Multiple-and-single-snapshot-compressive-beamformi

MultipleandsinglesnapshotcompressivebeamformingPeterGerstofta)ScrippsInstitutionofOceanography,UniversityofCaliforniaSanDiego,LaJolla,California92093{0238AngelikiXenakiDepartmentofAppliedMathematicsandComputerScience,TechnicalUniversityofDenmark,Kgs.Lyngby,2800DenmarkChristophF.MecklenbraukerChristianDopplerLab,Inst.ofTelecommunications,TUWien,Gusshausstr.25/389,1040Vienna,Austria(Dated:August25,2015)Forasoundeldobservedonasensorarray,compressivesensing(CS)reconstructsthedirection-of-arrival(DOA)ofmultiplesourcesusingasparsityconstraint.TheDOAestimationisposedasanunderdeterminedproblembyexpressingtheacousticpressureateachsensorasaphase-laggedsuperpositionofsourceamplitudesatallhypotheticalDOAs.Regularizingwithan`1-normconstraintrenderstheproblemsolvablewithconvexoptimization,andpromotingsparsitygiveshigh-resolutionDOAmaps.Here,thesparsesourcedistributionisderivedusingmaximumaposteriori(MAP)estimatesforbothsingleandmultiplesnapshots.CSdoesnotrequireinversionofthedatacovariancematrixandthusworkswellevenforasinglesnapshotwhereitgiveshigherresolutionthanconventionalbeamforming.Formultiplesnapshots,CSoutperformsconventionalhigh-resolutionmethods,evenwithcoherentarrivalsandatlowsignal-to-noiseratio.ThesuperiorresolutionofCSisdemonstratedwithverticalarraydatafromtheSWellEx96experimentforcoherentmulti-paths.PACSnumbers:43.60.Pt,43.60.Jn,43.60.FgI.INTRODUCTIONDirection-of-arrival(DOA)estimationreferstothelo-calizationofseveralsourcesfromnoisymeasurementsofthewaveeldwithanarrayofsensors.DOAestimationcanbeexpressedasalinearunderdeterminedproblemwithasparsityconstraintenforcedonitssolution.Thecompressivesensing1,2(CS)frameworkassertsthatthisissolvedecientlywithaconvexoptimizationprocedurethatpromotessparsesolutions.InDOAestimation,CSachieveshigh-resolutionacous-ticimaging3{5,outperformingtraditionalmethods6.Unlikethehigh-resolutionsubspace-basedDOAestimators7,8,DOAestimationviaCSisreliableevenwithasinglesnapshot9{11.Theleastabsoluteshrinkageandselectionoperator(LASSO)12hasbeenextendedtomultiplemeasurementvectors(heremultiplesnapshots)3,13.TheymodifytheLASSOobjectivefunctionbyintroducingamixed-normpenaltytermthatpromotesspatialsparsity.Morespecif-ically,thesnapshotsarecombinedwiththe`2-norm,whereasthespatialsamplesarecombinedwiththe`1-norm.Multiple-snapshotCSoersseveralbenetsoverotherhigh-resolutionDOAestimators3,4,13:1)Ithandlespartiallycoherentarrivals.2)Itcanbeformulatedwithanynumberofsnapshots,incontrastto,e.g.,theMini-mumVarianceDistortion-freeResponse(MVDR)beam-former.3)Itsexibilityinformulationenablesextensionstosequentialprocessing,andonlinealgorithms10.Here,weshowthatCSachieveshigherresolutionthanMUSICa)Correspondingauthor.Electronicmail:[e-mail],eveninscenariosthatfavortheseclassicalhigh-resolutionmethods.Inoceanacoustics,CShasfoundseveralapplicationsinmatchedeldprocessing14,15andincoherentpassivefathometryforinferringsedimentinterfacesdepthsandtheirnumber16.Variouswavepropagationphenomenafromasinglesource(refraction,diraction,scattering,ducting,reection)leadtomultiplepartiallycoherentar-rivalsreceivedbythearray.High-resolutionbeamformerscannotresolvethesecoherentarrivals.CSforsinglesnapshothashigh-resolutioncapabilitiesandcontrarytoeigenvalue-basedbeamformersworksforcoherentarrivals3,4,11.CSislimitedbybasismismatch17whichoccurswhentheDOAsdonotcoincidewiththelookdirectionsoftheangularspectrum,andbybasisco-herence.Solutionstobasismismatchinvolveforexam-pleusingatomicnormandsolvingthedualproblem5,18thatarenotaddressedhere.Gridrenementalleviatesbasismismatchforhighsignaltonoiseratio(SNR)attheexpenseofincreasedcomputationalcomplexity.Adensergridcausesincreasedcoherenceamongthesteer-ingvectors(basiscoherence)whichtranslatestobiasandspreadintheDOAestimatesasdemonstratedhere.Thisisespeciallytrueinlargetwo-dimensionalorthree-dimensionalgeo-acousticinversionproblemsase.g.seis-micimaging19{21.Weuseleastsquaresoptimizationwithan`1-normreg-ularizationterm,alsoknownastheLASSO12,toformu-latetheDOAestimationproblemforsingleandmultiplesnapshots.TheLASSOformulationcomplieswithsta-tisticalmodelsasitprovidesamaximumaposteriori(MAP)estimate,assumingaGaussiandatalikelihoodandaLaplacianpriordistributionforthesourceacous-ticpressure22,23forbothsingle(Sec.II.B)andmultipleCompressivebeamforming1arXiv:1503.02339v2[math.ST]21Aug2015snapshots13(Sec.III).TheLASSOisknowntobeacon-vexminimizationproblemandsolvedecientlybyin-teriorpointmethods.IntheLASSOformulation,Sec.IV.A,thereconstructionaccuracydependsonthechoiceoftheregularizationparameterthatcontrolsthebalancebetweenthedatatandthesparsityofthesolution.WeindicatethattheregularizationparametercanbefoundfromthepropertiesoftheLASSOpath24,25,i.e.,theevo-lutionoftheLASSOsolutionversustheregularizationparameter.Themainfocusofthepaperisonperformanceevalu-ationforsingleandmultiplesnapshotsusingbothsim-ulated(Sec.V)andrealdata(Sec.VI).Otherexcel-lentpapers11havealreadyperformedperformanceeval-uationforsinglesnapshot,consistentwithoursimula-tions.Wearenotawareofperforman