Array Signal Processing Algorithms for - User Web Areas

170 Pages·2009·1.25 MB·English

Array Signal Processing Algorithms
for Beamforming and Direction Finding
This thesis is submitted in partial fulfilment of the requirements for
Doctor of Philosophy (Ph.D.)
LeiWang
CommunicationsResearchGroup
DepartmentofElectronics
UniversityofYork
December 2009
ABSTRACT
Array processing is an area of study devoted to processing the signals received from
an antenna array and extracting information of interest. It has played an important role
in widespread applications like radar, sonar, and wireless communications. Numerous
adaptivearrayprocessingalgorithmshavebeenreportedintheliteratureinthelastseveral
decades. Thesealgorithms,inageneralview,exhibitatrade-offbetweenperformanceand
requiredcomputationalcomplexity.
In this thesis, we focus on the development of array processing algorithms in the ap-
plication of beamforming and direction of arrival (DOA) estimation. In the beamformer
design, we employ the constrained minimum variance (CMV) and the constrained con-
stantmodulus(CCM)criteriatoproposefull-rankandreduced-rankadaptivealgorithms.
Specifically, for the full-rank algorithms, we present two low-complexity adaptive step
sizemechanismswiththeCCMcriterionforthestepsizeadaptationofthestochasticgra-
dient (SG) algorithms. The convergence and steady-state properties are analysed. Then,
the full-rank constrained conjugate gradient (CG) adaptive filtering algorithms are pro-
posed according to the CMV and CCM criteria. We introduce a CG based weight vector
toincorporatetheconstraintinthedesigncriteriaforsolvingthesystemofequationsthat
arises from each design problem. The proposed algorithms avoid the covariance matrix
inversionandprovideatrade-offbetweenthecomplexityandperformance.
In reduced-rank array processing, we present CMV and CCM reduced-rank schemes
based on joint iterative optimization (JIO) of adaptive filters. This scheme consists a
bank of full-rank adaptive filters that forms the transformation matrix, and an adaptive
reduced-rank filter that operates at the output of the bank of filters. The transformation
matrixandthereduced-rankweightvectorarejointlyoptimizedaccordingtotheCMVor
CCM criteria. For the application of beamforming, we describe the JIO scheme for both
thedirect-formprocessor(DFP)andthegeneralizedsidelobecanceller(GSC)structures.
For each structure, we derive SG and recursive least squares (RLS) type algorithms to
iteratively compute the transformation matrix and the reduced-rank weight vector for the
reduced-rank scheme. An auxiliary vector filtering (AVF) algorithm based on the CCM
design for robust beamforming is presented. The proposed beamformer decomposes the
adaptive filter into a constrained (reference vector filter) and an unconstrained (auxiliary
vectorfilter)component. Theweightvectorisiteratedbysubtractingthescalingauxiliary
vectorfromthereferencevector.
For the DOA estimation, the reduced-rank scheme with the minimum variance (MV)
power spectral evaluation is introduced. A spatial smoothing (SS) technique is employed
in the proposed method to improve the resolution. The proposed DOA estimation algo-
rithms are suitable for large arrays and to deal with direction finding for a small number
ofsnapshots,alargenumberofusers,andwithouttheexactinformationofthenumberof
sources.
CONTENTS
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
ListofSymbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
ListofFigures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
ListofTables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 PriorWork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Beamforming . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.2 DOAEstimation . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 ThesisOutline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.6 ListofPublications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2. AdaptiveStepSizeCCMSGAlgorithms
forAdaptiveBeamforming. . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 ArrayStructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.1 AdaptiveArrayStructure . . . . . . . . . . . . . . . . . . . . . . 13
2.2.2 AntennaArrayandDirectionofArrival . . . . . . . . . . . . . . 14
2.3 SystemModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 AdaptiveSGAlgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.5 ProposedAdaptiveStepSizeMechanisms . . . . . . . . . . . . . . . . . 17
2.5.1 ModifiedAdaptiveStepSize(MASS) . . . . . . . . . . . . . . . 18
2.5.2 TimeAveragingAdaptiveStepSize(TAASS) . . . . . . . . . . . 18
2.5.3 ComputationalComplexity . . . . . . . . . . . . . . . . . . . . . 19
2.6 AnalysisoftheProposedAlgorithms . . . . . . . . . . . . . . . . . . . . 20
2.6.1 ConvergenceAnalysis . . . . . . . . . . . . . . . . . . . . . . . 21
SufficientConditionfortheConvergenceoftheMeanWeightVector 21
Steady-StateStepSizeValueforMASS . . . . . . . . . . . . . . 22
Steady-StateStepSizeValueforTAASS . . . . . . . . . . . . . 23
2.6.2 Steady-stateAnalysis . . . . . . . . . . . . . . . . . . . . . . . . 24
2.6.3 TrackingAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.7 SimulationResults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3. Constrained Adaptive Filtering Algorithms Based on the Conjugate Gradient
MethodforBeamforming . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 ProblemStatement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3 ProposedAdaptiveCGalgorithms . . . . . . . . . . . . . . . . . . . . . 41
3.3.1 ConjugateGradientAlgorithms . . . . . . . . . . . . . . . . . . 41
3.3.2 ProposedConventionalConjugateGradient(CCG)Algorithms . . 42
TheCMV-CCGAlgorithm . . . . . . . . . . . . . . . . . . . . . 42
TheCCM-CCGAlgorithm . . . . . . . . . . . . . . . . . . . . . 44
3.3.3 ProposedModifiedConjugateGradient(MCG)Algorithms . . . . 45
TheProposedCMV-MCGAlgorithm . . . . . . . . . . . . . . . 45
TheProposedCCM-MCGAlgorithm . . . . . . . . . . . . . . . 48
3.4 AnalysisoftheProposedMethods . . . . . . . . . . . . . . . . . . . . . 49
3.4.1 GlobalConvergenceandProperties . . . . . . . . . . . . . . . . 49
3.4.2 ComputationalComplexity . . . . . . . . . . . . . . . . . . . . . 49
3.4.3 ConvergenceAnalysis . . . . . . . . . . . . . . . . . . . . . . . 50
3.5 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4. AdaptiveReduced-rankCMVBeamforming
andDOAAlgorithmsBasedonJointIterativeOptimizationofFilters. . . . . . 60
4.1 IntroductionforBeamforming . . . . . . . . . . . . . . . . . . . . . . . 60
4.2 ProblemStatement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.3 ProposedReduced-rankMethod . . . . . . . . . . . . . . . . . . . . . . 62
4.4 AdaptiveAlgorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.4.1 StochasticGradientAlgorithm . . . . . . . . . . . . . . . . . . . 65
4.4.2 RecursiveLeastSquaresAlgorithms . . . . . . . . . . . . . . . . 65
4.4.3 ComplexityofProposedAlgorithms . . . . . . . . . . . . . . . . 67
4.4.4 AutomaticRankSelection . . . . . . . . . . . . . . . . . . . . . 67
4.5 AnalysisofAlgorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.5.1 StabilityAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.5.2 MSEConvergenceAnalysis . . . . . . . . . . . . . . . . . . . . 70
4.6 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.6.1 MSEAnalysisPerformance . . . . . . . . . . . . . . . . . . . . 74
4.6.2 SINRPerformance . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.7 IntroductionforDOAEstimation . . . . . . . . . . . . . . . . . . . . . . 79
4.8 ProblemStatement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.9 TheJIOSchemeforDOAEstimation . . . . . . . . . . . . . . . . . . . 81
4.10 ProposedReduced-RankAlgorithms . . . . . . . . . . . . . . . . . . . . 82
4.11 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.12 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5. AdaptiveReduced-rankCCMAlgorithmsBasedonJointIterativeOptimization
ofFiltersandAuxiliaryVectorFilteringforBeamforming. . . . . . . . . . . . 88
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.2 PreliminaryWorks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.2.1 Full-rankBeamformerDesignfortheDFP . . . . . . . . . . . . 90
5.2.2 Full-rankBeamformerDesignfortheGSC . . . . . . . . . . . . 91
5.3 Reduced-rankBeamformerDesign . . . . . . . . . . . . . . . . . . . . . 92
5.3.1 BeamformerDesignfortheDFP . . . . . . . . . . . . . . . . . . 93
5.3.2 BeamformerDesignfortheGSC . . . . . . . . . . . . . . . . . . 93
5.4 ProposedCCMReduced-rankScheme . . . . . . . . . . . . . . . . . . . 94
5.4.1 ProposedCCMReduced-rankSchemefortheDFP . . . . . . . . 94
5.4.2 ProposedCCMReduced-rankSchemefortheGSC . . . . . . . . 97
5.5 AdaptiveAlgorithmsoftheCCMReduced-rankScheme . . . . . . . . . 98
5.5.1 StochasticGradientAlgorithms . . . . . . . . . . . . . . . . . . 99
TheSGAlgorithmfortheDFP . . . . . . . . . . . . . . . . . . . 99
TheSGAlgorithmfortheGSC . . . . . . . . . . . . . . . . . . 100
5.5.2 RecursiveLeastSquaresAlgorithms . . . . . . . . . . . . . . . . 100
TheRLSAlgorithmfortheDFP . . . . . . . . . . . . . . . . . . 101
TheRLSAlgorithmfortheGSC . . . . . . . . . . . . . . . . . . 102
5.5.3 Gram-SchmidtTechniqueforProblem2 . . . . . . . . . . . . . . 105
5.5.4 AutomaticRankSelection . . . . . . . . . . . . . . . . . . . . . 105
5.6 AnalysisoftheProposedAlgorithms . . . . . . . . . . . . . . . . . . . . 106
5.6.1 ComplexityAnalysis . . . . . . . . . . . . . . . . . . . . . . . . 107
5.6.2 AnalysisoftheOptimizationProblem . . . . . . . . . . . . . . . 110
5.7 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.8 ProposedCCM-AVFAlgorithm . . . . . . . . . . . . . . . . . . . . . . 119
5.8.1 ProposedCCM-AVFScheme . . . . . . . . . . . . . . . . . . . 119
5.8.2 ProposedCCM-AVFAlgorithm . . . . . . . . . . . . . . . . . . 119
5.8.3 InterpretationsaboutProposedCCM-AVFAlgorithm . . . . . . . 122
5.9 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6. ConclusionsandFutureWork . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.1 SummaryofWork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.2 FutureWork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
Appendix 130
A. Derivationof(2.28) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
B. ConvexityConditionfortheCCMCriterion . . . . . . . . . . . . . . . . . . . 132
C. PreservationofMVandExistenceofMultipleSolutions . . . . . . . . . . . . 134
D. AnalysisoftheOptimizationoftheJIOCMVScheme . . . . . . . . . . . . . 135
E. DerivationofTransformationMatrix . . . . . . . . . . . . . . . . . . . . . . 138
F. Derivationof(5.31) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141