DigitalSignalProcessingChapter7FilterDesignTechniques7.1DesignofDiscrete-TimeIIRFiltersfromContinuous-TimeFiltersInthechapter,wedealwiththedigitalfilterdesignmethodsinwhichadesiredfrequencyresponseofthesystemisapproximatedbyasystemfunctionconsistingofaratioofpolynomials.Generally,thedesignofanIIRdigitalfilteriscarriedoutinthreestepsasfollows.SpecificationsApproximationsRealization27.1DesignofDiscrete-TimeIIRFiltersfromContinuous-TimeFiltersIIR數(shù)字濾波器旳設(shè)計(jì)措施從模擬濾波器設(shè)計(jì)IIR數(shù)字濾波器直接設(shè)計(jì)IIR數(shù)字濾波器沖激響應(yīng)不變法雙線(xiàn)性變換法零極點(diǎn)累試法頻域逼進(jìn)法時(shí)域逼進(jìn)法3Figure1Idealmagnitudespecificationsfordigitallowpassfilter理想濾波器旳幅度特征有理想、陡截止旳通帶和無(wú)窮大衰減旳阻帶兩個(gè)范圍，如圖1所示，這顯然是無(wú)法實(shí)現(xiàn)旳，因?yàn)樗鼈儠A單位取樣響應(yīng)均是非因果和無(wú)限長(zhǎng)旳。實(shí)踐中只能用一種因果可實(shí)現(xiàn)旳濾波器去與之逼近，使其滿(mǎn)足給定旳誤差容限。一種實(shí)際濾波器旳幅度特征在通帶中允許有一定旳波動(dòng)，阻帶衰減則應(yīng)不小于給定旳衰減要求，且在通帶與阻帶之間允許有一定寬度旳過(guò)渡帶，如圖2所示。

w-2p-p0p2p4Figure2Magnitudespecificationsfordigitallowpassfiltertransitionbandpassbandstopband5LowpassfilterspecificationPassband（通帶）

Thefrequencyrangeof0≤ω≤ωpciscalledthepassband;ωpcisthepassbandcutofffrequency（通帶截止頻率）；δpisthepassbandtolerance,thatisStopband（阻帶）

Thefrequencyrangeofωsc≤ω≤π

iscalledthestopband;ωscisthestopbandcutofffrequency（阻帶截止頻率）；δsisthestopbandtolerance,thatisTransitionbandThefrequencyrangeofωpc

≤ω≤ωsciscalledthetransitionband;6LowpassfilterspecificationMaximumpassbandattenuationαpMinimumstopbandattenuationαpCommonly,themaximummagnituderesponseisassumedtobenormalizedtounity.Foralowpassfilter,wehave7DesignStagesforDigitalFilterDesignstagesAnalogfilterapproximations,includingButterworth,Chebyshevandelliptic.Continuous-timetodiscrete-timetransformation,includingimpulseinvarianceandbilineartransformation.Frequencytransformations,thatis,transformingalowpassfilterintoahighpassorbandpassorbandstopfilter.81AnalogButterworth（巴特沃思）LowpassFiltersTheButterworthlowpassfilterhasseveralproperties:Allpoles,andnozero.Noripples（波紋）inthepassbandandstopband.TheButterworthlowpassfilterisdefinedinamagnitude-squaredfunction（幅度平方函數(shù)）: whereNisa

positiveintegerandiscalledtheorderofthefilter,Ωcisthepassbandcutofffrequency91AnalogButterworthLowpassFiltersBecauseThecutofffrequencyΩciscalledthehalf-powerfrequencypointofthefilter.ThemaximumpassbandattenuationisthefrequencyΩcisalsocalled3-dBcutofffrequencyor3-dBbandwidthofthefilter.101AnalogButterworthLowpassFiltersInpracticalapplications,theanaloglowpassfilterisspecifiedbythespecificationsasfollows:Ωpc:thepassbandcutofffrequencyinrad/s;αp:themaximumpassbandattenuationindB;Ωsc:thestopbandcutofffrequencyinrad/s;αs:theminimumstopbandattenuationindB;TousetheButterworthlowpassfiltertoapproximatealowpassfilter,weshouldobtaintheorderNandthe3-dBcutofffrequencyΩc.111AnalogButterworthLowpassFiltersThemagnituderesponseofthefilterisThemaximumpassbandattenuationαparrivesatΩ=Ωpc,whichisrepresentedasTheminimumstopbandattenuationαsarrivesatΩ=Ωsc,whichisrepresentedas(7.141)(7.142)(7.140)121AnalogButterworthLowpassFiltersSolvingtheabovetwoequations,wehave131AnalogButterworthLowpassFiltersTherearetwochoicestodetermineΩcSubstitutingNinEq.(7.141),wehaveUsingthisformulatodetermineΩc,αpisexactlymetatΩ=Ωpcandαsisexceededforthestopband,providedthatαp≤3dB.SubstitutingNinEq.(7.142),wehaveUsingthisformulatodetermineΩc,αsisexactlymetatΩ=Ωscandαpisexceededforthepassband,providedthatαs≤3dB.141AnalogButterworthLowpassFiltersTodeterminethetransferfunctionHa(s)ofthefilter,wesubstitutes=jΩ,thereforeThepolesofthemagnitude-squaredfunctionHa(s)Ha(-s)aregivenbyFigure3showsallthepoleswhenN=3.15Figure3Poleplotforathird-orderButterworthfilterk=0k=1k=2k=3k=4k=5σjΩ161AnalogButterworthLowpassFiltersThe2Npolesequallyspacedinangleonacircleinthes-plane.Theyaresymmetricabouttheimaginaryaxis.Inordertoobtainastablesystem,wechoosethenpolesontheleft-sideofthes-plane.Then,wegetthetransferfunctionasfollowswhere171AnalogButterworthLowpassFiltersTheattenuationoftheButterworthapproximationincreasesmonotonically（單調(diào)地）withfrequency.Anditincreasesveryslowlyinthepassbandandquicklyinthestopband.Ifonewantstoincreasetheattenuationonehastoincreasethefilterorder.The3-dBbandwidthisunrelatedtothefilterorder.18Figure4Themagnitude-frequencyresponseofButterworthLowpassFilters

192AnalogChebyshev（切貝雪夫）LowpassFiltersTheChebyshev-Ilowpassfiltershaveequiripple（等波紋旳）magnituderesponseinthepassbandandmonotonic（單調(diào)旳）magnituderesponseinthestopband.Themagnitude-squaredresponseofananalogChebyshev-IlowpassfilterisgivenbywhereNistheorderofthefilter,εisapositiveandlessthanunitynumberwhichisthepassbandripplefactor,Ωcisthepassbandcutofffrequencyatwhichtheattenuationofthemagnituderesponseisnotnecessarytobe3-dB.ThefunctionTN(x)isanNth-orderChebyshevpolynomialdefinedby(7.147)202AnalogChebyshevLowpassFiltersThecharacteristicsoftheChebyshev-Ifiltersareasfollows:AtΩ=0,|Ha(j0)|=1forNoddand|Ha(j0)|=forNeven;AtΩ=Ωc,|Ha(jΩc)|=forallN;Withinthepassbandof0≤Ω≤Ωc,|Ha(jΩ)|oscillatesbetween1and;ForΩ>Ωc,|Ha(jΩ)|approacheszeromonotonicallyandrapidly;AtΩ=Ωs(thestopbandcutofffrequency),|Ha(jΩs)|=1/A.21Figure5AnalogChebyshevLowpassFilters222AnalogChebyshevLowpassFiltersInthedesignofChebyshev-Ilowpassfilter,thespecificationsaregivenby:Ωc:thepassbandcutofffrequencyinrad/s;αp:thepassbandrippleindB;Ωs:thestopbandcutofffrequencyinrad/s;αs:theminimumstopbandattenuationindB;Todesignthefilter,theorderNandripplefactorεshouldbedetermined.232AnalogChebyshevLowpassFiltersSinceThenAndTherefore242AnalogChebyshevLowpassFiltersToobtainthetransferfunctionHa(s)oftheChebyshev-Ifilter,wesubstituteΩ=s/jintoEq.(7.147)(p19),andthenwegetThereare2Npolesofthemagnitude-squaredfunctionHa(s)Ha(-s),Theyarespacedonaellipseinthes-planeandsymmetricabouttheimaginaryaxis.Inordertoobtainastablesystem,wechoosetheNpolesontheleft-sideofthes-planeandgetthetransferfunctionasfollows.252AnalogChebyshevLowpassFilterswhereand

26兩種經(jīng)典模擬濾波器兩種經(jīng)典模擬濾波器：Butterworth（巴特沃思）濾波器：幅頻特征單調(diào)下降，但衰減特征較差；Chebyshev（切貝雪夫）濾波器：在通帶（或阻帶）中幅頻特征單調(diào)下降，在阻帶（或通帶）中有波紋，衰減特征好于巴特沃思濾波器；27Analog-to-DigitalFilterTransformationsThecontinuous-timetodiscrete-timetransformationsincludethreesteps:Transformationsofspecificationsindiscrete-timedomainintoonesincontinuous-timedomain.Designingtheanalogfilteraccordingtothespecificationsincontinuous-timedomain.Transformthefilterinsdomainintotheoneinzdomain.Themainmethodsoftransformationshavetwokinds:Impulse-invariancemethodBilineartransformationmethod28ztransformandLaplacetransformLaplacetransformztransformTheLaplacetransformofx(nT)isThereforetherelationshipbetweentheLaplacetransformandztransformofx(nT)is29ztransformandLaplacetransformLetandTherelationshipbetweenrandσIfσ=0（s平面旳虛軸）,r=1（z平面單位圓上）;Ifσ>0（s平面旳右半平面）,r>1（z平面單位圓外）;Ifσ<0（s平面旳左半平面）,r<1（z平面單位圓內(nèi)）.TherelationshipbetweenωandΩ:ω=ΩT

IfΩ=0（s平面旳實(shí)軸）,ω=0（z平面正實(shí)軸）

Ω=Ω0（s平面平行于實(shí)軸旳直線(xiàn)）,ω=Ω0T（z平面始于原點(diǎn)角度為ω=Ω0T旳輻射線(xiàn)）.30ztransformandLaplacetransformΩ

從–π/T

增長(zhǎng)到π/T，相應(yīng)旳，ω

從–π增長(zhǎng)到π，即s平面寬為2π/T旳一種水平條帶相當(dāng)于z平面輻角轉(zhuǎn)了一周，即整個(gè)z平面。所以Ω

每增長(zhǎng)一種抽樣角頻率Ωs=2π/T，則

ω

增長(zhǎng)2π。所以從s平面到z平面旳映射是多值映射。317.1.1FilterDesignbyImpulseInvarianceThetransferfunctionoftheanalogfiltercanbeexpressedintermsofapartial-fractionexpansionasfollowsThecorrespondingimpulseresponseis327.1.1FilterDesignbyImpulseInvarianceSamplingtheanalogimpulseresponse,wecanobtainthediscrete-timeimpulseresponseThecorrespondingdiscrete-timetransferfunctionis33s平面旳單極點(diǎn)s=pl變換為z平面上z=eplT處旳單極點(diǎn)；Ha(s)與H(z)旳部分分式旳系數(shù)是相同旳；假如模擬濾波器是穩(wěn)定旳，即全部極點(diǎn)s=pl旳實(shí)部不大于零，則全部z=eplT均在單位圓內(nèi)，即變換后旳數(shù)字濾波器也是穩(wěn)定旳。7.1.1FilterDesignbyImpulseInvariance347.1.1FilterDesignbyImpulseInvarianceInordertoobtainthesamepassbandgainforthecontinuous-anddiscrete-timefilters,weshouldusethefollowingexpressionforHd(z):35Example Transformthecontinuous-timelowpassfiltertransferfunctiongivenby intoadiscrete-timetransferfunctionusingtheimpulseinvariancetransformationmethodwithΩs=10rad/s.Solution

36Example(cont.)37Example(cont.)387.1.2BilinearTransformationThebilineartransformationmethodavoidstheproblemofaliasing.Inthebilineartransformationmethod,theentires-planeismappedintotheentirez-plane.Thelefthalfs-planemapsintotheinterioroftheunitcircleinthez-plane;Therighthalfs-planemapsintotheexterioroftheunitcircleinthez-plane;Theimaginaryaxisofthes-planemapsontotheunitcircle;397.1.2BilinearTransformationThebilineartransformationisdefinedasIfthecontinuous-timetransferfunctionisHa(s),thenFrequencytransformationrelation407.1.2BilinearTransformationThatisSinceWeshouldchoose417.1.2BilinearTransformationInconclusion,thebilineartransformationofacontinuous-timetransferfunctionintoadiscrete-timetransferfunctionisThereforethebilineartransformationmapsanalogfrequenciesintodigitalfrequenciesasfollows:Forhighfrequencies,thisrelationshipishighlynonlinear.Thebilineartransformationmethodavoidstheproblemofaliasing,butthepricepaidforthisistheintroductionofadistortioninfrequencyaxis,knownasthewarping.427.1.2BilinearTransformationThewarpingeffectcanbecompensatedbyprewarpingthefrequencyspecifications.ThestepsarePrewarpthepassbandandstopbandfrequenciesandobtainΩapand

Ωarthroughthefollowingmapping:GenerateHa(s)satisfyingthespecificationsforthefrequenciesΩapandΩar;ObtainHd(z)byreplacingswithinHa(s). 437.2DesignofFIRFiltersbyWindowingBasicconception Inordertodesignthefrequencyresponsessatisfyingtheprescribedspecifications,thefilterorderandmultipliercoefficientsshouldbedetermined.CharacteristicsofFIRfiltersItispossibletoobtainexactlinearphase.FIRsystemsarealwaysstable.FastalgorithmssuchasFFTcanbeused.Ahigherordermeansmoredelays,multipliersandadders.Designapproaches frequencysampling,windowfunctions,maximallyflatapproximation.44IdealcharacteristicsofstandardfiltersFourcommonlyusedFIRfiltersLowpassfiltersHighpassfiltersBandpassfiltersBandstopfiltersFouriertransformpair45Fouridealfilters—lowpassFrequencyresponseImpulseresponse146Figure--impulseresponseoflowpassfilter47Fourstandardfilters—highpassFrequencyresponseImpulseresponse148Figure--impulseresponseofhighpassfilter49Fourstandardfilters—bandpassFrequencyresponseImpulseresponse150Figure--impulseresponseofbandpassfilter51Fourstandardfilters—bandstopFrequencyresponseImpulseresponse152Figure--impulseresponseofbandstopfilter53NotesThedurationsoftheimpulseresponsesofthefourkindsoffiltersareinfinite.Theimpulseresponsesarenoncausal.So,allthesefourkindsoffiltersareidealonesandcannotberealized.54PropertiesofLinearPhaseFIRFiltersThelinear-phaseFIRfilters:Thisequationshowsthattheh(n)ofalinear-phaseFIRfilterissymmetricorantisymmetricaboutM/2.Therearefourkindsoflinear-phaseFIRfilters,whichisshowninthefigures.55PropertiesofLinearPhaseFIRFiltersnh(n)210345678nh(n)2103456789nh(n)210345678nh(n)2103456789symmetricantisymmetricMevenModdTypeITypeIITypeIIITypeIV56Frequencysampling（頻率采樣法）工程上，常給定頻域上旳技術(shù)指標(biāo)，所以采用頻域設(shè)計(jì)更直接。基本思想：使所設(shè)計(jì)旳FIR數(shù)字濾波器旳頻率特征在某些離散頻率點(diǎn)上旳值精確地等于所需濾波器在這些頻率點(diǎn)處旳值，在其他頻率處旳特征則有很好旳逼近。Step1Step2Step3Step4采樣IDFTFT57FrequencysamplingLetHd(ω)bethedesiredfrequencyresponse.ThedesignapproachoffrequencysamplingisjusttosampletheHd(ω).SupposethatH(k)aresamplesoftheHd(ω),i.e.LetthenA(k)=|H(k)|

θ(k)=arg[H(k)]

A(k)isthemagnitudeoftheH(k)

andθ(k)itsphase.58FrequencysamplingWecangettheimpulseresponseh(n)fromH(k)usingIDFT

ThentheztransformofthedesignedFIRfilter

wecanalsogetthefrequencyresponseH(ejω)

59ForexampleRipplesinthepassbandandstopband.60FrequencysamplingIfthelinearphaseisrequired,A(k)andθ(k)mustsatisfytheconditionsforlinearphase.Fourtypesoflinear-phasefiltersTypeI:theorderMisevenandtheh(n)issymmetric.TypeII:theorderMisoddandtheh(n)issymmetric.TypeIII:theorderMisevenandtheh(n)isantysymmetric.TypeIV:theorderMisoddandtheh(n)isantysymmetric.Relationbetweenthelengthofh(n),N,andorderM

N=M+161Frequencysampling—typeIInthiscase,theorderMisevenandtheh(n)issymmetric.ReviewThefrequencyresponseH(ω)issymmetricaboutω=0andω=π.62Frequencysampling—typeIPhaseMagnitude63Frequencysampling—typeIIInthiscase,theorderMisoddandtheh(n)issymmetric.ReviewThefrequencyresponseH(ω)issymmetricaboutω

=

0andantisymmetricaboutω=π.So,H(ω)=0,atω=π.64Frequencysampling—typeIIPhaseMagnitudeBecauseH(ω)=0,atω=π,highpassandbandstopfilterscannotberealizedinTypeIIfilters.65Frequencysampling—typeIIIInthiscase,theorderMisevenandtheh(n)isantisymmetric.ReviewThefrequencyresponseH(ω)isantisymmetricaboutω

=

0andω=π.So,H(ω)=0,atω

=

0andω=π.66Frequencysampling—typeIIIPhaseMagnitudeBecauseH(ω)=0,atω=0andω=π,lowpass,highpassandbandstopfilterscannotberealizedinTypeIIIfilters.67Frequencysampling—typeIVInthiscase,theorderMisoddandtheh(n)isantisymmetric.ReviewThefrequencyresponseH(ω)isantisymmetricaboutω

=

0andsymmetricatω=π.So,H(ω)=0,atω

=

0.68Frequencysampling—typeIVPhaseMagnitudeBecauseH(ω)=0,atω=0,lowpassandbandstopfilterscannotberealizedinTypeIVfilters.

6970h(n)MH(ω)旳對(duì)稱(chēng)性可實(shí)現(xiàn)旳濾波器I型偶對(duì)稱(chēng)偶數(shù)在ω=0及ω=π處偶對(duì)稱(chēng)四種濾波器都可設(shè)計(jì)II型偶對(duì)稱(chēng)奇數(shù)在ω=0處偶對(duì)稱(chēng)，在ω=π處奇對(duì)稱(chēng)不能設(shè)計(jì)高通和帶阻，可設(shè)計(jì)低、帶通濾波器III型奇對(duì)稱(chēng)偶數(shù)在ω=0及ω=π處奇對(duì)稱(chēng)不能設(shè)計(jì)低通、高通和帶阻，只能設(shè)計(jì)帶通濾波器IV型奇對(duì)稱(chēng)奇數(shù)在ω=0處奇對(duì)稱(chēng)，在ω=π處偶對(duì)稱(chēng)不能設(shè)計(jì)低通和帶阻，可設(shè)計(jì)高通、帶通濾波器DesignofLinear-PhaseFIRfiltersusingwindowsAllidealfiltershaveinfiniteduration,sotheycannotberealized.Truncatingtheimpulseresponseh(n),wecangetitsapproximationwithfiniteduration. whereMisthefilterorderandassumingthatitiseven.Thetransferfunctionis72DesignofLinear-PhaseFIRfiltersusingwindowsAfterh(n)istruncated,thesystemisstillnoncausal.Inordertomakeitcausal,wecanshiftitrightbyM/2,withouteitherdistortingthefiltermagnituderesponseordestroyingthelinear-phaseproperty.Thereareripplesclosetothebandedgesinthemagnituderesponse.TheseripplesarereferredtoasGibbs’oscillations.AndtheamplitudesofGibbs’oscillationsdonotdecreaseeventhefilterorderMisincreased.73DesignofLinear-PhaseFIRfiltersusingwindowsMultiplyingtheimpulseresponseh(n)byawindowfunctionw(n),thatis, wecanimprovethemagnituderesponse.ThemultiplicationinthetimedomaincorrespondstoperiodicconvolutionintegralofH(ejω)andW(ejω)inthefrequencydomain,thatis74Magnituderesponsesofawindowfunction|W(ejω)|ωMainlobeSidelobe（主瓣）（旁瓣）Magnituderesponsesofawindowfunction75正肩峰負(fù)肩峰76DesignofLinear-PhaseFIRfiltersusingwindows最大旁瓣旳相對(duì)幅度越小（即能量越集中在主瓣上），起伏振蕩旳幅度越小，阻帶衰減越多；窗函數(shù)頻譜旳主瓣越窄，過(guò)渡帶越陡；這兩者是矛盾旳。|W(ejω)|ωMainlobeSidelobe（主瓣）（旁瓣）771Rectangularwindow（矩形窗）TherectangularwindowfunctionThefrequencyresponseoftherectangularwindow781Rectangularwindow792TriangularwindowandBartlettwindowThetriangularwindowfunctionTheBartlettwindowfunction802Triangularwindow（三角形窗）81Bartlettwindow823HammingandHanningwindowsThewindowfunction

with0≤α≤1.Whenα=0.54,itiscalledtheHammingwindow（海明窗）.Whenα=0.5,itiscalledtheHanningwindow（漢寧窗）.83Hammingwindows（海明窗）843HammingandHanningwindowsThetransitionband（過(guò)渡帶）oftheHammingwindowislargerthanthatoftherectangularwindows,duetoitswidermainlobe.TheratiobetweentheamplitudesofthemainandsecondarylobesoftheHammingwindowismuchlargerthanfortherectangularwindow.Sothestopbandattenuation（衰減）fortheHammingwindowislargerthantheattenuationfortherectangularwindow.85Hanningwindows（漢寧窗）864Blackmanwindow（布萊克