数字信号处理-ch8

Chapter8DigitalFilterStructures•ALTIsystemcanbeexpressedbyH(z).Inthischapter,westudyseveralstructuresusedtorealizeagiventransferfunctionassociatedwithaspecificdifferenceequationthroughtheuseoftheztransform.*transferfunctionsconsideredherewillbe•polynomialform(nonrecursivefilters)•rational-polynomialform(recursivefilters)•8.1BlockDiagramRepresentation•8.2EquivalentStructures•8.3BasicFIRDigitalFilterStructures•8.4BasicIIRDigitalFilterStructuresChapter8DigitalFilterStructuresUnitDelayMultiplierAddersymbolsforsignalflowgraph1z-[]xn[1]xn-A[]xn[]Axn×12[][][]jxnxnxn++?2[]xn[]jxn1[]xn•8.1BlockDiagramRepresentation•8.2EquivalentStructures•8.3BasicFIRDigitalFilterStructures•8.4BasicIIRDigitalFilterStructuresChapter8DigitalFilterStructures•8.1BlockDiagramRepresentation•8.2EquivalentStructures•8.3BasicFIRDigitalFilterStructures•8.4BasicIIRDigitalFilterStructuresChapter8DigitalFilterStructures8.3BasicFIRDigitalFilterStructures*nonrecursivefiltersarecharacterizedbyadifferenceequationintheform0[][]Nllynbxnl==-å[]lmakebhl=*blcoefficientsaredirectlyrelatedtothesystemimpulseresponse,thatis0[][][]Nlthenynhlxnl==-å0()()NlllYzbzXz-==å00()[]NNlllllHzbzhlz--==?=邋8.3.1Directform(transversalform,convolutionform)*ThisissimplestrealizationofaFIRdigitalfilterwhichderivedfromequation00[][][]()[]NNlllynhlxnlHzhlz-===-=邋andxn[]0h[]1z-1h[]1z-yn[]1h-[N]h[N]suchastructureisalsoreferredtoasthecanonicdirectform-----realizesagiventransferfunctionwiththeminimumnumberofdelays,multipliersandadders.1z-2h[]120()[][0][1][2][]NlNlHzhlzhhzhzhNz----===++++å*analternativecanonicdirectform110()[][0]([1]([2][]))NllHZhlZhZhZhhN---===++++å1z-1z-1z-xn[]yn[]0h[]1h[]2h[]1h-[N]h[N]1z-8.3.1Directform(transversalform,convolutionform)where:ifNisthefilterorder,then0120121()[]()NllKkkkkHzhlzrrzrz-=--==?=++åÕ221(0)2kNKisevenNKisoddoneofgìïï=Nïïïíï+ï=N=ïïïî8.3.2Cascadeformxn[]1z-01r11r21r1z-02r12r22r1z-1z-0Kr1Kr2Kryn[]1z-1z-secondorder*everyblockcontrolsapairofzeros*thisstructureusesmoremultipliersthandirectform8.3.2Cascadeform1.:[][]TypeIhnhNneven=-andN0123()hnn456789102N10N=22210012()[][][][]NNNNNnnnnnnNHzhnzhnzhzhnz-----===+==++邋?22211()002[][][]NNNnNnnnNhnzhzhNnz------===++-邋221()02[][][]NNnNnnNhnzzhz-----==++å8.3.4Linear-PhaseFIRStructuresxn[]0h[]1z-1h[]1z-yn[]1h-[N]hN[]1z-2h[]Directform:Linearphaseform:1Z-1Z-xn[]yn[]1Z-1Z-1Z-1Z-0h[]1h[]2h[]2Nh[-1]2Nh[]1Z-1Z-8.3.4Linear-PhaseFIRStructures1.:[][]TypeIhnhNneven=-andN2.:()()TypeIIhnhNnNodd=-and121200()[][][]NNNNnnnnnnHzhnzhnzhnz-+---=====+邋?0123()hnn456789101111N=1122()00[][]NNnNnnnhnzhNnz-----===+-邋12()0[]()NnNnnhnzz----==+å8.3.4Linear-PhaseFIRStructuresLinearphaseform:1Z-1Z-xn[]yn[]1Z-1Z-1Z-1Z-0h[]1h[]2h[]N-3h[]2N-1h[]21Z-1Z-1Z-8.3.4Linear-PhaseFIRStructures2.:()()TypeIIhnhNnNodd=-and3.:[][]TypeIIIhnhNnNeven=--and0123()hnn4567891010Nmustbezero22101()[][]NNNnnnnHzhnzhnz---==+=+邋21()0[]()NnNnnhnzz----==-å2[]0Nthenh=8.3.4Linear-PhaseFIRStructuresLinearphaseform:1Z-1Z-xn[]yn[]1Z-1Z-1Z-1Z-0h[]1h[]2h[]Nh[-1]2Nh[]21Z-1Z-1-1-1-1-1-8.3.4Linear-PhaseFIRStructures3.:[][]*[]02TypeIIIhnhNnNevenNh=--=and4.:()()TypeIVhnhNnNodd=--and0123()hnn456789101111N112212()00()[][][]()NNNNnnnNnnnnHzhnzhnzhnzz--+-----====+=-邋?8.3.4Linear-PhaseFIRStructures1Z-1Z-xn[]yn[]1Z-1Z-1Z-1Z-0h[]1h[]2h[]N-3h[]2N-1h[]21Z-1Z-1Z-1-1-1-1-1-1-4.:()()TypeIVhnhNnNodd=--and8.3.4Linear-PhaseFIRStructuresLinearphaseform::[][]NisevenhnhNn=?:[][]NisoddhnhNn=?costalmosthalfmultiplicationofdirectform1Z-1Z-xn（）yn[]1Z-1Z-1Z-1Z-0h[]1h[]2h[]Nh[-1]2Nh[]2±1±1±1±1±11Z-1Z-1Z-1Z-xn[]yn[]1Z-1Z-1Z-1Z-0h（）1h（）2h（）±1±1±1±1±1N-3h()2N-1h()21Z-1Z-±11Z-typeIandtypeIIItypeIIandtypeIVh(n)={1,2,3,4,3,2,1}Classexerciseplotdirectformandlinear-phaseform,andexplaintheadvantageoflinear-phaseform.123456()123432Hzzzzzzz------=++++++AalternativeconditionAnalyzing:h(n)=h(N-n)andN=6,typeIlinearphaseform8.3.4Linear-PhaseFIRStructuresAnswerofclassexerciseh(0)=1h(1)=2h(2)=3h(3)=4h(4)=3h(5)=2h(6)=1N=6h(n)=h(N-n)xn[]11z-21z-yn[]31z-31z-41z-1z-211Z-1Z-xn[]yn[]1Z-1Z-12341Z-1Z-Directform:Linearphaseform:Advantage:costalmosthalfmultiplicationofdirectform8.3.4Linear-PhaseFIRStructures•8.1BlockDiagramRepresentation•8.2EquivalentStructures•8.3BasicFIRDigitalFilterStructures•8.4BasicIIRDigitalFilterStructuresChapter8DigitalFilterStructures8.4.1DirectformsH(z)ofarecursivefilterisgivenby01()()()()1MiiiNiiibzNzHzNMDzaz-=-===?+åå1()()()HzNzDz=?divideintotwopartscascaded0where()MiiiNzbz-==å1()1NiiiDzaz-==+å0()MiiiNzbz-==å111()1NiiiDzaz-==+å1a-2a-1Na--Na-[1]yn-[2]yn-[1]ynN-+[]ynN-[]yn1z-1z-1z-realizezerosrealizepolesai’ssign‘+’or‘-’cascade1z-[1]xn-[2]xn-[1]xnM-+[]xnM-1b2b1Mb-Mb[]xn1z-1z-0b8.4.1Directforms(noncanonic)[]xn1a-2a-1Na--Na-1z-1z-1z-*needsN+Mdelays,isnoncanonicdirectform11*()()()()NzNzDzDz??1z1b2b1Mb-Mb1z1z[]yn0b8.4.1Directforms*accordingtothetheoremofsignalflowgraph1z-1b2b1Mb-Mb1z-1z-[]yn[]xn1a-2a-1Na--Na-1z-1z-1z-0b8.4.1Directforms1b2b1Mb-Mb[]yn[]xn1a-2a-1Na--Na-1z-1z-1z-0b8.4.1Directforms*needsNdelays,*calleddirectformII(canonic)alternativestructure1b2b1MbMb()yn()xn1a2a1NaNa1z1z1z1zMa0