θ − a (1 − e2)), (2.26)

or

r = a (1 − e2)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 − e2) (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 PT1 = F2 PT2. To prove this, reflect F1on the tangent T1T2 at P (Figure 2.4) to produce F ′

1 and join F2 F ′1 to

intersect the tangent at P1 (see Figure 2.5).Then F2 P1 F ′

1 is the shortest path from F2 through a point onthe tangent to F ′

1. For any other point P2 on the tangent we have


F1F2

P1P2

T1

T2

F ′1FIGURE 2.5Reflection of F1 ontangent T1T2.

F2 P2 F ′1 > F2 P1 F ′

1 and hence

F2 P2 F1 = F2 P2 F ′1 > F2 P1 F ′

1 = F2 P1 F1. (2.29)

Thus P1 on T1T2 satisfies the property that for any other point P2 onT1T2

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 thetangent (Figure 2.4) since every other point on the tangent lies outsidethe ellipse which is a closed convex surface. Hence P1 in Figure 2.5coincides with P in Figure 2.4, and as a result in Figure 2.5

F ′1 P1T1 = F1 P1T1 = F2 P1T2 (2.31)

and this proves the claim. Notice that we have made use of the closedconvexity property of the ellipse in proving (2.31).

From Figure 2.5 we also obtain F1 P1 = P1 F ′1. But from (2.7)

F2 P1 + F1 P1 = 2a, (2.32)

so that

F2 P1 + P1 F ′1 = F2 F ′

1 = 2a, (2.33)

i.e., F ′1 (the reflection of F1 about the tangent at P) lies on the circum-

ference of a circle with center at the other focus F2 and radius equalto 2a (see Figure 2.6).

Yet another useful interpretation of an ellipse is also possible: InFigure 2.7, let P be any point on the ellipse with center-based Cartesiancoordinates (x, y). Draw an eccentric (auxiliary) circle of radius a withcenter at O. Draw a perpendicular line to the major axis OA throughP to cut the circle at Q and the major axis at R. Let QOR = E and itis known as the eccentric angle of P .


F2 F12a

P

F2 F12a

P

(a) (b)

2a2a

F ′1

F ′2

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)

Also

QR = a sin E . (2.36)

E q

O F1

(x,y)

Q

P

AB

R

(0,b)

(a,0)

PRQR

ba

=

r

a

FIGURE 2.7Alternatedefinition of anellipse using aneccentric (major)circle.


Hence for any point P on the ellipse, we obtain

PRQR

= b sin Ea sin E

= ba

, a constant less than 1. (2.37)

From (2.37), an ellipse can be viewed as the locus of points P (onthe line QR) such that PR/QR is a constant less than 1, where Q is anypoint on the circle of radius a and QR is perpendicular to the diameterAOB. P will trace out an ellipse with semi-axes a and b (<a). From(2.37) and Figure 2.7 it also follows that the area of an ellipse equalsπab.

A useful formula exists relating the true anomaly angle θ at P andits eccentric anomaly E in Figure 2.7. To see this, we can use (2.34) and(2.17). Thus

OF1 = OR + RF1 = a cos E − r cos θ = c = ae, (2.38)

so that

r cos θ = a (cos E − e) (2.39)

and

PR = r sin θ = b sin E . (2.40)

Squaring and adding (2.39) and (2.40), and using (2.16) we get

r = a (1 − e cos E). (2.41)

From (2.39) and (2.41)

r (1 − cos θ ) = a (1 + e)(1 − cos E) (2.42)

and

r (1 + cos θ ) = a (1 − e)(1 + cos E) (2.43)

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

tan(θ/2) =√

1 + e1 − e

tan(E/2). (2.44)

Another (minor) eccentric circle can be drawn with center at O andradius b as shown in Figure 2.8. Let P be any point on the ellipsewith coordinates (x, y). Draw a perpendicular line to the minor axisOC 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)


O F1

P(x,y )

AB

RQ

C

D

b

PR a

QR b=

(a,0)

(0,b)

E ′

FIGURE 2.8Minor eccentriccircle.


x = a cos E ′ = PR. (2.46)

Also

QR = b cos E ′. (2.47)

Hence for any point on the ellipse we get (see also (2.37))

PRQR

= a cos E ′

b cos E ′ = ab

> 1. (2.48)

Hence similar to (2.37), an ellipse can be viewed as the locus of pointsP on the line QR such that PR/QR is a constant greater than one.

In summary, for any ellipse there are two eccentric circles—oneinscribing and the other circumscribing the ellipse—and any one ofthem can be used to generate the ellipse.

2.1.2 ParabolaFor a parabola the eccentricity e = 1 in (2.4) and hence often it canbe thought of as the limiting case of an ellipse with e → 1. Since thesemilatus rectum p = F1 D is finite for a parabola, if we let e → 1 in anellipse, then from (2.28) a → ∞ for a parabola. From (2.4) with e = 1we obtain

r = p2

sec2(θ/2) = q sec2(θ/2) (2.49)

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 parabolacentered at A is

y2 = 2px, (2.50)


F1

D

E

y-axis

rq

qθ

P

A

FIGURE 2.9 Parabola.

since D has coordinates ( p/2, p). Parabola has only one (finite) focusF1 at ( p/2, 0) since F1 A = p/2. The second focus is at infinity since thedistance F1 F2 = 2ae → ∞ as e → 1 for an ellipse. Since F1 P and F2 Pmake equal angles with its tangent at P , it follows that the tangentat P makes equal angles with F1 P and the line parallel to the x-axisthrough P . Hence a ray of light traveling parallel to the x-axis getsfocused at F1 after reflection by the parabola as shown in Figure 2.9.

2.1.3 HyperbolaFor a hyperbola the eccentricity e > 1 and hence r is not bounded in(2.4). Proceeding as in (2.7)–(2.16), we have a2(1 − e2) to be negative,and hence if we define

b2 = a2(e2 − 1) (2.51)

then

x2

a2 − y2

b2 = 1 (2.52)

represents the equation of a hyperbola. Hyperbola can be defined asthe locus of points P such that the difference of distances from the twofoci 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 givesthe semilatus rectum p = F1 D to be

p = a (e2 − 1). (2.54)

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

+ yb

)(xa

− yb

)= 0 (2.55)


F1

P

O

D

E

p

AF2

B

FIGURE 2.10Hyperbola.

which represents the equation of two lines that pass through the ori-gin with slope ±b/a . Hence the larger (x, y) becomes in (2.52), themore nearly the hyperbola resembles these two lines, known as theasymptotes 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), wehave

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

ba

= tan(cos−1(1/e)) = sin(cos−1(1/e))1/e

=√

1 − cos2(cos−1(1/e))1/e

=√

e2 − 1, (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 hencethe 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, theother focus being empty. Earth moves around the Sun in an ellipti-cal orbit with eccentricity e = 0.016726. The Moon moves around the


Earth in an elliptical orbit (the Earth is at one focus) with eccentricitye = 0.05490. Both orbits are nearly circular. In fact all planetary orbitsare nearly circular except for Mercury and Pluto which have slightlyprolonged eccentric elliptical orbits with eccentricities 0.250 and 0.205respectively. The orbits of all planets are more or less in the same planecalled the ecliptic (plane of the Earth’s orbit around the Sun) except thatof Pluto which has a 17 inclination with respect to the ecliptic. Theecliptic itself is inclined only at 7 from the plane of the Sun’s equator.All planets in the Solar system revolve in the counterclockwise direc-tion around the Sun (as seen from above the North Pole of the Sun). Allplanets except Venus, Uranus, and Pluto also rotate in that same sense.

Comets are a collection of rocks and ice that travel around the Sunin highly eccentric orbits surrounded by a haze of gases and dustthat for some reason did not get included into the planets during theformation of the Solar system. Comets are thought to be stored in aregion outside the orbit of Pluto in the outer regions of the Solar systemcalled the Oort cloud. From time to time, due to perturbations frompassing stars, some of them enter the inner part of the Solar systemtraveling in parabolic orbits, and many return to the Oort cloud aftera close encounter with the Sun. There are many periodic comets inthe Solar system moving in elliptic orbits including Halley’s comet.Chance encounters close to Jupiter are thought to be the main causethat perturbs their orbits from parabolic to highly eccentric ellipses.

Comets have a nucleus that is relatively solid and stable. Made ofice and gas it is surrounded by a hydrogen cloud extending to millionsof kilometers in diameter. The nucleus is followed by a dust tail upto ten million kilometers long composed of particles escaping fromthe nucleus that get “heated up” as the comet passes around the Sun.Many comets have pronouncedly elongated elliptical orbits, often be-ing nearly parabolic with periods on the order of 10–3,000 years withthe Sun at their focus. Of the 1,000 comets whose orbits have beencomputed, fewer than 100 have periods less than 100 years. Manycomet orbits extend considerably farther out into space beyond Pluto(trans-plutonian space), that is still dominated by the gravitationalfield of the Sun. The Sun in fact is able to hold bodies as far as twoto three light years away from its center. The average period of thefamous Halley’s Comet is about 77 years and this gives the major axisof its elliptical orbit to be about 2.4933 light hours or 18.07 A.U. Aftera finite number of passes (500–1,000) around the Sun, a comet losesmost of its ice and gas leaving a rocky object similar to an asteroid.Meteor showers occur when the Earth passes through the dust tail ofa comet.

There are millions of asteroids in the Solar system. Most of them arein stable elliptical orbits in the asteroid belt between Mars and Jupiter.However, some of them have Earth crossing orbits as well. As theEarth moves around the Sun, these objects can occasionally strike the


Earth and other planets causing destruction. The degree of destructiondepends on the size and speed of the striking object. For example, a50-meter object can destroy a city; an object of size 1–2 km striking at45,000 km/h can practically wipe out life on Earth and “sterilize” theplanet. About 2,000 such objects circulate around the Sun. Less catas-trophic attacks are more often. For example, the Tunguska explosionover Siberia in 1908 was caused by a 60-meter object that explodedover the atmosphere flattening thousands of kilometers of forest.

It is easy to estimate the probability of Earth impact due to suchan object. Recall that Earth travels around the Sun in approximatelya circle of radius Ro = 150 million km. Most of the objects of interestthat loop around the Sun cut through this “Ro radius sphere” that hasa surface area of 4π R2

o (total exposed area). Earth’s surface area 4πr2e

represents the desired area. Thus probability of Earth impact is givenby (Laplace’s definition)

pH = Earth′s areaTotal area

= 4πr2e

4π R2o

=(

re

Ro

)2

=(

6,378150 × 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 Sunand an estimate of probability of impact on Earth due to each class.

N: Number of ObjectsNEO Size Around the Sun/Year Prob. of Hit: p =N× pH

10 km ∼30 2 × 10−8 ∼ once every50 Million years∗

1 km 2,000 3.6 × 10−6 ∼ once every200,000 years

100 m 300,000 5.4 × 10−5 ∼ once every20,000 years

50 m ∼1 − 10 Million 2 × 10−2− 5 × 10−3 ∼ onceevery 50–100 years♣

∗(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 Sunevery 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 theSun and the estimated probability of impact on Earth.


2.2 The Solar SystemA long time ago, a cloud of interstellar gas and dust began to collapseunder its own gravity, possibly prompted by shock wave disturbancesfrom nearby exploding supernovas. As the cloud collapsed it heatedup vaporizing gas and the center began to compress forming a starin about 100,000 years. The rest of the gas flowed toward the centerin 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 andbegan to condense into metal, rock, and other particles. These particlescollided forming larger particles and asteroids. With nontrivial gravityeventually they pulled in more particles and after about 10–100 millionyears planets were formed around the Sun.

The Sun is at the center of the Solar system with about 99.2% ofthe total mass of the Solar system in it (Ms 1.99 × 1030 kg) andJupiter contains most of the rest. The Sun is giant with its radius about109 times the radius of the Earth. In terms of sphere packing, about1,300,000 Earths could fit easily inside the Sun. Jupiter and Saturn arenext 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 thatof 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 suchas the Moon, and a large number of comets and meteors in the Solarsystem.

The Sun is about 70% hydrogen, 28% helium, and everything elseamounts to less than 2% by mass. This composition changes slowlyas the Sun converts hydrogen to helium through nuclear fusion at itscore. This process also generates gamma rays that eventually travel to-ward the Sun’s surface. As the gamma rays progress toward the Sun’ssurface, energy is continuously absorbed and reemitted at lower andlower temperatures within the Sun’s body, resulting in visible lightat the surface. It took about 4.5-billion light years for the Sun to getto its current state. By now the Sun has used up about half of the hy-drogen supply in its core. The Sun will continue to shine peacefullyfor another 5 billion years with its intensity roughly doubling duringthat time. In the end, the Sun will exhaust all of its hydrogen fuel andit will collapse triggering a nuclear fusion reaction of the elements atits core. This in turn will push the surrounding hydrogen outward to


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

(b) Then there was bright light.

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

(d) At the end, back to darkness.

Earth

Sun

FIGURE 2.11Darkness to lightto 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).

1The “immediate” cause for concern is the (partial/total) destruction of lifeon Earth that can be caused by an impact of an Earth-crossing asteroid or acomet 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 aster-oids can cause tremendous destruction. Hopefully the SBR technology will bein place on time to detect, track, and if necessary to deflect these Earth-crossingobjects.


c

Ca

b

CB

A

A

B

FIGURE 2.12Ordinary triangle.

Appendix 2-A: Spherical TrianglesIn an ordinary planar triangle ABC (Figure 2.12), the following rela-tion holds among the angles A, B, C and opposite sides a , b, c:

asin A

= bsin B

= csin C

. (2A.1)

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

sin B = pc

, sin C = pb

(2A.2)

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

Spherical TriangleIf three points are chosen on the surface of a sphere, then three uniquegreat circles can be drawn through them by selecting two of thesepoints at a time. These three great circles intersect generating threearcs on the surface of the sphere that form the boundaries of a spher-ical triangle (see Figure 2.13). The planes of these great circles form atrihedral at the center of the sphere as in Figure 2.13. The tangents to thearcs AB and AC at A along these planes are perpendicular to the radiusOA and the interior angle defined by these tangents gives the sphericalangle A.

In a spherical triangle located on a unit sphere, the angles A, B, Cso defined and the opposite side (arcs) a , b, c satisfy the followingrelation [4]:

The Law of Sines

sin asin A

= sin bsin B

= sin csin C

. (2A.3)


A

B

C

O a

bc

A

FIGURE 2.13Spherical triangle.

Proof We shall first prove (2A.3) for a right spherical triangle thathas one angle equal to 90 as shown in Figure 2.14.

Draw a plane through A perpendicular to the line OB that cuts thespherical pyramid in the triangle ADE. Since OD is perpendicular toDA and DE, the angle ADE equals the spherical angle B. From triangleAOE, by ordinary trigonometry we have

AE = sin b. (2A.4)

From triangle AOD, we have

AD = sin c. (2A.5)

B

B

A

A

C90°

90°

90°B

90°

b

C

a

Oc

a

b

D

E

90°

1

1

FIGURE 2.14 Right spherical triangle.


Hence from triangle ADE, we get

sin B = AEAD

= sin bsin c

(2A.6)

or

sin c = sin bsin B

. (2A.7)

Next, consider a general spherical triangle ABC in Figure 2.15, wherep 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) tothe spherical triangles ABD and ACD, we obtain

sin c = sin psin B

, (2A.8)

and

sin b = sin psin C

. (2A.9)

This gives

sin c sin B = sin b sin C (2A.10)

or

sin bsin B

= sin csin C

. (2A.11)

Similarly, considering another great circle arc from B that is per-pendicular to the opposite side b we get

sin asin A

= sin csin C

. (2A.12)

A

CBB C

A

c

a

b

D

p

90° ρ

FIGURE 2.15General sphericaltriangle.


Combining (2A.11) and (2A.12), we get (2A.3) and that completesthe proof for the law of sines.

Finally, if the sphere is of radius r, then a great arc of length a extendsan angle a/r at the origin, and hence (2A.3) holds good in that casewith a , b, c replaced by α = a/r , β = b/r , and γ = c/r , i.e.,

sin α

sin A= sin β

sin B= sin γ

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 ofmagnitude of the sides of a spherical triangle is the same as the orderof magnitude of the respective opposite angles, i.e.,

a < b < c ↔ A < B < C . (2A.14)

The Law of CosinesFor 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)

and

DE = OE sin a = cos b sin a . (2A.18)

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 = DEAD

= cos c tan asin c

(2A.20)

or

tan a = tan c cos B. (2A.21)

Equations (2A.19)–(2A.21) are valid for right spherical trianglesTo prove (2A.15) in the general case, refer to the general spheri-

cal triangle in Figure 2.15, and let the great arc DC = ρ. Then arc


BD = a −ρ, and applying (2A.19) to the right spherical triangles ABDand ACD, we get

cos c = cos p cos(a − ρ) (2A.22)

and

cos b = cos p cos ρ (2A.23)

or

cos ccos b

= cos(a − ρ)cos ρ

= cos a + sin a tan 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.

C H A P T E R 3Two Body Orbital Motion

and Kepler’s Laws

The problem of two body motion was first solved by Newton around1685 and the solution is given in his Principia. From studying Kepler’slaws, the gravity at the Earth’s surface,1 and the motion of the Moonaround the Earth, Newton was led to the universal law of gravitationwhich states that “every two bodies of matter in the universe attract eachother with a force that acts in the line joining them, and whose intensity variesas the product of their masses and inversely as the square of their distance.”

The law of gravitation involves considerably more than planetarymotion; however, Newton had nothing more than Kepler’s laws tobuild it on. Nevertheless, by a master stroke of genius Newton statedthe law of gravitation in immense generality applying it to the wholeuniverse and it has stood in its entirety without change for the last300 years. It is however noteworthy to observe that the question ofwhether the law of gravitation is truly universal has been proved onlyto hold in the Solar system and in the motion of double stars.

3.1 Orbital MechanicsOrbital motion of planets and satellites can be derived from Newton’slaw of gravitation [1], [2], [3]. Let (x1, y1, z1) and (x2, y2, z2) denotethe coordinates of the Sun and the planet with respect to an inertialreference frame centered at O as in Figure 3.1. Further, let M and mdenote the masses of the Sun and the planet.

By Newton’s second law, the planet P is attracted to the Sun withforce G Mm/r2 where r is the distance SP. The component of this force

1Among other things, the interesting anecdote of an apple falling on Newton’shead.



P

S

Z

X

OY

r

(x1,y1,z1)

(x2,y2,z2)

qx

FIGURE 3.1 TheSun and a planet.

in the positive x-direction equals

− GMmr2 cos θx = −GMm

r2

x2 − x1

r. (3.1)

Thus if d2x2

dt2 denotes the acceleration of the planet P parallel to thex-axis, we have

md2x2

dt2 = −GMmr3 (x2 − x1). (3.2)

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

Md2x1

dt2 = GMmr3 (x2 − x1). (3.3)

From (3.2) and (3.3) we get

d2(x2 − x1)

dt2 = −G(M + m)r3 (x2 − x1). (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 MassTo investigate the motion of the center of mass under the inversesquare law, consider the center of mass of the two-body system inFigure 3.1 is given by

x = Mx1 + mx2

M + m

y = My1 + my2

M + m(3.5)

z = Mz1 + mz2

M + m

C h a p t e r 3 : T w o B o d y O r b i t a l M o t i o n a n d K e p l e r ’ s L a w s 53

and it lies on the line joining M and m. To determine its motion underinverse square law, we can make use of (3.2) and (3.3). From there weget

Md2x1

dt2 + md2x2

dt2 = 0. (3.6)

Similarly for the y and z coordinates we get

Md2 y1

dt2 + md2 y2

dt2 = 0

Md2z1

dt2 + md2z2

dt2 = 0. (3.7)

Consecutive integration of these equations yield

Mdx1

dt+ m

dx2

dt= α1

Mdy1

dt+ m

dy2

dt= α2 (3.8)

Mdz1

dt+ m

dz2

dt= α3

and

Mx1 + mx2 = α1t + β1

My1 + my2 = α2t + β2 (3.9)

Mz1 + mz2 = α3t + β3.

Using (3.1)–(3.3) in (3.6) we get

(M + m)x(t) = α1t + β1

(M + m) y(t) = α2t + β2 (3.10)

(M + m)z(t) = α3t + β3

or(M + m)x − β1

α1= (M + m) y − β2

α2= (M + m)z − β3

α3, (3.11)

i.e., the coordinates of the center of mass move in a straight line. Takingthe derivatives in (3.10), squaring and adding we also obtain

V(t) =(

dxdt

)2

+(

d ydt

)2

+(

dzdt

)2

= α21 + α2

2 + α23

(M + m)2 = c, (3.12)

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


What is really important is to determine the relative motion of onebody 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 MotionIn this context, let

x = x2 − x1, y = y2 − y1, z = z2 − z1 (3.13)

represent the coordinates of the planet P referred to the rectangularaxes passing through the Sun (heliocentric coordinate system). Thenwith

µ = G(M + m) (3.14)

from (3.4) we obtain

d2x

dt2 = −µxr3 , (3.15)

and in a similar manner we get

d2 y

dt2 = −µyr3 , (3.16)

d2z

dt2 = −µz

r3 . (3.17)

Equations (3.15)–(3.17) represent the motion of the planet with ref-erence to the Sun. From (3.15)–(3.16) we get

xd2 y

dt2 − yd2x

dt2 = 0 (3.18)

or

ddt

(x

dydt

− ydxdt

)= 0 (3.19)

which gives

xdydt

− ydxdt

= A. (3.20)

Similarly from (3.16)–(3.17) we get

ydzdt

− zdydt

= B, (3.21)

and from (3.15) and (3.17) we have

zdxdt

− xdzdt

= C. (3.22)


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 inversesquare 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) representthe equations of motion for the planet.

To analyze this two-dimensional (2D) motion further, let (r, θ ) rep-resent the heliocentric polar coordinates of the planet with referenceto 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)

Hence

dxdt

= drdt

cos θ − r sin θdθ

dt(3.25)

and

dydt

= drdt

sin θ + r cos θdθ

dt. (3.26)

Also from (3.25)

d2x

dt2 = d2r

dt2 cos θ − 2drdt

sin θdθ

dt− r cos θ

(dθ

dt

)2

− r sin θd2θ

dt2

=(

d2r

dt2 − r(

dθ

dt

)2)

cos θ −(

rd2θ

dt2 + 2drdt

dθ

dt

)sin θ (3.27)

r

(r,q )

qx

y

P

S

FIGURE 3.2 TheSun centered polarcoordinate system.


and from (3.26)

d2 y

dt2 = d2r

dt2 sin θ + 2drdt

cos θdθ

dt− r sin θ

(dθ

dt

)2

+ r cos θd2θ

dt2

=(

d2rdt2 − r

(dθ

dt

)2)

sin θ +(

rd2θ

dt2 + 2drdt

dθ

dt

)cos θ. (3.28)

Let αr and αθ represent the components of the acceleration of theplanet along the radial direction SP and at right angles to SP respec-tively. Then

αr = d2x

dt2 cos θ + d2 y

dt2 sin θ , (3.29)

and

αθ = −d2x

dt2 sin θ + d2 y

dt2 cos θ. (3.30)

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

αr = d2r

dt2 − r(

dθ

dt

)2

. (3.31)

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

αr = − µ

r3 (x cos θ + y sin θ ) = − µ

r2 . (3.32)

From (3.31)–(3.32) we get

αr = d2r

dt2 − r(

dθ

dt

)2

= − µ

r2 . (3.33)

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

αθ = rd2θ

dt2 + 2drdt

dθ

dt= 1

rddt

(r2 dθ

dt

). (3.34)

However (3.15)–(3.16) in (3.30) gives

αθ = µ

r3 (x sin θ − y cos θ) = µ

r3 (r cos θ sin θ − r sin θ cos θ ) = 0(3.35)

and hence from (3.34)–(3.35) we get

ddt

(r2 dθ

dt

)= 0 (3.36)


rdA

dq

S

P

Q rdqFIGURE 3.3 Areaswept by theplanet P aroundthe Sun S.

or

r2 dθ

dt= h(constant). (3.37)

But 12r2dθ = 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),

dAdt

= 12

r2 dθ

dt= h/2 (3.38)

or

A(t) = ht/2. (3.39)

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 in-tervals of time. Thus depending on the size of the radius vector, theplanet has to move faster or slower to satisfy this requirement. Thisremarkable result is the mathematical expression of Kepler’s secondlaw.

3.2 Kepler’s LawsAs we have seen, Equations (3.37)–(3.39) are the mathematical expres-sion of Kepler’s second law that states that “the radius vector SP of aplanet sweeps equal areas in equal time intervals.”

To obtain a parametric expression for planetary motion, we need toeliminate t in (3.33) and (3.38). We define

u = 1r

(3.41)


so that from (3.37)

dθ

dt= hu2 (3.42)

and

drdt

= − 1u2

dudt

= − 1u2

dudθ

dθ

dt= −h

dudθ

(3.43)

and

d2rdt2 = d

dθ

(drdt

)dθ

dt= −h2u2 d2u

dθ2 . (3.44)

Also

r(

dθ

dt

)2

= h2u3. (3.45)

Substituting (3.44) and (3.45) into (3.33) we obtain

− h2u2 d2udθ2 − h2u3 = −µu2, (3.46)

or

d2udθ2 + u = µ

h2 (3.47)

which represents the path of the planet around the Sun in polar coor-dinates.

The general solution of d2udθ2 + u = 0 is easily seen to be c1 cos θ +

c2 sin θ and hence the particular solution to (3.47) is given by

u = µ

h2 + c1 cos θ + c2 sin θ = µ

h2 + co cos(θ − θo ), (3.48)

or

r = h2/µ

1 + e cos(θ − θo )(3.49)

with

e = h2co/µ. (3.50)

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

r = p1 + e cos θ

(3.51)


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) andit represents an ellipse. Hence all closed planetary orbits around theSun 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

µ= p = a (1 − e2) (3.53)

and hence we have

h2 = µa (1 − e2). (3.54)

From (3.39) if T represents the orbital period of a planet then sinceπab represents the total area of an ellipse, we have

A = πab = hT/2 (3.55)

or

h2 = 4π2 a2b2

T2 = 4π2a4(1 − e2)T2 . (3.56)

From (3.54)–(3.56), every planet obeys

a3

T2 = µ

4π2 = G(M + m)4π2 . (3.57)

Hence if ai , a j and Ti , Tj represent the semimajor axes of two plan-etary orbits and their corresponding orbital periods, then(

ai

a j

)3

= M + mi

M + m j

(Ti

Tj

)2

, (3.58)

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

Ti

Tj

)2

(

ai

a j

)3

(3.59)

2Notice that (3.49)–(3.51) can generate parabolic and hyperbolic orbits as well.


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 sizeof 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 inversesquare law leads to ellipses. The converse is also true. Thus assumethat orbital motion is governed by an ellipse. Then from (3.51)–(3.54)

u = 1 + e cos θ

a (1 − e2)(3.60)

and hence

d2udθ2 = − e cos θ

a (1 − e2)= 1

a (1 − e2)− u (3.61)

or

d2udθ2 + u = 1

a (1 − e2)(3.62)

which agrees with (3.47), an equation derived from the inverse squarelaw, thus proving the claim that for closed orbits inverse square lawand elliptical orbits are equivalent.

Notice that from the single law of gravitation, Kepler’s three lawscan be derived.

3.3 Synchronous and Polar OrbitsPositioning of synchronous satellites gives an interesting applicationof Kepler’s third law. For a satellite to be geosynchronous, its periodT needs to be 24 h. From Kepler’s third law this gives a = 42, 241 km(use (3.57) with G = 6.673×10−11m3 kg−1s−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 planeof the equator traveling in the same direction as the Earth, then boththe satellite and the Earth will complete one revolution in every 24 hand they both will rotate about the same axis. As a result, the satellitewill remain over the same spot over the equator at all times. Noticethat 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 max-imum Earth coverage. A satellite placed in a polar orbit (which is atright angles to the equator and passes through the Earth’s poles) on theother hand provides maximum Earth coverage for a single satellite.For example, a polar satellite at 500 km above the Earth’s surface has a


Antarctica0 4447 km

North America

Atlantic Ocean

Europe Asia

AustraliaIndian Ocean

South AmericaPacific Ocean

B

A

Geosynchronousorbit Polar orbit

EquatorEquator 0° 90°45°

(a) Geosynchronous and polar orbits (b) Mercator’s projection chart

B

A

Polar satellite paths

Africa

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

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

3.4 Satellite VelocityThe objective here is to determine the planetary velocity as a functionof the instantaneous radius vector. Toward this, let VP denote theplanet 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 vectorand r dθ

dt at right angles to the radius vector as in Figure 3.5 [1].This gives

V2P =

(drdt

)2

+ r2(

dθ

dt

)2

= h2

((dudθ

)2

+ u2

)(3.63)

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

u = µ

h2 (1 + e cos(θ − θo )) (3.64)


VPdrdt

dqrdt

P

r

O F1F2

FIGURE 3.5 Planet velocity.

and hence

V2P = µ2

h2 (1 + 2e cos(θ − θo ) + e2) = 2µu − µ2

h2 (1 − e2)

= µ

(2r

− 1 − e2

p

), (3.65)

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

V2P =

µ

(2r

− 1a

), elliptical orbit

µ

r, circular orbit

2µ

r, parabolic orbit

µ

(2r

+ 1a

), hyperbolic orbit

, µ = G(M + m). (3.66)

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


F2 F12a

P

P1

2a

rAB

FIGURE 3.6 Freefall of a mass fromthe circle to theellipse.

The acceleration of the mass m at distance s equals

d2sdt

= −G(M + m)s2 = − µ

s2 . (3.67)

Multiplying (3.67) by 2 dsdt and integrating we obtain(

dsdx

)2

= 2µ

s+ c = 2µ

(1s

− 1so

), (3.68)

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

V2p = µ

(2r

− 2so

)= µ

(2r

− 1a

), (3.69)

for so = 2a . From (3.69) and (3.66) we conclude that the speed of a bodyat any point in an elliptical orbit is the same as that of a stationary bodyunder free fall from the circumference of a circle along its radius to thesurface of the ellipse; here the radius of the circle equals to the majoraxis of the ellipse and the center of the circle is located at F1 as inFigure 3.6. Clearly the planet at its perihelion A moves the fastest andat 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 apoint 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 thatthe major axis and the period of the resulting orbit depend only onr and VP and not on the direction of projection. From (3.66), an ellip-tical orbit VP is greatest when r is the smallest (r = a (1 − e)), which


corresponds to the perihelion. Similarly the velocity is minimum whenthe planet is in aphelion3 (r = a (1 + e)).

The direction of projection, however, is very important in determin-ing the shape of the orbit [4]. Together with the speed with which thesatellite is finally launched into its orbit, it determines the type andcharacter of the satellite orbit. If the speed is Vc = √

µ/r at the launchheight 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 3.7shows three orbits all launched from P that is at a height a from thecenter of the Earth with speed equal to= √

µ/a .The circular orbit corresponds to a horizontal launch. The two ellip-

tical paths in Figure 3.7 correspond to launch directions that are high(ellipse E1) and low (ellipse E2) compared to the horizontal. Notethat both ellipses have semimajor axis equal to a , same as the radiusof the circular orbit, and if both satellites are launched at the samesmall angle α to the horizontal at P one upward and one downwardthe resulting orbits will have the same eccentricity and will be sym-metrically located with respect to the projection point. Their perigee

3If we assume the orbits of planets around the Sun are approximately circularand those of the comets parabolic, then from (3.66), the orbits of comets aroundthe Sun cross the planets’ orbits with velocities 1.414 times those of the respectiveplanets.

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

a H

a E

)=(

TH

TE

)2/3=(

771

)2/3= 18.09

or a H = 18.09a E . Using the elliptical orbit formula in (3.66), this gives the speed ofthe Halley’s comet at a distance r = a E from the Sun to be (i.e., at Earth’s orbit)

V2P = µ

a E

(21

− 118

)= 35

18µ

a E= 35

18V2

E

or VP =√

3518 VE , since VE =

√µ

a E. Assuming that Halley’s comet intersects with

the Earth’s orbit in a perpendicular direction, the relative speed between the Earthand Halley’s comet equals

V =√

12 + 3518

VE =√

5318

VE = 1.715VE = 1.715 × 29.772 = 51.08 km/sec

since VE = 2πa ETE

= 2π×1.4952×108 km365.24×24×3600 sec = 29.772 km/sec.

Kepler himself never applied his remarkable laws to comets, and it remained forHalley (1705) to prove that one comet at least, and hence many comets, moved inhighly eccentric periodic elliptical orbits around the Sun. Halley noticed that severalobservations that lead to nearly identical orbital parameters such as perihelion werealways apart by a 75–77 years interval, and he concluded that a single comet hasmade repeated returns to the perihelion at about 0.5871 A.U. Halley was able totrace the comet as far back as 1305 (Halley’s comet).


2a2a

2a

a

Earth

E1

E2

C

Circular

PVc

VcVc

Ellipticalα

FIGURE 3.7 Effectof direction ofprojection forcircular orbitvelocity.

(point closest to Earth) will be less than the projected height a . To al-low for launch velocity vector errors the launch height must be abovea minimum safety distance.

The eccentricity e of the elliptical orbits in Figure 3.7 is a functionof 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 theradius vector a at P . Hence (see Figure 3.5)

Vp cos α = adθ

dt= h

a=√

µa (1 − e2)a

, (3.70)

where we have used (3.37) and (3.54). But Vp = √µ/a = Vc and hence

we get

cos α =√

1 − e2 (3.71)

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 thatis 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 satelliteclears above the ground to be

(Re + H)(1 − sin α) − Re ≥ hmin (3.74)


Earth

r

PSatellite

E1

Elliptical

Circular

C

E2

Elliptical E3

Elliptical

Hyperbola

Parabola

Vc

FIGURE 3.8 Satellite orbits for various horizontal projection velocities.

or

α ≤ sin−1(

H − hmin

Re + H

), (3.75)

where hmin is chosen to be about 325 km to minimize the atmosphericdrag and prevent burn out. From (3.75), this gives α ≤ 1.5 for a satel-lite orbital height of 500 km and it shows that velocity deviation ineither up or down directions from the horizontal are equally undesir-able and the margin of error is very small.

Figure 3.8 shows the various orbits that result for horizontallaunches when the launch speed VP is different from Vc , the speedrequired for circular orbit [4].

If VP < Vc , the orbit is a subcircular ellipse E1 with its perigeeless than the projected height r . If Vc < VP <

√2µ/r , the orbit

will be elliptical as in E2 and E3 but the height of apogee (point far-thest away from the Earth) will be greater than the projected height.4

4Finally consider the case where the launch speed is different from Vc and thelaunch direction is different from the horizontal at an angle α away from the hori-zontal (up or down) as in Figure 3.7. Thus let Vp = Vc (1 + ε) with ε < 0.414 so thatit leads to an elliptical orbit with semimajor axis given by (use (3.66)-a)

a = r/(1 − 2ε − ε2) > r.The eccentricity e of this ellipse however depends on the launch angle, and

proceeding as in (3.70) we get√1 − e2 =

√1 − ε2(2 + ε)2 cos α < cos α

which shows that to maintain the same eccentricity the elliptical orbits have greatertolerance for launch angle errors compared to circular orbits.


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, andonce again the satellite never returns, and consequently hyperbolicorbits are important for spaceships that are attempting to leave theEarth for voyages into space.

Earth itself moves around the Sun in nearly a circular orbit (orbitC in Figure 3.8 with the center representing the Sun) with speed of29.772 km/s that is directed at right angles to the line connecting theSun and the Earth. If the Earth’s velocity is slowed down to 15 km/s,then the Earth will follow the subcircular elliptic path E1 in Figure 3.8.As the Earth falls toward the Sun its speed increases so that at perigeeits centrifugal force will exceed the gravitational pull of the Sun (eventhough that has also increased), and the Earth will pull away from theSun, slowing down as it goes until it arrives at P with the same velocityin the same direction. If we speed up the Earth in the 30–40 km/srange, the Earth will move away from the Sun along ellipses E2 andE3 slowing down near the apogee where the centrifugal force is stillinsufficient to overcome the weak gravitational pull of the Sun, andconsequently the Earth will fall back toward the Sun regaining speeduntil it reaches P with the same velocity. Increasing Earth’s velocityto 41.84 km/s will make the Earth move in the parabolic orbit inFigure 3.8 away from the Sun forever since the gravitational attractionof the Sun will be insufficient to slow down the Earth and cause it toreturn.

Appendix 3-A: Kepler’s EquationKepler’s second law that states “equal areas are swept in equal intervalsof time by the radius vector of a planet centered at a focus (Sun)” can beused to calculate the position of a planet in its elliptical orbit at anytime. Let τ represent the time at perihelion at A in Figure 3.9 and Pthe 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 = πab(t − τ )T

= 12

nab(t − τ ) (3A.3)


EO F1

Q

P

AB

R

rq

a

F2 Perihelion at t

FIGURE 3.9 Area swept by a planet.

where

n = 2π

T(3A.4)

represents the mean angular motion of the planet.From Figure 3.9,

Area AF1 P = Area F1PR + Area RPA. (3A.5)

But the area of the triangle F1PR is given by

Area F1 P R = 12

(F1 R)(PR) = 12

r cos θ r sin θ

= 12

a (cos E − e)b sin E = 12

ab sin E(cos E − e),(3A.6)

where we have used (2.39) and (2.40). We can make use of (2.37) tosimplify the elliptical segment area RPA as well. Since

PRQR

= ba

, (3A.7)

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

Area RPAArea RQA

=

∑i

bxi∑i

axi= b

a, (3A.8)


or

Area RPA = ba

(Area OQA − Area OQR)

= ba

(12

a2 E − 12

a cos E a sin E)

= 12

ab(E − cos E sin E). (3A.9)

Substituting (3A.6) and (3A.9) into (3A.5) we get

Area AF1 P = 12

ab(E − e sin E) (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, realsolution 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)

and

φ((m + 1)π ) = (m + 1)π − N > 0 (3A.16)

so that there are an odd number of solutions for φ(E) inside the inter-val (mπ, (m + 1)π ). But

dφ(E)dt

= 1 − e cos E > 0 (3A.17)

and hence φ(E) monotonically increases and takes the value zero onlyonce inside the interval (mπ, (m + 1)π ), thus proving the uniquenessclaim.


aa0

a0 + e sina

a1 a2a3

a0

a1

a2a3

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 . Substitutingthese values into the Kepler’s equation

E = N + e sin E, (3A.18)

we obtain

α = α0 + e sin α. (3A.19)

Equation (3A.19) represents the intersection of two convex func-tions α and α0 + e sin α and to find the solution the method of alter-nating 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 > α1and the graphical procedure is shown in Figure 3.10. From the methodof alternating projections, the above iteration converges to the truevalue 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 theeccentricity e of the planet’s orbit.5

5The solution of Kepler’s equation was first discovered by Kepler himself fol-lowed by Newton in his Principia. For the next 200 years, every prominent math-ematician gave his/her attention to the solution of Kepler’s equation and a verylarge number of analytic and graphical solutions have been discovered on thistopic.


Appendix 3-B: Euler’s Equation and theIdentification of Comets

Comets move reasonably rapidly through the sky and it is possible tomake intermittent observations about their positions. Let (r1, θ1) and(r2, θ2) correspond to two comet sightings (radii and true anomalies) atinstants t1 and t2 respectively (see Figure 3.11). An important problemin this context is to determine whether these observations correspondto the same comet or different comets.

An equation derived by Euler connecting the two radii and theircommon chord s for parabolic orbits can be used to answer this ques-tion. 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 leadsto (3.37), and using (3B.1) for r , it simplifies to

q 2(sec2(θ/2) + sec2(θ/2) tan2(θ/2))dθ = h =√

2µqdt, (3B.2)

since from (3.52) h2 = µp = 2µq , or

(sec2(θ/2) + sec2(θ/2) tan2(θ/2))dθ =√

2µ

q 3/2 dt. (3B.3)

The integral of this expression is

tan(θ/2) + 13

tan3(θ/2) = 1q 3/2

√µ/2(t − τ ), (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 joiningthe extremities of r1 and r2 as in Figure 3.11. From (3B.4) and withκ = √

µ/2, we have

κ

q 3/2 (t1 − τ ) = tan(θ1/2) + 13

tan3(θ1/3), (3B.5)

κ

q 3/2 (t2 − τ ) = tan(θ2/2) + 13

tan3(θ2/3). (3B.6)


F1

P1

A

P2

q1

r1 r2

s

q2

Comet

q

FIGURE 3.11 Identificationof comets using two observations.

Thusκ

q 3/2 (t2 − t1)

= tanθ2

2− tan

θ1

2+ 1

3

(tan3 θ2

2− tan3 θ1

2

)

=(

tanθ2

2− tan

θ1

2

)[1 + 1

3

(tan2 θ2

2+ tan

θ2

2tan

θ1

2+ tan2 θ1

2

)]

=(

tanθ2

2− tan

θ1

2

)[1 + tan

θ1

2tan

θ2

2+ 1

3

(tan

θ1

2− tan

θ2

2

)2].

(3B.7)

The equation for the chord P1 P2 in Figure 3.11 is

s2 = r21 + r2

2 − 2r1r2 cos(θ2 − θ1) = (r1 + r2)2 − 4r1r2 cos2 θ2 − θ1

2(3B.8)

which gives

2√

r1r2 cosθ2 − θ1

2= ±√

(r1 + r2 + s)(r1 + r2 − s), (3B.9)

with the plus sign corresponding to θ2 − θ1 < π and the negative signcorresponding to θ2 − θ1 > π .

From (3B.1) we have

r1 = q sec2 θ1

2, r2 = q sec2 θ2

2(3B.10)


so that

r1 + r2 = q(

2 + tan2 θ1

2+ tan2 θ2

2

)(3B.11)

and

√r1r2 = q sec

θ1

2sec

θ2

2= q

cos θ12 cos θ2

2

. (3B.12)

Substituting (3B.12) into (3B.9) we get

1 + tanθ1

2tan

θ2

2= ±

√(r1 + r2 + s)(r1 + r2 − s)

2q. (3B.13)

Finally by direct expansion6

(√r1 + r2 + s ∓ √

r1 + r2 − s)2

2= r1 + r2 ∓

√(r1 + r2 + s)(r1 + r2 − s)

= q(

tanθ1

2− tan

θ2

2

)2

, (3B.14)

where we have used (3B.11) and (3B.13). From (3B.14) we obtain

√r1 + r2 + s︸︷︷︸

a1

∓ √r1 + r2 − s︸︷︷︸

a2

=√

2q(

tanθ1

2− tan

θ2

2

). (3B.15)

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

κ(t2 − t1)q 3/2 = (a1 ∓ a2)√

2q

(±a1a2

2q+ 1

3(a1 ∓ a2)2

2q

)(3B.16)

or

6√

2κ(t2 − t1) = (a1 ∓ a2)(±3a1a2 + (a1 ∓ a2)2)

= (a1 ∓ a2)(a2

1 + a22 ± a1a2

) = a31 ∓ a3

2 (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) shouldbe used. When θ2 − θ1 = π , the second term in (3B.18) is zero and forlarger difference the plus sign should be used.

6If θ2 − θ1 < π , then a plus sign corresponds to (3B.13) and in that case minussign in (3B.14), and vice-versa.


Euler’s equation is remarkable since it does not involve q and itonly involves the observations (r1, θ1) and (r2, θ2) at instants t1 andt2. Clearly if the given observations belong to the same comet, thenthey must satisfy (3B.18) and hence Euler’s equation is useful to testthe hypothesis of whether two given observations correspond to thesame comet or not.7

There is a corresponding equation for elliptic orbits due to Lambert(see Appendix 3C).

Appendix 3-C: Lambert’s Equation forElliptic Orbits

Let E1 and E2 be the eccentric anomalies corresponding to two po-sitions P1 and P2 of a planet (or a comet) in an elliptic orbit withrespective radial distances r1 and r2 as in Figure 3.12. Let t1 and t2denote the associated time instants. Suppose E2 > E1 and define

G = E2 + E1

2, g = E2 − E1

2. (3C.1)

Then from (2.41)

r1 = a (1 − e cos E1), r2 = a (1 − e cos E2) (3C.2)

so that

r1 + r2 = a (2 − e(cos E1 + cos E2))

= 2a (1 − e cos G cos g). (3C.3)

Using (2.34) and (2.35), the chord length s in Figure 3.12 equals

s2 = (x2 − x1)2 + (y2 − y1)2

= a2(cos E2 − cos E1)2 + b2(sin E2 − sin E1)2 (3C.4)

= 4a2(sin2 G sin2 g + (1 − e2) cos2 G sin2 g).

7A more interesting situation is when a comet on its routine visit around theSun undergoes a chance encounter with an outer planet such as Jupiter that resultsin altered orbital parameters. How is the comet to be recognized with any certaintyin its subsequent visits?, i.e., how does one test the hypothesis that two comets ap-pearing several years apart are the same comet considering that random planetaryperturbations might have altered their orbital parameters?

A condition derived by Tisserand in connection with the three body problemgives an invariant relation that is unchanged by such perturbations [2].


P2

r1

O F1

r2s

E2E1

Planet

P1

F2

a

FIGURE 3.12 Identification of planets/comets using two observations.

Let

e cos G = cos c. (3C.5)

Then (3C.4) simplifies to

s2 = 4a2 sin2 g(1 − cos2 c), (3C.6)

or

s = 2a sin g sin c. (3C.7)

From (3C.3), we also have

r1 + r2 = 2a (1 − cos g cos c). (3C.8)

Let

α = c + g (3C.9)

and

β = c − g. (3C.10)

Then from (3C.7) and (3C.8) we get

r1 + r2 + s = 2a (1 − cos(g + c)) = 4a sin2(α/2) (3C.11)

and

r1 + r2 − s = 2a (1 − cos(g − c)) = 4a sin2(β/2). (3C.12)


Finally from Kepler’s equation in (3A.12) and (3A.13) we get

n(t2 − t1) = (E2 − E1) − e(sin E2 − sin E1)

= (α − β) − 2e cos G sin g

= (α − β) − 2 cos c sin g (3C.13)

or

n(t2 − t1) = (α − β) − (sin α − sin β), (3C.14)

Lambert’s equation for elliptic motion. As remarked earlier, t1 and t2in (3C.14) correspond to the time instants at which the planet passesthrough P1 and P2 in Figure 3.12. Notice that α and β are given in termof 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.

C H A P T E R 4Space Based

Radar----Kinematics

A space based radar (SBR) located at some height above the Earth’ssurface points its mainbeam at a point of interest on the ground whereit generates a certain grazing angle. Surrounding the mainbeam, theantenna projects sidelobes all of which contribute toward the datacollected for that specific point of interest. The SBR continues to collectdata in this fashion as it travels along its orbit around the Earth. Theobjective is to detect slowly moving ground/air targets by suppressingthe clutter returns and perform other tasks such as target identificationand imaging.

In this context, the SBR-Earth geometrical relationships are firstderived for an ideal spherical Earth, with various correction factorsdue to an ellipsoidal (oblate spheroid) Earth model considered in theappendices. Another important issue addressed here is the shift inDoppler due to Earth’s rotational effects and a detailed analysis ofthis and related phenomena are carried out in this chapter.

4.1 Radar-Earth GeometryAn SBR located at an orbital height H above its nadir point1 has itsmainbeam focused to a point of interest on the ground located atrange R. Figure 3.4 and Figure 4.1 (a) show an SBR on a polar orbitmoving in the North-South direction. In general, the SBR can be in anorbit that is inclined at an angle to the equator. The inclination of theSBR orbit is usually given at the point where it crosses the equatorfrom which its local inclination at other latitudes can be determined.The range is measured from the nadir point B (that is directly below

1Nadir point is on the Earth’s surface directly below the satellite and it is obtainedby joining a line between the SBR and the center of the Earth (see Figure 4.1 (b)).



North America

Atlantic

Pacific Ocean

0 4447 km

D

B

C

SBR

qAZ

Vp

2

Grazing angle

Point ofinterest on ground

Range

Slant range

Center of the Earth

Nadir point

Ground

HRs

R

ReRe

A

(b)(a)

B

SBR

Region of interest

qEL

qe

y

p

SouthAmerica

Reference map (c) 2000 ESRI

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

the satellite) to the antenna mainbeam footprint along a great circleon Earth (see Figure 4.1 (b)). Alternatively, instead of specifying therange 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 Dmaybe given (see Appendix 4-A for a detailed derivation of rangefrom these parameters).

For example, a polar orbit satellite at 506 km above the Earth’s sur-face has a period of 1.57 h. While it completes a circle around the Earththat is fixed with respect to the stars, the Earth turns through 22.5 or1/16 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 moves2500 km eastward giving an “automatic” scan of the surface belowto the onboard radar. As shown in Figure 4.1 (a), the radar is able toscan the Earth in both latitude and longitude by virtue of the Earth’srotation.

In Figure 4.1 (b), the SBR is located at A, and B represents the nadirpoint. The point of interest D is located at range R from B along thegreat circle that goes through B and D with C representing the centerof 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 interestalong a great circle on the surface of the Earth.

H : SBR orbit height above the nadir point.

C h a p t e r 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 79

Rs : Radar slant range from the satellite to the antenna footprintcenter at range R.

ψ : Grazing angle at the antenna footprint at range R (i.e., the angleat which the surface is illuminated by the radar beam).

Re : Spherical Earth’s radius (3,440 miles or 6,373 km).

θE L : Mainbeam elevation from the vertical line associated withrange R.

θAZ : Azimuth point angle measured between the plane of the array(generally also the SBR velocity vector) and the elevation planeAB D.

φE 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 pointmeasured 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 theequator).

From Figure 4.1 (b), the core angle subtended at the center of Earthby the range arc BD is given by

θe = R/Re (4.1)

and from triangle ACD we get

R2s = R2

e + (Re + H)2 − 2Re (Re + H) cos θe . (4.2)

Thus the slant range Rs equals

Rs =√

R2e + (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)(Re + H)

= sin θe

Rs= sin θEL

Re, (4.4)

or

cos ψ

(Re + H)= sin θe

Rs= sin θEL

Re. (4.5)


Slant Range Rs vs Range

Rs,max

Rs,max

Rs,max

Rmax Rmax Rmax

Range R (km)

Slan

t Ran

ge R

s (km

)

500

1,000

1,500

2,000

2,500

3,000

3,500

4,000

4,500

5,000

5,500

0 1,000 2,000 3,000 4,000 5,000

H = 1,000 km

H = 506 km

H = 2,000 km

FIGURE 4.2 Slant range vs. range.

Thus we obtain the grazing angle at range R to be2

ψ = cos−1(

Re + HRs

sin(R/Re ))

, (4.6)

and the corresponding elevation angle is given by

θEL = sin−1(

Re

Rssin(R/Re )

). (4.7)

Notice that both the grazing angle ψ and the elevation angle θEL arerange dependent.

From (4.5) we also have

θEL = sin−1(

11 + H/Re

cos ψ

). (4.8)

Similarly from triangle ACD we obtain an alternate formula

θEL = π/2 − θe − ψ = π/2 − ψ − R/Re (4.9)

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


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

Range R (km)

Gra

zing

Ang

le y

(deg

ree)

Ele

vati

on A

ngle

qE

L (d

egre

e)

Elevation Angle qEL vs RangeGrazing Angle vs Range

00

10

20

30

40

50

60

70

80

90

1,000 2,000 3,000 4,000

Range R (km)

00

10

20

30

40

50

60

70

1,000 2,000 3,000 4,000

Rmax Rmax Rmax

H = 2,000 km

Rmax Rmax Rmax

(a) Grazing angle vs range (b) Elevation angle vs range

qEL,max

qEL,max

qEL,max

H = 1,000 km

H = 506 km

FIGURE 4.3 Grazing angle and elevation angle vs. range.

for the angle of elevation as well. The slant range, grazing angle, andelevation 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 givenheight, the curvature of the Earth sets limits on the maximum availablerange, slant range, and elevation angle. This phenomenon is discussednext.

4.2 Maximum Range on EarthThe curvature of Earth limits the maximum range achievable by asatellite located at height H as shown in Figure 4.4.

At maximum range, the slant range becomes tangential to the Earthso 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

Re + H= 1

1 + H/Re(4.10)

or

θmax = cos−1(

11 + H/Re

). (4.11)

The maximum range on Earth for an SBR located at height H isgiven by

Rmax = Reθmax = Re cos−1(

11 + H/Re

). (4.12)


A

DB

C

SBR

qEL

p/2

qmax

H Rs,max

Re

Rmax

Re

FIGURE 4.4Maximum rangeon ground.

Similarly maximum slant range at the same height is given by

Rs,max = (Re + H) sin θmax = (Re + H) sin

cos−1(

11 + H/Re

)

=√

(Re + H)2 − R2e =√

H(2Re + H), (4.13)

and the maximum elevation angle equals

θEL,max = π

2− θmax = π

2− cos−1

(1

1 + H/Re

). (4.14)

For low Earth orbit (LEO) satellite located at 506 km above theground, the maximum range is 2,460 km and θEL,max = 67.9 (seeFigure 4.5).

Rmax vs Height

Rm

ax (k

m)

Height (km)

1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 9,000 10,000 1,000 2,000 3,000 4,000 5,000 6,000 7,000 8,000 9,000 10,0001,000

2,000

3,000

4,000

5,000

6,000

7,000

8,000

qEL,max vs Height

q EL

,max

(deg

ree)

Height H (km)

20

30

40

50

60

70

80

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


4.3 Mainbeam Footprint SizeThe mainbeam of the radar generates a footprint on the ground whosesize depends upon the actual range R. Let φEL represent the 3 dBmainbeam width of the antenna pattern in the elevation plane. Fur-ther let RT and RH denote the ranges of the “toe” and “heel” of themainbeam footprint whose center is at range R as shown in Figure 4.6.

Let ψT and ψH represent grazing angles at the “toe” and “heel” ofthe mainbeam footprint. Thus from triangle ACE in Figure 4.7 thatcorresponds to the footprint “toe”, we have

sin(π/2 + ψT )Re + H

= sin(θEL + φEL/2)Re

, (4.15)

where θEL represents the elevation at range R. This gives the grazingangle at the “toe” to be

ψT = cos−1(

1 + HRe

)sin(

θEL + φEL

2

), (4.16)

qEL

qH

R

fEL (Beamwidth)

Mainbeam

RT

Mainbeamfootprint

H

A

D

B

C

L

“Toe”“Heel” E

RH

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


A

B

2fELqEL +

p/2

yT

qT

H

Re

Re

C

Mainbeam

ERT

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

and similarly the grazing angle at the “heel” is given by

ψH = cos−1(

1 + HRe

)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

2− θEL − φEL

2− ψT (4.18)

and the range to the mainbeam “toe” equals

RT = ReθT = Re

(π

2− θEL − φEL

2− ψT

). (4.19)

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

RH = ReθH = Re

(π

2− θEL + φEL

2− ψH

). (4.20)

This gives the length of the footprint of the mainbeam at range Rto be

L = RT − RH = Re (ψH − ψT − φEL). (4.21)


Range (km)

500 1,000 1,500 2,000 2,500 3,000 3,5000

50

100

150

200

250

300

350

400

450L

engt

h of

Mai

nbea

m F

ootp

rint

(km

)

Range (km)

500 1,000 1,500 2,000 2,500 3,000 3,50010

20

30

40

50

60

70

80

Wid

th o

f Mai

nbea

m F

ootp

rint

(km

)

H = 2,000 km

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

H = 506 km

H = 506 km

H = 1,000 km

FIGURE 4.8 Length and width of mainbeam footprint vs. range. Mainbeam3 dB beamwidths in both elevation and azimuth directions are assumedto be 1.

Let φAZ represent the 3 dB beamwidth in the azimuth direction, thenthe horizontal mainbeam beamwidth equals

W = RsφAZ. (4.22)

As a result π LW/4 represents the area of the elliptically shapedmainbeam on the ground. As Figure 4.8 shows, both the length andwidth of the footprint are functions of the range and SBR height. Insummary, when the antenna mainbeam is focused along θEL, returnsfrom the illuminated region of the corresponding mainbeam footprintwill contribute toward clutter from that range [2] (see Figure 4.9). Forexample, an SBR located at a height of 506 km generates a mainbeamsize of 55 km × 20 km at a range of 1,000 km.

The mainbeam footprint size L = RT − RH and W = RsφAZ arefunctions of the range R and height H, and they need to be recomputedfor different ranges.

Notice that the region of coverage on the Earth’s surface is annularin shape since the region around the nadir point is inaccessible (nadirhole) and the farthest range is limited by the Earth’s curvature as inFigure 4.10.

L = RT − RH

W = RsfAZ

Illuminated regionsFIGURE 4.9Mainbeamfootprint on theground.


SBR

Nadir hole

Annular regionof coverage

Rmin

qELmax

Rmax

Equator

Nadir point

FIGURE 4.10 Annular region of coverage and mainbeam footprints.

4.4 Packing of Mainbeam FootprintsNext, we examine the interesting question of determining mid-footprint range points R1, R2, · · · along the conical strips in theannular region in Figure 4.10 such that the corresponding mainbeamfootprints are non-overlapping.

Toward this, Figure 4.11 shows a conical section corresponding toa fixed azimuth angle from the annular region. Let Li represent the

R1 R2 R3 Ri

RTi–1

yTi –1

qAZ

yi

RHi

R

...

L2

Rmin Rmax

...Wi

FIGURE 4.11 Mainbeam footprint vs. range.


mainbeam footprint size at range Ri . Thus

Li = RTi − RHi . (4.23)

To have non-overlapping mainbeam footprints as in Figure 4.11, therange points R1, R2, · · · at the centers of the footprints must be selectedsuch that the ith bin satisfies

RHi = RTi−1 , i = 2, 3, · · · (4.24)

and

ψHi = 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 theelevation plane by the 3 dB elevation beamwidth φEL. Since,

RHi = Re

(π

2− θELi + φEL

2− ψHi

)(4.26)

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

θELi = π

2+ φEL

2− ψTi−1 − RTi−1

Re. (4.27)

This gives the grazing angle at range Ri to be

ψi = cos−1(

1 + HRe

)sin θELi

(4.28)

and from (4.9)

Ri = Re

(π

2− ψi − θELi

). (4.29)

As a result the grazing angle and range at the toe of the ith binequals

ψTi = cos−1(

1 + HRe

)sin(

θELi + φEL

2

)(4.30)

and the corresponding range at the toe is given by

RTi = Re

(π

2− θELi − φEL

2− ψTi

). (4.31)

From (4.23), this gives

Li = RTi − RHi = Re (ψHi − ψTi − φEL). (4.32)

Starting with the initial point, RH1 = Rmin, and ψH1 = ψmin, weobtain RT1 = Re ( π

2 − θELmin − φEL − ψT1 ). These initial values are used


in (4.24)–(4.25) and the procedure is repeated to generate the rangevalues R1, R2, . . . that guarantee full overage of the ground using aminimum number of mainbeams.

Table 4.1 shows the range points Ri and elevation angles corre-sponding to the centers of the non-overlapping mainbeam footprints

Beam θE Li Ri ψi RHi = RTi−1 Li R# i () (km) () (km) (km) (km) Na,i

1 37 390.6 49.5 383.2 14.9 107.4 12 38 405.7 48.4 398.1 15.4 105.6 13 39 421.3 47.2 413.4 15.9 103.8 14 40 437.5 46.1 429.4 16.5 102.2 15 41 454.3 44.9 445.8 17.1 100.6 16 42 471.8 43.8 463.0 17.8 99.0 17 43 489.9 42.6 480.7 18.5 97.5 18 44 508.8 41.4 499.2 19.3 96.1 19 45 528.5 40.2 518.5 20.2 94.7 1

10 46 549.1 39.1 538.7 21.1 93.4 111 47 570.8 37.9 559.8 22.2 92.2 112 48 593.5 36.7 582.0 23.3 91.0 113 49 617.4 35.4 605.3 24.6 89.8 114 50 642.7 34.2 629.9 26.0 88.7 115 51 669.5 33.0 655.9 27.6 87.6 116 52 697.9 31.7 683.5 29.3 86.6 117 53 728.2 30.5 712.8 31.3 85.6 118 54 760.7 29.2 744.2 33.6 84.6 119 55 795.5 27.8 777.8 36.2 83.7 120 56 833.2 26.5 814.0 39.2 82.9 121 57 874.1 25.1 853.2 42.7 82.0 122 58 918.8 23.7 895.9 46.8 81.2 123 59 967.9 22.3 942.7 51.8 80.4 124 60 1,022.6 20.8 994.5 57.8 79.7 125 61 1,084.0 19.3 1,052.3 65.4 79.0 126 62 1,153.9 17.6 1,117.7 75.0 78.3 127 63 1,234.9 15.9 1,192.8 88.0 77.7 228 64 1,331.1 14.0 1,280.7 106.2 77.1 229 65 1,449.6 12.0 1,386.9 134.3 76.5 2

TABLE 4.1 Ground range locations and elevation angles fornon-overlapping antenna mainbeams corresponding to H = 506 km,Rmin = RH1

= 383 km, and Rmax = RT29= 1,521 km.


for a LEO satellite located at a height of 506 km. In this case the graz-ing angle is allowed to vary between 50 and 10, and it correspondsto ground range of 383–1521 km. In a pulsed radar situation, con-secutive pulse returns from different range points overlap generatingrange ambiguity. With Tr representing the pulse repetition interval,the range ambiguity is given by R cTr/2, where c represents thevelocity of light (see also (4.35) and (4.37)). A 2 KHz pulse repetitionfrequency (PRF) is used here to compute the range ambiguity R.Notice that the antenna mainbeam footprint sizes on the ground varyfrom 15–135 km and the number of range ambiguities in this casevaries from 1 to 2 within the mainbeam footprint. If a 10 kHz PRF isused instead, the number of range ambiguities in a footprint wouldvary from 1 to 9 within the mainbeam.

The footprints corresponding to ranges R1, R2, . . . in Figure 4.11 arenon-overlapping in the range direction and cover the entire regionof interest. However, their widths Wi = Rsi φAE will increase as Riincreases. This situation is shown in Figure 4.12 where the footprintsare bounded in a conical surface along the range direction. The endwidths of the conical sections are W1 and Wend = RsJ φAZ respectively.After finishing a range sweep along a particular θAZ, the azimuthangle can be incremented by the beamwidth φAZ and the procedure isrepeated to cover the entire annular region of interest in Figure 4.12.

Rmax

R1 R2RJ

Nadir hole

Annular region of coverage

Wend

…

………

…

RsfAZ

qAZ

FIGURE 4.12 Region of coverage for SBR.


4.5 Range Foldover PhenomenonTo detect targets, radar transmits pulses periodically. Range foldoveroccurs when clutter returns from previously transmitted pulses, re-turning from farther range bins, get combined with returns from thepoint of interest (see Figure 4.13). Depending on the size of the main-beam footprint, the two-dimensional (2D) antenna array pattern andthe radar pulse repetition frequency, range foldover can occur bothfrom within the mainbeam as well as from the entire 2D region. Theeffect of mainbeam foldover is discussed first, followed by its exten-sion to the entire 2D region.

4.5.1 Mainbeam FoldoverLet τ represent the radar output pulse length and Tr the pulse repeti-tion interval. Pulses travel along the slant range and interact with theground through the mainbeam as well as the sidelobes of the antennaarray as shown in Figure 4.14.

Each pulse travels along the slant range and hence the slant rangethat can be recovered unambiguously is of size cτ

2 . Thus slant rangeresolution is given by

δSR = cτ2

. (4.33)

D

R-1R- 2

R2R1 R3

SBR

Range point of interest at R

Returnsdue to later pulses

Returnsdue to earlierpulses

Forward

range

foldovers

Backward

range foldovers

…Transm

itted

pulses

Return

wavefront

A

FIGURE 4.13Common returnwavefrontshowing all rangeambiguity returnscorresponding to apoint of interest atrange R.


A

D

B

H

Re

Re

Vp

qEL

Mainbeam

Range ambiguities

Sidelobes

Tr

C

qAZ

t

y

Pulse impression

FIGURE 4.14 Mainbeam range ambiguities.

Translating to the ground plane, since the pulse wave front is per-pendicular to the slant range direction, we get the range resolution onground to be (see Figure 4.15)

δR = cτ2 cos ψ

= cτ2

sec ψ. (4.34)

Thus δR represents the ground-plane spatial resolution that can berealized by the SBR. Two objects that are separated by a distance lessthan δR will be indistinguishable by the radar. Notice that only theoutput pulse length contributes to the range resolution and it can beorders of magnitude smaller than the actual pulse length because ofpulse compression effects. For example, using chirp waveforms it ispossible to realize 1:100 or higher order compression. From (4.34) forshort range regions where the grazing angle ψ is closer to π/2, therange resolution is very poor, and for long range the resolution ap-proaches its limiting value δSR as ψ → 0. Radar transmits pulsesevery Tr seconds and for high PRF situations, following (4.34), the dis-tance R between range ambiguities on the ground (distance between

dR

ct /2

Pulse wavefront

y

FIGURE 4.15Ground rangeresolution.


A

D

B

qe

H

R

y

ReRe

RsRs

E

cTr/2

C

Re

∆R

∆qe

FIGURE 4.16Distance betweenrange ambiguities.

consecutive pulse shadows) is given by

R = cTr

2sec ψ. (4.35)

Equation (4.35) assumes a high PRF situation where the grazingangles at various range ambiguities are assumed to be equal. Thegeneral situation that takes the change in grazing angle into accountis shown in Figure 4.16.

If Rs represents the slant range in (4.3) at the end of one pulse (sayat D), then Rs + cTr/2 is the new slant range at the end of the nextpulse at E . Let R1 = R + R represent the new range correspondingto the second pulse shadow on the ground at E . From triangle ACE inFigure 4.16

(Rs + cTr/2)2 = R2e + (Re + H)2 −2Re (Re + H) cos

(R + R

Re

)(4.36)

or

R = Re cos−1(

R2e + (Re + H)2 − (Rs + cTr/2)2

2Re (Re + H)

)− R. (4.37)

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


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

(a) PRF = 500 Hz

∆ R (k

m)

Range (km)500

300

350

400

450

500

550

600

650

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

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

(b) PRF = 2 kHz

∆R vs Range∆R vs Range

∆ R (k

m)

Range (km)500

60

80

100

120

140

160

180

220

200

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

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

the mainbeam footprint. The situation in the mainbeam is shown inFigure 4.18(b).

If we assume R to be constant within a mainbeam footprint, thenthe 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 rangeRH , sequentially R1 , R2 , · · · are calculated within the mainbeamfootprint till the toe is reached. Thus the number of range ambiguitiesNa 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 thatNa is also a function of range. To have no range ambiguities within the

Mainbeamfootprint

(a) Short range (b) Long range (y ~ 0)

secy2

ctct /2

∆R2

RT – RH RT – RH

∆R1

RangeambiguitiescTr/2

RSfAZRSfAZ

FIGURE 4.18 Mainbeam footprint and range ambiguities.


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

Na

0

0.5

1

1.5

2

2.5

3

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

Number of Range Ambiguities vs Range

(a) PRF = 500 Hz

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

Na

0

2

1

3

4

5

6

7

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

Number of Range Ambiguities vs Range

(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 higherPRF is used resulting in Na > 1.

From Figure 4.18, we also obtain the total area illuminated on theEarth’s surface by the mainbeam to be

Ac = RsφAZ Nacτ2

sec ψ (4.40)

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

4.5.2 Total Range FoldoverThe total number of range ambiguities can be much more than thatgiven in (4.38) if the sidelobe pattern of the antenna beam in the eleva-tion direction is significant compared to the mainbeam. In that case,the pulse returns from sidelobe regions also must be considered. Thissituation is shown in Figure 4.20.

In Figure 4.20, the point of interest (D) is within the mainbeam, andthe return of the radar pulse from there represents the main clutter.However, because of the 2D antenna pattern, previous pulse returnsreturning from adjacent “range ambiguity points”—both forward andbackward—that have been appropriately scaled by the array gainpattern get added to the mainbeam return causing additional rangefoldover.

To compute the immediate forward and backward range ambiguitypoints (E and F respectively), the geometry in Figure 4.21 can be used.

Notice that the forward range ambiguity increments R+ are thesame as R in (4.36) and (4.37). From there, if R1 represents the first


…

Mainbeam

…

…

qAZ

Elevationsidelobes

Rangeambiguities

D : Point of interest

Range ofinterest

(qEL,m, qAZ,j)

∆R−

∆R+

Azimuthsidelobes

FIGURE 4.20 “Range foldover” phenomenon.

forward range ambiguity point then

R1 = R+R+ = Re cos−1(

R2e + (Re + H)2 − (Rs + cTr/2)2

2Re (Re + H)

)(4.41)

and this procedure can be repeated till Rmax is reached to determineall forward range ambiguity points R1, R2, · · · for a given R.

Similarly, the backward range ambiguity increments R− are givenby the new relation (see Figure 4.21)

(Rs −cTr/2)2 = R2e +(Re +H)2−2Re (Re +H) cos

(R − R−

Re

). (4.42)


A

D

B

H

Re

Re

Rs

E

cTr/2

C

F

y

∆R-

∆qe-∆qe +

qe

∆R+

Forwardrangeambiguitypoint

Backward range ambiguity point

Point of interest

Rs – cT

r /2

Rs

R

FIGURE 4.21 Forward and backward range ambiguity points.

This gives

R− = R − Re cos−1(

R2e + (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(

R2e + (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 ambiguitypoints are given by

R±k = Re cos−1(

R2e + (Re + H)2 − (Rs ± kcTr/2)2

2Re (Re + H)

), k = 1, 2, . . .

(4.45)


500 1,000 1,500 2,000

5

10

15

20

25

30

35

40

PRF (Hz)

Na

H = 1,000 km

H = 1,500 km

H = 506 km

FIGURE 4.22 Number of range ambiguities as function of SBR height andPRF.

where R+k = Rk . Notice that the actual range R enters (4.45) throughRs given in (4.3). Figure 4.14 shows the return wavefront from all rangeambiguities corresponding to a point of interest D at range R.

Let Na refer to the total number of range ambiguities (both forwardand backward) corresponding to a range bin of interest. The clutterreturns from forward and backward range ambiguities get scaled bythe array gain corresponding to those locations and get added to thereturns from the point of interest. Figure 4.22 shows the total numberof range ambiguities in the 2D region as a function of SBR height andPRF. On comparing this figure with Figure 4.19, we notice that thetotal number of range ambiguities at 500 Hz PRF jumps from 2 in themainbeam to 7 in the elevation direction.

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


D

B

C

SBR

qAZ

y

p

qEL

Vp

2

Grazing angle

Point of intereston ground

Range

Slant range

qe

Center of the Earth

Nadirpoint

Ground

HRs

R

Re

A

Re

FIGURE 4.23 The parameters of an SBR pointing its mainbeam to a groundpoint 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 atan azimuth angle θAZ with respect to the flight path and an elevationangle θEL with respect to the nadir line as shown in Figure 4.23. Toderive (4.47), from Figure 4.23 we notice that Vp cos θAZ represents therelative velocity of the SBR along the array azimuth direction, andVp cos θAZ sin θEL represents the relative velocity along the slant rangedirection AD.

3In a circular orbit the gravitational pull due to the inverse square law GmMe/r2

must equal the centripetal force mV2p /r . Thus

mV2p

r = GmMer2 gives Vp =

√G Me

rwhere Vp represents the satellite orbit speed, m its mass, Me mass of Earth, Guniversal gravitational constant (= 6.673×10−11 m3 kg−1s−2) and r = Re + H. For asatellite located at height H = 506 km, this gives Vp = 7.61 km/s and it correspondsto an orbital period of 1.579 h. Notice that (3.66) gives the more accurate formulagiven by VP =

√G(Me + m)/r .


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

ωd = 2VpTr

λ/2sin θEL cos θAZ (4.48)

and (4.48) accounts for the Doppler frequency of the stationary groundreturn due to the SBR motion.

Given R, θAZ, and H from (4.1)–(4.5), we obtain

sin θEL = Re

Rssin (R/Re )

= Re sin (R/Re )√R2

e + (Re + H)2 − 2Re (Re + H) cos(R/Re ), (4.49)

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

ωd = 2VpTr

λ/2Re sin (R/Re ) cos θAZ√

R2e + (Re + H)2 − 2Re (Re + H) cos(R/Re )

. (4.50)

From (4.50), the Doppler frequency is clearly range dependent.4 Forshort ranges, (4.50) reduces to

ωd (R) ≈ 2VpTr

λ/2RH

cos θAZ. (4.51)

Figure 4.24 shows the Doppler dependency on range as a functionof the azimuth angle. Clearly, the Doppler is an increasing function ofrange R as the azimuth angle moves away from 90. From (4.51), theDoppler increment ωd = ωd (R2)−ωd (R1) due to the range differenceR = R2 − R1 increases as azimuth angle moves away from 90, sinceωd ∝ R cos θAZ. This is seen in the iso-Doppler plots shown inFigure 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, theDoppler difference due to range generates an undesirable “Dopplerfilling” effect when data samples from different range bins are usedto estimate the covariance matrix.

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


−1 −0.5 0 0.5 1

−200

−100

0

100

200

Dop

pler

cos qAZ

(a)

R1 = 250 km

0 500 1,5001,000 2,000

0

20

40

60

80

100

120

Dop

pler

Range (km)

(b)

qAZ = 60°

qAZ = 75°

qAZ = 85°

qAZ = 90°

R2 = 1,000 km

R3 = 1,750 km

FIGURE 4.24 Doppler dependency on range vs. azimuth angle.

In practice the effect of difference in Doppler is even more severebecause of the clutter ridge slope parameter (Doppler foldover pa-rameter)

βo = 2VpTr

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 re-sult ωd can be much larger than unity at various azimuth angles. Sincethe Doppler term ωd appears in complex sinusoidal form (Appendix4-C), ωd > 1 causes the Doppler to foldover as shown in Figure 4.26.

To demonstrate the effect of βo on Doppler spread, Figure 4.26shows the Doppler-azimuth pattern in Figure 4.24 for βo = 6. Noticethat Doppler foldover corresponding to various range bins occurs atdifferent azimuth angles. For operating conditions corresponding to

200

-200

0

Range (km)

00

20

40

60

80

100

120

140

160

180

500 1,000 1,500 2,000

Azi

mut

h (d

eg)

200

-200

0

Range (km)

01

0.5

0

−0.5

−1

500 1,000 1,500 2,000

cos

(qA

Z)

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


−1−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Azimuth

Dop

pler

R1 = 1,000 km R2 = 500 km

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 azimuthangle 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 thataffects the clutter data is the effect of Earth’s motion around its ownaxis. At various locations on Earth this contributes differently toDoppler, and the effect is modeled here.

For any point on Earth at range R that is at an elevation angle θELand azimuth angle θAZ with respect to an SBR at height H, the Dopplershift due to the SBR motion equals [1]

ωd = 2VpTr

λ/2sin θEL cos θAZ, (4.53)

as derived in Section 4.6. Let ηi denote the inclination of the SBR orbitwith respect to the equator (see Figures 4.27–4.28).

As the SBR moves around the Earth, the Earth itself is rotatingaround its own axis on a 23.9345 hour basis in a west-to-east direction.


A

B

C

D

SBR

Z

Equator

R

North pole

Latitude a1

Longitude b1

Longitude b2

Point of contact on Earth (a2, b2)

SBR path

Ve cos a2 (eastward)

VpqAZ

qEL

g

y

a1

a

b2 – b1

a2

b1

hi

b

FIGURE 4.27 Doppler contributions from SBR velocity and Earth rotation.

B1

B

R

Z

SBR path

Ve cos a2D

qAZ

Equator

b1 b 2

Longitudes

Latitudes

Z

North pole

Y2

Y1

SBR inclination at equator

B0

b1

a2

a1 d

hi

g

b p/2

Region of interest

FIGURE 4.28 Effect of Earth rotation on Doppler frequency.


This contributes an eastward motion with equatorial velocity of

Ve = 2π Re

23.9345 × 3600= 0.4651 km/s. (4.54)

Let (α1, β1) refer to the latitude and longitude of the SBR nadir pointB and (α2, β2) those of the point of interest D as shown in Figure 4.27and Figure 4.28.

As a result, the point of interest D on the Earth at latitude α2 rotateseastward with velocity Ve cos α2 and this will contribute to the Dopplerin (4.48) as well. To compute this new component in Doppler shift,from Figure 4.28 the angle BDY2 between the ground range vector Rand 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 rangedirection toward B. Since the grazing angle ψ represents the slantrange 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 thepoint of interest D due to the Earth’s rotation toward the SBR. Com-bining (4.47) and (4.56) as in (4.48), we obtain the modified Dopplerfrequency that also accounts for the Earth’s rotation to be

ωd = 2Tr

λ/2

(Vp sin θEL cos θAZ − Ve cos α2 sin β cos ψ

). (4.57)

From triangle ACD in Figure 4.23, we have

sin (π/2 + ψ)Re + H

= sin θEL

Re(4.58)

so that

cos ψ =(

1 + HRe

)sin θEL (4.59)

and hence (4.57) becomes

ωd = 2VpTr

λ/2sin θEL

(cos θAZ − Ve

Vp

(1 + H

Re

)cos α2 sin β

)

= 2VpTr

λ/2sin θEL (cos θAZ − cos α1 sin γ ) , (4.60)


where we define

= Ve

Vp

(1 + H

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 alongthe great circle relative to the north measured at B.

To simplify (4.60) further, from Figure 4.28 we have

ZBD = γ , ZBBo = δ, B1BD = θAZ (4.63)

and hence

γ = ZBB1 + B1BD = 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 theSBR locations at the equator and at (α1, β1), we get (see Figure 4.29)

ZBo B = π

2− ηi , BoZB = β1, ZBBo = δ. (4.66)

Bo

Z (North pole)

B

b1

a1p

qod

p

p

-

hi

hi

-2

2

2 B1

SBR path

FIGURE 4.29Spherical triangleZBoB formed bythe north pole,SBR locations atequator (Bo) and(α1, β1) at B.


Hence using (2A.3), we obtain

sin δ

sin π/2= sin(π/2 − ηi )

sin(π/2 − α1)= cos ηi

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

sin γ = cos ηi cos θAZ + sin θAZ√


cos α1(4.70)

and finally with (4.70) in (4.60) we obtain the modified Doppler fre-quency to be

ωd = 2VpTr

λ/2sin θEL

((1 − cos ηi ) cos θAZ

−√

cos2 α1 − cos2 ηi sin θAZ)

= 2VpTr

λ/2ρc sin θEL cos(θAZ + φc) (4.71)

where

φc = tan−1

(√


1 − cos ηi

)(4.72)

and

ρc =√

1 + 2 cos2 α1 − 2 cos ηi . (4.73)

In (4.71)–(4.73), φc represents the crab angle and ρc represents thecrab magnitude. In summary, the effect of Earth’s rotation on the

5In (4.68), α1 ≤ ηi since the maximum latitude of the SBR never exceeds its in-clination at the equator. Equation (4.64) together with (4.68) compute the azimuthangle corresponding to the point of interest (α2, β2) on Earth from the SBR param-eters (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 andthe point of interest in terms of (α1, β1), (α2, β2), and ηi , the instantaneous azimuthangle θAZ, elevation angle θEL, and grazing angle ψ can be determined. The lattertwo require the SBR height as well.


Doppler velocity is to introduce two distortions—a crab angle andcrab magnitude into the SBR azimuth angle and modify it accord-ingly [7], [8]. Interestingly both these quantities depend only on theSBR orbit inclination and its latitude, and not on the latitude or longi-tude of the clutter patch of interest.

Equation (4.71) corresponds to the case where the region of interestD is to the east of the SBR path as shown in Figure 4.28. If on theother hand, the region of interest is to the west of the SBR path as inFigure 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)).

B1

B

R

Z

SBRpath

D

qAZ Y1

Equator

Longitudes

a2

a1

d

g

b2b1

hi

Latitudes

Z

North pole

Y2

SBR inclinationat equator

Bo

b1

b

b – p/2 p –d

Region ofinterest

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


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

ωd = 2Tr

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

= 2Tr

λ/2sin θEL(cos θAZ + cos α1 sin γ ). (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, (4.77)

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

ωd = 2VpTr

λ/2sin θEL((1 − cos ηi ) cos θAZ

+√

cos2 α1 − cos2 ηi sin θAZ)

= 2VpTr

λ/2ρc sin θEL cos(θAZ − φc), (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 = 2VpTr

λ/2ρc sin θEL cos(θAZ ± φc). (4.79)

In (4.79), the plus sign is to be used when the region of interest isto the east of the SBR path and the minus sign is to be used when thepoint of interest is to the west of the SBR path.

Figure 4.31 shows the crab angle and crab magnitude variation asa function of SBR height for a satellite located at the equator on threedifferent orbits. For example, on a polar orbit (Figure 4.31 (a)) at a lowaltitude of 506 km, the crab angle is about 3.77 and the crab magnitudeis about 1.002. However, at a medium Earth orbit of 5,000 km, the crabangle increases to 7.98 and the crab magnitude becomes 1.01. FromFigure 4.31 (a), both these parameters are increasing functions of theSBR 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


00

50

40

30

20

10

1 2 3 4

Cra

b A

ngle

(deg

)

H (km) ×104

(i) fc

0−1

1

0.5

0

−0.5

1 2 3 4

Cra

b A

ngle

(deg

)

H (km) ×104

(i) fc

01

1.1

1.8

1.2

1.3

1.4

1.5

1.6

1.7

1 2 3 4

Cra

b M

agni

tud

e

H (km) ×104

(ii) rc

00

1

0.2

0.4

0.6

0.8

1 2 3 4

Cra

b M

agni

tud

e

H (km) ×104

(ii) rc

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

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

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

00

80

60

40

20

1 2 3 4

Cra

b A

ngle

(deg

)

H (km) ×104

(i) fc

Geosynchronousorbit

00

1

0.2

0.4

0.6

0.8

1 2 3 4

Cra

b M

agni

tud

e

H (km) ×104

(ii) rc

H0

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


monotonically decreases till it becomes zero at Ho = 35,868 km, whichcorresponds 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

)(4.80)

and ρc = 0 gives the SBR height Ho to be

Re + Ho = Vp

VeRe =

√G Me√

Re + Ho

T2π Re

Re (4.81)

or

Re + Ho =(

T√

G Me

2π

)2/3

= 42,241 km (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. Theimportance of crab magnitude ρc in the modified Doppler frequencygiven in (4.79) is clearly evident from the above argument, and it mustbe taken into account when its contribution is significant. For example,referring back to Figure 4.31 (a), the effect of ρc maybe negligible on alow 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 orbitinclinated at 45. In this case, the crab angle increases whereas the crabmagnitude decreases up to about 3,000 km and then increases as theSBR altitude increases.

In what follows for illustrative purposes, the SBR height is set at506 km. Figure 4.32 shows the crab angle and crab magnitude for anSBR 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 magnitudewe can equate

dφc

dα1= 0, (4.84)

and

dρc

dα1= 0. (4.85)

After some simplification, Equation (4.84) and (4.85) together with(4.72) and (4.73) yield

sin 2α1 = 0. (4.86)


4

3.5

3

2.5

2

1.5

1

0.5

0

deg

0 10 20 30 40 50 60 70 80 90

(a) Crab angle

a1

fc 1.0025

1.002

1.0015

1.001

1.0005

10 10 20 30 40 50 60 70 80 90

(b) Crab magnitude

a1

rc

FIGURE 4.32 Crab angle and crab magnitude as a function of α1 for an SBRon 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 corre-sponds to the minimum. Hence irrespective of the SBR inclinationand 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 globallywhen it is above the equator and its minimum (zero) occurs when itis above the poles.

Figure 4.33 shows the crab angle and crab magnitude as functionsof the SBR latitude and orbit inclination. From Figure 4.33, for an SBRlocated at an altitude of 506 km, the crab angle varies between ±3.77and it has maximum effect for an SBR on a polar orbit located at theequator. The crab magnitude on the other hand has maximum effect

00

0

1

2

3

2040

6080

2040

6080

(a) Crab angle

hi a1

fc

00

0.95

0.990.980.970.96

0.94

1

2040

6080

2040

6080

(b) Crab magnitude

hia1

rc

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


−100 −50 0 50 100−4

−3

−2

−1

0

1

2

3

4C

rab

Ang

le (d

eg)

Latitude

(a) Crab angle

hi = 60°hi = 30°

hi = 90°

hi = 0°

−100 −50 0 50 1000.93

0.94

0.95

0.96

0.97

0.98

0.99

1

1.01

Cra

b M

agni

tud

e

Latitude

(b) Crab magnitude

hi= 60°

hi = 30°

hi = 90°

hi = 0°

FIGURE 4.34 Crab angle and crab magnitude vs. SBR latitude for differentinclination 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 hasmaximum effect on a polar orbit and crab magnitude has maximumeffect on an equatorial orbit.

Figure 4.34 shows the crab angle and crab magnitude as functionsof SBR latitude for different inclination angles. Once again about 3.77error can be expected for the crab angle in the worst case.

On comparing (4.48) and (4.79), we notice that the effect of theEarth’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)

and

uo = cos (θAZ + φc) − cos θAZ, (4.89)

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 angleθAZ, orbit inclination ηi , and SBR latitude αi .

Figure 4.35 (a)–(c) show the error uo introduced only by the crabangle as a function of the SBR latitude and azimuth angle for orbitinclinations ηi = 90, 45, and 0. Similarly Figure 4.35 (d)–(f) showthe overall error u due to Earth’s rotation as a function of α1 andθAZ. From these figures for any orbit inclination, the maximum erroroccurs when the SBR is directly above the equator (α1 = 0). However,as Table 4.2 shows, the azimuth angle at which these errors peak aredifferent for u and uo . From Table 4.2, the maximum crab error occurs


00

−0.08

−0.06

−0.04

0.02

0

−0.02

50100

150

2040

6080

qAZ

qAZ

a1

u 0u 0

00

−0.08

−0.06

−0.04

0.02

0

−0.02

50100

150

2040

6080

qAZ a1

uu

00

−0.05−0.04−0.03

0

−0.02−0.01

50100

150

1020

3040

qAZ

a1

u 0

00

−0.1

−0.05

0.1

0.05

0

50100

150

1020

3040

qAZ a1

0 15010050

1

−1

−0.5

0

0.5

qAZ

u

0 15010050

0.08

−0.08

−0.06

−0.04

−0.02

0

0.04

0.06

0.02

hi = 0° (Equatorial orbit)

hi = 45° (Inclined orbit)

hi = 90° (Polar orbit)

(c)

(a)

(b)

(d)

(f)

(e)

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

when the SBR is vertically above the equator (α1 = 0) and the point ofinterest is directly ahead on the equator in which case the error peaksat 6.6% irrespective of the SBR orbit inclination.

This phenomenon seems to be true when the SBR is at other latitudesas well. Table 4.3 shows the maximum crab error and the correspond-ing SBR azimuth for various orbit inclinations and SBR latitudes. Fromthere the crab error peaks when the azimuth for the point of interest


Crab Angle andCrab Angle Only Crab Magnitude

SBR Orbit Max Error SBR Azimuth Max Error SBR AzimuthInclination η, (u0) θAZ (u) θAZ

90 0.066 86.11 0.066 9060 0.059 88.38 0.066 6045 0.049 88.56 0.066 4530 0.035 88.92 0.066 3015 0.018 89.46 0.066 150 0 0 0.066 0

TABLE 4.2 Maximum error in (4.88) and (4.89) for various orbitinclinations ηi (global maximum occurs for α1 = 0)

is such that the mainbeam is directly ahead along the local latitude. Inthat case, as seen from Figure 4.36 and Table 4.3, we have θo+θAZ = π/2for all inclinations. Once again the maximum error depends only onthe SBR latitude and is independent of the orbit inclination.

Table 4.4 shows the minimum crab error and the corresponding SBRazimuth for various orbit inclinations and SBR latitudes. From therethe crab angle is zero when the azimuth for the point of interest isperpendicular to the local latitude (i.e., along the local longitude).

These results are summarized in Figure 4.36 which shows threepoints of interest D1, D2, and D3 for an SBR located at B. From theabove discussion, the beam B D1 that looks directly ahead along thelocal latitude generates the maximum crab error. The beam B D2 alongthe local longitude has minimum (zero) crab error, and the beamB D3 that is off the bore-side of the array has nonzero crab error andprovides a tradeoff between crab error and array azimuth ambigu-ity. However, the overall effect of crab error on the estimated clut-ter covariance matrix and the corresponding adaptive weight vectoris much more complicated because of the Doppler-Azimuth depen-dency on range and the range foldover phenomenon.

The effect of crab angle on Doppler as a function of azimuthangle 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’srotation is to shift the azimuth angle appearing in the Doppler by ap-proximately φc = 3.77 as shown in Figure 4.37 and simultaneouslymodify the Doppler magnitude as well. As a result, even for θAZ = 90,the Doppler peak values occur away from ωd = 0 depending on therange.

This shift in Doppler with and without crab effect is illustrated inFigure 4.38 for various azimuth angles. The actual Doppler frequen-cies of the incoming clutter will depend upon the Doppler foldoverparameter in (4.52) for the array configuration.

114Space

Based

Radar

ηi = 90 ηi = 45 ηi = 30Angle Between Angle Between Angle Between

Max Err Azimuth Local Longitude Max Err Azimuth Local Longitude Max Err Azimuth Local Longitudeα1 u θAZ and SBR Path θ0 u θAZ and SBR Path θ0 u θAZ and SBR Path θ00 0.066 90 0 0.066 45 45 0.066 30 6015 0.064 90 0 0.064 42.94 47.06 0.064 26.28 63.7230 0.057 90 0 0.057 35.27 54.73 0.057 0 9045 0.047 90 0 0.047 0 90 – – –60 0.033 90 0 – – – – – –75 0.017 90 0 – – – – – –90 0 – 0 – – – – – –

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


B

Z

SBR path

Y1

Equator

Longitude

a2

a1

hi

b1 b2

q0

Latitude

Z

North pole

Y2

D2

D1

D3

Large crab effect,good azimuthqAZ

Small crab effect, poor azimuth

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

Without crab effect

With crab effect

R2

R1

R1 < Ro < R2

Ro

−200

−100

0

200

100

Dop

pler

−1 −0.5 0 0.5 1

Azimuth (cos qAZ)

qAZ2

qAZ1qAZ3

fc

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

116Space

Based

Radar

ηi = 90 ηi = 45 ηi = 30Angle Between Angle Between Angle Between

Max Err Azimuth Local Longitude Min Err Azimuth Local Longitude Min Err Azimuth Local Longitudeα1 u θAZ and SBR Path θ0 u θAZ and SBR Path θ0 u θAZ and SBR Path θ00 0 0, 180 0 0 135 45 0 120 60

15 0 0, 180 0 0 132.94 47.06 0 116.28 63.7230 0 0, 180 0 0 125.28 54.73 0 90 9045 0 0, 180 0 0 90 90 – – –60 0 0, 180 0 – – – – – –75 0 0, 180 0 – – – – – –80 0 – 0 – – – – – –

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


−20

−10

0

10

20

30

40

50

60

Dop

pler

Range (km)

0 500 1,000 1,500 2,000

qAZ = 75°

qAZ = 80°

qAZ = 90°

Without crab effect ( ).

With crab effect ( ).

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

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

W/ Earth’s rotation W/O Earth’s rotation

180

160

140

120

100

80

60

40

20

00 500 1,000 1,500

Range (km)

(a) a1 = 0°

Azi

mut

h (d

eg)

180

160

140

120

100

80

60

40

20

00 500 1,000 1,500

Range (km)

(b) a1 = 30°

Azi

mut

h (d

eg)

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


Longitude b2−b1

Lat

itud

ea

2

0 5 10 15−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1


Longitude b2−b1

Lat

itud

ea

2

0 5 10 15−5

0

5

Longitude b2−b1

Lat

itud

ea

2

0 5 10 15−1

−0.5

0

0.5

1

W/ Earth’s rotation W/O Earth’s rotationIso-azimuth Iso-range

Longitude b2−b1

Lat

itud

ea 2

0 5 10 15−5

0

5

(a) (b)

(c) (d)

FIGURE 4.40 Iso-Doppler profile in the latitude-longitude domain for anSBR 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-azimuthand iso-range profiles as well (SBR height = 506 km).

Figures 4.42–4.43 show the iso-Doppler plots for an SBR on a 60inclination 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 do-main with details as indicated there. From these figures, there is sig-nificant difference in Doppler when Earth’s rotation in taken into ac-count, and its effect on target detection performance is taken up inChapters 6 and 7.

4.7.1 Range Foldover and Crab PhenomenonWhen crab phenomenon and range foldover are present simultane-ously, multiple Doppler frequencies are generated from the associatedrange foldover points as in Figure 4.20. These Doppler frequencies


Longitude b2−b1

Lat

itud

ea

2

29

29.2

29.4

29.6

29.8

30

30.2

30.4

30.6

30.8

31

0 5 10 15

(a)


Longitude b2−b1

Lat

itud

ea

2

25

30

35

0 5 10 15

(b)

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

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

00

20

40

60

80

100

120

140

160

180

1,5001,000500

Azi

mut

h (d

eg)

Range (km)

(a) a1 = 0°


00

20

40

60

80

100

120

140

160

180

1,5001,000500

Azi

mut

h (d

eg)

Range (km)

(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) SBRat latitude 30 (SBR height = 506 km).


(a)

Longitude b2−b1

Lat

itud

ea

2

0 15105−1

−0.5

0

0.5

1


(b)

Longitude b2−b1

Lat

itud

ea

2

0 15105−5

−4

−3

−2

−1

0

1

2

3

4

5

FIGURE 4.43 Iso-Doppler profiles in the latitude-longitude domain for anSBR 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 withthese modified Doppler frequencies. The effect of crab phenomenonon clutter Doppler spread and limitations imposed on processinggains are discussed in detail in Chapter 6.

0500 1,000 1,500 2,000

20

40

60

80

100

120

Main return

Without crab effect

With crab effect

Foldover returns

Dop

pler

Range (km)

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


A

B

C

D

SBR

X

Y

Z

E F

Equator

R

North

Latitude a1

Longitude b1 Longitude b2

Point of contacton Earth

y (Grazing angle)

Vernal equinox

a1a2

a

b1

g

ÆECB = a1, ÆXCE = b1, Æ FCD = a 2, ÆXCF = b 2ÆBCD = a, R = Rea

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

Appendix 4-A: Ground Range from Latitudeand Longitude Coordinates

Let (α1, β1) and (α2, β2) represent the latitude and longitude of thenadir point B and the point of interest D on Earth respectively asshown in Figure 4.45.

To determine the range R from point B to point D, and the coreangle α between the radii CB and CD, we proceed as follows:

We have

EC B = α1, XC E = β1, F C D = α2, 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)


A

B

C

Re

D

SBR

North

Vernal equinox

Re

Z

H

R

sin a1

sin a2

E F

cos a2

cos a1

a1

Point ofinterest on ground at (a 2, b 2)

Nadir point(a1, b1)

G

North pole

X

Y

Equator

b 2- b1

b 2- b1

b 2

b

b 1

a

a2

g

FIGURE 4.46 Nadir point—point of interest plane.

Further EC F = β2 − β1, so that from triangle ECF (seeFigure 4.46),

E F 2 = R2e (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 = 2R2e (1 − cos α). (4A.4)

Let BG be parallel to EF so that BG = EF and FG = EB as inFigure 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)


and hence

R = Reα = 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 tri-angular law applied to the spherical triangle ZBD in Figure 4.46, weobtain

sin γ

sin(π/2 − α2)= sin β

sin(π/2 − α1)= sin(β2 − β1)

sin α(4A.9)

which gives

sin γ = cos α2sin(β2 − β1)

sin α(4A.10)

and

sin β = cos α1sin(β2 − β1)

sin α. (4A.11)

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

Appendix 4-B: Nonsphericity of Earth and theGrazing 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 likean 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

b2 = a2(1 − e2) (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 ellipsoidas shown in Figure 4.47.

Let A represent the location of the SBR with its nadir point B atlatitude α1, C the center of the ideal Earth, D the point of interest onEarth with latitude α2, and E its counterpart on the ideal Earth. Let ψeand ψ represent the grazing angles at D and E . Clearly ψe representsthe grazing angle of interest based on the actual ellipsoidal model, and


A

D

B

H

C

E

G

Rs1Elliptical

Earth

SphericalEarth

SBR

a

b

Rs2

X

North pole

F

yye

ufa2

a1

a r

r1

FIGURE 4.47 Nonspherical Earth and grazing angle correction.

ψ that based on the ideal spherical model given by (4.6). The objectiveis to determine the correction factor of the ideal grazing angle ψ so asto obtain the actual grazing angle ψe at D.

Let ρ1 and ρ represent the radial distances CB and CD to the nadirpoint and point of interest respectively. Further, let6

α = α1 − α2 (4B.3)

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

Rs1 =√

a2 + (H + ρ1)2 − 2a (H + ρ1) cos α (4B.4)

and

cos ψ

H + ρ1= sin α

Rs1

(4B.5)

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


so that the ideal grazing angle equals

ψ = cos−1(

H + ρ1

Rs1

sin α

). (4B.6)

Similarly from ACD, the slant range AD equals

Rs2 =√

ρ2 + (H + ρ1)2 − 2ρ(H + ρ1) cos α (4B.7)

and with CDG = υ, where DG represents the perpendicular to thetangent at D, we have

cos(ψe − υ)H + ρ1

= sin α

Rs2

(4B.8)

so that the actual grazing angle is given by

ψe = υ + cos−1(

H + ρ1

Rs2

sin α

). (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 =√

R2e + (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) toaccommodate the nonsphericity of Earth. To determine υ, we proceedas follows.

Determination of the Correction Factor υThe equation of an ellipse is given by

x2

a2 + y2

b2 = 1, (4B.13)


where (x, y) represents the coordinates of the point of interest D. Dif-ferentiating (4B.13) the slope of the target point at D is given by

dydx

= − xy

b2

a2 . (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 tangentequals tan(π/2 + φ). Hence

dydx

= tan(π/2 + φ) = −cotφ = − xy

b2

a2 (4B.15)

or

tan φ = yx

a2

b2 . (4B.16)

Substituting (4B.16) into (4B.13) we get

x2

a2 + x2 b2

a4 tan2 φ = 1 (4B.17)

or

x2 = a4 cos2 φ

a2 cos2 φ + b2 sin2 φ= a2 cos2 φ

1 − e2 sin2 φ(4B.18)

and

y2 = a2(1 − e2)2 sin2 φ

1 − e2 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 − e2 sin2 φ

(4B.20)

y = ρ sin α2 = a (1 − e2) sin φ√1 − e2 sin2 φ

(4B.21)

which gives

ρ2 = a2 1 − (2e2 − e4) sin2 φ

1 − e2 sin2 φ= a2 1 + (1 − e2)2 tan2 φ

1 + (1 − e2) tan2 φ. (4B.22)

Using the relation

tan α2 = yx

= b2

a2 tan φ = (1 − e2) tan φ (4B.23)


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

ρ2 = a2(1 − e2)

sin2 α2 + (1 − e2) cos2 α2= a2(1 − e2)

1 − e2 cos2 α2(4B.24)

or

ρ = a

√1 − e2

1 − e2 cos2 α2. (4B.25)

Here ρ represents the Earth’s local radius at D, and it is given interms of the equatorial radius a , Earth’s eccentricity e and the locallatitude α2 at D. Notice that the local radius at latitude α2 is smallerthan that at the equator.

Since ρ1 corresponds to latitude α1, we also have

ρ1 = a

√1 − e2

1 − e2 cos2 α1. (4B.26)

Further from (4B.20)–(4B.21), we have

x sin φ − y cos φ = ρ sin(φ − α2) = a e2 sin φ cos φ√1 − e2 sin2 φ

. (4B.27)

But from CDG

υ = φ − α2. (4B.28)

Hence

ρ sin υ = a e2 sin(2φ)

2√

1 − e2 sin2 φ. (4B.29)

Similarly

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

1 − e2 sin2 φ.

(4B.30)

Thus

tan υ = e2 sin(2φ)

2(1 − e2 sin2 φ)= e2 sin(2α2)

2(1 − e2 cos2 α2). (4B.31)

Equations (4B.25)–(4B.31) can be used in (4B.4)–(4B.9) to determinethe 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(

e2 sin(2α2)2(1 − e2 cos2 α2)

)(4B.32)


where

ψ = cos−1(

Re + HRs

sin(R/Re ))

(4B.33)

represents the grazing angle corresponding to a spherical Earth asin (4.6). Thus the correction factor to the grazing angle at latitude α2equals

υ = tan−1(

e2 sin(2α2)2(1 − e2 cos2 α2)

)(4B.34)

and it is plotted in Figure 4.48. Notice that the maximum correctionoccurs at 45 latitude and the grazing angle at that latitude computedusing the spherical Earth must be incremented by 0.1932.

Finally suppose (α1, β1) and (α2, β2) represent the latitude-longitude pair for the nadir point B and the point of interest D re-spectively and assume that β1 = β2 as shown in Figure 4.49.

In this case consider the great circle that goes through B and D andlet it intersect the equator at X at latitude αo = 0 and longitude βo (seeFigure 4.49). The angles BCD and DCX are in the plane of this greatcircle and they need to be re-evaluated.

0 10 20 30 40 50 60 70 80 90

Latitude a 2 (deg)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Cor

rect

ion

Fact

or (d

eg)

Correction Factor vs a2

FIGURE 4.48 Correction factor for grazing angle at different latitudes.


A

B

C

D

SBR

X

Z

E F

Equator

R

North

Longitude b 2

Point of contacton Earth

Vernal equinox

Great circle through B and D

b1

y

aa2a1 (0, bo)

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)


and this equation is first used to determine the longitude of interestβo . Knowing βo , the corrected versions of α1 and α2 are given by

α′1 = cos−1 (cos α1 cos(β1 − βo )) (4B.40)

and

α′2 = cos−1 (cos α2 cos(β2 − βo )) . (4B.41)

These values and the projected versions of α1 and α2 onto the planecontaining the great circle that goes through B and D. From (4B.39),notice that α′

1 − α′2 = 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 determinethe correction factor for the actual grazing angle ψe .

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

What Is Doppler Effect?Suppose V represents the relative velocity between the transmit sources(t) and the target along their line of sight, and let D1 be the rangeof the target from the source at time t = to , when the first pulse istransmitted toward the target for a finite duration τo (see Figure 4.50).Assume the source to be stationary and the target moves toward thesource with velocity V.

Let τ1 represent the time it takes for the transmit signal to arrive atthe moving target. During that time interval, the waveform travels adistance 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 atthe stationary transmitter after another τ1 seconds at t = to + 2τ1 andthe corresponding received signal equals

x1(t) = s(to + 2τ1). (4C.3)


First pulse

D1

Target

V

Vt1

2Vt1

Vt2ct2

2Vt2

ct1 V

V

Second pulse Target

V

V

V

(a) Propagation of the first pulse

(b) Propagation of the second pulse

s(to)

s(to + T)

x1(t )

x2(t )

D2 = D1- VT

t = to

t = to + t1

t = to + 2t1

t = to + T + t2

t = to + T + 2t2

t = to + T

FIGURE 4.50 Propagation of two consecutive pulses.

Notice that the delay τ1 contains combined information about therange D1 and target velocity V, both of which are unknown.

Let T represent the radar pulse repetition interval. Thus a secondpulse is transmitted at t = to + T . At that instant, the distance totarget equals

D2 = D1 − VT. (4C.4)

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

τ2 = D2

c + V= D1

c + V− VT

c + V= τ1 − VT

c + V. (4C.5)


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(

to + 2τ1 + T − 2VTc + 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− j2ωoVT

c+V . (4C.8)

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

2ωo VTc + V

= π

(1

1 + V/c

)2VTλ/2

≈ π2VTλ/2

= πωd , (4C.9)

where we have used c V, which is the case in radar (but not insonar since sound travels much slower than light).

In (4C.9), we have defined

ωd = 2VTλ/2

. (4C.10)

Substituting (4C.9) into (4C.8), we get

x2(t) = s(to + 2τ1 + T)e− jπωd . (4C.11)

On comparing (4C.3) and (4C.11), the envelope of the second pulsereturn is a shifted version of the first pulse return (shifted by a knownamount T) and hence knowing the first pulse return, the envelope ofthe second return is also known. In particular, if these envelopes areconstant over the pulse duration, then the first and second returnshave the form

x1(t) = A, (4C.12)

and

x2(t) = Ae− jπωd . (4C.13)

In (4C.13), the factor ωd appearing in the phase shift factor repre-sents 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 pulsesof 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


conveniently expressed in vector form as

x1(t)

x2(t)...

xM(t)

= A

1

e− jπωd

...

e− j (M−1)πωd

. (4C.14)

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

y(ωd ) =M∑

i=1

xi (t)e jπ(i−1)ωd = MA. (4C.15)

Thus the target echoes can be coherently amplified in magnitude,provided the Doppler factor ωd used in (4C.15) coincides with the truetarget 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 Dopplervariable ωd in (4C.15) coincides with the true target Doppler frequency.

−1 −0.5 0 0.5 1wd0wd

y (w

d) in

dB

−30

−25

−20

−15

−10

−5

0

FIGURE 4.51 FIR Doppler filter.


Appendix 4-D: Oblate Spheroidal Earthand Crab Angle Correction

The Earth is not an exact sphere and it is more like an oblate spheroidwhich introduces a correction factor υ to the grazing angle ψ that isbased 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 + HRe

)sin θEL (4D.2)

and

υ = tan−1(

e2 sin(2α2)2(1 − e2 cos2 α2)

). (4D.3)

To account for the Earth’s oblate spheroidal shape, the actual graz-ing 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

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

ωde = 2Tr

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

+ 2Tr

λ/2υVe cos α2 sin β sin ψ

= ωd + 2Tr

λ/2υVe cos α1 sin γ sin ψ, (4D.6)

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


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

ξ = 2Ve Tr

λ/2υ sin ψ(cos ηi cos θAZ + sin θAZ

√cos2 α1 − cos2 ηi )

= 2VpTr

λ/2ε cos α1 cos (θAZ − u) υ sin ψ (4D.7)

where

u = tan−1

√cos2 α1

cos2 ηi− 1 (4D.8)

and

ε = Ve

Vp. (4D.9)

Substituting (4D.7) into (4D.6) and using (4.71) and (4.59), the actualDoppler simplifies to

ωde

= 2VpTr

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

= 2VpTr

λ/2sin θEL

(ρc + ευ cos α1 cos(φc + u)

sin ψ

sin θEL

)cos (θAZ + φc)

+ ευ cos α1 sin(φc + u)sin ψ

sin θELsin (θAZ + φc)

= 2VpTr

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

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

= 2VpTr

λ/2ρ sin θEL cos(θAZ + φc − φe ) (4D.10)

where

φe = tan−1(

υ cos α1 sin(φc + u) tan ψ

ρc + υ cos α1 cos(φc + u) tan ψ

)(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)


−20 −10 0 10 20−4

−2

0

2

4

a2 (deg)

(a) fe

f e(d

eg)

×10−4

−20 −10 0 10 201.0022

1.0022

1.0022

a2 (deg)

(b) r

r

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

Here, is as defined in (4.61). From (4D.10) and (4D.11), the quantityφe represents the correction to the crab angle φc due to the oblatespheroidal shape of the Earth. Figure 4.52 shows φe and ρ as functionsof 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 variedby moving the point of interest along that longitude. From there, thecorrection factor φe is between −0.0004 and 0.0004 and the correctionfactor ρ stays at 1.0022 as α2 changes. Interestingly, ρ is the same as ρcin (4.73) that corresponds to the spherical Earth. Figure 4.53 shows ρ

a1 (deg)−20 −10 0 10 20

1.0022

1.0021

1.002

r,r c

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


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 oblatespheroidal 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 Aero-nautics and Astronautics, Massachusetts Institute of Technology, Lexington,MA, May 2000.

[3] Mark E. Davis, Braham Himed, and David Zasada, “Design of Large SpaceRadar 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 Techniquesfor 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 Foldoveron Space-Based Radar Performance”, Proc. IEEE Transactions on Aerospaceand Electronic Systems, Vol. 42, no. 3, July 2003.

[6] G.A. Andrews and K. Gerlach, “SBR Clutter and Interference,” Ch. 11, Space-Based 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.


C H A P T E R 5Space-Time Adaptive

Processing

Space-Time Adaptive Processing (STAP) refers to spatially and tem-porally distributed data collection and processing. For spatial pro-cessing, an array of sensors that are distributed in space (in one ortwo dimensions) can be used; for temporal processing, a sequence ofpulses 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 pro-cessing gain that can be realized using spatio-temporal processing,the simpler situation—spatial (array) processing is considered first inthis chapter.

This is followed by a discussion on an optimal weight vector forcombining the array outputs so as to maximize the output signal tointerference plus noise ratio (SINR) in the spatial domain as well asthe spatio-temporal domain.

The STAP issues are considered next for a side-looking airborneradar. Angle-Doppler clutter power spectral properties, various tech-niques to illustrate clutter cancellation are discussed next includingthe eigen-structure based methods and their variations.

5.1 Spatial Array ProcessingArray processing refers to transmitting/receiving using a set of sen-sors rather than a single sensor. The use of multiple sensors calls foradditional processing both prior to transmitting and/or after receiv-ing signals.



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 ofsensors instead of a signal sensor at the receiver?

The advantage in using an array can be best illustrated in a narrow-band 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 inFigure 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 makeswith the line of the array. If the second sensor is d apart from the firstsensor, then d cos θ represents the additional distance to be covered bythe wavefront to reach the second sensor. Hence the wavefront arrivesat the second sensor after

τ = d cos θ

c(5.3)

seconds. Hence the second sensor output with reference to the firstsensor is given by

x(t) = s(t − τ ) = s(t)e− jωod cos θ

c

= s(t) e− jπ dλ/2 cos θ = s(t) e− jπd cos θ (5.4)

where

d = dλ/2

(5.5)

s(to)

s(t )

……

……

……

……x1(t ) x2(t ) xi (t ) xN (t )

Planarwavefront

s(t – t)

d cos q

d

Referencesensor

q

t

∼

∼

FIGURE 5.1 Spatial processing.

C h a p t e r 5 : S p a c e - T i m e A d a p t i v e P r o c e s s i n g 141

represents the interelement distance normalized to half-wavelength.In general, let di represent the ith sensor location normalized to λ/2with respect to the reference sensor. Then following the above argu-ment, 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

x(t) =

x1(t)

x2(t)...

xi (t)...

xN(t)

= s(t)

1

e− jπd2 cos θ

...

e− jπdi cos θ

...

e− jπdN cos θ

= s(t)a (θ ) (5.7)

where

a (θ) =

1

e− jπd2 cos θ

...

e− jπdi cos θ

...

e− jπdN cos θ

(5.8)

represents the spatial steering vector associated with the direction ofarrival 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

a (θ ) =

1

e− jπ cos θ

...

e− jπ(i−1) cos θ

...

e− jπ(N−1) cos θ

. (5.10)

In general, the sensor measurements are buried in noise and letn1(t), n2(t), · · · refer to the input noise signals at the sensors. Thus the


ith sensor output equals

xi (t) = s(t)e− jπdi 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)

where

n(t) = [n1(t), n2(t), . . . , nN(t)]T (5.13)

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 thefirst one, then the signal to noise ratio (SNR) at its input equals

(SNR)i = E|s(t)|2E|n1(t)|2 = P

σ 2n

(5.14)

where

σ 2n = 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 outputin (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 beby the amount e jπdi cos θ as shown in Figure 5.2.

Combining the phase shifted outputs, we obtain the final outputy(t) to be

y(t) =N∑

i=1

xi (t)e jπdi cos θ . (5.16)

. . .. . .

. . . . . .

y(t)

Σ

x1(t ) x2(t ) xi(t ) xN(t )

e jpd1cosq e jpd2cosq e jpdNcosqe jpdicosq

FIGURE 5.2 Phase shifted array.


From (5.11), we have

y(t) = Ns(t) +N∑

i=1

ni (t)e jπdi cos θ = Ns(t) + no (t) (5.17)

where

no (t) =N∑

i=1

ni (t)e jπdi cos θ . (5.18)

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

(SNR)o = E|Ns(t)|2E|no (t)|2 = N2 E|s(t)|2

E∑

i∑

k ni (t)n∗k (t)e jπ(di −dk ) cos θ

= N2 P∑N

i E|ni (t)|2= N2 P

Nσ 2n

= NPσ 2

n= N · (SNR)i . (5.19)

In (5.19), we have assumed the various sensor noise outputs to beuncorrelated, i.e., Eni (t)n∗

j (t) = 0 for i = g. The advantage of usingan array of sensors is obvious. From (5.19), in the case of uncorrelatednoise the simple phase shifting and sum operation improves the arrayoutput SNR for the desired signal by a factor equals to the number ofsensors!

From (5.17)–(5.19), the desired signal gets added coherently in am-plitude, whereas the noise gets added in power only; the above oper-ation 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 gainalong that direction.

Uniform ArrayAn interesting question in the context of beamforming is the nature ofthe gain pattern along other angles while the array is steered along aspecific 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 normalizedarray pattern is given by (xi (t) ≡ 1)

G(θ ) =∣∣∣∣∣ 1

N

N∑i=1

e jπdi cos θ

∣∣∣∣∣2

, (5.20)


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 ) =(

sin( Nπ cos θ

2

)N sin

(π cos θ

2

))2

. (5.21)

From (5.21), the width of the mainbeam is proportional to2N

N→∞−→ 0. (5.22)

cos (q )−1 1

−50

0.5−0.5

−10−13.46 dB

0

0

−20

−30

−40

G(q

) in

dB

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


Thus the array mainbeam width can be decreased and made nar-rower by increasing the number of sensor elements. However, thisis not true for the sidelobe levels. The sidelobe levels of a uniformlyplaced array get settled around −13.46 dB, and cannot be improvedfurther 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 ω′o = 2π/N, so that the first dominant

sidelobe level occurs approximately at the center of these two nulls at

ω1 = ωo + π

2N= 3π

2N. (5.26)

The corresponding gain at ω = ω1 is given by

G(ω1 = 3π/N) =(

sin(3π/2)N sin(3π/2N)

)2

= 1

[N sin(3π/2N)]2

= 1[N(

3π2N − (3π/2N)3

3! + · · ·)]2

= 1(3π2 − 9π2

16N2 + · · ·)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 inthe array.1 However, the sidelobe levels in the array gain pattern inFigure 5.4 can be further suppressed by introducing shading factorsat the sensor outputs [1].

1See Appendix 5-A for a detailed treatment on the sidelobe levels of a uniformarray.


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∑

i=1

w∗i xi (t). (5.29)

Equation (5.29) can be expressed more conveniently in matrix formas

y(t) = w∗x(t) (5.30)

where

w =

w1(θ )

w2(θ )...

wN(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 steeringvector. In general, x(t) is more complex than a single signal buried inuncorrelated noise (e.g., signal plus clutter or multiple signals arrivingsimultaneously).

Let

R = Ex(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 ithand j th sensor elements. R is in general a positive-definite hermitian(R = R∗ > 0) matrix. For example, for a single source situation as in

…

∗……

…

y(t )

Σ

x1(t ) x2(t ) xi (t ) xN(t )

w1 w2 wi wN∗ ∗ ∗

FIGURE 5.5 Arraywith direction-dependentweights.


(5.12), with θ = θo and p = E|s(t)|2, we get

R = E|s(t)|2a (θo )a∗(θo ) + σ 2n I

= Pa (θo )a∗(θo ) + σ 2n 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 uncorre-lated and of equal variance σ 2

n (or more generally, noise elements areindependent and identically distributed (i.i.d.) random variables). Ifthe noise term is taken to be broad enough to include interferences andclutter, then the element to element uncorrelated assumption may nolonger be true for noise and in that case (5.35) generalizes to

R = Pa (θo )a∗(θo ) + Rn (5.36)

where using (5.13),

Rn = En(t)n∗(t) > 0 (5.37)

represents the array input interference plus noise covariance matrix.Although the signal part of (5.35) and (5.36) has rank one, the noisecovariance matrix is invariably full rank and hence R is also generallyof full rank. Using (5.30) and (5.33), the array output power for lookdirection θ equals

P(θ ) = E|y(t)|2 = E|w∗(θ )x(t)|2= Ew∗(θ )x(t)x∗(t)w(θ ) = w∗(θ )Ex(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 everylook 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

PB(θ) = P|a∗(θ )a (θo )|2 + Nσ 2n = N2 P

(sin(Nω)N sin ω

)2

+ Nσ 2n , (5.40)

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


(a) q0 = 90°q in deg q in deg

(b) q 1 = 70°, q 2 = 75°

P B(q

) in

dB

P B(q

) in

dB

−2 −2

−4

−6

−8

−10

−12

−4

−6

−8

−10

−12

2

0 0

0

150150 100100 5050

2

0

FIGURE 5.6 Beamformer.

As Figure 5.6 (b) shows, the finite size of the mainbeam width cancause merging of peeks when multiple signals are present simultane-ously along close, but different directions, and it becomes necessaryto use more complicated weight vectors that suppress the unwantedsignals for better resolution when the array is focused along a specificdirection.

One approach in this context is to choose the weight vector so as tomaximize the array output SNR.

5.1.2 Maximization of Output SNRConsider a desired source s(t) located along the direction θ in presenceof other possible sources, interference, and noise. The array outputvector 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 = En(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 maximizationof 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)


This gives the output SNR to be

SNRo = E|s(t)|2|w∗a (θ )|2E|w∗n(t)|2

= P|w∗a (θ )|2w∗En(t)n∗(t)w = P|w∗a (θ )|2

w∗Rnw. (5.44)

To maximize (5.44), we can use Schwarz’s inequality that states forany 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

w∗a (θ ) = w∗R1/2n R−1/2

n a (θ ) = (R1/2n w

)∗(R−1/2

n a (θ )) = u∗v (5.47)

with u = R1/2n w and v = R−1/2

n a (θ ) so that the numerator in (5.44)becomes

|w∗a (θ )|2 = |u∗v|2 ≤ (u∗u)(v∗v). (5.48)

But in this case

u∗u = (R1/2n w

)∗(R1/2

n w) = w∗Rnw (5.49)

and

v∗v = (R−1/2n a (θ )

)∗(R−1/2

n a (θ )) = a∗(θ )R−1

n a (θ ). (5.50)

Substituting these into (5.48), we get

|w∗a (θ )|2 ≤ (w∗Rnw)(a∗(θ )R−1

n a (θ ))

(5.51)

and hence (5.44) becomes

SNRo ≤ Pa∗(θ)R−1n a (θ ) = SNRmax. (5.52)

From (5.46), clearly the maximum SNR in (5.52) can be realized if

R1/2n w = kR−1/2

n a (θ ). (5.53)

2If Rn is an N × N Hermitian positive-definite matrix, then Rn = UU∗ = R∗n,

where U is unitary (U U∗ = I ) and is diagonal with positive real entries λ1,λ2, . . . , λN. In that case, R1/2

n = U1/2U∗ and R−1/2n = U−1/2U∗ where ±1/2 is

again diagonal with entries λ±1/21 , λ

±1/22 , . . . , λ

±1/2N , etc.


Beamformer

−1/2Rn

y(t ) = s(t )b(q ) + wn(t )x (t )z (t ) = b∗(q )y(t ) = w ∗x(t)b(q )

FIGURE 5.7 Matched filteras whitening followed by beamformer.

or3

w = R−1n a (θ ). (5.54)

Equation (5.54) represents the optimum desired weight vector thatmaximizes the output SNR for the look direction θ . Notice that inthe case of a single source present in i.i.d. noise, (5.54) reduces tothe ordinary beamformer in (5.32); however, if the noise is not i.i.d.(colored noise), the optimum weight vector is not the beamformereven in a single source scene and is given by (5.54).

Interestingly, the optimum weight vector in (5.54) that maximizesthe output SNR can be given another interpretation as well.

Suppose the data x(t) in (5.41) is first passed through a whiteningfilter so that the colored noise n(t) becomes white noise wn(t). From(5.42), R−1/2

n whitens the noise, since

y(t) = R−1/2n x(t) = s(t)R−1/2

n a (θ ) + R−1/2n n(t)

= s(t)b(θ) + wn(t), (5.55)

and

Rwn = E

wn(t)w∗n(t) = R−1/2

n En(t)n∗(t)R−1/2n

= R−1/2n RnR−1/2

n = I. (5.56)

Thus the noise component wn(t) in (5.55) indeed represents whitenoise, and the signal component b(θ) is given by

b(θ ) = R−1/2n a (θ ). (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 filteris the ordinary beamformer b(θ). These two operations are shown inFigure 5.7.

From Figure 5.7, the final output

z(t) = b∗(θ )y(t) = a∗(θ )R−1/2n y(t)

= a∗(θ )R−1n x(t) = (R−1

n a (θ))∗

x(t) = w∗x(t), (5.58)

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


where w is as given in (5.54). In general,

w = R−1n a (θo ) (5.59)

represents the optimum weight vector that maximizes the output SNRfor a target located along θo . Thus the optimum weight vector is equiv-alent to whitening followed by matched filtering operation.

The whitening operation in Figure 5.7 shows that the adaptive pro-cessor cancels out any dominant interference signal present in Rn bygenerating nulls along their respective directions of arrival, and thematched filtering operation boosts the output along the desired direc-tion a (θo ).

For example, let the interference/noise signal scene consist of threespatially discrete uncorrelated signals of equal power along directionsθ1, θ2, θ3, and white noise. This gives

Rc = a (θ1)a∗(θ1) + a (θ2)a∗(θ2) + a (θ3)a∗(θ3) + σ 2n I (5.60)

and the maximum SNR output

SNRmax = a∗(θ )R−1n a (θ) (5.61)

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

Interferences

SIN

Rm

ax (d

B)

15010050

qAZ (deg)

−15

−5

−10

−25

−20

0

0

FIGURE 5.8 Three uncorrelated arrivals along θ1 = 55, θ2 = 75, andθ3 = 105. Interference to noise ratio = 20 dB. A 12-sensor array is used here.


The adaptive processor output

P(θ ) = |w∗a (θ)|2 =∣∣a∗(θo )R−1

n a (θ )∣∣2 (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 threeinterference sources and boosts the target output along θo .

To implement (5.54), Rn needs to be known. Generally, the idealcovariance matrices are unknown, and the next best thing (in termsof maximum likelihood (ML) estimation) is to use its ML estimate ofRn using data simples x1, x2, . . . , xK , where xi refers to a particularobservation of the array output “noise” vector. If the “noise” repre-sents jointly Gaussian random variables, then the ML estimate fortheir covariance matrix Rn is given by

Rn = 1K

K∑k=1

xk x∗k (5.63)

that is based on K samples. Clearly, to invert Rn, we need K > N, andwhen Rn is invertible, its inverse may be used in (5.54) as an estimatefor 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

wSMI = R−1n a (θ). (5.64)

Interferences

Target direction

SIN

R (d

B)

15010050qAZ (deg)

−30

−20

−10

−60

−50

−40

10

0

0qo

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


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

5.2 Space-Time Adaptive ProcessingSpace-Time-Adaptive Processing (STAP) refers to joint spatio-temporal processing where spatial diversity is realized utilizing a setof sensors configured in one or two dimensions and temporal process-ing is achieved using returns from a sequence of periodically trans-mitted pulses. The primary reason for repeated temporal pulse trans-mission is to detect target Doppler (i.e., moving targets), and in thepresence of an array of sensors, each transmitted pulse produces adata vector. Stacking these vectors corresponding to different pulses,we can generate a space-time data vector. The situation correspondingto 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 asingle 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 thesame reference sensor are given by (4C.11)–(4C.13). From Figure 5.10and (5.12), the N array output vector due to first pulse equals

x1(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

x2(t) = s(t)a (θ )e− jπωd + n2(t), (5.66)

1 2 N

……

[xM]

[x2]

[x1]

x =… …

(a) (b) STAP vector (MN × 1)

x(t + (M − 1)T ) = xM

x(t + T ) = x2

x(t ) = x1

FIGURE 5.10Space-time datacorresponding toN sensors and Mpulses in twoconfigurations.


where the identical nature of the transmit waveform from pulse topulse is utilized. Here ωd represents the Doppler frequency from thedesired target of interest. In general, the return due to the ith pulseequals

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

x =

x1x2...

xi...

xM

= s(t)

a (θ )a (θ)e− jπωd

...

a (θ)e− jπ(i−1)ωd

...

a (θ )e− jπ(M−1)ωd

+ n(t) (5.68)

where

n(t) =

n1(t)n2(t)

...

nM(t)

(5.69)

represents the external noise vector. Following (4C.14), (5.8), we definethe temporal steering vector (M × 1)

b(ωd ) =

1e− jπωd

...

e− jπ(i−1)ωd

...

e− jπ(M−1)ωd

. (5.70)

Using (5.70) in (5.68), the spatio-temporal data vector at time tequals

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)


represents the space-time data vector at time t = tk and such K ob-servations corresponding to time instants tk , k = 1, 2, . . . , K representthe space-time data cube.

Equation (5.73) has the same form as (5.12) and hence all the pro-cessing techniques described earlier are applicable here as well. Asthe following application shows, the specific form of the Doppler willdepend on the nature of the problem.

5.3 Side-Looking Airborne RadarIn an airborne radar, the platform is moving with velocity V, and pe-riodic pulses are transmitted toward the ground to perform groundmoving target indication (GMTI). In this case, the ground is station-ary, however, the returns do generate a Doppler frequency becauseof the relative velocity of the ground with respect to the movingplatform.

For a one-dimensional side-looking radar with N uniformly placedsensors that are spaced d apart (with d representing the normalized

Airborne radarV

q k

No bins

d~

R

FIGURE 5.11 Airborne side-looking radar.


interelement distance with respect to half-wavelength (Figure 5.11)),the spatial steering vector has the form (see (5.8))

a (θ) =

1e− jπd cos θ

...

e− jπ(i−1)d cos θ

...

e− jπ(N−1)d cos θ

(5.74)

corresponding to the look direction θ from the line of the array. Herea stationary ground patch in the direction θ has a relative velocityV cos θ with respect to the moving platform and hence from (4C.10),it generates a Doppler frequency

ωd = 2VTλ/2

cos θ , (5.75)

where T represents the pulse repetition interval. In this case the tem-poral steering vector b(ωd ) is given by (5.70). On comparing the expo-nents in the spatial and temporal steering vector, we obtain

ωd = 2VTλ/2

cos θ = 2VTdλ/2

d cos θ

= 2VTd

d cos θ = β(d cos θ ) (5.76)

where

β = 2VTd

= VTd/2

(5.77)

refers to the Doppler foldover factor.4 From (5.77), the Dopplerfoldover factor represents the number of half-interelement spacingtraveled by the platform during one pulse repetition interval T . From(5.76), the Doppler frequency ωd for a side-looking radar is propor-tional to the spatial angular parameter d cos θ appearing in (5.74), theproportionality term given by the Doppler foldover factor β.

If β = 1, the clutter fills the angle-Doppler space exactly once; ifβ > 1, the clutter folds over (aliasing) into the Doppler space creatingclutter Doppler ambiguity.

The space-time steering vector corresponding to ground clutter re-turns for a side-looking radar is given by (5.72) with a (θ ) and b(ωd )

4β is also known as the Brennan factor.


1

0.8

0.6

0.4

0.2

−0.5 0 1

0

0

0.5

cos (q )

−0.2

−25

−20

−15

−10

−5

−0.4

−0.6

−0.8

−1−1

PB(q, wd) in dBw

d

FIGURE 5.12 Angle-Doppler profile.

given by (5.74), (5.70) and ωd as in (5.75)–(5.76). Using these valuesand defining

φ = πd cos θ , (5.78)

the space-time steering vector corresponding to ground clutterpatches takes the explicit form

s(θ , ωd ) =[

1, e− jφ , · · · e− j (N−1)φ ,....

e− jβφ , · · · e− j(β+(N−1))φ ,

· · · ....· · · e− j((M−1)β+(N−1))φ

]T

. (5.79)

The peculiar single variable dependency in (5.79) is noteworthy,and it gives special structure to the clutter spectra when viewed inthe angle-Doppler domain (see Figure 5.12), that can be exploited forbetter clutter suppression through STAP.

In a simplified model, the ground clutter data vector correspondingto a particular range has the form

xc =No∑

k=1

cks(θk , ωdk ), (5.80)


where the summation is carried out over all ground patches at thesame range and s(θk , ωdk ) is as given in (5.79) with θ replaced by θk(see Figure 5.11). Here ck represents the scatter strength from the kthpatch for the particular range of interest. If we assume the clutterreturns are uncorrelated from patch to patch, the clutter covariancematrix has the form

Rc = E

xcx∗c

=No∑

k=1

Pksks∗k (5.81)

where

Pk = E|ck |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 thenoise term into (5.80), the total covariance matrix (in the case of i.i.d.noise) has the form

Rc =No∑

k=1

Pksks∗k + σ 2

n I. (5.84)

Here σ 2n represents the independent and identically distributed

common noise variance at the array input. Thus,

CNR =∑

k Pk

σ 2n

(5.85)

defines the clutter power to noise power ratio (CNR). To compute theangle-Doppler power distribution, we can employ a two-Dimensional(2D) beamformer with

wB = s(θ , ωd ). (5.86)

Thus

PB(θ , ωd ) = E∣∣w∗

Bx∣∣2 = s∗(θ , ωd )Rcs(θ , ωd ) (5.87)

represents the angle-Doppler power distribution. Figure 5.12 showsthe beamformer output using the ideal clutter covariance matrix Rcin (5.87).

The concentration of clutter power along the angle-Doppler diago-nal ridge in Figure 5.12 is not surprising since from (5.75), the Doppleris proportional to cos θ . The slope of the ridge is determined by theclutter foldover factor β appearing in (5.77). Figure 5.13 shows theclutter power distribution for β < 1, β = 1, and β > 1. Notice that


(a) b = 0.5 (b) b = 1

(c) b = 2

1

1

PB(q, wd) in dBPB(q, wd) in dB

PB(q, wd) in dB

0.50 1−0.5−0.5

−0.5

0.50

10.50

0

0

−0.2

−0.4

−0.8

−1−1

−1

−1

−0.6−25

−20

−10

−15

−5

0

−25

−20

−10

−15

−5

0

cos (q )cos (q )

cos (q )

−25

−20

−10

−15

−5

0.20.40.60.8

1

0

−0.2

−0.4

−0.8

−1

−0.6

0.20.40.60.8

1

0

−0.2

−0.4

−0.8

−1

−0.6

0.2

0.4

0.6

0.8

wd

wdw

d

FIGURE 5.13 Clutter power distribution for different β.

β > 1 generates clutter foldover and weak targets present at theseclutter ridges are impossible to identify.

Generally any target present is buried in clutter, and to identifythese targets the dominant clutter needs to be suppressed. We canmake use of the maximum output SINR strategy discussed in Section5.1.2 to determine the desired weight vector in this case. Let H1 andHo represent the two hypotheses corresponding to the presence orabsence of the target. Thus, the data vector under these two hypotheseshas 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 param-eters. The corresponding covariance matrix in this case is given by

R = Ex x∗ =

Pts(θt , ωdt )s∗(θt , ωdt ) + Rc , H1

Rc , H0,(5.89)


1

1cos (q ) cos (q )

PB(q, wd) in dBPB(q, wd) in dB

0.50

0

0−0.2−0.4

−0.8−1

−1 −0.5 −0.5 10.50−1

−0.6−25

−20

−10

−15

−5

0

−80

−120

−100

−40

−60

−20

0.20.40.60.8

wd

1

0−0.2−0.4

−0.8−1

−0.6

0.20.40.60.8

wd

Target

(b) PMSNR(q, wd)(a) PB(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 asin (5.84). From (5.54), it follows that the desired space-time adaptivevector that maximizes the output signal to interference plus noise ratio(SINR) is given by

wopt = R−1c s(θt , ωdt ). (5.90)

The 2D adaptive pattern

P(θ , ωd ) =∣∣w∗

opts(θ , ωd )∣∣2 =

∣∣s∗(θt , ωdt )R−1c s(θ , ωd )

∣∣2 (5.91)

can be used to evaluate the performance of the adaptive weight vectorin (5.90) for suppressing the clutter and enhancing target detection.

Figure 5.14 corresponds to a 14-element array, 16-pulse configura-tion with a target buried in clutter and noise with the target located atθt = 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 tohypothesis 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) andFigure 5.14 (b) respectively. Notice that the target is undetectable usingthe beamformer, whereas the optimum weight vector detects the tar-get by suppressing the clutter. Notice that the dominant clutter ridgein 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 withmeasured data, the ideal clutter covariance matrix is replaced by itsML estimate; thus

Rc = 1K

K∑k=1

xkx∗k = 1

KYK YK (5.92)


Range bin of interest

Training cells

FIGURE 5.15 Range bin of interest and training cells.

where

YK = [ x1, x2, . . . , xK ]. (5.93)

Here K training cells around the range bin of interest5 (seeFigure 5.15) are used to estimate Rc , under the assumption that theyall correspond to the same ideal covariance matrix.

The sample clutter power beamformer is given by

PB(θ , ωd ) = s∗(θ , ωd )Rcs(θ , ωd ) (5.94)

and this is illustrated in Figure 5.16 along with the ideal case.

1

1−0.5cos (q )

0.50

0

0−0.2−0.4

−0.8−1

−1

−0.6 −25

−30

−20

−10

−15

−5

0

−25

−30

−20

−10

−15

−5

0.20.40.60.8

1

0−0.2−0.4

−0.8−1

−0.6

0.20.40.60.8

wd wd

(a) PB(q, wd)

1−0.5cos (q )

0.50−1

(b) PB(q, wd)^

FIGURE 5.16 Ideal vs. Estimated clutter power using 10 data samples.

5The range bin of interest by hypothesis may contain target, and hence it is notused in estimating Rc in (5.92).


5.3.1 Minimum Detectable Velocity (MDV)To study the clutter suppression capabilities of an algorithm, one mayassume a hypothetical target at θ with unknown Doppler ωd , whichgives the output SINR to be [2]

SINR = |w∗s(θ , ωd )|2w∗Rcw

≤ s∗(θ , ωd )R−1c s(θ , ωd ) = SINRmax. (5.95)

Clearly, any weight vector other than

wopt = R−1c s(θ , ωd ) (5.96)

in (5.95) performs inferior to (5.96) resulting in output SINR that islower compared to the matched filter output

SINRmax = s∗(θ , ωd )R−1c 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 ofthe Mountain top program [3]. Figure 5.17 shows the ideal matched fil-ter output in (5.97) as a function of the target velocity for θ = 90. Fromthere, for unambiguous detection up to 5 dB loss, the target velocityhas to exceed 7 m/s (25.2 km/h) and this gives the minimum de-tectable 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 estimateR

−1c is required. Obviously, for this estimate to be meaningful, the

number of samples K ≥ MN, which can violate the underlying sta-tionarity assumption in (5.92).

Number of sensors (N) 14

Number of pulses (M) 16

Platform velocity (m/s) (Vp) 100

Operating frequency (MHz) ( fc) 435

Interelement spacing (m) (d) 0.33

Normalized interelement spacing with respect (d) 0.957to half-wavelength

Pulse repetition frequency (Hz) (PRF) 625

TABLE 5.1 Array parameters


Velocity (m/s)

MF

−100 −50

−50

−10

0

−20

−30

−40

0 50 100

SIN

Rm

ax

7

FIGURE 5.17 MDV performance using ideal matched filter (MF) output.

When Rc is invertible, the SMI can be employed using (5.92). Usingthis in (5.90) we obtain

wSMI = R−1c s(θt , ωdt ) (5.98)

and the corresponding estimate for (5.91) is given by

PSMI (θ , ωd ) =∣∣w∗

SMI s(θ , ωd )∣∣2 =

∣∣s∗(θt , ωdt )R−1c s(θ , ωd )

∣∣2. (5.99)

Figure 5.18 shows this sample estimator output in the angle-Doppler domain. The array parameters are given in Table 5.1. Thetarget is at range bin 200 with arrival angle of 90 and Doppler = 0.3.

(b) Side view(a) Top view

Pow

er (d

B)

Dop

pler

Doppler−1 −1

−0.5

−20

−30

−50

−40

−10

−0.5

−25

−20

−15

−10

−5

0

0

0.5

1

cos (q )cos (q )

0

00.5 1

1

0−1−1 −0.5 0.5 10

FIGURE 5.18 Angle-Doppler performance for SMI using (5.99).


V (m/s)

Ideal

50−50 100−100

−10

−20

−30

−40

−50

−600

SINR vs Velocity

SIN

R (d

B)

SMI

SMIMF

FIGURE 5.19 MDV performance for SMI.

The CNR is 40 dB and target-to-noise ratio is 0 dB. Here, 400 samplesare used to estimate the clutter covariance matrix. The MDV perfor-mance of this algorithm obtained by replacing R−1

c by R−1c in (5.97) is

shown in Figure 5.19.Figure 5.20 shows the 2D adaptive pattern in (5.99) in the range-

Doppler domain for a fixed arrival angle θ . The arrival angle is cho-sen to be the target arrival angle θt . To compute the adaptive patternfor range bin i , the corresponding clutter covariance matrix Rc is esti-mated using the identify

Ri = 1K

K∑k =i

xkx∗k (5.100)

and the desired weight vector equals

wi = R−1i s(θt , ωdt ). (5.101)

This gives the adaptive pattern to be

P(i, ωd ) =∣∣w∗

i s(θ , ωd )∣∣2. (5.102)


−5

−10

−15

−20

−25

−1

0

0

1

200

220210

190180 Range

Pow

er (d

B)

Doppler

FIGURE 5.20 Range-Doppler for SMI.

5.3.3 Sample Matrix with DiagonalLoading (SMIDL)

The number of data samples K that can be used in (5.92) or (5.100) maybe small because of non-stationarity issues. Heterogeneous terrain,large discretes such as lakes or paved roads, multiple targets can alsocontribute to non-stationarity. Thus, if K < MN, then the samplematrix Rc is singular and the SMI method cannot be implementeddirectly. One way to circumvent this issue is to diagonally load thesample covariance matrix by an appropriate amount ε as

Rε = Rc + εI, ε > 0 (5.103)

so that Rε is invertible. Notice that Rε being invertible, it can be used in(5.98) and (5.99) in place of Rc . Figure 5.21 shows the adaptive patternfor SMIDL using a diagonal loading factor of ε = 10−7.

5.4 Eigen-Structure Based STAPThe clutter covariance matrix corresponding to a uniformly spacedarray that receives signals in a constant PRF (pulse repetition fre-quency) mode exhibits additional structural properties, provided theclutter data is uncorrelated. To see this, we start with the clutter co-variance matrix in (5.81) that corresponds to an uncorrelated clutterscene. Thus

Rc =No∑

k=1

Pksks∗k (5.104)


−5

−10

−15

−20

−25

−1

0

0

1

200

220210

190180 Range

Pow

er (d

B)

Doppler

FIGURE 5.21 Adaptive range-Doppler pattern for SMIDL using 80 samples.

where No represents the number of clutter bins used. Note that thenumber of data samples No can be very large compared to MN, thesize of the matrix. As a result, it is tempting to conclude that Rc isalways a full rank matrix. However, this is often not the case becauseof the special structure of the space-time steering vector in (5.79) cor-responding to the clutter patches [4].

From (5.79), if β is an integer less than N, then clearly, some of theentries 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 sks∗k has two identical rows

and two identical columns. As a result, the 6 × 6 matrix∑No

k=1 Pksks∗k

also has two identical rows and two identical columns irrespective ofthe actual entries and the large number of terms used in the summa-tion, thus making it rank deficient.

5.4.1 Brennan’s RuleReturning to the general case in (5.104), the space-time steering vectoris given by (5.79). Clearly the first segment in (5.79) has N independententries, and thereafter each segment generates only β new entriesprovided β is an integer less than N. To see this, notice that the integerexponent in the last entry of the first segment corresponding to the


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−1such segments which make the total number of distinct entries in anysk to be (M − 1)β + N. Thus each sks∗

k and hence the ideal covariancematrix 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, andhence (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 referto [5]

rc = (M − 1)β + N (5.109)

as the Brennan’s rule for computing the rank of a clutter covariancematrix when the clutter foldover factor is an integer. Thus, the rank ofa clutter covariance matrix is approximately given by (5.109), where β

in (5.77) represents the number of half-interelement spacing traveledby the platform within one pulse interval.

Brennan’s rule is an effective indicator of both the severity of theclutter scene, and the number of degrees of freedom required to pro-duce an effective clutter cancellation. The eigen decomposition of Rcgives further insight into this phenomenon.

5.4.2 Eigencanceler MethodsFrom (5.108) and (5.109), we have Rc in (5.104) is nonnegative-definitehermitian matrix with rank rc . Hence its eigen decomposition gives

Rc = UU∗ (5.110)

where

U = [u1, u2, . . . , urc , . . . , uMN] (5.111)


represents a unitary matrix and

=

µ1µ2

. . .

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∑

i=1

µi ui u∗i . (5.113)

From (5.113), it is clear that the clutter covariance matrix has onlyrc degrees of freedom.

If we combine the noise covariance matrix σ 2n I along with the clut-

ter covariance matrix to define the total clutter plus noise covariancematrix as in (5.84), we have

Rc = UU∗ + σ 2n I = U

( + σ 2

n I)U∗

=rc∑

i=1

λi ui u∗i +

MN∑i=rc+1

σ 2n ui u∗

i (5.114)

where

λi = (µi + σ 2n

), i = 1, 2, · · · rc . (5.115)

The ideal eigenvalues when plotted exhibits the “brick wall” shapeshown in Figure 5.22.

In term of (5.114), the maximum output SINR in (5.97) can be rewrit-ten as

SINRmax =rc∑

i=1

|u∗i s(θ , ωd )|2µi + σ 2

n+

MN∑i=rc+1

|u∗i s(θ , ωd )|2

σ 2n

. (5.116)

We can use the above eigen-decomposition of Rc to explain thesharp null found in the MDV performance of the ideal matched filteroutput in Figure 5.17.


Flat bed

50

50

40

30

20

10

00

60

70

100 200 250150

Brick wall

Index

l in

dB

rc

FIGURE 5.22 Eigenvalues of the ideal covariance matrix.

Sharp Null in MDV PerformanceThe SINR output in Figure 5.17 shows the ideal matched filter outputin (5.97) for a fixed look angle θ = θ1 while spanning the Doppler fre-quency ωd . In that case s(θ1, ωd1 ) = s1 is the only steering vector gener-ated in (5.97) that coincides with the clutter scatter set s1, s2, . . . , sNo in (5.104). From (5.109) the rank rc of the clutter covariance matrix Rc ismuch less than MN and hence from (5.104)–(5.114), the clutter steeringvector set s1, s2, . . . , sNo spans the same subspace as u1, u2, . . . , urc .But the clutter subspace and noise subspace are orthogonal since theycorrespond to distinct eigenvalues, i.e.,

u1, u2, . . . , urc ⊥ urc+1, urc+2, . . . , uMN. (5.117)

As a result, the clutter steering vector set is also orthogonal to thenoise subspace, i.e.,

s1, s2, . . . , sNo ⊥ urc+1, urc+2, . . . , uMN, (5.118)

and hence

s∗1R−1

c s1 =rc∑

i=1

∣∣u∗i s1∣∣2

µi + σ 2 ≤ s∗(θ1, ωd )R−1c s(θ1, ωd ). (5.119)

In (5.119), the inequality follows for all other steering vectorss(θ1, ωd ) generated by spanning the Doppler frequencies as inFigure 5.17, since these steering vectors are not orthogonal to the


noise subspace eigenvectors as in (5.118). Consequently, all othersteering vectors s(θ1,ωd ) = s1 contribute to the noise subspace termsin (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 corre-sponds to the clutter ridge.

Eigencanceler [4], [6]Equation (5.114) can be used to derive an alternate expression for theoptimum weight vector in (5.90). From (5.114), we obtain

R−1c = U

( + σ 2

n I)−1

U∗ =rc∑

i=1

1λi

ui u∗i +

MN∑i=rc+1

1σ 2

nui u∗

i

= 1σ 2

n

MN∑

i=rc+1

ui u∗i −

rc∑i=1

(1 − σ 2

n

λi

)ui u∗

i

= 1σ 2

n

I −

rc∑i=1

(1 − σ 2

n

λi

)ui u∗

i

. (5.120)

Thus w opt in (5.90) simplifies to

wopt = R−1c s(θt , ωdt )

= s(θt , ωdt ) −rc∑

i=1

(1 − σ 2

n

λi

)(u∗

i s(θt , ωdt ))ui

= s(θt , ωdt ) −rc∑

i=1

bi ui (5.121)

where

bi =(

1 − σ 2n

λi

)(u∗

i s(θt , ωdt )). (5.122)

Equation (5.121) shows that the optimum weight vector is the qui-escent steering vector st = s(θt , ωdt ) corresponding to the target thathas been freed from the clutter subspace components. The clutter sub-space components of the quiescent steering vector is subtracted outfrom the quiescent steering vector corresponding to the target to ob-tain the optimum weight vector.

Direct implementation of (5.121) represents the eigencanceler (EC)approach. In the ideal case, there is no difference in performance wheneither formula (5.90) or (5.121) is used for clutter suppression. How-ever, this is not the case when data samples are used to estimatethe total clutter plus noise covariance matrix Rc . If λi , ui , σ 2

n , and


Ideal

50

40

20

−20

0

0

60

80

100 200 250150Estimated

Index

l in

dB

FIGURE 5.23 Estimated eigenvalue plot.

rc represent the estimate of λi , ui , σ 2n , and rc in (5.121) respectively,

then

wec = st −rc∑

i=1

(1 − σ 2

n

λi

)(u∗

i st)ui = st −rc∑

i=1

bi ui (5.123)

represents the EC weight vector. Figure 5.23 shows the estimatedeigenvalue plot using 400 data samples. Notice that the brickwall/flatbed ideal nature in Figure 5.22 is no longer present here.

Figure 5.24 shows the EC adaptive pattern given by

Pec(θ , ωd ) =∣∣w∗

ecs(θ , ωd )∣∣2 . (5.124)

and the 2D adaptive pattern in the range-Doppler domain for thetarget arrival angle is shown in Figure 5.25.

When the numbers of data sample K in (5.92) for estimating Rc isless than MN, as we have seen, diagonal loading can be used as in(5.103) with interesting results. The limiting case of this method asε → ∞ deserves special attention.

5.4.3 Hung-Turner Projection (HTP)From (5.103), the optimum weight vector in the diagonal loaded caseequals

wε = R−1ε s(θt , ωdt ) (5.125)



Pow

er (d

B)

Dop

pler

Doppler−1−1 −1 −1

−0.5

−20

−30

−50

−60

−40

−10

−0.5 −0.5

−25

−20

−15

−10

−5

0

0

0

0.5

0.5

1

1 cos (q )cos (q )

0

0

00.5

11

FIGURE 5.24 Eigencanceler (EC) angle-Doppler pattern with 80 samples.

where

Rε = 1K

YK Y∗K + εI. (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))

(P−1 + MQ−1M∗)−1 = P − PM(M∗PM + Q

)−1M∗P. (5.127)

This gives

R−1ε = 1

ε

I − YK

(Y∗

K YK + εK I)−1

Y∗K

(5.128)

−5

−10

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−20

−25

−1

0

0

1

200

220210

190180 Range

Pow

er (d

B)

Doppler

FIGURE 5.25 Eigencanceler (EC) range-Doppler pattern with 80 samples.



Pow

er (d

B)

Dop

pler

Doppler−1−1

−0.5

−20

−30

−50

−60

−40

−10

−0.5

0

0

0.5

0.50

1

1cos (q )

cos (q )−1 −1−0.5

−25

−20

−15

−10

−5

0

0

00.5

11

FIGURE 5.26 HTP angle-Doppler pattern with 80 samples.

so that

whtp = limε→∞ εR−1

ε st = I − YK (Y∗K YK + εK I)−1Y∗

K st

= st − YK (Y∗K YK )−1Y∗

K st. (5.129)

From (5.123) and (5.129), the projection operator

φK = I − YK (Y∗K YK + εK I)−1Y∗

K (5.130)

approximates the EC approach, without an explicit eigen-decompo-sition of Rc . From (5.129), the optimum weight vector is obtained bysubtracting out K “clutter subspace” components from the quiescentsteering vector st . Clearly, for this method to be effective, K rc .If K > rc , then additional K − rc eigen-subspace components aresubtracted out in (5.129). Hence, using a large number of samplesin (5.129) can cause degradation in performance and the method issuitable when the number of samples available are small due to sta-tionarity and logistic issues.

Figures 5.26–5.27 show the estimated angle-Doppler pattern using

Phtp(θ , ωd ) =∣∣w∗

htps(θ , ωd )∣∣2. (5.131)

and the adaptive pattern in the range-Doppler domain.

5.5 Subaperture Smoothing MethodsSubaperture smoothing methods refer to subdividing the outputs ofa large array into (overlapping or non-overlapping) smaller subar-rays, and averaging their outputs either in the data domain or in thecovariance domain in some appropriate manner.


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Doppler

FIGURE 5.27 HTP range-Doppler pattern with 80 samples.

In traditional array processing, maximally overlapping subarraysare formed and their output covariance matrices are averaged to gen-erate a smoothed covariance matrix that has improved decorrelationproperties compared to the original array. This procedure has directapplications in implementing direction finding algorithms in coherentsources scenes. In such cases, to obtain performance similar to thosein uncorrelated or correlated source scenes, the subarray smoothingprocedure can be implemented [7].

To illustrate this, consider an N element uniformly spaced lineararray receiving signals from its field of view from K sources locatedalong directions θ1, θ2, . . ., θK . Following (5.80), we have

x(n) =K∑

k=1

ck(n)a (θk) + w(n), (5.132)

where a (θk) refers to the direction vector associated with the kth sourceas in (5.8), and ck(n) = ck(nT) refers to the kth source signal at t = nT ,and w(n) represents the noise vector as in (5.13). Define

c(n) = [c1(n), c2(n), . . . , ck(n)]T (5.133)

and

A = [a (θ1), a (θ2), . . . , a (θK )] (5.134)

to represent the sources and the direction vectors of interest. Thus

x(n) = Ac(n) + w(n) (5.135)


and hence the N × N array output covariance matrix has the form

R = Ex(n)x∗(n) = ARsA∗ + Rw (5.136)

where

Rs = Ec(n)c∗(n), (5.137)

and

Rw = Ew(n)w∗(n) (5.138)

represent the K × K source covariance matrix and the N × N noisecovariance matrix respectively. If all the K sources are uncorrelated,then Rs is diagonal with the kth diagonal entry representing the kthpower level, i.e.,

Pi = E|ci (n)|2 (5.139)

and

Rs =

P1 0 0 00 P2 0 0

0 0. . . 0

0 0 0 PK

(5.140)

and it represents a full rank matrix. In general, the K sources can be cor-related, in which case the nondiagonal entries of Rs are nonzero. Thedegree of source correlation determines the rank of Rs . One extremecase is represented by (5.140) where all sources are completely uncor-related with each other. The other extreme case is where all sourcesare fully correlated as in a multipath scene shown in Figure 5.28. In

c1(t )

x1(t ) x2(t )

c2(t)ck(t )

xi(t ) xN(t)

. . . . . .

. . . . . .

. . . . . .

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

q1qkq2

FIGURE 5.28 Multipath scene.


this case

ck(n) = αkc1(n) (5.141)

and hence

c(n) = [α1, α2, . . . , αK ]T c1(n) = αc1(n) (5.142)

which gives the source covariance matrix in (5.137) to be

Rs = P1α 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 outthe undesired interferences by generating nulls along their arrivalangles (see (5.60) and Figure 5.8). In a coherent interference scenewith uncorrelated noise of equal variance σ 2

n , the interference plusnoise covariance matrix has the form

R = ARsA∗ + σ 2n I = P1s s∗ + σ 2

n I (5.145)

where from (5.136) and (5.143)

s = Aα =K∑

k=1

αka (θk). (5.146)

Hence the optimum processor R−1a (θ ) is only able to generate a nullalong the vector s. But s in (5.145) and (5.146) is a linear combinationof all steering vectors, and it doesn’t correspond to an actual steeringvector. Consequently, the adaptive processor is unable to generatenulls along the true coherent arrival directions θ1, θ2, . . . , θK . This isshown in Figure 5.29 for a three-source scenario that is the same asin Figure 5.8, where the two sources along 55 and 105 are coherentand the third one along θ2 = 75 is uncorrelated with the rest. Noticethat 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 portionof the array output covariance matrix R is also reduced to rank one,thereby making it ineffective in determining the associated directionsof arrival. In the case of a uniform array, the rank structure of R in

6The rank structure of Rs forces the same structure on R as well.


Angle (deg)50

−25

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−15

−5

−20

0

0 150100

SIN

R in

dB

FIGURE 5.29 SINR output for two coherent sources with one uncorrelatedsource.

(5.136) can be improved in a fully coherent source scene by employingsubarray smoothing techniques [7].

5.5.1 Subarray SmoothingThe uniform linear array with N sensors is subdivided into over-lapping subarrays with Ns sensors, with sensors 1, 2, . . . , Nsforming the first subarray, sensors 2, 3, . . . , Ns + 1 forming thesecond subarray, etc., up to the last subarray formed by sensorsN − Ns + 1, N − Ns + 2, . . . , N (see Figure 5.30).

In (5.135), the first subarray output has the form (suppressing thetime index)

x1 = Asc + n1, (5.147)

Ns Ns + 1 N − 1 N21

1st subarray2nd subarray

Lth subarray

FIGURE 5.30 Forward subarrays.


where As is Ns × K (see (5.134)). In the second subarray, the directionvectors are scaled by γk = e− jπ cos θk , k = 1, 2, . . . , K respectively sothat with

B =

γ1γ2

. . .

γK

, (5.148)

we get the second subarray output vector to be

x2 = AsBc + n2 (5.149)

and in general the ith subarray output vector is given by

xi = AsBi−1c + ni , i = 1, 2, . . . , L , (5.150)

where ni represents the noise input vector for the ith subarray. Withuncorrelated noise of equal variance σ 2, this gives the correspondingarray output covariance matrices to be

Ri = Exi x∗i = AsBi−1RsB(i−1)∗

A∗s + σ 2I, i = 1, 2, . . . , L . (5.151)

Although each Ri is of rank one in the fully coherent case, that isnot the case for their average R f given by

R f = 1L

L∑i=1

Ri = AsDA∗s + σ 2I (5.152)

where

D = 1L

L∑i=1

Bi−1RsB(i−1)∗. (5.153)

In the fully coherent case, D simplifies to

D = 1L

L∑i=1

Bi−1α α∗B(i−1)∗ = 1L

E E∗ (5.154)

where

E = [α, Bα, B2α, . . . , BL−1α]

=

α1α2

. . .

αK

1 γ1 · · · γ L−11

1 γ2 · · · γ L−12

......

......

1 γK · · · γ L−1K

= E1V. (5.155)


Angle (deg)50

−30

−25

−10

−15

−5

−20

0

0 150100

SIN

R in

dB

FIGURE 5.31 Optimum SINR using subarray smoothing scheme. Twocoherent interferences along 55 and 105, and one uncorrelated signal along75. Spatial smoothing is used to decorrelate the coherent sources.

Clearly, E and D are of full rank and hence the forward smoothedcovariance 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 ; how-ever, its rank has increased from one to K by the above smoothingoperation.

Interestingly, we can use this procedure to null out coherentinterference signals by generating the optimum adaptive processorin (5.54) using the smoothed covariance matrix R f rather than usingR. Figure 5.31 shows the adaptive processor output correspondingto the coherent scheme in Figure 5.29 where subarray smoothing hasbeen employed on the original covariance matrix using two subarrays(L = 2). Clearly, the two coherent interference signals are resolved andconsequently the adaptive processor is able to null them out.

Starting from the far end of the array at sensor N, the reversed andcomplex 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 formthe second subarray, etc., as shown in Figure 5.32.

Let yl

denote the complex conjugate of the lth backward subarrayelement outputs, i.e.,

yl= [x∗

N−l+1, x∗N−l , . . . , x∗

L−l+1]T = As(BN − l c) + nl

= AsBl−1(BN−1c) + nl = AsBl−1c + nl , l = 1, 2, . . . , L , (5.156)


N − Ns N − Ns + 1 N − 1 N21

1st backwardsubarray2nd backward subarray

Lth backward subarray

FIGURE 5.32 Backward subarray.

where we have used (5.150). Here the top bar refers to ordinary com-plex conjugation operation. This gives the corresponding backwardcovariance matrix to be

Eyly∗

l = AsBl−1RsBl−1∗

A∗s + σ 2I, l = 1, 2, . . . , L , (5.157)

where the effective source covariance matrix equals

Rs = B(N−1)∗Ec c∗B(N−1) = B(N−1)∗

RsB(N−1). (5.158)

In the case of a coherent scene,

Rs = B(N−1)∗α αT BN−1 = β β∗ (5.159)

where

β = B(N−1)∗α, (5.160)

i.e.,

βk = γ−(N−1)k 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

Rb = 1L

L∑l=1

Eyiy∗

i = AsF A∗

s + σ 2I (5.162)

where

F = 1L

L∑i=1

Bl−1RsB(l−1)∗. (5.163)


Ns Ns + 1 N − 1 N21

1st forward subarray 2nd forward subarray

Lth forward subarray

N − Ns N − Ns + 1

1st backward subarray

2nd 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, B2β, · · · BL−1β]

=

β1β2

. . .

βK

1 γ1 · · · γ L−11

1 γ2 · · · γ L−12

......

......

1 γK · · · γ L−1K

= E2V (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 re-duced by combining the forward and backward smoothing meth-ods. Thus define the forward/backward smoothed covariance (seeFigure 5.33) to be [7]

R = R f + Rb

2. (5.167)


Using (5.152)–(5.164), we obtain

R = 12

As(D + F)A∗s + σ 2I = AsPA∗

s + σ 2I (5.168)

where

P = 12

(D + F) = 12L

(E E∗ + G G∗)

= 12L

[ E G ][

E∗

G∗

]= 1

2LQ Q∗. (5.169)

But from (5.155) and (5.165)

Q = [ E G ] = [ E1V E2V ] = E1[ V εV ] = E1H (5.170)

where ε is a diagonal matrix with

ε = E−11 E2, εk = βk/αk = 0. (5.171)

Clearly, H is of size K × 2L and if all the K sources in H are linearlyindependent, then H and hence Q, P, and R are of rank K provided

2L ≥ K or L ≥ K/2. (5.172)

In this case, compared to (5.166) the number of required subarrayshas been reduced from K to K/2 and hence the total number of sensorssatisfy

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 uncor-

related, then each forward subarray covariance matrix equals

Ri = AsBi−1RsB(i−1)∗A∗

s + σ 2I = ARsA∗ + σ 2I = R (5.174)

and same is true for backward subarrays as well. In this context, wecan visualize these subarray data vectors to represent a sample datavector corresponding to the unknown covariance matrix R. Thus start-ing 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 gener-ated. This scheme is particularly attractive in sparse sample situationssuch as in a nonstationary clutter scene where only a few measured


data vectors maintain stationarity. This is explored next for STAPapplications.

5.6 Subaperture Smoothing Methodsfor STAPIn STAP, data is available both in spatial and temporal domains andhence subapertures can be generated in both domains using subarraysand subpulses to simulate additional data vectors.

5.6.1 Subarray MethodIn this case, Ns elements are grouped together in an overlapping man-ner to generate L = N − Ns + 1 subarray data vectors in the spatialdomain. Let x(ti ) refer to the N element output vector at t = ti , andy

l(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)y2(ti)

yL(ti)…

… (5.175)

Thus

yl(ti ) = [xl (ti ), xl+1(ti ), . . . , xl+Ns−1(ti )]T , (5.176)

and

ul,i =

yl(ti )

yl(ti + T)

...

yl(ti + (M − 1)T)

, i = 1, 2, . . . , K , l = 1, 2, . . . , L

(5.177)

represents the MNs × 1 data vector formed by M such pulses; theircommon covariance matrix can be expressed as

Ry =∑

k

Pkss(k)s∗s (k) + σ 2I (5.178)

where

ss(k) = b(ωdk ) ⊗ a Ns(θk) (5.179)


with a Ns(θk) representing the top Ns × 1 subvector of the spatial steer-

ing vector a (θk) in (5.8). Ry is independent of i and l implying that ev-ery data vector in (5.177) has the same covariance matrix. Further, theircomplex conjugated and transposed (backward) data vector also havethe 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 MethodHere, Ms pulses are grouped together in an overlapping manner toform subpulse vectors. Let x(ti ) refer to the M pulses output vector att = ti and x(ti ), the array vectors of size N × 1. We have

x(ti)

x(ti + T)

xi = x(ti + (Ms − 1)T )

x(ti + (M − 1)T )

……

…

…

…

y1(i)

y2(i)

yJ(i)

(5.180)

Then

yj,i

=

x(ti + ( j − 1)T)

x(ti + jT)...

x(ti + ( j + Ms − 2)T)

, i = 1, 2, . . . , K , j = 1, 2, . . . , J

(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×1have the same covariance matrix when the clutter components areuncorrelated. Thus, every original data vector of size MN×1 generates2J = 2(M − Ms + 1) reduced data vectors of size Ms N × 1.

5.6.3 Subarry-Subpluse-MethodTo jointly exploit the spatio-temporal characteristics, notice that eachsubarray vector in (5.177) contains M pulses and the subpulse oper-ation in (5.181) can be performed on each one of them to generateJ = (M − Ms + 1) subpulses.

Conversely, each subpulse in (5.181) consists of N sensors andthe subarray operation in (5.177) can be performed to generate


L = N − Ns + 1 subarrays. Either method generates JL = (N −Ns + 1)(M − Ms + 1) data samples and together with their backwarddata, we obtain 2J L = 2(N − Ns + 1)(M − Ms + 1) reduced data sam-ples of size Ms Ns × 1 for each original sample of size MN × 1. ThusNs = N − 1, Ms = M − 1 generates two subarrays and two subpulseswith a total of 8 data samples for every original sample vector.

To complete the task, we can perform the subpulse operation on thesubarray vector in (5.177). We have

yl(ti)

yl(ti + T)

ul,i = yl(ti + (Ms − 1)T)

yl(ti + (M − 1)T)

…… …

…

…

z(i)l,1

z(i)l,2

z(i)l,J

(5.182)

Define

z(i)l, j =

yl(ti + ( j − 1)T)

yl(ti + jT)

...

yl(ti + ( j + Ms − 2)T)

, l = 1, 2, . . . , L , j = 1, 2, . . . , J ,

(5.183)

where yl

is given by (5.176); (5.183) represents Ms Ns × 1 data vectorsfrom the subarrays and subpulses. The common covariance matrix ofthe data vectors in (5.183) and their backward data form can be shownto be

Rs,t =∑

k

Pkss,t(k)s∗s,t(k) (5.184)

where the reduced spatio-temporal vector

ss,t(k) = bMs(ωdk ) ⊗ a Ns

(θk) (5.185)

with bMs(ωdk ) representing the top Ms × 1 subvector of b(ωdk ).


Thus, for every MN×1 original data, the forward reduced size datavectors are given by

Z f =[

z(i)1,1 z(i)

1,2 · · · z(i)1, J

....z(i)

2,1 z(i)2,2 · · · z(i)

2, J....

· · · z(i)L, J

](5.186)

and the corresponding backward data vectors are given by

Zb = Jo Z f (5.187)

where Jo equals

Jo=

0 0 0 1

0 0 1 0

0 1 0 0

1 0 0 0

. (5.188)

This gives

Z =Z f

...

...

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

Rs,t = 12L J

Z Z∗. (5.190)

Interestingly, to estimate the subaperture smoothed covariance ma-trix Rs,t , it is not necessary to employ (5.190) when the original datahas 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 effi-ciently on the sample covariance matrix Rc in (5.81) and its complexconjugate by identifying the appropriate subblocks. This procedure isillustrated in Figure 5.34 for M = 3, N = 2, Ms = 2, and Ns = 2, whichgives L = J = 2.

Figure 5.35 shows the estimated adaptive pattern of ordinary HTPand HTP with forward/backward subarray subpulse (HTPSASPFB)smoothing method using ten data samples for a 14-element arraywith 16 pulses with C NR = 40 dB with Ns = 13 and Ms = 15. In-jected 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, whereasHTPSASPFB has in effect 80 samples and it is able to detect the targetunambiguously.


1

43

2

1

43

2

(a) Rc of size 9 × 9

(b) Subaperture Rsp of size 4 × 4

(3 × 3)

(3 × 3)

(3 × 3) (3 × 3)

(3 × 3)

(3 × 3) (3 × 3) (3 × 3)

(3 × 3)(3 × 3)

(2 × 2)

(2 × 2)

Rc =

=

+

43

21(2 × 2)

(2 × 2)(2 × 2)

(2 × 2)

1

3 4

2

1

3 4

2

FIGURE 5.34 Subaperture in covariance matrix domain. Region 1 in (a) getsaveraged and goes over to region 1 in (b), etc.

Figure 5.36 shows the estimated adaptive pattern for ordinaryEC 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, sub-aperture smoothing methods simulate a large number of samples atthe expense of a reduction in the array aperture.

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Doppler

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−10

−15

−20

−1

0

0 200220

210190

180 RangeDoppler

(a) HTP (b) HTPSASPFB

Pow

er (d

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FIGURE 5.35 Adaptive pattern for HTP and HTP with forward/backwardsubarray subpulse (HTPSASPFB) smoothing.


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1 1

200220

210190180 Range

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Doppler

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−20

−1

0

0 200220

210190

180 RangeDoppler

(b) ECSASPFB(a) EC

Pow

er (d

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FIGURE 5.36 Adaptive pattern for EC and EC with forward/backwardsubarray subpulse (ECSASPFB) smoothing.

5.7 Array Tapering andCovariance Matrix TaperingTraditionally, array tapering is used to decrease the sidelobes byweighting the array outputs by individual tapering coefficients asshown in Figure 5.37 (b).

Thus if

α = [α1, α2, . . . , αN]T (5.191)

represents a set of tapering coefficients to be applied to the array out-put vector x(ti ), we have

y(ti ) = [α1x1(ti ), α2x2(ti ), . . . , αNxN(ti )]T

= α x(ti ),(5.192)

a1 a2 aN

x1(t )

x1(t ) x2(t ) xN (t )

x2(t ) y2(t )y1(t )xN (t ) yN (t )

(a) Traditional array (b) Tapered array

. . . . . .

. . .

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

. . . . . .

FIGURE 5.37 Tapering an array.


where represents the Schur product (element-wise multiplication)as in (1.61)–(1.62). Similarly, if

β = [β1, β2, . . . , βM]T (5.193)

represents the temporal tapering to be applied to the various pulseoutputs x(ti ), x(ti − T), · · · x(ti − (M − 1)T), we have

yi =

β1x(ti )

β2x(ti − T)...

βMx(ti − (M − 1)T)

= (β ⊗ α

)

x(ti )

x(ti − T)...

x(ti − (M − 1)T)

= 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 taperedcovariance matrix Ry is given by

Ry = Eyi y∗i = Exi x∗

i c c∗ = Rx c c∗, (5.196)

and similarly, the corresponding sample estimates are given by

Ry = 1K

K∑i=1

yi y∗i = 1

K

K∑i=1

xi x∗i c c∗ = Rx c c∗ (5.197)

where

Rx = 1K

X X∗ (5.198)

and

X = [ x1 x2 . . . xK ] (5.199)

represents the original data vector matrix. It easily follows that if weapply different tapering vector c1, c2, . . . , cp to the data set X, we havethe enhanced data set

Z = [ Z1 Z2 · · · Zp ] (5.200)

where

Zi = [ci x1, ci x2, . . . , ci xK ] , i = 1, 2, . . . , p. (5.201)

Notice that X and Zi are of size MN×K , whereas the enhanced dataset if of size MN × pK , implying that the number of data samples hasincreased by a factor of p. However, unlike X, the samples in Z are not


independent, and hence statistically the improvement will not be bya factor of p.

The estimated covariance matrix corresponding to Z in (5.200) isgiven by

Rz = 1pK

ZZ∗ = 1pK

p∑i=1

Zi Z∗i = Rx 1

p

p∑i=1

ci c∗i = Rx T (5.202)

where

T = 1p

p∑j=1

c j c∗j ≥ 0 (5.203)

represents the equivalent covariance matrix tapering (CMT). Observethat T is a non-negative Hermitian matrix of rank p (provided all thetapering vectors are linearly independent) and since Rz correspondsto pK effective number of samples, compared to Rx , the rank of Rzmust have improved. This also follows from the matrix rank identityin (1.66) where equality is possible.

In particular, by adjusting p so that pK > MN, Rz in (5.202) canbe made to be full rank by a judicious choice of the tapering vectorc1, c2, . . . , cp . Rz is a better compensated estimate compared to Rx interms of their condition numbers. This also follows from the eigen-value 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

applied to Rx can be equivalently represented as an effective incrementin the available data samples. To see this, since

Rz = Rx T, (5.205)

and if T represents a nonnegative-definite tapering matrix, then itseigenvalue decomposition gives

T =p∑

k=1

λkeke∗k = 1

p

p∑k=1

ckc∗k (5.206)

where

ck =√

pλkek . (5.207)

It now follows from (5.200) and (5.203) that Rz in (5.205) correspondsto K p samples as in (5.200). Conceptually, to incorporate a givennon-negative matrix T in (5.205), one may equivalently construct thenew data set Z as in (5.200) and (5.201) with ci ’s obtained as in (5.206)


and (5.207). However, this approach will not be computationallyefficient.

Other methods such as forward/backward smoothing, subarray-subpulse methods can be applied to this data set to further enhancethe data set. Since subarray-subpulse smoothing can be efficientlycarried out using diagonal block matrices of Rx , tapering togetherwith subaperture smoothing can be efficiently incorporated in thecovariance domain using Rz.

To illustrate this, consider subaperture smoothing applied to dataX 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 matrixand consequently, the extended data set

Xs = [X1...X2]

(5.209)

can be used to estimate the smoothed covariance matrix

Rx,s = 12K

XsX∗s = 1

2K

(X1X∗

1 + X2X∗2)

(5.210)

that has twice the number of samples compared to X. Further, if R1and R2 represent the corresponding upper and lower blocks of Rx in(5.198), i.e.,

Rx = =R1 R2

^^ ^ (5.211)

then we also have

Rx,s = R1 + R2

2. (5.212)

Equation (5.212) turns out to be useful to analyze the joint imple-mentation of CMT and subaperture smoothing methods. To see this,observe that for a given nonnegative-definite T as in (5.203), the CMTrepresentation in (5.202) corresponds to the enhanced data set Z in(5.200). Subarray smoothing on Z as in (5.208) generates the data setsZ1 and Z2 given by

Z = =Z1 Z2. (5.213)

The corresponding estimated smoothed covariance matrix equals

Rz,s = 12pK

(Z1Z∗

1 + Z2Z∗2). (5.214)


Notice that Z1 and Z2 have the same structure as in (5.200) and(5.201) with c j replaced by c(1)

j and c(2)j , where c(1)

j , c(2)j represent the

“top” and the “bottom” portions of c j respectively, i.e.,

cj = =c(1)j c(2)

j· (5.215)

This gives

12pK

Zi Z∗i = 1

2pRi

p∑j=1

c(i)j c(i)∗

j = 12

Ri Ti , i = 1, 2, . . . (5.216)

with Ri as in (5.211) and

Ti= 1

p

p∑j=1

c(i)j c(i)∗

j , i = 1, 2, . . . (5.217)

On comparing (5.217) and (5.203), it follows that T1 and T2 corre-spond to the upper-left and lower-right principal block matrices ofT, i.e.,

T = =T1 T2 · (5.218)

Substituting (5.216) into (5.214), we obtain

Rz,s = R1 T1 + R2 T2

2. (5.219)

From (5.219), the tapered and smoothed covariance matrix Rz,s canbe obtained directly from Rx and T by applying tapering on the ap-propriate subblocks of Rx and summing them. Clearly, the procedurecan be extended to spatial and temporal domains, and additional pro-cessing such as forward-backward can be applied to Rz,s in (5.219) aswell. Observe that (5.219) avoids working directly on the extendeddata set Z, Z1, and Z2; instead, all additional preprocessing such asCMT, smoothing, etc., are carried out on the original estimate Rx andits partitioned blocks only.

5.7.1 Diagonal Loading as TaperingThe diagonal loading scheme (SMIDL) discussed in Section 5.3.3 canbe shown to be a special case of data-dependent tapering. To see this,observe that diagonal loading has the form

RDL = Rx + εI = R Tx (5.220)


where

Tx =

r1,1+ε

r1,11 1 · · · 1

1 r2,2+ε

r2,21 · · · 1

1 1 r3,3+ε

r3,3· · · 1

......

.... . . 1

1 1 1 1 rN, N+ε

rN, N

=

εr1,1

0 · · · 0

0 εr2,2

0

... 0. . .

0 0 εrN, N

+

1 1 · · · 1

1 1 · · · 1

1 1 · · · 1

1 1 · · · 1

= Dx + ee∗ (5.221)

where

e = [ 1, 1, . . . , 1 ]T . (5.222)

Eigen-decomposition of Tx gives

Tx = Dx + ee∗ =MN∑l=1

ckc∗k , (5.223)

since Dx is a data-dependent full rank diagonal matrix, and hence theequivalent tapering vectors ck in (5.223) are also data-dependent. Fromthe earlier discussion, it follows that the effect of diagonal loadingas 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 withexcellent performance as has been well documented in [5].

Interestingly, when viewed abstractly, other preprocessing schemessuch as subarray subpulses, relaxed projection method discussed inSection 5.8 can be viewed as data-dependent covariance matrix taper-ing as well. This follows since the final covariance matrix R obtainedfrom the original estimate Rx can always be expressed as

R = Rx T, (5.224)

where T is an appropriate data-dependent CMT. However, it is im-possible to know a-priori such a T before the actual processing.

Figure 5.38 shows the comparison between a conventional STAPmethod (SMIDL) with that of a covariance matrix tapering approach(SMIDLCMT). In this example, a 14-element array with 16 pulses re-ceives data with C NR = 40 dB. Injected target with SNR = 0 dB is


(b) SMIDLCMT

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0

0

1 1

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180 Range

Pow

er (d

B)

Doppler

−5

−10

−15

−20

−1

0

0 200220

210190

180 RangeDoppler

(a) SMIDL

Pow

er (d

B)

FIGURE 5.38 Adaptive pattern responses of (a) SMIDL and (b) SMIDLCMTmethods using 30 snapshot data samples.

located at range 220 at an angle of 90 and with a Doppler of 0.3. Thetapering matrix T used here is given by [8]

T = T f M⊗ TθN , (5.225)

where T f Mis of the size M × M and given by

T f M(i, j) = sinc

( |i − j | f

π

)(5.226)

and TθN is of size N × N and given by

TθN (i, j) = sinc( |i − j |θ

π

). (5.227)

The parameters f and θ are set at f = θ = 0.01 to generatethe tapering matrix in (5.226) and (5.227).

5.8 Convex Projection TechniquesAlgorithms based on ideal clutter covariance matrix Rc always out-perform the same algorithms based on their estimated counterpartRc . The ideal covariance matrix generally has well-defined structuralproperties—such as block Toeplitz nature in an uncorrelated scene,positive-definiteness, etc. The corresponding estimate on the otherhand need not possess any of these properties; for example, low num-ber of data samples can make the estimate ill-conditioned. The idealcovariance matrix might assume stationary behavior for the underly-ing data and in reality the data collected may be nonstationary leadingto poor performance. In such situations, given Rc one approach is toobtain a better estimate prior to processing that shares some of the


f1

f2

f

FIGURE 5.39Convex set.

properties of Rc . In other words, given Rc , find a matrix B that is“closer” to it and possesses the desired properties of Rc :∥∥Rc − B

∥∥ ≤∥∥Rc − Rc

∥∥ . (5.228)

Obviously, the above minimization needs to be performed in theabsence of Rc .

One approach in this context is to examine whether the structuralproperties satisfied by Rc exhibit any convexity constraints. Membersof a convex set satisfy the convexity constraint that states if C is aconvex set and if f1 ∈ C and f2 ∈ C , then (see Figure 5.39)

f = α f1 + (1 − α) f2 ∈ C, 0 ≤ α ≤ 1. (5.229)

5.8.1 Convex SetsConvex sets have the remarkable property that for any point x outsideC , 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 . Thusin Figure 5.40

xo = Px, (5.231)

where P is the projection operator associated with the closed convexset C .

Interestingly, many useful notions in signal processing form con-vex sets. For example, the set of all signals that are band limited toBo form a convex set CB , since f1(t) and f2(t) are band limited to Bo

x

xo

C

Pxx1

FIGURE 5.40“Unique”neighbor property.


implies α f1(t) + (1 − α) f2(t) is also band limited to Bo . As a result,given any signal f (t), it is easy to show that its projection onto CBis given by band limiting its Fourier transform F (ω) to Bo (settingF (ω) = 0, |ω| > Bo ). Thus

PB f (t) ↔

F (ω), |ω| ≤ Bo0, |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 negativepart to zero, i.e.,

P+ f (t) =

f (t), f (t) ≥ 00, f (t) < 0 . (5.233)

Another example is the set of all signals with common phase func-tion ϕ(ω). All such signals form a convex set since if f1(t) ↔ F1(ω) =A1(ω)e jϕ(ω) and f2(t) ↔ F2(ω) = A2(ω)e jϕ(ω) , then

f (t) = α f1(t) + (1 − α) f2(t) ↔ αF1(ω) + (1 − α)F2(ω)

= αA1(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 PropertyToeplitz matrices of the same size form convex sets CT 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 nearestToeplitz neighbor with to , t± 1, t± 2, t± 3 , representing the diagonalentries of the Toeplitz matrix. Then ‖T − R‖2 needs to be minimized.This gives

= ‖T − R‖2 =n∑

i=1

|to − ri, j |2 +n∑

i = j

|tj−i − ri, j |2

=n∑

i=1

|to − ri,i |2 +n−|k|∑i=1

|tk − ri,i+k |2. (5.235)

Hence

∂

∂to= 2

n∑i=1

(to − ri,i ) = 0 (5.236)

and

∂

∂tk= 2

n−|k|∑i=1

(tk − ri,i+k) = 0, k = ±1, ±2 · · · (5.237)


gives

to = 1n

n∑i=1

ri,i (5.238)

and

tk = 1n − |k|

n−|k|∑i=1

ri,i+k , k = ±1, ±2 · · · (5.239)

and the Toeplitz matrix so formed is the “nearest neighbor” Toeplitzsolution to the given matrix R . Thus the projection operator in this caseamounts to averaging the respective diagonal entries of R to obtainthe corresponding Toeplitz entry. Notice that if R is Hermitian, thenso will be its projection T.

Suppose R is Hermitian positive-definite, then its projection ontothe Toeplitz domain gives a Toeplitz matrix T that is Hermitian. How-ever, T need not be nonnegative-definite. It can have negative eigen-values. An interesting question in this context is given a Hermitianmatrix A, how to obtain the closest nonnegative-definite matrix B?

5.8.3 Positive-Definite PropertyInterestingly, the nonnegative-definite (positive-definite) propertyalso 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 − A‖2 = 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)


since UU∗ = U∗U = I. Clearly, (5.244) is minimized by setting

Ci, j = 0, i = j (5.245)

and7

Ci,i =

λi , λi > 00 λi ≤ 0 . (5.246)

Recall that the constraint B nonnegative-definiteness implies C isnonnegative-definite as well. From (5.245) and (5.246), C is diagonalwith non-negative entries. Let + define the diagonal matrix definedin (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 matrixdomain. The projection operator simply resets the negative eigenval-ues of A to zero as in (5.246) and recompute the matrix, and is quitesimple to implement.

If the ideal covariance matrix R is positive-definite and blockToeplitz, then it belongs to both the convex sets CT (block Toeplitz)and CP , i.e., to their intersection convex set

C = CT ∩ CP (5.249)

and in that case, the problem is to find a “nearest neighbor” to Cstarting from an estimate R that is neither block Toeplitz nor positive-definite. In this context, the method of alternating projections can beemployed.

5.8.4 Methods of Alternating ProjectionsConsider the closed convex sets C1, C2, . . . , Cn and their associatedprojection operators P1, P2, . . . , Pn. If the desired function belongs toeach Ci , then it belongs to their intersection

Co = C1 ∩ C2 ∩ · · · Cn. (5.250)

Starting from an outside point x (usually an approximant), the prob-lem is to determine its “nearest neighbor” xo within the derived con-vex set Co ; if not at least a point within Co . If the projection operator

7If the original covariance matrix is known to have lowest eigenvalues equal toσ 2, then this quantity can be used in (5.246) in place of zeros when λi < σ 2.


Po corresponding to the new convex set Co is known, then that nearestneighbor to x equals xo = Po x as in (5.231). However, Po is usually un-known since Co is defined only through (5.250), and only P1, P2, . . . , Pnare known. Thus the problem reduces to finding a neighbor in Co start-ing at x given P1, P2, . . . , Pn [9], [10], [11].

The method of alternating projections originally proposed by JohnNeumann [12] has been generalized to address this problem [13].Define

P = Pn Pn−1 · · · P2 P1 (5.251)

and consider the iteration

xi = Pxi−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 =CT , C2 = CP and x = R in Figure 5.41, then starting with an estimatedcovariance matrix R, the iteration

Rn = P2 P1Rn−1, n = 1, 2, · · · (5.253)

with Ro = R gives the best Toeplitz approximation that is also positive-definite. This procedure can be extended in STAP to obtain a betterestimate for R by preserving the block Toeplitz and positive definiteproperty.

The iterations in (5.252) and (5.253) can be accelerated in their con-vergence by making use of relaxed projection operators as shown inFigure 5.42.

x

xn

C1P1x

P2P1x

C2

Co

(a)

x

P1x C1

C2Co

(b)

FIGURE 5.41 Method of iterative projections.


x

y1 = Q1xC1

P1x

C2

Co

P2 y1

Q2 y1

FIGURE 5.42Relaxed projectionmethod.

5.8.5 Relaxed Projection OperatorsThe 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 representingthe identity operator. The quantity λk in (5.254) represents the relax-ation parameter and it can be chosen to form a sequence of adaptivecoefficients for faster convergence as in Figure 5.42.

Thus, relaxed projection operators can be substituted in (5.252) and(5.253) and by controlling λ′

ks, the convergence rate can be controlledas well.

Either the projection operators or the relaxed projection operatorscorresponding to the block Toeplitz convex set and positive-definiteconvex set can be used in STAP for better clutter covariance matrixestimation since in an uncorrelated clutter scene the ideal covariancematrix possesses these two properties.

(a) SMI (b) SMIPROJ

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0

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1

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Pow

er (d

B)

Doppler

−5

−10

−15

−20

−1

0

0

1

200220

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180 RangeDoppler

Pow

er (d

B)

FIGURE 5.43 Adaptive pattern responses of (a) SMI and (b) SMIPROJmethods using 30 snapshot data samples.

8If T is a nonexpansive operator, then ‖Tx − Ty‖ ≤ ‖x − y‖. If ‖Tx − Ty‖ <‖x − y‖ then T represents a contraction (strict inequality).


The new covariance matrix so obtained can be used in conjunc-tion with any other standard STAP algorithm such as SMIDL, EC,etc. Figure 5.43 shows the performance of the projection-based STAPalgorithm for the SMI case.

5.9 Factored Time-Space ApproachThe factored time-space (FTS) is also known as post-Dopplerdimension-reducing beamforming [5], [14], [15]. It is applied adap-tively after Doppler filtering has been first applied to each channel. Thesimplest form of the FTS is the single-bin method, which transformsan MN dimensional space-time filtering problem into N dimensionalspatial-only adaptive beamforming as shown in Figure 5.44 [14].

In FTS, the kth range cell MN × 1 space-time data vector xk is re-configured to an M × N signal matrix [14]

Xk =

xk(1, 1) xk(1, 2) · · · xk(1, N)

xk(2, 1) xk(2, 2) · · · xk(2, N)...

......

...

xk(M, 1) xk(M, 2) · · · xk(M, N)

= [x1,k , x2,k , . . . , xN,k]

(5.255)where

xn,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 process-ing is then performed on each column of Xk giving the transformed

FFT FFT FFT

Adaptive processor Adaptive spatial nulling

Y1

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

FIGURE 5.44Factoredtime-spaceapproach (FTS).


signal matrix to be

Yk =

yk(1, 1) yk(1, 2) · · · yk(1, N)

yk(2, 1) yk(2, 2) · · · yk(2, N)

......

......

yk(M, 1) yk(M, 2) · · · yk(M, N)

= [y

1,k, y

2,k, . . . , y

N,k],

(5.257)

where vector yn,k

is the DFT of the vector xn,k . The desired Dopplerfilter corresponding to a given row of Yk is selected and the N spa-tial samples are adaptively combined. The output of the adaptivelyfiltered mth Doppler filter for the kth range cell is given by

zk(m) = a∗R−1k,m y

k,m, (5.258)

where a is the N×1 spatial steering vector, yk,m

is the transpose of themth row of Y given by

yk,m

= [yk(m, 1), yk(m, 2), . . . , yk(m, N)]T (5.259)

and Rk,m is the N × N spatial covariance matrix

Rk,m = 1K

k+K/2∑i=k−K/2

yk,m

y∗k,m

, i = k. (5.260)

Notice only 2N i.i.d. samples are needed for estimation of Rk,m.The 1D Doppler filtering can be represented as [14]

yk =

yk,1

yk,2

...

yk, M

=

I(w∗

o

)1I · · · (

w∗o

)M−1I

I(w∗

1

)1I · · · (

w∗1

)M−1I

......

. . ....

I(w∗

M−1

)1I · · · (

w∗M−1

)M−1I

xk

=

f ∗t,o

⊗ I

f ∗t,1

⊗ I

...

f ∗t, M−1

⊗ I

xk =

o

1

...

M−1

xk,1

xk,2

...

xk, M

(5.261)


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 fm

is the mthDoppler DFT steering vector given by

ft,m

=

1

w1m

...

wM−1m

= b(ωd )

1

τ 1t,m

...

τ M−1t,m

(5.263)

where

τt,m = e− j2πm/M. (5.264)

The mth Doppler bin yk,m

is then processed as follows:

zk(m) = w∗m y

k,m= (R−1

k,msm

)∗y

k,m

= s∗m

(mR∗

m

)−1y

k,m(5.265)

where

sm = ms, (5.266)

and

yk,m

= mxk . (5.267)

Here s in the space-time steering vector as defined in (5.72). Oncomparing (5.265) with the regular SMI approach

zk = s∗R−1xk , (5.268)

the effective steering vector and data vector are given by (5.266) and(5.267) respectively. The effective covariance matrix equals

Rk,m = mR∗m. (5.269)


Equation (5.265) shows the single-bin FTS approach; however, itsperformance is usually poor. A better strategy is to use two or moreDoppler bines [15]. In general, let Ms be the number of Doppler bins,then the transformation matrix is given by [14]

m =

m−(Ms−1)/2

...

m−1

m

m+1

...

m+(Ms−1)/2

(5.270)

and the effective steering vector, data vector, and covariance matrixare as given by (5.266), (5.267), and (5.269) respectively. This methodis also known as the extended factored time-space (EFA) approach.

Figure 5.45 shows the adaptive angle-Doppler pattern of the FTSusing one Doppler bin. Fifty snapshot data samples are used here.However, as pointed out earlier, the performance of a single Dopplerbin FTS is poor [15]. Figure 5.46 shows the adaptive angle-Dopplerpattern of EFA using two Doppler bins and the number of snapshotdata samples used here is also 50. As seen from Figure 5.46, the tar-get is clearly visible. The improvement over a single bin approach issignificant even when only two bins are used.


Pow

er (d

B)

Dop

pler

−1−1

−0.5

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−30

−50

−60

−40

−10

−0.5

0

0

0

0.5

0.5

1

1

cos (q )

Doppler−1 −1

−0.5

−25

−20

−15

−10

−5

cos (q )

0

0

00.5

11

FIGURE 5.45 Adaptive angle-Doppler pattern of FTS with one Doppler bin.Fifty snapshot data samples are used here.


(a) Top view (b) Side view

Pow

er (d

B)

Dop

pler

−1−1

−0.5

−20

−30

−50

−40

−10

−0.5

0

0

0

0.5

0.5

1

1 Doppler −1 −1 −0.5

−25

−20

−15

−10

−5

cos (q )cos (q )

0

0

00.5

11

FIGURE 5.46 Adaptive angle-Doppler pattern of EFA with two Dopplerbins. Fifty snapshot data samples are used here.

5.10 Joint-Domain Localized ApproachJoint-Domain Localized (JDL) processing that adapts over a lo-cal processing region consists of adjacent angle and Doppler bins[16]. The MN × 1 space-time data vector is first transformed toangle-Doppler domain via the two-dimensional discrete Fouriertransform (2D-DFT). To achieve this, the kth range cell MN × 1 space-time data vector xk is reconfigured to a M × N signal matrix as in(5.255). Thus

Xk =

xk(1, 1) xk(1, 2) · · · xk(1, N)

xk(2, 1) xk(2, 2) · · · xk(2, N)...

......

...

xk(M, 1) xk(M, 2) · · · xk(M, N)

, (5.271)

where xk(m, n) represents the data from the mth pulse and the nth sen-sor. 2D-FFT is then performed on the space-time domain data matrixXk to create the angle-Doppler domain data matrix

Zk =

zk(1, 1) zk(1, 2) · · · zk(1, N)

zk(2, 1) zk(2, 2) · · · zk(2, N)...

......

...

zk(M, 1) zk(M, 2) · · · zk(M, 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


processed adaptively for clutter cancellation. Let the number of angle-bins be Ns and the number of Doppler-bins be Ms , then the total num-ber of angle-Doppler bins is given by L = Ms Ns . Notice that the num-bers of angle bins Ns and Doppler bins Ms (usually only 3–5 bins areneeded) are usually much smaller than the number of sensors N andpulses M [16]. As a result, L is much less than MN and the numberof independent data samples required for clutter covariance estima-tion and computation requirement are reduced. Figure 5.47 shows anexample 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,nxk (5.273)

where fm,n is the MN × 1 transformation vector given by

fm,n = ft,m

⊗ fs,n

. (5.274)

In (5.274), the M × 1 vector ft,m

is the mth Doppler (temporal) DFTsteering vector as given in (5.263). Similarly, the N × 1 vector f

s,nis

the nth angle (spatial) DFT steering vector given by

fs,n

= a (θ)

1

τ 1s,n

...

τ N−1s,n

(5.275)

Clutter ridge

Angle bins

Dop

pler

bin

s

FIGURE 5.47 Angle-Doppler bins grouping in JDL processing.


where

τs,n = e− j2πn/N. (5.276)

Thus when Ms Doppler bins and Ns angle bins are grouped, theMN× Ms Ns Doppler-angle transformation matrix at mth Doppler binand nth angle bin is given by

Fms ,ns =

fTm−(Ms−1)/2,n−(Ns−1)/2

...

fTm,n−(Ns−1)/2

...

fTm+(Ms−1)/2,n−(Ns−1)/2

...

fTm−(Ms−1)/2,n+(Ns−1)/2

...

fTm,n+(Ns−1)/2

...

fTm+(Ms−1)/2,n+(Ns−1)/2

T

. (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:

pk,ms ,ns = w∗ms ,ns

zk,ms ,ns = (K−1k,ms ,ns

sms ,ns

)∗zk,ms ,ns

= s∗ms ,ns

(F∗

ms ,nsR Fms ,ns

)−1zk,ms ,ns (5.279)

where the effective steering vector and effective clutter covariancematrix are given by

sms ,ns = F∗ms ,ns

s, (5.280)

and

Kk,ms ,ns = F∗ms ,ns

R Fms ,ns (5.281)

respectively. Figures 5.48–5.49 show the adaptive angle-Dopplerpattern and range-Doppler pattern of the JDL approach using fiveangle bins and three Doppler bins. Fifty snapshot data samples areused here and the target is clearly detected.



Pow

er (d

B)

Dop

pler

−1

−0.5

0

0.5

1

Doppler−1 −1

−0.5

−25

−20−15

−10

−5

cos (q )cos (q )

0

0

00.5

11

−1 −0.5 0 0.5 1

−20

−30

−50

−60

−40

−10

0

FIGURE 5.48 Adaptive angle-Doppler pattern of JDL with fiveangle andthree Doppler bins. Fifty snapshot data samples are used here.


Pow

er (d

B)

Dop

pler

−1

−0.5

0

0.5

1

Doppler −1

−25

−20−15

−10

−5

RangeRange

0

0

1

180 180190 190200200

210

210

220

220

−20

−30

−50

−60

−40

−10

0

FIGURE 5.49 Adaptive range-Doppler pattern of JDL with fiveangle andthree Doppler bins. Fifty snapshot data samples are used here.

A variety of STAP techniques are discussed above for adaptive clut-ter cancellation and target detection. The list of methods discussedhere is not exhaustive, and is used here for illustration purposes.

Appendix 5-A: Uniform Array Sidelobe LevelsFor an N element uniform array with interelement spacing equal tohalf-wavelength, the array gain pattern is given by

G(θ ) =∣∣∣∣∣ 1

N

N∑k=1

e jπ(k−1) cos θ

∣∣∣∣∣2

=(

sin(

N π cos θ2

)N sin

(π cos θ

2

))2

, (5A.1)


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 uniformarray with half-wavelength spacing.

From (5A.3), the mainlobe width 2ωo is dictated by the first zero inthe numerator and hence

sin(Nωo ) = 0 ⇒ Nωo = π ⇒ ωo = π

N(5A.5)

indicating that the mainlobe width tends to zero as the number ofsensors increases. Beyond the mainlobe, there are (N − 1) sidelobeswith distinct peaks occurring at frequencies ω1, ω2, . . .. To determinean approximate location of these peaks and their actual values, wecan proceed as follows:

To start with, from (5A.5), the first zero of G(ω) occurs at π/N andsimilarly the second zero occurs at 2π/N so that the first sidelobe

G(w

) in

dB

G(w1)−10

−20

−30

−40

−50−p/2 −p/2w1 w20

0

w

−13.46 dB

FIGURE 5.50 Array gain pattern for a 10-element uniform array withhalf-wavelength spacing.


peak located in between these zeros occurs at approximately at theirmidpoint 3π/2N. Substituting this frequency into (5A.3) we get thefirst sidelobe peak value to be (approximately)

G(ω = 3π/2N) =(

sin(3π/2)N sin(3π/2N)

)2

= 1N(

3π2N − (3π/2N)3

3! + · · ·)2

→(

23π

)2

= −13.46 dB. (5A.6)

Thus for a uniform array, the maximum sidelobe level stays around13.46 dB below the mainlobe peak and it is not possible to reduce itany further by increasing the number of sensors (unlike the mainlobewidth that tends to be zero as N → ∞).

It is possible to refine the above argument to obtain a more accu-rate location of the sidelobe peaks and their values. To start with, thefirst sidelobe level peaks at frequency ω1 with a peak value of G(ω1).To obtain these peak values and their locations, we can proceed asfollows: At a peak, we have

dG(ω)dω

∣∣∣∣ω=ωk

= 0 (5A.7)

so that from (5A.3)

dG(ω)dω

= 2(

sin Nω

N sin ω

)N sin ω cos(Nω) − sin(Nω) cos ω

(N sin ω)2

∣∣∣∣ω=ωk

= 0

(5A.8)

gives9

N sin ωk cos(Nωk) = sin(Nωk) cos ωk (5A.9)

or

N tan ωk = tan(Nωk). (5A.10)

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)π2N

− k , (5A.11)

where k > 0.

9The other solution sin(Nω) = 0 corresponds to the locations of the zeros of thegain pattern.


3p2N

N tan w

N tan w

tan Nw

tan Nw

(b) First sidelobe location(a) Sidelobe locations

0

1010

0

−10w1

w1

p

p p w2

N2N

∆13p2N

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)π2N

− k

)= tan

((2k + 1)π

2− Nk

)= cot(Nk)

= 1tan(Nk)

≈ 1N tan k

. (5A.12)

Expanding the left side of (5A.12) we get

N(

tan (2k+1)π2N − tan k

)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

N2x2 − (N2 − 1) tan(2k + 1)π

2Nx + 1 = 0 (5A.15)

whose roots are given by

x1,2 =(N2 − 1) tan (2k+1)π

2N ±√(

(N2 − 1) tan (2k+1)π2N

)2− 4N2

2N2

(5A.16)


First Sidelobe Peak Locations and Peak ValuesComputed 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 sidelobelocation and value

and the smallest among these roots gives the desired solution. Hencewe obtain

tan k =(N2 − 1) tan (2k+1)π

2N −√(

(N2 − 1) tan (2k+1)π2N

)2− 4N2

2N2

(5A.17)

which for k = 1 gives

1 = tan−1

(N2 − 1) tan 3π

2N −√(

(N2 − 1) tan 3π2N

)2 − 4N2

2N2

.

(5A.18)

Thus the first sidelobe peaks at

ω1 = 3π

2N− 1

= 3π

2N− tan−1

(N2 − 1) tan 3π

2N −√(

(N2 − 1) tan 3π2N

)2 − 4N2

2N2

(5A.19)

and for any N, the peak sidelobe value is given by

G(ω1) = G(

3π

2N− 1

)=(

cos(N1)N sin

( 3π2N − 1

))2

. (5A.20)


Table 5.2 shows the computed peak values and their location us-ing (5A.18)–(5A.20) and their actual values. To determine the limitingvalue of the peak sidelobe level, from (5A.18) we get (for large N)

tan 1 1 =N2 3π

2N −√(

N2 3π2N

)2− 4N2

2N2 = 3π

4N− 1

2N

√(3π/2)2 − 4

= 3π

4N

(1 −√

1 − 4(2/3π )2)

23π N

(5A.21)

so that

ω1 = 3π

2N− 2

3π N(5A.22)

which when substituted into (5A.3) gives

G(ω1) =(

cos(2/3π )N sin (3π/2N − 2/3π N)

)2

→(

cos(2/3π )3π/2 − 2/3π

)2

= (0.217228)2 = −13.26 dB. (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 de-creases as N → ∞, the sidelobe levels stay at −13.26 dB.

In general, the kth sidelobe peak location is given by

ωk = (2k + 1)π2N

− 2(2k + 1)π N

→ 0 (5A.25)

and the corresponding peak level equals

G(ωk) → cos

(2

(2k+1)π

)(2k+1)π

2 − 2(2k+1)π

2

. (5A.26)

From (5A.26), the second sidelobe level saturates at −17.83 dB andthe third sidelobe level saturates at −20.79 dB below the mainlobepeak, etc. In summary, the peak sidelobe level of a uniform arraysaturates at about 13.26 dB below the mainlobe peak and to lowerthem further additional weighting factor must be introduced at thearray output.

References[1] B. Friedlander, “The MDVR Beamformer for Circular Arrays,” Proc. 34th Asilo-

mar 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 Sys-tems, Vol. 30, No. 3, pp. 660–670, July 1994.


[3] G.W. Titi, D.F. Marshall, “The ARPA/NAVY Mountain top Program: Adap-tive Signal Processing for Airborne Early Washing Radar,” 1996 IEEE Inter-national Conference on Acoustics, Speech and Signal Processing, Atlanta, Geor-gia, 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 Meth-ods,” 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 Report1015, MIT Lincoln Laboratory, Lexington, MA, December 1994.

[6] A.M. Haimovich, “Eigenanalysis Based Space-Time Adaptive Radar,” IEEETransaction 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, April1999.

[9] D.C. Youla, “Generalized Image Restoration by the Method of AlternatingProjections,” 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 Findingthe Common Point of Convex Sets,” U.S.S.R. Computational Mathematics andMathematical Physics, Vol. 7, No. 6, pp. 1–24, 1967.

[11] Z. Opial, “Weak Convergence of the Sequence of Successive Approximationsfor 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 ofConvex Projections,” Chapter 2, Image Recovery Theory and Application, HenryStark, ed., Academic Press, Inc., New York, NY, 1987.


[15] R.C. DiPietro, ”Extended Factored Space-Time Processing for AirborneRadar,” Proc. 26th ASILOMAR Conf., Pacific Grove, CA, pp. 425–430, Octo-ber 1992.

[16] H. Wang and L.J. Cai, “On Adaptive Spatial-Temporal Processing for AirborneSurveillance Radar Systems,” IEEE Transaction on Aerospace and Electronic Sys-tems, Vol. 30, No. 3, July 1994.

http://spib.rice.edu/spib/mtn_top.html

http://spib.rice.edu/spib/mtn_top.html

C H A P T E R 6STAP for SBR

This chapter deals with clutter data modeling from an SBR platform bytaking into consideration the various phenomena that affect the datamodeling, and examining how they affect the target detection perfor-mance. These phenomena are imperfections present in any realisticdata scene compared to the ideal conditions. They include the arraypattern due to the finite antenna size, effect of Earth’s rotation on theDoppler shift (Section 4.6–4.7), range foldover generated by the pres-ence of multiple pulse returns on the received data (Section 4.5), andthe dependence of the scatter power profile on the local terrain type.In addition, factors such as the effect of wind and altitude informationcan be added to the model for higher fidelity clutter modeling.

In real life, all these items affect the SBR data and hence any realisticdata modeling must consider and account for these items. An impor-tant aspect in this context will be to understand the impact on perfor-mance of each such phenomenon separately, as well as when takentogether in various combinations. Performance measures to quantifyclutter nulling effects are reviewed in this context, and performanceevaluations are carried out both in the matched filter case and the esti-mated case using the various STAP algorithms discussed in Chapter 5.

Transmit beams scale the transmit pulses according to the trans-mit array pattern, and they get reflected from various range foldoverpoints depending upon the local terrain reflectivity and respecting theradar equations (1.2)–(1.5). The returns once again get weighted by thereceiver array pattern, and generate the receiver output. The followingaspects must be taken into consideration for clutter modeling:

Scatter Power Profile: Reflectivity of the terrain as a function ofelevation and azimuth. Local terrain dependence can be accom-modated using a Knowledge Aided Sensor Signal Processing andExpert Reasoning (KASSPER) like approach [1].

Array Gain Pattern: Both transmitter and receiver array configura-tions with weights can be used to control the sidelobe pattern.



Range–Foldover Return (Elevation): Returns due to earlier/latertransmitted pulses that arrive from range points other than the pointof interest must be taken into account for data generation.

Azimuth Return: Scatter returns from isorange spanning all az-imuth angles.

Earth’s Rotation: Modifies the Doppler due to SBR motion resultingin a crab angle effect, and should be accounted in data modeling.

Performance measures to quantify clutter nulling effect are re-viewed, and performance evaluations are carried out both in the idealcase and estimated case using various STAP algorithms discussed inChapter 5.

Performance degradation is severe when both range foldover andEarth’s rotation are present in the data. However, when either one ispresent, the degradation in performance can be corrected. Orthogonaltransmit pulsing schemes are used to minimize range foldover effect,and this results in improved overall performance.

6.1 SBR Data ModelingConsider an SBR array with N sensors and M pulses. If the incomingwavefront makes an azimuth angle θAZ and elevation angle θEL withreference to the array (Figure 4.1 (b)), then for two sensors that are dapart, the propagation delay τ at the second sensor (with respect tothe first sensor) is

τ = d sin θEL cos θAZ

cυ

, (6.1)

where cυ represents the velocity of light. Hence the second sensoroutput x2(t) is given by (see (5.3)–(5.4))

x2(t) = x1(t − τ ) = x1(t)e− jωoτ = x1(t)e− jπ dλ/2 sin θE L cos θAZ , (6.2)

where x1(t) refers to the first sensor output. If the interelement distanced is normalized with respect to λ/2, then the normalized interelementspacing is given by

d = dλ/2

. (6.3)

Let

c = sin θEL cos θAZ (6.4)

represent the “cone angle” associated with the spatial point (θEL, θAZ)for the SBR array. In that case, from (6.2), the output vector (spatial)

C h a p t e r 6 : S T A P f o r S B R 217

for a uniform N-sensor array becomes

x(t) =

x1(t)

x2(t)...

xN(t)

= a (c)x1(t). (6.5)

Here a (c) represents the spatial steering vector given by

a (c) =

1

e− jπdc

e− j2πdc

...

e− j (N−1)πdc

. (6.6)

Similarly, if M pulses are transmitted with a certain pulse repetitionfrequency (PRF), then the returns due to different pulses have a similarrelation as in (6.5) and (6.6). To be specific [2], [3], with Vp denoting theSBR platform velocity and T the pulse repetition period, we have ωdin (4.53) or (4.71) represents the clutter Doppler for that specific rangebin. The return vector due to the various pulses at the first sensor isgiven 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− jπωd

e− j2πωd

...

e− j (M−1)πωd

. (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 hasthe form

x(t) =

y1(t)

y2(t)...

yM

(t)

= s(c, ωd )x1(t) (6.9)


where

s(c, ωd ) = b(ωd ) ⊗ a (c) (6.10)

represents the spatio-temporal steering vector with ⊗ representingthe Kronecker product.

From (4.34) the ground resolution is represented by δR and hencethe range cells must be separated by at least δR. Let rk , k = 1, 2, . . . rep-resent the actual range bin locations along the azimuth and elevationangles θAZ and θEL,k, respectively. The clutter corresponding to anyrange bin has contributions from the mainbeam antenna footprint lo-cated along (θEL,k, θAZ) as well as from the sidelobe footprints locatedalong (θEL,k, θAZ, j ), where θAZ, j covers various azimuth angles alongthe field of view of the SBR antenna that is pointed along (θEL,k, θAZ).

6.1.1 Mainbeam and Sidelobe ClutterFrom Figures 4.10–4.12, if rk falls within the ith footprint then thereare Nai distinct range ambiguities within that footprint and they allcontribute to the clutter for that range. Here Nai represents the numberof range ambiguities within the ith mainbeam footprint as in (4.38)–(4.39). Thus if

RHi < rk < RTi , (6.11)

then the Nai range ambiguities located at

RHi ≤ rk ± mR ≤ 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

rk rk +1

Range ambiguities that contribute to clutter from range rk

Range ambiguities that contribute to clutter from range rk+1

dR

∆R

FIGURE 6.1Range ambiguitiesthat contribute tothe mainbeamclutter fromrange rk.


…

Mainbeam

…

…

qAZ

Elevationsidelobes

Rangeambiguities

D : Point of interest, cone angle C0

Range ofinterest

(qEL,m’ qAZ,j)

Iso-coneclutter

discretes

Azimuthsidelobes

A

SBR

Nadirhole

FIGURE 6.2 Mainbeam and sidelobe clutter for SBR antenna pointing alongazimuth direction θAZ and range rk (elevation angle θEL,k).

all range foldover points rk ± mR, m = 0, 1, . . . , Na (see Figure 6.2).Let θAZj = θAZ + jθ , j = 0, 1, 2, . . . , No represent the azimuth anglesassociated with the field of view, and θELm , m = 0, 1, 2, . . . , Na the ele-vation angles corresponding to the total number of range ambiguitiesin the field of view. Further let

cm, j = sin θELm cos θAZj (6.13)

represent the cone angle and A(θm, j ) the overall 2D array amplitudepattern associated with the spatial point of interest where

θm, j = (θELm , θAZj

). (6.14)


In general, the overall array pattern A(θm,n) is a product of thetransmit array pattern AT (θm,n) and the receiver pattern AR(θm,n),and hence

A(θm,n) = AT (θm,n) AR(θm,n). (6.15)

When omnidirectional sensors are used, the receiver pattern ariseswhen subarrays are formed at the receiver for data collection. In thatcase, sensor elements in the subarray are combined using beamformerand the array pattern so generated by the subarray plays the role ofAR(θm,n) in (6.15). Each of these subarray outputs generates the entriesof the spatial data vector in (6.5). With ai,k representing the appropri-ate array element weights (transmit array or receiver subarray withomnidirectional sensors), we have

Ax(θm,n) =NE L−1∑

i=0

NAZ−1∑k=0

ai,k e− jπ [i dE L (cos θE Lm −cos θE L )+kdAZ(cos θAZn −cos θAZ)] ,

x = T or R, (6.16)

where (θEL, θAZ) represents the point of interest on the ground. For aseparable array, we have

ai,k = αiβk , (6.17)

where αi and βk represent the weights in the azimuth and ele-vation direction, respectively. Let um, j represent the random clutterscatter returns from location θm, j for the kth range bin. From the radarequation (1.2), the transmit waveform as well as the scatter returns areattenuated 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

R2s j

, (6.18)

where Rs j represents the slant range from the SBR to the ground loca-tion θm, j as in (4.3). Then the total clutter return from range rk is givenby (see Figure 6.2)

xk =No∑j=0

Na∑m=0

αm, j A(θm, j )s(cm, j , ωdm, j ) + n, (6.19)

with the inner summation representing the various range foldovers(m = 0 represents the actual range of interest rk), and the outer sum-mation spanning over all azimuth angles including the sidelobes inthe field of view.


This gives the ensemble average clutter covariance matrix associ-ated with range rk to be

Rk = E

xkx∗k

=No∑j=0

Na∑m=0

Pm, j G(θm, j )sm, j s∗m, j + σ 2

n I. (6.20)

Here

Pm, j = E|αm, j |2

, (6.21)

represents the scatter power from θm, j and

G(θm, j ) =∣∣A(θm, j )

∣∣2 (6.22)

represents the array gain pattern and

sm, j = s(cm, j , ωdm, j ). (6.23)

To obtain realistic scatter returns um, j , “site-specific” informationcan be incorporated to determine the terrain types. In this approach,a land cover map with specific grazing angle dependent mean radarcross section (RCS) values can be used to generate the clutter scatterpower levels according to specific Weibull distribution. A further re-finement can be realized by introducing wind induced internal cluttermotion (ICM). A detailed account of this approach along with perfor-mance analysis is given in Section 6.5.

In (6.19) and (6.23), ωdm, j represents the Doppler associated with therange bin θm, j and is given by (see (4.53), (4.71)) [4]

ωdm, j =

βocm, j , no Earth′s rotation,

ωdm, j = βoρc sin θELm cos(θAZj + φc), with Earth′s rotation.

(6.24)

Using (5.76), we obtain

βo = 2VTλ/2

. (6.25)

From (5.76) and (5.77), it also follows that

β = 2VT

d= βo

d(6.26)

represents the Brennan factor. From (5.5), d and d represent the actualinterelement spacing as well as the normalized interelement spacing


# of sensors (N) 32# of pulses (M) 16Radius of Earth (km) (Re ) 6,373Altitude of SBR (km) (H) 506SBR velocity (m/s) (Vp) 7,160Normalized interelement spacing with respect (d) 13.4to half wavelengthPulse repetition frequency (Hz) (PRF) 500–2,000

TABLE 6.1 SBR parameters

with respect to half wavelength, respectively. With φ = dc repre-senting the array factor exponent in (6.6), in the absence of Earth’srotation,ωd = βφ as in (6.24). For example, for the SBR configuration inTable 6.1, we obtain the Brennan factor β = 19.47.

Figures 6.3–6.4 show Doppler as a function of the cone angle withand without Earth’s rotation present. Note that in the absence of Earth’srotation, there is a one-to-one linear correspondence between Dopplerand cone angle. As a result, points that project the same cone anglein the range-azimuth domain generate the same Doppler. Thus inthe absence of Earth’s rotation, all points on the isocone contour inFigure 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 Doppler-cone angle domain.

However, this is no longer true when Earth’s rotation is present,since different range points on a given cone angle contour generatedifferent Doppler frequencies as shown in Figures 6.3–6.4. Figure 6.3shows Doppler frequencies corresponding to R = 500 km and its

(b)(a)

Doppler spread due to Earth’s

rotation

Cone Angle−1

−100

100

50

150

−50

−1501−0.5 0.5

0

0

Dop

pler

(kH

z)

No Doppler spread without Earth’s rotation

Doppler spread/splitting due to Earth’s rotation

Cone Angle

15

10

−5

5

−0.15 0.15−0.05 0.05−0.1 0.1

−10

−15

0

0

Dop

pler

(kH

z)

No Doppler spread without Earth’s rotation

FIGURE 6.3 Doppler spread due to Earth’s rotation and range foldover forrange 500 km. (a) All cone angles (b) “Zoomed in” view around zero coneangle.


Doppler spread/splitting due to Earth’s rotation

Cone Angle

−5

−0.04 0.04−0.02 0.02

−10

−15

0

0

Dop

pler

(kH

z)No Doppler spread without Earth’s rotation

FIGURE 6.4 Doppler spread due to Earth’s rotation for all points of interestin the fieldof view.

seven foldovers. Since the seven range foldovers intersect any iso-cone contour at seven distinct points (see Figure 6.2), from (6.24) theygenerate seven distinct Doppler frequencies for the same cone an-gle. Figure 6.3(a) shows the Doppler spread generated by the sevenfoldovers for all cone angles, and Figure 6.3(b) shows a magnifiedview around zero cone angle.

Figure 6.4 shows the band of Doppler frequencies corresponding toall range points of interest as a function of the cone angle, and in thelimit they generate a continuous band of Doppler frequencies. In sum-mary, when Earth’s rotation is present, every point on Earth projects adifferent Doppler that is confined within a band of frequencies.

Figure 6.5 shows the Doppler frequency variation along the isoconecontour (co = 0.3) shown in Figure 6.2 without and with Earth’s rotationas a function of range. From (6.24), without Earth’s rotation, as thedashed curve shows, Doppler along an isocone contour is a constant;whereas with Earth’s rotation present, as the solid curve shows, eachrange point on the isocone projects a different Doppler.

6.1.2 Ideal Clutter SpectrumTo appreciate the effective clutter suppression capabilities of thematched filter receiver, it is instructive to examine the true clutter spec-trum. The clutter spectrum corresponding to the beamformer outputis given by [2], [5], [6]

PB(Ro, ωd ) = E|x∗s(c, ωd )|2 = s∗(c, ωd )Rs(c, ωd ) (6.27)


Range (km)

75

70

65

602,5002,0001,5001,0005000

80

Dop

pler

Without Earth’s rotation

With Earth’s rotation

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 ofthe Doppler, the range-Doppler clutter power profile is plotted inFigure 6.6 for a fixed azimuth angle θAZ = 89.5 in the absence ofboth Earth’s rotation and range foldover by varying the range param-eter k and the Doppler ωd in (6.27). As Table 6.1 shows, a 32-elementarray with 16 pulses, and normalized interelement spacing d = 13.4with PRF = 500 Hz is used in the simulations. From Figure 6.6, theDoppler is an increasing function of the range and this is in agreementwith (4.53). This should not be confused with Figure 6.5 that showsthe Doppler along the iso-cone contour (Figure 6.2).

Figures 6.7–6.8 show the angle-Doppler profile of the clutter spec-trum using the parameters listed in Table 6.2. Here the clutter powerspectrum

PB,k(θAZ, ωd ) = s∗(ck , ωd )Rks(ck , ωd ) (6.28)

and the elevation angle θEL,k correspond to the kth range bin and itremains fixed in (6.28). The azimuth angle θAZ and the Doppler pa-rameter 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 clutterspectrum for range rk equal to 500 km. Notice that for a particularazimuth angle, the clutter covers almost the entire Doppler region.As a result the clutter-free region in the Doppler domain is at a pre-mium. Obviously to detect a target, the target Doppler must fall in theclutter-free region. Figure 6.8 shows the “zoomed in” version of the


Range-Doppler PatternR

ange

(km

)

Doppler−1

−1

−4

−9

−10

−11

−8

−7

−6

−5

−3

−2

10

0

−0.5 0.5400

1,100

1,000

900

800

700

600

500

FIGURE 6.6 Clutter spectrum using 2D beamformer without range foldoverand Earth’s rotation for θAZ = 89.5.

angle-Doppler profile for two different ranges. As expected, the clut-ter Doppler in proportional to cos θAZ. From Figure 6.8, the “slow”range dependence of the clutter Doppler can be observed from therelative position of the clutter ridge as a function of range. As rangeincreases (Figure 6.8 (a)–(b)) the slope of the clutter ridge increasesand this is consistent with (4.53)–(4.50) since in that case θEL increaseswith range R.

Table 6.2 and Figure 6.9 list another set of SBR parameters corre-sponding to a hypothetical array studied in the literature [7], [8], [9].A planar array consisting of 384 elements in azimuth and 12 elementsin elevation is considered here. The azimuth and elevation elementspacings are taken to be 0.54λ and 0.695λ, respectively. It is often im-practical to place an analog to digital converter behind each element.As a result, in the receive mode 32 column subarrays of 12 by 12 el-ements separated by 6.48λ are considered. Although each subarrayis phased to point to the “target,” from a spatial sampling basis, theclutter will be weighted by the corresponding antenna pattern, andwill exhibit grating lobes based on the spacing between the subarrays.

All sensors in the azimuth and elevation directions transmit at thesame time generating the transmit pattern in Figure 6.10. However,

226Space

Based

Radar

Altitude of SBR (H) 506 kmSBR velocity (Vp) 7160 m/sNumber of pulses (M) 16Pulse Repetition Frequency PRF 500 Hz

Transmit Receive

Number of sensors in azimuth (NTx−AZ) 384 Number of sensors in azimuth (NRx−AZ) 12direction direction/subarrayInterelement spaceing (dTx−AZ) 1.08 Interelement spaceing (dRx−AZ) 1.08in azimuth direction normalized in azimuth direction normalizedto half wavelength to half wavelengthNumber of sensors in elevation (NTx−E L ) 12 Number of sensors in elevation (NRx−E L ) 12direction direction/subarrayInterelement spaceing (dTx−E L ) 1.39 Interelement spaceing (dRx−E L ) 1.39in elevation direction normalized in elevation direction normalizedto half wavelength to half wavelength

Number of subarrays in azimuth 32directionNumber of subarrays in elevation 1direction

TABLE 6.2 Realistic SBR array parameters


Angle-Doppler Pattern

Azimuth (cosq )

Dop

pler

−1−1

0

0.8

0.6

0.4

0.2

−0.2

−0.4

−0.6

−0.8

−10

−15

−20

−25

−5

1

1

0

0

−0.5 0.5

FIGURE 6.7 Ideal angle-Doppler profileof the clutter spectrum forR = 500 km.

on receive, 12 sensors in the azimuth and 12 sensors in the elevationdirection 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

10.80.60.40.2

−0.05 0

0

0

0.05Azimuth (cosq ) Azimuth (cosq )

Angle-Doppler Pattern Angle-Doppler Pattern

(a) R = 500 km (b) R = 1,400 km

Dop

pler

−0.2

−25

−20

−15

−10

−5

−0.4−0.6−0.8

−1

10.80.60.40.2

−0.05 0

0

0

0.05

Dop

pler

−0.2

−25

−20

−15

−10

−5

−0.4−0.6−0.8

−1

FIGURE 6.8 Zoomed in ideal angle-Doppler profilefor different ranges(PRF = 500 Hz).


Vp

Toward the center of the Earth

qAZ

qEL

384 sensors

12sensors12 by 12

sensorblock …… …

1st

block2nd

block32nd

block

(a)

(b)

FIGURE 6.9 Array configuration.

Table 6.2 shows the rest of the array parameters. All simulations thatfollow use these parameters unless otherwise specified.

Figure 6.10 shows the transmit patterns in the azimuth andelevation directions and the overall transmit pattern for the ar-ray pointing at 90 azimuth and 90 elevation. Figure 6.11 showsthe corresponding receiver pattern also with uniformly weights.Figure 6.12 shows the clutter spectrum generated using the set ofparameters shown in Table 6.2 with θAZ = 90 for range 500 km;Figure 6.12(a) shows the spectrum in the full azimuth domain andFigure 6.12(b) shows detailed version of the spectrum around 90azimuth.

Finally Figure 6.13 shows the clutter spectrum for a fixed azimuthangle θAZ = 90 in the range-Doppler domain, corresponding to thefour situations with and without Earth’s rotation and range foldover.Note that when the Earth’s rotation is enabled, the Doppler depen-dency on range becomes highly nonlinear, making target detectionmuch more difficult.


(b) Transmit pattern in elevation direction

(c) Overall transmit pattern

(a) Transmit pattern in azimuth direction

Azimuth Angle

Azimuth Angle

Elevation Angle

Elevation Angle

Tx

in d

B

Tx

Patt

ern

in d

B

Tx

in d

B

0

0

0 0

−10−20

0 20 40 60 18016014012010080 0 20 40 60 180160140120100

100100

80−110−100

−100

150150

50 50

−150

−90−80−70−60−50

−50

−40−30

0

−10

−20

−60

−50

−40

−30

FIGURE 6.10 Transmit pattern—uniform weights.

Transmit Array WeightsOne approach to reduce the transmit sidelobe levels in Figure 6.10is to use array weights ai,k as shown in (6.16). Taylor weights thatare suitably scaled can be used to lower sidelobes, at the expense ofthe mainlobe width, and sidelobe levels can be reduced as shown inFigure 6.14. Figure 6.14 shows the azimuth transmit pattern with andwithout applying Taylor weights in the azimuth direction only. Fromthere, sidelobe levels of the transmit pattern in the azimuth directioncan be further reduced by 20 dB when Taylor weights are used. Inthis case, elevation direction has uniform weight so that the transmitpattern is same as that in Figure 6.10 (b). Similarly the receiver patternis also uniformly weighted for the array configuration described here.


(b) Receiver pattern in elevation direction

(c) Receiver transmit pattern

(a) Receiver pattern in azimuth direction

Azimuth Angle

Azimuth Angle

Elevation Angle

Elevation Angle

Rx

in d

B

Rx

Patt

ern

in d

B

Rx

in d

B

0

0

0 0

−10

−20

0 20 40 60 18016014012010080 0 20 40 60 180160140120100

100 100

80

−100

150150

50 50

−120

−70

−60

−50

−80−60−40−20

−40

−30

0

−10

−20

−60

−50

−40

−30

FIGURE 6.11 Receiver pattern—uniform weights.

Azimuth

(b)(a)

Dop

pler

Dop

pler

0.05

11

0.5

−0.05−1

−0.5

−45

−40

−30

−35

−25

−20−15

−10

−5

0

0

0.5

−1

−0.5

0

0

−45

−40

−30

−35

−25

−20−15

−10

−50

Azimuth10.5−1 −0.5 0

FIGURE 6.12 Ideal clutter spectrum without Earth’s rotation and rangefoldover in angle-Doppler domain for range = 500 km. (a) All azimuthangles (b) Around 90 azimuth angle.


(a) w/o range foldover, w/o Earth’s rotation

(c) w/o range foldover, w/ Earth’s rotation

(b) w/ range foldover, w/o Earth’s rotation

−2

−12−14

−18−20

−16

−10−8−6−4

02,400

Normalized Doppler

Ran

ge (k

m)

800−1 −0.5 0 10.5

1,000

1,200

1,400

1,600

1,800

2,000

2,200

−2

−12−14

−18−20

−16

−10−8−6−4

02,400

Normalized Doppler

Ran

ge (k

m)

800−1 −0.5 0 10.5

1,000

1,200

1,400

1,600

1,800

2,000

2,200 −2

−12−14

−18−20

−16

−10−8−6−4

02,400

Normalized Doppler

Ran

ge (k

m)

800−1 −0.5 0 10.5

1,000

1,200

1,400

1,600

1,800

2,000

2,200

Ran

ge (k

m)

(d) w/range foldover, w/ Earth’s rotation

2,400

Normalized Doppler

800−1

−2

−12−14

−18−16

−10−8−6−4

−0.5

0

0 10.5

1,000

1,200

1,400

1,600

1,800

2,000

2,200

FIGURE 6.13 Clutter spectrum with/without range foldover and with/withoutEarth’s rotation, azimuth angle = 90.

Azimuth Angle (deg) Azimuth Angle (deg)

(a) Uniform weights (b) Taylor weights

Arr

ay G

ain

in d

B

Arr

ay G

ain

in d

B

0

−20

0 20 40 60 18016014012010080 0 20 40 60

Tx Array Gain in AZTx Array Gain in AZ

18016014012010080−160

−140

−120

−100

−80

−60

−40

0

−20

−160

−140

−120

−100

−80

−60

−40

FIGURE 6.14 Azimuth transmit pattern with and without Taylor weights.


6.2 Minimum Detectable Velocity (MDV)In general the total received signal consists of returns from target (ifany), mainbeam clutter, sidelobe clutter, and noise. Let xk represent thedata vector at an N element array corresponding to M pulses receivedfrom range rk . This gives

xk =

qt + ck + nk H1

ck + nk Ho, k = 1, 2, . . . (6.29)

Here qt = αts(ct, ωdt ) corresponds to the target (if any) present atrange rk (see Figure 6.15). Notice that both xk and qt are MN × 1 datavectors.

The optimum adaptive weight vector corresponding to (6.29) isgiven by

wk = R−1k s(ct, ωdt ) (6.30)

and the adaptive matched filter power output is given by

Popt,k(θAZ, ωd ) =∣∣w∗

k s(ck , ωd )∣∣2. (6.31)

In actual practice, the covariance matrix in (6.30) corresponding torange rk is unknown and it can be estimated from the actual data fromneighboring range bins, using the expression

Rk =∑

j

xk+ j x∗k+ j . (6.32)

In (6.32) the number of range bins over which the summation iscarried out is chosen so as to maintain stationary behavior for Rk . The

Rx Tx

H

R

Mainbeamfootprint

H1

FIGURE 6.15Monostaticairborne targetdetection usingSBR STAP.


estimated adaptive weight vector corresponding to (6.30) is given bythe sample matrix inversion (SMI) method as

wk = R−1k s(ct, ωdt ) (6.33)

and the corresponding estimated power output is given by

PSMI,k(θAZ, ωd ) = |w∗k s(ck , ωd )|2. (6.34)

Another useful metric for evaluating the performance of a particularSTAP algorithm is the signal power to interference plus noise ratio(SINR) defined in (5.95)

SINR = |w∗s|2w∗Rw

, (6.35)

where w is the associated adaptive weight vector and R is the idealclutter plus noise covariance matrix defined in (6.20). In the case ofSMI, for example, (6.35) can be written as (use (6.33))

SINR = |s∗R−1s|2s∗R−1RR−1s

. (6.36)

Clearly the performance of (6.36) is bounded by the ideal matchedfilter output SINRideal obtained by letting R = R in (6.36). This gives

SINRideal = s∗(c, ωd )R−1s(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 withoutusing Taylor weights on transmit. From there, when uniform weightsare used on transmit, the target velocity has to exceed 28 m/s for un-ambiguous detection up to 5-dB loss (or an appropriate user definedthreshold). When Taylor weights are used on transmit, the target ve-locity has to exceed 15 m/s for unambiguous detection and this givesthe Minimum Detectable Velocity (MDV) bound for this SBR config-uration [9], [10]. See also Section 5.3.1 for a discussion on MDV.

MDV defined here can be used to evaluate the performance of anyalgorithm. In what follows we examine the ideal performance in termsof clutter suppression for an SBR by considering the two phenomena—Earth’s rotation and range foldover—that invariably affect the SBRdata collection process.


(b) Taylor weights

V (m/s)

SIN

R in

dB

SIN

R in

dB

−30 −10−20 3020100

0

−10

−20

−60

−50

−40

−30

(a) Uniform weights

V (m/s)−30 −10−20 3020100

0

−10

−20

−60

−50

−40

−30

MDV MDV

FIGURE 6.16 Matched filteroutput and Minimum Detectable Velocity(MDV) (PRF = 500 Hz).

6.3 MDV with Earth’s Rotationand Range FoldoverThe two phenomena—Earth’s rotation (crab angle and crab magni-tude; Section 4.6–4.7) and range foldover (Section 4.5)—contribute tothe degradation in performance for clutter suppression and target de-tection. To quantify their effects in target detection performance, weconsider the following four possible scenarios:

A. Without Earth’s rotation and without range foldover

B. Without Earth’s rotation and with range foldover(6.38)

C. With Earth’s rotation and without range foldover

D. With Earth’s rotation and with range foldover

These four situations are first analyzed using the ideal clutter co-variance matrix Rc and then using the estimated covariance matrix Rc(Section 6.8) for various STAP algorithms.

Recall that range foldover occurs when pulse returns from previ-ously (later) transmitted pulses that are returning from farther (near)range points contribute to the current pulse return. For example, inTable 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 comple-mented 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 gainsfor these locations are shown in Table 6.3 as well. In addition, thenormalized Doppler frequencies associated with these ranges arealso tabulated in Table 6.3 for θAZ = 90 and 60. Notice that the

Chapter

6:

STAPfor

SBR

235

Ranges that contribute to clutter return (km) 500 858.73 1180.5 1489.8 1793.7 2095 2395.2Corresponding elevation angle (degree) 43.541 56.635 62.346 65.289 66.852 67.618 67.88Array gain in elevation direction (dB) 0 −26.163 −43.881 −35.159 −42.879 −52.267 −57.565Normalized Azimuth Angle = 90 0.4723 0.0238 −0.8226 0.7980 0.6129 0.5263 0.4974clutter Doppler Azimuth Angle = 60 −0.6463 −0.2164 −0.5364 0.0086 −0.7490 −0.1680 0.0257

TABLE 6.3 Range foldover information for range = 500 km


(b) Azimuth angle = 60°(a) Azimuth angle = 90°

A, B, C

D

V(m/s)

SIN

R in

dB

SIN

R in

dB

−30−40 −10−20 30 4020100−30−40 −10−20 30 4020100

0

−10

−20

−60

−50

−40

−30

0

−10

−20

−60

−50

−40

−30

V(m/s)

No Crab, No Foldover No Crab, Foldover Crab, No Foldover Crab, Foldover

A, B, C, D

FIGURE 6.17 Matched filteroutput SINR in (6.37) vs. velocity with/withoutEarth’s motion and with/without range foldover for range = 500 km. (A) NoEarth’s rotation, no foldover, (B) No Earth’s rotation, foldover, (C) Earth’srotation, no foldover, and (D) Earth’s rotation, foldover.

most significant range foldover occurs at −26 dB that corresponds toωd = 0.0238 (velocity = 1.04 m/s) for θAZ = 90 and its effect canbe seen in Figure 6.17 (a) as widening the notch when both Earth’srotation (crab) and range foldover are present. In another example,Table 6.4 corresponds to range of interest 1,200 km with correspond-ing matched filter performance details as shown in Figure 6.18. FromTable 6.4, the most significant clutter foldover return occurs at −4 dBthat corresponds to ωd = 0.7823 (velocity = 26.4 m/s) for θAZ = 90and ωd = 0.1144 (velocity = 3.9 m/s) for θAZ = 60. Their effect isvisible in Figure 6.18 in terms of wider notches and lower SINR levelswhen both Earth’s rotation and range foldover are present.

From Figures 6.17–6.18, when Earth’s rotation and range foldoverare present together, the SINR performance substantially degradescompared to the three other case (A, B, C) in (6.38).

To illustrate the limitations imposed on processing gains due toEarth’s rotational phenomenon and range foldover, Figure 6.19 showsthe matched filter output in terms of SINR loss defined in (6.37) without(ideal) and with range foldover and Earth’s rotation present whenTaylor weights are used on transmit. The resulting range dependentDoppler shift in Figure 6.19 (c) can be adjusted to reflect the normal-ized performance as shown in Figure 6.19 (a). However, as seen inFigure 6.19 (d), the performance is significantly degraded when bothrange foldover and Earth’s rotation are jointly present and range-dependent Doppler compensation is not possible.

Chapter

6:

STAPfor

SBR

237

Ranges that contribute to clutter return (km) 524.87 879.42 1200 1508.9 1812.5 2113.7 2413.9Corresponding elevation angle (degree) 44.819 57.125 62.589 65.419 66.919 67.647 67.883Array gain in elevation direction (dB) −57.035 −16.458 0 −4.076 −10.46 −15.283 −17.208Normalized Azimuth Angle = 90 0.2046 −0.0544 −0.8555 0.7823 0.6053 0.5232 0.4971clutter Doppler Azimuth Angle = 60 −0.8496 −0.3083 −0.3162 0.1144 −0.6974 −0.1465 0.0285

TABLE 6.4 Range foldover information for range = 1,200 km


(b) Azimuth angle = 60°(a) Azimuth angle = 90°

D

V(m/s) V(m/s)

SIN

R in

dB

SIN

R in

dB

−30 −10−20 3020100 −30 −10−20 3020100

0

−10

−20

−60

−50

−40

−30

0

−10

−20

−60

−50

−40

−30

D

A, B, C

No Crab, No Foldover No Crab, Foldover Crab, No Foldover Crab, Foldover

A, B, C

FIGURE 6.18 Matched filteroutput SINR in (6.37) vs. Velocity with/withoutEarth’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.

2,400

800 8001,0001,2001,4001,6001,8002,0002,200

Ran

ge (k

m)

−1 −0.5 0 0.5 1Normalized Doppler

0

−10

−20

−30

−40

−50

−60

2,400

1,0001,2001,4001,6001,8002,0002,200

Ran

ge (k

m)

Ran

ge (k

m)

Ran

ge (k

m)


0

−10

−20

−30

−40

−50

−60

800

2,400

1,0001,2001,4001,6001,8002,0002,200


0

−10

−20

−30

−40

−50

−60800

2,400

1,0001,2001,4001,6001,8002,0002,200


0

−10

−20

−30

−40

−50

−60

(a) w/o range foldover, w/o Earth’s rotation

(b) w/range foldover, w/o Earth’s rotation

(c) w/o range foldover, w/Earth’s rotation

(d) w/range foldover, w/Earth’s rotation

FIGURE 6.19 STAP matched filteroutput with and without range foldoverand Earth’s rotation for an SBR located at height 506 km above ground withPRF = 500 Hz and θAZ = 90. Taylor weights (azimuth direction) are used ontransmit.


D

A

V(m/s)

SIN

R in

dB

−30 −10−20 3020100

0

−10

−20

−60

−50

−40

−30

FIGURE 6.20 Matched filteroutput with and without (ideal) range foldoverand Earth’s rotation for range 2,400 km and azimuth angle of 90. (A) NoEarth’s rotation, no foldover, and (D) With Earth’s rotation, and foldoverpresent.

This effect can be seen in Figure 6.20 as well, which corresponds tothe SINR loss in (6.37) as range equals to 2,400 km. The performancein terms of clutter nulling is seen to be significantly inferior when bothof these phenomena are present. This is evidently the case in practice,which in turn presents significant challenges for target detection andtracking [8], [11].

To understand why the clutter notch widens and degrades in thepresence of Earth’s rotation, as shown in Figure 6.20, it is necessaryto review (6.24), (6.20)–(6.21), and (6.37) simultaneously. Figure 6.20is plotted for a fixed (Ro, θAZo ) which fixes the cone angle c in (6.4),while varying ωd in (6.37). As a result, the cone angle

co = sin θE Lo cos θAZo (6.39)

remains fixed. However, referring to (6.20)–(6.23) and (6.37), there aremany other points (Rk, θAZk ) or equivalent locations at (θE Lk , θAZk )that satisfy the identity

co = sin θE Lk cos θAZk (6.40)

for the co in (6.39) and the dotted line in Figure 6.21 shows the iso-cone contour plot in (6.40). These isocone contour can also be seen inFigure 6.22 that shows the azimuth-range profile as seen from the SBR.


Range

Azi

mut

h

Dop

pler

0 2,5002,0001,5001,000Rk R1500

52

54

56

58 qAZkqAZ0

qAZ1

50

48

44

42

46

60 80

40

30

10

20

50

60

7075°

Projected doppler w/ Earth’s rotationProjected doppler w/o Earth’s rotation

wd1 = wdk

= wd0

wd1≠ wdk

≠ wd0 wd1

wd1

wdk

wdk

wd0

wd0

~ ~~

~ ~ ~

R0

Iso-cone

Iso-cone

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 spa-tial part of the steering vector in (6.10), and hence they contribute themaximum toward the null in Figure 6.20. However, the Doppler pro-jected by these points satisfy (4.48) or (4.79) depending on whetherEarth’s rotation is absent or present (see also (6.24)). Thus, whenEarth’s rotation is absent, the projected Doppler ωdk at these selectedpoints agree exactly with ωdo , the Doppler associated with (Ro, θAZo ),giving a perfect match, together with the temporal part of the steeringvector in (6.10) (see dashed line in Figure 6.21). This results in a singlenull in the output SINR plot. However, in the presence of Earth’s rota-tion, the projected Doppler values ωdkm = βoρc sin θE Lm cos(θAZk + φc)at these selected points are different from the Doppler associated with(Ro, θAZo ), as seen from the solid curve in Figure 6.21. Hence, the adap-tive processor tries to null out the “secondary sources” at ωd1 , ωd2 , . . . ,

c0 = 0 c0 = 0

ck−ck

Iso-cone(Iso-Doppler)

All points on an isocone line project a single scatter

ck

−ck

Iso-cone

Iso-range

Points on an isocone line project different scatters

(a) Earth’s rotation absent (b) Earth’s rotation present

Iso-Doppler

Iso-range

FIGURE 6.22 Clutter with and without Earth’s rotation.


and taken together, the corresponding SINR plot results in a widerclutter null.

Clutter Notch WidthTo quantitatively evaluate the width of the clutter notch in the pres-ence of Earth’s rotation, consider a point of interest at range Ro withazimuth angle θAZo , and let R1 represent its dominant foldover range.Let θE Lo and θE L1 represent the elevation angles at these range points.Thus,

co = sin θE Lo cos θAZo = sin θE L1 cos θAZ1 (6.41)

represents the cone angle of interest, and θAZ1 represents the azimuthangle at range R1 that projects the same cone angle co . Then, with ωdo

and ωd1 representing the Doppler at points (Ro, θAZo ) and (R1, θAZ1 ),we have in the absence of Earth’s rotation (Figure 6.23 (a))

ωdo = ωd1 = βoco (6.42)

and with Earth’s rotation present, we have (Figure 6.23 (b))

ωdo = βoρc sin θE Lo cos(θAZo + φc), (6.43)

ωd1 = βoρc sin θE L1 cos(θAZ1 + φc). (6.44)

SINR

wd0= wd1

= wdk w~d0

SINR

(a) Without Earth’s rotation (b) With Earth’s rotation present

s0* R−1s0

s~0* R −1s~0

s~1* R −1s~1

wd wd

w~dk

w~d1

FIGURE 6.23 Widening of the clutter notch with Earth’s rotation (a) withoutEarth’s rotation, (b) with Earth’s rotation.


From (6.41) and (6.42) and in the absence of Earth’s rotation, bothpoints (Ro, θAZo ) and (R1, θAZ1 ) project the same steering vector

so = s(co , ωdo ) = s(co , βoco ). (6.45)

More generally, it follows that in the absence of Earth’s rotationall points on the isocone contour in Figure 6.22 (a) project the samespace-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 plotsdo not coincide. As a result, every point on the isocone contour projectsa different Doppler and hence they generate distinct steering vectors.Thus for the situation in (6.43) and (6.44), we obtain the distinct steer-ing vectors

so = s(co , ωdo ), s1 = s(co , ωd1 ). (6.46)

This results in the total covariance matrix in (6.20), due to rangesRo and R1, to be given by

R =

Qo + sos∗o without Earth’s rotation

Q1 + so s∗o + gs1s∗

1 with Earth’s rotation,(6.47)

where g < 1 represents the normalized array gain at (θE L1 , θAZ1 ).Using the matrix identity (see Appendix 6-A for a proof)

(Q + ss∗)−1 = Q−1 − Q−1ss∗Q−1

1 + s∗Q−1s, (6.48)

and with the specific eigenstructure of R in (6.47), it is easy to showthat in the absence of Earth’s rotation (see (6.64) for a proof)

s∗R−1s ≥ s∗o R−1so = s∗

o Q−1o so

1 + s∗o Q−1

o so(6.49)

for any s corresponding to ωd = ωdo indicating the existence of aunique null at ωdo as shown in Figure 6.23 (a). However, in the presenceof Earth’s rotation (see paragraph after (6.64))

s∗R−1s > s∗o R−1so , s∗R−1s > s∗

1R−1s1 (6.50)

are satisfied locally, indicating multiple local nulls as shown inFigure 6.23 (b). Here, repeated use of (6.48) in (6.47) gives (whenEarth’s rotation is present, see Appendix 6-B)

s∗o R−1so = A+ ( AB − |C |2)g

1 + A+ (B + AB − |C |2)g(6.51)


and

s∗1R−1s1 = B + ( AB − |C |2)

1 + A+ (B + AB − |C |2)g(6.52)

where

A = s∗o Q−1

1 so , B = s∗1Q−1

1 s1, and C = s∗o Q−1

1 s1. (6.53)

It is reasonable to assume that A B. However, since the normal-ized gain g < 1, from (6.51) and (6.52) we obtain

s∗o R−1so < s∗

1R−1s1. (6.54)

The situation is shown in Figure 6.23 (b), that exhibits two dipscorresponding to ωdo and ωd1 with the dominant one at ωdo , and hencea wider overall null.

The sharp null shown in Figure 6.23 (a) and its absence inFigure 6.23 (b) can be explained by expressing the covariance matrixR in terms of cone angles. From (6.42) and in the absence of Earth’srotation, all range foldover points project the same Doppler for thesame cone angle. Hence, using (6.20) we get

R =No∑

i=0

Pi si s∗i + σ 2

n I (6.55)

where

si = s(ci , βoci ), i = 1, 2, . . . (6.56)

as in (6.45), and from (6.20), we have

Pi =Na∑

m=0

Pi,mG(θ i,m). (6.57)

The summation in (6.57) is along the isocone curve c = ci inFigure 6.23 (a) and it represents the total power reflected along thecone angle ci . Notice that the summation in (6.55) covers the entirecone angle set (the field of view of all scattering points).

However, in the presence of Earth’s rotation and for the same coneangle ci , different range foldovers generate different Doppler frequen-cies ωdi,o , ωdi,1 , . . ., that are obtained as in (6.43) and (6.44), with coreplaced by ci in (6.41).

Define

s2i = s(ci , ωdi,o ), s2i+1 = s(ci , ωdi,1 ), i = 0, 1, 2, . . . (6.58)


for the two range foldovers, and (6.55) becomes

R =No∑

i=0

(P2i s2i s∗

2i + P2i+1s2i+1s∗2i+1)+ σ 2

n I. (6.59)

Note that using the particular structure (linear dependency betweenDoppler 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)

rB = minMN, N + β(M − 1), No (6.60)

which is much smaller than the total number of scatters No . How-ever, no such relation exists in (6.59) and hence, in general the cluttersubspace has full rank in presence of Earth’s rotation. This allows thefollowing eigendecomposition for R in (6.55)–(6.59)

R =

rB∑k=1

(λk + σ 2n )uku∗

k + σ 2n

MN∑i=rB+1

ui u∗i , without Earth’s rotation,

MN∑k=1

(λk + σ 2n )uk u∗

k , with Earth’s rotation,

(6.61)

where λk , uk , and λk , uk represent the clutter subspace eigenval-ues and eigenvectors without and with Earth’s rotation, respectively.Figure 6.24 shows the eigenvalue spread of the clutter covariance

Eig

enva

lues

in d

B

Index

20

60

50

30

40

100 600500400300200−10

10

0

0

With Earth’s rotation

Without Earth’s rotation

FIGURE 6.24 Clutter subspace rank with and without Earth’s rotation. A32-sensor array with 16 pulses is used with Brannan factor β = 10.


matrix R with and without Earth’s rotation. Note that in the absenceof Earth’s rotation and using Brennan’s rule, the clutter subspace is ofsmaller dimension compared to the whole space. However, the cluttersubspace occupies the whole space in the presence of Earth’s rotation.Thus, irrespective of range foldover effect, the clutter subspace rankis determined by the presence or absence of Earth’s rotation.

In particular, when Earth’s rotation is absent, from (6.55) and (6.61),the noise subspace eigenvectors ui MN

i=rB+1 are orthogonal to the clutterscatter steering vector set so , s1, s2, . . . in (6.55) and (6.56); i.e.,

so , s1, s2, . . . , sNo ⊥urB+1, urB+2, . . . , uMN. (6.62)

Using (6.61), the output SINR in (6.37) simplifies to

s∗R−1s =

rB∑k=1

∣∣u∗k s∣∣2

λk + σ 2n

+ σ 2n

MN∑i=rB+1

∣∣u∗i s∣∣2 , without Earth’s rotation

MN∑k=1

∣∣u∗k s∣∣2

λk + σ 2n

, with Earth’s rotation.

(6.63)

This is plotted in Figure 6.20 and Figure 6.23, by fixing the steeringvector cone angle to that corresponding to the desired location (c = co )and spanning over all Doppler frequencies. As a result, in the absenceof Earth’s rotation, so in (6.45) is the only steering vector generatedin this manner that corresponds to the clutter scatter set in (6.62) andhence, from (6.62) and (6.63) we get,

s∗o R−1so =

rB∑k=1

∣∣u∗k so∣∣2

λk + σ 2n

≤ s∗R−1s. (6.64)

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’srotation is absent.

A similar reasoning shows that in the presence of Earth’s rotation,with a fixed cone angle co for s in (6.63), so and s1 given in (6.46) arethe only two steering vectors generated that correspond to the clutterscatter set in (6.58), and they span the same subspace as that of theclutter subspace eigenvectors u1, u2, . . .. Hence, at both ωdo and ωd1 ,the output SINR tends to be lower than at other Doppler frequenciesas 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 wholespace even in the presence of Earth’s rotation, and using a similar


reasoning as in (6.64), sharper nulls can be expected at both ωdo andωd1 thus, resulting in a wider clutter notch.

6.4 Range Foldover Minimization UsingOrthogonal PulsingWhen range foldover and Earth’s rotation are present simultaneously,distinct Doppler frequencies are generated from the associated rangefoldover points (see Figure 6.25). These range foldover points willproject different gains depending on their position with respect to thearray gain pattern (Figure 6.25 (b)). For every range point of interest,the two phenomena together generate a sequence of Doppler frequen-cies. For example, at PRF = 500 Hz, there are seven range foldoverDoppler frequencies for an SBR located at a height of 506 km aboveEarth’s surface. The clutter corresponding to these range bins will beassociated with these modified Doppler frequencies.

As Figures 6.17–6.20 show, having both range foldover and Earth’srotation present at the same time results in unacceptable performancedegradation. Interestingly, the performance can be resorted to anacceptable level if only either effect is present in the data.

As remarked earlier, the linear relationship between the spatial andDoppler frequencies is lost when a crab angle, induced by the Earth’s

(b) Top view from SBR(a) Side view

Range point of interest

Returns due to earlier pulses

SBR

Returns due to later pulses

Return

wavefront

Forward

range

foldovers

Transmit

pulses

Backward

foldovers

...

Mainbeam

qAZ

Elevationsidelobes

Rangeambiguities

D: Point of interest,cone angle c0

Range of interest

Iso-cone

(qEL,m’qAZ,j)

Azi

mut

hsi

del

obes

…

……

FIGURE 6.25 Range foldover phenomenon.


rotation around its own axis, appears in the formulation of the Dopplerfrequency. As we have seen, this in turn, significantly degrades theperformance of adaptive processing algorithms. Several solutionshave been proposed to mitigate the impact of the crab angle. One suchsolution involves inclining the array with an angle commensurate (inthe negative direction) with the induced crab angle. This solution ap-pears to be highly sensitive to errors in such compensation [8]. As aresult, a signal processing solution involving waveform diversity isgenerally preferred over the mechanical correction solution.

Waveform diversity, in general, refers to the use of various wave-forms in a transmitter/receiver design, with the objective of improvingthe overall system performance, including detection, estimation, andidentification of targets embedded in clutter, jamming, and noise. De-pending upon the platform structure (monostatic, bistatic, multistatic)and the use of single vs. multiple apertures, waveform diversity canspatially augment the dimensionality of the processing space, whichthen permits the transmission of different waveforms, with the goalof achieving a finer separation of the target from the interference.Waveform diversity also allows for distinct waveforms of differentdurations over different spectral bands to be used in both time andfrequency domains. The extended multidimensionality allows for im-proved target detection and interference cancellation since the targetand interference signals are well localized in this extended space. Inpresence of additional Doppler component induced by Earth’s rota-tion around its axis and the presence of range ambiguities, waveformdiversity can effectively enhance the performance of adaptive pro-cessing techniques.

In the present context, waveform diversity can be used on thesequence of transmitted radar pulses to realize the above goal by min-imizing the effects of range foldover returns. As a result, the data con-tains mainly the effect of Earth’s rotation only and the performance canbe restored. Recall that in ordinary practice, a set of identical pulses aretransmitted as in Figure 6.26 (a). To suppress the returns due to rangefoldover, for example, individual pulses f1(t), f2(t), . . . , as shown inFigure 6.26 (b) can be made orthogonal to each other so that

∫ To

ofi (t) f j (t)dt = δi, j , i, j = 1, 2, . . . , Na , (6.65)

with To representing the common pulse length and Na correspondingto the maximum number of distinct range foldovers present in thedata. Here δi, j is the standard Kronecker delta product. Then, withappropriate matched filtering as shown in Figure 6.27, the range ambi-guous returns can be minimized from the main return corresponding


Tr

f (t ) f (t)

t

Tr

t. . .. . .

Conventional radar pulses

f1 f2 f3

Rectangularorthogonal pulses

(Na pulses)

(a) (b)

T0

FIGURE 6.26 Radar pulse stream with and without waveform diversity.

to the range of interest. The decision instant T satisfies T ≥ To tomaintain (6.65). In this case, performance will be closer to that shownin Figure 6.19 (c). Note that for range foldover elimination, waveformdiversity needs to be implemented only over Na pulses as shownin Figure 6.26 (b). For an SBR located at a height of 506 km and anoperating PRF = 500 Hz, we get Na = 7.

To understand how waveform diversity helps to minimize the rangefoldover effect, in Figure 6.28 assume that we have Na = 3 differentrange foldover points and the range R1 is the main return. Let thecorresponding clutter scatter returns for these three different rangefoldovers be c1, c2, and c3. Also assume that M = 6.

At t = 0, the array sends out pulse f1(t) and it arrives at R1 at t =T . At this time instant as shown in Figure 6.28, the earlier waveformsf3(t) and f2(t) returning from locations R2 and R3 are also present at

(a) Conventional matched filter (b) Bank of matched filters

Data f1(t -T )

t = kT

xk

f4(t −T )

f2(t −T )

f1(t −T )

f3(t −T )

T

t = kTr

xkData

t = (k + M − 1)Tr

t = (k + 2)Tr

t = (k +1)Tr

Σ

FIGURE 6.27 Matched filterswith and without waveform diversity.


c2 f3c1 f1

c3 f2

R3

R2R1

Tx f3f2

f1f3

f2f1

at t = ∆T

c1 f1 + c2 f3 + c3 f2

FIGURE 6.28 Wavefront present at R1 for t = T.

R1. Thus, the wavefront at R1 at t = T equals

c1 f1(t) + c2 f3(t) + c3 f2(t). (6.66)

At t = 2T , the first pulse wavefront returns to the array and thearray uses the output of the receiver h1(t) matched to the pulse f1(t).Thus the array output received data equals

x1 = c1ρ11a R1e− j0ωd1 + c2ρ31a R2e− j0ωd2 + c3ρ21a R3e− j0ωd3 (6.67)

where

ρi j =Tr∫

0

fi (t)h∗j (t)dt (6.68)

and a Ricorresponds 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 +2T , the second pulse returns

to the receiver and the output is given by the receiver h2(t) that ismatched to f2(t).

Thus, the received data for second pulse is

x2 = c1ρ22a R1e− jωd1 + c2ρ12a R2

e− jωd2 + c3ρ32a R3e− jωd3 . (6.69)

In a similar manner, the received data for the third, fourth, fifth, andsixth pulses are given by

x3 = c1ρ33a R1e− j2ωd1 + c2ρ23a R2

e− j1ωd2 + c3ρ13a R3e− j1ωd3 , (6.70)

x4 = c1ρ11a R1e− j3ωd1 + c2ρ31a R2e− j3ωd2 + c3ρ21a R3e− j3ωd3 , (6.71)


c2 f1c1 f2

c3 f3

Tx

at t = Tr + ∆T

c1 f2 + c2 f1 + c3 f3

R3

R2R1

f3f2

f1f3

f2f1

FIGURE 6.29 Wavefront present at R1 for t = Tr + T.



respectively. If we stack up the data for all the pulses, we get

x =

x1x2...

xM

= c1

ρ11ρ22...

ρ33

b(ωd1 )

⊗ a R1

+c2

ρ31ρ12...

ρ23

b(ωd2 )

⊗ a R2

+ c3

ρ21ρ32...

ρ13

b(ωd3 )

⊗ a R3

(6.74)

Note that the correlation coefficient ρi i = 1 so that the space-timesteering vector corresponding to R1 will not be changed. For trulyorthogonal waveforms, ρi j = 0, i = j and hence the inner summa-tion in (6.19) and (6.20) that represent the range foldovers returnsis 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 samewhen i = j . This amounts to an amplitude modulation for the tem-poral part of the steering vector and results in inferior performance ifthe waveforms have different correlations (see also Section 6.6).

However, if all ρi j can be made equal for i = j , the steering vec-tor will not change for the range foldover data. In that case with


ρi j = ρ , i = j , we have

x =

x1x2...

xM

= c1

11...

1

b(ωd1 )

⊗ a R1

+c2

ρ

ρ...

ρ

b(ωd2 )

⊗ a R2

+ c3

ρ

ρ...

ρ

b(ωd3 )

⊗ a R3

= c1b(ωd1 ) ⊗ a R1+ ρc2b(ωd2 ) ⊗ a R2

+ ρc3b(ωd3 ) ⊗ a R3

= c1s(θ1, ωd1 ) + ρc2s(θ2, ωd2 ) + ρc3s(θ3, ωd3 ).(6.75)

In summary, the effect of waveform diversity is to scale down thescatter return power from range foldover return points so that theimpact of the inner summation terms in (6.20) is minimized except forthe dominant first return term.

Figures 6.30–6.31 show the improvement in SINR obtained by usingeight rectangular orthogonal pulses as shown in Figure 6.26 (b). Notethat the performance is restored since the eight waveforms are able tosuccessfully eliminate the seven range foldover ambiguities presentthere.

Figure 6.33 shows the SINR improvement using a more practicalset of waveforms—four up/down chirp waveforms all with equal

(a) Range = 500 km (b) Range = 1,200 km

V (m/s) V (m/s)

SIN

R in

dB

SIN

R in

dB

−40 −20 40200 −30 −10−20 3020100

0

−10

−20

−60

−50

−40

−30

0

−10

−20

−60

−50

−40

−30

IdealConventional8-Ortho. Conventional

Ideal

8-Ortho.

(i), (iii)(i), (iii)

(ii)

(ii)

FIGURE 6.30 SINR performance improvement with and without using eightrectangular waveforms. (i) Ideal (no Earth’s rotation, no range foldover), (ii)Earth’s rotation and range foldover with conventional pulsing, (iii) Earth’srotation and range foldover with eight-orthogonal pulsing.


(a) Conventional

02,400

Normalized Doppler

Ran

ge (k

m)

800−1

−10

−20

−60

−50

−40

−30

−0.5 0 10.5

1,000

1,200

1,400

1,600

1,800

2,000

2,200

(b) Eight rectangular waveforms

02,400

Normalized Doppler

Ran

ge (k

m)

800−1

−10

−20

−60

−50

−40

−30

−0.5 0 10.5

1,000

1,200

1,400

1,600

1,800

2,000

2,200

FIGURE 6.31 Matched filteroutput SINR with range foldover and Earth’srotation for two different pulsing schemes (a) Conventional pulsing (same asFigure 6.20 (d)), (b) Eight rectangular orthogonal waveforms.

bandwidth and whose instantaneous frequencies are as shown inFigure 6.32. Quadrature phase shifting of these waveforms will gen-erate an additional set of four waveforms, resulting in a pool of eightwaveforms. Note that these eight waveforms are only approximatelyorthogonal. In this case, although the performance has improved overthe conventional pulsing scheme considerably, the remaining degra-dation compared to the ideal case can be attributed to the approximateorthogonal nature of these waveforms.

Figure 6.34 (b) shows the improvement in SINR as a function ofrange and Doppler obtained by using these chirp waveforms. Forcomparison purposes, Figure 6.34(a) shows the performance usingconventional pulsing when both range foldover and Earth’s rotationare present. Note that using waveform diversity at transmit, the per-formance 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 inFigure 6.32 is about −20 dB, their correlation is not uniformly low.The off-diagonal entries of their correlation matrix R1 are unequal as

Tt

Tt

B0B0 B0B0

Tt

Tt

(a) (b) (c) (d)

w4(t )w3(t )w2(t )w1(t )

FIGURE 6.32 Up/down chirp waveforms in frequency domain.



V (m/s) V (m/s)

SIN

R in

dB

SIN

R in

dB

−40 −20 40200 −30 −10−20 3020100

0

−10

−20

−60

−50

−40

−30

0

−10

−20

−60

−50

−40

−30

Ideal

8-ChirpConventional Ideal

8-ChirpConventional

(i), (iii)

(ii)

(ii)

(i)

(iii)

FIGURE 6.33 SINR performance improvement with and without orthogonalpulsing using eight up/down chirp waveforms for range = 1,200 km.(i) Ideal (no Earth’s rotation, No range foldover), (ii) Earth’s rotation andrange foldover with conventional pulsing, (iii) Earth’s rotation and rangefoldover with eight up/down chirp pulsing.

shown below.

R1=

1 0.0164 0.0121 0.0007 0.0062 0.0057 0.0006 0.00520.0164 1 0.0073 0.0049 0.0054 0.0079 0.0036 0.00300.0121 0.0073 1 0.0140 0.0179 0.0203 0.0087 00.0007 0.0049 0.0140 1 0.0055 0.0027 0.0196 00.0062 0.0054 0.0179 0.0055 1 0.0167 0.0049 0.00010.0057 0.0079 0.0203 0.0027 0.0167 1 0.0034 0.00430.0006 0.0036 0.0087 0.0196 0.0049 0.0034 1 00.0052 0.0030 0 0 0.0001 0.0043 0 1

.

(6.76)

(a) Conventional

02,400

Normalized Doppler

Ran

ge (k

m)

800−1

−10

−20

−60

−50

−40

−30

−0.5 0 10.5

1,000

1,200

1,400

1,600

1,800

2,000

2,200

(b) Eight chirp waveforms

02,400

Normalized Doppler

Ran

ge (k

m)

800−1

−10

−20

−60

−50

−40

−30

−0.5 0 10.5

1,000

1,200

1,400

1,600

1,800

2,000

2,200

FIGURE 6.34 Matched filteroutput SINR with range foldover and Earth’srotation for two different pulsing schemes (a) Conventional pulsing(same as Figure 6.19 (d)), (b) Eight-chirp waveforms that areapproximately orthogonal.


As a result, the residual leakages at the matched filter output inFigure 6.27 (b) are different and the steering vector model for STAPneeds to be modified to accommodate them. Of course, equal corre-lations among the waveforms will retain the standard STAP steeringvector, and hence an interesting problem is to design a set of wave-forms whose correlations are approximately equal.

The hybrid-chirp waveforms fk(t) cos(ωo t+bt2), fk(t) sin(ωo t+bt2),k = 1, . . . , 4, with fk(t)4

k=1 represents an orthogonal set satisfyingthis “equal correlation” property. With R2 representing their correla-tion matrix, we have

R2 =

1 0.0060 0.0064 0.0059 0.0062 0.0063 0.0064 0.00630.0060 1 0.0065 0.0056 0.0063 0.0062 0.0064 0.00700.0064 0.0065 1 0.0055 0.0064 0.0064 0.0062 0.00640.0059 0.0056 0.0055 1 0.0063 0.0070 0.0064 0.00620.0062 0.0063 0.0064 0.0070 1 0.0061 0.0065 0.00590.0063 0.0062 0.0064 0.0070 0.0061 1 0.0066 0.00570.0064 0.0064 0.0062 0.0064 0.0065 0.0066 1 0.00560.0066 0.0070 0.0064 0.0062 0.0059 0.0057 0.0056 1

.

(6.77)

Notice that the off-diagonal entries are approximately equal in thiscase. Figures 6.35–6.36 show the performance improvement for vari-ous range levels. On comparing Figures 6.35–6.36 and Figures 6.33–6.34, we observe that additional improvements can be realized byusing hybrid-chirp waveforms.


V (m/s) V (m/s)

SIN

R in

dB

SIN

R in

dB

−40 −20 40200 −30 −10−20 3020100

0

−10

−20

−60

−50

−40

−30

0

−10

−20

−60

−50

−40

−30

Ideal

8-HybridConventional

(i), (iii) (ii)

(ii)

(i)

(iii)

Ideal

8-HybridConventional

FIGURE 6.35 SINR performance improvement with and without orthogonalpulsing using eight hybrid-chirp waveforms for different ranges. (i) Ideal (noEarth’s rotation, no range foldover), (ii) Earth’s rotation and range foldoverwith conventional pulsing, (iii) Earth’s rotation and range foldover witheight hybrid-chirp pulsing (coincides with (i) for range = 500 km).


(a) Conventional

02,400

Normalized Doppler

Ran

ge (k

m)

800−1

−10

−20

−60

−50

−40

−30

−0.5 0 10.5

1,000

1,200

1,400

1,600

1,800

2,000

2,200

(b) Eight hybrid-chirp waveforms

02,400

Normalized Doppler

Ran

ge (k

m)

800−1

−10

−20

−60

−50

−40

−30

−0.5 0 10.5

1,000

1,200

1,400

1,600

1,800

2,000

2,200

FIGURE 6.36 Matched filteroutput SINR with range foldover and Earth’srotation for two different pulsing schemes (a) Conventional pulsing (same asFigure 6.19 (d)), (b) Eight hybrid-chirp waveforms.

Thus, the waveforms only modify the scatter return for the rangefoldover point. Hybrid-chirp waveforms give better performancecompared to up/down chirp waveforms because the off-diagonal en-tries of the correlation matrix for the hybrid-chirp waveforms are al-most the same whereas the off-diagonal entries of the correlation ma-trix for the up/down chirp waveforms are more dispersed (see (6.76)and (6.77)).

6.5 Scatter Return ModelingTo obtain realistic scatter return amplitudes for clutter data in (6.19),“site-specific” information can be incorporated into the clutter datamodeling problem. In this approach, a land cover map using NASA’sterra satellite based images with 1 km2 resolution categorizes the Earthinto 16 land types (desert, lake, forest, etc.) with specific grazing an-gle dependent mean radar cross section (RCS) values. Making useof Weibull-type modeling for each type of terrain, random scatter re-turns can be generated for the entire field of view. A further refinementcan be realized by introducing wind induced internal clutter motion(ICM) [12].

Terrain classification using NASA’s terra satellite image map witha 1 km2 resolution that shows various land cover types and theirarea of coverage using a 16 type land/water classification scheme isconsidered here [13]. Knowing the type of terrain from the NASA map,specific terrain statistical data can be used to determine the mean RCSand the actual backscatter amplitude return um, j in (6.18) and (6.19) for


that specific location [14]. At low grazing angles Weibull distributionsand at large grazing angles Rayleigh distributions have been found toeffectively model the radar backscattering [12], [14]. The parametersrequired to model these random variables can be tabulated from themean RCS data available for various terrain types.

ICM due to wind is also studied here. The effect of wind is to am-plitude modulate the various pulse returns. By modeling the effect ofwind as a covariance tapering matrix that modifies the clutter covari-ance matrix, the performance degradation is studied with site-specificexamples [15]. To illustrate the effect of wind on detection perfor-mance, Billingsley’s windblown clutter model is used to generate the“wind” random variables that modulate the various pulse returns. Al-though Billingsley’s wind model gives the autocorrelation functionsin an airborne context, that model is used in the SBR case here toevaluate the first order effects of wind on target detection. Paramet-ric modeling of the wind spectrum allows to draw some interestingconclusions in this case.

6.5.1 Terrain ModelingAs we have seen earlier, SBR array gain pattern modulates the transmitwaveform and depending upon the terrain mean RCS, backscatteredreturns are generated from the points of interest on the ground. WithPT representing the SBR transmitter power, for the clutter patch lo-cated at slant range Rs j , the input power density is given by PT

4π R2s j

.

If am, j represents the planar scattering area of the clutter patch, thenPi = PT

4π R2s j

am, j represents the input power incident on the patch. With

σ om, j representing the normalized mean RCS for the (m, j)th clutter

patch, in this case Piσom, j represents the scatter power off the clutter

patch of interest. This gives the average power received at the receiversensors located at a distance Rs j from the (m, j)th clutter patch to be

P (m, j)r = Piσ

om, j

4π R2s j

ar= κo

σ om, j

R4s j

, (6.78)

where ar 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 Satel-lite [13]. The map has 1 km patches of the actual Earth categorizedinto 16 land cover types—forest, urban, croplands, lakes, etc. Themean RCS value σ o

m, j in (6.78) for each terrain type for moderate tolarge grazing angles is listed in Table 6.5 along with their shape para-meter β that is useful for parametric modeling later (see (6.83)–(6.85)).


(m, j )th patch

SBR

FIGURE 6.37Land cover map(NASA’s terrasatellite).

For instance, urban has higher reflectivity, desert has lower reflectivity,and water has significantly lower reflectivity. Equation (6.78) repre-sents the average backscatter return at the sensor from the (m, j)thpatch, and to determine the corresponding random return, let um, jrepresent the random return from the (m, j)th bin such that

E|um, j |2

= σ om, j , (6.79)

so that the received power (6.78) at the reference sensor can be ex-pressed as

P (m, j)r = κo E

∣∣∣∣∣um, j

R2s j

∣∣∣∣∣2 = κo E

|αm, j |2

, (6.80)

with αm, j is as defined in (6.18). Assuming that the clutter patchesare 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 returnsignal from the (m, j)th patch that is received at the reference receiversensor. As a result, after taking into account the array factor and thespace-time steering vector, the received data vector has the form in(6.19). Observe that for data modeling, the random clutter return um, jhas been made site-specific using (6.79) as discussed below.


Low Grazing High GrazingAngles (0–6) Angles (45)

Terrain Type σ (dB) Shape σ dB (β = 1)Parameter β

Urban −20 .5 −6Deciduous Broadleaf −21 .77 −10Evergreen Broadleaf −23 .77 −12Evergreen Needleleaf −26 .77 −15Deciduous Needleleaf −25 .77 −14Mixed Forest −24 .77 −13Vegetation Mosaic −28 .42 −17Cropland −30 .42 −19Grassland −34 .77 −19Savanna −38 .77 −23Woody Savanna −36 .67 −21Closed Shrubland −37 .67 −22Open Shrubland −39 .67 −24Wetland −40 .77 −25Snow −50 .91 −32Barren −45 .91 −27Water −60 .33 −36

(see also [13])

TABLE 6.5 Mean RCS for various terrain types

In general, the mean RCS value σ o = σ om, j in (6.79) depends on the

grazing angle, and various models have been proposed to accommo-date 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 for10– 60 grazing angles, barring high grazing angle values that corre-spond to near nadir points. At high grazing angles, the return powerincreases significantly and to accommodate this an additional termcan be introduced to the constant gamma model in (6.81). With anextra constant term added to determine the plateau region, the nearRCS equation takes the form [16]

σ o (ψ) = A+ B sin ψ + Ce−D( π2 − ψ)E

. (6.82)

Notice that five parameters (A, B, C , D, E) are required to representthis model and hence it is known as the five parameter model [16]. Theseparameters in turn are determined by the terrain type. Figure 6.38


0 10 20 30 40 50 60 70 80 90−60

−50

−40

−30

−20

−10

0

10

20

Grazing Angle

RC

S (d

B) s

o (y)

UrbanDeciduous Broadleaf

Vegetation MosaicCroplandGrasslandSavannaWoody SavannaClose ShrublandOpen ShrublandWetlandSnowBarrenWater

Evergreen BroadleafEvergreen NeedleleafDeciduous NeedleleafMixed Forest

Water

Urban

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 thosein Table 6.5 for high grazing angles.

By fitting various statistical models to experimental data, it has beenwell documented that Weibull distributions can effectively model theamplitude or power levels of the backscattered signal, especially atlow grazing angles [14]. Recall that Weibull random variables havethe following probability density function [17]

fX(x) =

αxβ−1e−αxβ/β x ≥ 0

0 otherwise.(6.83)

Let

X = |um, j |2 (6.84)

represent the random backscattered power associated with thebackscatter return um, j . From (6.79) its mean value is given by thenormalized RCS. The random backscatter power return X in (6.84) isgenerally modeled as a Weibull random variable with parameters α

and β [15]. This gives

EX = µX =(

β

α

) 1β

(1 + 1/β) = σ om, j , (6.85)

where the last equality follows from (6.79) and (6.84). If we let β = 2in (6.83), we obtain the Rayleigh distribution. In general, knowing


0 1 2 3 4 5 6−70

−60

−50

−40

−30

−20

−10

0

Grazing Angle (deg)

Mea

n R

CS

0 1 2 3 4 5 60.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

Grazing Angle (deg)

Wei

bull

-sha

pe P

aram

eter

b

Rural

Desert

Urban

Rural

Desert

Urban

(a) Mean RCS (b) Weibull parameter b

Mountain

FarmMountain

Farm

RuralForestFarmDesertMountainUrban

RuralForestFarmDesertMountainUrban

FIGURE 6.39 (a) Mean RCS and (b) Weibull shape parameter β forlow-angle clutter over different terrains [12]. See also Table 6.5.

the shape parameter β and the mean terrain RCS σ om, j from Table 6.5,

the other Weibull parameter α in (6.83) can be computed from (6.85).Figure 6.39 shows the mean RCS σ o

m, j and the Weibull shape parameterβ for low values of grazing angle [12].

In general Figure 6.38 together with Figure 6.39 give an accuratemean RCS for various terrain types that takes the grazing angle de-pendency into consideration. Consequently, knowing the locations ofthe point of interest on Earth and that of the SBR, the terrain types,the mean RCS, and the Weibull random variable parameters for thebackscattered power return can be computed for the entire field ofview using this approach.

Interestingly, the return amplitude (magnitude) random variable|um, j | = √

X that is useful for simulation is also Weibull with pa-rameters 2α and 2β. This follows since Y = √

X, and with fX(x) in(6.83) representing the probability density function of X, we obtainthe probability density function of Y to be1

fY(y) = 1∣∣ dydx

∣∣ fX(y2) = 2yfX(y2) = 2αy2β−1e−αy2β/β

=

α1 yβ1−1e−α1 yβ1 /β1 , y ≥ 0

0, otherwise(6.86)

with α1 = 2α, β1 = 2β. This procedure is adapted here to simulatethe backscatter amplitude return random variables |um, j | for all pointsin the field of view of the point of interest. Using a uniform random

1More generally, if X ∼ w(α, β), then Y = Xa is also Weibull with parametersα/a and β/a .


phase for um, j , scatter returns can be then faithfully simulated. Thisapproach 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 ModelingA realistic clutter model should include other secondary effects inaddition to accounting for terrain type variations. For example, forestsand lakes are constantly modulated by wind and they affect the pulsereturns by suitably amplitude modulating the temporal returns thusaffecting the Doppler. For a uniform pulse sequence with PRF = 1/Tr ,the temporal steering vector b (ωd ) corresponding to M pulses is givenby (6.8).

The wind modulated temporal steering vector has the form [2]

b (ωd ) = b (ωd ) w (6.87)

where

w = [w1, w2, . . . wM]T , (6.88)

with w1, w2, . . . representing the “wind” random variables and rep-resenting the Schur Product as in (1.61) and (1.62). In other words, thewind random variables amplitude modulate the pulse returns. Let

rw(kTr ) = E

wi w∗i+k

(6.89)

represent the autocorrelation coefficients of the “wind” random vari-ables in (6.88) with Tr representing the pulse repetition interval. Forthe airborne case, Billingsley has modeled these windblown autocor-relations using the real symmetric function [12]

rw(τ ) = µ

1 + µ+ r (τ ) = co + r (τ ) (6.90)

where

co = µ

1 + µ, (6.91)

and the time-dependent term r (τ ) in (6.90) is given by

r (τ ) = 11 + µ

(cλ)2

(cλ)2 + (4πτ )2 . (6.92)

In (6.90)–(6.92), µ represents the DC to AC ratio defined by

µ = 489.8 v−1.55w f −1.21

o (6.93)


0 50 100 150 200 250 300 350 400 450 5000.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

110 mph20 mph30 mph40 mph50 mph60 mph70 mph80 mph

r w(k

Tr)

k

10

20

40

80

FIGURE 6.40 Windblown clutter autocorrelation function with PRF = 500Hz and a carrier frequency of 1.25 GHz.

with vw representing the wind speed in mph and fo the carrier fre-quency 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 fo = 1.25 GHzat 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 time-dependent portion r (τ ) in (6.92) needs to be modeled, and a variety oftechniques can be used for this purpose [18]. A direct approach in thiscase is to sample the autocorrelations at the desired PRF and therebydefine

rk = r (kTr ), k = 0, 1, 2, . . . (6.95)

in (6.92) and compute the discrete-time spectrum

S(ω) =+∞∑

k=−∞rke− jkω (6.96)

and identify the underlying real minimum phase system H(z) givenby [18]

S(ω) = |H(e jω)|2 (6.97)


where

H(z) =∞∑

k=0

hk z−k . (6.98)

Identifying the minimum phase system H(z) involves a spectralfactorization, and knowing the system H(z) in (6.98) the desired “windrandom variables” in (6.87) and (6.88) can be generated using therelation

wk =∑

i

hk−i vi + √co , (6.99)

where vi represents a white noise sequence of unit spectral densityand co = µ/(1 + µ) as in (6.91), which compensates for the con-stant term in (6.90). We can use the classic Bauer-type factorization[19] to identify the underlying minimum phase system in (6.98) (seeAppendix 6-C). Interestingly in this particular case the system so gen-erated being real can be further modeled using a set of damped realsinusoids. This gives the system response to be

hk = h(kTr ) (6.100)

where

h(t) =no∑

i=−no

ci e−(αi + jωi )t =no∑

k=0

|ck | e−αk t cos(ωkt − φk), t ≥ 0

(6.101)

corresponds to a real ARMA minimum phase system model

H(z) = b0 + b1z−1 + · · · + bm−1z−(m−1)

a0 + a1z−1 + · · · + amz−m = B(z)A(z)

(6.102)

in (6.98), where the model order m = 2no + 1. Using (6.102), the itera-tion in (6.99) can be expressed as

wk = −m∑

i=1

ai wk−i +m−1∑i=0

bi vk−i + √c0 , (6.103)

where ai and bi represent the real coefficients of A(z) and B(z) in(6.102). To illustrate this approach, Table 6.6 gives the damped sinu-soidal components and Table 6.7 gives the corresponding ARMA sys-tem coefficients for three typical wind speeds (10, 40, and 80 mph) atcarrier frequency 1.25 GHz and PRF = 500 Hz [20]. To obtain a similar


Damped Sinusoid ParametersWind (mph) |ci | αi fi = ωi

2πφi

0.068 6.5 0 010

0.079 17.21 3.49 −1460.0223 4.0500 0 00.3225 14.3500 0 0

400.3552 40.0000 3.68 −120.620.0417 61.9000 11.25 73.480.026 4.37 0 00.412 15.98 0 0

800.812 43.34 4.33 −1260.095 65.39 13.07 60

TABLE 6.6 Wind modeling using damped sinusoidal components forBillingsley model at carrier frequency = 1.25 GHz, PRF = 500 Hz

parametric form at other frequency and PRF sets, a separate spectralfactorization 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

a0 → am b0 → bm−1Wind (mph) Order (Top to Bottom) (Top to Bottom)

3 1.000000000 0.002425544−2.919100657 −0.005076445

102.842025760 0.002797401

−0.9228872826 1.000000000 0.006565227

−5.557239739 −0.03021140712.879352421 0.055984419

40 −15.935236822 −0.05217613511.102506392 0.024462356−4.130502782 −0.004624028

0.6411205866 1.000000000 0.0074985190

−5.522320350 −0.034192292012.723036421 0.0629394360

80 −15.655963091 −0.058337806010.853636468 0.0272206780−4.019910201 −0.005127624

0.621520855

TABLE 6.7 ARMA system coefficientsfor Billingsley model at carrierfrequency = 1.25 GHz, PRF = 500 Hz. (see Figure 6.42)


Exact

Simulated

0.5

0.5 1.5 2t (sec)

0

0.6

0.7

0.8

0.9

1

1.1

1

Recomputed

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 exponentiallydecaying components. Interestingly, this trend seems to be true for amuch larger set of frequencies (100 MHz–10 GHz). Figure 6.41 showsthe exact autocorrelations in (6.90)–(6.95), those recomputed from thesystem model in (6.100) using the identity

rw(kTr ) = µ

1 + µ+ rk (6.104)

with

rk =∑i=0

hi+kh∗i (6.105)

and the unbiased estimated autocorrelations rw generated from the“wind random variables” in (6.99), using the relation

rw = 1n

n∑i=0

wi w∗i+k . (6.106)

We have used these random variables wi in (6.87)–(6.88) to simu-late the effect of wind on the SBR clutter data. Figures 6.42–6.43 showthe original spectrum given by the Billingsley model and its variousrational approximations for two different situations [20]. Figure 6.42


Freq. (Hz)

Spec

trum

in d

B

−50 −25−120

50250

0

−20

−100

−80

−60

−40

Rational Model (Rank 6)

BillingsleySpectrum

Discrete-TimeSpectrum

FIGURE 6.42 Wind spectrum and its rational approximation—Billingsleymodel, discrete-time spectrum obtained from the exact autocorrelationssampled at PRF, and spectrum corresponding to a sixth order rationalapproximation. Wind speed = 40 mph, carrier frequency = 1.25 GHz,PRF = 500 Hz.

Freq. (Hz)

Spec

trum

in d

B

−150 −50 50 100 1500

0

−20

−100−100

−80

−60

−40

Billingsleyspectrum

Rational model (Rank 6)

FIGURE 6.43 Wind spectrum and its rational approximations—Billingsleymodel and spectrum corresponding to a sixth order rational approximation.Breezy wind conditions (6 mph), carrier frequency = 10 GHz, PRF =690.8 Hz.


corresponds to a wind speed of 40 mph, carrier frequency of 1.25GHz, and a PRF of 500 Hz. The original spectrum corresponding tothe Billingsley model in (6.90)–(6.92) given by

SB(ω) = µ

1 + µδ(ω) + 1

1 + µ

cλ4

e−cλ|ω|/2, (6.107)

the spectrum in (6.96) associated with the discrete-time autocorre-lations in (6.95), and the spectrum given by (6.97)–(6.98) correspond-ing to the sixth order rational model in Table 6.6 and Table 6.7 areshown there. In this case the spectrum associated with the sixth orderrational approximation is able to faithfully reproduce the true windspectrum in (6.107) up to about −60 dB. Figure 6.43 corresponds toanother set of parameter values (wind speed = 6 mph, carrier fre-quency = 10 GHz, and PRF = 690.8 Hz). The spectrum associatedwith the rational system of order six with filter coefficients as given inTable 6.8 is also shown in Figure 6.43 [21]. In this case, the proposedsixth order system, consisting of two damped sinusoids and two ex-ponentially decaying terms, is able to faithfully reproduce the windspectrum up to about −80 dB.

From (6.107), the slope in Figures 6.42–6.43 is given by −cλ/2 andhence it has different values in those two cases. Consequently therational approximations are valid over different frequency bands (20Hz vs. 125 Hz). This can also be explained using the location of the twosets of complex poles of the rational system in each case. In Figure 6.42they are located around frequencies 3.7 Hz, 11 Hz, whereas Figure 6.43corresponds to 16 Hz and 50 Hz and hence the pole near 50 Hz is ableto stretch the spectral match to about 125 Hz in Figure 6.43.

a0 → a6 b0 → b51.000000000 0.006086379

−4.448829511 −0.0148679258.262476857 0.020668209

−8.228964421 −0.0156365554.654376421 0.005389510

−1.423964559 −0.0013824180.184975558

TABLE 6.8 ARMA(6,5) system coefficientsfor Billingsley model at windspeed = 6 mph, carrier frequency = 10 GHz, PRF = 690.8 Hz (seeFigure 6.43).


6.6 MDV with Terrain Modelingand Wind EffectSite-specific terrain modeling and the wind phenomenon (ICM) stud-ied in Section 6.5 also affect the clutter suppression performance. Toquantify 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 theideal covariance matrix is given by

Pm, j = E|αm, j |2

= E|um, j |2

R4

s j

= σ om, j

R4s j

(6.108)

where we have used (6.21) and (6.79)–(6.80). From (6.87), the effectof wind is to modulate the pulse returns and as a result the windmodulated clutter data vector xk has the same form as in (6.19) withthe space-time steering vector si, j replaced by si, j where

sm, j = b(ωdm, j ) ⊗ a (θm, j ) (6.109)

with b(ωd ) as in (6.87). This gives the wind modulated clutter covari-ance matrix R to be

R = E

xk x∗k

= E

xkx∗k

Eyy∗ = R Eyy∗ (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 1N represents the spatial component.Let

Ty = Eyy∗ (6.112)

and

T = Ew w∗ (6.113)

so that Ty = T ⊗ 1N×N and

R = R Ty = R T ⊗ 1N×N. (6.114)

Here, 1N×N represents an “all one” M × M matrix. Notice that Trepresents the wind autocorrelation matrix with

Ti, j = rw((i − j)Tr ) (6.115)


−40 −30 −20 −10 0 10 20 30 40

−55−50−45−40−35−30−25−20−15−10

Velocity (m/s)

SIN

R

−40 −30 −20 −10 0 10 20 30 40−55−50−45−40−35−30−25−20−15−10−5

Velocity

SIN

R

Multi-Terrain

(b) Location 2 (Ontario)(a) Location 1 (Delmarva)

Single TerrainMultiple Terrain

Single TerrainMultiple Terrains

SingleTerrain

Multi-Terrain

SingleTerrain

FIGURE 6.44 The Effect of Terrain Modeling: Performance at Rt = 500 kmfor 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 of50, longitude −92 (Ontario, Canada). CNR = 40 dB.

as in (6.89)–(6.94). T is Toeplitz in nature indicating the wide sensestationary nature of the random process associated with the windphenomenon.

Figure 6.44 shows the effect of terrain modeling (uniform vs. mul-tiple terrains) on the SINR performance computed using the ideal co-variance matrix R in (6.20) for two different locations on Earth shownin Figure 6.45. Figure 6.44 (a) corresponds to the Delmarva region con-sidered 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 multiterrainperformance in Figure 6.44(a) is worse than the single terrain perfor-mance and the opposite is true in Figure 6.44(b). In Figure 6.44 (a),the mainlobe covers a terrain with significantly lower RCS comparedto the nearest sidelobes. Therefore, the sidelobe clutter is much moredifficult to suppress. In the Delmarva region considered here, landjuts out into the Chesapeake Bay, and the degradation is due to themainbeam covering a section of water, leading to smaller returns inthe mainbeam. The relatively high power of the sidelobes (caused byneighboring coastline/islands) leads to severe performance degrada-tion. Conversely, in Figure 6.44 (b) the sidelobes are weaker as theycover areas with low reflectivity such as lakes, leading to superiornulling capabilities for the processor in the multiterrain case [20].

Thus for a fixed CNR, incorporating site-specific terrain effect intothe clutter model does not always lead to inferior performance com-pared to a uniform terrain model. Site-specific terrain performancedepends on the exact combination and location of these differentterrains with respect to the mainlobe and their RCS characteristics.


2

1A

B

Southwest

Delmarva

New YorkCity

Ontario

30°N

45°N

60°N120°W 90°W 60°W

FIGURE 6.45 Two locations for clutter simulation: (a) Location 1 is atlatitude of 39, longitude of −76 (Delmarva region) and (b) Location 2 is atlatitude of 50, longitude −92 (Ontario).

Depending on the location, performance can either improve or worsencompared 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 modelwith wind speeds varying from 0 to 80 mph. Figure 6.47 shows thecombined terrain/wind effect for a single terrain as well as multi-terrain (Delmarva region shown in Figure 6.45) with wind speedsagain varying from 0 to 80 mph. In both cases, wind deteriorates theSINR performance by increasing the width of the clutter notch, thusmaking target detection much more difficult [20].

6.6.1 Effect of Wind on DopplerTo understand why the “wind effect” deteriorates the SINR perfor-mance by widening the clutter notch, we can make use of analysissimilar to the clutter notch width in (6.47)–(6.64). Let so in (6.45) repre-sent the space-time steering vector associated with the point of interest(io , jo ). In that case, (6.49) and (6.50) hold true for steering vectors withωd = ωdo , indicating the existence of a unique null at ωdo as shown inFigure 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, andas a result, the wind modulated clutter data vector has the same formas in (6.19) with the space-time steering vector sm, j replaced by sm, jgiven by (6.109). This gives the wind modulated covariance matrix R


−40 −30 −20 −10 0 10 20 30 40

−55

−50

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−25

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−15

−10

Velocity (m/s)

0 mph

80 mph70 mph60 mph50 mph40 mph30 mph20 mph10 mphSI

NR

Windy(80 mph)

No wind

FIGURE 6.46 The effect of wind on single terrain SINR: Windspeed = 0–80 mph.

−40 −30 −20 −10 0 10 20 30 40

−55

−50

−45

−40

−35

−30

−25

−20

−15

−10

Velocity (m/s)

Singleterrain

Multiterrain

SIN

R

No wind

Windy(80 mph)

FIGURE 6.47 Combined effect: Wind speeds of 0–80 mph applied to a singleterrain and multiple terrains (Delmarva region).


to be as in (6.110)–(6.114) with T representing the wind autocorrelationmatrix.

The parametric nature of the underlying system in (6.98) can beused to make further progress. In fact, from (6.100) the effect of windis equivalent to two damped sinusoids together with one to three de-caying exponentials. Substituting (6.100) into (6.104) or by directlyfitting a Prony-type model into the autocorrelations in (6.92), it is pos-sible to show that the Billingsley autocorrelation model in (6.92) itselfcan be expressed as a sum of two damped sinusoids together with oneor 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 ) = co + 22∑

i=1

qi cos(ωi k) = co +2∑

i=1

qi (e− jωi k+e+ jωi k) (6.116)

with ωi representing the frequencies given in Table 6.6, qi > 0, k ≥ 0,and co denoting the positive constant term in (6.90). With (6.116) in(6.113)–(6.115) we get

T = c01M×M +2∑

i=1

qi(bi b

∗i + b−i b

∗−i

)(6.117)

where

bi = b(ωi ) = [1, e− jπωi , e− j2πωi , . . . e− j (M−1)πωi ]T (6.118)

and

b−i = b(−ωi ) = [1, e jπωi , e j2πωi , . . . e j (M−1)πωi ]T (6.119)

represent temporal steering vectors and 1M×M represents an “all one”M× M matrix as in (6.114). Substituting (6.116)–(6.117) into (6.114) andmaking use of (6.46) and (6.47) for R we obtain

R = Qo + cosos∗o +

2∑i=1

qi(si s∗

i + s−i s∗−i

). (6.120)

Here

Qo = Qo Ty (6.121)

and

si = so (bi ⊗ 1N) = b(ωdo ) b(ωi ) ⊗ a (θ o )

= b(ωdo + ωi ) ⊗ a (θ o ), i = 1, 2 (6.122)


and

s−i = so (b−i ⊗ 1N) = 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 au-tocorrelations is to modify the original clutter component at Dopplerfrequency ωdo to four additional components with nearby Dopplerfrequencies ωd±1 = ωdo ± ω1 and ωd±2 = ωdo ± ω2. Furthermore, from(6.120), these new components are uncorrelated with the original com-ponent at ωdo and the rest. Interestingly, for the location of interestθ io , jo , from (6.117) and (6.120), co + 2(q1 + q2) = ro represents the totalbackscattered power factor, and the terrain RCS σ o

io , jo gets scaled bythis factor. However, ro = 1 and hence the RCS is not modified due tothe wind effect.

From (6.48) and (6.49) in the absence of wind, the clutter ridge passesthrough the Doppler ωdo associated with the point of interest. How-ever, this is no longer true when the modified Doppler frequenciesωd±1 and ωd±2 are also present. In that case the effect of the adaptiveprocessor is to generate local nulls at all these five frequencies. Thisresults in an overall wider null and, consequently, the clutter nullingeffect with wind present will be inferior to that in the absence of wind.

To see this explicitly let q1 > q2 in (6.120). This allows us to rewrite(6.120) as

R = q1(Q1 + s1s∗

1 + gs2s∗2)

(6.124)

where

Q1 = 1q1

(Qo + cosos∗

o +2∑

i=1

qi s−i s∗−i

), g = q2/q1 < 1. (6.125)

Equation (6.124) is similar to the situation where Earth’s rotation ispresent in (6.47), and following the same analysis we obtain

s∗1R−1s1 = 1

q1

A+ ( AB − |C |2)g1 + A+ (B + AB − |C |2)g

(6.126)

and

s∗2R−1s2 = 1

q1

B + ( AB − |C |2)1 + A+ (B + AB − |C |2)g

(6.127)

where

A = s∗1Q−1

1 s1, B = s∗2Q−1

1 s2, C = s∗1Q−1

1 s2. (6.128)


with A B. However, since the normalized gain g < 1 from (6.126),and (6.127), we obtain

s∗1R−1s1 < s∗

2R−1s2, (6.129)

indicating that the dominant component in (6.120) introduces a deepernotch (lower SINR).

To compare the widths of the clutter notches without and with windpresent, 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∗1R−1s1 > s∗

o R−1so . (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 (seeFigure 6.48). Equation (6.126) gives the corresponding SINR in thepresence of wind. To simplify this expression assume that the secondfrequency component in (41) is much weaker compared to the firstone (q1 = 1, q2 0). In that case g 0 in (6.126) and we get

s∗1R−1s1 = 1

q1

A1 + A

= 1q1

s∗1Q−1

1 s1

1 + s∗1Q−1

1 s1. (6.131)

Because of the local null at ωd1 due to the adaptive processor

s∗1R−1s1 < s∗

1R−1s1 (6.132)

(With Wind)

wdowd1 wd

wd3

SINR(Without Wind)

s3∗ R−1s3 = s1

∗ R−1s1~

s1∗ R−1s1

FIGURE 6.48 Clutter notch widening due to wind.


i.e., the wind induced SINR at ωd1 projects a smaller value than theone without the wind effect. This is shown in Figure 6.48. As a resultthe wind induced SINR is potentially wider than the one without thewind effect.

The real nature of the covariance in (6.90) guarantees conjugatesymmetry for the frequencies in (6.116). Thus the shifted frequenciesoccur at ωdo ± ω1 and ωdo ± ω2 forcing symmetry in the SINR output.Although this symmetry is evident in Figure 6.47 (a)–(b), the suggestedsecondary level sharp nulls are not present there. To understand theabsence of these specific nulls due to wind, it is necessary to examinethe wind model in further detail (Section 6.6.2).

The model in (6.114)–(6.116) assumes the wind autocorrelations tobe the sum of two pure sinusoids plus a constant term. From Table 6.6and Figure 6.41, the wind autocorrelations are more accurately repre-sented by the sum of damped sinusoids together with damped expo-nentials and a constant. Next, we analyze this model to understandthe effect of wind damping factor on Doppler.

6.6.2 General Theory ofWind Damping Effect on Doppler

If we assume the wind autocorrelations to be represented as a sumof m damped sinusoids together with damped exponentials and aconstant, the model in (6.116) can be rewritten as

rw(kT) = co +m∑

i=1

qi e−αi k e− jωi k , (6.133)

where the damped exponentials can be represented with ωi = 0. Sub-stituting (6.133) into (6.115), the temporal tapering matrix T in (6.113)takes the form

T = co1M×M +m∑

i=1

qi Hi bi b∗i (6.134)

where Hi is a real symmetric positive-definite Toeplitz matrix givenby

Hi =

1 ρi ρ2i · · · ρM−1

i

ρi 1 ρi · · · ρM−2i

ρ2i ρi 1 · · · ρM−3

i

......

.... . .

...

ρM−1i ρM−2

i ρM−3i · · · 1

> 0 (6.135)


where

ρi = e−αi > 0 (6.136)

and bi s are as in (6.118). Notice that ωi = 0 gives bi = 1M.In general, Hi has full rank and let

Hi =M∑

k=1

µi,kei,keTi,k (6.137)

represent its eigen-decomposition. Hi is real positive definite and sym-metric implies that µi,k , k = 1, 2, . . . , M are positive and ei,k are real.Substituting (6.137) into (6.134), it simplifies to

T = co1M×M +m∑

i=1

M∑k=1

qiµi,k(ei,k bi )(ei,k bi )∗. (6.138)

With (6.138) in (6.114), we obtain the wind modified clutter covari-ance matrix to be

R = R (T ⊗ 1N×N) = (Qo + sos∗o

) (T ⊗ 1N×N)

= Qo + cosos∗o +

m∑i=1

M∑k=1

µi,k si,k s∗i,k

= Qo + cosos∗o +

m∑i=1

Ui , (6.139)

where the last summation represents m uncorrelated covariance ma-trices with each one corresponding to one damped sinusoidal term in(6.133). Thus (6.139) suggests that the returns due to the sinusoidalterms in (6.133) are uncorrelated with each other. In (6.139)

Qo = Qo (T ⊗ 1N×N) (6.140)

Ui =M∑

k=1µi,k si,k s∗

i,k , µi,k = qiµi,k > 0, (6.141)

and

si,k = so (ei,k bi ) ⊗ 1N. (6.142)

Observe that (6.141) itself represents a bundle of M uncorrelatedreturns, each of them associated with the same damped sinusoidalterm in (6.133). Together with (6.139), it now follows that eachdamped sinusoidal term present in the wind generates a bundle that


is uncorrelated with the remaining sinusoidal returns, with each suchbundle consisting of M uncorrelated returns. To analyze the returnsin (6.141) further, notice that the vector si,k in (6.142) can be rewrittenas (use (6.138))

si,k = so (ei,k bi ) ⊗ 1N= b(ωdo ) ⊗ a (θ o ) (ei,k bi ) ⊗ 1N= ei,k b(ωdo ) b(ωi ) ⊗ a (θ o ) 1N= ei,k b(ωdo ) b(ωi ) ⊗ a (θ o ) = ei,k b(ωdo + ωi ) ⊗ a (θ o )

= ei,k b(ωdi ) ⊗ a (θ o ) = bi,k(ωdi ) ⊗ a (θ o ) (6.143)

where

bi,k(ωdi ) = ei,k b(ωdi ), k = 1, 2, . . . M (6.144)

represents amplitude modulated temporal steering vectors corre-sponding to the shifted Doppler frequency

ωdi = ωdo + ωi , i = 1 → m. (6.145)

As a result, si,k in (6.139)–(6.143) represents a temporally modulatedspace-time steering vector associated with the Doppler frequency ωdi

and spatial location θ o . On comparing (6.120) and (6.139), the effectof each sinusoidal wind frequency ωi is to shift the original Dopplerωdo to ωdo +ωi , and the effect of damping is to generate M new ampli-tude modulated temporal steering vectors as in (6.144) for each suchfrequency. From (6.141), these amplitude modulated steering vectorscorrespond to M uncorrelated returns all originating from the samespatial location θ o .

In summary, the effect of each damped sinusoidal componentpresent in the wind autocorrelation is to generate an additionalDoppler frequency as in (6.145), and then generate M new uncor-related scattered returns each corresponding to distinct amplitudemodulated temporal steering vectors as in (6.144).

Finally, to see the effect of amplitude modulation in (6.144), let fi,k

represent the DFT vector associated with each real eigenvector ei,k .The entries in ei,k correspond to a sampling period of Tr . Hence itsDFT coefficients fi,k(n) are sampled at = 1

MTrapart in the frequency

domain, we get

ei,k =M/2∑

n=−M/2

fi,k(n)

1e− j2πn/M

...

e− j2π(M−1)n/M

=

M/2∑n=−M/2

fi,k(n)b( 2n

M

),

(6.146)


where b is defined as in (6.118). Finally substituting (6.146) into (6.144)we get

bi,k(ωdi ) =M/2∑

n=−M/2

fi,k(n)b( 2n

M

) b(ωdi )

=M/2∑

n=−M/2

fi,k(n)b(ωdi + 2n

M

)(6.147)

and using this in (6.143) we obtain

si,k =M/2∑

n=−M/2

fi,k(n)b(ωdi + 2n

M

)⊗ a(θ o

)

=M/2∑

n=−M/2

fi,k(n)si,k(θ o , ωdi + 2n

M

). (6.148)

Equation (6.148) represents a scaled sum of space-time steering vec-tors 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 assuch they represent a coherent set of return signals at these incremen-tally shifted Doppler frequencies.

Together with (6.148), Equation (6.139) can be given the followinginterpretation:

The effect of wind can be represented as a constant term togetherwith a finite number of damped sinusoids. For each spatial location θ oon Earth with inherent clutter Doppler ωdo , each sinusoidal wind com-ponent with frequency ωi , i = 1 → m generates an additional clutterDoppler frequency ωdi = ωdo +ωi and the return bundle for each suchfrequency is uncorrelated with the returns from other frequencies. Theeffect of damping is two fold: Each such Doppler frequency ωdi gen-erates a bundle of M uncorrelated returns as in (6.141). Further from(6.148), each such uncorrelated return in that bundle contains at mostM coherent returns at new Doppler frequencies

ωdi ± 2nM

, n = 0, 1, . . . , M/2. (6.149)

This situation is illustrated in Figure 6.49, where the wind inducedfrequencies ω1 and ω2 generate two additional Doppler returns at fre-quencies ωd1 = ωdo + ω1 and ωd2 = ωdo + ω2 that are uncorrelatedwith each other and also with the return at the original Doppler fre-quency ωdo . The effect of wind damping factor is to generate a bundleof M uncorrelated returns for each new Doppler frequency ωd1 and


(a) Without wind (b) With wind

2nM

2nM

Coh

eren

t

C3

C1

C2

A

O

B

A

O

B D1 D2

D3

Uncorrelatedreturns

Uncorrelated

Bundlefor wd1

wd2±

wdo

wdowd1

±

Uncorrelatedreturns

wd1

wd2

FIGURE 6.49 Scatter returns without and with wind effect. (a) Withoutwind, a single Doppler return frequency is generated, (b) With wind, thereturn consist of m uncorrelated bunches, with each bunch consisting of Muncorrelated 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 ofM returns.

A sum of coherent returns does not correspond to a steering vectorand consequently, they cannot be nulled out by the standard adaptiveweight vector w = R−1s. This is illustrated in Figure 6.50 where the

Angle (deg)

15010050

−10

−20

0

0

Uncorrelated returns

Coherent returns

SIN

R in

dB

FIGURE 6.50 SINR loss in an uncorrelated scene vs. a coherent scene.Dashed curve represents three uncorrelated returns of equal power arrivingfrom 75, 95, and 112 at a seven-sensor element array. The solid curverepresents two identical coherent returns from 75 and 112 that areuncorrelated with the return from 95. SNR = 15 dB.


dotted line corresponds to three uncorrelated sources in white noise,and the solid line corresponds to two coherent sources and an uncor-related source in white noise.

In the uncorrelated case, the array output covariance matrix R1equals

R1 = s1s∗1 + s2s∗

2 + s3s∗3 + σ 2I (6.150)

and in the second case, the sources corresponding to steering vectorss1 and s3 are coherent. Hence

R2 = aa∗ + s2s∗2 + σ 2I (6.151)

where

a = s1 + s3. (6.152)

Clearly, the adaptive processor

w1 = R−11 s (6.153)

is able to null out the three uncorrelated sources as shown in theSINR output as the steering vector s spans through the field of view.However, the adaptive processor

w2 = R−12 s (6.154)

in the coherent case is only able to null out the uncorrelated components2 present in R2.

It follows that the wind induced coherent components cannot benulled out by the STAP processor and as a result, specific nulls donot show up at these frequencies ωd1 , ωd2 , . . .. Thus the wind inducedSINR performance will be potentially wider compared to the no-windsituation as evident in Figures 6.46–6.47.

6.7 Joint Effect of Terrain, Wind, RangeFoldover, and Earth’s Rotation onPerformanceFinally, to illustrate the combined effect of terrain, wind, rangefoldover, and Earth’s rotation on clutter nulling performance for theoptimum adaptive processor, two locations A and B in Texas shown inFigure 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 withconventional waveform and hybrid-chirp waveform. In these two


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V (m/s)

SIN

R in

dB

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SIN

R in

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(c)

(a) (b)

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SIN

R in

dB

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0

10 20 30−60

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−10

V (m/s)

SIN

R in

dB

(d)

Ideal

Ideal

FIGURE 6.51 Location A in Texas, SBR range 1,200 km, using conventionalwaveform. (a) Terrain only, (b) Terrain and wind, (c) Terrain and rangefoldover, (d) Terrain, wind, range foldover, Earth’s rotation present.

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0

10 20 30−60

−50

−40

−30

−20

−10

V (m/s)

Ideal

SIN

R in

dB

FIGURE 6.52 Location A in Texas. SBR using hybrid-chirp waveforms.Terrain, wind, range foldover, and Earth’s rotation are present.


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SIN

R in

dB

(a)

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SIN

R in

dB

(b)

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V (m/s)

SIN

R in

dB

(d)

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0

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−10

V (m/s)

SIN

R in

dB

(c)

Ideal

Ideal

FIGURE 6.53 Location B in Texas. SBR range 1,200 km, using conventionalwaveform. (a) Terrain only; (b) Terrain and wind; (c) Terrain and rangefoldover; (d) Terrain, wind, range foldover, Earth’s rotation present.

figures, the dashed lines show the performance of each case underconsideration, and the solid lines show the ideal performance (uni-form terrain, no wind, no Earth’s rotation, and no range foldover) forcomparison purposes.

Notice that when all these effects are present, conventional wave-form output has extremely poor performance, whereas waveform di-versity improves the performance to a considerable level. Similar con-clusions follow from Figures 6.53–6.54 that correspond to location Bin Texas.

In summary, when terrain effects, ICM due to wind, range foldovereffect, and Earth’s rotational effects are present in the clutter data;conventional waveforms are unable to detect targets. However, sig-nificant performance improvement can be realized using waveformdiversity.


−30 −20 −10 0

0

10 20 30

−50

−40

−30

−20

−10

V (m/s)

Ideal

SIN

R in

dB

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 SBRIn Section 4.7, we have modeled the effect of Earth’s rotation on SBRclutter data in terms of crab angle and crab magnitude that affectsthe clutter Doppler. The matched filter (MF) performance in terms ofSINR loss is analyzed in Section 6.3. From there, the performance issignificantly degraded when both range foldover and Earth’s rota-tion are jointly present. To understand the performance of the STAPalgorithms described in Chapter 5, simulated clutter data is used toestimate 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 of500 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 us-ing the traditional SMI algorithm for each of the four cases above areshown here. Recall that traditional SMI algorithm requires at leastMN = 32×16 = 512 samples in the present setup for satisfactory per-formance. As a result, 800 data samples are used for clutter covariancematrix estimation.

From Figure 6.55, the MF performance is the same as the ideal MFperformance when either Earth’s rotation or range foldover is absent(cases A, B, and C). However, the performance is significantly de-graded when both Earth’s rotation and range foldover are present to-gether (case D). This is in agreement with the results from Section 6.3.In addition, the estimated performance using SMI with 800 data


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20 40−60

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R in

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(a) Without Earth’s rotation, without range foldover

Ideal

SMIMF

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R in

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(b) Without Earth’s rotation, with range foldover

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R in

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(d) With Earth’s rotation, with range foldover

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−10

V (m/s)

SIN

R in

dB

(c) With Earth’s rotation, without range foldover

Ideal

SMIMF

Ideal

SMIMF

Ideal

SMIMF

FIGURE 6.55 SINR vs. velocity on clutter data without/with Earth’srotation and range foldover for ground range of 500 km. SMI with 800 datasamples is shown here.

samples is close to the MF performance for cases A, B, and C. How-ever, 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 rangefoldover return and result in improved performance. The performanceevaluation with waveform diversity in the data case is carried out laterin this section. When Earth’s rotation is present (case C), eventhoughthe MF performance is close to the ideal performance, the estimatedoutput is inferior to the MF output. This is due to the Doppler spreadcaused by the Earth’s rotation. The impact of Doppler spread andmethod to compensate it is later studied.

To analyze the clutter nulling performance of different STAP algo-rithms, the “clean” data corresponding to range 500 km for case A(without Earth’s rotation and range foldover) is used.


From Figure 6.55, SMI with 800 data samples gives good perfor-mance. However, the number of samples required is quite large andresults in higher computation requirement. Furthermore, large datasamples are not always available in a nonstationary environment. Inthis context other STAP algorithms introduced in Section 5.3.3 and 5.4can be used. Figure 6.56 shows the estimated performance with 200samples using SMIDL, EC, and HTP methods. From there, these STAPalgorithms perform close to the MF output using only 200 samples.

The number of samples required can be further reduced usingthe forward/backward, subarray, subpulse smoothing methods in-troduced in Section 5.5 and 5.6. Figure 6.57 shows the performancefor SMIDL, SMIDLSASPFB, EC, ECSASPFB with 40 samples. Fromthere, the traditional STAP methods (SMIDL, EC) give poor per-formance using a small number of samples. However, when theforward/backward, subarray, subpulse smoothing methods are used,the resulting performance is improved. Observe that the number of

−40 −20 0

0

20 40−60

−50

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−20

−10

V (m/s)

SIN

R in

dB

(a) SMIDL with 200 samples

SMIDLMF

−40 −20 0

0

20 40−60

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−10

V (m/s)

SIN

R in

dB

(c) HTP with 200 samples

HTPMF

−40 −20 0

0

20 40−60

−50

−40

−30

−20

−10

V (m/s)

SIN

R in

dB

(b) EC with 200 samples

ECMF

FIGURE 6.56 SINR vs. velocity on clutter data without Earth’s rotation andrange foldover for ground range of 500 km. Two hundred data samples areused here. Different STAP algorithm (a) SMIDL, (b) EC, and (c) HTP areshown here.


−40 −20 0

0

20 40−60

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V (m/s)

SIN

R in

dB

(d) ECSASPFB with 40 samples

ECSASPFBMF

−40 −20 0

0

20 40−60

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V (m/s)

SIN

R in

dB

(b) SMIDLSASPFB with 40 samples

SMIDLSASPFBMF

−40 −20 0

0

20 40−60

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−10

V (m/s)

SIN

R in

dB

(a) SMIDL with 40 samples

−40 −20 0

0

20 40−60

−50

−40

−30

−20

−10

V (m/s)

SIN

R in

dB

(c) EC with 40 samples

ECMF

SMIDLMF

FIGURE 6.57 SINR vs. velocity on clutter data without Earth’s rotation andrange 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 factoredtime-space (FTS), and Joint-Domain Localized (JDL) discussed inSection 5.9 and 5.10 also give excellent performance results.

Figure 6.58 shows the performance for the FTS method using oneDoppler bin and Figure 6.59 shows the performance of extended fac-tored time-space (EFA) method using two and three Doppler bins.The improvement over a single bin approach is significant even whenonly two bins are used. However, as the number of Doppler bins isincreased, the number of samples required for clutter covariance ma-trix estimation also increases. For the two bin approach, the size ofthe matrix is 2N × 2N and the number of data samples required is4N which turns out to be 128 for the current SBR setup. Interestingly,methods such as SMIDLSASPFB give similar performance even withsmaller number of samples (around 40 samples).


−40 −20 0

0

20 40−60

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−10

V (m/s)

FTS, K = 60, N2 = 1

SIN

R in

dB

(a) One bin with 60 samples (b) One bin with 200 samples

−40 −20 0

0

20 40−60

−50

−40

−30

−20

−10

V (m/s)

FTS, K = 200, N2 = 1

SIN

R in

dB

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 binsand various Doppler bins and Figure 6.61 shows the performance forJDL using five angle bins and various Doppler bins. As Figure 6.60shows, the performance is poor when number of angle bins used isthree irrespective of how many Doppler bins are used.

As Figure 6.61 (b)–(c) shows, JDL method using five angle bins andtwo Doppler gives acceptable performance with as few as 40 samples.

Computational Effect of SMI, EFA, and JDLIn FTS, dimension reduction is achieved by solving a different spatial-only adaptive beamforming problem for each Doppler bin. Thus onlyinversion of an N × N matrix (Ms N × Ms N for EFA) is required in

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V (m/s)

EFA, K = 200, N2 = 2

SIN

R in

dB

(a) Two bins with 200 samples (b) Three bins with 200 samples

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0

20 40−60

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−10

V (m/s)

EFA, K = 200, N2 = 3

SIN

R in

dB

FIGURE 6.59 Performance of EFA with two and three Doppler bin.Range = 500 km. Clutter without Earth’s rotation and range foldover.


−40 −20 0

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20 40−60

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V (m/s)

JDL, K = 40, N2 = 3, M2 = 3SI

NR

in d

B

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0

20 40−60

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−10

V (m/s)

JDL, K = 50, N2 = 3, M2 = 7

SIN

R in

dB

(b) Three angle bins and seven doppler bins with 50 samples

(b) Three angle bins and three doppler bins with 40 samples

FIGURE 6.60 Performance of JDL using three angle bins and variousDoppler bins. Range = 500 km. Clutter without Earth’s rotation and rangefoldover.

contrast to NM × NM in the case of full degree of freedom STAPproblem. This results in computational savings for one look angleand one Doppler of interest. From Section 5.3 and 5.4, there is only onematrix inversion needed for different look angles and Doppler in thecase of SMI, or SMIDL, etc. However, the effective covariance matricesfor FTS and EFA depends on the temporal steering vector b(ωd ) asseen from (5.269). Hence multiple matrix inversions are needed whendifferent Doppler values are searched.

−40−60

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0

−20 0 20 40

SIN

R in

dB

V (m/s)

(a) Five angle bins and one Doppler bin with 20 samples

JDL, K = 20, N2 = 5, M2 = 1

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0

−20 0 20 40

SIN

R in

dB

V (m/s)

(b) Five angle bins and two Doppler bins with 40 samples

JDL, K = 40, N2 = 5, M2 = 2

FIGURE 6.61 Performance of JDL with fiveangle bins and various Dopplerbins. Range = 500 km. Clutter without Earth’s rotation and range foldover.


As mentioned in Section 5.10, for JDL the size of the effective co-variance matrix is only Ms Ns × Ms Ns and it is also smaller than thefull size case of MN × MN. As Figure 6.61 (d) shows, in this case onlyfive angle bins and two Doppler bins are required for good perfor-mance for our SBR simulation setup. The resulting covariance matrixto be inverted is 10 × 10 instead of 512 × 512. Thus there is a hugecomputational saving for one look angle and Doppler of interest. Thecomputational requirement for JDL is even smaller compared to theFTS 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 angleand Doppler of interest and only one matrix inversion is requiredfor different angles and Dopplers; on the other hand, in the case ofJDL, the effective clutter covariance matrix depends both on the an-gle and Doppler of interest. The number of matrix inversion neededequals the product of number of angles and number of Doppler ofinterest.

Effect of Earth’s Rotation on Clutter Nulling PerformanceThe crab angle φc appearing in (4.71) generates additional undesirableeffects on the estimated clutter covariance matrix. As Figure 6.55 (c)shows, when Earth’s rotation is present and a large number of sam-ples is used to estimate the clutter covariance matrix, the resultingperformance is very poor. This is due to the Doppler spread causedby the Earth’s rotation. In addition, the performance degradation atrange 500 km is worse than at far ranges. To see this, from Figure 6.21in Section 6.3, when the data samples from range 500 km are used forestimating the clutter covariance matrix, the Doppler spread there islarger than the Doppler spread at range 1,200 km. Thus, the perfor-mance degradation at closer range is worse than the performance atfar range when Earth’s rotation is present in the data.

This Doppler spreading effect is illustrated in Figure 6.62 whichshows the estimated clutter spectrum s∗Rks at range equal to 500 kmin the Doppler–azimuth domain for θAZ = 90. Once again whenthe crab effect is absent (Figure 6.62 (a)), the clutter Doppler peaks atωd = 0 for θAZ = 90. Moreover, the clutter Doppler spread aroundωd = 0 is finite, irrespective of the number of samples used to estimatethe clutter covariance matrix. However, as Figure 6.62 (b) shows thesetwo conclusions are not true when crab effect is present in the clutterdata. From there, in addition to the shift in the Doppler peak, as thenumber of samples used in estimating R increases, the clutter Dopplerbegins to spread wider around its peak value (along the y-axis). Toillustrate this, the clutter spectrum using 4,000 samples is shown inFigure 6.62 (b). From there, the clutter covers the entire useful Doppler


(b) Clutter data with crab effect

100 Samples. Range: 499–501 km 4,000 Samples. Range: 450–550 km

(a) Clutter data without crab effect

Dopplerspreadcovers theentire region

cos qAZ

Dopplerspreads in afinite region

100 Samples. Range: 499–501 km 4,000 Samples. Range: 450–550 km

Dop

pler

1

−1

−0.5

−0.05 0 0.05

0

0.5

0

−50

−40−45

−35−30−25−20−15−10−5

cos qAZ

Dop

pler

1

−1

−0.5

−0.05 0 0.05

0

0.5

0

−50

−40−45

−35−30−25−20−15−10−5

cos qAZ

Dop

pler

1

−1

−0.5

−0.05 0 0.05

0

0.5

0

−40−35−30−25−20−15−10−5

cos qAZ

Dop

pler

1

−1

−0.5

−0.05 0 0.05

0

0.5

0

−50

−40−45

−35−30−25−20−15−10−5

FIGURE 6.62 Clutter power in the angle-Doppler domain using estimatedclutter covariance matrix as a function of number of samples forrange = 500 km (θAZ = 90).

region making target detection difficult for that look direction whena large number of samples are used to estimate the clutter covariancematrix.

This effect (spreading of clutter Doppler as the number of samplesincreases) is also validated in Figure 6.63 using the MDV analysis.The performance of the STAP algorithms using various number ofsamples are shown there. In Figure 6.63 (a), where the crab effectis absent, the performance of the STAP algorithms improves as thenumber of samples is increased. However, in Figure 6.63 (b) wherethe crab effect is present, increasing the number of samples degradesthe performance. For example, performance using 800 samples is in-ferior to that using 200 samples.

Recall that traditional STAP algorithms require at least MN =32 × 16 = 512 samples for reasonable performance. However, in thepresence of Earth’s rotation, the above analysis shows that using over200 samples will lead to inferior performance. Interestingly, the for-ward/backward, subarray, subpulse smoothing methods introducedin Section 5.5 and 5.6 only require a small set of data samples compared


MFSMIDL

V (m/s)

200 Samples

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R in

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MFSMIDL

V (m/s)

200 Samples

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SIN

R in

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MFSMIDL

V (m/s)

800 Samples

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SIN

R in

dB

(a) Clutter data without crab effect

MFSMIDL

V (m/s)

800 Samples

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SIN

R in

dB

(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 ad-dress the clutter data containing Earth’s rotation. Figure 6.64 showsthe SINR performance without and with Earth’s rotation for 500 kmrange. Performance using SMIDLSASPFB with 40 data samples areshown here. From there, the performance with crab effect is improvedcompared with Figure 6.63(b). However, the performance is still in-ferior compared to Figure 6.64(a) and additional processing must bedone to reduce the remaining performance degradation.

Doppler WarpingDoppler warping can be used on the secondary data set to reducethe Doppler spread caused by Earth’s rotation when range foldoveris absent. When Doppler warping is used, the Doppler frequency ofthe secondary data with the same cone angle as the primary data


V (m/s)

(a) Without crab effect

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SIN

R in

dB

MFSMIDLSASPFB

MFSMIDLSASPFB

V (m/s)

(b) With crab effect

−40−60

0

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−20 0 20 40

SIN

R in

dB

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 inFigure 6.65.

In this case, ωdo and ωdk are the Doppler frequencies of the primarydata and secondary data corresponding to the cone angle of interestrespectively. Define the differential Doppler between the primary dataand the secondary data to be

ωd = ωdo − ωdk . (6.155)

−4

−7−5 0 5

−6.5

−6

−5.5

−5

−4.5

Cone Angle ×10−3wdo~

wdk~

Dop

pler

(KH

z)

Secondary dataDopplerPrimary dataDopplerSecondary dataDoppler

FIGURE 6.65 Doppler warping.


The Doppler on the secondary data can be realigned by performingthe following operation:

xk = xk s(0, ωd ). (6.156)

Here, xk represents the secondary data and s(0, ωd ) is the space-time steering vector given by (6.10). The role of s(0, ωd ) is to re-focus the Doppler on the secondary data by applying the differentialDoppler to the data. Thus, the Doppler on the adjusted secondary dataxk is same as that in the primary data and the Doppler spread is mini-mized. The adjusted data xk is then used for clutter covariance matrixestimation. The Doppler spread can be eliminated when only Earth’srotation is present because there is only one Doppler frequency presentin the data for a fixed cone angle. However, when range foldoveris also present, there are multiple Doppler frequencies present inthe data for a fixed cone angle and all of them cannot be corrected.Figure 6.66 shows the SINR output without and with Doppler warp-ing. From there, when Doppler warping is applied to the secondarydata, the performance is restored close to the ideal case performance(see Figure 6.56(b) and Figure 6.57). Thus, Doppler warping can re-duce the Doppler spread resulting in improved performance.

STAP Performance with Waveform DiversityWhen both Earth’s rotation and range foldover are present at the sametime, the performance degrades significantly and it cannot be cor-rected 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 matchedfilter 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 thesecondary 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. Clutterdata is generated using orthogonal waveforms with Earth’s rota-tion and range foldover present. Figure 6.67(a) shows the perfor-mance of SMIDLSASPFB with Doppler warping and Figure 6.67(b)shows performance of HTPSASPFB with Doppler warping. FromFigure 6.67, the MF output is restored to the ideal case perfor-mance with orthogonal waveform diversity. In addition, with Dopplerwarping, various STAP algorithms give close to MF performanceusing 40 data samples. About 25-dB improvement can be achieved


MFSMIDL

V (m/s)

SMIDL with 200 samples

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0

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−20 0 20 40

SIN

R in

dB

(a) Without Doppler warping

(b) With Doppler warping

MFSMIDL

V (m/s)

SMIDL with 200 samples

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SIN

R in

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MFSMIDLSASPFB

V (m/s)

SMIDLSASPFB with 40 samples

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SIN

R in

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MFSMIDLSASPFB

V (m/s)

SMIDLSASPFB with 40 samples

−40−60

0

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SIN

R in

dB

FIGURE 6.66 SINR without and with Doppler warping. Clutter data withEarth’s rotation. Range = 500 km.

with waveform diversity and Doppler warping compared to that inFigure 6.55 (d).

Figure 6.68 shows the performance when hybrid-chirp waveformsare used for range 1,200 km. Performance of ECSASPFB with Dopplerwarping using 40 samples is shown here. From there, the MF perfor-mance is restored close to the ideal case output by waveform diversityand the estimated performance achieves more than 35 dB improve-ment compared to the performance in Figure 6.35 (b).

In summary, high resolution STAP algorithms together with for-ward/backward, subarray, and subpulse smoothing methods reducethe number of secondary data required for clutter covariance matrixestimation and improves SINR performance. When Earth’s rotationis present in the clutter data, Doppler spread occurs. In that case,Doppler warping can be used to realign the clutter Doppler frequency.


MFIdeal

SMIDLSASPFB

V (m/s)

(a) SMIDLSASPFB

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SIN

R in

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MFIdeal

HTPSASPFB

V (m/s)

(b) HTPSASPFB

−40−60

0

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−40

−50

−20 0 20 40

SIN

R in

dB

FIGURE 6.67 SINR performance with waveform diversity. Clutter data withEarth’s rotation and range foldover. Orthogonal waveforms are used togenerate 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 rangefoldover returns. High resolution STAP algorithms, Doppler warpingand waveform diversity together result in acceptable SINR perfor-mance for the current SBR setup.

Selection of a particular STAP algorithm may be irrelevant com-pared to addressing all these other issues—Earth’s rotational effect,

MFIdeal

ECSASPFB

V (m/s)−30

−60

0

−10

−20

−30

−40

−50

−20 −10 0 2010 30

SIN

R in

dB

FIGURE 6.68 SINR performance with waveform diversity. Clutter data withEarth’s rotation and range foldover. Hybrid-chirp waveforms are used togenerate clutter data. Range = 1, 200 km. ECSASPFB with Doppler warpingusing 40 samples is shown.


site specific clutter modeling, effect of wind, etc., and then taking thenecessary steps to correct/minimize these undesired effects prior toactual processing.

Appendix 6-A: Matrix Inversion IdentityConsider 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)

In (6A.2), let

(I + ss∗Q−1)−1 = I − A, (6A.3)

or

(I − A)(I + ss∗Q−1) = I (6A.4)

which gives

A + Ass∗Q−1 = ss∗Q−1. (6A.5)

Thus

As + Ass∗Q−1s = ss∗Q−1s (6A.6)

or

As(1 + s∗Q−1s) = ss∗Q−1s, (6A.7)

and hence we obtain the identity

As = ss∗Q−1

1 + s∗Q−1ss (6A.8)

from which A = ss∗Q−1

1+s∗Q−1sis clearly a solution in (6A.3). Substituting

this value of A in (6A.2)–(6A.3) we get the desired result

(Q + ss∗)−1 = Q−1 − Q−1ss∗Q−1

1 + s∗Q−1s. (6A.9)


Appendix 6-B: Output SINR DerivationIn the presence of Earth’s rotation, the clutter covariance matrix hasthe form (refer to (6.47))

R = so s∗o + gs1s∗

1 + Q1 = so s∗o + Q (6B.1)

so that using (6A.9) we get

R−1 = Q−1 − Q−1so s∗o Q−1

1 + s∗o Q−1so

(6B.2)

where

Q = gs1s∗1 + Q1. (6B.3)

Hence

Q−1 = Q−11 − gQ−1

1 s1s∗1Q−1

1

1 + gs∗1Q−1

1 s1. (6B.4)

Thus

s∗o Q−1so = s∗

o Q−11 so − g

∣∣s∗o Q−1

1 s1∣∣2

1 + gs∗1Q−1

1 s1. (6B.5)

Define

A = s∗o Q−1

1 so , B = s∗1Q−1

1 s1, C = s∗o Q−1

1 s1, (6B.6)

so that (6B.5) becomes

s∗o Q−1so = A− g|C |2

1 + gB= A+ g( AB − |C |2)

1 + gB. (6B.7)

From (6B.2), we also obtain

s∗o R−1so = s∗

o Q−1so −∣∣s∗

o Q−1so∣∣2

1 + s∗o Q−1so

= s∗o Q−1so

1 + s∗o Q−1so

= A+ g( AB − |C |2)1 + A+ g(B + AB − |C |2)

, (6B.8)

where we have made use of (6B.7). Similarly we also obtain

s∗1R−1s1 = s∗

1Q−1s1 −∣∣s∗

o Q−1s1∣∣2

1 + s∗o Q−1so

. (6B.9)


But using (6B.4) we get

s∗1Q−1s1 = s∗

1Q−11 s1 − g

∣∣s∗1Q−1

1 s1∣∣2

1 + gs∗1Q−1

1 s1= s∗

1Q−11 s1

1 + gs∗1Q−1

1 s1= B

1 + gB.

(6B.10)

Also

s∗o Q−1s1 = s∗

o Q−11 s1 − gs∗

o Q−11 s1s∗

1Q−11 s1

1 + gs∗1Q−1

1 s1= C − gC B

1 + gB= C

1 + gB.

(6B.11)

Substituting (6B.7), (6B.10), and (6B.11) into (6B.9) we get

s∗1R−1s1 = B

1 + gB− |C |2/(1 + gB)2

(1 + A+ g(B + AB − |C |2))/(1 + gB)

= B(1 + A+ gB + g AB − g|C |2) − |C |2(1 + gB)(1 + A+ g(B + AB − |C |2))

= (1 + A)(1 + gB)B − (1 + gB)|C |2(1 + gB)(1 + A+ g(B + AB − |C |2))

= B + ( AB − |C |2)1 + A+ g(B + AB − |C |2)

. (6B.12)

Appendix 6-C: Spectral FactorizationFor a stable system with transfer function

H(z) =∞∑

k=0

hk z−k , |z| > 1, (6C.1)

define its power spectrum S(ω) as

S(ω) =∣∣H(e jω)

∣∣2 =+∞∑

k=−∞rke− jkω ≥ 0 (6C.2)

so that its autocorrelations satisfy

rk =∞∑

i=0

hk+i h∗i = r∗

k , k = 0, 1, 2, . . . (6C.3)


Notice that from (6C.3),∑∞

i=0 |hk |2 = ro < ∞ implies hi → 0 asi → ∞. Every stable system transfer function H(z) can be expressedin 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) all-pass function(|A(e jω)| = 1).

For example, the rational stable system

H(z) = z + 23z + 1

(6C.5)

corresponds to the minimum phase factor2

Ho (z) = 2z + 13z + 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 ofautocorrelations. Thus from (6C.2), given the power spectrum S(ω),the corresponding stable system transfer function H(z) is unique onlyupto an all-pass factor A(z) since


∣∣2 =∣∣Ho (e jω) A(e jω)

∣∣2 =∣∣Ho (e jω)

∣∣2 (6C.7)

and hence the system and the corresponding impulse response hk∞k=0in (6C.1) represent a minimum phase system uniquely.

Equation (6C.3) represents the relation between the impulse re-sponse of a system and its autocorrelations. Given the impulse re-sponse, (6C.3) can be used to compute its autocorrelations. However,to obtain the impulse response sequence from the autocorrelations in-volves a spectral factorization. As (6C.7) shows, spectral factorizationdoes not lead to a unique factor. However, the minimum phase factorassociated with the spectral factorization is unique, and all other fac-tors can be obtained by cascading it with a regular all-pass function.Interestingly, the unique minimum phase factor can be obtained fromthe autocorrelation sequence using an iterative algorithm.

2Poles and zeros of a minimum phase system with series representation as in(6C.1) are inside the unit circle.


To make further progress, define the two infinite dimensionalmatrices

T =

ro r1 · · · rn · · ·r∗

1 ro · · · rn−1 · · ·...

.... . .

... · · ·r∗

n r∗n−1 · · · ro · · ·

......

......

. . .

(6C.8)

and

H =

ho h1 h2 · · · hn · · ·0 ho h1 · · · hn−1 · · ·0 0 ho · · · hn−2 · · ·...

......

. . .... · · ·

0 0 0 · · · ho · · ·...

......

......

. . .

. (6C.9)

Notice that T is a Hermitian Toeplitz matrix whose first row is givenby (ro , r1, r2, . . . rn, . . .) and H is an upper triangular Toeplitz matrixwhose first row equals (ho , h1, h2, . . . hn, . . .). In terms of T and H, theinfinite 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 matricesof T and H respectively. Then

Tn =

ro r1 · · · rn

r∗1 ro · · · rn−1

......

. . ....

r∗n r∗

n−1 · · · ro

=

ro r1 · · · rn

r∗1... Tn−1

r∗n

=

rn

Tn−1...

r1

r∗n · · · r∗

1 ro

> 0

(6C.11)

and

Hn =

ho h1 h2 · · · hn

0 ho h1 · · · hn−1

0 0 ho · · · hn−2

......

.... . .

...

0 0 0 · · · ho

=(

ho h1 · · · hn

0 Hn−1

)=(

ho b

0 Hn−1

).

(6C.12)


Using (6C.12), we can rewrite H in (6C.9) as

H =(

Hn Xn

0 Yn

)(6C.13)

where the semi-infinite matrix

Xn =

hn+1 hn+2 hn+3 · · ·hn hn+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 = HnH∗n + Xn X∗

n. (6C.15)

Equation (6C.15) represents an exact identity where the key obser-vation 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,hi → 0 as i → ∞. The Bauer-type factorization exploits this fact andshows that the iterative matrix factorization procedure above leads toa minimum phase system impulse response sequence [19]. This resultis based on Jensen’s inequality [23], which in the general matrix casestates that for an m × m transfer function

F(z) =∞∑

k=0

Fk z−k , det Fo = 0, (6C.16)

we have

12π

π∫−π

ln∣∣det F(e jθ )

∣∣ dθ ≥ ln |det Fo | > −∞, (6C.17)

with equality in the first part of (6C.17) impling that F(z) in (6C.16) isminimum phase, i.e., det F(z) is free of zeros in |z| > 1.

In fact from (6C.15), as n → ∞, we obtain

Tn → HnH∗n, (6C.18)


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 · · · L (n)

2n

......

.... . .

...

0 0 0 · · · L (n)nn

(6C.19)

represent the unique upper triangular factor with positive diagonalentries of the positive definite Toeplitz matrix Tn in (6C.11). Thus

Tn = LnL∗n. (6C.20)

Then (6C.15)–(6C.18) imply that

limn→∞ L (n)

i j = h j−i (6C.21)

and in particular the entries of the first row of Ln gives

limn→∞ L(n)

0 j = h j , j = 0, 1, 2, . . . (6C.22)

i.e., as n → ∞, the first row of Ln gives the desired impulse responsesequence. Interestingly, the upper triangular factor in (6C.19) can becomputed iteratively. Let

Ln =(

c b0 A

). (6C.23)

Then

Tn =

ro r1 · · · rnr∗

1... Tn−1

r∗n

= LnL∗

n =(

d bA∗

Ab∗ AA∗

)(6C.24)

where

d = c2 + b∗b. (6C.25)

But from (6C.24)

AA∗ = Tn−1 = Ln−1L∗n−1 (6C.26)


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

c = +√

ro − b∗b. (6C.29)

Using (6C.27)–(6C.29) in (6C.23), we obtain the iterative algorithm

Ln =(

c b0 Ln−1

), n ≥ 1. (6C.30)

Thus at every stage, only the first row of the upper triangular matrixneeds 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 desiredminimum phase impulse response.

Finally to establish that the sequence h j ∞j=0 in (6C.21) leads to aminimum phase factor, from Jensen’s inequality in (6C.17), it is nec-essary to show that

12π

π∫−π

ln S(ω)dω = ln |ho |2 . (6C.31)

This key identity is established in [19].

Appendix 6-D: Rational SystemRepresentation

Let

H(z) = bo + b1z−1 + · · · + bmz−m

1 + a1z−1 + · · · + anz−n= B(z)

A(z)(6D.1)

represent an ARMA(n, m) rational system, whose poles z1, z2, . . . , znare given by the zeros of its denominator polynomial. Thus

A(z) = ao + a1z−1 + · · · + anz−n =n∏

k=1

(1 − zk z−1), zk = e−(αk+ jωk ).

(6D.2)


Partial fraction expansion of (6D.1) gives

H(z) =n∑

k=1

ck

1 − zk z−1 =n∑

k=1

ck

∞∑i=0

(zk z−1)i

=∞∑

i=0

(n∑

k=1

ck zik

)z−i =

∞∑i=0

hi z−i (6D.3)

where

hi=

n∑k=1

ck zik , i ≥ 0 (6D.4)

represents the impulse response of the system. Equation (6D.4) repre-sents a sum of complex sinusoids since with

zk = e−(αk+ jωk ) , k = 1, 2, . . . (6D.5)

we obtain

hi =n∑

k=1

cke−(αk+ jωk )i , k = 0, 1, 2, . . . (6D.6)

If (6D.1) represents a system with real coefficients, then the polesand residues in (6D.4) occur in complex conjugate pairs so that

hi = 2n/2∑k=1

|ck | e−αk i cos(ωki + φi ). (6D.7)

For a stable system representation as in (6D.1) and (6D.2), the polesare 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 fre-quencies ω1, ω2, . . . .

It is easy to show that in the rational case, the hi s in (6D.4) arelinearly dependent beyond a certain stage. This follows by equating(6D.1) and the right side of (6D.3). Thus

(ao + a1z−1 + · · · + anz−n)( ∞∑

i=0

hi z−i

)= bo + b1z−1 + · · · + bmz−m.

(6D.8)Equating coefficients of like terms on both sides, we get

aoho = bo ,

aoh1 + a1ho = b1,

... (6D.9)

aohm + a1hm−1 + · · · + amho = bm,


and, in general,

aohk + a1hk−1 + · · · + anhk−n = 0, k > m. (6D.10)

Notice that (6D.10) represents an infinite set of equations in theunknowns ao , a1, . . . , an. With ao = 1 (see (6D.1)), Equation (6D.10)gives

hk = − (a1hk−1 + · · · + anhk−n) , k > m (6D.11)

showing the linear dependency of the response coefficients beyond acertain stage. From (6D.10) for k = n, n + 1, . . . , 2n − 1 we get

ho h1 · · · hn−1

h1 h2 · · · hn

...... · · · ...

hn−1 hn · · · h2n−2

an

an−1

...

a1

= −

hn

hn+1

...

h2n−1

. (6D.12)

Equation (6D.12) uses the first 2n impulse response coefficientshi 2n

i=0 of the system, and it can be used to determine the denominatorcoefficients ai n

i=0 and hence the poles zi ni=1 of the system. Know-

ing the ai , the numerator coefficients bi in (6D.1) can be computedusing (6D.9).

More generally, define the Hankel matrix

Hk =

ho h1 h2 · · · hk

h1 h2 h3 · · · hk+1

......

... · · · ...

hk−1 hk hk+1 · · · h2k−1

hk hk+1 hk+2 · · · h2k

=

hkHk−1 hk+1

...

hk hk+1 · · · h2k

(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 linearlydependent 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)


an old result due to Kronecker (1881). Equation (6D.15) representsthe classic necessary and sufficient condition for an infinite sequencehk∞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

Hn

an

an−1

...

a1

ao

= 0. (6D.16)

Notice that Hn is of size (n+1)×(n+1) and has rank n (see (6D.14)).Hence

a = [an, an−1, . . . , a1, ao ]T (6D.17)

formed from the denominator coefficients represents the uniqueeigenvector associated with the zero eigenvalues of Hn.

The power spectrum associated with any system transfer functionH(z) is given by


∣∣2 =∞∑

k=−∞rke− jkω ≥ 0, (6D.18)

where rk = r∗−k , k ≥ 0 represents its autocorrelation sequence. Substi-

tuting H(z) =∑∞i=0 hi z−i into (6D.18), we obtain

S(ω) =∞∑

m=0

hme− jmω

∞∑i=0

h∗i e jiω =

∞∑m=0

∞∑i=0

hmh∗i e− j(m−i)ω

=∞∑

k=−∞

( ∞∑i=0

hi+kh∗i

)e− jkω =

∞∑k=−∞

rke− jkω (6D.19)

which gives (see (6C.3))

rk =∞∑

i=0

hk+i h∗i , k = 0 → ∞. (6D.20)

For a rational system as in (6D.1), its impulse response is givenby (6D.4). To show that, a similar relation holds among its


autocorrelations, we can use (6D.4) in (6D.20). This gives

rk =∞∑

i=0

(n∑

m=1

cmzi+km

)(n∑

l=1

c∗l (z∗

l )i

)=

n∑m=1

n∑l=1

cmc∗l

( ∞∑i=0

(zmz∗

l

)izk

m

)

=n∑

m=1

cmzkm

n∑l=1

c∗l

1 − zmz∗l︸︷︷︸

dm

=n∑

m=1

cmdm︸︷︷︸qm

zkm =

n∑m=1

qmzkm

=n∑

m=1

qme−(αm+ jωm)k , k = 0 → ∞ (6D.21)

where

qm = cmdm = cm

n∑l=1

c∗l

1 − zmz∗l. (6D.22)

Thus the autocorrelations of a rational system also has a represen-tation similar to (6D.4) as the sum of damped exponentials.

On comparing (6D.6)–(6D.15) with (6D.21), it follows that Hankelmatrices generated using the autocorrelations also satisfy similar rankconditions [18].

References[1] Knowledge Aided Sensor Signal Processing and Expert Reasoning

(KASSPER), Third Annual Workshop, Clearwater, Florida, February 22–24,2005.


[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 onSpace-Based Performance”, Proc. IEEE Transactions on Aerospace and ElectronicSystems, 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 Alterna-

tives,” 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.


[12] J.B. Billingsley, Low-Angle Radar Land Clutter, William Andrew PublishingNorwich, 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 Trans-actions 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 andInternal Clutter Motion on Space-Based Radar Performance”, 2006 IEEE RadarConference, Verona, NY, April 24–27, 2006.

[21] P. Mountcastle, “New Implementation of the Billingsley Model for GMTI DataCube Generation,” IEEE Radar Conference, Philadelphia, PA, 2004.

[22] G.K. Borsari, “Mitigating Effects on STAP Processing Caused by an InclinedArray,” Proceedings of IEEE National Radar Conference, Dallas, TX, May 1998.

[23] R.P. Boas, Entire Functions, Academic Press, New York, NY, 1954.

http://terra.nasa.gov

C H A P T E R 7Performance Analysis

Using Cramer-RaoBounds

A theorem independently developed by Rao (1945) and Cramer (1946)states that the variance of any unbiased estimator for a nonrandomparameter can be always lower bounded under some very generalconditions. Known as the Cramer-Rao bound it is related to the Fisherinformation content about the unknown parameter contained in thejoint density function of the data set (observations). When the parame-ter set contains more than one unknown, the bounds become coupledand can be expressed in matrix form.

7.1 Cramer-Rao Bounds forMultiparameter CaseLet θ = [θ1, θ2, . . . , θm]T represents the unknown set of nonrandomparameters contained in the data set x under the probability densityfunction (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

ET(x) = θ (7.2)

and the covariance matrix for T(x) is given by

covT(x) = E(T(x) − θ )(T(x) − θ )∗. (7.3)



The covT(x) is of size m × m and it represents a positive-definitematrix. The Cramer-Rao bound in this case has the form1

covT(x) ≥ J−1(θ ) (7.4)

where J(θ ) represents the m × m Fisher information matrix associatedwith the parameter set θ under the p.d.f. f (x, θ ). The entries of theFisher information matrix are given by

Ji j= E

∂ log f (x; θ )

∂θi

∂ log f (x; θ )∂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

∫f (x, θ) dx =

∫∂ f (x, θ )

∂θidx = 0, (7.6)

and

∂

∂θi

∫T(x) f (x, θ ) dx =

∫T(x)

∂ f (x, θ )∂θi

dx. (7.7)

Here the integrals represent n-fold integration.

Proof. Using the unbiased property, for any parameter θi we have

ETi (x) − θi =+∞∫

−∞(Ti (x) − θi ) f (x; θ ) dx = 0. (7.8)

Differentiate the later part of (7.8) with respect to θi on both sidesto obtain

+∞∫−∞

(Ti (x) − θi )∂ f (x; θ )

∂θidx −

+∞∫−∞

f (x; θ ) dx = 0, (7.9)

where we have made use of the regularity conditions given in (7.6)and (7.7). Thus

+∞∫−∞

(Ti (x) − θi )∂ f (x; θ )

∂θidx = 1. (7.10)

1In (7.4), the notation A ≥ B is used to indicate that the matrix A − B is anonnegative-definite matrix. Strict inequality would imply positive-definiteness.

C h a p t e r 7 : P e r f o r m a n c e A n a l y s i s U s i n g C r a m e r - R a o B o u n d s 311

But

∂ log f (x; θ )∂θi

= 1f (x; θ )

∂ f (x; θ )∂θi

, (7.11)

so that2

∂ f (x; θ )∂θi

= f (x; θ )∂ log f (x; θ )

∂θi(7.12)

and (7.10) becomes

+∞∫−∞

(Ti (x) − θi ) f (x; θ )∂ log f (x; θ )

∂θidx = 1, 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

(Ti (x) − θi )∂ log f (x; θ )

∂θ j

=

+∞∫−∞

(Ti (x) − θi )∂ f (x; θ )

∂θ jdx

= ∂

∂θ j

+∞∫−∞

Ti (x) f (x; θ ) dx − θi∂

∂θ j

+∞∫−∞

f (x; θ) dx

= ∂θi

∂θ j= 0, i = j = 1 → m. (7.15)

2From (7.11), we get the identity E

∂ log f (x;θ )∂θi

=∫ +∞

−∞∂ f (x;θ )

∂θidx = ∂

∂θi

∫ +∞−∞

f (x; θ ) dx = 0.


To exploit the key identities obtained in (7.14) and (7.15), define the2m × 1 vector

z =

T1(x) − θ1

T2(x) − θ2...

Tm(x) − θm

∂ log f (x;θ )∂θ1

∂ log f (x;θ )∂θ2...

∂ log f (x;θ )∂θm

= =

y1

y2

. (7.16)

Then using (7.2) and (7.11), we get

Ez = 0 (7.17)

and hence

covz = Ezz∗ = E(

y1y∗1 y1y∗

2

y2y∗1 y2y∗

2

). (7.18)

But

Ey1y∗1 = covT(x) (7.19)

and from (7.14) and (7.15)

Ey1y∗2 = Im (7.20)

the identity matrix. Also from (7.5)

Ey2y∗2 = J =

J11 J12 · · · J1m

J21 J22 · · · J2m

......

. . ....

Jm1 Jm2 · · · Jmm

(7.21)

represents the Fisher information matrix with entries Ji j = J j i as in(7.5). Thus

covz =(

cov(T(x)) I

I J

)≥ 0. (7.22)


To make further progress, define the matrix

S =(

I 0

−D−1C I

)(7.23)

and obtain the matrix identity

S∗(

A B

C D

)S =

(A − BD−1C B − C∗D∗−1D

0 D

). (7.24)

With A = covT(x), B = C = I, and D = J in (7.24), it represents apositive-defined matrix that reduces to(

A − BD−1C B − C∗D∗−1D

0 D

)=(

covT(x) − J−1 0

0 J

)≥ 0

(7.25)

which gives

covT(x) − J−1 ≥ 0 (7.26)

or

covT(x) ≥ J−1(θ ), (7.27)

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∫

f (x, θ )∂ log f (x, θ )

∂θidx = ∂

∂θi

∫f (x, θ) dx = 0. (7.28)

Differentiate this expression with respect to θ j to get∫∂ f (x, θ )

∂θ j

∂ log f (x, θ )∂θi

dx +∫

f (x, θ)∂2 log f (x, θ )

∂θi ∂θ jdx = 0, (7.29)

or using (7.12) again, we get the desired identity

Ji j =∫

f (x, θ)∂ log f (x, θ )

∂θi

∂ log f (x, θ )∂θ j

dx

= −∫

f (x, θ)∂2 log f (x, θ )

∂θi ∂θ jdx, (7.30)

the desired result. In particular, from (7.27)

varTi (x) ≥ J i i = (J−1)i i , i = 1 → m. (7.31)

Thus in the multiparameter case, the Cramer-Rao bound is givenby the diagonal entries of the inverse of the Fisher information matrix.


When the unknown set consists of a single parameter θi , the corre-sponding CR bound in (7.27), (7.31) becomes

varTi (x) ≥ 1Ji i

= 1

E

(∂ log f (x; θi )

∂θi

)2 = −1

E

∂2 log f (x; θi )∂θ2

i

.

(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. Inter-estingly (7.27)–(7.32) can be used to evaluate the degradation factorfor the CR bound caused by the presence of additional unknown pa-rameters in the scene. For example, when θ1 is the only unknown, theCR bound is given by 1/J11, whereas it is given by J11 in presence ofother parameters θ2, · · · θ3, · · · θm. With

J =(

J11 b∗

b G

)(7.33)

in (7.21), from (7.24) we have

J11 = 1

J11 − b∗G−1b= 1

J11

1

1 − b∗G−1b/J11. (7.34)

Thus 1/(1 − b∗G−1b/J11) > 1 represents the effect of the remainingunknown parameters on the bound for θ1. As a result, with one addi-tional parameter, we obtain the increment on the bound for θ1 to be

J11 − 1J11

= 1J11

J212

J11J22 − J212

≥ 0. (7.35)

Referring back to (7.21) and (7.33), if the Fisher information matrixturns out to be diagonal, then the estimators are independent, andtheir bounds are not degraded by the presence of other parameters.Conversely, the freedom present in setting up an experiment can beused sometimes to realize diagonal Fisher information matrices.

To illustrate how the performance degrades when an additionalunknown parameter is introduced, next we consider some examples.

Example 7.1 Let x1, x2, · · · , xn represent i.i.d. gamma distributed random variableswith parameters α and β. Then their joint probability density function is givenby [3]

f (x; α, β) = βnα

n(α)

(n∏

i=1

xα−1i

)e−β∑n

i=1xi

. (7.36)

Determine the Cramer-Rao bounds for α and β when either or both parametersare unknown.


Solution From (7.36), we have

L= log f (x; α, β) = nα log β − n log (α) + (α − 1)

n∑i=1

log xi − β

n∑i=1

xi . (7.37)

The parameters α and β may be unknown one at a time or together. In eithercase, we have

∂L∂α

= n log β − n′(α)(α)

+n∑

i=1

log xi , (7.38)

and

∂L∂β

= nα

β−

n∑i=1

xi . (7.39)

This gives

∂2 L∂α2 = −n

(

′ ′(α)

(α)−(

′(α)(α)

)2)

= nϕ(α), (7.40)

∂2 L∂α ∂β

= nβ

, (7.41)

∂2 L∂β2 = − nα

β2 , (7.42)

and hence from (7.5)

J11 = −E

∂2 L∂α2

= nϕ(α) , (7.43)

J22 = −E

∂2 L∂β2

= nα

β2 , (7.44)

J12 = −E

∂2 L

∂α ∂β

= − n

β2 . (7.45)

Hence if either α or β is the only unknown, then their respective CR bounds aregiven by

σ 2CR(α) = 1

J11= 1

nϕ(α)(7.46)

and

σ 2CR(β) = 1

J22= β2

nα. (7.47)

However, if both α and β are unknown simultaneously, we have

J =(

J11 J12

J21 J22

)= n

(ϕ(α) − 1

β

− 1β

α

β2

)(7.48)

so that

J−1 = 1n(αϕ(α) − 1)

(α β

β β2ϕ(α)

). (7.49)


This gives the corresponding bounds to be

σ 2CR(α) = J11 = α

n(αϕ(α) − 1)>

1nϕ(α)

(7.50)

and

σ 2CR(β) = J22 = β2ϕ(α)

n(αϕ(α) − 1)>

β2

nα. (7.51)

Clearly when both parameters are unknown, the bounds are inferior com-pared to the case where only either parameter is unknown.

As the next example shows, the presence of an extra parameter doesnot always lead to inferior bounds, however, at times it can lead toinferior estimators.

Example 7.2 Let x1, x2, · · · , xn be i.i.d. Gaussian data samples with common meanµ and common variance σ 2. Find the Cramer-Rao bounds for µ and σ 2 wheneither 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(2πσ 2)n/2 e

−∑n

i=1(xi −µ)2

2σ2 (7.52)

which gives

L = log f (x; µ, σ 2) = − n2

log(2πσ 2) −n∑

i=1

(xi − µ)2

2σ 2 . (7.53)

Hence

∂L∂µ

=n∑

i=1

(xi − µ)σ 2 (7.54)

and

∂L∂σ 2 = − n

2σ 2 +n∑

i=1

(xi − µ)2

2(σ 2)2 . (7.55)

As a result, we obtain

∂2 L∂µ2 =

n∑i=1

−1σ 2 = − n

σ 2 , (7.56)

∂2 L∂µ ∂σ 2 = −

n∑i=1

(xi − µ)(σ 2)2 , (7.57)

and

∂2 L∂(σ 2)2 = n

2(σ 2)2 −n∑

i=1

(xi − µ)2

(σ 2)3 . (7.58)


Hence

J11 = −E

∂2 L∂µ2

= n

σ 2 , (7.59)

J12 = −E

∂2 L

∂µ ∂σ 2

=

n∑i=1

Exi − µσ 4 = 0, (7.60)

J22 = −E

∂2 L

∂(σ 2)2

= − n

2σ 4 +n∑

i=1

E(xi − µ)2σ 6

= − n2σ 4 + nσ 2

σ 6 = n2σ 4 . (7.61)

Hence

J =(

J11 J12

J21 J22

)=( n

σ 2 0

0n

2σ 4

). (7.62)

The diagonal nature of the Fisher information matrix in (7.62) indicates thatin the case of i.i.d. Gaussian data samples, the bounds on the variance of anyunbiased estimator for parameters µ and σ 2 are the same whether only oneparameter is unknown or both parameters are unknown simultaneously. In par-ticular, these bounds for µ and σ 2 are given by

σ 2CR(µ) = 1

J11= σ 2

n(7.63)

and

σ 2CR(σ 2) = 1

J22= 2σ 4

n. (7.64)

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 thestatus of the other parameter; however, the actual unbiased estimatorsin these cases may not exhibit similar behavior. We shall illustrate thisfor the i.i.d. Gaussian data case under discussion here.

In that case, for the unknown parameter µ, the sample mean

µ = 1n

n∑i=1

xi = x (7.65)

is an efficient estimator since

Eµ = Ex = µ (7.66)

and

varµ = E(x − µ)2 = 1n2

n∑i=1

E(xi − µ)2 = nσ 2

n2 = σ 2

n(7.67)


that agrees with (7.63). Thus the sample mean estimator is always ef-ficient whether σ 2 is known or unknown.

However, the situation with the parameter σ 2 is quite different. Ifµ is known, then

σ 2 = 1n

n∑i=1

(xi − µ)2 (7.68)

is an efficient estimator for σ 2 since

Eσ 2 = 1n

n∑i=1

E(xi − µ)2 = nσ 2

n= σ 2 (7.69)

and3

varσ 2 = 2σ 4

n(7.70)

that agrees with the bound in (7.64). Notice that in this case the boundis proportional to the square of the unknown parameter of interest.However if µ is unknown, the estimator in (7.68) is not useful. In thatcase, it can be shown that [1]

θ2 = 1n

n∑i=1

(xi − x)2 (7.71)

is the best unbiased estimator for σ 2 in terms of attaining the low-est possible variance. Moreover, after some algebra, its variance isgiven by [1], [2], [3]

varθ2 = 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 unknownparameter can depend on whether the other parameter is known orunknown.

Next we address the general multichannel Gaussian case.

3We have yi = xi − µ ∼ N(0, σ 2). Hence Ey2i = σ 2, Ey4

i = σ 4. In term of yi ,

we have σ 2 = 1n

∑ni=1 y2

i and hence varσ 2 = Ey4i −Ey2

i n = 3σ 4−σ 4

n = 2σ 4

n .


Gaussian DataWhen n independent and identically distributed observations

x = [x1, x2, . . . , xn] (7.73)

are jointly Gaussian with covariance matrix R, the Fisher informationmatrix J in (7.4) and (7.5) can be further simplified. In that case [4]

f (x; θ) = 1|πR|n e−

∑n

i=1x∗

i R−1xi = 1|πR|n e−n tr (R−1S) (7.74)

where

S = 1n

n∑i=1

xi x∗i (7.75)

represents an unbiased estimate for R. Thus

log f (x; θ ) = −n(log |πR| − trR−1S) (7.76)

so that

∂ log f (x; θ )∂θi

= −n

|R−1|∂|R|∂θi

+ tr

∂R−1

∂θiS

= −n

tr

R−1 ∂R∂θi

− tr

R−1 ∂R∂θi

R−1S

= −n tr

R−1 ∂R∂θi

(I − R−1S)

(7.77)

since

|R−1|∂|R|∂θi

= tr

R−1 ∂R∂θi

=∑

k

1λk

∂λk

∂θi, (7.78)

and

∂R−1

∂θi= −R−1 ∂R

∂θiR−1 (7.79)

where λi mi=1 represent the eigenvalues of R. Differentiating (7.77)

again with respect to θ j we get

∂2 log f (x; θ )∂θi ∂θ j

= −n tr−R−1 ∂R

∂θ jR−1 ∂R

∂θi(I − R−1S)

+R−1 ∂2R∂θi ∂θ j

(I−R−1S)+(

R−1 ∂R∂θi

)(R−1 ∂R

∂θ jR−1S

).

(7.80)


Taking expectation on both sides we obtain

Ji j = −E

∂2 log f (x; θ )∂θi ∂θ j

= n tr

R−1 ∂R

∂θiR−1 ∂R

∂θ j

(7.81)

since ES = R. Equation (7.81) is also known as the Slepian-Bang’sformula [5]. We can use (7.81) to determine the Fisher informationmatrix by substituting the appropriate R in each case.

7.2 Cramer-Rao Bounds for Target Dopplerand Power in Airborne and SBR CasesConsider a target located at (θELt , θAZt ) with power level Pt andDoppler parameter ωdt . In general, all these four target parametersare unknown. In practice, however, the data from a specific location isanalyzed for the presence or absence of a target. Hence we will assumethe target location (θELt , θAZt ) and its associated cone angle

ct = sin θELt cos θAZt (7.82)

are known, and the power level Pt and the Doppler ωdt are unknownparameters. The simpler situation is to assume that only one parame-ter is unknown at a time, and in the general case both parameters areunknown simultaneously.

Both in the airborne and the space based radar (SBR) case, the tar-get is buried in clutter and noise; although the clutter model dependsupon the type of platform being used. For example in the SBR case, aswe have seen in Chapter 6, the clutter covariance matrix Rc takes dif-ferent forms depending on whether or not Earth’s rotation and rangefoldover are modeled into the formulation. Naturally with increasingclutter complexity, one expects poor performance and a quantitativemeasure of what can be expected in terms of target parameter estima-tion accuracy is of crucial importance both in the airborne as well asthe SBR case.

The theory developed here is platform independent and can be usedto evaluate the performance of either platforms by substituting theclutter covariance matrix and steering vector models that are appro-priate for the case under consideration.

In all these situations, a baseline can be established by consideringthe total interference to be consisting only of independent and identi-cally distributed sensor noise with common input variance σ 2

n . Notethat the total interference covariance matrix in that case becomes

Rc = σ 2n I. (7.83)


The performance in all other situations can be compared with thisreference case to determine the degradation due to each factor presentin that specific situation. In that context, the array configuration aswell as the array reference points—both spatial and temporal—caninfluence the estimation accuracy and the bounds in question may beused to design optimum array configurations as well as new protocolsfor determining the best data reference points.

The target at location (θELt , θAZt ) that projects an unknown Dopplerωdt at the array can be represented using the steering vector

st = b(ωdt ) ⊗ a (ct) (7.84)

where ct is as in (7.82) and a and b represent the spatial and temporalportions of the steering vector (see (6.6)–(6.8)). In that case the covari-ance matrix associated with the received data vector x (containingtarget) is given by

R = Exx∗ = Ptsts∗t + Rc . (7.85)

Here Pt represents the target power and Rc represents the totalinterference (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 =∑

i

Pi G(θi )s(θi )s∗(θi ) + σ 2n I (7.86)

and in the SBR case, from (6.20)

Rc =No∑j=1

Na∑m=1

Pm, j G(θm, j )sm, j s∗m, j + σ 2

n I. (7.87)

Here the inner summation is over the Na range foldovers atR1, R2, · · · , RNa , and the outer summation is over No azimuth anglesof interest including sidelobes. Further, Pm, j and G(θm, j ) correspondto the clutter power and array gain, respectively, and

sm, j = s(cm, j , ωdm, j ) (7.88)

represents the steering vector for the (m, j)th clutter patch. In (7.88),cm, j represents the cone angle for the (m, j)th patch given by

cm, j = sin θELm cos θAZj (7.89)

and

ωdm, j =

βocm, j , no Earth’s rotation,

ωdm, j =βoρc sin θELm cos(θAZj ± φc), with Earth’s rotation,

(7.90)

depending on whether Earth’s rotation is absent or present in (7.87).


Let

θ1 = ωdt and θ2 = Pt (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−1/2c RR−1/2

c (7.93)

where R−1/2c represents any square root of the positive-definite matrix

Rc . Then using (7.85), takes the convenient form

= I + Ptα α∗ (7.94)

where

α = R−1/2c st. (7.95)

From (7.93) we have

R = R1/2c R1/2

c (7.96)


Ji j = n tr(

R−1/2c −1R−1/2

c R1/2c

∂

∂θiR1/2

c R−1/2c −1R−1/2

c R1/2c

∂

∂θ jR1/2

c

)

= n tr(

−1 ∂

∂θi−1 ∂

∂θ j

). (7.97)

We shall use the later form in (7.97) together with (7.94) to computeJi j . From (7.94)

−1 = I − Ptα α∗

1 + Ptα∗α= I − Ptα α∗

1 + γ(7.98)

where we define

γ = Ptα∗α = Pts∗

t R−1c st = Pt

σ 2 (7.99)

and

σ 2 = 1

s∗t R−1

c st(7.100)

represents the “effective interference pulse noise power” at the targetsite. Notice that γ represents the effective SINR at the target site since


in the noise only case we have Rc = σ 2n I and from (7.100)

s∗t R−1

c st = MN/σ 2n (7.101)

and4 γ = MN Pt/σ2n .

From (7.94) we also obtain

∂

∂ωdt

= Pt

(∂α

∂ωdt

α∗ + α∂α∗

∂ωdt

)= Pt (7.102)

where

=(

∂α

∂ωdt

α∗ + α∂α∗

∂ωdt

)= R−1/2

c

(∂st

∂ωdt

s∗t + st

∂s∗t

∂ωdt

)R−1/2

c . (7.103)

With st as in (7.84) we get

∂st

∂ωdt

= ∂b(ωdt )∂ωdt

⊗ a (ct) = − jπ

0e− jπωdt

2e− j2πωdt

...

(M − 1)e− j (M−1)πωdt

⊗ a (ct).

(7.104)

Using (7.98) and (7.102) we have

−1 ∂

∂ωdt

=(

I − Ptα α∗

1 + γ

)Pt = Pt

( − Pt

α α∗1 + γ

)(7.105)

and hence from (7.97)

J11 = Jωdt ,ωdt= n tr

(−1 ∂

∂ωdt

)2

= nP2t tr

2 − Ptα α∗2

1 + γ− Pt

α α∗1 + γ

+ P2t

α α∗α α∗(1 + γ )2

= nP2t

(tr2 − 2Pt

α∗2α

1 + γ+ P2

t(α∗α)2

(1 + γ )2

). (7.106)

Similarly

∂

∂ Pt= α α∗ (7.107)

4In the white noise case, let σ 2n represents the noise at each sensor element input.

Then Pt/σ2n represents the input SNR at each sensor input. Thus γ = Pt/σ

2 =MNPt/σ

2n = 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.


so that

−1 ∂

∂ Pt= α α∗− Pt

α α∗α α∗

1 + γ=(

1 − γ

1 + γ

)α α∗ = α α∗

1 + γ(7.108)

and hence

J22 = n tr

(−1 ∂

∂ Pt

)2

= n tr

α α∗

1 + γ

α α∗

1 + γ

= n(α∗α)2

(1 + γ )2 = n(

γ

1 + γ

)2 1P2

t. (7.109)

Finally from (7.97)

J12 = n tr

−1 ∂

∂ωdt

−1 ∂

∂ Pt

= n tr

Pt

( − Pt

α α∗1 + γ

)α α∗

1 + γ

= nPt

1 + γtr

α α∗ − Ptα α∗α α∗

1 + γ

= nPt

1 + γ

(α∗α − Pt

α∗α1 + γ

α∗α

)

= nPt

1 + γ

(1 − γ

1 + γ

)α∗α = nPt

(1 + γ )2 α∗α

= nγ

(1 + γ )2

α∗α

α∗α= n

γ

(1 + γ )2 α∗oα (7.110)

where we have defined

o=

α∗α=

s∗t R−1

c st. (7.111)

Now using (7.95) and (7.103) we have

α∗α = s∗t R−1

c

(∂st

∂ωdt

s∗t + st

∂s∗t

∂ωdt

)R−1

c st

= s∗t R−1

c st

(s∗

t R−1c

∂st

∂ωdt

+ ∂s∗t

∂ωdt

R−1c st

)(7.112)

and hence using (7.111)

α∗oα = α∗α

α∗α= s∗

t R−1c

∂st

∂ωdt

+ ∂s∗t

∂ωdt

R−1c st = 2Re

s∗

t R−1c

∂st

∂ωdt

.

(7.113)


Define

ρ = s∗t R−1

c∂st

∂ωdt

= a + jb (7.114)

to represent the correlation between the target steering vector and itsDoppler derivative weighted with respect to the inverse of the cluttercovariance matrix. Then from (7.113)

α∗oα = 2a (7.115)

represents twice its real part and using this in (7.110) we get

J12 = n(

γ

1 + γ

)2 2aγ

. (7.116)

To simplify (7.106) further, using (7.111) we can rewrite it as

J11 = nP2t (α∗α)2

(tr2

o

− 2Ptα∗2

oα

1 + γ+ P2

t(α∗oα)2

(1 + γ )2

)

= n(

γ

1 + γ

)2((1 + γ )2tr

2

o

− 2Pt(1 + γ )α∗2oα + P2

t (α∗oα)2).

(7.117)

Now using (7.95) and (7.103) we get

α∗2α = s∗t R−1

c

(∂st

∂ωdt

s∗t + st

∂s∗t

∂ωdt

)R−1

c

(∂st

∂ωdt

s∗t + st

∂s∗t

∂ωdt

)R−1

c st

= (s∗t R−1

c st)(

ρ2 + (ρ∗)2 + |ρ|2 + s∗t R−1

c st∂s∗

t

∂ωdt

R−1c

∂st

∂ωdt

),

(7.118)

with ρ is as defined in (7.114). Let

µ =(

∂s∗t

∂ωdt

R−1c

∂st

∂ωdt

)s∗

t R−1c st (7.119)

represents the weighted (Mahalanobis) distance of the gradient steer-ing vector normalized with respect to the “effective interference plusnoise power”σ 2 defined in (7.100). In that case using (7.114) and (7.119)in (7.118), we get

α∗2oα = α∗2α

(α∗α)2 = α∗2α(s∗

t R−1c st)2 = σ 2(3a2 − b2 + µ). (7.120)


Similarly

tr2 = tr

R−1/2c

(∂st

∂ωdt

s∗t + st

∂s∗t

∂ωdt

)R−1

c

(∂st

∂ωdt

s∗t + st

∂s∗t

∂ωdt

)R−1/2

c

= 2(

∂s∗t

∂ωdt

R−1c

∂st

∂ωdt

)(s∗

t R−1c st)+ ρ2 + (ρ∗)2 = 2(µ + a2 − b2),

(7.121)

so that

tr2

o

= tr2(α∗α)2 = 2σ 4(µ + a2 − b2). (7.122)

Finally substituting (7.115), (7.120), and (7.122) into (7.117) we get

J11 = n(

γ

1 + γ

)2 [(1 + γ )22σ 4(µ + a2 − b2)

− 2Pt(1 + γ )σ 2(3a2 − b2 + µ) + P2t 4a2]

= 2n(

γ

1 + γ

)2

σ 4[(1 + γ )µ + a2(1 − γ ) − b2(1 + γ )]. (7.123)

From (7.32), if either the target Doppler ωdt or target power Pt is theonly unknown, then 1/J11 or 1/J22 acts as the respective lower boundfor their variance and (7.123) or (7.109) can be used to compute them.However, when both the target parameters are unknown, the 2 × 2Fisher information matrix must be used to compute the correspondingbounds. In that case [6]

J = 2n(

γ

1 + γ

)2

σ 4[(1 + γ )µ + a2(1 − γ ) − b2(1 + γ )]aγ

aγ

12P2

t

(7.124)

so that

J−1 = 12no

(1 + γ

γ

)2

×

12P2

t− a

γ

− aγ

σ 4[(1 + γ )µ + a2(1 − γ ) − b2(1 + γ )]

(7.125)


where

o = σ 4

2P2t

[(1 + γ )µ + a2(1 − γ ) − b2(1 + γ )] − a2

γ 2

= 1 + γ

2γ 2 (µ − (a2 + b2)) = 1 + γ

2γ 2 (µ − |ρ|2). (7.126)

Hence

J−1 = 1n

1 + γ

µ − |ρ|2

12P2

t− a

γ

− aγ

σ 4[(1 + γ )µ + a2(1 − γ ) − b2(1 + γ )]

.

(7.127)

Equation (7.127) can be used to determine the Cramer-Rao boundsfor target Doppler and power estimates when both these quantitiesare simultaneously unknown. From (7.127) the variance of the targetDoppler estimate is lower bounded by [6]

σ 2CR(ωdt ) = J 11 = 1

n1 + γ

µ − |ρ|21

2P2t

= 12nσ 2

1 + 1/γ

Pt(µ − |ρ|2)(7.128)

and the variance of the target power estimate is lower bounded by

σ 2CR( Pt) = J 22 = 1 + 1/γ

nPtσ

2[(1 + γ )µ + a2(1 − γ ) − b2(1 + γ )]µ − |ρ|2 .

(7.129)

Equations (7.128) and (7.129) represent the Cramer-Rao lowerbounds for target Doppler and power estimates when both are un-known. Recall that µ represents the normalized Mahalanobis distanceof the Doppler gradient of the target steering vector as in (7.119),and ρ represents the “weighted cross correlation” between the tar-get 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 areunknown, array design freedom may be utilized to minimize theweighted correlation factor ρ so that the bounds are as low as possible.

From (7.123), J11 represents the bound on target Doppler esti-mate when the power level is also unknown; whereas from (7.123),1/J11 represents the same bound when the power level is also known.


Hence their difference

η1 = J11 − 1J11

≥ 0 (7.131)

can be used as a measure of the degradation in target Doppler estima-tion due to not knowing the target power. Similarly

η2 = J22 − 1J22

≥ 0 (7.132)

represents the degradation factor for target power when targetDoppler is unknown.

To make meaningful comparisons using these bounds, the noiseonly case (clutter is absent) can be used as a reference frame.

Noise Only CaseIn this case, the total interference matrix is diagonal so that

Rc = σ 2n I (7.133)

and substituting this into (7.86), (7.114) and (7.119), we get

σ 2 = 1

s∗t R−1

c st= σ 2

n

MN, (7.134)

γ = Pt/σ2 = MN

Pt

σ 2n

= MN(SNR), (7.135)

ρ = 1σ 2

ns∗

t∂st

∂ωdt

(7.136)

and

µ = MNσ 4

n

∂s∗t

∂ωdt

∂st

∂ωdt

= 1σ 4

n

∥∥∥∥ ∂st

∂ωdt

∥∥∥∥2

. (7.137)

In (7.135), SNR represents the input signal to noise ratio at eachsensor input4. Using these in (7.128) we obtain the target Dopplerbound in the noise only case to be

σ 2CR(ωdt ) = MN

2nσ 2n

1 + 1/γ

Pt(µ − |ρ|2)= (MN)2

2n(1 + γ )/γ 2(

MN∥∥ ∂st

∂ωdt

∥∥2 −∣∣s∗

t∂st∂ωdt

∣∣2) .(7.138)

We can readily compute (7.138) for various specific array configu-rations.


Uniform Linear ArrayIn this case the Doppler derivative of the target steering vector is givenin (7.104) and from there5

∥∥∥∥ ∂st

∂ωdt

∥∥∥∥2

= (a∗(ct)a (ct))(

∂b∗(ωdt )∂ωdt

∂b(ωdt )∂ωdt

)= Nπ2

M−1∑k=1

k2

= Nπ2(

(M − 1)3

3+ (M − 1)2

2+ M − 1

6

)

= MNπ2(

M2

3− M

2+ 1

6

). (7.139)

Similarly the cross correlation in (7.136) is given by

s∗t

∂st

∂ωdt

= (a∗(ct)a (ct))(

b∗(ωdt )∂b(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) iszero. As a result, the Fisher information matrix in (7.124) is diagonal,implying that in the noise only case, the two target parameter estima-tors 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

σ 2CR(ωdt ) = 6

π2n1 + γ

γ 2

1M2 − 1

. (7.141)

From (7.135) and (7.141), higher values of input SNR improve thetarget 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 tobe

σ 2CR( Pt) = (1 + γ )2

nγPtσ

2 = (1 + γ )2

nγ

Ptσ2n

MN= 1

n

(1 + γ

γ

)2

P2t

(7.143)

that agrees with (7.109). Next, we examine a centro-symmetric array.

5The identity∑n

k=1 k2 = n3

3 + n2

2 + n6 can be easily verified by induction.


Centro-Symmetric Uniform Linear ArrayFor a centro-symmetric uniform linear array, the reference pulse ischosen to be the center one, so that the temporal vector [7]

b(ωd ) = [ e jπ( M−12 )ωd , . . . , e jπωd , 1, e− jπωd , . . . , e− jπ( M−1

2 )ωd ]T (7.144)

and using this as in (7.104) we get

s∗t

∂st

∂ωdt

= − jπ

M−12∑

k=− M−12

k = 0. (7.145)

Hence ρ = 0, however, in this case

∥∥∥∥ ∂st

∂ωdt

∥∥∥∥2

= 2Nπ2

M−12∑

k=1

k2

= 2Nπ2

(13

(M − 1

2

)3

+ 12

(M − 1

2

)2

+ 16

M − 12

)

= MNπ2 M2 − 112

(7.146)

and once again using (7.138) we get

σ 2CR(ωdt ) = 6

π2n1 + γ

γ 2

1M2 − 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

σ 2CR(ωdt ) = 6

n1 + γ

γ 2

1M2 − 1

6nγ (M2 − 1)

(7.148)

that agrees with [7]. In this case, ρ = 0, and hence from (7.129) weobtain the bound for the power estimate to be

σ 2CR( Pt) = (1 + γ )2

nγ

Ptσ2n

MN= 1

n

(1 + γ

γ

)2

P2t . (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 hencethey can be used as a reference baseline for comparison with cluttersituations. Observe that the centro-symmetric array does not result inany improvement in the noise only case. Figure 7.1 shows the various


CR Bounds

White Noise Only

Airborne or SBR

Single or MultipleParameters

Centro-symmetricArray (C)

RegularLinear

Array (L)

Clutter and Noise

Airborne SBR

SingleParameter

MultipleParameters

SingleParameter

MultipleParameters

Ro

C L C L

C LC L R1 R2 R4R3 R1 R2 R4R3

… … … … … … …

FIGURE 7.1 Various possible options for Cramer-Rao.

possible options when clutter is present. In the case of SBR, the pres-ence/absence of range foldover and/or Earth’s rotation introducesfour different situations with associated clutter covariance matricesR1, 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, σ 2R4

(ωd ) will refer to the Cramer-Rao boundfor the target Doppler in presence of both Earth’s rotation and range-foldover effects for multiparameter case using a regular linear array.

7.3 Simulation ResultsIn what follows the Cramer-Rao bounds are computed for targetDoppler and power, and compared with their variance estimates ob-tained from simulation results corresponding to various airborne andSBR situations shown in Figure 7.1. In the airborne case, a 14-sensorarray with 16 pulses is used, and the parameter set for the SBR case isas shown in Table 6.2. In all these cases the array look angle is set atboreside (θAZ = 90).

Figures 7.2 and 7.3 show the CR bounds for target Doppler andpower and their variance estimates for the airborne case. Figure 7.2corresponds to the noise only case and Figure 7.3 corresponds to theclutter and noise case with CNR = 40 dB.


SNR = −10 dBSNR = 0 dBSNR = 20 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

Target Doppler

−1 −0.5 0 0.5 1−10

−8

−6

−4

−2

0

Var

. in

dB

−10−10 −5 0 5 10 15 20

−8

−6

−4

−2

0

Var

. in

dB

CR

bou

ndV

ar. E

st.

SNR in dB

wd = 0.1

wd = 0.55Var. Est.

CR bound

wd = 0.25

FIGURE 7.2 Airborne case (noise only): CR bounds for target Doppler andtheir variance estimates as function of (a) target Doppler and (b) SNR.

Significant degradation in target Doppler estimation is to be ex-pected around the clutter dominant neighborhood, which is diagonalin the angle-Doppler domain. For a side-looking array with look anglealong boreside (θAZ = 90), clutter is dominant at ωd = 0, and the CRbound and the variance estimates have worst performance aroundthat region as is evident in Figure 7.3. From Figures 7.2 and 7.3, theCR bound and the variance estimates for target Doppler decreases asSNR increases.

Figures 7.4 and 7.6 correspond to the SBR situations Ro → R4 shownin Figure 7.1 for Range = 500 km. Figures 7.7 and 7.8 refer to target


(b) CR bound and variance estimates for target Doppler as a function of SNR

Target Doppler

−1 −0.5 0 0.5 1−10

−8

−6

−4

−2

0

Var

. in

dB

−10−10 −5 0 5 10 15 20

−8

−6

−4

−2

0

Var

. in

dB

CR

bou

ndV

ar. E

st.

CR

bou

ndV

ar. E

st.

SNR in dB


wd = 0.1wd = 0.25wd = 0.55

FIGURE 7.3 Airborne case (clutter and noise): CR bounds for target Dopplerand their variance estimates as a function of (a) target Doppler and (b) SNR.


−1−9

−8

−7

−6

−5

−4

−3

−2

−1

−0.5 0 0.5 1

CR

Bou

nd fo

r D

oppl

er in

dB

Target Doppler

(a) Target power is unknown

SNR = 0 dB

Ro

R1 → R3

R4

R0R1R2R3R4

R0R1R2R3R4

−1−9

−8

−7

−6

−5

−4

−3

−2

−1

−0.5 0 0.5 1

CR

Bou

nd fo

r D

oppl

er in

dB

Target Doppler

(b) Target power is known

SNR = 0 dB

Ro

R1 → R3

R4

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) Targetpower is unknown, (b) Target power is known.

Doppler CR bounds and its estimates in the noise only case (Ro ) asfunctions of target Doppler and SNR. Similar situations for cases R1and R4 in (7.150) are shown in Figures 7.9–7.11 for ranges 500 km and1,200 km.

In the SBR case performance degradation occurs in the clutter dom-inant region also (see Figures 7.4, 7.6, 7.9). In addition the CR boundsfor Doppler improve as SNR increases (Figure 7.5). However, as crabeffect and/or range foldover are included into the modeling (cases

Power is Unknown, Doppler = 0.1

R4

R0R1R2R3R4

Ro

R1 → R3

wd = 0.1

SNR (dB)−10 −5 0 5 10 15 20

(a) Target Doppler = 0.1

−12

−10

−8

−6

−4

−2

0

CR

Bou

nd fo

r D

oppl

er in

dB

R0R1R2R3R4

Power is Unknown, Doppler = 0.6

R4

Ro

R1 → R3

wd = 0.6

SNR (dB)−10 −5 0 5 10 15 20

(b) Target Doppler = 0.6

−11

−10

−8

−9

−6

−7

−5

−4

−3

CR

Bou

nd fo

r D

oppl

er in

dB

FIGURE 7.5 SBR case: CR bounds for target Doppler as function of SNR forthree 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.


R4

Ro

R1 → R3

Target Doppler−1 −0.5 0 0.5 1

(a) CR bound for target power as a function of Doppler

2

3

4

5

6

7

8

9

CR

Bou

nd fo

r D

oppl

er in

dB

SNR = 0 dB

R4

Ro → R3

wd = 0.6

SNR (dB)−10 −5 0 5 1510 20

(b) CR bound for target power as a function of SNR

0

1

2

3

4

5

6

7

CR

Bou

nd fo

r D

oppl

er in

dB

Doppler = 0.6

R0R1R2R3R4

R0R1R2R3R4

FIGURE 7.6 SBR case: CR bounds for target power as a function of Dopplerand 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 predom-inant. Notice that the CR bounds and the variance estimates havevery 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 (seeSection 6.3; Figures 6.17–6.20) showing that when either crab effector range foldover is present, it is possible to compensate for that effectand obtain accurate estimates, whereas when both effects are present,their effect cannot be compensated.


−1−1

−0.5

0

0.5

1

−0.5 0 0.5 1

Target Doppler

(b) Mean of estimated target Doppler as a function of Doppler

Mea

n of

Est

imat

ed D

oppl

er


−1−11

−10

−9

−8

−7

−2

−3

−4

−5

−6

−0.5 0 0.5 1

Target Doppler


Var

. in

dB

CR

bou

ndV

ar. E

st.

FIGURE 7.7 SBR case: Noise only case (Ro ). Range = 500 km. (a) CR boundsfor target Doppler and its estimates as a function of Doppler and (b) Mean ofestimated target Doppler as a function of Doppler.


−100

0.5

0.4

0.3

0.2

0.1

0.7

0.6

−5 0 5 10 15 20

SNR in dB

(b) Mean of estimated target Doppler as a function of SNR

Mea

n of

Est

imat

ed D

oppl

er

wd = 0.1wd = 0.25wd = 0.55

wd = 0.1wd = 0.25wd = 0.55

−10−11

−6

−7

−8

−9

−10

−4

−5

−5 0 5 10 15 20

SNR in dB

(a) CR bound and variance estimates for target Doppler as a function of SNR

Var

. in

dB

CR bound

Var. Est.

FIGURE 7.8 SBR case: Noise only case (Ro ). Range = 500 km. CR bounds fortarget Doppler and its estimates as function of SNR and (b) Mean ofestimated target Doppler as a function of SNR.

To demonstrate the effect of clutter on parameter estimation, themean value of the estimated target Doppler is plotted as a function ofDoppler and SNR in Figures 7.7(b)–7.10(b). The estimates are unbiasedin the noise only case (Ro ), whereas when either the range foldovereffect or Earth’s rotation is present, the estimates are not very accuratein the clutter dominant region (ωd ≈ 0). The degree of bias depends ona variety of parameters and gets worse when both effects are present(Figure 7.10(b)).

−1−1

−0.5

0

0.5

1

−0.5 0 0.5 1

Target Doppler


Mea

n of

Est

imat

ed D

oppl

er



−1−10

−8

−6

−4

−2

0

−0.5 0 0.5 1

Target Doppler


Var

. in

dB

CR

bou

ndV

ar. E

st.

FIGURE 7.9 SBR case: Clutter and noise case (R1), w/o range foldover, w/oEarth’s rotation. Range = 500 km. CR bounds for target Doppler and itsestimate as a function of Doppler and (b) Mean of estimated target Doppleras a function of Doppler.


−1−1

−0.5

0

0.5

1

−0.5 0 0.5 1Target Doppler


Mea

n of

Est

imat

ed D

oppl

er



−1−8

−6

−4

−2

2

0

−0.5 0 0.5 1Target Doppler


Var

. in

dB

CR

bou

ndV

ar. E

st.

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 itsestimate as a function of Doppler and (b) Mean of estimated target Doppleras a function of Doppler.

The performance degradation becomes further pronounced as afunction of range. This is exhibited in Figure 7.11 that correspondsto range 1,200 km. From there, it is clear that when both crab effectand range foldover effect are present in the data at far ranges, targetdetection is difficult, and it is necessary to introduce waveform diver-sity into transmit design to minimize the effect of clutter and otherinterference.


−1−10

−8

−6

−4

−2

0

−0.5 0 0.5 1

Target Doppler

(a) Without range foldover, without Earth’s rotation

Var

. in

dB


−1−6

−4

−5

−3

−2

−1

2

1

0

−0.5 0 0.5 1

Target Doppler

(b) With range foldover, with Earth’s rotation

Var

. in

dB

CR

bou

nds

Var

. Est

.

Var

. Est

.

CR bounds

FIGURE 7.11 SBR case: CR bounds for target Doppler and its estimate as afunction 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.


Figures 7.12 and 7.13 show the effect of waveform diversityon the CR bounds for ranges 500 km and 1,200 km. Noticethat the bound associated with any waveform diversity methodis smaller than that associated with the conventional waveformwhen Earth’s rotation and range foldover effects are present(case R4). This is generally true for both Doppler and power esti-mates indicating that introducing waveform diversity improves per-formance. As Figure 7.13 shows, at higher range (R = 1,200 km),different waveforms give different performance, but all of them aresuperior to the conventional waveform for case R4. In general, the

−1−9

−8

−7

−6

−5

−4

−3

−2

−1

−0.5 0 0.5 1

CR

Bou

nd fo

r D

oppl

er in

dB

CR

Bou

nd fo

r D

oppl

er in

dB

Target Doppler

(a) CR bounds as a function of target Doppler

SNR (dB)

(b) CR bounds as a function of SNR

SNR = 0 dB

Ro

R4: Conv

R0R1R4, Conv.R4, Orth.R4, ChirpR4, Hybrid

−1−8

−6

−4

−2

2

0

−0.5 0 0.5 1

CR

Bou

nd fo

r D

oppl

er in

dB

Target Doppler

(a) CR bounds as a function of target Doppler

SNR = 0 dB



−10−12

−10

−8

−6

−4

−2

−5 0 5 10 15 20

Doppler = 0.25

Ro

R1, R4: Orth,Chirp, Hybrid

R1, R4: Orth,Chirp, Hybrid

R4: Conv

wd = 0.25

CR

Bou

nd fo

r D

oppl

er in

dB

SNR (dB)

(b) CR bounds as a function of SNR

−10−12

−10

−8

−6

−4

−2

0

−5 0 5 10 15 20

Doppler = 0.25

wd = 0.25

Range = 500 km

Range = 1,200 km


R

FIGURE 7.12 CR bound for target Doppler with waveform diversity forrange = 500 km and 1,200 km.


−1−8

−7

−6

−5

−4

−3

−2

−1

−0.5 0 0.5 1

(a) Range = 500 km

R4 with waveformdiversity

R4: Conv

R1 R1

R4, Conv.R4, Orth.R4, Chirp

R1

R4, Hybrid

R4, Conv.R4, Orth.R4, Chirp

R1

R4, Hybrid

−1−8

−7

−6

−5

−4

−3

−2

1

0

−1

−0.5 0 0.5 1

(b) Range = 1,200 km

R4 with waveformdiversity

R4: Conv

FIGURE 7.13 CR bound for target Doppler without and with waveformdiversity 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 tothat of case R1.

In summary, for target detection using SBR, when bothrange foldover and Earth’s rotation effects are jointly present, theperformance bounds are inferior to those associated with when onlyeither one of the effect is present. Waveform diversity can be usedto minimize the effect of range foldover and consequently leads tosuperior performance. These conclusions are also supported by theCramer-Rao bounds associated with various scenarios presented here.As a result, waveform diversity should be considered to improve tar-get detection especially for large ground range values [6].

Interestingly, the Cramer-Rao bounds for target Doppler and powerhave asymmetric behavior with respect to SNR, i.e., the Dopplerbound decreases as SNR increases whereas the corresponding boundfor the power increases with SNR. This is in agreement with the gen-eral Gaussian data case with unknown variance as in (7.64) and (7.72)[1]. The asymmetric behavior suggests an optimum SNR for the jointparameter 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.


[6] S.U. Pillai, K.Y. Li, B. Himed, “Cramer-Rao Bounds for Target Parametersin Space Based Radar Applications,” To appear in Proc. IEEE Transactions onAerospace and Electronic Systems, Vol. 44, No. 4, October 2008.

[7] J. Ward, “Cramer-Rao Bounds for Target Angle and Doppler Estimationwith Space-Time Adaptive Processing Radar,” 29th Asilomar Conference onSignals, Systems and Computers (2-Volume Set), Pacific Grove, CA, October 30–November 2, 1995.


C H A P T E R 8Waveform Diversity

Waveform diversity refers to the use of various waveforms (signals) inboth transmitter (Tx) and receiver (Rx) design for improving the over-all performance such as detection and/or identification of targets ininterference and noise. Waveform diversity can be exploited spatiallyusing a multiple set of sensors for both transmission and/or reception,and in the time-frequency domain using distinct waveforms of differ-ent durations over different spectral bands. In addition, other featuressuch as polarization, energy distribution of various transmit signalscan be used for further diversity.

Spatial diversity can be realized from a single platform or mul-tiple platforms for both transmission and reception. For example, inmonostatic mode the same set of sensors are used for transmission andreception, whereas in the bistatic mode, different sensors are used forthese two functions (Figure 8.1). Thus in the bistatic case, a dedicatedsensor is used for transmission and a multiple set of sensor outputscan be used as the receiver. In both cases, the platform involved maybe stationary or in motion. In a typical airborne or Space Based Radar(SBR) situation, the platform is moving and both transmitter and re-ceiver may be time sharing the same set of antennas in different con-figurations giving rise to a monostatic situation. If we envision a setof unmanned aerial vehicles (UAVs) into this situation used in thereceiver mode, then we have a multisensor bistatic situation at hand.

The objective in such situations also may comprise of multiple taskssuch as Ground Moving Target Indication (GMTI), Airborne MovingTarget Indication (AMTI), and Synthetic Aperture Radar (SAR). Inthe case of GMTI, the task is to improve the overall system design sothat slowly moving targets can be detected and identified in the pres-ence of clutter, jamming signals, and noise. Once the transmitter andreceiver array configurations and their relative positions are known,



Rx-3(UAV)

Tx-Rx 2 (UAV)

SBR or Airborne Radar (Tx−Rx 1)

Rx-4

GMTI

Beam #1

Beam #4

Temporal Tx diversity

ww0

Spectraldiversity

t

Polarizationdiversity

Spatialdiversity

Temporal Tx diversity

f ′2(t) f ′1(t)

f2(t) f1(t) F1(w)

t

AMTI

Multitasks

Spatialdiversity

FIGURE 8.1 Multichannel waveform diversity.

the problem is to select various transmitter and receiver waveformsfor target return enhancement and simultaneous suppression of in-terference and noise. Target scene may consist of single or multipletargets which may be stationary or moving under various conditions.The system goal is to detect and identify all of them, and estimatethe relevant parameters such as their range, Doppler, elevation, andazimuth angles, etc. Usually target return is buried in competing inter-ference which consists of stochastic return signals from the environ-ment and noise. These random returns are transmit signal dependentand hence they directly compete with the target returns usually dom-inating them. One task of waveform diversity is to distinguish the in-terference spectral characteristics and design waveforms accordinglyto 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 regionswhere target characteristics are dominant, and minimize transmissionat those regions where interference and noise are dominant. However,this is to be attained using finite duration waveforms in some opti-mal 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 communi-cation scene as well, where the channel response plays the role of the

C h a p t e r 8 : W a v e f o r m D i v e r s i t y 343

Multiple transmitter waveforms Multi receiver set for

coherent processing

∑OutputMoving

Target

Interference

...

...

...

Noise

Noise

Rx −1

Ry −1

Ry −2

Rx −2

Rx −m1

Ry −m2

fi (t)

f1 (t)

f2 (t)

fN (t) SAR

wdo

qo

MTI

∑t = to

t = to

Output

SAR

MTI Rx

SAR Rx

FIGURE 8.2 Transmitter–receiver design.

target. In that case, the transmitted signal passes through the chan-nel to generate an output signal that also gets modified by interfer-ence and noise. Once again, the goal is to enhance the signal part ofthe channel output and minimize the interference and noise compo-nents through proper joint transmitter–receiver design. Clearly, thetarget or channel characteristics must play a role in the transmitteras well as the receiver design—to enhance the corresponding finaldesired output. At the same time, the design must deal with sup-pressing the interference and noise output components. In this con-text, the receiver output signal to interference plus noise ratio (SINR)may be used as an optimization goal. This is illustrated in Figure 8.2where the receiver outputs are combined to make a decision at t = to .Thus

SINR = Receiver output signal (target) power at t = toAverage interference pulse noise output power

. (8.1)

To maximize the output SINR, the following design componentsmust be simultaneously taken into account:

Transmitter: Set of finite duration waveforms fi j (t) ↔ Fi j (ω) (i :spatial and j : temporal) with an energy constraint on the whole setor subsets of waveforms. Known transmitter output filters Pi j (ω)can be used to control the transmitter bandwidth Bo . The spatialset of waveforms can be realized by partitioning the transmit array,and the temporal set by utilizing a pulse train of waveforms.


Target (channel): Impulse response q (t) ↔ Q(ω). Simplest modelis 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)thentry Qi j (ω) is the transfer function between the ith transmitterand the j th receiver.

Noise: Stationary stochastic process wn(t) with power spectrumGn(ω) = σ 2

n that corresponds to the white (flat) noise case. Ingeneral, Gn(ω) is an arbitrary spectrum. (High interference (clut-ter dominant) situation may be represented as Gn(ω) ≈ 0.) Inthe multichannel case, Gn(ω) is represented by a positive-definitespectral matrix.

Interference (clutter): Interference can be modeled as a station-ary stochastic process wc(t) with power spectrum Gc(ω) = σ 2

c thatcorresponds to flat interference. In general Gc(ω) is an arbitraryspectrum with specific bandwidth. (Noise dominant situation maybe represented as Gc(ω) ≈ 0.) In the multichannel case, Gc(ω) rep-resents a positive-definite spectral matrix.

Receiver filter: A multiple set of waveforms hi (t) ↔ Hi (ω) thatmay be causal or noncausal.

Decision instant to : The overall goal is to maximize the receiveroutput SINR at the decision instant.

The simplest situation is to consider a single transmitter and a singlereceiver with a target (channel) and no interference signal. To startwith assume the transmitter signal and the target to be known. Hencethe goal is to design the optimum receiver. The receiver in this caseis presented with the target output (fixed signal) that is corrupted bynoise. The simplest noise situation is white noise (equal strength at allfrequencies). Thus the simplest waveform design problem is to designa receiver to maximize the receiver output signal to noise ratio (SNR)for a given incoming signal that is buried in white noise. The solutionis the well-known “matched filter” receiver.

8.1 Matched Filter ReceiversConsider the problem of designing a receiver to detect an incomingsignal s(t) that is buried in noise. This situation occurs—in radar, sonar,and communication scenarios. In a communication scene, a transmit-ted 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


(a) Communication scene (b) Radar/sonar scene

⊗f (t)s(t)

n(t)

r (t)q(t) Q(w)

q(t)f (t)

s(t)

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

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 innoise, 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 receiverwith transfer function H(ω) to minimize the effect of noise as shownin Figure 8.4. The problem is how to design a good receiver?

Toward this, let y(t) represent the receiver output. Then

y(t) = s(t) + w(t), (8.4)

where s(t) represents the signal part of the output and w(t) the outputnoise. Clearly

s(t) = s(t) ∗ h(t) (8.5)

and

w(t) = n(t) ∗ h(t) (8.6)

r(t) = s(t) + n(t) y(t) = s(t) + w(t)H(w)

FIGURE 8.4 Receiver.


where

h(t) ↔ H(ω) (8.7)

represents the unknown receiver characteristics. Obviously the re-ceiver transfer function H(ω) should be selected so as to maximizethe output signal and minimize the output noise effect.

8.1.1 Matched Filter Receiversin White Noise

Consider the detection problem where the receiver output must beused to decide in favor of the target being present or absent in thedata. Toward this the output SNR of the received signal evaluated atsome desired time instant t = to can be used to select the optimumreceiver so as to maximize the receiver output SNR at t = to . Note thatthe output SNR at t = to is given by the instantaneous signal powerat t = to divided by the average noise power.1 Thus (see (8.1))

SNR|t=to= |s(to )|2

E|w(t)|2 . (8.8)

From (8.5)

s(t) ↔ S(ω) = S(ω)H(ω) (8.9)

so that

s(t) = 12π

+∞∫−∞

S(ω)e jωtdω, (8.10)

and using (8.10), we obtain

s(to ) = 12π

+∞∫−∞

S(ω)H(ω)e jωto dω. (8.11)

Let Gn(ω) and Gw(ω) represent the noise power spectral densitiesof n(t) and w(t) respectively. Hence

E|w(t)|2 = 12π

+∞∫−∞

Gw(ω)dω. (8.12)

1The definition in (8.8) uses the instantaneous output signal power vs. averageoutput noise power. The numerator here is not an average value of the signal poweras in the case of the denominator.


w

s 2

Gw(w)FIGURE 8.5White noise.

But from (8.6)

Gw(ω) = Gn(ω) |H(ω)|2 (8.13)

and hence

E|w(t)|2 = 12π

+∞∫−∞

Gn(ω) |H(ω)|2 dω. (8.14)

If we assume Gn(ω) to be white noise as shown in Figure 8.5, wehave

Gn(ω) = σ 2, (8.15)

and using this in (8.14), we get

E|w(t)|2 = σ 2

2π

+∞∫−∞

|H(ω)|2dω. (8.16)

Substituting (8.11) and (8.16) into (8.8), we get the output SNR att = to to be [1]

SNRo =

∣∣∣∣+∞∫−∞

S(ω)H(ω)e jωto dω

∣∣∣∣22πσ 2

+∞∫−∞

|H(ω)|2 dω

. (8.17)

Obviously, the unknown receiver H(ω) in (8.17) should be chosenso 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

∣∣∣∣∫

A(ω)B(ω)dω

∣∣∣∣2 ≤∫

|A(ω)|2 dω

∫|B(ω)|2 dω (8.18)


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)

and

B(ω) = H(ω). (8.21)

We can choose the constant multiplier µ in (8.19) to be unity here,since that gets canceled out from the numerator and the denominatorof (8.17). Using (8.20) and (8.21) in (8.17), from (8.18) we obtain∣∣∣∣∣∣

+∞∫−∞

S(ω)H(ω)e jωto dω

∣∣∣∣∣∣2

≤+∞∫

−∞|S(ω)|2 dω

+∞∫−∞

|H(ω)|2 dω. (8.22)

With (8.22) in (8.17), we get

SNRo ≤ 12πσ 2

+∞∫−∞

|S(ω)|2 dω. (8.23)

Thus the maximum value of the output SNR is given by

SNRmax =1

2π

+∞∫−∞

|S(ω)|2 dω

σ 2 =

+∞∫−∞

|s(t)|2 dt

σ 2 = Eσ 2 , (8.24)

a quantity independent of H(ω), and from (8.21) this maximum valueis 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 addi-tive white noise, the optimum receiver that maximizes the output SNR


r(t) = s(t) + n(t ) h(t) = s(to − t)

150

0

0 0.20.2t (sec) t (sec)

r(t)

0.10

0

2

5

0.50.40.3

r(t) = s(t) + w(t)

0.8 10.60.4

50

−50

−5

−2 100

FIGURE 8.6 Optimum receiver (matched filter).

at t = to is given by (8.25) and (8.26). Notice that h(t) depends onlyon the receiver input signal waveform s(t), or the receiver is matchedto its input signal s(t). To summarize, if s(t) + n(t) is received and thenoise n(t) is white, then the optimum receiver is given by the classicalmatched filter solution as in Figure 8.6.

In Figure 8.6, the time index to represents the time instant at whichthe output SNR is maximized and hence that instant (t = to ) must beused to make the decision. Note that the optimum filter in (8.26) neednot represent a causal solution. However, if s(t) is a finite durationsignal, then by suitably selecting to , the receiver h(t) can be madecausal.

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 andchannel characteristics. From (8.24), the maximum value of outputSNR is also given by

SNRmax =

+∞∫−∞

|s(t)|2 dt

σ 2 =1

2π

+∞∫−∞

|S(ω)|2 dω

σ 2

=1

2π

+∞∫−∞

|F (ω)Q(ω)|2 dω

σ 2 . (8.30)


The same conclusions can also be reached by making use of a timedomain analysis. To see this, referring back to (8.8) and (8.5)–(8.6), wehave

SNRt =

∣∣∣∣+∞∫−∞

h(τ )s(t − τ )dτ

∣∣∣∣2

E

∣∣∣∣+∞∫−∞

h(τ )n(t − τ )dτ

∣∣∣∣2

=

∣∣∣∣+∞∫−∞

h(τ )s(t − τ )dτ

∣∣∣∣2+∞∫−∞

+∞∫−∞

h(τ1)h∗(τ2)E n(t − τ1)n∗(t − τ2) dτ1dτ2

. (8.31)

If we assume n(t) to be a stationary white noise, then

E

n(t − τ1)n∗(t − τ2) = Rnn(τ1 − τ2) = σ 2δ(τ1 − τ2), (8.32)

and using this, (8.31) becomes

SNRt=to =

∣∣∣∣+∞∫−∞

h(τ )s(to − τ )dτ

∣∣∣∣2+∞∫−∞

+∞∫−∞

h(τ1)h∗(τ2)σ 2δ(τ1 − τ2)dτ1dτ2

=

∣∣∣∣+∞∫−∞

h(τ )s(to − τ )dτ

∣∣∣∣2σ 2

+∞∫−∞

|h(τ )|2 dτ

. (8.33)

Once again, using Schwartz’ inequality, the numerator in the aboveexpression gives

∣∣∣∣∣∣+∞∫

−∞h(τ )s(to − τ )dτ

∣∣∣∣∣∣2

≤+∞∫

−∞|h(τ )|2 dτ

+∞∫−∞

|s(to − τ )|2 dτ (8.34)

and (8.33) reduces to

SNRmax|t=to ≤ 1σ 2

+∞∫−∞

|s(to − τ )|2 dτ = 1σ 2

+∞∫−∞

|s(t)|2 dt = Eσ 2 (8.35)


f(t)

Tt

1

FIGURE 8.7Triangular pulse.

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 throughan 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)

Here

S(ω) = F (ω)Q(ω) = F (ω)e− jωtc (8.40)

so that

s(t) = f (t − tc ) (8.41)

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).

(a) s(t) (b) s(−t) (c) h(t)

T + tc

ttc

s(t )

1 1 1

−(T + tc)t

−tc

s(−t) h(t) = s(to − t )

−(T + tc)t

−tc + to

to

FIGURE 8.8 Optimum receiver for decision instant t = to.


h(t) = s(tc + T − t ) = f (t)h(t ) = s(tc − t )

−Ttt

11

(b) Optimum receiver for decision instant to = tc + T .

(a) 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 causalfilter.

From (8.24) and (8.30), the maximum output SNR at t = to is given by

SNRmax =

+∞∫−∞

|s(t)|2 dt

σ 2 =2

T/2∫0

(2t/T)2 dt

σ 2 = 83T2σ 2

(T2

)3= T

3σ 2 .

(8.42)

To summarize, if r (t) = s(t) + n(t), where s(t) has the form shown inFigure 8.10 (a), then if n(t) is white noise, the optimum receiver impulseresponse is given by

h(t) = s(to − t), (8.43)

and this is illustrated in Figure 8.10 (a)–(c) for some arbitrary to .

t t tT

(a)

−T

(b) (c)

−T −(T − to) to

s(t ) s(−t) s(−t)

FIGURE 8.10 Optimum receiver for some arbitrary to.


Matched Filter as a Correlator ReceiverSuppose the received data r (t) is available in the interval (0, t). In thatcase using (8.26), the matched filter output is given by

r (t) = r (t)∗h(t) =t∫

0

r (τ )h(t − τ )dτ =t∫

0

r (τ )s∗(to − t + τ )dτ , (8.44)

since h(t − τ ) = s∗(to − (t − τ )) = s∗(to − t + τ ). The matched filtermaximizes its output SNR at t = to , and hence the output r (t) mustbe used at that instant to make further decision. Thus from (8.44), wehave

r (to ) =to∫

0

r (τ )s∗(τ )dτ =to∫

0

r (t)s∗(t)dt. (8.45)

Notice that (8.45) can be interpreted as a correlator receiver and itcan be represented as in Figure 8.11.

From Figure 8.11, the received waveform is correlated with s(t) andintegrated until t = to to generate the output r (to ). Since the correlatorreceiver in (8.45) follows from the matched filter receiver in (8.43), weconclude that, in the case of additive white noise, both matched filterreceiver as well as the correlator receiver are equivalent realizations.

8.1.2 Matched Filter Receiversin Colored Noise

What if the receiver noise is of nonwhite nature? How does one designthe optimum causal filter in the colored noise case? Interestingly, byincreasing to alone the optimum solution cannot be made causal inthis case. To see this, let Gn(ω) represent the input noise spectrum so

0∫(⋅)dt

s(t)

r(to)

t = to

t t

r(t)

r(t) ⊗

r(t)

to

to

FIGURE 8.11 Correlator receiver.


that the receiver output noise power is given by

Ew2(t) = 12π

+∞∫−∞

Gn(ω) |H(ω)|2dω. (8.46)

Every spectrum G(ω) that is non-negative and satisfying the Paley-Wiener condition [1]

+∞∫−∞

|ln G(ω)|1 + ω2 dω < ∞ (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 halfplane of the complex s = σ + jω plane (minimum phase system). LetLn(s) represent the Wiener factor associated with Gn(ω) so that

Gn(ω) = |Ln( jω)|2 (8.49)

and

SNR|t=to =

∣∣∣∣ 12π

+∞∫−∞

H(ω)S(ω)e jωto dω

∣∣∣∣21

2π

+∞∫−∞

Gn(ω) |H(ω)|2 dω

=

∣∣∣∣ 12π

+∞∫−∞

H(ω)S(ω)e jωto dω

∣∣∣∣21

2π

+∞∫−∞

|Ln(ω)H(ω)|2 dω

.

(8.50)

To accommodate the new denominator, let us rewrite the termH(ω)S(ω) in the numerator as

Ln( jω)H(ω)L−1n ( jω)S(ω)

. (8.51)

Substituting this into (8.50) and a straightforward application ofSchwarz’ inequality gives

ρ(to ) ≤ 12π

+∞∫−∞

∣∣L−1n ( jω)S(ω)

∣∣2 dω = 12π

+∞∫−∞

|S(ω)|2Gn(ω)

dω, (8.52)

with equality if and only if

Ln( jω)H(ω) = (L−1n ( jω)

)∗S∗(ω)e− jωto (8.53)


or

H(ω) = S∗(ω)e− jωto

Gn(ω). (8.54)

Thus in the time domain if

linv(t) ↔ L−1n ( jω) (8.55)

then the optimum receiver impulse response is given by

h(t) = linv(t) ∗ l∗inv(−t) ∗ s(to − t). (8.56)

Clearly (8.56) represents a noncausal waveform, and a simple finiteshift by to alone (however large) may not be enough to maintain thecausal nature of the receiver.

Optimum Causal Matched Filter in Colored NoiseTo obtain the optimum causal receiver in this context, it is necessaryto proceed differently. Toward this, let [2]

υ(t) ↔ H(ω)Ln( jω) (8.57)

and let g(t) represent the inverse transform of L−1n ( jω)S(ω)

g(t) ↔ L−1n ( jω)S(ω) = L−1

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 Ln(s) and L−1n (s) are analytic in Re s > 0, we have υ(t) and

g(t) are causal waveforms and by Parseval’s theorem

+∞∫−∞

υ(t)g(to − t)dt =+∞∫

−∞υ(t)g∗(to − t)∗dt

= 12π

+∞∫−∞

H(ω)Ln( jω)L−1n ( jω)S(ω)e jωto

dω

= 12π

+∞∫−∞

H(ω)S(ω)e jωto dω= Q (8.61)


same as the numerator factor in (8.50). But from (8.61)

Q =+∞∫

−∞υ(t)g(to − t)dt =

+∞∫0

υ(t)g(to − t)dt, (8.62)

because of the causal nature of υ(t). However, by making use of theunit step function u(t), we can rewrite the later integral in (8.62) asfollows

Q =+∞∫0

υ(t)g(to − t)dt =+∞∫

−∞υ(t)g(to − t)u(t)dt. (8.63)

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

Q =+∞∫

−∞υ(t)g∗(to − t)u(t)

∗dt

= 12π

+∞∫−∞

H(ω)Ln( jω) K ∗(ω)dω. (8.65)

Using this in (8.50), we get

ρ(to ) = SNR|t=to =

∣∣∣∣ 12π

+∞∫−∞

H(ω)Ln( jω)K ∗(ω)dω

∣∣∣∣21

2π

+∞∫−∞

|Ln(ω)H(ω)|2 dω

≤ 12π

+∞∫−∞

|K (ω)|2 dω, (8.66)

with equality if and only if

H(ω)Ln( jω) = K (ω) (8.67)

or equivalently

H(ω) = L−1n ( jω)K (ω) ↔ linv(t) ∗ g∗(to − t)u(t). (8.68)


Note that the impulse response h(t) of this matched filter is neces-sarily causal by design. Thus

hc(t) = linv(t) ∗ g∗(to − t)u(t). (8.69)

In addition, using (8.64) in (8.66) we obtain

ρmax(to ) = 12π

+∞∫−∞

|K (ω)|2 dω =to∫

0

|g(t)|2 dt. (8.70)

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 shiftedversion of the noncausal solution in (8.56), nor can it be obtained di-rectly from (8.56). Using (8.54)–(8.55) and (8.60), we can rewrite (8.56)as

hnc(t) = linv(t) ∗ g∗(to − t) (8.71)

and on comparing (8.71) with (8.68), we notice that the causal solutionin (8.68) and (8.69) drops the noncausal portion of g∗(to − t) prior to itsconvolution with linv(t) as in Figure 8.12. We also have

hnc(t) = hc(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 impossibleto obtain the optimum causal filter hc(t) from the noncausal one at thefinal state.

The causal matched filter receiver in (8.68) that corresponds to thecolored noise case can be given an interesting interpretation as shownin Figure 8.13.

From Figure 8.13, the input noise n(t) is first “whitened” using thefilter L−1( jω) to generate white noise w(t). During this transforma-tion, the input signal s(t) becomes g(t) that is given by (8.58). Sinceg(t) is contaminated by white noise, from (8.26), g∗(to − t) representsthe optimum noncausal matched filter receiver for the input g(t)+w(t).

g(to − t) g(to − t)u(t)

(a) (b) (c)

t t t

linv (t)

FIGURE 8.12 Causal solution of the matched filter.


r(t) = s(t) + n(t) g(t) + w(t )g (to − t)u(t ) K(w)

y(t)t = to

L−1( jw) r (to)

FIGURE 8.13 Matched filter in colored noise.

However from (8.68) and (8.69), the causal matched filter receiver forthe input g(t)+w(t) is given by g∗(to −t)u(t) ↔ K (ω). Hence taken to-gether the concatenated overall receiver has the form in (8.68), wherethe input is first whitened followed by matched filtering to generatethe desired output.

Finally, in the white noise case Gn(ω) = σ 2 so that from (8.58),g(t) = s(t)/σ and substituting this into (8.70), we obtain

SNRmax = ρ(to ) = 1σ 2

to∫0

|s(t)|2 dt. (8.73)

As a result, the causal matched filter in the white noise case is givenby (compared with the noncausal receiver in (8.26))

hc(t) = s∗(to − t)u(t). (8.74)

From (8.70) and (8.73), in the causal case the SNR at the matchedfilter output at t = to is proportional to the energy in the input signalup to that instant. This relation is unlike (8.35), where the SNR is pro-portional to the total energy. Thus in the case of causal receiver, themaximum SNR output is a monotonically nondecreasing function ofthe observation instant to , and as we shall see later, this property hascertain interesting implications.

Since the matched filter generates outputs with large peaks, theycontain the potential for generating time-compressed waveforms thatmay lead to accurate target range estimation. Next, we investigatethis possibility and show that the chirp waveform indeed possessesthe desired time-compression property.

8.2 Chirp and Pulse Compression [3]A receiver matched to the incoming signal produces large peaks atthe output and hence it has the potential for time compression (seeFigure 8.14). For example, the matched filter receiver correspondingto the incoming signal s(t) is given by s∗(to −t), where to represents theoutput observation instant and for finite duration input signals, to is


t = toto

s(t )

h(t ) = s(to − t )y(t)

y(t )

tt

FIGURE 8.14 Matched filter and time compression.

usually chosen to be the signal duration T itself, so that the matchedfilter represents a causal filter (see Figure 8.9 (b)). In that case, from(8.26) the matched filter output y(t) equals

y(t) =t∫

0

s∗(τ )s(to − t + τ )dτ (8.75)

which peaks at t = to and the peak value is given by

y(to ) =to∫

0

|s(t)|2 dt. (8.76)

It follows that if the input waveform is delayed by To due to targetecho, the matched filter output also is delayed by the same amount,and hence the method can be used for range estimation as well, pro-vided the output is time compressed enough to give the desired rangeresolution.

To analyze this concept further, consider a finite energy signal s(t)with corresponding transform S(ω). Then s(t) is square integrable andhence

Es =+∞∫

−∞|s(t)|2 dt = 1

2π

+∞∫−∞

|S(ω)|2 dω < ∞. (8.77)

Assume that |s(t)|M, the maximum value of s(t) is finite. Then wemay 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)


s(t)

|s(t)|M

Ts

|s(t)|M

so(t)

Ts

tt

FIGURE 8.15 An arbitrary signal and its equal energy equivalentrectangular 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 Tshas 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) withto = 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

y(t) = 12π

+∞∫−∞

|S(ω)|2 e jωtdω, (8.81)

whose peak value is obtained at t = 0. Hence, its peak value is given by

|y(t)|M = y(0) = 12π

+∞∫−∞

|S(ω)|2 dω = Es . (8.82)

|s(w )|M |s(w )|M

so(w )s(w )

Bs-Bs

w w

FIGURE 8.16 An arbitrary spectrum and its equal energy equivalentrectangular spectrum.


The output signal energy Ey equals

Ey =+∞∫

−∞|y(t)|2 dt = 1

2π

+∞∫−∞

|S(ω)|4 dω

≤ |S(ω)|2M1

2π

+∞∫−∞

|S(ω)|2 dω = |S(ω)|2M Es . (8.83)

Hence

Ey = |S(ω)|2Mη

Es, (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 givenby (see Figure 8.17)

Ty = Ey

|y(t)|2M= |S(ω)|2M

η

Es

E2s

= |S(ω)|2Mη

1Es

= 1ηBs

, (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 outputeffective pulse length ratio Ts/Ty. From (8.85), [3]

Tinput

Toutput= Ts

Ty= TsηBs = ηTs Bs ≥ Ts Bs (8.86)

since η > 1. Thus large values of time compression can be realizedat the matched filter output by any input signal with large time-bandwidth product Ts Bs . In that case, the matched filter output willappear as a narrow pulse whose peak corresponds to the input delay.

As Appendix 8-A shows, the chirped pulse signal (both modulatedas well as centered at baseband) has the capacity to realize largetime-bandwidth products. For example, from (8A.35)–(8A.37) the

h(t) = s(to − t)

s(t) y(t)

y(t)

ttTs

to

ty

FIGURE 8.17 Effective duration of the matched filter output y(t).


transform of the causal chirped signal

s(t) =

cos(ωo t + βt2), 0 < t < T,

0, otherwise(8.87)

is given by [1]

|S(ω)| ≈ 1

2√

π/β, ωo < |ω| < ωo + 2βT,

0, otherwise(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 canbe increased by increasing the duration of the chirp signal. Finally,substituting these values into (8.86) we get

Ts

Ty= Tη2βT = 2ηβT2 ≥ 1 (8.89)

since βT 1. From (8.83)–(8.84) and (8.88),

η = |S(ω)|2M1

2π

+∞∫−∞

|S(ω)|4 dω

Es ≈ (π/β)2

(π/β)2 = 1. (8.90)

From (8.89) by increasing the pulse duration of the chirp signal, itis possible to realize output pulse compression to any degree. This isillustrated in Figure 8.18. This remarkable property of the chirp signalis exploited in radar for accurate target range estimation.

The above analysis assumes the channel to be impulsive so that thetransmit signal s(t) is returned with an unknown delay τo as s(t−τo ). Inthat case in the presence of additive noise, the matched filter achievesexcellent pulse compression when s(t) is a chirp signal. However ifthe target (channel) interaction is significant enough to change theshape of the return signal, chirp waveform may not be the optimum

3

1

−1−2−3 −50

0

0 0.1 0.2 0.2

150

50

100

0

0

10.80.60.40.5t (sec) t (sec)

0.40.3

2

h(t) = s(to − t)y(t)

FIGURE 8.18 Chirp signal and its matched filter output y(t).


transmit signal. Clearly in that case the designer must pay attentionto the target (channel) characteristics as well.

Before addressing this general design problem, we show that if thetarget/channel effect is simply to modulate the chirp signal with abaseband waveform a (t) so that

b(t) = a (t)e− j (ωo t+βt2) , 0 < t < T (8.91)

is received at the receiver input, then remarkably enough the originalmatched filter matched to the chirp signal still retains the desirablepulse compression property.

The target interaction can result in either modulating the transmitchirp signal by a time-varying gain function a (t) as in (8.91), or it canresult in convolving the chirp by a target impulse response functionao (t). Interestingly, the output bo (t) has the same form as in (8.91),since

bo (t) = ao (t) ∗ e− jβt2 =T∫

0

ao (τ )e− jβ(t−τ )2dτ

= e− jβt2

T∫0

ao (τ )e− jβτ (τ−2t)dτ = Ao (t)e− jβt2. (8.92)

Thus the target interaction either in the form of convolution ormultiplicative gain can be modeled as in (8.91) and as we show belowin both cases it is still possible to attain pulse compression property.

Modulated Chirp SignalSuppose a suitable baseband signal a (t) is used to modulate the chirpsignal as in (8A.30) to generate the output in (8.91).

Clearly, a (t) and b(t) have the same duration but the bandwidth ofb(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 receiverho (t) that is matched to only the chirp signal (use (8.26) with to = 0),i.e.,

ho (t) = s∗(−t) = e j (−ωo t+βt2) (8.94)


a(t )e- j(wot+b t 2) ho(t ) = e j(-wot+b t 2) y(t )FIGURE 8.19Modulated chirpsignal andmatched filteroutput.

so that the matched filter output y(t) equals (see Figure 8.19)

y(t) =+∞∫

−∞b(τ )ho (t − τ )dτ =

+∞∫−∞

a (τ )e− j (ωoτ+βτ 2)e j (−ωo (t−τ )+β(t−τ )2)dτ

= e j (−ωo t+βt2)

+∞∫−∞

a (τ )e− j2βtτ dτ = e j (−ωo t+βt2) A(2βt). (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 largeenough, 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-dependentamplification gains. Equation (8.91) can be given another interpreta-tion as well. From a signal design view point, the freedom present inselecting a (t) ↔ A(ω) can be used for enhancing the target featuresand suppressing the interferences.

8.3 Joint Transmitter–ReceiverDesign in NoiseAdditive White Noise CaseConsider the problem of transmitting a signal f (t) over a channelwith impulse response q (t). The channel output s(t) gets corruptedby additive white noise n(t), and the received signal is given by (seeFigure 8.3)

r (t) = s(t) + n(t). (8.97)

If maximization of the output SNR at the decision instant t = to ischosen to be the criterion for receiver design, then as we have seen


in the previous section, for a given s(t) the matched filter is the opti-mum 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

s(t) = f (t) ∗ q (t) =+∞∫

−∞f (τ )q (t − τ )dτ =

t∫0

f (τ )q (t − τ )dτ , (8.99)

where s(to ) depends only on the segment of q (t) up to t = to . Sincethe maximum SNR also depends only on s(t) up to t = to , the signaldesign problem reduces to designing the optimum pulse f (t), for0 < t < to that maximizes (8.73). Notice that the SNR in (8.73) canbe obviously increased by simply scaling f (t), and hence it makessense to restrict the energy in f (t) to a prescribed constant for thisoptimization problem. Thus the problem reduces to finding an f (t)for 0 < t < to , such that

to∫0

| f (t)|2 dt = 1, (8.100)

and

ρ(to ) =to∫

0

|s(t)|2 dt (8.101)

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 carrierused to modulate the information bearing symbols. Usually a rectan-gular pulse is used as the baseband carrier. That formulation assumesthat we use the same rectangular pulse irrespective of the actual chan-nel characteristics. But the rectangular pulse may not be the best pulseshape for all channels. Thus, given some additional information aboutthe channel, the problem is to find a meaningful way to incorporatethat information in selecting the best baseband pulse. Notice that sinceinformation symbols arrive at every To seconds in this model, at the


s

jw

s = s + jwZeros

FIGURE 8.20Zeros of aminimum phasesignal transform.

output, decisions must be made also at every To seconds, and henceto 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) thatmaximizes ρ(to ) in (8.101) must be a minimum phase signal. A signalf (t) is said to be minimum phase, if its Laplace transform F (s) isfree of zeros and poles in the open right half plane (Res > 0) as inFigure 8.20. To establish the minimum phase condition, assume thatthe finite pulse f (t) is not minimum phase. Then

F (s) =to∫

0

f (t)e−stdt (8.102)

has at least one zero in Re s > 0, say s = so . Rewrite F (s) as

F (s) = Fo (s)(

s − so

s + s∗o

)= Fo (s) A(s), σo = Re s > 0, (8.103)

where A(s) is a regular all-pass function.2 Let fo (t) represent theinverse Laplace transform corresponding to Fo (s). By extracting allan-pass, the remaining portion Fo (s) is guaranteed to have the sameenergy as the original pulse f (t). However, for fo (t) to act as a possi-ble candidate in this optimization problem, it should also be a finitepulse of duration to . To examine this, notice that

Fo (s) = F (s)(

s + s∗o

s − so

)= F (s) + 2σo

F (s)s − so

(8.104)

2A regular all-pass function has all its poles in the open left-half plane, andrepresents a causal system.


so that, its inverse transform in given by

fo (t) = f (t) + 2σo f (t) ∗ eso tu(t) = f (t) + 2σo

∞∫0

f (τ )eso (t−τ )u(t − τ )dτ

=

f (t) + 2σoeso tt∫

0f (τ )e−soτ dτ, t < to

f (t) + 2σoeso tto∫0

f (τ )e−soτ dτ, t ≥ to

=

f (t) + 2σoeso tt∫

0f (τ )e−soτ dτ, t < to

2σoeso t F (so ) = 0, t ≥ to

. (8.105)

Thus fo (t) is also of finite duration to , and it possesses the same en-ergy as f (t), establishing the fact that extraction of an all-pass factorfrom a finite pulse transform still retains its finite pulse shape charac-teristics.

Toward establishing the minimum phase character of the optimumpulse f (t) that maximizes (8.101), let s(t) and so (t) represent the chan-nel output due to inputs f (t) and fo (t) respectively (see Figure 8.21).Recall that fo (t) is also a finite pulse with the same energy, but itstransform contains one less zero in the right-half plane compared tothat of f (t). Thus

s(t) = f (t) ∗ q (t); so (t) = fo (t) ∗ q (t). (8.106)

We will show that the running energy up to t = to in so (t) is greaterthan that in s(t); i.e.,

to∫0

|s(t)|2dt ≥to∫

0

|s(t)|2dt, (8.107)

indicating that extraction of a right-half plane zero, or more precisely aregular all-pass function, from the input excitation pulse will result ingreater running energy for the system output. Clearly, by continuingthis process of regular all-pass extraction, maximum running energyat the output will be realized by a pulse whose transform is free of

f (t) q(t) q(t)s(t ), fo(t) so(t),

FIGURE 8.21 Transmit waveform f(t) and fo(t).


so(t)

so(t) s(t)t

a(t) ↔ A(s)

FIGURE 8.22 Minimum phase input signal.

zeros in the right-half plane; i.e., among all pulses with the same inputenergy, the minimum phase pulse generates maximum running energyat 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 inputso (t).

By making use of the regular all-pass nature of A(s) and Parseval’srelation, we have

∞∫0

|so (t)|2 dt =∞∫

0

|s(t)|2 dt. (8.109)

Let x(t) represent the output of the same all-pass filter A(s) due tothe truncated input s1(t) as shown in Figure 8.23. Thus

s1(t) =

so (t), 0 < t < to0, otherwise

. (8.110)

s1(t) so(t)

s1(t)to

x(t)A(s)t

FIGURE 8.23 Truncated input signal.


Then once again using Parseval’s relation, by equating the totalinput and output energy, we get

∞∫0

|s1(t)|2 dt =to∫

0

|so (t)|2 dt =∞∫

0

|x(t)|2 dt. (8.111)

But, because of the causal nature of A(s), we have

x(t) =t∫

0

s0(τ )a (t − τ )dτ = s(t), 0 < t < to , (8.112)

(i.e., the outputs in Figures 8.22 and 8.23 agree up to t < to ) andsubstituting this into the later half of (8.111) we get

to∫0

|so (t)|2 dt =to∫

0

|s(t)|2 dt +∞∫

to

|x(t)|2 dt, (8.113)

orto∫

0

|so (t)|2 dt ≥to∫

0

|s(t)|2 dt, (8.114)

establishing our original claim. Thus to maximize the running energyof any system output, the input pulse f (t) must be necessarily mini-mum phase.

Optimum SolutionReturning to the original optimization problem in (8.101), to makefurther progress, we can substitute (8.99) into (8.101). This gives

ρ(to ) =to∫

0

∣∣∣∣∣∣∞∫

0

f (τ )q (t − τ )dτ

∣∣∣∣∣∣2

dt

=to∫

0

∞∫

0

∞∫0

f ∗(τ1)q∗(t − τ1) f (τ2)q (t − τ2)dτ1dτ2

dt

=∞∫

0

f ∗(τ1)dτ1

∞∫0

to∫

0

q∗(t − τ1)q (t − τ2)dt

︸︷︷︸K (τ1, τ2)

f (τ2)dτ2.

(8.115)


Define

K (τ1, τ2) =to∫

0

q∗(t − τ1)q (t − τ2)dt (8.116)

to represent the kernel associated with the channel impulse responseq (t). Note that only the channel response up to t = to takes part indefining the above kernel. Further, q (t) causal implies that

K (τ1, τ2) = 0, if |τ1| > to , or |τ2| > to . (8.117)

Using (8.116) in (8.115), it simplifies to

ρ(to ) =∞∫

0

f ∗(τ1)dτ1

∞∫0

K (τ1, τ2) f (τ2)dτ2

=to∫

0

f ∗(τ1)dτ1

to∫0

K (τ1, τ2) f (τ2)dτ2,

(8.118)

where the later limits in the above integral follow form (8.117). Theabove integral defines an integral operator T( f ) given by

T( f ) =to∫

0

K (τ1, τ2) f (τ2)dτ2= ψ(τ1), (8.119)

and using this, (8.118) can be expressed also as an inner product givenby

ρ(to ) =to∫

0

f ∗(τ1)ψ(τ1)dτ1. (8.120)

However, using Schwartz’ inequality, the above expression yields

ρ(to ) ≤ to∫

0

| f (τ1)|2 dτ1

to∫0

|ψ(τ1)|2 dτ1

1/2

= to∫

0

|ψ(t)|2 dt

1/2

(8.121)

and (8.120) is maximized if and only if ψ(τ1) is chosen to be propor-tional to f (τ1) in 0 < τ1 < to , i.e., for maximum SNR f (τ1) must be a


solution of the integral equation3

ψ(τ1) =to∫

0

K (τ1, τ2) f (τ2)dτ2 = λ f (τ1), 0 < τ1 < to . (8.122)

Substituting (8.122) into (8.120), we get

ρ(to ) = λ

to∫0

| f (τ1)|2 dτ1 = λ ≤ λ1, (8.123)

where λ1 is the largest (positive) eigenvalue of the above integral equa-tion. 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 largesteigenvalue of the above integral equation. In other words, ρ(to ) is max-imized by selecting f (t) to correspond to the eigenfunction associatedwith the largest eigenvalue λ of the integral equation (8.122), and themaximum possible SNR is given by

ρ(to ) = λmax(to ), (8.124)

where λmax(to ) is the largest value of λ that satisfies (8.122). Fromthe previous argument, the eigenfunction f (t) corresponding to thelargest eigenvalue of the kernel in (8.122) also represents a minimumphase function.

Optimum Input and the Matched Filter ReceiverIt is easy to show that in the white noise case, the optimum solutionfor f (t) that satisfies the integral equation (8.122) is identical to thematched 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) =to∫

0

q∗(to − t − τ ) f ∗(τ )dτ , (8.126)

3Every 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 kernelK (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)dxdy > 0 for any f (x).


and consider the integral

to∫0

K (τ1, τ2)h(τ2)dτ2, (8.127)

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

to∫0

K (τ1, τ2)

to∫0

q∗(to − τ2 − τ ) f ∗(τ )dτ2dτ

=to∫

0

to∫0

q∗(t − τ1)q (t − τ2)

to∫0

q∗(to − τ2 − τ ) f ∗(τ )dτ2dτdt

=to∫

0

q∗(t − τ1)

to∫0

to∫0

q (t − τ2)q∗(to − τ2 − τ ) f ∗(τ )dτ2dτdt.

(8.128)

Let to − τ2 = u, so τ2 = to − u, and the above integral becomes

to∫0

q∗(t − τ1)

to∫0

to∫0

q (t − to + u)q∗(u − τ ) f ∗(τ )du dτ dt. (8.129)

But by definition

to∫0

q (u − (to − t)) q∗(u − τ )du = K ∗(to − t, τ ). (8.130)

Thus (8.129) becomes

to∫0

q∗(t − τ1)

to∫

0

K (to − t, τ ) f (τ )dτ

∗

dt

=to∫

0

q∗(t − τ1)(λ f ∗(to − t))dt, (8.131)


where we have made use of (8.122). Finally, let to − t = v, then t =to − v, and hence

λ

to∫0

q∗(t − τ1) f ∗(to − t)dt = λ

to∫0

q∗(to − τ1 − v) f ∗(v)dv = λh(τ1),

(8.132)

where we have made use of (8.126). Thus from (8.127)–(8.132) we haveshown that

to∫0

K (τ1, τ2)h(τ2)dτ2 = λh(τ1), 0 < τ1 < to , (8.133)

where f (t) in (8.126)–(8.131) is the actual eigenfunction in (8.122) cor-responding to any eigenvalue λ there. Obviously, (8.133) is true evenwhen f (t) corresponds to the largest eigenvalue λmax, and in that caseho (t) is the desired matched filter.

Figures 8.24 and 8.26 show the optimal input waveforms for variouschannel characteristics with details as indicated there. In Figure 8.24,the channel is assumed to be the usual rectangular pulse with widthTo , and the optimum transmit waveform f (t) for to = To obtainedby solving (8.122) and the corresponding matched filter responses areshown in Figure 8.24 (b)–(d). Figure 8.24 (e) shows the optimum inputpulse shape f (t), for different values of to .4 Similarly, Figures 8.25 and8.26 exhibit the desired transmit pulse f (t) as well as the correspond-ing matched filters for various channel characteristics as shown there.

This leads us to an interesting question: for a given channel, is there abest value for to at which decision should be made? If we use the outputSNR ρ(to ) as the optimality criterion, this question can be answeredby examining ρ(to ) vs. to .

It is easy to see that ρ(to ) is in fact a monotonically nondecreasingfunction 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 somet1 > to , the optimum eigenvector f1(t) associated with t1 leads to asmaller SNR compared to the optimum eigenvector fo (t) associatedwith t = to . In that case, since so (t) = fo (t) ∗ q (t), s1(t) = f1(t) ∗ q (t),

4For the decision instant to , since f (t) exists only in the interval 0 < t < to ,it should be easy to identify the corresponding input pulse in Figure 8.24 (e) forvarious value of to .


q(t) f(t)

f(t)

To to = To

to + To

To 2To 3To 4To To 2To 3To 4To

to

to

to

t t

tt

t

(a) Channel impulse response (b) Optimum pulse f(t) for to = To

s(t) = q(t) * f(t) h(t) = s(to - t)u(t)

(c) Channel output (d) Matched filter for to = To

(e) Optimum pulse f(t) for different values of to

(f) Output SNR for different values of to

10

15

20

25r (to)

dB

FIGURE 8.24 Optimum transmit pulse and matched filter for a rectangularchannel response.

and if we have

ρ(to ) =to∫

0

|so (t)|2 dt > ρ(t1) =t1∫

0

|s1(t)|2 dt, (8.135)

by redefining

f1(t) =

fo (t), 0 < t ≤ to ,

0, to < t ≤ t1,(8.136)


To 2To 3To 4To

(d) Output SNR for different values of to

10

15

20

25

r(to)

dB

t

t t

(c) Matched filter for to = To

h(t)

q(t) f(t)

(a) Target impulse response (b) Optimum pulse f(t) for to = To

to = ToTo

To

to

FIGURE 8.25 Optimum transmit pulse and matched filter for anexponentially decaying target response.

t

q(t)

h(t)

tTo

(a) Target impulse response (b) Optimum pulse f(t) for to = To

to = To

t

f(t)

ToTo 2To 3To 4To

−505

101520

dB

(d) Output SNR for different values of to(c) Matched filter for to = To

r (to)

to

FIGURE 8.26 Optimum transmit pulse and matched filter for an arbitrarytarget.


the inequality in (8.134) can be satisfied with equality. Thus (8.134) isalways true, indicating that under the joint optimality condition, de-laying the decision instant to cannot make it worse in terms of discrim-inating 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 ofto . These figures indicate that ρ(to ) → ρ(∞), the maximum possiblevalue 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 aspecific waveform s(t) and maximization of the output SNR at t = toleads to the matched filter receiver. Since the output SNR dependson f (t), further improvement in the output SNR can be obtained byexploiting the freedom present in selecting f (t) at the transmitter.Thus for a given channel, by combining the matched filter designtogether with the input pulse design problem, we have been able toachieve joint optimal transmitter–receiver design. The optimum inputpulse will always be minimum phase, and moreover in the white noisecase the matched filter is identical to the optimum input pulse.

The key idea in this section is to make use of the kernel of thechannel characteristics to define an optimal input waveform, and aswe show in the next section, this approach has been used to solveanother interesting problem in communication theory.

8.4 Joint Time Bandwidth OptimizationSuppose a pulse e(t) ↔ E(ω) of duration to needs to be transmittedthrough a channel with impulse response q (t) ↔ Q(ω). Let r (t) ↔R(ω) denote the channel output.

Then

r (t) = e(t) ∗ q (t) =to∫

0

e(τ )q (t − τ )dτ , (8.137)

or, in terms of their transforms

R(ω) = E(ω)Q(ω). (8.138)

Suppose −Bo < ω < Bo represents the desired spectral band atthe receiver as shown in Figure 8.27(b). We would like to choose theinput signal e(t) such that the output signal r (t) has maximum energyconcentrated in the above band.

This formulation makes sense in communication problems for mini-mizing cochannel interference, since the above design criterion makessure that most of the energy of the transmitted signal will be concen-trated in the desired band (−Bo, Bo ), and hence very little energy willspill over to the adjacent bands. In a different context, we may be


t

e(t)

to

(a) (b)

e(t) q(t) r(t)Q(w)

w

|E(w)|2

Bo-Bo

Desiredband

FIGURE 8.27 Joint time-bandwidth optimization.

interested in finding the best time-limited signal that also has max-imum energy in a given band. Since time-limited signals cannot beband limited, the next best thing that one could hope for in terms ofbandlimiting a time-limited signal is to maximize the energy within agiven bandwidth [2].

Returning to the original problem, the objective is to select the bestpulse e(t) such that the output energy

E = 12π

Bo∫−Bo

|R(ω)|2 dω = 12π

Bo∫−Bo

|Q(ω)|2 |E(ω)|2 dω (8.139)

in the spectral band (−Bo, Bo ) is maximized. Since simple scaling ofe(t) can increase the value of the above E , it is necessary to normalizee(t) by requiring it to possess unit (or, constant) energy. Thus ourproblem reduces to maximizing (8.139) by selecting a suitable time-limited signal e(t), 0 < t < to under the energy constraint

to∫0

|e(t)|2 dt = 1. (8.140)

To solve this problem, once again we can make use of the conceptof the channel kernel discussed in the previous section. However, itis necessary to bring in the desired bandwidth information into thiskernel in some suitable manner.

To express this problem in the time domain, we can make use ofParseval’s theorem. If we define

(ω) = |Q(ω)|2 E(ω), −Bo < ω < Bo

0, otherwise,(8.141)


Bo−Bo

|Q(w)|2

t

q(t)

w

FIGURE 8.28 Inverse Fourier transform of the channel restricted to(−Bo < ω < Bo).

then using Parseval’s theorem in (8.139) we obtain

E = 12π

+∞∫−∞

(ω)E∗(ω)dω =+∞∫

−∞ψ(t)e∗(t)dt =

to∫0

ψ(t)e∗(t)dt,

(8.142)

where ψ(t) represents the inverse transform of (ω) defined above. Tosimplify ψ(t), define K (τ ) to represent the inverse Fourier transformof the channel transfer function gain |Q(ω)|2 restricted to the frequencyband (−Bo < ω < Bo ) as in (8.141). Thus (Figure 8.28)

K (τ ) = 12π

Bo∫−Bo

|Q(ω)|2 e jωτ dω = K ∗(−τ ). (8.143)

The function K (τ ) acts as the kernel associated with the channeltransfer function gain |Q(ω)|2 over the desired bandwidth (−Bo, Bo ).Note that K (τ ) represents a stationary kernel (autocorrelation func-tion), since it is a function of only one argument.

Using (8.141)–(8.143) and ψ(t) ↔ (ω) we get

ψ(t) = K (t) ∗ e(t) =+∞∫

−∞K (t − τ )e(τ )dτ =

to∫0

K (t − τ )e(τ )dτ , (8.144)

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 usingthis, the positive quantity E in (8.142) can be represented as an innerproduct. Thus

E =to∫

0

ψ(t)e∗(t)dt, (8.145)


and once again using Schwartz’ inequality, the above expression gives

E ≤ to∫

0

|ψ(t)|2 dt

to∫0

|e(t)|2 dt

1/2

= to∫

0

|ψ(t)|2 dt

1/2

, (8.146)

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

to∫0

K (t − τ )e(τ )dτ = λe(τ ), 0 ≤ t ≤ to . (8.148)

In this case

E = λ

to∫0

|e(t)|2 dt = λ. (8.149)

Since the right side of the above equation is independent of e(t), tomaximize E , the eigenvalue λ in (8.148) must be chosen as the largestpossible value. Thus the optimal input waveform e(t) that maximizesE is the solution of the integral equation in (8.148) corresponding tothe largest eigenvalue of λ, where the positive-definite kernel K (t −τ )is as defined in (8.143). From (8.146),

Emax=

√√√√√ to∫0

|ψmax(t)|2 dt = λmax (8.150)

represents the maximum value for the energy, where ψmax(t) is asdefined in (8.144) with e(t) representing the eigenfunction associatedwith the largest eigenvalue in (8.148). Since this situation is identi-cal to that in (8.101), using the arguments developed in that sectionit follows that to maximize Emax, the input waveform e(t) must benecessarily minimum phase. Thus the above optimum solution givenby the eigenfunction associated with the largest eigenvalue of (8.148)must be necessarily minimum phase. In addition, it is easy to showthat if e(t) satisfies the integral equation (8.148), then so does

f (t) = e∗(to − t), 0 ≤ t ≤ to . (8.151)


This follows easily, since

to∫0

K (t − τ ) f (τ )dτ =to∫

0

K (t − τ )e∗(to − τ )dτ

= to∫

0

K (to − t − x)e(x)dx

∗

= λe∗(to − t) = λ f (t), 0 < t < to , (8.152)

where we have used the substitution to − τ = x, and made use ofthe symmetric nature of the kernel given in (8.143). Thus if λ rep-resents 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 thesense that it agrees with its matched filter with respect to to . Alterna-tively, 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 eigenfunctioncorresponding to the largest eigenvalue has no zeros in the strict righthalf plane. Consequently it is also free of zeros in the strict left halfplane. Thus all of its zeros must lie on the jω − axis.

To summarize, the best input pulse e(t) of finite duration to thatalso maximizes the channel output energy over the band (−Bo, Bo )is given by the eigenfunction associated with the largest eigenvalueλmax(to ) of the integral equation in (8.148). It represents a minimumphase symmetric function with all the zeros of its Laplace transformE(s) distributed along the jω − axis. The maximum possible valueof the energy in (−Bo, Bo ) is given by λmax(to ). Note that the channelcharacteristics comes into play in the kernel through |Q(ω)|2 alongwith the desired frequency band (−Bo, Bo ).

Once again, it is easy to show that λmax(to ) is a monotone nonde-creasing function of to . To see this, using (8.146), we have

Emax = λmax(to ) =

√√√√√ to∫0

|ψ(t)|2 dt, (8.155)


where ψ(t) is given by (8.144). With t1 > to , let e1(t) and e(t) representthe solution of (8.148) for t1 and to respectively. Thus

λmax(t1) =

√√√√√ t1∫0

|ψ1(t)|2 dt (8.156)

where ψ1(t) = ∫ t10 K (t − τ )e1(τ )dτ , and if λmax(t1) < λmax(to ), then it

is easy to satisfy this inequality with an equality by redefining e1(t) as

e1(t) =

e(t), 0 < t < to0, otherwise.

(8.157)

Thus if t1 > t0, then λmax(t1) ≥ λmax(to ), which establishes the mono-tone nondecreasing nature of λmax(to ).

To illustrate the scheme presented here, Figures 8.29 and 8.30 showthe optimum waveform e(t) for a typical low-pass channel gain func-tion |Q(ω)|2 given by

|Q(ω)|2 = 11 + ω2 (8.158)

for different values of to and Bo . In this case, from (8.143) we have

K (τ ) = 1π

Bo∫0

cos ωτ

1 + ω2 dω, (8.159)

1.5

1

0.5

0 0.5 1 1.5 2 2.5t

e(t)

FIGURE 8.29 Optimum e(t) for to = 0.5, 1, 1.5, 2, and 2.5 with Bo fixed to0.5 normalized bandwidth.


1.1

1

0.9

0 0.2 0.4 0.6 0.8 1t

e(t)

Bo

FIGURE 8.30 Optimum e(t) for Bo = 0.2, 0.4, 0.6, 0.8, and 1 with to fixedequal 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 Bofixed equal to the normalized bandwidth of 0.5. Similarly, Figure 8.30shows the corresponding e(t) for different values of Bo with to fixedequal to unity. Notice the symmetric nature of these waveforms. Asnoted earlier, in all these cases the transforms of these waveforms haveall their zeros on the jω − axis.

Figure 8.31 shows the square magnitude of the transforms of theoptimum waveforms in Figure 8.30 as a function of the normalizedfrequency 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-limitedsignal that is also maximally band-limited. In other words, this isequivalent to finding the best time-limited signal of unit energy, thathas maximum energy within a given bandwidth Bo as in Figure 8.32.

Thus, the problem is to maximize

E = 12π

Bo∫−Bo

|E(ω)|2 dω (8.160)


0

−20

0 0.5−0.5−60

−1 1

−40

|E(w)|2 forBo = 0.2, 0.4, 0.6, 0.8, and 1

|Q(w)|2

dB

w

FIGURE 8.31 Transforms of optimum waveform forBo = 0.2, 0.4, 0.6, 0.8, and 1 with to = 1 vs. |Q(ω)|2.

subject to

to∫0

|e(t)|2 dt = 12π

+∞∫−∞

|E(ω)|2 dω = 1. (8.161)

Notice that E in (8.160) represents the energy compaction ratio in-side the frequency band (−Bo, Bo ) for e(t), and it has the same formas (8.139), with |Q(ω)|2 as shown in Figure 8.33.

From (8.143), this gives the corresponding kernel to be

K (τ ) = 12π

Bo∫−Bo

e jωτ dτ = 1π

sin Boτ

τ, (8.162)

to

e(t) |E(w)|2

tBo−Bo

w

FIGURE 8.32 Optimum pulse of unit energy that has maximum energywithin a given bandwidth.


–Bo

wBo

1

|Q(w)|2FIGURE 8.33 Flatspectrum for|Q(ω)|2.

and the desired signal e(t) is given by the eigenfunction associatedwith the largest eigenvalue of the integral equation [6]

to∫0

1π

sin Bo (t1 − t2)t1 − t2

e(t2)dt2 = λe(t1), 0 < t1 < t0. (8.163)

By a simple change of variables (t = Bot1, τ = Bot2), we can rewritethe above equation as

1π

to Bo∫0

sin(t − τ )t − τ

e(τ )dτ = λe(τ ), 0 < t < t0 Bo, (8.164)

which shows that the time-bandwidth product η= to Bo is the only

independent parameter in this problem. Solutions of the integralequation

1π

η∫0

sin(t − τ )t − τ

e(τ )dτ = λ(η)e(τ ), 0 < t < η (8.165)

for various values of the time-bandwidth product η represent the wellknown prolate–spheroidal functions [5]. These functions are plotted in

FIGURE 8.34 Energy compaction ratio for prolate–spheroidal functions.


0

0

–1

–2

–3

–4

2 4

dB

6 8 10h

lmax (h)FIGURE 8.35λmax(η) as afunction of η withto fixed equal to 1.

Figure 8.34 for various values of η. As pointed out earlier, the Laplacetransform 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 tounity.

As η → ∞, λmax(η) → 1, since this is equivalent to fixing t = to andletting Bo → ∞. In that case, E in (8.160) tends to unity as well.

Appendix 8-A: Transform of a Chirp SignalConsider the chirped rectangular symmetric pulse (noncausal)

fo (t) =

e jβt2, −T/2 < t < T/2,

0, otherwise,(8A.1)

whose transform is given by

Fo (ω) =T/2∫

−T/2

e jβt2e− jωtdt = e− jω2/4β

T/2∫−T/2

e jβ(

t− ω2β

)2

dt. (8A.2)

Let√

β(t − ω

2β

) = τ . Then

Fo (ω) = 1√β

e− jω2/4β

(βT−ω)/2√

β∫−(βT+ω)/2

√β

e jτ 2dτ , (8A.3)


or

Fo (ω) = 1√β

e− jω2/4β

(βT−ω)/2√

β∫0

e jτ 2dτ +

(βT+ω)/2√

β∫0

e jτ 2dτ

. (8A.4)

Define the Fresnel integral

K (ω) =√

2π

ω∫0

e jt2dt (8A.5)

so that

K (∞) =√

2π

∞∫0

e jt2dt = 1

2

√2π

∞∫−∞

e jt2dt

= 1√2π

√jπ =

√j2

= 1√2

e jπ/4, (8A.6)

and hence

∞∫0

cos t2dt =∞∫

0

sin t2dt = 12

√π/2. (8A.7)

In terms of (8A.5), (8A.4) becomes

Fo (ω) = e− jω2/4β

√π

2β

K(

βT + ω

2√

β

)+ K

(βT − ω

2√

β

)(8A.8)

and it represents the transform of the pulsed symmetric chirp signalin (8A.1). Notice that near the baseband region, (i.e., for

√βT 1)

K(

βT + ω

2√

β

)≈ K

(√βT2

)≈ K (∞) =

√j/2. (8A.9)

Hence for large values of√

βT , in the baseband region (see (8A.6))

Fo (ω) ≈√

π

2β

(√j2

+√

j2

)(8A.10)

or

|Fo (ω)| ≈√

π

β, |ω| < βT (8A.11)


bT−bT

|Fo(w)|

w

p | b

Bs = 2bT

FIGURE 8.36Transform of thechirp signalfo(t) = ejβt2

,−T/2 < t < T/2.

and for ω > βT (use K (−ω) = −K (ω))

K(

βT − ω

2√

β

)= −K

(ω − βT

2√

β

)≈ −K (ω) (8A.12)

and hence

Fo (ω) = 0, ω > βT. (8A.13)

Hence

|Fo (ω)| =√

π/β, |ω| < βT,

0, |ω| > βT.(8A.14)

As Figure 8.36 shows, the chirp signal in (8A.1) has constant mag-nitude in the baseband region, and it is essentially zero outside thatregion. Notice that even in the passband the magnitude goes down as1/√

β for the chirp.Next consider the modulated chirped pulse signal

f (t) = e j (ωo t+βt2) , − T2

< t <T2

. (8A.15)

Then

F (ω) = Fo (ω − ωo ). (8A.16)

Hence we get

F (ω) = e− j (ω−ωo )2/4β

√π

2β

[K(

ω − ω1

2√

β

)+ K

(ω2 − ω

2√

β

)](8A.17)

where

ω1 = ωo − βT, ω2 = ωo + βT. (8A.18)


|F(w)|

wwo w2 = wo + b Tw1 = wo − bT

p | b

Bs = 2b T

FIGURE 8.37Transform of thechirp signalf(t) = ej(ωot+βt2) ,−T

2 < t < T2 .

Once again for large values of√

βT compared to ω, we have

|F (ω)| ≈ 2√

π

2β

∣∣∣∣K(√

βT2

)∣∣∣∣ ≈√

π

2β2

∣∣∣∣∣√

j2

∣∣∣∣∣ =√

π

β, (8A.19)

and for |ω − ωo | βT , we have

F (ω) ≈√

π

2βK (ω) + K (−ω) ≈ 0, for ωo − βT > ω > ωo + βT.

(8A.20)Hence (see Figure 8.37)

|F (ω)| =√

π/β, ωo − βT < ω < ωo + βT,

0, otherwise.(8A.21)

To obtain the transform of a causal chirp, define using (8A.15)

f1(t) = f (t − T/2) = e jωo (t−T/2)+β(t−T/2)2

= e− j (2ωo T−βT2)/4e j (ωo−βT)t+βt2, 0 < t < T. (8A.22)

Notice that f1(t) has a new carrier at

o = ωo − βT (8A.23)

rather than at ωo . Hence with φo = (2ωo T − βT2)/

4, we have

f1(t) = e− jφo e j (o t+βt2) , 0 < t < T (8A.24)

has a transform equal to F (ω)e− jωT/2. Hence

F1(ω) = F (ω)e− jωT/2. (8A.25)


|F2(w)|

wΩo + bTΩo Ωo + 2bT

p|b

FIGURE 8.38Transform of thechirp signal

f2(t) = ej(ot+βt2) ,

0 < t < T.

Let (see Figure 8.38)

f2(t) = e j (o t+βt2) , 0 < t < T. (8A.26)

From (8A.24) we have

f2(t) = e jφo f1(t), 0 < t < T, (8A.27)

and hence

F2(ω) = e jφo F1(ω) = e jφo e− jωT/2 F (ω)

= e j(2ωo T−βT2)/4e− jωT/2e− j (ω−ωo )2/4βG(ω)

= e− j( (ω−ωo )2

4β+ βT2

4 + (ω−ωo )T2

)G(ω)

= e− j(

ω−ωo2√

β+

√βT2

)2

G(ω)

= e− j(

ω−(ωo−βT)2√

β

)2

G(ω) = e− j(

ω−o2√

β

)2

G(ω)

(8A.28)

where from (8A.17)

G(ω) =√

π

2β

K(

ω − o

2√

β

)+ K

(o + 2βT − ω

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 zeroelsewhere.

As Figure 8.38 shows, when a carrier modulated chirp as in (8A.26)is transmitted in 0 < t < T (causal waveform), its nonzero transformregion is in (o , o + 2βT) where o represents the carrier frequency.Thus, in standard form (see Figure 8.39)

f3(t) =

e j (ωo t+βt2) , 0 < t < T,

0, otherwise(8A.30)


|F3(w)|

wwo wo + b T wo + 2b T

p|b

Bs = 2bT

FIGURE 8.39Transform of thechirp signal

f3(t) = ej(ωot+βt2) ,

0 < t < T.

has the transform (see (8.29) and (8.30))

F3(ω) = e− j(

ω−ωo2√

β

)2√π

2β

[K(

ω − ωo

2√

β

)+ K

(ωo + 2βT − ω

2√

β

)].

(8A.31)

To down convert F3(ω) to baseband, one must translate it by thecenter frequency ωo + βT , and not by the carrier frequency ωo alone.Equation (8A.31) may be also rewritten as

F3(ω) = e− j(

ω−ωo2√

β

)2√1β

x∫x−√

βT

e jt2dt (8A.32)

where

x = ω − ωo

2√

β. (8A.33)

Finally since

cos(ωo t + βt2) = f3(t) + f ∗3 (t)

2, (8A.34)

from (8A.30)–(8A.34) we have the transform pair (see Figure 8.40)

f4(t) = cos(ωo t + βt2), 0 < t < T ↔ F4(ω) = F3(ω) + F ∗3 (ω)

2.

(8A.35)From (8A.32)

F ∗3 (−ω) = e

− j(

ω+ωo2√

β

)2√1β

y+√βT∫

y

e− j t2dt (8A.36)


|F4(w)|

w

p|b| 2

Bs = 2bT

−wo wo

wo + b T

−(wo + 2bT )

−(wo + b T )

cos (wot + b t 2 )

wo + 2bTt

FIGURE 8.40 Transform of a real and causal chirp signalf4(t) = cos(ωot + βt2), 0 < t < T.

where y = ω+ωo2√

β. Using (8A.30)–(8A.34) we get

|F4(ω)| = 1

2

√π/β, ωo < |ω| < ωo + 2βT,

0, otherwise.(8A.37)

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 Signal-

Dependent Interference and Channel Noise,” IEEE Transactions on InformationTheory, Vol. 46, No. 2, pp. 577–584, March 2000.

[3] D.C. Youla, “Chirp and Pulse Compression,” Private Communications, Poly-technic 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 Analysisand 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.


C H A P T E R 9Advanced Topics

The motion of an infinitesimal body when it is attracted by two finitebodies that revolve around their common center of mass is an impor-tant configuration for deep space-based platforms such as satellites,space stations, and SBRs. Applications of such a configuration includedetermining locations to park space stations and Space Based Radarsin the Sun-Earth, or the Earth-Moon frame for long-time surveillanceover a considerable period. Obviously stability of these artificial ob-jects is an important goal and in that context preferential stable orbitsare of great importance.

This situation falls under the problem of three bodies, where a num-ber of important results have been established rigorously under spe-cific initial conditions on the positions and velocities of the three bod-ies. For example, the particular solutions of the motion of three finitebodies such that the ratios of their mutual distances are constants werefirst given by Lagrange in a prize memoir in 1772. The theorem dueto Lagrange states that it is possible to start three finite bodies in sucha manner that their orbits will be similar ellipses, all described in thesame time [1].

The general three-body problem is insolvable, i.e., given the posi-tions and velocities of three mass points at a certain time that attracteach other according to Newton’s second law, it is impossible to pre-dict the future progress of their motion. Thus in general, for such asystem it is no longer possible to state with certainty that no memberwill drift away to infinity (stability of the system), or that collisionsbetween two or more mass points will not occur. However, a specialcase is of considerable interest—the motion of an infinitesimal bodyaround two finite bodies, and more definite statements can be maderegarding their motion.



9.1 An Infinitesimal Body Around TwoFinite BodiesAn infinitesimal body is attracted by two finite bodies, but it does notattract the finite bodies. Consider two finite bodies (such at the Sunand Earth pair, Sun and Jupiter pair, or Earth and Moon pair) revolvingin circles1 around their common center of mass with the infinitesimalbody subject to their combined attraction. Assume that the unit ofmass is chosen so that the sum of the masses of the two finite bodiesis 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 distancebetween the finite bodies is unity, and the unit of time is adjusted sothat the gravitational constant G is unity.

Assume the center of mass of the finite bodies represent the origin ofthe coordinate system, with the α-β plane representing the plane of themotion of these finite bodies. Let the coordinates of the finite bodies1 − 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 bodycan be written as

d2α

dt2 = −(1 − µ)α − α1

r31

− µα − α2

r32

, (9.1)

d2β

dt2 = −(1 − µ)β − β1

r31

− µβ − β2

r32

, (9.2)

andd2γ

dt2 = −(1 − µ)γ

r31

− µγ

r32

(9.3)

where

r1 =√

(α − α1)2 + (β − β1)2 + γ 2, (9.4)

r2 =√

(α − α2)2 + (β − β2)2 + γ 2. (9.5)

From (3.57), the mean angular motion of the finite bodies is given by

n = 2π

T=

√G(M + m)

a3/2 = 1, (9.6)

since G ≡ 1, M + m = 1, and a = 1. Thus the finite bodies are ro-tating in the α-β plane with uniform angular velocity of unity and

1For more realistic elliptical orbits treatment, refer to F. R. Moulton, “PeriodicOrbits”.

C h a p t e r 9 : A d v a n c e d T o p i c s 395

y

r1r2

x1 = −m x2 = 1 − m

m1 = 1 − m m2 = m

Infinitesimal body(x, y )

x

FIGURE 9.1Infinitesimal bodyaround two finitebodies.

let (x, y, z) represent the coordinates of the infinitesimal body in thisrotating system with the new axes having the same origin as theold ones. Then the coordinates of the new system are given by (seeFigure 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 ofthe finite bodies in the rotating system. From (9.7)

dα

dt= dx

dtcos t − x sin t − dy

dtsin t − y cos t (9.11)

so that

d2α

dt2 = d2xdt2 cos t − 2

dxdt

sin t − x cos t

−d2 ydt2 sin t − 2

dydt

cos t + y sin t. (9.12)


Substituting (9.7), (9.10), and (9.12) into (9.1) we get(d2xdt2 − 2

dydt

− x)

cos t −(

d2 ydt2 + 2

dxdt

− y)

sin t

= −

(1 − µ)x − x1

r31

+ µx − x2

r32

cos t

+

(1 − µ)y − y1

r31

+ µy − y2

r32

sin t (9.13)

where

r1 =√

(x − x1)2 + (y − y1)2 + z2 (9.14)

and

r2 =√

(x − x2)2 + (y − y2)2 + z2. (9.15)

Similarly using the derivatives of (9.8) and (9.10)–(9.12) in (9.2) weget (

d2xdt2 − 2

dydt

− x)

sin t +(

d2 ydt2 + 2

dxdt

− y)

cos t

= −

(1 − µ)x − x1

r31

+ µx − x2

r32

sin t

−

(1 − µ)y − y1

r31

+ µy − y2

r32

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 toobtain

d2xdt2 − 2

dydt

− x = −(1 − µ)x − x1

r31

− µx − x2

r32

(9.17)

and

d2 ydt2 + 2

dxdt

− y = −(1 − µ)y − y1

r31

− µy − y2

r32

. (9.18)

Finally from (9.9), (9.10), and (9.3) we also obtain

d2zdt2 = −(1 − µ)

zr3

1− µ

zr3

2. (9.19)


If the x-axis is chosen to continuously pass through the centers ofthe finite bodies, we have

y1 = 0, y2 = 0, (9.20)

and we get the simplified equations of motion

d2xdt2 − 2

dydt

= x − (1 − µ)x − x1

r31

− µx − x2

r32

, (9.21)

d2 ydt2 + 2

dxdt

= y − (1 − µ)y

r31

− µy

r32

, (9.22)

and

d2zdt2 = −(1 − µ)

zr3

1− µ

zr3

2(9.23)

where

r1 =√

(x − x1)2 + y2, r2 =√

(x − x2)2 + y2. (9.24)

From (9.21)–(9.23), in general the motion of the infinitesimal body isa sixth-order problem, and it becomes fourth order if the body movesin the plane of the motion of the finite bodies.

The only general integral solution to (9.21)–(9.23) was first given byJacobi (1843) by definiting

U(x, y, z) = 12

(x2 + y2) + 1 − µ

r1+ µ

r2. (9.25)

In terms of U(x, y, z), (9.21)–(9.23) becomes

d2xdt2 − 2

dydt

= ∂U∂x

, (9.26)

d2 ydt2 + 2

dxdt

= ∂U∂y

, (9.27)

d2zdt2 = ∂U

∂z. (9.28)

Following Jacobi, multiply (9.26)–(9.28) by 2 dxdt , 2 dy

dt , and 2 dzdt respec-

tively, and sum those equations to obtain

ddt

(dxdt

)2

+(

dydt

)2

+(

dzdt

)2

= 2(

∂U∂x

dxdt

+ ∂U∂y

dydt

+ ∂U∂z

dzdt

)

= 2dU(x, y, z)

dt(9.29)


and upon integrating, we get the desired general integral to be(dxdt

)2

+(

dydt

)2

+(

dzdt

)2

= 2U − C = x2 + y2 + 2(1 − µ)r1

+ 2µ

r2− C.

(9.30)

But (dxdt

)2

+(

dydt

)2

+(

dzdt

)2

= V2 (9.31)

represents the square of the velocity of the infinitesimal body, so thatthe integral solution due to Jacobi reduces to [1]

V2 = 2U − C = x2 + y2 + 2(1 − µ)r1

+ 2µ

r2− C, (9.32)

and it represents a relation between the velocity and coordinates ofthe infinitesimal body in the rotating frame. Once the initial conditionsdetermine the constant of integration C , equation (9.32) determinesthe velocity of the infinitesimal body for all locations, or for a given ve-locity the corresponding locations of the infinitesimal body are givenby (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 realand consequently the region where the infinitesimal body can movearound, and the other side represents prohibited region for the bodysince the velocity is imaginary (V2 < 0) there.

From (9.32), the equation of surface of zero velocity equals

x2 + y2 + 2(1 − µ)r1

+ 2µ

r2= C (9.33)

with r1 and r2 as in (9.14)–(9.15), and (9.20). As z → ∞, (9.33) reduces to

x2 + y2 = C (9.34)

which represents a cylinder with radius√

C and whose axis coin-cides with the z-axis. Hence all surfaces represented by (9.33) are con-tained within the above cylinder. These surface are symmetric withrespect to the x-y and x-z planes since only the square of y and z ap-pear in (9.33). In the x-y plane, we have z = 0, and the surface curvesin the x-y plane are given by

x2 + y2 + 2(1 − µ)√(x − x1)2 + y2

+ 2µ√(x − x2)2 + y2

= C. (9.35)


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 leftside of (9.35) are unimportant, and the equation reduces to

x2 + 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 valuesof C , corresponding large values of x and y satisfy (9.36), and henceε1 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, thefirst two terms are negligible and the equation reduces to

1 − µ√(x − x1)2 + y2

+ µ√(x − x2)2 + y2

= C2

− ε2(x, 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 whichreads (x − x1)2 + y2 C ′, etc.), and for small values of C , these ovalsmerge forming dumbbell-shaped figures as in Figures 9.3 and 9.4.

Finally, for even smaller values of C the dumbbell shape enlarges tobecome an oval that covers both the bodies. These surfaces for variousvalues of C are shown in Figure 9.4.

Interestingly, for large values of C , since the ovals around the finitebodies are closed and if the velocity is positive inside, it will be posi-tive everywhere inside those ovals. As a result, an infinitesimal body


100

50

10

0.4

0

−0.4

y

x0 0.5−0.5

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 theresince it could not cross a surface of zero velocity. Motion of the Moonin the Sun-Earth frame corresponds to this situation and it was inthis manner that Hill proved that the Moon cannot recede indefinitelyfrom Earth and its distance from Earth has a finite upper limit.

FIGURE 9.4 Equation of surface for small values of x and y.


9.1.1 Particular Solutions ofthe Three-Body Problem

If the infinitesimal body is placed on a surface of zero velocity, depend-ing on the value of the acceleration terms in (9.21)–(9.23), the body willmove inward into the permissible region unless the acceleration termsthemselves are zeros. In that case, the infinitesimal body with zero ve-locity will remain forever relatively at rest, unless distorted by someexternal force.

To examine these particular solutions, where the infinitesimal bodyin a three body system stays relatively at rest forever, we can equateboth 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 theJacobi function in (9.25) to zero, i.e.,

∂U∂x

= 0,∂U∂y

= 0,∂U∂x

= 0. (9.38)

This gives (use (9.25))

x − (1 − µ)x − x1

r31

− µx − x2

r32

= 0, (9.39)

y − (1 − µ)y

r31

− µy

r32

= 0, (9.40)

(1 − µ)z

r31

+ µz

r32

= 0. (9.41)

Clearly, (9.41) is satisfied only for z = 0 which shows that all theseparticular solutions lie in the x-y plane. They are special cases of theLagrangian solutions known as the Lagrange libration points, wherethe 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 substitutedinto (9.39) gives

ψ(x) = x − (1 − µ)x − x1

|x − x1|3− µ

x − x2

|x − x2|3= 0. (9.42)

As Figure 9.5 shows, (9.42) is positive for x = +∞, negative atx = 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 atL1, once between x2 and +∞ at L2, once between −∞ and x1 at L3.These three Lagrange points are all on the x-axis, and the infinitesimalbody under proper initial conditions will remain stationary at theselocations.


xx2x1 L1 L2L3

m1 = 1 − m m2 = m

y (x)

FIGURE 9.5 Three co linear Lagrange points on the x-axis.

Since x1 = −µ, x2 = 1 − µ, the three real Lagrange solutions L1,L2, and L3 of (9.42) can be expressed in terms of µ. To obtain explicitexpressions for them, consider the solution L1 located between x1 andx2, and let ρ represent the actual distance of L1 from the second massm2 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) = ρ5 − (3 − µ)ρ4 + (3 − 2µ)ρ3 − µρ2 + 2µρ − µ = 0. (9.44)

Equation (9.44) has five sign variations among its coefficients, andhence applying Descartes’ theorem2 it has at least one real positiveroot. Further, for µ = 0, the above equation reduces to

ρ3(ρ2 − 3ρ + 3) = 0 (9.45)

which has three roots at ρ = 0 and two complex roots. Hence forsufficiently small values of µ, the three roots of (9.44) are expressibleas powers of µ1/3, one of which is real and positive and the other twoare complex conjugate pairs. Thus the only real and positive root of

2Descartes’ Theorem: If ν represents the sign variations among the coefficientsof a polynomial with real coefficients, and p the number of its positive real roots,then ν = p + k, where k is an even number.


(9.44) has the form

ρ = aµ1/3 + bµ2/3 + · · · (9.46)

Substituting this into (9.44) and equating the coefficients of the low-est powers of µ in the expansion to zero, we obtain

(3a3 − 1)µ = 0, (−3a3 + 2 + 9ab)aµ4/3 = 0 (9.47)

which gives

a =(

13

)1/3

; b = − 19a

= −13

(13

)2/3

. (9.48)

Hence (9.46) yields [1]

ρ (

µ

3

)1/3− 1

3

(µ

3

)2/3+ · · · (9.49)

to be the approximate distance of L1 from the second mass located atx2.

Using a similar approach, it is easy to show that the second solutionL2 is located approximately at a distance of [1]

ρ (

µ

3

)1/3+ 1

3

(µ

3

)2/3+ · · · (9.50)

from the second mass at x2 in the other direction (toward infinity).Similarly, the distance of the L3 solution from the first mass at x1 canbe shown to be [1]

ρ 1 − 712

µ + · · · . (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 locationof the Lagrange points L1 and L2 to be (first-order approximation)

ρ a(

µ

3

)1/3 1, 496, 477 km 1.5 million km (9.52)

from Earth on either side along the Sun-Earth line. Clearly, the Moonlocated at around 384,403 km from Earth is well within the closed ovalsaround the Earth and hence as Hill has shown the Moon cannot re-cede beyond a certain limiting distance from Earth [2] (see Figures 9.3and 9.4).

For the Sun-Earth configuration, these linear solutions are of specialsignificance to NASA and several missions have been planned aroundthem. For example, the Solar Heliospheric Observatory (SOHO) andthe Advanced Composition Explorer (ACE) are currently positionedat the Sun-Earth L1 point that is ideally suited to make observations


Visiblegegenschein

MoonEarth

Earth’s orbit

L1L3

L2

Sun

FIGURE 9.6 The Gegenschein phenomenon.

about the Sun since these satellites are never shadowed by either theEarth or the Moon.

The Sun-Earth L2 point is a good location for Space Based Radarssince an object at L2 will maintain the same orientation with respect tothe Sun-Earth system and hence, calibration and shielding are muchsimpler in this case. The Wilkinson Microwave Anisotropy probe isalready at the Sun-Earth L2 location, and NASA plans to launch the fu-ture Herschel space observatory and the next generation James Webbspace telescope to the Sun-Earth L2 location. The Earth-Moon L2 lo-cation can be used for a communication satellite that covers the farside of Moon.

Interestingly, the linear solution L2 in the Sun-Earth system has beenpointed out to explain the hazy patch of light observed at the side ofEarth away from Sun, around the ecliptic, above the horizon knownas the Gegenschein (German for “counter-glow”), that was indepen-dently discovered by astronomers Brorsen (1855), Backhouse (1868),and Barnard (1875). It is a faint glow of light stretching about 10 onthe opposite side of the Sun, and hence it rises when the Sun sets reach-ing its peak at midnight (Figure 9.6). According to a suggestion madeby astronomer Gylden, meteors under the right initial conditions thatare passing near the Sun-Earth L2 point might get trapped aroundthat point for a while, and if a very large number of such meteorsare trapped in this manner around L2, their collective glow originat-ing from back-scattered sunlight would appear from Earth as a hazypatch of light with its center approximately at the anti-Sun location(very near to L2) similar to the observed Gegenschein phenomenon.No certain answers can be given to this explanation, although the


low brightness glow area at the anti-Sun location above the horizonis unmistakable on a dark night.

It is quite possible that there is such a collection at the Sun-Earth L3point as well. However, its verification is difficult from Earth since L3is behind the Sun on the other side away from the Earth.

To determine the location of other Lagrange points in the x-y planethat are not along the x-axis, we return to (9.40) and assume y = 0. Inthat case, (9.40) reduces to

1 − 1 − µ

r31

− µ

r32

= 0. (9.53)

Upon multiplying (9.53) with (x − x2) and (x − x1) respectively andsubtracting them from (9.39) we obtain

x2 − (1 − µ)x2 − x1

r31

= 0 (9.54)

and

x1 − µx1 − x2

r31

= 0. (9.55)

Since x1 = −µ, x2 = 1 − µ, x2 − x1 = 1, from (9.54)–(9.55) we get

1 − 1r3

1= 0 ⇒ r1 = 1 (9.56)

and

1 − 1r3

2= 0 ⇒ r2 = 1. (9.57)

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 L4 and L5 formequilateral triangles with the finite bodies (see Figure 9.7). Once again,under the right initial conditions, the infinitesimal body at these loca-tions will remain trapped there forever, provided they represent stablesolutions.

9.1.2 Stability of the Particular SolutionsThere are three particular solutions of the motion of the infinitesimalbody along the x-axis (L1, L2, and L3) and two off the x-axis in the x-yplane (L4 and L5) where it can remain relatively at rest in the absenceof external forces (Figure 9.7). Known as the Lagrange libration points,these solutions are potential places to build Space Based Radars andspace stations.


xL2x2x1

L5

L4

L1

r 1 =

1 r2 = 1

L3

m1 = 1 − m m2 = m

y (x )

FIGURE 9.7 The two Lagrange points L4 and L5 that form equilateraltriangles with the finite bodies.

An important question in this context is the stability of these par-ticular solutions that correspond to (9.21)–(9.23) with the velocity andacceleration terms there equal to zero. Hence, with

f (x, y, z) = x − (1 − µ)x − x1

r31

− µx − x2

r32

, (9.58)

g(x, y, z) = y − (1 − µ)y

r31

− µy

r32

, (9.59)

and

h(x, y, z) = −(1 − µ)z

r31

− µz

r32

, (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 partic-ular solutions L1 through L5 mentioned earlier.

If the infinitesimal body is located at a stable solution, then a slightdisplacement from the exact solution along with a small velocity andacceleration will only cause the small body to oscillate around thestable point for a considerable amount of time; however, if the solution


is unstable, the small body will eventually depart from the exact so-lution forever and drift away.

To examine the nature of stability of these particular solutions, thesmall body is given a small displacement and a small velocity so thatits coordinates and velocities are [1]

x = xo + u, y = yo + v, z = zo + w (9.62)

and

dxdt

= dudt

,dydt

= dvdt

,dzdt

= dwdt

;

d2xdt2 = d2u

dt2 ,d2 ydt2 = d2v

dt2 ,d2zdt2 = d2w

dt2 . (9.63)

Thus (9.21)–(9.23) can be rewritten as

d2udt2 − 2

dvdt

= f (xo + u, yo + v, zo + w), (9.64)

d2vdt2 + 2

dudt

= g(xo + u, yo + v, zo + w), (9.65)

and

d2wdt2 = h(xo + u, yo + v, zo + w). (9.66)

Using Taylor’s formula

f (xo + u, yo + v, zo + w)

= f (xo , yo , zo ) + ∂ f∂x

∣∣∣∣xo, yo ,zo

u + ∂ f∂y


v + ∂ f∂z


w + · · · , (9.67)

g(xo + u, yo + v, zo + w)

= g(xo , yo , zo ) + ∂g∂x


u + ∂g∂y


v + ∂g∂z


w + · · · , (9.68)

and

h(xo + u, yo + v, zo + w)

= h(xo , yo , zo ) + ∂h∂x


u + ∂h∂y


v + ∂h∂z


w + · · · (9.69)

If the disturbances u, v, w are very small at least initially, the secondand 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


reduce tod2udt2 − 2

dvdt

= ∂ f∂x

u + ∂ f∂y

v + ∂ f∂z

w, (9.70)

d2vdt2 + 2

dudt

= ∂g∂x

u + ∂g∂y

v + ∂g∂z

w, (9.71)

andd2wdt2 = ∂h

∂xu + ∂h

∂yv + ∂h

∂zw, (9.72)

where the partial derivatives are evaluated at x = xo , y = yo , z = zoas 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 theirpartial derivatives to be

∂ f∂x

= 1 − (1 − µ)r3

1 − (x − x1)( 3

2r1)2(x − x1)

r61

−µr3

2 − (x − x2)( 3

2r2)2(x − x2)

r62

, (9.73)

∂ f∂y

= −(1 − µ)(x − x1)

(− 32

)2y

r51

− µ(x − x2)

(− 32

)2y

r52

, (9.74)

and

∂ f∂z

= −(1 − µ)(x − x1)

(− 32

)2z

r51

− µ(x − x2)

(− 32

)2z

r52

. (9.75)

Similarly, we obtain

∂g∂x

= −(1 − µ)y(− 3

2

)2(x − x1)

r51

− µy(− 3

2

)2(x − x2)

r52

, (9.76)

∂g∂y

= 1 − (1 − µ)r3

1 − y( 3

2r1)2y

r61

− µr3

2 − y( 3

2r2)2y

r62

, (9.77)

∂g∂z

= −(1 − µ)y(− 3

2

)2z

r51

− µy(− 3

2

)2z

r52

, (9.78)

∂h∂x

= −(1 − µ)z(− 3

2

)2(x − x1)

r51

− µz(− 3

2

)2(x − x2)

r52

, (9.79)

∂h∂y

= −(1 − µ)z(− 3

2

)2y

r51

− µy(− 3

2

)2y

r52

, (9.80)

∂h∂z

= −(1 − µ)r3

1 − z( 3

2r1)2z

r61

− µr3

2 − z( 3

2r2)2z

r62

. (9.81)


These partial derivatives (9.73)–(9.81) can be evaluated for eachparticular solution and substituted into (9.70)–(9.72) to determine thenature of their stability.

9.1.3 Stability of Linear SolutionsIn the case of the linear solutions, L1, L2, and L3 in Figures 9.5–9.7, let(xo , yo , zo ) represent a particular solution. Then

yo = zo = 0 (9.82)

and xo > 0 for L1 and L2 and xo < 0 for L3. Also

r1 = |xo − x1| , r2 = |xo − x2| (9.83)

so that from (9.73)–(9.75)

∂ f∂x


= 1 −(

1 − µ

r31

+ µ

r32

)+ 3(

1 − µ

r31

+ µ

r32

)

= 1 + 2(

1 − µ

r31

+ µ

r32

)= 1 + 2α (9.84)

where we define [3]

α = 1 − µ

r31

+ µ

r32

> 0 (9.85)

and

∂ f∂y


= 0,∂ f∂z


= 0. (9.86)

Similarly, from (9.76)–(9.81)

∂g∂x


= 0,∂g∂z


= 0, (9.87)

∂g∂y


= 1 −(

1 − µ

r31

+ µ

r32

)= 1 − α, (9.88)

∂h∂x


= 0,∂h∂y


= 0, (9.89)

and

∂h∂z


= −(

1 − µ

r31

+ µ

r32

)= −α. (9.90)


Substituting (9.84)–(9.90) in (9.70)–(9.72) we obtain

d2u(t)dt2 − 2

dv(t)dt

= (1 + 2α)u(t), (9.91)

d2v(t)dt2 + 2

du(t)dt

= (1 − α)v(t), (9.92)

andd2w(t)

dt2 = −αw(t). (9.93)

Equations (9.91)–(9.92) are coupled, whereas (9.93) is independentof (9.91) and (9.92). Hence (9.93) gives the stable solution

w(t) = a cos(√

α t) + b sin(√

α t) (9.94)

that is periodic with period 2π/√

α. From (9.83) and (9.85), the periodis different for the three linear solutions. To examine the solutionsof the coupled set of linear equations (9.91) and (9.92), consider theLaplace transforms

U(s) =∞∫

0

u(t)e−stdt (9.95)

and

V(s) =∞∫

0

v(t)e−stdt. (9.96)

Taking Laplace transforms of (9.91) and (9.92), we obtain the cou-pled set of equations[

s2 − (1 + 2α) −2s

2s s2 − (1 − α)

]︸︷︷︸

A(s)

[U(s)

V(s)

]

=[

su(0) + u′(0) − 2v(0)

sv(0) + v′(0) + 2u(0)

]=[

a1(s)

a2(s)

](9.97)

whose solution is given by[U(s)

V(s)

]= A−1(s)

[a1(s)

a2(s)

]

= 1det A(s)

[s2 − (1 − α) 2s

−2s s2 − (1 + 2α)

] [a1(s)

a2(s)

]

= 1det A(s)

[x1(s)

x2(s)

], (9.98)


where x1(s) and x2(s) are two polynomials of degree three at most.Hence

U(s) = x1(s)a (s)

, (9.99)

V(s) = x2(s)a (s)

(9.100)

where the denominator

a (s) = det A(s) = (s2 − (1 − α))(s2 − (1 + 2α)) + 4s2

= s4 + (2 − α)s2 + (1 + 2α)(1 − α) (9.101)

is a biquadratic polynomial of degree four. Hence if s1, s2, s3, and s4represent the four roots of the determinantal polynomial in (9.101),then from (9.99)–(9.100) their partial fraction expansion followed byinverse Laplace transform gives

u(t) =4∑

i=1

ai e−si t , (9.102)

and

v(t) =4∑

i=1

bi e−si t , (9.103)

where ai , bi are constants that depend on the initial disturbancesin (9.97). The locations of these roots si , i = 1, 2, 3, 4 determine thenature 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) (9.105)

and in terms of its roots λ1 and λ2 we have

s1 =√

λ1, s2 = −√

λ1, s3 =√

λ2, s4 = −√

λ2. (9.106)

Clearly for (9.102) and (9.103) to represent stable solutions, bothλ1 and λ2 must be negative. In that case (9.106) represents purelyimaginary solutions and (9.103) and (9.104) represent stable periodicsolutions. If λ1 and λ2 are both negative, then the product λ1λ2 > 0and from (9.105) this leads to the condition (1 + 2α)(1 − α) must bepositive, or we must have

1 − α > 0 (9.107)

since α in (9.85) is positive.


To determine whether (9.107) is satisfied at the desired solutionsL1, L2, and L3, from (9.42) with (xo , 0, 0) representing any one suchsolution, we obtain (use (9.85))

ψ(xo ) = xo − (1 − µ)xo − x1

|xo − x1|3− µ

xo − x2

|xo − x2|3

= xo (1 − α) − µ(1 − µ)(

1r3

1− 1

r32

)= 0, (9.108)

since x1 = −µ, x2 = 1 − µ. Hence we obtain

1 − α = µ(1 − µ)xo

(1r3

1− 1

r32

). (9.109)

For L1 and L2, we have xo > 0 and r1 > r2 so that 1 − α < 0. ForL3, xo < 0, and r1 < r2 so that 1 − α < 0, and hence all three straightline solutions L1, L2, and L3 are unstable! As a result, infinitesimalbodies at these solutions under small perturbations eventually willdrift away to great distances.

9.1.4 Stability of Equilateral SolutionsIn the case of the equilateral solutions L4 and L5, we have

r1 = r2 = 1 (9.110)

and from Figure 9.7 and 9.8 for L4 and L5

xo = 12

− µ, yo = ±√

32

, zo = 0, (9.111)

so that

xo − x1 = 12

, xo − x2 = 12

− µ − (1 − µ) = −12. (9.112)

Substituting (9.110)–(9.112) into (9.73)–(9.75), we obtain

∂ f∂x


= 1 − (1 − µ)(

1 − 12

· 32

· 2 · 12

)− µ

(1 −(−1

2

)· 3

2· 2 ·(−1

2

))

= 1 − (1 − µ)4

− µ

4= 3

4, (9.113)

∂ f∂y


= −(1 − µ)12

·(

−32

)· 2 ·

√3

2− µ

(−1

2

)·(

−32

)· 2 ·

√3

2

= 3√

34

(1 − 2µ), (9.114)

∂ f∂z


= 0. (9.115)


x

L5

L4

x2 = 1 − mx1 = −m

1/2

√3/2

(xo,yo,zo)

y

0

FIGURE 9.8Stability ofLagrange solutionsL4 and L5.

Similarly (9.76)–(9.78) give

∂g∂x


= −(1 − µ)

√3

2·(−3

2

)· 2 ·(

12

)− µ

√3

2·(−3

2

)· 2 ·(−1

2

)

= 3√

34

(1 − 2µ), (9.116)

∂g∂y


= 1 − (1 − µ)(

1 −√

32

· 32

· 2 ·√

32

)− µ

(1 −

√3

2· 3

2· 2 ·

√3

2

)

= 1 − (1 − µ + µ)(

1 − 94

)= 9

4(9.117)

∂g∂z


= 0, (9.118)

and finally (9.79)–(9.81) gives

∂h∂x


= 0,∂h∂y


= 0,∂h∂z


= −(1 − µ) − µ = −1. (9.119)

Using (9.113)–(9.115) in (9.70) we obtain

d2u(t)dt2 − 2

dv(t)dt

= 34

u(t) + 3√

34

(1 − 2µ)v(t). (9.120)


Similarly (9.116)–(9.118) in (9.71) gives

d2v(t)dt2 + 2

du(t)dt

= 3√

34

(1 − 2µ)u(t) + 94

v(t) (9.121)

and (9.119) in (9.72) gives

d2w(t)dt2 = −w(t). (9.122)

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 ofthe 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

s2 − 3

4−(

2s + 3√

34

(1 − 2µ))

2s − 3√

34

(1 − 2µ) s2 − 94

︸︷︷︸B(s)

[U(s)

V(s)

]

=[

su(0) + u′(0) − 2v(0)

sv(0) + v′(0) + 2u(0)

]=[

b1(s)

b2(s)

](9.124)

whose solution is given by[U(s)

V(s)

]= B−1(s)

[b1(s)

b2(s)

]=

1det B(s)

s2 − 94

2s + 3√

34

(1 − 2µ)

−(

2s − 3√

34

(1 − 2µ))

s2 − 34

[

b1(s)

b2(s)

]

= 1det B(s)

[c1(s)

c2(s)

], (9.125)

where c1(s) and c2(s) are two polynomials of degree three at most.Hence

U(s) = c1(s)b(s)

, (9.126)

V(s) = c2(s)b(s)

(9.127)


where the denominator

b(s) = det B(s) =(

s2 − 94

)(s2 − 3

4

)+(

(2s)2 −(

3√

34

(1 − 2µ))2)

= s4 + s2 + 274

µ(1 − µ) (9.128)

is a biquadratic polynomial of degree four.As before if s1, s2, s3, s4 represent the four roots of the determinan-

tal polynomial in (9.128), then from (9.126) and (9.127), after partialfraction expansion their inverse transform gives

u(t) =4∑

i=1

ci e−si t (9.129)

and

v(t) =4∑

i=1

di e−si t (9.130)

where ci , di are constants that depend on the initial disturbances in(9.124). The locations of the roots s1, s2, s3, s4 clearly determine thenature 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 + 274

µ(1 − µ) = (λ − λ1)(λ − λ2) (9.132)

whose roots are given by

λ1,2 = −1 ± √1 − 27µ(1 − µ)

2. (9.133)

From (9.131), clearly

s1 =√

λ1, s2 = −√

λ1, s3 =√

λ2, 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 benegative. In that case, si , i = 1, 2, 3, 4 represent purely imaginary so-lutions and (9.129) and (9.130) represent stable periodic disturbances.From (9.133), the condition for periodic stability is that the discrimi-nant there be positive, i.e.,

1 − 27µ(1 − µ) > 0, (9.135)


or

µ2 − µ + 127

> 0. (9.136)

But µ2 − µ + 127 = 0 gives the solution

µ1,2 = 12

± 12

√1 − 4

27= 1

2±√

23108

(9.137)

and since µ < 12 , we obtain

µ2 = 12

−√

23108

= 0.03852 (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 normalizedperiodic 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

T1 = 1ω1

, and T2 = 1ω2

. (9.141)

In summary, the Lagrange libration points L4 and L5 represent(linear) stable solutions, and an infinitesimal body situated at thoselocations under small disturbances will continue to evolve aroundthose points in stable orbits that are a combination of the periodic or-bits with both periods given by (9.141), provided the mass of one ofthe finite bodies is less than 0.03852 of the mass of their sum.

Interestingly, this condition is satisfied by the Sun-Jupiter, the Sun-Earth combination as well as Earth-Moon combination. The mass ofthe 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 havebeen found at the L4 and L5 locations on the Jupiter’s orbit around theSun. There are several thousands of asteroids at these Trojan locationsas well. With µ = 1/1,000, their normalized (Jupiter years) orbitalperiods are

T1 = 1.0034, and T2 = 12.1363 (9.142)

that correspond to 11.9 years and 144 years respectively. Although asimilar asteroid Eureka was found in 1990 in the orbit of Mars, no


such Trojan objects have been found either in the Sun-Earth systemor in Earth-Moon system. Interestingly, large concentrations of dustglowing fainter than the Gegenschein has been reported in the L4 andL5 points in Earth-Moon system by Kordylewski (1960), and there isstill controversy as to its existence due to the extreme faintness. ForEarth-Moon system, µ = 0.0123 which is less than µ2 = 0.03852 sothat the L4 and L5 solutions in Earth-Moon system are also stable.With µ = 0.0123 in (9.141), we obtain the normalized orbital periodsto be

T1 = 1.04835 and T2 = 3.33099. (9.143)

Thus space stations and Space Based Radars located at these L4 andL5 points will float around those points along paths that are combi-nations of periodic solutions with periods 1 and 3.33 sidereal monthsrespectively. Interestingly, the L4 and L5 locations both in the Sun-Earth system as well as Earth-Moon system are possible candidatesfor future permanent space station colonies. It should be possible tohave any number of infinitesimal bodies revolving around the samepoint without interfering with each other.

Appendix 9-A: Hill SphereThe Hill sphere refers to the gravitational sphere of influence of oneastronomical body such as Earth in presence of perturbations fromanother heavier body (Sun) around which its orbits. In the restrictedthree-body problem containing a heavier mass and a lighter mass, theHill sphere refers to the region around the lighter mass located at x2within which the total gravitational influence on an infinitesimal masswill be directed toward the lighter mass.

Let rH represent the radius of the Hill sphere. Then according toHill [2]

rH = a(

µ

3(1 − µ)

)1/3

, (9A.1)

where a represents the distance between the two dominant massesand 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 termsof r1 and r2 as

U = 12

(1 − µ)(

r21 + 2

r1

)+ 1

2µ

(r2

2 + 2r2

)− µ(1 − µ)

2. (9A.2)

Here r1 and r2 refer to the distance of the infinitesimal body fromthe heavier and lighter masses respectively. Equation (9A.2) follows


xL2

r1 r2

rH

L1L3

1 − m mA

(x,y)

FIGURE 9.9 Hill Sphere.

by noticing that

r21 = (x − x1)2 + y2 = x2 + y2 + 2µx + µ2, (9A.3)

r22 = (x − x2)2 + y2 = x2 + y2 − 2(1 − µ)x + (1 − µ)2, (9A.4)

so that

(1 − µ)r21 + µr2

2 = x2 + 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, whenthe infinitesimal body is at location A as in Figure 9.9, we have inparticular r1 + r2 = 1 and the equilibrium condition

∂U∂r1

= ∂U∂r2

. (9A.6)

From (9A.2)

∂U∂r1

= (1 − µ)(r1 − r−21 ) (9A.7)

∂U∂r2

= µ(r2 − r−22 ) (9A.8)

so that (9A.6) gives [4]

µ

1 − µ= r1 − r−2

1

r2 − r−22

= r22

(1 − r3

1

)(1 − r3

2

)r2

1= r2

2

(3r2 − 3r2

2 + r32

)(1 − r3

2

)(1 − r2)2

= 3r32

(1 − r2 + 1

3r2

2

)(1 + r3

2 + · · · )×

1 + (2r2 − r22)+ (2r2 − r2

2)2 + · · ·


= 3r32

(1 − r2 + 1

3r2

2

)(1 + r3

2 + · · · )(1 + 2r2 + 3r22 − 4r3

2 + · · ·)= 3r3

2

(1 + r2 + 4

3r2

2 + · · ·)

. (9A.9)

Define

µ

1 − µ= 3α3. (9A.10)

From (9A.9) we get

α = r2

(1 + r2 + 4

3r2

2 + · · ·)1/3

= r2

1 + 1

3

(r2 + 4

3r2

2 + · · ·)

+13

( 13 − 1

)2

(r2 + 4

3r2

2 + · · ·)2

+ · · ·

= r2

(1 + 1

3r2 + 1

3r2

2 + · · ·)

, (9A.11)

or

r2 = α

1 + 13r2 + 1

3r22 + · · ·

= α

1 − 1

3r2

(1 + 1

3r2 + · · ·

)+ 1

9r2

2

(1 + 1

3r2 + · · ·

)2

+ · · ·

= α

1 − 1

3α

(1 − 1

3α + · · ·

)(1 − 1

3α + · · ·

)

+ 19α2(

1 + 13α + · · ·

)2

+ · · ·

= α

(1 − 1

3α − 2

9α2 + · · · + 1

9α2 + · · ·

)

= α − α2

3− α3

9+ · · · (9A.12)

where (use (9A.10))

α =(

µ

3(1 − µ)

)1/3

. (9A.13)


But in this limiting case r2 = rH . Thus to a first-order approximationfrom (9A.12) and (9A.13) the normalized Hill radius equals

r2 = rH = α =(

µ

3(1 − µ)

)1/3

. (9A.14)

For the Sun-Earth system (1 − µ 1) so that the Hill radius rH in(9A.14) is the same as the distance to the Lagrange solutions L1 andL2 from Earth.

Beyond the Hill sphere the infinitesimal body will be more and moreinfluenced by the larger mass and would eventually end up orbitingthe 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 Publica-tions, New York, NY, 1960.

Index

2D beamformer, 224, 2252D (two-dimensional) motion, 552D-DFT (two-dimensional discrete

Fourier transform), 205

Aadaptive beamforming, 201, 287adaptive clutter cancellation, 205,

208adaptive processor output, 152,

176, 179, 201additive white noise case, 364–366Airborne Moving Target Indicator

(AMTI), 341–342airborne systems

Cramer-Rao bounds, 320–331ground coverage, 2–3noise only case bounds, 330side-looking, 155–165target Doppler/power, 320–338vs. space-based systems, 3

all-pass function, 299, 366–368alternating projections, 198–201AMTI (Airborne Moving Target

Indicator), 341–342angle bins, 206–208angle-Doppler dependency, 224angle-Doppler domain, 230, 290angle-Doppler pattern, 204angle-Doppler performance, 163,

164angle-Doppler profile, 157, 225,

227angle-Doppler space, 157–163

ARMA systemsBillingsley model, 264, 267rational system representation,

303–307wind speeds, 263–264, 267

array processing, 139–153array tapering, 187–194arrays. See also subarrays

beamforming and, 143–150centro-symmetric uniform

linear array, 330direction-dependent weights,

146gain patterns, 215, 220mainbeam width, 144–145noise and, 146–152N-sensor, 216–218overall amplitude patterns,

219–220phase shifted, 142–143sensors, 140–148separable, 220sidelobe patterns, 143–145SMI method, 152–153SNR and, 142–143subdividing into subarrays,

173–187uniform. See uniform arraysuses for, 140–148weight vector, 139, 148–153weights, 229–231

asteroids. See also comets; meteorsimpact on Earth, 43, 45number of, 42, 44



asteroids. (Cont.)overview, 42–43tracking, 7Trojan, 416–417

asymptotes, 41, 399attraction, central force of, 1azimuth angles, 100, 111–113,

216–219azimuth domain, 230, 290azimuth return, 216azimuth transmit pattern, 229, 231

Bbackscatter amplitude return, 256backward subarrays, 179–182baseband carrier, 365baseband region, 386baseband signal, 363beamforming. See also mainbeam

2D, 158adaptive, 201, 287arrays and, 143–150clutter power, 161coherent, 6gain pattern and, 143–144output power, 160overview, 143uniform arrays and, 143–148

Billingsley’s wind model, 256, 261,263–267, 272

bistatic constants, 8bistatic mode, 341bistatic SBRs, 6–7Boltzmann’s constant, 8Brennan’s rule

Earth rotation and, 221–222,244–245

overview, 166–167

CCanadian Space Agency (CSA), 4Cartesian coordinate system

ellipses, 33–36hyperbolas, 41parabolas, 39

Cassini orbiter, 5

Cassini radar, 5Cassini-Huygens radar, 5causal filters, 352, 357causal receivers, 355, 365, 371causal waveforms, 355–356, 389center of mass, 52–54centripetal force, 98centro-symmetric uniform linear

array, 330channel kernel, 370–371, 376–380,

383chirp compression, 358–364chirp signal, 364, 385–391chirp waveforms, 91, 251–255,

281–283, 294chirped rectangle symmetric

pulse, 385–387circles, 32circular orbit velocity, 65closed orbits, 59, 60clutter. See also interference

corresponding to rangebins, 218

Doppler frequency and, 156–162Earth rotation and, 234–246mainbeam, 218–223range ambiguities and, 218–219sidelobe, 218–223waveform diversity, 344

clutter covariance matrixBrennan rank, 167–168Eigen-structure based STAP,

165–173ground patches and, 158

clutter dataEarth rotation, 234–246range foldover, 234–246scatter return modeling, 255–267

clutter Doppler ambiguity, 156clutter foldover, 158–159, 167, 236clutter modeling, 215clutter notch width

Earth rotation and, 239, 241–246overview, 241wind effect on, 270–275

clutter nulling effects, 215, 216,273, 280, 289–292

I n d e x 423

clutter nulling performanceEarth rotation and, 280–283,

289–292effect of combined influences

on, 280–283effect of terrain on, 280–283range foldover, 280–283STAP algorithms, 283–296wind effect on, 280–283

clutter power beamformer, 161clutter power distribution,

158–160clutter power to noise power ratio

(CNR), 160clutter ridge slope, 100, 101clutter scene

Brennan’s rule, 166–167nonstationary, 182

clutter spectrum, 223–231clutter steering vector, 169clutter subspace, 169, 170, 173,

244–245clutter suppression

minimum detectable velocityand, 162, 232–234

range-Doppler response and, 3terrain modeling, 268–280wind effect, 268–280

CMT (covariance matrix tapering),188–194

colored noise, 353–358comets. See also asteroids; meteors

altered orbital parameters, 74Halley’s comet, 64identification of, 71–74number of, 44orbits of, 42, 64overview, 42

communication scene, 345compression

chirp, 358–364pulse, 91, 358–364

conics, 31–50circle, 32eccentricity of, 31, 34ellipse, 32, 33–39gravitation and, 1

hyperbola, 32, 40–43overview, 31–33parabola, 32, 39–40sections of, 32solar system, 44–46spherical triangles, 46–50

covariance matrix tapering,188–194

convex projection techniques,194–201

convex sets, 195–196correlator receivers, 353cosines, law of, 49–50covariance matrix

smoothed, 174, 179–180subaperture in, 186–187tapered, 187–189, 192

crab angle, 101–120Doppler frequencies and, 117,

119, 246–247Earth rotation and, 216oblate Earth, 133–136range foldover and, 118–120

crab angle correction, 134–137crab error, 111–113crab magnitude, 101–115crab phenomonen, 117–120Cramer-Rao bounds, 309–338

airborne platform, 320–331Fisher information matrix,

310–320, 326, 329Gaussian data, 319–320multiparameter case, 309–320SBR platform, 320–331simulation results, 331–338target Doppler/power, 320–338

CSA (Canadian Space Agency), 4

DDARPA (Defense Advanced

Research Project Agency), 6decision instant, 344, 351–352, 373,

376Defense Advanced Research

Project Agency (DARPA), 6DFT (Doppler temporal) steering

vector, 169, 202–206


diagonal loading schemes, 165,192–193

Doppler bins, 204–208Doppler (temporal) DFT steering

vector, 169, 202–206Doppler effect, 130–133“Doppler filling” effect, 99Doppler filters, 133, 202Doppler frequencies

airborne systems, 320–338along isocone contour, 224angle-Doppler dependency, 224angle-Doppler domain, 230, 290angle-Doppler pattern, 204angle-Doppler performance,

163, 164angle-Doppler profile, 225, 227angle-Doppler space, 157–163clutter and, 156–162crab angle and, 117, 119, 246–247Cramer-Rao bounds and, 320–331detecting with temporal pulse

transmission, 153–154effect of crab angle on, 113, 115,

117effect of Earth rotation on, 102–107effect of wind on, 270–275foldover factor, 117–120, 156iso-Doppler plots, 99–100,

117–120range bins, 221range dependency of, 99–101, 224range foldover points, 118–120target Doppler/power, 320–338wind dampening effect on,

275–280Doppler shift, 97–101Doppler spread, 100–102, 222–223,

294–295Doppler warping, 292–294Doppler-azimuth pattern, 100Doppler-azimuth profile, 224–229

EEarth

destruction of, 44–45eccentricity of, 123

grazing angle correction factor,123–130

impact of asteroids on, 43, 45maximum range on, 81–82nonsphericity of, 123–130oblate spheroidal shape of,

134–137place in solar system, 44size of, 44

Earth observing radars, 3–5Earth rotation

Brennan’s rule, 221–222, 244–245clutter data, 234–246clutter notch width, 239clutter nulling performance,

280–284, 289–292crab angle effect, 216Doppler frequency and,

102–107, 216Doppler spread, 100–102,

222–223, 294–295MDV, 234–246range foldover due to, 222–223SINR performance, 236, 291

Earth-Moon system, 393, 404, 416,417

EC (eigencanceler) methods,167–171

EC with forward/backwardsubarray subpulse(ECSASPFB) smoothing,187–188

eccentricityconics, 31, 34Earth, 123

ecliptic orbits, 42ECSASPFB (EC with

forward/backwardsubarray subpulse)smoothing, 187–188

EFA (extended factoredtime-space) approach,204–205, 286

eigencanceler (EC) methods,167–171

eigen-structure based methods,165–173

I n d e x 425

eigenvaluesmatched filter receiver, 371–374,

379–380, 384matrices, 10–12

eigenvectors, 10–12, 202–206elevation angle, 81, 87, 88ellipses

conics, 32, 33–39eccentric, 42inverse square law, 60iso-range plots and, 8semimajor axis, 64Sun, 59, 67

elliptic orbits, 42, 74–76elliptical orbits, 41–43, 62–67energy compaction ratio, 383, 384equations

Euler’s, 71–74Kepler’s, 67–70Lambert’s, 74–76Lyapunov, 23–24radar, 7–8relative motion, 54–57

equatorial orbit, 107–112escape velocity, 63Euler’s equation, 71–74extended factored time-space

(EFA) approach, 204–205,286

Ffactor time-space (FTS) approach,

201–205filters

causal, 352, 357Doppler, 133, 202matched filter output, 162–163matched filter receivers,

344–358, 371–376noncausal, 352receiver, 344, 346–352

finite bodies, 393–420finite duration waveforms, 343finite impulse response (FIR)

Doppler filter, 133FIR (finite impulse response)

Doppler filter, 133

Fisher information matrix,310–320, 326, 329

Five parameter RCS model,258–259

foldover. See also range foldoverclutter, 158–159, 167, 236Doppler, 117–120, 156mainbeam, 90–94

foliage penetration (FOPEN)operation, 2

footprint, mainbeam, 82–89FOPEN (foliage penetration)

operation, 2free fall motion, 62–63FTS (factor time-space) approach,

201–205

Ggamma rays, 44Gaussian data, 319–320Gaussian data samples, 316–317Gegenschein phenomenon,

404–405, 417geosynchronous satellites, 60–61GMTI (ground moving target

indication), 155, 341–342gravitation, law of, 1, 31, 51, 52–55gravitational pull, 98grazing angle

correction factor for, 123–130MEO/LEO satellites, 88–89radar-Earth geometry, 77–81scatter return modeling, 255–260vs. range, 81

ground moving target indication(GMTI), 155, 341–342

ground rangefrom latitude/longitude

coordinates, 120–123between nadir point and point

of interest, 121–122ground range resolution, 91

HHalley’s comet, 64heliocentric coordinate system,

54–55


helium, 44Hermitian matrices, 12–16, 146,

149, 197Hermitian Toeplitz matrix, 26–27High Range Resolution Ground

Moving Target Indication(HRR-GMTI), 6

high resolution photography(IMINT), 6

Hill sphere, 417–420HRR-GMTI (High Range

Resolution Ground MovingTarget Indication), 6

HRR-GMTI coverage, 6HRR-GMTI surveillance, 6HTP (Hung-Turner projection),

171–173HTP angle-Doppler pattern,

173–174HTP with forward/backward

subarray subpulse(HTPSASPFB) smoothingmethod, 186–187

HTPSASPFB (HTP withforward/backwardsubarray subpulse)smoothing method,186–187

Hung-Turner projection (HTP),171–173

Huygens probe, 5–6hybrid-chirp waveforms, 254–255,

280–283, 294–295hydrogen, 44hyperbola

conics, 32, 40–43planet velocity and, 62satellite orbits, 66

hyperbolic orbits, 1

IICM (internal clutter motion)

effect of wind on, 221, 255–257,261–267

scatter return modeling, 261–267white noise and, 263

ICM modeling, 261–267

IMINT (high resolutionphotography), 6

infinitesimal bodies, 393–420Innovative Space Based Radar

Antenna Technology(ISAT), 6

interelement spacing, 221–222interference, 342. See also clutterinternal clutter motion. See ICMInverse Fourier transform, 378inverse square law of gravitation,

1, 31, 51, 52–55ISAT (Innovative Space Based

Radar AntennaTechnology), 6

isocone contour, 223, 224, 239–242iso-Doppler plots, 99–100, 117–120iteration projections, 199

JJacobi function, 397–398, 401, 417JDL (Joint-Domain Localized)

approach, 205–208Joint-Domain Localized (JDL)

approach, 205–208Joint STARS (Joint Surveillance

Target Attack RadarSystem), 2

Joint Surveillance Target AttackRadar System (JointSTARS), 2

joint time-bandwidthoptimization, 376–385

joint transmitter-receiver design,364–376

Jupiter, 44

KKASSPER (Knowledge Aided

Sensor Signal Processingand Expert Reasoning), 215

Kepler’s equation, 67–70Kepler’s laws, 57–60kernel, channel, 370–371, 376–380,

383Khatri-Rao product, 19–25kinematics, 77–137

I n d e x 427

Knowledge Aided Sensor SignalProcessing and ExpertReasoning (KASSPER), 215

Kronecker product, 18–19

LLacrosse Series, 6Lagrange libration points, 401,

405, 416Lagrange theorem, 393, 401–409Lambert’s equation, 74–76Laplace transforms, 366, 380, 385,

410–411launch height, 65launch speed, 66launch velocity vector errors, 65law of cosines, 49–50law of gravitation, 1, 31, 51, 52–55law of sines, 46–49lemmas, 14–16, 25–26LEO (low-earth orbit), 3LEO/MEO satellites, 88–89line spectra matrices, 26–28low-earth orbit (LEO), 3Lyapunov equation, 23–24

MM pulses, 216–218Mac Donald, Dettwiler and

Associates (MDA), 4Magellan radar, 5Mahalanobis distance, 325, 327mainbeam footprints, 82–89mainbeams. See also beamforming

arrays, 144–145clutter, 218–223foldover, 90–94non-overlapping, 88range ambiguities, 89–97

Mars, 44Mars Express, 6Mars Global Surveyor, 6Mars Odyssey, 6Mars reconnaissance orbiter, 5Marsis radar, 5matched filter (MF) receivers,

344–358

chirp signal, 361–364eigenvalues, 371–374, 379–380,

384optimum input and, 371–376overview, 344–345performance, 283–284

matricesclutter covariance matrix, 158,

165–173covariance. See covariance

matrixdescribed, 9eigenvalues, 10–12eigenvectors, 10–12Fisher information matrix,

310–320, 326, 329Hermitian, 12–16, 146, 149, 197line spectra, 26–28Sample Matrix Inversion (SMI),

162–165singular covariance, 26–28Toeplitz, 13, 196–197wind autocorrelation matrix,

268–269matrix inversion identity, 296matrix inversion lemmas, 25–26MDA (Mac Donald, Dettwiler and

Associates), 4MDV (minimum detectable

velocity)angle-Doppler performance for,

164clutter suppression and,

161–162, 232–234Earth’s rotation, 234–246overview, 161–162performance, 169–170range foldover, 234–246signal to interference plus noise

ratio, 233terrain modeling, 268–280wind effect, 268–280

medium-earth orbit (MEO), 3MEO (medium-earth orbit), 3MEO/LEO satellites, 88–89Mercator’s projection chart, 61Mercury, 44


meteor showers, 42meteors. See also asteroids; comets

glow from, 404number of, 44trapped, 404

MF receivers. See matched filter(MF) receivers

military satellites, 6minimum detectable velocity. See

MDVminimum phase pulse, 368minimum phase signal, 366–369modulated chirp pulse signal,

387–389modulated chirp signal, 363–364monostatic constants, 8monostatic mode, 341monostatic SBRs, 6–7Moon

Earth-Moon system, 393, 404,416, 417

elliptical orbits and, 41Gegenschein phenomonen, 404orbit, 1, 44receding of, 400, 403Saturn’s moons, 5

motionfree fall, 62–63internal clutter motion.

See ICMmean angular, 67–68relative, 54–57two-dimensional, 55

motion of center of mass, 52–54multichannel waveform diversity,

341–344multiparameter case, 309–320multipath scenes, 175–176multi-static SBRs, 6–7

Nnadir hole, 86, 96nadir point

great circle through, 128–130inaccessible, 85latitude/longitude of, 120–121overview, 77–79

NASA terra satellite image map,255–261

Neptune, 44noise. See also SNR

angle-Doppler powerdistribution, 158–160

arrays and, 146–152colored, 353–358joint transmitter-receiver

design, 364–376matched filter receivers, 344–358waveform diversity, 344white. See white noise

noise bandwidth, 8noise only case, 328noise power spectral densities,

346–347noise subspace eigenvectors, 170,

245noise temperature, 8noncausal filters, 352noncausal waveforms, 355nonnegative-definite property,

197–198north pole, 104N-sensor array, 216–218

Ooblate spheroidal Earth, 133–136omnidirectional sensors, 220Oort cloud, 42Opportunity rover, 6optimization problem, 365–371optimum transmit pulse, 374–376optimum waveforms, 382–383orbital mechanics, 51–57orbits

closed, 59, 60comets, 42, 64elliptic, 41, 74–76elliptical, 41–43, 62–67equatorial, 107–112grativation and, 1hyperbola, 66hyperbolic, 1inclination of, 77low-earth, 3

I n d e x 429

maximum error, 111–114medium-earth, 3minimum error, 116Moon, 1, 44parabola, 66parabolic, 1planets, 51–57polar, 60–61, 77–78, 107–109sun and, 51–57synchronous, 60–61two-body orbital motion, 51–57

orthogonal pulsingminimizing range foldover

with, 246–255SINR performance and, 253

orthogonal transmit pulsingschemes, 216

output SINR, 297–298output SNR, 143, 148–153Ovals of Cassini, 8

PPaley-Wiener condition, 354parabola

conics, 32, 39–40satellite orbits, 66

parabolic orbits, 1parabolic speed, 63Parseval’s relation, 368–369Parseval’s theorem, 355–356, 378phase shifted arrays, 142–143planar wavefront, 140planetary radars, 5–6planetary velocity, 61–67planets. See also specific planets

elliptical orbits and, 41–42,58–60

location of, 70mean angular motion of, 67–68orbital motion of, 51–57relative motion and, 54–57

polar coordinates, 34, 35polar orbits, 60–61, 77–78, 107–109positive-definite property, 197–198PRF (pulse repetition frequency),

162, 165projection operators, 198–201

prolate spheroidal functions,382–385

pulse compression, 358–364pulse repetition frequency (PRF),

162, 165pulse repetition interval, 89, 90pulse wave front, 91pulsed radar, 89pulses

finite impulse response (FIR),133

M, 216–218minimum phase, 368optimum transmit, 374–376rectangular, 360, 365, 373temporal, 217–218vectors corresponding to,

153–154

Qquadrature phase shifting, 252quiescent steering vectors, 170

Rradar. See also SBR (space based

radar)airborne. See airborne systemsCassini, 5Cassini-Huygens, 5Earth observing, 3–5ISAT, 6Magellan, 5mainbeam of, 83–89Marsis, 5planetary, 5–6pulsed, 89side-looking airborne, 155–165synthetic aperture. See SAR

radar cross section (RCS)described, 7Five parameter RCS model,

258–259grazing angle dependent, 221scatter return modeling, 255–261

radar equation, 7–8radar pulse repetition rate, 99radar-Earth geometry, 77–81


RadarSat-1 satellite, 4–5RadarSat-2 satellite, 5radar/sonar scene, 345random returns, 342range

Doppler dependency on, 99–100mainbeam footprints along,

86–89maximum range on Earth, 81–82vs. slant range, 80–81

range ambiguitiesclutter and, 218–219mainbeam footprints, 89–97

range bin of interest, 161range bins

clutter corresponding to, 218Doppler frequencies, 221

range foldover, 90–97. See alsofoldover

clutter data, 234–246clutter nulling performance,

280–283crab phenomenon and,

118–120Doppler frequencies, 118–120due to Earth rotation, 222–223minimizing with orthogonal

pulsing, 246–255minimum detectable velocity,

234–246performance degradation and,

216scatter return power, 251SINR peformance, 236total range foldover, 94–97waveform diversity and,

248–251range-azimuth domain, 100,

117–119, 222range–foldover return, 215Rao bounds. See Cramer-Rao

boundsrational system representation,

303–307Rayleigh distributions, 256, 259RCS (radar cross section)

described, 7

Five parameter RCS model,258–259

grazing angle dependent, 221scatter return modeling,

255–261receiver subarrays, 220receivers

causal, 355, 365, 371correlator, 353filters, 344, 346–352joint transmitter-receiver

design, 364–376matched filter. See matched filter

(MF) receiversrectangular channel response,

373–374rectangular pulse, 360, 365, 373reference sensor, 140–141relative motion, 54–57relaxed projection operators,

200–201return signal, 140

SSample Matrix Inversion. See SMISample Matrix Inversion with

Diagonal Loading (SMIDL),165, 192–194

SAR (synthetic aperture radar)imaging support, 6, 341–343

SAR sensor, 4SAR systems, 1, 4satellite image map, 255–261satellite velocity, 61–67satellites

geosynchronous, 60–61MEO/LEO, 88–89military, 6number of, 44orbits. See orbitspolar, 60–61RadarSat-1, 4–5RadarSat-2, 5Sesat, 4synchronous, 60–61velocity, 61–67

Saturn, 44

I n d e x 431

SBR (space based radar). See alsoradar

array parameters, 226challenges, 1–2Cramer-Rao bounds, 320–331data modeling, 216–231described, 1Doppler shift, 97–101ground coverage, 2–3modeling Earth’s rotation for,

101–120orbits. See orbitsparameters, 222region of coverage, 89requirements, 1STAP algorithms for, 283–296target Doppler/power, 320–331

SBR systemsdesign considerations, 2introduction of, 2main parameters of, 78–81overview, 3–7types of, 3–7vs. airborne systems, 3

scalars, 9scatter power, 220scatter power profile, 215scatter return modeling

clutter data, 255–267grazing angle, 255–260ICM modeling, 261–267Rayleigh distributions, 256, 259RCS (radar cross section),

255–261terrain modeling, 256–261Weibull distributions, 255–256,

259–260windblown autocorrelations,

256, 261–277scatter return power, 251scatter returns, 220Schur product, 17–18, 261Schwartz’ inequality, 347–351, 370,

379sensors

array of, 140–148input noise and, 142

N-sensors, 216–218omnidirectional, 220reference, 140–141SAR, 4SNR, 142

Sesat satellite, 4sidelobes

array gain patterns, 215arrays, 143–145including clutter contribution

from, 218–219saturation levels, 208–213

side-looking airborne radar,155–165

SIGINT (signal intelligence), 6signal intelligence (SIGINT), 6signal to interference plus noise

ratio. See SINRsignal to noise ratio. See SNRsines, law of, 46–49singular covariance matrices,

26–28singular value decomposition

(SVD), 16–17SINR (signal to interference plus

noise ratio)clutter data, 160–164, 168–170MDV and, 233optimum, 179output SINR derivation, 297–298receiver output signal, 343

SINR loss, 236, 279, 283SINR performance

Earth’s rotation, 236, 291orthogonal pulsing, 253range foldover, 236terrain modeling, 269–270waveform diversity, 251–253,

295slant range, 79–80, 220SMI (Sample Matrix Inversion),

162–165angle-Doppler performance for,

163arrays and, 152–153computational effect of, 289overview, 162–163


SMIDL (Sample Matrix Inversionwith Diagonal Loading),165, 192–194

SMIDLCMT method, 193–194SMIPROJ methods, 201SNR (signal to noise ratio). See also

noiseadditive white noise case,

364–365, 370–376arrays and, 142–143, 148–153described, 8matched filter receivers, 345–353multiple sensors and, 142output, 143, 148–153

solar system, 44–46space based radar. See SBRspace-time adaptive processing.

See STAPspace-time steering vectors,

153–157, 166, 169, 270space-time vectors, 153–154spatial array processing, 139–153spatial arrays, 216–218spatial diversity, 153–154spatial-steering vectors, 140–148,

169, 204–206, 216–218spatio-temporal processing, 139,

153–154spectral factorization, 298–303spherical triangles, 46–50Spirit rover, 6STAP (space-time adaptive

processing)algorithms, 3, 283–296array tapering, 188–194covariance matrix tapering,

188–194convex projection techniques,

194–201eigen-structure based methods,

165–173FTS approach, 201–205JDL approach, 205–208overview, 139, 153–155performance, 293–296side-looking airborne radar,

155–165

spatial array processing,139–153

subaperture smoothingmethods, 173–187

waveform diversity, 293–296STAP steering vectors, 254stationary stochastic process, 344steering vectors. See also vectors

clutter, 169distinct, 242Doppler temporal, 169quiescent, 170space-time, 153–157, 166, 250,

270STAP, 254

stochastic process, 344subaperture smoothing methods,

173–187subarray vectors, 183–184subarrays. See also arrays

backward, 179–182receiver, 220

subarray-subpulse method,184–187

subpulse smoothing method, 184Sun

ellipses, 59, 67elliptical orbits and, 41gravitational force of, 42orbital motion of, 51–57overview, 44–45relative motion and, 54–57size of objects around, 43

SVD (singular valuedecomposition), 16–17

synchronous orbits, 60–61synchronous satellites, 60–61synthetic aperture radar.

See SAR

Ttapered covariance matrix,

187–189, 192target Doppler, 320–338target impulse response, 374target power, 320–338target return, 342

I n d e x 433

Taylor weights, 229–231TechSat 21, 6temporal processing, 153–154temporal pulse processing, 139,

153–154temporal pulses, 217–218temporal steering vector, 156terra satellite image map, 255–261terrain

clutter nulling performance,280–283

NASA terra satellite image map,255–261

reflectivity of, 215terrain classification, 255–261terrain modeling

MDV and, 268–280scatter return modeling,

256–261SINR performance, 269–270

three-body problem, 393, 401–405time compression, 359, 361time-bandwidth products,

361, 384Toeplitz matrices, 13, 196–197Toeplitz property, 196–197total range foldover, 94–97training cells, 161transmit array weights, 229–231transmit beams, 215transmit signal, 140transmit waveform, 154, 220, 342,

367, 373transmitter output filters, 343transmitter-receiver design, 343,

364–376transmitters

joint transmitter-receiverdesign, 364–376

waveform diversity, 341–344triangles, spherical, 46–50Trojan asteroids, 416–417truncated input signal, 368two-body orbital motion, 51–57two-dimensional discrete Fourier

transform (2D-DFT), 205two-dimensional (2D) motion, 55

UUAVs (unmanned aerial vehicles),

341uniform arrays

beamforming and, 143–148centro-symmetric, 330linear, 329–330sidelobe saturation levels,

208–213spatial steering vector, 141subarray smoothing and,

176–177uniform linear array, 329universal gravitational constant,

98unmanned aerial vehicles (UAVs),

341Uranus, 44

VVan Allen radiation belts, 2–3vectors

corresponding to pulses,153–154

described, 9eigenvectors, 10–12, 202–206errors, 65input noise, 142launch velocity, 65space-time, 153–154steering. See steering vectorssubarray, 183–184weight, 139, 148–153

velocityescape, 63launch, 65minimum detectable, 161–162planetary, 61–67satellite, 61–67

Venus, 44

Wwaveform diversity, 341–391

bistatic mode, 341chirp signal, 364, 385–391clutter, 344


decision instant, 344, 351–352,373, 376

interference, 344joint time-bandwidth

optimization, 376–385joint transmitter-receiver

design, 364–376matched filter receivers,

344–358, 371–376monostatic mode, 341multichannel, 341–344noise, 344overview, 247, 341–344prolate spheroidal functions,

382–385range foldover and, 248–251SINR performance,

251–253, 295STAP performance, 293–296target (channel), 344target Doppler, 335–338transmitters, 341–344

waveformscausal, 355–356, 389chirp, 91, 251–255, 281–283, 294noncausal, 355optimum, 382–383transmit, 154, 220, 342, 367, 373

wavefront, planar, 140Weibull distributions, 255–256,

259–260Weibull-type modeling, 255weight vector, 139, 148–153

white noiseadditive white noise case,

364–366, 370–376ICM modeling and, 263matched receiver filters in,

346–352, 358output SNR, 150, 151sensor element input, 323uncorrelated sources in, 280

wide area surveillance systems, 2Wiener factor, 354wind

Billingsley’s wind model, 256,261, 263–267, 272

clutter nulling performance,280–283

dampening effect, 270–275effect on clutter notch width,

270–275effect on Doppler frequency,

270–280effect on ICM, 221, 255–257,

261–267MDV and, 268–280random variables, 256, 261–262,

263Schur product, 261

wind autocorrelation matrix,268–269

wind modeling, 256, 264, 275wind spectrum, 256, 266windblown autocorrelations, 256,

261–277

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Space Based Radar Theory & Applications