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# Quadratic Chirp Signals Using Product Cubic Phase Function

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Quadratic Chirp Signals Using Product Cubic Phase Function

Cubic Phase Function C:hineseJournalofElectronics

Vo1.16,No.1,Jan.2007

ParameterEstimationofMulticomponent

CubicPhaseFunction

WANGPuandYANGJianyu

lSchoolofElectronicEngineering,University01ElectronicScienceandTechnologyofChin

a,Chengdu610054,China

Keywords——Statisticalsignalprocessing,Chirpsig- nal,Parameterestimation.

I.Introduction

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-

attentionaswellforitsimprovednoiserejectionandability

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

()::?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)

128ChineseJournaloyElectronics2007

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

II.TheIdentifiabilityProblemofthe

CPFforMulticomponentSignals

Toillustratetheidentifiabilityproblemarisingfrommulti

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

(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

,

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-

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