ON-EMPIRICAL-MODE-DECOMPOSITION-AND-ITS-ALGORITHMS

ONEMPIRICALMODEDECOMPOSITIONANDITSALGORITHMSGabrielRilling∗,PatrickFlandrin∗andPauloGon¸calv`es∗∗∗LaboratoiredePhysique(UMRCNRS5672),´EcoleNormaleSup´erieuredeLyon46,all´eed’Italie69364LyonCedex07,France{grilling,flandrin}@ens-lyon.fr∗∗ProjetIS2,INRIARhˆone-Alpes,655,avenuedel’Europe38330Montbonnot,[e-mail]’sdata-driventechniqueofEmpiricalModeDecomposition(EMD)ispresented,andissuesre-latedtoitseffectiveimplementationarediscussed.Anumberofalgorithmicvariations,includingnewstoppingcriteriaandanon-lineversionoftheal-gorithm,areproposed.Numericalsimulationsareusedforempiricallyassessingperformanceelementsrelatedtotoneidentificationandseparation.Theobtainedresultssupportaninterpretationofthemethodintermsofadaptiveconstant-Qfilterbanks.1INTRODUCTIONAnewnonlineartechnique,referredtoasEmpiricalModeDecomposition(EMD),hasrecentlybeenpio-neeredbyN.E.Huangetal.foradaptivelyrepresent-ingnonstationarysignalsassumsofzero-meanAM-FMcomponents[2].Althoughitoftenprovedre-markablyeffective[1,2,5,6,8],thetechniqueisfacedwiththedifficultyofbeingessentiallydefinedbyanalgorithm,andthereforeofnotadmittingananalyt-icalformulationwhichwouldallowforatheoreticalanalysisandperformanceevaluation.Thepurposeofthispaperisthereforetocontributeexperimen-tallytoabetterunderstandingofthemethodandtoproposevariousimprovementsupontheoriginalformulation.Somepreliminaryelementsofexperi-mentalperformanceevaluationwillalsobeprovidedforgivingaflavouroftheefficiencyofthedecompo-sition,aswellasofthedifficultyofitsinterpretation.2EMDBASICSThestartingpointoftheEmpiricalModeDecompo-sition(EMD)[2]istoconsideroscillationsinsignalsataverylocallevel.Infact,ifwelookattheevolu-tionofasignalx(t)betweentwoconsecutiveextrema(say,twominimaoccurringattimest−andt+),wecanheuristicallydefinea(local)high-frequencypart{d(t),t−≤t≤t+},orlocaldetail,whichcorre-spondstotheoscillationterminatingatthetwomin-imaandpassingthroughthemaximumwhichnec-essarilyexistsinbetweenthem.Forthepicturetobecomplete,onestillhastoidentifythecorrespond-ing(local)low-frequencypartm(t),orlocaltrend,sothatwehavex(t)=m(t)+d(t)fort−≤t≤t+.Assumingthatthisisdoneinsomeproperwayforalltheoscillationscomposingtheentiresignal,thepro-cedurecanthenbeappliedontheresidualconsistingofalllocaltrends,andconstitutivecomponentsofasignalcanthereforebeiterativelyextracted.Givenasignalx(t),theeffectivealgorithmofEMDcanbesummarizedasfollows[2]:1.identifyallextremaofx(t)2.interpolatebetweenminima(resp.maxima),endingupwithsomeenvelopeemin(t)(resp.emax(t))3.computethemeanm(t)=(emin(t)+emax(t))/24.extractthedetaild(t)=x(t)−m(t)5.iterateontheresidualm(t)Inpractice,theaboveprocedurehastoberefinedbyasiftingprocess[2]whichamountstofirstiterat-ingsteps1to4uponthedetailsignald(t),untilthislattercanbeconsideredaszero-meanaccordingtosomestoppingcriterion.Oncethisisachieved,thedetailisreferredtoasanIntrinsicModeFunction(IMF),thecorrespondingresidualiscomputedandstep5applies.Byconstruction,thenumberofex-tremaisdecreasedwhengoingfromoneresidualtothenext,andthewholedecompositionisguaranteedtobecompletedwithafinitenumberofmodes.Modesandresidualshavebeenheuristicallyintro-ducedon“spectral”arguments,butthismustnotbeconsideredfromatoonarrowperspective.First,itisworthstressingthefactthat,eveninthecaseofharmonicoscillations,thehighvs.lowfrequencydiscriminationmentionedaboveappliesonlylocallyandcorrespondsbynowaytoapre-determinedsub-bandfiltering(as,e.g.,inawavelettransform).Se-signalEmpiricalModeDecompositionimf1imf2imf3imf4imf5imf6imf7imf8imf92004006008001000120014001600180020000246x10-3res.timefrequencysignaltimefrequencymode#1timefrequencymode#2timefrequencymode#3Figure1:EMDofa3-componentsignal.Thean-alyzedsignal(firstrowofthetopdiagram)isthesumof2sinusoidalFMcomponentsand1Gaus-sianwavepacket.ThedecompositionperformedbyEMDisgiveninthe8IMF’splottedbelow,thelastrowcorrespondingtothefinalresidue.Thetime-frequencyanalysisofthetotalsignal(topleftofthe4bottomdiagrams)reveals3time-frequencysigna-tureswhichoverlapinbothtimeandfrequency,thusforbiddingthecomponentstobeseparatedbyanynon-adaptivefilteringtechnique.Thetime-frequencysignaturesofthefirst3IMF’sextractedbyEMDevidencethatthesemodesefficientlycapturethe3-componentstructureoftheanalyzedsignal.(Alltime-frequencyrepresentationsarereassignedspec-trograms[3,9].)lectionofmodesrathercorrespondstoanautomaticandadaptive(signal-dependent)time-variantfilter-ing.Anexampleinthisdirection,whereasignalsignalEmpiricalModeDecompositionimf1imf2imf3res.Figure2:EMDofa3-componentsignal—Nonlin-earoscillations.Theanalyzedsignal(firstrowofthediagram)isthesumof3components:asinuso¨ıdofsomemediumperiodTsuperimposedto2triangu-larwaveformswithperiodssmallerandlargerthanT.ThedecompositionperformedbyEMDisgiveninthe3IMF’splottedbelow,thelastrowcorrespondingtothefinalresidue.composedof3componentswhichsignificantlyover-lapintimeandfrequencyissuccessfullydecomposedbythemethod,isgiveninFigure1.ThisfigurehasbeenobtainedbyrunningtheMatlabscriptemd_fmsin2.m.1Anotherexample(emd_sawtooth.m)thatputsem-phasisonthepotentially“non-