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Source Localization and Power Estimation in Aeroacoustic Noise Measurements

Permanent Link: http://ufdc.ufl.edu/UFE0024868/00001

Material Information

Title: Source Localization and Power Estimation in Aeroacoustic Noise Measurements
Physical Description: 1 online resource (148 p.)
Language: english
Creator: Yardibi, Tarik
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: acoustic, adaptive, aeroacoustics, arrays, beamforming, coherent, correlated, deconvolution, incoherent, localization, microphone, noise, power, processing, signal, source, sparse
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: Using microphone arrays for noise source localization and power estimation has become common practice in aeroacoustic measurements, with the ultimate goal being the development of acoustic treatments to reduce overall airframe noise. This dissertation discusses the challenges involved in aeroacoustic testing with microphone arrays and develops a number of new signal processing techniques to overcome these challenges. The proposed algorithms are validated using both simulations and experimental data acquired at the University of Florida Aeroacoustic Flow Facility (UFAFF) with a 63-element microphone array. The standard delay-and-sum (DAS) beamformer is the most widely employed beamforming algorithm due to its simplicity and robustness, although it suffers from high sidelobe level and low resolution problems. Deconvolution can be used to eliminate the effects of the array response function from the DAS estimates. In this dissertation, the deconvolution problem is carried onto the sparse signal representation area and a sparsity constrained deconvolution approach (SC-DAMAS) as well as a sparsity preserving covariance matrix fitting approach (CMF) area presented. These algorithms are shown to offer better performance than several existing methods. Next, a systematic experimental analysis of DAS, deconvolution approach for the mapping of acoustic sources (DAMAS), SC-DAMAS, CMF, and CLEAN based on spatial source coherence (CLEAN-SC) is presented using uncorrelated and coherent sources as well as a NACA Mod 63-215 Mod B airfoil model. The source localization and absolute signal power estimation performance of the aforementioned algorithms are analyzed. To deal with correlated sources, the CMF-C algorithm, which is an extension to CMF, is proposed as an alternative to DAMAS-C, which is the extension of DAMAS to the correlated case. Since DAMAS-C and CMF-C are computationally impractical, an alternative algorithm, named mapping of acoustic correlated sources (MACS), is also presented. MACS is shown to work with simulated and experimental data containing correlated (or coherent) sources within a reasonable amount of time. Furthermore, a systematic uncertainty analysis of the DAS beamformer and a widely used array calibration procedure is presented. It is shown using experimental data that the uncertainties in the DAS beamformer integrated levels can be expected to be larger than about 1 dB. It is also shown that the array calibration procedure is essential when the assumed steering vectors are expected to contain errors. Most existing array processing algorithms for aeroacoustic noise measurement applications assume the presence of monopole sources. The last chapter of the dissertation addresses the problem of directive sources with unknown steering vectors. An algorithm for estimating non-diagonal measurement noise covariance matrices is also presented in this chapter as an alternative to diagonal removal.
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 Tarik Yardibi.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Li, Jian.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024868:00001

Permanent Link: http://ufdc.ufl.edu/UFE0024868/00001

Material Information

Title: Source Localization and Power Estimation in Aeroacoustic Noise Measurements
Physical Description: 1 online resource (148 p.)
Language: english
Creator: Yardibi, Tarik
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: acoustic, adaptive, aeroacoustics, arrays, beamforming, coherent, correlated, deconvolution, incoherent, localization, microphone, noise, power, processing, signal, source, sparse
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: Using microphone arrays for noise source localization and power estimation has become common practice in aeroacoustic measurements, with the ultimate goal being the development of acoustic treatments to reduce overall airframe noise. This dissertation discusses the challenges involved in aeroacoustic testing with microphone arrays and develops a number of new signal processing techniques to overcome these challenges. The proposed algorithms are validated using both simulations and experimental data acquired at the University of Florida Aeroacoustic Flow Facility (UFAFF) with a 63-element microphone array. The standard delay-and-sum (DAS) beamformer is the most widely employed beamforming algorithm due to its simplicity and robustness, although it suffers from high sidelobe level and low resolution problems. Deconvolution can be used to eliminate the effects of the array response function from the DAS estimates. In this dissertation, the deconvolution problem is carried onto the sparse signal representation area and a sparsity constrained deconvolution approach (SC-DAMAS) as well as a sparsity preserving covariance matrix fitting approach (CMF) area presented. These algorithms are shown to offer better performance than several existing methods. Next, a systematic experimental analysis of DAS, deconvolution approach for the mapping of acoustic sources (DAMAS), SC-DAMAS, CMF, and CLEAN based on spatial source coherence (CLEAN-SC) is presented using uncorrelated and coherent sources as well as a NACA Mod 63-215 Mod B airfoil model. The source localization and absolute signal power estimation performance of the aforementioned algorithms are analyzed. To deal with correlated sources, the CMF-C algorithm, which is an extension to CMF, is proposed as an alternative to DAMAS-C, which is the extension of DAMAS to the correlated case. Since DAMAS-C and CMF-C are computationally impractical, an alternative algorithm, named mapping of acoustic correlated sources (MACS), is also presented. MACS is shown to work with simulated and experimental data containing correlated (or coherent) sources within a reasonable amount of time. Furthermore, a systematic uncertainty analysis of the DAS beamformer and a widely used array calibration procedure is presented. It is shown using experimental data that the uncertainties in the DAS beamformer integrated levels can be expected to be larger than about 1 dB. It is also shown that the array calibration procedure is essential when the assumed steering vectors are expected to contain errors. Most existing array processing algorithms for aeroacoustic noise measurement applications assume the presence of monopole sources. The last chapter of the dissertation addresses the problem of directive sources with unknown steering vectors. An algorithm for estimating non-diagonal measurement noise covariance matrices is also presented in this chapter as an alternative to diagonal removal.
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 Tarik Yardibi.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Li, Jian.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024868:00001


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SOURCELOCALIZATIONANDPOWERESTIMATION INAEROACOUSTICNOISEMEASUREMENTS By TARIKYARDIBI ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2009 1

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c r 2009TarikYardibi 2

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Tomybelovedwifeandmyparents 3

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ACKNOWLEDGMENTS Iwouldliketoexpressmymostsinceregratitudetomyadviso rProf.JianLifor theenormoushelpandguidanceshehasprovidedtomebothtec hnicallyandpersonally inthelastthreeyearsofmylife.Herendlessenergyandplea santpersonalitytogether withhervasttechnicalknowledgemadethisgraduatestudya productiveandenjoyable journey.Iverymuchappreciatetheopportunitiesandthepe acefulworkingenvironment sheprovidedmeduringmystudies. Iamverythankfultomyco-advisorProf.LouisNCattafestaI IIforhisincredible supportandadvices.Iamgratefulforthemanyocehourshes pentondiscussingmy researchproblems.Hehasalwaysbeenanexcellentleaderan daveryunderstanding person,whichmadetheexperimentalpartofmyresearchacul tivatingexperience. IwouldliketothankProf.MarkSheplakforservinginmycomm itteeandforhis advicesonvariousissues.IwouldalsoliketothankProf.He nryZmudaandProf.Clint Slattonforservinginmycommittee. IamgratefultoProf.PetreStoicaforhisguidanceinawidev arietyoftopics.It wasapleasurehavingtheopportunitytoworkwithsuchadist inguishedscholarinarray processing. IwouldliketothanktomylabmatesfromtheSpectralAnalysi sLab,ZhaofuChen, YuboCheng,LinDu,Dr.BinGuo,HaoHe,ArsenIvanov,JunLing ,WilliamRoberts, EnriqueSantiago,XiangSu,XingTan,DucVu,Dr.LuzhouXu,M ingXue,Dr.Xiayu Zheng,andDr.XuminZhu.IwouldalsoliketothankChrisBahr ,Dr.FeiLiu,Drew Wetzel,andNikolasZawodnyfromtheInterdisciplinaryMic rosystemsGroup. IamalsothankfultomyMaster'sthesisadvisorProf.EzhanK arasan,andtoProf. DefneAktas,Prof.TolgaDumanandProf.TugrulDayarforthe irsupportsandhelps thatmadeitpossibleformetoentergraduateschoolinther stplace. 4

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Mylove,Ozlem,youknowIcouldnothavesurvivedhere(orany where)withoutyou andyoursupport.Ihopethatonedaywecanreadtheselinesan dsmiletothechallenges andhardshipswehadtofaceinacountryfarfarawayfromourh omes.Don'tforgetthat I(Ben)love(seni)you(seviyorum)! Icannotforgetaboutmydearfriendsbackathome:EmreAlsah an,Erman Kayakesen,UtkuHarpaslan,FatihDemir,NidaBerberoglu,S inanTasdelen,andmany morethatwerealwaysthereformeeventhoughIcouldonlysee themeveryonceina while.IamalsothankfultoMuratKeceliandSevnurKomurluK ecelifortheirfriendships inGainesville. Finally,aspecialthanksgoestomymomHaticeYardibi,myda dCengizYardibi, mysisterEmineNurYardibiandmyparents-in-lawFatmaandO smanSubakan.Also,a belatedthankyoutoErkanYardibiforhishelpsinmyundergr aduateseniorproject. ThisworkwassupportedbytheNationalAeronauticsandSpac eAdministration (NASA)underGrantNo.NNX07AO15A. 5

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 9 LISTOFFIGURES .................................... 10 ABSTRACT ........................................ 13 CHAPTER 1INTRODUCTION .................................. 15 1.1Notation ..................................... 16 1.2Background ................................... 16 1.3OrganizationoftheDissertation ........................ 20 2FUNDAMENTALSOFBEAMFORMING ..................... 24 2.1DataModel ................................... 24 2.2Delay-and-SumBeamformer .......................... 25 2.3ArrayCalibration ................................ 27 3DECONVOLUTIONWITHUNCORRELATEDSOURCES ........... 32 3.1AnExistingDeconvolutionApproach ..................... 32 3.2DiagonalRemoval ................................ 34 3.3SparsityConstrainedDeconvolution ...................... 35 3.3.1SparsityConstrainedFormulation ................... 35 3.3.2EstimatingtheUserParameter .................... 36 3.3.3AMoreEcientImplementation ................... 37 3.4CovarianceMatrixFitting ........................... 38 3.5NumericalExamples .............................. 38 3.6Conclusions ................................... 41 4EXPERIMENTALRESULTS ............................ 48 4.1MicrophoneArray ................................ 48 4.2ExperimentalSetup ............................... 49 4.3Software ..................................... 50 4.4AbsoluteLevels ................................. 51 4.5ArrayCalibrationPerformance ........................ 51 4.6SingleSource .................................. 52 4.7TwoUncorrelatedSources ........................... 52 4.8TwoCoherentSources ............................. 54 4.9ComputationalComplexity ........................... 57 6

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4.10AnExamplewithanAirfoilModel ...................... 58 4.11Conclusions ................................... 58 5DECONVOLUTIONWITHCORRELATEDSOURCES ............. 76 5.1AnExistingDeconvolutionApproach ..................... 76 5.2CovarianceMatrixFittingwithCorrelatedSources ............. 77 5.3AFastBeamformerforCorrelatedSources .................. 77 5.4MeasurementNoise ............................... 81 5.5NumericalExamples .............................. 81 5.5.1Simulations ............................... 81 5.5.2ExperimentalResults .......................... 84 5.6Conclusions ................................... 84 6UNCERTAINTYANALYSIS ............................ 91 6.1UncertaintyAnalysisTechniques ........................ 92 6.1.1MultivariateUncertaintyAnalysis ................... 92 6.1.2Monte-CarloUncertaintyAnalysis ................... 94 6.2ApplicationofUncertaintyAnalysistotheDelay-and-S umBeamformer .. 95 6.3NumericalandExperimentalResults ..................... 97 6.3.1CalibrationUncertainty ......................... 97 6.3.2Delay-and-SumBeamformerUncertainty ............... 100 6.3.2.1ComparisonofMultivariateandMonte-CarloAnalys es .. 100 6.3.2.2UncertaintyAnalysiswithSimulations ........... 102 6.3.2.3UncertaintyAnalysiswithExperimentalData ....... 106 6.4Conclusions ................................... 108 7DIRECTIVESOURCESANDSPATIALLYNON-WHITENOISE ....... 122 7.1ProblemFormulation .............................. 123 7.2NumericalExamples .............................. 126 7.3Conclusions ................................... 128 8CONCLUSIONSANDFUTUREWORK ...................... 131 8.1Conclusions ................................... 131 8.2FutureWork ................................... 132 APPENDIX AJACOBIANMATRIX ................................ 134 A.1JacobianMatrixfortheCSM ......................... 134 A.2JacobianMatrixfortheCalibrationFactors ................. 134 A.3JacobianMatrixforMicrophoneLocations .................. 134 A.4JacobianMatrixforTemperature ....................... 135 BCOVARIANCEMATRIXOFTHECSM ...................... 136 7

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CGENERATINGCORRELATEDGAUSSIANRANDOMVARIABLES ..... 138 DMOREONMICROPHONELOCATIONERRORS ................ 139 REFERENCES ....................................... 142 BIOGRAPHICALSKETCH ................................ 148 8

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LISTOFTABLES Table page 1-1Mathematicalnotationusedinthedissertation. .................. 23 3-1Characteristicsoftheacousticimagingalgorithms. ................ 47 3-2ComputationtimesofDAMAS,SC-DAMASandCMF. .............. 47 3-3Speeding-upofSC-DAMAS. ............................. 47 4-1Computationtimes. .................................. 75 5-1PseudocodeofMACS. ................................ 90 6-1DASerrorsources. .................................. 121 6-2CovariancesoftheCSMelements. .......................... 121 9

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LISTOFFIGURES Figure page 1-1Themicrophonelayoutsoftwoarrays. ....................... 23 2-1Amicrophonearrayinthepresenceoftwosources. ................ 31 2-2Usingascanningregion. ............................... 31 3-1Methodologyofvariousbeamformingtechniques. ................. 42 3-2Beamformingmapsobtainedforcase1. ....................... 43 3-3Beamformingmapsobtainedforcase2. ....................... 44 3-4Beamformingmapsobtainedforcase3. ....................... 45 3-5FasterversionofSC-DAMAS. ............................ 46 4-1LAMDAcharacteristics. ............................... 60 4-2Experimentalsetup. ................................. 60 4-3Picturesfromtheexperiments ............................ 61 4-4Snapshotfromsoftwarepackage1. ......................... 62 4-5Snapshotfromsoftwarepackage2. ......................... 62 4-6Eectsofcalibration. ................................. 63 4-7Beamformingmapswithasinglesource. ...................... 63 4-8Integratedlevelswithasinglesource. ........................ 64 4-9Sourcesofsimilaranddierentstrengths. ..................... 64 4-10Beamformingmapswithtwouncorrelatedsourcesofsimi larpowers. ....... 65 4-11Beamformingmapswithtwouncorrelatedsourcesofdie rentpowers. ...... 66 4-12Integratedlevelsfortwouncorrelatedsources. ................... 67 4-13Integrationregions. .................................. 67 4-14IndividualsignallevelestimationwithDASandSC-DAM AS. .......... 68 4-15IndividualsignallevelestimationwithDASandCLEANSC. ........... 68 4-16Referencemicrophonelevelsforuncorrelatedandcohe rentsources ........ 69 4-17Beamformingmapswithtwocoherentsourcesat2kHz. .............. 70 10

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4-18Beamformingmapswithtwocoherentsourcesat4kHz. .............. 71 4-19Integratedlevelsfortwocoherentsources. ..................... 72 4-20Interferenceduetosourcecoherence. ........................ 72 4-21PerformanceofthefastversionofSC-DAMAS. .................. 73 4-22NACA63-215Mod-Bairfoilexperimentalsetup. .................. 74 4-23PictureoftheNACA63-215Mod-Bairfoil. .................... 74 4-24BeamformingmapswiththeNACA63-215Mod-Bairfoilat2 .6kHz. ...... 75 5-1ComparisonofMACSwithDAS-CusingSADAandforcase1. ......... 86 5-2ComparisonofMACSwithDAS-CusingSADAandforcase2. ......... 87 5-3ComparisonofMACSwithDAS-CusingLAMDA. ................ 88 5-4PerformanceofMACSwhen ^ L isvaried. ...................... 89 5-5BeamformingwithDAS,DAMASandCMFinthepresenceofcoh erentsources. 89 5-6AnalysisofMACSusingexperimentaldatawithtwocohere ntsources. ..... 90 6-1Calibrationuncertaintywhenindividualsensitivitie sareperturbed. ....... 109 6-2CalibrationuncertaintywhentheCSMisperturbed. ............... 109 6-3Calibrationuncertaintywhenthereferencemicrophone levelisperturbed. ... 109 6-4Calibrationuncertaintywithvaryingfrequency. .................. 110 6-5ComparisonofmultivariateandMonte-Carlomethodsfor case1. ........ 110 6-6ComparisonofmultivariateandMonte-Carlomethodsfor case2. ........ 111 6-7Eectsofmicrophonelocationerrorsforcase1. .................. 112 6-8Eectsofmicrophonelocationerrorsforcase2. .................. 112 6-9Furtheranalysisofmicrophonelocationerrors. ................... 113 6-10Eectsofmicrophonelocationerrorswithonesource. ............... 114 6-11Eectsofmicrophonelocationerrorswithtwosources. .............. 115 6-12Eectsofmicrophonelocationerrorswithcalibration ............... 116 6-13The95%condenceintervalswhenbroadbanddistanceis perturbed. ...... 117 6-14Eectsofarraybroadbanddistance. ........................ 117 11

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6-15The95%condenceintervalswhentheCSMisperturbed. ............ 118 6-16The95%condenceintervalswhensensitivitiesareper turbed. .......... 118 6-17The95%condenceintervalswhenphasesareperturbed. ............. 119 6-18The95%condenceintervalswhentemperatureispertur bed. .......... 119 6-19The95%condenceintervalswhenallinputvariablesar eperturbed. ...... 120 6-20Analysisofexperimentaldatawithasinglesource. ................ 120 6-21AnalysisoftheNACA63-215Mod-Bairfoil. .................... 121 7-1Amicrophonearrayinthepresenceofadirectivesource. ............. 129 7-2Thedirectivitiesobservedatthearraymicrophones. ................ 129 7-3Beamformingmapswithtwodirectivesources. ................... 130 7-4Beamformingmapswithtwodirectivesourcesandbandedm easurementnoise. 130 D-1Biasinestimatedsignalpowerswhenmicrophonelocatio nsareperturbed. ... 141 12

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy SOURCELOCALIZATIONANDPOWERESTIMATION INAEROACOUSTICNOISEMEASUREMENTS By TarikYardibi August2009 Chair:JianLiMajor:ElectricalandComputerEngineering Usingmicrophonearraysfornoisesourcelocalizationandp owerestimationhas becomecommonpracticeinaeroacousticmeasurements,with theultimategoalbeingthe developmentofacoustictreatmentstoreduceoverallairfr amenoise.Thisdissertation discussesthechallengesinvolvedinaeroacoustictesting withmicrophonearraysand developsanumberofnewsignalprocessingtechniquestoove rcomethesechallenges.The proposedalgorithmsarevalidatedusingbothsimulationsa ndexperimentaldataacquired attheUniversityofFloridaAeroacousticFlowFacility(UF AFF)witha63-element microphonearray. Thestandarddelay-and-sum(DAS)beamformeristhemostwid elyemployed beamformingalgorithmduetoitssimplicityandrobustness ,althoughitsuersfrom highsidelobelevelandlowresolutionproblems.Deconvolu tioncanbeusedtoeliminate theeectsofthearrayresponsefunctionfromtheDASestima tes.Inthisdissertation, thedeconvolutionproblemiscarriedontothesparsesignal representationareaanda sparsityconstraineddeconvolutionapproach(SC-DAMAS)a swellasasparsitypreserving covariancematrixttingapproach(CMF)areapresented.Th esealgorithmsareshownto oerbetterperformancethanseveralexistingmethods. Next,asystematicexperimentalanalysisofDAS,deconvolu tionapproachforthe mappingofacousticsources(DAMAS),SC-DAMAS,CMF,andCLE ANbasedonspatial sourcecoherence(CLEAN-SC)ispresentedusinguncorrelat edandcoherentsourcesas 13

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wellasaNACAMod63-215ModBairfoilmodel.Thesourcelocal izationandabsolute signalpowerestimationperformanceoftheaforementioned algorithmsareanalyzed. Todealwithcorrelatedsources,theCMF-Calgorithm,which isanextensionto CMF,isproposedasanalternativetoDAMAS-C,whichistheex tensionofDAMASto thecorrelatedcase.SinceDAMAS-CandCMF-Carecomputatio nallyimpractical,an alternativealgorithm,namedmappingofacousticcorrelat edsources(MACS),isalso presented.MACSisshowntoworkwithsimulatedandexperime ntaldatacontaining correlated(orcoherent)sourceswithinareasonableamoun toftime. Furthermore,asystematicuncertaintyanalysisoftheDASb eamformerandawidely usedarraycalibrationprocedureispresented.Itisshownu singexperimentaldatathatthe uncertaintiesintheDASbeamformerintegratedlevelscanb eexpectedtobelargerthan about 1dB.Itisalsoshownthatthearraycalibrationprocedureis essentialwhenthe assumedsteeringvectorsareexpectedtocontainerrors. Mostexistingarrayprocessingalgorithmsforaeroacousti cnoisemeasurement applicationsassumethepresenceofmonopolesources.Thel astchapterofthedissertation addressestheproblemofdirectivesourceswithunknownste eringvectors.Analgorithm forestimatingnon-diagonalmeasurementnoisecovariance matricesisalsopresentedin thischapterasanalternativetodiagonalremoval. 14

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CHAPTER1 INTRODUCTION Duetotheever-increasingdemandforcommercialairtransp ortation,airportshave toaccommodatemoreairplaneseveryyear.Thisresultsinin creasednoisepollutionand disturbancetothenearbyresidents,and,accordingly,str icterregulationsonairplane noiselevelsarebeingimposedtoaddressthisproblem.Alth oughthedominantnoise componentofanairplaneisthejetenginesduringtake-o,t heairframenoiseisespecially signicantwhileanairplaneislandingsincetheenginesar eusuallyinlowthrustmode duringthisphase.Asnoisecanbethedeterminingfactoront henumberofairplanes thatcanbeaccommodatedduringacertaintimeperiodinsome airports,itisessential toaccuratelylocalizeandestimatethestrengthsofdomina ntairframenoisecomponents, inanattempttodevelopacoustictreatmentsforreducingov erallairframenoise.Forthis purpose,itisimportanttobothprovidesystematicanalyse sofexistingmeasurementand datareductionmethodologiesforaeroacousticmeasuremen tsanddevelopnovelsignal processingtechniquesthatcanmitigatetheshortcomingso ftheexistingtechniques. Theuseofmicrophonearraysfornoisesourcelocalizationa ndpowerestimationhas becomecommonpracticeinaeroacousticmeasurementsinthe recentyears.Asopposed tousingasinglemicrophone,microphonearrayscanbeelect ronicallysteeredintodesired regionsinspacetocreateanimageofacousticsourcesatagi venfrequency[ 1 2 ].This imageconsistsoftheestimatedsoundpressurelevelofeach scanninggridpointonthe modelunderinvestigationandcanbeusedtoidentifythemod elcomponentsprimarily responsibleforthegeneratednoise.Atypicalmicrophonea rrayconsistsofanumberof microphonesarrangedinacertaingeometry,whichiscarefu llydesignedtomeetgiven 3-dBbeamwidthandsidelobespecicationswithinafrequen cyrangeofinterest.Figure 1-1 Ashowsanexamplemicrophonearray,thesmallaperturedire ctionalarray(SADA) [ 1 3 4 ]whichconsistsof33microphones;amicrophoneisplacedat thecenterofthe arrayandtheothersarearrangedinfourcircularringsof8m icrophoneseach.Thisarray, 15

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beingrelativelysmallinaperture,ismainlyusedformeasu ringsourcedirectivityandis designedforthefrequencyrangeof10kHzto40kHz.Sincethe resolutionofanarray decreaseswithdecreasingfrequencyforagivenaperturesi ze,alargerarrayisrequired inordertoachievesucientresolutionatlowerfrequencie s.Anexampleofsuchalarge aperturearrayisthelargeaperturemicrophonedirectiona larray(LAMDA)shownin Figure 1-1 B,whichconsistsof63microphonesarrangedinalogarithmi cspiralstructure. LAMDAcanbeeectivelyutilizedatfrequenciesaslowas1kH z.Thelogarithmicspiral layoutofthemicrophonesresultsinlowersidelobelevelsc omparedtoacircularlayout (see[ 5 6 ]foradetailedtreatmentonarraydesign). 1.1Notation Thefollowingnotationswillbeusedthroughoutthisdisser tation.Vectorsand matricesaredenotedbyboldfacelowercaseandboldfaceupp ercaseletters,respectively. The k th componentofavector x iswrittenas x k .The k th diagonalelementofamatrix P iswrittenas P k and ^ s denotestheestimateof s .SeeTable 1-1 forothersymbolsandtheir meanings.Allsoundpressurelevels(SPLs)presentedinthi sdissertationareindBref.20 Pa. 1.2Background Themostcommonlyusedbeamformingalgorithminpracticeis thedelay-and-sum (DAS)beamformer[ 1 2 4 6 { 8 ]whichsumsthedelayedandweightedversionsofeach microphonesignalsothattheactualsourcesignalsarerein forcedandtheunwantednoise signalstendtocancel.Awell-knownissuewiththeclassica ldelay-and-sumapproachis thatthebeamformingmapsareusuallycontaminatedwithlar gesidelobes.Sidelobescan causeboththesmearingandleakageofthesources[ 8 9 ].Considerascenariowithtwo closelyspacedsourcessothattheresponseofthearraytoth erstsourcedoesnotwither awaybeforetheresponsetothesecondsourcestarts.Thiswi llresultinthesmoothing, orsmearing,ofthespectruminthesensethatthetwopeakswi llbemergedintoasingle broadpeak.Lookingfromadierentperspective,astrongso urcecanyieldpoweratother 16

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locations,wherenosourceispresent,throughtheconvolut ionwiththesidelobes.This latterformofdegradationiscalledleakage.Therefore,it isusuallydiculttoidentify thetruesourcelocationsandstrengthsthroughthebeamfor mingmapobtainedviathe DASmethod.AnotherproblemwithDASisthatthebeamwidthte ndstoincreasewith decreasingfrequency. Variousothermicrophonearrayprocessingmethodshavebee ndevelopedinorder tomitigatethedrawbacksoftheDASbeamformer.Weightings chemesthatmaintain aconstantbeamwidthoverafrequencyrangewhenusedinconj unctionwiththeDAS beamformerhavebeendiscussedintheliterature[ 10 11 ].Also,severalrobustadaptive beamformingtechniqueshavebeenproposedasalternatives toDAS[ 12 { 16 ].Mostofthese techniquesareparametricapproachesthatrequirethenumb erofsourcestobeknown [ 15 16 ],andCapontypebeamformerscannotprovideasparserepres entationofthe imagingsceneandfailtoworkforcoherentsources[ 12 ]. InordertoremedythesidelobeproblemoftheDASbeamformer ,apostprocessing techniquecalledthedeconvolutionapproachforthemappin gofacousticsources (DAMAS)wasdeveloped[ 17 18 ].Thisapproachisconsideredabreakthroughin aeroacousticmeasurementsandhasbeenusedwidelyinpract ice.Assumingthatthe sourcewaveformsareuncorrelated,itcanbeshownthattheD ASdatareductionequation isalinearfunctionoftheactualsignalpowersandthecoec ientsofthislinearfunction areknown.DAMASsolvesthisinverseproblemforthesignalp owersafterevaluating theDASdatareductionequationateveryscanningpoint.Due tothematrixinvolved inthislinearsystembeingill-conditioned(matrixinvers edoesnotexist),DAMASuses theiterativeGauss-Seidelmethod.Onedrawbackofusingth eGauss-Seidelmethodfor solvingthislinearsystemofequationsiscomputationtime .(Thenumberofiterations requiredbytheGauss-Seidelwasreportedtobeontheordero fthousands[ 18 ].)Many otheralgorithmsrelatedtoDAMAShavebeenproposedinthel iterature[ 19 ].Localization andoptimizationofarrayresults(LORE)[ 20 ]isadeconvolutionmethodthatusesa 17

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twostageprocesstosolvetheinverseproblem.Intherstst epamethodsimilarto DAMASisusedtoreducetheproblemdimensionsandintheseco ndstepanoptimization schemeisusedtosolvethereducedinverseproblem.DAMAS2a ndDAMAS3[ 21 ]oer alternativesandextensionstoDAMASbyusingthefastFouri ertransform(FFT)and assumingthatthepointspreadfunctionisshift-invariant .DAMAShasalsobeenapplied tothethreedimensionalsourcelocalizationproblem[ 22 ].Asidefromtheaforementioned deconvolutionalgorithms,CLEANbasedonspatialsourceco herence(CLEAN-SC)is anotherwidelyusedalgorithmwhich,unliketheabovealgor ithms,doesnotassumethe truesteeringvectorstobeknown.Instead,CLEAN-SCiterat ivelybuildsupthesteering vectorscorrespondingtothedominantsourcesusingthepre viouslyestimatedsignal powersandassumingthatthesidelobesarecoherentwiththe peakforagivensource. Afterestimatingthesteeringvectorsandthesignalpowers ,CLEAN-SCconstructsa cleanimageofthescanningregionsimilartotheoriginalCL EANalgorithm[ 23 ].One drawbackofCLEAN-SCisthatitrequirestheselectionoffou ruserparameters(awrong selectioncouldresultinpoorperformance),whereasDAMAS isauserparameter-free algorithm.Alloftheaforementionedalgorithmsassumetha tthesourcesareuncorrelated andtheliteratureonthedeconvolutionofcorrelatedsourc esissparsetothebestofthe author'sknowledge.DAMAS-C[ 24 ]extendsDAMAStothecorrelatedcaseusingavery similarmethodologyandassumingthatthecross-correlati onbetweenanytwosourcesis real-valued.Althoughrequiringhighcomputationalresou rces,thisapproachistherst deconvolutionalgorithmconsideringcorrelatedsourcesi naeroacousticmeasurements. Inthisdissertation,wemakeuseofthesparsesignalrepres entationframework severaltimesandthereforewenditusefultoreviewthebac kgroundonthissubject beforedescribingourcontributions.Sparsesignalrepres entationisanextensivelystudied topicinmanyareasincludingstatisticsandsignalprocess ingfortherecoveryofsignals consistingofonlyfewnonzeroelementswithaknownlinearr elationtothemeasured data.Thesparsesignalrepresentationproblemmainlyaims tondthesparsest x such 18

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that y = A x issatised.Statedmoreformally,theobjectiveistominim ize k x k 0 such that y = A x where A isknownand y ismeasured.Theprobleminitsoriginalformis acombinatorialproblemandhasanexponentialcomplexitym akingitimpractical[ 25 ]. Fortunately,when x issucientlysparse[ 25 ], k x k 0 canbereplacedby k x k 1 whichleadsto aconvexoptimizationproblemthatcanbesolvedmuchmoreea silyusingleastabsolute shrinkageandselectionoperator(LASSO)[ 26 ]orbasispursuit(BP)[ 27 28 ]typeof algorithms.Therearemanystudiesintheliteratureelabor atingonthisrelaxationfrom the ` 0 -normtothe ` 1 -normbothinthenoiselessandnoisycase[ 25 29 { 31 ].Itisshown thatwhen x containsasmallnumberofnonzeroelementswithrespecttoi tssize,the solutionswiththe ` 0 -and ` 1 -normscoincideundersomemildconditions.Alternatively thefocalunderdeterminedsystemsolution(FOCUSS)[ 32 { 35 ]algorithmcanbeusedto iterativelysolvethesparseproblembyminimizingacostfu nctionthatpromotessparsity. Thismethodrequirestheproperselectionoftwouserparame ters.Nevertheless,FOCUSS algorithmshavebeensuccessfullyappliedtobrainelectro encephalography(EEG)signals andfar-eldsourcelocalizationproblems.ABayesianappr oach,suchassparseBayesian learning(SBL)[ 36 { 38 ]ortheapproachin[ 39 ],canalsobeusedtoestimate x using variouspriorprobabilitydistributionstoenforcesparsi ty.Althoughtheseapproachesdo notrequireauserparameter,theyusuallyhavehighcomputa tionalcomplexity.Notethat LASSOcanalsobethoughtofasaBayesianapproachassuminga Gaussianlikelihoodfor y andaLaplacianpriorfor x whichisknowntoenforcesparsity.Anotheralgorithm,the ` 1 -SVDalgorithm[ 40 { 43 ],whereSVDstandsforthematrixsingularvaluedecomposit ion, isproposedforestimatingsourcelocationsinamannersimi lartoBPbutforthemultiple snapshotandcomplexcase.Thisalgorithmrequiresanestim ateforthenumberofsources. Althoughthisestimatedoesnothavetobeexact,asmallerro risrequiredforgood performance.Moreimportantly,thisalgorithmcontainsau serparameterwhoseselection isverydicultandcouldresultinsignicantperformancel osswhentunedimproperly [ 44 ].Notethatthismethodisdesignedtoworkwiththetimesign als.Inaeroacoustic 19

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applications,however,hugeamountsoftimedataarecollec tedanditispreferredto workwiththecovariancematrixsinceitrequiresmuchlesss toragespaceandismore convenienttodealwith.Finally,Fuchs[ 45 ]alsodiscussesmethodsfortherecoveryof sourcelocationsforthefar-eldlineararraycasebyappen dinganoisebasisvectorto thesteeringmatrixandusingdeconvolutiontogetherwiths parsity.Itisinterestingto notethattheseapproachesinthesignalprocessingliterat urearesimilartoDAMAS. Fuchsalsoprovidesanextensiontohismethodbyusingaunif ormcirculararrayanda sparsealgorithmsimilartoBP[ 46 ].Inthismethod,theuserparameterisselectedtobea constant,whichcanintroduceerrors. 1.3OrganizationoftheDissertation Thedissertationbeginsbyformulatingtheaeroacousticso urcelocalizationandpower estimationprobleminChapter 2 .ThewidelyusedDASbeamformerandapopulararray calibrationmethod[ 2 ]isalsopresentedinthischapter. Chapter 3 describesdeconvolutionapproachesforuncorrelatedsour ces.After reviewingDAMAS,wedescribethesparsityconstrainedDAMA Salgorithm(SC-DAMAS), whichisanapplicationofaslightlymodiedversionofLASS Otothespecicinverse problemofDAMAS.Incontrasttoselectingaconstant[ 46 ],theuserparameterof SC-DAMASisselectedbyanadaptiveandsimplemethodwhichu sestheeigenvalues ofthecrossspectralmatrix(CSM).Wealsodiscussawaytosi gnicantlyspeedup SC-DAMAS.Moreover,weproposeanotheralgorithmcalledth ecovariancematrixtting (CMF),whichalsoexploitssparsityfordeconvolutionbuti nadierentwaythanthe DAMASformulation.BothSC-DAMASandCMFareformulatedasc onvexoptimization problems.Thesealgorithmsareguaranteedtoconvergetoth egloballyoptimalsolution andtheytakelesscomputationtimethanDAMAS.TheCMFalgor ithmismorerobust tonoisethanbothDAMASandSC-DAMAS.Weprovidenumericale xamplesthat demonstratetheperformanceoftheproposedalgorithms. 20

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Chapter 4 providesasystematiccomparisonofDAS,DAMAS,SC-DAMAS, CMFandCLEAN-SCusingexperimentaldataacquiredattheUni versityofFlorida AeroacousticFlowFacility(UFAFF)usingLAMDA[ 47 ].Thetestcasesconsidered includeasinglesource,twouncorrelatedsourceswithsimi laranddierentpowers,and twocoherentsources.Thesourcelocalizationcapabilityo fthealgorithmsaswellastheir accuracyinestimatingtheabsolutesignalpowerswillbean alyzed. InChapter 5 ,weextendouranalysistodealwithcorrelated(orevencohe rent) sources.WerstproposeCMF-C,whichisanextensionofCMFt othecorrelatedcase. However,DAMAS-CandCMF-Crequireextremecomputationtim esforevensmall numberofscanningpoints.Therefore,wepresentanewcovar iancettingapproachfor themappingofacousticcorrelatedsources(MACS).MACSisa cyclicalgorithmbasedon convexoptimizationandsparsity,andcanworkwithuncorre lated,partiallycorrelatedor evencoherentsourceswithareasonablylowcomputationalc omplexity.Itisshownvia simulationsaswellasexperimentaldatathatMACScansucce ssfullylocalizecorrelated acousticsourcesandestimateboththeauto-correlationle velofeachsourceandthe cross-correlationlevelsbetweenthesources. Chapter 6 providesasystematicuncertaintyanalysisoftheDASbeamf ormerandthe arraycalibrationprocedureundertheassumptionthattheu nderlyingmathematicalmodel ofuncorrelated,monopolesourcesiscorrect.Ananalytica lmultivariatemethodbased onarst-orderTaylorseriesexpansionandanumericalMont e-Carlomethodbasedon assumeduncertaintydistributionsfortheinputvariables areconsidered.Theuncertainty ofcalibrationisanalyzedusingtheMonte-Carlomethod,wh ereastheuncertaintyofthe DASbeamformerisanalyzedusingboththecomplexmultivari ateandtheMonte-Carlo methods.Itisshownthatthearraycalibrationprocedureis essentialwhenerrorsinthe assumedsteeringvectorsareexpected.Itisalsoshownusin gexperimentaldatathatthe uncertaintyintheDASpowerestimatescanbeaslargeas 1dB(andevenlargerathigh frequencies). 21

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Mostexistingarrayprocessingalgorithmsforaeroacousti cnoisemeasurement applicationsassumethepresenceofmonopolesources.InCh apter 7 ,aneigenvalue decompositionbasedalgorithmforthelocalizationofdire ctivesourceswithunknown steeringvectorsispresented.Sincesubspacemethodsares ensitivetomeasurement noise,aniterativealgorithmthatmakesuseofconvexprogr ammingforextractingthe measurementnoisecovariancematrix,whichcanbeeitherdi agonalornon-diagonal,from thearraycovariancematrixisalsopresented.Numericalex amplesareprovidedtovalidate theproposedalgorithms. Finally,thedissertationisconcludedinChapter 8 whereadiscussiononpotential futureresearchdirectionsisalsoprovided. 22

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-10 -5 0 5 10 -10 -5 0 5 10 x, cmy, cm -80 -60 -40 -20 0 20 40 60 80 -80 -60 -40 -20 0 20 40 60 80 x, cmy, cm A B Figure1-1.Themicrophonelayoutsoftwoarrays.A)SADAwit h33microphones.B) LAMDAwith63microphones. Table1-1.Mathematicalnotationusedinthedissertation. NotationExplanation E[ ]Expectationoperation jj Absolutevalueofascalar kk 0 ` 0 -norm,i.e.,thenumberofnon-zeroelementsofavector kk 1 ` 1 -norm,i.e.,thesumoftheabsolutevalueofeachelementina vector kk 2 ` 2 -norm kk F Frobeniusnormofamatrix TheHadamard(elementwise)matrixproduct tr( )Traceofamatrix ( ) Complexconjugateofascalar ( ) T and( ) H Transposeandconjugatetransposeofavectorormatrix P 0 P isapositivesemi-denitematrix vec( )Thecolumn-by-columnvectorizationofamatrix R n r n c and C n r n c Realandcomplexmatriceswith n r rowsand n c columns I Identitymatrixofappropriatedimension Re[ ]andIm[ ]Realandimaginarypartsoftheargument diag( )Adiagonalmatrixwiththevectorargumentinitsdiagonal 23

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CHAPTER2 FUNDAMENTALSOFBEAMFORMING Inthischapterweintroducethetraditionaldatamodelused inaeroacoustic measurements,anddescribethestandardDASbeamformingal gorithmandthearray calibrationprocedureintroducedbyDougherty[ 2 ]. 2.1DataModel Considerthewaveeldgeneratedby L monopoleacousticsourceswherethe three-dimensionallocationofthe l th sourceisdenotedby(~ x l ; ~ y l ; ~ z l )for l =1 ;:::;L .The datareductionprocessforfrequencydomainbeamformingst artswiththecomputation oftheCSMateachfrequencyofinterest.Forthispurpose,th epressuredatarecordedat eachmicrophonefor T acq secondsisdividedinto v percentoverlappingblocksoflength H where0 v< 100.Theresultingnumberofblocks, B ,canbecomputedasfollows: B = 1+ T acq f s =H 1 1 v= 100 ; (2{1) where bc denotesthenearestintegerlessthanorequaltotheargumen tand f s denotes thesamplingfrequency.Next,the H -pointFFTofeachblockiscomputed(anappropriate spectralwindowcanbeappliedtothedatabeforetakingtheF ouriertransforms),where H isapowerof2.Thisresultsinafrequencyresolutionof f s =H .The h th elementofeach frequencydomainblockcorrespondstothenarrow-bandfreq uency f h = hf s =H ,where h =0 ;:::;H= 2andonlythesingle-sidedspectrumisconsidered. Consideran M -elementmicrophonearraywiththe m th microphonelocatedat ( x m ;y m ;z m ),where m =1 ;:::;M .Letthefrequencyofinterestbe f .Followingthe sphericalwavepropagationmodel[ 48 ],thefrequencydomainpressuredataatallthe microphonescanbeusedtoobtainthefollowingsetofequati ons(seeFigure 2-1 )[ 7 { 9 ] y ( b )= L X l =1 a l s l ( b )+ e ( b ) ;b =1 ;:::;B; (2{2) 24

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where y ( b )= 266666664 y 1 ( b ) y 2 ( b ) ... y M ( b ) 377777775 ; a l = 266666664 e jkr l; 1 =r l; 1 e jkr l; 2 =r l; 2 ... e jkr l;M =r l;M 377777775 ; e ( b )= 266666664 e 1 ( b ) e 2 ( b ) ... e M ( b ) 377777775 ; (2{3) y m ( b )isthefrequencydomainpressuredataofmicrophone m atblock b a l isthesteering vector(orthewavepropagationvector)correspondingtoth e l th monopolesource, r l;m = p (~ x l x m ) 2 +(~ y l y m ) 2 +(~ z l z m ) 2 istheEuclideandistancebetweenthe l th sourceandthe m th microphone, k =2 f=c isthewavenumber, c isthespeedofsound inair, s l ( b )istheacousticwaveformofthe l th sourceatblock b ,and e m ( b )istheadditive contamination(ormeasurement)noise(duetoelectronicno iseandacousticsourcesother thanthe L sourcesconsideredaswellasrerectionsandscattering)at the m th microphone atblock b .Notethat f y ( b ) g a l and f e ( b ) g areallcomplexvectorsofsize M 1,and f s l ( b ) g arecomplexscalars.Inaddition, f y ( b ) g and a l areknown,whereas f s l ( b ) g and f e ( b ) g areunknownquantities.Notealsothattheindices l m and b runfrom1to L M and B ,respectively. Eq. 2{2 canalsobewritteninamorecompactformas y ( b )= As ( b )+ e ( b ) ;b =1 ;:::;B; (2{4) where A =[ a 1 ;:::; a L ] 2 C M L and s ( b )=[ s 1 ( b ) ;:::;s L ( b )] T 2 C L 1 2.2Delay-and-SumBeamformer TheDASbeamformerbasicallysumsthedelayedandweightedv ersionsofeach microphonesignalinordertoreinforcethesignalfromthes ourceofinterestwhile suppressingthecontributionfromothersourcesandcontam inationnoise.Thedelaysand weightsaredesignedaccordingtotherelativedistancesbe tweenthemicrophones.Inthe frequencydomain,thiscorrespondstoapplyingappropriat ephaseshiftsandweighting factors.Beamformingisusuallydoneindependentlyateach narrow-bandfrequency 25

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ofinterestandthereforeinthefollowinganalysis,wewill consideronlyoneparticular frequency f or,equivalently,oneparticularwavenumber k (thesameanalysisisrepeated atallfrequenciesofinterest).TheDASestimateofthe l th sourcewaveformisdenedas [ 1 17 ] ~ s l ( b )= 1 M ~ a Hl y ( b ) ;l =1 ;:::;L;b =1 ;:::;B; (2{5) where ~ a l = 1 r l; 0 r l; 1 e jkr l; 1 ;r l; 2 e jkr l; 2 ;:::;r l;M e jkr l;M T ; (2{6) r l; 0 = p (~ x l x ) 2 +(~ y l y ) 2 +(~ z l z ) 2 isthedistancefromthe l th scanningpointtothe arraycenter,and x isthe x componentofthearraycenter( y and z aredenedsimilarly). Notethat ~ a l ,whichisan M 1complexvectorthatisknown,isusedtoaccountforthe dierentdistancestraveledbythewavebeforereachingeac hmicrophone.Thepurposeof normalizing ~ a l by r l; 0 istomatchtheestimatedsignalpowertowhatasinglemicrop hone inthecenterofthearraywouldmeasure.Theunderlyingassu mptionbehindDASisthat while r l; 0 ~ a Hl a l = M r l 0 ; 0 ~ a Hl 0 a l isrelativelysmallfor l 0 6 = l l;l 0 =1 ;:::;L .Thisassumption isevaluatedbyanalyzingtheso-calledpointspreadfuncti on,psf( l ),denedas j ~ a Hl a 0 j =M 2 forthe l th scanningpointwhere a 0 denotesthesteeringvectorofasourcelocatedatthe centerofthescanningregion.Thepsfisalsousedtocompute the3-dBbeamwidth(and theresolution)ofthearray. Consequently,DASestimatesthepowerlevelofthe l th source(asmeasuredatthe arraycenter)asfollows ^ P (D) l = 2 0 B B X b =1 j ~ s l ( b ) j 2 = 1 M 2 ~ a Hl G ~ a l ;l =1 ;:::;L; (2{7) where 0 isthespectralwindowcorrectionfactorand G = 2 0 B B X b =1 y ( b ) y H ( b )(2{8) 26

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istheCSM.(Thefactorof2isduetotheuseofthesingle-side dspectrum.)TheCSM isa M M complexsymmetricmatrixandhenceconsistsof M 2 real-valueddistinct components.Ingeneral,itispreferabletoworkwiththeCSM G ,andEq. 2{7 rather thanthefrequencydomainpressuredata, f y ( b ) g ,andEq. 2{5 astheCSMsrequiremuch lessstoragespaceandaremoreconvenientforanalysis. Inpractice,ascanninggridthatcoverstheregionofintere stwithacertainresolution isconstructedandeverypointofthisgridisconsideredasa potentialsourcewhosepower willbeestimated(seeFigure 2-2 ).Asaresult,abeamformingmap(orimage)thatshows thesignalpowersateachscanningpointwillbeobtained.Co nsequently, L isconsidered tobethenumberofscanningpointsinsteadofthenumberofso urces. Ingeneral,eachofthearraymicrophonesdonotpossessratf requencyresponse withzerophase,andthisisaccountedforviaafrequency-de pendentdiagonalcalibration matrix ~ D =diag( ~ D 1 ;:::; ~ D M ) 2 C M M ,where ~ D m denotesthecomplexcorrectionfactor formicrophone m .ThecalibratedDASdatareductionequationthenbecomes ^ P (D) l = 1 M 2 ~ a Hl ~ DG ~ D H ~ a l ;l =1 ;:::;L: (2{9) Notethatifamicrophoneisknowntobeproblematic,simplyp lacinga0inthe correspondingdiagonalentryof ~ D andchanging M to M 1willpreventitfrom propagatingthroughthedatareductionequation.Thenexts ubsectiondescribeshow ~ D canbeobtainedinpractice. 2.3ArrayCalibration InorderforDAStogiveaccuratesourcelocationandstrengt hestimates,theassumed steeringvectorshavetobeclosetothetrueones.Errorsinm icrophonelocationsand temperature(throughitseectonthesoundspeed)canhavem ajoreectsontheDAS signalpowerestimates,especiallyatrelativelyhighfreq uencies,sincetheseerrorsare multipliedbythewavenumberbeforepropagatingthroughth eDASdatareduction equation.Thissectiondescribesawidelyusedcalibration procedureintroducedby 27

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Dougherty[ 2 ]toremedythisproblem.Thecalibrationsetupconsistsofa speakerthat resemblesapointsourceandisdrivenwithabroadbandsigna l(oratonalsignalwhere thetonefrequencyisvaried).Thespeakerisplacednearthe modellocationanda temporaryanechoicenclosureisrecommendedforahardwall windtunneltominimize sourcererections[ 2 ].ArraydataiscollectedwithnorowandtheresultingCSMis analyzedateachfrequencytoobtainfrequency-dependentc omplexcorrectionfactorsfor allthemicrophones. Theoretically,themeasurementsinthepresenceofasingle sourcearemodeledas(see Eq. 2{2 ) y ( b )= a cal s cal ( b ) ;b =1 ;:::;B; (2{10) where a cal istheactualsteeringvector(unknown)correspondingtoth elocationofthe calibrationspeaker, s cal ( b )isthecalibrationspeakerwaveform(unknown)andthenois e term e ( b )isneglected.AccordingtoEq. 2{8 ,theCSMbecomes G cal = P cal a cal a Hcal ; (2{11) where P cal isthesignalpowerofthecalibrationspeaker.Since G cal isanouterproduct (andhencerank-1),ithasonlyasinglenon-zeroeigenvecto requalto cal = a cal e j = k a cal k 2 andasinglenon-zeroeigenvalueequalto v cal = P cal k a cal k 22 ,where0 2 is anarbitraryphasevalue.Inpractice,althoughtheremaini ngeigenvalueswillnotbe identicallyzero,thedominanteigenvalueisexpectedtobe noticeablylargerwitha goodqualityspeakerthatproducessucientsound[ 2 ].Themeasurementofthe m th microphoneisthenscaledbythecomplexcoecient ~ D m = ( theory ) m ( cal ) m ;m =1 ;:::;M; (2{12) where( ) m denotesthe m th elementofthevectorargument, theory = a theory = k a theory k 2 and a theory isgivenbyEq. 2{3 where r l;m isreplacedbythedistancebetweenthe calibrationspeakerandthe m th microphone.Itisassumedthat k a cal k 2 k a theory k 2 28

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andtherefore y calibrated ( b )= ~ Dy ( b )= k a cal k 2 e j k a theory k 2 a theory s cal ( b ) e j a theory s cal ( b ) ;b =1 ;:::;B: (2{13) Theconstantphaseoset e j willdisappearwhenthepowerlevelsareconsidered.The reasonforthenormalization k a cal k 2 = k a theory k 2 isbecauseonly v cal isknowninpractice andnot a cal or k a cal k 2 (i.e.,thereisascalingambiguity).Animportantaspectof the calibrationisthatitwillcorrectthephasemismatchperfe ctlyforthecalibrationdata.As thedistancebetweenthebeamformingsource(duringtestin g)andthecalibrationspeaker locationsincrease,thebenetofcalibrationisexpectedt odegrade[ 2 ]. Thecalibrationproceduredescribedabovecorrectsthefre quencyresponsedierences betweenthemicrophones.Anadditionalstepcanbeemployed whereanoverallarray correctionfactorisobtainedwithrespecttoareferencemi crophonethatisassumed tobecalibratedseparately.Notethatthedominanteigenva lueofthecalibratedCSM, ~ DG cal ~ D H ,isapproximately v = P cal M=r 2 c ,where r c isthedistancefromthecalibration speakertothearraycenter,andthereferencemicrophonewi llmeasure P ref = P cal =r 2 ref where r ref isthedistancefromthecalibrationspeakertothereferenc emicrophone.The overallarraycorrectionfactoristhengivenas P ref r ref r c 2 M v : (2{14) Therefore,ateachfrequency, M complex-valuedcorrectionfactors(seeEq. 2{12 )will beusedtoscalethemicrophonemeasurementsandasinglerea l-valuedcorrectionfactor (seeEq. 2{14 )willbeusedtoscalethenalarrayestimatedpowerlevels. Whenalltheassumptionsmentionedabovearemet,calibrati onwillprovideaccurate correctionfactorsforasourcenearthecalibrationspeake rlocation.However,inpractice, manysourcesofuncertaintyarepresentduringcalibration .ErrorsinCSMandreference microphonelevelsaretwosuchuncertaintysources.Inaddi tion,itmightbeeasierto measurethedistancebetweenthecalibrationspeakerandth earraythanitistomeasure 29

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thedistancebetweenacomplextestmodelandthearray.This willcauseuncertaintiesin thearraybroadbanddistance,whichisthedistancefromthe arraycentertothecenter ofthescanningregion,forwhichcalibrationcannotaccoun t.Wewillconsiderthese uncertaintiesindetailinChapter 6 30

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array x y z 1 st source 2 nd source Figure2-1.Amicrophonearrayextendinginthe xy -planewith M microphones(shownby thecircles)andinthepresenceoftwonear-eldacousticso urces.The microphoneatthearraycenterisassumedtobeindexedby m =1inthis gure. z array x y scanning region Figure2-2.Ascanningregionwithagivenresolutionisused toobtainabeamforming imagebysteeringthearrayateachofthe L gridpoints. 31

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CHAPTER3 DECONVOLUTIONWITHUNCORRELATEDSOURCES Assumethatthemeasurementnoise f e ( b ) g andacousticsignals f s ( b ) g areindependent andhavezero-meanvalues.Then, E[ s ( b ) e H ( b )]=E[ e H ( b ) s ( b )]=0 : (3{1) Furthermore,whenthemeasurementnoiseiswhitewithpower 2 ,i.e.,E[ e ( b ) e H ( b )]= 2 I thearraycovariancematrix,denotedas G 2 C M M ,canbewrittenas(seeEq. 2{4 ): G =E[ y ( b ) y H ( b )]= APA H + 2 I ; (3{2) where P =E[ s ( b ) s H ( b )] 2 C L L isthesignalcovariancematrix.Since G isnotavailablein practice,itisreplacedbytheCSM, ^ G ,whichwasdenedinEq. 2{8 Theacousticsignalsareassumedtobeuncorrelatedwhenthe correlationbetweenany twosourcesissmallcomparedtotheauto-correlationofthe sources,i.e.,when 1 B B X b =1 s l ( b ) s l 0 ( b ) 1 B B X b =1 j s l ( b ) j 2 ;l 6 = l 0 ;l;l 0 =1 ;:::;L; (3{3) andinthiscase, P inEq. 3{2 isapproximatelyadiagonalmatrixwiththeunknown signalpowers f P l g Ll =1 onitsdiagonal.Theproblemofinterest,then,istoestimat e these L real-valuednon-negativesignalpowers.Whensourcesarep artiallycorrelated (orcoherent),Eq. 3{3 isnolongervalid.Deconvolutionalgorithmsthatcandealw ith correlatedsourceswillbepresentedinChapter 5 ,whereasthealgorithmspresentedinthis ChapterwillbebasedontheassumptiongiveninEq. 3{3 3.1AnExistingDeconvolutionApproach Inthissection,theinverseequationsolvedinDAMASwillbe obtainedfromaslightly dierentperspectivethanthederivationintheoriginalpa per[ 18 ].DAMASusesthe DASbeamformerresultstoobtainthedeconvolvedsourcestr engths.Hence,substituting 32

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Eq. 2{8 inEq. 2{7 ,weobtain ^ P (D) l = 1 M 2 ~ a Hl 2 0 B B X b =1 y ( b ) y H ( b ) ~ a l ; (3{4) andbysubstitutingEq. 2{4 inEq. 3{4 ,weobtain ^ P (D) l = 1 M 2 ~ a Hl 2 0 B B X b =1 ( a 1 s 1 ( b )+ ::: + a L s L ( b ))( a 1 s 1 ( b )+ ::: + a L s L ( b )) H + 2 I # ~ a l (3{5) for l =1 ;:::;L ,wherethenumberofblocksisassumedtobesucientlylarge suchthat themeasurementnoisecovariancematrixisapproximatelye qualtotheensembleaverage. ThecrosstermsinEq. 3{5 arenegligiblewhenthesourcesareuncorrelated(seeEq. 3{3 ). InDAMAS,Eq. 3{5 isapproximatedbyalsoneglectingthemeasurementnoisete rm 2 I toobtain ^ P (D) l = 1 M 2 ~ a Hl L X l 0 =1 2 0 B B X b =1 j s l 0 ( b ) j 2 a l 0 a Hl 0 # ~ a l = L X l 0 =1 ~ A l;l 0 P l 0 ; (3{6) where P l 0 = 2 0 B P Bb =1 j s l 0 ( b ) j 2 istheensembleestimateforthesignalpowerofsource l and ~ A l;l 0 = 1 M 2 j ~ a Hl a l 0 j 2 ;l;l 0 = l;:::;L: (3{7) Stackingupall ^ P (D) l l =1 ;:::;L ,generatesthefollowinglinearsystemofequations: 266664 ^ P (D) 1 ... ^ P (D) L 377775 | {z } ^ p (D) = 1 M 2 266664 j ~ a H1 a 1 j 2 j ~ a H1 a 2 j 2 ::: j ~ a H1 a L j 2 ... ... ... j ~ a HL a 1 j 2 j ~ a HL a 2 j 2 ::: j ~ a HL a L j 2 377775 | {z } ~ A 266664 P 1 ... P L 377775 | {z } p ; (3{8) where ^ p (D) 2 R L 1 and ~ A 2 R L L areknown, ~ A l;l 0 ,whichwasdenedinEq. 3{7 ,denotes theelementinthe l th rowand l 0 ,th columnof ~ A ,andthegoalistoestimate p 2 R L 1 whichconsistsoftheunknownsignalpowersateachscanning point. DAMASestimates p iterativelyusingtheGauss-Seidelmethod[ 49 ]asfollows: ^ P ( i ) l = 1 ~ A l;l max 0 ; ^ P (D) l l 1 X l 0 =1 ~ A l;l 0 ^ P ( i ) l 0 + L X l 0 = l +1 ~ A l;l 0 ^ P ( i 1) l 0 #! ; (3{9) 33

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where i isthecurrentiterationnumberlimitedabovebyauserdene dmaximumnumber ofiterations, ^ P ( i ) l istheDAMASestimateof P l atthe i thiteration, ^ P (0) l =0, l =1 ;:::;L andthenon-negativityofeach ^ P ( i ) l isenforcedsince ^ P l l =1 ;:::;L ,representpower. DAMASrequiresmanyiterations,ontheorderofthousands,t oshowgood performance,andhenceDAMAScanbecomeverytimeconsuming dependingonthe scanningresolution,i.e., L .Inaddition,theGauss-Seidelmethodisnotguaranteedto convergeunless ~ A isdiagonallydominant,i.e.,foreachrow,theabsoluteval ueofthe diagonaltermisgreaterthanthesumofabsolutevaluesofot hertermswhichisusually nottrue[ 19 ].Nevertheless,DAMAShasbeensuccessfullyemployedinma nypractical applications[ 18 ]. 3.2DiagonalRemoval OmittingthenoiseterminEq. 3{6 canbejustiedbytheuseofthediagonalremoval (DR)technique,whicheliminatesthewhitemeasurementnoi sethatappearsinthe diagonalof ^ G bysimplyremovingthediagonalelements,i.e.,makingthem zero[ 2 18 ].In thiscase,theDASoutputiscalculatedas ^ P (D) l = 1 M 2 M ~ a Hl ~ G DR ~ a l ; (3{10) where ~ G DR isobtainedbyremovingthediagonalelementsof ^ G .Then,Eq. 3{8 becomes ^ p (D) = ~ A DR p where, ~ A DRl;l 0 = 1 M 2 M ~ a Hl [ a l 0 a Hl 0 ] diag.=0 ~ a l ;l;l 0 = l;:::;L; (3{11) and[ ] diag.=0 meansthatthediagonalofthematrixargumentissettozero. Intherest ofthischapter,DAMASisappliedwithDRinallcases.Otherw ise,theperformance becomesworseas 2 increases.Consequently,wewilldenote ~ A DRl;l 0 as ~ A l;l 0 (and ~ A DR as ~ A )inordertosimplifythenotation.ItcanbeshownthatDRdoe snotaectthe signal-of-interesttermintheDASoutput,butithasasligh teectontheinterference term[ 2 ]. 34

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3.3SparsityConstrainedDeconvolution 3.3.1SparsityConstrainedFormulation Assumethatameasuredvector y existswhichisknowntosatisfythelinearrelation y = A x where A isknownand x isanunknownquantitythatistobeestimated(see Eq. 3{8 ).Initssimplestform,sparsemodelingcanbestatedasfoll ows: min k x k 0 s.t. y = A x : (3{12) Usually,thisproblemisacombinatorialproblemwhichbeco mesintractablequicklyas thedimensionof x increases[ 25 ].Ifthesolutionissucientlysparse,the ` 0 -normcan bereplacedwiththe ` 1 -normtomaketheproblemconvex[ 25 ].Theimportantpointin Eq. 3{12 isthatthematrix A isusuallyill-conditionedandnotinvertible.Otherwise, the solutioncouldbeobtainedbytakingtheinverse,had A beenasquarematrix,orbythe leastsquaresmethodotherwise. Followingthediscussionabove,theprobleminEq. 3{8 isdirectlyapplicableinthe sparsesignalrepresentationcontextbyobservingthat p issparsesincethenumberof scanningpointsismuchlargerthantheactualnumberofsour cespresent.Thus,wecan immediatelythinkofapplyingaslightlymodiedversionof LASSO[ 26 ]tothisproblem, min p k ^ p (D) ~ Ap k 22 s.t. L X l =1 j P l j ;P l 0 ;l =1 ;:::;L; (3{13) wherethemodicationistoenforceeveryelementof p tobenon-negative.Thefollowing sectiondescribesasimplemethodforchoosingtheparamete r automatically.Also,it isempiricallyobservedthatthemethodinEq. 3{13 isnotverysensitivetotheselection of .Infact,ifwelet !1 ,theformulationwillreducetoanon-negativeleast squaresproblem.However,usingpriorknowledgeaboutthes parsityof p willimprove theestimateinmostcases.TheprobleminEq. 3{13 isaquadraticconvexoptimization problemwhichcanbesolvedecientlyviareadilyavailable interiorpointmethods[ 50 51 ] tondthegloballyoptimalsolutionfor p .Self-dualminimization(SeDuMi)[ 50 ]isan 35

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extensivelyusedpublicdomainsoftwarepackageinthesign alprocessingcommunityfor solvingoptimizationproblemsoversymmetricconesinclud inglinear,quadratic,second orderconicandsemi-deniteprograms.Thedetaileddiscus sionofadvancedoptimization methodsisbeyondthescopeofthisdissertationandtheinte restedreaderisreferredtoa standardtextbookonconvexoptimization[ 52 ]. 3.3.2EstimatingtheUserParameter IntheformulationEq. 3{13 constrainsthe ` 1 -normof p ,i.e.,thetotalpower ofthesignals.Sincethisvalueisunknown,awayofdetermin ing hastobefound forpracticality.Withoutlossofgenerality,assumethate achcolumnof A hasbeen normalizedsuchthatithasunitEuclideannorm[ 53 ].ConsiderEq. 3{2 again.The eigenvaluedecomposition(EVD)of APA H canbewrittenas APA H = UU H ; where thecolumnsoftheunitarymatrix U denotetheeigenvectorsof APA H andthediagonal elementsofthediagonalmatrix arethecorrespondingeigenvalues,denotedas 1 2 ::: L 0 0= ::: =0,where L 0 isthetruenumberofsources.Notethat 2 R M M where M L 0 .Then, G canbewrittenas: G = U + 2 I U H = UU H ; (3{14) wherethediagonalelementsofthediagonalmatrix 2 R M M are r 1 r 2 ::: r L 0 2 = ::: = 2 .Notethat tr( )=tr( UU H )=tr( APA H )=tr( PA H A )=tr( P )= L X l =1 P l ; (3{15) whichisthetotalpowerofthesources.Here,wehaveusedthe factthatthecolumns in A haveunitnormand P isdiagonal.(Notethatwehavealsousedthefactthat tr( APA H )=tr( A H AP )=tr( PA H A ),whichfollowsfromthepropertiesofthematrix traceoperation[ 8 ].) Inpractice,weonlyhavetheestimatedcovariancematrixgi veninEq. 2{8 insteadof thetruecovariancematrix G .Let ^ G = ^ U ^ ^ U H denotetheEVDof ^ G withtheeigenvalues 36

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of ^ G ,i.e.,thediagonalelementsof ^ ,arrangedinnon-increasingorder.Then,wecan determinetheuserparameter inEq. 3{13 as: =tr( ^ ^ r M I ) ; (3{16) where^ r M isthesmallestdiagonalelementof ^ .Werefertotheapproachofsolving Eq. 3{13 usingthe determinedinEq. 3{16 asthesparsityconstrainedDAMAS (SC-DAMAS).WewillshowusingnumericalexamplesthatSC-D AMASiscomputationally moreecientthanDAMAS.NotethatSC-DAMASalsousesDRinEq 3{13 ,whereas whenestimatingtheuserparameter,thefullCSM(withoutre movingthediagonal)is used,andthenoisetermsareremovedasinEq. 3{16 byusingthesmallesteigenvalueof ^ G 3.3.3AMoreEcientImplementation TheDASestimate ^ p (D) usuallycontainsredundantinformationduetothewide3-dB beamwidthoftheDASbeamformer.Therefore,itisnotalways necessarytobeamform atall L scanningpointsfordeconvolutionwithSC-DAMAS.Beamform ingatfewer pointsreducesthesizeof p andincreasesthespeedofSC-DAMAS.Inthiscase,Eq. 3{8 becomes, 266664 ^ P (D) 1 ... ^ P (D) L 0 377775 = 1 M 2 266664 j ~ a H1 a 1 j 2 j ~ a H1 a 2 j 2 ::: j ~ a H1 a L j 2 ... ... ... j ~ a HL 0 a 1 j 2 j ~ a HL 0 a 2 j 2 ::: j ~ a HL 0 a L j 2 377775 266664 P 1 ... P L 377775 ; (3{17) where L 0
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3.4CovarianceMatrixFitting InsteadofusingthesamplecovariancematrixtoobtaintheD ASestimatesandthen tryingtodeconvolvetheresults,weintroduceamethodfore stimatingsourcelocations andstrengthsbaseddirectlyontheCSM,i.e., ^ G .Specically,thecovariancematrix ttingapproachdetermines 2 and f P l g via, min P ; 2 k ^ G APA H 2 I k 2F ; s.t. L X l =1 j P l j ;P l 0 ;l =1 ;:::;L; (3{18) whichisaquadraticconvexoptimizationproblemwhere isdenedinEq. 3{16 and P is adiagonalmatrixwith f P l g Ll =1 onitsdiagonal. TheideabehindCMFisquiteintuitiveinthesensethatitbas icallytriestotthe unknownsignalpowersandthenoisepowertothemodelinEq. 3{2 suchthatthesolution issparse.Incontrasttodeletingthediagonalsof ^ G ,CMFtriestoextractthenoiseand usethesignalcomponentsinthediagonal.Moreover,thisfo rmulationdoesnotrequirethe implementationoftheDASbeamformerasaninitialstepandi tconvergesquicklythanks totheconvexformulation.SimilartoSC-DAMAS,CMFisquite insensitiveto Figure 3-1 showsapictorialrepresentationofEq. 3{8 ,andDAS,DAMAS,SC-DAMAS andCMF.Asmentionedpreviously,DAMASandSC-DAMASdeconv olvethetruesignal powersfromtheDASresults,whereasCMFdeconvolvesthesig nalpowersusingtheCSM directly.Notethatthedeconvolutionapproacheswillesti matethesignalpowersatthe sourcelocations.Therefore,inordertomatchthearrayout putlevelstowhatasingle microphonewouldmeasureatthearraycenter, P l shouldbedividedby r 2 l; 0 for l =1 ;:::;L afterbeingestimatedwithDAMAS,SC-DAMASandCMF. 3.5NumericalExamples Inthissection,weevaluatetheperformanceofDAMAS,SC-DA MASandCMFusing SADA(seeFigure 1-1 (a))[ 1 3 ].WeassumethatSADAisatadistanceof1.50mfrom theregionofinterestsimilartotheDAMASpaper[ 17 18 ].Thescanningregionextends from-25.40cmto25.40cminboththe x -and y -axesandtheresolutioninbothdirections 38

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is2.54cm.Wesimulatethereceivedsignalateachmicrophon eaccordingtoEq. 2{4 Thenoisecomponents f e ( n ) g inEq. 2{4 areassumedtobeuncorrelatedwiththesource signalsanddistributedascircularlysymmetricindepende ntandidenticallydistributed (i.i.d.)complexGaussianrandomvariableswithzero-mean valuesandvariance 2 [ 8 ]. Thesignalwaveformsarealsodistributedascircularlysym metrici.i.d.complexGaussian randomvariableswithzero-meanvaluesandacertainpowerl evelwhichisassumedtobe 25dB.Thisvalueischosenwithoutlossofgeneralitysince 2 willbevariedthroughout theexperiments. Therearethreeparameterswhichaecttheperformancesoft healgorithmsdirectly: numberofFFTblocks B ,noisevariance 2 ,andfrequencyofinterest f .Theincoherence assumptionbreaksdownasthenumberofFFTblocksdecreases sincethecrossterms inEq. 3{5 becomesignicantandalsotheadditionalnoiseterminEq. 3{5 isnolonger 2 I butanon-diagonalmatrix.Noiseaectsallthealgorithmsn egativelyasinall applications.Finally,thefrequencydeterminestheresol utionofthealgorithmssinceas thefrequencyincreases,thedierenceinthesteeringvect orsfornearbysourcesincreases andhenceitiseasierforthealgorithmstodiscriminatethe m. Westartoursimulationswitharelativelyeasyscenariowhe rethenumberofFFT blocksislarge( B =10 ; 000)and 2 =0,inwhichcasetheassumptionsofthealgorithms arealmostcorrect.Theresultingbeamformingmapsareshow ninFigures 3-2 AandB, wherethehorizontalaxisandtheverticalaxisrepresentth e2-Dscanningplaneandthe powerlevelsarerepresentedinagraycolorscaleoveraspan of10dB.Asexpected,DAS mergesthepeaksasiftherewereasinglesource.CMFworksqu itewellandSC-DAMAS andDAMASshowgoodperformancewhenthefrequencyishigh.N ext,thenoisevariance 2 isincreasedto100.AsshowninFigure 3-2 C,CMFismorerobusttonoisethan DAMASandSC-DAMAS.Notethatforalltheexamplesinthissec tion,DAMAShas beenrunfor10,000iterationsafterwhichnosignicantimp rovementwasobserved. 39

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Next,wedecreasethenumberofFFTblocksto500andincrease thefrequencyto20 kHztoresolveamorecomplicatedsourcedistribution.Ther esultsareshowninFigure 3-3 for 2 =0and5.ThelocationestimatesofSC-DAMASforsomeoftheso urcesare inaccurateandDAMASisnotabletodiscriminatethesources .CMF,however,isableto recoverthesourcesreasonablywellinbothcases.Ifthefre quencyisdecreasedto f =5 kHz,DAMASperformancedegradessignicantlyasalsoobser vedinotherstudies[ 22 ]. Figure 3-4 showsthebeamformingmapsfor 2 =0and4with B =500.DAMASisnot abletorecoverthesourcelocationfor f =5kHzevenafter50 ; 000iterations.Weagain observeosetsinthelocationestimatesofSC-DAMASandwhe nthenoiseisincreased, theresultsbecomeworse.ThisisduetothefactthatSC-DAMA Sissolvingthesame DAMASinverseproblem.InFigure 3-4 A,thehighestoutlierforCMFis5dBbelow theactualsignallevelsandinFigure 3-4 B,CMFprovidesthebestperformance.This performancegaincanbeattributedtothedierentformulat ionofCMFthanDAMASand SC-DAMAS.RecallthatCMFdoesnotrelyontheDASbeamformer estimatesanddoes notdeletethediagonals.Notethatthesourcedistribtutio nconsideredforthelastcaseis simplerthantheothertwosincethefrequencyislow. AqualitativeassessmentofthealgorithmsisgiveninTable 3-1 .Thecomputational complexityofDAMASwashigherthanthatofSC-DAMASandCMFi noursimulations. Notethatthepublicdomainsolver[ 50 ]weuseforndingtheoptimalsolutionstothe SC-DAMASandCMFproblemsworksupto L =1 ; 000scanningpointswhenthesignals arecomplex-valued.However,acommercialsoftwaredesign edforthispurposecangoup tomanymorevariablesandhencetheformulationsareapplic abletohigherresolutions ifdesired.Table 3-2 showsthecomputationtimesofeachalgorithmfortheexampl es consideredinthissection.WeobservethatSC-DAMASisthef astestandCMFtakes almostthricethetimeofSC-DAMAS.DAMAStakesalmosttwice thetimeofCMFand eighttimesthetimeofSC-DAMAS.Furthermore,asmentioned inSection 3.3.3 ,the sparseformulationofSC-DAMASallowsustoreducethesizeo f ^ p (D) inEq. 3{8 andthis 40

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inturnprovidesmoreimprovementsinspeed.InTable 3-3 ,wenotedthecomputation timeofSC-DAMASfordierentvaluesof L 0 fortheexampleconsideredinFigure 3-2 B. InFigure 3-5 ,theperformanceofSC-DAMASusingfourdierent L 0 valuesisshownfor theexampleconsideredinFigure 3-2 B(notethatinthisgure,1Drepresentationsofthe imagesareshownwherethehorizontalaxisrepresentsthegr idpointsandtheverticalaxis representsthepowerestimatedateachgridpoint).Itisobs ervedthattheperformance ofSC-DAMASundergoesonlyaminordegradationwhen L 0 > 36.Belowthat,the algorithmdoesnotprovideaccurateresults.Thesavingsar ehugeandtheproblemcan besolvedinalmost1secondwhen L 0 =36whichshouldbecomparedwith167seconds forDAMASand62secondsforCMFkeepinginmindthatCMFprovi desbetterresults thanSC-DAMASandSC-DAMASprovidesbetterresultsthanDAM AS.Inanycase, SC-DAMASoersafastwayofsolvingEq. 3{8 3.6Conclusions Inthischapter,sparsitycontrainedconvexoptimizationm ethods,namelySC-DAMAS andCMF,forthedeconvolutionofuncorrelatedsourceshave beenpresented.SC-DAMAS isanextensionofDAMASandtriestosolvethesamebasicequa tionbyexploiting sparsity.SimilarlytoDAMAS,SC-DAMASemploysDRtomitiga tetheeectsofnoise, whereastheCMFalgorithmeliminatesnoisewithoutdeletin gthediagonalsoftheCSM completely.Also,DAMASandSC-DAMASalgorithmsrequireth eimplementationofDAS andimplicitlydependontheperformanceofthismethod.Ont heotherhand,CMFis independentofDAS.ItwasshownwithsimulationsthatCMFsh owsbetterperformance thanDAS,DAMASandSC-DAMASandSC-DAMASshowsbetterresul tsthanDAMAS. AnalternativeimplementationofSC-DAMASwasprovidedwhi choersafastalgorithm ascomparedtoDAMASandCMF. 41

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nr nr rr rr rnr Figure3-1.MethodologyofDAS,DAMAS,SC-DAMASandCMF. 42

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x-axis (cm.)y-axis (cm.)Actual -24.1 1.3 24.1 -24.1 1.3 24.1 15 20 25 x-axis (cm.)y-axis (cm.)Actual -24.1 1.3 24.1 -24.1 1.3 24.1 15 20 25 x-axis (cm.)y-axis (cm.)Actual -24.1 1.3 24.1 -24.1 1.3 24.1 15 20 25 x-axis (cm.)y-axis (cm.)DAS -24.1 1.3 24.1 -24.1 1.3 24.1 20 22 24 26 28 30 x-axis (cm.)y-axis (cm.)DAS -24.1 1.3 24.1 -24.1 1.3 24.1 20 22 24 26 28 30 x-axis (cm.)y-axis (cm.)DAS -24.1 1.3 24.1 -24.1 1.3 24.1 20 22 24 26 28 30 x-axis (cm.)y-axis (cm.)DAMAS -24.1 1.3 24.1 -24.1 1.3 24.1 15 20 25 x-axis (cm.)y-axis (cm.)DAMAS -24.1 1.3 24.1 -24.1 1.3 24.1 15 20 25 x-axis (cm.)y-axis (cm.)DAMAS -24.1 1.3 24.1 -24.1 1.3 24.1 15 20 25 x-axis (cm.)y-axis (cm.)SC-DAMAS -24.1 1.3 24.1 -24.1 1.3 24.1 15 20 25 x-axis (cm.)y-axis (cm.)SC-DAMAS -24.1 1.3 24.1 -24.1 1.3 24.1 15 20 25 x-axis (cm.)y-axis (cm.)SC-DAMAS -24.1 1.3 24.1 -24.1 1.3 24.1 15 20 25 x-axis (cm.)y-axis (cm.)CMF -24.1 1.3 24.1 -24.1 1.3 24.1 15 20 25 x-axis (cm.)y-axis (cm.)CMF -24.1 1.3 24.1 -24.1 1.3 24.1 15 20 25 x-axis (cm.)y-axis (cm.)CMF -24.1 1.3 24.1 -24.1 1.3 24.1 15 20 25 ABC Figure3-2.Thebeamformingmapsoftheactualsources,DAS, DAMAS,SC-DAMASand CMFwiththreedierentsettingsasfollows: B =10 ; 000,A) f =10kHzand 2 =0,B) f =15kHzand 2 =0,andC) f =15kHzand 2 =100.The2-D plotsrepresentthescanningregionandthepowerlevelsare indB. 43

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x-axis (cm.)y-axis (cm.)Actual -24.1 1.3 24.1 -24.1 1.3 24.1 15 20 25 x-axis (cm.)y-axis (cm.)DAS -24.1 1.3 24.1 -24.1 1.3 24.1 20 22 24 26 28 30 x-axis (cm.)y-axis (cm.)DAMAS -24.1 1.3 24.1 -24.1 1.3 24.1 15 20 25 x-axis (cm.)y-axis (cm.)SC-DAMAS -24.1 1.3 24.1 -24.1 1.3 24.1 15 20 25 x-axis (cm.)y-axis (cm.)CMF -24.1 1.3 24.1 -24.1 1.3 24.1 15 20 25 A x-axis (cm.)y-axis (cm.)Actual -24.1 1.3 24.1 -24.1 1.3 24.1 15 20 25 x-axis (cm.)y-axis (cm.)DAS -24.1 1.3 24.1 -24.1 1.3 24.1 20 22 24 26 28 30 x-axis (cm.)y-axis (cm.)DAMAS -24.1 1.3 24.1 -24.1 1.3 24.1 15 20 25 x-axis (cm.)y-axis (cm.)SC-DAMAS -24.1 1.3 24.1 -24.1 1.3 24.1 15 20 25 x-axis (cm.)y-axis (cm.)CMF -24.1 1.3 24.1 -24.1 1.3 24.1 15 20 25 B Figure3-3.Thebeamformingmapsoftheactualsources,DAS, DAMAS,SC-DAMASand CMFwith f =20kHzand B =500.A) 2 =0.B) 2 =5.The2-Dplots representthescanningregionandthepowerlevelsareindB. 44

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x-axis (cm.)y-axis (cm.)Actual -24.1 1.3 24.1 -24.1 1.3 24.1 10 15 20 25 x-axis (cm.)y-axis (cm.)DAS -24.1 1.3 24.1 -24.1 1.3 24.1 26 26.5 27 27.5 28 28.5 29 x-axis (cm.)y-axis (cm.)DAMAS -24.1 1.3 24.1 -24.1 1.3 24.1 10 15 20 25 x-axis (cm.)y-axis (cm.)SC-DAMAS -24.1 1.3 24.1 -24.1 1.3 24.1 10 15 20 25 x-axis (cm.)y-axis (cm.)CMF -24.1 1.3 24.1 -24.1 1.3 24.1 10 15 20 25 A x-axis (cm.)y-axis (cm.)Actual -24.1 1.3 24.1 -24.1 1.3 24.1 10 15 20 25 x-axis (cm.)y-axis (cm.)DAS -24.1 1.3 24.1 -24.1 1.3 24.1 26 26.5 27 27.5 28 28.5 29 x-axis (cm.)y-axis (cm.)DAMAS -24.1 1.3 24.1 -24.1 1.3 24.1 10 15 20 25 x-axis (cm.)y-axis (cm.)SC-DAMAS -24.1 1.3 24.1 -24.1 1.3 24.1 10 15 20 25 x-axis (cm.)y-axis (cm.)CMF -24.1 1.3 24.1 -24.1 1.3 24.1 10 15 20 25 B Figure3-4.Thebeamformingmapsoftheactualsources,DAS, DAMAS,SC-DAMASand CMFwith f =5kHzand B =500.A) 2 =0.B) 2 =4.The2-Dplots representthescanningregionandthepowerlevelsareindB. 45

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A B C D Figure3-5.PerformanceofSC-DAMASwhen L 0 isvariedfortheexampleconsideredin Figure 3-2 B.A) L 0 =441.B) L 0 =121.C) L 0 =36.D) L 0 =16.Notethat L =441.Ineachplot,thehorizontalaxisrepresentsthegridp ointsandthe verticalaxisrepresentsthepowerestimatedateachgridpo int.Thecircles indicatethetruesourcelocationsandpowers. 46

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Table3-1.Characteristicsoftheacousticimagingalgorit hms. DASDAMASSC-DAMASCMF ResolutionLowMediumHighHighSensitivitytonoiseLowMediumMediumLowComputationtimeLowHighMediumMediumMax.numberofvariablesHighMediumLowLow Table3-2.ComputationtimesofDAMAS,SC-DAMASandCMFinse condsonpersonal computer,2.0GHzdualcoreprocessor,2GBofRAM. Fig.1Fig.2Fig.3 (a)(b)(c)(a)(b)(a)(b) DAMAS160165161160160180162SC-DAMAS24242423242122CMF66657169657573 Table3-3.SpeedingupofSC-DAMASfortheexampleofFigure 3-2 (b).Timingvaluesare inseconds. L 0 4411961218136 Time24.012.74.22.81.3 47

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CHAPTER4 EXPERIMENTALRESULTS Thepurposeofthischapter 1 istoprovideacomparisonofDAS,DAMAS,SC-DAMAS, CMFandCLEAN-SCusingexperimentsconsistingofasingleso urce,twouncorrelated sourceswithsimilaranddierentpowersandtwocoherentso urces.Theexperimental datawasacquiredattheUFAFFusingLAMDA.Thesourcelocali zationcapabilityof thealgorithmsaswellastheiraccuracyinestimatingtheab solutesignalpowerswillbe analyzed.Theabsolutesignalpowersestimatedbythebeamf ormingalgorithmswillbe comparedwiththosemeasuredbyareferencemicrophoneplac edatthecenterofthe microphonearray. 4.1MicrophoneArray LAMDAisazeroredundancyspiralaperturearraybuiltona1. 82mdiameterrigid aluminumplate,thatconsistsof90rush-mountedPanasonic WM-61Amicrophones,and ithasbeenusedinalltheexperimentsinthischapter.LAMDA wasdesignedbythe proceduresdescribedbyUnderbrink[ 5 6 ]andwasfabricatedforuseattheUFAFF[ 47 ]. LAMDAcontainstwonestedspiralarrays: i) asmallapertureinnerarrayconsistingof 45microphonesand ii) alargerapertureouterarrayconsistingof63microphones. We consideronlytheouterarray,whichisshowninFigure 4-1 A(notethatthisisthesame plotshowninFigure 1-1 Bbutwithareferencemicrophoneplacedatthecenter),inth is dissertationduetoitshigherresolutionatlowerfrequenc iesofoperation(werefertothe outerLAMDAarraysimplyasLAMDA).Figure 4-1 Bshowsthe3-dBbeamwidthofthe arrayversusfrequencyatabeamformingdistanceof1.48mfr omthearrayplate.A0.03 mdiameterBruelandKjaer(B&K)microphoneisplacedatthec enterofthearrayand referredtoasthereferencemicrophonethroughoutthischa pter.Thereferencemicrophone 1 TheauthorwouldliketothanktoNikolasZawodny,Christoph erBahr,Dr.FeiLiu, AlbertoGordon,TomKennedyandAdamEdstrandoftheUnivers ityofFloridafortheir assistanceonwindtunneltesting. 48

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iscalibratedusingapistonphoneat1kHz.Thearrayoutputl evelswillbecomparedwith thelevelsmeasuredbythisreferencemicrophone.Therefer encemicrophoneisalsousedin thesecondstepofarraycalibration. 4.2ExperimentalSetup Thedataanalyzedinthischapterwasacquiredbyusinga68-c hannelNational InstrumentsPXI-1045chassiswith17NIPXI-4462dataacqui sition(DAQ)cards.Each channelhas24-bitresolutionwith118dBdynamicrange.All measurementswereac coupledwitha-3dBcut-onat3.4Hz,andappropriateanti-al iasinglterswereapplied. Unlessspeciedotherwise,thefollowingsetofparameters wereusedfordataanalysis andreduction.Thesamplingfrequencyusedinthemeasureme ntswas65,536Hzandthe blocklengthwassetto4096samples,resultingina16Hznarr owbandbinwidth.The dataacquisitiontimewas15seconds.Hanningwindowswith7 5%overlapwereappliedto eachblockofdatabeforetakingtheFFTs.Theresultingnumb erofblockswas957(498 eectiveblocks[ 54 ]). Theacousticsourcesusedintheexperimentswerecustombui ltusingJBLtype 2426Hspeakers.Analuminiumtubeofdiameter0.03mandleng th0.52mwasattached attheoutputofeachJBLspeakertofacilitateplanewavepro pagationandeliminate higherordermodes,andacousticfoamwasusedtomitigatere rectionsasshowninFigure 4-2 A.Themeasurementswereconductedwithoutrow.Twosetsofe xperimentswere conducted;onewithasinglesourceandonewithtwosourcesa sshowninFigures 4-2 B andC,respectively.Inthelattersetofexperiments,thesi gnalpowersandthecoherence betweenthesourceshavebeenvaried.Figure 4-3 showspicturesofthetwospeakersand LAMDAduringtesting. Unlessotherwisespecied,thebeamformingscanningregio nsaresetfrom 0 : 3m to0 : 3mwitharesolutionof0 : 03minboththe x and y directionsforthesinglesource experiments,andfrom 0 : 4mto0 : 2minthe x directionandfrom 0 : 3mto0 : 3m 49

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inthe y directionwitharesolutionof0 : 025minboththe x and y directionsforthe experimentscontainingtwosources. 4.3Software CustomsoftwarehasbeenwritteninMATLAB 2 andLabVIEW 3 fordataacquisition andanalysis.ThedatawereacquiredusingLabVIEWandtheti meseriesofeach microphonewasstoredinbinaryformat.MATLABwasusedfort hereductionand post-processingofthetimedata. Figure 4-4 andFigure 4-5 showsnapshotsfromthedataanalysissoftwarewhichhas beencodedinMATLAB.Therstgraphicaluserinterface(GUI ),showninFigure 4-4 ,is usedtocomputetheCSMsfromtherawmicrophonemeasurement sintime.Thisprogram outputsthefrequency-dependentCSMsandaninformationl econtainingtheparameters usedforcomputingtheCSMs.ThesecondGUI,whichisshownin Figure 4-5 ,isusedto beamformusingtheCSMsandtheinformationleproducedbyt herstuserinterface. ThebeamformingGUIallowstheusertoenterthecenterfrequ encyofinterest,height ofthebeamformingplaneandtheresolution,anditcanbeuse dtoimplementDAS, DAMAS,SC-DAMAS,CMFandCLEAN-SCaswellassomeotheralgor ithmswhich areinthedevelopmentphase.Adierentscanningresolutio ncanbeusedforthebasic algorithmssuchasDASandthemoreadvancedonessuchasDAMA S,SC-DAMASand CMF.OtheroptionsoeredbythebeamformingGUIarethearra ycalibrationprocedure describedinSection 2.3 ,theshearlayercorrection(SLC)proceduredescribedin[ 1 55 ] andDR(seeSection 3.2 ).Thebeamformingresultscanbeobtainedinnarrow-band,1 = 3 rd or1 = 12 th octavebandsusingtheproceduredescribedin[ 20 ],i.e.,bysummingupthe CMSsateachnarrow-bandfrequencyinagivenoctavebandand thenbeamformingonly onceusingthesteeringvectorscorrespondingtothecenter frequency. 2 MATLABisaregisteredtrademarkofTheMathWorks. 3 LabVIEWisaregisteredtrademarkofNationalInstruments. 50

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4.4AbsoluteLevels Theabsolutesignalpowersareestimatedusingtheintegrat ionmethodwithDAS, DAMAS,SC-DAMAS,andCMF,whereasCLEAN-SCusesthecleansp ectrumit constructstoestimatethesignalpowers.WithDAS,theinte gratedSPLiscomputed bysummingtheDASpowerestimatesinsidetheintegrationre gion(whichiswithinthe scanningregion)andnormalizingtheresultbyascalingfac torobtainedbysumming thepsfvaluesoverthesameintegrationregion.Statedmath ematically,theintegrated DASlevelisdenedas P l 2L P l = P l 2L psf( l ),where L isasetcontainingtheindicesofthe scanninggridpointswithintheintegrationregion[ 3 56 ]andpsfwasdenedinSection 2.2 .WithDAMAS,SC-DAMASandCMF,thereisnoneedforthenormal izationsince thearrayresponseisalreadyeliminatedfromtheresultsan donlysummingtheestimated powerlevelswithintheintegrationregionsuces.CLEAN-S Cusestheaverageofthe diagonalofthecleancrossspectralmatrixthatitconstruc tsbyconsideringonlythe contributionsfromsourceswithintheintegrationregion[ 57 ]. 4.5ArrayCalibrationPerformance Consideracalibrationsetupwithasinglespeakerplacedat ( x;y;z )=(0 ; 0 ; 1 : 48) m,wherethearraycenterisat( x;y;z )=(0 ; 0 ; 0)mandthearrayplateextendsin the xy -plane.TheintegratedDASlevelsareshowninFigure 4-6 Aalongsidewith thereferencemicrophonelevelswhenarraycalibrationisn otappliedandthenominal sensitivities(30mV/PaasusedattheUFAFF)ofthemicropho nesareusedfor beamforming.Theintegrationregionisasquarecenteredat ( x;y )=(0 ; 0)mand eachsideofthesquareis0.4mlong.Itisobservedthatthere aredierences,aslarge as5dB,betweenthearrayestimatedlevelsandthereference microphonelevels.On theotherhand,whenarraycalibrationisapplied,theDASin tegratedlevelsmatch thereferencemicrophonelevelsasshowninFigure 4-6 B.Thissimpleexampleshows thatarraycalibrationisessentialformatchingthearrayo utputlevelstothereference microphonelevels.Calibrationprocedurewasalsoshownto benecessaryforreducingthe 51

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uncertaintiesinthebeamforminglevelswhenerrorsareexp ectedinmicrophonelocations and/ortemperature[ 58 ].Therefore,intheresultspresentedbelow,arraycalibra tionis alwaysapplied. 4.6SingleSource ConsidertheexperimentalsetupshowninFigure 4-2 Bwithasinglespeaker generatingbroadbandnoiseatadistanceof1.48mfromthear ray.Figure 4-7 shows thebeamformingplotsobtainedwithDAS,DAMAS,SC-DAMAS,C MFandCLEAN-SC togetherwiththearraypsfat2kHz.Theintegrationregioni sindicatedwiththesolid rectangleandtheintegrated(Int.)andmaximum(Max.)leve lsarenotedintheupper rightcornerofeachplot.Thetruesourcelocationisindica tedbythecross.Weobserve thatallthealgorithmsrecoverthesourcelocationsuccess fullyandalsoindicateconsistent integratedlevelswitheachother.InFigure 4-8 weplottheintegratedSPLsobtained withallthebeamformingalgorithmsoverafrequencyrangeo f0.5kHzto12kHzwith bothsimulatedandexperimentaldata.Inthesimulations,c ircularlysymmetrici.i.d. complexGaussianrandomvariablesareusedtogeneratethes ignalandmeasurementnoise waveformswhicharethenusedtoobtainthesyntheticmicrop honemeasurementsusing Eq. 2{2 andEq. 2{3 .Thesimulatedsignalpowerissetto50dBand458eectivebl ocks areusedsimilartotheexperimentalscenario.Themeasurem entnoisepowerissetto30 dB.WeobservethattheintegratedSPLsofallthealgorithms areingoodagreementwith thereferencemicrophoneSPLswithbothsimulatedandacqui reddata. 4.7TwoUncorrelatedSources ConsidertheexperimentalsetupshowninFigure 4-2 Cwithtwospeakersgenerating uncorrelatedbroadbandnoise.Uncorrelatedbroadbandnoi seisgeneratedbyfeedingthe twospeakerswithindependentwhitenoisesignalsfromtwod ierentoutputsofafunction generator.Weconsidertwoscenarios;intherstcase,thet wospeakersgeneratesignals ofsimilarpowersandinthesecondcase,thesecondspeakerg eneratesaweakersignal thantherstone.InFigure 4-9 thereferencemicrophonelevelswhen i) onlyspeaker1 52

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ison, ii) onlyspeaker2ison, iii) bothspeakersareonand iv) thecomputedsumfrom i) and ii) areplotted.Thenoiseroorofthereferencemicrophoneisal soshowninFigure 4-9 .Figure 4-9 AandBareforthecaseswhenthespeakersgeneratesignalsof similar anddierentpowers,respectively.FromFigure 4-9 A,weobservethatthepowersofthe twosourcesareverysimilar;theaveragesignalpowerovert heentirefrequencyrangeis 47.7dBforspeaker1,47.6dBforspeaker2and50.8dBforboth speakers(computed sumis50.9dB).FromFigure 4-9 B,ontheotherhand,weobservethatsource2has lowerpowerthansource1atmostofthefrequencies;theaver agesignalpoweroverthe entirefrequencyrangeis49.0dBforspeaker1,43.8dBforsp eaker2and50.3dBforboth speakers(computedsumis50.4dB).FromFigure 4-9 ,wealsoobservethatthecomputed summatchesthemeasuredlevels,especiallyforfrequencie saboveapproximately800Hz. Thisshowsthatthesourcesareindeeduncorrelatedforthos efrequencies. InFigures 4-10 and 4-11 ,thebeamformingmapsobtainedwithequalanddierent signalpowers,respectively,areshownatafrequencyof2kH z.ItisobservedthatDASis unabletodistinguishthetwosourcesinbothofthecases,wh ereasDAMAS,SC-DAMAS andCMFcandistinguishthetwosourcessuccessfully.Thees timatedpowerlevelsforthe twosourcesaresimilarinFigure 4-10 andtheestimatedpowerlevelforthesecondsource islowerthanthatoftherstoneinFigure 4-11 withDAMAS,SC-DAMASandCMF. CLEAN-SC,ontheotherhand,identiesthelocationofthese condsourceinaccurately (itwasobservedinresultsnotshownherethatathigherfreq uencies,CLEAN-SCwas abletorecoverthelocationsofbothofthesourcesaccurate ly).Theintegratedlevels obtainedwiththebeamformingalgorithmsareshowninFigur e 4-12 .Itisobservedthat allthealgorithmsyieldverysimilarintegratedlevelswit heachotherandthereference microphone.WeobservethatalthoughCLEAN-SCisunabletor ecoverthesecondsource locationveryaccurately,itsintegratedlevelestimatesl ineupwiththoseoftheother algorithms. 53

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Next,weconsidertheabilityoftheadvancedalgorithmsine stimatingthesignal powersofthetwospeakersindividually.Forthispurpose,t heintegrationregionfor estimatingthesignalpoweroftherstsource(Reg.1)andth eintegrationregionfor estimatingthesignalpowerofthesecondsource(Reg.2)are denedasshowninFigure 4-13 .InFigure 4-14 Aweshowtheintegratedlevels(computedusingReg.1)obtai ned withDASandSC-DAMASwhenonlyspeaker1isonandwhenbothof thespeakers areon.Similarly,inFigure 4-14 Bweshowtheintegratedlevels(computedusingReg. 2)whenonlyspeaker2isonandwhenbothofthespeakersareon .Wealsoshowthe referencemicrophonelevelsobtainedwheneitheroneofthe speakersisoninthesetwo gures.FromFigure 4-14 AitisobservedthatSC-DAMASlevelsobtainedwhenonly oneofthespeakersisonisconsistentwiththelevelsobtain edwhenbothofthespeakers areonandtheintegrationregioncoversonlythesourceofin terest.Ontheotherhand, forDAS,duetolowresolution,theselevelsdonotcoincidew ellforfrequencieslower thanabout2kHz.FromFigure 4-14 B,weobservethatDASperformanceisevenworse whenestimatingthepowerofthesecond(theweaker)sources incethesidelobesfromthe strongersourcearecausingtheDASestimatestobelargerth anthetruesignalpower. Itisalsoobservedthatthearraylevelsmatchthereference microphonelevelsbetterin Figure 4-14 A(withthestrongersource)thaninFigure 4-14 B(withtheweakersource). TheresultswithDAMASandCMFaresimilartothoseobtainedw ithSC-DAMAS, whereastheCLEAN-SCresult(showninFigure 4-15 )isslightlyworsethanDAMAS, SC-DAMASandCMFbutbetterthanDAS. 4.8TwoCoherentSources ConsidertheexperimentalsetupshowninFigure 4-2 Cagainwithtwospeakers butnowgeneratingcoherentbroadbandnoise.Coherentbroa dbandnoiseisgenerated byfeedingthetwospeakerswithasinglewhitenoisewavefor m(thisisdonebyusinga Tconnectionattheoutputofthefunctiongenerator).Figur es 4-16 AandBshowthe SPLsmeasuredbythereferencemicrophoneandtheLAMDAmicr ophoneswhenthetwo 54

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sourcesareuncorrelatedwithsimilarpowers(thesameexam pleconsideredinFigures 4-9 Aand 4-10 )andwhenthetwosourcesarecoherent,respectively.Weobs ervethatthe soundspectraoftheLAMDAmicrophonesbecomeverydierent atfrequenciesabove2 kHzinthelattercase.Adiscussiononthisissueispresente dbelow. Alloftheaforementionedbeamformingalgorithmsarebased ontheassumptionthat thesourcesareuncorrelated.InFigure 4-17 weshowthebeamformingmapsobtained withtwocoherentsourcesat2kHz.Asexpected,thealgorith msfailtodistinguishthe twosources.DASandCLEAN-SCpointasinglesourceapproxim atelyinthemiddleof thetruesourcelocations.AlthoughDAMAS,SC-DAMASandCMF identifytwosources, thelocationsareinaccurate.InFigure 4-18 weshowthebeamformingmapsat4kHz forthecoherentsourcescase.Itisobservedthatallthebea mformingalgorithmsexcept CLEAN-SCcannowidentifythesourcesrelativelymoreaccur ately.SinceCLEAN-SCis basedonremovingthecorrelatedsourcecomponentswiththe peaksinthebeamforming map,itonlyidentiesoneofthecoherentsourcesregardles softhefrequencyandtreats theothercoherentsourcesasthesidelobesduetotheidenti edsource. InFigure 4-19 theintegratedlevelsareshownusingbothsimulatedandexp erimental datawithcoherentsources.Inthesimulations,thesignals originatingfromthetwo speakersaregeneratedasidenticalwaveformsandthesigna lpowerofeachsourceisset to50dB.Weobservethatallthebeamformingalgorithmsstil lyieldconsistentresults witheachotherbutdierentfromthereferencemicrophone, especiallyabove5kHz.In fact,thereferencemicrophoneleveldecreaseswithfreque ncy(withbothsimulationsand experiments),whereasthearrayestimatesdonot. Tounderstandthisphenomenon,considerthemeasurementof the m th microphone, modeledas(seeEq. 2{2 andEq. 2{3 ) y m ( b )=exp( jkr 1 ;m ) =r 1 ;m s 1 ( b )+exp( jkr 2 ;m ) =r 2 ;m s 2 ( b ) ;b =1 ;:::;B; (4{1) 55

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inthepresenceofonlytwosourcesandnocontaminatingnois e.Whenthetwospeaker waveformsareidentical,i.e.,when s 1 ( b )= s 2 ( b )for b =1 ;:::;B (thetwowaveformsmight dierinpracticeduetodisparitiesinwiring,speakersand soon),then y m ( b )=(exp( jkr 1 ;m ) =r 1 ;m +exp( jkr 2 ;m ) =r 2 ;m ) s 1 ( b ) =(exp( jkr 1 ;m ) =r 1 ;m +exp( jkr 2 ;m ) =r 2 ;m ) s 2 ( b ) : (4{2) InFigure 4-20 A,theautospectraofalltheLAMDAmicrophonesandtherefer ence microphone,i.e., 1 B P Bb =1 j y m ( b ) j 2 for m =0 ;:::;M ,where m =0correspondstothe referencemicrophone,isplotted.Inthisgure, s 1 ( b ), b =1 ;:::;B ,(whichisequalto s 2 ( b ))aresimulatedasi.i.d.Gaussianrandomvariableswithze romeanandunitvariance for B =498.Notethat 1 B P Bb =1 j s 1 ( b ) j 2 (and 1 B P Bb =1 j s 2 ( b ) j 2 )isnormalizedsothat thesignalpoweris50dB.Itisobservedthatduetothecohere nceofthesources,each microphoneobservessignicantcancellationatdierentf requencies.Thereasonwhythe arraylevelsdonotdecreaseasfastasthereferencemicroph onelevelinFigure 4-19 is becausethereisalwaysasetofmicrophonesinthearraywhic hdonotencountersevere cancellationandthesemicrophoneshelpkeepthearrayoutp utestimatelargerthanthe referencemicrophonelevel. InFigure 4-20 B,thesimulatedandmeasuredautospectrumofasingleLAMDA microphonelocatedat( x;y;z )=( 0 : 56 ; 0 : 02 ; 0)misshowntogetherwiththemicrophone noiseroor.Itisobservedthattheexperimentaldatamatche sthesimulatedpattern, especiallyforfrequenciesabove2kHz.Thisservesasajust icationthattheinterference patternobservedatthearraymicrophonesisinfactduetoth ecoherenceofthesources. Whenthetwosourcesareuncorrelated,theautospectrumof y m ( b )(seeEq. 4{1 )does notcontainthecrosstermbetweenthetwosourcesandhencet heexponentialtermsdo nothaveaneectontheoutcome(sincetheywillbecancelled outafterbeingmagnitude squared),whereaswhenthesourcesarehighlycorrelated,t heexponentialtermscomeinto 56

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play.Infact,allthecurvesinFigure 4-20 Abecomestraightlinesatthesamelevelwhen thesourcesareuncorrelated(resultsnotshownforbrevity ). Coherentsourcesviolatethefundamentalincoherenceassu mptionofthebeamforming algorithms.Approachesthatcandealwithcoherentsources willbepresentedinthenext chapter. 4.9ComputationalComplexity Wenowelaborateonthecomputationalcomplexitiesofthebe amformingalgorithms withthescanningresolutionsusedinthischapter.Table 4-1 showsthecomputation timesrequiredbyeachalgorithmforthetwoscanningregion settingsusedintheanalysis presentedabove.WeobservethatDASandCLEAN-SCarethefas testalgorithms followedbySC-DAMAS,CMFandDAMAS(5000iterationshavebe enemployedwith DAMAS).NotethatalthoughCLEAN-SCisfasterthantheother advancedalgorithms, itsperformancedependsontheselectionoffouruserparame ters(signicantdegradations inperformancemightbeencounteredwhentheseparametersa renotselectedproperly). OneadvantageofSC-DAMASoverDAMASandCMFisthatwhilesol vingEq. 3{8 ^ p (D) canbeevaluatedatfewerpointsthan L whilestillbeingabletoestimatethepower atallthe L gridpointsasdescribedinSection 3.3.3 .Aruleofthumbistoselectthe scanninggridresolutiontobeatmosthalfthe3-dBbeamwidt hofthearrayatagiven frequency.TheresultsobtainedusingthefastversionofSC -DAMASforthesinglesource ( L 0 =L =0 : 38)andtwouncorrelatedsources( L 0 =L =0 : 27)withsimilarpowerscaseare showninFigure 4-21 (comparewiththeSC-DAMASimagesinFigures 4-7 and 4-10 ). NotethattheperformanceofSC-DAMASdidnotdegradesigni cantlycomparedto usingallthe L gridpointswhenevaluatingDAS.Thecomputationtimerequi redby SC-DAMASbysettingtheresolutionto0.05m(whichisapprox imately1 = 5 th the3-dB beamwidthofthearrayat2kHz)isalsogiveninTable 4-1 57

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4.10AnExamplewithanAirfoilModel Asanalcase,weanalyzetheperformanceofthebeamforming algorithmswiththe NACA63-215ModBairfoil[ 47 59 ].TheschematicofthetestsetupisgiveinFigure 4-22 andapictureoftheairfoilisshowninFigure 4-23 .(Thedetailsofthisaeroacoustic experimentaregivenbyBahretal.[ 47 ]andhenceomittedhereforbrevity.)Notethat intheseexperiments,thereferencemicrophoneatthearray centerwasnotpresent.The beamformingimagesoftheairfoilat2.6kHzisshowninFigur e 4-24 ,wheretwolocations withdominantnoisecanbeidentiedonthetrailingedge(T. E.)whichislocatedat x =0 m.Theleadingedgeislocatedat x =0 : 74mandnotshowninthebeamformingimage sincetheT.E.isthedominantnoisesource.Notethatintheb eamformingmap,the scanningregionextendsfrom-0.5mto0.5minthe x directionandfrom-0.6mto0.6m inthe y directionwithacommonresolutionof0.04m,andthemodelis atabroadband distanceof1.30mwithrespecttothearrayplane.TheMachnu mberis0.17.Dueto thepresenceofrowduringtheairfoiltesting,DRisapplied (seeSection 3.2 ).Moreover, SLChasalsobeenemployed[ 1 55 ].Thedataacquisitiontimewas5seconds,sampling frequencywas65,536Hzandtheblocklengthwas2048samples (frequencyresolutionof 32Hz).AHanningwindowwith75%overlaphasbeenemployedle adingto331eective averages[ 47 ].Weobservethattheestimatednoisesourcesarewellalign edwiththe T.E.oftheairfoil.Thedeconvolutionalgorithmsindicate dominantsourcesatconsistent locationsandtheintegratedlevelsofthesealgorithmsare ingoodagreement.Notethat theintegrationregionisfrom-0.2to0.2minthe x -axisandfrom-0.5to0.5minthe y -axisasshownintheplots. 4.11Conclusions InthischapterwecomparedtheperformanceofDAS,DAMAS,SC -DAMAS,CMF andCLEAN-SCusingexperimentaldataconsistingofseveral testcasesincludingasetup withasinglesource,anothersetupwithtwouncorrelatedso urcesofsimilaranddierent powersandasetupwithtwocoherentsources.Fromtheresult spresentedabove,itwas 58

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observedthatDAMAS,SC-DAMASandCMFyieldthemostreliabl eestimatesinterms ofsourcelocationsandpowers.Theintegratedlevelsobtai nedbyallthealgorithms wereshowntocollapsewiththereferencemicrophonelevels overafrequencyrangefrom 0.5kHzto12kHz.Itwasshownthatwithcoherentsources,non eofthealgorithms candistinguishthesourcesunlessthefrequencyishigh(in whichcaseallalgorithms exceptCLEAN-SCwasshowntoperformreasonablywell).Itwa salsoshownthatthe coherenceofthesourcesresultsinsevereinterferencelos sesoverthearrayaperture. Finally,DASandCLEAN-SCwereshowntobefastestintermsof computationfollowed bySC-DAMAS,CMFandDAMAS. 59

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-80 -60 -40 -20 0 20 40 60 80 -80 -60 -40 -20 0 20 40 60 80 x, cmy, cm LAMDA with 63 mics Reference B&K microphone 1 2 3 4 5 6 8 10 12 16 0 0.1 0.2 0.3 0.4 0.5 0.6 Frequency (kHz)3-dB Beamwidth (m) A B Figure4-1.LAMDAcharacteristics.A)Themicrophonelayou tofLAMDA.Thesolid circleshowsthealuminumplateofLAMDA.AreferenceB&Kmic rophone (notanelementofLAMDA)isincludedinthearraycenterforc omparison purposes.B)The3-dBbeamwidthofLAMDAversusfrequency. ABC Figure4-2.Experimentalsetup.A)Speakersusedintheexpe rimentswerecustombuilt usingJBLtype2426Hspeakers.B)Setup1consistsofasingle sourceplaced 1.48mabovetheLAMDAplate.C)Setup2consistsoftwosource splaced 0.20mapartfromeachother. 60

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A B Figure4-3.Picturesfromtheexperiments.A)Apictureofth etwospeakersasseenfrom nearthearrayplateduringtesting.B)ApictureofLAMDAdur ingtesting. 61

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Figure4-4.TheGUIusedforconstructingtheCSMsfromthera wtimedata. Figure4-5.ThebeamformingGUIusedforthepost-processin goftheCSMs. 62

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1 2 3 4 5 6 7 8 9 10 12 30 35 40 45 50 55 Frequency (kHz)SPL (dB ref. 20 m Pa) BnK DAS 1 2 3 4 5 6 7 8 9 10 12 30 35 40 45 50 55 Frequency (kHz)SPL (dB ref. 20 m Pa) BnK DAS A B Figure4-6.ComparisonofDASintegratedlevelswiththeref erenceB&Kmicrophone levelsA)withoutarraycalibrationandB)witharraycalibr ation. -40 -20 0 20 40 -40 -20 0 20 40 x, cmy, cmPSF, f = 2 kHz -15-15-15-15-8-8-8-8-3-3 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 x, cmy, cmDAS Max. = 49.3 dBInt. = 49.3 dB -40 -20 0 20 40 -40 -20 0 20 40 40 41 42 43 44 45 46 47 48 49 x, cmy, cmDAMAS Max. = 48.8 dBInt. = 49.4 dB -40 -20 0 20 40 -50 -30 -10 10 30 50 40 41 42 43 44 45 46 47 48 49 x, cmy, cmSC-DAMAS Max. = 48.8 dBInt. = 49.3 dB -40 -20 0 20 40 -50 -30 -10 10 30 50 40 41 42 43 44 45 46 47 48 49 x, cmy, cmCMF Max. = 48.8 dBInt. = 49.4 dB -40 -20 0 20 40 -50 -30 -10 10 30 50 40 41 42 43 44 45 46 47 48 49 x, cmy, cmCLEAN-SC Max. = 49.3 dBInt. = 49.1 dB -40 -20 0 20 40 -50 -30 -10 10 30 50 40 41 42 43 44 45 46 47 48 49 Figure4-7.Thearraypointspreadfunction(levelsinnorma lizeddB)andthe beamformingimages(levelsindB)obtainedusingDAS,DAMAS SC-DAMAS,CMFandCLEAN-SCforasinglesourcelocatedatadi stanceof 1.48mfromthearraycenter(setup1).Theintegrationregio nisindicated withthesolidrectangle,andtheintegrated(Int.)andmaxi mum(Max.)levels areshownintheupperrightcornerofeachplot.Thetruesour celocationis indicatedbythecross.Beamformingfrequencyis2kHzandth ereference microphonelevelis49.4dB.Theresultsareobtainedusinge xperimentaldata. 63

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1 2 3 4 5 6 7 8 9 10 12 35 40 45 50 55 Frequency (kHz)SPL (dB ref. 20 m Pa) BnK DAS DAMAS SC-DAMAS CMF CLEAN-SC 1 2 3 4 5 6 7 8 9 10 12 35 40 45 50 55 Frequency (kHz)SPL (dB ref. 20 m Pa) BnK DAS DAMAS SC-DAMAS CMF CLEAN-SC A B Figure4-8.Comparisonofthebeamformerintegratedlevels withthereferenceB&K microphonelevelsforasinglesourcelocatedatadistanceo f1.48mfromthe arraycenter(setup1).A)SimulateddataandB)experimenta ldata. 1 2 3 4 5 6 7 8 9 10 12 20 25 30 35 40 45 50 55 60 Frequency (kHz)SPL (dB ref. 20 m Pa) Spk 1 only Spk 2 only Computed sum Both spks on Noise Floor 1 2 3 4 5 6 7 8 9 10 12 20 25 30 35 40 45 50 55 60 Frequency (kHz)SPL (dB ref. 20 m Pa) Spk 1 only Spk 2 only Computed sum Both spks on Noise Floor A B Figure4-9.Thereferencemicrophonelevelswhenthespeake rsgeneratesignalswithA) similarpowersandB)dierentpowers.Thelevelswheneithe roneofthe speakersisonandbothofthespeakersareonareshowntogeth erwiththe computedsumofthelevelsobtainedwheneitheroneofthespe akersison. Theresultsareobtainedusingexperimentaldata. 64

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-40 -20 0 20 40 -40 -20 0 20 40 x, cmy, cmPSF, f = 2 kHz -15-15-15-15-8-8-8-8-3-3 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 x, cmy, cmDAS Max. = 49.7 dBInt. = 51.2 dB -40 -20 0 20 40 -40 -20 0 20 40 40 41 42 43 44 45 46 47 48 49 x, cmy, cmDAMAS Max. = 47.2 dBInt. = 51.9 dB -40 -20 0 20 40 -50 -30 -10 10 30 50 40 41 42 43 44 45 46 47 48 49 x, cmy, cmSC-DAMAS Max. = 47.4 dBInt. = 51.9 dB -40 -20 0 20 40 -50 -30 -10 10 30 50 40 41 42 43 44 45 46 47 48 49 x, cmy, cmCMF Max. = 47.3 dBInt. = 51.9 dB -40 -20 0 20 40 -50 -30 -10 10 30 50 40 41 42 43 44 45 46 47 48 49 x, cmy, cmCLEAN-SC Max. = 45.6 dBInt. = 51.7 dB -40 -20 0 20 40 -50 -30 -10 10 30 50 40 41 42 43 44 45 46 47 48 49 Figure4-10.Thearraypointspreadfunction(levelsinnorm alizeddB)andthe beamformingimages(levelsindB)obtainedusingDAS,DAMAS SC-DAMAS,CMFandCLEAN-SCfortwouncorrelatedsourceswit hsimilar powerslocated0.20mapartfromeachother(setup2).Theint egration regionisindicatedwiththesolidrectangle,andtheintegr ated(Int.)and maximum(Max.)levelsareshownintheupperrightcornerofe achplot.The truesourcelocationsareindicatedbythecrosses.Beamfor mingfrequencyis 2kHzandthereferencemicrophonelevelis52.6dB.Theresul tsareobtained usingexperimentaldata. 65

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-40 -20 0 20 40 -40 -20 0 20 40 x, cmy, cmPSF, f = 2 kHz -15-15-15-15-8-8-8-8-3-3 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 x, cmy, cmDAS Max. = 50.5 dBInt. = 50.9 dB -40 -20 0 20 40 -40 -20 0 20 40 41 42 43 44 45 46 47 48 49 50 x, cmy, cmDAMAS Max. = 48.1 dBInt. = 51.5 dB -40 -20 0 20 40 -50 -30 -10 10 30 50 41 42 43 44 45 46 47 48 49 50 x, cmy, cmSC-DAMAS Max. = 48.7 dBInt. = 51.5 dB -40 -20 0 20 40 -50 -30 -10 10 30 50 41 42 43 44 45 46 47 48 49 50 x, cmy, cmCMF Max. = 48.6 dBInt. = 51.5 dB -40 -20 0 20 40 -50 -30 -10 10 30 50 41 42 43 44 45 46 47 48 49 50 x, cmy, cmCLEAN-SC Max. = 49.7 dBInt. = 51.3 dB -40 -20 0 20 40 -50 -30 -10 10 30 50 41 42 43 44 45 46 47 48 49 50 Figure4-11.Thearraypointspreadfunction(levelsinnorm alizeddB)andthe beamformingimages(levelsindB)obtainedusingDAS,DAMAS SC-DAMAS,CMFandCLEAN-SCfortwouncorrelatedsourceswit h dierentpowerslocated0.20mapartfromeachother(setup2 ).The integrationregionisindicatedwiththesolidrectangle,a ndtheintegrated (Int.)andmaximum(Max.)levelsareshownintheupperright cornerof eachplot.Thetruesourcelocationsareindicatedbythecro sses. Beamformingfrequencyis2kHzandthereferencemicrophone levelis52.3 dB.Theresultsareobtainedusingexperimentaldata. 66

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1 2 3 4 5 6 7 8 9 10 12 30 35 40 45 50 55 60 Frequency (kHz)SPL (dB ref. 20 m Pa) BnK DAS DAMAS SC-DAMAS CMF CLEAN-SC 1 2 3 4 5 6 7 8 9 10 12 30 35 40 45 50 55 60 Frequency (kHz)SPL (dB ref. 20 m Pa) BnK DAS DAMAS SC-DAMAS CMF CLEAN-SC A B Figure4-12.Comparisonofthebeamformerintegratedlevel swiththereference microphonelevelsfortwouncorrelatedsourceslocated0.2 0mapartfrom eachother(setup2).A)Thetwosourcesareofsimilarpoweri nA)andthe sourceat( x;y )=(0 ; 0)misstrongerthanthesourceat( x;y )=( 0 : 2 ; 0)m inB).Theresultsareobtainedusingexperimentaldata. x, cmy, cm -40 -20 0 20 40 -40 -20 0 20 40 41 42 43 44 45 46 47 48 49 50 Reg. 2 Reg. 1 Figure4-13.Integrationregion1(Reg.1)andintegrationr egion2(Reg.2)areusedwhen estimatingthepowerofsources1and2,respectively. 67

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1 2 3 4 5 6 7 8 9 10 12 30 35 40 45 50 55 Frequency (kHz)SPL (dB ref. 20 m Pa) B&K, only spk 1 on DAS, only spk 1 on SC-DAMAS, only spk 1 on DAS, both spks on SC-DAMAS, both spks on 1 2 3 4 5 6 7 8 9 10 12 30 35 40 45 50 55 Frequency (kHz)SPL (dB ref. 20 m Pa) B&K, only spk 2 on DAS, only spk 2 on SC-DAMAS, only spk 2 on DAS, both spks on SC-DAMAS, both spks on A B Figure4-14.ComparisonoftheDASandSC-DAMASintegratedl evelswiththereference microphonelevelsfortwouncorrelatedsourceswithdiere ntpowerslocated 0.20mapartfromeachother(setup2).A)DASandSC-DAMASint egrated levelsforsource1(calculatedoverReg.1)areshownwhenon lyspeaker1is onandwhenbothspeakersareon.ThereferenceB&Kmicrophon elevelsare shownwhenonlyspeaker1ison.B)DASandSC-DAMASintegrate dlevels forsource2(calculatedoverReg.2)areshownwhenonlyspea ker2isonand whenbothspeakersareon.B&Klevelsareshownwhenonlyspea ker2ison. Theresultsareobtainedusingexperimentaldata. 1 2 3 4 5 6 7 8 9 10 12 30 35 40 45 50 55 Frequency (kHz)SPL (dB ref. 20 m Pa) B&K, only spk 1 on DAS, only spk 1 on CLEAN-SC, only spk 1 on DAS, both spks on CLEAN-SC, both spks on 1 2 3 4 5 6 7 8 9 10 12 30 35 40 45 50 55 Frequency (kHz)SPL (dB ref. 20 m Pa) B&K, only spk 2 on DAS, only spk 2 on CLEAN-SC, only spk 2 on DAS, both spks on CLEAN-SC, both spks on A B Figure4-15.ComparisonoftheDASandCLEAN-SCintegratedl evelswiththereference microphonelevelsfortwouncorrelatedsourceswithdiere ntpowerslocated 0.20mapartfromeachother(setup2).A)DASandSC-DAMASint egrated levelsforsource1(calculatedoverReg.1)areshownwhenon lyspeaker1is onandwhenbothspeakersareon.ThereferenceB&Kmicrophon elevelsare shownwhenonlyspeaker1ison.B)DASandSC-DAMASintegrate dlevels forsource2(calculatedoverReg.2)areshownwhenonlyspea ker2isonand whenbothspeakersareon.B&Klevelsareshownwhenonlyspea ker2ison. Theresultsareobtainedusingexperimentaldata. 68

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1 2 3 4 5 6 7 8 9 10 12 0 10 20 30 40 50 60 Frequency (kHz)SPL (dB ref. 20 m Pa) LAMDA microphones B&K microphone Noise floor (LAMDA mics) Noise floor (B&K mic) 1 2 3 4 5 6 7 8 9 10 12 0 10 20 30 40 50 60 Frequency (kHz)SPL (dB ref. 20 m Pa) LAMDA microphones B&K microphone Noise floor (LAMDA mics) Noise floor (B&K mic) A B Figure4-16.Thesoundpressurelevelsmeasuredbytherefer enceB&Kmicrophoneand theLAMDAmicrophoneswhenthetwosourcesareA)uncorrelat edwith similarpowersandB)coherent. 69

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-40 -20 0 20 40 -40 -20 0 20 40 x, cmy, cmPSF, f = 2 kHz -15-15-15-15-8-8-8-8-3-3 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 x, cmy, cmDAS Max. = 55.7 dBInt. = 55.9 dB -40 -20 0 20 40 -40 -20 0 20 40 46 47 48 49 50 51 52 53 54 55 x, cmy, cmDAMAS Max. = 48.4 dBInt. = 56.3 dB -40 -20 0 20 40 -50 -30 -10 10 30 50 46 47 48 49 50 51 52 53 54 55 x, cmy, cmSC-DAMAS Max. = 52.7 dBInt. = 56.1 dB -40 -20 0 20 40 -50 -30 -10 10 30 50 46 47 48 49 50 51 52 53 54 55 x, cmy, cmCMF Max. = 51.6 dBInt. = 56.1 dB -40 -20 0 20 40 -50 -30 -10 10 30 50 46 47 48 49 50 51 52 53 54 55 x, cmy, cmCLEAN-SC Max. = 55.7 dBInt. = 55.8 dB -40 -20 0 20 40 -50 -30 -10 10 30 50 46 47 48 49 50 51 52 53 54 55 Figure4-17.Thearraypointspreadfunction(levelsinnorm alizeddB)andthe beamformingimages(levelsindB)obtainedusingDAS,DAMAS SC-DAMAS,CMFandCLEAN-SCfortwocoherentsourceslocated 0.20m apartfromeachother(setup2).Theintegrationregionisin dicatedwiththe solidrectangle,andtheintegrated(Int.)andmaximum(Max .)levelsare shownintheupperrightcornerofeachplot.Thetruesourcel ocationsare indicatedbythecrosses.Beamformingfrequencyis2kHzand thereference microphonelevelis58.2dB.Theresultsareobtainedusinge xperimental data. 70

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-40 -20 0 20 40 -40 -20 0 20 40 x, cmy, cmPSF, f = 4 kHz -15-15-8-8-3 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 x, cmy, cmDAS Max. = 53.7 dBInt. = 56.7 dB -40 -20 0 20 40 -40 -20 0 20 40 44 45 46 47 48 49 50 51 52 53 x, cmy, cmDAMAS Max. = 49.9 dBInt. = 56.9 dB -40 -20 0 20 40 -50 -30 -10 10 30 50 44 45 46 47 48 49 50 51 52 53 x, cmy, cmSC-DAMAS Max. = 50.6 dBInt. = 56.7 dB -40 -20 0 20 40 -50 -30 -10 10 30 50 44 45 46 47 48 49 50 51 52 53 x, cmy, cmCMF Max. = 49.9 dBInt. = 56.7 dB -40 -20 0 20 40 -50 -30 -10 10 30 50 44 45 46 47 48 49 50 51 52 53 x, cmy, cmCLEAN-SC Max. = 53.7 dBInt. = 56.4 dB -40 -20 0 20 40 -50 -30 -10 10 30 50 44 45 46 47 48 49 50 51 52 53 Figure4-18.Thearraypointspreadfunction(levelsinnorm alizeddB)andthe beamformingimages(levelsindB)obtainedusingDAS,DAMAS SC-DAMAS,CMFandCLEAN-SCfortwocoherentsourceslocated 0.20m apartfromeachother(setup2).Theintegrationregionisin dicatedwiththe solidrectangle,andtheintegrated(Int.)andmaximum(Max .)levelsare shownintheupperrightcornerofeachplot.Thetruesourcel ocationsare indicatedbythecrosses.Beamformingfrequencyis4kHzand thereference microphonelevelis58.3dB.Theresultsareobtainedusinge xperimental data. 71

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1 2 3 4 5 6 7 8 9 10 12 30 35 40 45 50 55 60 Frequency (kHz)SPL (dB ref. 20 m Pa) BnK DAS DAMAS SC-DAMAS CMF CLEAN-SC 1 2 3 4 5 6 7 8 9 10 12 30 35 40 45 50 55 60 Frequency (kHz)SPL (dB ref. 20 m Pa) BnK DAS DAMAS SC-DAMAS CMF CLEAN-SC A B Figure4-19.Comparisonofthebeamformerintegratedlevel swiththereference microphonelevelsfortwocoherentsourceslocated0.20map artfromeach other(setup2).A)SimulateddataandB)experimentaldata. 1 2 3 4 5 6 7 8 9 10 12 30 35 40 45 50 55 60 Frequency (kHz)SPL (dB ref. 20 m Pa) LAMDA microphones B&K microphone 1 2 3 4 5 6 7 8 9 10 12 16 -10 0 10 20 30 40 50 60 Frequency (kHz)SPL (dB ref. 20 m Pa) Simulation Experiment Noise Floor A B Figure4-20.Interferenceduetosourcecoherence.A)Thein terferenceinducedbythe coherenceofthesourcesateachmicrophone(usingsimulate ddata).B)The comparisonofthesimulatedwithmeasuredinterferenceee ct. 72

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x, cmy, cmSC-DAMAS Max. = 48.7 dBInt. = 49.4 dB -40 -20 0 20 40 -50 -30 -10 10 30 50 40 41 42 43 44 45 46 47 48 49 x, cmy, cmSC-DAMAS Max. = 45.9 dBInt. = 51.6 dB -40 -20 0 20 40 -50 -30 -10 10 30 50 40 41 42 43 44 45 46 47 48 49 AB Figure4-21.ThebeamformingmapsobtainedusingthefastSC -DAMAS.A)Forasingle sourcelocatedatadistanceof1.48mfromthearraycenter(s etup1). ComparetoFigure 4-7 .B)Fortwouncorrelatedsourceswithsimilarpowers located0.20mapartfromeachother(setup2).ComparetoFig ure 4-10 73

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Test Section Inlet Jet Collector Flow Direction Shear Layer 0.862 mA c o u s t i c R a y P a t h 0.74 m 0.84 m 0.30 m 0.44 m LAMDA 0.91 m x y z 0.91 m 1.30 m 0.74 m Figure4-22.TheexperimentalsetupfortheNACA63-215ModBairfoilacoustic measurements.CourtesyofChrisBahr. Figure4-23.PictureoftheNACA63-215Mod-Bairfoil.Court esyofChrisBahr. 74

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-40 -20 0 20 40 -40 -20 0 20 40 x, cmy, cmPSF, f = 2.6 kHz -15-15-15-8-8-8-3-3 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 x, cmy, cmDAS T.E. Max. = 45.3 dBInt. = 51.5 dB -40 -20 0 20 40 -60 -40 -20 0 20 40 60 36 37 38 39 40 41 42 43 44 45 x, cmy, cmDAMAS T.E. Max. = 41.3 dBInt. = 51.0 dB -40 -20 0 20 40 -60 -40 -20 0 20 40 60 36 37 38 39 40 41 42 43 44 45 x, cmy, cmSC-DAMAS T.E. Max. = 40.8 dBInt. = 51.1 dB -40 -20 0 20 40 -60 -40 -20 0 20 40 60 36 37 38 39 40 41 42 43 44 45 x, cmy, cmCMF T.E. Max. = 42.0 dBInt. = 51.8 dB -40 -20 0 20 40 -60 -40 -20 0 20 40 60 36 37 38 39 40 41 42 43 44 45 x, cmy, cmCLEAN-SC T.E. Max. = 43.5 dBInt. = 50.8 dB -40 -20 0 20 40 -60 -40 -20 0 20 40 60 36 37 38 39 40 41 42 43 44 45 Figure4-24.Thearraypointspreadfunction(levelsinnorm alizeddB)andthe beamformingimages(levelsindB)obtainedusingDAS,DAMAS SC-DAMAS,CMFandCLEAN-SCfortheNACA63-215airfoilandwi tha Machnumberof0.17.Theintegrationregionisindicatedwit hthesolid rectangle,andtheintegrated(Int.)andmaximum(Max.)lev elsareshownin theupperrightcornerofeachplot.Thesolidlineat x =0mindicatesthe trailingedge(T.E.)andtheleadingedgeisat x =0.74m(notshown). Beamformingfrequencyis2.6kHz.Theresultsareobtainedu sing experimentaldata. Table4-1.Computationtimes(inseconds)onapersonalcomp uter(2.53GHzprocessor and3GbytesofRAM). No.ofgridsDASDAMASSC-DAMASCMFCLEAN-SCSC-DAMAS(Fast) 4410.378.012.369.21.24.26250.6138.031.8123.51.611.6 75

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CHAPTER5 DECONVOLUTIONWITHCORRELATEDSOURCES Whentheacousticsignalsgeneratedbythesourcesarecorre lated,Eq. 3{3 isno longervalidand,asaresult,thenumberofreal-valuedunkn ownsin P =E[ s ( b ) s H ( b )] (seeEq. 3{2 )increasesto L 2 (notethat P isnolongeradiagonalmatrixnowbutitis Hermitiansymmetric,i.e., P H = P ).Inthefollowing,themeasurementnoiseisassumed toberemovedfrom ^ G .Discussionsonhowthiscanbedoneinpracticeareprovided in Section 5.4 TheDASpowerestimatesforcorrelatedsourcesbecomes[ 24 ] ^ P (D) l;l 0 = 1 M 2 ~ a Hl ^ G ~ a l 0 ;l;l 0 =1 ;:::;L; (5{1) andthisalgorithmisreferredtoasDAS-C. 5.1AnExistingDeconvolutionApproach ThecounterpartofDAMASforcorrelatedsources,DAMAS-C,s olves 2666666666666664 ^ P (D) 1 ; 1 ... ^ P (D) 1 ;L ^ P (D) 2 ; 2 ... ^ P (D) L;L 3777777777777775 | {z } ^ p (D)c = 1 M 2 2666666666666664 j ~ a H1 a 1 j 2 ~ a H1 a 1 a H2 ~ a 1 ::: ~ a H1 a L a HL ~ a 1 ... ... ... ~ a H1 a 1 a H1 ~ a L ~ a H1 a 1 a H2 ~ a L ::: ~ a H1 a L a HL ~ a L ~ a H2 a 1 a H1 ~ a 2 ~ a H2 a 1 a H2 ~ a 2 ::: ~ a H2 a L a HL ~ a 2 ... ... ... ~ a HL a 1 a H1 ~ a L ~ a HL a 1 a H2 ~ a L ::: j ~ a HL a L j 2 3777777777777775 | {z } ~ A c 2666666666666664 P 1 ; 1 ... P 1 ;L P 2 ; 2 ... P L;L 3777777777777775 | {z } p c ; (5{2) for p c 2 C L ( L +1) = 2 1 given ^ p (D)c 2 C L ( L +1) = 2 1 and ~ A c 2 C L ( L +1) = 2 L ( L +1) = 2 .Note thatsince P isHermitiansymmetric,only P l;l 0 for l 0 l l;l 0 =1 ;:::;L ,shouldbe estimated.DAMAS-CalsousestheGauss-Seidelmethod(seeE q. 3{9 ),asdescribed previously,tosolveEq. 5{2 .AmajordrawbackofDAMAS-Cisthatitassumesthat thecross-correlationlevelofanytwosourcesisrealandno n-negative,i.e., P l;l 0 0for l;l 0 =1 ;:::;L 76

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5.2CovarianceMatrixFittingwithCorrelatedSources WedeviseCMF-C,whichisanextensionofCMFtothecorrelate dsourcecase,asan alternativetoDAMAS-C.CMF-Csolves[ 53 ]: min P k ^ G APA H k 2F ; s.t.tr( P ) ; P 0 ; (5{3) whichisaconvexsemi-deniteprogram(SDP)thatcanbesolv edwithSeDuMi.CMF-C allowsforcomplex-valuedcross-correlationvaluesandhe nceismoregeneralthan DAMAS-C.NotethatCMF-CreducestoCMFwhenthesourcesareu ncorrelated. Duetothesignicantincreaseinthenumberofunknownsinth epresenceof correlatedsources(from L to L 2 ),DAMAS-CandCMF-Careimpracticalcomputation-wise evenforasmallnumberofscanninggrids(anexampleusingon ly36scanningpointswas shownin[ 53 ]).Therefore,beamformingalgorithmsthatcanworkwithco rrelated(even coherent)sourceswithmuchlowercomputationalcomplexit iesaredesirable.Wepresent suchanapproachinthefollowinganddemonstrateitsabilit ytoseparatepartially correlatedorcoherentsourcesusingsimulationsaswellas experimentaldata. 5.3AFastBeamformerforCorrelatedSources Thealgorithmpresentedinthissectionmakesuseofthefact thatthenumberof non-zeroeigenvaluesofthe(noise-free)CSMissmallertha norequaltothenumberof sources[ 8 9 ],whichismuchsmallerthanthenumberofscanningpoints.T herefore,if aroughestimateofthenumberofsourcesisavailable,thepr oblemdimensioncanbe reducedsignicantlyaswillbeshownbelow.Moreover,assh ownintheresultssection, theproposedalgorithmisquiteinsensitivetothe(over)es timationofthenumberof sources. Therststepinreducingtheproblemdimensionistotruncat etheeigenvaluesofthe CSMthatarealmostzero.Forthispurpose,let ^ G = UU H betheEVDof ^ G ,where thecolumnsoftheunitarymatrix U 2 C M M aretheeigenvectorsof ^ G andthediagonal elementsofthediagonalmatrix 2 R M M arethecorrespondingeigenvalues f m g Mm =1 77

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suchthat 1 ::: M 0.Furthermore,let G ^ L 2 C M M denotethecovariance matrixobtainedbykeepingonlythe ^ L M largesteigenvaluesof ^ G (sincetherestofthe eigenvalueswillbeverysmallwhenthenoiseisremovedfrom theCSM),i.e., G ^ L = U 264 ^ L 0 00 375 U H = U ^ L ^ L U H ^ L ; (5{4) where ^ L isanestimateofthenumberofsources, U ^ L 2 C M ^ L consistsoftherst ^ L columnsof U U H ^ L U ^ L = I ,and ^ L = 266664 1 0 0 ^ L 377775 : (5{5) Replacing ^ G by G ^ L ,Eq. 5{3 canbewrittenas: min P k G ^ L APA H k 2F ; s.t.tr( P ) ; P 0 ; (5{6) where =tr( )[ 53 ]. Thesecondstepforreducingtheproblemdimensionistorepl ace P intheabove optimizationwithasmallermatrix C 2 C L ^ L ofranklessthanorequalto ^ L ,suchthat CC H = P .Thisensuresthat P isHermitiansymmetric,positivesemi-deniteandofrank atmost ^ L ,asdesired.Then,Eq. 5{6 becomes min C k G ^ L AC ( AC ) H k 2F ; s.t. k C k 2F : (5{7) However,evenafterthesesimplications,itisstilldicu lttosolvethequarticproblemin Eq. 5{7 .Therefore,inanattempttofurthersimplifyEq. 5{7 ,wewriteitinthefollowing equivalentform: min C k ( ~ GQ H )( ~ GQ H ) H AC ( AC ) H k 2F ; s.t. k C k 2F ; Q H Q = I ; (5{8) 78

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where ~ G = U ^ L ( ^ L ) 1 = 2 2 C M ^ L suchthat ~ G ~ G H = G ^ L ,and Q 2 C ^ L ^ L isanarbitrary unitarymatrixsuchthat Q H Q = QQ H = I .Theauxiliaryvariable Q hasbeenintroduced inEq. 5{8 toobtainamoreconvenientcostfunctionwhoseminimizatio nshouldgive similarresultstothoseobtainedfromEq. 5{7 [ 60 { 62 ]: min Q ; C k ~ GQ H AC k 2F ; s.t. k C k 2F ; Q H Q = I : (5{9) Indeed,ifthecostfunctioninEq. 5{9 canbeminimizedtoasmallvalue,thenthecost functioninEq. 5{8 willalsobesmallandviceversa(moredetailsonthistransi tionis providedin[ 62 ]). SolvingtheprobleminEq. 5{9 requiresthejointestimationofboth Q and C ,which canbedonebymeansofacyclicmethodologyinwhichthecostf unctionisminimized inanalternatingfashionwithrespectto Q or C ,whiletheothervariableisassumed given.Afterperformingacertainnumberofiterations,the estimateof P ,denotedas ^ P willbegivenby ^ C ^ C H ,where ^ C isthenalestimateof C .Thisapproachismuchfaster thantryingtosolveEq. 5{3 asthenumberofunknownswhenestimating Q or C ismuch smallerthanthenumberofunknownswhenestimating P However,theFrobeniusnormconstrainton C doesnotleadtogoodperformancein generalsince C isexpectedtobesparse(as P issparse)andtheFrobeniusnormdoesnot promotesparsity[ 28 ].Accordingly,onecanreplacetheFrobeniusnorminEq. 5{9 with the ` 1 -norm,whichiswidelyusedintheliteratureonrecoveryofs parsesignals,see,e.g., [ 28 ],toachieveasparseestimateof C (andconsequently P ): min Q ; C k ~ GQ H AC k 2F ; s.t. k c k 1 ; Q H Q = I ; (5{10) where c =vec( C )andtheuserparameter isselectedbyusingtheCauchy-Schwartz inequality[ 63 ]asfollows(recallthat =tr( )): L ^ L X l =1 j c l j vuut L ^ L X l =1 j c l j 2 ( L ^ L ) q L ^ L = : (5{11) 79

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Forgiven Q ,theprobleminEq. 5{10 isasecond-orderconeprogram(SOCP)that canbesolvedfor C ecientlybyusingSeDuMi.Furthermore,forgiven C ,Eq. 5{10 can besolvedfor Q byusingthemethoddescribedin[ 64 ].Notethat k ~ GQ H AC k 2F = 2Re h tr( ~ G H ACQ ) i ; (5{12) where issomeconstanttermnotdependingon Q .Let ~ G H AC = ^ U ~ ~ U H bethe singular-valuedecomposition(SVD)of ~ G H AC ,where ^ U 2 C ^ L ^ L and ~ U 2 C ^ L ^ L .Then, thesolutiontoEq. 5{10 isgivenby[ 64 ]: ^ Q = ~ U ^ U H : (5{13) Thealgorithmthatestimates C viaEq. 5{10 and Q viaEq. 5{13 inaniterative mannerisreferredtoasMACS.ForimplementingMACS,rst I isusedasaninitial estimatefor Q and C isestimated.Thengiventheestimated C Q isupdatedand soon.Table 5-1 outlinestheMACSalgorithm.MACSisacyclicalgorithm,i.e .,the costfunctionisguaranteedtodecreaseateachiteration,a ndthisensuresthatMACS willconvergeatleastlocally.Itwasempiricallyobserved thatusuallynosignicant improvementinperformanceisachievedwithMACSafter5ite rations(itisalsopossible toimplementthealgorithmuntilnosignicantimprovement in ^ P isachievedintwo consecutiveiterations). ThecomputationalcomplexityofMACSperiterationismainl ydictatedbythe complexityoftheSOCPinEq. 5{10 ,whichisgivenby O ( L 3 ^ L 3 )[ 65 ](thecomplexity ofCMF-Cis O ( L 6 )andthatofDAMAS-Cis O ( L 4 )timesthenumberofDAMAS-C iterationsrequiredforconvergencewhichisusuallyonthe orderoftenthousands[ 24 ]). Itisimportanttonotethattheactualcomputationtimedoes notdependsolelyon thenumberofmultiplicationanddivisionoperationsascon sideredwhenreportingthe complexityofthealgorithmbutratherisafunctionoftheme moryaccesstime,the implementationsoftwareandhardware,andthenumberofcom putationscombined 80

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together.SamplerunningtimesforMACSwillbeprovidedint henumericalexamples section. Notethat P l;l 0 estimatedwithDAMAS-C,CMF-CandMACSshouldbedividedby r l; 0 r l 0 ; 0 for l;l 0 =1 ;:::;L tomatchthelevelsestimatedwithDAS-C. 5.4MeasurementNoise Notethatasintheuncorrelatedcase,DAS-CandDAMAS-Cempl oyDR,whereas CMF-Cminimizes k ^ G APA H 2 I k 2F withrespecttoboth P and 2 ,thusestimating 2 alongsidewith P .SosimilartoCMF,CMF-Conlyextractsthenoisefrom ^ G ,while keepingthesignalportionofthediagonalof ^ G intact.WhenusingMACS,thewhitenoise powerisestimatedusing^ 2 = 1 M ^ L P Mm = ^ L +1 m .Then, G ^ L (denedinEq. 5{4 )isreplaced by G ^ L = U 264 ^ L ^ 2 I0 00 375 U H = U ^ L ~ ^ L U H ^ L : (5{14) and ^ L isreplacedby ~ ^ L inTable 5-1 5.5NumericalExamples ThissectiondemonstratestheperformanceofDAS-CandMACS usingSADAand LAMDA(seeFigure 1-1 ).RecallthatLAMDAhasamuchlargeraperturesizecompared toSADAandhencehashigherresolution.Bothsimulatedande xperimentaldatawillbe usedtoevaluatetheperformanceofMACS.5.5.1Simulations Inthesimulations,thenoisewaveforms,i.e., f e ( b ) g inEq. 2{2 ,aregeneratedas zero-meancircularlysymmetrici.i.d.complexGaussianra ndomprocesses.Thecorrelated signalwaveformsaregeneratedas ~ s ( b )= T 1 = 2 s ( b ), b =1 ;:::;B ,where T isthesource correlationmatrixsuchthat T l;l 0 = j l l 0 j ,where l;l 0 =1 ;:::;L 0 L 0 istheactualnumber ofsources,0 1determinesthecorrelationamongsourcesand f s ( b ) g aregenerated aszero-meancircularlysymmetrici.i.d.complexGaussian randomprocesseswithunit variance.Theresultingcorrelatedsignalwaveforms f ~ s ( b ) g arethennormalizedsothat 81

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1 B P Bb =1 j ~ s l ( b ) j 2 = P l for l =1 ;:::;L 0 toyieldthesignalwaveforms f s ( b ) g ,whichareused togeneratethemeasurementvector f y ( b ) g usingEq. 2{2 .Thesignal-to-noiseratio(SNR), whichisdenedastheratiobetweenthesignalpowerandthen oisepower,issetto0dB and B =500FFTblocksareusedinalltheexamples. WerstconsidertheperformanceofMACSwithSADAwhenthear raycenter ofSADAisat( x;y;z )=(0 ; 0 ; 0)mandthearrayextendsalongthe xy -plane.Four monopolesourcesareplacedat( x;y;z )=( 0 : 20 ; 0 : 20 ; 1 : 48)m(source#1),( 0 : 20 ; 0 : 20 ; 1 : 48) m(source#2),(0 : 20 ; 0 : 20 ; 1 : 48)m(source#3)and(0 : 20 ; 0 : 20 ; 1 : 48)m(source#4)and eachsourceisassumedtohaveasignalpowerof20dB.Thescan ningregioninboth the x -and y -axesrangefrom-0.4mto0.4mandtheresolutioninbothdire ctionsis 0.05m.Notethatwiththisscanninggrid, L =289andCMF-CorDAMAS-Cwould beimpracticaltoimplementasthenumberoftheirreal-valu edunknownsisequalto L 2 =83 ; 521.(Notethattheheightofthesourceswasselectedtobe1. 48mbecausethis wasthecasewithexperimentaldataanalyzedlateron.) Figure 5-1 showsthebeamformingimagesobtainedusingDAS-CandMACSw hen thecorrelationbetweenthesourcesis =0 : 2andthefrequencyis15kHz.Inthegures wecomparetheestimatedsignalauto-correlationlevels,i .e.,thediagonalof ^ P ,withthe truesignalauto-correlationlevels(ortruesignalpowers ),i.e.,thediagonalof P .Letthe scanninggridcorrespondingtothesourceat( x;y;z )=( 0 : 20 ; 0 : 20 ; 1 : 48)mbedenoted as l 0 .Then,wealsocomparethemodulusofthe l th 0 rowsof ^ P and P ,i.e.,wecompare thecross-correlationbetweenallthescanningpointsandt he l th 0 scanningpointwiththe actualcross-correlationvalues.The\x"marksinthegure sindicatethetruesource locations.ThevaluesestimatedwithMACSatthefoursource locationsarealsoshown inthegures.ItisobservedthatDAS-Cfailstoprovideanyc learinformationandsuers severelyfromhighsidelobelevels,whereasMACSisabletod istinguishthesourcesand estimatetheirsignalpowerssuccessfully.Next,weincrea sethecorrelationlevelbetween thesourcesto =0 : 8andshowtheresultingbeamformingimagesinFigure 5-2 .Itis 82

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observedthatMACSisstillabletoresolvethesourcesandpr ovideaccurateestimatesfor boththeauto-andcross-correlationlevels. Inthesecondexampleofthissection,weconsidertheuseofL AMDA.Thearray centerofLAMDAisat( x;y;z )=(0 ; 0 ; 0)mandthearrayextendsalongthe xy -plane. Fourmonopolesourcesareplacedat( x;y;z )=( 0 : 30 ; 0 ; 1 : 48)m(source#1), ( 0 : 10 ; 0 ; 1 : 48)m(source#2),(0 : 10 ; 0 ; 1 : 48)m(source#3)and(0 : 30 ; 0 ; 1 : 48)m(source #4)toresemblealinesource.Linesources,forwhichthecor relationsbetweensources diminishwithincreasedspatialdistance,arefrequentlye ncounteredinaeroacoustic applications.Inthisexample, =0 : 9wasused.Thesignalpowers,SNRandthenumber ofblocksarekeptthesameasinthepreviousexample.Figure 5-3 showsthebeamforming imagesobtainedusingDAS-CandMACSatafrequencyof3kHz(n otethatthefrequency islowercomparedtothepreviousexamplesinceLAMDAhasala rgerapertureandhence higherresolutionthanSADA).Similarobservationsasinth epreviousexamplescanbe madehereaswell.MACSisabletolocalizethesourcesandest imatetheirauto-and cross-correlationlevelsaccuratelyevenwhenthesources arehighlycorrelated. Intheexamplesabove,weusedtheminimumdescriptionlengt h(MDL)information criterionasdescribedin[ 66 ]forestimatingthenumberofsources ^ L .Itwasinfact empiricallyobservedthatMACSisquiteinsensitivetothes electionof ^ L .However, choosingan ^ L muchlargerthanthenumberofsourceswillresultinunneces sarily increasedcomputationtimes.Therefore, ^ L shouldbechosenonlyslightlylargerthana roughestimateofthenumberofsources.Figure 5-4 showstheestimationperformanceof MACSwithdierentvaluesof ^ L fortheexamplewithfoursourcesarrangedinaline(see Figure 5-3 ).Itisobservedthatunderestimating ^ L ismoreharmfulthanoverestimating it(asexpected)andthaterrorsin ^ L donotresultindrasticperformancedegradation. NotethatthecomputationtimesforMACSwere17.2,79.9,126 .2,and245.1secondson a32-bitpersonalcomputerwitha2.53GHzprocessorand3Gby tesofRAMrunning MATLABwhen ^ L =1 ; 3 ; 4and6,respectively. 83

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5.5.2ExperimentalResults ConsidertheexperimentalsetupshowninFigure 4-2 Cwithtwospeakersgenerating coherentbroadbandnoise.Wesawinthepreviouschapterhow DAS,DAMAS,SC-DAMAS, CMFandCLEAN-SCallfailedwiththisexampleat2kHz(seeFig ure 4-17 ).InFigure 5-5 ,weshowthebeamformingresultsobtainedwithDAS,DAMAS,a ndCMFwiththe scanningresolutionusedinthischapterforamorefaircomp arisonat2kHz.InFigure 5-6 ,theauto-andcross-correlationestimatesobtainedbyDAS -CandMACSareshown againat2kHz.ItisobservedthatMACSestimatesthelocatio nsofthetwosources accurately.Sincethetwosourcesarecoherent,thecross-c orrelationbetweenthesource at( x;y;z )=( 0 : 20 ; 0 ; 1 : 48)mandallthescanningpointsshouldbezeroexceptat thesourcelocations.MACSsuccessfullyestimatesthecros s-correlationbetweenthetwo sourcestobeofapproximatelythesamestrengthasshowninF igure 5-6 .Theresulting correlationcoecient,whichisobtainedbydividingthecr oss-correlationbetweenthe twosourcestothesquarerootoftheauto-correlationofeac hsource,is0.96.(Notethat eventhoughthesamewaveformisfedtothespeakers,thetran smittedacousticsignals mightbeslightlydierentduetohardwaredisparitiesandh encethesourcesmightnotbe perfectlycoherent). 5.6Conclusions Inthischapter,wehaverstpresentedtheCMF-Calgorithm, whichistheextension ofCMFtothecorrelatedsourcecase.WearguedthatDAMAS-Ca ndCMF-Cwere impracticalcomputation-wiseevenwhenthenumberofscann ingpointsissmall. Therefore,wehavealsopresentedanewbeamformingapproac h,calledMACS,for theecientmappingofcorrelatedacousticsignalsinaeroa cousticmeasurements. MACSconsistsofbasicallytwosteps:onesteprequiressolv ingaconvexoptimization problem,andtheotherstepisbasedonthematrixsingularva luedecompositionand hasaclosed-formsolution.Thesetwostepsareimplemented inacyclicmanneruntil convergenceisachieved(MACSconvergesinabout5iteratio ns).Duetoitscyclic 84

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property,MACSisguaranteedtoconvergeatleastlocally.I twasshownviaboth simulationsandexperimentaldata,andbyusingtwodieren ttypesofarrays(with dierentaperturesizesandnumbersofmicrophones),thatM ACSisabletoeectively locateandestimatetheauto-andcross-correlationlevels betweenuncorrelated,partially correlatedandcoherentsources.ThecomputationtimeofMA CSwasshowntobe around1to2minuteswith289scanninggridpointsand2to4co rrelatedsources, respectively,usinga63-elementmicrophonearray.Thisis asignicantimprovement intermsofcomputationalcomplexityovertheexistingdeco nvolutionapproachesfor correlatedsources,suchasDAMAS-CandCMF-C,whichareimp racticalforareasonable scanningresolutionwithtoday'scomputingtechnology.Ev aluationofMACSusingfurther experimentaldata,especiallywithrow,appearstobeanint erestingtopicthatisleftfor futureresearch. 85

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20.020.020.0 20.0 x, cmy, cmActual auto-correlation -40 -20 0 20 40 -40 -20 0 20 40 10 12 14 16 18 20 x, cmy, cmDAS-C, auto-correlation -40 -20 0 20 40 -40 -20 0 20 40 10 12 14 16 18 20 19.5 19.3 19.319.2 x, cmy, cmMACS, auto-correlation -40 -20 0 20 40 -40 -20 0 20 40 10 12 14 16 18 20 A 20.013.38.3 1.3 x, cmy, cmActual cross-correlation -40 -20 0 20 40 -40 -20 0 20 40 10 12 14 16 18 20 x, cmy, cmDAS-C, cross-correlation -40 -20 0 20 40 -40 -20 0 20 40 10 12 14 16 18 20 19.313.6 8.27.0 x, cmy, cmMACS, cross-correlation -40 -20 0 20 40 -40 -20 0 20 40 10 12 14 16 18 20 B Figure5-1.Fourcorrelatedsourceswith =0 : 2at15kHzandusingSADA.500FFT blocksareusedandSNR=0dB.A)Comparisonoftheauto-corre lation levels.B)Comparisonofthecross-correlationlevelsbetw eenallthescanning pointsandthesourceat( x;y;z )=( 0 : 2 ; 0 : 2 ; 1 : 48)m(theoneontheupper leftcornerofeachgure).Thetruesignalauto-andcross-c orrelationlevels areshownintheleftmostplots.InbothA)andB),\x"indicat esthetrue sourcelocations.TheMACSestimatedlevelsareshownonthe gure.All levelsareindB. 86

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20.0 20.0 20.0 20.0 x, cmy, cmActual auto-correlation -40 -20 0 20 40 -40 -20 0 20 40 10 12 14 16 18 20 x, cmy, cmDAS-C, auto-correlation -40 -20 0 20 40 -40 -20 0 20 40 10 12 14 16 18 20 19.3 19.0 18.8 18.6 x, cmy, cmMACS, auto-correlation -40 -20 0 20 40 -40 -20 0 20 40 10 12 14 16 18 20 A 20.019.017.8 17.1 x, cmy, cmActual cross-correlation -40 -20 0 20 40 -40 -20 0 20 40 10 12 14 16 18 20 x, cmy, cmDAS-C, cross-correlation -40 -20 0 20 40 -40 -20 0 20 40 10 12 14 16 18 20 18.818.317.0 16.4 x, cmy, cmMACS, cross-correlation -40 -20 0 20 40 -40 -20 0 20 40 10 12 14 16 18 20 B Figure5-2.Fourcorrelatedsourceswith =0 : 8at15kHzandusingSADA.500FFT blocksareusedandSNR=0dB.A)Comparisonoftheauto-corre lation levels.B)Comparisonofthecross-correlationlevelsbetw eenallthescanning pointsandthesourceat( x;y;z )=( 0 : 2 ; 0 : 2 ; 1 : 48)m(theoneontheupper leftcornerofeachgure).Thetruesignalauto-andcross-c orrelationlevels areshownintheleftmostplots.InbothA)andB),\x"indicat esthetrue sourcelocations.TheMACSestimatedlevelsareshownonthe gure.All levelsareindB. 87

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20.020.0 20.020.0 x, cmy, cmActual auto-correlation -40 -20 0 20 40 -40 -20 0 20 40 10 12 14 16 18 20 x, cmy, cmDAS-C, auto-correlation -40 -20 0 20 40 -40 -20 0 20 40 10 12 14 16 18 20 19.9 19.919.8 19.7 x, cmy, cmMACS, auto-correlation -40 -20 0 20 40 -40 -20 0 20 40 10 12 14 16 18 20 A 20.019.519.018.5 x, cmy, cmActual cross-correlation -40 -20 0 20 40 -40 -20 0 20 40 10 12 14 16 18 20 x, cmy, cmDAS-C, cross-correlation -40 -20 0 20 40 -40 -20 0 20 40 10 12 14 16 18 20 19.919.419.018.5 x, cmy, cmMACS, cross-correlation -40 -20 0 20 40 -40 -20 0 20 40 10 12 14 16 18 20 B Figure5-3.Fourcorrelatedsourceswith =0 : 9at3kHzandusingLAMDA.500FFT blocksareusedandSNR=0dB.A)Comparisonoftheauto-corre lation levels.Thetruesignalauto-correlationlevelsareshowni ntheleftmostplot. B)Comparisonofthecross-correlationlevelsbetweenallt hescanningpoints andthesourceat( x;y;z )=( 0 : 3 ; 0 ; 1 : 48)m(theleft-mostsourceineach gure).InbothA)andB),\x"indicatesthetruesourcelocat ions.The MACSestimatedlevelsareshownonthegure.Alllevelsarei ndB. 88

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16.6 16.2 15.8 15.7 x, cmy, cmMACS, auto-correlation -40 -20 0 20 40 -40 -20 0 20 40 10 12 14 16 18 20 19.7 19.5 19.3 19.5 x, cmy, cmMACS, auto-correlation -40 -20 0 20 40 -40 -20 0 20 40 10 12 14 16 18 20 19.8 19.9 19.7 19.6 x, cmy, cmMACS, auto-correlation -40 -20 0 20 40 -40 -20 0 20 40 10 12 14 16 18 20 16.6 16.4 16.2 16.1 x, cmy, cmMACS, cross-correlation -40 -20 0 20 40 -40 -20 0 20 40 10 12 14 16 18 20 19.6 18.9 18.5 18.2 x, cmy, cmMACS, cross-correlation -40 -20 0 20 40 -40 -20 0 20 40 10 12 14 16 18 20 19.9 19.3 19.1 18.6 x, cmy, cmMACS, cross-correlation -40 -20 0 20 40 -40 -20 0 20 40 10 12 14 16 18 20 ABC Figure5-4.TheperformanceofMACSwhen ^ L isvaried.A) ^ L =1,B) ^ L =3,andC) ^ L =6.ComparewithFigure 5-3 x, cmy, cmDAS -40 -20 0 20 40 -40 -20 0 20 40 34 36 38 40 42 44 46 48 50 52 x, cmy, cmDAMAS -40 -20 0 20 40 -40 -20 0 20 40 36 38 40 42 44 46 48 50 52 54 x, cmy, cmCMF -40 -20 0 20 40 -40 -20 0 20 40 36 38 40 42 44 46 48 50 52 54 Figure5-5.Beamformingimages(levelsindB)obtainedusin gDAS,DAMASandCMF fortwocoherentsourceslocated0.20mapartfromeachother .Thetrue sourcelocationsareindicatedbythe\x"marks.Beamformin gfrequencyis2 kHz.Theresultsareobtainedusingexperimentaldata. 89

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x, cmy, cmDAS-C, auto-correlation -40 -20 0 20 40 -40 -20 0 20 40 34 36 38 40 42 44 46 48 50 52 x, cmy, cmMACS, auto-correlation -40 -20 0 20 40 -40 -20 0 20 40 34 36 38 40 42 44 46 48 50 52 x, cmy, cmDAS-C, cross-correlation -40 -20 0 20 40 -40 -20 0 20 40 34 36 38 40 42 44 46 48 50 52 x, cmy, cmMACS, cross-correlation -40 -20 0 20 40 -40 -20 0 20 40 34 36 38 40 42 44 46 48 50 52 Figure5-6.Theauto-correlationateachscanningpointand thecross-correlationofeach scanningpointwiththesourceat( x;y;z )=( 0 : 2 ; 0 ; 1 : 48)mestimatedwith DAS-CandMACSareshown(alllevelsareindB).Thesetupcons istsoftwo coherentsourceslocated0.20mapartfromeachother.Thetr uesource locationsareindicatedbythe\x"marks.Beamformingfrequ encyis2kHz. Theresultsareobtainedusingexperimentaldata. Table5-1.PseudocodeofMACS. CalculatetheEVDof ^ G : ^ G = UU H =tr( ), = q L ^ L Q = I ~ G = U ^ L ( ^ L ) 1 = 2 repeat(5times) Solve ^ C =argmin C k ~ G ^ Q H AC k 2F ; s.t. k c k 1 CalculatetheSVDof ~ G H A ^ C : ~ G H A ^ C = ^ U ~ ~ U H ^ Q = ~ U ^ U H endrepeat ^ P = ^ C ^ C H 90

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CHAPTER6 UNCERTAINTYANALYSIS Uncertaintyanalysisanswersthequestionofhowgoodthere sultsofanexperiment areand,withoutsuchananalysis,itisdiculttostatethec ondenceintheobtained estimates[ 67 ].Astandardmethodtocalculatetheoutputuncertaintiesi stopropagate theuncertaintiesoftheinputvariablesthroughthedatare ductionequation(theequation usedtoestimatethequantitiesofinterestfromthemeasure ments).Thedatareduction equationisafunctionofmultipleinputvariables,mostofw hichareobtainedfrom separatemeasurements.Notethattheuncertaintiesofthei nputvariablesarenot necessarilyuncorrelated.Forinstance,theDASdatareduc tionequation(seeEq. 2{9 ) containsbothreal-andcomplex-valuedinputvariablesand ,ingeneral,therealand imaginarycomponentsofthecomplex-valuedcomponentsare correlated.Aswillbeshown below,thisleadstoincreasedcomplexityintheuncertaint yanalysis. BothcomplexmultivariateandMonte-Carlouncertaintyana lyseswillbeconsidered inthischapter.Themultivariateanalysisisbasedonarst -orderTaylorseriesexpansion ofthequantitiesofinterestandassumesthattheperturbat ionsarerelativelysmall,and hence,thenonlineartermsintheTaylorseriesexpansionar enegligible.Italsoassumes thattheoutputdistributionsareGaussianinordertocompu tethecondenceintervals. Themultivariateuncertaintyanalysisdiersfromtheclas sicaluncertaintytechniques inthatitestimatesthecorrelationoftheoutputvariables [ 68 ].TheMonte-Carlo method,ontheotherhand,usesassumeddistributionsforth einputvariables,which maybecorrelated.Randomperturbationsfortheinputvaria blesaredrawnfromthese distributionsandthedatareductionequationisevaluated usingtheperturbedinput variables.Thisprocessisrepeateduntilthedistribution softheoutputvariableshave converged,afterwhichtheuncertaintyestimatecanberead ilyobtainedfromthese distributions.Theadvantageofthemultivariateanalysis isthatitisanalyticalandcan estimatetheuncertaintiesrelativelyquickly.However,m oreoftenthannot,closed-form 91

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expressionsforthederivativesinvolvedintheTaylorseri esexpansionarenotavailableor verycumbersome,andtheinputperturbationsarenotsmalle noughtoassumelinearity. Inaddition,theGaussian-typeassumptionsmadewhenestim atingthecondenceintervals fromthesamplecovariancematricesmightbeviolatedinpra ctice.Monte-Carloanalysis providesmuchmorerexibilityintermsofdesigningtheexpe rimentssincethedata reductionequationisalreadyimplementedfortheexperime ntalanalysisandembedding theperturbationsoftheinputvariablestothisequationis ingeneralstraightforward. Castellinietal.[ 69 ]studytheuncertaintyoftheDASbeamformerfora2Dlinear arrayandfar-eldnoisepropagationwherethesourcelocat ionsareparameterizedby anglesrangingfrom0to ratherthanbeingparameterizedby3Dlocations.Moreover, theanalysisprovidedisnottargeteddirectlyforaeroacou sticapplicationsanddoesnot considertheuncertaintiesintheCSM,calibrationorinteg ratedDASsoundpressure levels.Inthischapter,wespecicallyanalyzetheuncerta intyoftheDASbeamformeras implementedinaeroacousticmeasurements[ 1 2 4 17 ]. 6.1UncertaintyAnalysisTechniques Asmentionedabove,weconsidertwouncertaintyanalysiste chniques: i) multivariate uncertaintyanalysis,whichisbasedonarst-orderTaylor seriesexpansion,and ii) Monte-Carlouncertaintyanalysis,whichisbasedonassumi ngdistributionsforeachinput variable.WeapplybothmultivariateandMonte-Carlouncer taintyanalysestotheDAS beamformer,whereasweapplyMonte-Carlouncertaintyanal ysistocalibration(dueto thecomplexityofthenonlineareigen-decompositioninvol vedintheprocedure).The followinganalysesconsideronlyasinglebeamformingloca tion(inparticular,the l th one) andshouldberepeatedforeverypointinthescanninggrid,i .e., L times. 6.1.1MultivariateUncertaintyAnalysis Theclassicaluncertaintyanalysistechniqueestimatesth euncertaintyoftheoutput variablesbymakinguseofarst-orderTaylorseriesexpans ion.Theuncertaintiesofthe inputvariablesshouldbesucientlysmallsothatthelinea rapproximationremainsvalid. 92

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Theresultingsamplestandarddeviationorthestandardunc ertaintyofavariable, P l in ourcase(seeEq. 2{9 ),isthencomputedby g P l = vuut T 0 X t =1 @P l @V t 2 V t ;V t +2 T 0 1 X t =1 T 0 X u = t +1 @P l @V t @P l @V u V t ;V u ; (6{1) where T 0 isthenumberofinputvariables, V t ;V t isthestandarduncertaintysquaredof the t th inputvariable, @P @V t iscalledthesensitivitycoecientofthe t th inputvariable,and V t ;V u isthesamplecovariancebetweenthe t th and u th inputvariables.Notethat t and u runfrom1to T 0 and t +1to T 0 ,respectively. IntheDASdatareductionequation,thepowerestimate P l isreal-valuedandthe inputvariablesarecomplex-valued.However,Eq. 6{1 isderivedforrealvariablesand thereforethecomplexinputvariablesshouldbeseparatedi ntotheirrealandimaginary componentsbeforebeingpropagatedthroughthedatareduct ionequation[ 70 { 72 ]. Oneimportantreasonfortreatingtherealandimaginarypar tsoftheinputvariables separatelyisbecausethesecomponentscanbecorrelatedin manyapplications.(Note thatthesecondterminthesquarerootinEq. 6{1 accountsforthecorrelationbetween suchcomponents.)FortheDASbeamformer,sinceonlyonerea l-valuedpowerlevelis consideredatatime,themultivariateandclassicaluncert aintyanalysismethodsare similar[ 72 ]. Eq. 6{1 canalsobewritteninmatrixformasfollows[ 70 { 72 ] g P l = p JJ T ; (6{2) where isthe2 T 0 2 T 0 real-valuedandsymmetricsamplecovariancematrixofther eal andimaginarypartsofalltheinputvariables,i.e.,thevar iables f Re f V 1 g ;:::; Re f V T 0 g ; Im f V 1 g ;:::; Im f V T 0 gg (6{3) 93

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and J isthe1 2 T 0 real-valuedJacobianmatrix(avectorinourcase)denedas J = @P l @ Re f V 1 g ;:::; @P l @ Re f V T 0 g ; @P l @ Im f V 1 g ;:::; @P l @ Im f V T 0 g : (6{4) Considerasimpleexamplewith2realvariables, V 1 and V 2 ,where V 1 and V 2 are assumedtobeuncorrelatedandlettheoutputvariable P l beafunctionof V 1 and V 2 ThenEq. 6{1 orEq. 6{2 becomes g P l = vuuuut @P l @V 1 @P l @V 2 264 V 1 ;V 1 0 0 V 2 ;V 2 375 264 @P l @V 1 @P l @V 2 375 = s @P l @V 1 2 V 1 ;V 1 + @P l @V 2 2 V 2 ;V 2 ; (6{5) wherethesquareduncertaintiesofthevariables V 1 and V 2 arescaledbythesensitivity coecients @P l @V 1 and @P l @V 2 squared,respectively,andsummeduptogeneratethenal uncertaintysquared. Ingeneral,95%condenceintervalsareusedwhenreporting theuncertaintyresults. Inordertoobtainthecondenceintervals, g P l shouldbemultipliedbyacoveragefactor whichissimplytakenas2inourcaseassumingaGaussiandist ributionfortheunivariate outputvariable[ 72 ].(Notethat( P l P l ) =S P l ,where P l and S P l arethesamplemeanand samplestandarddeviationof P l ,respectively,followsthetdistributionwithnumberof Monte-Carlotrialsminusonedegreesoffreedom.Itisrecom mendedthatacoveragefactor of2isusedwhenthedegreesoffreedomislargerthan31[ 67 ].) 6.1.2Monte-CarloUncertaintyAnalysis Whentheperturbationsarerelativelylarge(sothatthelin earassumptionofthe multivariateanalysisisviolated)and/ortheoutputdistr ibutionsarenon-Gaussian,the multivariatemethodcannolongeryieldreliableuncertain tyestimates.Inaddition, thesensitivitycoecientsareoftendiculttoevaluatein closed-form.Therefore,a Monte-Carlouncertaintyanalysisispreferable.InMonteCarlouncertaintyanalysis,a distributionisassumedforalloftheinputvariablesandth eneachvariableisrandomly perturbedusingaperturbationvaluedrawnfromitsuncerta intydistribution(notethat 94

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theinputvariablesarenotnecessarilyuncorrelated)[ 67 72 ].Next,theperturbedinput variablesarepropagatedthroughthedatareductionequati oninordertoobtainthe perturbedoutput.Thisprocessisrepeateduntilthedistri butionoftheoutputvariables converge(itwasobservedthat1000iterationsweresucien tforconvergenceinour examples)[ 68 ].Theresultingdistributionisthenusedtoobtainthemean ,variance (covariance)and95%condenceintervalsforthequantitie sofinterest. 6.2ApplicationofUncertaintyAnalysistotheDelay-and-S umBeamformer Inthissectionwedescribehowtheaforementioneduncertai ntyanalysistechniques canbeappliedtotheDASdatareductionequationgiveninEq. 2{9 .Let G = 266666664 G 11 C 12 + jQ 12 :::C 1 M + jQ 1 M C 12 jQ 12 G 22 :::C 2 M + jQ 2 M ... ... ... C 1 M jQ 1 M C 2 M jQ 2 M :::G MM 377777775 ; (6{6) where C mn =Re f G mn g and Q mn =Im f G mn g m 6 = n m;n =1 ;:::;M .Similarly,let ~ D = 266666664 D 1 + jE 1 0 ::: 0 0 D 2 + jE 2 ::: 0 ... ... ... 00 :::D M + jE M 377777775 ; (6{7) where ~ D m = D m + jE m D m =Re n ~ D m o and E m =Im n ~ D m o m =1 ;:::;M TheinputvariablescontainedinEq. 2{9 canbeexpressedas V =[ V CSM ; V Calib ; V Locs ;V Temp ] ; (6{8) where V CSM =[ G 11 ;:::;G MM ;C 12 ;:::;C 1 M ;C 23 ;:::;C M 1 ;M ;Q 12 ;:::;Q 1 M ;Q 23 ;:::;Q M 1 ;M ] ; (6{9) 95

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V Calib =[ D 1 ;:::;D M ;E 1 ;:::;E M ] ; (6{10) V Locs =[ x 1 ;:::;x M ;y 1 ;:::;y M ;z 1 ;:::;z M ] ; (6{11) and,nally, V Temp =[ T ] : (6{12) Table 6-1 liststhefourcategoriesofinputvariablesasconsidereda bove. TheJacobianmatrixfor P l isdenedas(seeEq. 6{4 ) J = 26664 @P l @G 11 ;:::; @P l @Q M 1 ;M | {z } J CSM ; @P l @D 1 ;:::; @P l @E M | {z } J Calib ; @P l @x 1 ;:::; @P l @z M | {z } J Locs ; @P l @T |{z} J Temp 37775 (6{13) andthereforethe95%condenceintervalfor P l isgivenby(seeEq. 6{2 ) 2 g P l =2 Jg (CSM ; Calib ; Locs ; Temp) J T 1 = 2 =2( J CSM g CSM J TCSM + J Calib g Calib J TCalib + J Locs g Locs J TLocs + J Temp g Temp J T Temp ) 1 = 2 ; (6{14) where g (CSM ; Calib ; Locs ; Temp) isthesamplecovariancematrixof V and g CSM isthesample covariancematrixof V CSM ( g Calib g Locs and g Temp aredenedinasimilarmanner).(It isassumedthattheCSM,calibration,microphonelocationa ndtemperatureerrorsare independentofeachother.) InordertoevaluateEq. 6{14 ,weneedtocomputethesamplecovariancematricesof theinputvariables, g CSM g Calib g Locs ,and g Temp ,aswellastheJacobianmatrices, J CSM J Calib J Locs and J Temp .WeneedthesamplecovariancematricesalsofortheMonte-C arlo analysis,sincethesewillbethecovariancematricesofthe Gaussiandistributionsfrom whichtherandomperturbationsaredrawn.TheJacobianmatr icesforeachcategoryof inputvariables,i.e.,thetermsinEq. 6{13 ,arederivedinAppendix A .Whencomputing g CSM ,weconsidertherandomerrorsassociatedwithusingtheni teaveragingmethod inEq. 2{8 .Theexpressionforeachcomponentof g CSM isgiveninTable9.1ofBendat 96

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andPiersol[ 54 ]fortwomicrophones( M =2).Weextendthisanalysistothecaseof M microphones,where M canbeanynumbergreaterthan1,andlistourndingsinTable 6-2 .Appendix B providesthedetailsonhowthecovariancesinTable 6-2 arecomputed. NotethatwhenoverlappingblocksareusedtocomputetheCSM s,thenumberofblocks, B ,shouldbereplacedbytheeectivenumberofblocks, 1 B ,inTable 6-2 ,where 1 isusedtoaccountforthecorrelationbetweenoverlappingb locks.Forinstance,fora Hanningwindowwith50%or75%overlap, 1 =0 : 947or 1 =0 : 520,respectively[ 54 ]. Thecovariancematricesduetocalibrationandlocationerr orsaretakenasdiagonal matriceswiththecorrespondinguncertaintiesalongthedi agonals. Whenperturbing G intheMonte-Carlouncertaintyanalysis,aGaussianrandom vector,say ~ V CSM ,withcovariancematrix g CSM asgiveninTable 6-2 andazeromean vectorisgeneratedeverytrial(AppendixCbrierydiscusse sonewayofdoingthis)to obtaintheperturbationsofeachvariablecontainedin V CSM .Theperturbationvaluesare thenusedtoformaperturbationmatrix G p (byproperlyindexingthevariables)andthe perturbedCSMiscomputedas G + G p .Whenperturbingtheinputvariablescontained in V Calib V Locs and V Temp ,i.i.d.Gaussianrandomvariableswithzeromeanandgiven uncertaintyvaluesaregenerated,andtheseperturbations areaddedtothenominalvalues. 6.3NumericalandExperimentalResults Thissectionpresentstheuncertaintyanalysisofthecalib rationprocedureandthe DASbeamformerusingnumericalaswellasexperimentaldata .Boththeindividualand thecumulativeeectsoftheinputparametersareanalyzedt ounderstandthedominant sourcesofuncertainty.LAMDAisusedfortheanalysis.6.3.1CalibrationUncertainty AsmentionedinChapter 2 ,whenalltheassumptionsaremet,thearraycalibration procedurewillprovideaccuratecorrectionfactorsforaso urcenearthecalibrationspeaker location.However,inpractice,manysourcesofuncertaint yarepresentduringcalibration suchastheuncertaintiesintheCSMandreferencemicrophon elevels.Thissection 97

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willanalyzethesensitivityofthecalibrationproceduret osucherrors.Thecalibration performancewillalsodegradewhenthecalibrationsourcei snotaperfectmonopole and/ortherearererectionsinthecalibrationsetup.Howev er,wedonotconsidersuch modelingerrorsandinsteadfocusourattentionontheuncer taintyofthecalibration procedurewhentheunderlyingdatamodeliscorrect.(Simil arly,whenevaluatingthe uncertaintyoftheDASbeamformer,thedatamodelisassumed tobecorrect.) Theuncertaintyanalysisofthecalibrationprocedureisco nductedusingMonte-Carlo simulations(aTaylorseriesbasedanalysisisomittedduet othecomplexityofthe eigen-decompositionandduetotheincreasedrexibilitypr ovidedbytheMonte-Carlo method).Theinputvariablesthatareperturbedincludethe CSM,thesoundpressure levelofthereferencemicrophone,andtheindividualmicro phonesensitivities(real-valued andinmV/Pa).Theindividualmicrophonesensitivitiesand phasesareusuallyobtained frommanufacturerspecicationsorfromindividualcalibr ationswithrespecttosome highqualitymicrophone.Weassumethatafrequencyindepen dentsensitivityvalueis usedforallthemicrophoneswithanominalvalueof30mV/Pa( whichisthesensitivity usedwiththePanasonicWM-61AmicrophonesattheUFAFF),an dweassumethat thenominalphaseofeachmicrophoneis0 .TheSNRissetto25dBinthecalibration setup,whereSNRisdenedas10log 10 ( P Bb =1 k a cal s cal ( b ) k 22 ) 10log 10 ( P Bb =1 k e ( b ) k 22 ).In ordertomodeltheuncertaintiesinthecalibrationprocedu re,wesimulatemicrophone pressuremeasurementsfromanidealmonopolesourcelocate dat(0,0,1.48)m.Thearray centerislocatedat(0,0,0)m(nominally).Thesamplingfre quency f s =65 ; 536Hz andtheblocklength H =4096(seeSection 2.1 ).All B valuesshownrepresenteective numberofblocks.Thefrequencyissetat5kHz,and1000Monte -Carlotrialshavebeen implemented.Duetotheobservationthattheoutputdistrib utionsareGaussian,2times thesamplestandarddeviationisconsideredastheuncertai ntyintheplotsofthissection. First,weconsidertheeectsofperturbingtheindividualm icrophonesensitivities whilekeepingtheotherinputvariablesattheirnominalval ues.AsobservedfromFigure 98

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6-1 ,thereisaone-to-onerelationshipbetweentherelativeun certaintiesinthemicrophone sensitivitiesandthemagnitudeofthecalibrationfactors .Inthegurespresentedinthis section,theaverage(overallmicrophones)uncertaintyin themagnitudeandphaseofthe microphonecorrectionfactorsareplotted.Notethattheav eragesaretakenafternding theuncertaintyofeachindividualmicrophone.Oneimporta ntnotehereisthatahigher uncertaintyisnotnecessarilydetrimentalsincethegoalo fcalibrationistocorrectfor sucherrors.Therefore,wenditmoreappropriatetoconsid ertheeectsofmicrophone sensitivityandphaseerrors,temperatureerrorsandmicro phonelocationerrorsinthe followingsectionswhereweanalyzetheoverallDASuncerta inty. Theeectofthenumberofblocks, B ,ontheuncertaintyofthecalibrationprocedure canbeobservedfromFigure 6-2 ,wherealltheothervariablesarekeptattheirnominal values.Thereisapproximatelyaone-to-tworatiointheunc ertaintiesasexpectedsince theerrorintheCSMdropswith p B .Althoughtheuncertaintiesinthemagnitude appearstobelarge(10%)foraconventional B suchas1000,thenaleectontheDAS estimateiswithinreasonablelimitsaswillbeshownbelow. Forinstance,withanominal sensitivityof30mV/Pa,apositive10%perturbationinallt hemicrophonesensitivities willyield j 20log 10 (30 = 33) j =0 : 83dBdierenceinthesourcelevels.Weobservethatthe phaseuncertaintyisrelativelylowevenforsmall B Next,weexaminetheuncertaintiesinthereferencemicroph onesoundpressure levelsonlyinFigure 6-3 .Sucherrorsmightrisefromtheimperfectcalibrationsoft he referencemicrophone.Asexpected,thereisaone-to-onere lationbetweenthecalibration uncertaintyandthereferencemicrophoneleveluncertaint y.Thephaseisnotaectedsince thesecondstageofcalibrationisformagnitudecorrection only. Finally,inFigure 6-4 ,wheretheuncertaintiesinthemicrophonesensitivitiesa nd referencemicrophonelevelaresetto10%and B =1000,theuncertaintyofcalibration withvaryingfrequencyisplotted.Itisobservedthattheun certaintyisindependentof frequency.Thisisbecausethemicrophonesensitivityandr eferencemicrophonelevel 99

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uncertaintiesareassumedtobefrequency-independentino uranalysis.(Notethat inpractice,theuncertaintiesmightvarysomewhatwithfre quency,inwhichcasethe calibrationuncertaintywillalsovarywithfrequency.)In thenextsection,wewillseethat whenthemicrophonelocationsorthetemperaturearepertur bed,thefrequencywillbe importantsincetheseperturbationswillbemultipliedbyt hewavenumber. Basedontheaboveobservations,theaccuratecalibrationo fthereferencemicrophone isveryimportantsincethiswilldeterminethearraypowere stimatesdirectly.Moreover,it appearsthatduringcalibration,itisbenecialtoacquire dataforaslongaspossible.As mentionedabove,theuncertaintiesincalibrationduetomi crophonesensitivityandphase errors,temperatureerrorsandmicrophonelocationerrors arebetteranalyzedwithinthe contextofbeamforming.6.3.2Delay-and-SumBeamformerUncertainty Unlessotherwisestated,inallthesimulationsinthissect ion,amonopolesourcewith 50dBsignalpower(powerisdenedatthenominalarraycente r)issimulatedat(0,0, 1.48)mwherethearraycenterisnominallylocatedat(0,0,0 )masinthecalibration case.(NotethatthecalibrationspeakeratUFAFFproducesa pproximately50dBsignal poweratthearraycenter.)1000Monte-Carlotrialshavebee nimplementedandthe frequencyissetat5kHz.TheSNRis25dB.Thescanningregion issetfrom-0.50mto 0.50mwith0.02mresolutioninboththe x and y directions.Theroomtemperatureis T 0 =293Kandthenominalsoundspeedis343m/s. 6.3.2.1ComparisonofMultivariateandMonte-CarloAnalys es InthissectionweshowthatthemultivariateandMonte-Carl ouncertaintyanalysis oftheDASbeamformeryieldconsistentresultswhenthepert urbationsarerelatively lowandthatthetwomethodsdierwhentheperturbationsbec omelarger.Weconsider theperturbationsinthecalibrationfactorsandthemicrop honelocations,andsimilar conclusionscanbemadewhentheCSMandmicrophonecorrecti onfactorsareperturbed. 100

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Inordertoanalyzetheeectsofthecalibrationuncertaint yontheDASestimates, weperturb ~ D (seeEq. 2{9 )usingthevaluesobtainedinSection 6.3.1 asaguideline. Theuncertaintiesintherealandimaginarycomponentsofth ecalibrationfactorscanbe foundbyeitherusingasimpleMonte-Carloanalysisorusing multivariateuncertainty propagationgiventheuncertaintiesinmagnitudeandphase [ 72 ].Theresultingsample covariancematrixfromthisprocedureisthenusedtogenera te(possiblycorrelated) perturbationvaluesfor D m and E m ateachMonte-Carloiteration,where m =1 ;:::;M Figure 6-5 showsthedierenceindBbetweenthetruesourcepower, P 0 ,and P 0 +2 l where l isthesamplestandarddeviationestimatedviaeachofthetw omethodsatthe l th scanningpoint.Theresultsofthetwomethodsmatchwelland itwasobservedthatthis isthecaseevenforlargeperturbationvaluesinmicrophone calibrationfactors(resultsnot shown). Next,weconsidertwoperturbationsettingsforthemicroph onelocations.The microphonelocationsareperturbedwithi.i.d.Gaussianra ndomvariablesofstandard deviations1mmand10mminFigures 6-6 AandB,andFigures 6-6 CandD,respectively. Itisobservedthatthetwouncertaintyanalysismethodsgiv edierentresultswhenthe perturbationsarelarger.Notethat1mmand10mmperturbati onstranslateintorelative uncertaintiesof0.2%and2.3%,respectively,intermsofmi crophonetosourcedistances. Astheuncertaintiesintheinputvariablesincrease,ther st-orderlinearapproximation withtheTaylorseriesdoesnotsucetomodeltheoverallunc ertaintyduetothe nonlinearities.Toincreasetheaccuracyofthismethod,mo retermsneedtobeconsidered intheTaylorseriesexpansion[ 68 ].However,thealgebracanquicklybecomecumbersome forEq. 2{9 .EveniftheTaylorseriesexpansioninvolvedasmanytermsa sneeded,when theresultingdistributionsarenotGaussian,thestandard deviationestimatesofthe multivariatemethodcannotbeusedtoobtain95%condencei ntervals.Theselimitations andsomeotherreasoningsprovidedbelowmakeMonte-Carloa nalysisabettercandidate foranalyzingtheDASbeamformeruncertainty. 101

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6.3.2.2UncertaintyAnalysiswithSimulations ThissectionconsiderstheuncertaintyoftheDASbeamforme rusingMonte-Carlo simulations.Weconsidertheuncertaintiesinmicrophones ensitivityandphase,microphone location,arraybroadbanddistance,temperatureandCSM.I nthemultivariateuncertainty method,thecalibrationeectshadtobeanalyzedthrough ~ D .However,intheMonte-Carlo method, ~ D willbeestimatedfromcalibration,whichisdoneateachMon te-Carloiteration usingtheperturbedinputs,anddirectlysubstitutedinthe datareductionequation.In theMonte-Carlomethod,weestimatethedistributionofthe DASpowerestimatesat eachscanningpointandthenobtainthe95%condenceinterv alsandmeanvalues.Then, thesevaluesareconvertedintodB.Thereasonforshowing95 %condenceintervals insteadofsamplestandarddeviationsisthattheresulting distributionsareingeneral asymmetricalabouttheirmeanvalues.Forinstance,forasc anningpointwheretheDAS estimateisrelativelylow,sincethepowerestimateiscons trainedtobepositive, 2times thestandarddeviationcannotbeusedtoobtainthecondenc eintervalswhenthemean islessthantwicethestandarddeviation.The95%condence intervalsarethereforebest estimatedfromthedistributionsandstandarddeviationsm ightbeinsucientinmodeling theuncertainties. First,weinvestigatethemicrophonelocationuncertainty indetailwhilekeeping theotherinputvariablesattheirnominalvalues.Figure 6-7 andFigure 6-8 showthe 95%condenceintervalsateachscanningpointwhenthe x y and z componentsofthe microphonelocationsareperturbedusingi.i.d.Gaussianr andomvariableswithzero meansandstandarddeviationsof Locs =10mmand Locs =1mm,respectively. Theseperturbationvaluescanbenormalizedbythewaveleng th(at5kHz)toobtain dimensionlessvaluesof0.146(for10mm)and0.015(for1mm) .Notethatcalibration isnotappliedintheseplots.The3Dplots(see,e.g.,Figure 6-7 A)showthemean,and theupperandlowerlimitsofthe95%condenceintervalsate achscanninglocation. Thetruesourcelocationandpowerareindicatedwiththedas hedlineandthedotat 102

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itstip,respectively.The2Dplots(see,e.g.,Figure 6-7 B),ontheotherhand,showtwo slicesfromthe3Dplotstakenat x =0mand x =0 : 06m.Notethatinthe2Dplots, thecondenceintervalsintheregionfrom y =0 : 2mto0.5mareomittedsincethey resemblecloselythecondenceintervalsintheregionfrom y = 0 : 5mto0.2m.Instead, azoomedinviewofthemainbeam,whichisofrelativelymorei nterest,isprovided(the nominalcurveisomittedinthezoomedinplots).Oneimporta ntobservationthatcan bemadefromFigure 6-7 isthatthepowerestimatesarebiaseddownwardswithrespec t tothenominalvalue.AppendixDprovidesanexplanationfor thisrathernon-intuitive phenomenon.Tofurtherelaborateonthebiasissue,weshowt hepeaklocationofthe DASbeamformingimageateachMonte-Carlotrialwhen Locs =10mmtogetherwith thehistogramofthepeaklocationinFigures 6-9 Aand 6-9 B.NotethattheDASpeak locationexhibitsadiscretepatternduetothenitescanni ngresolutionwhichissetto 5mminFigure 6-9 A.Themeanofthepeaklocationsoverallthetrialsisindica ted withtheemptycircle.ItisobservedthattheDASpeakoccurs eitheratthetruesource locationorinitsvicinityandthatthemeanlocationofthep eakscoincideswiththetrue sourcelocation.However,evenwhenthepeakappearsatthet ruesourcelocation,the estimatedpowervalueislessthanthenominalvalue(asdisc ussedinAppendixD).This canalsobeobservedfromFigure 6-9 Cwhereslicesfromthebeamformingmapatx=0 mfrom4dierentsamplesareshowntogetherwiththenominal value.Weobservethat therearelargeructuationsatalmosteveryscanningpointd uetothelocationerrors, consistentwiththeplotsinFigure 6-7 Asobservedabove,microphonelocationerrorscancausesig nicantproblemsifnot accountedfor.Sincecalibrationisspecicallydesignedf orsucherrors,weexpectitto improvetheresults.InFigure 6-10 ,weagainshowthe95%condenceintervalsateach scanningpointbutnowwithcalibrationapplied.Itisobser vedthatcalibrationgreatly reducesthevariationsoftheDASpowerestimatesduetoloca tionerrors.InFigure 6-9 D) isshownthatthepowerestimatesfromtrial-to-trialnowli neupnicelyasopposedto 103

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Figure 6-9 C.Whenndingthecalibrationfactors,weassumedthatallt heinputvariables exceptthemicrophonelocationsareattheirnominalvalues Toanalyzetheperformanceofthecalibrationinthepresenc eofsourcesatdierent locationsthanthecalibrationspeaker,weconsiderascena riowheretwomonopolesources ofequalstrength(50dB)areplacedat(0 ; 0 ; 1 : 48)mand(0 ; 0 : 20 ; 1 : 48)m.InFigures 6-11 and 6-12 ,the95%condenceintervalsareshownwhencalibrationisn otappliedand whencalibrationisapplied,respectively.Similarobserv ationstothesinglesourcecase canbemade.Itseemsthatalthoughthesecondsourceisnotat thesamelocationasthe calibrationspeaker,calibrationstillhelpstoreducethe uncertainties. Theapplicationofcalibrationthusseemsessentialwhenwe anticipateerrorsinour locationmeasurements.Eventhoughthelocationerrorsoft hemicrophonesonthearray plane,i.e.,inthe x and y directions,canbemeasuredveryaccurately,thenon-unifo rmity ofthearraysurfacecanresultinunknownlocationerrors.I ntheexamplesconsidered above,weassumedthatthetemperaturewasthesameinthecal ibrationandtestdata. However,inthepresenceofrow,thenon-uniformityoftempe ratureinthetestsectionwill causesoundspeeddierencesinthecalibrationandtestcas es.Furthermore,inpractice, thereisacertainuncertaintyassociatedwiththearraybro adbanddistance,especially withcomplextestmodels. Notethatintheexamplesthatfollow,weonlyshowthe2Dslic esfromthebeamforming imagessincetheyappeartobemoreinformativethanthe3Don eswhenthecondence intervalsarerelativelysmall. Torepresenttheuncertaintyinthemodeltoarraydistance, weperturbthearray broadbanddistancetogetherwiththearraymicrophoneloca tions,andapplycalibration asbefore.Figures 6-13 Aand 6-13 Bshowthe95%condenceintervalswhentherelative uncertaintiesinthearraybroadbanddistancearesetto2.5 %and5%,respectively.Note thatinourcase,thesecorrespondtonetuncertaintiesof0. 04mand0.07m.Keepingin mindthatthenominalbroadbanddistanceofthesourceis1.4 8m,sucherrorsmightbe 104

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realizableinpractice,especiallywhentestingmodelswit hcomplexgeometries.Wealso analyzethedeterministicerrorintheestimatedpowerleve lswhenthearraybroadband distanceisvaried.Figure 6-14 showsthepowerestimatedat( x;y )=(0 ; 0)mwhenthe arraybroadbanddistanceisvariedfrom1.1mto1.9mwithinc rementsof0.5mm.Itis observedthattheestimatedsourcelevelsexhibitaconcave behaviorwithapeakatthe truesourceheight. InFigure 6-15 ,weanalyzethe95%condenceintervalswhentheblocksize B used incomputingtheCSMisvariedandtheothervariablesarekep tattheirnominalvalues. TheCSMofthecalibrationisalsoperturbedassumingthatth eblocksizeofcalibration was1000.Weobservethat B =1000casereducestheuncertaintybyabout0.5dB comparedtothe B =200case. Figure 6-16 andFigure 6-17 showthe95%condenceintervalswhentheindividual microphonesensitivitiesandphasesareperturbed.Itisas sumedthatthemicrophone sensitivitiesandphasevaluesremainthesameduringcalib rationandtesting.Weobserve thattheuncertaintiesaresomewhatlargefor15%relativeu ncertaintyinthemicrophone sensitivitiesandthatwhenthecalibrationinputvariable sareattheirnominalvalues,the phaseerrorsarecorrectedaccurately. Figure 6-18 showsthe95%condenceintervalswhenthetemperatureispe rturbed. Inpractice,theerrorsintemperaturewillbenegligibledu ringcalibrationduetothe absenceofrow.However,duringmodeltesting,thetemperat ureuncertaintycouldbe signicant.Hereweconsider0.1 C(Figure 6-18 A)and3 C(Figure 6-18 B)uncertainty intemperatureduringtestingand0.1 Cuncertaintyduringcalibration.Itappearsthat thebeamformingprocedureisquiteinsensitivetotemperat ureuncertaintiesprovidedthat calibrationisapplied.Notethata3 Cuncertaintyintemperaturewillcausearelative perturbationof0.5%insoundspeed.Weemphasizethatthete mperatureuncertainty hasonlybeenconsideredthroughitseectonthesoundspeed .Inpractice,microphone 105

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transferfunctionsaswellasmicrophonelocations(duetot heexpansion/contractionofthe arrayplate)couldbeaectedbytemperature,resultinginl argeruncertainties. Finally,inFigure 6-19 weconsidertheoveralluncertaintywhenthemicrophone locationuncertaintiesare10mm,relativearraybroadband distanceuncertaintyis5%, temperatureuncertaintyis3 Cfortestingand0.1 Cforcalibration,CSMuncertaintyis calculatedusing1000blocksforbothcalibrationandtesti ng,andmicrophonesensitivity andphaseuncertaintiesare15%and10 ,respectively.Itisobservedthatthe95% condenceintervalatthesourcelocationisaround[-0.84, 0.45]dBofthemeanvalue. 6.3.2.3UncertaintyAnalysiswithExperimentalData TheMonte-Carlomethodisnowdemonstratedonexperimental datatakenatthe UFAFFusingLAMDAwiththepurposeofinvestigatingtheunce rtaintyintheintegrated DASlevels(seeSection 4.4 forthedescriptionofintegratedlevels).IntheMonte-Car lo trials,thepsfiscalculatedateachiterationwiththepert urbedvalueswhencalculating thenormalizationfactorforDAS. Thersttestsetupconsistsofasinglespeakerplacedat(0 ; 0 ; 1 : 48)msimilartothe scenarioconsideredearlierwithsimulations.Thedataana lysisparametersareasfollows: aHanningwindowwith75%overlaphasbeenappliedtoblockso fsize4096samples,the samplingfrequencyis65,536Hzandthedataacquisitiontim eis15secondsresultingin 498eectiveblocksandafrequencyresolutionof16Hz.Thes canningregionextends from-0.50mto0.50mwitharesolutionof0.02minboththe x and y directions.The beamformingmapat2kHzisshowninFigure 6-20 A.1000Monte-Carlotrialshave beenrunandtheresulting95%condenceintervalsoftheint egratedDASlevelsversus frequencyhavebeenplottedinFigure 6-20 Bforafrequencyrangeof1kHzto10kHz. Inthisgure,theuncertaintiesfortheCSMarecalculatedf oraneectiveblocksize of498fortestingand1000forcalibration,theuncertainti esforindividualmicrophone sensitivitiesandphasesaresetto15%and15degrees,respe ctively,thetemperature uncertaintyissetto1%fortestingand0.1%forcalibration ,themicrophonelocation 106

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uncertaintiesinallthe x y and z directionsaresetto10mm,andthearraybroadband distanceuncertaintyissetto2.5%.(10mmstandarddeviati oninmicrophonelocations correspondstodimensionlessperturbationsof0.029and0. 292at1kHzand10kHz, respectively,whennormalizedbythewavelength).Theunce rtaintiesaredenedwith respecttotheassumednominalvaluesandwithexperimental data,the\nominal"values mightnotbeidenticaltotheunknowntruevalues.Itisobser vedthattheestimatedlevels arewithin 0.5dBofthemeanvalue. Asanalcase,weanalyzetheuncertaintyintheintegratedD ASlevelsofthe NACA63-215ModBairfoil[ 47 59 ].(ThisisthesamemodelconsideredinSection 4.10 .) Thebeamformingimageoftheairfoilat2.5kHzisshowninFig ure 6-21 A,wheretwo locationswithdominantnoisecanbeidentied.Notethatin thebeamformingmap,the scanningregionextendsfrom-0.5mto0.5minthe x directionandfrom-0.6mto0.6m inthe y directionwithacommonresolutionof0.02m,andthemodelis atabroadband distanceof1.30mwithrespecttothearrayplane.TheMachnu mberis0.17.Dueto thepresenceofrowduringtheairfoiltesting,DRisapplied ,i.e.,thediagonalof G is removedintheDASdatareductionequation(seeSection 3.2 ).Moreover,SLChasalso beenemployed[ 1 55 ].ThedataacquisitionparametersarethesameasinSection 4.10 andrepeatedhereforcompleteness.Thedataacquisitionti mewas5seconds,sampling frequencywas65,536Hzandtheblocklengthwas2048samples (frequencyresolutionof 32Hz).AHanningwindowwith75%overlaphasbeenemployedle adingto331eective averages[ 47 ].Theresultinguncertaintiesintheintegratedlevelsare showninFigure 6-21 Bwheretheinputuncertaintiesaresettothesamevaluesuse dintheprevious example.(10mmstandarddeviationinmicrophonelocations correspondstodimensionless perturbationsof0.022and0.073at0.75kHzand2.5kHz,resp ectively,whennormalized bythewavelength).Weobservethattheestimatedlevelsare within 1dBofthemean valuesoverafrequencyrangeof0.75kHzto2.5kHz.(Notetha ttheuncertaintiesdueto shearlayercorrectionshavenotbeenconsidered.) 107

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6.4Conclusions Thischapterhaspresentedtheuncertaintyanalysisofthea rraycalibrationtechnique andtheDASbeamformer.ItwasshownthattheDASuncertainty obtainedfromthe multivariateandMonte-Carlomethodsaresimilarwhenthep erturbationsarerelatively small.However,whenthecomponentuncertaintiesarerelat ivelylarge,thetwomethods dierduetothebreakdownoftherst-orderassumptionofth emultivariatetechnique.It wasalsoshownthattheMonte-Carlomethodissimplertoimpl ementandprovidesmore rexibilityintermsofanalyzingtheDASdatareductionequa tionalongwithcalibration. However,theMonte-Carlomethodrequiresapproximately4t imesmorecomputation thantheanalyticmultivariatemethodfor1000iterationsw iththescanningresolutions implementedinthenumericalexamplesabove. Thecalibrationprocedurewasshowntobeessentialwhenerr orsareexpectedin microphonefrequencyresponses,microphonelocationsand /ortemperaturemeasurements. Withcalibration,theDASbeamformerwasshowntobeaected mostlybytheuncertainty inthearraybroadbanddistancefollowedbytheuncertainti esintheCSMandindividual microphonesensitivities.Inaddition,theuncertaintyin theintegratedDASlevelsof experimentaldatawasalsoconsidered.Inparticular,the9 5%condenceintervalswere foundtobearound 0.5dBforasinglemonopolesourcealsousedforcalibration whereaswiththeNACA63-215ModBairfoilmodel,the95%con denceintervals oftheintegratedlevelswerefoundtobelargerthan 1dB.Itshouldbenotedthat iftheconditionsofcalibrationandtestingaresignicant lydierent(forinstance,if themicrophonetransferfunctionschangefromcalibration totesting),thecalibration procedurewillbelesseective.Therefore,theuncertaint iesandthecondenceintervals providedinthischaptercouldbeconsideredaslowerbounds ontheerrorsthatwillbe encounteredinpractice. 108

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10 0 10 1 10 -1 10 0 10 1 10 2 Relative uncertainty of mic. sensitivity (%)Average relative uncertainty of the magnitude (%) 10 0 10 1 10 -4 10 -3 10 -2 10 -1 Relative uncertainty of mic. sensitivity (%)Average uncertainty of the phase (deg) A B Figure6-1.Theaverage(overallmicrophones)uncertainty oftheA)magnitudeandB) phasetermsofthemicrophonecorrectionfactorswhenthein dividual microphonesensitivitiesareperturbed.Frequencyis5kHz 10 2 10 3 10 4 10 0 10 1 10 2 No of blocksAverage relative uncertainty of the magnitude (%) 10 2 10 3 10 4 10 -1 10 0 10 1 No of blocksAverage uncertainty of the phase (deg) A B Figure6-2.Theaverage(overallmicrophones)uncertainty oftheA)magnitudeandB) phasetermsofthemicrophonecorrectionfactorswhentheCS Misperturbed. Frequencyis5kHz. 10 0 10 1 10 -1 10 0 10 1 10 2 Relative uncertainty of the ref. mic. level (%)Average relative uncertainty of the magnitude (%) 10 0 10 1 10 -15 10 -14 10 -13 Relative uncertainty of the ref. mic. level (%)Average uncertainty of the phase (deg) A B Figure6-3.Theaverage(overallmicrophones)uncertainty oftheA)magnitudeandB) phasetermsofthemicrophonecorrectionfactorswhenthere ference microphonelevelisperturbed.Frequencyis5kHz. 109

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10 3 10 4 10 0 10 1 10 2 Frequency (Hz)Average relative uncertainty of the magnitude (%) 10 3 10 4 10 -1 10 0 10 1 Frequency (Hz)Average uncertainty of the phase (deg) A B Figure6-4.Theaverage(overallmicrophones)uncertainty oftheA)magnitudeandB) phasetermsofthemicrophonecorrectionfactorsforvaryin gfrequencywhere therelativeuncertaintyinmicrophonesensitivitiesandr eferencemicrophone levelare10%andthenumberofblocksis1000. Taylor-series x, cmy, cm -40 -20 0 20 40 -40 -20 0 20 40 0 0.5 1 1.5 2 2.5 Monte-Carlo x, cmy, cm -40 -20 0 20 40 -40 -20 0 20 40 0 0.5 1 1.5 2 2.5 A B Figure6-5.ComparisonofA)multivariateandB)Monte-Carl omethodswhenmicrophone correctionfactorsareperturbed.ThedierenceindBbetwe enthetruesource power, P 0 ,and P 0 +2 l ,where l isthesamplestandarddeviationestimated viaeachofthetwomethodsatthe l th scanningpoint.Therelativeuncertainty ofthemicrophonecorrectionfactormagnitudeis5%andtheu ncertaintyof themicrophonecorrectionfactorphaseis1 .Frequencyis5kHz. 110

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Taylor-series x, cmy, cm -40 -20 0 20 40 -40 -20 0 20 40 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Monte-Carlo x, cmy, cm -40 -20 0 20 40 -40 -20 0 20 40 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 A B Taylor-series x, cmy, cm -40 -20 0 20 40 -40 -20 0 20 40 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Monte-Carlo x, cmy, cm -40 -20 0 20 40 -40 -20 0 20 40 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 C D Figure6-6.ComparisonofA)andC)multivariateandB)andD) Monte-Carlomethods whenmicrophonelocationsareperturbed.Thedierenceind Bbetweenthe truesourcepower, P 0 ,and P 0 +2 l ,where l isthesamplestandarddeviation estimatedviaeachofthetwomethodsatthe l th scanningpoint.The microphonelocationsareperturbedwithi.i.d.Gaussianra ndomvariablesof standarddeviations1mminA)andB),and10mminC)andD).Fre quency is5kHz. 111

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-50 -40 -30 -20 -10 0 10 20 10 20 30 40 50 Slice from x = 0 cmPower (dB)y, cm -10 -5 0 5 10 36 38 40 42 44 46 48 y, cm -50 -40 -30 -20 -10 0 10 20 10 20 30 40 50 Slice from x = 6 cmPower (dB)y, cm -10 -5 0 5 10 34 36 38 40 42 44 46 y, cm A B Figure6-7.Microphonelocationsareperturbedwithastand arddeviationof10mm.A) 3Dplotshowingthemeanandthe95%condenceintervals.The truesource locationandpowerareindicatedwiththedashedlineandthe dotatitstip, respectively.B)TwoslicesfromtheplotinA)tofurtherill ustratethe95% condenceintervals.Theblacksolidlineandthebluedashe dlineindicatethe meanvaluesandthenominalvalues,respectively.Azoomedi nviewofthe mainbeamregionisalsoprovided.Frequencyis5kHz. -50 -40 -30 -20 -10 0 10 20 10 20 30 40 50 Slice from x = 0 cmPower (dB)y, cm -10 -5 0 5 10 40 42 44 46 48 50 52 y, cm -50 -40 -30 -20 -10 0 10 20 10 20 30 40 50 Slice from x = 6 cmPower (dB)y, cm -10 -5 0 5 10 36 38 40 42 44 46 48 y, cm A B Figure6-8.Microphonelocationsareperturbedwithastand arddeviationof1mm.A)3D plotshowingthemeanandthe95%condenceintervals.Thetr uesource locationandpowerareindicatedwiththedashedlineandthe dotatitstip, respectively.B)TwoslicesfromtheplotinA)tofurtherill ustratethe95% condenceintervals.Theblacksolidlineandthebluedashe dlineindicatethe meanvaluesandthenominalvalues,respectively.Azoomedi nviewofthe mainbeamregionisalsoprovided.Thenominalandthemeanva luesare indistinguishableatmost y values.Frequencyis5kHz. 112

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-40 -30 -20 -10 0 10 20 30 40 -40 -30 -20 -10 0 10 20 30 40 x, mmy, mm A B -50 -40 -30 -20 -10 0 10 20 30 40 50 20 25 30 35 40 45 50 y, cmDAS power estimate (dB) Nominal Trial 1 Trial 2 Trial 3 Trial 4 -50 -40 -30 -20 -10 0 10 20 30 40 50 20 25 30 35 40 45 50 y, cmDAS power estimate (dB) Nominal Trial 1 Trial 2 Trial 3 Trial 4 C D Figure6-9.Microphonelocationsareperturbedwithastand arddeviationof10mm.A) ThelocationoftheDASpeakestimateateachMonte-Carlotri alismarked withadotandthemeanlocationofthepeaksismarkedwiththe empty circle.B)Thehistogramofthelocationsofthepeaks.C)&D) TheDASpower estimatesat4arbitrarytrialstogetherwiththenominales timate.No calibrationisappliedinA)-C)andcalibrationisappliedi nD).Slicesfromthe beamformingimagesat x =0mareshowninC)andD).Notethattheunits inA)andB)areinmm.Frequencyis5kHz. 113

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-50 -40 -30 -20 -10 0 10 20 10 20 30 40 50 Slice from x = 0 cmPower (dB)y, cm -10 -5 0 5 10 40 42 44 46 48 50 52 y, cm -50 -40 -30 -20 -10 0 10 20 10 20 30 40 50 Slice from x = 6 cmPower (dB)y, cm -10 -5 0 5 10 36 38 40 42 44 46 48 y, cm A B Figure6-10.Microphonelocationsareperturbedwithastan darddeviationof10mmand calibrationisapplied.A)3Dplotshowingthemeanandthe95 %condence intervals.Thetruesourcelocationandpowerareindicated withthedashed lineandthedotatitstip,respectively.B)Twoslicesfromt heplotinA)to furtherillustratethe95%condenceintervals.Theblacks olidlineandthe bluedashedlineindicatethemeanvaluesandthenominalval ues, respectively.Azoomedinviewofthemainbeamregionisalso provided.The nominalandthemeanvaluesareindistinguishableatmost y values. Frequencyis5kHz. 114

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-50 -40 -30 -20 -10 0 10 20 30 35 40 45 50 Slice from x = 0 cmPower (dB)y, cm -30 -20 -10 0 10 36 38 40 42 44 46 48 y, cm -50 -40 -30 -20 -10 0 10 20 30 35 40 45 50 Slice from x = 6 cmPower (dB)y, cm -30 -20 -10 0 10 34 36 38 40 42 44 46 y, cm A B Figure6-11.Twosourcesareplacedat(0,0,1.48)mand(0,-0 .20,1.48)mwithequal strengthsof50dB.Microphonelocationsareperturbedwith astandard deviationof10mmandcalibrationisnotapplied.A)3Dplots howingthe meanandthe95%condenceintervals.Thetruesourcelocati onandpower areindicatedwiththedashedlineandthedotatitstip,resp ectively.B)Two slicesfromtheplotinA)tofurtherillustratethe95%cond enceintervals. Theblacksolidlineandthebluedashedlineindicatethemea nvaluesand thenominalvalues,respectively.Azoomedinviewofthemai nbeamregion isalsoprovided.Frequencyis5kHz. 115

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-50 -40 -30 -20 -10 0 10 20 30 35 40 45 50 Slice from x = 0 cmPower (dB)y, cm -30 -20 -10 0 10 40 42 44 46 48 50 52 y, cm -50 -40 -30 -20 -10 0 10 20 30 35 40 45 50 Slice from x = 6 cmPower (dB)y, cm -30 -20 -10 0 10 36 38 40 42 44 46 y, cm A B Figure6-12.Twosourcesareplacedat(0,0,1.48)mand(0,-0 .20,1.48)mwithequal strengthsof50dB.Microphonelocationsareperturbedwith astandard deviationof10mmandcalibrationisapplied.A)3Dplotshow ingthemean andthe95%condenceintervals.Thetruesourcelocationan dpowerare indicatedwiththedashedlineandthedotatitstip,respect ively.B)Two slicesfromtheplotinA)tofurtherillustratethe95%cond enceintervals. Theblacksolidlineandthebluedashedlineindicatethemea nvaluesand thenominalvalues,respectively.Azoomedinviewofthemai nbeamregion isalsoprovided.Thenominalandthemeanvaluesareindisti nguishableat most y values.Frequencyis5kHz. 116

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-50 -40 -30 -20 -10 0 10 20 10 20 30 40 50 Slice from x = 0 cmPower (dB)y, cm -10 -5 0 5 10 40 42 44 46 48 50 52 y, cm -50 -40 -30 -20 -10 0 10 20 10 20 30 40 50 Slice from x = 6 cmPower (dB)y, cm -10 -5 0 5 10 36 38 40 42 44 46 y, cm -50 -40 -30 -20 -10 0 10 20 10 20 30 40 50 Slice from x = 0 cmPower (dB)y, cm -10 -5 0 5 10 40 42 44 46 48 50 52 y, cm -50 -40 -30 -20 -10 0 10 20 10 20 30 40 50 Slice from x = 6 cmPower (dB)y, cm -10 -5 0 5 10 36 38 40 42 44 46 y, cm A B Figure6-13.The95%condenceintervalsoftheDASpowerest imateswhenthearray broadbanddistanceisperturbed.Twoslicesfromthebeamfo rmingimageat x =0cmand x =6cmareconsidered.Therelativeuncertaintyinarray broadbanddistanceisA)2.5%,andB)5%.Theblacksolidline andtheblue dashedlineindicatethemeanvaluesandthenominalvalues, respectively.A zoomedinviewofthemainbeamregionisalsoprovided.Theno minaland themeanvaluesareindistinguishableatsome y values.Frequencyis5kHz. 1.1 1.2 1.3 1.4 1.48 1.6 1.7 1.8 1.9 45 45.5 46 46.5 47 47.5 48 48.5 49 49.5 50 Power (dB)z, m Figure6-14.Thepowerestimatedat( x;y )=(0 ; 0)mwhenthearraybroadbanddistance (inparticular, z )isvariedfrom1.1mto1.9mwithincrementsof0.5mm. Thetruesourcedistancetoarrayis1.48mandthetruesource poweris50 dB.Frequencyis5kHz. 117

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-50 -40 -30 -20 -10 0 10 20 10 20 30 40 50 Slice from x = 0 cmPower (dB)y, cm -10 -5 0 5 10 40 42 44 46 48 50 52 y, cm -50 -40 -30 -20 -10 0 10 20 10 20 30 40 50 Slice from x = 6 cmPower (dB)y, cm -10 -5 0 5 10 36 38 40 42 44 46 y, cm -50 -40 -30 -20 -10 0 10 20 10 20 30 40 50 Slice from x = 0 cmPower (dB)y, cm -10 -5 0 5 10 40 42 44 46 48 50 52 y, cm -50 -40 -30 -20 -10 0 10 20 10 20 30 40 50 Slice from x = 6 cmPower (dB)y, cm -10 -5 0 5 10 36 38 40 42 44 46 y, cm A B Figure6-15.The95%condenceintervalsoftheDASpowerest imateswhentheCSMis perturbed.Twoslicesfromthebeamformingimageat x =0cmand x =6 cmareconsidered.NumberofblocksareA) B =200,andB) B =1000.The blacksolidlineandthebluedashedlineindicatethemeanva luesandthe nominalvalues,respectively.Azoomedinviewofthemainbe amregionis alsoprovided.Thenominalandthemeanvaluesareindisting uishableat most y values.Frequencyis5kHz. -50 -40 -30 -20 -10 0 10 20 10 20 30 40 50 Slice from x = 0 cmPower (dB)y, cm -10 -5 0 5 10 40 42 44 46 48 50 52 y, cm -50 -40 -30 -20 -10 0 10 20 10 20 30 40 50 Slice from x = 6 cmPower (dB)y, cm -10 -5 0 5 10 36 38 40 42 44 46 y, cm -50 -40 -30 -20 -10 0 10 20 10 20 30 40 50 Slice from x = 0 cmPower (dB)y, cm -10 -5 0 5 10 40 42 44 46 48 50 52 y, cm -50 -40 -30 -20 -10 0 10 20 10 20 30 40 50 Slice from x = 6 cmPower (dB)y, cm -10 -5 0 5 10 36 38 40 42 44 46 y, cm A B Figure6-16.The95%condenceintervalsoftheDASpowerest imateswhenthe individualmicrophonesensitivitiesareperturbed.Twosl icesfromthe beamformingimageat x =0cmand x =6cmareconsidered.Therelative inputuncertaintiesareA)5%,andB)15%.Theblacksolidlin eandtheblue dashedlineindicatethemeanvaluesandthenominalvalues, respectively.A zoomedinviewofthemainbeamregionisalsoprovided.Theno minaland themeanvaluesareindistinguishableatmost y values.Frequencyis5kHz. 118

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-50 -40 -30 -20 -10 0 10 20 10 20 30 40 50 Slice from x = 0 cmPower (dB)y, cm -10 -5 0 5 10 40 42 44 46 48 50 52 y, cm -50 -40 -30 -20 -10 0 10 20 10 20 30 40 50 Slice from x = 6 cmPower (dB)y, cm -10 -5 0 5 10 36 38 40 42 44 46 y, cm -50 -40 -30 -20 -10 0 10 20 10 20 30 40 50 Slice from x = 0 cmPower (dB)y, cm -10 -5 0 5 10 40 42 44 46 48 50 52 y, cm -50 -40 -30 -20 -10 0 10 20 10 20 30 40 50 Slice from x = 6 cmPower (dB)y, cm -10 -5 0 5 10 36 38 40 42 44 46 y, cm A B Figure6-17.The95%condenceintervalsoftheDASpowerest imateswhenthe individualmicrophonephasesareperturbed.Twoslicesfro mthe beamformingimageat x =0cmand x =6cmareconsidered.Therelative inputuncertaintiesareA)1 ,andB)10 .Theblacksolidlineandtheblue dashedlineindicatethemeanvaluesandthenominalvalues, respectively.A zoomedinviewofthemainbeamregionisalsoprovided.Theno minaland themeanvaluesareindistinguishableatmost y values.Frequencyis5kHz. -50 -40 -30 -20 -10 0 10 20 10 20 30 40 50 Slice from x = 0 cmPower (dB)y, cm -10 -5 0 5 10 40 42 44 46 48 50 52 y, cm -50 -40 -30 -20 -10 0 10 20 10 20 30 40 50 Slice from x = 6 cmPower (dB)y, cm -10 -5 0 5 10 36 38 40 42 44 46 y, cm -50 -40 -30 -20 -10 0 10 20 10 20 30 40 50 Slice from x = 0 cmPower (dB)y, cm -10 -5 0 5 10 40 42 44 46 48 50 52 y, cm -50 -40 -30 -20 -10 0 10 20 10 20 30 40 50 Slice from x = 6 cmPower (dB)y, cm -10 -5 0 5 10 36 38 40 42 44 46 y, cm A B Figure6-18.The95%condenceintervalsoftheDASpowerest imateswhenthe temperatureisperturbed.Twoslicesfromthebeamformingi mageat x =0 cmand x =6cmareconsidered.TherelativeinputuncertaintiesareA )0.1 C,andB)3 C.Theblacksolidlineandthebluedashedlineindicatethe meanvaluesandthenominalvalues,respectively.Azoomedi nviewofthe mainbeamregionisalsoprovided.Thenominalandthemeanva luesare indistinguishableatmost y values.Frequencyis5kHz. 119

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-50 -40 -30 -20 -10 0 10 20 10 20 30 40 50 Slice from x = 0 cmPower (dB)y, cm -10 -5 0 5 10 40 42 44 46 48 50 52 y, cm -50 -40 -30 -20 -10 0 10 20 10 20 30 40 50 Slice from x = 6 cmPower (dB)y, cm -10 -5 0 5 10 36 38 40 42 44 46 y, cm Figure6-19.The95%condenceintervalsoftheDASpowerest imateswhenalltheinput variablesareperturbed.Seethetextforspecicperturbat ionvalues.Two slicesfromthebeamformingimageat x =0cmand x =6cmareconsidered. Theblacksolidlineandthebluedashedlineindicatethemea nvaluesand thenominalvalues,respectively.Azoomedinviewofthemai nbeamregion isalsoprovided.Thenominalandthemeanvaluesareindisti nguishableat most y values.Frequencyis5kHz. x, cmy, cm Max. = 47.8 dBInt. = 47.8 dB -40 -20 0 20 40 -40 -20 0 20 40 38 40 42 44 46 1 2 3 4 5 6 7 8 9 10 40 45 50 55 Frequency (kHz)Integrated SPL (dB) Nominal value Monte-Carlo A B Figure6-20.Analysisofexperimentaldatawithasinglesou rce.A)Thebeamforming imageat2kHz.Theintegrationregionisindicatedwiththed ashedsquare andthetruesourcelocationisindicatedwiththe\x".Thema ximum(Max.) andintegrated(Int.)levelsareindicatedontheplot.B)Th e95%condence intervalsoftheintegratedDASlevelsversusfrequency.Se ethetextforthe inputuncertainties.Thenominalandthemeanvaluesareind istinguishable atmostofthefrequencies. 120

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x, cmy, cm T.E. Max. = 48.5 dBInt. = 54.3 dB -40 -20 0 20 40 -60 -40 -20 0 20 40 60 40 42 44 46 48 0.75 1 2 3 4 30 35 40 45 50 55 60 65 Frequency (kHz)Integrated SPL (dB) Nominal value Monte-Carlo 1.5 2 2.5 55 56 57 58 59 60 Frequency (kHz) A B Figure6-21.AnalysisoftheNACA63-215Mod-Bairfoil.A)Th ebeamformingimageat 2.5kHz.Theintegrationregionisindicatedwiththedashed rectangleand thetrailingedge(T.E.)isshownwiththesolidline.Themax imum(Max.) andintegrated(Int.)levelsareindicatedontheplot.B)Th e95%condence intervalsoftheintegratedDASlevelsversusfrequency.Se ethetextforthe inputuncertainties.Azoomedinviewofthecondenceinter valsinthe frequencyrangefrom1.5kHzto2.5kHzisalsoprovided.Then ominaland themeanvaluesareindistinguishableatmostofthefrequen cies. Table6-1.ErrorsourcesfortheDASbeamformer. NameErrorsourceNo.ofvariables V CSM (variablesin G )Randomaveragingerror M 2 V Calib (variablesin C )Calibrationerrors2 M V Locs (microphonelocations)Distancemeasurementerrors3 M V Temp (temperature)Temperaturemeasurementerrors1 Table6-2.CovariancesoftheCSMvariables(theelementsin g CSM ). VariablesCovariance G mm G mm j G mm j 2 =B G mm G nn j G mn j 2 =B G mm C np ( C mn C mp + Q mn Q mp ) =B G mm Q np ( C mn Q mp Q mn C mp ) =B C mn C pq ( C mp C nq + Q mp Q nq + C mq C np + Q mq Q np ) = (2 B ) C mn Q pq ( C mp Q nq + Q mq C np Q mp C nq C mq Q np ) = (2 B ) Q mn Q pq ( C mp C nq + Q mp Q mq C mq C np Q mq Q np ) = (2 B ) 121

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CHAPTER7 DIRECTIVESOURCESANDSPATIALLYNON-WHITENOISE Mostexistingarrayprocessingalgorithmsforaeroacousti cnoisemeasurement applicationsassumethepresenceofmonopolesourcesinwhi chcasethesteeringvector foragivenscanningpointisafunctionofonlythedistanceb etweeneachsensorandthe scanningpoint(seeEq. 2{3 ).However,whenthearrayapertureislargeandthefrequenc y isrelativelyhigh,directivitymightnolongerbenegligib leandthesteeringvectorsnow becomeafunctionofboththeimpinginganglesfromthescann ingpointstoeachsensor andthedistancesinbetween.Beamformingwithdirectiveso urcesischallengingsince thereisnopriorknowledgeonthedirectivitypatternsofth esources;thismakesthe actualsteeringvectorsunknowntothebeamformingalgorit hm.Moreover,withdirective sources,eachsensormightencountersubstantiallydiere ntsignallevelsasopposedtothe non-directivecase.AlthoughDAScanstillbeusedtogetaro ughestimateofthesource distribution,itsperformancemightdegradesignicantly Beamformingwithothertypesofsourcesbesidesmonopolesh asbeenanalyzedby Suzuki[ 73 ].Thismethodusestheeigen-modesoftheCSMtogetherwitha sparsity constrainttoresolvethesourceparameters.Yet,thismeth odassumesthatthesteering vectorsofthenon-monopolesourcesareknown.CLEAN-SCiso neofthefewalgorithms consideringthecaseofunknownsteeringvectors[ 57 ].AsmentionedinChapter 1 ,a disadvantageofCLEAN-SCisthatitrequirestheselectiono ffouruserparameters. DoughertysuggeststhattheEVDoftheCSMcanbeusedtoestim atetheunknown steeringvectorsandthesourcepowers[ 2 ].Itiswell-knownthatsubspacebased beamformingalgorithmsaresensitivetomeasurementnoise [ 8 ].Inthischapter,wepresent aniterativealgorithmusingconvexprogrammingforestima tingthenoisecovariance matrix.Thisprogramisabletoworkwithbothdiagonal(i.e. ,thenoiseisuncorrelated amongsensorsbutmaybeofdierentpowerlevels)andnon-di agonal(i.e.,thenoisecan becorrelatedorcoherentamongsensors)noisecovariancem atrices.Theestimatedsignal 122

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covariancematrix,determinedbyremovingtheestimatedno isecovariancematrixfrom thearraycovariancematrix,canthenbeusedtoestimatethe sourceparametersusingthe aforementionedeigen-decompositionbasedtechnique. 7.1ProblemFormulation ConsiderthedatamodelintroducedinEq. 2{4 .Underthemonopolesources assumption,thearraysteeringvectorcorrespondingtothe l th sourcewasdenedas (seeEq. 2{2 ): a (mono)l =[ e jkr l; 1 =r l; 1 ;:::;e jkr l;M =r l;M ] T ;l =1 ;:::;L 0 : (7{1) Recallthat L 0 isthenumberofsources.Inthepresenceofdirectivesource s,thesteering vectorforthe l th sourcebecomes: a l = a (mono)l d l ,where d l =[ D l ( l; 1 ) ;:::;D l ( l;M )] T consistsofthedirectivityterms( d l =[1 ; 1 ;:::; 1] T foramonopolesource).Forinstance, whenthe l th sourceisapistoninaninnitebae,thefar-eldexpressio nfor f D l ( l;m ) g is: D l ( l;m )= 2 J 1 ( kr sin( l;m )) kr sin( l;m ) ; (7{2) where r istheradiusoftheplanepiston, J 1 ( )denotestheBesselfunctionoftherstkind and m;l denotestheemissionangleofthe l th sourceatthe m th sensor(seeFigure 7-1 ) [ 48 ].(Thedirectivity D l ( l;m )isassumedtobeindependentoftheazimuthangle,i.e., m;l isaxisymmetricforagiven m and l .)Weassumethatthemodulusofthemaximum directivityforeachsourceisnormalizedto1inordertoavo idambiguitywhenestimating thesourcepowerlevels. DirectiveSourceLocalization RecallfromEq. 3{2 that G = APA H inthenoiselesscase,where P isadiagonal matrixwiththeunknownsourcepowersonitsdiagonalassumi ngthatthesourcesare uncorrelated.Consequently,weobtain[ 2 ] Ga l =( P l k a l k 22 ) a l + L 0 X l 0 =1 ;l 0 6 = l P l 0 a l 0 ( a Hl 0 a l ) ;l =1 ;:::;L 0 : (7{3) 123

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Thisequationshowsthattheactualsteeringvectorscanbeo btainedfromtheEVDof G .SincethesecondtermontherighthandsideofEq. 7{3 isrelativelysmallforwidely separatedsourcesandcanbeneglected, a l = k a l k 2 isapproximatelyaneigenvectorof G and P l k a l k 22 isthecorrespondingeigenvalue[ 2 ].Consequently,let G = UU H betheEVD of G ,wherethecolumnsoftheunitarymatrix U denotetheeigenvectorsof G andthe diagonalelementsofthediagonalmatrix denotethecorrespondingeigenvalues f m g Mm =1 suchthat 1 ::: M .Let u l denotethe l th columnof U .Themonopolesteering vectorsareusedtoestimatethelocationsofthedominantdi rectivesourcesintheeld. Thelocationofthe l th sourcecorrespondstoargmax l 0 =1 ;:::;L j ( a (mono)l 0 ) H u l j k a (mono)l 0 k 22 .Recallthat L isthenumberofscanningpointsintheimageplane. Wecanestimatethe l th directivityvectoras ^ d l = u l := a (mono)l l =1 ;:::;L 0 ,where \ := "denotestheelement-wisedivision.Notethat ^ d l isanestimateofascaledversionof thetrue d l dueto u l beinganorthonormaleigenvectorof G .Inordertoestimatethe powerlevelsaccurately, ^ d l shouldbenormalizedappropriately.Oneoptionistonormal ize ^ d l sothatitslargestelementhasunitmodulus.Analternative istotapolynomial(a 4 th -orderpolynomialissatisfactory)tothemodulusof ^ d l andnormalize ^ d l sothatthe maximumvalueofthispolynomialovertheentireanglerange equals1.Itwasempirically observedthatthesetwomethodsyieldalmostidenticalresu lts(sinceatleastonesensor islikelytosamplethedirectivitypatternnearitsmaximum value),andhencetheformer easierapproachisusedinourexamples.After ^ d l isnormalized(andstilldenotedas ^ d l ), thepowerofthe l th sourceisestimatedas ^ P l = l k ( a (mono)l ^ d l k 22 l =1 ;:::;L 0 .Wereferto thisalgorithmasthedirectivesourcelocalizer(DSL).Not ethatDSLissimilartothe proceduredescribedbyDougherty[ 2 ]butitismoregeneralsinceitcanestimatethe powerlevelsevenwhen k a l k 22 6 =1, l =1 ;:::;L 0 TheperformanceofDSLcanbedegradedwhenthearraycovaria ncematrixis contaminatedwithmeasurementnoiseunlessthenoisecovar iancematrixisascaled identitymatrix(inwhichcasetheeigenvectorsarenotaec ted).Inpractice,thenoise 124

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covariancematrixmightbediagonalwithunequaldiagonale lementsornon-diagonal sincethenoiseinsensorsthatareclosetoeachothercanbeh ighlycorrelatedoreven coherent.Thiswillaectthesubspaceof G ,andtheeigenvectorsof G maynolonger correspondtothetruesourcesteeringvectors.Notethatwh enthemeasurementnoiseis nolongerdiagonal,Eq. 3{2 becomes G = G L 0 + G E ,where G L 0 = APA H denotesthe signalcovariancematrixand G E denotesthemeasurementnoisecovariancematrix(note earlier,wehaveassumedthat G E = 2 I ).Toaccountfornon-whitemeasurementnoise,a covariancematrixttingmethodologysimilartoCMF(seeSe ction 3.4 )canbeused,where thenoisecovariancematrixandthesourcecovariancematri xareestimatedfrom G using aniterativeprocedure. Assumethat G L 0 ,whichisarankL 0 matrix,isgiven. G E canthenbeestimatedby solvingthefollowingconvexoptimizationproblemusingSe DuMi[ 50 ]: ^ G E =argmin G E k G G L 0 G E k 2F + k g E k 1 ; (7{4) where g E =vec( G E )and isauserparameter.InEq. 7{4 ,thesparsityofthe non-diagonalnoisecovariancematrixisenforcedsinceiti sassumedthatthenoiseinonly closelyspacedsensorswillbehighlycorrelatedwitheacho ther.Notethattheminimizer ofEq. 7{4 mustsatisfy ^ G E 0and G ^ G E 0since ^ G E and G ^ G E arecovariance matrices.Imposingtheconstraints G E 0and G G E 0totheprogramdenedin Eq. 7{4 resultsintheso-calledSDP[ 52 ],whichisconvex.Althoughtheseconstraints willpreservetheconvexityoftheoptimizationprobleminE q. 7{4 ,theywillincreasethe computationalcomplexitysignicantly,especiallyforla rge M .Itisempiricallyobserved thatthesolutiontoEq. 7{4 satisestheseconstraintsautomatically.(Alternativel y, thenegativeeigenvaluesof ^ G E and G ^ G E canbesettozerotoenforcethepositive semi-denitenessofthecovariancematrices.) Given G E ,thesignalcovariancematrix G L 0 canbeestimatedasthebestrankL 0 approximationto ~ G = G G E intheFrobeniusnormsense.Thesolutionis ^ G L 0 = 125

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~ U ~ ~ U H ,where ~ containsonlythelargest L 0 eigenvaluesof ~ G onitsdiagonaland ~ G = ~ U ~ U H istheEVDof ~ G Asmentionedabove,estimating G E requires G L 0 andviceversa.Therefore,the problemcanbesolvediterativelybyrstxing G E andestimating G L 0 andthenxing G L 0 andestimating G E G E canbeinitializedasallzeros.Afteracertainnumberof iterations(25inourexamples),theso-obtained ^ G L 0 isusedinsteadof G inDSL.At eachiteration, canbechosensimplyas k G ^ G L 0 ^ G E k 2F = k ^ g E k 1 ,where ^ G L 0 and ^ G E arethelatestestimatesof G L 0 and G E ,respectively.Theinitialvalueof k g E k 1 canbe computedas C P Ml = L 0 +1 l ,where C isaconstant.Since G E canbenon-diagonal, C is usedtocompensatefortheo-diagonaltermsof G E .( C =20inourexamples.)This procedureisreferredtoasDSLwithnoiseextraction(DSL-N E).DSL-NEisnotvery sensitivetotheselectionof C .Moreover,itisempiricallyobservedthatafteracertain numberofiterations, convergestoaxedvalueandhencethealgorithmbecomescyc lic, i.e.,thecostfunctioninEq. 7{4 isguaranteedtonotincreaseateachiteration(notethat aniterativealgorithmisnotnecessarilycyclic,whereasa cyclicalgorithmisiterative). ThecyclicpropertyofDSL-NEensuresthatitwillconvergea tleastlocally.Finally,note thatsince G isnotavailableinpractice,itisreplacedbytheCSM,i.e., ^ G ,asusual. 7.2NumericalExamples ThissectionevaluatestheperformanceofDAS,CLEAN-SC,DS LandDSL-NEfor twouncorrelateddirectivesourcesandwithLAMDA.Figure 7-2 Ashowsthesimulated directivity(computedbyEq. 7{2 with r =0 : 03mand kr =9 : 9)ofasource,which isplaced0.2mo-centerinboththe x -and y -axesandataheightof1m,versusits emissionangle.ThecirclesinFigure 7-2 Adenotethedirectivityvaluesencounteredby thearraysensorsforthissource.Figures 7-2 Band 7-2 Cshowthereductionrelativeto thesourceaxis(indB)incurredateacharraysensorbyamono polesourceandadirective source,respectively,whichareplacedat0.2mo-centerin boththe x -and y -axesand 1mabovethearrayplane.Itisobservedthatdirectivityres ultsinsevereamplitude 126

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dierencesamongdierentsensors(evenverycloselyspace dones)andthatthemonopole weightingappliedbyDAS,whichisdesignedtocompensatefo rthelossinFigure 7-2 B, willbeunabletocompensateforthelossduetodirectivity. Werstconsidertwodirectivesources(withtheirdirectiv itypatternshowninFigure 7-2 A):oneplacedattheorigin(with60dBpower)andtheotheron eat x = y =0 : 2m (with55dBpower).Bothsourcesareplaced1mabovethearray plane.Thesignaland noisewaveformsaregeneratedaszero-meancircularlysymm etrici.i.d.complexGaussian randomprocesses.Thenoisecovariancematrixisascaledid entitymatrix,theSNRis 0dBand B =500.(TheSNRisdenedas10log 10 oftheratiooftheminimumsource powertothenoisevariance.)Figure 7-3 showsthebeamformingimagesobtainedbyDAS, CLEAN-SCandDSL(DSL-NEimageisidenticaltotheDSLimagea ndhenceisnot shown).WeobservethatDASsuersfrompoorresolutionwher easCLEAN-SC,DSL andDSL-NEcanclearlyidentifythetwodominantsources.Th epowerestimatesofthe algorithmsarealsonotedinthegures.DASandCLEAN-SCare unabletoestimatethe powerlevelscorrectly. Inthesecondexample,wekeepthesources,aswellastheirlo cationsandpowers, thesameasinthepreviousexamplebutletthenoisecovarian cematrixbenon-diagonal: thenon-diagonalnoisecovariancematrixisgeneratedas Q H Q ,where Q isasymmetric M M matrixwithits( m;m 0 ) th entryequalto1ifsensors m and m 0 ( m;m 0 =1 ;:::;M ) areclosertoeachotherthan0.15m.SNRissetat-5dBand B =500.(Withthe non-diagonalnoisecovariancematrix,theSNRisdenedas1 0log 10 oftheratioofthe minimumsourcepowertothelargestdiagonalelementofthen oisecovariancematrix.)We observethatinthiscase,DSLcannotidentifythelocationo ftheweakersourcecorrectly whereasDSL-NEcanstillaccuratelyidentifybothofthesou rces.CLEAN-SCalsofailsto resolvetheo-centersourceclearlyandthebackgroundclu tterishigh.Thecomputation timesofDSL,DSL-NEandCLEAN-SCinthisexamplewere0.3,31 .2and2.1seconds, respectively.DSL-NEtakesthelongesttimeduetosolvingE q. 7{4 ateachiteration. 127

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Notethatwehaveassumedthatthenumberofsources L 0 isknownwhilerunning DSLandDSL-NE.Inresultsnotshown,itwasobservedthatthe algorithms,especially DSL-NE,canstillprovidereasonableperformancewhenthea ssumednumberofsourcesis largerthantheactualnumberofsources. 7.3Conclusions Abeamformingalgorithm(namedDSL)basedontheeigendecom positionofthe arraycovariancematrixcanbeusedforthelocalizationofd irectivesourceswithunknown directivitypatterns[ 2 ].AnextensionofDSL,DSL-NE,hasbeenpresentedtoextract thenoisecovariancematrixfromthearraycovariancematri xusinganiterativealgorithm andconvexoptimizationtoachievebetterestimationperfo rmance.Ithasbeenshown vianumericalexamplesthatDSLandDSL-NEshowbetterperfo rmancethanDAS andCLEAN-SCandthatDSL-NEshowsbetterperformancethanD SLwithspatially non-whitemeasurementnoise. 128

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n m l ,q r m lr, Figure7-1.Aplanarmicrophonearrayextendinginthe xy -planewith M microphones (shownbythecircles)andinthepresenceofadirectivesour ce. -100 -50 0 50 100 -30 -25 -20 -15 -10 -5 0 q Directivity, dB -60 -40 -20 0 20 40 60 -60 -40 -20 0 20 40 60 0 0 0 1 1 1 1 0 0 1 1 1 1 1 0 0 1 1 1 1 1 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 1 1 x, cmy, cm Monopole source loss (dB) -60 -40 -20 0 20 40 60 -60 -40 -20 0 20 40 60 2 4 10 10 13 17 25 4 12 9 11 14 17 22 8 12 10 11 11 12 13 17 10 9 9 9 9 9 16 1112 23 12 10 11 18 21 5 2 2 4 7 11 6 0 1 5 15 10 6 2 1 8 11 9 11 3 1 10 10 10 13 17 x, cmy, cm Directive source loss (dB) ABC Figure7-2.Thedirectivitiesobservedatthearraymicroph ones.A)Absolutevalueofthe directivityversustheemissionangle.Thecirclesshowthe directivityvalues thatthearraysensorsencounterwiththedirectivesourcep lacedat x = y =0 : 2mand1mabovethearrayplane.Absolutevalue(roundedandi n dB)ofthepropagationlossateachmicrophoneforasourceat x = y =0 : 2m and1mabovethearrayplane:B)monopolesource,andC)direc tivesource. 129

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-40 -24 -8 8 24 40 -40 -24 -8 8 24 40 47 35 x, cmy, cmDAS 30 35 40 45 50 -40 -24 -8 8 24 40 -40 -24 -8 8 24 40 48 38 x, cmy, cmCLEAN-SC 30 35 40 45 50 -40 -24 -8 8 24 40 -40 -24 -8 8 24 40 60 55 60 55 x, cmy, cmDSL 30 35 40 45 50 55 60 Figure7-3.Twodirectivesourceslocatedat x = y =0m(60dB)and x = y =0 : 2m(55 dB)(asindicatedbythecircles).Bothsourcesarelocated1 mabovethearray plane.Noisewithscaledidentitycovariancematrixisappl ied.SNR=0dB. Thepowerestimatesofthealgorithmsarenotedinthegures .DSL-NEimage isidenticaltotheDSLimageandhenceisnotshown. -40 -24 -8 8 24 40 -40 -24 -8 8 24 40 46 36 x, cmy, cmDAS 30 35 40 45 50 -40 -24 -8 8 24 40 -40 -24 -8 8 24 40 48 40 x, cmy, cmCLEAN-SC 30 35 40 45 50 -40 -24 -8 8 24 40 -40 -24 -8 8 24 40 62 61 62 61 x, cmy, cmDSL 30 35 40 45 50 55 60 -40 -24 -8 8 24 40 -40 -24 -8 8 24 40 60 55 60 55 x, cmy, cmDSL-NE 30 35 40 45 50 55 60 Figure7-4.Twodirectivesourceslocatedat x = y =0m(60dB)and x = y =0 : 2m(55 dB)(asindicatedbythecircles).Bothsourcesarelocated1 mabovethearray plane.Noisewithanon-diagonalcovariancematrixisappli ed.SNR=-5dB. Thepowerestimatesofthealgorithmsarenotedinthegures 130

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CHAPTER8 CONCLUSIONSANDFUTUREWORK 8.1Conclusions Inthisdissertationwehavediscussedvariousaspectsofmi crophonearrayprocessing foraeroacousticmeasurements.Wehaveshownthechallenge sinvolvedinnoisesource localizationandpowerestimation,anddevelopedanumbero fnewarrayprocessing techniques.Thesetechniqueswereevaluatedusingbothsim ulatedandexperimentaldata. First,wehavediscusseddeconvolutionmethodsforelimina tingtheeectsofthearray responsefromtheDASbeamformingresultassumingthatthes ourcesareuncorrelated. Wehaveproposedtwodeconvolutionapproaches,namelySC-D AMASandCMF,based onsparsityandconvexoptimization.Wehaveshownthatbyle veragingsparsity, improvementsinestimationaccuracyandcomputationtimec anbeachieved.Wehave thenevaluatedtheproposedalgorithmswithexperimentalt estcasesandaNACA63-215 ModBairfoiltestedinthepresenceofrow.Itwasshownthatt heproposedalgorithms areaseectivewithexperimentaldataastheyarewithsimul ateddata. Next,wehavediscusseddeconvolutionmethodsforcorrelat edsources.Wehave proposedanewdeconvolutionalgorithm,namelyCMF-C,byex tendingCMF.However, thisalgorithmiscomputationallyverydemanding.Therefo re,wehavealsopresented anotherapproach,calledMACS,fortheecientmappingofco rrelatedacousticsources inaeroacousticmeasurements.Thisalgorithmwasdemonstr atedtobeeectivewith simulatedandexperimentaldata. Wehavealsopresentedasystematicuncertaintyanalysisof theDASbeamformerand thearraycalibrationprocedure.Ouranalysisshowedthatw ithexperimentaldataand inthepresenceofrow,errorsintherangeof 1dBaretobeexpectedwhenreporting integratedlevelswithDAS. Finally,wehavepresentedabeamformingmethod,namedDSLNE,thatcandeal withdirectivesourcesandnon-diagonalmeasurementnoise covariancematrices.Itwas 131

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shownwithnumericaldatathatDSL-NEcanlocalizedirectiv esourcesgiventhattheyare well-separated. 8.2FutureWork Thereareanumberofpossiblefutureresearchdirectionsth atcouldbepursued.We listsomeideasbelow. Anuncertaintyanalysisofthecalibrationprocedurehasbe enpresentedinthis dissertation.Inadditiontothisanalysis,thesensitivit yofthecalibrationproceduretothe validityofitsassumptions(monopolesource,norerection sandsoon)couldbeanalyzed. Thisisofpracticalinterestsincetheindividualcalibrat ionofhundredsofmicrophonesis anexpensivetask. ThefastbeamformingmethodproposedinChapter 5 canbeanalyzedbyfurther experimentsresemblingthesimulatedexamples.Thisisimp ortantasmostoftheexisting deconvolutiontechniquesassumethatthesourcesareuncor relatedandtheeectsof sourcecoherenceonthebeamformingmapsobtainedwiththes emethodsisunclear. WehavepresentedtheuncertaintyanalysisoftheDASbeamfo rmer.Thesame analysiscanbeextendedtoestimatetheuncertaintyofmore advancedalgorithmssuch asDAMAS.Thismightinvolvethedevelopmentofnewuncertai ntyanalysistechniques asthemultivariateandMonte-Carlobasedmethodswillmost likelybecomputationally infeasiblewiththeadvancedbeamformingalgorithms. Moreover,thefurtherdevelopmentandanalysisofthepropo sedalgorithminChapter 7 fordirectivesourcelocalizationmightbeofinterest.Mea surementnoisecovariance matrixestimationisanothertopicofbiginterest[ 74 ]asthecurrentmethodofdiagonal removalisnotveryappealingfromatheoreticalpointofvie w. Finally,wenotethataniterativeadaptiveapproach(IAA)w asdevelopedduring thisdissertationstudyforapplicationssimilartoaeroac ousticnoisemeasurements.IAA wasshowntobepromisinginanumberofapplicationsincludi ngpassivefar-eldarray processing[ 44 75 ],radar/sonarrange-Dopplerimaging[ 44 ],underwatermulti-input 132

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multi-output(MIMO)communications[ 76 ],andMIMOrange-Doppler-angleimaging [ 77 ].IAAwasshowntobeanapproximationtoanotheriterativet echniquebasedon likelihoodmaximization(calledIAA-ML)[ 78 79 ].Furthermore,DAMAScanbeshownto beanapproximationtoIAA-ML.Therefore,thesealgorithms canbeanalyzedwithinthe aeroacousticmeasurementsframeworkandcomparedwiththe methodspresentedinthis dissertation. 133

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APPENDIXA JACOBIANMATRIX Thisappendixderivestheclosed-formexpressionsfortheJ acobianmatriceswith respecttoalloftheinputvariablesin V (seeSection 6.2 ).NotethattheJacobiansare evaluatedusingthenominalvaluesoftheinputvariables. A.1JacobianMatrixfortheCSM Thederivativesof P l withrespecttotheCSMelementsare @P l @G mm = 1 M 2 j ~ D m j 2 r l;m r l; 0 2 ;m =1 ;:::;M; (A{1) @P l @C mn = 2 M 2 r l;m r l;n r 2 l; 0 Re n ~ D m ~ D n e jk ( r l;m r l;n ) o ;m;n =1 ;:::;M;m 6 = n; and (A{2) @P l @Q mn = 2 M 2 r l;m r l;n r 2 l; 0 Im n ~ D m ~ D n e jk ( r l;m r l;n ) o ;m;n =1 ;:::;M;m 6 = n: (A{3) A.2JacobianMatrixfortheCalibrationFactors Thederivativeof P l withrespectto D m canbewrittenas @P l @D m = 2 M 2 r 2 l; 0 r 2 l;m D m G mm +Re ( r l;m e jkr l;m M X p =1 ;p 6 = m r l;p e jkr l;p ~ D p G pm )! ;m =1 ;:::;M: (A{4) Thederivativewithrespectto E m isobtainedbyreplacing D m by E m andthereal componentbytheimaginarycomponentoftheargumentinEq. A{4 A.3JacobianMatrixforMicrophoneLocations Thederivativeof P l withrespectto x m isgivenby @P l @x m = 1 M 2 j ~ D m j 2 @ @x m r 2 l;m r 2 l; 0 +2Re ( M X p =1 ;p 6 = m ~ D p ~ D m r l;p e jkr l;p @ @x m 1 r 2 l; 0 r l;m e jkr l;m !)# ; (A{5) where @ @x m ( r l;m e jkr l;m )= (~ x l x m )(1 =r l;m + jk ) e jkr l;m ; (A{6) @ @x m 1 r 2 l; 0 = 2( x l x ) Mr 4 l; 0 ; and @r 2 l;m @x m = 2(~ x l x m ) ; (A{7) 134

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for m =1 ;:::;M .Aclosed-formexpressioncanbeobtainedforEq. A{5 byusingthe productrulefordierentiationtogetherwithEq. A{6 andEq. A{7 .Similarexpressions canbeobtainedfor y m and z m byreplacing~ x l with~ y l or~ z l and x m with y m or z m A.4JacobianMatrixforTemperature Thederivativeof P l withrespecttotemperaturecanbecalculatedusing @P l @T = @P l @k @k @T ; (A{8) where @P l @k = 2 M 2 r 2 l; 0 Im ( M 1 X m =1 M X n = m +1 ~ D m ~ D n r l;m r l;n ( r l;m r l;n ) e jk ( r l;m r l;n ) G mn ) ; (A{9) @k @T = k 2 T 0 ; (A{10) and T 0 istheroomtemperature. Notethatitisdiculttocommentonthescalingofthesensit ivitycoecientsdue tothecomplexityofthecorrespondingexpressionsandtheh ighcorrelationbetweenthe inputvariables.Forinstance,althoughitappearsthatthe sensitivitycoecientsofthe CSMdecreaseby M 2 ,thecontributionsfromalltheterms( J CSM g CSM J TCSM inEq. 6{14 ) willalsoscalewith M 2 andhencetheeectof M willbecancelledout. 135

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APPENDIXB COVARIANCEMATRIXOFTHECSM Thisappendixderivesthecovariancesbetweenallthereala ndimaginarycomponents ofthe M M complexsymmetricCSM, ^ G (seeSection 6.2 ). Letthepressuresmeasuredatmicrophones m and n bedenotedas p 0m ( t )and p 0n ( t ), respectively,where t denotestime.Notethatunlessotherwisestated,theindice s m and n bothrunfrom1to M .TheniteFouriertransformsof p 0m ( t )and p 0n ( t )arethendenedas [ 54 ] y m ( f )= Z T 0 p 0m ( t ) e j 2 ft dt = y m;R ( f ) jy m;I ( f ) ; and y n ( f )= Z T 0 p 0n ( t ) e j 2 ft dt = y n;R ( f ) jy n;I ( f ) ; (B{1) where y m;R ( f )and y m;I ( f )denotetherealandimaginarypartsof y m ( f ),respectively, (similarlyfor y n ( f ))and T = H=f s istheniteblocklengthintime.Therawestimatefor thecross-spectrumisthengivenby[ 54 ] G mn ( f h )= 2 T y m ( f h ) y n ( f h ) ;h =0 ; 1 ;:::;H= 2 : (B{2) Notethatwhen p 0m ( t )and p 0n ( t )areassumedtobenormallydistributedwithzeromean, sowillbe y m ( f )and y n ( f ).Thefrequencyvariable f isomittedintherestfornotational simplicity.FromEq. B{1 andEq. B{2 weobtain[ 54 ] G mm = 2 T ( y 2 m;R + y 2 m;I ) ;G nn = 2 T ( y 2 n;R + y 2 n;I ) ; (B{3) and C mn = 2 T ( y m;R y n;R + y m;I y n;I ) ;Q mn = 2 T ( y m;R y n;I y m;I y n;R ) : (B{4) 136

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Moreover,fromEq. B{1 evaluatedat f = f 0 ;f 1 ;:::;f H= 2 ; orequivalently,at f = 0 ; 1 =T;:::;H= (2 T ) ; weobtain[ 54 ] E [ y m;R y m;I ]= E [ y n;R y n;I ]=0 ; E [ y 2 m;R ]= E [ y 2 m;I ]= T 4 G mm ; E [ y 2 n;R ]= E [ y 2 n;I ]= T 4 G nn ; (B{5) and E [ y m;R y n;R ]= E [ y m;I y n;I ]= T 4 C mn ;E [ y m;R y n;I ]= E [ y m;I y n;R ]= T 4 Q mn : (B{6) Inordertocomputethecovariancebetween C mn and Q pq ,weneedtond E [ C mn Q pq ]= 4 T 2 E [( y m;R y n;R + y m;I y n;I )( y p;R y q;I y p;I y q;R )] = 4 T 2 ( E [ y m;R y n;R y p;R y q;I ] E [ y m;R y n;R y p;I y q;R ] + E [ y m;I y n;I y p;R y q;I ] E [ y m;I y n;I y p;I y q;R ]) = C mn Q pq + 1 2 ( C mp Q nq + Q mq C np Q mp C nq C mq Q np ) ; (B{7) wherewehaveusedthefactthatforanyfourGaussianvariabl es a 1 ;a 2 ;a 3 ;a 4 withzero meanvalues[ 54 ] E [ a 1 ;a 2 ;a 3 ;a 4 ]= E [ a 1 ;a 2 ] E [ a 3 ;a 4 ]+ E [ a 1 ;a 3 ] E [ a 2 ;a 4 ]+ E [ a 1 ;a 4 ] E [ a 2 ;a 3 ] : (B{8) Since E [ G mm ]= G mm E [ G nn ]= G nn E [ C mn ]= C mn and E [ Q pq ]= Q pq (seeEqs. ( B{3 )-( B{6 )),itfollowsthat Cov( C mn ;Q pq )= 1 2 ( C mp Q nq + Q mq C np Q mp C nq C mq Q np ) ;m;n;p;q =1 ;:::;M: (B{9) TheothercovarianceformulaslistedinTable 6-2 canbeobtainedinasimilarmanner. 137

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APPENDIXC GENERATINGCORRELATEDGAUSSIANRANDOMVARIABLES ThisappendixprovidesashortdescriptiononhowGaussianr andomvariableswitha givencovariancematrixcanbegenerated(seeSection 6.2 ). TogenerateGaussianrandomvectorswithzeromeanandcovar iancematrix werstgenerateani.i.d.Gaussianrandomvectorwithzerom eanandunitvariance. Denotingthisrandomvectorwith willyieldaGaussianrandomvectorwith covariancematrix andzeromeanprovidedthat H = canbeobtainedfromthe Choleskydecompositionof [ 49 ]. LetthenumberofMonte-Carlotrialsbedenotedby N trial .Werecommended that N trial perturbationvectorsbegeneratedsimultaneouslybeforei mplementing theMonte-Carloanalysisratherthangeneratingasinglepe rturbationvectorateach Monte-Carloiteration(i.e., N trial times)sincecalculating canbeatime-consuming process. 138

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APPENDIXD MOREONMICROPHONELOCATIONERRORS ThisappendixelaboratesonthebiasobservedintheDASpowe restimateswhenthe microphonelocationsareperturbed(seeSection 6.3.2.2 ). Assumethatthereisonlyasinglesourcepresentandthatthe reisnocontaminating noise.Moreoverassumethatonlythemicrophonelocationsa reperturbed.Consequently (seeEq. 2{8 ), G = P 0 aa H ; (D{1) where P 0 isthepowerofthesourceand a isthesteeringvectorcorrespondingtothe source,i.e., a = e jkr 0 ; 1 =r 0 ; 1 ;:::;e jkr 0 ;M =r 0 ;M T ,with r 0 ;m denotingthedistance betweenthe m th microphoneandthesourcelocation.Whenthemicrophoneloc ations areperturbed,thedistancebetweenthe m th microphoneandthesourcelocationbecomes r 0 ;m + r m ,where r m denotestheperturbationanditcanbenegativeorpositive. The perturbedDASestimateofthesourcepoweristhen(byomitti ngcalibrationerrorsin Eq. 2{9 ) ^ P 0 = 1 M 2 ~ a Hp G ~ a p ; (D{2) where ~ a p istheperturbedversionofEq. 2{6 andisdenedas ~ a p = 1 r 0 ; 0 ( r 0 ; 1 + r 1 ) e jk ( r 0 ; 1 + r 1 ) ;:::; ( r 0 ;M + r M ) e jk ( r 0 ;M + r M ) T : (D{3) Notethat r 0 ; 0 (thedistancebetweenthesourceandthearraycenter)isnot goingtobe aectedmuchbytheperturbationssincethearraycenterwil lremainapproximatelyatthe samelocationwhenthemicrophonelocationsareperturbedw ithi.i.d.Gaussianrandom variableswithzeromeanvalues.Therefore, ^ P 0 = 1 M M X m =1 1+ r m r 0 ;m e jkr m 2 ~ P 0 ; (D{4) where ~ P 0 = P 0 =r 2 0 ; 0 isthesourcepoweratthearraycenter.Notethatwhenthe microphonelocationsarenotperturbed, r m =0, m =1 ;:::;M ,andhence ^ P 0 = ~ P 0 139

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However,when r m 6 =0,since f isrelativelylargeandtheperturbationsappear onthephaseterms,evensmallperturbationscanaecttheov erallresult.Usingthe Cauchy-Schwarzinequalitygives[ 63 ] 1 M M X m =1 1+ r m r 0 ;m e jkr m 2 1+ 1 M M X m =1 2 r m r 0 ;m + 1 M M X m =1 r m r 0 ;m 2 1 ; wheresince r m r 0 ;m ,thesecondandthirdtermshavebeenneglectedinthelast equality.So,wecanapproximatelyclaimthat ^ P 0 ~ P 0 .Weplot 1+ r m r 0 ;m e jkr m m =1 ;:::;M ,forasingleMonte-CarlotrialinFigure D-1 ,wherethestandarddeviation oftheperturbation r m ischosentobe1and10mm.Weobservethatas fr m gets larger,thephase kr m = 2 f c r m startstodeviatefromthenominalvalueof0 ,whereas theamplitudeisalwaysaround1regardlessof f since 1+ r m r 0 ;m 1.Therefore,we concludethatforrelativelylarge fr 0 ;m ,theaverageofthesamples(markedwithacross intheplots)willhaveamplitudemuchsmallerthan1whichme ansthattheaverage squared,i.e., ^ P 0 = ~ P 0 ,willbeevensmaller. Anotherrathersimpleargumentisthatsincethenon-pertur bed ~ a l aredesignedsoas tomaximizethebeamformingoutputforthe l th scanningpoint,whenthelocationsare perturbed,themismatchbetween ~ a l and a l willresultinequalorsmallerpowerestimates thanthetruepower.Therefore,thiswillcreateanegativeb iasinthepowerestimates. 140

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0 90 180 -90 1 f = 5000 Hz, s = 0.001 0 90 180 -90 1 f = 5000 Hz, s = 0.01 A B FigureD-1.Thepolarplotof 1+ r m r 0 ;m e jkr m m =1 ;:::;M ,atoneMonte-Carlotrial, wherethestandarddeviationoftheperturbation r m ischosentobeA)1 mmandB)10mm. f =5kHzand z =1 : 48masusual.Thecrossmarks indicatetheaveragesofthe M valuesineachplot. 141

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[68] T.Schultz,M.Sheplak,andL.N.Cattafesta,\Uncertaintya nalysisofthe two-microphonemethod," JournalofSoundandVibration ,vol.304,no.1-2, pp.91{109,2007. [69] P.CastelliniandM.Martarelli,\Acousticbeamforming:An alysisofuncertaintyand metrologicalperformances," MechanicalSystemsandSignalProcessing ,vol.22,no.3, pp.672{692,2008. [70] B.D.Hall,\Calculatingmeasurementuncertaintyforcompl ex-valuedquantities," MeasurementScienceandTechnology ,vol.14,pp.368{375,2003. [71] B.D.Hall,\Onthepropagationofuncertaintyincomplex-va luedquantities," Metrologia ,vol.41,pp.173{177,2004. [72] T.Schultz,M.Sheplak,andL.N.Cattafesta,\Applicationo fmultivariate uncertaintyanalysistofrequencyresponsefunctionestim ates," JournalofSound andVibration ,vol.305,no.1-2,pp.116{133,2007. [73] T.Suzuki,\Generalizedinversebeamformingalgorithmres olving coherent/incoherent,distributedandmultipolesources, 14thAIAA/CEASAeroacousticsConference,AIAA-2008-2954, BritishColumbia,Canada,May2008. [74] C.Bahr,T.Yardibi,F.Liu,andL.N.Cattafesta,\Analysiso fsourcedenoising techniques," TheJournaloftheAcousticalSocietyofAmerica ,vol.125, pp.2538{2538,April2009. [75] L.Du,T.Yardibi,J.Li,andP.Stoica,\Reviewofuserparame ter-freerobust adaptivebeamformingalgorithms," DigitalSignalProcessing ,vol.19,pp.567{582, July2009. [76] J.Ling,T.Yardibi,X.Su,H.He,andJ.Li,\Enhancedchannel estimationand symboldetectionforhighspeedmulti-inputmulti-outputu nderwateracoustic communications," TheJournaloftheAcousticalSocietyofAmerica ,vol.125, pp.3067{3078,May2009. [77] W.Roberts,P.Stoica,J.Li,T.Yardibi,andF.A.Sadjadi,\I terativeadaptive approachestoMIMOradarimaging," IEEEJournalofSelectedTopicsinSignal Processing, toappear [78] T.Yardibi,J.Li,andP.Stoica,\Nonparametricandsparses ignalrepresentationsin arrayprocessingviaiterativeadaptiveapproaches," 42ndAsilomarConferenceon Signals,SystemsandComputers, PacicGrove,CA,October2008. [79] T.W.Anderson,\Asymptoticallyecientestimationofcova riancematriceswith linearstructure," TheAnnalsofStatistics ,vol.1,pp.135{141,January1973. 147

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BIOGRAPHICALSKETCH TarikYardibireceivedhisB.S.degreeinelectricalengine eringfromHacettepe University,Ankara,TurkeyinJune2004.Hegraduatedwitht hehighestGPAinthe FacultyofEngineeringatHacettepeUniversity.Hereceive dhisM.S.degreeinelectrical engineeringfromBilkentUniversity,Ankara,TurkeyinJul y2006andhisPh.D.degree inelectricalandcomputerengineeringfromtheUniversity ofFlorida,Gainesville,FL, USAinAugust2009.Hisresearchinterestsincludestatisti calsignalprocessing,array processing,aeroacousticnoisemeasurements,multiple-i nputmultiple-output(MIMO) communications,radar/sonarsignalprocessing,sparsesi gnalrepresentationandwireless sensornetworks. 148