数字信号处理a（双语）chapter 6-z transform b

Chapter6z-Transform.6.4TheInversez-Transform6.5z-TransformTheorems

PartB:InverseZTandZTTheorems.6.4TheInversez-Transform6.4.1GeneralExpression6.4.2Inversez-TransformbyTableLook-UpMethod6.4.3Inversez-TransformbyPartial-FractionExpansion6.4.4Partial-FractionUsingMATLAB6.4.5Inversez-TransformviaLongDivision6.4.6Inversez-TransformUsingMATLAB.6.4.1GeneralExpression.6.4.1GeneralExpression.6.4.1GeneralExpression逆z变换是一个对X(Z)Zn-1进行的围线积分，积分路径C是一条在X(Z)收敛环域（Rx-，Rx+）内以逆时针方向绕原点一周的单围线。围线积分路径.直接计算围线积分比较麻烦，一般不采用此法求逆z变换，求解逆z变换的常用方法有：留数定律法查表法部分分式法★长除法6.4.1GeneralExpression.6.4.1GeneralExpression.利用留数定理计算围线积分•一阶极点的留数

•N阶极点的留数.1.1GeneralExpression.6.4.1GeneralExpression.6.4.1GeneralExpression.6.4.1GeneralExpression.6.4.1GeneralExpression.6.4.1GeneralExpression.6.4.1GeneralExpression.Example6.126.4.2TableLook-upMethodLookupTable6.1onPage253.6.4.3Inversez-TransformbyPartial-FractionExpansionArationalz-transformG(z)withacausalinversetransformg(n)hasanROCthatisexteriortoacircleHereitismoreconvenienttoexpressG(z)inapartial-fractionexpansionformandthendetermineg(n)bysummingtheinversetransformoftheindividualsimplertermsintheexpansion.6.4.3Inversez-TransformbyPartial-FractionExpansion.6.4.3Inversez-TransformbyPartial-FractionExpansionSolutions:Step1--ConvertingG(z)intotheformofproperfractionsbylongdivisionStep2--Summingtheinversetransformoftheindividualsimplertermsintheexpansion.6.4.3Inversez-TransformbyPartial-FractionExpansion.6.4.3Inversez-TransformbyPartial-FractionExpansion.6.4.4Partial-FractionUsingMATLAB[r,p,c]=residuez(num,den)developsthepartial-fractionexpansionofarationalz-transformwithnumeratoranddenominatorcoefficientsgivenbyvectorsnumanddenVectorrcontainstheresiduesVectorpcontainsthepolesVectorccontainstheconstantsηl.[num,den]=residuez(r,p,c)

convertsaz-transformexpressedinapartial-fractionexpansionformtoitsrationalform6.4.4Partial-FractionUsingMATLAB.6.4.5Inversez-TransformviaLongDivision.6.4.5Inversez-TransformviaLongDivision.6.4.6Inversez-TransformUsingMATLABThefunctionimpzcanbeusedtofindtheinverseofarationalz-transformG(z)ThefunctioncomputesthecoefficientsofthepowerseriesexpansionofG(z)Thenumberofcoefficientscaneitherbeuserspecifiedordeterminedautomatically.6.4TheInversez-Transform6.5z-TransformTheorems

PartB:InverseZTandZTTheorems.6.5z-TransformTheorems..6.5z-TransformTheorems.6.5z-TransformTheorems.6.5z-TransformTheorems.6.5z-TransformTheorems.6.6ComputationoftheConvolutionSumofFinite-LengthSequences6.6.1LinearConvolutionUsingPolynomialMultiplication6.6.2CircularConvolutionUsingPolynomialMultiplication

PartC:Convolution

.6.6.1LinearConvolutionUsingz-Transform

Letdenote{x[n]},0nL,

afinite-lengthsequenceoflengthL+1Letdenote{h[n]}0nM,afinite-lengthsequenceoflengthM+1Weshallevaluatey[n]=x[n]*h[n]usingz-transformNote:{y[n]}

isasequenceoflengthL+M+1.6.6.1LinearConvolutionUsingz-Transform

LetX(z)denotethez-transformof{x[n]}

whichisapolynomialofdegreeLinz-1,i.e.,X(z)=x[0]+x[1]z-1+x[2]z-2+…+x[L]z-L

LetH(z)denotethez-transformof{h[n]}whichisapolynomialofdegreeMinz-1H(z)=h[0]+h[1]z-1+h[2]z-2+…+h[M]z-M.6.6.1LinearConvolutionUsingz-Transform

Fromtheconvolutionpropertyofthez-transform,itfollowsthatthez-transformof{y[n]}issimplygivenby

Y(z)=X(z)H(z)whichisapolynomialofdegreeL+Minz-1,i.e.,Y(z)=y[0]+y[1]z-1+y[2]z-2+…+y[L+M]z-(L+M).6.6.1LinearConvolutionUsingz-Transform

WhereIntheabovewehaveassumed

x[n]=0forn>L

h[n]=0forn>M.6.6.1LinearConvolutionUsingz-Transform

(pp.278)Example6.30LinearConvolutionofOne-SidedSequencesUsingthePolynomialMultiplicationMethod

.6.6.2CircularConvolutionUsingz-Transform

Let{x[n]}and{h[n]}betwolength-Nsequencesdefinedfor0nN-1

withX(z)andH(z)denotingtheirz-transformsLetyc[n]=x[n]Nh[n]denotetheN-pointcircularconvolutionofx[n]

andh[n]LetyL[n]=x[n]*h[n]denotethelinearconvolutionofx[n]

andh[n].6.6.2CircularConvolutionUsingz-Transform

LetYc(z)andYL(z)denotethez-transformsofyc[n]andyL[n]Itcanbeshownthat

Yc(z)=<YL(z)>(z-N

-1)Themodulooperationwithrespecttoz-N

-1istakenbysettingz-N=1

<z-N

-1)>(z-N

-1)=z-1

<z-N

-2)>(z-N

-1)=z-2

.6.6.2CircularConvolutionUsingz-Transform

(pp.279)Example6.32CircularConvolutionofCausalSequencesUsingthePolynomialMultiplicationMethod.6.6.2CircularConvolutionUsingz-Transform

.6.6.2CircularConvolutionUsingz-Transform

.6.7TheTransferFunctionThetransferfunctionisageneralizationofthefrequencyresponsefunction.TheconvolutionsumdescriptionofanLTIdiscrete-timesystemwithanimpulseresponseh[n]isgivenby.6.7TheTransferFunctionTakingthez-transformsofbothsidesweget.6.7TheTransferFunctionThus,Y(z)=H(z)X(z)

Or,Therefore,.6.7.1DefinitionHence,H(z)=Y(Z)/X(z)ThefunctionH(z),whichisthez-transformoftheimpulseresponseh[n]oftheLTIsystem,iscalledthetransferfunctionorthesystemfunctionTheinversez-transformofthetransferfunctionH(z)yieldstheimpulseresponseh[n].6.7.2TransferFunctionExpressionConsideranLTIdiscrete-timesystemcharacterizedbyadifferenceequationItstransferfunctionisobtainedbytakingthez-transformofbothsidesoftheaboveequationThus.6.7.2TransferFunctionExpressionOr,equivalentlyasAnalternateformofthetransferfunctionisgivenby.6.7.2TransferFunctionExpressionOr,equivalentlyas1,2,…,

Marethefinitezeros,and1,2,…,

NarethefinitepolesofH(z)IfN>M,thereareadditional(N-M)zerosatz=0IfN<M,thereareadditional(M-N)polesatz=0

.6.7.2TransferFunctionExpressionForacausalIIRdigitalfilter,theimpulseresponseisacausalsequenceTheROCofthecausaltransferfunctionisthusexteriortoacirclegoingthroughthepolefurthestfromtheoriginThustheROCisgivenby.6.7.2TransferFunctionExpressionExample-ConsidertheM-pointmoving-averageFIRfilterwithanimpulseresponse

Itstransferfunctionisthengivenby.6.7.2TransferFunctionExpressionThetransferfunctionhasMzerosontheunitcircleatz=ej2k/M,0kM-1ThereareM-1polesatz=0andasinglepoleatz=1Thepoleatz=1exactlycancelsthezeroatz=1TheROCistheentirez-planeexceptz=0M=8.6.7.2TransferFunctionExpressionExample-AcausalLTIIIRdigitalfilterisdescribedbyaconstantcoefficientdifferenceequationgivenby

y[n]=x[n-1]-1.2x[n-2]+x[n-3]+1.3y[n-1]-1.04y[n-2]+0.222y[n-3]Itstransferfunctionisthereforegivenby.6.7.2TransferFunctionExpressionAlternateforms:ROC:Note:Polesfarthestfromz=0haveamagnitude.6.7.3FrequencyResponsefromTransferFunctionIftheROCofthetransferfunctionH(z)includestheunitcircle,thenthefrequencyresponseH(ej)oftheLTIdigitalfiltercanbeobtainedsimplyasfollows:

ForarealcoefficienttransferfunctionH(z)itcanbeshownthat.6.7.3FrequencyResponsefromTransferFunctionForastablerationaltransferfunctionintheformthefactoredformofthefrequencyresponseisgivenby.6.7.3FrequencyResponsefromTransferFunctionItisconvenienttovisualizethecontributionsofthezerofactor(z-k)andthepolefactor(z-k)fromthefactoredformofthefrequencyresponseThemagnitudefunctionisgivenby.6.7.3FrequencyResponsefromTransferFunctionThephaseresponseforarationaltransferfunctionisoftheformThemagnitude-squaredfunctionofareal-coefficienttransferfunctioncanbecomputedusing.6.7.4GeometricInterpretationofFrequencyResponseComputation

Thefactoredformofthefrequencyresponse

isconvenienttodevelopageometricinterpretationofthefrequencyresponsecomputationfromthepole-zeroplotasωvariesfrom0to2πontheunitcircle.6.7.4GeometricInterpretationofFrequencyResponseComputationThegeometricinterpretationcanbeusedtoobtainasketchoftheresponseasafunctionofthefrequencyAtypicalfactorinthefactoredformofthefrequencyresponseisgivenby(ejω

-ρejΦ

)whereρejΦ

isazeroifitiszerofactororisapoleifitisapolefactor.6.7.4GeometricInterpretationofFrequencyResponseComputationAsshownbelowinthez-planethefactor(ejω-ρejΦ)representsavectorstartingatthepointz=ρejΦ

andendingontheunitcircleatz=ejω.Asωisvariedfrom0to2π,thetipofthevectormovescounterclockisefromthepointz=1tracingtheunitcircleandbacktothepointz=16.7.4GeometricInterpretationofFrequencyResponseComputation.Asindicatedby

themagnituderesponse|H(ejω)|ataspecificvalueofωisgivenbytheproductofthemagnitudesofallzerovectorsdividedbytheproductofthemagnitudesofallpolevectors6.7.4GeometricInterpretationofFrequencyResponseComputation.Likewise,fromweobservethatthephaseresponseataspecificvalueofωisobtainedbyaddingthephaseofthetermp0/d0andthelinear-phasetermω(N-M)tothesumoftheanglesofthezerovectorsminustheanglesofthepolevectors6.7.4GeometricInterpretationofFrequencyResponseComputation.Thus,anapproximateplotofthemagnitudeandphaseresponsesofthetransferfunctionofanLTIdigitalfiltercanbedevelopedbyexaminingthepoleandzerolocationsNow,azero(pole)vectorhasthesmallestmagnitudewhenω

=Φ

6.7.4GeometricInterpretationofFrequencyResponseComputation幅度最小幅度最大.Tohighlyattenuatesignalcomponentsinaspecifiedfrequencyrange,weneedtoplacezerosveryclosetoorontheunitcircleinthisrange（零点——谷值）Likewise,tohighlyemphasizesignalcomponentsinaspecifiedfrequencyrange,weneedtoplacepolesveryclosetoorontheunitcircleinthisrange（极点——峰值）6.7.4GeometricInterpretationofFrequencyResponseComputation..Example:y(n)=x(n)-x(n-4).Question：1、极点的位置对系统的稳定性会有影响吗？2、系统在Z域的稳定性条件是什么？.AcausalLTIdigitalfilterisBIBOstableifandonlyifitsimpulseresponseh[n]

isabsolutelysummable,i.e.,

WenowdevelopastabilityconditionintermsofthepolelocationsofthetransferfunctionH(z)6.7.5StabilityConditioninTermsofthePoleLocations

.6.7.5StabilityConditioninTermsofthePoleLocationsTheROCofthez-transformH(z)oftheimpulseresponsesequenceh[n]isdefinedbyvaluesof|z|=rforwhichh[n]r-nisabsolutelysummableBIBOstable.Thus,iftheROCincludestheunitcircle|z|=1,thenthedigitalfilterisstable,andviceversaInaddition,forastableandcausaldigitalfilterforwhichh[n]isaright-sidedsequence,theROCwillincludetheunitcircleandentirez-planeincludingthepointz=∞AnFIRdigitalfilterwithboundedimpulseresponseisalwaysstable6.7.5StabilityConditioninTermsofthePoleLocations.Ontheotherhand,anIIRfiltermaybeunstableifnotdesignedproperlyInaddition,anoriginallystableIIRfiltercharacterizedbyinfiniteprecisioncoefficientsmaybecomeunstablewhencoefficientsgetquantizedduetoimplementation6.7.5StabilityConditioninTermsofthePoleLocations.Example-ConsiderthecausalIIRtransferfunctionTheplotoftheimpulseresponsecoefficientsisshownonthenextslide6.7.5StabilityConditioninTermsofthePoleLocations.6.7.5StabilityConditioninTermsofthePoleLocationsh[n]decaysrapidlytozerovalueasnincreasesTheabsolutesummabilityconditionofh[n]issatisfied.Hence,H(z)isastabletransferfunction..Now,considerthecasewhenthetransferfunctioncoefficientsareroundedtovalueswith2digitsafterthedecimalpoint:6.7.5StabilityConditioninTermsofthePoleLocations.6.7.5StabilityConditioninTermsofthePoleLocations

increasesrapidlytoaconstantvalueasnincreasesHence,theabsolutesummabilityconditionofisviolated.Thus,isanunstabletransferfunction.ThestabilitytestingofaIIRtransferfunctionisthereforeanimportantproblemInmostcasesitisdifficulttocomputetheinfinitesumForacausalIIRtransferfunction,thesumScanbecomputedapproximatelyas6.7.5StabilityConditioninTermsofthePoleLocations.6.7.5StabilityConditioninTermsofthePoleLocationsThepartialsumiscomputedforincreasingvaluesofKuntilthedifferencebetweenaseriesofconsecutivevaluesofSKissmallerthansomearbitrarilychosensmallnumber,whichistypically10-6ForatransferfunctionofveryhighorderthisapproachmaynotbesatisfactoryAnalternate,easy-to-test,stabilityconditionisdevelopednext.ConsiderthecausalIIR