IEEEJOURNALONSELECTEDAREASINCOMMUNICATIONS,VOL.17,NO.11,NOVEMBER19991829AngleandTimeofArrivalStatisticsforCircularandEllipticalScatteringModelsRichardB.ErtelandJeffreyH.Reed,SeniorMember,IEEEAbstract—Withtheintroductionofantennaarraysystemsintowirelesscommunicationnetworkscomestheneedtobetterunderstandthespatialcharacteristicsofthechannel.Scatteringmodelsprovidebothangleofarrival(AOA)andtimeofarrival(TOA)statisticsofthechannel.Anumberofdifferentscatteringmodelshavebeenproposedintheliteratureincludingellipticalandcircularmodels.Thesemodelsassumethatscatterersliewithinanellipticalandcircularregioninspace,respectively.Inthispaper,thejointTOA/AOA,themarginalTOA,andthemarginalAOAprobabilitydensityfunctions(pdf’s)arederivedfortheellipticalandcircularscatteringmodels.Thesepdf’sprovideinsightintothepropertiesofthespatialwirelesschannel.IndexTerms—Antennaarrays,propagation.Fig.1.Scatterergeometry.I.INTRODUCTIONderivedfortheellipticalandcircularscatteringmodels.TheORbeamformingandemitterlocalizationapplicationsinpdf’sfortheellipticalmodelwerepreviouslyderivedin[7].Fwhichanarrayofsensorsisusedtoreceivecommuni-Hereamoregeneralapproachisusedthatallowsthepdf’scationsignals,itisnecessarytohavechannelmodelsthatforbothmodelstobederivedusingacommonapproach.Theprovideangleofarrival(AOA)andtimeofarrival(TOA)jointTOA/AOApdfandthemarginalTOApdfforthecircularinformationaboutthemultipathcomponents.Tomeetthismodelareoriginalcontributionsofthispaper.challenge,geometricallybasedsinglebouncechannelmodelshavebeenproposed.ThesemodelsassumethatscatterersareII.PROBLEMGEOMETRYANDASSUMPTIONSrandomlylocatedintwo-dimensionalspaceaccordingtoaFig.1showsthegeometryandnotationusedtoderivethejointprobabilitydensityfunction(pdf).Forconvenience,thisTOAandAOApdf’softhescatteringmodels.Themobileisjointpdfwillbereferredtoasthescattererdensityfunctionseparatedfromthebasestationbythedistance.Althoughorjustscattererdensity.Fromtheassumedscattererdensity,onlyonescattererisshowninthe