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TEAM LinG Robust Adaptive Beamforming Robust Adaptive Beamforming Edited by Jian Li and Petre Stoica A JOHN WILEY & SONS, INC., PUBLICATION Copyright#2006byJohnWiley&Sons,Inc.Allrightsreserved. PublishedbyJohnWiley&Sons,Inc.,Hoboken,NewJersey. PublishedsimultaneouslyinCanada. Nopartofthispublicationmaybereproduced,storedinaretrievalsystem,ortransmittedinany formorbyanymeans,electronic,mechanical,photocopying,recording,scanning,orotherwise, exceptaspermittedunderSection107or108ofthe1976UnitedStatesCopyrightAct,without eitherthepriorwrittenpermissionofthePublisher,orauthorizationthroughpaymentofthe appropriateper-copyfeetotheCopyrightClearanceCenter,Inc.,222RosewoodDrive,Danvers, MA01923,978-750-8400,fax978-750-4470,oronthewebatwww.copyright.com.Requeststo thePublisherforpermissionshouldbeaddressedtothePermissionsDepartment,JohnWiley&Sons,Inc., 111RiverStreet,Hoboken,NJ07030,201-748-6011,fax201-748-6008, oronlineathttp://www.wiley.com/go/permission. 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TK7871.67.A33R632005 621.38204--dc22 2004065908 PrintedintheUnitedStatesofAmerica 10 9 8 7 6 5 4 3 2 1 CONTENTS Contributors ix Preface xi 1 Robust MinimumVariance Beamforming 1 RobertG.LorenzandStephenP.Boyd 1.1 Introduction 1 1.2 APractical Example 8 1.3 Robust Weight Selection 12 1.4 ANumerical Example 23 1.5 Ellipsoidal Modeling 28 1.6 Uncertainty Ellipsoid Calculus 31 1.7 BeamformingExamplewith Multiplicative Uncertainties 41 1.8 Summary 44 Appendix:Notationand Glossary 44 References 45 2 Robust Adaptive BeamformingBased on Worst-Case Performance Optimization 49 AlexB.Gershman,Zhi-QuanLuo,andShahramShahbazpanahi 2.1 Introduction 49 2.2 Background and TraditionalApproaches 51 2.3 Robust Minimum Variance Beamforming Based on Worst-Case Performance Optimization 60 2.4 NumericalExamples 74 2.5 Conclusions 80 Appendix2.A: Proof of Lemma 1 81 Appendix2.B: Proof of Lemma 2 81 Appendix2.C:Proof of Lemma 3 82 Appendix2.D:Proof of Lemma 4 84 Appendix2.E: Proof of Lemma 5 85 References 85 v vi CONTENTS 3 Robust CaponBeamforming 91 JianLi,PetreStoica,andZhisongWang 3.1 Introduction 91 3.2 Problem Formulation 93 3.3 StandardCaponBeamforming 95 3.4 Robust Capon Beamforming with SingleConstraint 96 3.5 Capon Beamforming with Norm Constraint 112 3.6 Robust Capon Beamforming with DoubleConstraints 116 3.7 Robust Capon Beamforming with Constant Beamwidth andConstant Powerwidth 133 3.8 Rank-Deficient Robust Capon Filter-Bank Spectral Estimator 148 3.9 AdaptiveImagingfor Forward-LookingGround Penetrating Radar 166 3.10 Summary 185 Acknowledgments 185 Appendix3.A: RelationshipbetweenRCB andthe Approach in [14] 185 Appendix3.B: Calculating theSteering Vector 188 Appendix3.C: RelationshipbetweenRCB and the Approachin [15] 189 Appendix3.D: Analysisof Equation (3.72) 190 Appendix3.E: Rank-DeficientCapon Beamformer 191 Appendix3.F: Conjugate Symmetry ofthe Forward-Backward FIR 193 Appendix3.G: Formulations ofNCCF andHDI 194 Appendix3.H: Notationsand Abbreviations 195 References 196 4 DiagonalLoading for Finite SampleSize Beamforming: AnAsymptotic Approach 201 XavierMestreandMiguelA.Lagunas 4.1 Introduction andHistorical Review 202 4.2 Asymptotic OutputSINR with DiagonalLoading 213 4.3 Estimating theAsymptoticallyOptimum Loading Factor 225 4.4 Characterizationof theAsymptoticallyOptimum Loading Factor 236 4.5 Summary andConclusions 243 Acknowledgments 243 Appendix4.A: Proofof Proposition 1 243 Appendix4.B: Proofof Lemma 1 246 Appendix4.C: Derivation ofthe Consistent Estimator 247 Appendix4.D: Proof of Proposition 2 249 References 254 CONTENTS vii 5 Mean-Squared Error Beamforming forSignalEstimation: ACompetitive Approach 259 YoninaC.EldarandAryeNehorai 5.1 Introduction 259 5.2 Background and Problem Formulation 261 5.3 Minimax MSE Beamforming for Known Steering Vector 271 5.4 RandomSteering Vector 281 5.5 PracticalConsiderations 284 5.6 NumericalExamples 285 5.7 Summary 294 Acknowledgments 295 References 296 6 ConstantModulusBeamforming 299 Alle-JanvanderVeenandAmirLeshem 6.1 Introduction 299 6.2 The Constant ModulusAlgorithm 303 6.3 Prewhitening and RankReduction 307 6.4 MultiuserCMATechniques 312 6.5 The Analytical CMA 315 6.6 AdaptivePrewhitening 325 6.7 AdaptiveACMA 328 6.8 DOA Assisted Beamforming of ConstantModulus Signals 338 6.9 Concluding Remarks 347 Acknowledgment 347 References 347 7 Robust Wideband Beamforming 353 ElioD.DiClaudioandRaffaeleParisi 7.1 Introduction 353 7.2 Notation 357 7.3 Wideband Array SignalModel 358 7.4 Wideband Beamforming 363 7.5 Robustness 369 7.6 Steered AdaptiveBeamforming 381 7.7 Maximum Likelihood STBF 389 7.8 ML-STBF Optimization 393 7.9 SpecialTopics 399 7.10 Experiments 401 7.11 Summary 410 Acknowledgments 411 References 412 Index 417 CONTRIBUTORS STEPHEN P. BOYD, Information Systems Laboratory, Stanford University, Stanford,CA 94305 ELIO D. DI CLAUDIO, INFOCOM Department, University of Roma “La Sapienza,” Via Eudossiana 18, I-00184 Roma, Italy YONINA C. ELDAR, Department of Electrical Engineering, Technion—Israel Institute ofTechnology,Haifa 32000, Israel ALEX B. GERSHMAN, Darmstadt University of Technology, Institute of Tele- communications,Merckstrasse25, 64283 Darmstadt,Germany MIGUEL A. LAGUNAS, Centre Tecnolo`gic de Telecomunicacions de Catalunya, NEXUS1 Building, Gran Capita 2–4,08034Barcelona, Spain AMIR LESHEM, School of Engineering, Bar-Ilan University, 52900 Ramat-Gan, Israel JIAN LI, Department of Electrical and Computer Engineering, Engineering Bldg., CenterDrive, Universityof Florida, Gainesville, FL 32611 ROBERT G. LORENZ,Beceem Communications, Santa Clara,CA 95054 ZHI-QUANLUO,DepartmentofElectricalandComputerEngineering,University of Minnesota,Minneapolis, MN 55455 XAVIER MESTRE, Centre Tecnolo`gic de Telecomunicacions de Catalunya, NEXUS1 Building, Gran Capita 2–4,08034Barcelona, Spain ARYE NEHORAI, Department of Electrical Engineering and Computer Science, University ofIllinois at Chicago, Chicago, IL60607 RAFFAELEPARISI,INFOCOMDepartment,UniversityofRoma“LaSapienza,” Via Eudossiana 18, I-00184 Roma, Italy SHAHRAM SHAHBAZPANAHI, McMaster University, Hamilton, Ontario L8S 4L8,Canada PETRE STOICA, Division of Systems and Control, Department of Information Technology, UppsalaUniversity, SE-75105 Uppsala, Sweden ALLE-JANVANDERVEEN,DepartmentofElectricalEngineering,DelftUniver- sity ofTechnology, 2628 Delft, TheNetherlands ZHISONGWANG,DepartmentofElectricalandComputerEngineering,Engineer- ing Bldg.,CenterDrive, Universityof Florida, Gainesville, FL 32611 ix