时频分析新方法：吕分布

1LvDistributionPresentedby:****2Outline•Introduction•Lvdistribution(LVD)–Definition,properties–Performances•Conclusion3Introduction•Time-frequencyanalysis–Linearfrequencymodulated(LFM)signal:manysignalscanbemodeledasLFMsignals;widelyexistinmanyfields,e.g.,sonar,radar,imaging,etc.121exp2KkkKkkkkststAjftjtcentroidfrequencychirprateamplitudeTime-frequencyanalysis4Introduction•Time-frequencyanalysis–Time-frequencyanalysismethods(twocategories):•Time-frequencyrepresentations(TFRs):showthefrequencycurvesvaryingwithtime.Short-timeFouriertransform(STFT),Wigner-Villedistribution(WVD),etc.•Frequency-chirpraterepresentations(FCRs):showthejointinformationoffrequencyandchirprate.PolynomialFT(PFT),fractionalFT(FrFT),etc.–Requirementsforagoodmethod:•Highresolution•Highconcentration•Lowrepresentationerrors•Lowcross-terms•Lowcomputationalcomplexity5Introduction•Time-frequencyrepresentations(TFRs)TFRsproducetime-frequencymaps,including:WindowedFouriertransform(FT)based(linear):Short-timeFT(STFT),localpolynomialperiodogram(LPP),etc.Cohenclass(bilinear):Wigner-Villedistribution(WVD),smoothedpseudoWVD(SPWVD),S-method,etc.6Introduction•Time-frequencyrepresentations(TFRs)–TheSTFT:2,jfSTFTStfswtedwindow2,STFTStfSpectrogramofathree-componentLFMsignalNocross-term,poorresolution/concentration7Introduction•Time-frequencyrepresentations(TFRs)–The2ndorderLPP,ade-chirpSTFT:2,expexp2LPTFTStfstwjjfdEstimatedchirprate2LPPLPTFTSSLPPofthe3-componentLFMsignalBetterresolutionthanSTFTbyusingestimatedchirpratewithmorecomplexity;Cross-termscausedbyde-chirpprocessing8Introduction•Time-frequencyrepresentations(TFRs)–TheWVD:*2,22jfWVDStfststedHighresolution/con-centration;Heavycross-termsWVDofthe3-componentLFMsignal9Introduction•Time-frequencyrepresentations(TFRs)–TheSPWVD,awindowedWVD:*2,22jfSPWVDStfgttststwedtdReducecross-termsinfrequencydomainReducecross-termsintimedomainReducedresolutionandcross-terms;Trade-offbetweenauto-termsandcross-termsSPWVDofthe3-componentLFMsignal10Introduction•Frequency-chirpraterepresentations(FCRs)Provideinformationofparametersofsignal,including:MethodsbasedonFTandsearching:polynomialFouriertransform(PFT),fractionalFT(FrFT),etc.MethodsbasedonTFRsandintegration:LPP-Houghtransform(LHT),Wigner-Houghtransform(WHT),Radon-Wignertransform,etc.11Introduction•Frequency-chirpraterepresentations(FCRs)–TheFrFT:22cotcsc2,2,,21cot,2utjjuttunKtutunjeotherwise,,FrFTSustKtudtSearchingangleFrFTofthe3-componentLFMsignalNocross-term;relativelyhighresolution/concentrationsignal12Introduction•Frequency-chirpraterepresentations(FCRs)–MethodsbasedonTFRsandintegration:basicallyeveryTFRcanbeintegratedalongthetime-frequencylinesbysomeintegrationalgorithms,suchasHoughtransform(HT)andRadontransform.Forexample,weshowLPP-HT(LHT):Reducedeffectofcross-terms;Highresolution/concentration;Highcomputationalcomplexity.signal13Introduction•Everymethodhasadvantagesandshortcomings.•Weaimtofindamethodhavingtheseadvantageswithouttheirshortcomings.LVD14Introduction•LVDLVDLVDNearlynocross-term;Highresolutionandconcentration.2Dplotting3Dplotting15Outline•Introduction•Lvdistribution(LVD)–Definition,properties–Performances•Conclusion16Lvdistribution(LVD)•TheLVD[1]–Anewtime-frequencyanalysismethod–GenerateanFCRplane–Merits:•Highresolution/concentration•Asymptoticlinear(verysmallcross-termscanbeignored)•Nomorecomplexity[1]X.Lv,G.Bi,C.Wan,andM.Xing,Lv'sdistribution:Principle,implementation,properties,andperformance,IEEETransactionsonSignalProcessing,vol.59,no.8,pp.3576--3591,Aug.2011.TheLVDisaperfectcombinationoflinearandbilinearmethods17Lvdistribution(LVD)•DefinitionoftheLVDWedefineaparametricsymmetricinstantaneousautocorrelationfunction(PSIAF)oftheinputsignal:Ascalingoperatorofaphasefunctionwithrespecttoas*,22CsaaRtstst21exp22KkkkkAjfajat111,ikkiKKCCssssikiRtRtst:constanttime-delayacross-termsG,t,,ntGtGha:ascalingfactorhauto-terms18Lvdistribution(LVD)•DefinitionoftheLVDPerformthescalingoperationonthePSIAFandobtainThenwetakethe2-DFTofthepreviousequationwithrespecttoandas21,exp22KCkskknkRtAjfajth111,,ikkiKKCCssssikiRtRtnt111,,,,,nkikKKKCstsssskikikiLfFFRtLfLf22,kjafkskkLfAeffhLVDauto-termcross-termsFCRplaneNegligible19Lvdistribution(LVD)•PropertiesoftheLVD–Asymptoticlinear:1222112212,,ssLVDcstcstcLVDfcLVDf11,,.kKKksskkststLfLfNegligiblecross-terms20Lvdistribution(LVD)•PropertiesoftheLVD–Asymptoticlinear:–Frequencyandchirprateshift:ifwehave–Scaling:–Resolution:–Others:translation,convolution,etc.1222112212,,ssLVDcstcstcLVDfcLVDf221000exp2expststtjftjt001222000,/jatjafsLVDsteeLVDftfh1132333111,sLVDsctLVDfccc12,ddfThT