版《数字信号处理(英)》课件Chap10-FIR-Digital-Filter-Design

1Introduction

☆

FIRfilter:directdesignofDTfilterwiththeoftenaddedlinear-phaserequirement(1)WindowedFourierseriesapproach(§10.2)(2)Frequencysamplingapproach(Problem10.31,10.32)

(3)Computer-basedoptimizationmethod(§10.3)

Chap.10FIRDigitalFilterDesign1IntroductionChap.10FIRDi210.1PreliminaryConsiderationsForFIRsystem:realpolynomialapproximationifalinearphaseisdesired

10.1.1BasicApproachestoFIRDigitalFilterDesign210.1PreliminaryConsiderati310.1.2EstimationoftheFilterOrderKaiser’sFormulaForlowpassFIRfilterdesign:P397-398Bellanger’sFormulaHermann’sFormulaParametersseeP398.310.1.2EstimationoftheFil410.2DesignofFIRFiltersbyWindowing(P400)

10.2.1LeastIntegral-SquaredErrorDesignofFIRFilters410.2DesignofFIRFiltersb510.2.2ImpulseResponsesofIdealFiltersIdeallinearphaselowpassfilterIdeallinearphase

highpassfilter510.2.2ImpulseResponsesof6ImpulseResponsesofIdealFilters(II)Ideallinearphasebandpassfilter

Ideallinearphasebandstopfilter

6ImpulseResponsesofIdealFi7ImpulseResponsesofIdealFilters(III)IdealmultibandfilterIdealdiscrete-timeHilberttransformer

Idealdiscrete-timedifferentiator

7ImpulseResponsesofIdealFi8Gibbsphenomenon:

OscillatorybehaviorinthemagnituderesponseofcausalFIRfiltersdesignedutilizingtruncation10.2.3GibbsPhenomenon8Gibbsphenomenon:9mainlobesidelobeMainlobewidth--truncationperiodiccontinuousconvolution=N/210.2.3GibbsPhenomenon(II)9mainlobesidelobeMainlobewidt10-2/(N+1)10-2/(N+1)1110.2.3GibbsPhenomenon(III)1110.2.3GibbsPhenomenon(II12

Noscillatemorerapidly,buttheamplitudesofthelargestripples=constantFor,Nm

,sidelobe,

10.2.3GibbsPhenomenon(IV)(2)Fortheintegral

,oscillationwilloccurateachsidelobeofmovespastthediscontinuity(3)

ThemethodstoreduceGibbsphenomenon:taperingthewindowsmoothlytozeroateachend,butmasmoothtransitioninmagnitudespecifications12Noscillatemorerapi1310.2.4FixedWindowFunctions(1)Hanningwindow:

A=－B=1/2,C=0;Hammingwindow:A=0.54,B=－0.46,C=0Blackmanwindow:A=0.42,B=－0.5,C=0.08.Rectangularwindow:w[n]=u[n]–u[n–

N

–

1]

Hanning,Hamming,Blackman:

Bartlettwindow:triangular1310.2.4FixedWindowFunctio14P406Fig.10.6Commonlyusedfixedwindows10.2.4FixedWindowFunctions(II)N/2NRectangular

Hamming

HanningBartlett

Blackmann

w[n]

114P406Fig.10.6Commonlyu1510.2.4FixedWindowFunctions(III)P407Fig.10.750N1510.2.4FixedWindowFunctio1610.2.4FixedWindowFunctions(IV)Parameterspredictingtheperformanceofawindowmainlobewidth

relativesidelobelevel(dB)Sameripplesinpassbandandstopbandwidthoftransitionband1610.2.4FixedWindowFunctioTypeofwindowRelativeSidelobeLevel(dB)Main-lobewidthMinimumStopbandAttenuation(dB)TransitionBandwidthRect.13.34π/(N+1)20.91.84π/NBartlett26.58π/NHanning31.58π/N43.96.22π/NHamming42.78π/N54.56.64π/NBlackman58.112π/N75.311.12π/N1710.2.4FixedWindowFunctions(V)P408Table10.2’TypeofwindowRelativeSidelob1810.2.4FixedWindowFunctions(VI)ExampletoillustratetheeffectofwindowsN=50P4091810.2.4FixedWindowFunctio1910.2.4FixedWindowFunctions(VII)Computeimpulseresponseofthedesiredfilter(accordingtotheinverseFourierequation)(2)Determinethesuitablewindowbytheminimumstopbandattenuationand(3)DeterminethelengthofFIRbythetransitionwidth(4)ObtainthedesignedFIRfilter:StepsforFIRfilterdesign:1910.2.4FixedWindowFunctio20Example10.6Page410

DesignanFIRlowpassdigitalfilterwithspecifications:theattenuationofthestopbandshouldmorethan40dB;.2)AccordingtoTable10.2,wecouldselectHanning,hamming,Blackmanwindow,thenthebandwidthofthetransition

bandshouldsatisfy(for

Hanning)TypeI:N=32;TypeII:N=3310.2.4FixedWindowFunctions(VIII)1)i.e.Pleaseselectasuitablewindowfunctionanddeterminethesmallestlengthofthewindow.20Example10.6Page410Des2110.2.4FixedWindowFunctionsExampleShowthattheidealhighpasstransformerwithafrequencyresponsedefinedby(1)Determinetheimpulseresponseh[n]，therelationofαandN?(2)Whattypeoflinear-phaseFIRfilter?(3)Writetheimpulseresponseh[n]

usingtheHannwindows-basemethod.Solution:2110.2.4FixedWindowFunctio2210.2.4FixedWindowFunctions2210.2.4FixedWindowFunctio2310.2.4FixedWindowFunctions（2）

IfNisevenwhen,thefilterhaslinearphaseisinteger，hd[n]isanti-symmetries，andh[n]=-h[N-n],thefilteristypeIII.IfNisoddisn’tinteger，hd[n]issymmetries，andh[n]=h[N-n],thefilteristypeII.2310.2.4FixedWindowFunctio2410.2.4FixedWindowFunctions（3）2410.2.4FixedWindowFunctio25with

=N/2.

β

controlstheside-lobeamplitudes(attenuation)

controlsthemainlobewidth

Predictionformula:–attenuation

s

=20log10δsβ–transitionregionwidthω=ωsωp

togetherwithattenuation

s

N(10.39’)10.2.5AdjustableWindowFunctions(P410)KaiserwindowN25with=N/2.(10.39’)10.2.526Amplitude0.305101520(10.41)(10.42)10.2.5AdjustableWindowFunctions(II)26Amplitude0.3051015227Kaiserwindowdesignexample(1)DeterminethewindowfunctionKaiserwindow:,Ni.e.,s=0.01,Assume:Question:IsitsuitableforNtobe23?27Kaiserwindowdesignex28Kaiserwindowdesignexample(II)(2)Thedesiredimpulseresponse28Kaiserwindowdesignex29Kaiserwindowdesignexample(III)(3)TheFIRfilterdesignedWhereN=24,=3.395TypeIlinearphaseFIR29Kaiserwindowdesignex3010.3CADofEquirippleLinear-PhaseFIRFiltersApproximationmethods:(2)LeastIntegral-SquaredapproximationWindowedFourierSeriesapproach(1)InterpolationFrequencysamplingapproach(3)ChebyshevapproximationEquirippleapproximationParks-McClellanAlgorithm3010.3CADofEquirippleLine3110.3CADofEquirippleLinear-PhaseFIRFilters(II)Weightederrorfunction:(10.47)or(10.62)(10.68)3110.3CADofEquirippleLine3210.3CADofEquirippleLinear-PhaseFIRFilters(III)ChebyshevorMinimaxcriterion:equirippleFIRfilterMinimizethepeakabsolutevalueofLinear-phaseFIRfiltersobtainedbythecriterionpolynomialapproximation3210.3CADofEquirippleLine3310.3CADofEquirippleLinear-PhaseFIRFilters(IV)AlternationTheorem:LetRbeaunionofdisjointclosedsubsetsofLetadesiredfunctionD(x)andweightingfunctionW(x)becontinuousonRDefinetheerrorfunction

E(x)=W(x)[PL(x)－D(x)]Maximumerror

3310.3CADofEquirippleLine3410.3CADofEquirippleLinear-PhaseFIRFilters(V)necessaryandsufficientconditionforPL(x)beingtheuniqueLthorderpolynomialundertheMinimax

criterioncanbeexpressedbythealternationtheorem:E(x)hasatleastL+2alterationsonF

,i.e.

xi,i=1,...,L’≥L+2suchthatxi

<xi+1,E(xi)=－E(xi+1),fori=1,...,L’－1andE(xi)

=±Emax,fori=1,...,L’3410.3CADofEquirippleLine3510.3CADofEquirippleLinear-PhaseFIRFilters(VI)Parks-McClellanAlgorithmIterativemethodtodeterminethealternationfrequenciesωi

andtheripple

1.initialize

ωi

pute—erpolateapolynomialbetweenthealternationpoints4.findthemaximum/minimumvaluesoftheerror5.if|E(ω)|≤

:stopelsecomputenewωi’asextremeofE(ω),andgoto2(elserecursive)3510.3CADofEquirippleLine3610.5FIRDigitalFilterDesignUsingMatlabOrderEstimation:kaiord()Kaiser’sFormulabellangord()Bellanger’sFormularemezord()Hermann’sFormulakaiserord()filterorderforKaiserwindow-baseddesign3610.5FIRDigitalFilterDes3710.5FIRDigitalFilterDesignUsingMatlab(II)Equiripplelinear-phaseFIRfilterdesign:remez()equirippleFIRfilterdesignusingParks-McClellanalgorithmExample10.15DesignanequirippleFIRfilterwithspecifications:3710.5FIRDigitalFilterDes3810.5FIRDigitalFilterDesignUsingMatlab(III)3810.5FIRDigitalFilterDes3910.5FIRDigitalFilterDesignUsingMatlab(IV)3910.5FIRDigitalFilterDes4010.5FIRDigitalFilterDesignUsingMatlab(V)4010.5FIRDigitalFilterDes4110.5FIRDigitalFilterDesignUsingMatlab(VI)4110.5FIRDigitalFilterDes42WindowingmethodforFIRfilterdesign:fir1()andfir2()Example10.15DesignaFIRlowpassfilterusingakaiserwindowwithspecifications:10.5FIRDigitalFilterDesignUsingMatlab(VI)42WindowingmethodforFIRfil4310.1,10.2,10.3estimationformula10.4multibandfilterimpulseresponse10.5truncationapproximation10.6,10.7idealdigitalHilberttransformation10.8idealdigitaldifferentiator10.9delay-complementarypair10.10,10.11,10.12,10.18inverseDTFT10.15,10.16,10.17windowingmethoddesign10.20fractionaldelayFIRfilter10.21idealcombfilter10.27,10.28differentfittingalgorithm10.29filtersharpening10.31~10.35frequencysamplingmethod10.40WDFT10.36~10.38Parks-McClellanalgorithmweightingfunctionExercises4310.1,10.2,10.3estimat44Introduction

☆

FIRfilter:directdesignofDTfilterwiththeoftenaddedlinear-phaserequirement(1)WindowedFourierseriesapproach(§10.2)(2)Frequencysamplingapproach(Problem10.31,10.32)

(3)Computer-basedoptimizationmethod(§10.3)

Chap.10FIRDigitalFilterDesign1IntroductionChap.10FIRDi4510.1PreliminaryConsiderationsForFIRsystem:realpolynomialapproximationifalinearphaseisdesired

10.1.1BasicApproachestoFIRDigitalFilterDesign210.1PreliminaryConsiderati4610.1.2EstimationoftheFilterOrderKaiser’sFormulaForlowpassFIRfilterdesign:P397-398Bellanger’sFormulaHermann’sFormulaParametersseeP398.310.1.2EstimationoftheFil4710.2DesignofFIRFiltersbyWindowing(P400)

10.2.1LeastIntegral-SquaredErrorDesignofFIRFilters410.2DesignofFIRFiltersb4810.2.2ImpulseResponsesofIdealFiltersIdeallinearphaselowpassfilterIdeallinearphase

highpassfilter510.2.2ImpulseResponsesof49ImpulseResponsesofIdealFilters(II)Ideallinearphasebandpassfilter

Ideallinearphasebandstopfilter

6ImpulseResponsesofIdealFi50ImpulseResponsesofIdealFilters(III)IdealmultibandfilterIdealdiscrete-timeHilberttransformer

Idealdiscrete-timedifferentiator

7ImpulseResponsesofIdealFi51Gibbsphenomenon:

OscillatorybehaviorinthemagnituderesponseofcausalFIRfiltersdesignedutilizingtruncation10.2.3GibbsPhenomenon8Gibbsphenomenon:52mainlobesidelobeMainlobewidth--truncationperiodiccontinuousconvolution=N/210.2.3GibbsPhenomenon(II)9mainlobesidelobeMainlobewidt53-2/(N+1)10-2/(N+1)5410.2.3GibbsPhenomenon(III)1110.2.3GibbsPhenomenon(II55

Noscillatemorerapidly,buttheamplitudesofthelargestripples=constantFor,Nm

,sidelobe,

10.2.3GibbsPhenomenon(IV)(2)Fortheintegral

,oscillationwilloccurateachsidelobeofmovespastthediscontinuity(3)

ThemethodstoreduceGibbsphenomenon:taperingthewindowsmoothlytozeroateachend,butmasmoothtransitioninmagnitudespecifications12Noscillatemorerapi5610.2.4FixedWindowFunctions(1)Hanningwindow:

A=－B=1/2,C=0;Hammingwindow:A=0.54,B=－0.46,C=0Blackmanwindow:A=0.42,B=－0.5,C=0.08.Rectangularwindow:w[n]=u[n]–u[n–

N

–

1]

Hanning,Hamming,Blackman:

Bartlettwindow:triangular1310.2.4FixedWindowFunctio57P406Fig.10.6Commonlyusedfixedwindows10.2.4FixedWindowFunctions(II)N/2NRectangular

Hamming

HanningBartlett

Blackmann

w[n]

114P406Fig.10.6Commonlyu5810.2.4FixedWindowFunctions(III)P407Fig.10.750N1510.2.4FixedWindowFunctio5910.2.4FixedWindowFunctions(IV)Parameterspredictingtheperformanceofawindowmainlobewidth

relativesidelobelevel(dB)Sameripplesinpassbandandstopbandwidthoftransitionband1610.2.4FixedWindowFunctioTypeofwindowRelativeSidelobeLevel(dB)Main-lobewidthMinimumStopbandAttenuation(dB)TransitionBandwidthRect.13.34π/(N+1)20.91.84π/NBartlett26.58π/NHanning31.58π/N43.96.22π/NHamming42.78π/N54.56.64π/NBlackman58.112π/N75.311.12π/N6010.2.4FixedWindowFunctions(V)P408Table10.2’TypeofwindowRelativeSidelob6110.2.4FixedWindowFunctions(VI)ExampletoillustratetheeffectofwindowsN=50P4091810.2.4FixedWindowFunctio6210.2.4FixedWindowFunctions(VII)Computeimpulseresponseofthedesiredfilter(accordingtotheinverseFourierequation)(2)Determinethesuitablewindowbytheminimumstopbandattenuationand(3)DeterminethelengthofFIRbythetransitionwidth(4)ObtainthedesignedFIRfilter:StepsforFIRfilterdesign:1910.2.4FixedWindowFunctio63Example10.6Page410

DesignanFIRlowpassdigitalfilterwithspecifications:theattenuationofthestopbandshouldmorethan40dB;.2)AccordingtoTable10.2,wecouldselectHanning,hamming,Blackmanwindow,thenthebandwidthofthetransition

bandshouldsatisfy(for

Hanning)TypeI:N=32;TypeII:N=3310.2.4FixedWindowFunctions(VIII)1)i.e.Pleaseselectasuitablewindowfunctionanddeterminethesmallestlengthofthewindow.20Example10.6Page410Des6410.2.4FixedWindowFunctionsExampleShowthattheidealhighpasstransformerwithafrequencyresponsedefinedby(1)Determinetheimpulseresponseh[n]，therelationofαandN?(2)Whattypeoflinear-phaseFIRfilter?(3)Writetheimpulseresponseh[n]

usingtheHannwindows-basemethod.Solution:2110.2.4FixedWindowFunctio6510.2.4FixedWindowFunctions2210.2.4FixedWindowFunctio6610.2.4FixedWindowFunctions（2）

IfNisevenwhen,thefilterhaslinearphaseisinteger，hd[n]isanti-symmetries，andh[n]=-h[N-n],thefilteristypeIII.IfNisoddisn’tinteger，hd[n]issymmetries，andh[n]=h[N-n],thefilteristypeII.2310.2.4FixedWindowFunctio6710.2.4FixedWindowFunctions（3）2410.2.4FixedWindowFunctio68with

=N/2.

β

controlstheside-lobeamplitudes(attenuation)

controlsthemainlobewidth

Predictionformula:–attenuation

s

=20log10δsβ–transitionregionwidthω=ωsωp

togetherwithattenuation

s

N(10.39’)10.2.5AdjustableWindowFunctions(P410)KaiserwindowN25with=N/2.(10.39’)10.2.569Amplitude0.305101520(10.41)(10.42)10.2.5AdjustableWindowFunctions(II)26Amplitude0.3051015270Kaiserwindowdesignexample(1)DeterminethewindowfunctionKaiserwindow:,Ni.e.,s=0.01,Assume:Question:IsitsuitableforNtobe23?27Kaiserwindowdesignex71Kaiserwindowdesignexample(II)(2)Thedesiredimpulseresponse28Kaiserwindowdesignex72Kaiserwindowdesignexample(III)(3)TheFIRfilterdesignedWhereN=24,=3.395TypeIlinearphaseFIR29Kaiserwindowdesignex7310.3CADofEquirippleLinear-PhaseFIRFiltersApproximationmethods:(2)LeastIntegral-SquaredapproximationWindowedFourierSeriesapproach(1)InterpolationFrequencysamplingapproach(3)ChebyshevapproximationEquirippleapproximationParks-McClellanAlgorithm3010.3CADofEquirippleLine7410.3CADofEquirippleLinear-PhaseFIRFilters(II)Weightederrorfunction:(10.47)or(10.62)(10.68)3110.3CADofEquirippleLine7510.3CADofEquirippleLinear-PhaseFIRFilters(III)ChebyshevorMinimaxcriterion:equirippleFIRfilterMinimizethepeakabsolutevalueofLinear-phaseFIRfiltersobtainedbythecriterionpolynomialapproximation3210.3CADofEquirippleLine7610.3CADofEquirippleLinear-PhaseFIRFilters(IV)AlternationTheorem:LetRbeaunionofdisjointclosedsubsetsofLetadesiredfunctionD(x)andweightingfunctionW(x)becontinuousonRDefinetheerrorfunction

E(x)=W(x)[PL(x)－D(x)]Maximumerror

3310.3CADofEquirippleLine7710.3CADofEquirippleLinear-PhaseFIRFilters(V)necessaryandsufficientconditionforPL(x)beingtheuniqueLthorderpolynomialundertheMinimax

criterioncanbeexpressedbythealternationtheorem:E(x)hasatleastL+2alterationsonF

,i.e.

xi,i=1,...,L’≥L+2suchthatxi

<xi+1,E(xi)=－E(xi+1),fori=1,...,L’－1andE(xi)

=±Emax,fori=1,...,L’3410.3CADofEquirippleLine7810.3CADofEquirippleLinear-PhaseFIRFilters(VI)Parks-McClellanAlgorithmIterativemethodtodeterminethealternationfrequenciesωi

andtheripple

1.initialize

ωi

to