Chapter7LTIDiscrete-TimeSystemsintheTransformDomain第1頁FIRfilterIIRfilterClassificationbasedonthelengthoftheimpulseresponseh(n)inthetimedomainThereareseveralothertypesofclassificationsoftransferfunctionsbasedontheirmagnitudecharacteristics.Likewise,thereareseveralothertypesofclassificationsoftransferfunctionsbasedontheirphasecharacteristics.第2頁7.1TransferFunctionClassificationBasedonMagnitudeCharacteristics第3頁7.1.1DigitalFilterswithidealMagnitudeResponse第4頁第5頁第6頁Exercise:IfH(z)isalowpassfilterwithpassbandedgefrequencywc,determinethetypeoffilterH(-z)anditsbandedgeofH(-z)intermsofthebnadedgeofH(z)第7頁PassbandStopbandCutofffrequencyTransitionbandThespecificationsinthetransformdomainiscorrespondingto第8頁7.1.2BoundedRealTransferFunctionspassivestructure第9頁7.1.3AllpassTransferFunctionTransferfunction?第10頁Supposeithasapoleattheremustbeazeroatmirror-imagesymmetrymirror-imagepolynomial第11頁Considera3-orderallpassfilter,andsupposeweknowittwopoles,complementtheremainingpolesandzerosofthesystem.

第12頁第13頁ShowthatisaAllpasssystemproof:第14頁Example:A3-orderAllpasssystem第15頁Exercise:ConsiderN(z)isthenumeratorfunctionofa3-ordercausalreal-coefficientallpasstransferfunction,andN(z)=0.32+0.075z-1+0.63z-2+z-3,determinethedenominatorfunctionofthisallpassfunction.1+0.63z-1+0.075z-2+0.32z-3D(z)=第16頁第17頁第18頁Asimpleapplication:delayequalizer第19頁7.2TransferFunctionClassificationBasedonPhaseCharacteristics第20頁7.2.1Zero-PhaseTransferFunctionnoncausalsystem第21頁第22頁7.2.2Linear-PhaseTransferFunctiony[n]=x[n-D]第23頁第24頁7.2.3Minimum-PhaseandMaximum-PhaseTransferFunctionAnotherusefulclassificationofatransferfunctionisbasedonthelocationofitszeros,which,inturn,influencesitsphaseresponse.第25頁第26頁第27頁第28頁7.3TypesofLinear-PhaseFIRTransferFunction第29頁Q1:ToinsureaFIRfilterhaslinear-phase,therestrictionofh[n]?Q2:TheformoffrequencyresponseofLinear-Phasefilter?第30頁7.3.1FrequencyResponseforaFIRFilterwithaLinear-PhaseWhere=±第31頁h[n]=±h[N-n],0≤n≤Nc=-N/2=±第32頁第33頁Examples:H1(z)=-0.0637-0.0691z-1+0.101z-2+0.2857z-3+0.4107z-4+0.2857z-5+0.101z-6-0.0691z-7-0.0637z-8H2(z)=-1+2z-1-3z-2+6z-3-3z-4+2z-5-z-6H3(z)=1-2z-1+3z-2-3z-4+2z-5-z-6第34頁7.3.2ZeroLocationsofLinear-PhaseFIRTransferFunctionMirror-ImageandAntimirror-ImagePolynomials第35頁Forsymmetrysequenceh[n]=h[N-n]WhichtypesofFIRFilterhaveMIPproperty?第36頁Forantisymmetrysequenceh[n]=-h[N-n]WhichtypesofFIRFilterhaveAIPproperty?第37頁Moreover,foraFIRfilterwithrealimpulseresponseh[n],itszerosoccurincomplexconjugatepairs第38頁·forcomplexzerothatisnotontheunitcircle,itsassociatedwithasetof4zerosForrealzero,it’sapairwithreciprocalz=ξ,z=1/ξ第39頁FortypeIfilter,itsMIPisevendegree,henceithasfactorsoftheform(1+az-1+bz-2+az-3+z-4)and(1+cz-1+z-2),soitcanhaveanevennumberofzerosornozerosatz=z-1oratz=-z-1,orbothlocations.第40頁H(z)musthaveazeroatz=-1第41頁Fortype3FIRfilter,NisevenH(z)musthaveazeroatz=-1andz=1第42頁Fortype4FIRfilter,NisoddH(z)musthaveazeroatz=1第43頁第44頁第45頁第46頁第47