数字信号处理chapter 6 FIR desig

Chapter6DesignofFIRDigitalFilter1.IIRvs.FIRIIRAnalogfilter—Designsimply.Spectrumspecificationuneasytocontrol.FIR:Strictlylinearspectrumcharacteristics.

h(n)isfinitelength,noproblemofstabilityandcausality.2.CONTENTS6.1CharacteristicsoflinearphaseFIRfilter6.2DesignofFIRfilterbyWindowing6.3Frequencysamplingmethod6.4ComparisonofIIRandFIRfilter3.Introduction

IIRfilter’sphasecharacteristicsishardtocontrol.FIRfilter’sadvantage：Exactlinerphase.(Arbitraryamplitude-frequencycharacteristics)FIRfilter’simpulseresponseh(n)isafinitelengthsequence

z-transformconvergentonwholez-plane(nostabilityproblem)candelaysometime(nocausalityproblem)FIRfilter’sstructureisnon-recursivegenerally,therewillnoproblemofinstabilitywithfiniteprecisioncomputation.4.6.1CharacteristicoflinearphaseFIRfilter（1）Conditionforlinearphase

Ifimpulseresponseh(n)ofFIRfilterisrealvalue,andmeetsconditionofevensymmetricoroddsymmetric,thatis:thenfilter’sphasecharacteristicisexactlinear.（2）LinearphaseFIRfilter’samplitude-frequencycharacteristicAmplitudePhase5.evensymmetricwith=0,,2Oddsymmetricwith=EvensymmetricimpulseresponsePhaseresponseNisoddNiseven6.Oddsymmetricwith=0,,2Evensymmetricwith=OddsymmetricimpulseresponseNisoddNisevenPhaseresponse7.（3）ZerolocationoflinearphaseFIRfilterSystemfunction:Ifz=ziiszeroofH(z),itsreciprocalzi-1isalsozeroofH(z).Becauseh(n)isrealsequence,H(z)’szeroareconjugatesymmetric.Thatis:zi*and(zi-1)*arealsozeroofH(z).

Characteristicofzerolocation:Zerosarereciprocalconjugatesymmetricpair.Exceptional:Zerosarerealvalue,thenthereareonlytwozeros.Zerosarepurecomplexandlieonunitcircle,thenthereareonlytwozeros.Zerosarerealvalueandlieonunitcircle,thenthereareonlytwozeros.8.BasicstructureStructureoflinearphaseFIRfilter（4）StructureoflinearphaseFIRfilter9.StructureoflinearphaseFIRfilterSavehalfmultiplierNisevenNisodd10.6.2Windowing1.BasicideaofwindowingmethodInfinitelylongNotabsolutelysummableWindowfunctionApproximateUnrealizable!11.单位脉冲响应矩形窗h(n)偶对称理想幅度特性矩形窗幅度特性时域相乘频域卷积？1.Basicideaofwindowingmethod12.时域相乘频域卷积理想幅度特性矩形窗幅度特性13.14.截止频率15.max16.min17.矩形窗对理想低通幅度特性的影响18.19.讨论：增大N对窗函数的影响N=10N=20增加窗函数的长度，能够减少窗函数频谱主瓣宽度；不能改变主瓣和旁瓣的相对特性，该值取决于窗函数的形状。GibbsPhenomenon吉布斯效应20.2.EffectsofwindowingonidealfilterspecificationAtdisconnectedpointsofcutofffrequency,Hd(w)becomescontinuouscurve

—transitionregion.It’swidthisequaltowindow’smainlobewidth.Widerwindow’smainlobewidertransitionregion4/NEffectsofwindow’ssidelobe:rippleappearinfilter’samplitude-frequencycharacteristics,amplitudeofrippledependsonsidelobe’srelativeamplitude.Largerareaofsidelobelargerrippleofpassbandandstopband.

Increasewindow’slengthcanonlydecreasemainlobewidth,notrelativecharacteristicbetweenmainlobeandsidelobe,thatvalueonlydependsonwindow’sshape.IncreaseNcanonlydecreasewidthoftransitionregion,notripplecharacteristics.21.3.Principleofwindowselection

Smallsidelobeamplitude,especiallythefirstsidelobeamplitude.Highspeeddecreasingofsidelobeamplitude,inordertoincrease

stopbandattenuation.Narrowmainlobewidth,inordertogetnarrowtransitionwidth.Trade-off

Smallsidelobeamplitude:Smoothingamplitude-frequencycharacteristics

Transitionwidthincrease

Narrowmainlobewidth:Narrowertransitionwidth.

Rippleofpassbandandstopbandislarge.

22.矩形窗三角窗哈明窗RectangleTriangleHamming常用窗函数幅频特性23.汉宁窗凯塞窗HannBlackmanKaiser常用窗函数幅频特性布莱克曼窗24.4.ProcessofFIRfilterdesignusingwindowmethod(1)Determinefilter’sunitimpulseresponse:

Iffilter’sfrequencyresponseisHd(ejw),then:

Ifedgefrequency,attenuationofpassband,stopbandisknown,wecanchooseidealfilterasapproximationfunction:

Hd(ejw)IDFThd(n)25.(2)DeterminewindowfunctionanditslengthbasedonrequirementSupposewidthoftransitionis,itisanapproximationofwindow’smainlobe.isreverseratiotowindow’slengthN:NA/,Awindowform;Forexample:Rectangular:A=4;Hamming:A=8;Principle:Selectwindowfunctionwithnarrowermainlobewhenstopbandattenuationmeetstherequirement.WindowPeakattenuationofsidelobe(dB)TransitionwidthMinimumattenuationofstopband(dB)Rectangular-134/N-21Hann-318/N-44Hamming-418/N-53Blackman-5712/N-74Kaiser-5710/N-8026.(3)Computefilter’sunitimpulseresponseh(n)（4）VerificationComputedesignedfilter’sfrequencyresponse:——Selectedwindow.FFTAlgorithm27.5KAISER（1）BasicpropertyAmostusefulandoptimumwindow.Zeroth-ordermodifiedBesselfunctionParametercancontrolshapeofthewindow.Characteristic:DifferentwidthoftransitionbandundersameN.（1）If=5.658，widthoftransitionband7.8/N,leastattenuationofstopband60dB,（2）If=4.538，widthoftransitionband5.8/N,leastattenuationofstopband50dB.28.I0()=1=5.44=8.5Kaiserwindowfordifferent

29.（2）DesignapproachKaiserobtainapairofformulasthatpermitdesignertopredictinadvanceparametersneededtomeetagivenfrequencyselectivefilterspecification.p,sandAsisknown,acquirethefollowingparameter:WidthoftransitionbandOrderoffilter30.Example:DesignadigitalFIRlowpassfilterwithKaiserwindow,specificationisasfollows:p=0.2，s=0.3，As=50dBDetermineimpulseresponseandfilter’sfrequencyresponsediagram.MATLABdemo:KAISERFIR_design.m31.6.3FrequencysamplingmethodSupposesystemfunctionofdesignedfilterdoesn’thaveclosedformequation.Npointsregularintervalsamplingbetween=0to2

Conditionforlinearphase32.（1）PerformanceIIRfilter:Requirelessmemoryandcalculationtoachieveagivenfilterresponsecharacteristic,highcostefficiency.Betterselectivity,phasenonlinearity.FIRfilter:

Requiremorememoryandcalculationtoachieveagivenfilterresponsecharacteristic.Delaytimeislarge.ExactlinearphaseUnderthesamerequirement,IIR’scomplexitywillincreasedramatically,soFIRoutweighIIRinperformanceandcost.6.4ComparisonofIIRandFIRfilter33.（2）StructureIIRfilter:Recursivestructure,polemustbeinsideunitcircle,orsysteminstabilitywillappear.

Finite-precisionarithmeticcancausesignificantproblemsduetotheuseoffeedback.FIRfilterNon-recursivestructure,noinstabilityproblemintheoryandpractice.Lowcomputationerror.FastFourierTransform(FFT),highspeedwithsameorder.34.（3）DesignapproachIIRfilter:

Avarietyoffrequency-selectivefilterscanbedesignedusingclosedformdesignformulas,soitcansavecomputationtime.Limitedtofrequency-selectivefilters,inflexible.FIRfilter:

ClosedformdesignequationdoesnotexitforFIRfilter.Althoughwindowmethodisstraightforwardtoapply,someiterationisnecessarytomeetaprescribedspecification.

Flexible,arbitraryfrequencyresponsewithexactlinearphase.35.（4）ApplicationIIRfilter:Applicationwithnorequirementforlinearphase.FIRfilter:

Applicationwithhighrequirementforlinearphase.Inpracticalapplication,requirement,toolsandcostmustbeconsidered.36.RIVIEWComparisonofFIRfilterandIIRfilterSpecification,structure,phasecharacteristics,designapproach