Ch7_LTI systems in the Transform Domain数字信号处理

Chapter7LTIDiscrete-TimeSystemsintheTransformDomainTransferFunctionClassificationTypesofLinear-PhaseTransferFunctionsSimpleDigitalFiltersTypesofTransferFunctionsThetime-domainclassificationofadigitaltransferfunctionbasedonthelengthofitsimpulseresponsesequence:-Finiteimpulseresponse(FIR)transferfunction.-Infiniteimpulseresponse(IIR)transferfunction.TypesofTransferFunctionsInthecaseofdigitaltransferfunctionswithfrequency-selectivefrequencyresponses,therearetwotypesofclassifications:(1)Classificationbasedontheshapeofthemagnitudefunction|H(ei)|.(2)Classificationbasedontheformofthephasefunctionθ().7.1TransferFunctionClassificationBasedonMagnitudeCharacteristicsDigitalFilterswithIdealMagnitudeResponsesBoundedRealTransferFunctionAllpassTransferFunction7.1.1DigitalFilterswithIdealMagnitudeResponsesAdigitalfilterdesignedtopasssignalcomponentsofcertainfrequencieswithoutdistortionshouldhaveafrequencyresponseequalto1atthesefrequencies,andshouldhaveafrequencyresponseequalto0atallotherfrequencies.Therangeoffrequencieswherethefrequencyresponsetakesthevalueof1iscalledthepassband.Therangeoffrequencieswherethefrequencyresponsetakesthevalueof0iscalledthestopband.Magnituderesponsesofthefourpopulartypesofidealdigitalfilterswithrealimpulseresponsecoefficientsareshownbelow:Thefrequenciesc,c1,andc2arecalledthecutofffrequencies.Anidealfilterhasamagnituderesponseequalto1inthepassbandand0inthestopband,andhasa0phaseeverywhere.TheinverseDTFTofthefrequencyresponseHLP(ej)oftheideallowpassfilteris:hLP[n]=sincn/n,-nWehaveshownthattheaboveimpulseresponseisnotabsolutelysummable,andhence,thecorrespondingtransferfunctionisnotBIBOstable.Also,hLP[n]isnotcausalandisofdoublyinfinitelength.Theremainingthreeidealfiltersarealsocharacterizedbydoublyinfinite,noncausalimpulseresponsesandarenotabsolutelysummable.Thus,theidealfilterswiththeideal“brickwall”frequencyresponsescannotberealizedwithfinitedimensionalLTIfilter.Todevelopstableandrealizabletransferfunctions,theidealfrequencyresponsespecificationsarerelaxedbyincludingatransitionbandbetweenthepassbandandthestopband.Thispermitsthemagnituderesponsetodecayslowlyfromitsmaximumvalueinthepassbandtothe0valueinthestopband.Moreover,themagnituderesponseisallowedtovarybyasmallamountbothinthepassbandandthestopband.•Typicalmagnituderesponsespecificationsofalowpassfilterareshownas:7.1.2BoundedRealTransferFunctionsAcausalstablereal-coefficienttransferfunctionH(z)isdefinedasaboundedreal(BR)transferfunctionif:1|)(|jeHforallvaluesofLetx[n]andy[n]denote,respectively,theinputandoutputofadigitalfiltercharacterizedbyaBRtransferfunctionH(z)withX(ejω)andY(ejω)denotingtheirDTFTs.1|)(|jeHThentheconditionimplies:22)()(jjeXeYIntegratingtheabovefrom-πtoπ,andapplyingParseval’srelationweget:nnnxny22][][Thus,forallfinite-energyinputs,theoutputenergyislessthanorequaltotheinputenergy.ItimpliesthatadigitalfiltercharacterizedbyaBRtransferfunctioncanbeviewedasapassivestructure.1|)(|jeHIf,thentheoutputenergyisequaltotheinputenergy,andsuchadigitalfilteristhereforealosslesssystem.1|)(|jeHAcausalstablereal-coefficienttransferfunctionH(z)withisthuscalledalosslessboundedreal(LBR)transferfunction.TheBRandLBRtransferfunctionsarethekeystotherealizationofdigitalfilterswithlowcoefficientsensitivity.BoundedRealTransferFunctionsExample:ConsiderthecausalstableIIRtransferfunction:-1KH(z)=,0||11-zwhereKisarealconstant.Itssquare-magnitudefunctionisgivenby:jω2j2-12z=e|H(e)|H(z)H(z)|(1)2cosKThus,forα0,(|H(ejω)|2)max=K2/(1-α)2|ω=0(|H(ejω)|2)min=K2/(1+α)2|ω=πOntheotherhand,forα0,(|H(ejω)|2)max=K2/(1+α)2|ω=π(|H(ejω)|2)min=K2/(1-α)2|ω=0isaBRfunctionforK≤±(1-α).-1KH(z)=,0||11-zHence,LowpassfilterHighpassfilter7.1.3AllpassTransferFunctionThemagnituderesponseofallpasssystemsatisfies:|A(ejω)|2=1,forallω.TheH(z)ofasimple1th-orderallpasssystemis:111)(azazzAWhereaisreal,and.Oraiscomplex,theH(z)shouldbe:10a1*11)(azazzAAnM-thordercausalreal-coefficientallpasstransferfunctionisoftheform:MkkkMzaazzA11*11)(MMMMMMMMzdzdzdzzdzdd11111111...1...Ifwedenotepolynomial:MMMMMzdzdzdzD1111...1)(Then:)()()(1zDzDzzAMMMMThenumeratorofareal-coefficientallpasstransferfunctionissaidtobethemirror-imagepolynomialofthedenominator,andviceversa.()MDzWeshallusethenotationtodenotethemirror-imagepolynomialofadegree-MpolynomialDM(z),i.e.,1()()MMMDzzDzTheexpression:)()()(zDzDzMMMMzA1impliesthatthepolesandzerosofareal-coefficientallpassfunctionexhibitmirror-imagesymmetryinthez-plane.32132132.018.04.014.018.02.0)(zzzzzzzAToshowthat|AM(ejω)|=1weobservethat:)()(11)(zDzDzMMMMzATherefore:)()()()(111)()(zDzDzzDzDzMMMMMMMMzAzAHence:1)()(|)(|12jezMMjMzAzAeAFigurebelowshowstheprincipalvalueofthephaseofthe3rd-orderallpassfunction:32132132.018.04.014.018.02.0)(zzzzzzzANote:Thephasefunctionofanyarbitrarycausalstableallpassfunctionisacontinuousfunctionofω.Properties:1.Acausalstablereal-coefficientallpasstransferfuncti