基于逆积分方程的正弦波频率估计

基于逆积分方程的正弦波频率估计 第35卷第10期 2007年10月 华南理工大学(自然科学版) JournalofSouthChinaUniversityofTechnology (NaturalScienceEdition) VO1.35No,10 October2007 ArticleID:1000-565X(2007)10—0147—05 SinusoidalFrequencyEstimationBasedonInversionIntegralEquation WeiLiDao—yuanChenFang-jiong (SchoolofElectronicandInformationEngineering,SouthChinaUniv,ofTech.,Guangzhou510640,Guangdong,China) Abstract:Intraditionalfrequencyestimationalgorithms,thereexistsageneralproblemarisingfromthediscre— pancybetweenthecostlycomputationandtheunsatisfactoryperformanceunderlowSNRs(Signal—to—NoiseRati— os).Inordertosolvethisproblem,thispaperproposesanewfrequencyestimationalgorithmbasedonthelIE (InversionIntegralEquation)approach,Inthisalgorithm,FFT(FastFourierTransform)isusedtoroughly estimatethefrequency,basedonwhichanlIEisconstructedaccordingtothenalTOW— bandsignalresultingfrom FFT.Then.thefinefrequencyisestimatedbycalculatingtheparametersandcharacteristicfrequencyofthe lIE.Simulatedresultsshowthattheproposedalgorithmisofgoodperformanceandmoderatecomputationalre— quirementunderlowSNRs. Keywords:frequencyestimation;integralequation;characteristicfrequency CLCnumber:TN911Documentcode:A 0Introduction Frequencyestimationisaclassicsignal—processing problemthathasattractedresearchers'attentionforde— cades.Ithasapplicationsinawiderangeofareassuch asradar,wirelesscommunicationsandspeechsignal processing.Thispaperaddressestheproblemofsinu— soidalfrequencyestimationmodeledby x(t)=CCOS(2t+)+n0(t)(1) whereC,fc,anddenoterespectivelytheamplitude, frequencyandinitialphaseofthesinusoidalsignal, andn0()istheadditivenoise.Forderivationsimpli— city,inEq.(1)weconsiderasingle—tonecase.We willextendtomultiple—tonecaseintheseque1.Theo— retically,thefrequencyfccanbedirectlyestimated fromtheFourierTransformofx(t).However.ade— tailedperformanceanalysisshowsthattheFourier TransforiBmethodsuffersfromlowresolutionwhenonlv Receiveddate:March1,2007 木Foundationitem:SupportedbytheNationalNatureScience FoundationofChina(60402014,60625101) Biography:WeiGang(bornin1963),male,professor,Ph.D. tutor,mainlyresearchesoncommunicationsystems.E—mail: ecgwei@scut.edu.an asmallsamplesetofx(t)isavailable…. Alotoffrequencyestimationmethodshavebeen putforwardwhichestimatethefrequencynotdirectly fromtheFourierspectrum,butfromsomesuitablepa- rameters.SofartheoreticallytheML(MaximumLikeli— hood)methodisknowntoshowthebestperforlnance. However,itneedsalotofcomputations[.Othersub— optimalmethodsareeitherdeterministic—based[一4]or statistics—basedRef.[5—6].Thestatistics—basedme— thods,whichusuallyexploitthesecond—ordercorrelation matrix,canachievestableestimation.butt}leyalso exhibit bound Bound significantperformancegaptotheperformance i.e.,theso—calledCRB(Cramer—Rao [7].Thedeterministicapproachesdonotneed toestimatethestatisticsand,therefore,usuallyrequire asmallerdataset.Forinstance.theLPfLiBearPre— diction)deterministic—basedapproachcanachievethe performanceboundwhenonly10samplesareavailable (howeveritsperformancebecomesworsethantheCRB whenalargedatasetisavailable).Recently,Fou— tierspectrum—basedestimatorswerere—examined[驯. whichmayobtaintheperformanceboundatmediumto highSNRs(Signal—to—NoiseRatios).KayandSaha haveproposedacomputation—reducedve~ionofthe , ,, 148JournalofSouthChinaUniversityofTechnology(NaturalScienceEdition)Vol_35 MLalg0rithm[. Ab0utani0sandMulgrewhavepre. sentedamethodthatattainstheperformanceboundat lowSNRsinthesingletonecase[. Generallyspeaking,currentfrequencyestimation methodsencountermainlytwoproblems:eitherthe performancedropsdramaticallyunderlowSNRs, orthe computationalrequirementistoocostly.Inthispapera newfrequencyestimationmethodisproposedbasedon theIIEapproach.Firstly,theFourierTransformis appliedtogiveacoarsefrequencyestimate.Then,an IIEisconstructedbasedonthenarrow.bandsignalre. constructedfromtheFourierspectrumcenteredatthe estimatedcoarsefrequency.Finally.thefrequencyis estimatedbycalculatingthecharacteristicfrequencyof theIIE.Experimentalresultsshowthatournewalg0. rithmhastheadvantagesofboththeMLmethodand theparametricmethods.Itisofgoodperformancean- derlowSNRsatthecostofmoderatecomputati0ns. 1TheProposedMethod Thetheoreticalbasisofouralgorithmisthatany sinusoidalsignalx(t)isasolutionofthefollowingin. tegralequation ()+口,f. ()d+口f. J一()dd:0JJ(2)一?一?J一?, wherea1,a2arerelatedwiththefrequenciesinthewav thatthefrequenciescanbecalculatedfr0mtheimagi. narypartoftherootsofitscharacteristicequati0nas 1+61s一+口2s一=0(3) Soifwecanconstructanintegralequationfromthe sampledataset,thatis,theIIE.wecanestimatethe frequencyfromitscharacteristicequationas :T~—4ac2一口(4) where=2istheangularfrequency.Itsh0uldbe remarkedthattheoreticallyadifferentialequati0n. in. steadoftheintegralequationshowninEq. (2).c0uld alsobeappliedtodescribethesinusoidalsigna1.H0w. 0er,inpractme.thesampledsignaliscorruptedby additivenoise,whichisusuallymoresensitivetothe differentialoperationthantheintegral0Derati0n . Therefore,weprefertheintegralequationmode1 .The keypointofouralgorithmistofindanefficientwayto constructtheIIE,i.e.toestimatea】,a,.Thisprob. 1emissolvedintwosteps. First,applyingFourierTransformtoEq.(1),we canget(),thespectrumofx(t).Fromtheprinci. pieofMLmethod,weknowthatthesinusoida1fre. quencycanbedeterminedbyfindingthepeakvalue pointof()ifthefrequencyresolutionandtheSNR ishighenough.Whenonlyasmallsampledatasetis available,whichisusuallythecaseinpractice.weatta. chtheavailablesampleswithzerostomakethelength apowerof2,soastocomputeaFFT.AfterF不. therewillbeanarrowbandenvelopethatincludesthe frequencypoint.Thoughinthiscasewecannotaccu. ratelyobtainthefrequency,wecanlocatetherangeof thefrequencypoint.Inotherwords ,wecangetana. rrowbandsignalcontainingthesinusoidalsignalasis shownbelow: l()=Acos(moo0t+咖)(5) mrno—Am where0isthefundamentalfrequencyoftheFourier Transform;m0denotesthepositionofthepeakva1ue: Amindicatesthefrequencyrange;Aanddenote theamplitudeandthephaseofthem.thharmonicsig. nal,respectively.Eq.(5)canalsobeinterpretedbv Fourierseries.Letx(t)denotethezer0.attached x(t).Performtheharmonicdecompositiononthepc- riodicalextensionof(t).Thespectrumlinesinthe frequencydomainwillhaveanenvelopecenteIledat m00,whichistheclosestspectrumlinetotherea1 value.InEq.(5)weinfactusethespectrumlines 盯."ndetoapproximate(t)?Wehavethefollowing remarksontheselectionofAmandm0. Remark1:Letf,Tdenotethelengthsofft) and(,)respectively.(,)canbepresentedasf,): x(t)w(t)(0?t<)wherew(,)istherectangular window.Calculatingtheharmonicdecompositi0nof(,) (0?,<T)withrespectto0=2rr/T,wehave c=_J01Tccos(,+咖. )edc. — wocos(w— oT)-j(mw0sin(wcT)+//20)0) Lc一//2090c+B/co0).ej咖c (6) Fortheclosestspectrumlineto~o c, i.e.themnth spectrumline,wehavef一m00f?0/2.N0te thatwealsohaveI.一(m0?4)0I~9oJ0/2.Fr0m Eq.(6),itiseasytoverifythattheenergy0fany spectrumlinedeclinesmorethan20dBwhenitsdis. No.10WeiGangetal:SinusoidalFrequencyEstimationBasedonInversionIntegralEquatio n149 tancetothemothspectrumlineisgreaterthan4atte' nuate.Therefore.itisarguedthat(t)canbeappro? ximatedbythespectrumbetween,一4tomo+4.Itis 0bviousthatthemainlodeofthespectrumofthewin. dowsigr~w(t)hasabandrangebetween[一o~T/r,+ o~T/r],thuswearguethat贾(t)canbesufficientlyap? proximatedbythespectrumlinesinthemainlode, whichapp.roximatelyhasabandwidthof[(mo一4一 T/r)o9o,(mo+4+T/v)?o].Therefore,itissuffi? cienttochooseAm=4+r.Notealsothattheaddi. tivenoiseeffectstheselectionofAm.Innoisycondi? tions.alargerAmresultsinmoreaccuratesinusoidal signals.However,insuchcases,morenoisesarealso included,whichwilldegradetheestimationaccuracy. SothechoiceofAmisatradeoffbetweentheenclosed usefulsignalandthenoise.Asdiscussedearlier,itis suggestedthatAminT/r?Am?4+T/rbeselected inpractice. Remark2:mocanbeselectedasthespectrum linethathasthelargestamplitude.However,inlow SNRcasestheremaybeafakepeakduetothenoise. Itisthusproposedthattheenergyinamovingwindow oflength2Am—lbedetectedinthefrequencydo? main.Thecenterofthewindowwiththelargestenergy isselectedasmo. Nowwewouldliketoshowhowtoestimatea1,a2 from1(t).Denote t (7) zeromeans,from mo+?mA :()=丽/'Imsin(,+咖m) 蔓cos()(8) (9) ThenextstepistousetheLS(LeastSquare)method toconstructanIIEbasedonthesamplesof1(t),(t) and"1(t),denotingas1(n),1(n),andx1(n), respectively.Let e(n)=1(n)+a1(n)+a2x(n)(10) (n=0,l,…,?一1) whereNisthelengthofthesampledataset.Thepa? rametersa1,a2canbedeterminedbyusingtheLS metIlod.Let ?一1 E=?e2(n)n;0 (11) BysettingOE/Oa1=0,OE/Oa2=0,wehavethefo' llowinglinearequationsystem, f<1(n)x:(n)>+a1<:(n)x:(n)>+ j.zl(n):(n)=0fl2)I<1(n)x(n)>+01<:(n)x(n)>+, 【a2<Xl(n)(n)>=0 ?一1 where<"(n)(n)>:?"(n)(n).FromEq.(12)n=o we caneasilyobtaintheestimatesofa1,a2.Witha1,a2 found.theIIEcanbedetermined,andthustheesti? matedfrequencycanbecalculatedbyEq.(4). Theproposedsingle?tonealgorithmcanbeeasily extendedtomultiple?tonecases.Inmultiple?toneca? ses,firstlyweneedtolocatemultiplepeaksinthe spectrum,thenfor"eachpeakweconstructanarrow? bandsignalshownasEq.(5),fromwhichwebuildan IIEandestimatetheassociateda1,a2.Inthiswayall thefrequenciescanbeestimatedseparatelybasedon Eq.(12). 2SimulationResuits Theproposedalgorithmwasevaluatedandeom? paredwiththeCRB(Cramer?RaoBound).TheMSE (MeanSquareError)ofestimationisdefinedas MSE=E[(f一)](13) wherefistheestimateandfistherealfrequency.It hasbeenshownthattheestimationerrorisbounded by[: (14) wehaveset thesamplingintervalto1.Theproposedalgorithmis comparedwiththeLP(LinearPrediction)basedapp? . roach[-]andtheESPRITalgorithm[引 whicharethe representativealgorithmsofthedeterministicapproa? chesandthestatistics?basedapproaches.Bothsingle? tonecasesandmultiple?tonescaseswereinvestigated. NotethattheLPapproachinRef.[3]isforsingle. tonecaseswhiletheapproachinRef.[4]isformulti. pie?tonecases.InRef.[9]aFFT?basedalgorithmfor single?tonefrequencyestimationisapplied,which No.10WeiGangetal:SinusoidalFrequencyEstimationBasedonInversionIntegralEquatio n151 WhencomparedwiththetraditionalFFTmethod, theproposedalgorithmrequirestheextracomputation ofEq.(8),(9)andEq.(12),whichisminorwhen comparedwiththecomputationofFFT.Therefore.itis concludedthatthenewalgorithmachievessatisfactory performancewithacceptablecomputation. 3Conclusions Anewfrequencyestimationalgorithmbasedon IIEhasbeenproposed.Wemodeledthesinusoidbyan integralequationandalternativelyestimatetheequation coefficients.AfterperformingtheFourierTrans~rmon theavailablesamples,anarrow—bandsignalextracted fromthespectrumisappliedtoestimatetheequation coefficientsfromwhichthefrequencyiscalculated.Its extensiontomultiple—tonecasesisalsoconsidered. Simulationresulthasverifiedtheefficiencyofthepro— posedalgorithm. References: [2] [3] [4] [5] [6] [7] [8] KaySM,JrMarpleSL.Spectrumanalysis—Amodern peIspective[J].Pr0ceedi"g.ftheIEEE,1981,69(11):[93 1380.1419. StoicaP,MosesRL,FriedlanderB,eta1.Maximumlikeli— hoodestimationoftheparametersofmultiplesinusoids fromnoisymeasurements[J].IEEETransAcoust, Speech,SignalProcessing,1989,37(3):378-392. SoHC.ChanKW.Reformulationofpisarenkoharmonic decompositionmethodforsingle—tonefrequencyestimation [J].IEEETransSignalProcessing,2004,52(4):1128— 1135. SoHC,ChanKitWing,ChanYT,eta1.Linearprediction approachforefficientfrequencyestimationofmultiplereal sinusoids:algorithmsandanalyses[J].IEEETranson SignalProcessing,2005,53(7):2290—2305. RoyR,PaulrajA,KailathT.ESPRIT:Asubspacerotation approachtoestimationofparametersofcisoidinnoise [J].IEEETranAcoustics,SpeechandSignalProc, 1986,34(5):1340—1342. MahataKaushik.Subspacefittingapproachesforfrequen- cyestimationusingreal—valueddata[J].IEEETranson SignalProcessing,2005,53(8):3099-3110. RifeDC,BoorstynRR.Singletoneparameterestimation fromdiscrete.timeobservations『J].IEEETransInform Theory,1974,20(5):591-598. KaySteven,SahaSupratim.Meanlikelihoodfrequencyes— timation[J].IEEETransonSignalProcessing,2000,48 (7):1937-1946. AboutaniosElias,MemberIEEE,MulgrewBernard.hera. tivefrequencyestimationbyinterpolationonFouriercoe— fficients[J].IEEETransonSignalProcessing,2005,53 1242. (4):1237— 基于逆积分方程的正弦波频率估计 韦岗李道远陈芳炯 (华南理工大学电信学院,广东广州510640) 摘要:频率估计算法的普遍问是计算量大并且在低信噪比时性能较差.文中提出一种基于逆积分方程 (InversionIntegralEquation,IIE)的频率估计新算法.首先利用快速傅立叶变换得到频率的粗估计,并从傅立 叶变换中提出一个窄带信号建立积分方程.然后通过对积分方程中参数和特征频率的估计得到最终的频率 估计.仿真结果显示文中算法以适中的计算量在低信噪比下达到了较好的性能. 关键词:频率估计;积分方程;特征频率 中图分类号:TN911文献标识码:A 文章编号:1000-565X(2007)10—0147—05 责任编辑:李嘉 收稿日期:2007.03.01 基金项目:国家自然科学基金资助项目(60402014,60625101) 作者简介:韦岗(1963一),男,教授,博士生导师,主要从事信息系统研究.E-mail:ecgwei@.t..d..