IEEETRANSACTIONSONSIGNALPROCESSINGDr.ChevalierisaCo-Recipientofthe2003 - PDF document

IEEETRANSACTIONSONSIGNALPROCESSINGDr.ChevalierisaCo-Recipientofthe2003ScienceandDefenseAwardfromtheFrenchMinistryofDefenceforworkonarrayprocessingformilitaryradiocommunications.HeiscurrentlymemberoftheEuropeanAssociationforSignalandImageProcessing(EURASIP)andanemeritusmemberoftheSo-desElectriciensetdesElectroniciens(SEE). AnneFerrwasborninLyon,France,in1964.ShereceivedtheM.Sc.degreefromICPI-Lyon,Lyon,France,theMastredegreefromEcoleNa-tionaleSuprieuredesTcommunications(ENST),Paris,France,andthePh.D.degreefromtheEcoleNormaleSuprieuredeCachan,Cachan,France,in1988,1989,and2005,respectively.Since1989,shehasbeenattheArrayProcessingDepartment,Thals-Communications,ColombesCedex,France.Hercurrentinterestsconcerndirec-ndingandblindsourceseparation.Dr.FerrolisaCo-Recipientofthe2003ScienceandDefenseAwardfromtheFrenchMinistryofDefenceforworkonaboutarrayprocessingformilitary LaurentAlberawasborninMassy,France,in1976.HereceivedtheD.E.S.S.degreeinmathematicsandtheD.E.A.degreeinautomaticandsignalprocessingfromtheUniversityofScience(ParisXI),Orsay,FranceandthePh.D.degreeinsciencefromtheUniversityofNice,Sophia-Antipolis,France,in2000,2001,and2003,respectively.Currently,heisanAssistantProfessorattheUniversityofRennesI,Rennes,France,andisliatedwiththeLaboratoireTraitementduSignaletdelImage(LTSI).Hisresearchinterestsincludehigh-orderstatistics,multidimensionalalgebra,blinddeconvolutionandequal-ization,digitalcommunications,statisticalsignalandarrayprocessing,andnumericalanalysis.Since2000,hehasbeeninvolvedwithblindsourcesepa-ration(BSS)andindependentcomponentanalysis(ICA)byprocessingboththecyclostationarysourcecaseandtheunderdeterminedmixtureidenti lBirotwasborninLaRoche-Sur-Yon,France,in1982.HereceivedtheMaster2researchdegreeinsignalprocessingfromtheTrampandImage(STI),UniversityofRennes1,Rennes,France,in2006,whereheiscurrentlyworkingtowardsthePh.D.degreeinsignalprocessingandtelecommunicationattheLaboratoireduTraitementduSignaletdelImage(LTSI)underthedirectionofL.AlberaandI.Merlet.Hisresearchinterestincludeshigherordersourceslocalizationmethodsinbothapplicationstelecom-municationsandbiomedicalengineering. CHEVALIERetal.:HODFFROMARRAYSWITHDIVERSELYPOLARIZEDANTENNAS[3]P.Chevalier,L.Albera,A.Ferreol,andP.Comon,Onthevirtualarrayconceptforhigherorderarrayprocessing,IEEETrans.SignalProcess.,vol.53,no.4,pp.12541271,Apr.2005.[4]P.ChevalierandA.Ferreol,Onthevirtualarrayconceptforthefourth-orderdirectionndingproblem,IEEETrans.SignalProcess.vol.47,no.9,pp.25922595,Sep.1999.[5]P.ChevalierandA.Ferreol,etdispositifdegoniomhauterunordrepairarbitraire,05.03180,Apr.2005.[6]P.Chevalier,A.Ferreol,andL.Albera,Highresolutiondirectionndingfromhigherorderstatistics:The-MUSICalgorithm,Trans.SignalProcess.,vol.54,no.8,pp.29862997,Aug.2006.[7]Q.ChengandY.Hua,PerformanceanalysisoftheMUSICandpencil-MUSICalgorithmsfordiverselypolarizedarrays,IEEETrans.SignalProcess.,vol.42,no.11,pp.31503165,Nov.1994.[8]H.H.ChiangandC.L.Nikias,TheESPRITalgorithmwithhighorderstatistics,Proc.WorkshopHigherOrderStatist.,Jun.1989,pp.163[9]R.T.Compton,Jr.,AdaptiveAntennas—ConceptsandPerformanceEnglewoodCliffs,NJ:Prentice-Hall,1988.[10]M.C.DoganandJ.M.Mendel,ApplicationsofcumulantstoarrayPartI:Apertureextensionandarraycalibration,Trans.SignalProcess.,vol.43,no.5,pp.12001216,May1995.[11]E.R.Ferrara,Jr.andT.M.Parks,ndingwithanarrayofantennashavingdiversepolarizations,IEEETrans.AntennasPropag.vol.AP-31,no.2,pp.231236,Mar.1983.[12]A.Ferreol,E.Boyer,andP.Larzabal,Low-costalgorithmforsomebearingestimationmethodsinpresenceofseparablenuisanceparam-Electron.Lett.,vol.40,no.15,pp.966967,Jul.2004.[13]B.FriedlanderandA.J.Weiss,Adirectionndingalgorithmfordi-verselypolarizedarrays,DigitalSignalProcess.,vol.2,no.3,pp.134,1992.[14]B.FriedlanderandA.J.Weiss,Performanceofdiverselypolarizedantennaarraysforcorrelatedsignals,IEEETrans.Aerosp.Electron.,vol.28,no.3,pp.869880,Jul.1992.[15]B.FriedlanderandA.J.Weiss,Theresolutionthresholdofadirection-ndingalgorithmfordiverselypolarizedarrays,IEEETrans.SignalProcess.,vol.42,no.7,pp.17191727,Jul.1994.[16]E.GnenandJ.M.Mendel,Applicationsofcumulantstoarraypro-PartVI:Polarizationanddirectionofarrivalestimationwithminimallyconstrainedarrays,IEEETrans.SignalProcess.,vol.47,no.9,pp.25892592,Sep.1999.[17]K.C.Ho,K.C.Tan,andB.T.G.Tan,Lineardependenceofsteeringvectorsassociatedwithtripolearrays,IEEETrans.AntennasPropag.vol.46,no.111,pp.17051711,Nov.1998.[18]K.C.Ho,K.C.Tan,andW.Ser,Aninvestigationonnumberofsignalswhosedirections-of-arrivalareuniquelydeterminablewithanelectro-magneticvectorsensor,SignalProcess.,vol.47,pp.4154,1995.[19]B.HochwaldandA.Nehorai,abilityinarrayprocessingmodelswithvector-sensorapplications,IEEETrans.SignalProcess.vol.44,no.1,pp.8395,Jan.1996.[20]Y.Hua,Apencil-MUSICalgorithmforndingtwo-dimensionalan-glesandpolarizationsusingcrosseddipoles,IEEETrans.AntennasPropag.,vol.41,no.3,pp.370376,Mar.1993.[21]J.Li,Directionandpolarizationestimationusingarrayswithsmallloopsandshortdipoles,IEEETrans.AntennasPropag.,vol.41,no.3,pp.379387,Mar.1993.[22]J.LiandR.T.Compton,AngleandpolarizationestimationusingESPRITwithapolarizationsensitivearray,IEEETrans.AntennasPropag.,vol.39,no.9,pp.13761383,Sep.1991.[23]J.LiandR.T.Compton,Angleestimationusingapolarizationsensitivearray,IEEETrans.AntennasPropag.,vol.39,no.10,pp.1543,Oct.1991.[24]J.LiandR.T.Compton,Two-dimensionalangleandpolarizationes-timationusingtheESPRITalgorithm,IEEETrans.AntennasPropag.vol.40,no.5,pp.550555,May1992.[25]J.LiandR.T.Compton,Angleandpolarizationestimationinaco-herentsignalenvironment,IEEETrans.Aerosp.Electron.Syst.,vol.29,no.3,pp.706716,Jul.1993.[26]J.Li,P.Stoica,andD.Zheng,cientdirectionandpolarizationestimationwithaCOLDarray,IEEETrans.AntennasPropag.,vol.44,no.4,pp.539547,Apr.1996.[27]A.NehoraiandE.Paldi,Vector-sensorarrayprocessingforelectro-magneticsourcelocalization,IEEETrans.SignalProcess.,vol.42,no.2,pp.376398,Feb.1994.[28]B.PoratandB.Friedlander,ndingalgorithmsbasedonhigherorderstatistics,IEEETrans.SignalProcess.,vol.39,no.9,pp.2024,Sep.1991.[29]D.Rahamim,J.Tabrikian,andR.Shavit,Sourcelocalizationusingvectorsensorarrayinamultipathenvironment,IEEETrans.SignalProcess.,vol.52,no.11,pp.30963103,Nov.2004.[30]R.O.Schmidt,Multipleemitterlocationandsignalparameterestima-IEEETrans.AntennasPropag.,vol.34,no.3,pp.276280,Mar.[31]S.ShamsunderandG.B.Giannakis,Modelingofnon-Gaussianarraydatausingcumulants:DOAestimationofmoresourceswithlesssen-SignalProcess.,vol.30,no.3,pp.279297,Feb.1993.[32]A.SwindlehurstandM.Viberg,ttingwithdiverselypolar-izedantennaarrays,IEEETrans.AntennasPropag.,vol.41,no.12,pp.16871694,Dec.1993.[33]J.Tabrikian,R.Shavit,andD.Rahamiim,Anefcientvectorsensorgurationforsourcelocalization,IEEESignalProcess.Lett.,vol.11,no.8,pp.690693,Aug.2004.[34]K.C.Tan,K.C.Ho,andA.Nehorai,Uniquenessstudyofmeasure-mentsobtainablewitharraysofelectromagneticvectorsensors,Trans.SignalProcess.,vol.44,no.4,pp.10361039,Apr.1996.[35]K.C.Tan,K.C.Ho,andA.Nehorai,Linearindependenceofsteeringvectorsofanelectromagneticvectorsensor,IEEETrans.SignalProcess.,vol.44,no.12,pp.30993107,Dec.1996.[36]A.J.WeissandB.Friedlander,Performanceanalysisofdiverselypo-larizedantennaarrays,IEEETrans.SignalProcess.,vol.39,no.7,pp.1603,Jul.1991.[37]A.J.WeissandB.Friedlander,ndingfordiverselypolar-izedsignalsusingpolynomialrooting,IEEETrans.SignalProcess.vol.41,no.5,pp.18931905,May1993.[38]A.J.WeissandB.Friedlander,Maximumlikelihoodsignalestimationforpolarizationsensitivearrays,IEEETrans.AntennasPropag.,vol.41,no.7,pp.918925,Jul.1993.[39]K.T.WongandM.D.Zoltowski,Uni-vector-sensorESPRITformul-tisourceazimuth,elevationandpolarizationestimation,IEEETrans.AntennasPropag.,vol.45,no.10,pp.14671474,Oct.1997.[40]K.T.WongandM.D.Zoltowski,Closed-formdirectionndingandpolarizationestimationwitharbitrarilyspacedelectromagneticvector-sensorsatunknownlocations,IEEETrans.AntennasPropag.,vol.48,no.5,pp.671681,May2000.[41]K.T.WongandM.D.Zoltowski,Self-initiatingMUSIC-basedndingandpolarizationestimationinspatio-polarizationalIEEETrans.AntennasPropag.,vol.48,no.8,pp.1245,Aug.2000.[42]K.T.Wong,L.Li,andM.D.Zoltowski,Root-MUSIC-baseddirec-ndingandpolarizationestimationusingdiverselypolarizedpos-siblycollocatedantennas,IEEEAntennasWirelessPropag.Lett.,vol.3,pp.129132,2004.[43]I.ZiskindandM.Wax,Maximumlikelihoodlocalizationofdiverselypolarizedsourcesbysimulatedannealing,IEEETrans.AntennasPropag.,vol.38,no.7,pp.1111114,Jul.1990.[44]M.D.ZoltowskiandK.T.Wong,ESPRIT-based2-Ddirectionndingwithasparseuniformarrayofelectromagneticvectorsensors,IEEETrans.SignalProcess.,vol.48,no.8,pp.21952203,Aug. PascalChevalierreceivedtheM.Sc.degreefromEcoleNationaleSuprieuredesTechniquesAvances(ENSTA),Paris,FranceandthePh.D.degreefromSouth-ParisUniversity,Paris,France,in1985and1991,respectively.Since1991hehasbeenwithThomson-CSF/RGS(nowThals-Communications),ColombesCedex,France,wherehehassharedindustrialactivities(studies,experimentations,expertise,andmanage-ment),teachingactivitiesbothinFrenchengineerschools(Supelec,ENST,andENSTA)andFrenchUniversities(Cergy-Pontoise)andresearchactivities.Since2000,hehasalsobeenactingasTechnicalManagerandArchitectofthearrayprocessingsubsystemaspartofanationalprogramofmilitarysatellitetelecommuni-cations.HehasbeenaThalsExpertsince2003.HehasbeenamemberoftheTHOMSON-CSFTechnicalandScienticalCouncilbetween1995and1998.Heisauthororcoauthorofabout20patentsand100papers.Hispresentresearchinterestsareinarrayprocessingtechniques,eitherblindorinformed,second-orderorhigherorder,spatial-orspatio-temporal,time-invariantortime-varyingespeciallyforcyclostationarysignals,linearornonlinearandpar-ticularlywidelylinearfornoncircularsignals,forapplicationssuchasTDMAandCDMAradiocommunicationsnetworks,satellitetelecommunications,spectrummonitoring,andpassivelisteninginHVUHFband. IEEETRANSACTIONSONSIGNALPROCESSING Fig.7.(a)MaximalRMSEand(b)minimalprobabilityofnonaberrantresultsofsourcesasafunctionof:1)4-MUSIC,2)KP-PD-4-MUSIC,3)UP-PD-4-MUSIC-1,4)UP-PD-4-MUSIC-2,5)6-MUSIC,6)KP-PD-6-MUSIC,7)UP-PD-6-MUSIC,and8)UP-PD-6-MUSIC-2.,UCA,SNR15dB, )=(15 45 ;75 ),( ; )=(45 ;45 ;0 ),( )=(95 22:5 ;75 ,and )=(122 ;45 ;150 ,withoutmod-elingerrors.KP-PD-4-MUSIC,UP-PD-4-MUSIC-1,UP-PD-4-MUSIC-2,6-MUSIC,KP-PD-6-MUSIC,UP-PD-6-MUSIC-1,andUP-PD-6-MUSIC-2algorithms,respectively,andtheper-formancearecomputedfrom300realizations.NotethecapabilityofPD- -MUSICmethodstoprocessunderdeter-minedmixturesofsourcesprovidedthat givenby(29)or(33).Notethepoorperformanceof -MUSICmethodsfortheconsideredscenarioduetothelowinputpoweroftheweakestsourceattheoutputofthesensors.Betterperformancewouldbeobtainedforhighervaluesof VII.CInthispaper,the -MUSICalgorithm, ,hasbeenextendedtoputupwitharrayshavingpolarizationdiversityandreceivingdiverselypolarizedsources,whichgivesriseto -MUSICalgorithms.ThreePD- -MUSICalgorithmshavebeenpresented,dependingontheaprioriknowledgeaboutthepolarizationofthesources.Therstversion,called -MUSIC,iswellsuitedforsourceswithknownpolarization,whilethetwoothers,calledUP-PD- andUP-PD- -MUSIC-2,areabletoestimatethesourcesDOAwithoutanyknowledgeabouttheirpolarizationandallowtoestimatethepolarizationofthesourcesifnecessary.Foragivenvalueof ,thesealgorithmsareshowninthispapertoincreasetheresolution,therobustnesstomodelingerrors(atleastforseveralpoorlyangularlyseparatedsources),andtheprocessingcapacity(atleastforVAwithoutanyHOambiguities)ofthe -MUSICmethodinthepresenceofdiverselypolarizedsourcesandfromanarraywithpolarizationdiversity.Moreover,despiteahighervarianceinthestatisticsestimation,performanceofUP-PD- -MUSICalgorithmshavebeenshowntogenerallyincreasewith whensomeresolutionisrequired.Thisoccurs,inparticular,forsourceswhicharepoorlyseparatedinbothDOAandpolarization.ThisresultshowsoffforthesescenariostheinteresttojointlyexploitpolarizationdiversityandHOstatisticsforDF.IdentiresultshavebeenpresentedforeachofthethreePD- methods,forVAwithoutHOambiguities.Nevertheless,forVAwithvectorialsensors,adeeperanalysisoftheidentiabilityof -MUSICalgorithmsisrequired.Weshowherethatitispossibletoobtainanestimate ofthepolarizationvector ofthesource fromtheestimate ofthevector .Todoso,weimplementthedifferentfollowingsteps.1)Compute,from(16),anestimate ofthe vector by 2)Decomposethevector into subvec- , ,suchthat 3)Mapthecomponentsofeachsubvector intoa matrix suchthat ,where and aretheelements and of and ,respectively4)Buildthematrices , ,denedbythefollowing: if ;b) if ;c) if 5)Jointlydiagonalizethematrices , .6) istheassociatedeigenvectorcorrespondingtothemax-imaleigenvalue.[1]J.F.Cardoso,LocalisationetIdenticationparlaquadricovariance,TraitementduSignal,vol.7,no.5,Jun.1990.[2]J.F.CardosoandE.Moulines,Asymptoticperformanceanalysisofndingalgorithmsbasedonfourth-ordercumulants,Trans.SignalProcess.,vol.43,no.1,pp.214224,Jan.1995. CHEVALIERetal.:HODFFROMARRAYSWITHDIVERSELYPOLARIZEDANTENNAS Fig.5.(a)RMSEand(b)probabilitynonaberrantresultsofsource2asafunctionof:1)2-MUSIC,2)KP-PD-2-MUSIC,3)UP-PD-2-MUSIC-1,4)UP-PD-2-MUSIC-2,5)4-MUSIC,6)KP-PD-4-MUSIC,7)UP-PD-4-MUSIC-1,8)UP-PD-4-MUSIC-2,9)6-MUSIC,10)KP-PD-6-MUSIC,11)UP-PD-6-MUSIC-1,and12)UP-PD-6-MUSIC-2.,UCA,SNR5dB, ; )=(50 45 ;0 ,and )=(60 ;45 180 ,withoutmodelingerrors. Fig.6.(a)RMSEand(b)probabilitynonaberrantresultsofsource2asafunctionof:1)2-MUSIC,2)KP-PD-2-MUSIC,3)UP-PD-2-MUSIC-1,4)UP-PD-2-MUSIC-2,5)4-MUSIC,6)KP-PD-4-MUSIC,7)UP-PD-4-MUSIC-1,8)UP-PD-4-MUSIC-2,9)6-MUSIC,10)KP-PD-6-MUSIC,11)UP-PD-6-MUSIC-1,and12)UP-PD-6-MUSIC-2.,UCA,SNR5dB, ; )=(50 ;45 ;0 ,and )=(60 ;45 ;180 ,withmodelingerrors.manceofUP-PD- -MUSIC-1,UP-PD- -MUSIC-2,and -MUSICalgorithmsas increases.ThisisduetoahighervarianceinthestatisticsestimationsincenoresolutionisrequiredforDFduetoahighseparationofsourcesinpolariza-tion.Finally,notethat,for ,KP-PD- -MUSICalgorithmmaybesurprisinglyworsethanUP-PD- -MUSICalgorithm.C.UnderdeterminedMixturesofSourcesToillustratethecapabilityofPD-4-MUSICandPD-6-MUSICalgorithmstoprocessunderdeterminedmixturesofsources,welimitthenumberofsensorsofthepre-viouscirculararrayto sensors.Thesesensorsaresuchthat and forPD- -MUSICmethods for -MUSICmethodsandweassumethat .Undertheseassumptions,wededucefrom[3,Tables1and2],(29),and(33)that and forPD- -MUSICmethodsandfrom[3,Tables6and7]that and for -MUSICmethods.WethenassumethatfourstatisticallyindependentQPSKsourceswitharaisecosinepulseshapedlterarere-ceivedbythearray.ThefourQPSKsourceshavethesamesymbolduration,thesamerolloff ,thesameinputSNRequalto15dB,andDOAandpolariza-tionparametersequalto , , ,and ,respectively.Undertheseassumptions,Fig.7showsthevariations,asafunc-tionofthenumberofsnapshots ,ofthehighestRMSEandthelowestprobabilityofnonabberantresults,amongallthesources,attheoutputofseveralDFmethodswithoutmodelingerrors.Thesemethodscorrespondto4-MUSIC, IEEETRANSACTIONSONSIGNALPROCESSING Fig.3.(a)RMSEand(b)probabilitynonaberrantresultsofsource2asafunctionof:1)2-MUSIC,2)KP-PD-2-MUSIC,3)UP-PD-2-MUSIC-1,4)UP-PD-2-MUSIC-2,5)4-MUSIC,6)KP-PD-4-MUSIC,7)UP-PD-4-MUSIC-1,8)UP-PD-4-MUSIC-2,9)6-MUSIC,10)KP-PD-6-MUSIC,11)UP-PD-6-MUSIC-1,and12)UP-PD-6-MUSIC-2,.,UCA,SNR5dB, )=(50 45 ;0 ,and )=(60 ;45 ;10 ,withoutmodelingerrors. Fig.4.(a)RMSEand(b)probabilitynonaberrantresultsofsource2asafunctionof:1)2-MUSIC,2)KP-PD-2-MUSIC,3)UP-PD-2-MUSIC-1,4)UP-PD-2-MUSIC-2,5)4-MUSIC,6)KP-PD-4-MUSIC,7)UP-PD-4-MUSIC-1,8)UP-PD-4-MUSIC-2,9)6-MUSIC,10)KP-PD-6-MUSIC,11)UP-PD-6-MUSIC-1,and12)UP-PD-6-MUSIC-2.,UCA,SNR5dB, ; )=(50 ;45 ;0 ,and ; )=(60 ;45 ;10 ,withmodelingerrors.givenvalueof ,thebetterperformanceofKP-PD- methodwithrespecttoUP-PD- -MUSICmethods,duetotheexploitationofthetrueaprioriknowledgeofthesourcespo-larization.Notenally,fortwosourceswithknownpolariza-tions,increasingperformancesofKP-PD- -MUSICmethods increasesinthepresenceofmodelingerrorsassoonasthenumberofsnapshotsgetsbeyond1300.Thisresultseemstobedirectlyrelatedtothedegreeofcouplingofthetwoesti-matedpseudospectra(oneforeachpolarization),computedbyagivenKP-PD- -MUSICmethod,whichincreaseswithmod-elingerrorsandwhenthepolarizationseparationofthesourcesdecreases.Moreprecisely,whenthiscouplingishigh(weakpolarizationseparationwithmodelingerrors),thetwosourcesinteractineachofthetwocomputedpseudospectraandresolu-tionisrequiredtoseparatethem;hence,theincreasingperfor-mancewith ofKP-PD- -MUSICmethods.However,whenthiscouplingisweak(strongpolarizationseparationorabsenceofmodelingerrors),sourcesnolongerinteractinagivenpseu-dospectrum.Then,onlyonesourcehastobefoundforagivenpseudospectrumandnoresolutionisrequired,hencedecreasingperformanceduetoahighervarianceinthestatisticsestimation WenowconsiderthescenarioofFigs.3and4butwenowassumethatthetwosourcesarestillpoorlyangularlyseparatedbutarewellseparatedinpolarization,suchthat and ,respec-tively.Undertheseassumptions,Figs.5and6showsim-ilarvariationsasforFigs.3and4,respectively.Westillnotethatwhateverthevalueof ,thereisabetterperfor-mancesofPD- -MUSICmethodswithrespectto ones,duetotheexploitationofpolarizationdiversityinadditiontospacediversity.Westillnoteverycloseperfor-mancesofUP-PD- -MUSIC-1andUP-PD- algorithms.Moreover,wenotethedecreasingperfor- CHEVALIERetal.:HODFFROMARRAYSWITHDIVERSELYPOLARIZEDANTENNASVI.CIMULATIONSTheresultsoftheprevioussectionsareillustratedinthissectionthroughcomputersimulations.Todoso,werstin-troduceaperformancecriterioninSectionVI-AanddescribethesimulationsinSectionsVI-BandVI-Cforoverdeterminedandunderdeterminedmixturesofsources,respectively.Thesourcesareassumedtohaveazeroelevationangle andtobezero-meanstationarysourcescorrespondingtoquaternaryphase-shiftkeying(QPSK)sourcessampledatthesymbolrate.A.PerformanceCriterionForeachofthe consideredsourcesandforagivenDFmethod,twocriteriaareusedinthefollowingtoquantifythequalityoftheassociatedDOAestimation.Foragivensource,rstcriterionisaprobabilityofaberrantresultsgeneratedbyagivenmethodforthissource.Thesecondoneisanaveragedrootmeansquareerror(RMSE),computedfromthenonaberrantresults,generatedbyagivenmethodforthissource.Thesetwocriteriawerepreciselydenedin[6]andarenotrecalledinthisB.OverdeterminedMixturesofSourcesToshowtheinterestoftakingintoaccountboththepolariza-tionofthesourcesandHOstatisticsforDF,weconsideraUCA crosseddipoleswitharadius suchthat Onedipoleisparalleltothe -axiswhereastheotherisparalleltothe -axis.Threeofthesecrosseddipolesarecombinedtogeneratearightsensecircularpolarizationinthe -axiswhilethethreeotherdipolesarecombinedtogeneratealeftsensecir-cularpolarizationinthe -axis.Thearrayisthencomposedoftwoorthogonallypolarizedoverlapped(noncolocated)circularsubarraysof sensorssothatadjacentsensorsalwayshavedifferentpolarizationsasdepictedinFig.2(a).Undertheseassumptions,thesensorsoftherstandsecondsubarrayhaveacomplexresponsetoaunitelectriceldcomingfromDOA withpolarization equalto and ,respectively.Intheseexpres- and ,whichcorrespondtothecomplexresponsesofthetwodipoles,aregivenby[4],[9] (36) Inotherwords,thecomplexresponse ofsensor , ,toaunitelectriceldcomingfromDOA withpolarization isgivenby Wethendeducefrom(2),(3),and(38)that,inthiscase,the cientsofmatrix aredenedby (39a) (39b)where andwhere isdenedby Notethatthiscorrespondstochoosingthevectors and suchthat and .Notethatthechosenarrayofsensorspresentsambiguitiesfor ,where isaninteger,andthuspreventsfromestimatingDOAofsourcescomingfrom .In- , and arenotfullrank, isalwayssolutionof(10),(23),and Inthiscontext,twoQPSKsourceswiththesamesymboldu-ration,thesameraisecosinepulseshapedlterwitharolloff ,andthesameinputsignal-to-noiseratio(SNR),whichwouldbereceivedbyanomnidirectionalsensor,equalto5dB,areassumedtobereceivedbythearray.Thesourcesareassumedtobeweaklyseparatedinbothspaceandpolarizationandsuchthat and ,respectively.Undertheseassumptions,Figs.3and4showthevariations,asafunctionofthenumberofsnap- ,oftheRMSEforthesource2,RMSE ,andtheasso-ciatedprobabilityofnonabberantresultsforDFmethodswithandwithoutmodelingerrors,respectively.TheDFmethodscor-respondto2-MUSIC,KP-PD-2-MUSIC,UP-PD-2-MUSIC-1,UP-PD-2-MUSIC-2,4-MUSIC,KP-PD-4-MUSIC,UP-PD-4-MUSIC-1,UP-PD-4-MUSIC-2,6-MUSIC,KP-PD-6-MUSIC,UP-PD-6-MUSIC-1,andUP-PD-6-MUSIC-2algorithmsforar-rangementoftheconsideredstatisticsindexedby .Theperformancesarecomputedfrom300realizationsandsimilarresultsareobtainedforthesource1.Inthepresenceofmod-elingerrors,thesteeringvectorofthesource attheoutputofthesensorsbecomes ,wherethevectors , ,areassumedtobezero-meanstatisticallyindepen-dentcircularGaussianvectorssuchthat Forthesimulations, ,whichcorresponds,forex-ample,toaphaseerrorwithastandarddeviationof0.76 anamplitudeerrorwithastandarddeviationof0.1dB.For2-MUSIC,4-MUSIC,and6-MUSICalgorithms,thesixsensorsoftheUCAareassumedtobeidenticalwithcomplexresponses , Figs.3and4show,forsourceswhichareweaklyseparatedbothinDOAandpolarization,withandwithoutmodelinger-rorsandforagivenvalueof ,thebestbehaviorofDP- -MUSICmethodswithrespectto -MUSIConesassoonaspolarizationsofthesourcesaredifferent.Thisshowsthebetterresolutionandrobustnesstomodelingerrors,what-everthevalueof ,ofmethodsexploitingbothpolarizationandspacediversitywithrespecttomethodsexploitingspacediver-sityonly.Moreover,wenote,foragivenvalueof ,similarper-formancesofUP-PD- -MUSIC-1andUP-PD- algorithms,whichseemtodifferonlyfromacomplexitypointofview.Besides,wenoteincreasingperformanceswith ofUP-PD- -MUSICmethods,forsituationswhereresolutionisrequired.ThisisduetoanincreasingresolutioninbothDOAandpolarizationoftheassociated th-orderVAandthisshowstheinterestofexploitingbothpolarizationdiversityandHOstatisticsforDFofpoorlyseparatedsources.Notealso,fora IEEETRANSACTIONSONSIGNALPROCESSINGTABLEIV ASAUNCTIONOFALUESOFANDFORANRRAYRTHOGONALLYOLARIZEDANDOLOCATEDNIFORMLYPACEDUBARRAYS ,denedby(8),bringnoinformation.Thismeans canbewrittenas (30)where isafullrankandconstant matrixand isthenonredundant steeringvectorofasourcecomingfromDOA withpolarization fortheVAassociatedwithparameters .Expression(30)showsthatanarbitrarysteeringvector necessarilybelongstothespacespannedbythe columnsof ,noted .Noting full-rankmatrixwhosecolumnsspanthespaceorthogonalto ,wededucethatallthevectors of ,where isanarbi- vector,areorthogonalto arbitraryvaluesof .Adirectconsequenceofthisresultisthatwhateverthenumber ofstatisticallyindependentsourcessuchthat ,andwhatevertheirDOAandpolariza- ,thismeansthat columnsof arenotdiscriminant.Inotherwords,wededucefrom(30)andthepreviousresultsthatonly columnsof arediscriminant,whileexpression(9)takestheform (31)where isthe orthogonalprojector .Replacing(9)by(31)inthedevelopmentsofSectionIII,wededucethat,forgivenvaluesof , and arealsosolutionof(23)where hasbeenreplacedby nedby Asthequantity ,denedby(23)with insteadof ,hastobenulledonlyfortheDOAofthesourcesandnotforotherDOAs,the matrix hastobefullrankwhen notcorrespondtoasourcesDOA.Using(32),thismeansthatrankof cannotbelowerthan Thismeansthatthenumberofcolumnsof hastobegreaterthanorequalto .Moreover,inthepresenceof statisticallyindependentsourcessuchthat ,thenumberofcolumnsof isequalto forassociatedVAwithnoambiguitiesuptotheorder .Asaconsequence,themaximalnumberofsources thatmaybeprocessedbyUP-PD- -MUSICalgorithmsforthearrangementindexedby hasto,forsuchVAs,verify .Conversely,fora VAwithoutanyambiguitiesuptotheorder , comingfrom differentdirectionswithdifferentpolarizationsandsuchthat aresuchthattheirDOAaretheonlysolutionsof .Fromthepreviousresults,assuminga th-orderVAforthearrangementindexed with differentVSsandwithnoambiguitiesuptothe ,wededucethatUP-PD- -MUSICalgorithmsforthearrangement areabletoprocessupto sources.Thisisstrictlylowerthan(29)andthisgives for andarrayswithscalarsensors,resultalreadyobtainedin[11].NotethatforVAwithHOambiguities, hastobereplacedby in(33).V.VIRTUALRRAYTogetmoreinsightsintothegaininresolutionobtainedwithHOVAwithpolarizationdiversity,letuscomputethenormalizedinnerproductofthesteeringvectors and fortwoarbitrarycouples and Thisquantityisdenotedby andissuchthat .Usingtheresultsof[3],weobtain whichshowsanincreasingresolutionas increases.Toshowalsotheinterestofexploitingpolarizationdiversity,weconsideranarrayof sensorscomposedoftwocolocalizedandorthogonallypolarizedsubarraysof sensors.Wedenethe vector ,where and arethetwocomponentsof andwhere and arethecomplexresponsesofthetwoorthogonallypolarizedcomponentsofavectorsensortoaunitarysourcecomingfromDOA withadaptedpolariza-tions,respectively.Undertheseassumptions,itisstraightfor-wardtoshowthat(34)becomes (35)where ,suchthat ,isthenor-malizedinnerproductofthesteeringvectorsofthetwosourcesforthearray,withspacediversityonly,composedof nidirectionalsensorslocatedatthepositionsofthevectorsen-sorsoftheinitialarray.Inaddition, ,suchthat ,isthenormalizedinnerproductofthevectors and ,where correspondsto with replacing .Forelementarysensorssuchthat , assoonasthetwosourceshavedif-ferentpolarization.Forelementarysensorssuchthat , assoonasthetwosourceshaveei-therdifferentpolarizationordifferentDOA.Thisshowsanin-creasingresolutionobtainedwithanarraywithpolarizationdi-versity,atleastforsourceswithdifferentpolarization. CHEVALIERetal.:HODFFROMARRAYSWITHDIVERSELYPOLARIZEDANTENNAS Fig.2.Circulararrayofsixequispacedsensorscomposedoftwooverlappedorthogonallypolarizedsubarraysofthreesensors:(a)noncollocatedsubarraysand(b)collocatedsubarrays.TABLEI q;lASAUNCTIONOFALUESOFRRAYSPACEANDOLARIZATION samephasecenter.TablesIIandIIIshow,fornoncolocatedandcolocatedsubarrays,respectively,andfrom(27)and(28),theexpressionof asafunctionof for severalvaluesof .Notethatthisupperboundcorrespondsto inmostcasesofarraygeometrywithnoparticularsym-metry,whichis,inparticular,thecaseforuniformcirculararray(UCA)of vectorialsensorswithtwocomponents,when isaprimenumber,asdepictedinFig.2(b)for .How-ever,whilefourth-orderVAassociatedwithnoncolocatedsub-arrayscontainsnovectorsensorfor ,itcontainsseveralscalarsensorsandonevectorsensorwithtwocompo-nentsfor .Besides,thefourth-orderVAassoci-atedwithcolocatedsubarrayscontainsonlyvectorsensorswithtwoorthreecomponentsfor andwithtwoorfourcomponentsfor .Asaconsequence,fornon-colocatedsubarrays,(29)holdsfor andprob-ablyfor butmaynotholdfor andcolo-catedsubarrays,butthispotentialresulthastobeveried.Fi-TABLEII q;lASAUNCTIONOFALUESOFANDFORRRAYSRTHOGONALLYONCOLOCATEDUBARRAYS TABLEIII q;lASAUNCTIONOFANDFORRRAYSRTHOGONALLYOLLOCATEDUBARRAYS nally,TableIVshowstheexpressionof asafunctionof for andseveralvaluesof ,foranarraycom-posedoftwocolocalizedandorthogonallypolarizeduniformlyspacedlineararray(ULA)of identicalsensors.Notethat,inthiscase,fourth-orderVAcontains vectorsensorswiththreeandfourcomponentsfor and ,respectively.B.UP-PD- -MUSICAlgorithmsThedevelopmentsofSectionIV-Aarestillvalidfor -MUSICalgorithms.Inparticular,themaximalnumberofstatisticallyindependentsourcesthatmaybepro-cessedbyUP-PD- -MUSICalgorithmsforthearrangementindexedby cannotexceeds iftheassociatedVAhasnoambiguitiesuptotheorder .Moreover,wededucefromtheHOVAtheory[3]that componentsof IEEETRANSACTIONSONSIGNALPROCESSINGDOAof statisticallyindependentsourcesfromanarrayof sensorsprovidedthathypothesesH1)H4)areveriedandtheDOAandthepolarizationofthesourcesaretheonlysolutionsof(9).Ithasbeenshownin[3]that,intheabsenceofcouplingbetweensensors,thevector canbeconsideredasatruesteeringvectorbutforaHOVAof virtualsensors(VSs)withcoordinates andcomplexamplitudepatterns , for ,givenby (27) Assomeofthese VSsmaycoincide,wenote , ,thenumberofdifferentVSsoftheassociatedVA.Inthese componentsofallthevectors areredundantcomponentsthatbringnoinformation.Asacon-sequence,therankof cannotbegreaterthan .Wethendeducethatthe matrixmayhavearankequalto onlyif .Conversely,fora th-orderVAforthearrange- withoutanyambiguitiesuptoorder , comingfrom differentdirectionswithdifferentpolarizationsgeneratean matrixwithafullrank aslongas Letusrecallthatthe th-orderVAforthearrangementin-dexedby hasnoambiguitiesoforder ifanysetof vectors withdistinctparameters , ,arelinearlyindependent.Thus,providedthe th-orderVAforthearrangementindexedby hasnoambigu-itiesuptotheorder ,themaximalnumberofstatisti-callyindependentsourcesabletogenerateamatrix withrank is .However,when ,anarbitraryvector associatedwithanarbitrarysetofparameters isnecessarilyalinearcombinationofthesourcesteeringvec- , ,sincematrix cannothavearankgreaterthan .Then,allthesetsofparameters aresolutionsof(9),whichdoesnotallowthesourcesDOAestimation.Thus,anecessaryconditionfortheDOAsandpo-larizationsofthesourcestobetheonlysolutionsof(9)isthat andthisconditionbecomessufcientforHOVAwithnoambiguitiesuptotheorder .Fromthepreviousre-sults,assuminga th-orderVAforthearrangementindexed with differentVSsandwithnoambiguitiesuptoorder ,wededucethattheKP-PD- -MUSICalgorithmforthearrangementindexedby isabletoprocessupto sources.As,foragivenarrayof sensors, isafunctionof and [3];theprocessingcapacityoftheKP-PD- algorithmisalsoafunctionof and .Thisshowsoff,inpartic-ular,theexistenceofanoptimalarrangementofthe datastatisticsforagivenvalueof ,whichisdiscussedin[3].Notethattheproblemof th-orderambiguitiesofHOVAisanimportantopenproblemwhichdeservestobeanalyzedinde-tailbutwhichisbeyondthescopeofthispaper.ForaVAassoci-atedwiththeparameters andwithoutcolocalizedorvectorsensors,i.e.,withscalarsensorsonly,allthe differentVSsbringinformation.Then,despitethepotentialexistenceofHOambiguitiesoftheVAforsomesourcesgurations,therealwaysexist,ingeneral,somesourcescongurationsforwhich hasarankequalto andforwhichKP-PD- -MUSICalgorithmforthearrangementindexedby isableto sources.However,foraVAassociatedwiththe suchthatsomeofthe differentsensorsarecolocalized,someofthecolocalizedsensorsmaybringnoin-formationasitisshownanddiscussed,for ,in[17][17][27],[33][35]forelectricandelectromagneticvectorsensors,respectively.Inthiscase, matrixmaynothaveamaximalrankequalto andtheKP-PD- -MUSICalgorithmforthearrangementindexedby mayonlybeabletoprocess sources,atleastinsomesituations,where isthemaximalpossiblerankof .Toquantifyexpressionof asafunc-tionof ,somevaluesof arepresentedinSectionsIV-A2andIV-A3fordifferentarraysandsensors2)CaseofanArrayWithDifferentSensors:Forgivenvalues , ,and andforarraysof sensorswithbothspaceandpolarizationdiversities,ithasbeenshownin[3]that necessaryupperboundedbyaquantity,noted ,such .TableIshows,forageneralarraywithspaceandpolarizationdiversitieshavingsensorsarbitrarylo-catedwithdifferentresponses,theexpressionof afunctionof for andseveralvaluesof .Thisupperboundcorrespondsto inmostcasesofsensorssponsesandarraygeometry.Moreover,[3,Tables1and3]showthatfor ,or andforarrayswithnoparticularsymmetry,theassociatedVAhasnovectorsensors.Inthesecases,expression(29)generallyholds.Onthecontrary,[3,Tables1and3]showthatfor ,or ,theassociatedVAhasatleastonevectorsensorwithatleast differentcomponents.Inthesecases, maybestrictlylowerthan andexpression(29)maynothold,espe-ciallyforhighvaluesof 3)CaseofanArrayWithTwoSubarraysofSensorsHavingOrthogonalPolarizations:Aparticularcaseofpracticalinterestcorrespondstothecaseofanarrayof sensorscom-posedoftwosubarraysof sensorshavingorthogonalpolar-izations.Twokindsofsucharraysareconsideredinthissectionandcorrespondtoarraysforwhichthesensorsofthetwosub-arraysareeithercolocatedornot.Examplesofnoncolocatedandcolocatedsubarraysof sensorsarepresentedinFig.2for .Fig.2(a)showsacirculararrayofsixequispacedscalarsensorscomposedoftwooverlappedor-thogonallypolarizedcircularsubarraysofthreescalarsensorssuchthattwoadjacentsensorsofthearrayhavedifferentpo-larizations.Fig.2(b)showsacirculararrayofthreeequispacedandidenticalvectorialsensorssuchthateachvectorialsensoriscomposedoftwoorthogonallypolarizedsensorshavingthe CHEVALIERetal.:HODFFROMARRAYSWITHDIVERSELYPOLARIZEDANTENNASInsomecases,thenumberofsources isknown,suchthat ,buttheirstatisticaldependenceisnotknown.Insuchacase, andaconservativeapproachmaybetouseonlythe eigenvectorsassociatedwiththesmallesteigenvaluestobuilt ,whichimplicitlyassumesthestatisticaldependenceofallthesourcesandthefullrankof fortheassociatedgroup.Finally,notethatsimilarly -MUSICalgorithm,PD- -MUSICalgorithmscannothandleperfectlycoherentsources.2)CaseofSourcesWithUnknownPolarization(UP-PD- -MUSICAlgorithms):Forsourceswithunknownpolariza-tion,thecomplexityofthesearchingprocedure,describedinSectionIII-C1,ofthePD- -MUSICalgorithmwithrespecttoDOAandpolarizationparametersisdramaticallyhigh.Asimplewaytoremovethesearchingprocedurewithrespecttothepolarizationparameterconsists,foranyxedDOA,ofmin-imizingtheleft-handsideof(17)withrespecttopolarizationparameter,asitisproposedin[11]for .ThisgivesrisetotheunknownpolarizationPD- -MUSIC(UP-PD- algorithmwhosepseudospectrum,forthearrangementindexed ,isgivenby Itiswellknown[11]thattheright-handsideof(22)correspondstotheminimumeigenvalue ofthe matrix inthemetric thattheminimizingvector ,noted ,correspondstotheassociatedeigenvector.Inotherwords, and satisfythefollowing: Thus,arstversionoftheUP-PD- -MUSICalgorithmforthearrangementindexedby ,calledUP-PD- -MUSIC consistsofndingthe setsofparameters , ,forwhichthepseudospectrum iszero.Thisalgorithmcorrespondstoa th-orderextension,forthearrangementindexedby ,ofthealgorithmproposedin[11]for .Then,itisshownintheAppendixthatitbecomespossibletoestimatethepolarizationofeachsource fromtheassociatedeigenvector whichissolutionof(23)for .Notethatonewayinwhichtheeigenvalue canbecomputedisbydeterminingtheminimumrootofthefollowing: (25)where meansdeterminantof .Thus,foreachvalueof ,searchinginpolarizationspacehasbeenavoidedbytherootsofanequationoforder ,whichcor-respondstoasubstantialreductionincomputation,atleastforsmallvaluesof .Wededucefrom(25)and[12]thatforinvert-iblematrix ,nding suchthat iszeroisequivalentto suchthat .Asecondversionofthe -MUSICalgorithmforthearrangementindexedby ,calledUP-PD- -MUSIC -2,thenconsistsofndingthe setsofparameters , ,forwhichthe iszero,whichallowsacomplexitydecreasewithrespecttothecomputationof(24).Inpracticalsituations,matrix hastobeestimatedfromtheobservationsand,assumingsourceswithunknownpo-larization,theDOAofthesourcesmaybefoundbysearchingfortheminimaoftheright-handsideof(24)or(26).Thedif-ferentstepsofthetwoversionsoftheUP-PD- -MUSICalgo-rithmforthearrangement aresummarizedasfollows.1)Estimation ofthematrix from snap-shots , ,usingasuitableestimatorofthe th-ordercumulantsofobservations.2)Eigenvaluedecompositionofthematrix andex-tractionofanestimate ofthe matrix.Thisstepmayinvolverankdeterminationincaseswherethenumberofsourcesand/ortheirmutualstatisticaldepen-dencearenotknownapriori3)Computationofmatrices , ,andoneofthetwoestimatedpseudospectra overasuitablychosengrid.Then,searchforthelocalminimaof or (includinginterpolationateachlocalminimum),where isthemin-imumeigenvalueof inthemetric 4)Ifneeded,computationofboththeassociatedestimatedvectors andthepolarizationvectoroftheIV.IAlthoughalgorithmspresentedinSectionIIImaybeusedinthepresenceofcouplingbetweensensors,providedthatthe isknownorcanbeestimatedbycalibration,theabilityanalysispresentedinthissectionassumestheab-senceofcouplingbetweensensors.Moreover,asthemaximalnumberofsourcesthatcanbeprocessedbyagivenversionofthePD- -MUSICalgorithmisobtainedwhenallthesourcesarestatisticallyindependent,welimittheidentiabilityanal-ysisofthethreealgorithmsintroducedinSectionIIItothelatterA.KP-PD- -MUSICAlgorithm1)GeneralResults:FollowingthedevelopmentsofSectionIII,wededucethattheKP-PD- -MUSICalgo-rithmforthearrangementindexedby isabletoestimatethe IEEETRANSACTIONSONSIGNALPROCESSING .Equation(9)correspondstotheheartofthePD- -MUSICalgorithmsforthearrange- andcanalsobewritten,using(3)and(8),as (10)where and arethe vectorand matrixdenedby (11) respectively.Forsomevaluesof ,somecomponentsof areequalanditmaybeusefultoatleastreducethecom-plexityofthecomputationoftheleft-handsideof(10),butalsotoimprovetheperformanceofthealgorithmspresentedinSectionIII-C2,toremovetheredundantcomponentsof Thiscanbedonebyremovingtheredundantcomponentsof and .Itisstraightforwardtoshowthat canbewrittenas (13)where isthe realmatrixsuchthat (14a) (14b)where isthe identitymatrix, and arethe vectorsdenedby and , isthe zerovector,and isthe vectorwith nedby (15)where and arethecomponentsofthepolarizationvector .From(11)and(13),wededucethat (16)where isa vector.Noteadimensionreductionof withrespectto formostvaluesof .Toensure,intheabsenceofsources,i.e.,when ,aconstantvalue,independentofparameters and ,oftheleft-handsideof(10),itisnecessarytonormalizethelatterbythequantity .Using(16)into(10),theproblemofsourcesDOAestimationbythe -MUSICalgorithmforthearrangement thenconsistsndingthe setsofparameters , ,whicharesolutionsof(17)orwhichminimizetheleft-handsideofthelatter,denedby wherethe matrices and aredenedby (18) Forsourceswithknownpolarization,thesetofparametersforagivensourcereducestothesetofitsDOAandthecomplexityofthePD- -MUSICalgorithm,calledinthiscaseknownpolarizationPD- -MUSIC(KP-PD- -MUSIC)algorithm,correspondstothatofthe -MUSICalgorithm.However,forsourceswithunknownpolarization,thesetofparametersforagivensourcehastotakeintoaccountpolarizationparametersinadditiontoDOAparametersandthecomplexityofthesearchingprocedureofthePD- -MUSICalgorithmdramati-callyincreasesbeyondwhatisgenerallypracticallyreasonable.Forthisreason,ourchoiceinthispaperistolimittheuseofthepreviousalgorithmtothecasewheresourcespolarizationisknown.Otherwise,weconsideralternativealgorithmswhichdonotrequirethesearchingprocedurewithrespecttothepolariza-tionparametersandwhicharepresentedinSectionIII-C2.Notethatforunknownpolarizations,despitethefactthatitisnotourchoiceinthispaper,solutionsof(17)mayalsobefoundfromasearchingprocedureinbothpolarizationandDOAparameters.Removingtheredundancyof by(16)thenallowsinthiscasetodecreasethecomplexityofthesearchingprocedure.Inpracticalsituations,matrices and havetobeestimatedfromtheobservationsandassumingsourceswithknownpolarization,theDOAofthesourcesmaybefoundbysearchingfortheminimaoftheestimatedleft-handsideof(17).ThedifferentstepsoftheKP-PD- -MUSICalgorithmforthe aresummarizedasfollows.1)Estimation ofthematrix from snap-shots , ,usingasuitableestimatorofthe th-ordercumulantsofobservations.2)Eigenvaluedecompositionofthematrix andex-tractionofanestimate ofthe matrix.Thisstepmayinvolverankdeterminationincaseswherethenumberofsourcesand/ortheirmutualstatisticaldepen-dencearenotknownapriori3)Computation,foreachknownvector , oftheestimatedpseudospectrum overasuitablychosengrid.Then,searchforthelocalminima(includinginterpolationateachlocalminimum),wherethe matrix isdenedby (21)where . CHEVALIERetal.:HODFFROMARRAYSWITHDIVERSELYPOLARIZEDANTENNAS .Vector istheunitnorm vectorwithcompo- and .Thisvectorcanbewritten,towithinaphaseterm,as where and aretwoanglescharacterizingthepolarizationofsource andsuchthat and .Notethatforanarraywithspacediversityonly, , ,and arecolinear,whichmeansthat,towithinaconstant, doesnotde-pendonthepolarizationofthesource B.StatisticsoftheData1)Presentation: th-order, ,DFmethodsconsideredinthispaperexploittheinformationcontainedinthe th-ordercovariancematrix whoseentriesarethe th-ordercumulantsofthedata , , ,where correspondstothecomplexconjugation.However,thepreviousentriescanbearrangedinthe matrixindifferentways,indexedbyaninteger suchthat ,asitisexplainedin[6].Thisgivesrise,underhypothesesofSectionII-A,tothe matrixgivenby[6] (4)where isthemeanpowerofthenoisepersensor, isthe spatialcoherencematrixofthenoiseforthearrangementindexedby ,suchthat , meanstrace,and istheKroneckersymbol.The matrix containsthe th-ordercumulantsof forthearrangementindexedby andcanbewrittenas (5)where isthe matrixofthe th-ordercumu-lantsof forthearrangementindexedby , totheconjugatetransposition, istheKroneckerproduct,and isthe matrixdenedby withanumberofKroneckerproductequalto .Notethatitisshownin[3]andveriedinthispaperthattheparam- determines,inparticular,themaximalprocessingpowerofPD- -MUSICalgorithms.2)Estimation:Insituationsofpracticalin-terests,the th-orderstatisticsofthedata arenotknownaprioriandhavetobeestimatedfrom samplesof , ,where isthesampleperiod,inawaythatiscompletelydescribedin[6]andwhichisnotrecalledhere.III.PD-2q-MUSICAInthissection,weanalyzethepropertiesofmatrix andwededucefromthelatterthreeversions,dependingontheaprioriinformationaboutthepolarizationofthesources,ofthe -MUSICalgorithmforthearrangementindexedby A.HypothesesTodevelopthePD- -MUSICalgorithmsforthearrange-mentindexedby ,wehavethefollowinghypotheses: , H2)matrix hasfullrank , ;H3) H4)matrix hasfullrank B.Propertiesof Althoughcomponentsof arestatisticallydependent, matrix ,whichcontainsthe cumulantsof forthearrangementindexedby ,maynotbefullrankforsomecouples .Thisresultwasunknownbeforethepublicationof[6].Indeed,assuming,forexample, ,itiseasytoverifythatthemaximalrank is3(andnot4)for and6(andnot9)for .Inthiscontext,noting ,therankof , ,wededucefromH1)andH2)thatmatrix for hasalsorank .Hence,usingH4)andfor ,matrix hasarank equalto andsuchthat fromH3).Asmatrix isHermitian,wededucethat has real-valuednonzeroeigenvaluesand zeroeigenvaluesfor C.PD- -MUSICAlgorithms1)CaseofSourcesWithKnownPolarization(KP-PD- -MUSICAlgorithm):TobuiltaMUSIC-likealgorithmfrom ,for ,werstcomputetheeigendecom-positionofthelatter,givenby (7)where isthe diagonalmatrixofthenonzeroeigenvaluesof , isthe unitarymatrixoftheassociatedeigenvec- isthe diagonalmatrixofthezeroeigenvaluesof ,and isthe unitarymatrixoftheassociatedeigenvectors.As isHermitian,allthecolumnsof areorthogonaltoallthecolumnsof .Moreover, whenmatrices , ,arefullrankwhereas ,otherwise.Wedenethe vector by Then,noting ,theDOAandpolarizationparam-etersofthe thsourceinthe thgroup,itcanbeeasilyveriedthat,inallcases,thevector alwaysbelongsto .Consequently,allvectors areorthogonaltothecolumnsof andaresolutionsofthefollowing: (9) IEEETRANSACTIONSONSIGNALPROCESSINGispresentedin[7].Nevertheless,HODFmethodsavailableforarrayswithdiverselypolarizedsensorsareveryscarce,amongwhichwendanFOESPRIT-likealgorithmdevelopedforveryspecicarraycongurations[16].Inthiscontext,inordertoincreasetheperformanceofthe -MUSICalgorithminthepresenceofsourceshavingdifferentpolarizations,thepurposeofthispaperistointroduce,forarbitraryvaluesof threeextensionsofthe -MUSICmethodabletoputupwitharrayshavingdiverselypolarizedsensors.Thisgivesrisetothepolarizationdiversity -MUSIC(PD- -MUSIC)algorithms.Foragivenvalueof ,thesealgorithmsareshowninthispapertoincreasetheresolution,therobustnesstomodelingerrors(atleastforpoorlyangularlyseparatedsources),andtheprocessingcapacityofthe -MUSICmethodinthepresenceofdiverselypolarizedsources.Besides,despiteahighervarianceofHODFmethods,somePD- -MUSICalgorithmsareshowninthispapertoofferincreasingperformanceswith whenresolutioninDOAandpolarizationisrequired.Afteranintroductionofsomenotations,hypotheses,anddatastatisticsinSectionII,threeversionsofthePD- methodarepresentedinSectionIIIforparticulararrangementsofthe th-orderdatastatisticsina th-orderstatisticalmatrix.abilityissuesforseveralkindsofarrayconareaddressedinSectionIV.Considerationsaboutresolutionof -MUSICmethodsareinvestigatedinSectionV.SomesimulationsaboutthebehaviorofPD- -MUSICalgorithmsforbothoverdeterminedandunderdeterminedmixturesofsourcesarepresentedinSectionVI,showingoff,inparticular,thegreatinterestofPD- -MUSICmethodsfor .Finally,SectionVIIconcludesthispaper.Notethatthecontentofthispaperhasbeenpatentedin[5].II.HYPOTHESESOTATIONSTATISTICSOFTHEATAA.HypothesesandNotationsWeconsideranarrayof narrowband(NB)potentiallydif-ferentsensorsandwecall thevectorofcomplexampli-tudesofthesignalsattheoutputofthesesensors.Eachsensorisassumedtoreceivethecontributionof zero-meanstationaryNBsources,whichmaybestatisticallyindependentornot,cor-ruptedbyanoise.Weassumethatthe sourcescanbedivided groups,with sourcesinthegroup ,suchthatthesourcesineachgroupareassumedtobestatisticallydependent,butnotperfectlycoherent,whilesourcesbelongingtodifferentgroupsareassumedtobestatisticallyindependent.Ofcourse, isthesumofalltheparameters , .Undertheseassumptions,theobservationvectorcanbewrittenasfollows: (1)where isthenoisevector,assumedzero-mean,stationary,andGaussian,whereas ,independentof ,isthecom-plexenvelopeofthesource .Couple nesthe andelevationangle ofsource (Fig.1).Vector isa vectorcharacterizingthestateofpolarizationof Fig.1.Incomingsignalin3-D. andwhosecomponentswillbedenedhereafter.Vector , ,isthesteeringvectorofthesource whichcontains,inparticular,theinformationabouttheDOAandthepolarizationofthelatterjointlywiththecharacteristicsofthesensorsandarray.Matrix isthe matrixcon-tainingthesteeringvectorsofthesourcesbelongingtothe groupofsources,whereas isthecorresponding vectorofcomplexenvelopesand .Methodsdevelopedinthispapermaybeimplementedinthepresenceofcouplingbetweensensors.Nevertheless,intheabsenceofmu-tualcouplingbetweensensors,assumingaplane-wavepropaga-tion,component ofvector ,denoted ,canbewritten,inthegeneralcaseofanarraywithspaceandpolar-izationdiversity,as[9] (2)where isthewavelength, and arethecoordinatesofsensor ofthearray,and isacomplexnumbercorrespondingtotheresponseofsensor toaunitelectriccomingfromthedirection andhavingthestateofpolarization [9].Letusrecallthatanarrayofsensorshasspacediversityifallthesensorsdonothavethesamephasecenter.Thearrayhaspolarizationdiversityifallthesensorsdonothavethesame and betwodistinctpolarizationsforthesource (forexample,verticalandhorizontal)and and bethecorrespondingsteeringvectorsforDOA .Weassumethatthevectors and becalculatedanalyticallyormeasuredbycalibrationwhateverthevalueof .Consideringanarbitrarypolarization forthe ,thecomplexelectriceldofthelattercanbebrokendownintothesumoftwocomplexelds,eacharrivingfromthesamedirection,andhavingthepolarizations and Thesteeringvector ofthesource isthentheweightedsumofthesteeringvectors and givenby (3)where isthe matrixofvectors and ,whereas and arecomplexnumberssuchthat -MUSICmethodswith mayofferbetterperformancesthan2-MUSICor4-MUSICmethodswhensomeresolutionisrequired.Thisis,inparticular,thecaseinthepresenceofseveralsources,whenthelatterarepoorlyangularlyseparatedorinthepresenceofmodelingerrorsinherentinoperationalcontexts.However,both4-MUSIC[28]and -MUSIC, algorithmshavebeenmainlydevelopedforarrayswithiden-ticalsensors,andcannotputup,inthepresenceofarbitrarypolarizedsources,witharraysofdiverselypolarizedsensors.Theexploitationofarrayswithdiverselypolarizedsensorsis , .Thisgivesrisetotheso-called -MUSICmethod,whichexploitstheinformationcontainedinthe th-ordersta-tisticsoftheobservations.Thismethodisshownin[6]tohaveresolution,robustnesstomodelingerrors(forseveralpoorlyan-gularlyseparatedsources),andprocessingcapacityincreasing .Theseresultsaredirectlyrelatedtothehigherorder -MUSIC(PD-2 -MUSIC)algorithms.Foragivenvalueof ,thesealgo-rithmsareshowntoincreasetheresolution,therobustnesstomodelingerrors,andtheprocessingcapacityofthe methodinthepresenceofdiverselypolarizedsources.Besides,somePD- -MUSICalgorithmsareshowntoofferincreasingperformanceswith whenresolutioninbothdirectionofarrivalandpolarizationisrequired.IndexTerms— ,arethemostpopular.Thesemethodsareasymp-toticallyrobusttoaGaussianbackgroundnoisewhosespatialcoherenceisunknownandofferincreasingresolutionandro-bustnesstomodelingerrorsjointlywithanincreasingprocessingcapacityas increases.However,thesemethodshavebeenmainlydevelopedforarrayswithidenticalsensorsonlyandcannotputupwitharraysofdiverselypolarizedsensorsinthepresenceofdiverselypolarizedsources.Inthiscontext,thepurposeofthispaperistointroduce,forarbitraryvaluesof , ,threeextensionsofthe -MUSICmethod,abletoputupwitharrays IEEETRANSACTIONSONSIGNALPROCESSINGHigherOrderDirectionFindingFromArraysWithDiverselyPolarizedAntennas:ThePD-2q-MUSICAlgorithmsPascalChevalier,AnneFerréol,LaurentAlbera,andGwénaëlBirotFourth-order(FO)and,ashortwhileago, th-order, ,high-resolutionmethodsexploitingtheinformationcon-tainedintheFOandthe th-order, ,statisticsofthedata,respectively,arenowavailablefordirectionÞndingofnon-Gaussiansignals.Amongthesemethods,the -MUSIC