large dimensional random matrix theory for signal detection and estimation in array processing

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1、LARGEDIMENSIONALRANDOMMATRIXTHEORYFORSIGNALDETECTIONANDESTIMATIONINARRAYPROCESSINGJ.W.SilversteinyandP.L.CombetteszyDepartmentofMathematics,NorthCarolinaStateUniversity,Raleigh,NC27695,USA.zDepartmentofElectricalEngineering,CityCollegeandGraduateSchool,CityUniversityofNewYork,Ne

2、wYork,NY10031,USA.ABSTRACTsensorsaremodeledasobservationsoftherandomvec-torX(t)=AS(t)+N(t),t2[0;+1[,whereAisaInthispaper,webringintoplayelementsofthespec-pqcomplexmatrixdependingonthegeometryoftraltheoryoflargedimensionalrandommatricesandthearrayandtheparametersofthesignals,andi

3、sdemonstratetheirrelevancetosourcedetectionandassumedtohaverankq.bearingestimationinproblemswithsizablearrays.Thedetectionproblemistoestimateqfromtheob-Theseresultsareappliedtothesamplespatialcovari-servationofnsnapshots(X(ti))1inofthedatapro-ancematrix,Rb,ofthesenseddata.Itiss

4、eenthatde-cess.Undertheaboveassumptions,therandomvec-tectioncanbeachievedwithasamplesizeconsiderablytors(X(t))t2[0;+1[arei.d.withspatialcovariancema-lessthanthatrequiredbyconventionalapproaches.AstrixR=EX(0)X(0)=ARA+2I,whereIde-Sppregardstodeterminingthedirectionsofarrivals,it

5、isnotestheppidentitymatrix.Moreover,thep−qarguedthatmoreaccurateestimatescanbeobtainedsmallesteigenvaluesofRareequalto2.Theseeigen-byconstrainingRbtobeconsistentwithvariousapri-valuesarereferredtoasthenoiseeigenvaluesandtheoriconstraints,includingthosearisingfromlargedi-remaind

6、erofthespectrumisreferredtoasthesignalmensionalrandommatrixtheory.Asettheoreticfor-eigenvalues.SinceRisnotknownitsspectrummustmalismisusedtoformulatethisfeasibilityproblem.beinferredfromobservingthatofthesamplecovari-PnUnsolvedissuesarediscussed.ancematrixRb=(1=n)i=1X(ti)X(ti).L

7、ooselyspeaking,onemustthendecidewheretheobservedPROBLEMSTATEMENTspectrumsplitsintonoiseandsignaleigenvalues.TheestimationproblemistodeterminethedirectionWeconsidertheproblemofdetectingthenumberqofarrivals(i)1ipofthesources.Understandardofsourcesimpingingonanarrayofp(q

8、potheses,thisproblemcanbesolvedviatheMUS