2REVIEWOFFREQUENCYESTIMATORS2.1MaximumLikelihoodEstimatorRife&Boorstyn - PDF document

2REVIEWOFFREQUENCYESTIMATORS2.1MaximumLikelihoodEstimatorRife&Boorstyn[1]havederivedthemaximumlikelihoodestimatorforthesignalmodelgivenby(1).Itisthefrequencyatwhichtheperiodogramofthesamplesismaximised,or^!MLE=argmaxN�1Xn=0xne�jn2 (3) where^!MLEisthemaximumlikelihoodestimate.Rife&BoorstynproposeanumericalmethodsimilartotheMLE,whichinvolvesacoarseandanesearch.ThecoarseestimateisobtainedbychoosingthefrequencyhavinggreatestmagnitudeintheFastFourierTransform(FFT).Anerestimateisobtainedusingamethodsuchasthesecantmethod.ThecomputationoftheFFTrequiresO(NlogN)steps,andthustheestimatorisnotcomputationallyecient.However,itisstatistically(andthereforeSNR)ecientwheneitherNorrislarge.2.2Kay'sWindowEstimatorFollowingfromtheworkofTretter[2],Kayformulatedanestimatorwhichuseslinearregressioninanindirectfashiontoestimatefrequency[3].ToperformlinearregressionaccordingtoTretter'smethod,itisnecessarytomeasurethephase,n,ofeachobservation,xn.Thephasestaketheformn=[n!+0+un]2; (4) wheretheun=\e�j(!n+0)znarethephasenoiseand\denotesthecomplexargument.Further,inordertoperformlinearregressiononthephasesdirectly(asTretterproposes),itisnecessarytoperform\phaseunwrapping"|theprocessofattemptingtoundothemodulooperationof(4).Toavoidphaseunwrapping,Kayproposesworkingonlywithphasedierences,
n,where
n=\xnx?n�1=[!+un�un�1]2.Thesolutiontothegeneralisedmaximumlikelihoodproblemofthecorrelated
nisaweightingofthe
nsothattheestimatoris^!KW=N�1Xn=1wn
n (5) wherewn=6n(N�n)=N�N2�1
aretheweightsofKay'swindow.1Asrevealedin[4],Kay'swindowestimatorisbiassed.Thebiasiscausedbythebranchpointoftheargument.Ifthebranchpointischosensothattheargumentfallswithintherange[;2+),thentheestimatorwillexhibitalargebias(andhencealargemeansquareerror)when!iscloseto.Undercertainconditions,suchasthoseusedinSection4,Kay'swindowestimatorexhibitsverypoorperformance.Itiscomputationally,butnotstatistically(orSNR),ecient.2.3OtherEcientEstimatorsAnumberofecientestimatorshaveappearedintheliterature.AmongstthesearetheLank-Reed-Pollonestimator[5],Kay'scircularestimator(referredtobyKayasthe\weightedlinearpredictor")[3],andtheParabolicSmoothedCentralFiniteDierenceEstimator(PSCFD)[6].Thesehavetheforms^!LRP=\N�1Xn=1xnx?n�1; (6) ^!KC=\N�1Xn=1wnxnx?n�1and (7) ^!PSCFD=\N�1Xn=1wnxnx?n�1 jxnj
jxn�1j; (8) respectively.Ofthese,allarecomputationallyecientandallareSNRecientexcepttheLank-Reed-Pollonestimator.TheLank-Reed-Pollonestimatorcanbeshowntohaveameansquareerroroftheform2^!LRP=1=r(N�1)2. 1Kayreferstothisestimatorasthe\weightedphaseaverager",butitisreferredtoheresimplyas\Kay'swindowestimator." Figure1:ComparisonofKay'swindowestima-tortootherfrequencyestimators,showingitsine-ciencywhen!U(�;),0U(�;),N=24. Figure2:RatioofmeansquareerrortoCramer-RaolowerboundforthePSCFDestimatorindeci-bels. Figure3:RatioofmeansquareerrortoCramer-RaolowerboundfortheimprovedKay'swindowestimatorindecibels. Figure4:10dBthresholdsoftheratioofmeansquareerrortoCramer-Raolowerbound. 6ACKNOWLEDGEMENTSIwouldliketothankJohnKitchen,BarryQuinnandDougGrayforthemanyfruitfuldiscussionsIhavehadwiththemconcerningthispaper.IamalsogratefultoCiTR,UniversityofQueensland,fortheirgenerousdonationofprocessortimeontheirMasParcomputer.References [1] D.C.RifeandR.R.Boorstyn,\Single-toneparameterestimationfromdiscrete-timeobservations,"IEEETrans.Inform.Theory,vol.IT{20,pp.591{598,Sept.1974. [2] S.A.Tretter,\Estimatingthefrequencyofanoisysinusoidbylinearregression,"IEEETrans.Inform.Theory,vol.IT{31,pp.832{835,Nov.1985. [3] S.M.Kay,\Afastandaccuratesinglefrequencyestimator,"IEEETrans.Acoust.,Speech,SignalPro-cessing,vol.37,pp.1987{1990,Dec.1989. [4] B.G.Quinn,\OnKay'sfrequencyestimator."Tobepublished. [5] G.W.Lank,I.S.Reed,andG.E.Pollon,\Asemicoherentdetectionanddopplerestimationstatistic,"IEEETransactionsonAerospaceandElectronicSystems,vol.AES{9,pp.151{165,Mar.1973. [6] B.C.Lovell,P.J.Kootsookos,andR.C.Williamson,\Thecircularnatureofdiscrete-timefrequencyestimators,"inProc.ICASSP,1991.