数字信号处理英文版课件：Chap 8 Digital Filter Structures

Chat8DigitalFilterStructuresBlockDiagramRepresentationBasicFIRDigitalFilterStructureBasicIIRDigitalFilterStructure18.1BlockDiagramRepresentation1)Therepresentationoftheinput-outputrelationwithanalyticalexpression

convolutionsumLinearconstantcoefficientdifferenceequation22)

TheimplementationofanLTIfilter

---------aValidcomputationalalgorithmToillustratewhatwemeanbyacomputationalalgorithm,considerthecausalfirst-orderLTIdigitalfiltershownbelow8.1BlockDiagramRepresentation3•Thefilterisdescribedbythedifferenceequationy[n]=-d1y[n-1]+p0x[n]+p1x[n-1]•

Usingtheaboveequationwecancomputey[n]forn0knowingtheinitial

conditiony[n-1]andtheinputx[n]forn-1y[0]=-d1y[-1]+p0x[0]+p1x[-1]y[1]=-d1y[0]+p0x[1]+p1x[0]y[2]=-d1y[1]+p0x[2]+p1x[1].…8.1BlockDiagramRepresentation4WecancontinuethiscalculationforanyvalueofthetimeindexnwedesireAsaresult,thefirstorderdifferenceequationcanbeinterpretedasavalidcomputationalalgorithm8.1BlockDiagramRepresentation58.1.1BasicBuildingBlocksThecomputationalalgorithmofanLTIdigitalfiltercanbeconvenientlyrepresentedinblockdiagramformusingthebasicbuildingblocksshownbelowx[n]y[n]w[n]Ax[n]y[n]y[n]x[n]x[n]y2[n]y1[n]AdderUnitdelayMultiplierPick-offnode6Advantagesofblockdiagramrepresentation

(l)Easytowritedownthecomputationalalgorithmbyinspection(2)Easytoanalyzetheblockdiagramtodeterminetheexplicitrelationbetweentheoutputandinput8.1.1BasicBuildingBlocks7(3)Easytomanipulateablockdiagramtoderiveother"equivalent,'blockdiagramsyieldingdifferentcomputationalalgorithms(4)Easytodeterminethehardwarerequirements(5)Easytodevelopblockdiagramrepresentationsfromthetransferfunctiondirectly8.1.1BasicBuildingBlocks88.1.2AnalysisofBlockDiagrams

Carriedoutbywritingdowntheexpressionsfortheoutputsignalsofeachadderasasumofitsinputsignals,anddevelopingasetofequationsrelatingthefilterinputandoutputsignalsintermsofallinternalsignals

EliminatingtheunwantedinternalvariablesthenresultsintheexpressionfortheoutputsignalasafunctionoftheinputsignalandthefilterparametersthatarethemultipliercoefficientsAnalysisMethod9Example(1)Considertheshownbelowsingle-loopfeedbackStructureTheoutputE(z)oftheadderisE(z)=X(z)+G2(z)Y(z)ButfromthefigureY(z)=G1(z)E(z)8.1.2AnalysisofBlockDiagrams10(2)Analyzethecascadedlatticestructureshownbelowwherethez-dependenceofsignalvariablesarenotshownforbrevityEliminatingE(z)fromtheprevioustwoequationswearriveat[1-G1(z)G2(z)]Y(z)=G1(z)X(z)whichleadsto

8.1.2AnalysisofBlockDiagrams11TheoutputsignalsaregivenbyW1=X-S2W2=W1-S1W3=S1-W2Y=W1-S2FromthefigureweobserveS2=z-1W3S1=z-1W28.1.2AnalysisofBlockDiagrams12EliminatingW1,W2,

W3,S1andS2wefinally

arriveat8.1.2AnalysisofBlockDiagrams138.1.3TheDelay-freeLoopProbIemToillustratethedelay-freeloopproblemconsiderthestructurebelowForphysicalrealizabilityofthedigitalfilterstructure,itisnecessarythattheblockdiagramrepresentationcontainsnodelay-freeloops(containdelayloops)14Analysisofthisstructureyieldsu[n]=w[n]+y[n]y[n]=B(v[n]+Au[n])whichwhencombinedresultsiny[n]=B(v[n]+A(w[n]+y[n]))

Thedeterminationofthecurrentvalueofy[n]requirestheknowledgeofthesamevaluey[n]8.1.3TheDelay-freeLoopProbIem15

However,thisisphysicallyimpossibletoachieveduetothefinitetimerequiredtocarryoutallarithmeticoperationsonadigitalmachine

Methodexiststodetectthepresenceofdelay-freeloopsinanarbitrarystructure,alongwithmethodstolocateandremovetheseloopswithouttheoverallinput-outputrelation8.1.3TheDelay-freeLoopProbIem16Figurebelowshowssucharealizationoftheexamplestructuredescribedearlier8.1.3TheDelay-freeLoopProbIem178.1.4CanonicandNoncanonicStructuresDefinition:Adigitalfilterstructureissaidtobecanonicifthenumberofdelaysintheblockdiagramrepresentationisequaltotheorderofthetransferfunction.Otherwise,itisanoncanonicstructure•Thestructureshownbelowisnoncanonicasitemploystwodelaystorealizeafirst-orderdifferenceequationy[n]=-d1y[n-1]+p0x[n]+p1x[n-1]188.3BasicFIRDigitalFilterStructure

ExpressionofFIRFilterwithTransferFunctionandConvolutionSumAcausalFIRfilteroforderNischaracterizedbyatransferfunctionH(z)givenby

whichisapolynomialinz-1Inthetime-domaintheinput-outputrelationoftheaboveFIRfilterisgivenby198.3.1DirectFormFIRDigitalFilterStructuresDefinition

AnFIRfilteroforderN

ischaracterizedbyN+1coefficientsand,ingeneral,requireN+1multipliersandN

two-inputadders

Structuresinwhichthemultipliercoefficientsarepreciselythecoefficientsofthetransferfunctionarecalleddirectformstructures20•AdirectformrealizationofanFIRfiltercanbereadilydevelopedfromtheconvolutionsumdescriptionasindicatedbelowforN=4

Ananalysisofthisstructureyieldsconvolutionsumdescription8.3.1DirectFormFIRDigitalFilterStructures21•Thetransposeofthedirectformstructureshownearlierisindicatedbelow•Bothdirectformstructuresarecanonicwithrespecttodelays8.3.1DirectFormFIRDigitalFilterStructuresx[n]h[0]y[n]z–1z–1z–1h[1]h[2]h[3]228.3.2CascadeFormFIRDigitalFilterStructures

Ahigher-orderFIRtransferfunctioncanalsoberealizedasacascadeofsecondorderFIRsectionsandpossiblyafirst-ordersection

TothisendweexpressH(z)aswhere:

k=N/2ifNiseven, k=(N+1)/2ifNisodd,with2k=023

AcascaderealizationforN=6isshownbelow

Eachsecond-ordersectionintheabovestructurecanalsoberealizedinthetransposeddirectform8.3.2CascadeFormFIRDigitalFilterStructuresx[n]y[n]z–1z–1h[0]1121z–1z–1z–1z–112221323248.3.4Linear-PhaseFIRStructures

IntroductionTypeI:h[n]=h[N-n],NisevenThesymmetry(orantisymmetry)propertyofalinear-phaseFIRfiltercanbeexploitedtoreducethenumberofmultipliersintoalmosthalfofthatinthedirectformimplementations•Consideralength-7Type1FIRtransferfunctionwithasymmetricimpulseresponse:25RewritingH(z)intheform

weobtaintherealizationshownbelow8.3.4Linear-PhaseFIRStructures

x[n]h[0]y[n]h[1]h[2]z–1z–1z–1z–1h[3]z–1z–126Type2:h[n]=h[N-n],NisoddThecorrespondingrealizationisshownasright•Forexample,alength-8Type2FIRtransferfunctioncanbeexpressedas8.3.4Linear-PhaseFIRStructures

278.3.4Linear-PhaseFIRStructures

x[n]h[0]y[n]h[1]h[2]z–1z–1z–1z–1h[3]z–1z–1z–128•Note:TheType1linear-phasestructureforalength-7FIRfilterrequires4multipliers,whereasadirectformrealizationrequires7multipliers•Note:TheType2linear-phasestructureforalength-8FIRfilterrequires4multipliers,whereasadirectformrealizationrequires8multipliers•SimilarsavingsoccursintherealizationofType3(h[n]=-h[N-n],Nisodd)andType4(h[n]=-h[N-n],Niseven)

linear-phaseFIRfilterswithantisymmetricimpulseresponses8.3.4Linear-PhaseFIRStructures

298.4

BasicIIRdigitalfilterStructure

DescriptionofIIDDigitalFilterwithTransferFunctionandDifferenceEquation

Fromthedifferenceequationrepresentation,itcanbeseenthattherealizationofthecausalIIRdigitalfiltersrequiressomeformoffeedback30

AnN-thorderIIRdigitaltransferfunctionischaracterizedby2N+1uniquecoefficients,andingeneral,requires2N+1multipliersand2Ntwo-inputaddersforimplementation8.4

BasicIIRdigitalfilterStructure318.4.1DirectForm

DirectFormIIRfilters:Filterstructuresinwhichthemultipliercoefficientsarepreciselythecoefficientsofthetransferfunction

Example:Considerforsimplicitya3rd-orderIIRfilterwithatransferfunctionWecanimplementH(z)asacascadeoftwofiltersectionsasshownonthenext32Note:ThedirectformIstructureisnoncanonicasitemploys6delaystorealizea3rd-ordertransferfunction8.4.1DirectFormDirectform133VariousothernoncanonicdirectformstructurescanbederivedbysimpleblockdiagrammanipulationsasshownbelowTheorderofthecascadelinearsystemcanbechanged8.4.1DirectForm341Observeinthedirectformstructureshownright,thesignalvariableatnodesandarethesame,andhencethetwotopdelayscanbesharedLikewise,thesignalvariablesatnodesandarethesame,permittingthesharingofthemiddletwodelays8.4.1DirectForm35

Followingthesameargument,thebottomtwodelayscanbeshared

Sharingofalldelaysreducesthetotalnumberofdelaysto3resultinginacanonicrealizationshownonthenextalongwithitstransposestructureCanonicrealization(DirectFormII)8.4.1DirectForm36DirectFormII(Left)DirectFormII

(right)

8.4.1DirectForm378.4.2CascadeFormIIRFilterRealizations

TheAnalyticalExpressionFormByexpressingthenumeratorandthedenominatorpolynomialsofthetransferfunctionasaproductofpolynomialsoflowerdegree,adigitalfiltercanberealizedasacascadeoflow-orderfiltersectionsConsider,forexample,H(z)=P(z)/D(z)expressedas38•Examplesofcascaderealizationsobtainedbydifferentpole-zeropairingsareshownbelow

Therearealtogetheratotalof36differentcascaderealizationsofH(z)basedonpole-zero-pairingsandordering8.4.2CascadeFormIIRFilterRealizations39Duetofinitewordlengtheffects,eachsuchcascaderealizationbehavesdifferentlyfromothers

WordlengthEffectsUsually,thepolynomialsarefactoredintoaproductof1st-orderand2nd-orderpolynomialspolynomialfactorIntheabove,forafirst-orderfactor8.4.2CascadeFormIIRFilterRealizations40Considerthe3rd-ordertransferfunctionOnepossiblerealizationisshownbelow8.4.2CascadeFormIIRFilterRealizations41Example:areshownonthenextDirectformIICascadeform8.4.2CascadeFormIIRFilterRealizations428.4.3ParallelFormIIRDigitalFilterStructuresTheAnalyticalExpressionFormApartial-fractionexpansionofthetransferfunc