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Quadratic Chirp Signals Using Product Cubic Phase Function

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Quadratic Chirp Signals Using Product Cubic Phase Function

    Quadratic Chirp Signals Using Product

    Cubic Phase Function C:hineseJournalofElectronics

    Vo1.16,No.1,Jan.2007

    ParameterEstimationofMulticomponent

    Linear/QuadraticChirpSignalsUsingProduct

    CubicPhaseFunction

    WANGPuandYANGJianyu

    lSchoolofElectronicEngineering,University01ElectronicScienceandTechnologyofChin

    a,Chengdu610054,China

    AbstractTheidentifiab.1ityproblemariseswhenthe Cubicphasefunction(CPF)dealswithmulticomponent linear/quadraticsignals.Ibremovethisproblem,anim- provedmethodonthebasisofCPFisproposedformul- ticomponentlinear/quadraticchirpsignals.Theproposed algorithmprovidesaauraberofadvantages,suchaslow thresholdSNR,suppressionofthecrossterms/spurious peakandrelativelyfastimplementation.Thenumerical simulationsareevaluatedforverifyingtheproposedalgo- rithm.

    Keywords——Statisticalsignalprocessing,Chirpsig- nal,Parameterestimation.

    I.Introduction

    Linearfrequencymodulated(LFM)signalsorchirpsignals areoftenfoundinsignalprocessingandcommunicationappli- cationssuchasradar,sonar,biomedicine,seismicanalysis,and mobilecommunications.Thispaperfocusesontheanalysisof

    thelinearandquadraticchirpsigna1.Twopracticalapplica- tionsofthiskindofsignalcanbefoundinRefs.I131.The

    firstapplicationisthepassiveintelligentradarsurveillance, whereonetriestodeterminewhetheralinearchirp.quadratic chirp,orothertypeofradarpulseisbeingtransmitted.The otherapplicationisintheprocessingofecholocationsignals frombrownbats.Thesesignalsaremulticomponentquadratic chirpsignals,withtheparametersvaryingaccordingtotheac- tivityofthebat.

    Intheliterature,thetime.frequencydistributions,such astheWigner.villedistributionfWVD)anditsrelatedbi- linearclasst,.

    areemcienttorevealtheInstantaneousfre-

    quency(IF)overthetime-frequencyplaneandthenestimate theparametersusingtheestimatedIF.Astotheparamet- ricestimation,theMa~ximumlikelihood(ML)estimationis themostaccuratetoestimatetheparametersofthechirp signalt6,'.

    Thedirectimplementation.however,requires

    multi.dimensionalmaximization.i.e.two-dimensionalsearch f2.D1forlinearchirpandthree-dimensional(3-D)searchfor quadraticchirp.Moreover,iftheobjectivefunctionisnot convex.theMLestimationislikelytoconvergetolOCalmax- ima.Toavoidthemulti.dimensionalgridsearch,thesub- optimaltechniques,suchasthephasean-wrapping_oJ'Poly- nomialphasetransform(PPT)l1,Producthigh-orderam-

    biguityfunctionfPHAF)JandRadon-ambig:uitytransform (RAT)[,arepresented.Besidestiona1FouriertransformfFrFT)[.thesemethods,theFrac

    hasreceivedconsiderable

    attentionaswellforitsimprovednoiserejectionandability

    toresolveclosely-spacedchirpsignals.Recently,P.O'Shea proposedabilineartransform.whichisknownastheCubic phasefunction(CPF)(alsocalledasTime-frequencyratedis

    tributionrTFRD))[2,3Jfortheanalysisoflinearandquadratic chirp.Forachirpsignal,

    ()::?P-...n'

    (?一1)(?一1)

    n一—一,,—一(1)

    whereA,(n)andaiaretheamplitude,phaseandtheith- ordercoefficientrespectively,andNisodd,theCPFisdefined cPF(n,)=+..(

    n+r).(nr)e-j~'-r2dr(2)

    Substitutingx(n)inEq.(2)withEq.(1)andusingtheidentity theresultis

    CPF(n,)=

    jvt2dt=eJ(/(3)

    2ej[2(n)+/厂————A

    \/

    ifIFR(n)

    A2ej[2(n)/厂————一

    \/

    ifIFR(n1<

    (4)

    whereIFR(n)istheInstantaneousfrequencyrate(IFR)de- finedas

    IFR(n)=d2~(n)/dn.(5)

    Inspecific,forlinearchirpcase(P=2),theIFRis2a2, whereastheIFRis2a2+6aanforquadraticchirpcase(P=3). ItisobviousthattheCPFwillmaximizealongtheIFRof thechirpsigna1.TheperformanceoftheCPFisanalyzedin

    thetermsofestimatebiasandvarianceinRefI3Iforthea2 anda3estimate.Formulticomponentchirpsignals,thespu- riouspeaksariseandthustheidentifiabilityproblemoccurs. ManuscriptReceivedOct.2005;AcceptedAug.2006

128ChineseJournaloyElectronics2007

    Toremovethisidentifiabilityproblem,animprovedalgorithm basedontheProductCPF(PCPF1,whichexploitsthedif- ferenttimedependenceoftheautotermsandthecrossterms orthespuriouspeaks,ispresented.Itisworthynotingthat thesimilartechnique,thePHAF,isproposedinR,ef-l101.The PHAFdiscernstheautotermsandcrosstermsbyexploiting thedifferentlagdependence.DifferentfromthePHAF,the PCPFutilizesthedifferenttimedependencies.

    Thepaperisorganizedasfollows.InSectionII.theiden- tifiabilityproblemisanalyzedforbothlinearandquadratic chirpcases.ThePCPFisthenpresentedinSectionIIIand SectionIVprovidesanumberofsimulationexamples.Finally, conclusionsaredrawninSectionV.

    II.TheIdentifiabilityProblemofthe

    CPFforMulticomponentSignals

    Toillustratetheidentifiabilityproblemarisingfrommulti

    componentsignals,thediscussionwillbecategorizedintotwo parts:P=2(1inearchirpcase)andP=3(quadraticchirp case,13.

    1.CaseofP=2

    Con$idertwolinearchirpsignalsas

    z(n)=AleJ(dl,.+al,ln+dl,2n'+A2ej(d2,.+d2,ln+d2,2n'(6) SubstitutingEq.(6)toEq.(2)yields

    +A2ej(d2,.+a2.1r~+a2~2n2+a2,

3~3)(10)

    SubstitutingEq.(10)intoEq.(2),thebilineartransform x(n+7-)z(n7-)hasfouritems:twoautotermsandtwocross terms.Theresultsoftheautotermsare:

    Aej(2ai,o+2ai.1r~+2ai,2n2+2al,3n3)ej(2ai,2+6al,3n)f ,:1,2

    (11)

    FromEq.(11),eachautotermoccursalongrespectiveIFR as=2al,

    2+6ai,3n.Incontrast,thecrosstermsarederived as:

    rA1A2z(n)expj{[(al,la2,1)+2(al,2a2,2)n

    l+3(0l,3a2,3)n27-+[(0l,2+02,2)+3(0l,3+a2,3)7-2

    J+(0l,3a2,3)7-.)

    lA1A2z(n)expj{[(a2,lal,1)+2(a2,2al,2)n+3(a2,3

    Ial,3n27-+[(0l,2+02,2)+3(al,3+a2,3)7-2

    L+(02,3al,3)7-.)

    (12)

    where

    (n)=eJ((dl,.+d2,.)+(dl,l+d2,1)n+(al,2-}-a2,2)n+(al,3+a2,3)n.)

    FromEq.(12),thetroublesomecrosstermsOCCuralonganon- linearfunctionintime.Inparticular,if

    r(61,162,1)+2(al,2a2,2)n=0

    al,3a2,30

    GPF,):A.(.l,.+ln+zn)+ooeJ(2ai,2-D)r2d7-tw.cr.sstermsmergeint..neemaLsJ0

    +Aa2,0-}-a2,1n-}-a2,2n2).zrJ0

    +AA.zc,+oo

    e

    J(d2+.z,2n)f

    .e

J{(.l,l.2,1)+2(.l,2.2,20n)dr

    +A1A2z(n)

    .

    e

    j{(.2,1.l,

    Itisfoundthat.if

    e

    j(al,2+a2,2n)f

    ,

    2.l,2)n)d7-(7)

    (a2,1al,1)+2(a2,2al,2)n=0(8)

    thespuriouspeakarises,forexample

    (13)

    AlA2(n)eJ{(dl,2+d2,2)+3(dl,3+d2,3)nl)(14)

    Inthiscase,thespuriouspeakarisesat=(al,2+62,2)+ 3(al,3+a2,z)n,wherenissubjecttoEq.(13).

    3.Summary

    Extensiontothemulticomponentcase(thenumberofcom- portentsismorethan2)canbesimilarlyconductedinthepro- cedureabove.Ingeneral,themorecomponentsexist,themore crosstermsarise,andthemorespuriouspeaksmayOCCur. Basedontheaboveanalysis,thealgorithminRef.[2], Ref.[3isnotsuitabletoestimatemulticomponentsignalsdue totheexistingofthecrosstermsanditsresultingspurious peaks.Furthermore,thereisnotaprinciplewhichavoidthe spuriouspeaks.Toexploitthedifferentdependenceontime, theproductformoftheCPFisproposedinthefollowingsec

    tion.

    GPF(n.,)=A2(al,0+al,lne+al,2n2)~~

    eJ(2al,2--I2)~-2d1-III.Pr.ductcubicPhaseFuncti.n

+A;eJ2(a2,0+a2,1ne+a2,2n2)+ooe

    J(2a2,2-D)r2d7-

    0

    +2AA.z(n.)fO+~~eJ(al,2+a2,2--I2)"r2dr(9)

    wherencsat~fiesEq.(8).Thespuriouspeakcorrespondingto thelastterminEq.(9)locatesat=al,2+a2,2.

    2.CaseofP=3

    InthecaseofP=3,wefirstconsidertwoquadraticFM signals:

    (n):A1eJ(l,.+dl,ln+dl,2n+al,3r~3)

    Theabovesectionestablishestheexistingofcrossterms andthespuriouspeaks.Formulticomponentsignal,theCPF hastofacetheidentifiabilityproblemandhenceanimproved algorithmisneed.Inthissection,theProductCPF(PCPF) ispresentedforbothlinearandquadraticchirpsignals. 1.ProductCPFforP=2

    Forlinearchirpsignal,itisfoundthatthecrossterms disperseacrosstheIFRdomainandturnintospuriouspeak, whereastheautotermsconcentrateonastraightlinethatlo- cateat:2ak.

    2,k=1,,.Inotherwords,theauto

    termsareindependentontimewhilethecrosstermsarere- latetotime.Hence,itprovidesthebasisfordiscerningthe ?

    2

    ,

ParameterEstimationofMulticomponentLinear/QuadraticChirpSignalsUsingProductCu

    bicPhaseFunction129

    autotermsfromcrossterms,evenforthecrosstermsgiverise

tospuriouspeaks.

    GiventhesetofLdifierenttimepositionsni,ni?f(?一

    1)/2:(N1)/2],theCPF(ni,)correspondingtothetime setsiscomputedbyEq.(2)andthenthePCPFastheproduct oftheCPFsatdifferenttimepositionsisdefinedas PCPF(f1)=L_1CPF(n,)(15)

    FromEq.(15),thePCPFwillpeakat=2ak,2.Itpro-

    videsanumberofadvantages:(1)themultiplyoperationam- plifiestheautotermsduetothefactthattheautotermsalign, andweakenscrosstermsduetoitsdispersioninthedomain; (2)thespuriouspeaksaresuppressedandeveneliminated;(3) thenoisere]'ectionisimprovedwithrespecttoCPF.Obviously, themoresetsoftimeused,thebetterthecrosstermssuppres- sioncapability.butthehigherthecomputationalload. 2.ProductCPFforP=3

    ThedirectextensionofthePCPFforP=2cannotdeal withmulticomponentquadraticchirpsignals,sinceboththe autotermsandcrosstermsarerelatedtotime.Fortunately, wefurtherfindthatthelocationsofcrosstermsisnonlinearto thetimepositions.whereastheautotermsarelinearrelatedto thetimepositions.Thispropertyprovidesthecapabilityfor decouplingtheautotermsandcrosstermsorspuriouspeaks. ThePCPFforP=3isfirsttoextractthea2tolettheauto termsacrosstheoriginintheTime-frequencyrate(TFR)d main.thenscaletheIFRspectraltoaligntheautotermsand finallymultiplytheCPFsatdifferenttimepositions.There- fore.thePCPFforquadraticchirpsignalwillbeintroducedin threesteps:(1)Reassignmentoperator;(2)IFRscaling;(3) ProductformofCPF.

    (1)Reassignmentoperator

ThereassignmentisusedintheTime-frequency(TF)anal

    ysisbyrelocatingtheenergydistributionintheTFdomain}"J. Ingeneral,thisoperatormovesthedistributiontothecenter ofgravityoftheseenergycontributions.Herein,thismethod isusedtoremovethecontributionofthea2fromtheIFR. Ref.[3showsthatthea2estimateatn=0usingtheCPFap- proachestheCRLBandhasalowthresholdSNR.ie-3dB.

    Therefore,wefirstestimatethea2atn=0.Oncethea2is obtained,thereassignmentoperatorisappliedintheTFRD, yields,

    RCPF(n,)=CPF(n,hilt)(16)

    whereRCPFdenotesthereassignmentoftheCPFand hilt=252.

    SubstitutingEq.(4)toEq.(16)yields,

    RCPF(n,)l=A\//(17)

    Itisobviousthatthereassignmentmovestheautoterms acrosstheoriginattheTFRdomain.Thus.thereassigned IFRwilldistributealong=6a3n.Notethateachreas

    signoperationresultsinonlyonecomponentpassthroughthe originintheTFRdomain.Therefore.thePCPFsequentially extractsthecomponentandtheestimatestheextractedcom- ponent.

    f2)ScaleintheIFRdomain

    Thereassignmentoperatorresultsintheautotermslo- catealongthestraightline=6a3n.Inthiscase,theauto termsattwotimepositionsnlandn2peakat16a3nl and2=6a3n2,respectively.Toaligntheautoterms, weemploysthespectralscalingtechniqueintroducedinthe PHAF[10asl:ln2/n1.Usingthescaleoperation,the autotermsat16a3nlwillmovetol6a3n2,which

    istheexactlocationsoftheautotermsat2=6a3n2.Since thecrosstermsisnonlinearlyrelatedtotimepositions,both reassignmentandscaleoperationsmisalignthecrossterms. Thus,thescaleoperationcandecoupletheautotermsand crosstermsintheIFRdomain.

    (3)DefinitionofPCPF

    Afterthereassignmentandscaleoperations.wecandefine thePCPFforP=3inthesimilarwayofthePCPFforP=2. GiventhesetofLdifferenttimepositionsni,theCPF(ni,) correspondingtothetimesetiscomputedandthenPCPFas theproductoftheCPFsatdifferenttimepositionsisdefined as

    PCPF(~;n3)=L_1RCPF()(18)

    FromEq.(18),eachreassignedCPFslice=6a3niwill bescaledat=6a3n3.Thustheautotermswillbeamplified intheproduct.whilethecrosstermsandspuriouspeakwill besuppressed.Fig.1illustratestheprocedureforthePCPF forP=3.Inthisfigure.weselecttheCPFslicesatthree timepositions,nl,2,n3.Toimplementthescalingoperation easily,wespecifythatn32n2=4n1.

    3.Computationcomplexityandtimepositionse- lection

    Ingeneral,thePCPFrequiresaboutLtimesmorecom- putationsthantheCPF.However,thecomputationofCPF canbereducedtotheorderofNlog2Nusingthesubbandde- compositiontechniques[2,3,whichisequaltothecomputation ofanNpointsFastFouriertransformfFFT).Moreover,Lis alwaysasmallnumber.whichmeanstheadditionalcostisnot excessiveandthePCPFcanberelativelyfastimplemented. AstotheselectionoftimepositionofCPF,thereisnot

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