Augmented EMD for complex‐valued univariate signals Beom‐Seok Oh.pdf

该【Augmented EMD for complex‐valued univariate signals Beom‐Seok Oh 】是由【dt83088549】上传分享，文档一共【10】页，该文档可以免费在线阅读，需要了解更多关于【Augmented EMD for complex‐valued univariate signals Beom‐Seok Oh 】的内容，可以使用淘豆网的站内搜索功能，选择自己适合的文档，以下文字是截取该文章内的部分文字，如需要获得完整电子版，请下载此文档到您的设备，方便您编辑和打印。IETSignalProcessingResearchArticleISSN1751-9675plex-valuedReceivedon6thJanuary2018Revised17thSeptember2018eptedon17thJanuary2019E-Firston25thMarch2019doi:--SeokOh1,HuipingZhuang1,Kar-AnnToh2,ZhipingLin11SchoolofElectricalandElectronicEngineering,NanyangTechnologicalUniversity,50NanyangAvenue,639798Singapore,Singapore2SchoolofElectricalandElectronicEngineering,YonseiUniversity,50Yonsei-ro,Seodaemun-gu,Seoul03722,RepublicofKoreaE-mail:******@:Inthisstudy,position(EMD)forcomplex--valuedunivariatesignalintoalongerreal-valuedsignalbyaugmentingtherealpartwiththeflippedimaginarypart,poseitintointrinsicmodefunctions(IMFs)(BEMD),theproposedmethodshowsbettermicro-Dopplersignatureanalysisperformanceonphysicallymeasuredcontinuous-?(EMD)[1]isapowerfultoolAlthoughtheBEMDhasshownpromisingperformanceinfortime–-posingthebivariatesignals[20,21],itiswellknownthatunivariatesignalintoasetofamplitude-andfrequency-putationalcost[14,15].Thisis(AM–FM)functionscalledtheintrinsicmodefunctions(IMFs).duetothenatureofprojectingbivariatesignals,wherethenumberDifferentfromtheFourieranalysis,esatrade-,putationalcost[10].lower-ponentfromtheinputsignaladaptivelyandInthispaper,,plex--linearandnon-keyideaistoaugmenttherealpartwiththeflippedimaginarypartstationarysignals([2,3]plex-valuedunivariatesignal,andthenusethe[4,5]).EMDontheresultantlongerreal-(plex-valuedbivariateIMFs(BIMFs).Wenoteherethattheproposedmethodisunivariate)[6–10],trivariate[11,12]orevenmultivariate[13–18]differentfromthechannel-,severalmethodshave[6]andmentionedin[16].Particularly,thesamenumberofmodebeenrecentlyproposed,plexEMD(CEMD)[6],functionsforeachrealandimaginarychannelsisguaranteedsincerotation-invariantEMD(RI-EMD)[7],bivariateEMD(BEMD),thesame-indexedBIMF[8],non-uniformlysampledBEMD(NS-BEMD)[9](DS-BEMD)[10].IntheCEMD,theThemaincontributionsofthispapercanbeenumeratedascomplexvaluesarefirstconvertedintoananalyticformofthefollows:(i)proposalofanefficientextensionofthestandardEMDsignalbasedontheHilberttransform[1].plex-position;(ii);and(iii)provisionofanalysisresultsshowingthatsomeinconsistentnumberofIMFsforeachchannel[10,16,19],,suchchannel-,thepositioncannotguaranteethatthesame-indexedIMFsproposedtreatmentofsignalfeaturescanhaveabetterresolutioncontainsimilarscalesacrossdatachannels[16,19].,boththeRI-anisedasfollows:thebackgroundknowledgefullbivariateextensionsoftheEMDmethod[10].Abivariate-/ontheEMDandtheBEMDmethodsisprovidedinSection2forcomplex-,-likethree-plex--EMDandtheBEMDmethods[10].-EMDtakestwodirections,,theBEMDismoreappropriateforreal-worldsignalsthantheRI-?BriefreviewofEMDandBEMDmethods,NS-BEMDandDS-BEMD,areextensionsoftheBEMDThemainideaoftheEMDistotreatareal-.,2019,,-433424?,oncethemeanoftheenvelopecurveisobtained(seeLines#7–#14ofAlgorithm2)thesimilarsiftingprocesstothatoftheEMDprocessisperformed().3?ProposedmethodInthissection,weproposeanextensionoftheEMDmethod,namelyanaugmentedEMD(AEMD)plex--valuedsignalisconvertedintoalongerreal-valuedsignalbyconcatenatingtherealandtheimaginaryparts,?Algorithm1::Let1×dbeas=s1,s2,…,sd∈?complex-∈?×as[22]1,ifb=d?a+1E(a,b)=(1)0,ifb≠d?a+1where(a,b)-reversedversionoftheidentitymatrix[22].Usingthisexchangematrix,theinputsignalsisconvertedtos~asfollows:s~=Re(s),Im(s)+ηE,(2)dwheres~∈?1×2,theoperation?,?indicatesarowvectorconcatenation,Re(s)andIm(s),respectively,,(Im(s)+η)E∈?1×indicatesthetime()flippedIm(s)aftershiftingbyaconstantη,whichisdefinedasfollows:Re(s)?Im(s),ifRe(s)≥Im(s)η=,(3)?Algorithm2:BEMD?Re(s)?Im(s),ifRe(s)<Im(s)extensionofthenotion,theBEMDtreatsabivariatesignalasfastwhereResdenotesthemeanofResand?:Asmentionedearlierinthis1×dGivenareal-valuedunivariatesignalx∈?,thestandardEMDsection,theaugmentedsignals~obtainedusing(2)istobeposedintoasetofIMFsusingthestandardEMDmethod.[1]asLwhere1×ddenotesx=∑l=1ml+qL,ml∈?,l=1,…,L,However,itisworthnotingherethattheaugmentedsignalmayor~1×∈?[1]:simpleconcatenationoftherealpartandtheflippedimaginary(i)thenumberoflocalextremaandzero-,,respectively,showtheaugmentedsignalobtainedusingsyntheticsignals(9)andordifferbyatmostoneand(ii)atanypointoftime,themeans21s22valueoftheenvelopedefinedbythelocalmaximaandthe(10).,theendingpointoftherealpartandthestartingpointcomputingasetofIMFsiscalleda‘siftingprocess’,bothsummarisedinAlgorithm1().,,,posesareal-,,iftheinputbivariatesignals∈?2×(plex-valuedunivariatesignal1×dsignalssatisfieseitherofthefollowingtwoconditions,weproceeds∈?;)ponents,namelyBIMFsck,k=1,…,K,as[8]:?maxf0,f1Isf0?f1≥?Here,thesymbolsf0andf1,K10sckpKrespectively,indicatetheestimatedinstantfrequency[23]atthe=∑k=1+,.,2019,,-433425?TheInstitutionofEngineeringandTechnology2019?max[Res,Ims]IsResd?Imsd≥?Recallthatsd10ddenotesthelastelementofs∈?1×.(Conditionalstep)removalofthesignaldiscontinuitybasedonaninterpolationusingachirpsignal:Signalsmoothingintime-/frequency-domain[24,25]andlinear/non-linearinterpolation[25],theformerisaninappropriateoptiontobeincorporatedwiththeAEMDbecauseofthelocalityandadaptivitypropertiesoftheEMDmethod().Themainreasonisthatthesmoothingchangestheoriginalsignalaroundtheedge,(),,interpolation,,weproposetointerconnectoneendofbothrealandimaginarypartsusingafrequency-modulatedsinusoidalwave()-paddingeffectasinothersignalprocessingtechniques(),?Illustrationoftheproposedsignalaugmentationandthepotentialpensatingtheissueofasignaldiscontinuityfrequencydifferencesexistingbetweentherealandtheimaginary(a)Obtainedusingasyntheticsignals21shownin(9),(b)=(1),(2),…,()signals22shownin(10)follows:P(4)v(t)=Acos2πf(t)tt=1,wheref(t)=(k/2)t+f0denotesthetime-varyingfrequency;k=f1?f0/Pindicatesthechangingrateofthefrequency;f0andf1indicatethefrequenciesatthebeginning(=1)andattheend(=P)≥1kHz;otherwise,P==maxsmax,smin,wheresmax=maxmaxRe(s),maxIm(s)andsmin=min[minRe(s),minIm(s)].BothendsofthegeneratedchirpsignalvarethenadjustedtoensurethelocalsignalcontinuityasdetailedinAlgorithm3()(,mvandmbshowninLines#2,#4,#11and#14ofAlgorithm3()).Wepthensegmentv~∈?1×,whichisadjustedtobothRe(s)andPIm(s)+ηE,fromv∈?1×,wherep≤P(seeLine#23ofAlgorithm3()fordetail).Asaresultofthisboundaryadjustmentstep,~,(2)isnowre-definedasfollows:s~=Re(s),v~,Im(s)+ηE,(5)?Algorithm3:EdgeAdjust(Re(s),v,(Im(s)?+?η)E))where~1×2d+∈?:(2)and(5),plex-valuedsignalsaforementioned,iftheconcatenatedsignal~1×2dby(2)doess∈esareal-valuedsignal~1×2d+pby(5)(ornotcontainalargediscontinuity,putings∈?~1×2d~,thes∈?by(2)).TheaugmentedsignalsistheninputtedtothedpstandardEMD,whereListhenumberofIMFsputedusingthesignals~∈?1×2+obtainedbyTTTL×2d+p~1×(2d+p)(5).M=m1,…,mL∈?ifs∈?by(5)(or426IETSignalProcess.,2019,,-433??Illustrationoftheproposedsignalaugmentationanddiscontinuityremovalstepsonasyntheticsignals22in(10)LddM∈?×2ifs~∈?1×2by(2)),thefirstdcolumnsofMaretheIMFsobtainedusingtherealpartofs,whilethemiddlepcolumnsandthelastdcolumnsaretheEMDresultsobtainedusingthepaddedchirpsignalandtheimaginarypart,,MisdividedintothreesubsetsMreal,MchirpandMimagasfollows:realL×dM=M1,M2,…,Md∈?,chirpL×p(6)M=Md+1,Md+2,…,Md+p∈?,imagL×dM=M(d+p)+1,M(d+p)+2,…,M2d+p∈?,L×1whereMi∈?~∈?1×2by(2)posed(seeLines#14–#16ofAlgo