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Fourier

By Ernest Woods,2014-02-18 20:23
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Fourier

    Fourier

    380ChineseChemicalLettersVo1.17,No.3,PP380382,2006

    http://www.imm.ac.cn/journaFcc1.html

    FourierSelf-deeonvolutionUsingApproximationObtainedfrom

    FrequencyDomainWaveletTransformasaLinearFunction

    YuanZhenZHOU,JianBinZHENG*

    InstituteofAnalyticalScience/ShaanxiProvincialKeyLabofElectroanalyticalChemistry, NorthwestUniversity,Xi'an710069

    Abstract:Anewmethodofresolvingoverlappedpeak,Foufierself-deconvolufion(FSD)using

    approximationCuobtalnedfromfrequencydomainwaveletVansformof

    yieldedbyFourier

    vansformofoverlappedpeaksignalsf(tJasthelinearfunction.waspresentedinthispaper~ ComparedwithclassicalFSD,thenewmethodexhibitsexcellentresolutionfordifferent overlappedpeaksignalssuchasHPLCsignals,andhavesomecharacteristicssuchasanextensive

    applicabilityforanyoverlappedpeakshapesignalsandasimpleoperationbecauseofnoselection

    procedureoftheJinearfunction+Itsexcellentresolutionforthosedifferentoverlappedpeak signalsismainlybecauseFtobtainedfromFouriertransformoffit)andCqobtainedfrom wavelettransfotinofhavethesimilarlinearityandpeakwid.Theeffectofthosefake

    peakscanbeeliminatedbythealgorithmproposedbyauthors.Thismethodhasgoodpotentialin

    theprocessofdifferentoverlappedpeaksignals.

    Keywords:Fourierdeconvolution,wavelettransform,signalprocessing,HPLCsignal Fourierself-deconvolution(FSD)forresolvingoverlappedbandswasoriginally developedbyKauppinenteta1.in19811-3.Unfortunately,theconceptualsimplicityof

    FSDiscounterbalancedbyanumberoffailings4.Firstly,linearfunctionfor deconvolutioncannotcompletelyreflectunresolvedpeakshapeandtheprocessof selectingthefunctioniscomplicatedbecauseitimmixedsomeartificialfactors. Secondly,filterfunction,selectingprocessoftruncationpointandfilterresulthave certainerrorwhichlimitedtheapplicationofFSD.Thosedifficultiescanberectified byallapodizationfunction,butitreducedresolution.Therefore,thereweresome methodstoovercometheinherentdrawbacksinFSDand.indeed.thathadbeen proventobesuperiortoFSDinprocessingsomecertainchemicalsignals.Inthispaper, anewmethodofresolvingoverlappedpeaksFourierself-decOnvolution.whichuse approximationCuobtainedfromfrequencydomainwavelettransformofto)yieldedby Fouriertransformofoverlappedpeaksignalsf)asthelinearfunction,wasproposed.

    ForanysignalofpeakshapeJ,itsFSDcanbeexpressedasfollows:

    F'()=F(oJ)(/G(

    E-mail:zhengjb@nwu.edu.cn

YuanZhenZH0U口正381

    WhereF'(og)indicatesthedeconvolutionresultofF(.F(,Dg(og)andG(og)were

    frequencyfeaturesofthepeaksignalf(t),theapodizationfunctionf)andthelineshape functiong(f),respectively.Then_(,FSDresultof,Canbederivedfromthe

    equation(2).

    []

    Furthermore,discreteapproximationco(anddiscretedetailsON(Canbeobtained fromthewavelettransformof.InEq.

    (2),G(CanbereplacedwithC~oo)and

    Dg(usedasfilterfunction.So,Eq.(2)Canbeexpressedasfollowing: f'(t)=Y-i[]

    Wherenequals1.,(islikelytocontainsomefakepeaks.Theeffectofthesefake peaksCanbeeliminatedbythealgorithmproposedbyauthors.DefineasetT, H

T=U=UUU,

    f_l

    +

    Thenthefunctioneliminatingfakepeakswillbe

    q(t)=-(),(E,k=l,2,,,1)(5)

    Where3~(hm)indicatespeakvalueinf(t);nisthenumberofpeaks;hmandWarethepeak positionandthepeakwidthin(ofzerobaselines,respectively.

    calledthismethodofresolvingoverlappedpeakFourierself-deconvolution. whichuseapproximationCNobtainedfromfrequencydomainwavelettransfo1-mofF( yieldedbyFouriertransformofoverlappedpeaksignalsfit)asthelinearfunction.The half-widthofthosepeaksinwouldbecomemuchnarrowbecausethefunctionco(to) CanchangethevibrationamplitudeoffunctionF(,andthepositionofpeaksalmost doesn'tchange.Eq.(5)Canbeappliedtoeliminatefakepeaksin,(f).Hence,this methodCanexhibitanexcellentresolutionresultsfortheoverlappedpeaksignais. ThemethodproposedinthisPapercanprocessdifferentsignalsofCE,IRand HPLC.WeusedthismethodintheexperimenttoprocesstheoverlappedHPLCsignais FigurelashowstheoverlappedHPLCsignalsofprotocatechuicacid(1),cafficacid(2) andferulicacid(3),andFigurelbshowsthecorrespondingprocessedresults.Clearly, thebaselineseparationofthoseHPLCsignaiscouldbeachievedbytheproposedmethod (Figurela,lb).ThecorrelationcoemcientofthreeacidsWas0.9981.0.9991and 0.9987,respectively.

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