A New Phase Unwrapping Algorithm.pdf

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1、1IOIEEETRANSACTIONSONACOUSTICS,SPEECH,ANDSIGNALPROCESSING,VOL.ASSP-25,NO.2,APRIL1911ANewPhaseUnwrappingAlgorithmJOSEM.TRIBOLET,STUDENTMEMBER,IEEEAbstract-Anewphaseunwrappingalgorithmisproposedthatarg[X(*)]i:itsphase.Let?(n)bethecomplexcepstrumcombinestheinformationcontainedinboththephasederivativ

2、eandofx(n)andX(ei’)itsFouriertransform.Thentheprincipalvalueofthephaseintoanadaptivenumericalintegrationscheme.Thisnewalgorithmhasprovenitselftobeveryreliableandit2(eiW>=logx(eiw>=logIX(eia>Itjarg[~(eiw)](2)cpbeeasilyincorporatedinstandardhomomorphicsignalprocessors,wherewehaveassumedthatthecompl

3、exlogarithmisdefinedsothat(2)representsAaproperFouriertransform.InthatI.INTRODUCTIONcase,thederivativeofX(eiw)iswelldefined:OMOMORPHICsignalprocessing,anonlineartechniqueHbasedonlinearfilteringoperationsonthecomplex~_d2(eiw)--dlogX(eiw)dX(efw)/dw--(3)cepstrum[11,wasintroducedin1965byOppenheimandh

4、asdwdwX(eia)’beenquitesuccessfullyappliedtoavarietyofproblems,anditfollowsthatthederivativeofarg[X(eiw)]isgivenbynamely,speech,seismic,andEEGdataprocessing.theimaginarypartof(3):Thecomputationofthecomplexcepstrumrequiresthedefinitionofacontinuousphasecurvefortheinputdata[11.darg[x(eiW)]--XR(eiw)X

5、;(eiw)-Xz(eiw)xi(e’”)ThisphasecurveiscommonlycalledtheunwrappedphaseanddwIX(eiw)12itcanberigorouslydefinedintermsoftheintegralofitsderivative[11.(4)Conventionalphaseunwrappinghasbeendoneeitherbywheretheprimedenotesd/dwandappropriateprocessingoftheprincipalvalueofthephaseX’(eiw)=Xi(eiw)tjXj(eiw)=-

6、jFT{nx(n)}.(5)(Schafer’salgorithm)orbynumericalintegrationofthephasederivative.Thephasearg[X(eiw)]canthusbeunambiguouslydefinedThealgorithmweproposehereisanadaptivenumericalintermsofitsderivativearg‘[X(eiw)]asintegrationschemethatcombinestheinformationcontainedWinboththephasederivativeandtheprinc

7、ipalvalueofthearg[X(eiw)]=arg’[X(ejs)]dv(6)phase,asfollows:ateachfrequency,asetofpermissiblephasevaluesisdefinedbyaddingintegermultiplesof2~towithinitialconditionsgivenbytheprincipalvalueofthephase.Theselectionofoneoft