版《数字信号处理(英)》课件ch8-Digital-Filter-Structures

1ch8DigitalFilterStructures1ch8DigitalFilterStructures2IntroductionllRsystemcan’tbeimplementedusingthe

convolutionsum,because

theimpulseresponseisofinfinitelength

FIRsystemcanbeimplementedusingtheconvolutionsumwhichisafinitesumofproducts2IntroductionllRsystemcan’t3Introduction

However,adirectimplementationofthellRfinite-dimensionalsystemispractical

FormsofImplementation:TheactualimplementationofanLTIdigitalfiltercanbeeitherinsoftwareorhardwareform,dependingonapplications

NoteThat:Ineithercase,thesignalvariablesandthefiltercoefficientscannotberepresentedwithinfiniteprecision.3IntroductionHowever,adir4

So,adirectimplementationofadigitalfilterbasedoneitherthedifferenceequationorthefiniteconvolutionsummaynotprovidesatisfactoryperformanceduetothefiniteprecisionarithmeticKEYPROBLEM:ItisthusofpracticalinteresttodevelopalternaterealizationsandchoosethestructurethatprovidessatisfactoryperformanceunderfiniteprecisionarithmeticIntroduction4So,adirectimplementati5TheImportanceofthestructuralrepresentation:--------thefirststepinthehardwareorsoftwareimplementationofanLTIdigitalfilter

Thestructuralrepresentationprovidesthekeyrelationsbetweensomepertinentinternalvariableswiththeinputandoutputthatinturnprovidesthekeytotheimplementation.Introduction5TheImportanceofthestructu68.1BlockDiagramRepresentation1)Therepresentationoftheinput-outputrelationwithanalyticalexpression

convolutionsumLinearconstantcoefficientdifferenceequation68.1BlockDiagramRepresentat72)

TheimplementationofanLTIfilter

---------aValidcomputationalalgorithmToillustratewhatwemeanbyacomputationalalgorithm,considerthecausalfirst-orderLTIdigitalfiltershownbelow8.1BlockDiagramRepresentation72)TheimplementationofanL8•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].…8.1BlockDiagramRepresentation8•Thefilterisdescribedby9WecancontinuethiscalculationforanyvalueofthetimeindexnwedesireAsaresult,thefirstorderdifferenceequationcanbeinterpretedasavalidcomputationalalgorithm8.1BlockDiagramRepresentation9Wecancontinuethiscalculat108.1.1BasicBuildingBlocksThecomputationalalgorithmofanLTIdigitalfiltercanbeconvenientlyrepresentedinblockdiagramformusingthebasicbuildingblocksshownbelowx[n]y[n]w[n]Ax[n]y[n]y[n]x[n]x[n]y2[n]y1[n]AdderUnitdelayMultiplierPick-offnode108.1.1BasicBuildingBlocks11Advantagesofblockdiagramrepresentation

(l)Easytowritedownthecomputationalalgorithmbyinspection(2)Easytoanalyzetheblockdiagramtodeterminetheexplicitrelationbetweentheoutputandinput8.1.1BasicBuildingBlocks11Advantagesofblockdiagram12(3)Easytomanipulateablockdiagramtoderiveother"equivalent,'blockdiagramsyieldingdifferentcomputationalalgorithms(4)Easytodeterminethehardwarerequirements(5)Easytodevelopblockdiagramrepresentationsfromthetransferfunctiondirectly8.1.1BasicBuildingBlocks12(3)Easytomanipulateab138.1.2AnalysisofBlockDiagrams

Carriedoutbywritingdowntheexpressionsfortheoutputsignalsofeachadderasasumofitsinputsignals,anddevelopingasetofequationsrelatingthefilterinputandoutputsignalsintermsofallinternalsignals

EliminatingtheunwantedinternalvariablesthenresultsintheexpressionfortheoutputsignalasafunctionoftheinputsignalandthefilterparametersthatarethemultipliercoefficientsAnalysisMethod138.1.2AnalysisofBlockDia14Example(1)Considertheshownbelowsingle-loopfeedbackStructureTheoutputE(z)oftheadderisE(z)=X(z)+G2(z)Y(z)ButfromthefigureY(z)=G1(z)E(z)8.1.2AnalysisofBlockDiagrams14Example(1)Considerthesho15(2)Analyzethecascadedlatticestructureshownbelowwherethez-dependenceofsignalvariablesarenotshownforbrevityEliminatingE(z)fromtheprevioustwoequationswearriveat[1-G1(z)G2(z)]Y(z)=G1(z)X(z)whichleadsto

8.1.2AnalysisofBlockDiagrams15(2)Analyzethecascadedla16TheoutputsignalsaregivenbyW1=X-S2W2=W1-S1W3=S1-W2Y=W1-S2FromthefigureweobserveS2=z-1W3S1=z-1W28.1.2AnalysisofBlockDiagrams16Theoutputsignalsaregiven17EliminatingW1,W2,

W3,S1andS2wefinally

arriveat8.1.2AnalysisofBlockDiagrams17EliminatingW1,W2,W3,S1an188.1.3TheDelay-freeLoopProbIemToillustratethedelay-freeloopproblemconsiderthestructurebelowForphysicalrealizabilityofthedigitalfilterstructure,itisnecessarythattheblockdiagramrepresentationcontainsnodelay-freeloops(containdelayloops)188.1.3TheDelay-freeLoop19Analysisofthisstructureyieldsu[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-freeLoopProbIem19Analysisofthisstructure20

However,thisisphysicallyimpossibletoachieveduetothefinitetimerequiredtocarryoutallarithmeticoperationsonadigitalmachine

Methodexiststodetectthepresenceofdelay-freeloopsinanarbitrarystructure,alongwithmethodstolocateandremovetheseloopswithouttheoverallinput-outputrelation8.1.3TheDelay-freeLoopProbIem20However,thisisphysical21Figurebelowshowssucharealizationoftheexamplestructuredescribedearlier8.1.3TheDelay-freeLoopProbIem21Figurebelowshowssuchare228.1.4CanonicandNoncanonicStructuresDefinition:Adigitalfilterstructureissaidtobecanonicifthenumberofdelaysintheblockdiagramrepresentationisequaltotheorderofthetransferfunction.Otherwise,itisanoncanonicstructure•Thestructureshownbelowisnoncanonicasitemploystwodelaystorealizeafirst-orderdifferenceequationy[n]=-d1y[n-1]+p0x[n]+p1x[n-1]228.1.4CanonicandNoncanoni8.3BasicFIRDigitalFilterStructure

23ExpressionofFIRFilterwithTransferFunctionandConvolutionSumAcausalFIRfilteroforderNischaracterizedbyatransferfunctionH(z)givenby whichisapolynomialinz-1Inthetime-domaintheinput-outputrelationoftheaboveFIRfilterisgivenby8.3BasicFIRDigitalFilterS8.3.1DirectFormFIRDigitalFilterStructures24Definition

AnFIRfilteroforderN

ischaracterizedbyN+1coefficientsand,ingeneral,requireN+1multipliersandN

two-inputadders

Structuresinwhichthemultipliercoefficientsarepreciselythecoefficientsofthetransferfunctionarecalleddirectformstructures8.3.1DirectFormFIRDigital25•AdirectformrealizationofanFIRfiltercanbereadilydevelopedfromtheconvolutionsumdescriptionasindicatedbelowforN=4

Ananalysisofthisstructureyieldsconvolutionsumdescription8.3.1DirectFormFIRDigitalFilterStructures25•Adirectformrealization26•Thetransposeofthedirectformstructureshownearlierisindicatedbelow•Bothdirectformstructuresarecanonicwithrespecttodelays8.3.1DirectFormFIRDigitalFilterStructuresx[n]h[0]y[n]z–1z–1z–1h[1]h[2]h[3]26•Thetransposeofthedirec278.3.2CascadeFormFIRDigitalFilterStructures

Ahigher-orderFIRtransferfunctioncanalsoberealizedasacascadeofsecondorderFIRsectionsandpossiblyafirst-ordersection

TothisendweexpressH(z)aswhere:

k=N/2ifNiseven, k=(N+1)/2ifNisodd,with2k=0278.3.2CascadeFormFIRDigit28

AcascaderealizationforN=6isshownbelow

Eachsecond-ordersectionintheabovestructurecanalsoberealizedinthetransposeddirectform8.3.2CascadeFormFIRDigitalFilterStructuresx[n]y[n]z–1z–1h[0]1121z–1z–1z–1z–11222132328Acascaderealizationfor298.3.4Linear-PhaseFIRStructures

IntroductionTypeI:h[n]=h[N-n],NisevenThesymmetry(orantisymmetry)propertyofalinear-phaseFIRfiltercanbeexploitedtoreducethenumberofmultipliersintoalmosthalfofthatinthedirectformimplementations•Consideralength-7Type1FIRtransferfunctionwithasymmetricimpulseresponse:298.3.4Linear-PhaseFIRStru30RewritingH(z)intheform

weobtaintherealizationshownbelow8.3.4Linear-PhaseFIRStructures

x[n]h[0]y[n]h[1]h[2]z–1z–1z–1z–1h[3]z–1z–130RewritingH(z)intheform w31Type2:h[n]=h[N-n],NisoddThecorrespondingrealizationisshownasright•Forexample,alength-8Type2FIRtransferfunctioncanbeexpressedas8.3.4Linear-PhaseFIRStructures

31Type2:h[n]=h[N-n],Niso328.3.4Linear-PhaseFIRStructures

x[n]h[0]y[n]h[1]h[2]z–1z–1z–1z–1h[3]z–1z–1z–1328.3.4Linear-PhaseFIRStru33•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

33•Note:TheType1linear-ph8.4

BasicIIRdigitalfilterStructure34

DescriptionofIIDDigitalFilterwithTransferFunctionandDifferenceEquation

Fromthedifferenceequationrepresentation,itcanbeseenthattherealizationofthecausalIIRdigitalfiltersrequiressomeformoffeedback8.4BasicIIRdigitalfilter35

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

BasicIIRdigitalfilterStructure35AnN-thorderIIRdigital368.4.1DirectForm

DirectFormIIRfilters:Filterstructuresinwhichthemultipliercoefficientsarepreciselythecoefficientsofthetransferfunction

Example:Considerforsimplicitya3rd-orderIIRfilterwithatransferfunctionWecanimplementH(z)asacascadeoftwofiltersectionsasshownonthenext368.4.1DirectFormDirectFo37Note:ThedirectformIstructureisnoncanonicasitemploys6delaystorealizea3rd-ordertransferfunction8.4.1DirectFormDirectform137Note:ThedirectformIstru38VariousothernoncanonicdirectformstructurescanbederivedbysimpleblockdiagrammanipulationsasshownbelowTheorderofthecascadelinearsystemcanbechanged8.4.1DirectForm38Variousothernoncanonicdir391Observeinthedirectformstructureshownright,thesignalvariableatnodesandarethesame,andhencethetwotopdelayscanbesharedLikewise,thesignalvariablesatnodesandarethesame,permittingthesharingofthemiddletwodelays8.4.1DirectForm391Observeinthedirectform40

Followingthesameargument,thebottomtwodelayscanbeshared

Sharingofalldelaysreducesthetotalnumberofdelaysto3resultinginacanonicrealizationshownonthenextalongwithitstransposestructureCanonicrealization(DirectFormII)8.4.1DirectForm40Followingthesameargumen41DirectFormII(Left)DirectFormII

(right)

8.4.1DirectForm41DirectFormII(Left)Direct428.4.2CascadeFormIIRFilterRealizations

TheAnalyticalExpressionFormByexpressingthenumeratorandthedenominatorpolynomialsofthetransferfunctionasaproductofpolynomialsoflowerdegree,adigitalfiltercanberealizedasacascadeoflow-orderfiltersectionsConsider,forexample,H(z)=P(z)/D(z)expressedas428.4.2CascadeFormIIRFilte43•Examplesofcascaderealizationsobtainedbydifferentpole-zeropairingsareshownbelow

Therearealtogetheratotalof36differentcascaderealizationsofH(z)basedonpole-zero-pairingsandordering8.4.2CascadeFormIIRFilterRealizations43•Examplesofcascaderealiz44Duetofinitewordlengtheffects,eachsuchcascaderealizationbehavesdifferentlyfromothers

WordlengthEffectsUsually,thepolynomialsarefactoredintoaproductof1st-orderand2nd-orderpolynomialspolynomialfactorIntheabove,forafirst-orderfactor8.4.2CascadeFormIIRFilterRealizations44Duetofinitewordlengtheff45Considerthe3rd-ordertransferfunctionOnepossiblerealizationisshownbelow8.4.2CascadeFormIIRFilterRealizations45Considerthe3rd-ordertrans46Example:areshownonthenextDirectformIICascadeform8.4.2CascadeFormIIRFilterRealizations46Example:areshownonthenex478.4.3ParallelFormIIRDigitalFilterStructuresTheAnalyticalExpressionFormApartial-fractionexpansionofthetransferfunctioninz-1leadstotheparallelformIstructure

Assumingsimplepoles,thetransferfunctionH(z)canbeexpressedas

Intheaboveforarealpole478.4.3ParallelFormIIRDigi48

Adirectpartial-fractionexpansionofthetransferfunctioninzleadstotheparallelformIIstructure

Assumingsimplepoles,thetransferfunctionH(z)canbeexpressedas

Intheaboveforarealpole8.4.3ParallelFormIIRDigitalFilterStructures48Adirectpartial-fractione49•Thetwobasicparallelrealizationsofa3rd-orderIIRtransferfunctionareshownbelowParallelformIParallelformII8.4.3ParallelFormIIRDigitalFilterStructures49•Thetwobasicparallelrea50ThecorrespondingparallelformIrealizationisshownbelowExample:8.4.3ParallelFormIIRDigitalFilterStructures50Thecorrespondingparallelf51•Likewise,apartial-fractionexpansionofH(z)inzyields•ThecorrespondingparallelformIIrealizationisshownontheright8.4.3ParallelFormIIRDigitalFilterStructures51•Likewise,apartial-fracti52Exercise:8.3(stableproblem),8.7(transferfunction),8.14(structureofsymmetricFIR),8.15(structureofsymmetricFIR),8.19(canonicandtransposeconfiguration),8.20(a)(cascadecanonicstructure),8.24(Transferfunction,differenceequation)realization,impulseresponse,inputandoutput),M8.1,M8.252Exercise:53ch8DigitalFilterStructures1ch8DigitalFilterStructures54IntroductionllRsystemcan’tbeimplementedusingthe

convolutionsum,because

theimpulseresponseisofinfinitelength

FIRsystemcanbeimplementedusingtheconvolutionsumwhichisafinitesumofproducts2IntroductionllRsystemcan’t55Introduction

However,adirectimplementationofthellRfinite-dimensionalsystemispractical

FormsofImplementation:TheactualimplementationofanLTIdigitalfiltercanbeeitherinsoftwareorhardwareform,dependingonapplications

NoteThat:Ineithercase,thesignalvariablesandthefiltercoefficientscannotberepresentedwithinfiniteprecision.3IntroductionHowever,adir56

So,adirectimplementationofadigitalfilterbasedoneitherthedifferenceequationorthefiniteconvolutionsummaynotprovidesatisfactoryperformanceduetothefiniteprecisionarithmeticKEYPROBLEM:ItisthusofpracticalinteresttodevelopalternaterealizationsandchoosethestructurethatprovidessatisfactoryperformanceunderfiniteprecisionarithmeticIntroduction4So,adirectimplementati57TheImportanceofthestructuralrepresentation:--------thefirststepinthehardwareorsoftwareimplementationofanLTIdigitalfilter

Thestructuralrepresentationprovidesthekeyrelationsbetweensomepertinentinternalvariableswiththeinputandoutputthatinturnprovidesthekeytotheimplementation.Introduction5TheImportanceofthestructu588.1BlockDiagramRepresentation1)Therepresentationoftheinput-outputrelationwithanalyticalexpression

convolutionsumLinearconstantcoefficientdifferenceequation68.1BlockDiagramRepresentat592)

TheimplementationofanLTIfilter

---------aValidcomputationalalgorithmToillustratewhatwemeanbyacomputationalalgorithm,considerthecausalfirst-orderLTIdigitalfiltershownbelow8.1BlockDiagramRepresentation72)TheimplementationofanL60•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].…8.1BlockDiagramRepresentation8•Thefilterisdescribedby61WecancontinuethiscalculationforanyvalueofthetimeindexnwedesireAsaresult,thefirstorderdifferenceequationcanbeinterpretedasavalidcomputationalalgorithm8.1BlockDiagramRepresentation9Wecancontinuethiscalculat628.1.1BasicBuildingBlocksThecomputationalalgorithmofanLTIdigitalfiltercanbeconvenientlyrepresentedinblockdiagramformusingthebasicbuildingblocksshownbelowx[n]y[n]w[n]Ax[n]y[n]y[n]x[n]x[n]y2[n]y1[n]AdderUnitdelayMultiplierPick-offnode108.1.1BasicBuildingBlocks63Advantagesofblockdiagramrepresentation

(l)Easytowritedownthecomputationalalgorithmbyinspection(2)Easytoanalyzetheblockdiagramtodeterminetheexplicitrelationbetweentheoutputandinput8.1.1BasicBuildingBlocks11Advantagesofblockdiagram64(3)Easytomanipulateablockdiagramtoderiveother"equivalent,'blockdiagramsyieldingdifferentcomputationalalgorithms(4)Easytodeterminethehardwarerequirements(5)Easytodevelopblockdiagramrepresentationsfromthetransferfunctiondirectly8.1.1BasicBuildingBlocks12(3)Easytomanipulateab658.1.2AnalysisofBlockDiagrams

Carriedoutbywritingdowntheexpressionsfortheoutputsignalsofeachadderasasumofitsinputsignals,anddevelopingasetofequationsrelatingthefilterinputandoutputsignalsintermsofallinternalsignals

EliminatingtheunwantedinternalvariablesthenresultsintheexpressionfortheoutputsignalasafunctionoftheinputsignalandthefilterparametersthatarethemultipliercoefficientsAnalysisMethod138.1.2AnalysisofBlockDia66Example(1)Considertheshownbelowsingle-loopfeedbackStructureTheoutputE(z)oftheadderisE(z)=X(z)+G2(z)Y(z)ButfromthefigureY(z)=G1(z)E(z)8.1.2AnalysisofBlockDiagrams14Example(1)Considerthesho67(2)Analyzethecascadedlatticestructureshownbelowwherethez-dependenceofsignalvariablesarenotshownforbrevityEliminatingE(z)fromtheprevioustwoequationswearriveat[1-G1(z)G2(z)]Y(z)=G1(z)X(z)whichleadsto

8.1.2AnalysisofBlockDiagrams15(2)Analyzethecascadedla68TheoutputsignalsaregivenbyW1=X-S2W2=W1-S1W3=S1-W2Y=W1-S2FromthefigureweobserveS2=z-1W3S1=z-1W28.1.2AnalysisofBlockDiagrams16Theoutputsignalsaregiven69EliminatingW1,W2,

W3,S1andS2wefinally

arriveat8.1.2AnalysisofBlockDiagrams17EliminatingW1,W2,W3,S1an708.1.3TheDelay-freeLoopProbIemToillustratethedelay-freeloopproblemconsiderthestructurebelowForphysicalrealizabilityofthedigitalfilterstructure,itisnecessarythattheblockdiagramrepresentationcontainsnodelay-freeloops(containdelayloops)188.1.3TheDelay-freeLoop71Analysisofthisstructureyieldsu[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-freeLoopProbIem19Analysisofthisstructure72

However,thisisphysicallyimpossibletoachieveduetothefinitetimerequiredtocarryoutallarithmeticoperationsonadigitalmachine

Methodexiststodetectthepresenceofdelay-freeloopsinanarbitrarystructure,alongwithmethodstolocateandremovetheseloopswithouttheoverallinput-outputrelation8.1.3TheDelay-freeLoopProbIem20However,thisisphysical73Figurebelowshowssucharealizationoftheexamplestructuredescribedearlier8.1.3TheDelay-freeLoopProbIem21Figurebelowshowssuchare748.1.4CanonicandNoncanonicStructuresDefinition:Adigitalfilterstructureissaidtobecanonicifthenumberofdelaysintheblockdiagramrepresentationisequaltotheorderofthetransferfunction.Otherwise,itisanoncanonicstructure•Thestructureshownbelowisnoncanonicasitemploystwodelaystorealizeafirst-orderdifferenceequationy[n]=-d1y[n-1]+p0x[n]+p1x[n-1]228.1.4CanonicandNoncanoni8.3BasicFIRDigitalFilterStructure

75ExpressionofFIRFilterwithTransferFunctionandConvolutionSumAcausalFIRfilteroforderNischaracterizedbyatransferfunctionH(z)givenby whichisapolynomialinz-1Inthetime-domaintheinput-outputrelationoftheaboveFIRfilterisgivenby8.3BasicFIRDigitalFilterS8.3.1DirectFormFIRDigitalFilterStructures76Definition

AnFIRfilteroforderN

ischaracterizedbyN+1coefficientsand,ingeneral,requireN+1multipliersandN

two-inputadders

Structuresinwhichthemultipliercoefficientsarepreciselythecoefficientsofthetransferfunctionarecalleddirectformstructures8.3.1DirectFormFIRDigital77•AdirectformrealizationofanFIRfiltercanbereadilydevelopedfromtheconvolutionsumdescriptionasindicatedbelowforN=4

Ananalysisofthisstructureyieldsconvolutionsumdescription8.3.1DirectFormFIRDigitalFilterStructures25•Adirectformrealization78•Thetransposeofthedirectformstructureshownearlierisindicatedbelow•Bothdirectformstructuresarecanonicwithrespecttodelays8.3.1DirectFormFIRDigitalFilterStructuresx[n]h[0]y[n]z–1z–1z–1h[1]h[2]h[3]26•Thetransposeofthedirec798.3.2CascadeFormFIRDigitalFilterStructures

Ahigher-orderFIRtransferfunctioncanalsoberealizedasacascadeofsecondorderFIRsectionsandpossiblyafirst-ordersection

TothisendweexpressH(z)aswhere:

k=N/2ifNiseven, k=(N+1)/2ifNisodd,with2k=0278.3.2CascadeFormFIRDigit80

AcascaderealizationforN=6isshownbelow

Eachsecond-ordersectionintheabovestructurecanalsoberealizedinthetransposeddirectform8.3.2CascadeFormFIRDigitalFilterStructuresx[n]y[n]z–1z–1h[0]1121z–1z–1z–1z–11222132328Acascaderealizationfor818.3.4Linear-PhaseFIRStructures

IntroductionTypeI:h[n]=h[N-n],NisevenThesymmetry(orantisymmetry)propertyofalinear-phaseFIRfiltercanbeexploitedtoreducethenumberofmultipliersintoalmosthalfofthatinthedirectformimplementations•Consideralength-7Type1FIRtransferfunctionwithasymmetricimpulseresponse:298.3.4Linear-PhaseFIRStru82RewritingH(z)intheform

weobtaintherealizationshownbelow8.3.4Linear-PhaseFIRStructures

x[n]h[0]y[n]h[1]h[2]z–1z–1z–1z–1h[3]z–1z–130RewritingH(z)intheform w83Type2:h[n]=h[N-n],NisoddThecorrespondingrealizationisshownasright•Forexample,alength-8Type2FIRtransferfunctioncanbeexpressedas8.3.4Linear-PhaseFIRStructures

31Type2:h[n]=h[N-n],Niso848.3.4Linear-PhaseFIRStructures

x[n]h[0]y[n]h[1]h[2]z–1z–1z–1z–1h[3]z–1z–1z–1328.3.4Linear-PhaseFIRStru85•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

33•Note:TheType1linear-ph8.4

BasicIIRdigitalfilterStructure86

DescriptionofIIDDigitalFilterwithTransferFunctionandDifferenceEquation

Fromthedifferenceequationrepresentation,itcanbeseenthattherealizationofthecausalIIRdigitalfiltersrequiressomeformoffeedback8.4BasicIIRdigitalfilter87

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

BasicIIRdigitalfilterStructure35AnN-thorderIIRdigital888.4.1DirectForm

DirectFormIIRfilters:Filterstructuresinwhichthemultipliercoefficientsarepreciselythecoefficientsofthetransferfunction

Example:Considerforsimplicitya3rd-orderIIRfilterwithatransferfunctionWecanimplementH(z)asacascadeoftwofiltersectionsasshownonthenext368.4.1DirectFormDirectFo89Note:ThedirectformIstructureisnoncanonicasitemploys6delaystorealizea3rd-ordertransferfunction8.4.1DirectFormDirectform137Note:ThedirectformIstru90VariousothernoncanonicdirectformstructurescanbederivedbysimpleblockdiagrammanipulationsasshownbelowTheorder