数字信号处理课件第11章-数字滤波器结构v1.0

Chapter8DigitalFilterStructuresHeretheinput-outputrelationinvolvesafinitesumofproducts:§8.1IntroductionOntheotherhand,anFIRsystemcanbeimplementedusingtheconvolutionsumwhichisafinitesumofproducts:TheactualimplementationofanLTIdigitalfiltercanbeeitherinsoftwareorhardwareform,dependingonapplicationsIneithercase,thesignalvariablesandthefiltercoefficientscannotberepresentedwithfiniteprecision§8.1Introduction★§8.1.1BlockDiagramRepresentationFortheimplementationofanLTIdigitalfilter,theinput-outputrelationshipmustbedescribedbyavalidcomputationalalgorithmToillustratewhatwemeanbyacomputationalalgorithm,considerthecausalfirst-orderLTIdigitalfiltershownbelow§8.1.1BlockDiagramRepresentationThefilterisdescribedbythedifferenceequationy[n]=-d1y[n-1]+p0x[n]+p1x[n-1]Usingtheaboveequationwecancomputey[n]forn0knowingtheinitialconditiony[n-1]andtheinputx[n]forn-1§8.1.1BlockDiagramRepresentation

y[0]=-d1y[-1]+p0x[0]+p1x[-1]y[1]=-d1y[0]+p0x[1]+p1x[0]

y[2]=-d1y[1]+p0x[2]+p1x[1].…Wecancontinuethiscalculationforanyvalueofthetimeindexnwedesire§8.1.1BlockDiagramRepresentationEachstepofthecalculationrequiresaknowledgeofthepreviouslycalculatedvalueoftheoutputsample(delayedvalueoftheoutput),thepresentvalueoftheinputsample,andthepreviousvalueoftheinputsample(delayedvalueoftheinput)Asaresult,thefirst-orderdifferenceequationcanbeinterpretedasavalidcomputationalalgorithm§8.1.2BasicBuildingBlocksThecomputationalalgorithmofanLTIdigitalfiltercanbeconvenientlyrepresentedinblockdiagramformusingthebasicbuildingblocksshownbelowx[n]y[n]w[n]Ax[n]y[n]y[n]x[n]x[n]x[n]x[n]AdderUnitdelayMultiplierPick-offnode§8.1.2BasicBuildingBlocksAdvantagesofblockdiagramrepresentation(1)Easytowritedownthecomputationalalgorithmbyinspection(2)Easytoanalyzetheblockdiagramtodeterminetheexplicitrelationbetweentheoutputandinput★★√√§8.1.2BasicBuildingBlocks(3)Easytomanipulateablockdiagramtoderiveother“equivalent”blockdiagramsyieldingdifferentcomputationalalgorithms(4)Easytodeterminethehardwarerequirements(5)Easiertodevelopblockdiagramrepresentationsfromthetransferfunctiondirectly√√√§8.1.3AnalysisofBlockDiagramsCarriedoutbywritingdowntheexpressionsfortheoutputsignalsofeachadderasasumofitsinputsignals,anddevelopingasetofequationsrelatingthefilterinputandoutputsignalsintermsofallinternalsignalsEliminatingtheunwantedinternalvariablesthenresultsintheexpressionfortheoutputsignalasafunctionoftheinputsignalandthefilterparametersthatarethemultipliercoefficients★★§8.1.3AnalysisofBlockDiagramsTheoutputE(z)oftheadderisE(z)=X(z)+G2(z)Y(z)ButfromthefigureY(z)=G1(z)E(z)Example-Considerthesingle-loopfeedbackstructureshownbelow§8.1.3AnalysisofBlockDiagramsEliminatingE(z)fromtheprevioustwoequationswearriveat

[1-G1(z)G2(z)]Y(z)=G1(z)X(z)

whichleadsto§8.1.3AnalysisofBlockDiagramsExample-Analyzethecascadedlatticestructureshownbelowwherethez-dependenceofsignalvariablesarenotshownforbrevity§8.1.3AnalysisofBlockDiagramsTheoutputsignalsofthefouraddersaregivenbyW1=X-S2W2=W1-S1W3=S1-W2Y=W1-S2FromthefigureweobserveS2=z-1W3S1=z-1W2

§8.1.3AnalysisofBlockDiagramsSubstitutingthelasttworelationsinthefirstfourequationsweget

W1=X-z-1W3

W2=W1-z-1W2

W3=z-1W2+W2Y=W1+z-1

W3

FromthesecondequationwegetW2=W1/(1+z-1)andfromthethirdequationwegetW3=(+z-1)W2

§8.1.3AnalysisofBlockDiagramsCombiningthelasttwoequationswegetwefinallyarriveatSubstitutingtheaboveequationin§8.2EquivalentStructuresTwodigitalfilterstructuresaredefinedtobeequivalentiftheyhavethesametransferfunctionWedescribenextanumberofmethodsforthegenerationofequivalentstructuresHowever,afairlysimplewaytogenerateanequivalentstructurefromagivenrealizationisviathetransposeoperation★★§8.2EquivalentStructuresTransposeOperation(1)Reverseallpaths(2)Replacepick-offnodesbyadders,andviceversa(3)InterchangetheinputandoutputnodesAllothermethodsfordevelopingequivalentstructuresarebasedonaspecificalgorithmforeachstructure★★§8.2EquivalentStructuresThereareliterallyaninfinitenumberofequivalentstructuresrealizingthesametransferfunctionItisthusimpossibletodevelopallequivalentrealizationsInthiscoursewerestrictourattentiontoadiscussionofsomecommonlyusedstructures§8.2EquivalentStructuresUnderinfiniteprecisionarithmeticanygivenrealizationofadigitalfilterbehavesidenticallytoanyotherequivalentstructureHowever,inpractice,duetothefinitewordlengthlimitations,aspecificrealizationbehavestotallydifferentlyfromitsotherequivalentrealizations§8.2EquivalentStructuresHence,itisimportanttochooseastructurethathastheleastquantizationeffectswhenimplementedusingfiniteprecisionarithmeticOnewaytoarriveatsuchastructureistodeterminealargenumberofequivalentstructures,analyzethefinitewordlengtheffectsineachcase,andselecttheoneshowingtheleasteffects★★§8.2EquivalentStructuresIncertaincases,itispossibletodevelopastructurethatbyconstructionhastheleastquantizationeffectsWedeferthereviewofthesestructuresafteradiscussionoftheanalysisofquantizationeffectsHere,wereviewsomesimplerealizationsthatinmanyapplicationsarequiteadequate§8.3BasicFIRDigitalFilterStructures whichisapolynomialinz-1Inthetime-domaintheinput-outputrelationoftheaboveFIRfilterisgivenbyAcausalFIRfilteroforderNischaracterizedbyatransferfunctionH(z)givenby§8.3.1DirectFormFIRDigitalFilterStructuresAnFIRfilteroforderNischaracterizedbyN+1coefficientsand,ingeneral,requireN+1multipliersandNtwo-inputaddersStructuresinwhichthemultipliercoefficientsarepreciselythecoefficientsofthetransferfunctionarecalleddirectformstructures

★★§8.3.1DirectFormFIRDigitalFilterStructuresAdirectformrealizationofanFIRfiltercanbereadilydevelopedfromtheconvolutionsumdescriptionasindicatedbelowforN=4★★§8.3.1DirectFormFIRDigitalFilterStructures whichispreciselyoftheformoftheconvolutionsumdescriptionThedirectformstructureshownonthepreviousslideisalsoknownasatappeddelaylineoratransversalfilterAnanalysisofthisstructureyields§8.3.1DirectFormFIRDigitalFilterStructuresThetransposeofthedirectformstructureshownearlierisindicatedbelow★★§8.3.2CascadeFormFIRDigitalFilterStructuresAhigher-orderFIRtransferfunctioncanalsoberealizedasacascadeofsecond-orderFIRsectionsandpossiblyafirst-ordersectionTothisendweexpressH(z)aswherek=N/2ifNiseven,andk=(N+1)/2ifNisodd,with

2k=0§8.3.2CascadeFormFIRDigitalFilterStructuresAcascaderealizationforN=6isshownbelowEachsecond-ordersectionintheabovestructurecanalsoberealizedinthetransposeddirectform★★§8.3.3Linear-PhaseFIRStructuresThesymmetry(orantisymmetry)propertyofalinear-phaseFIRfiltercanbeexploitedtoreducethenumberofmultipliersintoalmosthalfofthatinthedirectformimplementationsConsideralength-7Type1FIRtransferfunctionwithasymmetricimpulseresponse§8.3.3Linear-PhaseFIRStructuresRewritingH(z)intheform weobtaintherealizationshownbelow★★§8.3.3Linear-PhaseFIRStructuresAsimilardecompositioncanbeappliedtoaType2FIRtransferfunctionForexample,alength-8Type2FIRtransferfunctioncanbeexpressedasThecorrespondingrealizationisshownonthenextslide§8.3.3Linear-PhaseFIRStructuresNote:TheType1linear-phasestructureforalength-7FIRfilterrequires4multipliers,whereasadirectformrealizationrequires7multipliers§8.3.3Linear-PhaseFIRStructuresNote:TheType2linear-phasestructureforalength-8FIRfilterrequires4multipliers,whereasadirectformrealizationrequires8multipliersSimilarsavingsoccursintherealizationofType3andType4linear-phaseFIRfilterswithantisymmetricimpulseresponses§8.4BasicIIRDigitalFilterStructuresThecausalIIRdigitalfiltersweareconcernedwithinthiscoursearecharacterizedbyarealrationaltransferfunctionofz-1or,equivalentlybyaconstantcoefficientdifferenceequationFromthedifferenceequationrepresentation,itcanbeseenthattherealizationofthecausalIIRdigitalfiltersrequiressomeformoffeedback§8.4BasicIIRDigitalFilterStructuresAnN-thorderIIRdigitaltransferfunctionischaracterizedby2N+1uniquecoefficients,andingeneral,requires2N+1multipliersand2Ntwo-inputaddersforimplementationDirectformIIRfilters:Filterstructuresinwhichthemultipliercoefficientsarepreciselythecoefficientsofthetransferfunction★★§8.4.1DirectFormIIRDigitalFilterStructuresConsiderforsimplicitya3rd-orderIIRfilterwithatransferfunctionWecanimplementH(z)asacascadeoftwofiltersectionsasshownonthenextslide§8.4.1DirectFormIIRDigitalFilterStructures§8.4.1DirectFormIIRDigitalFilterStructuresThefiltersectionH1(z)canbeseentobeanFIRfilterandcanberealizedasshownright§8.4.1DirectFormIIRDigitalFilterStructuresThetime-domainrepresentationofH2(z)isgivenby Realizationoffollowsfromtheaboveequationandisshownontheright§8.4.1DirectFormIIRDigitalFilterStructuresAcascadeofthetwostructuresrealizingH1(z)andH2(z)leadstotherealizationofH(z)shownbelowandisknownastheDirectFormIstructure★★§8.4.1DirectFormIIRDigitalFilterStructuresNote:ThedirectformIstructureisnoncanonicasitemploys6delaystorealizea3rd-ordertransferfunctionAtransposeofthedirectformIstructureisshownontherightandiscalledthedirectformIstructure★§8.4.1DirectFormIIRDigitalFilterStructuresVariousothernoncanonicdirectformstructurescanbederivedbysimpleblockdiagrammanipulationsasshownbelow§8.4.1DirectFormIIRDigitalFilterStructures1Observeinthedirectformstructureshownright,thesignalvariableatnodesandarethesame,andhencethetwotopdelayscanbeshared§8.4.1DirectFormIIRDigitalFilterStructuresFollowingthesameargument,thebottomtwodelayscanbesharedSharingofalldelaysreducesthetotalnumberofdelaysto3resultinginacanonicrealizationshownonthenextslidealongwithitstransposestructureLikewise,thesignalvariablesatnodesandarethesame,permittingthesharingofthemiddletwodelays§8.4.1DirectFormIIRDigitalFilterStructuresDirectformrealizationsofanN-thorderIIRtransferfunctionshouldbeevident★★§8.4.2CascadeFormIIRDigitalFilterStructuresByexpressingthenumeratorandthedenominatorpolynomialsofthetransferfunctionasaproductofpolynomialsoflowerdegree,adigitalfiltercanberealizedasacascadeoflow-orderfiltersectionsConsider,forexample,H(z)=P(z)/D(z)expressedas§8.4.2CascadeFormIIRDigitalFilterStructuresExamplesofcascaderealizationsobtainedbydifferentpole-zeropairingsareshownbelow★★§8.4.2CascadeFormIIRDigitalFilterStructuresExamplesofcascaderealizationsobtainedbydifferentorderingofsectionsareshownbelow§8.4.2CascadeFormIIRDigitalFilterStructuresbasedonpole-zero-pairingsandorderingDuetofinitewordlengtheffects,eachsuchcascaderealizationbehavesdifferentlyfromothersTherearealtogetheratotalof36differentcascaderealizationsof§8.4.2CascadeFormIIRDigitalFilterStructuresUsually,thepolynomialsarefactoredintoaproductof1st-orderand2nd-orderpolynomials:Intheabove,forafirst-orderfactor★★§8.4.2CascadeFormIIRDigitalFilterStructuresConsiderthe3rd-ordertransferfunctionOnepossiblerealizationisshownbelow★§8.4.2CascadeFormIIRDigitalFilterStructuresExample-DirectformIIandcascadeformrealizationsof areshownonthenextslide§8.4.2CascadeFormIIRDigitalFilterStructuresDirectformIICascadeform§8.4.3ParallelFormIIRDigitalFilterStructuresApartial-fractionexpansionofthetransferfunctioninz-1leadstotheparallelformIstructureAssumingsimplepoles,thetransferfunctionH(z)canbeexpressedasIntheaboveforarealpole★★§8.4.3ParallelFormIIRDigitalFilterStructuresThetwobasicparallelrealizationsofa3rd-orderIIRtransferf