Wavelet frequency spectrum and its application in

Waveletfrequencyspectrumanditsapplicationin

analyzinganoscillatingchemicalsystem

XiaoquanLu∗,HongdeLiu,JingwanKang,JinCheng

DepartmentofChemistry,NorthwestNormalUniversity,Lanzhou730070,Gansu,ChinaReceived12June2002;receivedinrevisedform7March2003;accepted10March2003

Abstract

Basedonthecontinuouswavelettransform(CWT),threetypesfrequencyspectra,waveletfrequencyspectrum(WFS),pointfrequencyspectrum(PFS)andtimefrequencyspectrum(TFS),weredeveloped.Twodatasetsweresimulatedandtreatedwiththeproposedspectra,theresultsindicatedthatWFScouldextractthefrequencyinformation,whichwaslikeFourieranalysisbutmoreaccurate,PFScouldobtainthefrequencyatanymoment,TFScouldshowfrequencychangewithtime.TheseabilitiesofPFSandTFSwereimpossibleforFourieranalysis.AnoscillatingchemicalsignalwasprocessedwithWFSandTFS.Fromtheprocessedresults,twopointscouldbelearnedabouttheoscillatingchemicalreaction:onewastheoscillatingchemicalreactionwasamixtureoneincludingtwoormorecomplexkineticsprocesses,thevelocityoftheswitchfromthereducedstate(RS)totheoxidizedstate(OS)wasfasterthanthereverseswitch(fromOStoRS);theotherwasincreaseofKBrO3coulddecreasethevelocitiesofbothswitches,whichledtotheoscillatingperiodbecamelonger.©2003ElsevierScienceB.V.Allrightsreserved.

Keywords:Continuouswavelettransform;Waveletfrequencyspectrum;Timefrequencyspectrum;Oscillatingchemicalsignal

1.Introduction

Fourieranalysis(FA)hadalwaysbeenamathe-maticaltoolinprocessingsignalbeforewaveletthe-orycameout[1].ButFAhastwomostlimitations[2,3]:oneistheinformationbothintimeandinfre-quencydomaincannotbeobtainedsimultaneously;theotherisFAhasdifficultiesindealingwiththesingularsignal.Withwavelettransform(WT)thelimitationscanbeovercome.ThemaincharacteristicofWTisthatittransformsthesignalintoatwo-dimensionaltime-frequencyformandretainsbothtimeandfre-quencyinformation.ContinuousWT(CWT)anddis-Correspondingauthor.Tel.:+86-931-3360668;fax:+86-931-7971697.

E-mailaddress:luxq@nwnu.edu.cn(X.Lu).

∗creteWT(DWT)aretwomostimportantpartsofwavelettheory[4].Inthepastyears,DWTwaswidelyusedindenoising,smoothing,fittingcurve,compress-ingdataandextractingdetailsthroughmallat’sfastde-composedalgorithm[5–8].ButreportsonapplicationsofCWTwereless,whichmayattributetothecom-plicatedcomputationsandtheredundanttransformeddata.Wuetal.[9]andNieetal.[10]usedCWTtoresolvetheoverlappingvoltammetrypeaks.Intheirpapers,CWTwasemployedinfindingpeakpositionsbydetectingthemaximumofCWT.Infact,signal’sCWTcontainsmuchimportantinformation[4],ofwhichonlyasmallpartshavebeenexplored,soitisnecessarytodevelopnewtoolstoextractmoreusefulinformationfromCWT.Ontheotherhand,studyingsomecomplicatedchemicalspectrumalsoneedad-vancedmethods,herewemadeatry.

0003-2670/03/$–seefrontmatter©2003ElsevierScienceB.V.Allrightsreserved.doi:10.1016/S0003-2670(03)00309-X

202X.Luetal./AnalyticaChimicaActa484(2003)201–210

Aswehaveknown,insomefields,signal’sfre-quencycharactersareveryimportant,suchasinvoiceanalysis,velocitydetectionofthemovingparticles[11,12],andsoon.Inthosemeasurements,thefre-quencyofthesignalatanymomentisneededtoknow,andFAoftenfailstogetthem[2,3].Inthispaper,threefrequencyspectramethods,whichwerederivedfromCWTandnamedasWFS,PFSandTFSbyus,weredevelopedandusedinanalyzingacomplicatedsignalwhichwasobservedinoscillatingchemicalreaction.Oscillatingchemicalreactiondirectlyreflectsthephenomenaofthenonlinearandthenon-equilibriuminnature[13],andithasacloserelationshipwiththelivingorganisms.Withthedeepstudiesoftheoscillat-ingchemical,moreandmoreattentionswereattractedinthearea,aclassicofoscillator,metal-ion-catalyzedoxidationoforganiccompoundsbybromateion,calledBZoscillationreaction,hasbecomeanexperi-mentalmodelinstudyingthenonlinearphenomena.In1972,Fieldetal.developedawellknownFKNmechanismtoexplainBZreaction[14].InBZsys-tem,metal-ionexhibitsfluctuationsinitsconcen-tration,suchfluctuationsareusuallyperiodicunderthespecificreactionconditionsandcanbereflectedonchangesofcolor,pHvalue,redoxpotential,etc.Changesoftemperature,concentrationandcata-lystcanaffecttheoscillation’speriodandintensity.Trackingtheperiodchangesisakeytounderstandthemechanismofoscillation,i.e.itisnecessarytoextractfrequencyinformationforstudyingoscillatingchemicalreaction.

Foranoscillatingchemicalsystem,ifsomeotherreactantsareaddedintoit,theoriginaloscillatingpe-riodandamplitudewillhaveachange,thischangeof-tenhasarelationshipwiththeadditionconcentration[15],whichwasthebaseofquantitativedeterminationusingoscillatingchemicalreaction.Now,theproblemishowtogettheexactvalueoftheoscillatingperiodandamplitude.Asweknow,theoscillatingperiodisthetimespentinacycle,anoscillatingchemicalsig-nalmayincludemanycycles.Inthepast,tominimizethemeasurementerroroftheperiod,abettertreatmentwastocalculatetheaveragedperiodvalue,thesimi-lartreatmentwasusedtogettheamplitudevalue.Butthatwouldstillcausesomeerrors[16,17].Moreover,ithasnotreportedyethowthesignalbehavesinacy-cleinfrequencydomain.Here,WFS,PFSandTFSwereusedtodetecttheaccurateoscillatingperiodand

theeffectoftheadditionontheperiod,acomparisonbetweenWFSandFAwasalsopresented.2.Theory

Forasignalf(t)intimedomain,itsFouriertrans-formandWTcanbedescribedasEqs.(1)and(2),respectively:

󰀈F(ω)=+∞

−∞

eiωtf(t)dt(1)

󰀈

󰀄

Wf(a,b)=√1+∞a−∞

f(t)ψ

∗

t−b󰀅adt

(2)

wheretheasteriskdenotescomplexconjugate,ψiswavelet,athescalefactor,bthetransitionfactor,thesefactorsareusedtocontrolthedilationandthetransi-tionofthemotherwavelet.WTisperformedbymov-ingashortpieceofwaveform(‘wavelet’)scaledwithaalongthetimeaxisoff(t)andexpressingthegoodnessoffitateverylocationinthecoefficientsWf(a,b),thenextstepstillrepeatsprovidingthecoefficientsusingafurtherscaling[18,19].TypicalrepresentationsofaWTpresentthegoodnessoffit(thevalueofWf(a,b))inatwo-dimensionalplotwithtimeontheabscissaandthescaleontheordinate.Thescaleisproportionaltothefrequency,thesmallerthescaleis,thehigherthefrequencyis.Ifthescalingwasdonewithfactorsinacontinuousfashion,suchaWTiscalledCWT.BasedonCWT,threetypesoffrequencyspectraweredeveloped.

2.1.Waveletfrequencyspectrum(WFS)

ThecoefficientsWf(a,b)arefunctionofthescaleandtransitionfactors,summatingtheabsolutevalueofthesecoefficientsonthetimeaxiscanobtaintheprojectionofthesignaloneveryscale,thispro-cessconvertsathree-dimensionaldata(Wf(a,b))intotwo-dimensionaldata(Wf(a)),ifthescalevalueistransformedintothecorrespondingfrequencyvalue,WFSofthesignalcanbeestablished,ofwhichtheabscissaisthefrequencyandtheordinateisthesum-matedabsolutevalueofthecoefficients,thiscanbeexpressed󰀈as(3)or(4):

Wf(a)=t2

t|Wf(a,t)|dt(3)

1

X.Luetal./AnalyticaChimicaActa484(2003)201–210203

Wf

󰀂ω󰀃

0

󰀈=

t2

at1

󰀁󰀂ω󰀃󰀁

0󰀁󰀁,t󰀁dt󰀁Wfa(4)

2.3.Timefrequencyspectrum(TFS)

DetectthemodulemaximumofCWTandsaveitscorrespondingscalevalueateverytransition(time),andplotthesavedscalevaluetothetime,thenanotherspectrumcanbeobtained,itsabscissaistime,andtheordinateisthesavedscalesvalue,thedetailalgorithmareasthefollowing:

(i)Fortime=1tothelengthofthesignal:

DetecttheCWTmaximumandsavethecorre-spondingscaletothearrayS.Nexttime:

(ii)PlotthesavedscalevalueinarrayStothetime.Ifthesavedscalevalueistransformedintothefre-quencyvalue,anewspectrum,calledTFS,willbeob-tained,whichisabletodisplaythefrequencyalongwiththetime.

Itisclearthatthethreetypesoffrequencyspectrahavecloserelationships,hereisasummarization:EachtypeofspectrumisbasedontheCWTandcanbederivedfromCWT.WFScandisplaythefre-quencyinanytimespan,itissuitablefordetectingthefrequencycomponents.InWFS,everypeakrepre-sentsafrequencycomponent,thepeakpositionisthefrequency,thepeakheightrelateswiththeintensityofthesignalatthefrequencyposition.WFSwillbecomePFSwhenthesummatedtimespanisshortenedtoatimepoint.PFSissuitableforobtainingthefrequencyinformationataanygiventimepoint.TFScanshowthefrequencychangeswiththetime,andcanalsobederivedfromPFS.

whereWf(a)istheintensityofthesignalfrequencycorrespondingtothescalea,t1andt2formatimespan,onwhichthewaveletcoefficientsaresummated;ω0thefundamentalfrequencyoftheusedwavelet,ω0/athefrequencyvalue.Inpractice,signalisoftenexpressedindiscretedata,(3)or(4)canbewrittenasformula(3󰀆)and(4󰀆):Wf(a)=󰀂ω󰀃

0t2󰀇t1

|Wf(a,t)|

(3󰀆)(4󰀆)

Wf

at2󰀁󰀂ω󰀃󰀁󰀇0󰀁󰀁=,t󰀁󰀁Wf

at

1

2.2.Pointfrequencyspectrum(PFS)

In(3󰀆)or(4󰀆),timespanformedbyt1andt2can

beshortenedorwidenedinvalidrangeofthesignal,i.e.throughWFS,thefrequencyspectrumofthesig-nalinanytimespan,eventhespanisveryshort,canbeacquired,whichisoneoftheadvantageofWFSoverFourierfrequencyspectrum.Whenthetimespanisshortenedtoatimepoint,WFSbecomespointfre-quencyspectrum(PFS),whichcandisplaythefre-quencyatthetimepoint.Itsabscissaisthescalevalueorthecorrespondingfrequencyvalue,andtheordi-nateisthesummatedabsolutevalueofthewaveletcoefficients.

Fig.1.WaveformofMorletwaveletintimedomain(a)andinfrequencydomain(b).

204X.Luetal./AnalyticaChimicaActa484(2003)201–210

ManytypesofwaveletscanbeusedtoperformCWT,suchasDauchechieswavelet,Biorthogo-nalswavelet,Symletswavelet,Mexicanhat,Morletwavelet,Dogfunction,andsoon[18].Amongthem,theMorletwaveletissimplyamonochromaticwavewithinaGaussianenvelopeandverysuitableforCWTanalysis[19,20].So,itwaschosenasthean-alyticalwaveletinthispaper.Fig.1showedMorletwaveletintimedomain(Fig.1(a))andfrequencydomain(Fig.1(b)),thefundamentalfrequencyofMorletwaveletwas0.8Hz.Thetransformfromscaletofrequencycanbecompletedwiththefollowingequation:freq=0.8Hz/a,whereaisthescalevalue,andfreqthefrequency.3.Experiment

Totesttheeffectoftheproposedmethods,twosim-ulateddatasetsandarealchemicaloscillatingsignalwerebothprocessed.

1.Twodatasetsweresimulatedbyfunction(5)and(6).Function(5)wasusedtogeneratesignal1(s1).Accordingtopropertyoffunction(5),s1includedtwofrequencycomponents,onewas0.01592Hz,theotherwas0.00796Hz.Lengthofs1was500;

Signal2(s2)wasderivedfromfunction(6),itcouldbeseparatedintotwoparts,theforepartwasbetween1and300s,ofwhichthefrequencyis0.01592Hz,thefrequencyoftheresidualpart(timespanwasbetween300and600s)is0.03184Hz,s1(t)=sin(0.1t)+sin(0.05t)(t=1,2,3,...,500)

(5)󰀆

s2(t)=

sin(0.1t)t∈[1,300]sin(0.2t)t∈[300,600]

(6)

2.Allchemicalsusedwereofanalyticalreagentgradeexceptmalonicacid.SolutionofKBrO3(0.2M),Ce(SO4)2(0.04M),CH2(COOH)2(0.5M)werepreparedin0.8MH2SO4.Doublydistilledwaterwasusedthroughout.

A50mlwater-jacketedglassvessel,amagneticstirrer(PujiangAnalyticalInstrumentinShaihai),aCs-501thermostatcomposedtheexperimental

set-up.Atypeof213Ptelectrodeservedasworkelectrode,whichwasusedtomonitortheoscillat-ingreactioninthevessel,aPtdiskascounterelec-trodeandasaturatedcalomelelectrode(SCE)asreferenceelectrode.AllelectrodeswereconnectedwithCHI900(CHInstrumentalCo.Ltd.,Austin,USA)andplacedinthevessel.

Theoscillatingreactionwascarriedoutinthewater-jacketedglassvesselplacedonthemagneticstirrer,thetemperaturewascontrolledat303±2KbytheCs-501thermostat.Asolutioncomposedof0.05mlH2SO4,6.25mlKBrO3,1mlCe(SO4)2and6.7mlCH2(COOH)2wasaddedintotheves-sel.AcomputerconnectedwithCHI900startedtorecordthedataafterthesolutionhadreachedthethermalequilibrium.Whentimewasat160s,1.5mlKBrO3(0.2M)wasinjectedintotheves-sel.Thesolutionwasstirredcontinuouslyinwholemeasurement.

3.CWTwasperformedbytheWavelettoolbox2.0ofMatlab(TheMathworksInc.,Natick,MA,USA)onaPentiumIIIPCwith128MBofmemory,othercomputationprogramwasdevelopedbyuswithVisualBasic6.0,someplotswereplottedwithOrigin5.0.4.Resultsanddiscussion

4.1.Processionofthesimulateddata

Fig.2(a)and(b)displayeds1anditsCWT,respec-tively.ForFig.2(b),thescalewasfrom1to150,thestepwasStep1.Thelightnesswasproportiontothemagnitudeofthewaveletcoefficients.Itcouldbeseenthewaveletcoefficientswerebiggeratscales50and100,whichindicatedthats1hadmoreprojectiononthetwoscales.Thatistosaythatthefrequenciesofs1wereatabout0.01592and0.0079Hz,thisconclusionwasprovedinFig.3(a).

Fig.3(a)wasWFSofs1onwholetimespan(1–600s),thereweretwofrequencypeakslocatedat0.0159and0.0079Hz,respectively,whichwasagreedwiththepropertyofthefunction(5),soitcouldbeconcludedthatWFSwasabletoextractthefrequencycomponentsexactly.

UsingPFS,thefrequencyinformationatanymo-mentcouldbegained.Asanexample,PFSofs1

X.Luetal./AnalyticaChimicaActa484(2003)201–210205

Fig.2.Thes1(a)anditsCWT(b).Lengthofthesignalwas500;thescalewasfrom1to150withStep1.

at250swaspresentedinFig.3(b),thereweretwofrequencypeaksinit,whichlocatedat0.0159and0.0079Hz,theheightofthepeakat0.0159Hzwasmorethanthatat0.0079Hz,whichindicatedthatwhentimewasat250sthesignalintensityat0.0159Hzwasmorethanthatat0.0079Hz.

Thes2wasusedtosimulateanothertypeofos-cillatingsignalshowninFig.4(a),itsCWTplotwasshowninFig.4(b),fromwhichitcouldbeseenthats2hadmoreprojectiononscale25and50.Thescalerangewasbetween1and80,thestepwasStep1.

Fig.5(a)wasWFSofs2forthewholetime(1–600s),andFig.5(c)wasWFSforthetimespanbetween1and300s,Fig.5(d)wasWFSforthetimespanbetween300and600s.Fig.5(b)wasFourierfrequencyspectrumofs2.InFig.5(a),twofrequencypeakscouldbeseen,fromlefttoright,theirposi-tionswereat0.01595and0.03188Hz,respectively,whichwereconsistentwiththetheoreticalpositions

Fig.3.Subplot(a)wasWFSofs1forthewholetime;subplot(b)wasPFSofs1at250s.

206X.Luetal./AnalyticaChimicaActa484(2003)201–210

Fig.4.Thes2(a)anditsCWT(b).Lengthofs2was600;thescalerangedfrom1to80withStep1.

(0.01592and0.03184Hz).InFourierfrequencyspec-trum(Fig.5(b)),thetwopeakscouldbealsoseen,theirpositionwereat0.01601and0.03169Hz,respec-tively.TherelativeerrorsofthetwopeakspositionsinWFSwere0.19and0.13%,inFourierfrequencyspectrumtherelativeerrorswere0.56and0.47%,whichindicatedthatFourierfrequencyspectrumhadaloweraccuracythanWFS.

Therewasapeakat0.01595HzinFig.5(c),whichindicatedthatthefrequencyofs2between1and300swas0.01595Hz.InFig.5(d),apeakwaspresentedat0.03188Hz,whichindicatedthefrequencyofs2be-tween300and600swas0.03188Hz.Itwasfoundthattherewasaveryweakpeakat0.01595HzinFig.5(d),whichwasduetothattheforepartofs2(be-tween1and300s)stillhadsomeweakprojectionsonCWTafter300s,butthisdidnotaffecttherecogni-tionofthemajorfrequency0.03188Hz.TheseresultsinFig.5(a),(c)and(d)wereallconsistentwiththepropertyoffunction(6).

Fig.6wasTFSofsignals2,itplottedthechangeofthefrequencywithtime.Whenthetimespanwasbetween1and300s,theprincipalfrequencycompo-nentwascenteredatabout0.0159Hz,andwhenthetimewasfrom300to600s,theprincipalfrequencywasabout0.0318Hz,whichwereconsistentwiththeresultsinFig.5(a),(c)and(d).

4.2.Processionoftheoscillatingchemicalsignal

Fig.7(a)wastheoscillatingchemicalsignalob-tainedbythetechniqueofopencircuitpotential–time(OCPT)onCHI900electrochemicalworkstation.At160s,thesolutionof1.5mlKBrO3wasaddedintothesystem.Thelengthofthesignalwas500.ItcouldbeseenthatthepotentialofCe4+/Ce3+changedperiodically.AccordingtoFKNmechanismforBZreaction[12],thewholereactionwasincontrolofBr−,thesystemswitchedbetweenthereducedstate(RS)andtheoxidizedstate(OS).ForRS,Br−hadahighconcentrationandtheCeionwasinorap-proacheditsreducedstate;OSwascharacterizedbyhighvalueof[HBrO2],[Ce4+]andorganicradi-calconcentrationandbythesimultaneousoxidationandbrominationofmalonicacid.Theoverallre-actioncouldbeexpressedasthefollowingthreesteps:

Step1:BrO3−+2Br−+3CH2(COOH)2+3H+→3BrCH(COOH)2+3H2O

Step2:BrO3−+4Ce3++CH2(COOH)2→BrCH(COOH)2+3H2O

Step3:4Ce4++BrCH(COOH)2+2H2O→Br−+HCOOH+2CO2+4Ce3++5H+

X.Luetal./AnalyticaChimicaActa484(2003)201–210207

Fig.5.Subplot(a)wasWFSofs2forthewholetime;subplot(b)wasFourierfrequencyspectrumofs2;subplot(c)wasWFSofs2inthetimespanbetween1and300s;subplot(d)wasWFSofs2inthetimespanbetween300and600s.

When[Br−]washighsufficiently,Step1domi-nated;withthereduceofBr−,thesystemswitchedtoStep2,Br−couldberegeneratedinStep3.

Thepotentialofthesystemwasrelatedwithratioof[Ce4+]/[Ce3+].TheconcentrationsCe4+andCe3+werechangedbyeachreactioninvolvedinthesystem,thustracingthechangesofratio[Ce4+]/[Ce3+]wasanentranceofstudyingthemechanisms.

FromFig.7(a),itcouldbeseenthattherewereadistinctkineticsdifferencebetweentheprocessfromOStoRSandthereverseprocess(fromRStoOS).Fig.7(b)wasCWTofthedatainFig.7a,thescalewasfrom1to80,thestepwas1.

Todetectthefrequencyinformationandtheeffectoftheadditionof1.5mlKBrO3onthesystem,threetime

Fig.6.Timefrequencyspectrum(TFS)ofsignals2.

208X.Luetal./AnalyticaChimicaActa484(2003)201–210

Fig.7.Thechemicaloscillatingsignal(a)anditsCWT(b).Lengthoftheoscillatingsignalwas500;scalewasfrom1to60withStep1.

spanswerechosentostudy.Thefirsttimespanwasforthewholetime,thesecondonewasbetween1and150s,andthethirdonebetween250and400s,theirWFSwereshowninFig.8andlabeledwith“a”,“b”and“c”,respectively.Curvebdisplayedthefrequencycomponentsinthesecondtimespan,inwhichthereweretwofrequencypeaks,onewasat0.02581Hz,the

Fig.8.CurveawasWFSforthefirsttimespan(thewholetime);CurvebwasWFSforthesecondtimespan(between1and150s);CurvecwasWFSforthethirdtomespan(between250and400s).

otherwasat0.04440Hz.Theseresultssuggestedthatthereweretwocomplexkineticsprocessesinvolvedinthereaction,andthesekineticsprocesseshaddifferenteffectsontheratio[Ce4+]/[Ce3+].FromtheanalysisassociatedwithFig.7(a),itcouldbeinferredthatfre-quency0.02581HzcharacterizedtheswitchvelocityfromOStoRS,andfrequency0.04440Hzcharacter-izedthereverseprocess(i.e.fromRStoOS).ItwasclearthattheswitchvelocityfromRStoOSwasfasterthanthereverseswitch.

Curvecdisplayedthefrequencyofthesignalaf-teradding1.5mlKBrO3solution,thereweretwofre-quencypeaksinit,theirpositionswere0.02162and0.04211Hz,respectively.ComparingwithCurveb,thepeakspositionsbothshiftedtonegative,whichsuggestedthattheincreaseofKBrO3interferedtheoriginalkineticsprocessesandmadetheoriginalos-cillatingperiodbecomelonger,i.e.theincreaseofKBrO3slowedthechangeofratio[Ce4+]/[Ce3+]inbothswitches.

Curveadisplayedtheaveragedfrequencycompo-nentsofthesignal,twofrequencypeakspositionswereat0.02353and0.04322Hz.ItcouldbeviewedastheoverlappedplotofCurvesbandc,thusthefrequencypeaksat0.02581,0.04440,0.02162and0.04211HzinCurvesbandcemergedattheaveragedpositions(0.02353and0.04322Hz)inCurvea.

X.Luetal./AnalyticaChimicaActa484(2003)201–210209

Fig.9.Timefrequencyspectrum(TFS)oftheoscillatingsignalinFig.7(a).

ItcouldbeseenthattheheightsofthefrequencypeaksinCurvecwerebothlowerthanthoseinCurveb,whichindicatedthattheincreaseofKBrO3decreasedtheoscillatingamplitude.Sincethegoalofthispaperwastoprovideafrequencyextractingmethod,themoredetailsontheoscillatingreactionwerenotpresentedhere,itcouldbefoundintheliterature[21,22].

Fig.9wasTFSofFig.7(a),whichdisplayedtheprincipalfrequencyateachtimepoint.ItcouldbeseenthereweremanyunexpectedfrequencycomponentsinFig.9,whichsuggestedthatthekineticsprocessoftheoscillatingchemicalreactionwasacomplexandcom-plicatedone,andsomeunknownkineticsprocesseswereinvolvedinthewholereaction.

Fig.10wastheFourierfrequencyforFig.7(a),itwasregretthatitwashardtodiscern,thisprovedthattheproposedmethod(WFS)wassuperiortoFA.Accordingtotheaboveresults,WFSandTFScon-tainedalmostalltheinformationoftheoscillatingchemicalsignalanddirectlydisplayeditsfrequencycharacteristics.So,anoscillatingsignalcouldbecom-pletelyexpressedasWFSorTFSinsteadofOCPTsignal.

Fig.10.FourierfrequencyspectrumoftheoscillatingsignalinFig.7(a).

210X.Luetal./AnalyticaChimicaActa484(2003)201–210

5.Conclusion

Theproposedspectra,WFS,PFSandTFSwereeffectiveinprocessingsignal’sfrequency.Compar-ingwithFA,WFShadhigheraccuracyinobtainingsignal’sfrequencycomponents,moreover,WFScouldanalyzesignal’sfrequencyatanytimespan.PFSandTFScouldanalyzesignal’sfrequencyatanymomentanddetectfrequencychangewithtime,whichwasthemostadvantageofthem.Thesespectracouldbeusedtomeasurethevelocityofmovingparticles,ana-lyzevoiceandanyotherperiodicalsignalinchemical,physicsandotherfields.Acknowledgements

ThisworkwassupportedbyNationalNaturalSci-enceFoundationofChina(No.:20275031),Founda-tionofStateKeyLaboratoryofChemo/biosensingofHunanUniversity(No.:2002-16),KJCX-01ofNorthWestNormalUniversity,theTeachingandResearchAwardProgramforoutstandingYoungTeachersinHigherEducationInstitutionsofMDE,P.R.C.References

[1]X.Q.Lu,X.W.Wang,J.W.Kang,J.Z.Gao,Anal.Chim.Acta

404(2000)249–255.

[2]M.Vetterli,C.Herley,IEEETransSP40(1992)2207–2232.[3]S.Mallat,AWaveletTourofSignalProcessing,Academic

Press,NY,1998.Seealso:http://cas.ensmp.fr/∼chaplais/Wavetourpresentation/WavetourpresentationUS.html.

[4]B.K.Alsberg,A.M.Woodward,D.B.Kell,Chemometr.Intel.

Lab.Syst.37(1997)215–239.

[5]X.Q.Lu,J.Y.Mo,J.W.Kang,J.Z.Gao,Anal.Lett.31(3)

(1998)529–540.

[6]X.Q.Lu,X.W.Wang,J.Y.Mo,J.W.Kang,J.Z.Gao,Analyst

124(1999)739–744.

[7]V.J.Barclay,R.F.Bonner,Anal.Chem.69(1997)78–

90.

[8]K.Jetter,U.Depczynski,K.Molt,A.Niemöller,Anal.Chim.

Acta420(2000)169–180.

[9]S.G.Wu,L.Nie,J.W.Wang,X.Q.Lin,L.Z.Zheng,L.Rui,

J.Electroanal.Chem.508(2001)11–27.

[10]L.Nie,S.G.Wu,J.W.Wang,L.Z.Zheng,X.Q.Lin,L.Rui,

Anal.Chim.Acta450(2001)185–192.

[11]Y.Kwok,N.Jeffery,A.Manz,Anal.Chem.71(1999)

2130.

[12]S.Kitagewa,O.Nozaki,T.Tsuda,Electrophoresis20(1999)

2560.

[13]B.P.Belousov,APeriodicReactionandItsMechanism,

SbornikReferatovpoRadiatsionniMeditsine,Medgiz,Moscow,1958,pp.145.

[14]R.J.Field,E.Koros,R.M.Noyes,J.Am.Chem.Soc.94

(1972)8649.

[15]K.B.Yatsmirskii,P.E.Strizhak,T.S.Ivaschenko,Talanta40

(1993)1227.

[16]P.E.Strizhak,O.Z.Didenko,T.S.Ivaschenko,Anal.Chim.

Acta428(2001)15.

[17]J.Z.Gao,H.Yang,X.H.Liu,J.Ren,Q.Z.Li,J.W.Kang,

Talanta50(2002)105.

[18]T.Y.Wang,C.M.Luo,L.Duan,S.X.Zheng,Z.G.Yang,J.

TsinghuaUniv.Sci.Technol.39(1999)9.

[19]J.Lewallw,Tutorialoncontinuouswaveletanalysisofex-perimentaldata,http://www.ecs.syr.edu/faculty/lewalle/tutor/tutor.html,1995.

[20]Y.H.Peng,WaveletTransformanditsApplicationin

Engineering,SciencePress,China,2000,pp.21–27.[21]P.Ruoff,R.M.Noyes,J.Phys.Chem.84(1986)1413.[22]R.J.Field,H.D.Forsterling,J.Phys.Chem.90(1986)

5400.