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Robust and iterative adaptive signal processing

Permanent Link: http://ufdc.ufl.edu/UFE0041486/00001

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Title: Robust and iterative adaptive signal processing
Physical Description: 1 online resource (135 p.)
Language: english
Creator: Du, Lin
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: adaptive, aeroacoustic, beamforming, spectrogram
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Ph.D.
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theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

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Abstract: Adaptive signal processing plays an important role in many applications including radar, sonar, acoustics, communications, image processing, speech processing, medical imaging and other fields. The goal of this dissertation is to investigate several adaptive signal processing techniques and their related applications. We focus on adaptive beamforming, ultrasound imaging, Doppler spectrogram analysis and aeroacoustic noise analysis. We first consider the adaptive signal processing techniques in the context of beamforming. Numerous approaches have been proposed in the literature to improve the robustness of the data-adaptive standard Capon beamformer (SCB). One of the most popular and widely used robust adaptive beamforming methods is the diagonal loading approach (as well as its extended versions). However, most of these schemes determine the diagonal loading level either in an ad-hoc way or need user parameters that might be hard to determine in practice. Therefore, user parameter-free approaches are desirable. We present a fully automatic approach to compute the diagonal loading level. In our diagonal loading algorithm, the conventional sample covariance matrix used in the SCB formulation is replaced by an enhanced covariance matrix estimate based on a shrinkage method. The enhanced estimate can be achieved by a general linear combination (GLC), of the sample covariance matrix and an identity matrix in a minimum mean-squared error (MSE) sense. The shrinkage parameters, which are related to the diagonal loading levels of the beamformers, can be calculated from the measurements automatically without the need to specify any user parameters. We have demonstrated that the GLC is very useful in the case of small sample sizes - the case in which the users of adaptive arrays are most interested. We then present a comprehensive review of user parameter-free robust adaptive beamforming algorithms. We provide a thorough evaluation of GLC, its special case convex combination (CC) method, ridge regression Capon beamformers (RRCB), the mid-way (MW) algorithm and several iterative approaches including the iterative adaptive approach (IAA), the maximum likelihood based IAA (referred to as IAA-ML) and the multi-snapshot sparse Bayesian learning (M-SBL) under various scenarios such as coherent, non-coherent and distributed sources, steering vector mismatches, snapshot limitations and low signal-to-noise ratio (SNR) levels. Furthermore, we discuss the computational complexities of the algorithms and provide insights into which algorithm is the best choice under which circumstances. We also consider applying adaptive signal processing techniques to ultrasound imaging. We discuss the challenges in ultrasound imaging applications including the wideband, near-field environment and limited data samples. We then extend GLC and IAA to accommodate those requirements, which result in wideband GLC (WGLC) and wideband IAA (WIAA). Both approaches have been shown to have high resolution, and are robust to the finite sample size problems and other model errors. We then consider Doppler spectrogram analysis. We propose a short-time iterative adaptive approach (ST-IAA) based on IAA to form the Doppler spectrogram. Due to its adaptive character, ST-IAA has much higher frequency resolution and lower sidelobes than its data-independent counterpart, i.e., the conventional short-time Fourier transform (STFT) based approach, and thus ST-IAA provides much more accurate spectrograms. Moreover, a model-order selection tool, the generalized information criterion (GIC) can be used in conjunction with ST-IAA to further improve the spectrogram quality. Finally, we present several iterative adaptive signal processing approaches to aeroacoustic noise analysis. One of the approaches is based on optimizing the maximum likelihood (ML) criterion via using the Newton's method. The other approaches, referred to as the Frobenius norm (FN) and Rank-1 methods, employ the cyclic optimization algorithm to solve the problem. We also derive the Cramer-Rao Bounds (CRB) of the unbiased source power estimates. The proposed methods are evaluated using both simulated and measured data. The numerical examples show that these algorithms significantly outperform the existing least squares approach and provide accurate power estimates even under low SNR conditions. Furthermore, the MSEs of the so-obtained estimates are close to the corresponding CRB, especially when the number of data samples is large. The experimental results show that the power estimates obtained by the proposed approaches are consistent with one another.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Lin Du.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Li, Jian.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041486:00001

Permanent Link: http://ufdc.ufl.edu/UFE0041486/00001

Material Information

Title: Robust and iterative adaptive signal processing
Physical Description: 1 online resource (135 p.)
Language: english
Creator: Du, Lin
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: adaptive, aeroacoustic, beamforming, spectrogram
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Adaptive signal processing plays an important role in many applications including radar, sonar, acoustics, communications, image processing, speech processing, medical imaging and other fields. The goal of this dissertation is to investigate several adaptive signal processing techniques and their related applications. We focus on adaptive beamforming, ultrasound imaging, Doppler spectrogram analysis and aeroacoustic noise analysis. We first consider the adaptive signal processing techniques in the context of beamforming. Numerous approaches have been proposed in the literature to improve the robustness of the data-adaptive standard Capon beamformer (SCB). One of the most popular and widely used robust adaptive beamforming methods is the diagonal loading approach (as well as its extended versions). However, most of these schemes determine the diagonal loading level either in an ad-hoc way or need user parameters that might be hard to determine in practice. Therefore, user parameter-free approaches are desirable. We present a fully automatic approach to compute the diagonal loading level. In our diagonal loading algorithm, the conventional sample covariance matrix used in the SCB formulation is replaced by an enhanced covariance matrix estimate based on a shrinkage method. The enhanced estimate can be achieved by a general linear combination (GLC), of the sample covariance matrix and an identity matrix in a minimum mean-squared error (MSE) sense. The shrinkage parameters, which are related to the diagonal loading levels of the beamformers, can be calculated from the measurements automatically without the need to specify any user parameters. We have demonstrated that the GLC is very useful in the case of small sample sizes - the case in which the users of adaptive arrays are most interested. We then present a comprehensive review of user parameter-free robust adaptive beamforming algorithms. We provide a thorough evaluation of GLC, its special case convex combination (CC) method, ridge regression Capon beamformers (RRCB), the mid-way (MW) algorithm and several iterative approaches including the iterative adaptive approach (IAA), the maximum likelihood based IAA (referred to as IAA-ML) and the multi-snapshot sparse Bayesian learning (M-SBL) under various scenarios such as coherent, non-coherent and distributed sources, steering vector mismatches, snapshot limitations and low signal-to-noise ratio (SNR) levels. Furthermore, we discuss the computational complexities of the algorithms and provide insights into which algorithm is the best choice under which circumstances. We also consider applying adaptive signal processing techniques to ultrasound imaging. We discuss the challenges in ultrasound imaging applications including the wideband, near-field environment and limited data samples. We then extend GLC and IAA to accommodate those requirements, which result in wideband GLC (WGLC) and wideband IAA (WIAA). Both approaches have been shown to have high resolution, and are robust to the finite sample size problems and other model errors. We then consider Doppler spectrogram analysis. We propose a short-time iterative adaptive approach (ST-IAA) based on IAA to form the Doppler spectrogram. Due to its adaptive character, ST-IAA has much higher frequency resolution and lower sidelobes than its data-independent counterpart, i.e., the conventional short-time Fourier transform (STFT) based approach, and thus ST-IAA provides much more accurate spectrograms. Moreover, a model-order selection tool, the generalized information criterion (GIC) can be used in conjunction with ST-IAA to further improve the spectrogram quality. Finally, we present several iterative adaptive signal processing approaches to aeroacoustic noise analysis. One of the approaches is based on optimizing the maximum likelihood (ML) criterion via using the Newton's method. The other approaches, referred to as the Frobenius norm (FN) and Rank-1 methods, employ the cyclic optimization algorithm to solve the problem. We also derive the Cramer-Rao Bounds (CRB) of the unbiased source power estimates. The proposed methods are evaluated using both simulated and measured data. The numerical examples show that these algorithms significantly outperform the existing least squares approach and provide accurate power estimates even under low SNR conditions. Furthermore, the MSEs of the so-obtained estimates are close to the corresponding CRB, especially when the number of data samples is large. The experimental results show that the power estimates obtained by the proposed approaches are consistent with one another.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Lin Du.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Li, Jian.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041486:00001


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ROBUSTANDITERATIVEADAPTIVESIGNALPROCESSINGByLINDUADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2010

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c2010LinDu 2

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Tomyparentsandmyhusband 3

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ACKNOWLEDGMENTS Ifeelextremelyfortunateandgratefultohavereceivedsupportandhelpfromnumerousteachers,colleaguesandfriendsduringtheresearch.Firstandforemost,Iwouldliketoexpressmysinceregratitudetomyadvisor,Dr.JianLi,forheradvice,encouragementandsupportinguidingtheresearch.Iamgratefulfortheopportunityshehasofferedmetopursuemyresearchunderhersupervision.Igreatlyappreciateherenormoushelpbothtechnicallyandpersonally.IamgratefultoDr.PetreStoicaatUppsalaUniversity,Sweden,forhisguidanceinavarietyofinterestingtopics.IwouldliketoconveymyappreciationtoDr.HenryZmuda,Dr.YijunSunandDr.MingzhouDinginthedepartmentofbiomedicalengineeringforservinginmysupervisorycommitteeandfortheirvaluablecommentsandsuggestions.IgratefullyacknowledgeDr.LuzhouXuforhisgenerosityindevotinghistimetohelpmewiththeresearch.IwishtothankallmylabmatesinSpectralAnalysisLab.Itwasagreatpleasuretoworkwiththem.IalsothankallmydearfriendsinGainesvilleandbackathomeinChina.Theirfriendshipandhelpmademylifemoreenjoyable.Finally,Iamdeeplythankfultomymom,mydadandmyhusband,JianZou,fortheirendlessandselesslove,theircareandconstantsupport. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 8 LISTOFFIGURES ..................................... 9 ABSTRACT ......................................... 11 CHAPTER 1INTRODUCTION ................................... 14 1.1AdaptiveBeamforming ............................. 14 1.2UltrasoundImagingApplication ........................ 17 1.3DopplerSpectrogramAnalysisoftheHumanGait ............. 19 1.4AeroacousticNoiseAnalysis ......................... 20 1.5OutlineofThisDissertation .......................... 21 1.6Notation ..................................... 22 2FULLYAUTOMATICCOMPUTATIONOFDIAGONALLOADINGLEVELSBASEDONSHRINKAGE ................................... 24 2.1Introduction ................................... 24 2.2Background ................................... 25 2.2.1SCB ................................... 25 2.2.2DL .................................... 27 2.2.3HKB ................................... 28 2.3GLC-BasedRobustCaponBeamforming .................. 29 2.3.1GLC-BasedCovarianceMatrixEstimation .............. 29 2.3.2GLC-BasedRobustCaponBeamformer ............... 31 2.4ConvexFormulationsofGLC ......................... 32 2.5NumericalExamples .............................. 34 2.6Conclusions ................................... 37 3PERFORMANCECOMPARISONOFUSERPARAMETER-FREEROBUSTADAPTIVEBEAMFORMERS ............................ 43 3.1Introduction ................................... 43 3.2DataModelandProblemFormulation .................... 43 3.2.1DAS ................................... 45 3.2.2SCB ................................... 46 3.3DiagonalLoadingApproaches ........................ 46 3.3.1RRCB .................................. 47 3.3.2ShrinkageBasedRobustCaponBeamforming ........... 47 3.3.3MW ................................... 49 5

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3.4Iterativeapproaches .............................. 51 3.4.1IAA .................................... 51 3.4.2IAA-ML ................................. 52 3.4.3M-SBL .................................. 54 3.5NumericalExamples .............................. 55 3.5.1SimulationDetails ........................... 55 3.5.2PointSources .............................. 56 3.5.3DistributedSources ........................... 58 3.5.4ComplexityAnalysis .......................... 60 3.5.5OverallAssessments .......................... 60 3.6Conclusions ................................... 61 4USERPARAMETERFREEROBUSTADAPTIVEAPPROACHESTOULTRASOUNDIMAGING ....................................... 71 4.1Introduction ................................... 71 4.2DataModel ................................... 72 4.3WidebandGLCApproaches .......................... 73 4.3.1Preprocess ............................... 73 4.3.2WGLC-1 ................................. 74 4.3.3WGLC-2 ................................. 78 4.3.4WGLC-C ................................. 79 4.4WidebandIAAapproach ............................ 80 4.4.1Preprocess ............................... 81 4.4.2IAA .................................... 82 4.5ExperimentalExamples ............................ 83 4.6Conclusions ................................... 85 5DOPPLERSPECTROGRAMANALYSISOFTHEHUMANGAITVIAANITERATIVEADAPTIVEAPPROACH ............................... 88 5.1Introduction ................................... 88 5.2ProblemFormulation .............................. 89 5.3ST-IAA ...................................... 90 5.4ExamplesonHumanGaitAnalysis ...................... 91 5.5Conclusions ................................... 92 6COVARIANCE-BASEDAPPROACHESTOAEROACOUSTICNOISESOURCEANALYSIS ...................................... 96 6.1Introduction ................................... 96 6.2DataModelandProblemFormulation .................... 96 6.3LSApproach .................................. 99 6.4Covariance-BasedTechniques ........................ 100 6.4.1FrobeniusNormMethod ........................ 100 6.4.2Rank-1Method ............................. 102 6.4.3MaximumLikelihoodMethod ..................... 103 6

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6.5Cramer-RaoBound .............................. 106 6.6NumericalandExperimentalExamples ................... 107 6.6.1NumericalExamples .......................... 108 6.6.2ExperimentalExamples ........................ 109 6.7Conclusions ................................... 109 7CONCLUSIONSANDFUTUREWORK ...................... 115 7.1Conclusions ................................... 115 7.2FutureWork ................................... 117 APPENDIX ADERIVATIONOFPARAMETER^FORGLCAPPROACH ............ 119 BSTFTANDREASSIGNEDMETHOD ........................ 121 CDERIVATIONOFMLAPPROACH ......................... 122 C.1First-orderDerivatives ............................. 122 C.2Second-orderDerivatives ........................... 124 REFERENCES ....................................... 127 BIOGRAPHICALSKETCH ................................ 135 7

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LISTOFTABLES Table page 1-1Mathematicalnotationusedinthedissertation. .................. 23 3-1TheIAAalgorithm. .................................. 52 3-2TheIAA-MLalgorithm. ................................ 54 3-3TheM-SBLalgorithm. ................................ 55 3-4Summaryofresultsforcomparingvariousbeamformingalgorithms. ...... 61 4-1TheWGLC-1approach. ............................... 78 4-2TheWGLC-2approach. ............................... 79 4-3TheWGLC-Capproach. ............................... 80 6-1TheFNmethod. ................................... 101 6-2TheRank-1method. ................................. 103 6-3TheMLmethod. ................................... 106 6-4Computationtimesofvariousmethods. ...................... 109 8

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LISTOFFIGURES Figure page 2-1BeamformeroutputSINRversusthesnapshotnumberintheabsenceofarraysteeringvectorerrors. ................................ 38 2-2Comparisonofaveragediagonalloadinglevels. .................. 39 2-3Performancecomparisonintheabsenceofsteeringvectorerrors. ....... 40 2-4Performancecomparisoninthepresenceofa1steeringangleerror. ..... 41 2-5Performancecomparisoninthepresenceofarraycalibrationerrors. ...... 42 3-1Alineararray. ..................................... 44 3-2Performanceofvariousapproachesversusthenumberofsnapshots. ..... 62 3-3PerformanceofvariousapproachesversusSNR. ................. 62 3-4PerformanceofvariousapproachesversustheperturbationvariancewhenN=20. ........................................ 63 3-5PerformanceofvariousapproachesversustheperturbationvariancewhenN=100. ....................................... 63 3-6Performanceofvariousapproachesversusthecorrelationcoefcient. ..... 64 3-7Spatialpowerestimatesforuncorrelatedsources. ................ 65 3-8Spatialpowerestimatesforuncorrelatedsourceswhenthesourcesarenotonthescanninggrid. ................................. 66 3-9Spatialpowerestimatesforuncorrelatedsourcesinthepresenceofarraycalibrationerrors. ......................................... 67 3-10Spatialpowerestimatesfortwocoherentsources. ................ 68 3-11Performanceofvariousapproachesversusthenumberofsnapshotsforuncorrelatedspatiallydistributedsources. ............................ 69 3-12PerformanceofvariousapproachesversusSNRforuncorrelatedspatiallydistributedsources. ................................. 69 3-13Spatialpowerestimatesfortwouncorrelatedspatiallydistributedsources. ... 70 4-1Themultistaticdatacubeforagivenfocalpoint. ................. 75 4-2DatapreprocessingforWIAA. ............................ 81 4-3Thereconstructedimagesforwirephantom. ................... 86 9

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4-4Thereconstructedimagesforheartphantom. ................... 87 5-1Spectrogramsofthesimulatedhumangaitdata. ................. 94 5-2Spectrogramsofthemeasuredhumangaitdata. ................. 95 6-1Thesingle-inputmultiple-outputsystemforaeroacousticnoiseanalysis. .... 97 6-2Themeanvaluesandthe90%CIoftheestimatedSOIandnoisepowersusingsimulateddata. .................................... 111 6-3TheMSEsoftheSOIpowerestimatesusingsimulateddata. .......... 112 6-4TheexperimentalsetupfortheNACA63-215Mod-Bairfoil. ........... 113 6-5TheSOIpowerestimatesusingthemeasureddata. ............... 113 6-6TheSOIpowerspectrumestimatesusingthemeasureddata. .......... 114 10

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyROBUSTANDITERATIVEADAPTIVESIGNALPROCESSINGByLinDuMay2010Chair:JianLiMajor:ElectricalandComputerEngineering Adaptivesignalprocessingplaysanimportantroleinmanyapplicationsincludingradar,sonar,acoustics,communications,imageprocessing,speechprocessing,medicalimagingandotherelds.Thegoalofthisdissertationistoinvestigateseveraladaptivesignalprocessingtechniquesandtheirrelatedapplications.Wefocusonadaptivebeamforming,ultrasoundimaging,Dopplerspectrogramanalysisandaeroacousticnoiseanalysis. Werstconsidertheadaptivesignalprocessingtechniquesinthecontextofbeamforming.Numerousapproacheshavebeenproposedintheliteraturetoimprovetherobustnessofthedata-adaptivestandardCaponbeamformer(SCB).Oneofthemostpopularandwidelyusedrobustadaptivebeamformingmethodsisthediagonalloadingapproach(aswellasitsextendedversions).However,mostoftheseschemesdeterminethediagonalloadingleveleitherinanad-hocwayorneeduserparametersthatmightbehardtodetermineinpractice.Therefore,userparameter-freeapproachesaredesirable.Wepresentafullyautomaticapproachtocomputethediagonalloadinglevel.Inourdiagonalloadingalgorithm,theconventionalsamplecovariancematrixusedintheSCBformulationisreplacedbyanenhancedcovariancematrixestimatebasedonashrinkagemethod.Theenhancedestimatecanbeachievedbyagenerallinearcombination(GLC),ofthesamplecovariancematrixandanidentitymatrixinaminimummean-squarederror(MSE)sense.Theshrinkageparameters,whicharerelatedtothe 11

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diagonalloadinglevelsofthebeamformers,canbecalculatedfromthemeasurementsautomaticallywithouttheneedtospecifyanyuserparameters.WehavedemonstratedthattheGLCisveryusefulinthecaseofsmallsamplesizes-thecaseinwhichtheusersofadaptivearraysaremostinterested. Wethenpresentacomprehensivereviewofuserparameter-freerobustadaptivebeamformingalgorithms.WeprovideathoroughevaluationofGLC,itsspecialcaseconvexcombination(CC)method,ridgeregressionCaponbeamformers(RRCB),themid-way(MW)algorithmandseveraliterativeapproachesincludingtheiterativeadaptiveapproach(IAA),themaximumlikelihoodbasedIAA(referredtoasIAA-ML)andthemulti-snapshotsparseBayesianlearning(M-SBL)undervariousscenariossuchascoherent,non-coherentanddistributedsources,steeringvectormismatches,snapshotlimitationsandlowsignal-to-noiseratio(SNR)levels.Furthermore,wediscussthecomputationalcomplexitiesofthealgorithmsandprovideinsightsintowhichalgorithmisthebestchoiceunderwhichcircumstances. Wealsoconsiderapplyingadaptivesignalprocessingtechniquestoultrasoundimaging.Wediscussthechallengesinultrasoundimagingapplicationsincludingthewideband,near-eldenvironmentandlimiteddatasamples.WethenextendGLCandIAAtoaccommodatethoserequirements,whichresultinwidebandGLC(WGLC)andwidebandIAA(WIAA).Bothapproacheshavebeenshowntohavehighresolution,andarerobusttothenitesamplesizeproblemsandothermodelerrors. WethenconsiderDopplerspectrogramanalysis.Weproposeashort-timeiterativeadaptiveapproach(ST-IAA)basedonIAAtoformtheDopplerspectrogram.Duetoitsadaptivecharacter,ST-IAAhasmuchhigherfrequencyresolutionandlowersidelobesthanitsdata-independentcounterpart,i.e.,theconventionalshort-timeFouriertransform(STFT)basedapproach,andthusST-IAAprovidesmuchmoreaccuratespectrograms.Moreover,amodel-orderselectiontool,thegeneralizedinformationcriterion(GIC)canbeusedinconjunctionwithST-IAAtofurtherimprovethespectrogramquality. 12

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Finally,wepresentseveraliterativeadaptivesignalprocessingapproachestoaeroacousticnoiseanalysis.Oneoftheapproachesisbasedonoptimizingthemaximumlikelihood(ML)criterionviausingtheNewton'smethod.Theotherapproaches,referredtoastheFrobeniusnorm(FN)andRank-1methods,employthecyclicoptimizationalgorithmtosolvetheproblem.WealsoderivetheCramer-RaoBounds(CRB)oftheunbiasedsourcepowerestimates.Theproposedmethodsareevaluatedusingbothsimulatedandmeasureddata.ThenumericalexamplesshowthatthesealgorithmssignicantlyoutperformtheexistingleastsquaresapproachandprovideaccuratepowerestimatesevenunderlowSNRconditions.Furthermore,theMSEsoftheso-obtainedestimatesareclosetothecorrespondingCRB,especiallywhenthenumberofdatasamplesislarge.Theexperimentalresultsshowthatthepowerestimatesobtainedbytheproposedapproachesareconsistentwithoneanother. 13

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CHAPTER1INTRODUCTION Signalprocessingdealswiththeproblemofrepresentation,transformationandmanipulationofsignalsandinformationtheycontain[ 1 ].Alltheseoperationsareperformedinasignalprocessingsystem.Inparticular,adaptivesignalprocessing(see,e.g.,[ 2 4 ])adjustsitsoperationadaptivelybasedontheavailabledata.Adaptivesignalprocessinghasbeenextensivelyusedinwideapplicationsincludingradar,sonar,acoustics,communications,imageprocessing,speechprocessing,medicalimagingandotherelds.Insomeoftheseapplications,theobjectiveistoenhanceorrestoresignalsthathavebeendegraded,likeinmobilecommunications.Whileinotherapplications,thegoalistoextractspecicinformationfromthemeasuredsignals,likeinaeroacousticnoiseanalysis. Inthisdissertation,wepresentseveraladaptivesignalprocessingalgorithmsinthecontextofbeamformingthatcanbeusedinbothtypesofapplications.Theproposedapproachesarerobustandmoreimportantly,userparameter-free,i.e.,wedonotneedtospecifyanyuserparameters.Wealsodevelopseveraladaptivesignalprocessingmethodsinwidebandscenarioandapplythemtoultrasoundimagingapplications.Inaddition,weintroduceseveraladaptivesignalprocessingtechniquesforDopplerspectrogramanalysisofthehumangaitandforaeroacousticnoiseanalysis. 1.1AdaptiveBeamforming Beamformingreferstotheprocessofcombiningthemeasurementsfromanarrayofsensors,e.g.,antennas,microphones,withthegoalofestimatingthespatialandtemporalinformationofthesources(e.g.,thenumberofsources,sourcewaveforms,theirspatiallocations)presentinacertainenvironment[ 5 ].Morespecically,itappliesasetofweightsatthearrayoutputtopassthesignal-of-interest(SOI)fromadesiredlocationandsuppressthebackgroundnoiseanddirectionalinterferences[ 6 ]. 14

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Dependingonhowthearrayweightsareselected,beamformerscanbeclassiedaseitherdata-independentordata-dependent.Adata-independentbeamformerselectstheweightsregardlessoftheincomingdatastatistics.Thesimplestyetwidelyuseddata-independentbeamformeristhedelay-and-sum(DAS)approach,wheretheoutputofeachsensoristime-delayedappropriatelypriortosummation.Duetothedata-independentnature,thisclassofbeamformersusuallysufferfromlowresolutionandhighsidelobeproblems[ 5 7 ].Adata-dependentbeamformer,oradaptivebeamformer,adjuststheweightsadaptivelyaccordingtotheincomingdatastatistics.Themostwell-knowndata-dependentbeamformeristhestandardCaponbeamformer(SCB),whereitsweightsaredesignedtominimizethearrayoutputpowersubjecttothelinearconstraintthattheSOIispassedundistorted[ 8 ]. TheSCB[ 8 ]isanoptimalspatiallterthatmaximizesthearrayoutputsignal-to-interference-plus-noiseratio(SINR)underidealconditions,wherethetruecovariancematrixandthesignalsteeringvectorareaccuratelyknown.However,inpractice,noneoftheseassumptionshold.Thetruecovariancematrixisunknownandduetolimitedamountofdata,itsestimatemaybequitepoorandevenill-conditionedinwhichcasetheSCBcannotfunctionatall.Moreover,arraymiscalibrationandinaccuratedatamodelsmaycausenonnegligiblearraysteeringvectorerrors.Infact,limitedsampleerrorsareessentiallyequivalenttosteeringvectorerrors[ 9 ].Whenevertheseerrorsexist,thereisaclearperformancedegradationforSCB. TheseverityoftheperformancedegradationdependsonwhetherthecovariancematrixisestimatedwhiletheSOIispresentindatasamples[ 9 ].Insomeapplications,e.g.,pulseradarapplications,signal-freesamplesareavailableforcovariancematrixestimation.However,thisisnottrueinnumerousapplications,including,e.g.,mobilecommunications,acousticarrayprocessingandmedicalimaging.Whenthesignalispresentinthedatasamples,theimpreciseknowledgeofeithercovariancematrixestimateorSOIsteeringvectorwillresultinSOIcancelationproblem[ 10 ]andthus 15

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signicantlydegradestheperformanceofSCB.Sometimes,theperformanceofSCBmaybecomeworsethanthatofDAS.Therefore,adaptivebeamformingapproachesrobusttotheaforementionedproblemsareneeded. Numerousrobustadaptivebeamformingapproacheshavebeenproposedinrecentdecades(see,e.g.,[ 5 9 11 15 ],andthereferencestherein).Oneofthemostpopularrobustadaptivebeamformingapproachesisdiagonalloading,whereascaledidentitymatrixisaddedtothesamplecovariancematrix[ 12 13 16 ].Themaindrawbackofthismethodisthatthediagonalloadinglevelischosenmanually,orsometimesbasedonthenoiseleveloranormconstraintoftheweightvector(neitherofwhichmaybeeasilydetermined).SeveralrecentrobustCaponbeamformers(RCB)havebeenproposedin[ 14 17 20 ](seealso[ 21 ]).Ithasbeenshownin[ 22 ]thatthesebeamformersareequivalentintermsofSINRandcanberegardedasadiagonalloadingapproach,withthediagonalloadinglevelcalculatedbasedontheuncertaintysetofthearraysteeringvector.However,westillneedtospecifytheparameterrelatedtothesizeoftheuncertaintysetofthearraysteeringvectoranditmaynotbeasimpletasktochoosethisparameterinpractice.Forinstance,itisdifculttodeterminethecorrespondinguncertaintysetparameterofthearraysteeringvectortodealwiththesmallsampleproblems,sincetheamountoferrorscausedbythesmallsamplesizeisdatadependentandhardtospecify. Indeed,mostoftheexistingrobustadaptivebeamformersareuserparameterdependent.Inthisdissertation,weproposeanalgorithmthatcanbeusedtocomputethediagonalloadinglevelfullyautomaticallyfromthegivendatawithouttheneedofspecifyinganyuserparameter[ 23 24 ].Inthisalgorithm,thesamplecovariancematrixusedinSCBisreplacedwithanenhancedcovariancematrixestimatebasedonashrinkagemethod(see[ 25 27 ]andthereferencestherein).Theenhancedcovariancematrixestimatecanbeobtainedbyagenerallinearcombination(GLC)ofthesample 16

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covariancematrixandtheidentitymatrixinaminimummean-squarederror(MSE)sense. Inaddition,weconsiderseveralotheruserparameter-freerobustadaptivebeamformingalgorithmsincludingconvexcombination(CC)method,whichisaspecialcaseofGLC,ridgeregressionCaponbeamformers(RRCB)[ 28 ]andthemid-way(MW)algorithm[ 29 ].RRCBs,whicharebasedonthegeneralizedsidelobecanceler(GSC)formulationofSCB(see,e.g.,[ 5 ])usedifferentridgeregression(RR)techniquessuchasHKB(proposedbyHoerl,KennardandBaldwinin[ 30 ])toimprovetherobustnessofSCB.MW,ontheotherhand,makesuseofthePisarenkoframework[ 31 ]forestimatingthespatialpowerspectrumwherethepowerestimatesalwaysliebetweenthoseobtainedfromSCBandDAS.MWisshowntobemorerobustthanSCBandtohavebetterresolutionthanDAS[ 29 ].Wealsoconsiderseveraliterativealgorithms,namelyiterativeadaptiveapproach(IAA)[ 32 ],maximumlikelihoodbasedIAA(referredtoasIAA-ML)[ 33 ]andmulti-snapshotsparseBayesianlearning(M-SBL)[ 34 35 ].M-SBLemploysaBayesianapproachtogetherwiththeexpectationmaximization(EM)algorithm.IAAiterativelyupdatesthespatialpowerestimatesbasedonaweightedleastsquaresapproachanditisabletoworkwithfewsnapshots(evenonesnapshot).IAA-ML,ontheotherhand,isbasedonthelikelihoodmaximizationprinciples.Weprovideathoroughevaluationoftheaforementioneduserparameter-freebeamformingalgorithmsundervariousscenarios[ 36 ]. 1.2UltrasoundImagingApplication Ultrasoundimagingisinexpensive,fastandradiation-free,whichmakeitverysuitableformedicalapplications.DASisthestandardtechniqueforultrasoundimagingapplications,wherethereceivedsignalsaredelayedappropriatelyaccordingtothelocationsofthetransmitter,receiverandthetargetofinterest,andthenaresummedup.However,thisdata-independentapproachsuffersfromlowresolutionandpoorinterferencesuppressioncapability.Adata-adaptiveapproach,ontheotherhand, 17

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usestheinformationofthedatareceivedbyanarrayofsensorstodetermineasetofweightsthatoptimizethebeamformeroutput,andhencehasthepotentialtoimprovetheimagingqualitysignicantly.Theproblemisthat,inpractice,thereisaclearperformancedegradationformanyadaptivebeamformersincluding,e.g.,SCBinthepresenceofnitesampleerrorsandothermodelerrors(asstatedinSection 1.1 ). MuchworkhasbeendonetoimprovetherobustnessofSCBasmentionedearlierinSection 1.1 .Inultrasoundimagingapplications,onlylimitedstatisticsareavailableforcovariancematrixestimation.Therefore,theaforementionedrobustadaptivebeamformerscanbeusedtoimprovetheoverallimagingquality.However,duetothepropertiesofultrasoundimaging,includingwidebandsignalsandnear-eldenvironment,wecannotapplythosebeamformingtechniquesdirectly,sincetheyaredesignedfornarrowbandsignals. Theconceptsoriginallydevelopedinnarrowbandscenariomaynotbeapplicableforwidebandcase,withoutappropriateextensions[ 14 ].Severalsuchextensionshavebeenproposedforsourcelocalizationapplications.In[ 37 ],focusingmatricesareconstructedtoreducethecovariancematricesatdifferentfrequenciestoasinglematrix,andthistechniqueisfurtherdevelopedin[ 38 ].Theproblemwiththismethodisthepresenceofsourcelocationbiasduetotheerrorsinestimatingthefocussingmatrices.Toavoidthebiasproblem,asteeredcovariancematrix(STCM)methodhasbeenproposedin[ 39 ].ThebasicideaofSTCMistoalignintimethewaveformsreceivedbydifferentsensorsforeachdirectionofinterest,byinsertingaproperdelaytoeachsensoroutput.Anotherapproachisbasedontheconceptofspatialresamplingorinterpolation[ 40 ],wherethesamearraysteeringvectorisobtainedatdifferentfrequenciesbysuitablyinterpolatingthedata.However,spatialresamplingcannotbeusedinultrasoundimagingapplicationsdirectlysinceitisconcernedwithfar-eldproblems. TheideainSTCMisusefultoallowthenarrowbandmethodstobeusedonwidebanddata.Thatis,time-aligningthereceivedsignalstofocusatagiventarget 18

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position,whichissimilartothedelaystepinDAS.Thentheadaptivebeamformerscanbeusedtoestimatethereectedsignalfromthefocalpoint.Followingthisidea,theSCBbeamformeranditsrobustvariationshavebeenappliedtoultrasoundimaginginseveralworks[ 41 44 ].Specically,theconventionaldiagonalloadingalgorithmhasbeenappliedtoultrasoundimagingin[ 42 ],wherethediagonalloadinglevelwassettobeproportionaltothereceivedpower.In[ 44 ],RCBhasbeenapplied,andtheresultsshowedthatRCBcanprovidemuchbetterimagingqualitythanDAS.However,theuncertaintysetparameterusedinRCBmaybehardtodetermineinpractice.Thismotivesustousetheuserparameter-freealgorithms.Inthedissertation,weconsiderseveralmultiple-stagewidebandGLC(WGLC)ultrasoundimagingapproachesbasedontheGLCmethod. Oneofthedisadvantagesofthetime-shiftingalgorithmsisthatthedirectionalinterferencesaremessedupandbehavelikedistributedsources.Theycannolongerbeviewedasrank-onesignals.Therefore,itwillcostmanymoredegreesoffreedom(DOF)tosuppresseachoneofthem.AnalternativewaytoprocessthewidebandsignalsistodividethearrayoutputsintomanynarrowbandfrequencybinsbyusingtheFouriertransform(FT).Thenthenarrowbandbeamformerscanbeappliedtoeachfrequencybin.Inthisdissertation,wepresentawidebandIAA(WIAA)approachbyusingIAAineachnarrowbandfrequencybintoestimatethesignalwaveformandthusdeterminethebackscatteredenergytorecovertheultrasoundimage. 1.3DopplerSpectrogramAnalysisoftheHumanGait TheDopplerspectrogramanalysisofthehumangait[ 45 47 ]isusefulinsomeimportantapplications,e.g.,securityapplications,whereonewouldliketoobservehumansviathroughwallimaging. Radarcanbeusedtoobserveawalkinghumanbehindtheobstacles,suchaswalls.Thereturnedradarsignalfromamovinghumanprovidestheinformation,e.g.,distanceandradialvelocityofthehumanasafunctionoftime.Sincehuman 19

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bodypartsusuallydonotmovewithconstantspeed,Dopplercharacteristicsofthereceivedsignalaretime-varyingandthuscanbeexaminedbytime-frequencyanalysismethods[ 48 49 ].Amongthesemethods,thedata-independentshort-timeFouriertransform(STFT)hasbeenextensivelyused[ 48 ].STFTassumesthatthesignalisstationaryduringashorttimeintervalandthenappliesFTtothatdataintervalforspectralanalysis.Theso-obtainedspectrogram,i.e.,theestimatedsignalpowerdistributioninthetime-frequencyplane,containsthetime-varyingDopplertracksduetothetorsoreturns,aswellastheweakermicroDopplersduetothemotionofthearmsandlegs.However,itiswell-knownthatSTFTfacesadifcultcompromise:ashorteranalysiswindowleadstobettertimeresolutionbutalsotoworsefrequencyresolution,andviceversa.ThistradeoffmakesithardtodiscriminatemicroDopplersintheSTFTspectrogram,especiallywhentheradaroperatingfrequencyisdecreasedtoachievewallpenetration[ 45 ]. Inthedissertation,weproposeanalternativemethodtoimprovetheSTFTspectrogram.InsteadofusingSTFT,anovelshort-timeiterativeadaptiveapproach(ST-IAA)isusedtoformthespectrogram[ 50 ].Duetoitsadaptive(i.e.,data-dependent)character,ST-IAAhasmuchhigherfrequencyresolutionandlowersidelobelevelsthanSTFTandthusST-IAAprovidesmuchmoreaccuratespectrograms.Moreover,amodel-orderselectiontool,thegeneralizedinformationcriterion(GIC)(seee.g.,[ 33 51 ])canbeusedinconjunctionwithST-IAAtofurtherimprovethespectrogramquality. 1.4AeroacousticNoiseAnalysis Aeroacousticnoisesourceanalysiscanbefoundinmanyapplicationsandhasattractedmuchresearchinterestforthepastdecades.Oneoftheimportantapplications,forinstance,isintheefforttoreducetheaircraftnoisesincetheexcessnoisefromairtravelisamajorconcernforcommunitieslivinginthevicinityofairports.Forthispurpose,manyexperimentalstudieshavebeenconductedtoanalyzetheacousticcharacterofanaircraftmodel[ 52 57 ],wheremicrophonemeasurementsare 20

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usedforaeroacousticnoisesourceanalysis.Unfortunately,themeasuredacousticdatafromasinglemicrophoneconsistofnotonlytheaeroacousticnoisesourceofinterestbutalsovariouscontaminationorextraneousnoisesources,e.g.,theelectronicnoise,whichisinevitableinanytestfacility.Therefore,themainchallengefortheaeroacousticnoisesourceanalysisproblemistoproperlyseparatethesourcepowerofinterestfromtheextraneousnoisesources. Torejecttheextraneousnoiseandprovideaccurateestimationofthesourcepowerofinterest,numerousmethodologiesusingmultiplemicrophonemeasurementshavebeenproposed[ 54 57 59 ].Asimpletechnique,thecoherentoutputpower(COP)method,involvesonlytwo-microphonemeasurements[ 54 59 ].However,itspowerestimationisnotaccurate,especiallywhenthesignal-to-noiseratio(SNR)islow[ 60 ].Inthegeneralcaseofmorethanthreemicrophones,aleast-squares(LS)algorithmcanbeusedtosolvetheproblem.However,asdiscussedin[ 60 ],lowSNRconditionsatoneormoremicrophonesmaydegradetheLSperformancesignicantly. Inthisdissertation,wepresentseveraliterativeapproachestosourcepowerestimationbasedonthespectraldensitycovariancematrixofthemicrophoneoutputs.Oneoftheapproachesisbasedonoptimizingthemaximumlikelihood(ML)criterionviausingtheNewton'smethod.Theotherapproaches,referredtoastheFrobeniusnorm(FN)andRank-1methods,employthecyclicoptimizationalgorithm[ 61 ]tosolvetheproblem.Asdemonstratedbyoursimulations,theproposedtechniquesprovideaccuratepowerestimatesforthenoisesourceofinterestevenunderlowSNRconditions. 1.5OutlineofThisDissertation InChapter 2 ,wepresentanapproachtothefullyautomaticcomputationofdiagonalloadinglevels.Inourdiagonalloadingalgorithm,theconventionalsamplecovariancematrixusedintheSCBformulationisreplacedbyanenhancedcovariancematrixestimatebasedonashrinkageapproach. 21

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InChapter 3 ,weprovideacomprehensivereviewofuserparameter-freerobustadaptivebeamformingalgorithms,includingthediagonalloadingapproachesofCC,GLC,MWanditerativeapproachesofIAA,IAA-MLandM-SBL.WepresentathoroughevaluationofthesebeamformingmethodsintermsofSOIpower,theoutputSINRandspatialspectrumestimationaccuraciesundervariousscenarios. InChapter 4 ,wepresentseveralwidebandrobustadaptiveapproachesbasedonGLCandIAA,namely,WGLCandWIAA,forultrasoundimagingapplications. InChapter 5 ,adata-dependentalgorithm,whichwerefertoasST-IAA,ispresentedforDopplerspectrogramanalysistodiscriminatevariousmicroDopplertracksduetothemovementsofdifferenthumanbodyparts. InChapter 6 ,weproposeseveralcovariance-basedapproaches,includingFNmethod,Rank-1method,andMLmethod,foraeroacousticnoisesourceanalysis.WealsoderivetheCramer-RaoBounds(CRB)oftheunbiasedsourcepowerestimates. Finally,wesummarizethedissertationandprovideadiscussiononpotentialfutureresearchdirectionsinChapter 7 1.6Notation Wedenotevectorsandmatricesbyboldfacelowercaseandboldfaceuppercaseletters,respectively.SeeTable 1-1 fornotationweusethroughoutthisdissertation. 22

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Table1-1. Mathematicalnotationusedinthedissertation. NotationExplanation ()Tand()TransposeandcomplexconjugatetransposeofavectorormatrixR)]TJ /F10 7.97 Tf 6.59 0 Td[(1InverseofamatrixRR1=2AHermitiansquarerootoftheamatrixRR0(R0)Risapositivesemidenite(denite)matrixx0(x0)Eachelementofavectorxisnonnegative(positive)diag(x)Adiagonalmatrixwithitsdiagonalformedbytheelementsofavectorxvecd(R)AcolumnvectorformedbythediagonalelementsofamatrixRE()Expectationoperatortr()Traceofamatrixdet()DeterminantofamatrixkkEuclideannormforavectorortheFrobeniusnormforamatrixIIdentitymatrixofappropriatedimensionRe()andIm()Realandimaginarypartsofacomplex-valuedvectorormatrix 23

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CHAPTER2FULLYAUTOMATICCOMPUTATIONOFDIAGONALLOADINGLEVELSBASEDONSHRINKAGE 2.1Introduction Oneofthemostpopularrobustadaptivebeamformingapproachesisdiagonalloading(DL)anditsextendedversions(see,e.g.,[ 12 14 ]).TheproblemwiththemostoftheDLapproachesisthatthereisnoclearguidelineonhowtochoosethediagonalloadinglevelreliablyorhowtoselectuserparametersappropriatelyinpractice.Indeed,simpleuserparameter-freerobustadaptivebeamformersarescarceinliterature.OneexampleistheclassofridgeregressionCaponbeamformers(RRCBs)[ 28 ],whicharebasedonthegeneralizedsidelobecanceler(GSC)[ 5 ]formulationofthestandardCaponbeamformer(SCB).RRCBsbelongtotheclassofdiagonalloadingapproaches,withtheloadinglevelautomaticallycomputedfromtheavailabledata.AmongRRCBs,theHKBmethod(proposedbyHoerl,KennardandBaldwinin[ 30 ])providesaclosed-formsolutionandisrecommendedin[ 28 ].However,weshowbelowthattheHKBalgorithmmayhaveaninherentprobleminchoosinganappropriatediagonalloadinglevel,whichmakesitsusefulnesssomewhatlimited. Inthischapter,weconsideranalternativeandsimpleapproachtothefullyautomaticcomputationofthediagonalloadinglevel.WereplacetheconventionalsamplecovariancematrixusedinSCBbyanenhancedestimateobtainedviaashrinkagemethod(see[ 25 26 ]andthereferencestherein),whichisobtainedbyagenerallinearcombination(GLC)ofthesamplecovariancematrixandtheidentitymatrixinaminimummean-squarederror(MSE)sense. Theremainderofthechapterisorganizedasfollows.Section 2.2 givessomebackgroundonSCB,theconventionalDLmethodsandananalysisoftheproblemofHKB.InSection 2.3 ,wedescribehowtoobtainanenhancedcovariancematrixestimateviaashrinkageapproach.WealsodiscusstheresultingrobustCaponbeamformerandpresentitsdiagonalloadinginterpretation.InSection 2.4 ,we 24

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formulatetheMSEminimizationproblemofGLCasconvexoptimizationproblems.NumericalexamplescomparingtheperformanceoftheGLC-basedbeamformerwiththatofHKB,theconventionalDLmethodsandSCBintermsoftheoutputsignal-to-interference-plus-noiseratio(SINR)andsignal-of-interest(SOI)powerestimationareprovidedinSection 2.5 .Finally,Section 2.6 containstheconclusions. 2.2Background ConsideranarraycomprisingMsensorsandletRdenotethetheoreticalcovariancematrixofthearrayoutputvector.WeassumethatR0hasthefollowingform: R=20a0a0+Q,(2) where20denotesthepoweroftheSOI,a0isthearraysteeringvectoroftheSOIwithka0k2=M,whichisafunctionoftheSOIsourcelocationparameter,e.g.,itsdirection-of-arrival(DOA),andQistheinterference-plus-noisecovariancematrix. 2.2.1SCB TheSCBdeterminesthearrayweightvectorbyminimizingthearrayoutputpowersubjecttotheconstraintthattheSOIispassedundistorted: minwwRwsubjecttowa0=1.(2) Thesolutionto( 2 )is: w0=R)]TJ /F10 7.97 Tf 6.59 0 Td[(1a0 a0R)]TJ /F10 7.97 Tf 6.59 0 Td[(1a0,(2) andanestimateoftheSOIpoweris: w0Rw0=1=(a0R)]TJ /F10 7.97 Tf 6.58 0 Td[(1a0).(2) Theabovesolutionisobtainedunderidealconditions,i.e.,theactualarraysteeringvectorfortheSOI(a0)andtruecovariancematrix(R)areaccuratelyknown.Inwhatfollows,wedemonstratethatthesolutionmaximizestheoutputSINR.Firstconsiderthe 25

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SINRmaximizationproblem: maxwjwa0j2 wQw,(2) thesolutiontowhichisw=Q)]TJ /F10 7.97 Tf 6.59 0 Td[(1a0.isacomplexconstantandcanbechosentomaintainthedistortionlessresponsetowardtheSOIsothatwa0=1,whichyields: w=Q)]TJ /F10 7.97 Tf 6.59 0 Td[(1a0 a0Q)]TJ /F10 7.97 Tf 6.58 0 Td[(1a0.(2) From( 2 )andthematrixinversionlemma(see,e.g.,[ 7 ]),wecanshowthatthesolutionsin( 2 )and( 2 )areequivalent.TheoptimalvalueofSINRcanbeobtainedusingeithersolution: SINRopt=20a0Q)]TJ /F10 7.97 Tf 6.58 0 Td[(1a0.(2) Inpractice,theexactcovariancematrixRisunavailable.Therefore,Rin( 2 )isreplacedbythesamplecovariancematrix^R,where ^R=1 NNXn=1y(n)y(n),(2) withNdenotingthenumberofsnapshotsandy(n)representingthenthsnapshot.Moreover,themismatchbetweenthetrueandassumedsteeringvectors(letussay,a0anda)canexist.TheweightvectorofSCB,whenusing^Randa,isthengivenby: wSCB=^R)]TJ /F10 7.97 Tf 6.59 0 Td[(1a a^R)]TJ /F10 7.97 Tf 6.59 0 Td[(1a,(2) andthecorrespondingSINRandSOIpowerestimateare,respectively, SINR=20jwSCBa0j2 wSCBQwSCB,(2) and ^20=wSCB^RwSCB=1 a^R)]TJ /F10 7.97 Tf 6.59 0 Td[(1a.(2) AsNincreases,^RconvergestoR,andthevalueofthecorrespondingSINRwillapproachtheoptimalone( 2 )eventually.However,whentheSOIispresentin 26

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thesamplecovariancematrix(thecasewhichwewillfocusoninthischapter),theconvergencerateofSCBcanbeveryslow(NMisrequired),andthisisparticularlysoforstrongSOI[ 9 ].Consequently,theperformanceofSCBdegradessubstantiallyinthepresenceofsmallsamplesizeproblems,evenwhenthearraysteeringvectorfortheSOIisexactlyknown.Themismatchbetweena0andaalsosignicantlydeterioratestheperformanceofSCB,especiallywhentheSOIispresentinthesamplecovariancematrix. 2.2.2DL ToimprovetheperformanceofSCBintheaforementionedcases,theDLapproacheshavebeenproposed(see,e.g.,[ 12 13 ]),whichhavethefollowingform: wDL=^R+I)]TJ /F10 7.97 Tf 6.59 0 Td[(1a a^R+I)]TJ /F10 7.97 Tf 6.59 0 Td[(1a,(2) whereisthediagonalloadingleveltobedetermined.Onewaytodetermineistosetittoaxedvalueaccordingtothenoisepower,e.g.,0dBwithrespecttothenoiseoor.WerefertothisapproachasxedDL(FDL).Alternatively,thediagonalloadingformin( 2 )canbeobtainedbyimposinganadditionalconstraintontheEuclideannormoftheweightvectorwintheSCBformulation.WerefertothisapproachasthenormconstrainedCaponbeamforming(NCCB)[ 12 13 ]: minww^Rwsubjecttowa=1, (2) kwk2, (2) whereisauserparameterconstrainingthemaximumvalueofthesquareoftheweightvectornorm.Thelargerthe,thecloserNCCBistoSCB.Notethatwhenthenormconstraintin( 2 )isinactive,NCCBisequivalenttoSCB,andthecorrespondingdiagonalloadinglevelis=0.Otherwise,canbecalculatedbasedonthefollowing 27

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equation[ 12 62 ]: a^R+I)]TJ /F10 7.97 Tf 6.58 0 Td[(2a a^R+I)]TJ /F10 7.97 Tf 6.59 0 Td[(1a2=.(2) TheproblemwithFDLandNCCBisthatitisnotclearhowtomakethebestchoicesoftheuserparametersand. 2.2.3HKB HKBdistinguishesitselffromtheotherrobustadaptivebeamformersduetothefactthatitisaparameter-freerobustbeamformer[ 28 ].However,asweexplainbelow,thismethodappearstohavesomedeciencies. TheformulationofHKBisbasedontheGSC[ 5 ]reparameterizationofSCB: w=a M)]TJ /F9 11.955 Tf 11.96 0 Td[(B,(2) whereBisanM(M)]TJ /F3 11.955 Tf 12.08 0 Td[(1)semi-unitarymatrixthatisorthogonaltoa,i.e.,Ba=0andBB=I.ThentheSCBproblem( 2 )becomes(byusing^Randa): minB)]TJ /F9 11.955 Tf 15.76 8.09 Td[(a M^RB)]TJ /F9 11.955 Tf 15.76 8.09 Td[(a M=min^R1=2B)]TJ /F3 11.955 Tf 13.02 2.66 Td[(^R1=2a M2(2) LetX=^R1=2Bandb=^R1=2a M.Theaboveequationcanbeinterpretedasaleast-squares(LS)solutiontothefollowinglinearregressionproblem: b=X+e,(2) whereeissomeresidualvector.Tosolvethisproblem,SCBusesthestandardLSestimatortoobtainthesolutionLS: LS=(XX))]TJ /F10 7.97 Tf 6.59 0 Td[(1Xb.(2) Ontheotherhand,theridgeregression-basedHKBmethodprovidesasolutionforas: HKB=(XX+HKBI))]TJ /F10 7.97 Tf 6.59 0 Td[(1Xb,(2) 28

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where HKB=(M)]TJ /F3 11.955 Tf 11.95 0 Td[(1)^2LS kLSk2,(2) with ^2LS=kXLS)]TJ /F9 11.955 Tf 11.95 0 Td[(bk2.(2) ItturnsoutthatHKBisinfactadiagonalloadingapproach(seethederivationsin[ 28 ]): wHKB=(^R+HKBI))]TJ /F10 7.97 Tf 6.59 0 Td[(1a a(^R+HKBI))]TJ /F10 7.97 Tf 6.58 0 Td[(1a,(2) whereHKBisthediagonalloadinglevel,whichiscomputedfullyautomatically. SomecommentsontheHKBin( 2 ),intheabsenceofsteeringvectorerrors,arenowinorder: (1) Thenumeratorof( 2 )containsthetermkXLS)]TJ /F9 11.955 Tf 11.96 0 Td[(bk2,whichistheSOIpowerestimateobtainedviaSCB(see( 2 )).ThisSOIpowerestimatetendstoincreaseasN!1since,forniteN,SCBtendstosuppresstheSOI. (2) Thedenominatorof( 2 )isequaltokLSk2.AsN!1,kLSk2tendstodecrease.Indeed,forniteN,likethenormofwSCB,kLSk2tendstoberatherlarge(see[ 5 ]). (3) Basedontheaforementionedtwofacts,HKBin( 2 )maybeverylargeforarelativelylargeN.YetintheabsenceofsteeringvectorerrorsandasN!1,SCBtendstoworkoptimallyanddiagonalloadingshouldnolongerbeneeded.ThisanalysissuggeststhatHKB'sperformanceshoulddegradewhenNisrelativelylarge,whichisconrmedbythenumericalexamplesinSection 2.5 2.3GLC-BasedRobustCaponBeamforming WeprovidebelowtheGLC-basedcovariancematrixestimateandthendescribetheresultingbeamformerbasedontheestimate.Thediagonalloadinginterpretationofthebeamformerisgivenattheendofthesection. 2.3.1GLC-BasedCovarianceMatrixEstimation Thesamplecovariancematrix^RisapoorestimateofthetruecovariancematrixRwhenthesamplesizeNissmallrelativetothearraydimensionM.Toalleviatethisproblem,intheGLC-basedcovariancematrixestimation,whichisashrinkagemethod 29

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[ 25 ],weconsideragenerallinearcombination(GLC)ofthesamplecovariancematrix^RandtheidentitymatrixItoobtainpreferablyamoreaccurateestimateofRthan^R: ~R=I+^R,(2) where~R,whichshouldsatisfy~R0,istheenhancedestimateofR.andin( 2 )aretheshrinkageparameterswhicharechosenbyminimizing(anestimateof)theMSEof~R,whereMSE(~R)=Efk~R)]TJ /F9 11.955 Tf 12.6 0 Td[(Rk2g,assuggestedin[ 25 ],andalsoin[ 63 ]forcomplex-valueddatacase.Notethattheconstraints0,0canbeimposedtoguaranteethat~R0. Inwhatfollows,weconsidertheMSEminimizationproblemforGLCandcalculatetheshrinkageparameters.From( 2 ): MSE(~R)=EfkI)]TJ /F3 11.955 Tf 11.96 0 Td[((1)]TJ /F4 11.955 Tf 11.96 0 Td[()R+(^R)]TJ /F9 11.955 Tf 11.95 0 Td[(R)k2g (2) =kI)]TJ /F3 11.955 Tf 11.95 0 Td[((1)]TJ /F4 11.955 Tf 11.95 0 Td[()Rk2+2Efk^R)]TJ /F9 11.955 Tf 11.96 0 Td[(Rk2g=2M)]TJ /F3 11.955 Tf 11.96 0 Td[(2(1)]TJ /F4 11.955 Tf 11.96 0 Td[()tr(R)+(1)]TJ /F4 11.955 Tf 11.96 0 Td[()2kRk2+2Efk^R)]TJ /F9 11.955 Tf 11.95 0 Td[(Rk2g, (2) Consequently,the(unconstrained)optimalvaluesforandcanbeobtainedas: 0= +.(2) 0=(1)]TJ /F4 11.955 Tf 11.96 0 Td[(0)= +,(2) where,Efk^R)]TJ /F9 11.955 Tf 11.23 0 Td[(Rk2g,=tr(R) Mand=kI)]TJ /F9 11.955 Tf 11.22 0 Td[(Rk2.Wenotethat02[0,1]and00.Notealsothat0and0dependontheunknowncovariancematrixR.Toestimate0and0fromthegivendata,weneedanestimateof,whichcanbecalculatedas[ 63 ](tobeself-contained,thedetailsforthederivationarealsopresentedintheAppendix A ). ^=1 N2NXn=1ky(n)k4)]TJ /F3 11.955 Tf 14.71 8.09 Td[(1 Nk^Rk2.(2) 30

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Inaddition,wehave +=kI)]TJ /F9 11.955 Tf 11.95 0 Td[(Rk2+Efk^R)]TJ /F9 11.955 Tf 11.95 0 Td[(Rk2g=Efk^R)]TJ /F4 11.955 Tf 11.96 0 Td[(Ik2g,(2) anestimateofwhichisgivenbyk^R)]TJ /F3 11.955 Tf 12.77 0 Td[(^Ik2,where^canbeobtainedbyreplacingRbyitsunbiasedestimate^R.Consequently,wecangetanestimateof0as: ^^ k^R)]TJ /F3 11.955 Tf 12.41 0 Td[(^Ik2,(2) andof0as: 1)]TJ /F3 11.955 Tf 36.19 8.08 Td[(^ k^R)]TJ /F3 11.955 Tf 12.41 0 Td[(^Ik2.(2) Theestimateof0in( 2 )maybenegative.Toguaranteethattheestimateof0isnonnegative,assuggestedin[ 25 ],wecanusethealternativeestimateof0: ^0=min^^ k^R)]TJ /F3 11.955 Tf 12.41 0 Td[(^Ik2,^,(2) andof0: ^0=1)]TJ /F3 11.955 Tf 14.1 8.09 Td[(^0 ^.(2) 2.3.2GLC-BasedRobustCaponBeamformer Wenowhaveadiagonally-loadedenhancedestimateofthecovariancematrixasfollows: ~RGLC=^0I+^0^R,(2) where^0and^0areobtainedfrom( 2 )and( 2 ),respectively.Usingtheenhancedestimate~RGLCinlieuof^RintheSCBformulationyieldsthefollowingGLC-basedrobustadaptivebeamformer: wGLC=~R)]TJ /F10 7.97 Tf 6.59 -.01 Td[(1GLCa a~R)]TJ /F10 7.97 Tf 6.59 0 Td[(1GLCa.(2) TheoutputSINRoftheresultingbeamformerisgivenby: SINR=20jwGLCa0j2 wGLCQwGLC,(2) 31

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andtheSOIpowerestimateis: ^20=wGLC~RGLCwGLC=1 a~RGLCa.(2) Rewriting( 2 )asfollows(assumingthat^06=0): ~wGLC=h^0 ^0I+^Ri)]TJ /F10 7.97 Tf 6.59 0 Td[(1a ah^0 ^0I+^Ri)]TJ /F10 7.97 Tf 6.59 0 Td[(1a,(2) weobservethattheGLC-basedrobustadaptivebeamformerisadiagonalloadingapproachwiththediagonalloadingleveldeterminedeasilyandautomaticallyfromtheobserveddatasnapshotsfy(n)gNn=1. 2.4ConvexFormulationsofGLC TheGLCapproach(see( 2 )-( 2 ))couldbesuboptimalsincetheconstraints0,0wereimposedaposterior.Furthermore,itisthecondition~R0thatmaybeofrealinterestinpracticalapplications.BelowtheMSEminimizationproblemforGLC(see( 2 ))isformulatedasconvexoptimizationproblems,wherealltheaforementionedconstraintscanbeimposeddirectlyandthegloballyoptimalsolutionscanbedeterminedefciently. Werewrite( 2 )as: MSE(~R)=2M+2tr(R)+2kRk2+2)]TJ /F3 11.955 Tf 11.95 0 Td[(2tr(R))]TJ /F3 11.955 Tf 11.96 0 Td[(2kRk2+const,(2) wherehasbeendenedinSection 2.3.1 .Let=[]T.Then,thetermof( 2 )thatisalinearfunctionofcanbewrittenmorecompactlyas)]TJ /F3 11.955 Tf 9.3 0 Td[(2bT,where b=tr(R)kRk2T,(2) andthequadratictermof( 2 )canbewrittenas: kI+Rk2+2=TA,(2) 32

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where A=264Mtr(R)tr(R)kRk2+375.(2) Hence, MSE(~R)=TA)]TJ /F3 11.955 Tf 11.96 0 Td[(2bT+const.(2) NotethatA0.Thesimplestwaytoprovethisfollowsfrom( 2 )sinceTA>0,86=0.InviewofA0,( 2 )hasaunique(unconstrained)minimumsolutiongivenby: 0=[00]T=A)]TJ /F10 7.97 Tf 6.59 0 Td[(1b,(2) whichisinfactequivalenttotheoptimalsolutionin( 2 )and( 2 ).Next,werewrite( 2 )as: )]TJ /F9 11.955 Tf 11.96 0 Td[(A)]TJ /F10 7.97 Tf 6.59 0 Td[(1bTA)]TJ /F9 11.955 Tf 11.95 0 Td[(A)]TJ /F10 7.97 Tf 6.59 0 Td[(1b+const.(2) UsingthestandardtoolofSchurcomplements,theMSEminimizationproblemforGLCundertheconstraint~R0canbeformulatedasthefollowingsemideniteprogram(SDP),whichcanbesolvedinpolynomialtimeusingpublicdomainsoftware[ 64 66 ]: min,subjectto264h)]TJ /F3 11.955 Tf 13.2 2.65 Td[(^A)]TJ /F10 7.97 Tf 6.58 0 Td[(1^biTh)]TJ /F3 11.955 Tf 13.2 2.66 Td[(^A)]TJ /F10 7.97 Tf 6.58 0 Td[(1^bi^A)]TJ /F10 7.97 Tf 6.59 0 Td[(13750~R()0. (2) NotethatAandbarereplacedbytheirestimates^Aand^b,respectivelyin( 2 ).Toobtaintheestimates,notingthat +kRk2=Efk^Rk2g,(2) sowecanestimate+kRk2inAbyk^Rk2,andestimatekRk2inbbyk^Rk2)]TJ /F3 11.955 Tf 13.6 0 Td[(^.WealsoreplaceRby^Rintr(R).Consequently,wecanobtainestimates^(1)0and^(1)0from 33

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( 2 )forGLC,whichwerefertoasGLC-1.GLC-1couldbedifferentfromGLC,duetoGLC-1enforcing~R0directlywhileminimizing( 2 )(withAandbreplacedby^Aand^b).GLC,ontheotherhand,minimizes( 2 )(withAandbreplacedbythesame^Aand^b)withoutimposinganyconstraints,andthenclipthesolutions(ifnecessary)tosatisfy^00and^00(see( 2 )-( 2 )).NotethatthefullyoptimalversionofGLCistheGLC-2discussedbelow. Insteadofenforcing~R()0,wecanusetheconstraints0and0in( 2 )toobtainaconvexquadraticprogram(QP)forGLC: min)]TJ /F3 11.955 Tf 13.2 2.65 Td[(^A)]TJ /F10 7.97 Tf 6.59 0 Td[(1^bT^A)]TJ /F3 11.955 Tf 13.2 2.65 Td[(^A)]TJ /F10 7.97 Tf 6.58 0 Td[(1^bsubjecttoi0,i=1,2, (2) whereidenotestheithelementof.When^Aand^busedin( 2 )areobtainedinthesamewayasinGLC-1,wegetGLC-2,whichisanoptimizedversionofGLCandcouldbedifferentfrombothGLCandGLC-1. InallofthenumericalexamplespresentedinSection 2.5 ,weobservethattheGLCsolutionsdidnotneedclipping,andhenceGLC,GLC-1,andGLC-2providedthesamesolution.Consequently,forthesakeofconciseness,wepresentonlytheresultsobtainedwithGLCinSection 2.5 2.5NumericalExamples WepresentbelowseveralnumericalexamplescomparingtheperformanceoftheGLC-basedrobustadaptivebeamformerwiththatofHKB,FDL,NCCBandSCB.ForFDL,wesetthexeddiagonalloadingleveltothenoisepower(weassumethatthenoisepowerisperfectlyknownforFDL).ForNCCB,wesetthenormconstraintparameterequalto4=M.(Notethatwhen=1=M,NCCBreducestothedelay-and-sumbeamformer.)Inallexamples,weassumeauniformlineararraywithM=10sensorsandhalf-wavelengthinter-elementspacing.Thefar-eldnarrowbandsourcewaveformsandtheadditivenoiseareassumedtobetemporallywhitecircularlysymmetriccomplex 34

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Gaussianrandomprocesseswithzero-meanandacertainvariance.ThenoiseisfurtherassumedtobespatiallywhitewhosecovariancematrixistheidentitymatrixI.Threesourceswithpowers10dB,20dBand20dBareassumedtobepresentat0,20and60,respectively.WeconsidertherstsourceastheSOIandtheothertwosourcesasinterferences.Foreachscenario,1000Monte-Carlotrialsareperformed. First,weexaminetheperformanceofthebeamformersasthesnapshotnumberNincreaseswhentheperfectknowledgeofthearraysteeringvectorfortheSOIisavailable.Fig. 2-1 showsthemeanoftheoutputSINRsversusthenumberofsnapshotsN.Forreference,theoptimalSINR(obtainedusingthetrueR)isalsoincluded.Asshowninthegure,SCBconvergestotheoptimalSINRvalueveryslowly,since^RcontainstheSOIandtheSOIpowerisnotsmall.GLCconvergestotheoptimalvaluefasterthanalltheotheralgorithms.FDLgivesthesecondbestperformance.AlthoughNCCBshowsgoodperformanceforsmallN,itconvergestoSCBforrelativelylargeN,sincethenormconstraintin( 2 )tendstobeinactiveasNincreases.Figs. 2-2 (A)and 2-2 (B)showthemeanvaluesofthediagonalloadinglevelsofGLCandNCCB,respectively,asafunctionofthesnapshotnumberN.WeobservethatbothcurvesgodownasNincreases.However,thediagonalloadinglevelofGLCismuchlargerthanthatofNCCBforsmallNanddecreasesmoreslowlyasNbecomeslarger.AnotherobservationfromFig. 2-1 isthattheoutputSINRofHKBdecreaseswhenNisbeyondacertainnumber(about100inthisexample).AsshowninFig. 2-2 (C),themeanofHKBstartsfromaverysmallvalueandincreasestoalargevalueforlargeN.ThisbehaviorlimitsHKB'sperformanceimprovementoverSCBwhenNissmallanddegradesitsperformancewhenNislarge.Inconclusion,theautomaticallydetermineddiagonalloadinglevelofGLCadaptsthebesttotheavailabledata. Next,weexaminetherobustnessofthebeamformerstosmallsamplesizeproblemsandtosteeringvectorerrors.Fig. 2-3 -Fig. 2-5 showthebeamformeroutputSINRandSOIpowerestimatesversusthesnapshotnumberN(forNupto50)for 35

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variousscenarios.TheperformanceintheabsenceofsteeringvectorerrorsisshowninFig. 2-3 (Fig. 2-3 (A)isazoomed-inversionofFig. 2-1 ).WenotethatGLCoutperformsalltheotheralgorithms,especiallywhenthesamplesizeissmall.Moreover,HKBandSCBarenotapplicablewhen^Risrankdecient(N
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2.6Conclusions WehaveconsideredaGLC-basedrobustadaptivebeamformer,wheretheconventionalsamplecovariancematrix^RusedintheSCBformulationisreplacedbyanenhancedcovariancematrixestimate~Rbasedonashrinkagemethod.GLCisaDLapproachandonecanefcientlyobtaintheenhancedcovariancematrixestimatefullyautomatically,i.e.,withoutspecifyinganyuserparameter.SeveralnumericalexampleshavebeenusedtocomparetheperformanceofGLCwiththoseofSCB,FDL,NCCBandHKB.WehavedemonstratedthatGLCisveryusefulinthecaseofsmallsamplesizes-thecaseinwhichtheusersofadaptivearraysaremostinterested. 37

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Figure2-1. BeamformeroutputSINRversusthesnapshotnumberNintheabsenceofarraysteeringvectorerrors.TheSOIisat0with10dBpowerandthetwointerferenceseachwith20dBpowerarelocatedat20and60. 38

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AB C Figure2-2. Comparisonofaveragediagonalloadinglevels.(A)DiagonalloadinglevelofGLC,i.e.,^0=^0,versusthesnapshotnumberN.(B)DiagonalloadinglevelofNCCB,i.e.,,versusthesnapshotnumberN.(C)DiagonalloadinglevelofHKB,i.e.,HKB,versusthesnapshotnumberN. 39

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A B Figure2-3. Performancecomparisonintheabsenceofsteeringvectorerrors.(A)SINRversusN.(B)SOIpowerestimatesversusN.TheSOIisat0with10dBpowerandthetwointerferenceseachwith20dBpowerarelocatedat20and60. 40

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A B Figure2-4. Performancecomparisoninthepresenceofa1steeringangleerror.(A)SINRversusN.(B)SOIpowerestimatesversusN.TheSOIisat0(whileassumedtobeat1)with10dBpowerandthetwointerferenceseachwith20dBpowerarelocatedat20and60. 41

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A B Figure2-5. Performancecomparisoninthepresenceofarraycalibrationerrors.(A)SINRversusN.(B)SOIpowerestimatesversusN.TheSOIisat0with10dBpowerandthetwointerferenceseachwith20dBpowerarelocatedat20and60.ThearraysteeringvectorsofboththeSOIandtheinterferershavebeenperturbedtosimulatethecalibrationerrors. 42

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CHAPTER3PERFORMANCECOMPARISONOFUSERPARAMETER-FREEROBUSTADAPTIVEBEAMFORMERS 3.1Introduction Thischapterprovidesacomprehensivereviewofuserparameter-freerobustadaptivebeamformingalgorithms.WepresentridgeregressionCaponbeamformers(RRCB)[ 28 ],themid-way(MW)algorithm[ 29 ],andtheconvexcombination(CC)aswellasthegenerallinearcombination(GLC)approaches[ 23 24 ].ThepurposeofthesemethodsistomitigatetheeffectofsmallsamplesizeandsteeringvectorerrorsonthestandardCaponbeamformer(SCB).Wealsopresentseveraliterativebeamformingalgorithms,namelytheiterativeadaptiveapproach(IAA)[ 32 ],maximumlikelihoodbasedIAA(referredtoasIAA-ML)[ 33 ]andmulti-snapshotsparseBayesianlearning(M-SBL)[ 34 ].Weprovideathoroughevaluationofthesebeamformingmethodsintermsofpowerandspatialspectrumestimationaccuracies,outputsignal-to-interference-plus-noiseratio(SINR)andresolutionundervariousscenariosincludingcoherent,non-coherentanddistributedsources,steeringvectormismatches,snapshotlimitationsandlowsignal-to-noiseratio(SNR)levels.Furthermore,wediscussthecomputationalcomplexitiesofthealgorithmsandprovideinsightsintowhichalgorithmisthebestchoiceunderwhichcircumstances. Theremainderofthischapterisorganizedasfollows.First,inSection 3.2 ,thedatamodelusedinarrayprocessingisintroducedandthebasicbeamformingalgorithms,namelyDASandSCB,arepresented.NextinSection 3.3 ,RRCB,CC,GLCandMWarepresented.IAA,IAA-MLandM-SBLarepresentedinSection 3.4 andextensivenumericalexamplesareprovidedinSection 3.5 .Finally,thechapterisconcludedinSection 3.6 3.2DataModelandProblemFormulation ConsiderthewaveeldgeneratedbyKfar-eldnarrowbandsourceslocatedatwhere,[12...K]Tandkistheimpingingangleofthekthsignal,k=1,...,K. 43

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Inthemulti-snapshotcase,theM1arrayoutputvectorofanMelementarrayinthepresenceofadditivenoisecanberepresentedas[ 5 7 ]: y(n)=A()s(n)+e(n),n=1,...,N,(3) whereNisthenumberofsnapshots,A()istheMKsteeringmatrixdenedasA(),[a1a2...aK]ands(n),[s1(n)s2(n)...sK(n)]T,n=1,...,N,isthesourcewaveformvector.Thearraysteeringvectorhasdifferentexpressionsdependingonthearraygeometryandonwhetherthesourceisinthenear-eldorfar-eldofthearray.Forinstance,inthefar-eldcasetheassumedsteeringvectorcorrespondingtothekthsourceforalineararrayis, ak=he)]TJ /F8 7.97 Tf 6.59 0 Td[(j2f c0x1sin(k),...,e)]TJ /F8 7.97 Tf 6.58 0 Td[(j2f c0xMsin(k)iT,(3) wherefisthecenterfrequency,c0isthewavepropagationvelocityandxmisthepositionofthemthsensor,m=1,...,M(seeFig. 3-1 ). Figure3-1. Alineararray. IfoneofthesourcesisconsideredastheSOI,thearrayoutputcanbeexpressedas(see( 3 )): y(n)=a0s0(n)+i(n)+e(n),(3) 44

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wherea0isthearraysteeringvectoroftheSOIwithka0k2=M,s0(n)isthesignalwaveform,andi(n)ande(n)aretheinterferenceandnoisecomponents,respectively(notethata0canhaveadifferentformthantheassumedonein( 3 )duetomodelerrors).Therefore,undertheassumptionthattheinterferenceplusnoisetermandtheSOIareuncorrelated,thecovariancematrixoffy(n)gNn=1canbewrittenasfollows: R=P0a0a0+Q,(3) whereP0denotesthepoweroftheSOI,andQ=Ef(i(n)+e(n))(i(n)+e(n))gistheinterference-plus-noisecovariancematrix.Inpractice,Randa0areunavailableandhenceareusuallyreplacedbythesamplecovariancematrix^Randtheassumedsignalsteeringvectora(forinstance,( 3 )forafar-eldlineararray),respectively,where ^R=1 NNXn=1y(n)y(n).(3) AbeamformingalgorithmaimstodesignacomplexvectorwinthebestwaytocanceltheinterferencesandnoisewhilekeepingtheSOIundistorted.Accordingly,thesignalwaveformandpowerareestimatedby^s0(n)=wy(n)and^P0=wRw,respectively.Inaddition,aspatialpowerspectrumofthesourcespresentinaregioncanbeobtainedbyestimating^P0overthesetofallanglesofinterest.AnimportantmeasureforthebeamformerperformanceistheSINRwhichisdenedas,see,e.g.,[ 5 14 ]: SINR=P0jwa0j2 wQw.(3) 3.2.1DAS TheclassicalDASbeamformer,whichisnothingbutaspatialmatchedlter,selectstheweightsas: wDAS=a M.(3) Duetoitsdata-independentproperty,DASmaysufferfromlowerresolutionandworseinterferencesuppressioncapabilitythanthedata-adaptivemethods. 45

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3.2.2SCB Themostwell-knownadaptivebeamformingtechniqueSCB,asdescribedinChapter 2 ,determinesthearrayweightvectorbyminimizingthearrayoutputpowersubjecttotheconstraintthattheSOIispassedundistorted.Underidealconditions,i.e.,whenthetrueRanda0areknown,itsweightvectorisgivenby wopt=R)]TJ /F10 7.97 Tf 6.58 0 Td[(1a0 a0R)]TJ /F10 7.97 Tf 6.59 0 Td[(1a0,(3) whichmaximizesthearrayoutputSINRdenedin( 3 )andthecorrespondingoptimalvalueis(seederivationsinSection 2.2 ): SINRopt=P0a0Q)]TJ /F10 7.97 Tf 6.59 0 Td[(1a0.(3) Inpractice,( 3 )isreplacedby wSCB=^R)]TJ /F10 7.97 Tf 6.59 0 Td[(1a a^R)]TJ /F10 7.97 Tf 6.59 0 Td[(1a.(3) Inthiscase,SCBisnolongeroptimumanditisverysensitivetomodelerrorsincludinginaccuratecovariancematrixestimates(especiallywhenthesamplesizeissmall)andsteeringvectorerrors.Theuserparameter-freerobustadaptivebeamformerspresentedinthenexttwosectionsaredesignedtomitigatetheseproblems. 3.3DiagonalLoadingApproaches Theweightvectorusedbythediagonalloadingapproachisgivenby: wDL=(^R+I))]TJ /F10 7.97 Tf 6.58 0 Td[(1a a(^R+I))]TJ /F10 7.97 Tf 6.59 0 Td[(1a,(3) wherethediagonalloadinglevelischosenmanually,orsometimesbasedonthenoiselevelorwhitenoisegainconstraint[ 13 17 ].Insomemoreadvanceddiagonalloadingapproaches,e.g.,in[ 14 17 18 20 22 ],thediagonalloadinglevelisdeterminedbasedontheuncertaintysetofthesignalsteeringvector.However,theseapproachesstillrequireuserparameterswhichmaybehardtodetermineinpractice.Inthefollowing,we 46

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presentthreeuserparameter-freebeamformingalgorithmswhichbelongtotheclassofdiagonalloadingapproaches. 3.3.1RRCB RRCBisbasedontheGSCreparameterizationoftheSCBformulation(see( 2 )).Asaresult,SCBbecomesalinearregressionproblem(see( 2 )and( 2 )).WhileSCBgivesthestandardleast-squares(LS)solution(see( 2 )),RRCBappliesridgeregressiontechniquestomitigatetheill-conditioningproblemsofSCB[ 28 ].Oneofthesetechniques,theHKBregularization[ 30 ](asintroducedinChapter 2 ),providesaclosed-formsolution(see( 2 ))byusingTikhonovregularization[ 67 ]. HKBhasadiagonalloadingformas: wHKB=(^R+HKBI))]TJ /F10 7.97 Tf 6.59 0 Td[(1a a(^R+HKBI))]TJ /F10 7.97 Tf 6.58 0 Td[(1a,(3) withitsdiagonalloadinglevelHKBdeterminedautomaticallyfromthegivendata(see( 2 )). Asshownin[ 23 ]andChapter 2 ,intheabsenceofsteeringvectorerrors,aninherentproblemofHKBisthatHKBmaybecomeverylargeasNincreases,whichdegradesitsperformance.Infact,whenHKBissufcientlylargesothatHKBIisthedominanttermin( 3 ),HKBbehavesmorelikeDAS,see( 3 ). 3.3.2ShrinkageBasedRobustCaponBeamforming Tocombatthesmallsamplesizeproblemintroducedbythesamplecovariancematrix^R,aclassofshrinkage-basedrobustCaponbeamformershavebeenpresentedin[ 23 24 ],wherethecovariancematrixestimate^RusedinSCBisreplacedbyanenhancedestimatebasedonashrinkagemethod;thisenhancedestimatecanbeobtainedbylinearlycombining^Randanidentitymatrixinaminimummean-squarederror(MSE)sense[ 25 ].Weconsidertwolinearcombinationshere,i.e.,agenerallinearcombination(GLC)aspresentedinChapter 2 : ~R=I+^R,(3) 47

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anditsspecialcaseconvexcombination(CC): ~R=I+(1)]TJ /F4 11.955 Tf 11.95 0 Td[()^R,(3) where~R,whichshouldsatisfy~R0,istheenhancedestimateofR.andin( 3 )and( 3 )aretheshrinkageparameterswhicharechosenbyminimizing(anestimateof)theMSEof~R,whereMSE(~R)=Efk~R)]TJ /F9 11.955 Tf 11.96 0 Td[(Rk2g. TheMSEminimizationproblemforGLCisconsideredrst.Toguaranteethat~R0,werstgettheunconstrainedsolutionsforand,andthenenforcethemtobenonnegative1.SincethedetailshavebeengiveninSection 2.3 ,onlythemainresultsareprovidedhere.MinimizingMSE(~R)givestheoptimalvaluesforandas(seeSection 2.3 ): 0= +,(3) 0=(1)]TJ /F4 11.955 Tf 11.96 0 Td[(0)= +,(3) where=Efk^R)]TJ /F9 11.955 Tf 10.86 0 Td[(Rk2g,=tr(R) M,and=kI)]TJ /F9 11.955 Tf 10.86 0 Td[(Rk2.Thenwecangetthenonnegativitiyenforcedestimatesof0and0fromthegivendataasfollows(seeSection 2.3 ): ^0=min^^ k^R)]TJ /F3 11.955 Tf 12.41 0 Td[(^Ik2,^,(3) ^0=1)]TJ /F3 11.955 Tf 14.1 8.09 Td[(^0 ^,(3) where^=1 N2PNn=1ky(n)k4)]TJ /F10 7.97 Tf 14.22 4.71 Td[(1 Nk^Rk2and^=tr(^R) M. Togettheshrinkageparameterestimate^0forCC,wenotethat0and0canberewrittenas0=0and0=1)]TJ /F4 11.955 Tf 12.25 0 Td[(0(see( 3 )and( 3 ),0= +),whichimplies 1NotethattheMSEminimizationproblemforGLCcanalsobeformulatedasconvexoptimizationproblems(seeSection 2.4 fordetails)wheretheconstraint~R0or,0canbereadilyincorporatedintotheconvexformulation.Inallofournumericalexamples,theconvexformulationand( 3 )-( 3 )gaveidenticalresultsastheconstraintswerefoundtobeinactive(notehoweverthattheycanbeactivedependingontheapplicationscenario,inwhichcasetheresultscouldbedifferent). 48

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thatGLCreducestoCCwhen=1.Therefore,setting^=1,wecanobtain^0from( 3 )forCC.Since=1isgenerallynottrueinpractice,GLCandCCarebasicallydifferent. Consequently,from( 3 )and( 3 )togetherwith( 3 )and( 3 ),wecanobtaintheenhancedestimatesofthecovariancematrix~RGLCand~RCC.Usingthemtoreplace^RintheSCBformulation(see( 3 ))yieldsthefollowingshrinkage-basedrobustadaptivebeamformers: wGLC=~R)]TJ /F10 7.97 Tf 6.59 0 Td[(1GLCa a~R)]TJ /F10 7.97 Tf 6.59 0 Td[(1GLCa=^0 ^0I+^R)]TJ /F10 7.97 Tf 6.59 0 Td[(1a a^0 ^0I+^R)]TJ /F10 7.97 Tf 6.59 0 Td[(1a,(3) and wCC=~R)]TJ /F10 7.97 Tf 6.58 0 Td[(1CCa a~R)]TJ /F10 7.97 Tf 6.59 0 Td[(1CCa=^0 1)]TJ /F10 7.97 Tf 7.32 0 Td[(^0I+^R)]TJ /F10 7.97 Tf 6.58 0 Td[(1a a^0 1)]TJ /F10 7.97 Tf 7.32 0 Td[(^0I+^R)]TJ /F10 7.97 Tf 6.58 0 Td[(1a.(3) Weobservethattheshrinkage-basedrobustadaptivebeamformersareinfactdiagonalloadingapproacheswiththeloadinglevelsdeterminedfromthemeasurementsfullyautomatically.Finally,theSOIpowerestimateforGLC(CC)isgivenbywGLC~RGLCwGLC(wCC~RCCwCC). 3.3.3MW MWpowerestimationmethodisdevelopedbasedonthePisarenkoframework[ 31 ],originallydevisedfortemporalpowerspectrumestimation,whichyieldsthefollowingclassofpowerestimates[ 29 ]: ^P0(r)=8><>:1 Ma^Rra M1=r,forr6=0,1 Mexpalog(^R)a M,forr=0.(3) TodealwiththeproblemsofDASandSCB,MWusesr=0in( 3 ),whichtakesamid-waypositionbetweenr=)]TJ /F3 11.955 Tf 9.29 0 Td[(1(correspondingtothepowerestimateofSCB)andr=1(correspondingtothepowerestimateofDAS).ThepowerestimateofMWcanbe 49

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expressedas: ^P0,MW=1 Mexp alog(^R)a M!.(3) Notethattakingthelogarithmof^Rreducesthedynamicrangeoftheeigenvaluesof^R,whichisalsooneofthemainobjectivesofthediagonalloadingapproaches.However,MWcompensatesthisdynamicrangereductionbyusingtheinverselogarithmicfunction,i.e.,theexponential,whenestimatingthepowersunliketheshrinkageapproaches[ 29 ]. MWcanobtainthepowerestimateswithoutobtaininganexplicitexpressionfortheweightingvector.ForSOIwaveformestimation,however,abeamformerhastobedesignedwhichcanbeobtainedbyminimizingthewhite-noisegainundertheconstraintsthattheoutputpowerisequaltotheMWpowerestimategivenby( 3 )andthattheSOIispassedundistorted: minimizewkwk2subjecttow^Rw=^P0,MW,wa=1. (3) TheaboveproblemcanbesolvedbyusingtheLagrangemethodandthesolutionis[ 29 ]: wMW=(I+^R))]TJ /F10 7.97 Tf 6.58 0 Td[(1a a(I+^R))]TJ /F10 7.97 Tf 6.59 0 Td[(1a,(3) wheretheLagrangeparametercanbecalculatedusingNewton'smethodbasedonthefollowingequation: a(I+^R))]TJ /F10 7.97 Tf 6.59 0 Td[(1^R(I+^R))]TJ /F10 7.97 Tf 6.58 0 Td[(1a (a(I+^R))]TJ /F10 7.97 Tf 6.59 0 Td[(1a)2=^P0,MW.(3) Notefrom( 3 )thattheMWbeamformercanalsobeconsideredasauserparameter-freediagonalloadingapproachwheretheloadingleveliscalculatedbysolving( 3 ). 50

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3.4Iterativeapproaches Inthissectionweconsideriterativeapproachestobeamforming,namelyIAA,IAA-MLandM-SBL.Forthispurpose,considerthedatamodelin( 3 )whereKisreplacedbythenumberofpotentialsourcelocationsintheeld(orthenumberofscanningpoints)toavoidtheneedforestimatingthetruenumberofsourceswhichisusuallyunknowninpractice.Asaresult,Kwillbemuchlargerthanthetruenumberofsourcesands(n),n=1,...,N,in( 3 )willcontainonlyafewnon-zeroelements,i.e.,itwillbesparse.Consequently,iterativeapproachescouldbeusedforsignalwaveform(andhencepower)estimation. 3.4.1IAA LetPbeaKKdiagonalmatrix,whosediagonal(Pk)containsthesignalpowerateachangleonthescanninggridanddenethecovariancematrixoftheinterferenceandnoiseas(see( 3 )) Qk=R)]TJ /F6 11.955 Tf 11.95 0 Td[(Pkakak,(3) whereR=A()PA()andkisthegridindexofthecurrentSOI.Then,theweightedleastsquarescostfunctionisgivenby,see,e.g.,[ 7 68 69 ], NXn=1ky(n))]TJ /F6 11.955 Tf 11.96 0 Td[(sk(n)akk2Q)]TJ /F17 5.978 Tf 5.75 0 Td[(1k,(3) wherekxk2Q)]TJ /F17 5.978 Tf 5.76 0 Td[(1k,xQ)]TJ /F10 7.97 Tf 6.58 0 Td[(1kx.Minimizing( 3 )withrespecttosk(n),n=1,...,N,yields, ^sk(n)=akQ)]TJ /F10 7.97 Tf 6.58 0 Td[(1ky(n) akQ)]TJ /F10 7.97 Tf 6.58 0 Td[(1kak (3) =akR)]TJ /F10 7.97 Tf 6.59 0 Td[(1y(n) akR)]TJ /F10 7.97 Tf 6.59 0 Td[(1ak, (3) wherethesecondequalityfollowsfrom( 3 )andthematrixinversionlemma,see,e.g.,[ 7 ].Byusing( 3 )inlieuof( 3 ),wesavecomputationalcostbyavoidingcalculatingQ)]TJ /F10 7.97 Tf 6.58 0 Td[(1kforeachgridpoint.TheIAApowerestimatesarethengivenby^Pk=1 NPNn=1j^sk(n)j2,k=1,...,K.SinceIAArequiresR,whichdependsontheunknown 51

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signalpowers,itmustbeimplementedasaniterativeapproach;theinitializationisdonebyDAS.TheIAAalgorithmissummarizedinTable 3-1 .NotethatwhenR=I,thepowerestimateofIAAreducestotheDASpowerestimate(see( 3 )and( 3 )).Also,IAAcanbethoughtintheformof( 3 )whereRisestimatedrstandthenthelterweightsarecalculatedusingthisestimate.InIAA,^PandhenceRareobtainedfromthesignalestimatesofthepreviousiterationandnotfromthedatasnapshots. Table3-1. TheIAAalgorithm. ^sk(n)=aky(n)=M,n=1,...,N,k=1...,K^Pk=1 NPNn=1j^sk(n)j2,k=1,...,KrepeatR=A()^PA()with^P=diag(^P1,...,^PK)fork=1,...,Kwk=R)]TJ /F10 7.97 Tf 6.59 0 Td[(1ak akR)]TJ /F10 7.97 Tf 6.58 0 Td[(1ak^Pk=wk^Rwkendforuntil(convergence) NotethatforIAA(aswellasIAA-MLandM-SBL,seebelow)ascanninggridhastobesetandthesignalparameterscorrespondingtothesegridsareestimatedjointly,i.e.,thesignalparameterestimateforeverygridpointisreadilyavailableoncethealgorithmisrun.Thisisunlikeinthepreviouslymentionedmethods(DAS,SCB,HKB,CC,GLCandMW)inwhichonlytheSOIpowerisestimated.TogetthesignalparameterestimatesattheSOI,wecandenetheSOIweightvectorofIAAaswIAA=wi,whereiistheindexofthescanninggridpointcorrespondingtotheassumeddirection-of-arrival(DOA)oftheSOI. 3.4.2IAA-ML IAA-ML[ 33 ]minimizesthenegativelog-likelihoodfunctionoffy(n)gNn=1,i.e., log[det(R)]+1 NNXn=1y(n)R)]TJ /F10 7.97 Tf 6.59 0 Td[(1y(n),(3) 52

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withrespecttotheunknownsinR,whereitwasassumedthatthereceivedsignalisacomplexmultivariatezero-meanGaussianrandomvectorwithcovariancematrixRandthatthesnapshotsareindependentandidenticallydistributed(i.i.d.)[ 5 ].AssumethatQkisknownandthatthesignalpoweratkistobeestimated.Usingthefactthatdet(I+AB)=det(I+BA)andthematrixinversionlemmatogetherwith( 3 ),itcanbeshownthatminimizing( 3 )withrespecttoPkisequivalenttominimizing, f(Pk),log(1+PkakQ)]TJ /F10 7.97 Tf 6.58 0 Td[(1kak))]TJ /F6 11.955 Tf 13.15 8.09 Td[(PkakQ)]TJ /F10 7.97 Tf 6.59 0 Td[(1k^RQ)]TJ /F10 7.97 Tf 6.59 0 Td[(1kak 1+PkakQ)]TJ /F10 7.97 Tf 6.58 0 Td[(1kak.(3) Settingtherstderivativeof( 3 )withrespecttoPktozerogives: ~Pk=akQ)]TJ /F10 7.97 Tf 6.58 0 Td[(1k(^R)]TJ /F3 11.955 Tf 13.57 2.66 Td[(Qk)Q)]TJ /F10 7.97 Tf 6.59 0 Td[(1kak (akQ)]TJ /F10 7.97 Tf 6.59 0 Td[(1kak)2.(3) Moreover,itcanbeshownthatthesecondderivativef(2)(~Pk)>0,whichmeansthat~Pkistheuniqueminimizeroff(Pk).Inprinciple~Pkmaybenegative;therefore,thenonnegativityofthepowerestimatesisenforcedateachiterationbysettingthenegativeestimatestozero.Accordingly,theIAA-MLpowerestimateisobtainedas ^Pk=max 0,Pk+akR)]TJ /F10 7.97 Tf 6.59 0 Td[(1(^R)]TJ /F3 11.955 Tf 13.02 2.66 Td[(R)R)]TJ /F10 7.97 Tf 6.59 0 Td[(1a (akR)]TJ /F10 7.97 Tf 6.59 0 Td[(1ak)2!,(3) wherewehaveusedthematrixinversionlemmain( 3 )toreplaceQkbyR.AsPk=~Pkistheuniqueminimizeroff(Pk),^Pkminimizesf(Pk)subjecttoPk0.Sincecomputing^PkrequiresknowledgeofPkandR,thealgorithmmustbeimplementediteratively;theinitializationof^PkisdonewithDAS.IAA-MLisoutlinedinTable 3-2 ,wheretheinversecovariancematrixiscalculatedefcientlyusingthematrixinversionlemma.Thesortingprocedurehelpsdrivingtheestimatesforthepotentiallysourcefreelocationstozerosothattheotherestimationscanbedonemoreaccurately.Finally,IAA-MLislocallyconvergentduetocyclicallymaximizingthelikelihoodfunction[ 33 ]. IAA-MLestimatesthesignalpowers,butifdesiredthewaveformscanbeobtainedaswellbyaminimummean-squarederror(MMSE)estimator.Sincefy(n)gNn=1and 53

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Table3-2. TheIAA-MLalgorithm. ^sk(n)=aky(n)=M,n=1,...,N,k=1...,K^Pk=1 NPNn=1j^sk(n)j2,k=1,...,KR)]TJ /F10 7.97 Tf 6.59 0 Td[(1=(A()^PA()))]TJ /F10 7.97 Tf 6.58 0 Td[(1repeatAdjust[i1,...,iK]suchthat^Pi1^Pi2...^PiKfork=1,...,K^Ppreviousik=^Pik^Pik=max 0,^Pik+aikR)]TJ /F10 7.97 Tf 6.59 0 Td[(1(^R)]TJ /F3 11.955 Tf 13.02 2.66 Td[(R)R)]TJ /F10 7.97 Tf 6.58 0 Td[(1aik (aikR)]TJ /F10 7.97 Tf 6.59 0 Td[(1aik)2!R)]TJ /F10 7.97 Tf 6.59 0 Td[(1=R)]TJ /F10 7.97 Tf 6.59 0 Td[(1)]TJ /F3 11.955 Tf 13.63 9.4 Td[((^Pik)]TJ /F3 11.955 Tf 13.24 2.66 Td[(^Ppreviousik)R)]TJ /F10 7.97 Tf 6.58 0 Td[(1aikaikR)]TJ /F10 7.97 Tf 6.59 0 Td[(1 1+(^Pik)]TJ /F3 11.955 Tf 13.24 2.66 Td[(^Ppreviousik)aikR)]TJ /F10 7.97 Tf 6.58 0 Td[(1aikendforuntil(convergence) fs(n)gNn=1arejointlyGaussiandistributedwithmeanszero,theMMSEestimateoffs(n)gNn=1giventheobservationsfy(n)gNn=1is[ 70 71 ] ^s(n)=Wy(n)=PA()(A()PA()))]TJ /F10 7.97 Tf 6.59 0 Td[(1y(n)(3) forn=1,...,N.TheIAA-MLsignalwaveformestimatesareobtainedbyusing^PobtainedwithIAA-MLinlieuofPin( 3 ).Accordingly,theSOIweightvectorofIAA-ML,namelywIAA-ML,isgivenbytheithcolumnofthematrixWin( 3 ),whereiisthegridindexcorrespondingtotheassumedDOAoftheSOI. 3.4.3M-SBL ABayesianapproachcanalsobeusedtoestimatethesignalwaveformsusingvariouspriorstoenforcesparsity.AnimportantalgorithminthiscontextisthesparseBayesianlearning(SBL)approach[ 72 73 ],andM-SBL,themulti-snapshotextensionofit[ 34 35 ],whichusesazero-meanGaussianpriorwithadistinctvarianceforeach~sk=[sk(1),...,sk(N)]T,i.e., p(~sk;k)=CN(0,kI),k=1,...,K(3) 54

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Table3-3. TheM-SBLalgorithm. =1,s>0repeat)]TJ /F3 11.955 Tf 10.26 0 Td[(=diag(k)t=A()A()+sI=)]TJ /F3 11.955 Tf 6.94 0 Td[((I)]TJ /F9 11.955 Tf 11.95 0 Td[(A())]TJ /F10 7.97 Tf 6.59 0 Td[(1tA())]TJ /F3 11.955 Tf 6.94 0 Td[()^s(n)=Wy(n)=A())]TJ /F10 7.97 Tf 6.59 0 Td[(1ty(n),n=1,...,Ns=1 NPNn=1ky(n))]TJ /F9 11.955 Tf 11.96 0 Td[(A()^s(n)k2 M)]TJ /F6 11.955 Tf 11.96 0 Td[(K+PKk=1(k=k)k=1 NPNn=1j^sk(n)j2=)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(1)]TJ /F4 11.955 Tf 11.96 0 Td[()]TJ /F10 7.97 Tf 6.59 0 Td[(1kk,k=1,...,Kuntil(convergence) where=[1,...,K]TisavectorofKhyperparameterscontrollingthepriorvarianceoftheelementsof~sk.TheBayesianestimateofthesignalwaveformscanbeobtainedbyusingatype-IIlikelihoodmaximizationaswellasanEMalgorithmresultinginthealgorithmsummarizedinTable 3-3 .Fromthetable,wecandenetheSOIweightvectorofM-SBL,namelywM-SBL,astheithcolumnofthematrixWwithidenotingthegridindexcorrespondingtotheassumedDOAoftheSOI. 3.5NumericalExamples InthissectionwepresentanumberofexperimentsdemonstratingthebenetsandlimitationsofSCB,HKB,MW,GLC,CC,M-SBL,IAAandIAA-ML.WeconsiderthebeamformeroutputSINR,SOIpowerestimateandspatialpowerspectrumestimateasourperformancemetrics.Theexampleswillcontain:i)uncorrelatedandcorrelatedpointsourcesanddistributedsources,ii)differentsnapshotnumbers(Ncanrangefromone,forinstance,inunderwateracousticsmeasurements[ 74 75 ],tohundreds,forinstance,inaeroacousticmeasurements[ 76 ]),iii)differentSNRvalues,andiv)steeringvectorerrors.Finally,wediscussthecomputationalcomplexitiesofeachalgorithmandprovidesomeinsightsintowhichalgorithmismoresuitableunderwhatcircumstances. 3.5.1SimulationDetails WeconsiderauniformlineararraywithM=10sensorsandhalf-wavelengthinter-elementspacing.Thefar-eldnarrowbandsignalwaveformsandtheadditivenoise 55

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signalsareassumedtobetemporallywhitecircularlysymmetriccomplexGaussianrandomprocesseswithzeromeanandacertainvariance.Thenoiseisfurtherassumedtobespatiallywhiteandindependentofthesources.TheSNRforeachsourceisdenedas: SNRk=10log10Pk 2dB,k=1,...,K,(3) wherePkand2denotethevariancesofthekthsourceandnoise,respectively.ThescanninggridforIAA,IAA-MLandM-SBLisuniformintherangefrom)]TJ /F3 11.955 Tf 9.3 0 Td[(90to90with1incrementbetweenadjacentgridpoints.Foreachexample,100Monte-Carlotrialsareperformedandaverageresultsarepresented.Thesteeringvectorerrorsaresimulatedbyperturbingeachelementoftheassumedsteeringvector(correspondingtoboththeSOIandinterferences)usingi.i.d.zero-meancircularlysymmetriccomplexGaussianrandomvariableswithvariance2.TheperturbedsteeringvectorsarethennormalizedsothattheirnormsquareequalsM(see( 3 )). 3.5.2PointSources First,weconsiderthescenarioofuncorrelatedsources.Threesourceswithpowers10,20and20dBareassumedtobepresentat0,20and60,respectively.WeconsidertherstsignalastheSOIandtheothertwosignalsasinterferences. Fig. 3-2 showsthebeamformeroutputSINRandSOIpowerestimatesversusthesnapshotnumberNintheabsenceofsteeringvectorerrorsforSNR=10dB(hereSNRmeansthepowerratiobetweentheSOIandnoise,andthenoisepoweris0dB).Amongthetestedmethods,SCB,HKBandMWcannotworkproperlywhenthesamplecovariancematrixisrankdecientandhencetheresultsofthesealgorithmsarenotshownwhenN
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thanSCBasNincreaseswithIAAhavingthebestperformance.Fig. 3-2 (B)showsthatIAAandGLCprovidemoreaccurateSOIpowerestimatesthantheotheralgorithmsforalmostallsamplesizesconsidered. Fig. 3-3 showstheoutputSINRandSOIpowerestimatesfordifferentSNRvaluesobtainedbyvaryingthenoisepowerforN=20.WeobservethatIAAshowsthebestSINRperformancewithintheSNRrangeconsideredanditsperformanceisquiteclosetotheoptimumvalue.Moreover,IAAtogetherwithGLCandMWprovidegoodSOIpowerestimatesforallSNRvaluesconsidered.IAA-MLandM-SBLalsoprovidegoodSINRperformance.However,theyunderestimatetheSOIpowerforrelativelylowSNR. Next,weexaminetherobustnessofthebeamformerstoarraysteeringvectorerrors.Fig. 3-4 showstheoutputSINRandSOIpowerestimatesversustheperturbationvariance2forSNR=10dBandN=20.Inthisscenario,GLCandMWshowthebestSINRperformance.GLCandIAAfollowedbyMWprovidemoreaccurateSOIpowerestimatesthantheothermethods.However,whenNisrelativelylarge,GLCtendstochooseasmalldiagonalloadinglevelandthusislesseffectiveincombattingthesteeringvectorerrors.AsshowninFig. 3-5 ,whichconsidersthesamescenarioasFig. 3-4 exceptthatNisincreasedto100,theperformanceofGLCdegradescomparedtothecasewhenN=20.Withmoresnapshots,MWshowsthebestSINRperformancewhileIAAstillprovidesthemostaccurateSOIpowerestimatesamongallthemethods. Wenowconsideranexampleconsistingoftwocorrelatedsources.TheSOIwith10dBpowerisat0andaninterferencewith20dBpowerisat20.Fig. 3-6 showsthebeamformeroutputSINRandSOIpowerestimatesversusthecorrelationcoefcientbetweentheSOIandtheinterferenceforN=20.Interestingly,weobservethattheperformances(bothSINRandSOIpowerestimates)ofIAA,IAA-MLandM-SBLremainalmostunchangedwhenthecorrelationcoefcientincreases.Ontheotherhand,theperformancesoftheothermethodsdegradeastheinterferencebecomes 57

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morecorrelatedwiththeSOIandthesealgorithmsfailalmostcompletelywhenthetwosourcesbecomecoherent.IAAshowsthebestperformanceinthiscaseaswell. Finally,wepresentthespatialpowerspectrumestimatesofthebeamformersforvariousscenarioswhenN=20.DASisalsoincludedinthiscomparisonasareference.First,weconsiderveuncorrelatedsourceswithpowers10,15,30,25and20dBat)]TJ /F3 11.955 Tf 9.3 0 Td[(45,)]TJ /F3 11.955 Tf 9.3 0 Td[(35,0,5and60,respectively.Thenoisepowerisassumedtobe0dB.FromFig. 3-7 ,wenotethatIAA-MLandM-SBLgivethebestresolutionfollowedbySCB,CCandHKB.However,thesealgorithmsunderestimatethepowersofsomesources,especiallythosewithrelativelylowSNR.Ontheotherhand,IAAalwaysprovidesaccuratepowerestimates.Thesecondexampleisthesameasthepreviousone,exceptthattheangularlocationsofthesourcesare)]TJ /F3 11.955 Tf 9.3 0 Td[(45.6,)]TJ /F3 11.955 Tf 9.3 0 Td[(35.1,0,5.2and60.5,i.e.,thesourcesarenolongeronthegridpointsoftheiterativealgorithms.TheresultsareshowninFig. 3-8 .ComparingFig. 3-8 withFig. 3-7 ,weobservethattheperformancesofthealgorithmswithrelativelylowresolution,i.e.,IAA,GLC,MWandDASarevirtuallyunchangedandtheperformancesoftheothermethodsdegradeslightly.Thirdly,wecomparethespatialpowerspectrumestimatesinthepresenceofsteeringvectorerrorsforacasewherethreeuncorrelatedsourceswithpowers15,30and20dBarelocatedat)]TJ /F3 11.955 Tf 9.3 0 Td[(15,0and60andtheperturbationvarianceis-10dB.TheperformancesofthebeamformersareshowninFig. 3-9 ,fromwhichwecanobservethatIAAgivesthebestpowerestimates.Finally,Fig. 3-10 showsthepowerandlocationestimatesfortwocoherentsourcesat)]TJ /F3 11.955 Tf 9.3 0 Td[(5and5withpowers10and20dB,respectively.Notethatonlytheiterativealgorithmscanfunctionwellinthiscase,whichisconsistentwiththeresultsshowninFig. 3-6 .Again,IAA-MLandM-SBLhavebetterresolutionthanIAA,whileIAAprovidesmoreaccuratepowerestimates,especiallyforrelativelylowSNR. 3.5.3DistributedSources Sofarwehavestudiedpointsources,i.e.,sourcesimpingingfromasinglelocationinspace.Inthefollowingexamples,weexaminetheperformanceofthebeamformers 58

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fordistributedsourcesaswell[ 77 81 ].ThearrayoutputvectorforKspatiallydistributedsourcescanberepresentedas[ 78 79 ](seealso( 3 )): y(n)=KXk=1Z=2)]TJ /F11 7.97 Tf 6.59 0 Td[(=2a()sk(n)gk(, k)d+e(n),=KXk=1c( k)sk(n)+e(n) (3) wherec( k)=R=2)]TJ /F11 7.97 Tf 6.58 0 Td[(=2a()gk(, k)disthegeneralizedsteeringvector,g(, k)istheangularsignaldensityand kisthelocationparametervectorofthekthsource[ 79 ].Forauniformlydistributedsource, k=[k,k]T,wherekdenotesthecentralangleandkdenotestheangularspreadofthekthsource.Accordingly,themthcomponentofthesteeringvectorc( k)duetothekthsourcecanbeapproximatedas(forsmallvaluesofk): c( k)me)]TJ /F8 7.97 Tf 6.58 0 Td[(j2f c0xmsin(k)sinc2f c0xmkcos(k)(3) wheresinc(x)=sin(x)=(x)[ 78 ]. Fig. 3-11 showstheSINRandSOIpowerestimatesofthealgorithmsversusNfortwouncorrelatedspatiallydistributedsources,simulatedusing( 3 )and( 3 ),thatarelocatedat1=0and2=20with1=2=2.Thesteeringvectorsarenormalizedsuchthatkc( k)k2=M,k=1,2.WeassumethattherstsourceistheSOIwith10dBpowerandtheotheroneistheinterferencewith20dBpower.Fig. 3-12 showsthecorrespondingresultsobtainedbyvaryingSNR.WenotethatIAAshowsthebestperformance.MWandGLCalsoprovidegoodSOIpowerestimates.Inaddition,theSINRperformancesofIAA-MLandM-SBLaregoodwhiletheirSOIpowerestimatesarenotsoaccurateespeciallyforrelativelylowSNR.Finally,thespatialpowerestimatesareshowninFig. 3-13 fortwouncorrelatedspatiallydistributedsources,whereoneofthesourcesislocatedat1=)]TJ /F3 11.955 Tf 9.3 0 Td[(5with10dBpowerandtheotherat2=5with20dBpowerand1=2=3.Ascanbeobservedfromtheplots,IAAprovidesthemoreaccuratepowerandlocationestimatesinthiscase. 59

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3.5.4ComplexityAnalysis ThecomputationalcomplexityofDASisO(M)whereasthecomplexitiesofSCB,HKB,CC,GLCandMWareO(M3),mainlybecauseofthematrixinversionoperation.AssumingKM,thecomplexityofeachIAA,IAA-MLandM-SBLiterationisO(M2K)periteration2.Notehoweverthatifthespatialestimateofthesourcesinthewholeregionisdesired,DAShascomplexityO(MK)andalltheothermethodshavecomplexityO(M2K),assumingKM.Inalloursimulations,ingeneral,IAAandIAA-MLconvergedinatmosthalfthenumberofiterationsnecessaryforM-SBLtoconverge. 3.5.5OverallAssessments Basedonallthenumericalexamplesabove,IAAshowsthebestbeamformeroutputSINRexceptinthepresenceofarraycalibrationerrors.Also,IAAprovidesthemostaccuratepowerestimatesinallthecases.IAA-MLandM-SBLprovidethehighestresolution,whichisusefulforDOAestimation.However,theSOIpowerestimationperformancesofthesetwomethodsaremoresensitivetolowSNRthanfortheothermethods.Onedesirablepropertyoftheiterativealgorithmsisthattheirperformancesarenotaffectedmuchbythepresenceofcorrelated(evencoherent)sources,whilealltheotheralgorithmsfailinthiscase.Exceptforthecorrelatedsourcecase,GLCandMWprovidegoodoverallperformance,especiallyinthepresenceofsteeringvectorerrors.However,GLCdoesnotworkwellforlargesnapshotnumberswhentherearesteeringvectorerrorsandinthespatiallydistributedsourcescase,whileMWcannotbeusedforlowsnapshotnumbercases.CCprovidessimilarperformanceasSCBexceptforsmallsamplesizesinwhichcaseitoutperformsSCB.Wesummarizeourempirical 2ThecomplexityofM-SBLdoesnotincludeNperthediscussionin[ 34 ].ThisstemsfromthefactthattheM-SBLiterationscanbeimplementedbyusing^Rinsteadoffy(n)gNn=1. 60

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observationsinTable 3-4 whichliststhealgorithmsthatshowgoodperformanceforagivenscenario. Table3-4. Summaryofresultsforcomparingvariousbeamformingalgorithms. CorrectsteeringvectorSteeringvectorerrorsCorrelatedSourcesDistributedSources SINR IAAMWaIAAIAA M-SBLGLCbM-SBLM-SBL IAA-MLIAAIAA-MLIAA-MLSOIPowerEstimate IAAIAAIAAIAA GLCbGLCbIAA-MLGLCb MWaMWaM-SBLMWaResolution IAA-MLIAA-MLIAA-MLIAA-ML M-SBLM-SBLM-SBLM-SBL SCBaMWaIAAIAA aNMorlarger,bNnotverylarge 3.6Conclusions Inthischapterwehavereviewedandcomparedthefollowingbeamformingmethods:DAS,SCB,thediagonalloadingapproachesofRRCBandofMW,theshrinkagebaseddiagonalloadingapproachesofCCandGLC,andtheiterativebeamformersofIAA,ofIAA-MLandofM-SBL.ThesealgorithmshavebeenevaluatedundervariousscenariosaccordingtotheirSINRaswellastotheirSOIpowerestimationandspatialpowerestimationaccuracies.Generalguidelineshavebeenofferedtoassistselectingthemostsuitablealgorithminagivenapplicationscenario. 61

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AB Figure3-2. PerformanceofvariousapproachesversusthenumberofsnapshotsNwhenSNR=10dB.(A)BeamformeroutputSINRversusN.(B)SOIpowerestimatesversusN.TheSOIisat0with10dBpowerandthetwointerferenceseachwith20dBpowerarelocatedat20and60. AB Figure3-3. PerformanceofvariousapproachesversusSNRwhenN=20.(A)BeamformeroutputSINRversusSNR.(B)SOIpowerestimatesversusSNR.TheSOIisat0with10dBpowerandthetwointerferenceseachwith20dBpowerarelocatedat20and60. 62

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AB Figure3-4. PerformanceofvariousapproachesversustheperturbationvariancewhenSNR=10dBandN=20.(A)BeamformeroutputSINRversustheperturbationvariance.(B)SOIpowerestimatesversustheperturbationvariance.TheSOIisat0with10dBpowerandthetwointerferenceseachwith20dBpowerarelocatedat20and60. AB Figure3-5. PerformanceofvariousapproachesversustheperturbationvariancewhenSNR=10dBandN=100.(A)BeamformeroutputSINRversustheperturbationvariance.(B)SOIpowerestimatesversustheperturbationvariance.TheSOIisat0with10dBpowerandthetwointerferenceseachwith20dBpowerarelocatedat20and60. 63

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AB Figure3-6. Performanceofvariousapproachesversusthecorrelationcoefcient.(A)BeamformeroutputSINRand(B)SOIpowerestimatesversusthecorrelationcoefcientbetweentheSOIat0with10dBpowerandaninterferenceat20with20dBpowerwhenSNR=10dBandN=20. 64

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AB C Figure3-7. Spatialpowerestimatesforuncorrelatedsources.Circlesrepresentsthetruesourcelocationsandpowers.Thesourcesarelocatedat1=)]TJ /F3 11.955 Tf 9.3 0 Td[(45,2=)]TJ /F3 11.955 Tf 9.3 0 Td[(35,3=0,4=5and5=60withSNR1=10dB,SNR2=15dB,SNR3=30dB,SNR4=25dBandSNR5=20dB.(A)SCB,IAA,GLCandDAS.(B)CC,IAA-ML,MWandDAS.(C)HKB,M-SBLandDAS. 65

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AB C Figure3-8. Spatialpowerestimatesforuncorrelatedsourceswhenthesourcesarenotonthescanninggrid.Circlesrepresentsthetruesourcelocationsandpowers.Thesourcesarelocatedat1=)]TJ /F3 11.955 Tf 9.3 0 Td[(45.6,2=)]TJ /F3 11.955 Tf 9.3 0 Td[(35.1,3=0,4=5.2and5=60.5withSNR1=10dB,SNR2=15dB,SNR3=30dB,SNR4=25dBandSNR5=20dB.(A)SCB,IAA,GLCandDAS.(B)CC,IAA-ML,MWandDAS.(C)HKB,M-SBLandDAS. 66

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AB C Figure3-9. Spatialpowerestimatesforuncorrelatedsourcesinthepresenceofarraycalibrationerrors.Circlesrepresentsthetruesourcelocationsandpowers.Thesourcesarelocatedat1=)]TJ /F3 11.955 Tf 9.3 0 Td[(15,2=0and3=60withSNR1=15dB,SNR2=30dBandSNR3=20dB.(A)SCB,IAA,GLCandDAS.(B)CC,IAA-ML,MWandDAS.(C)HKB,M-SBLandDAS. 67

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AB C Figure3-10. Spatialpowerestimatesfortwocoherentsources.Circlesrepresentsthetruesourcelocationsandpowers.Thesourcesarelocatedat1=)]TJ /F3 11.955 Tf 9.3 0 Td[(5and2=5withSNR1=10dBandSNR2=20dB.(A)SCB,IAA,GLCandDAS.(B)CC,IAA-ML,MWandDAS.(C)HKB,M-SBLandDAS. 68

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AB Figure3-11. PerformanceofvariousapproachesversusthenumberofsnapshotsforuncorrelatedspatiallydistributedsourceswithSNR=10dB.(A)BeamformeroutputSINRversusN.(B)SOIpowerestimatesversusN.TheSOIisat0withangularspread2and10dBpower,andtheinterferenceisat20withangularspread2and20dBpower. AB Figure3-12. PerformanceofvariousapproachesversusSNRforuncorrelatedspatiallydistributedsourceswithN=20.(A)BeamformeroutputSINRversusSNR.(B)SOIpowerestimatesversusSNR.TheSOIisat0withangularspread2and10dBpower,andtheinterferenceisat20withangularspread2and20dBpower. 69

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AB C Figure3-13. Spatialpowerestimatesfortwouncorrelatedspatiallydistributedsources.Circlesrepresentsthetruecentraldirectionsandpowersofthesources.Thecentralanglesofthesourcesare1=)]TJ /F3 11.955 Tf 9.29 0 Td[(5and2=5withangularspreads1=2=3,SNR1=10dBandSNR2=20dB.(A)SCB,IAA,GLCandDAS.(B)CC,IAA-ML,MWandDAS.(C)HKB,M-SBLandDAS. 70

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CHAPTER4USERPARAMETERFREEROBUSTADAPTIVEAPPROACHESTOULTRASOUNDIMAGING 4.1Introduction Delay-and-sum(DAS)beamformingisthestandardtechniqueforultrasoundimagingapplications,wherethereceivedsignalsaredelayedappropriatelyaccordingtothelocationsofthetransmitter,receiverandthetargetofinterest,andthenaresummedup.However,itsuffersfromlowresolutionandpoorinterferencesuppressioncapability.Adataadaptivebeamformer,e.g.thestandardCaponbeamformer(SCB)[ 8 ]hasthepotentialtoimprovetheimagingqualitysignicantly.Whereas,itsperformancemaybedegradedsignicantlyduetoerrorsincovariancematrixand/orarraysteeringvectors.MuchworkhasbeendonetoimprovetherobustnessofSCB,asexaminedinChapters 2 and 3 .Sinceallofthemaredesignedfornarrowbandcases,wecannotapplythosebeamformingtechniquesdirectlytoultrasoundimaging,whichdealswiththewidebandsignals. Inthischapter,weextendtwouserparameter-freerobustadaptivesignalprocessingtechniques,i.e.,thegenerallinearcombination(GLC)algorithm[ 23 ](seealsoChapter 2 )andtheiterativeadaptiveapproach(IAA)[ 32 ](seealsoChapter 3 )towidebandscenarioandapplythemtoultrasoundimagingapplications.ForthewidebandGLC(WGLC)approaches,wersttime-aligningthereceivedsignalstofocusatagiventargetpositionandthenapplyGLCtoestimatethereectedsignalfromthefocalpoint.Inoneofthemultiple-stageWGLCmethods,whichwerefertoasWGLC-1,GLCisusedinStageItoobtainasetofbackscatteredsignalestimatesforeachtransmitter.Basedontheseestimates,ascalarwaveformisrecoveredviaGLCinStageII,whichisthenusedtocomputethebackscatteredenergyinthenalstage.AnalternativewayofsignalprocessinginStageIistocomputeasetofbackscatteredwaveformsforeachtimeinstant.WerefertothismethodasWGLC-2.Inaddition,weconsideracombined 71

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method,referredtoasWGLC-C,whichusesthesignalestimatesfrombothWGLC-1andWGLC-2inStageIforthecomputationofbackscatteredenergy. ForwidebandIAA(WIAA),wepreprocessthewidebandsignalsbydividingthearrayoutputsintomanynarrowbandfrequencybinsusingtheFouriertransform(FT).ThenweuseIAAineachnarrowbandfrequencybintoestimatethesignalamplitudesinthefrequencydomain,obtainthereectedsignalwaveformsinthetimedomainbyapplyinginverseFTtothebeamformeroutput,andthusgetthebackscatteredenergy. Theremainderofthechapterisorganizedasfollows.InSection 4.2 ,wedescribethewidebanddatamodelinbothtimeandfrequencydomains.InSections 4.3 and 4.4 ,wepresenttheWGLCandWIAAapproachestoultrasoundimaging,respectively.ExperimentalexamplescomparingtheperformanceofWGLC,WIAAandDASapproachesaregiveninSection 4.5 .Finally,weconcludethechapterinSection 4.6 4.2DataModel ConsideranultrasoundimagingsystemwithMtransducers.Thetransducersprobetheunknownpropagatingmediumbytransmittingshortpulsesandrecordingthereectedsignalsbackfromthemedium.Inmultistatic(alsocalledMIMO(multi-inputmulti-output)[ 82 ])dataacquisitionsystem,eachoftheMtransducerstakesturnstotransmitthesameshortprobingpulseandallthetransducersrecordthebackscatteredechoes.Asaresult,thearrayresponsecontainsthedatasequencesforallpossibletransmitterandreceiverpairsofthearray.Let~yi,j(t),i,j=1,...,Mbethedatasequenceofthebackscatteredechoatthejthtransducerduetotransmittingapulsefromtheithtransducer.Itcanbemodeledas: ~yi,j(t)=KXk=1fi,j(k)~si,j(k,t)]TJ /F4 11.955 Tf 11.96 0 Td[(i,j(k))+~ei,j(t),(4) where~si,j(k,t)isthereectedsignalfromthekthpointwithKdenotingthenumberofgridpointswithintheimagingareaofinterest,and~ei,j(t)isthenoise.In( 4 ),i,j(k)is 72

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thetimedelayduetothepulsewavepropagationfromtheithtransmittertothekthgridpointandthenbacktothejthreceiver,anditcanbeapproximatedas: i,j(k)=kqi)]TJ /F6 11.955 Tf 11.96 0 Td[(pkk+kqj)]TJ /F6 11.955 Tf 11.96 0 Td[(pkk c,(4) wherecisthesoundpropagationspeedinthemediumunderinterrogation,qmisthelocationofthemthtransducer,andpkisthelocationofthekthgridpoint.fi,j(pk)istheattenuationfactoraccountingforthepropagationlossinthemedium,whichismainlyduetotheamplitudedecreaseofthesphericalwaveasitexpands,anditcanbeapproximatedas: fi,j(k)=1 kqi)]TJ /F6 11.955 Tf 11.96 0 Td[(pkkkqj)]TJ /F6 11.955 Tf 11.96 0 Td[(pkk.(4) ByapplyingFTtothetimedomainreceiveddatain( 4 ),wegetthefrequencydomaindatamodelanditcanbeexpressedasfollowsforaparticularfrequencybinl: ri,j(l)=KXk=1fi,j(k)xi,j(k,l)e)]TJ /F8 7.97 Tf 6.58 0 Td[(j!li,j(k)+zi,j(l),(4) whereri,j(l),xi,j(k,l),andzi,j(l)denotetheFTof~yi,j(t),~si,j(k,t)and~ei,j(t),respectively. Ourgoalistoestimatethebackscatteredenergyofallthegridpointsfromthereceiveddatasetf~yi,j(t)g,whichrepresentstheso-obtainedultrasoundimage. 4.3WidebandGLCApproaches TheWGLCapproachesemployarobustadaptivebeamformingalgorithmGLC(seeChapter 2 and[ 23 ]),whichisinitiallydesignedfornarrowbandscenario.InordertoapplyGLCtothewidebandultrasoundimagingdata,weneedtorstpreprocessthedatabyappropriatetime-shiftingandcompensation. 4.3.1Preprocess Wealignthereceivedsignalsinthedatasetf~yi,j(t)gtofocusateachgridpointbyinsertingappropriatetimedelays.Letpkbethelocationofthefocalpointofcurrentinterest.Thetimedelayscanbedeterminedby( 4 ).Then,thetimeshiftedsignalfor 73

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thefocalpointkafteramplitudecompensation,canberepresentedas yi,j(k,t)=1 fi,j(k)~yi,j(t+i,j(k)).(4) Foreachdatasequenceyi,j(k,t),weareonlyinterestedint=0,...,N)]TJ /F3 11.955 Tf 12.22 0 Td[(1,whereNisdeterminedbythetimedurationofthetransmittedpulse.Finally,thepreprocesseddatasetcanbewrittenas: yi,j(k,t)=si,j(k,t)+ei,j(t),i,j=1,...,M,t=0,...,N)]TJ /F3 11.955 Tf 11.96 0 Td[(1,(4) wheresi,j(k,t)isthesignalofinterest(SOI)reectedfromthefocalpointkandei,j(t)istheresidualsignalincludingtheinterferencefromotherreectionpointsandthenoise.Hereafter,weconcentrateonthefocalpointk.Thedependenceonkwillbeomittedfornotationalsimplicity. Thecompletedatasetforagivenfocalpointin( 4 )canberepresentedbythedatacubeasshowninFig. 4-1 .Inthefollowingthree-stageWGLCalgorithms,WGLC-1rstslicesthedatacubeforeachtransmitterinStageIandappliesGLCtoeachdataslice(receiver-timeslice)toobtainasetofbackscatteredsignalestimates.Basedontheseestimates,ascalarwaveformisrecoveredviaGLCinStageII,whichisthenusedtocomputethebackscatteredenergyforthefocalpointinStageIII.Alteratively,WGLC-2slicesthedatacubeforeachtimeinstantandcomputesasetofbackscatteredwaveformsinStageI.Inaddition,weconsideracombinedmethodWGLC-C,whichusesthesignalestimatesfrombothWGLC-1andWGLC-2inStageIforthecomputationofbackscatteredenergy. 4.3.2WGLC-1 TheWGLC-1algorithmconsidersthefollowingsignalmodelforthedatasetin( 4 ): yi(t)=aisi(t)+ei(t),i=1,...,M,t=0,...,N)]TJ /F3 11.955 Tf 11.96 0 Td[(1,(4) 74

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Figure4-1. Themultistaticdatacubeforagivenfocalpoint.WGLC-1processesthedatacubeforagiventransmitteri,whileWGLC-2processesthedatacubeforagiventimeinstanttatStageI. whereyi(t)=[yi,1(t)yi,M(t)]Trepresentsthealignedarraydatavectorfortheithtransmitter,si(t)denotesthesignalreectedfromthefocalpointduetotheithtransmitter,whichisassumedtodependonlyonthetransmitteributnotonthereceiverj,ei(t)denotestheresidualterm,andaidenotesthearraysteeringvector.aiisassumedtobeapproximatelyequalto1M1,with1M1beingaM1vectorofallones,sincethereceivedsignalshavebeenalignedandtheiramplitudeattenuationscompensated.Here,weassumethataimayvarywithi,butisconstantwithrespecttot. InStageI,werstconsideragiventransmitteri.Then,thebackscatteredsignalfromthefocalpointsi(t)canbeestimatedbyusingSCB.Thecorrespondingweightvectorisgivenby(see( 2 )): ^wi=R)]TJ /F10 7.97 Tf 6.59 0 Td[(1i1M1 1TM1R)]TJ /F10 7.97 Tf 6.58 0 Td[(1i1M1,(4) whereRi=E[yi(t)yi(t)]isthecovariancematrixofyi(t).Inpractice,Riisunavailable.Therefore,thesamplecovariancematrix^RiisusedinlieuofRiin( 4 ),whichcanbeexpressedas: ^Ri=1 NN)]TJ /F10 7.97 Tf 6.59 0 Td[(1Xt=0yi(t)yi(t).(4) 75

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Itiswell-knownthatthesamplecovariancematrixisapoorestimateofthetruecovariancematrixwhenthesamplesize(N)isnotsufcientlylargecomparedtothearraydimension(M).Asaresult,theperformanceofSCBmaysignicantlydegrade[ 9 ].Moreover,whenN
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Oncewegottheweightvector,wecanestimatethesignalsi(t)as: ^s1,i(t)=~wiyi(t),t=0,...,N)]TJ /F3 11.955 Tf 11.96 0 Td[(1.(4) RepeatingtheaboveprocessforalltransmittersyieldsatotalofMwaveformestimates,i.e.,^s1,1(t),,^s1,M(t).Let ^s1(t)=[^s1,1(t)^s1,M(t)]T.(4) Sinceallthetransducerstransmitthesamepulse,wecanassumethatthedesiredbackscatteredsignalwaveformsfromthefocalpointaresameforalltransmitters.ThentheoutputinStageIcanbemodelledasfollows: ^s1(t)=as(t)+e1(t),t=0,,N)]TJ /F3 11.955 Tf 11.95 0 Td[(1,(4) wherethesteeringvectoraisassumedtobe1M1,ande1(t)representstheresidualterm. AtStageII,weaimtoestimatethescalarwaveforms(t)from^s1(t)in( 4 ).SimilartoStageI,theknowledgeofamaybeimpreciseandthesamplesizeNmaybesmall.HencetheGLCrobustadaptivebeamformerisusedagaintoestimatethebackscatteredsignals(t).Let ^R1=1 NN)]TJ /F10 7.97 Tf 6.58 0 Td[(1Xt=0^s1(t)^s1(t)(4) bethesamplecovariancematrixof^s1(t).Replacingyi(t)and^Riin( 4 )-( 4 )by^s1(t)and^R1respectively,wecanobtaintheweightvector~w1forStageIIofWGLC-1.Then,thescalarwaveformestimateis: ^s(t)=~w1^s1(t).(4) 77

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Table4-1. TheWGLC-1approach. InitialStage:Preprocessthedatasetbytime-shiftingandamplitudecompensationasin( 4 ).StageI:Foreachtransmitter,applyGLConreceiver-timeslice.Asaresult,obtainMwaveformestimates^s1(t)( 4 ).StageII:Estimatethescalarwaveform^s(t)from^s1(t)viaGLC.StageIII:Obtainthebackscatteredenergyvia( 4 ). InStageIII,thesignalenergyforaparticularfocalpointkiscomputedas: E(k)=N)]TJ /F10 7.97 Tf 6.59 0 Td[(1Xt=0^s2(t).(4) TheWGLC-1approachforthekthgridpointissummarizedinTable 4-1 .Finally,theimageofthesignalenergyasafunctionofgridpointkcanbeobtainedbyvaryingkfrom1toK. 4.3.3WGLC-2 WGLC-2usesasignalmodelasfollowsfordatasetin( 4 ): yi(t)=a(t)si(t)+ei(t),i=1,...,M,t=0,...,N)]TJ /F3 11.955 Tf 11.95 0 Td[(1,(4) whereyi(t),si(t)andei(t)havebeendenedin( 4 ).a(t)denotesthearraysteeringvector,whichisalsoassumedtobeapproximatelyequalto1M1forthesamereasonasforWGLC-1.However,differentfromWGLC-1,herea(t)isassumedtochangewitht,butbeconstantwithrespecttoi. InStageI,werstfocusonagiventimeinstantt.Then,apseudosamplecovariancematrixcanbeformulatedbyconsideringthenumberoftransmittersasthenumberofsnapshots: ^R(t)=1 MMXm=1ym(t)ym(t).(4) Bycombining^R(t)andIfollowingasimilarprocedurein( 4 )-( 4 ),wecangetanenhancedestimate~R(t)andthusaweightvector~w(t)forStageIofWGLC-2.The 78

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Table4-2. TheWGLC-2approach. InitialStage:Preprocessthedatasetbytime-shiftingandamplitudecompensationasin( 4 ).StageI:Foreachtimeinstant,applyGLConreceiver-transmitterslice.Asaresult,obtainMwaveformestimates^s2(t)( 4 ).StageII:Estimatethescalarwaveform^s(t)from^s2(t)viaGLC.StageIII:Obtainthebackscatteredenergyvia( 4 ). signalvectorofthecorrespondingbeamformeroutputcanbewrittenas ^s2,i(t)=~w(t)yi(t),i=1,...,M.(4) Repeatingtheaboveprocessfort=0,,N)]TJ /F3 11.955 Tf 12.49 0 Td[(1,weagainobtainMestimatedsignalwaveformsforMtransmittersatthebeamformeroutputinStageI.Let ^s2(t)=[^s2,1(t)^s2,M(t)]T.(4) Then,followingasimilarprocedureasforStageIIofWGLC-1,wecangetarecoveredbackscatteredsignalwaveformf^s(t)gt=0,,N)]TJ /F10 7.97 Tf 6.59 0 Td[(1basedonf^s2(t)gt=0,,N)]TJ /F10 7.97 Tf 6.58 0 Td[(1.Thebackscatteredenergycanbeobtainedvia( 4 )inthenalstage.TheWGLC-2approachissummarizedinTable 4-2 4.3.4WGLC-C WGLC-1andWGLC-2usedifferentsamplecovariancematricesinStageI.HencetheimprovementsobtainedbyusingGLCmaybedifferent.ThisfactmotivatesustocombineWGLC-1andWGLC-2toachieveabetterperformance.WerefertothiscombinedmethodasWGLC-C.Let ^sC(t)=264^s1(t)^s2(t)375,(4) wheres1(t)ands2(t)aretheoutputsofStageIforWGLC-1andWGLC-2(see( 4 )and( 4 )),respectively.NotethatbothWGLC-1andWGLC-2obtainMsignalwaveformestimatesattheendofStageI,whileWGLC-Cobtains2Msignalwaveformestimates. 79

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Table4-3. TheWGLC-Capproach. InitialStage:Preprocessthedatasetbytime-shiftingandamplitudecompensationasin( 4 ).StageI:a.Foreachtransmitter,applyGLCtoobtainMwaveformestimates^s1(t)( 4 ).b.Foreachtimeinstant,applyGLCtoobtainMwaveformestimates^s2(t)( 4 ).c.Combinetheestimatesina.andb.toobtain2Mestimates^sC(t)( 4 ).StageII:Estimatethescalarwaveform^s(t)from^sC(t)viaGLC.StageIII:Obtainthebackscatteredenergyvia( 4 ). InStageIIofWGLC-C,thesignalvectors^sC(t),t=0,,N)]TJ /F3 11.955 Tf 12.3 0 Td[(1aretreatedasasnapshotfrom2M-elementvirtualarray ^sC(t)=aCs(t)+eC(t),t=0,,N)]TJ /F3 11.955 Tf 11.95 0 Td[(1,(4) wherethesteeringvectoraCisassumedtobe12M1,andeC(t)denotestheresidualterm.Weobtaintheweightvector~wCforWGLC-Cfollowingtheprocedurein( 4 )-( 4 ),wheretheenhancedcovariancematrixisbasedonthefollowingsamplecovariancematrix: ^RC=1 NN)]TJ /F10 7.97 Tf 6.59 0 Td[(1Xt=0^sC(t)^sC(t).(4) Thebeamformer~wCyieldsanestimateofthesignal: ^s(t)=~wC^sC(t).(4) Then,thebackscatteredenergyiscomputedvia( 4 ).TheWGLC-CapproachissummarizedinTable 4-3 4.4WidebandIAAapproach Inthissection,weextendtheIAAapproach(see[ 32 36 ])tothewidebandscenario.WerstapplyFTtothetime-domainarrayoutputtotransformthewidebanddataintonarrowbandfrequencybins.ThenIAAcanbeappliedtoeachfrequencybintoestimatethesignalwaveformandhencethebackscatteredenergy.Inwhatfollows,werstpreprocessthewidebanddatainfrequencydomainandthenshowhowtouseIAAtoformtheimageofbackscatteredenergy. 80

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4.4.1Preprocess Notethatthetimedurationofthetransmittedpulseforultrasoundimagingisveryshortcomparedtothelengthofthewholereceiveddatasequence.IfweapplyFTonthewholedatasequenceasexpressedin( 4 ),theSOImaybeburiedinnoiseandinterference.Therefore,werstuseaslidingrectangularwindowwithlength~L,whichisequaltothelengthofafewpulses,onthedatasequence,andthenperformL-pointFTforeachwindowedsegment(eachsegment,originallywith~Lpoints,iszero-paddedtoLpoints).NotethatLshouldbesufcientlylargesothatthenarrowbandassumptionforeachfrequencybinishold(see[ 5 ]).AdiagramdescribingthedatapreprocessingisshowninFig. 4-2 .Inthefollowing,weapplyWIAAtoeachsegmenttoformaimagecorrespondingtothatsegmentandthenpatchtheimagestoformthewholeimage. Figure4-2. DatapreprocessingforWIAA. Inthissection,weassumethebackscatteredwaveformsfromagivengridpointareidenticalfordifferenttransmittersandreceivers.Then,thefrequencydatamodelin( 4 )canberewrittenas ri,j(l)=~KXk=1fi,j(k)x(k,l)e)]TJ /F8 7.97 Tf 6.59 0 Td[(j!li,j(k)+zi,j(l),i,j=1,...,M,(4) 81

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where~Kisthenumberofgridpointswithintheimagingareaaccordingtothesegmentofcurrentinterest. Westackthearrayoutputsfromallthechannelstogetacompactformof( 4 ): r(l)=A(l)x(l)+z(l),(4) wherer(l)istheM21arrayoutputvector,andr(l)=[rT1(l)...rTM(l)]T,withri(l)=[ri,1(l)...ri,M(l)]T,i=1,...,M,zlisdenedsimilarly,andx(l)=[x(1,l)...x(K,l)]T.TheM2Ksteeringmatrixcanbewrittenas: A(l)=[a(1,l)a(K,l)],(4) wherethekthcolumnofA(l)isdenotedas: a(k,l)=266664a1(k,l)...aM(k,l)377775,(4) andthesteeringvectorai(k,l)correspondingtotheithtransmitterhasthefollowingform: ai(k,l)=266664fi,1(k)e)]TJ /F8 7.97 Tf 6.58 0 Td[(j!li,1(k)...fi,M(k)e)]TJ /F8 7.97 Tf 6.59 0 Td[(j!li,M(k)377775.(4) Hereafter,weestimatex(l)foreachfrequencybinindividually,andtherefore,thedependenceonlwillbedroppedfornotationalsimplicity.Notethatonlyasinglesnapshotisavailableforeachfrequencybin.Inthiscase,SCBfailstoworkcompletely. 4.4.2IAA WebrieydescribetheIAAalgorithminthecontextoftheultrasoundimagingproblem(seeChapter 3 and[ 32 ]formoredetails). LetP=diag(P1,...,PK),and Pk=jx(k)j2(4) 82

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representsthepowerofx(k).Furthermore,wedenethecovariancematrixoftheinterference(thebackscatteredsignalsfromthepointsotherthanthekthpoint)andnoiseas Qk=R)]TJ /F6 11.955 Tf 11.95 0 Td[(Pka(k)a(k),(4) whereR=APA,andkisthegridindexofthecurrentSOI.a(k)isthekthcolumnofthesteeringmatrixAasdenedin( 4 ).Then,givenP,x(k)canbeestimatedbyminimizingthefollowingweightedleastsquares(WLS)costfunction, kr)]TJ /F6 11.955 Tf 11.95 0 Td[(x(k)a(k)k2Q)]TJ /F17 5.978 Tf 5.75 0 Td[(1k.(4) Minimizing( 4 )withrespecttox(k)yields x(k)=a(k)Q)]TJ /F10 7.97 Tf 6.59 0 Td[(1kr a(k)Q)]TJ /F10 7.97 Tf 6.59 0 Td[(1ka(k) (4) =a(k)R)]TJ /F10 7.97 Tf 6.59 0 Td[(1r a(k)R)]TJ /F10 7.97 Tf 6.59 0 Td[(1a(k), (4) wherethesecondequalityfollowsfrom( 4 )andthematrixinversionlemma.AsstatedinChapter 3 ,theIAAalgorithmmustbeimplementedinaniterativewayanditsinitializationcanbedonebysettingR=Iin( 4 )(seeTable 3-1 fortheimplementationofIAA).ApplyingIAAtoeachfrequencybin,wecangetthesignalestimates^x(k,l),l=1,...,L,k=1,...,K.ThenthesignalwaveformforthekthpointcanbereconstructedbyapplyingtheL-pointinverseFToff^x(k,l)gLl=1,andhencethebackscatteredenergyforthekthpointcanbecalculatedusing( 4 ). 4.5ExperimentalExamples Inthissection,weprovideseveralexperimentalexamplestodemonstratetheperformanceofWGLCapproachesandWIAAapproach.Forcomparison,theimagingresultsofDASisalsoincluded.TheDASschemeestimatesthesignalwaveforms(t)accordingtothekthpointas ^s(t)=PMi=1PMj=1yi,j(k,t) M2,t=0,...N)]TJ /F3 11.955 Tf 11.95 0 Td[(1,(4) 83

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whereyi,j(k,t)isdenedasin( 4 ).Andthecorrespondingbackscatteredenergyisestimatedvia( 4 ). First,weconsideradatasetcontainingseveralwiretargetsarrangedinacomplicatedpattern.ThecompletemultistaticdatasetwasobtainedfromBioacousticsResearchLaboratoryoftheUniversityofIllinoisatUrbana-Champaign.Thedatasetarecollectedusinga64-elementlineararray.Thetransducercenterfrequencyis2.6MHzandthesamplingrateis25MHz.Thesoundvelocityis1450m/s. Fig. 4-3 showsthereconstructedultrasoundimagesobtainedviaDAS,WGLC-1,WGLC-2,WGLC-CandWIAAusingonlythecentral16elementsofthearray(M=16).Theimagesaredisplayedonalogarithmicscalewitha30dBdynamicrange.FromFig. 4-3 (a),wecanseethatDASsufferfrompoorresolutionandhighsidelobelevel.InFigs. 4-3 (A)-(D),wecomparetheimagesobtainedviadifferentWGLCalgorithms.NotethatWGLC-1imagehasalowerbackgroundclutterlevelthanthatobtainedviaWGLC-2.However,WGLC-1haspoorerresolutionandsomewiretargetscannotbeshownclearly.Ontheotherhand,theimageobtainedviaWGLC-ChaslowsidelobelevelsimilarlytoWGLC-1andhighresolutionsimilarlytoWGLC-2.Fig. 4-3 (E)showstheimageobtainedviaWIAA.AscanbeseenfromFigs. 4-3 (D)and(E),WIAAimagegiveshigherresolutionwhilehighersidelobelevelaswellthanWGLC-Cimage.Also,alltheWGLCapproachesandWIAAprovidemuchbetterperformancethanDAS. Wenextexamineaheartphantom,whichwasobtainedformtheBiomedicalUltrasonicsLaboratoryoftheUniversityofMichiganatAnnArbor.Thedataarecollectedusinga64-elementlineararray,whileonlythecentral16elementsareusedforthefollowingimagingresults.Thetransducerfrequencyis3.333MHzandthesamplingrateis17.76MHz.Thesoundvelocityis1540m/s.ThereconstructedimagesobtainedviaDAS,WGLC-CandWIAAareshowninFig. 4-4 .Theimagesaredisplayedonalogarithmicscalewitha40dBdynamicrange.Forthisexample,WGLC-1imagehasalowerbackgroundclutterlevelthanthatobtainedviaWGLC-2.However,WGLC-1is 84

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unabletoshowsomepartsoftheimage.Duetothecombinedeffect,WGLC-CprovidesthewholepictureoftheimageandshowslowerbackgroundclutterthanWGLC-2,whereasitsbackgroundclutterlevelisstillrelativelyhigh.Ontheotherhand,WIAAproducemuchbetterimagingresultthanbothDASandWGLC-C. 4.6Conclusions Wehavepresentedseveraluserparameter-freeapproachestoultrasoundimaging.Inparticular,wehaveextendedthenarrowbandrobustadaptivebeamformersGLCandIAAtowidebandscenario,whichresultinWGLC-1,WGLC-2,WGLC-CandWIAA.Theproposedapproachesarerobusttoarraysteeringvectorerrorsandlimitedsamplesizeproblems.Moreimportantly,theyareuserparameter-freealgorithmsasopposedtomostoftheotherexistingrobustadaptivebeamformingalgorithms,whichmakethemeasytouseinpractice.Theexperimentalexampleshavedemonstratedtheeffectivenessoftheproposedmethods. 85

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AB CD E Figure4-3. Thereconstructedimagesforwirephantom.(A)DAS.(B)WGLC-1.(C)WGLC-2.(D)WGLC-C.(E)WIAA. 86

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AB CD E Figure4-4. Thereconstructedimagesforheartphantom.(A)DAS.(B)WGLC-1.(C)WGLC-2.(D)WGLC-C.(E)WIAA. 87

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CHAPTER5DOPPLERSPECTROGRAMANALYSISOFTHEHUMANGAITVIAANITERATIVEADAPTIVEAPPROACH 5.1Introduction TheDopplerspectrogramanalysisofthehumangaitisusefulforphysicalsecurityandsurveillance[ 45 ].SinceDopplercharacteristicsofthereturnedradarsignalfromamovinghumanaretime-varying(nonstationary),time-frequencyanalysismethodscanbeusedtoobtainthespectrogram,i.e.,thetime-Dopplerfrequencydiagram.Themostwidelyusedmethodisthedata-independentshort-timeFouriertransform(STFT)[ 48 ],whichassumesthatthesignalisstationaryduringashorttimeintervalandthenusesFTtoperformspectralanalysisforthatdatainterval.However,theproblemwithSTFTisthatashorteranalysiswindowleadstobettertimeresolutionbuttoworsefrequencyresolutionaswell,andviceversa.ThistradeoffmakesithardtodiscriminatemicroDopplersintheSTFTspectrogram,especiallywhentheradaroperatingfrequencyisdecreasedtoachievewallpenetration[ 45 ].Toovercomethislimitation,thereassignedmethod[ 83 85 ]hasbeendeveloped,andhasbeenappliedrecentlytomicroDoppleranalysisofthehumangait[ 45 ].ThemethodmakesuseofthephaseinformationofSTFT,togettheinstantaneoustimeandinstantaneousfrequencyofthesignalasbelow: tins=t)]TJ /F3 11.955 Tf 16.68 8.09 Td[(1 2@ @f,(5) fins=1 2@ @t,(5) andthenreassignsthevalueateachpointintheSTFTspectrogramtothecorrespondingposition(tins,fins)1. 1Tobeself-contained,theSTFTandthereassignedapproachesarebrieydescribedinAppendix B 88

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Weproposeanalternativemethod,anovelshort-timeiterativeadaptiveapproach(ST-IAA)toimprovetheSTFTspectrogram.Moreover,amodel-orderselectiontool,thegeneralizedinformationcriterion(GIC)[ 33 51 ]canbeusedinconjunctionwithST-IAAtofurtherimprovethespectrogramquality. Theremainderofthechapterisorganizedasfollows.TheproblemformulationisgiveninSection 5.2 .InSection 5.3 ,wepresenttheST-IAAalgorithm.NumericalandexperimentalexamplescomparingtheperformanceofST-IAA,STFTandthereassignedmethodapproachesaregiveninSection 5.4 .Finally,weconcludethechapterinSection 5.5 5.2ProblemFormulation Letfy(n)gNn=1denotethedatasamplesofthereturnedradarsignalandlety(l)=[y(P(l)]TJ /F3 11.955 Tf 11.95 0 Td[(1)+1)...y(P(l)]TJ /F3 11.955 Tf 11.96 0 Td[(1)+M)]TbethelthdataframeoflengthM,wherePisapositiveintegerthatdenotesthenumberofnonoverlappingdatasamplesbetweentwoconsecutiveframes(intheexamplesectionweuseP=M=10).Assumingthatthefrequencycomponentsofthesignalareinvariantovertheintervalofeachdataframe,apossibledatamodelfory(l)hasthefollowingform: y(l)=As(l)+e(l),l=1,...,L(5) whereA,[a1a2...aK],e(l)denotesanoiseterm,andL=b(N)]TJ /F6 11.955 Tf 13.1 0 Td[(M)=Pc+1isthenumberofavailableframes.ThecolumnvectorsinAaredenedasak=1ej2fk...ej(M)]TJ /F10 7.97 Tf 6.58 0 Td[(1)2fkT,k=1,...,K,wherefkisthefrequencyvalueatthekthgridpoint,andKdenotesthenumberoftheuniformfrequencygridpointscoveringthefrequencyband[)]TJ /F6 11.955 Tf 9.3 0 Td[(fs=2,fs=2](fsisthesamplingfrequency).Finallys(l)=[s1(l)...sK(l)]Tisthevectorcomprisingtheamplitudesofthepossiblefrequencycomponentsinthel-thframedata.Ourgoalistoestimatefsk(l)gandformaspectrogramofthepowerdistributionPk(l)=jsk(l)j2,k=1,...,K,l=1,...,L,foreachfrequencygridpointkandeachtimeintervall.Inwhatfollows,wefocuson 89

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theanalysisofagenericdataframeandthereforetheindexlisomittedfornotationalsimplicity. 5.3ST-IAA TheST-IAAalgorithm2estimatesskusingtheweightedleastsquarescriterion[ 32 36 ]: [y)]TJ /F9 11.955 Tf 11.96 0 Td[(aksk]Q)]TJ /F10 7.97 Tf 6.59 0 Td[(1k[y)]TJ /F9 11.955 Tf 11.96 0 Td[(aksk],(5) whereQkistheinterference(thefrequencycomponentsinyatfrequenciesotherthanfk)andnoisecovariancematrixandisdenedasfollows: Qk=R)]TJ /F6 11.955 Tf 11.95 0 Td[(Pkakak.(5) IntheaboveequationR=APAwithPdenotingaKKdiagonalmatrixwhosediagonalelementsareequaltothepowersfPkgKk=1ateachfrequencygridpoint.Minimizing( 5 )withrespecttoskyields: ^sk=(akQ)]TJ /F10 7.97 Tf 6.58 0 Td[(1ky)=(akQ)]TJ /F10 7.97 Tf 6.59 0 Td[(1kak),(5) whichwouldrequiretheinversionofQkforeveryk.Toreducethecomputationalcomplexity,wecanmakeuseofthematrixinversionlemmatogetanalternativeexpressionfortheaboveestimate: ^sk=(akR)]TJ /F10 7.97 Tf 6.58 0 Td[(1y)=(akR)]TJ /F10 7.97 Tf 6.58 0 Td[(1ak).(5) Since^skin( 5 )dependsonRandthereforeitdependsontheveryquantitiesfskgthatwewanttoestimate,theequation( 5 )mustbeimplementedinaniterativemanner(refertoTable 3-1 ).Thisiterativeprocesscanbeinitializedviatheestimatein( 5 )correspondingtoR=I.ThisinitialestimateisinfacttheSTFTwitharectangular 2ItsstationarycounterpartIAA,isdescribedinChapter 3 inthecontextofbeamforming.Seealso[ 32 36 ]. 90

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window.Thedata-adaptiveST-IAAisexpectedtohavehigherfrequencyresolutionandlowersidelobesthanthedata-independentSTFT,assuggestedbytheanalysisofthestationarycasein[ 32 36 ]. Usually,thenumberoffrequencygridpointsKismuchlargerthantheactualnumberoffrequencycomponents.Therefore,theestimatesobtainedbyST-IAAforagiventimeintervalgenerallyyieldsomedominantpeaksaroundthefrequenciescorrespondingtotheactualcomponents.TodiscernthemicroDopplercharacteristicsmoreclearly,itmaybedesirabletoshowonlythevaluesofthosepeaks.Toachievethisobjective,weincorporateamodel-orderselectiontool,theGIC[ 33 51 ]intoST-IAA.TheresultiswhatwecallST-IAA-GIC.Todescribethisalgorithmbriey,letPdenotethesetofindicesofthepeaksoftheST-IAAspectrum,andletIdenoteasubsetofP.TheGICvalueforIiscalculatedas: GICI()=2Mln ky)]TJ /F13 11.955 Tf 11.95 11.35 Td[(Xj2Iaj^sjk2!+ln(2M),(5) whereisthesizeofI,kkdenotestheEuclideanvectornorm,^sjisprovidedbyST-IAA,andisaparameterthatcontrolsthepenaltyterm,i.e.,thesecondtermontherighthandsideof( 5 ),whichissetto0.1inthefollowingexamples.ThesetIofsignicantpeaksofthespectrumfPk=jskj2gisdeterminedbyminimizingGICI()withrespecttoIP. 5.4ExamplesonHumanGaitAnalysis Inthefollowingexamples,weuseST-IAAandST-IAA-GICforhumangaitanalysisandcomparetheresultingspectrogramswiththoseobtainedviaSTFTandthereassignedmethod[ 45 85 ]. Firstweconsidersimulateddata,inthecaseofwhichthehumanbodyisrepresentedby12connectedparts(modeledasspheres,cylindersandellipsoids)aslistedinFig. 5-1 andthekineticsofthehumanbodypartsareaccordingtotheThalmannmodel[ 47 86 ].Therelativepositionsofthebodypartsagainsttimearegivenbythemodel. 91

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Thenradarequationisusedtocalculatethereectionofeachbodypartandallthereectionsaresummedupateachtimesample.Weconsiderascenarioinwhichacontinuous-waveradaroperatingat2.4GHzisused,andahumanapproachestheradarataspeedof1.3m/s.SpectrogramsofthesedatahavebeenobtainedviavariousalgorithmsforN=1500,M=30(correspondingtotimeintervalsequalto3sand0.06sforfs=500Hz,respectively)andK=626.BurstymovementsthatrequirenetimeresolutionshouldbedetectableusingthissmallM.TheSTFTspectrograminFig. 5-1 (A)isalmostcompletelysmearedduetopoorfrequencyresolution,andthereassignedspectrograminFig. 5-1 (B)alsofailstodiscernvariousmicroDopplerstracks.Ontheotherhand,thespectrogramsinFig. 5-1 (C)and 5-1 (D),whichcorrespondtoST-IAAandST-IAA-GIC,respectively,aremuchclearerwiththelattershowingthebestperformance.Specically,ST-IAA-GICisabletoresolve11distinctmicroDopplertracks(ofthetotalof12tracks)asshowninFig. 5-1 (D)3.TheheadDopplerdoesnotshowupseparatelyinFig. 5-1 (D)becauseitisbasicallythesameasthetorsoDoppler,astheheadandthetorsomoveusuallytogether. Next,weconsiderasimilarscenariototheoneinthepreviousexamplebutformeasuredhumangaitdata.Fig. 5-2 isthecounterpartofFig. 5-1 (alsoforN=1500,M=30andK=626).Onceagain,thespectrogramsobtainedviaST-IAAandST-IAA-GICshowimprovedresolutionperformancefordiscerningthemicroDopplertracks. 5.5Conclusions Ashort-timeiterativeadaptiveapproach(ST-IAA)hasbeenproposedtoobtainaDopplerspectrogramforawalkinghuman.Inaddition,theGIChasbeenincorporated 3Wehavethesimulateddataforeachindividualbodypartonebyone.Byusingthesedata,wecanobtaintheDopplertrackforeachpartseparately.ThenwecanusethisinformationtoidentifythespecicDopplertrackinacompositespectrogram. 92

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intoST-IAAtoimprovethespectrogramquality.TheresultsobtainedviaST-IAAandST-IAA-GIChavebeencomparedwiththeSTFTandthereassignedspectrogramresultsusingbothsimulatedandmeasuredhumangaitdata.IthasbeendemonstratedthattheproposedapproacheshavethenecessaryresolutionfordiscriminatingthevariousmicroDopplertracksassociatedwiththemovementsofdifferentbodyparts. 93

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AB CD Figure5-1. Spectrogramsofthesimulatedhumangaitdata.(A)STFT.(B)Thereassignedtransform.(C)ST-IAA.(D)ST-IAA-GIC. 94

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AB CD Figure5-2. Spectrogramsofthemeasuredhumangaitdata.(A)STFT.(B)Thereassignedtransform.(C)ST-IAA.(D)ST-IAA-GIC. 95

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CHAPTER6COVARIANCE-BASEDAPPROACHESTOAEROACOUSTICNOISESOURCEANALYSIS 6.1Introduction Themainchallengefortheaeroacousticnoisesourceanalysisproblemistoproperlyseparatethenoisesourceofinterest(SOI)powerinformationfromtheextraneousnoisesourcesatthemicrophonemeasurements. TorejecttheextraneousnoiseandprovideaccurateSOIpowerestimation,numerousmethodologiesusingmultiplemicrophonemeasurementshavebeenproposed[ 54 57 59 ].Recently,aleast-squares(LS)algorithmhasbeenpresentedtosolvetheproblem[ 60 ].However,asdiscussedin[ 60 ],lowsignal-to-noiseratio(SNR)conditionsatoneormoremicrophonesmaydegradetheLSperformancesignicantly. Inthischapter,wepresentseveralalternativeapproachestoSOIpowerestimationbasedonthespectraldensitycovariancematrixofthemicrophoneoutputs.OneoftheapproachesisbasedonoptimizingthemaximumlikelihoodcriterionviausingtheNewton'smethod.Theotherapproaches,referredtoastheFrobeniusnormandRank-1methods,employthecyclicoptimizationalgorithm[ 61 ]tosolvetheproblem. Theremainderofthechapterisorganizedasfollows.ThedatamodelandtheproblemformulationaregiveninSection 6.2 .AfterintroducingtheLSmethodinSection 6.3 ,threecovariance-basednoisepowerestimationtechniquesarepresentedinSection 6.4 .Next,theCramer-RaoBound(CRB)oftheSOIpowerestimatesisderivedinSection 6.5 .InSection 6.6 ,severalnumericalandexperimentalexamplesareprovidedtoevaluatetheperformanceoftheproposedalgorithms.Finally,concludingremarksaregiveninSection 6.7 6.2DataModelandProblemFormulation Considerasingle-inputmultiple-output(SIMO)systemasshowninFig. 6-1 [ 60 ].ThemodelrepresentsanM-microphonemeasurementscenarioofanacousticeld 96

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inputcorruptedbyextraneousnoise, ~ym(t)=~um(t)+~nm(t)=~hm(t)~x(t)+~nm(t),m=1,...,M, (6) where~x(t)denotesthesourceofinterest,~hm(t),~um(t),~nm(t)and~ym(t)denotethechannelresponse,channeloutput,extraneousnoisecontaminationandobservedmeasurementforthemthchannel,respectively. Figure6-1. Thesingle-inputmultiple-outputsystemforaeroacousticnoiseanalysis. Inthepreprocessingstage,thetime-domainobserveddataofeachchannel~ym(t),m=1,...,M,isdividedintoKblocksandeachblockofdataisthenconvertedintonarrowbandfrequencybinsviatheFouriertransform.From( 6 ),themthchanneloutputforaparticularfrequencybinfcanbemodeledas: ym(k,f)=um(k,f)+nm(k,f)=hm(f)x(k,f)+nm(k,f),k=1,...,K, (6) whereym(k,f),um(k,f),andnm(k,f)aretheFouriertransformsof~ym(t),~um(t)and~nm(t)forthekthblock,respectively,andhm(f)istheFouriertransformof~hm(t).We 97

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willestimatetheSOIpowerlevelsindependentlyforeachfrequencybin.Fornotationalsimplicity,thedependenceonfwillbeomitted. Bystackingthedatafromallchannels(m=1,...,M)together,themodelin( 6 )canbewritteninavectorform: y(k)=u(k)+n(k)=hx(k)+n(k),k=1,...,K, (6) whereh=[h1...hM]Tistheunknownspatialchannel,y(k)=[y1(k)...yM(k)]Tistheobserveddatavector,andu(k)andn(k)aredenedsimilarlytoy(k)andarebothunknown. Weassumethattheextraneousnoisen(k)isspatiallyuncorrelatedandisuncorrelatedwiththesourcesignal,whichisreasonableaslongastheextraneousnoiseislimitedtodeviceelectronicnoiseandpressureuctuationsoverthemicrophonesgeneratedbysmall-scaleturbulence[ 60 ].Underthisassumption,weobtainthecovariancematrixofy(k)asfollows: R=U+=hh+diag(), (6) where R=Efy(k)y(k)g,(6) =Efn(k)n(k)g=diag(),(6) withbeingthenoisepowervectoratthechanneloutputs,and U=Efu(k)u(k)g=hh,(6) 98

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byassumingthatEfjx(k)j2g=1sincehisassumedunknownandcanthereforeabsorbthepowerofx(k).Inpractice,wedonothaveexactknowledgeofR.Thenthesamplecovariancematrix^R=1 KPKk=1y(k)y(k)isusedinlieuofR. TheproblemofinterestistoaccuratelyestimatetheSOIpowerateachchanneloutput,i.e.,toestimates=[s1...sM]Twithsm=jhmj2fromtheobserveddatasamplesor^R.Asabyproduct,isestimatedaswell. 6.3LSApproach Beforepresentingthecovariance-basedtechniques,webrieydescribetheexistingLSapproach.Forthediagonalandoff-diagonalelementsin( 6 ),wehave: Ri,i=si+i,i=1,...,M (6) Ri,j=Ui,j,i=1,...,M)]TJ /F3 11.955 Tf 11.95 0 Td[(1,j=i+1,...,M, (6) From( 6 ),weget jRi,jj2=jUi,jj2=sisj,i=1,...,M)]TJ /F3 11.955 Tf 11.95 0 Td[(1,j=i+1,...,M,(6) whereRi,j,Ui,jdenote(i,j)thelementofRandU,respectively,andwehaveusedthefactthatjUi,jj2=jhij2jhjj2=sisj. NotethatRi,iandRi,jareeasilyestimatedfromtheavailabledata.Therefore,( 6 )and( 6 )compriseasystemof[M+M(M)]TJ /F3 11.955 Tf 12.03 0 Td[(1)=2]equationsfor2Munknowns(siandi,i=1,...,M,arethereal-valuedunknowns).WhenM=2,theCOPmethodcanbeusedtoapproximatetheSOIpower[ 54 59 ].WhenM=3,wehavethesamenumberofequationsasthenumberofunknowns,andtheproblemcanbesolvedasdescribedin[ 59 ].ForthegeneralcaseofM>3,anonlinearleastsquares(LS)solutioncanbeobtained,forexample,byusinglsqnonlininMATLAB[ 60 ].WerefertothisapproachastheLSmethod.However,theperformanceofLSispooratlowSNR,asdemonstratedbythenumericalexamplesinSection 6.6 99

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6.4Covariance-BasedTechniques Inthissection,wepresentseveralcovariance-basedtechniquesforSOIpowerestimation.Eachofthesealgorithmstriestondachannelresponsehandadiagonalnoisecovariancematrix=diag()suchthathh+tsthesamplecovariancematrix^Rundercertaincriteria.TheSOIpowerfsmgMm=1canbeestimatedfromtheso-obtainedchannelestimatebyusingthefactthatsm=jhmj2. 6.4.1FrobeniusNormMethod ConsidertheproblemofminimizingtheFrobeniusnormofthedifferencebetween^Randhh+diag(): min,hk^R)]TJ /F9 11.955 Tf 11.96 0 Td[(hh)]TJ /F3 11.955 Tf 11.96 0 Td[(diag()k2s.t.0. (6) WerefertothisapproachastheFrobeniusNorm(FN)method. Wecanusethecyclicoptimizationapproach[ 61 ]tosolvetheaboveproblem.Foragivenh,theproblemin( 6 )canberewrittenas: mink)]TJ /F1 11.955 Tf 11.96 0 Td[(vecd(^R)]TJ /F9 11.955 Tf 11.95 0 Td[(hh)k2s.t.0, (6) whosesolutioniseasilyobtainedas m=max(0,[vecd(^R)]TJ /F9 11.955 Tf 11.95 0 Td[(hh)]m),m=1,...,M,(6) where[]mdenotesthemthelementofthevector. Byxing,( 6 )isreducedto minhkhh)]TJ /F3 11.955 Tf 11.95 0 Td[((^R)]TJ /F3 11.955 Tf 11.96 0 Td[(diag())k2,(6) 100

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andthesolutionis(seee.g.,[ 87 ]): h=q max[^R)]TJ /F3 11.955 Tf 11.96 0 Td[(diag()]vmax[^R)]TJ /F3 11.955 Tf 11.95 0 Td[(diag()],(6) wheremax[A]andvmax[A]denotethedominanteigenvalueandeigenvectorofamatrixA,respectively. Theinitialvalueofhcanbeobtainedbyassumingthatthenoisepowerlevelsarethesameforallchannels,i.e.,=I: h=q (max(^R))]TJ /F4 11.955 Tf 11.95 0 Td[()vmax(^R),(6) where =tr(^R))]TJ /F4 11.955 Tf 11.95 0 Td[(max(^R) M)]TJ /F3 11.955 Tf 11.96 0 Td[(1.(6) Thein( 6 )isessentiallythemaximum-likelihoodestimateofthenoisepower(see[ 88 ]formoredetails). Westopthecyclicprocessafteracertainnumberofiterationsorwhenthefollowingstopcriteriaaresatised: js(i+1)m)]TJ /F6 11.955 Tf 11.95 0 Td[(s(i)mj s(i+1)m<,m=1,...,M,(6) j(i+1)m)]TJ /F4 11.955 Tf 11.95 0 Td[((i)mj (i+1)m<,m=1,...,M,(6) wherethesuperscript()(i)denotestheithiteration,andisasmallpositivenumber,e.g.,10)]TJ /F10 7.97 Tf 6.59 0 Td[(6. TheFNalgorithmissummarizedinTable 6-1 Table6-1. TheFNmethod. Initializehby( 6 ). Repeat Givenh,ndtheoptimalsolutionofusing( 6 ). Given,ndtheoptimalsolutionofhusing( 6 ). Untilthestopcriteria( 6 )and( 6 )aresatisedoracertainnumberofiterationsisreached. 101

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6.4.2Rank-1Method Notethathhisarank-onematrix.Therefore,anotherpossiblewayofsolvingtheproblemistondasuchthat^R)]TJ /F3 11.955 Tf 11.31 0 Td[(diag()isclosetoarank-onematrix.Inotherwords,thesumoftheeigenvaluesofthematrix^R)]TJ /F3 11.955 Tf 11.74 0 Td[(diag(),exceptforthedominantone,whichcanbeexpressedas: tr[^R)]TJ /F3 11.955 Tf 11.95 0 Td[(diag()])]TJ /F4 11.955 Tf 11.95 0 Td[(max[^R)]TJ /F3 11.955 Tf 11.95 0 Td[(diag()],(6) shouldbeassmallaspossible.WerefertothisapproachastheRank-1method. Sincemax[^R)]TJ /F3 11.955 Tf 11.96 0 Td[(diag()]isgivenby(seee.g.,[ 87 ]): maxzz[^R)]TJ /F3 11.955 Tf 11.96 0 Td[(diag()]zs.t.kzk=1, (6) theproblemcanbeformulatedasfollows(byalsotakingintoaccountthefactthattheSOIcovariancematrixshouldbep.s.d.andthenoisepowersshouldbenonnegative): maxz,z[^R)]TJ /F3 11.955 Tf 11.96 0 Td[(diag()]z+tr[diag()]s.t.0^R)]TJ /F3 11.955 Tf 11.96 0 Td[(diag()0kzk=1. (6) Again,wecanuseacyclicoptimizationapproachtosolvethisproblem.Byxing,theaboveproblemisreducedto( 6 ),andthesolutionis(seee.g.,[ 87 ]): z=vmax[^R)]TJ /F3 11.955 Tf 11.95 0 Td[(diag()].(6) Then,byxingz,theproblemin( 6 )isreducedto minT[jzj2)]TJ /F9 11.955 Tf 11.96 0 Td[(1M1]s.t.0and^R)]TJ /F3 11.955 Tf 11.96 0 Td[(diag()0, (6) 102

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wherejjdenotestakingelement-wiseabsolutevaluesand1M1isanM1vectorofallones.Theproblemin( 6 )isasemi-deniteprogram(SDP)thatcanbesolvedefcientlyusingpublicoptimizationsoftwaresuchasSeDuMiandYALMIP[ 64 65 ].Oncewegot,hcanbeobtainedbyusing( 6 ). Theinitialvalueofcanbeobtainedbyusing( 6 )andbyassumingthathisgivenby( 6 ),i.e., =vecdh^R)]TJ /F3 11.955 Tf 11.95 0 Td[((max(^R))]TJ /F4 11.955 Tf 11.96 0 Td[()vmax(^R)vmax(^R)i.(6) whereeachelementofin( 6 )isnonnegative. ThestepsoftheRank-1methodaresummarizedinTable 6-2 Table6-2. TheRank-1method. Initializeby( 6 ). Repeat Given,ndtheoptimalsolutionofzusing( 6 ). Givenz,ndtheoptimalsolutionofviaSDPin( 6 ). Untilthestopcriterion( 6 )issatisedoracertainnumberofiterationsisreached. Computehusing( 6 ). 6.4.3MaximumLikelihoodMethod Alternatively,wecanusethemaximumlikelihood(ML)method,whichwillbereferredtoasMLforshort.AssumethattheSOIandthenoisearebothindependentandidenticallydistributed(i.i.d.)Gaussianrandomprocesses.Then, y(k)CN(0,hh+),k=1,...,K.(6) Wecanobtainandhbyminimizingthenegativelog-likelihoodfunctionf(h,): minh,f(h,)=Klogdet(hh+)+Ktr[^R(hh+))]TJ /F10 7.97 Tf 6.58 0 Td[(1]+const. (6) 103

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Aclosed-formsolutionto( 6 )doesnotappeartoexist.TheNewton'smethod[ 89 ]canbeusedinsteadforsolving( 6 ).Let a=264~h375,(6) with ~h=ReT(h)ImT(h)T.(6) Theunknownvectoracanbeestimatedviaiteratingthefollowingequation: a(i+1)=a(i))]TJ /F3 11.955 Tf 11.96 0 Td[([H(i)])]TJ /F10 7.97 Tf 6.58 0 Td[(1d(i),(6) wherea(i)istheestimateofaattheithiteration. In( 6 ),disthegradientvector,i.e., d=264@f @~h@f @375.(6) Wehave(seeAppendix C fordetailedderivations) @f @~h=2K~)]TJ /F10 7.97 Tf 6.59 0 Td[(1~h 1+~hT~)]TJ /F10 7.97 Tf 6.58 0 Td[(1~h)]TJ /F3 11.955 Tf 13.15 8.09 Td[(2K~)]TJ /F10 7.97 Tf 6.58 0 Td[(1~R~)]TJ /F10 7.97 Tf 6.59 0 Td[(1~h 1+~hT~)]TJ /F10 7.97 Tf 6.59 0 Td[(1~h+2K(~hT~)]TJ /F10 7.97 Tf 6.59 0 Td[(1~R~)]TJ /F10 7.97 Tf 6.59 0 Td[(1~h)~)]TJ /F10 7.97 Tf 6.58 0 Td[(1~h (1+~hT~)]TJ /F10 7.97 Tf 6.58 0 Td[(1~h)2, (6) and @f @=Kvecd(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1)]TJ /F6 11.955 Tf 11.95 0 Td[(Kvecdh(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1^R(hh+))]TJ /F10 7.97 Tf 6.58 0 Td[(1i. (6) where ~=26400375,(6) 104

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and ~R=264Re(^R))]TJ /F3 11.955 Tf 11.29 0 Td[(Im(^R)Im(^R)Re(^R)375.(6) Moreover,HistheHessianmatrix H=264@2f @~h@~hT@2f @~h@T@2f @@~hT@2f @@T375.(6) Wehavethat(seeAppendix C fordetails) @2f @~h@~hT=2K~)]TJ /F10 7.97 Tf 6.59 0 Td[(1 1+~hT~)]TJ /F10 7.97 Tf 6.59 0 Td[(1~h)]TJ /F3 11.955 Tf 13.15 8.09 Td[(4K~)]TJ /F10 7.97 Tf 6.59 0 Td[(1~h~hT~)]TJ /F10 7.97 Tf 6.59 0 Td[(1 (1+~hT~)]TJ /F10 7.97 Tf 6.59 0 Td[(1~h)2)]TJ /F3 11.955 Tf 20.46 8.09 Td[(2K~)]TJ /F10 7.97 Tf 6.59 0 Td[(1~R~)]TJ /F10 7.97 Tf 6.58 0 Td[(1 1+~hT~)]TJ /F10 7.97 Tf 6.58 0 Td[(1~h+4K~)]TJ /F10 7.97 Tf 6.59 0 Td[(1~R~)]TJ /F10 7.97 Tf 6.58 0 Td[(1~h~hT~)]TJ /F10 7.97 Tf 6.59 0 Td[(1 (1+~hT~)]TJ /F10 7.97 Tf 6.59 0 Td[(1~h)2+2K(~hT~)]TJ /F10 7.97 Tf 6.59 0 Td[(1~R~)]TJ /F10 7.97 Tf 6.58 0 Td[(1~h)~)]TJ /F10 7.97 Tf 6.59 0 Td[(1 (1+~hT~)]TJ /F10 7.97 Tf 6.59 0 Td[(1~h)2+4K~)]TJ /F10 7.97 Tf 6.59 0 Td[(1~h~hT~)]TJ /F10 7.97 Tf 6.58 0 Td[(1~R~)]TJ /F10 7.97 Tf 6.59 0 Td[(1 (1+~hT~)]TJ /F10 7.97 Tf 6.58 0 Td[(1~h)2)]TJ /F3 11.955 Tf 20.46 8.09 Td[(8K(~hT~)]TJ /F10 7.97 Tf 6.59 0 Td[(1~R~)]TJ /F10 7.97 Tf 6.58 0 Td[(1~h)~)]TJ /F10 7.97 Tf 6.59 0 Td[(1~h~hT~)]TJ /F10 7.97 Tf 6.59 0 Td[(1 (1+~hT~)]TJ /F10 7.97 Tf 6.59 0 Td[(1~h)3, (6) @2f @@T=)]TJ /F6 11.955 Tf 9.29 0 Td[(K(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1conj(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1+2KRenh(hh+))]TJ /F10 7.97 Tf 6.58 0 Td[(1^R(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1iconj(hh+))]TJ /F10 7.97 Tf 6.58 0 Td[(1o, (6) whereandconj()denotetheHadamardmatrixproductandthecomplexconjugate,respectively, @2f @~h@T="Re@g() @TTIm@g() @TT#T,(6) with @g() @T=)]TJ /F3 11.955 Tf 9.3 0 Td[(2K(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1diag(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1h+2K(hh+))]TJ /F10 7.97 Tf 6.58 0 Td[(1diagh(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1^R(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1hi+2Kh(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1^R(hh+))]TJ /F10 7.97 Tf 6.58 0 Td[(1idiag(hh+))]TJ /F10 7.97 Tf 6.58 0 Td[(1h. (6) 105

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ThestepsoftheMLmethodaresummarizedinTable 6-3 Table6-3. TheMLmethod. InitializeandhbyusingthesolutionsofFN. Repeat Updateandhusing( 6 ). Untilthestopcriteria( 6 )and( 6 )aresatisedoracertainnumberofiterationsisreached. 6.5Cramer-RaoBound CRBisalowerboundonthemean-square-error(MSE)ofanyunbiasedestimate.Inthissection,wederivetheCRBoftheSOIpowerestimatess,andwilluseitasaperformancebenchmarkforourproposedalgorithmsinSection 6.6 First,weconsidertheFisherinformationmatrix(FIM)of~hand,whichcanbewrittenas[ 90 ]: FIM(~h,)=E8><>:264@2f @~h@~hT@2f @~h@T@2f @@~hT@2f @@T3759>=>;,(6) wherethesecond-orderderivativesarethesameasthoseexpressedin( 6 )-( 6 ).Theexpectationin( 6 )canbeobtainedbyusingE(^R)=hh+. Lets=[jh1j2...jhMj2]Tand=[angle(h1)...angle(hM)]Tbethepowersandanglesofthecomplex-valuedchannels,respectively.TocomputetheFIMwith 106

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respecttos,weneedtocomputethefollowingJacobianmatrix: J=264@~h @sT@~h @T@~h @T@ @sT@ @T@ @T375=2666664@Re(h) @sT@Re(h) @T0@Im(h) @sT@Im(h) @T000I3777775=266664diag[Re(h).=(2s)])]TJ /F1 11.955 Tf 9.3 0 Td[(diag[Im(h)]0diag[Im(h).=(2s)]diag[Re(h)]000I377775, (6) where.=denoteselement-wisedivision. TheFIMwithrespecttos,andcanbewrittenas: FIM(s,,)=JTFIM(~h,)J.(6) Notethatthereisaphaseambiguityintheestimateofh.Hence,theso-obtainedFIMisrankdecient.Tosolvetheproblem,weconstrainoneoftheelementsin(e.g.,therstelement)tobezerotoeliminatetheambiguity,whichwillnotaffectthepowerestimationresults.Consequently,thereareonly3M)]TJ /F3 11.955 Tf 12.2 0 Td[(1unknownsleft.TheassociatedFIMofthe3M)]TJ /F3 11.955 Tf 12.24 0 Td[(1unknownscanbeobtainedbysimplydeletingthe(M+1)thcolumnand(M+1)throwof( 6 ),whichresultsinFIMr(s,,).Then,theCRB(s,,)canbeobtainedbyusingthefactthatCRB(s,,)=FIM)]TJ /F10 7.97 Tf 6.59 0 Td[(1r(s,,).Finally,theCRBforthemthelementofsisthemthdiagonalelementofCRB(s,,). 6.6NumericalandExperimentalExamples WeexaminetheperformanceofthealgorithmspresentedinSection 6.4 usingbothsimulatedandmeasureddata.TheMLmethodisinitializedbytheestimatesobtainedviaFN.MLcouldalsobeinitializedbytheRank-1methodtohaveasimilarperformance,butRank-1isslowerthanFN(seeTable 6-4 ).Inalltheexamples,weterminateRank-1 107

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accordingtothecriterion( 6 ),andterminateFNandMLaccordingtoboth( 6 )and( 6 ).Also,wesetthemaximumnumberofiterationsto20.TheLSmethod[ 60 ]isalsoevaluatedforcomparisonpurposes. 6.6.1NumericalExamples Weassumethateightmicrophones(M=8)areavailableinthesystem.Theaeroacousticmeasurementvectorsfy(k)gKk=1aregeneratedasi.i.d.zero-meancircularlysymmetriccomplexGaussianrandomvectorswithcovariancematrixR,wherethetruepowerlevelsfortheSOI(s)andthenoise()aredenotedbythecirclesinFig. 6-2 ,andthephasesofh(exceptfortherstonebeingzero)arechosedindependentlyfromuniformdistributionbetween0and2. Fig. 6-2 showsthemeanvaluesandthe90%condenceintervals(90%CI,shownasthesmallbarsaroundthemeanvalues)oftheSOIandthenoisepowerestimatesforvariousalgorithmsobtainedfrom1000Monte-Carlotrials,alongwiththetruepowersateachmicrophone.ThenumberofdatasamplesissettoK=2000.Fromthegures,wenotethattheSOIpowerestimationofLSispoor,especiallyforthemicrophoneswithlowSNRs.Ontheotherhand,thecovariance-basedalgorithmsprovidemoreaccurateestimatesintermsofmeanvaluesandCI,andtheresultsofMLandFNareslightlybetterthanthoseofRank-1.Wealsocomparethecomputationaltimeofeachapproachforthisexample,whereaPCwithCPUCore(TM)2DuoE6850(3GHz)wasusedtoperformthesimulations.Thenalresultsareaveragedover1000trialsandsummarizedinTable 6-4 .Asevidenced,FNtakestheleastamountoftimetorun,andMLtakesalittlelongertimesinceitisinitializedbyFN.Ontheotherhand,LSandRank-1demandmuchmorecomputationaltime. Fig. 6-3 showstheMSEsoftheSOIpowerestimatesobtainedfrom1000Monte-Carlotrialsforallthealgorithms,aswellasthecorrespondingCRB,asafunctionofthedatasamplenumberK,atmicrophones4(SNR4=-25dB)and8(SNR8=0dB).Theperformancesfortheothermicrophonesaresimilar.AlltheMSEsdecreaseas 108

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Kgoesup.NotethatMLconvergestotheCRBasKincreases,whichisasexpectedsinceMLisasymptoticallystatisticallyefcientforalargenumberofdatasamples.Also,theresultsforRank-1andFNmethodsareclosetotheCRBforlargedatasamplesizeswithFNperformingbetterthanRank-1. 6.6.2ExperimentalExamples TheexperimentaltrailingedgenoisedatawasobtainedfromaNACA63-215ModBairfoil[ 91 ]attheUniversityofFloridaAeroacousticFlowFacility[ 92 ].TheschematicofthetestsetupisgiveinFig. 6-4 .Atotalof23microphoneswereusedtoobtainthedata.Thesamplingratewassetto65.536KHz,andthelengthofeachdatablockwas4096,resultingina16Hznarrowbandfrequencybin.AHanningwindowwith75%overlapwasappliedtoeachblock,leadingto3326effectivedataaveragesintheconstructionofthesamplecovariancematrix^R.Fig. 6-5 showstheperformancecomparisonoftheSOIpowerestimatesforeachmicrophoneat992Hz.Asobserved,theSOIpowerestimatesofallthealgorithmsexceptLSareconsistentwithoneanother.TheruntimeofeachapproachforthisexampleisagainshowninTable 6-4 .FNandML,onceagain,takelessamountoftimethanRank-1andLS. Fig. 6-6 showstheSOIpowerestimatesobtainedviaallthealgorithmsasafunctionofthefrequencyformicrophones7and20.Asobserved,theresultsofRank-1,FNandMLareconsistentwithoneanotherinmostcases. Table6-4. Computationtimesofvariousmethods. ExampleLSRank-1FNML Simulateddata(average)0.227s0.439s0.005s0.017sMeasureddata1.84s4.65s0.88s0.99s 6.7Conclusions Severalcovariance-basedtechniqueshavebeenpresentedforaeroacousticnoisesourcepowerestimation.Thealgorithmsestimateachannelresponsevectorhandadiagonalnoisecovariancematrixsuchthat(hh+)tsthesamplecovariance 109

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matrix^Rsubjecttovariousconstraints,whichleadtoFN,Rank-1andMLalgorithms.TheaforementionedalgorithmshavebeenevaluatedandcomparedwitheachotherandwiththeLSmethodusingbothsimulatedandmeasureddata.Inthenumericalexamples,thecovariance-basedalgorithmswereshowntoprovideaccurateSOIpowerestimatesevenunderlowSNRconditions.TheCRBoftheSOIpowerestimateshasalsobeenderivedandevaluatedforthesimulateddata.IthasbeenshownthattheMLestimatesapproachtheCRB,andthatFNandRank-1methodsareclosetotheCRBforlargedatasamplesizes.Allthecovariance-basedalgorithmsoutperformLSsignicantly.ExperimentalexamplesshowthattheSOIpowerestimatesobtainedusingtheproposedalgorithmsareconsistentwithoneanother.WehavealsoshownthatFNandMLaremorecomputationallyefcientthanRank-1andLS. 110

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AB CD Figure6-2. Themeanvaluesandthe90%CIoftheestimatedSOIandnoisepowersusingsimulateddatawithM=8.(A)LS.(B)Rank-1.(C)FN.(D)ML. 111

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A B Figure6-3. TheMSEsoftheSOIpowerestimatesandthecorrespondingCRBusingsimulateddatawithM=8.(A)Microphone4,SNR4=-25dB.(B)Microphone8,SNR8=0dB. 112

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Figure6-4. TheexperimentalsetupfortheNACA63-215Mod-Bairfoilacousticmeasurements.CourtesyofFeiLiu. Figure6-5. TheSOIpowerestimatesusingthemeasureddatawithM=23atfrequency992Hz. 113

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A B Figure6-6. TheSOIpowerspectrumestimatesusingthemeasureddatawithM=23.(A)Microphone7.(B)Microphone20. 114

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CHAPTER7CONCLUSIONSANDFUTUREWORK 7.1Conclusions Wesummarizetheissuesaddressedinthedissertationandthecontributionswemade.Wehavedevelopedagenerallinearcombination(GLC)-basedrobustadaptivebeamformer,wheretheconventionalsamplecovariancematrixusedinthestandardCaponBeamforming(SCB)formulationisreplacedbyanenhancedcovariancematrixestimatebasedonashrinkagemethod.Theenhancedcovariancematrixisobtainedbyalinearcombinationofthesamplecovariancematrixandtheidentitymatrixinaminimummean-squarederror(MSE)sense.GLCcanbeinterpretedasadiagonalloadingapproachanditsdiagonalloadinglevelcanbecalculatedfullyautomatically,i.e.,withoutspecifyinganyuserparameter,asopposedtomostoftheotherexistingrobustadaptivebeamformingalgorithms.WehavedemonstratedthatGLCisveryusefulinthecaseofsmallsamplesizes-thecaseinwhichtheusersofadaptivearraysaremostinterested. Inaddition,wehavereviewedandcomparedthefollowinguserparameter-freeadaptivebeamformingmethods:thediagonalloadingapproachesofridgeregressionCaponbeamformers(RRCB)andofmid-way(MW),theshrinkagebaseddiagonalloadingapproachesofconvexcombination(CC)andofGLC,andtheiterativebeamformersoftheiterativeadaptiveapproach(IAA),ofthemaximumlikelihoodbasedIAA(referredtoasIAA-ML)andofmulti-snapshotsparseBayesianlearning(M-SBL).Thesealgorithmshavebeenevaluatedundervariousscenariosaccordingtotheirsignal-to-interference-plus-noiseratio(SINR)aswellastotheirsignalofinterest(SOI)powerestimationandspatialpowerestimationaccuracies.Generalguidelineshavebeenofferedtoassistselectingthemostsuitablealgorithminagivenapplicationscenario. 115

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Wehavepresentedseveraluserparameter-freeapproachestoultrasoundimaging.Inparticular,wehaveextendedtherobustadaptivebeamformersGLCandIAAtoaccommodatethespecicpropertiesofultrasoundimaging,i.e.,widebandsignals,near-eldenvironment,whichresultinwidenbandGLC(WGLC)includingWGLC-1,WGLC-2,WGLC-C,andwidebandIAA(WIAA).Theproposedapproachesarerobustagainstarraysteeringvectorerrorsandlimitedsamplesizeproblems.Moreimportantly,theyareuserparameter-freealgorithms,whichmakethemeasytouseinpractice.Theexperimentalexampleshavedemonstratedtheeffectivenessoftheproposedmethods. Moreover,wehaveproposedashort-timeiterativeadaptiveapproach(ST-IAA)toobtainaDopplerspectrogramforawalkinghuman.Inaddition,amodel-orderselectiontool,thegeneralizedinformationcriterion(GIC)hasbeenincorporatedintoST-IAAtoimprovethespectrogramquality.TheperformancesofST-IAAandST-IAA-GIChavebeendemonstratedusingbothsimulatedandmeasuredhumangaitdata.IthasbeenshownthattheproposedapproacheshavethenecessaryresolutionfordiscriminatingthevariousmicroDopplertracksassociatedwiththemovementsofdifferentbodyparts. Finally,wehavepresentedseveralcovariance-basedtechniquesforaeroacousticnoisesourcepowerestimationundervariouscriteria,whichleadtoFrobeniusnorm(FN)method,aRank-1method,andamaximumlikelihood(ML)method.WehavealsoderivedCramer-RaoBounds(CRB)oftheunbiasedsourcepowerestimates.Theaforementionedalgorithmshavebeenevaluatedandcomparedwitheachotherusingbothsimulatedandmeasureddata.Thenumericalexampleshaveshownthatthecovariance-basedalgorithmsareabletoprovideaccurateSOIpowerestimatesevenunderlowsignal-to-noise(SNR)conditions.Inaddition,ithasbeenshownthattheMLestimatesapproachtheCRB,andthatFNandRank-1methodsareclosetotheCRBforlargedatasamplesizes.TheexperimentalexampleshavedemonstratedthattheSOIpowerestimatesobtainedusingtheproposedalgorithmsareconsistentwithoneanother.WehavealsoshownthatFNandMLarequitecomputationallyefcient. 116

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7.2FutureWork Welistseveralpossiblefutureresearchdirectionsthatarecloselyrelatedtothetopicsdiscussedinthedissertation. InChapter 4 ,wehavepresentedseveraladaptivesignalprocessingapproachestoultrasoundimaging.Therapidlygrowingareaofsparsesignalrepresentationcanbeusedforimagingapplicationsaswell,especiallyiftheultrasoundimageisknowntobesparse(e.g.,forthewirephantomexampleinChapter 4 ).Theprominentmethodsinthisareaincludeconvexoptimizationmethodssuchastheleastabsoluteshrinkageselectionoperator(LASSO)[ 93 ]orbasispursuit(BP)[ 94 ],`1-SVD(singularvaluedecomposition)[ 95 96 ]anditerativemethodssuchasfocalunderdeterminedsystemsolution(FOCUSS)anditsvariants[ 97 102 ].Allofthesemethodsrequiretheproperselectionofoneormoreuserparameters.Also,thesparseBayesianlearning(SBL)approachescanbeusedtoenforcesparsityviaadoptingahierarchicalBayesianmodel[ 34 72 73 ].ThoughtheSBLmethodscanbeuserparameter-free,theyusuallyhavehighcomputationalcomplexities.Recently,aregularizedsparsesignalrecoveryalgorithm,referredtoassparselearningviaiterativeminimization(SLIM)hasbeenappliedtomultiple-inputmultiple-output(MIMO)RadarImaging[ 103 ].SLIMfollowsan`q-quasi-normconstraint(0
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x.ThemainchallengehereisthelargesizeofthematrixA((M2T)K).Wemayexploitthespecialstructure(e.g.,sparsity)ofAtodevelopfastimplementationtechniques. AspresentedinChapter 3 ,theweightedleastsquares(WLS)basedIAAapproachhasseveralattractiveproperties,includingroustness,userparameter-free,abilitytoworkwithfeworevensinglesnapshot.However,fastandlowmemoryimplementationofIAAisdesired.Insomeapplications,e.g.,ultrasoundimaging,thefastimplementationofIAAisveryimportant.Therefore,howtospeeduptheIAAcomputationsmaybeapromisingandimportantfutureresearchtopic. Finally,fortheaeroacousticnoisesourceanalysis(seeChapter 6 ),wemayconsiderextendingthesingle-inputmultiple-output(SIMO)system(withasinglesourceofinterest)toaMIMOsystemwithmultiplesourcesofinterest. 118

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APPENDIXADERIVATIONOFPARAMETER^FORGLCAPPROACH Thisappendixderivestheexpressionfor^in( 2 )(seeSection 2.3 ). Let^rmandrmdenotethemthcolumnsof^RandR,respectively.Then,wehave ^rm=1 NNXn=1y(n)ym(n)(A) and rm=Efy(n)ym(n)g(A) whereym(n)denotesthemthelementofy(n).Inwhatfollows,weomitthesubscriptmfornotationalsimplicity(wewillreinstatethesubscriptwhenneeded).Let x(n)=y(n)ym(n).(A) Fromtheassumptionsmadeonfy(n)gNn=1,fx(n)gNn=1areindependentandidenticallydistributed(i.i.d.)randomvectorswithmean=r.Since =Efk^R)]TJ /F9 11.955 Tf 11.95 0 Td[(Rk2g=MXm=1Efk^rm)]TJ /F9 11.955 Tf 11.96 0 Td[(rmk2g(A) ourgenericproblemistoestimateEfk^r)]TJ /F9 11.955 Tf 12.5 0 Td[(rk2g=Efk(1=N)PNn=1x(n))]TJ /F14 11.955 Tf 12.51 0 Td[(k2g,whichgives Efk1 NNXn=1x(n))]TJ /F14 11.955 Tf 11.96 0 Td[(k2g=Efk1 NNXn=1[x(n))]TJ /F14 11.955 Tf 11.95 0 Td[(]k2g=1 NEfkx(n))]TJ /F14 11.955 Tf 11.95 0 Td[(k2g. (A) ThevarianceEfkx(n))]TJ /F14 11.955 Tf 11.96 0 Td[(k2gin( A )canbeestimatedas 1 NNXn=1kx(n))]TJ /F3 11.955 Tf 13.04 0 Td[(^k2;^=1 NNXn=1x(n)=^r.(A) 119

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Consequently,wecanestimateEfk^rm(n))]TJ /F9 11.955 Tf 11.96 0 Td[(rmk2gas 1 N2NXn=1ky(n)ym(n))]TJ /F3 11.955 Tf 11.13 0 Td[(^rmk2.(A) Basedon( A )and( A ),wecanachievethefollowingestimatefor: ^=1 N2NXn=1MXm=1ky(n)ym(n))]TJ /F3 11.955 Tf 11.14 0 Td[(^rmk2=1 N2NXn=1ky(n)y(n))]TJ /F3 11.955 Tf 13.02 2.65 Td[(^Rk2. (A) Theabovecalculationof^maybesomewhatslow.Alesscomputationalcostcalculationof^canbeobtainedbyobservingthat ^=1 NMXm=11 NNXn=1ky(n)ym(n))]TJ /F3 11.955 Tf 11.13 0 Td[(^rmk2=1 NMXm=1"1 NNXn=1ky(n)k2kym(n)k2)-222(k^rmk2# (A) andthereforethat ^=1 N2NXn=1ky(n)k4)]TJ /F3 11.955 Tf 14.71 8.09 Td[(1 Nk^Rk2.(A) 120

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APPENDIXBSTFTANDREASSIGNEDMETHOD Theshort-timeFouriertransform(STFT)andthereassignedmethodforDopplerspectrogramanalysisaresummarizedinthisappendix(seeChapter 5 ). Givenasignaly(t),itsSTFTusingaGaussianwindowisgivenby[ 45 ]: (t,f)=Zy()e)]TJ /F10 7.97 Tf 6.59 0 Td[((t)]TJ /F11 7.97 Tf 6.59 0 Td[()2=22ej2f(t)]TJ /F11 7.97 Tf 6.58 0 Td[()d=jjej,(B) whereisthetimeextentparameterfortheGaussianwindow.jjandaremagnitudeandphaseoftheSTFT.TheSTFTmethodonlymakesuseofthemagnitudeinformationtoobtainthespectrogram.Whilethereassignedmethodemploysthephaseinformationaswell. Weobtaintwoquantitiesfrom,i.e.,theinstantaneoustimetinsandtheinstantaneousfrequencyfins[ 85 ]: tins=t)]TJ /F3 11.955 Tf 16.68 8.09 Td[(1 2@ @f,(B) fins=1 2@ @t.(B) Let (t,f)=1 Z()]TJ /F6 11.955 Tf 11.96 0 Td[(t)y()e)]TJ /F10 7.97 Tf 6.59 0 Td[((t)]TJ /F11 7.97 Tf 6.59 0 Td[()2=22ej2f(t)]TJ /F11 7.97 Tf 6.59 0 Td[()d.(B) Wecangetclosed-formsolutionsfortinsandfins[ 85 ]: tins=t+Ref g,(B) fins=f+1 2Imf g.(B) Toformthereassignedspectrogram,weneedatransformationF:(t,f)!(tins,fins).Thetransformationorreassignmentpresentedin[ 85 ]issummarizedasfollows.Generateanegridin(t,f)planeandmapeverypointsinto(tins,fins)plane.Thenforeachpixelinthelatterplane,wecounthowmanypointsfallingwithinthegridbinof(tins,fins),whereeachpointisweightedbyitscorrespondingSTFTspectrogram. 121

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APPENDIXCDERIVATIONOFMLAPPROACH Thisappendixgivestherst-orderandsecond-orderderivativesofthenegativelog-likelihoodfunctionf(h,)(see( 6 ))requiredbytheNewton'smethodforMLapproach(seeSection 6.4.3 ). C.1First-orderDerivatives First,weconsidertherst-orderderivativeofthenegativelog-likelihoodfunctionf(h,)withrespecttoh. From( 6 ),wehave f(h,)=Klogdet((hh+))+Ktrh^R(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1i+const. (C) =Klogdet()+Klog(1+h)]TJ /F10 7.97 Tf 6.59 0 Td[(1h)+Ktr(^R)]TJ /F10 7.97 Tf 6.59 0 Td[(1))]TJ /F6 11.955 Tf 11.95 0 Td[(Kh)]TJ /F10 7.97 Tf 6.59 0 Td[(1^R)]TJ /F10 7.97 Tf 6.59 0 Td[(1h 1+h)]TJ /F10 7.97 Tf 6.59 0 Td[(1h+const. (C) wherewehaveuseddet(I+AB)=det(I+BA),tr(AB)=tr(BA)andthematrixinversionlemma(see,e.g.,[ 7 ])forthesecondequality. Wecaneasilyverifythat h)]TJ /F10 7.97 Tf 6.59 0 Td[(1h=~hT~)]TJ /F10 7.97 Tf 6.59 0 Td[(1~h,(C) and h)]TJ /F10 7.97 Tf 6.59 0 Td[(1^R)]TJ /F10 7.97 Tf 6.59 0 Td[(1h=~hT~)]TJ /F10 7.97 Tf 6.59 0 Td[(1~R~)]TJ /F10 7.97 Tf 6.59 0 Td[(1~h,(C) where~h,~and~Rhavebeendenedin( 6 ),( 6 )and( 6 ),respectively. Hence,thenegativelog-likelihoodfunction( C )canbewrittenasafunctionof~h: f(~h,)=K 2logdet(~)+Klog(1+~hT~)]TJ /F10 7.97 Tf 6.59 0 Td[(1~h)+K 2tr(~R~)]TJ /F10 7.97 Tf 6.59 0 Td[(1))]TJ /F6 11.955 Tf 11.95 0 Td[(K~hT~)]TJ /F10 7.97 Tf 6.59 0 Td[(1~R~)]TJ /F10 7.97 Tf 6.58 0 Td[(1~h 1+~hT~)]TJ /F10 7.97 Tf 6.59 0 Td[(1~h+const. (C) 122

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Then,wehave @f @~h=2K~)]TJ /F10 7.97 Tf 6.59 0 Td[(1~h 1+~hT~)]TJ /F10 7.97 Tf 6.58 0 Td[(1~h)]TJ /F3 11.955 Tf 13.15 8.09 Td[(2K~)]TJ /F10 7.97 Tf 6.58 0 Td[(1~R~)]TJ /F10 7.97 Tf 6.59 0 Td[(1~h 1+~hT~)]TJ /F10 7.97 Tf 6.59 0 Td[(1~h+2K(~hT~)]TJ /F10 7.97 Tf 6.59 0 Td[(1~R~)]TJ /F10 7.97 Tf 6.59 0 Td[(1~h)~)]TJ /F10 7.97 Tf 6.58 0 Td[(1~h (1+~hT~)]TJ /F10 7.97 Tf 6.58 0 Td[(1~h)2. (C) Next,wecomputetherst-orderderivativewithrespectto.From( C ),thedifferentialoff(denotedby@f)withrespecttocanbewritten(seee.g.,[ 104 ])asfollows: @f=K@flogdet(hh+)g+K@ntr[^R(hh+))]TJ /F10 7.97 Tf 6.58 0 Td[(1]o=Ktr(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1@)]TJ /F6 11.955 Tf 11.95 0 Td[(Ktrh^R(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1@(hh+))]TJ /F10 7.97 Tf 6.58 0 Td[(1i. (C) Basedontheaboveequation,wecangetthedifferentialoffwithrespecttothemthelementof,denotedbym,asfollows: K(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1m,m@m)]TJ /F6 11.955 Tf 11.95 0 Td[(Kh(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1^R(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1im,m@m, (C) with[]m,mbeingthe(m,m)thelementofamatrix. Then,wereadilyget: @f @=Kvecd(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1)]TJ /F6 11.955 Tf 11.95 0 Td[(Kvecdh(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1^R(hh+))]TJ /F10 7.97 Tf 6.58 0 Td[(1i. (C) 123

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C.2Second-orderDerivatives Fromtherst-orderderivative( C ),wecalculatethesecond-orderderivativeoffwithrespectto~hasfollows: @2f @~h@~hT=2K~)]TJ /F10 7.97 Tf 6.59 0 Td[(1 1+~hT~)]TJ /F10 7.97 Tf 6.59 0 Td[(1~h)]TJ /F3 11.955 Tf 13.15 8.09 Td[(4K~)]TJ /F10 7.97 Tf 6.59 0 Td[(1~h~hT~)]TJ /F10 7.97 Tf 6.59 0 Td[(1 (1+~hT~)]TJ /F10 7.97 Tf 6.59 0 Td[(1~h)2)]TJ /F3 11.955 Tf 20.46 8.09 Td[(2K~)]TJ /F10 7.97 Tf 6.59 0 Td[(1~R~)]TJ /F10 7.97 Tf 6.58 0 Td[(1 1+~hT~)]TJ /F10 7.97 Tf 6.58 0 Td[(1~h+4K~)]TJ /F10 7.97 Tf 6.59 0 Td[(1~R~)]TJ /F10 7.97 Tf 6.58 0 Td[(1~h~hT~)]TJ /F10 7.97 Tf 6.59 0 Td[(1 (1+~hT~)]TJ /F10 7.97 Tf 6.59 0 Td[(1~h)2+2K(~hT~)]TJ /F10 7.97 Tf 6.59 0 Td[(1~R~)]TJ /F10 7.97 Tf 6.58 0 Td[(1~h)~)]TJ /F10 7.97 Tf 6.59 0 Td[(1 (1+~hT~)]TJ /F10 7.97 Tf 6.59 0 Td[(1~h)2+4K~)]TJ /F10 7.97 Tf 6.59 0 Td[(1~h~hT~)]TJ /F10 7.97 Tf 6.58 0 Td[(1~R~)]TJ /F10 7.97 Tf 6.59 0 Td[(1 (1+~hT~)]TJ /F10 7.97 Tf 6.58 0 Td[(1~h)2)]TJ /F3 11.955 Tf 20.46 8.09 Td[(8K(~hT~)]TJ /F10 7.97 Tf 6.59 0 Td[(1~R~)]TJ /F10 7.97 Tf 6.58 0 Td[(1~h)~)]TJ /F10 7.97 Tf 6.59 0 Td[(1~h~hT~)]TJ /F10 7.97 Tf 6.59 0 Td[(1 (1+~hT~)]TJ /F10 7.97 Tf 6.59 0 Td[(1~h)3. (C) Next,wecomputethesecond-orderderivativeoffwithrespectto(see( C )).Thedifferentialofvecd(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1withrespecttois: @vecd(hh+))]TJ /F10 7.97 Tf 6.58 0 Td[(1=vecd@(hh+))]TJ /F10 7.97 Tf 6.58 0 Td[(1=)]TJ /F1 11.955 Tf 9.3 0 Td[(vecd(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1@(hh+))]TJ /F10 7.97 Tf 6.58 0 Td[(1. (C) From( C ),thedifferentialofvecd(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1withrespecttothemthelementof(i.e.,m)is: )]TJ /F3 11.955 Tf 11.95 0 Td[([(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1].mconjf[(hh+))]TJ /F10 7.97 Tf 6.58 0 Td[(1].mg@m,(C) where[].mdenotethemthcolumnofamatrix.Hence,wehave @vecd(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1 @T=)]TJ /F3 11.955 Tf 9.3 0 Td[((hh+))]TJ /F10 7.97 Tf 6.58 0 Td[(1conj(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1.(C) Similarly,thedifferentialofvecdh(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1^R(hh+))]TJ /F10 7.97 Tf 6.58 0 Td[(1iwithrespecttois: @nvecdh(hh+))]TJ /F10 7.97 Tf 6.58 0 Td[(1^R(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1io=)]TJ /F1 11.955 Tf 9.29 0 Td[(vecdh(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1^R(hh+))]TJ /F10 7.97 Tf 6.58 0 Td[(1@(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1i)]TJ /F1 11.955 Tf 19.27 0 Td[(vecdh(hh+))]TJ /F10 7.97 Tf 6.58 0 Td[(1@(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1^R(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1i, (C) 124

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andconsequently, @vecdh(hh+))]TJ /F10 7.97 Tf 6.58 0 Td[(1^R(hh+))]TJ /F10 7.97 Tf 6.58 0 Td[(1i @T=)]TJ /F3 11.955 Tf 9.3 0 Td[(2Renh(hh+))]TJ /F10 7.97 Tf 6.58 0 Td[(1^R(hh+))]TJ /F10 7.97 Tf 6.58 0 Td[(1iconj(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1o. (C) From( C )and( C ),weobtainthesecond-orderderivativeoffwithrespectto: @2f @@T=)]TJ /F6 11.955 Tf 9.29 0 Td[(K(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1conj(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1+2KRenh(hh+))]TJ /F10 7.97 Tf 6.58 0 Td[(1^R(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1iconj(hh+))]TJ /F10 7.97 Tf 6.58 0 Td[(1o. (C) Nowweconsiderthesecond-orderderivativeswithrespectto~hand: @2f @~h@T="@2f @Re(h)@TT@2f @Im(h)@TT#T.(C) Forconvenience,let g()=@f @Re(h)+j@f @Im(h). (C) Therefore, @g() @T=@2f @Re(h)@T+j@2f @Im(h)@T, (C) i.e., @2f @Re(h)@T=Re@g() @Tand@2f @Im(h)@T=Im@g() @T.(C) From( C ),aftersomematrixmanipulations,wehave g()=2K(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1h)]TJ /F3 11.955 Tf 11.96 0 Td[(2K(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1^R(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1h.(C) 125

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Thedifferentialofg()in( C )withrespecttoisasfollows: @g()=)]TJ /F3 11.955 Tf 9.3 0 Td[(2K(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1@(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1h+2K(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1@(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1^R(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1h+2K(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1^R(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1@(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1h. (C) From( C ),theg()differentialwithrespecttothemthelementofis: )]TJ /F3 11.955 Tf 9.3 0 Td[(2K(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1.m(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1hm@m+2K(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1.mh(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1^R(hh+))]TJ /F10 7.97 Tf 6.58 0 Td[(1him@m+2Kh(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1^R(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1i.m(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1hm@m. (C) Hence, @g() @T=)]TJ /F3 11.955 Tf 9.3 0 Td[(2K(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1diag(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1h+2K(hh+))]TJ /F10 7.97 Tf 6.58 0 Td[(1diagh(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1^R(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1hi+2Kh(hh+))]TJ /F10 7.97 Tf 6.59 0 Td[(1^R(hh+))]TJ /F10 7.97 Tf 6.58 0 Td[(1idiag(hh+))]TJ /F10 7.97 Tf 6.58 0 Td[(1h. (C) From( C ),( C )and( C ),@2f @~h@Tcanbeobtainedimmediately. 126

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REFERENCES [1] A.V.Oppenheim,R.W.Schafer,andJ.R.Buck,Discrete-TimeSignalProcess-ing,2nded.UpperSaddleRiver,NJ:PrenticeHall,1999. [2] B.WidrowandS.D.Stearns,AdaptiveSignalProcessing.EnglewoodCliffs,NJ:Prentice-Hall,1985. [3] S.T.Alexander,AdaptiveSignalProcessing:TheoryandApplications.NewYork,NY:Springer-Verlag,1986. [4] J.BenestyandY.Huang,Eds.,AdaptiveSingalProcessing:ApplicationstoReal-WorldProblems.Berlin:Springer-Verlag,2003. [5] H.L.VanTrees,OptimumArrayProcessing:PartIVofDetection,Estimation,andModulationTheory.NewYork,NY:JohnWiley&Sons,2002. [6] B.D.V.VeenandK.M.Buckley,Beamforming:Aversatileapproachtospatialltering,IEEEASSPMagazine,pp.4,April1988. [7] P.StoicaandR.L.Moses,SpectralAnalysisofSignals.UpperSaddleRiver,NJ:Prentice-Hall,2005. [8] J.Capon,Highresolutionfrequency-wavenumberspectrumanalysis,Proceed-ingsoftheIEEE,vol.57,pp.1408,August1969. [9] D.D.FeldmanandL.J.Grifths,Aprojectionapproachforrobustadaptivebeamforming,IEEETransactionsonSignalProcessing,vol.42,pp.867,April1994. [10] B.Widrow,K.Duvall,R.Gooch,andW.Newman,Signalcancellationphenomenainadaptiveantennas:Causesandcures,IEEETransactionsonAntennasandPropagation,vol.AP-30,pp.469,May1982. [11] A.Steele,Comparisonofdirectionalandderivativeconstraintsforbeamformerssubjecttomultiplelinearconstraints,IEEProceedings,Pts.FandH,vol.130,pp.41,February1983. [12] J.E.Hudson,AdaptiveArrayprinciples.London,U.K.:PeterPeregrinus,1981. [13] H.Cox,R.M.Zeskind,andM.M.Owen,Robustadaptivebeamforming,IEEETransactionsonAcoustics,Speech,andSignalProcessing,vol.35,pp.1365,October1987. [14] J.LiandP.Stoica,Eds.,RobustAdaptiveBeamforming.NewYork,NY:JohnWiley&Sons,2005. [15] C.-C.LeeandJ.-H.Lee,Robustadaptivearraybeamformingundersteeringvectorerrors,IEEETransactionsonAntennasandPropagation,vol.45,pp.168,January1997. 127

PAGE 128

[16] B.D.Carlson,Covariancematrixestimationerrorsanddiagonalloadinginadaptivearrays,IEEETransactionsonAerospaceandElectronicSystems,vol.24,pp.397,July1988. [17] S.A.Vorobyov,A.B.Gershman,andZ.-Q.Luo,Robustadaptivebeamformingusingworst-caseperformanceoptimization,IEEETransactionsonSignalPro-cessing,vol.51,pp.313324,February2003. [18] P.Stoica,Z.Wang,andJ.Li,RobustCaponbeamforming,IEEESignalProcess-ingLetters,vol.10,pp.172,June2003. [19] J.Li,P.Stoica,andZ.Wang,OnrobustCaponbeamforminganddiagonalloading,IEEETransactionsonSignalProcessing,vol.51,pp.1702,July2003. [20] R.G.LorenzandS.P.Boyd,Robustminimumvariancebeamforming,IEEETransactionsonSignalProcessing,vol.53,pp.1684,May2005. [21] S.VerduandH.V.Poor,Minimaxrobustdiscrete-timematchedlters,IEEETransactionsonCommunications,vol.31,pp.208,Feburary1983. [22] J.Li,P.Stoica,andZ.Wang,OnrobustCaponbeamforminganddiagonalloading,IEEETransactionsonSignalProcessing,vol.51,pp.1702,July2003. [23] L.Du,J.Li,andP.Stoica,Fullyautomaticcomputationofdiagonalloadinglevelsforrobustadaptivebeamforming,toappearinIEEETransactionsonAerospaceandElectronicSystems,2010. [24] J.Li,L.Du,andP.Stoica,Fullyautomaticcomputationofdiagonalloadinglevelsforrobustadaptivebeamforming,The2008IEEEInternationalConferenceonAcoustics,Speech,andSignalProcessing,LasVegas,Nevada,USA,March2008. [25] O.LedoitandM.Wolf,Awell-conditionedestimatorforlarge-dimensionalcovariancematrices,JournalofMultivariateAnalysis,vol.88,pp.365,2004. [26] O.LedoitandM.Wolf,Improvedestimationofthecovariancematrixofstockreturnswithanapplicationtoportfolioselection,JournalofEmpiricalFinance,vol.10,pp.603,2003. [27] J.SchaferandK.Strimmer,Ashrinkageapproachtolarge-scalecovariancematrixestimationandimplicationsforfunctionalgenomics,StatisticalApplicationsinGeneticsandMolecularBiology,vol.4,Art.No.32,2005. [28] Y.Selen,R.Abrahamsson,andP.Stoica,Automaticrobustadaptivebeamformingviaridgeregression,SignalProcessing,vol.88,pp.33,2008. 128

PAGE 129

[29] P.Stoica,J.Li,andX.Tan,OnspatialpowerspectrumandsignalestimationusingthePisarenkoframework,IEEETransactionsonSignalProcessing,vol.56,pp.5109,October2008. [30] A.E.Hoerl,R.W.Kennard,andK.F.Baldwin,Ridgeregression:somesimulations,CommunicationinStatistics:TheoryandMethods,vol.4,pp.105,1975. [31] V.F.Pisarenko,Ontheestimationofspectrabymeansofnon-linearfunctionsofthecovariancematrix,GeophysicalJournaloftheRoyalAstronomicalSociety,vol.28,pp.511,June1972. [32] T.Yardibi,J.Li,P.Stoica,M.Xue,andA.B.Baggeroer,Sourcelocalizationandsensing:Anonparametriciterativeadaptiveapproachbasedonweightedleastsquares,toappearinIEEETransactionsonAerospaceandElectronicSystems,2010. [33] T.Yardibi,J.Li,andP.Stoica,Nonparametricandsparsesignalrepresentationsinarrayprocessingviaiterativeadaptiveapproaches,42ndAsilomarConferenceonSignals,SystemsandComputers,PacicGrove,CA,October2008. [34] D.P.WipfandB.D.Rao,AnempiricalBayesianstrategyforsolvingthesimultaneoussparseapproximationproblem,IEEETransactionsonSignalProcessing,vol.55,no.7,pp.3704,2007. [35] D.P.WipfandS.Nagarajan,Beamformingusingtherelevancevectormachine,ICML'07:Proceedingsofthe24thinternationalconferenceonMachinelearning,pp.1023,2007. [36] L.Du,T.Yardibi,J.Li,andP.Stoica,Reviewofuserparameter-freerobustadaptivebeamformingalgorithms,DigitalSignalProcessing,vol.19,no.4,pp.567,2009. [37] H.WangandM.Kaveh,Coherentsignal-subspaceprocessingforthedetectionandestimationofanglesofarrivalofmultiplewide-bandsources,IEEETransac-tionsonAcoustics,Speech,andSignalProcessing,vol.ASSP-33,pp.823,August1985. [38] H.HungandM.Kaveh,Focusingmatricesforcoherentsignal-subspaceprocessing,IEEETransactionsonAcoustics,Speech,andSignalProcessing,vol.ASSP-36,pp.1272,August1988. [39] J.KrolikandD.Swingler,Multiplebroad-bandsourcelocationusingsteeredcovariancematrices,IEEETransactionsonAcoustic,SpeechandSignalProcess-ing,vol.37,pp.1481,October1989. 129

PAGE 130

[40] J.KrolikandD.Swingler,Focusedwide-bandarrayprocessingviaspatialresampling,IEEETransactionsonAcoustic,SpeechandSignalProcessing,vol.38,pp.356,Feburary1990. [41] J.A.MannandW.F.Walker,Aconstrainedadaptivebeamformerformedicalultrasound:Initialresults,ProceedingsofIEEEUltrasnoicSymposium,vol.2,pp.1807,2002. [42] J.Synnevag,A.Austeng,andS.Holm,Adaptivebeamformingappliedtomedicalultrasoundimaging,IEEETransactionsonUltrasonics,Ferroelectrics,andFrequencyControl,vol.54,no.8,pp.1606,2007. [43] J.Synnevag,A.Austeng,andS.Holm,Minimumvarianceadaptivebeamformingappliedtomedicalultrasoundimaging,IEEEUltrasonicsSymposium,vol.2,pp.1199,September2005. [44] Z.Wang,J.Li,andR.Wu,Time-delayandtime-reversalbasedrobustCaponbeamformersforultrasoundimaging,IEEETransactionsonMedicalImaging,vol.24,no.10,pp.1308,2005. [45] S.S.RamandH.Ling,Analysisofmicrodopplersfromhumangaitusingreassignedjointtime-frequencytransform,ElectronicsLetters,vol.43,pp.1309,November2007. [46] J.L.G.E.F.GrenekerandW.S.Marshall,High-resolutiondopplermodelofthehumangait,ProceedingsofSPIE,RadarSensorTechnologyandDataVisualization,pp.8,2002. [47] D.P.VanandF.C.A.Groen,Humanwalkingestiamtionwithradar,IEEproceed-ingsRadar,SnonarandNavigation,pp.356,2003. [48] L.Cohen,Time-FrequencyAnalysis.EnglewoodCliffs,NJ:Prentice-Hall,1995. [49] V.C.ChenandH.Ling,Time-FrequencyTransformsforRadarImagingandSignalAnalysis.Boston,MA:ArtechHouse,2002. [50] L.Du,J.Li,P.Stoica,H.Ling,andS.S.Ram,Dopplerspectrogramanalysisofhumangaitviaiterativeadaptiveapproach,IETElectronicsLetters,vol.45,pp.186,January2009. [51] P.StoicaandY.Selen,Model-orderselection:areviewofinformationcriterionrules,IEEESignalProcessingMagazine,vol.21,pp.36,July2004. [52] R.W.Paterson,P.G.Vogt,M.R.Fink,andC.L.Munch,Vortexnoiseofisolatedairfoils,JournalofAircraft,vol.10,no.5,pp.296,1973. [53] J.C.YuandM.C.Joshi,Onsoundradiationfromthetrailingedgeofanisolatedairfoilinauniformow,5thAIAAAeroacousticsConference,Seattle,WA,USA,March1979. 130

PAGE 131

[54] T.F.BrooksandT.H.Hodgson,Trailingedgenoisepredictionfrommeasuredsurfacepressures,JournalofSoundandVibration,vol.78,no.1,pp.69,1981. [55] W.K.BlakeandJ.L.Gershfeld,Theaeroacousticoftrailingedges.InFrontiersinExperimentalFluidMechanics,editedbyM.Gad-el-Hak.NewYork,NY:Springer-Verlag,1989. [56] T.F.Brooks,D.S.Pope,andM.A.Marcolini,Airfoilself-noiseandprediction,NASAReferencePublication1218,1989. [57] F.V.HutchesonandT.F.Brooks,Measurementoftrailingedgenoiseusingdirectionalarrayandcoherentoutputpowermethods,8thAIAA/CEASAeroa-cousticsConference,Breckenridge,CO,USA,June2002. [58] J.Y.Chung,Rejectionofownoiseusingacoherencefunctionmethod,JournaloftheAcousticalSocietyofAmerica,vol.62,no.2,pp.388,1977. [59] J.S.BendatandA.G.Piersol,RandomDataAnalysisandMeasurementProce-dures,3rdedition.NewYork,NY:JohnWileyandSons,2000. [60] C.Bahr,T.Yardibi,F.Liu,andL.Cattafesta,Ananalysisofdifferentmeasurementtechniquesforairfoiltrailingedgenoise,14thAIAA/CEASAeroacousticsConfer-ence,Vancouver,BC,Canada,May2008. [61] P.StoicaandY.Selen,Cyclicminimizers,majorizationtechniques,andexpectation-maximizationalgorithm:Arefresher,IEEESignalProcessingMagazine,pp.112,January2004. [62] J.Li,P.Stoica,andZ.Wang,DoublyconstrainedrobustCaponbeamformer,IEEETransactionsonSignalProcessing,vol.52,pp.2407,September2004. [63] P.Stoica,J.Li,X.Zhu,andJ.R.Guerci,Onusingaprioriknowledgeinspace-timeadaptiveprocessing,IEEETransactionsonSignalProcessing,vol.56,pp.2598,June2008. [64] J.F.Sturm,UsingSeDuMi1.02,aMATLABtoolboxforoptimizationoversymmetriccones,OptimizationMethodsandSoftwareOnline,vol.11-12,pp.625,Oct.1999.Availablefromhttp://sedumi.ie.lehigh.edu. [65] J.Lofberg,YALMIP:AtoolboxformodelingandoptimizationinMATLAB,The2004IEEEInternationalSymposiumonComputerAidedControlSystemsDesign,Taipei,Taiwan,,pp.pp.284,September2004.Availablefromhttp://control.ee.ethz.ch/~joloef/yalmip.php. [66] M.Grant,S.Boyd,andY.Ye,CVX:Matlabsoftwarefordisciplinedconvexprogramming,Availablefromhttp://www.stanford.edu/~boyd/cvx/. 131

PAGE 132

[67] A.TikhonovandV.Arsenin,SolutionofIll-posedProblems.Winston,Washington,DC:Winston&Sons,1977. [68] J.LiandP.Stoica,AnadaptivelteringapproachtospectralestimationandSARimaging,IEEETransactionsonSignalProcessing,vol.44,pp.1469,June1996. [69] P.Stoica,H.Li,andJ.Li,AnewderivationoftheAPESlter,IEEESignalProcessingLetter,vol.6,pp.205,August1999. [70] H.StarkandJ.W.Woods,ProbabilityandRandomProcesseswithApplicationstoSignalProcessing.UpperSaddleRiver,NJ:Prentice-Hall,2002. [71] T.SoderstromandP.Stoica,SystemIdentication.London,U.K.:Prentice-HallInternational,1989. [72] M.E.Tipping,SparseBayesianlearningandtherelevancevectormachine,JournalofMachineLearningResearch,vol.1,pp.211,2001. [73] D.P.WipfandB.D.Rao,SparseBayesianlearningforbasisselection,SIAMJournalonScienticComputing,vol.52,no.8,pp.2153,2004. [74] A.B.BaggeroerandH.Cox,Passivesonarlimitsuponnullingmultiplemovingshipswithlargeaperturearrays,33thAsilomarConferenceonSignals,SystemsandComputers,vol.1,pp.103,1999. [75] A.L.KraayandA.B.Baggeroer,Aphysicallyconstrainedmaximum-likelihoodmethodforsnapshot-decientadaptivearrayprocessing,IEEETransactionsonSignalProcessing,vol.55,pp.4048,2007. [76] W.M.Humphreys,Jr.,T.F.Brooks,W.W.Hunter,Jr.,andK.R.Meadows,Designanduseofmicrophonedirectionalarraysforaeroacousticmeasurements,AIAAPaper98-0471,AIAA,36thAerospaceSciencesMeet-ingandExhibit,Reno,NV,January1998. [77] M.Bengtsson,AntennaArraySignalProcessingforHighRankModels.Ph.D.dissertation,RoyalInstituteofTechnology,Sweden,1999. [78] S.Valaee,B.Champagne,andP.Kabal,Parametriclocalizationofdistributedsources,IEEETransactionsonSignalProcessing,vol.43,pp.2144,September1995. [79] A.Zoubir,Y.Wang,andP.Charge,Thegeneralizedbeamformingtechniquesforestimatingthecoherentlydistributedsources,TheEuropeanConferenceonWirelessTechnology,pp.157,November2005. [80] O.BessonandP.Stoica,DecoupledestimationofDOAandangularspreadforaspatiallydistributedsource,IEEETransactionsonSignalProcessing,vol.48,no.7,pp.1872,2000. 132

PAGE 133

[81] O.BessonandP.Stoica,DecoupledestimationofDOAandangularspreadforspatiallydistributedsources,33thAsilomarConferenceonSignals,SystemsandComputers,PacicGrove,CA,vol.1,pp.253,October24-271999. [82] J.LiandP.Stoica,MIMOradarwithcolocatedantennas:Reviewofsomerecentwork,IEEESignalProcessingMagazine,vol.24,pp.106,September2007. [83] K.Kodera,R.Gendrin,andC.D.Villedary,Analysisoftime-varyingsignalswithsmallBTvalues,IEEETransactionsonAcoustics,Speech,andSignalProcessing,vol.ASSP-26,pp.64,February1978. [84] F.AugerandP.Flandrin,Improvingthereadabilityoftimefrequencyandtimescalerepresentationsbythereassignmentmethod,IEEETransactionsonSignalProcessing,vol.43,pp.1068,May1995. [85] T.J.GardnerandM.O.Magnasco,Sparsetime-frequencyrepresentations,ProceedingsoftheNationalAcademyofSciences,vol.103,pp.6094,April2006. [86] R.Boulic,M.N.Thalmann,andD.Thalmann,Aglobalhumanwalkingmodelwithreal-timekinematicpersonication,TheVisualComputer,vol.6,pp.344,December1990. [87] D.A.Harville,MatrixAlgebrafromaStatistician'sPerspective.NewYork,NY:Springer-Verlag,1997. [88] A.K.GuptaandY.Sheena,Estimationofamultivariatenormalcovariancematrixunderacertainstructure,Statitics,vol.38,pp.371,October2004. [89] S.BoydandL.Vandenberghe,ConvexOptimization.Cambridge,UnitedKingdom:CambridgeUniversityPress,2004. [90] S.M.Kay,FundamentalsofStatisticalSignalProcessing:EstimationTheory.UpperSaddleRiver,NJ:PrenticeHall,1993. [91] C.A.SzelazekandR.M.Hicks,Upper-surfacemodicationsforclmaximprovementofNACA6-seriesairfoils,NASATechnicalMemorandum78603,1979. [92] J.Mathew,C.Bahr,B.Carroll,M.Sheplak,andL.Cattafesta,Design,fabrication,andcharacterizationofananechoicwindtunnelfacility,11thAIAA/CEASAeroa-cousticsConference,Monterey,CA,USA,May2005. [93] R.Tibshirani,Regressionshrinkageandselectionviathelasso,JournaloftheRoyalStatisticalSociety,vol.58,no.1,pp.267,1996. [94] S.S.Chen,D.L.Donoho,andM.A.Saunders,Atomicdecompositionbybasispursuit,SIAMJournalonScienticComputing,vol.20,no.1,pp.33,1998. 133

PAGE 134

[95] D.M.Malioutov,ASparseSignalReconstructionPerspectiveforSourceLocaliza-tionwithSensorArrays.PhDthesis,MIT,July2003. [96] D.Malioutov,M.Cetin,andA.Willsky,Asparcesignalreconstructionperspectiveforsourcelocalizationwithsensorarrays,IEEETransactionsonSignalProcess-ing,vol.53,pp.3010,Aug.2005. [97] I.F.GorodnitskyandB.D.Rao,SparsesignalreconstructionfromlimiteddatausingFOCUSS:Are-weightedminimumnormalgorithm,IEEETransactionsonSignalProcessing,vol.45,no.3,pp.600,1997. [98] B.D.RaoandK.Kreutz-Delgado,Anafnescalingmethodologyforbestbasisselection,IEEETransactionsonSignalProcessing,vol.47,no.1,pp.187,1999. [99] K.Kreutz-Delgado,J.F.Murray,B.D.Rao,K.Engan,T.Lee,andT.J.Sejnowski,Dictionarylearningalgorithmsforsparserepresentation,NeuralComputation,vol.15,no.2,pp.349,2003. [100] M.A.T.Figueiredo,Adaptivesparsenessforsupervisedlearning,IEEETransac-tionsonPatternAnalysisandMachineIntelligence,vol.25,no.9,pp.1150,2003. [101] R.ChartrandandW.Yin,Iterativelyreweightedalgorithmsforcompressivesensing,The2008IEEEInternationalConferenceonAcoustics,Speech,andSignalProcessing,LasVegas,Nevada,USA,April2008. [102] J.J.Fuchs,Convergenceofasparserepresentationsalgorithmapplicabletorealorcomplexdata,IEEEJournalofSelectedTopicsinSignalProcessing,vol.1,no.4,pp.598,2007. [103] X.Tan,W.Roberts,J.Li,andP.Stoica,Mimoradarimaingviaslim,submittedtoIEEETransactionsonSignalProcessing,2009. [104] K.B.PetersenandM.S.Pedersen,Thematrixcookbook.2008.Availablefromhttp://matrixcookbook.com. 134

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BIOGRAPHICALSKETCH LinDureceivedherB.EngdegreeinelectricalengineeringfromXi'anJiaotongUniversity,Xi'an,Chinain2001andherM.EngdegreeinelectricalengineeringfromNationalUniversityofSingapore,Singaporein2004.SheisexpectedtoreceiveherPh.DdegreeinelectricalengineeringfromUniversityofFlorida,Gainesville,Florida,USAin2010.Herresearchinterestsincludestatisticalsignalprocessing,arraysignalprocessingandtheirapplications. 135