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1、720IEEETRANSACTIONSONACOUSTICS.SPEECH.ANDSIGNALPROCESSING.VOL17.NO5.hlAYIYKYMUSIC,MaximumLikelihood,andCramer-RaoBoundAbstract-TheproblemoffindingthedirectionsofmultipleplaneTherearethreemainproblemsassociatedwithfittingwaveswithnarrow-bandarraysofsensors,andtherelatedproblemofmodelsoftheform(1.1)to
2、thedata{y(l),*.,y(N)}.estimatingtheparametersofmultiplesuperimposedexponentialsig-a)Estimationofthenumberofsignalsn.Methodsfornalsinnoise,haveattractedconsiderableinterestrecently.Severalmethods,suchastheMUSICandmaximumlikelihood(ML),havebeenestimatingnarewelldocumentedintheliterature(see,proposedfo
3、rsolvingtheseproblems.e.g.,[31,[41,[21],and[22])andwillnotbediscussedThispaperstudiestheperformanceoftheMUSICandMLmethods,here.Inthispaperweassumethatthenumberofsignalsandanalyzestheirstatisticalefficiency.ItalsoderivestheCramer-Raonisgiven.bound(CRB)fortheestimationproblemsmentionedabove,andestab-b
4、)Estimationofthesignalamplitudes{~(r)}.OncelishessomeusefulpropertiesoftheCRBcovariancematrix.There-lationshipbetweentheMUSICandMLestimatorsisinvestigatedasanestimateof8isavailable,theestimationofx(t)re-well.Finally,thepapercontainsanumericalstudyofthestatisticalduces(underreasonableconditions)toasi
5、mpleleast-efficiencyoftheMUSICestimatorfortheproblemoffindingthedi-squaresfit.Wewillomitanyexplicitdiscussionontherectionsoftwoplanewavesusingauniformlineararray.problemofestimating{x(t)}.However,estimatesofAmoreexactdescriptionoftheresultsofthispapercanbefound{x(t)}willimplicitlyappearintheanalysis
6、thatfollows.intheconclusionsection.Notethatsinceitisrequiredtoestimate{x(t)};=(andnottheir''averagecharacteristics,"suchastheircovari-1.INTRODUCTIONANDPRELIMINARIESancematrix),wewillconsiderthesevariablestobede-EVERALimportantproblemsinthesignalprocessingterministic(i.e.,fixed).Thisassumptiondoesnot
7、excludeSfield,amongthemdirectionfindingwithnarrow-bandthepossibilitythatx(1),**.,x(N)aresamplesfromasensorarrays,estimationoftheparametersofmultiplesu-randomprocess.Ifso,thenthedistributionalresultsde
