Chat6z-tranformDefinitionz-TransformsRegionofConvergencez-TransformsTheinversez-Transformsz-TransformsPropertiesTheTransferFunctionChat6z-tranformDefinition6.1DefinitionandPropertiesTheDTFTprovidesafrequency-domainrepresentationofdiscrete-timesignalsandLTIdiscrete-timesystems.Becauseoftheconvergencecondition,inmanycase,theDTFTofasequencemaynotexist.Asaresult,itisnotpossibletomakeuseofsuchfrequency-domaincharacterizationinthesecase.6.1DefinitionandProperties6.1DefinitionandProperties(p227)

z-TransformmayexistformanysequenceforwhichtheDTFTdoesnotexist.Moreover,useofz-Transformtechniquespermitssimplealgebraicmanipulation.Consequently,z-Transformhasbecomeanimportanttoolintheanalysisanddesignofdigitalfilters.1.

Definition6.1DefinitionandProperties6.1DefinitionandProperties(p227)RezjImzz=rejr11jjUnitcircle06.1DefinitionandPropertiesForagivensequence,thesetRofvaluesofzforwhichitsz-transformconvergesiscalledthe

regionofconvergence(ROC).6.1DefinitionandProperties(p227)Theinterpretationofthez-transformG(z)astheDTFTofsequenceg[n]r-n.Wecanchoosethevalueofrproperlyeventhoughg[n]isnotabsolutelysummable.Ingeneral,ROCcanberepresentedasForagivensequence,thes6.1DefinitionandProperties(p227)Note:Thez-transformofthetwosequenceareidenticaleventhoughthetwoparentsequencearedifferent.

Onlywayauniquesequencecanbeassociatedwithaz-transformisbyspecifyingitsROC.

TheDTFTG(ejω)ofasequenceg[n]convergesuniformlyifandonlyiftheROCofthez-transformG(z)ofg[n]includestheunitcircle.

6.1DefinitionandProperties6.1DefinitionandProperties(p227)Table6.16.1DefinitionandProperties6.2Rationalz-Transforms(p231)M-----thedegreeofthenumeratorpolynomialP(z)N-----thedegreeofthedenominatorpolynomialD(z)6.2Rationalz-Transforms(p26.2Rationalz-Transforms(p231)InEq.(6.15),thereareMfinitezerosandNfinitepolesIfN>M,thereareadditionalN-Mzerosatz=0.IfN<M,thereareadditionalM-Npolesatz=0.6.2Rationalz-Transforms(p236.3ROCofRationalz-TransformsTheROCofarationalz-transformisboundedbythelocationofitspoles.TheROCofarationalz-Transformcannotcontainanypoles

Asequencecanbeoneofthefollowingtype:finite-length,right-sided,left-sidedandtwo-sided.Iftherationalz-transformhasNpoleswithRdistinctmagnitudes,thenithasR+1ROCs,R+1distinctsequencehavingthesamerationalz-transform.6.3ROCofRationalz-Transfoa)TheROCofthez-transformofafinite-lengthsequencedefinedforM≤n≤Nistheentirez-planeexceptpossiblyz=0and/orz=+∞6.3ROCofRationalz-TransformsWehavethefollowingobservationwithregardtotheROCofaRationalz-Transforma)TheROCofthez-transform6.3

ROCofRationalz-Transformsb)TheROCofthez-transformofaright-sidedsequencedefinedforM≤n≤∞istheexteriortoacircleinthez-planepassingthroughthepolefurthestfromtheoriginz=0.6.3ROCofRationalz-Transfor6.3ROCofRationalz-Transformsc)TheROCofthez-transformofaleft-sidedsequencedefinedfor-∞≤n≤Nistheinteriortoacircleinthez-planepassingthroughthepolenearestfromtheoriginz=0.6.3ROCofRationalz-Transfor6.3ROCofRationalz-Transformsd)TheROCofthez-transformofatwo-sidedsequenceofinfinitelengthisaringboundedbytwocircleinthez-planepassingthroughtwopoleswithnopolesinsidethering.6.3ROCofRationalz-Transfor6.4TheInversez-Transform(p238)6.4.1GeneralExpression-----Cauchy’sintegraltheorem6.4TheInversez-Transform(6.4.1GeneralExpressionIfthepoleatz=λ0ofG(z)zn-1isofmultiplicitym.6.4.1GeneralExpressionIfth6.4.3Partial-FractionExpansionMethodArationalz-transformG(z)withacausalinversetransformg[n]hasanROCthatisexterior----M≥N,P(z)/D(z)isanimproperfraction----M<N,P1(z)/D(z)isaproperfraction6.4.3Partial-FractionExpansi6.4.3Partial-FractionExpansionMethodSimplePoles6.4.3Partial-FractionExpansi6.4.3Partial-FractionExpansionMethodMultiplePolesIfthepoleatz=visofmultiplicityLandtheremainingN-Lpolesaresimple.6.4.3Partial-FractionExpansi6.5z-TransformProperties(p246)ConjugationPropertyTime-ReversalPropertyLinearityProperty6.5z-TransformProperties(p6.5z-TransformProperties(p246)MultiplicationbyanExponentialSequenceDifferentiationPropertyTime-ShiftingProperty6.5z-TransformProperties(p6.5z-TransformProperties(p246)ModulationtheoremParseval’sRelationConvolutionProperty6.5z-TransformProperties(p6.7TheTransferFunction(p258)h[n]x[n]y[n]6.7.1Definition6.7TheTransferFunction(p256.7.1Definition-----systemfunctionortransferfunction6.7.2TransferFunctionExpressionFIRDigitalFilterForacausalFIRfilter,0≤N1≤N2,theROCofH(z)istheentirez-plane,excludingthepointz=Definition-----systemfuFinite-DimensionalLTIIIRDiscrete-TimeSystem6.7.2TransferFunctionExpressionFinite-DimensionalLTIIIRDis6.7.2TransferFunctionExpression

ForacausalIIRfilter,h[n]isacausal,theROCofH(z)isexteriortothecirclegoingthroughthepolefurthestfromtheorigin.6.7.2TransferFunctionExpres6.7.3FrequencyResponsefrom

TransferFunctionIftheROCofH(z)includesthecircle6.7.3FrequencyResponsefrom6.7.3FrequencyResponsefrom

TransferFunctionMagnitudefunction6.7.3FrequencyResponsefrom6.7.3FrequencyResponsefrom

TransferFunctionPhaseresponseMagnitude-squaredfunctionforareal-coefficientrationaltransferfunction6.7.3FrequencyResponsefromAcausalLTIdigitalfilterisBIBOstableifandonlyifitsimpulseresponseh[n]isabsolutelysummable.

WenowdevelopastabilityconditionintermsofthepolelocationsofthetransferfunctionH(z)

IftheROCincludestheunitcircle|z|=1,thenthedigitalfilterisstable,andviceversa.6.7.5StabilityConditionintermsofpolelocationAcausalLTIdigitalfilteriAFIRdigitalfilterwithboundedimpulseresponseisalwaysstable.

Ontheotherhand,anIIRfiltermaybeunstableifnotdesignedproperly.

AnoriginallystableIIRfiltercharacterizedbyinfiniteprecisioncoefficientsmaybecomeunstablewhencoefficientsgetquantizedduetoimplementation6.7.5StabilityConditionintermsofpolelocationAFIRdigitalfilterwithbouExample6.38:consideracausalIIRtransferfunction.6.7.5StabilityConditionintermsofpolelocationExample6.38:consideracauTheabsolutesummabilityconditionofh[n]issatisfied.

Hence,H(z)isastabletransferfunction.6.7.5StabilityConditionintermsofpolelocationTheabsolutesummabilitycondiNow,considerthecasewhenthetransferfunctioncoefficientsareroundedtovalueswith2digitsafterthedecimalpoint:6.7.5StabilityConditionintermsofpolelocationNow,considerthecasewhenInthiscase,theimpulseresponsecoefficient increasesrapidlytoaconstantvalueasnincreases.

Hence,theabsolutesummabilityconditionofisviolated

Thus,isanunstabletransferfunction6.7.5StabilityConditionintermsofpolelocationInthiscase,theimpulseres6.7.5StabilityConditionintermsofpolelocation1)AllpolesofacausalstabletransferfunctionH(z)mustbestrictlyinsidetheunitcircle.3)TheROCoftransferfunctionofanLTIdigitalfilterincludestheunitcircle,thenthefilterisBIBOstable.2)AllpolesofaanticausalstabletransferfunctionH(z)mustbestrictlyoutsidetheunitcircle.Conclusions:6.7.5StabilityConditioninExample:

DeterminethestableorcausalofthefollowingtransferfunctionAnswer:1stableandcausal6.7.5StabilityConditionintermsofpolelocationExample:Determinethestable6.7.5StabilityConditionintermsofpolelocation1unstableandcausal1stableandanticausal6.7.5StabilityConditionin6.7.5StabilityConditionintermsofpolelocationstableandnocausal16.7.5StabilityConditioninExercises36.20(a)6.25(a)(b)6.376.40(a)6.416.43(a)(b)6.48Exercises36.20(Chat6z-tranformDefinitionz-TransformsRegionofConvergencez-TransformsTheinversez-Transformsz-TransformsPropertiesTheTransferFunctionChat6z-tranformDefinition6.1DefinitionandPropertiesTheDTFTprovidesafrequency-domainrepresentationofdiscrete-timesignalsandLTIdiscrete-timesystems.Becauseoftheconvergencecondition,inmanycase,theDTFTofasequencemaynotexist.Asaresult,itisnotpossibletomakeuseofsuchfrequency-domaincharacterizationinthesecase.6.1DefinitionandProperties6.1DefinitionandProperties(p227)

z-TransformmayexistformanysequenceforwhichtheDTFTdoesnotexist.Moreover,useofz-Transformtechniquespermitssimplealgebraicmanipulation.Consequently,z-Transformhasbecomeanimportanttoolintheanalysisanddesignofdigitalfilters.1.

Definition6.1DefinitionandProperties6.1DefinitionandProperties(p227)RezjImzz=rejr11jjUnitcircle06.1DefinitionandPropertiesForagivensequence,thesetRofvaluesofzforwhichitsz-transformconvergesiscalledthe

regionofconvergence(ROC).6.1DefinitionandProperties(p227)Theinterpretationofthez-transformG(z)astheDTFTofsequenceg[n]r-n.Wecanchoosethevalueofrproperlyeventhoughg[n]isnotabsolutelysummable.Ingeneral,ROCcanberepresentedasForagivensequence,thes6.1DefinitionandProperties(p227)Note:Thez-transformofthetwosequenceareidenticaleventhoughthetwoparentsequencearedifferent.

Onlywayauniquesequencecanbeassociatedwithaz-transformisbyspecifyingitsROC.

TheDTFTG(ejω)ofasequenceg[n]convergesuniformlyifandonlyiftheROCofthez-transformG(z)ofg[n]includestheunitcircle.

6.1DefinitionandProperties6.1DefinitionandProperties(p227)Table6.16.1DefinitionandProperties6.2Rationalz-Transforms(p231)M-----thedegreeofthenumeratorpolynomialP(z)N-----thedegreeofthedenominatorpolynomialD(z)6.2Rationalz-Transforms(p26.2Rationalz-Transforms(p231)InEq.(6.15),thereareMfinitezerosandNfinitepolesIfN>M,thereareadditionalN-Mzerosatz=0.IfN<M,thereareadditionalM-Npolesatz=0.6.2Rationalz-Transforms(p236.3ROCofRationalz-TransformsTheROCofarationalz-transformisboundedbythelocationofitspoles.TheROCofarationalz-Transformcannotcontainanypoles

Asequencecanbeoneofthefollowingtype:finite-length,right-sided,left-sidedandtwo-sided.Iftherationalz-transformhasNpoleswithRdistinctmagnitudes,thenithasR+1ROCs,R+1distinctsequencehavingthesamerationalz-transform.6.3ROCofRationalz-Transfoa)TheROCofthez-transformofafinite-lengthsequencedefinedforM≤n≤Nistheentirez-planeexceptpossiblyz=0and/orz=+∞6.3ROCofRationalz-TransformsWehavethefollowingobservationwithregardtotheROCofaRationalz-Transforma)TheROCofthez-transform6.3

ROCofRationalz-Transformsb)TheROCofthez-transformofaright-sidedsequencedefinedforM≤n≤∞istheexteriortoacircleinthez-planepassingthroughthepolefurthestfromtheoriginz=0.6.3ROCofRationalz-Transfor6.3ROCofRationalz-Transformsc)TheROCofthez-transformofaleft-sidedsequencedefinedfor-∞≤n≤Nistheinteriortoacircleinthez-planepassingthroughthepolenearestfromtheoriginz=0.6.3ROCofRationalz-Transfor6.3ROCofRationalz-Transformsd)TheROCofthez-transformofatwo-sidedsequenceofinfinitelengthisaringboundedbytwocircleinthez-planepassingthroughtwopoleswithnopolesinsidethering.6.3ROCofRationalz-Transfor6.4TheInversez-Transform(p238)6.4.1GeneralExpression-----Cauchy’sintegraltheorem6.4TheInversez-Transform(6.4.1GeneralExpressionIfthepoleatz=λ0ofG(z)zn-1isofmultiplicitym.6.4.1GeneralExpressionIfth6.4.3Partial-FractionExpansionMethodArationalz-transformG(z)withacausalinversetransformg[n]hasanROCthatisexterior----M≥N,P(z)/D(z)isanimproperfraction----M<N,P1(z)/D(z)isaproperfraction6.4.3Partial-FractionExpansi6.4.3Partial-FractionExpansionMethodSimplePoles6.4.3Partial-FractionExpansi6.4.3Partial-FractionExpansionMethodMultiplePolesIfthepoleatz=visofmultiplicityLandtheremainingN-Lpolesaresimple.6.4.3Partial-FractionExpansi6.5z-TransformProperties(p246)ConjugationPropertyTime-ReversalPropertyLinearityProperty6.5z-TransformProperties(p6.5z-TransformProperties(p246)MultiplicationbyanExponentialSequenceDifferentiationPropertyTime-ShiftingProperty6.5z-TransformProperties(p6.5z-TransformProperties(p246)ModulationtheoremParseval’sRelationConvolutionProperty6.5z-TransformProperties(p6.7TheTransferFunction(p258)h[n]x[n]y[n]6.7.1Definition6.7TheTransferFunction(p256.7.1Definition-----systemfunctionortransferfunction6.7.2TransferFunctionExpressionFIRDigitalFilterForacausalFIRfilter,0≤N1≤N2,theROCofH(z)istheentirez-plane,excludingthepointz=Definition-----systemfuFinite-DimensionalLTIIIRDiscrete-TimeSystem6.7.2TransferFunctionExpressionFinite-DimensionalLTIIIRDis6.7.2TransferFunctionExpression

ForacausalIIRfilter,h[n]isacausal,theROCofH(z)isexteriortothecirclegoingthroughthepolefurthestfromtheorigin.6.7.2TransferFunctionExpres6.7.3FrequencyResponsefrom

TransferFunctionIftheROCofH(z)includesthecircle6.7.3FrequencyResponsefrom6.7.3FrequencyResponsefrom

TransferFunctionMagnitudefunction6.7.3FrequencyResponsefrom6.7.3FrequencyResponsefrom

TransferFunctionPhaseresponseMagnitude-squaredfunctionforareal-coefficientrationaltransferfunction6.7.3FrequencyResponsefromAcausalLTIdigitalfilterisBIBOstableifandonlyifitsimpulseresponseh[n]isabsolutelysummable.

WenowdevelopastabilityconditionintermsofthepolelocationsofthetransferfunctionH(z)

IftheROCincludestheunitcircle|z|=1,thenthedigitalfilterisstable,andviceversa.6.7.5StabilityConditionintermsofpolelocationAcausalLTIdigitalfilteriAFIRdigitalfilterwithboundedimpulseresponseisalwaysstable.

Ontheotherhand,anIIRfiltermaybeunstableifnotdesignedproperly.

AnoriginallystableIIRfiltercharacterizedbyinfiniteprecisioncoefficientsmaybecomeunstablewhencoefficientsgetquantizedduetoimplementation6.7.5StabilityConditionintermsofpolelocationAFIRdigitalfilterwithbouExample6.38:consideracausalIIRtransferfunction.6.7.5Sta