Wireless CommunicationsPRENTICE HALLUpper Saddle River, NJ 07458www.phptr.com ISBN0-13-042232-09 780130 4223239 00 0 0Xiaodong wangH. Vincent PoorCommunicationWirelessPrentice Hall Communications Engineering and Emerging Technologies SeriesTheodore S. Rappaport, Series EditorADVANCEDTECHNI QUESF ORSI GNALRECEPTI ONWireless CommunicationSystemsWangPoorWireless Communication SystemsADVANCEDTECHNI QUESF ORSI GNALRECEPTI ONThe indispensable guide to wireless communicationsnow fully revised and updated!Wireless Communications: Principles and Practice, Second Edition is the definitive modern text for wireless communications technology and system design. Building on his classic first edition, Theodore S. Rappaport covers thefundamental issues impacting all wireless networks and reviews virtually every important new wireless standard andtechnological development, offering especially comprehensive coverage of the 3G systems and wireless local area networks (WLANs) that will transform communications in the coming years. Rappaport illustrates each key conceptwith practical examples, thoroughly explained and solved step-by-step. Coverage includes:s An overview of key wireless technologies: voice, data, cordless, paging, fixed and mobile broadband wireless systems, and beyonds Wireless system design fundamentals: channel assignment, handoffs, trunking efficiency, interference, frequency reuse, capacity planning, large-scale fading, and mores Path loss, small-scale fading, multipath, reflection, diffraction, scattering, shadowing, spatial-temporal channel modeling, and microcell/indoor propagation s Modulation, equalization, diversity, channel coding, and speech codings New wireless LAN technologies: IEEE 802.11a/b, HIPERLAN, BRAN, and other alternativess New 3G air interface standards, including W-CDMA, cdma2000, GPRS, UMTS, and EDGEs Bluetooth, wearable computers, fixed wireless and Local Multipoint Distribution Service (LMDS), and other advanced technologiess Updated glossary of abbreviations and acronyms, and a thorough list of referencess Dozens of new examples and end-of-chapter problemsWhether youre a communications/network professional, manager, researcher, or student, Wireless Communications:Principles and Practice, Second Edition gives you an in-depth understanding of the state-of-the-art in wireless technologytodays and tomorrows.About the AuthorTHEODORE S. RAPPAPORT holds The William and Bettye Nowlin Chair in Electrical Engineering at the University of Texas,Austin and is the Series Editor for Prentice Halls Communications Engineering and Emerging Technologies Series.In 1990 he founded the Mobile & Portable Radio Research Group at Virginia Tech, one of the first university researchand educational programs focused on wireless communications. Rappaport has developed dozens of commercialproducts now used by major carriers and manufacturers. He has also created fundamental research and teachingmaterials used in industry short courses and in university classrooms around the globe. His current research focuseson new methods for analyzing and developing wireless broadband and portable Internet access in emerging frequencybands, and on the development, modeling, and practical use of 3-D site-specific propagation techniques for futurewireless networks.SystemsWirelessCommunicationSystemsADVANCED TECHNIQUESFOR SIGNAL RECEPTIONXiaodong wangH. Vincent PoorFPOWirelessCommunicationSystems:AdvancedTechniquesforSignalReceptionXiaodongWangColumbiaUniversityH.VincentPoorPrincetonUniversityJune2,20022Contents1 Introduction 131.1 Motivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.2 TheWirelessSignalingEnvironment . . . . . . . . . . . . . . . . . . . . . . 151.2.1 Single-userModulationTechniques . . . . . . . . . . . . . . . . . . . 151.2.2 Multiple-accessTechniques. . . . . . . . . . . . . . . . . . . . . . . . 181.2.3 TheWirelessChannel . . . . . . . . . . . . . . . . . . . . . . . . . . 211.3 BasicReceiverSignalProcessingforWireless . . . . . . . . . . . . . . . . . . 271.3.1 TheMatchedFilter/RAKEReceiver . . . . . . . . . . . . . . . . . . 271.3.2 Equalization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311.3.3 MultiuserDetection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.4 OutlineoftheBook. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 BlindMultiuserDetection 432.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.2 LinearReceiversforSynchronousCDMA. . . . . . . . . . . . . . . . . . . . 452.2.1 SynchronousCDMASignalModel . . . . . . . . . . . . . . . . . . . . 452.2.2 LinearDecorrelatingDetector . . . . . . . . . . . . . . . . . . . . . . 472.2.3 LinearMMSEDetector . . . . . . . . . . . . . . . . . . . . . . . . . . 482.3 BlindMultiuserDetection: DirectMethods. . . . . . . . . . . . . . . . . . . 492.3.1 LMSAlgorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522.3.2 RLSAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.3.3 QR-RLSAlgorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 542.4 BlindMultiuserDetection: SubspaceMethods . . . . . . . . . . . . . . . . . 5934 CONTENTS2.4.1 LinearDecorrelatingDetector . . . . . . . . . . . . . . . . . . . . . . 602.4.2 LinearMMSEDetector . . . . . . . . . . . . . . . . . . . . . . . . . . 622.4.3 AsymptoticsofDetectorEstimates . . . . . . . . . . . . . . . . . . . 632.4.4 AsymptoticMultiuserEciencyunderMismatch . . . . . . . . . . . 652.5 PerformanceofBlindMultiuserDetectors . . . . . . . . . . . . . . . . . . . 682.5.1 PerformanceMeasures . . . . . . . . . . . . . . . . . . . . . . . . . . 682.5.2 AsymptoticOutputSINR . . . . . . . . . . . . . . . . . . . . . . . . 702.6 SubspaceTrackingAlgorithms . . . . . . . . . . . . . . . . . . . . . . . . . . 812.6.1 ThePASTdAlgorithm. . . . . . . . . . . . . . . . . . . . . . . . . . 822.6.2 QR-JacobiMethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 872.6.3 NAHJSubspaceTracking . . . . . . . . . . . . . . . . . . . . . . . . 892.7 BlindMultiuserDetectioninMultipathChannels . . . . . . . . . . . . . . . 922.7.1 MultipathSignalModel . . . . . . . . . . . . . . . . . . . . . . . . . 932.7.2 LinearMultiuserDetectors. . . . . . . . . . . . . . . . . . . . . . . . 952.7.3 BlindChannelEstimation . . . . . . . . . . . . . . . . . . . . . . . . 992.7.4 AdaptiveReceiverStructures . . . . . . . . . . . . . . . . . . . . . . 1052.7.5 BlindMultiuserDetectioninCorrelatedNoise . . . . . . . . . . . . . 1102.8 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1172.8.1 DerivationsinSection2.3.3 . . . . . . . . . . . . . . . . . . . . . . . 1172.8.2 ProofsinSection2.4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . 1192.8.3 ProofsinSection2.5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 1203 Group-BlindMultiuserDetection 1333.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1333.2 LinearGroup-BlindMultiuserDetectionforSynchronousCDMA . . . . . . 1343.3 PerformanceofGroup-BlindMultiuserDetectors . . . . . . . . . . . . . . . 1453.3.1 Form-IIGroup-blindHybridDetector. . . . . . . . . . . . . . . . . . 1453.3.2 Form-IGroup-blindDetectors . . . . . . . . . . . . . . . . . . . . . . 1543.4 NonlinearGroup-BlindMultiuserDetection . . . . . . . . . . . . . . . . . . 1593.4.1 Slowest-DescentSearch . . . . . . . . . . . . . . . . . . . . . . . . . 1603.4.2 NonlinearGroup-BlindMultiuserDetection . . . . . . . . . . . . . . 1633.5 Group-BlindMultiuserDetectioninMultipathChannels . . . . . . . . . . . 170CONTENTS 53.5.1 LinearGroup-BlindDetectors . . . . . . . . . . . . . . . . . . . . . . 1723.5.2 AdaptiveGroup-blindLinearMultiuserDetection . . . . . . . . . . . 1823.5.3 LinearGroup-BlindDetectorinCorrelatedNoise . . . . . . . . . . . 1853.5.4 NonlinearGroup-BlindDetector. . . . . . . . . . . . . . . . . . . . . 1903.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1953.6.1 ProofsinSection3.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 1953.6.2 ProofsinSection3.3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 1994 RobustMultiuserDetectioninNon-GaussianChannels 2054.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2054.2 MultiuserDetectionviaRobustRegression. . . . . . . . . . . . . . . . . . . 2084.2.1 SystemModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2084.2.2 Least-SquaresRegressionandLinearDecorrelator . . . . . . . . . . . 2094.2.3 RobustMultiuserDetectionviaM-Regression . . . . . . . . . . . . . 2104.3 AsymptoticPerformanceofRobustMultiuserDetector . . . . . . . . . . . . 2144.3.1 TheInuenceFunction. . . . . . . . . . . . . . . . . . . . . . . . . . 2144.3.2 AsymptoticProbabilityofError. . . . . . . . . . . . . . . . . . . . . 2174.4 ImplementationofRobustMultiuserDetectors. . . . . . . . . . . . . . . . . 2214.5 RobustBlindMultiuserDetection. . . . . . . . . . . . . . . . . . . . . . . . 2314.6 RobustMultiuserDetectionbasedonLocalLikelihoodSearch . . . . . . . . 2384.6.1 Exhaustive-SearchDetectionandDecorrelativeDetection. . . . . . . 2384.6.2 Local-SearchDetection. . . . . . . . . . . . . . . . . . . . . . . . . . 2404.7 RobustGroup-blindMultiuserDetection . . . . . . . . . . . . . . . . . . . . 2434.8 ExtensiontoMultipathChannels . . . . . . . . . . . . . . . . . . . . . . . . 2484.8.1 RobustBlindMultiuserDetectioninMultipathChannels. . . . . . . 2494.8.2 RobustGroup-BlindMultiuserDetectioninMultipathChannels . . . 2504.9 RobustMultiuserDetectioninStableNoise . . . . . . . . . . . . . . . . . . 2544.9.1 TheSymmetricStableDistribution . . . . . . . . . . . . . . . . . . . 2544.9.2 PerformanceofRobustMultiuserDetectorsinStableNoise. . . . . . 2594.10 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2614.10.1 ProofofProposition4.1inSection4.4 . . . . . . . . . . . . . . . . . 2614.10.2 ProofofProposition4.2inSection4.5 . . . . . . . . . . . . . . . . . 2656 CONTENTS5 Space-TimeMultiuserDetection 2675.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2675.2 AdaptiveArrayProcessinginTDMASystems . . . . . . . . . . . . . . . . . 2695.2.1 SignalModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2695.2.2 LinearMMSECombining . . . . . . . . . . . . . . . . . . . . . . . . 2715.2.3 ASubspace-basedTrainingAlgorithm . . . . . . . . . . . . . . . . . 2735.2.4 ExtensiontoDispersiveChannels . . . . . . . . . . . . . . . . . . . . 2805.3 OptimalSpace-TimeMultiuserDetection. . . . . . . . . . . . . . . . . . . . 2835.3.1 SignalModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2845.3.2 ASucientStatistic . . . . . . . . . . . . . . . . . . . . . . . . . . . 2865.3.3 MaximumLikelihoodMultiuserSequenceDetector . . . . . . . . . . 2895.4 LinearSpace-TimeMultiuserDetection. . . . . . . . . . . . . . . . . . . . . 2915.4.1 LinearMultiuserDetectionviaIterativeInterferenceCancellation . . 2925.4.2 Single-UserLinearSpace-TimeDetection. . . . . . . . . . . . . . . . 2955.4.3 CombinedSingle-user/MultiuserLinearDetection . . . . . . . . . . . 2995.5 AdaptiveSpace-TimeMultiuserDetectioninSynchronousCDMA. . . . . . 3075.5.1 OneTransmitAntenna,TwoReceiveAntennas . . . . . . . . . . . . 3095.5.2 TwoTransmitAntennas,OneReceiveAntenna . . . . . . . . . . . . 3165.5.3 TwoTransmitterandTwoReceiveAntennas. . . . . . . . . . . . . . 3195.5.4 BlindAdaptiveImplementations . . . . . . . . . . . . . . . . . . . . 3235.6 AdaptiveSpace-TimeMultiuserDetectioninMultipathCDMA . . . . . . . 3305.6.1 SignalModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3305.6.2 BlindMMSESpace-TimeMultiuserDetection . . . . . . . . . . . . . 3365.6.3 BlindAdaptiveChannelEstimation. . . . . . . . . . . . . . . . . . . 3366 TurboMultiuserDetection 3476.1 Introduction-ThePrincipleofTurboProcessing . . . . . . . . . . . . . . . 3476.2 TheMAPDecodingAlgorithmforConvolutionalCodes . . . . . . . . . . . . 3516.3 TurboMultiuserDetectionforSynchronousCDMA . . . . . . . . . . . . . . 3576.3.1 TurboMultiuserReceiver . . . . . . . . . . . . . . . . . . . . . . . . 3576.3.2 TheOptimalSISOMultiuserDetector . . . . . . . . . . . . . . . . . 3596.3.3 ALow-ComplexitySISOMultiuserDetector . . . . . . . . . . . . . . 362CONTENTS 76.4 TurboMultiuserDetectionwithUnknownInterferers . . . . . . . . . . . . . 3746.4.1 SignalModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3756.4.2 Group-blindSISOMultiuserDetector. . . . . . . . . . . . . . . . . . 3766.4.3 SlidingWindowGroup-BlindDetectorforAsynchronousCDMA. . . 3836.5 TurboMultiuserDetectioninCDMAwithMultipathFading. . . . . . . . . 3886.5.1 SignalModelandSucientStatistics . . . . . . . . . . . . . . . . . . 3886.5.2 SISOMultiuserDetectorinMultipathFadingChannel . . . . . . . . 3926.6 TurboMultiuserDetectioninCDMAwithTurboCoding. . . . . . . . . . . 3956.6.1 TurboCodeandSoftDecodingAlgorithm . . . . . . . . . . . . . . . 3966.6.2 Turbo Multiuser Receiver in Turbo-coded CDMA with Multipath Fading4016.7 TurboMultiuserDetectioninSpace-timeBlockCodedSystems . . . . . . . 4096.7.1 MultiuserSTBCSystem. . . . . . . . . . . . . . . . . . . . . . . . . 4116.7.2 TurboMultiuserReceiverforSTBCSystem . . . . . . . . . . . . . . 4146.7.3 Projection-basedTurboMultiuserDetection . . . . . . . . . . . . . . 4196.8 TurboMultiuserDetectioninSpace-timeTrellisCodedSystems . . . . . . . 4246.8.1 MultiuserSTTCSystem. . . . . . . . . . . . . . . . . . . . . . . . . 4256.8.2 TurboMultiuserReceiverforSTTCSystem . . . . . . . . . . . . . . 4276.9 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4336.9.1 ProofsinSection6.3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 4336.9.2 DerivationoftheLLRfortheRAKEReceiverinSection6.6.2 . . . . 4357 NarrowbandInterferenceSuppression 4397.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4397.2 LinearPredictiveTechniques. . . . . . . . . . . . . . . . . . . . . . . . . . . 4447.2.1 SignalModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4447.2.2 LinearPredictiveMethods . . . . . . . . . . . . . . . . . . . . . . . . 4477.3 NonlinearPredictiveTechniques . . . . . . . . . . . . . . . . . . . . . . . . . 4527.3.1 ACMFilter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4537.3.2 AdaptiveNonlinearPredictor . . . . . . . . . . . . . . . . . . . . . . 4567.3.3 NonlinearInterpolatingFilters. . . . . . . . . . . . . . . . . . . . . . 4597.3.4 HMM-BasedMethods . . . . . . . . . . . . . . . . . . . . . . . . . . 4637.4 Code-AidedTechniques. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4648 CONTENTS7.4.1 NBISuppressionViatheLinearMMSEDetector . . . . . . . . . . . 4657.4.2 TonalInterference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4677.4.3 Autoregressive(AR)Interference . . . . . . . . . . . . . . . . . . . . 4707.4.4 DigitalInterference . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4747.5 PerformanceComparisonsofNBISuppressionTechniques . . . . . . . . . . . 4777.6 Near-farResistancetoBothNBIandMAIbyLinearMMSEDetector . . . . 4847.6.1 Near-farResistancetoNBI . . . . . . . . . . . . . . . . . . . . . . . . 4847.6.2 Near-farResistancetoBothNBIandMAI . . . . . . . . . . . . . . . 4857.7 AdaptiveLinearMMSENBISuppression. . . . . . . . . . . . . . . . . . . . 4897.8 AMaximum-LikelihoodCode-AidedMethod. . . . . . . . . . . . . . . . . . 4927.9 Appendix: ConvergenceoftheRLSLinearMMSEDetector . . . . . . . . . 4977.9.1 LinearMMSEDetectorandRLSBlindAdaptationRule . . . . . . . 4977.9.2 ConvergenceoftheMeanWeightVector . . . . . . . . . . . . . . . . 4987.9.3 WeightErrorCorrelationMatrix . . . . . . . . . . . . . . . . . . . . 5027.9.4 ConvergenceofMSE . . . . . . . . . . . . . . . . . . . . . . . . . . . 5057.9.5 Steady-stateSINR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5067.9.6 ComparisonwithTraining-basedRLSAlgorithm . . . . . . . . . . . . 5078 MonteCarloBayesianSignalProcessing 5098.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5098.2 BayesianSignalProcessing. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5118.2.1 TheBayesianFramework . . . . . . . . . . . . . . . . . . . . . . . . . 5118.2.2 BatchProcessingversusAdaptiveProcessing . . . . . . . . . . . . . . 5128.2.3 MonteCarloMethods . . . . . . . . . . . . . . . . . . . . . . . . . . 5148.3 MarkovChainMonteCarlo(MCMC)SignalProcessing. . . . . . . . . . . . 5148.3.1 Metropolis-HastingsAlgorithm . . . . . . . . . . . . . . . . . . . . . 5158.3.2 GibbsSampler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5168.4 BayesianMultiuserDetectionviaMCMC. . . . . . . . . . . . . . . . . . . . 5188.4.1 SystemDescription . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5198.4.2 BayesianMultiuserDetectioninGaussianNoise . . . . . . . . . . . . 5218.4.3 BayesianMultiuserDetectioninImpulsiveNoise . . . . . . . . . . . . 5298.4.4 BayesianMultiuserDetectioninCodedSystems . . . . . . . . . . . . 533CONTENTS 98.5 SequentialMonteCarlo(SMC)SignalProcessing . . . . . . . . . . . . . . . 5458.5.1 SequentialImportanceSampling. . . . . . . . . . . . . . . . . . . . . 5458.5.2 SMCforDynamicalSystems. . . . . . . . . . . . . . . . . . . . . . . 5488.5.3 ResamplingProcedures. . . . . . . . . . . . . . . . . . . . . . . . . . 5518.5.4 MixtureKalmanFilter . . . . . . . . . . . . . . . . . . . . . . . . . . 5548.6 BlindAdaptiveEqualizationofMIMOChannelsviaSMC . . . . . . . . . . 5558.6.1 SystemDescription . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5568.6.2 SMCBlindAdaptiveEqualizerforMIMOChannels. . . . . . . . . . 5578.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5618.7.1 DerivationsinSection8.4.2 . . . . . . . . . . . . . . . . . . . . . . . 5618.7.2 DerivationsinSection8.4.3 . . . . . . . . . . . . . . . . . . . . . . . 5648.7.3 ProofofProposition8.1inSection8.5.2 . . . . . . . . . . . . . . . . 5658.7.4 ProofofProposition8.2inSection8.5.3 . . . . . . . . . . . . . . . . 5669 SignalProcessingTechniquesforFastFadingChannels 5699.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5699.2 StatisticalModellingofMultipathFadingChannels . . . . . . . . . . . . . . 5729.2.1 Frequency-non-selectiveFadingChannels . . . . . . . . . . . . . . . . 5749.2.2 Frequency-selectiveFadingChannels . . . . . . . . . . . . . . . . . . 5759.3 CoherentDetectioninFadingChannelsBasedontheEMAlgorithm . . . . . 5769.3.1 TheExpectation-MaximizationAlgorithm . . . . . . . . . . . . . . . 5769.3.2 EM-basedReceiverinFlat-FadingChannels . . . . . . . . . . . . . . 5779.3.3 Linear Multiuser Detection in Flat-Fading Synchronous CDMA Channels5809.3.4 TheSequentialEMAlgorithm. . . . . . . . . . . . . . . . . . . . . . 5829.4 Decision-FeedbackDierentialDetectioninFadingChannels . . . . . . . . . 5849.4.1 Decision-FeedbackDierentialDetectioninFlat-FadingChannels . . 5849.4.2 DecisionFeedbackSpace-TimeDierentialDecoding . . . . . . . . . 5879.5 AdaptiveDetection/DecodinginFlat-FadingChannelsviaSequentialMonteCarlo. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6019.5.1 SystemDescription . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6019.5.2 AdaptiveReceiverinFadingGaussianNoiseChannels-UncodedCase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60410 CONTENTS9.5.3 DelayedEstimation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 6099.5.4 AdaptiveReceiverinFadingGaussianNoiseChannels-CodedCase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6159.5.5 AdaptiveReceiversinFadingImpulsiveNoiseChannels. . . . . . . . 6209.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6249.6.1 ProofofProposition9.1inSection9.5.2 . . . . . . . . . . . . . . . . 62410AdvancedSignalProcessingforCodedOFDMSystems 62710.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62710.2 TheOFDMCommunicationSystem . . . . . . . . . . . . . . . . . . . . . . . 62810.3 Blind MCMC Receiver for Coded OFDMwith Frequency Oset andFrequency-selectiveFading . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63110.3.1 SystemDescription . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63210.3.2 BayesianMCMCDemodulator . . . . . . . . . . . . . . . . . . . . . 63410.4 Pilot-symbol-aided Turbo Receiver for Space-Time Block Coded OFDM Systems64810.4.1 SystemDescriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 65310.4.2 MLReceiverbasedontheEMAlgorithm . . . . . . . . . . . . . . . . 65710.4.3 Pilot-symbol-aidedTurboReceiver . . . . . . . . . . . . . . . . . . . 66110.5 LDPC-basedSpace-TimeCodedOFDMSystems . . . . . . . . . . . . . . . 67210.5.1 CapacityConsiderationsforSTC-OFDMSystems . . . . . . . . . . . 67410.5.2 Low-DensityParity-CheckCodes . . . . . . . . . . . . . . . . . . . . 68310.5.3 LDPC-basedSTC-OFDMSystem. . . . . . . . . . . . . . . . . . . . 68610.5.4 TurboReceiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68910.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69910.6.1 DerivationsinSection10.3. . . . . . . . . . . . . . . . . . . . . . . . 699CONTENTS 11PREFACEWireless communications, together withits applications andunderlying technologies, isamongtodaysmostactiveareasof technologydevelopment. Theveryrapidpaceof im-provementsinbothcustomandprogrammableintegratedcircuitsforsignalprocessingap-plications has led to the justable view of advanced signal processing as a key enabler of theaggressivelyescalatingcapacitydemandsofemergingwirelesssystems. Consequently,therehasbeenatremendousandverywidespreadeortonthepartof theresearchcommunitytodevelopnovel signal processingtechniquesthatcanfulll thispromise. Thepublishedliteratureinthisareahasgrownexplosivelyinrecentyears, andithasbecomequitedi-culttosynthesizethemanydevelopmentsdescribedinthisliterature. Thepurposeofthismonographistopresent, inoneplaceandinauniedframework, anumberofkeyrecentcontributions in this eld. Even though these contributions come primarily from the researchcommunity, the focus of this presentation is on the development, analysis, and understandingof explicit algorithms for performing advanced processing tasks arising in receiver design foremergingwirelesssystems.Althoughthisbookislargelyself-contained, itiswrittenprincipallyfordesigners, re-searchers, andgraduatestudentswithsomepriorexposuretowirelesscommunicationsys-tems. KnowledgeoftheeldatthelevelofTheodoreRappaportsbook,WirelessCommu-nications: Principles&Practice[397], forexample, wouldbequiteuseful tothereaderofthisbook, aswouldsomeexposuretodigitalcommunicationsatthelevelofJohnProakisbook,Digital Communications[388].AcknowledgementTheauthors wouldlike tothankthesupports fromtheArmyResearchLaboratory, theNationalScienceFoundation, theNewJerseyCommissiononScienceandTechnology, andtheOceofNavalResearch.12 CONTENTSChapter1Introduction1.1 MotivationWirelesscommunicationsisoneofthemostactiveareasoftechnologydevelopmentofourtime. Thisdevelopmentisbeingdrivenprimarilybythetransformationof whathasbeenlargely a medium for supporting voice telephony, into a medium for supporting other servicessuchasthetransmissionofvideo,images,textanddata. Thus,similarlytowhathappenedtowirelinecapacityinthe1990s, thedemandfor newwireless capacityis growingat averyrapidpace. Althoughthereareof coursestill agreatmanytechnical problemstobesolvedinwirelinecommunications,demandsforadditionalwirelinecapacitycanbefullledlargelywiththe additionof newprivate infrastructure, suchas additional optical ber,routers, switches, andsoforth. Onthe other hand, the traditional resources that havebeenusedtoaddcapacitytowirelesssystemsareradiobandwidthandtransmitterpower.Unfortunately, thesetworesourcesareamongthemostseverelylimitedinthedeploymentof modernwireless networks - radiobandwidthbecause of the verytight situationwithregardtouseful radiospectrum, andtransmitterpowerbecauseoftheprovisionofmobileorotherwiseportableservicesrequirestheuseof batterypower, whichislimited. Thesetworesourcesaresimplynotgrowingorimprovingatratesthancansupportanticipateddemandsforwirelesscapacity. Ontheotherhand, oneresourcethatisgrowingataveryrapidrateisthatofprocessingpower. MooresLaw, whichassertsadoublingofprocessorcapabilitieseveryeighteenmonths,hasbeenquiteaccurateoverthepasttwentyyears,and1314 CHAPTER1. INTRODUCTIONitsaccuracypromisestocontinueforyearstocome. Giventhesecircumstances, therehasbeenconsiderableresearcheortinrecentyearsaimedatdevelopingnewwirelesscapacitythroughthedeploymentof greaterintelligenceinwirelessnetworks. (See, forexample, in[142, 143, 268, 374, 383] for reviews of some of this work.)A key aspect of this movement hasbeenthedevelopmentofnovel signal transmissiontechniquesandadvancedreceiversignalprocessing methods that allow for signicant increases in wireless capacity without attendantincreases in bandwidth or power requirements. The purpose of this monograph is to presentsomeofthemostrecentofthesereceiversignalprocessingmethodsinasingleplaceandinauniedframework.Wirelesscommunicationstodaycoversaverywidearrayofapplications. Thetelecom-municationsindustryisoneof thelargestindustriesworldwide, withmorethanatrillionUSdollarsinannualrevenuesforservicesandequipment. (Toputthisinperspective,thisnumberiscomparabletothegrossdomesticproductof manyof theworldsrichestcoun-tries, includingFrance, ItalyandtheUnitedKingdom.) Thelargest, andmostnoticeable,partofthetelecommunicationsbusinessistelephony. Theprincipal wirelesscomponentoftelephonyismobile(i.e., cellular) telephony. Theworldwidegrowthrateincellular tele-phonyisveryaggressive, andmostanalystspredictthatthenumberof cellulartelephonysubscriptionsworldwidewillsurpassthenumberofwireline(i.e.,xed)telephonysubscrip-tionsbythetimethisbookisinprint. Moreover, thenumberof cellularsubscriptionsissimilarly expected to surpass one billion in the very near future. (At the time of this writingin2002, thenumberof xedtelephonysubscriptionsworldwideisreportedlyontheorderof900million.) Thesenumbersmakecellulartelephonyaveryimportantdriverofwirelesstechnologydevelopment,andinrecentyearsthepushtodevelopnewmobiledataservices,whichgocollectivelyunderthenamethird-generation(3G)cellular, hasplayedakeyroleinmotivatingresearchinnewsignal processingtechniquesforwireless. However, cellulartelephonyisonlyoneofaverywidearrayofwirelesstechnologiesthatarebeingdevelopedvery rapidly at the present time. Among other technologies are wireless pico-networking (asexempliedbytheBluetoothradio-on-a-chip)andotherpersonalareanetwork(PAN)sys-tems (e.g., the IEEE 802.15 family of standards), wireless local area network (LAN) systems(exempliedbytheIEEE802.11andHiperLANfamiliesofstandards-so-calledWi-Fisys-tems), wireless metropolitan area network (MAN) systems (exemplied by the IEEE 802.161.2. THEWIRELESSSIGNALINGENVIRONMENT 15familyofstandards),wirelesslocalloop(WLL)systems,andavarietyofsatellitesystems.Theseadditional wirelesstechnologiesprovideabasisforaveryricharrayofapplications,includinglocal telephonyservice, broadbandInternetaccess, anddistributionof high-rateentertainment content suchas high-denitionvideoandhigh-qualityaudiotothehome,within the home, to automobiles, etc. (See, e.g., [9, 40, 41, 129, 156, 158, 161, 163, 339, 356,358,361,385,386,387,420,428,440,448,499,550,551]forfurtherdiscussionoftheseandrelated applications.) Like 3G, these technologies have also spurred considerable research insignalprocessingforwireless.Thesetechnologiesaresupportedbyanumberoftransmissionandchannel-assignmenttechniques, includingtime-divisionmultiple-access(TDMA), code-divisionmultipleaccess(CDMA) andother spread-spectrumsystems, orthogonal frequency-divisionmultiplexing(OFDM)andothermulti-carriersystems,andhigh-ratesingle-carriersystems. Thesetech-niquesarechosenprimarilytoaddressthephysical propertiesof wirelesschannels, amongthe mostprominent of which are multipath fading,dispersion,and interference. In additiontothesetemporaltransmissiontechniques,therearealsospatialtechniques,notablybeam-forming and space-time coding, that can be applied at the transmitter to exploit the spatialandangulardiversityofwirelesschannels. Toobtainmaximalbenetfromthesetransmis-sion techniques, to exploit the diversity opportunities of the wireless channel, and to mitigatetheimpairmentsofthewirelesschannel,advancedreceiversignalprocessingtechniquesareofinterest. Theseincludechannel equalizationtocombatdispersion, RAKEcombiningtoexploit resolvablemultipath, multiuser detectiontomitigatemultiple-access interference,suppressionmethodsforco-channel interference, beamformingtoexploitspatial diversity,and space-time processing to jointly exploit temporal and spatial properties of the signalingenvironment. Thesetechniquesarealldescribedintheensuingchapters.1.2 TheWirelessSignalingEnvironment1.2.1 Single-userModulationTechniquesInordertodiscussadvancedreceiversignal processingmethodsforwireless, itisuseful torstspecifyageneralmodelforthesignalreceivedbyawirelessreceiver. Todoso,wecanrst think of a single transmitter, transmitting a sequence or frame b[0], b[1], . . . , b[M1] of16 CHAPTER1. INTRODUCTIONchannel symbols over a wireless channel. These symbols can be binary (e.g., 1), or they maytakeonmoregeneralvaluesfromanitealphabetofcomplexnumbers. Inthistreatment,wewillconsideronlylinearmodulationsystems,inwhichthesymbolsaretransmittedintothe channel by being modulated linearly onto a signaling waveform to produce a transmittedsignalofthisform:x(t) =M1i=0b[i]si(t), (1.1)wheresi()isthemodulationwaveformassociatedwiththeithsymbol. Inthisexpression,thewaveformscanbequitegeneral. Forexample, asingle-carriermodulationsystemwithcarrier frequency c, baseband pulse shape p(), and symbol rate 1/Tis obtained by choosingsi(t) =Ap(t iT)e(ct+), (1.2)whereA>0and (, ] denotecarrieramplitudeandphaseoset, respectively. Thebaseband pulse shape may, for example, be a simple unit-energy rectangular pulse of durationT,i.e.,p(t) =pT(t)z=

1T, 0 t < T0, otherwise(1.3)oritcouldbearaised-cosinepulse, abandlimitedpulse, etc. Similarly, adirect-sequencespread-spectrumsystemis producedbychoosingthewaveforms as in(1.2) but withthebasebandpulseshapechosentobeaspreadingwaveform:p(t) =N1=0a

(t Tc), (1.4)where Nis the spreading gain, a0, a1, . . . , aN1, is a pseudo-random spreading code (typicallya

1), () is the chip waveform, and Tcz= T/Nis the chip interval. The chip waveformmay,forexample,beaunit-energyrectangularpulseofdurationTc:(t) =pTc(t). (1.5)Otherchoicesof thechipwaveformcanalsobemadetolowerthechipbandwidth. Thespreading waveform of (1.4) is periodic when used in (1.2),since the same spreading code isrepeated in every symbol interval. Some systems (e.g., CDMA systems for cellular telephony)operate withso-calledlongspreading codes,forwhichthe periodicityismuch longerthana1.2. THEWIRELESSSIGNALINGENVIRONMENT 17singlesymbolinterval. Thissituationcanbemodelledby(1.1)byreplacing p(t)in(1.2)byavariantof(1.4)inwhichthespreadingcodevariesfromsymboltosymbol;i.e.,pi(t) =N1=0a(i)

(t Tc). (1.6)Spread spectrum modulation can also take the form of frequency hopping, in which the carrierfrequencyin(1.2)ischangedovertimeaccordingtoapseudorandompattern. Typically,the carrier frequency changes at a rate much slower than the symbol rate, a situation knownas slowfrequencyhopping; however, fast hopping, inwhichthecarrier changes withinasymbol interval, is alsopossible. Single-carrier systems, includingbothtypes of spread-spectrum, are widely used in cellular standards, in wireless LANs, Bluetooth, etc. (See, e.g.,[41,128,147,160,174,244,333,356,358,384,386,399,400,440,514,581].)Multicarrier systems canalsobemodelledintheframeworkof (1.1) bychoosingthesignaling waveforms si() to be sinusoidal signals with dierent frequencies. In particular,(1.2)canbereplacedbysi(t) =Ap(t)e(it+i), (1.7)wherenow, thefrequencyandphasedependonthesymbol numberi, butall symbolsaretransmitted simultaneously in time with baseband pulse shape p(). We can see that (1.2) isthe counterpart of this situation with time and frequency reversed: all symbols are transmit-tedatthesamefrequency, butatdierenttimes. (Ofcourse, inpracticemultiplesymbolsaresentintimesequenceovereachofthemultiplecarriersinmulti-carriersystems.) Theindividual carriers can also be direct-spread, and the baseband pulse shape used can dependonthesymbolnumberi.(Forexample,thelattersituationisusedinso-calledmulticarrierCDMA, in which a spreading code is used across the carrier frequencies.)A particular case of(1.7) is OFDM, in which the baseband pulse shape is a unit pulse pT, the intercarrier spacingis1/Tcyclespersecond, andthephasesarechosensothatthecarriersareorthogonal atthis spacing. (This is the minimal spacing for which such orthogonality can be maintained.)OFDMiswidelybelievedtobeamongthemosteectivetechniquesforwirelessbroadbandapplications, and is the basis for the IEEE 802.11a high-speed wireless LAN standard. (See,e.g.,[349]foradiscussionofmulti-carriersystems.)Anemergingtypeofwirelessmodulationschemeisultra-wideband(UWB)modulation,inwhichdatais transmittedwithnocarrier throughthemodulationof extremelyshort18 CHAPTER1. INTRODUCTIONpulses. Eitherthetimingoramplitudeofthesepulsescanbeusedtocarrytheinformationsymbols. Typical UWBsystemsinvolvethetransmissionof manyrepetitionsof thesamesymbol, possibly with the use of a direct-sequence type spreading code from transmission totransmission. (See,e.g.,[561]forabasicdescriptionofUWBsystems.)Furtherdetailsontheabovemodulationwaveformsandtheirpropertieswill beintro-ducedasneededthroughoutthistreatment.1.2.2 Multiple-accessTechniquesTheabovesectiondiscussedwaysinwhichasymbol streamassociatedwithasingleusercanbetransmitted. Manywirelesschannels, particularyinemergingsystems, operateasmultiple-accesssystems,inwhichmultipleuserssharethesameradioresources.Thereareseveral ways inwhichradioresources canbesharedamongmultipleusers.Thesecanbeviewedaswaysofallocatingregionsinfrequency,spaceandtimetodierentusers, asshowninFig. 1.1. Forexample, aclassicmultiple-accesstechniqueisfrequency-divisionmultiple-access(FDMA),inwhichthefrequencybandavailableforagivenserviceisdividedintosub-bandsthatareallocatedtoindividualuserswhowishtousetheservice.Users are given exclusive use of their sub-band during their communication session, but theyarenotallowedtotransmitsignalswithinothersub-bands. FDMAistheprincipal multi-plexingmethodusedinradioandtelevisionbroadcast, andintherst-generation(analogvoice) cellular telephony systems, such as Advanced Mobile Phone Systems (AMPS), NordicMobileTelephone(NMT), etc., developedinthe1980s (cf. [449]). FDMAis alsousedinsomeforminall other current cellular systems, intandemwithother multiple-accesstechniquesthatareusedtofurtherallocatethesub-bandstomultipleusers.Similarly, users can share the channel on the basis of time-division multiple-access(TDMA) inwhichtime is dividedintoequal-lengthintervals, whichare further dividedintoequal-lengthsub-intervals, ortimeslots. Eachuserisallowedtotransmitthroughouttheentireallocatedfrequencybandduringagivenslotineachinterval, butisnotallowedtotransmitduringothertimeslotswhenotherusersaretransmitting. So,whereasFDMAallows eachuser tousepart of thespectrumall of thetime, TDMAallows eachuser touseallofthespectrumpartofthetime. Thismethodofchannelsharingiswidelyusedinwirelessapplications, notablyinanumberofsecond-generationcellular(i.e., digital voice)1.2. THEWIRELESSSIGNALINGENVIRONMENT 19Time user #1user #2user #KFrequencyDivisionMultipleAccess(FDMA)Time FrequencyFrequencyuser #1user #2user #KFrequencyTime user #1user #2user #KFrequencyTime Users#1, #2, ..., #K...... ... ...... ...FrequencyHoppingCodeDivisionMultipleAccess(FHCDMA)TimeDivisionMultipleAccess(TDMA)DirectSequenceCodeDivsionMultipleAccess(DSCDMA)Figure1.1: Multiple-accesstechniques.20 CHAPTER1. INTRODUCTIONsytemsincludingthewidelyusedGlobal SystemforMobile(GSM)system[174,399,400],and in the IEEE 802.16 wirless MAN standards. A form of TDMA is also used in Bluetoothnetworks,inwhichoneoftheBluetoothdevicesinthenetworkactsasanetworkcontrollertopolltheotherdevicesintimesequence.FDMAandTDMAsystems are intendedtoassignorthogonal channels toall activeusersbygivingeachasliceoftheavailablefrequencybandoroftheavailabletransmissiontimefortheirexclusiveuse. Thesechannelsaresaidtobeorthogonal becauseinterferencebetween users does not, in principle, arise in such assignments. (Although, in practice, thereis often such interference, as will be discussed further below.) Code-division Multiple Access(CDMA) assigns channels inawaythat allows all users tosimultaneouslyuseall of theavailable time and frequency resources, through the assignment of a pattern or code to eachuserthatspeciesthewayinwhichtheseresourceswill beusedbythatuser. Typically,CDMA is implemented via spread-spectrum modulation, in which the pattern is the pseudo-randomcodethatdeterminesthespreadingsequenceinthecaseofdirect-sequence, orthehoppingpatterninthecaseoffrequency-hopping. Insuchsystems,achannelisdenedbyaparticularpseudo-randomcode,andsoeachuserisassignedachannelbybeingassignedapseudo-randomcode. CDMAisused,notably,inthesecond-generationcellularstandardIS-95(InterimStandard95), whichmakesuseof direct-sequenceCDMAtoallocatesub-channelsoflarger-bandwidth(125MHz)sub-channelsoftheentirecellularband. Itisalsoused, intheformoffrequency-hopping, inGSMinordertoprovideisolationamongusersinadjacentcells. ThespectrumspreadingusedinwirelessLANsystemsisalsoaformofCDMA in that it allows multiple such systems to operate in the same, lightly regulated, partoftheradiospectrum. CDMAisalsothebasisfortheprincipalstandardsbeingdevelopedanddeployedfor3Gcellulartelephony(e.g.,[127,356,358,399]).Anyof themultiple-access techniques discussedherecanbemodelledanalyticallybyconsideringmultipletransmittedsignalsoftheform(1.1). Inparticular,forasystemofKusers,wecanwriteatransmittedsignalforeachuserasxk(t) =M1i=0bk[i]sk,i(t), k = 1, 2, . . . , K, (1.8)wherexk(), bk[0], bk[1],, bk[M 1], andsk,i() represent thetransmittedsignal, thesymbol stream,andthe ithmodulationwaveform,respectively,of Userk. Thatis,eachuser1.2. THEWIRELESSSIGNALINGENVIRONMENT 21in a multiple-access system can be modelled in the same way as a single-user system, but with(usually) diering modulation waveforms (and symbol streams, of course). If the waveformssk,i()areof theform(1.2)butwithdierentcarrierfrequencies ksay, thenthisisFDMA. If theyareof theform(1.2) but withtime-slottedamplitudepulses pk()say,thenthisisTDMA.Andnally, iftheyarespread-spectrumsignalsofthisform, butwithdierentpseudo-randomspreadingcodesorhoppingpatterns, thenthisisCDMA. Detailsofthesemultiple-accessmodelswillbediscussedinthesequelasneeded.1.2.3 TheWirelessChannelFrom a technical point of view, the greatest distinction between wireless communications andwirelinecommunicationsliesinthephysicalpropertiesofwirelesschannels. Thesephysicalproperties can be described in terms of several distinct phenomena, including ambient noise,propagationlosses, multipath, interference, andpropertiesarisingfromtheuseofmultipleantennas. Here will review these phenomena only briey. Further discussion and details canbefound,forexample,in[37,45,145,213,397,441,449,456].Likeallpracticalcommunicationschannels, wirelesschannelsarecorruptedbyambientnoise. This noise comes from thermal motion of electrons on the antenna and in the receiverelectronics,andfrombackgroundradiationsources. Thisnoiseiswell-modelledashavingaverywidebandwidth(muchwiderthanthebandwidthofanyusefulsignalsinthechannel)and no particular deterministic structure (structured noise can be treated separately as inter-ference). AverycommonandusefulmodelforsuchnoisethatitisadditivewhiteGaussiannoise(AWGN), which, asthenameimplies, meansthatitisadditivetotheothersignalsinthereceiver, ithasaatpowerspectral density, anditinducesaGaussianprobabilitydistribution at the output of any linear lter to which it is input. Impulsive noise also occursinsomewirelesschannels. Suchnoiseissimilarlywideband, butinducesanon-Gaussianamplitudedistributionat theoutput of linear lters. Specicmodels for suchimpulsivenoisewillbediscussedinChapter4.Propagationlossesarealsoanissueinwirelesschannels. Theseareoftwobasictypes:diusive losses andshadowfading. Diusive losses arise because of the opennature ofwirelesschannels. Forexample, theenergyradiatedbyasimplepointsourceinfreespacewill spread over an ever-expanding spherical surface as the energy propagates away from the22 CHAPTER1. INTRODUCTIONsource. Thismeansthatanantennawithagivenaperaturesizewill collectanamountofenergy that decreases with the square of the distance between the antenna and the source. Inmostterrestrialwirelesschannels,thediusionlossesareactuallygreaterthanthis,duetotheeectsofground-wavepropagation,foilage,etc. Forexample,incellulartelephony,thediusion loss is inverse-square with distance within line-of-sight of the cell tower, and it fallsowithahigherpower(typically3or4)atgreaterdistances. Asitsnameimplies,shadowfadingresultsfromthepresenceofobjects(buildings,walls,etc.) betweentransmitterandreceiver. Shadow fading is typically modelled by an attenuation (i.e., a multiplicative factor)insignal amplitudethatfollowsalog-normal distribution. Thevariationinthisfadingisspeciedbythestandarddeviationofthelogarithmofthisattenuation.Multipath refers to the phenomenon in which multiple copies of a transmitted signal arereceivedatthereceiverduetothepresenceofmultipleradiopathsbetweenthetransmitterand receiver. These multiple paths arise due to reections from objects in the radio channel.Multipath is manifested in several ways in communications receivers, depending on the degreeof pathdierencerelativetothewavelengthof propagation, thedegreeof pathdierencerelativetothesignalingrate,andtherelativemotionbetweenthetransmitterandreceiver.Multipath from scatterers that are spaced very close together will cause a random change inthe amplitude of the received signal. Due to central-limit type eects, the resulting receivedamplitude is often modelled as being a complex Gaussian random variable. This results in arandom amplitude whose envelope has a Rayleigh distribution, and this phenomenon is thustermedRayleighfading. Otherfadingdistributionsalsoarise, dependingonthephysicalconguration. (See, e.g., [388].) Whenthescatterers arespacedsothat thedierencesintheircorrespondingpathlengthsaresignicantrelativetoawavelengthof thecarrier,thenthe signals arrivingat the receiver alongdierent paths canaddconstructivelyordestructively. This gives rise to fading that depends on the wavelength (or, equivalently, thefrequency) of radiation, which is thus called frequency-selective fading. When there is relativemotion between the transmitter and receiver, this type of fading also depends on time, sincethepathlengthisafunctionof theradiogeometry. Thisresultsintime-selectivefading.(Suchmotionalsocausessignal distortionduetoDopplereects.) Arelatedphenomenonarises when the dierence in path lengths is such that the time delay of arrival along dierentpaths is signicant relative to a symbol interval. This results in dispersion of the transmitted1.2. THEWIRELESSSIGNALINGENVIRONMENT 23signal, andcausesintersymbol interference(ISI); i.e., contributionsfrommultiplesymbolsarriveatthereceiveratthesametime.Many of the advanced signal transmission and processing methods that have been devel-opedforwirelesssystemsaredesignedtocontravenetheeectsofmultipath. Forexample,wideband signaling techniques, such as spread spectrum, are often used as a countermeasuretofrequency-selectivefading. Thisbothminimizestheeectsof deepfrequency-localizedfades, andalsofacilitates the resolvabilityandsubsequent coherent combiningof multi-plecopiesof thesamesignal. Similarly, bydividingahigh-ratesignal intomanyparallellower-ratesignals, OFDMmitigatestheeectsof channel dispersiononhigh-ratesignals.Alternatively,high-data-ratesingle-carriersystemsmakeuseofchannelequalizationatthereceivertocounteractthisdispersion. Someoftheseissueswillbediscussedfurtherinthenextsub-section.Interferenceis alsoasignicant issueinmanywireless channels. This interferenceistypicallyoneof twotypes: multiple-accessinterference(MAI)andco-channel interference(CCI). MAI refers to interference arising from other signals in the same network as the signalof interest. Forexample, incellulartelephonysystems, MAIcanariseatthebasestationwhen the signals from multiple mobile transmitters are not orthogonal to one another. ThishappensbydesigninCDMAsystems,andithappensinFDMAorTDMAsystemsduetochannel properties such as multipath or to non-ideal system characteristics such as imperfectchannelizationlters. CCIreferstointerferencefromsignalsfromdierentnetworks, butoperating in the same frequency band, as the signal of interest. An example is the interferencefrom adjacent cells in a cellular telephony system. This problem is a chief limitation of usingFDMAincellularsystems, andwasamajorfactorinmovingawayfromFDMAinsecondgeneration systems. Another example is the interference from other devices operating in thesamepartof theunregulatedspectrumasthesignal of interest, suchasinterferencefromBluetoothdevicesoperatinginthesame2.4GHzISMbandasIEEE802.11wirelessLANs.Interferencemitigationisalsoamajorfactorinthedesignoftransmissiontechniques(liketheabove-notedmovementawayfromFDMAincellularsystems), aswell asinthedesignofadvancedsignalprocessingsystemsforwireless,asweshallseeinthesequel.Thephenomenawehavediscussedabovecanbeincorporatedintoageneral analyticalmodel forawirelessmultiple-accesschannel. Inparticular, thesignal model inawireless24 CHAPTER1. INTRODUCTIONh(t)2 h(t)Kh(t) 1x(t)1x(t)2x(t)Kb[i]1b[i]2b[i]Ks(t)1s(t)2s(t)Kn(t)+ +r (t)y(t)y(t)y(t)12KFigure1.2: Signalmodelinawirelesssystem.systemisillustratedinFig.1.2. Wecanwritethesignalreceivedatagivenreceiverinthefollowingform:r(t) =Kk=1M1i=0bk[i]

hk(t, u)sk,i(u)du +i(t) +n(t), < t < , (1.9)wherehk(t, u)denotestheimpulseresponseof alinearlterrepresentingthechannel be-tweenthekthtransmitterandthereceiver,i()representsco-channelinterference,andn()representsambientnoise. Themodellingof thewirelesschannel asalinearsystemseemstoagreewellwiththeobservedbehaviorofsuchchannels. Allofthequantitieshk(, ),i()andn()are, ingeneral, randomprocesses. Asnotedabove, theambientnoiseistypicallyrepresented as a white process with very little additional structure. However, the co-channelinterferenceandchannel impulseresponsesaretypicallystructuredprocessesthatcanbeparameterized.Animportant special caseis that of apuremultipathchannel, inwhichthechannelimpulseresponsescanberepresentedintheform:hk(t, u) =Lk=1gk,(t u k,), (1.10)whereLkisthenumberofpathsbetweenUserkandthereceiver,gk,andk,arethegainanddelay,respectively,associatedwiththethpathofthekthuser,andwhere()denotesthe Dirac delta function. This model is an idealization of the actual behavior of a multipathchannel,whichwouldnothavesuchasharplydenedimpulseresponse. However,itservesas a useful model for signal processor design and analysis. Note that this model gives rise to1.2. THEWIRELESSSIGNALINGENVIRONMENT 25frequencyselectivefading, sincetherelativedelayswill causeconstructiveanddestructiveinterferenceatthereceiver,dependingonthewavelengthofpropagation. Oftenthedelaysk,areassumedtobeknowntothereceiver,ortheyarespaceduniformlyattheinverseof the bulk bandwidth of the signaling waveforms. A typical model for the path gains gk,isthattheyareindependent, complexGaussianrandomvariables, givingrisetoRayleighfading.Notethat,ingeneral,thereceiverwillseethefollowingcompositemodulationwaveformassociatedwiththesymbolbk[i] :fk,i(t) =

hk(t, u)sk,i(u)du. (1.11)If these waveforms are not orthogonal for dierent values of i, then ISI will result. Consider,forexample,thepuremultipathchannelof(1.10)withsignalingwaveformsoftheformsk,i(t) =sk(t iT), (1.12)where Tis the inverse of the single-user symbol rate. In this case, the composite modulationwaveformsaregivenbyfk,i(t) =fk(t iT), (1.13)withfk(t) =Lk=1gk,sk (t k,) . (1.14)If thedelayspread, i.e., themaximumof thedierencesof thedelays k,fordierentvaluesof, issignicantrelativetoT, ISImaybeafactor. Notethat, foraxedchannelthedelayspreadisafunctionofthephysicalgeometryofthechannel,whereasthesymbolrate depends on the date-rate of the transmitted source. Thus, higher-rate transmissions aremorelikelytoencounterISIthanarelower-ratetransmissions. Similarly, if thecompositewaveforms for dierent values of k are not orthogonal, then MAI will result. This can happen,forexample,inCDMAchannelswhenthepseudorandomcodesequencesusedbydierentusersarenotorthogonal. ItcanalsohappeninCDMAandTDMAchannelsduetotheeectsofmultipathorofasynchronoustransmission. Furtherdiscussionoftheseissueswillbeincludedinthesequelastheneedarises.26 CHAPTER1. INTRODUCTIONThismodelcanbefurthergeneralizedtoaccountformultipleantennasatthereceiver.Inparticular,wecanmodify(1.9)asfollows:r(t) =Kk=1bk[i]

hk(t, u)sk,i(u)du +i(t) +n(t), < t < , (1.15)where the boldface quantities denote (column) vectors with dimensions equal to the numberof antennas at the received array. For example,the pthcomponent of hk(t, u) is the impulseresponseofthechannelbetweenUserkandthepthelementofthereceivingarray. Ausefulsuchmodel istocombinethepuremultipathmodel of (1.10)withamodel inwhichthespatial aspects of the arraycanbe separatedfromits temporal properties. This yieldschannelimpulseresponsesoftheformhk(t, u) =Lk=1gk,ak,(t u k,), (1.16)wherethecomplexvectorak,describestheresponseofthearraytothethpathofUserk.Thesimplestsuchsituationisthecaseofauniformlineararray(ULA),inwhichthearrayelementsareuniformlyspacedalongaline, receivingasingle-carriersignal arrivingalongaplanar wavefront, andsatisfyingtheso-callednarrowbandarrayassumption. Thenar-rowband array assumption essentially assumes that the the signaling waveforms are carrierscarryingnarrowbandmodulation,andthatallofthevariationinthereceivedsignalacrossthearrayatanygiveninstantintimeisduetothecarrier(i.e., themodulatingwaveformischangingslowlyenoughtobeassumedconstantacrossthearray). Inthiscase,thearrayresponse depends only on the angle k,at which the corresponding paths signal is incidentonthearray. Inparticular,theresponseofaP-elementarrayisgiveninthiscasebyak,=

1e sin k,e2 sin k,...e(P1) sin k,, (1.17)where denotes the imaginary unit, and where z=2dwith the carrier wavelength and dtheinter-elementspacing. (See,[123,264,267,396,436,441,501]forfurtherdiscussionofsystemsinvolvingmultiplereceiverantennas.)1.3. BASICRECEIVERSIGNALPROCESSINGFORWIRELESS 27Itisalsoofinteresttomodelsystemsinwhichtherearemultipleantennasatboththetransmitter and receiver so-called multi-input/multi-output (MIMO) systems. In this case,the channel transfer functions are matrices, with the number of rows equal to the number ofreceivingantennas,andthenumberofcolumnsequaltothenumberoftransmittinganten-nasateachsource. Thereareseveralwaysofhandlingthesignalinginsuchcongurations,depending on the desired eects and the channel conditions. For example, transmitter beam-forming can be implemented by transmitting the same symbol simultaneously from multipleantennaelementsonappropriatelyphasedversionsofthesamesignalingwaveform. Space-timecodingcanbeimplementedbytransmittingframesof relatedsymbolsovermultipleantennas. Othercongurationsareof interestaswell. Issuesconcerningmultiple-antennasystemswillbediscussedfurtherinthesequelastheyarise.1.3 BasicReceiverSignalProcessingforWirelessThisbookisconcernedwiththedesignofadvancedsignal processingmethodsforwirelessreceivers, basedlargelyonthemodelsdiscussedintheprecedingsections. Beforemovingtothesemethods, however, itisof interesttoreviewbrieysomebasicelementsof signalprocessingforthesemodels. Thisisnotintendedtobeacomprehensivetreatment,andthereaderisreferredto[142,143,268,371,374,383,388,501,511,514]forfurtherdetails.1.3.1 TheMatchedFilter/RAKEReceiverTodoso, weconsiderrsttheparticularcaseof themodel of (1.9)inwhichthereisonlyasingleuser(i.e.,K= 1),thechannelimpulseh1(, )isknowntothereceiver,thereisnoCCI(i.e., i() 0), andtheambientnoiseisAWGNwithspectral height2. Thatis, wehavethefollowingmodelforthereceivedsignal:r(t) =M1i=0b1[i]f1,i(t) +n(t), < t < , (1.18)wheref1,i()denotesthecompositewaveformof(1.11),givenbyf1,i(t) =

h1(t, u)s1,i(u)du. (1.19)28 CHAPTER1. INTRODUCTIONLetusfurtherrestrictattention,forthemoment,tothecaseinwhichthereisonlyasinglesymboltobetransmitted(i.e.,M= 1),inwhichcase,wehavethereceivedwaveformr(t) =b1[0]f1,0(t) +n(t), < t < , (1.20)Optimal inferences about the symbol b1[0] in (1.20) can be made on the basis of the likelihoodfunctionoftheobservations,conditionedonthesymbolb1[0],whichisgiveninthiscasebyL

r() [ b1[0]

=exp

122 '

b1[0]

f1,0(t)r(t)dt

[b1[0][2

[f1,0(t)[2dt ,(1.21)wherethesuperscriptasteriskdenotescomplexconjugation,and 'denotestherealpartofitsargument.Optimalinferencesaboutthesymbolb1[0]canbemade,forexample,bychoosingmax-imumlikelihood(ML)ormaximumaposterioriprobability(MAP)valuesforthesymbol.TheMLsymbol decisionisgivensimplybytheargumentthatmaximizes L(r() [b1[0])overthesymbolalphabet, /,i.e.,b1[0] = arg

maxb.L

r() [b1[0] = b

= arg

maxb.2 '

b

f1,0(t)r(t)dt

[b[2

[f1,0(t)[2dt

. (1.22)Itiseasytoseethatthecorrespondingsymbolestimateisthesolutiontotheproblemminb.

bz

2, (1.23)wherezz=

f1,0(t)r(t)dt

[f1,0(t)[2dt. (1.24)Thus, the ML symbol estimate is the closest point in the symbol alphabet to the observablez.Notethat thetwosimplest andmost commonchoices of symbol alphabet areM-aryphaseshiftkeying(MPSK)andquadratureamplitudemodulation(QAM). InMPSK, thesymbolalphabetis/=

e2m/M[ m 0, 1,, M 1, (1.25)1.3. BASICRECEIVERSIGNALPROCESSINGFORWIRELESS 29orsome rotationof this setaround the unitcircle. ForQAM, a symbol alphabetcontainingM Nvaluesis/=

bR + bI[ bR /RandbI /I, (1.26)where /Rand /IarediscretesetsofamplitudescontainingMandNpoints,respectively;e.g.,forM= Neven,acommonchoiceis/R=/I=

12, 32,, M4

, (1.27)or a scaled version of this choice. A special case of both of these is that of binary phase-shiftkeying(BPSK), inwhich /= 1, +1. Thislattercaseistheonewewill considermostofteninthistreatment, primarilyforthesakeof simplicity. However, mostof theresultsdiscussedhereinextendstraightforwardlytothesemoregeneralsignalingalphabets.ML symbol estimation (i.e., the solution to (1.23) ) is very simple for MPSK and QAM. Inparticular, sincetheMPSKsymbolscorrespondtophasorsatevenlyspacedanglesaroundtheunit circle, theMLsymbol choice is that whose angle is closest totheangle of thecomplex number z. For QAM, the choices of the real and imaginary parts of the ML symbolestimatearedecoupled, with 'bbeingchosentobetheclosestelementof /Rto 'z,andsimilarlyfor b. ForBPSK,theMLsymbolestimateisbi[0] =sign 'z z=sign

'

f1,0(t)r(t)dt , (1.28)wheresigndenotesthesignumfunction:signx =

1 ifx < 00 ifx = 0+1 ifx > 0. (1.29)MAPsymboldetectionin(1.20)isalsobasedonthelikelihoodfunctionof(1.21),aftersuitable transformation. In particular, if the symbol b1[0] is a random variable, taking valuesin /withknownprobabilities, thentheaposterioriprobabilitydistributionofthesymbolconditionedonr(),isgivenviaBayesformulaasP

b1[0] = b [r()

=L(r() [b1[0] = b) P(b1[0] = b)a.L(r() [b1[0] = a) P(b1[0] = a), b /. (1.30)30 CHAPTER1. INTRODUCTIONTheMAPcriterionspeciesasymboldecisiongivenbyb1[0] = arg

maxb.P (b1[0] = b)

= arg

maxb.[L(r() [b1[0] = b) P(b1[0] = b)]

. (1.31)Notethat, inthissingle-symbol case, if thesymbol valuesareequiprobable, thentheMLandMAPdecisionsarethesame.Thestructureof theaboveMLandMAPdecisionrulesshowsthatthemainreceiversignal-processingtaskinthissingle-user, single-symbol, known-channel caseisthecompu-tationofthetermy1[0]z=

f1,0(t)r(t)dt. (1.32)Thisstructureiscalledacorrelatorbecauseitcorrelatesthereceivedsignal r()withtheknowncomposite signalingwaveformf1,0(). This structure canalsobe implementedbysamplingtheoutputofatime-invariantlinearlter:

f1,0(t)r(t)dt =(hr)(0), (1.33)wheredenotes convolution, and h is the impulse response of the time-invariant linear lter,givenbyh(t) = f1,0(t). (1.34)Thisstructureiscalledamatchedlter, sinceitsimpulseresponseismatchedtothecom-positewaveformonwhichthesymbolisreceived. Whenthecompositesignalingwaveformhasanitedurationsothath(t)=0fort< D 0,thenthematchedlterreceivercanbeimplementedbysamplingattimeDtheoutputof thecausal lterwiththefollowingimpulseresponse:hD(t) =

f1,0(D t) ift 00 ift < 0. (1.35)Forexample,ifthesignalingwaveforms1,0(t)hasduration[0, T]andthechannelhasdelayspreaddwith 1, thenthecompositesignalingwaveformwill havethispropertywithD = T+ d.Aspecialcaseofthecorrelator(1.32)arisesinthecaseofapuremultipathchannel,inwhichthechannel impulseresponseisgivenby(1.10). Thecompositewaveform(1.11)is1.3. BASICRECEIVERSIGNALPROCESSINGFORWIRELESS 31thiscaseisf1,0(t) =L1=1g1,s1,0(t 1,), (1.36)andthecorrelatoroutput(1.32)becomesy1[0]z=L1=1g1,

s1,0(t 1,)r(t)dt, (1.37)acongurationknownasaRAKEreceiver.Furtherdetailsonthisbasicreceiverstructurecanbefound,forexample,in[388].1.3.2 EqualizationWe nowturntothe situationinwhichthere is more thanone symbol inthe frame ofinterest;i.e.,whenM> 1.Inthiscase,wewouldliketoconsiderthelikelihoodfunctionoftheobservationsr()conditionedontheentireframeofsymbols, b1[0], b1[1],, b1[M 1],whichisgivenbyL

r() [b1[0], b1[1],, b1[M 1]

=exp

12

2 'bH1y1bH1H1b1

, (1.38)wherethesuperscript Hdenotes theconjugatetranspose(i.e, theHermitiantranspose),b1denotesacolumnvectorwhoseithcomponentisb1[i], i=0, 1, . . . , M 1, y1denotesacolumnvectorwhoseithcomponentisgivenbyy1[i]z=

f1,i(t)r(t)dt, i = 0, 1, . . . , M 1, (1.39)and H1 is an MMHermitian matrix, whose (i, j)thelement is the cross-correlation betweenf1,i(t)andf1,j(t),i.e.,H1[i, j] =

f1,i(t)f1,j(t)dt. (1.40)Since the likelihood function depends on r() only through the vector y1 of correlator outputs,thisvectorisasucientstatisticformakinginferencesaboutthevectorb1ofsymbols.Maximumlikelihooddetectioninthissituationisgivenbyb1=arg

maxb.M

2 'bHy1bHH1b

. (1.41)32 CHAPTER1. INTRODUCTIONNotethat, if H1isadiagonal matrix(i.e., all of itso-diagonal elementarezero), then(1.41) decouples into a set of Mindependent problems of the single-symbol type (1.22). Thesolutioninthiscaseiscorrespondinglygivenbyb1[i] =arg minb.

bz1[i]

2, (1.42)wherez1[i]z=yi[i]

[f1,i(t)[2dt. (1.43)However,inthe more general caseinwhichthere isintersymbol interference,(1.41)willnotdecoupleandtheoptimizationmusttakeplaceovertheentireframe, aproblemknownassequencedetection.Theproblemof (1.41)isanintegerquadraticprogram, whichisknowntobeanNP-completecombinatorial optimizationproblem[377]. This implies that thecomplexityof(1.41) is potentiallyquite high: exponential inthe frame lengthM, whichis essentiallythecomplexityorderof exhaustingoverthesequencealphabet /M. Thisisaprohibitivedegreeofcomplexityformostapplications,sinceatypicalframelengthmightbehundredsoreventhousandsofsymbols. Fortunately, thiscomplexitycanbemitigatedsubstantiallyforpractical ISI channels. Inparticular, if thecompositesignalingwaveformshavenitedurationD, thenthematrixH1isabandedmatrixwithnon-zeroelementsonlyonthosediagonalsthatarenomorethan =

DT

diagonalsawayfromthemaindiagonal(here |denotesthesmallestintegernotlessthanitsargument);i.e.,[H1[i, j][ =0, [i j[ > . (1.44)This structure on the matrix permits solution of (1.41) with a dynamic program of complexityorder O

[/[

, as opposed to the O

[/[M

complexity of direct search. In most situations 0, k =1,, K, and n[i] ^(0, 2IN). (Here ^(, ) denotes a real-valued Gaussian distribution.)Thesignalsubspacecomponent ss1canthenbewrittenas ss1=Kk=1ksk=S, (2.100)for some RKwith1>0. Acommonlyusedperformancemeasureforamultiuserdetectoristheasymptoticmultiusereciency(AME)[511],denedas11z= sup

0 r 1 :lim0P1()/QrA1

= 0

, (2.101)whichmeasurestheexponentialdecayrateoftheerrorprobabilityasthebackgroundnoiseapproacheszero. Arelatedperformancemeasure,thenear-farresistance,istheinmumofAMEastheinterferersenergiesareallowedtoarbitrarilyvary,1=infAk0k,=11 . (2.102)Since, as 0, thelineardecorrelatingdetectorandthelinearMMSEdetectorbecomeidentical, thesetwodetectorshavethesameAMEandnear-farresistance[292, 302]. Itis1P1() is the probability of error of the detector for noise level;Q(x)

=12

xexp

x22

.66 CHAPTER2. BLINDMULTIUSERDETECTIONstraightforwardtocomputetheAMEof thelineardecorrelatingdetector, sinceitsoutputconsistsofonlythedesireduserssignalandtheambientGaussiannoise. By(2.15)-(2.17),weconcludethattheAMEandthenear-farresistanceofbothlineardetectorsaregivenby1= 1=1

R1

1,1. (2.103)NextwecomputetheAMEandthenear-resistanceofthetwosubspacelineardetectorsunderspreadingwaveformmismatch. DenetheN Ndiagonalmatrices0z= diag

12,, K 2, 0,, 0

, (2.104)and 0z= diag

[12]1,, [K 2]1, 0,, 0

. (2.105)Denotethesingularvaluedecomposition(SVD)ofSbyS = WVT, (2.106)wheretheNKmatrix=[ij] hasij=0forall i =j, and11 22 KK.ThecolumnsoftheN NmatrixWaretheorthonormaleigenvectorsof

SST

,andthecolumns of the KKmatrix Vare the orthonormal eigevectors of R = STS. We have thefollowingresult,whoseproofisfoundintheAppendix(Section2.8.2).Lemma2.3Let theeigendecompositionof CrbeCr=UUT. ThenNNdiagonalmatrix0in(2.105)isgivenby0= UTWTVTA2V WTU. (2.107)whereisthetransposeofinwhichthesingularvaluesarereplacedbytheirreciprocals.Using the above result, we obtain the AME of the subspace linear detectors under spread-ingwaveformmismatch,asfollows.Proposition2.5The AMEof the subspace linear decorrelatingdetector givenby(2.74)andthat of thesubspacelinearMMSEdetectorgivenby(2.78)underspreadingwaveformmismatchisgivenby1=max2

0, [1[ Kk=2[k[A1/AkA41TA2R1A2. (2.108)2.4. BLINDMULTIUSERDETECTION:SUBSPACEMETHODS 67Proof: Sinced1andm1havethesameAME, weneedonlytocomputetheAMEford1.Because a positive scaling on the detector does not aect its AME, we consider the AME ofthefollowingscaledversionofd1underthesignaturewaveformmismatch.d1z= Us

s2IK

1UTs s1= Us

s2IK

1UTsss1= U0UTS, (2.109)wherethesecondequalityfollowsfromthefactthatthenoisesubspacecomponent sn1isorthogonal to the signal subspace Us. Substituting (2.106) and (2.107) into (2.109), we havedT1sk= TSTU0UTSek= T V TWT

WTVTA2V WT

WVT

ek= TA2ek=kA2k, (2.110)dT1d1= T V TWT

WTVTA2V WTU

(2.111)

UTWTVTA2V WT

WVT

(2.112)= TA2V TVT. .. .R1A2. (2.113)Theoutputofthedetectord1isgivenbyz[i]z=dT1r[i] =Kk=1Akbk

dT1sk

+ dT1n[i]=Kk=1kAkbk + v[i],wherev[i] ^

0,2|d1|2

. TheprobabilityoferrorforUser1isthengivenbyP1() =12K1(b2,,bk)|1,1K1Q

A1

1Kk=2kbkA1/Ak

A41TA2R1A2

. (2.114)ItthenfollowsthattheAMEisgivenby(2.108). 2It is seen from (2.114) that spreading waveform mismatch causes MAI leakage at the de-tector output. Strong interferers (Ak A1) are suppressed at the output, whereas weak in-terferers (Ak Kk=2[k[A1/Ak),thentheperformancelossisnegligible; otherwise, theeectivespreadingwaveformshouldbe estimatedrst. Moreover,since the mismatchedspreading waveform s1is rstprojectedontothesignalsubspace,itsnoisesubspacecomponent sn1isnulledoutanddoesnotcauseperformance degradation; whereas for the blind adaptive MOE detector discussed in Section2.3,suchanoisesubspacecomponentmayleadtocompletecancellationofboththesignalandMAIifthereisnoenergyconstraintonthedetector[179].2.5 PerformanceofBlindMultiuserDetectors2.5.1 PerformanceMeasuresInthe previous sections, we have discussedtwoapproaches toblindmultiuser detectionnamely, thedirectmethodandthesubspacemethod. Thesetwoapproachesarebasedprimarily on two equivalent expressions for the linear MMSE detector, i.e., (2.26) and (2.78).WhentheautocorrelationCrofthereceivedsignalsisknownexactly, thetwoapproacheshavethesameperformance. However, whenCrisreplacedbythecorrespondingsampleautocorrelation,quiteinterestingly,theperformanceofthesetwomethodsisverydierent.Thisisduetothefactthatthesetwoapproachesexhibitdierentestimationerrorsontheestimateddetector[188, 189, 193]. Inthissection, wepresentperformanceanalysisof thetwoblindmultiuserdetectorstheDMI blinddetectorandthesubspaceblinddetector.Forsimplicity, weconsideronlyreal-valuedsignals, i.e., in(2.4)Ak>0, k=1,, K, andn[i] ^(0, 2IN).Suppose a linear weight vector w1 RNis applied to the received signal r[i] in (2.5). Theoutput is given by (2.10). Since it is assumed that the user bit streams are independent, andthenoiseisindependentoftheuserbits, thesignal-to-interference-plus-noiseratio(SINR)attheoutputofthelineardetectorisgivenbySINR(w1) =EwT1r[i] [ b1[i]2E Var wT1r[i] [ b1[i]=A21

wT1s1

2Kk=2A2k

wT1sk

2+ 2|w1|2. (2.115)2.5. PERFORMANCEOFBLINDMULTIUSERDETECTORS 69Thebiterrorprobabilityofthelineardetectorusingweightvectorw1isgivenbyPe(w1) = P

b1[i] = b1[i]

=12K1[b2 bK]|1,+1K1Q

A1wT1s1 +Kk=2AkbkwT1sk|w1|

. (2.116)Nowsupposethatanestimate w1oftheweightvectorw1isobtainedfromthereceivedsignals r[i]M1i=0. Denotew1z= w1w1. (2.117)Obviously both w1and w1are random vectors and are functions of the random quantitiesb[i], n[i]M1i=0. Intypical adaptivemultiuserdetectionscenarios[179, 540], theestimateddetector w1isemployedtodemodulatefuturereceivedsignals, sayr[j], j>M. Thentheoutputisgivenby wT1r[j] = wT1r[j] + wT1r[j], j> M, (2.118)wherethersttermin(2.118)representstheoutputof thetrueweightvectorw1, whichhas the same form as (2.10). The second term in (2.118) represents an additional noise termcaused by the estimation error w1. Hence from (2.118) the average SINR at the output ofanyunbiasedestimatedlineardetector w1isgivenbySINR( w1) =A21

wT1s1

2Kk=2A2k

wT1sk

2+ 2|w1|2+ E

wT1r[j]

2, (2.119)withE

wT1r[j]

2= tr

EwT1r[j]r[j]Tw1= tr

Ew1wT1r[j]r[j]T= tr

Ew1wT1. .. .CwEr[j]r[j]T. .. .Cr=SA2ST+2IN=1Mtr (CwCr) , (2.120)70 CHAPTER2. BLINDMULTIUSERDETECTIONwhereCwz= MEw1wT1andCrz= Er[j]r[j]T. Notethatinbatchprocessing,ontheotherhand,theestimateddetectorisusedtodemodulatesignalsr[i], 1 i M. Sincew1isafunctionof r[i]Mi=1,forxedi,w1andr[i]areingeneralcorrelated. ForlargeM, suchcorrelationissmall. Thereforeinthiscasewestill use(2.119)and(2.120)astheapproximateSINRexpression.If we assume further that w1is actually independent of r[i], then the average bit errorrateofthisdetectorisgivenbyPe( w1) =

Pe( w1) f( w1) d w1, (2.121)wherePe( w1)isgivenby(2.116),andf( w1)denotestheprobabilitydensityfunction(pdf)oftheestimatedweightvector w1.Fromtheabovediscussion, itisseenthatinordertoobtaintheaverageSINRattheoutputoftheestimatedlineardetector w1,itsucestonditscovariancematrixCw. Ontheotherhand, theaveragebiterrorrateof theestimatedlineardetectordependsonitsdistributionthroughf( w1).2.5.2 AsymptoticOutputSINRWe rst present the asymptotic distribution of the two forms of blind linear MMSE detectors,for largenumber of signal samples, M. Recall that inthedirect-matrix-inversion(DMI)method,theblindmultiuserdetectorisestimatedaccordingtoCr=1MMi=1r[i]r[i]T, (2.122)and w1=C1rs1. [DMIblindlinearMMSEdetector] (2.123)Inthesubspacemethod,theestimateoftheblinddetectorisgivenbyCr=1MMi=1r[i]r[i]T=Uss UTs+Unn UTn, (2.124)and w1=Us1sUTss1, [subspaceblindlinearMMSEdetector] (2.125)wheresandUscontainrespectivelythe largest Keigenvalues andthe correspondingeigenvectorsofCr; andwherenandUncontainrespectivelytheremainingeigenvalues2.5. PERFORMANCEOFBLINDMULTIUSERDETECTORS 71andeigenvectorsofCr. ThefollowingresultgivestheasymptoticdistributionoftheblindlinearMMSEdetectorsgivenby(2.123)and(2.125). Theproof isfoundintheAppendix(Section2.8.3).Theorem2.1Letw1bethetrueweightvectorofthelinearMMSEdetectorgivenbyw1= C1rs1=Us1sUTss1, (2.126)and let w1be the weight vector of the estimated blind linear MMSE detector given by (2.123)or (2.125). Let the eigendecomposition of the autocorrelation matrix Crof the received signalbeCr= UssUTs+ 2UnUTn. (2.127)ThenM ( w1w1) ^(0, Cw), indistribution,asM ,withCw=

wT1s1

Us1sUTs+w1wT1 2Us1sUTsSDSTUs1sUTs+ UnUTn,(2.128)whereDz= diag

A41

wT1s1

2, A42

wT1s2

2,, A4K

wT1sK

2, (2.129)z=

12sT1Us1sUTss1, DMIblinddetector2sT1Us1s(s2IK)2UTss1, subspaceblinddetector. (2.130)Hence for large M, the covariance of the blind linear detector, Cw= M Ew1wT1,canbeapproximatedby(2.128). Dene,asbefore,Rz= STS. (2.131)ThenextresultgivesanexpressionfortheaverageoutputSINR,denedby(2.119),oftheblindlineardetectors. TheproofisgivenintheAppendix(Section2.8.3).72 CHAPTER2. BLINDMULTIUSERDETECTIONCorollary2.1TheaverageoutputSINRoftheestimatedblindlineardetectorisgivenbySINR( w1)=A21

wT1s1

2Kk=2A2k

wT1sk

2+ 2|w1|2+1M(K + 1)wT1s12Kk=1A4k

wT1sk

2

wTksk

+ (N K)2,(2.132)wherewTlsk=1A2l

R

R+ 2A2

1

k,l, k, l = 1,, K, (2.133)|w1|2=1A41

R+ 2A2

1R

R+ 2A2

1

1,1, (2.134)and 2=

wT1s1, DMIblinddetector4A41

R+ 2A2

1A2R1

1,1, subspaceblinddetector.(2.135)Itisseenfrom(2.132)thattheperformancedierencebetweentheDMIblinddetectorandthesubspaceblinddetectoriscausedbythesingleparameter givenby(2.130)-thedetectorwithasmaller hasahigheroutputSINR. Let1,, KbetheeigenvaluesofthematrixRgivenby(2.131). Denotemin=min1kKk,andmax=max1kKk.DenotealsoAmin=min1kKAk, andAmax=max1kKAk. Thenext result givessucientconditionsunderwhichoneblinddetectoroutperformstheother,intermsoftheaverageoutputSINR.Corollary2.2IfA2min2>max, thenSINRsubspace>SINRDMI; andifA2max2 22, k = 1,, K. (2.137)2.5. PERFORMANCEOFBLINDMULTIUSERDETECTORS 73Ontheotherhand,notethatCr=SA2ST+ 2IN_A2minSST+ 2IN. (2.138)Sincethenonzeroeigenvaluesof SSTarethesameof thoseof R=STS, itfollowsfrom(2.138)thatk A2mink + 2, k = 1,, K. (2.139)Therstpartof thecorollarythenfollowsbycombining(2.137)and(2.139). Thesecondpartofthecorollaryfollowsasimilarproof. 2Thenextresultgivesanupperandalowerboundontheparameter, intermsofthedesiredusersamplitudeA1,thenoisevariance2,andthetwoextremeeigenvaluesofCr.Corollary2.3Theparameter denedin(2.130)satises

1 2min

1A21 2

1 2max

1A21, DMIblinddetector,1max

max21 1A21 21min

min21 1A21, subspaceblinddetector.Proof: Theproofthenfollowsfrom(2.136)andthefollowingfactfoundinChapter4. [cf.Proposition4.2.]1A21=Kk=1

sT1uk

2k2 . (2.140)2Inordertogetsomeinsightsfromtheresult(2.132),wenextconsidertwospecialcasesforwhichwecomparetheaverageoutputSINRsofthetwoblinddetectors.Example1- Orthogonal Signals: In this case, we have uk= sk, R = IK, and k= A2k+2,k = 1,, K. Substitutingtheseinto(2.136),weobtain2=

1A21+2, DMIblinddetector

2A21

21A21+2, subspaceblinddetector. (2.141)74 CHAPTER2. BLINDMULTIUSERDETECTIONSubstituting(2.141) into(2.132), andusingthefact that inthiscasewk=1A2k+2sk, weobtainthefollowingexpressionsoftheaverageoutputSINRsSINR( w1) =

11+1M(1+1)(N+1)2211+1

, DMIblinddetector11+1M(1+1)

K+1+NK21

2211+1

, subspaceblinddetector, (2.142)where 1z=A212is the signal-to-noise ratio (SNR) of the desired user. It is easily seen that inthiscase,anecessaryandsucientconditionforthesubspaceblinddetectortooutperformtheDMIblinddetectoristhat1> 1,i.e.,SNR1> 0dB.Example2- EquicorrelatedSignalswithPerfectPowerControl: In this case, it is assumedthatsTksl=, fork =l, 1 k, l K. ItisalsoassumedthatA1==AK=A. ItisshownintheAppendix(Section2.8.3)thattheaverageoutputSINRsforthetwoblinddetectorsaregivenbySINR( w1) =1(K 1) + +1M

K+12 [1 + (K 1)] + (N K), (2.143)withz=

2A22A2+ (1 )[1 + (K 1)]

2, (2.144)z=2A2

1 + (K 1) +2A2

2[1 + (K 1)]

2 + (K 2) + 22A2

(1 )[1 + (K 1)] +2A2

2, (2.145)z=11 +2A2+1 + (K 1)K11 + (K 1) +2A211 +2A2, (2.146)andz=

1, DMIblinddetector12

2A2

2

1(1)2

1+ 2A2

+1+(K1)K1[1+(K1)]2

1+(K1)+ 2A2

1(1)2

1+ 2A2

subspaceblinddetector.(2.147)AnecessaryandsucientconditionforthesubspaceblinddetectortooutperformtheDMIblinddetectorisDMI> subspace,which,aftersomemanipulations,reducesto(12)33+ (12)22> 31 + 11 +3231K

, (2.148)2.5. PERFORMANCEOFBLINDMULTIUSERDETECTORS 75where z=A22, and where 1z= 1+(K1) and 2z= 1 are the two distinct eigenvalues ofR[cf. Appendix(Section2.8.3)]. TheregionontheSNR-planewherethesubspaceblinddetector outperforms the DMI blind detector is plotted in Fig. 2.2, for dierent values of K.Itisseenthatingeneral thesubspacemethodperformsbetterinthelowcross-correlationandhighSNRregion.TheaverageoutputSINRasafunctionofSNRandforbothblinddetectorsisshowninFig. 2.3. Itisseenthattheperformanceof thesubspaceblinddetectordeterioratesinthehighcross-correlationandlowSNRregion; whereastheperformanceoftheDMIblinddetectorislesssensitivetocross-correlationandSNRinthisregion. ThisphenomenonismoreclearlyseeninFig.2.4andFig.2.5,wheretheperformanceofthetwoblinddetectorsiscomparedasafunctionof andSNRrespectively. Theperformanceof thetwoblinddetectorsasafunctionof thenumberof signal samplesMisplottedinFig. 2.6, whereitisseenthat, forlargeM, bothdetectorsconvergetothetruelinearMMSEdetector, withthesubspaceblinddetectorconvergingmuchfasterthantheDMIblinddetector; andtheperformance gain oered by the subspace detector is quite signicant for small values of M.Finally, in Fig. 2.7, the performance of the two blind detectors is plotted as a function of thenumber of users K. As expected from (2.132), the performance gain oered by the subspacedetectorissignicantforsmallervalueofK,andthegaindiminishesasKincreasestoN.Moreover,itisseenthattheperformanceoftheDMIblinddetectorisinsensitivetoK.SimulationExamplesWeconsider asystemwithK=11users. Theusers spreadingsequencesarerandomlygenerated with processing gain N= 13. All users have the same amplitudes. Fig. 2.8 showsboththeanalytical andthesimulatedSINRperformancefortheDMIblinddetectorandthesubspaceblinddetector. Foreachdetector, theSINRisplottedasafunctionof thenumberof signal samples(M)usedforestimatingthedetector, atsomexedSNR. Thesimulated and analytical BER performance of these estimated detectors is shown in Fig. 2.9.TheanalyticalBERperformanceisevaluatedusingthetheapproximationPe= Q(SINR), (2.149)which eectively treats the output interference-plus-noise of the estimated detector as havinga Gaussian distribution. This can be viewed as a generalization of the results in [372], where76 CHAPTER2. BLINDMULTIUSERDETECTION0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1105051015202530SNR (dB)subspace detector DMI detector K=2K=10K=100Figure2.2: PartitionoftheSNR-planeaccordingtotherelativeperformanceofthetwoblinddetectors. ForeachK, intheregionabovetheboundarycurve, thesubspaceblinddetectorperformsbetter; whereasintheregionbelowtheboundarycurve, theDMIblinddetectorperformsbetter.2.5. PERFORMANCEOFBLINDMULTIUSERDETECTORS 7705101520253000.20.40.60.81105051015SNR (dB)SINR (dB)Figure 2.3: The average output SINR versus SNR and for the two blind detectors. N= 16,K=6, M=150. TheuppercurveinthehighSNRregionrepresentstheperformanceofthesubspaceblinddetector.78 CHAPTER2. BLINDMULTIUSERDETECTION0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1642024681012SINR (dB)N=16, k=6, M=150, SNR=15dBDMI blind detectorsubspace blind detectorFigure2.4: TheaverageoutputSINRversusforthetwoblinddetectors. N= 16,K= 6,M= 150,SNR = 15dB.5 0 5 10 15 20 25 30105051015SNR (dB)SINR (dB)N=16, K=6, M=150, =0.4DMI blind detectorsubspace blind detectorFigure2.5: Theaverageoutput SINRversusSNRfor thetwoblinddetectors. N=16,K= 6,M= 150, = 0.4.2.5. PERFORMANCEOFBLINDMULTIUSERDETECTORS 79200 400 600 800 1000 1200 1400 1600 1800 200067891011121314Number of signal samples (M)SINR (dB)N=16, K=6, SNR=15dB, =0.4DMI blind detectorsubspace blind detectorFigure2.6: TheaverageoutputSINRversusthenumberof signal samplesMforthetwoblinddetectors. N= 16,K= 6, = 0.4,SNR = 15dB.2 4 6 8 10 12 14 16891011121314Number of users (K)SINR (dB)N=16, M=150, SNR=15dB, =0.4DMI blind detectorsubspace blind detectorFigure 2.7: The average output SINRversus the number of users Kfor the twoblinddetectors. N= 16,M= 150, = 0.4,SNR = 15dB.80 CHAPTER2. BLINDMULTIUSERDETECTIONitisshownthattheoutputofanexactlinearMMSEdetectoriswell-approximatedwithaGaussian distribution. From Fig. 2.8 and Fig. 2.9, it is seen that the agreement between theanalytical performance assessment and the simulation results is excellent, for both the SINRandtheBER.Themismatchbetweentheanalyticalandsimulationperformanceoccursforsmall values of M, whichis not surprisingsince the analytical performance is basedonasymptoticananalysis.1021030246810DMI blind detectornumber of samples (M)SINR (dB)SNR=8 SNR=10SNR=12SNR=14SNR=161021030246810subspace blind detectornumber of samples (M)SINR (dB)SNR=8 SNR=10SNR=12SNR=14SNR=16Figure2.8: TheoutputaverageSINRversusthenumberofsignalsamplesMforDMIandsubspacedetectors. N=13, K=11. Thesolidlineistheanalyticalperformance,andthedashedlineisthesimulationperformance.Finally we note that although in this section we treated only the performance analysis ofblind multiuser detection algorithms in simple real-valued synchronous CDMA systems,theanalysis for the more realistic complex-valued asynchronous CDMA with multipath channelsandblindchannelestimationcanbefoundin[192]. Someupperboundsontheachievableperformanceof variousblindmultiuserdetectorsareobtainedin[190, 191]. Furthermore,large-system asymptotic performance analysis of blind multiuser detection algorithms is givenin[594].2.6. SUBSPACETRACKINGALGORITHMS 81102103103102101SNR=8 SNR=10SNR=12SNR=14SNR=16DMI blind detectornumber of samples (M)BER102103103102101subspace blind detectornumber of samples (M)BERSNR=8 SNR=10SNR=12SNR=14SNR=16Figure 2.9: The BER versus the number of signal samples Mfor DMI and subspace detectors.N=13, K=11. Thesolidlineistheanalytical performance, andthedashedlineisthesimulationperformance.2.6 SubspaceTrackingAlgorithmsItisseenfromtheprevioussectionthatthelinearmultiuserdetectorsareobtainedoncethesignalsubspacecomponentsareidentied. Theclassicapproachtosubspaceestimationisthroughbatcheigenvaluedecomposition(ED)of thesampleautocorrelationmatrix, orbatchsingularvaluedecomposition(SVD)ofthedatamatrix,bothofwhicharecomputa-tionallytooexpensiveforadaptiveapplications. Modernsubspacetrackingalgorithmsarerecursiveinnatureandupdatethesubspaceinasample-by-samplefashion. Anadaptiveblindmultiuserdetectorcanbebasedonsubspacetrackingbysequentiallyestimatingthesignal subspace components, and forming the closed-form detector based on these estimates.Specically, suppose that at time (i 1), the estimated signal subspace rank is K[i 1] andthe components are Us[i 1],s[i 1] and2[i 1]. Thenattime i,the adaptive detectorperformsthefollowingstepstoupdatethedetectorandtodetectthedata.Algorithm2.6[BlindadaptivelinearMMSEdetectorbasedonsubspacetracking-syn-chronousCDMA]Updatethesignal subspace: Usingaparticularsignal subspacetrackingalgorithm,up-82 CHAPTER2. BLINDMULTIUSERDETECTIONdatethesignalsubspacerankK[i]andthesubspacecomponentsUs[i],s[i]and2[i].Formthedetectorandperformdetection:m1[i] = Us[i]s[i]1Us[i]Hs1,z1[i] = m1[i]Hr[i],1[i] = sign '(z1[i]z1[i 1]) .Varioussubspacetrackingalgorithmsaredescribedintheliterature, e.g., [42, 83, 92, 398,403,452,484,578]. Herewepresenttwolow-complexitysubspacetrackingalgorithms, thePASTdalgorithm[578]andthemorerecentlydevelopedNAHJalgorithm[403].2.6.1 ThePASTdAlgorithmLet r[i] CNbe a random vector with autocorrelation matrix Cr= Er[i] r[i]H. Considerthescalarfunction.(W) = E

r[i] WWHr[i]

2= tr (Cr) 2 tr

WHCrW

+ tr

WHCrWWHW

, (2.150)withamatrixargumentW CNr(r < N). Itcanbeshownthat[578]:Wis a stationary point of .(W) if and only if W= UrQ, where Ur CNrcontainsanyrdistincteigenvectorsofCrandQ Crrisanyunitarymatrix;andAllstationarypointsof .(W)aresaddlepointsexceptwhenUrcontainstherdom-inanteigenvectorsofCr. Inthatcase, .(W)attainstheglobalminimum.Therefore, for r =1, the solutionof minimizing .(W) is givenbythe most dominanteigenvectorof Cr. Inreal applications, onlysamplevectorsr[i] areavailable. Replacing(2.150)withtheexponentiallyweightedsumsyields.(W[i]) =in=0in

r[n] W[i]W[i]Hr[n]

2. (2.151)Thekeyissueof thePASTd(projectionapproximationsubspacetrackingwithdeation)approachistoapproximateW[i]Hr[n]in(2.151), theunknownprojectionofr[n]ontothe2.6. SUBSPACETRACKINGALGORITHMS 83columnsofW[i],byy[n] = W[n 1]Hr[n],whichcanbecalculatedfor1 n iattimei. Thisresultsinamodiedcostfunction.(W[i]) =in=0in

r[n] W[i]y[n]

2. (2.152)Therecursiveleast-squares(RLS)algorithmcanthenbeusedtosolvefortheW[i] thatminimizestheexponentiallyweightedleast-squarescriterion(2.152).The PASTd algorithm for tracking the eigenvalues and eigenvectors of the signal subspaceisbasedonthedeationtechniqueandcanbedescribedasfollows. Forr=1, themostdominanteigenvectorisupdatedbyminimizing.(W[i])in(2.152). Thentheprojectionof thecurrentdatavectorr[i] ontothiseigenvectorisremovedfromr[i] itself. Nowthesecondmost dominant eigenvector becomes themost dominant oneintheupdateddatavectoranditcanbeextractedsimilarly. Thisprocedureisappliedrepeatedlyuntil all theKeigenvectorsareestimatedsequentially.Basedonthe estimatedeigenvalues, using informationtheoretic criteria suchas theAkaike information criterion (AIC) or the minimum description length (MDL) criterion [548],therankofthesignalsubspace, orequivalently, thenumberofactiveusersinthechannel,canbeestimatedadaptivelyas well [577]. Thequantities AICandMDLaredenedasfollows:AIC(k)z= (N k)M ln (k) + k(2N k), (2.153)MDL(k)z= (N k)M ln (k) +k2(2N k)ln M, (2.154)k = 1, 2,, N,whereMisthenumber of datasamplesusedintheestimation. Whenanexponentiallyweightedwindowwithforgettingfactorisappliedtothedata, theequivalentnumberofdatasamplesisM= 1/(1 ). (k)intheabovedenitionsisdenedas(k) =

Ni=k+1i

/(N k)

Ni=k+1i

1/(Nk). (2.155)TheAIC(resp. MDL)estimateofsubspacerankisgivenbythevalueofkthatminimizesthequantity(2.153)(resp. (2.154)). FinallythePASTdalgorithmforbothrankandsignal84 CHAPTER2. BLINDMULTIUSERDETECTIONsubspace tracking is summarized in Table 1. The computational complexity of this algorithmis(4K+ 3)N+ O(K) = O(NK) per update. Theconvergencedynamicsof thePASTdalgorithmarestudiedin[579]. Itisshowntherethatwithaforgettingfactor= 1,undermild conditions, this algorithm converges globally and almost surely to the signal eigenvectorsandeigenvalues.SimulationExamplesInwhat follows weprovidetwosimulationexamples toillustratetheperformanceof thesubspaceblindadaptivedetectoremployingthePASTdalgorithm.Example1: Thisexamplecomparestheperformanceofthesubspace-basedblindMMSEdetector with the performance of the minimum-output-energy (MOE) blind adaptive detectorproposedin[179]. Itassumesareal-valuedsynchronousCDMAsystemwithaprocessinggain N= 31 and six users (K= 6). The desired user is User 1. There are four 10 dB multiple-access interferers (MAIs) and one 20 dB MAI, i.e., A2k/A21= 10, for k = 2,, 5, and A2k/A21=100, for k = 6. The performance measure is the output signal-to-interference-plus-noise ratio(SINR), denedasSINRz=E2wTr/VarwTr, wheretheexpectationiswithrespecttothedatabitsof theMAIsandthenoise. Inthesimulation, theexpectationoperationisreplacedbythetimeaveragingoperation. ForthePASTdsubspacetrackingalgorithm,wefoundthatwitharandominitialization,theconvergenceisfairlyslow. Thereforeinthesimulations, the initial estimates of the eigencomponents of the signal subspace are obtainedbyapplyingaSVDtotherst50datavectors. ThePASTdalgorithmisthenemployedthereafterfortrackingthesignalsubspace. ThetimeaveragedoutputSINRversusnumberofiterationsisplottedinFig.2.10.As a comparison, the simulated performance of the recursive least-squared (RLS) versionoftheMOEblindadaptivedetectorisalsoshowninFig. 2.10. Ithasbeenshownin[381]that the steady-state SINR of this algorithm is given by SINR=SINR1+d+dSINR, where SINRistheoutputSINRvalueof theexactlinearMMSEdetector, anddz=12N(0 0and2> 0. By(3.108)and(3.109)wehave1g1g22< 0 =12g1g2. (3.111)Hence, (3.110) and (3.111) contradict each other. The same contradiction arises for the otherthreechoicesofpolaritiesfor1and2. 2Ingeneral,theslowest-descentsearchmethodchoosesthecandidatesetin(3.104)asfollows: = b Qq=1

bq, 1, +1K: bq,k=

sign (kgqk) , ifkgqk = 0bk, ifkgqk= 0,gqistheqthsmallesteigenvectorof 2

,

1gq1,,KgqK

, (3.112)wherekandgqkdenotethekthelementsoftherespectivevectorsandgq. Hence, bq,containstheKclosestneighborsof in 1, +1Kalongthedirectionof gq. NotethatgqQq=1represent the Q mutually orthogonal directions where the likelihood function p(y[b)drops the slowest from the peak point . Intuitively, the maximum likelihood solutionbML in(3.101)ismostlikelyfoundinthisneighborhood. Themultiuserdetectionalgorithmbasedon the slowest-descent-search method is summarized as follows (assuming that the signaturewaveformsSandthecomplexamplitudesAofallusersareknown).Algorithm3.3[Slowest-descent-searchmultiuserdetector]Compute the Hessianmatrix 2

givenby (3.103), andits Qsmallest eigenvectorsg1,, gQ;3.4. NONLINEARGROUP-BLINDMULTIUSERDETECTION 163Computethestationarypointgivenby(3.102);Solve the discrete optimization problem dened by (3.104) and (3.112) by an exhaustivesearch(over(KQ + 1)points).The rst step involves calculating the eigenvectors of a KKsymmetric matrix; the secondstep involves inverting a KKmatrix; and the third step involves evaluating the likelihoodvaluesat(KQ + 1)points. Notethatthersttwostepsonlyneedtobeperformedonceif ablockof Mdatabits needtobedemodulated. Hencethedominant computationalcomplexityoftheabovealgorithmis O(KQ)perbitforKusers.SimulationExamplesFor simulations, weassumeaprocessinggainN=15, thenumber of usersK=8, andequal amplitudes of user signals, i.e., [Ak[ =1, k =1,, K. ThesignaturematrixSandtheuserphaseosets AkKk=1arechosenatrandomandkeptxedthroughoutthesimulations. Fig. 3.11demonstratesthattheslowest-descentmethodwithonlyonesearchdirection (Q = 1) oers a signicant performance gain over the linear decorrelator. Searchingonemoredirection(Q=2)resultsinsomeadditional performanceimprovement. Furtherincreaseinthenumber of searchdirections onlyresults inadiminishingimprovement inperformance.3.4.2 NonlinearGroup-BlindMultiuserDetectionIngroup-blindmultiuserdetection,onlytherstKuserssignalsneedtobedemodulated.As before, denoteSandSas matrices containingrespectivelythe rstKandthe last(K K)columnsofS. SimilarlydenethequantitiesA,b,A, andb. Then(3.2)canberewrittenas(again,wedropthesymbolindexiforconvenience)r = SAb +n (3.113)=SAb +SAb +n. (3.114)LetDdenotethedecorrelatingdetectorsofthedesiredusers,givenbyD = [d1dK]=

S

STS

1

:,1: K, (3.115)164 CHAPTER3. GROUP-BLINDMULTIUSERDETECTION2 3 4 5 6 7 8 9 10104103102101BERSNR [dB]slowest descent: 2 directions slowest descent: 1 directiondecorrelator Figure3.11: Performanceoftheslowest-descent-basedmultiuserdetectorinasynchronousCDMAsystem. N=15, K=8. ThespreadingwaveformsSandthecomplexamplitudesAof all usersareassumedknowntothereceiver. Thebiterrorrate(BER)curvesof thelineardecorrelatorandtheslowest-descentdetectorwithQ = 1andQ = 2areshown.3.4. NONLINEARGROUP-BLINDMULTIUSERDETECTION 165where[X]:,n1:n2denotescolumnsn1ton2ofthematrixX. ItiseasilyseenthatDsatisesthefollowingDTS= 0, andDTS= IK. (3.116)Ingroup-blindmultiuserdetection,theundesireduserssignalsarerstnulledoutfromthereceivedsignal,bythefollowingprojectionoperation,zz=DTr =Ab +DTn, (3.117)wherethesecondequalityin(3.117)followsfrom(3.114)and(3.116). Denoteyz= 'zz, z= 'AA, and vz=DT'nDTn.Then(3.117)canbewrittenasy = b +v. (3.118)NotethatthecovariancematrixofvisgivenbyCovv =22Q 00 Q. .. ., (3.119)with Qz=DTD (3.120)=

STS

1

1: K,1: K.Inwhatfollowsweconsidernonlinearestimationofbfrom(3.118)basedontheslowest-descentsearch. WewillalsodiscusstheproblemofestimatingDandAfromthereceivedsignals.Themaximumlikelihoodestimateofbbasedonyin(3.118)isgivenbybML= arg maxb|+1,1Kp(y [ b)= arg minb|+1,1K

y b

T1

y b

. .. .(b)= arg minb|+1,1K() +

b

T2

b

, (3.121)166 CHAPTER3. GROUP-BLINDMULTIUSERDETECTIONwheretheHessianmatrixisgivenby2

= T1= 'AQ1'A +AQ1A; (3.122)andisthestationarypointof(b),i.e.,

() = 0 = =

T1

1T1y=

2

1

'AQ1'z +AQ1z

. (3.123)Asbefore,weapproximatethesolutionto(3.121)byb= arg maxb|+1,1Kp(y [ b)= arg minb

b

T2

b

, (3.124)wherecontains( KQ + 1)closestneighborsofin 1, +1Kalongtheslowest-descentdirectionsofthelikelihoodfunctionp(y[b),givenbybz= sign(), (3.125) = b Qq=1

bq, +1, 1K: bq,k=

sign

kgqk

, ifkgqk = 0bk, ifkgqk= 0,gqistheqthsmallesteigenvectorof 2

,

1gq1,,KgqK. (3.126)EstimationofDandAInorder toimplement theabovelocal-search-basedgroup-blindmultiuser detectionalgo-rithm, wemustrstestimatethedecorrelatormatrixDandthecomplexamplitudesA.Notethat thedecorrelatingdetectorsDfor thedesiredusersaresimplythegroup-blindlineardecorrelatingdetectorsdiscussedinSection3.2. Forexample, basedontheeigende-composition(3.37)oftheautocorrelationmatrixofthereceivedsignal,Disgivenintermsofthesignalsubspaceparametersby(3.38).We next consider the estimationof the complexamplitudesAof the desiredusers.Considerthedecorrelatoroutput(3.117),wenowhave[RecallthatAz= (A1,, AK).]zk= Ak bk + nk, k = 1,,K, (3.127)3.4. NONLINEARGROUP-BLINDMULTIUSERDETECTION 167where nkz=

DTn

k. Sincebk +1, 1, itfollowsfrom(3.127)thatthedecorrelatoroutputs c