189603-Multiple-Emitter-Location-and-Signal-Parame

276IEEETRANSACTIONSONANTENNASANDPROPAGATION,VOL.AP-34,NO.3,MARCH1986MultipleEmitterLocationandSignalParameter--EstimationRALPH0.SCHMIDT,MEMBER,EEEEAbsfracf-Processingthesignalsreceivedonanarrayofsensorsforthelocationoftheemitterisofgreatenoughinteresttohavebeentreatedundermanyspecialcaseassumptions.Thegeneralproblemconsiderssensorswitharbitrarylocationsandarbitrarydirectionalcharacteristics(gain/phase/polarization)inanoisehnterferenceenvironmentofarbi-trarycovariancematrix.Thisreportisconcernedfirstwiththemultipleemitteraspectofthisproblemandsecondwiththegeneralityofsolution.Adescriptionisgivenofthemultiplesignalclassification(MUSIC)algorithm,whichprovidesasymptoticallyunbiasedestimatesof1)numberofincidentwavefrontspresent;2)directionsofarrival(DOA)(oremitterlocations);3)strengthsandcrosscorrelationsamongtheincidentwaveforms;4)noiselinterferencestrength.Examplesandcomparisonswithmethodsbasedonmaximumlikelihood(ML)andmaximumentropy(ME),aswellasconventionalbeamformingareincluded.Anexampleofitsuseasamultiplefrequencyestimatoroperatingontimeseriesisincluded.TINTRODUCTIONHETERMMULTIPLEsignalclassification(MUSIC)isusedtodescribeexperimentalandtheoreticaltechniquesinvolvedindeterminingtheparametersofmultiplewavefrontsarrivingatanantennaarrayfrommeasurementsmadeonthesignalsreceivedatthearrayelements.Thegeneralproblemconsidersantennaswitharbitrarylocationsandarbitrarydirectionalcharacteristics(gaidphaselpolarization)inanoisehnterferenceenvironmentofarbitrarycovariancematrix.Themultiplesignalclassificationapproachisdescribed;itcanbeimplementedasanalgorithmtoprovideasymptoticallyunbiasedestimatesof1)numberofsignals;2)directionsofarrival(DOA);3)strengthsandcrosscorrelationsamongthedirectionalwaveforms;4)polarizations;5)strengthofnoisehnterference.Thesetechniquesareverygeneralandofwideapplication.SpecialcasesofMUSICare1)conventionalinterferometry;2)monopulsedirectionfinding(DF),i.e.,usingmultipleManuscriptreceivedAugust9,1985.TheauthoriswithSaxpyComputerCorporation,(formerlyGuilTechResearchCo.),255SanGeronimoWay,Sunnyvale,CA94086.Editor’sNote-ThispaperwasfirstpublishedintheProceedingsoftheRADCSpectrumEstimationWorkshop,heldinOctober1979atGriffissAirForceBase,NY.Thedocumentwaslimitedincirculation,buttheauthor’s“MUSIC”algorithmisaprincipalcandidateinthefieldofspectralestimationandratherwidelyreferencedintheliterature.Therefore,itseemedappropriatetoreprintitinthisspecialissue.IEEELogNumber8407014.colocatedantennas;3)multiplefrequencyestimation.THEDATAMODELThewaveformsreceivedattheMarrayelementsarelinearcombinationsoftheDincidentwavefrontsandnoise.Thus,themultiplesignalclassificationapproachbeginswiththefollowingmodelforcharacterizingthereceivedMvectorXasinorX=AF+W.(1)Theincidentsignalsarerepresentedinamplitudeandphaseatsomearbitraryreferencepoint(forinstancetheoriginofthecoordinatesystem)bythecomplexquantitiesF,,F2,-*,FD.Thenoise,whether“sensed”alongwiththesignalsorgeneratedinternaltotheinstrumentation,appearsasthecomplexvectorW.TheelementsofXandAarealsocomplexingeneral.Theaijareknownfunctionsofthesignalarrivalanglesandthearrayelementlocations.Thatis,aijdependsontheitharrayelement,itspositionrelativetotheoriginofthecoordinatesystem,anditsresponsetoasignalincidentfromthedirectionofthejthsignal.ThejthcolumnofAisa“mode”vector48;)ofresponsestothedirectionofarrival0,ofthejthsignal.Knowingthemodevectora(&)istantamounttoknowing0,(unlessa(&)=@(e2)withel#d2,anunresolvablesituation,atypeIambiguity).Ingeometricallanguage,themeasuredXvectorcanbevisualizedasavectorinMdimensionalspace.Thedirectionalmodevectorsa@,)=ai,fori=1,2,*-,M,i.e.,thecolumnsofA,canalsobesovisualized.Equation(1)statesthatXisaparticularlinearcombinationofthemodevectors;theelementsofFarethecoefficientsofthecombination.NotethattheXvectorisconfinedtotherangespaceofA.Thatis,ifAhastwocolumns,therangespaceisnomorethanatwo-dimensionalsubspacewithintheMspaceandXnecessarilyliesinthesubspace.Alsonotethata(@,thecontinuumofallpossiblemodevectors,lieswithintheMspacebutisquitenonlinear.Forhelpinvisualizingthis,seeFig.1.Forexample,inanazimuth-onlydirectionfindingsystem,19willconsistofasingleparameter.Inanazimuth/elevation/0018-926X/86/0300-0276$01.OOO1986IEEESCHMIDT:MULTIPLEEMITTERLOCATION277x3II9THESIGNALSUBSPACEIDETERMINEDBYTHEDATA1ISIGNALSPACEEIGENVECTOR1SIGNALEIGENVTHEde)CONTINUUMOFVECTORSIDETERMINEDDIRECTION-0F.ARRIVAL0YARRAYGEMETRYAN0CHARACTERLSTICSIel.ez.aMINARETHEEIGENVECTORSOFSCORRESPONDINGTOEIGENVALUEShlh2tlN30e,.ezSPANTHESIGNALSUBSPACEalrll.a1e21ARETHEINCIDENTSIGNALMODEVECTORSFig.1.Geometricportrayalforthree-antennacase.rangesystem,8willbereplacedby8,4,rforexample.Inanycase,a(@isavectorcontinuumsuchasa“snake”(azimuthonly)ora“sheet”(Az/EI)twistingandwindingthroughtheMspace.(Inpractice,theprocedurebywhichthea(@continuumismeasuredorotherwiseestablishedcorrespondstocalibratingthearray.)Inthesegeometricalterms(seeFig.l),theproblemofsolvingforthedirectionsofarrivalofmultipleincidentwavefrontsconsistsoflocatingtheintersectionsofthea(@continuumwiththerangespaceofA.Therangespac