CHAPTER7filterdesigntechniquesintroductionCHAPTER7filterdesigntechniquesintroductiondesignofdiscrete-timeIIRfiltersfromcontinuous-timefilters.2filterdesignbyimpulseinvariancefilterdesignbybilineartransformdesignofFIRsummaryfiltersbywindowing.7.0introduction1.idealfrequency-selectivefilter（理想選頻濾波器）Frequency-selectivefiltersuggestsasystemthatpasses7.0introduction1.idealfrequency-selectivefilter（理想選頻濾波器）Frequency-selectivefiltersuggestsasystemthatpassescertainfrequencycomponentsandtotallyrejectsallothers.IdeallowpassfilterIdealhighpassfilterIdealbandpassfilterIdealbandstopfilter2.Thedesignoffilters2.Thedesignoffiltersinvolvesthefollowingstages:1)propertiesofthesystem（由應(yīng)用定指標(biāo)）;Theapproximationofthespecificationsusingacausaldiscrete-timesystem（由指標(biāo)定類型，然后設(shè)計(jì)H(z)或h[n]，結(jié)果不唯一）;Therealizationstructureofthesystem（根據(jù)有限字長(zhǎng)效應(yīng)調(diào)整系統(tǒng)）.2)3)3.Thespecificationsforthefilteraretypicallygiveninthefrequencydomain:magnituderesponseand3.Thespecificationsforthefilteraretypicallygiveninthefrequencydomain:magnituderesponseandphaseresponse.Usually,adiscrete-timefilteristobeusedfordiscrete-timeprocessingofcontinuous-timesignalsasshowninFigure7.1.Figure7.1/TH(ejT)(j)Heff/T0So,thespecificationsinthefrequencydomaincanbegivenforeitherthediscrete-timefilterortheeffectivecontinuous-timefilter.幅度響應(yīng)以容限圖的形式給出等效模擬系統(tǒng)的幅度響應(yīng)Forspecificationsofeffectivecontinuous-timefilter,weshouldchangethemintodigitalspecifications:p幅度響應(yīng)以容限圖的形式給出等效模擬系統(tǒng)的幅度響應(yīng)Forspecificationsofeffectivecontinuous-timefilter,weshouldchangethemintodigitalspecifications:ppTssT幅度是1Passbandcutofffrequency通帶截止頻率3dB/半功率截止頻率通帶容差Stopbandcutofffrequency阻帶截止頻率1/2阻帶容差單調(diào)下降Figure7.2ca istobasicbetoa theof thetotheis1/T104Hza istobasicbetoa theof thetotheis1/T104Hz:the the0.99|H(ej)|1.01|H(ej)|0.01EXAMPLEItiscommontoexpressthe umpassItiscommontoexpressthe umpassbandandstopbandgainsinunitofdecibels:p 20*log10(1p)0|p|稱為通帶最大衰減s 20*log10s0，|s|稱為阻帶最小衰減Normally,the umamplitudeisscaledto1,thatis,0dB.For3dBcutofffrequency:|H(ejc)|1/ 2j20log10|H(e c)|3dB7.0節(jié)指標(biāo)總結(jié)：相位：線性或非線性；幅度：根據(jù)截止頻率分模擬或數(shù)字指標(biāo)，根據(jù)容差分絕對(duì)或相對(duì)指標(biāo)。在工程實(shí)際中，相對(duì)指標(biāo)最受歡迎。7.1designofdiscrete-timeIIRfiltersfromcontinuous-timefiltersFordesigningadiscrete-timehigh7.1designofdiscrete-timeIIRfiltersfromcontinuous-timefiltersFordesigningadiscrete-timehighpassfilter,wecanrealizeitbythreepaths.WeonlyconcentrateonA/Dtransformationwhichtransformsacontinuous-timefilterintoadiscrete-timefilter.ow pass/Ddigitalprototypealalogylowpassfrequencytransformhigh pass/Ddigitalprototypealalogyhighpass7.1.0introductionofanalogyfilter7.1.1filterdesignbyimpulseinvariance7.1.2filterdesignbybilineartransform7.1.0introductionofanalogy7.1.0introductionofanalogyfilterThemagnitudefunctionforthreeclassesoflowpassfilters區(qū)別：波動(dòng)不同相同階設(shè)計(jì)復(fù)雜度增(d) (b) (c) WeonlygivetheformulasforButterworthfilterdesign:1|H(j)|2AmplitudeFunction：c1(/)2Nc3dB截止頻率向上取整2Thenwehave：1pN11010logs101WeonlygivetheformulasforButterworthfilterdesign:1|H(j)|2AmplitudeFunction：c1(/)2Nc3dB截止頻率向上取整2Thenwehave：1pN11010logs101ps10spORcc11s2Np2N1011101010j2k112Thepolesare:se,k0N12NkcNc全極點(diǎn)型 1 H(s)Thereforewehavecsk0thesystemfunction：1k0(ssk)skdesignalowpassanalogyfilter:designalowpassanalogyfilter:p 21000rad/s,s22000rad/s,1dB,s15dBpSolution1:[N,Wc]=buttord(2000*pi,4000*pi,1,15,'s')[Bs,As]=butter(N,Wc,'s')[H,W]=freqs(Bs,As);plot(W/2/pi,20*(log10(abs(H))))axis([1000,2000,-16,0])gridonEXAMPLEOUTPUT:N=4Wc=8.1932e+003Bs=1.0e+015*0000.000004.5063AsOUTPUT:N=4Wc=8.1932e+003Bs=1.0e+015*0000.000004.5063As=1.0e+015*0.00000.00000.00144.5063bbs1bs2bs3bs4Hc(s) 0 1 2 3 4 aas1as2as3as401234Solution2:[N,Wc]=buttord(2000*pi,4000*pi,1,15,'s')[z,p,k]=butter(N,Wc,'s')Output:z=p=Emptymatrix:0-by-1Solution2:[N,Wc]=buttord(2000*pi,4000*pi,1,15,'s')[z,p,k]=butter(N,Wc,'s')Output:z=p=Emptymatrix:0-by-11.0e+003*-7.5695+3.1354i-7.5695-3.1354i-3.1354+7.5695i-3.1354-7.5695i4.5063e+015k=kH(s)c(sp)(sp)(sp)(sp)0123designalowpasschebyIanalogyfilter:designalowpasschebyIanalogyfilter:21000rad/s,s 22000rad/s,1dB,s15dBpp[N,Wc]=cheb1ord(2000*pi,4000*pi,1,15,'s')[Bs,As]=cheby1(N,1,Wc,[H,W]=freqs(Bs,As);'s')plot(W/2/pi,20*(log10(abs(H))))axis([0,4000,-30,0])gridonEXAMPLEEXAMPLEdesignalowpasschebyIIanalogyfilter:2EXAMPLEdesignalowpasschebyIIanalogyfilter:21000rad/s,s 22000rad/s,1dB,s15dBpp[N,Wc]=cheb2ord(2000*pi,4000*pi,1,15,'s')[Bs,As]=cheby2(N,15,Wc,[H,W]=freqs(Bs,As);'s')plot(W/2/pi,20*(log10(abs(H))))axis([0,4000,-30,0])gridondesignahighpassanalogyfilter:designahighpassanalogyfilter:22000rad/s,s 21000rad/s,1dB,s15dBpp[N,Wc]=buttord(4000*pi,2000*pi,1,15,'s')[Bs,As]=butter(N,Wc,'high','s')[H,W]=freqs(Bs,As);plot(W/2/pi,20*(log10(abs(H))))axis([0,4000,-16,0])gridon帶通帶阻濾波器的通帶和阻帶截止頻率分別是2維,帶阻用參數(shù)’stop’EXAMPLEInthetransformationofacontinuous-timefilter(Hc(s))intoadiscrete-timefilter(H(z)),wegenerallyrequireInthetransformationofacontinuous-timefilter(Hc(s))intoadiscrete-timefilter(H(z)),wegenerallyrequirethattheessentialpropertiesofthecontinuous-timefrequencyresponsebepinthefrequencyresponseoftheresultingdiscrete-timefilter.Specifically,thisimpliesthatwewanttheimaginaryaxisofthes-planetomapontotheunitcircleofthez-plane.Asecondconditionsisthatastablecontinuous-timefiltershouldbetransformedtoastablediscrete-timefilter.Thismeansthatifthecontinuous-timesystemhaspolesonlyinthelefthalfofthes-plane,thenthediscrete-timefiltermusthavepolesonlyinsidetheunitcircleinthez-plane.Theseconstraintsarebasictoallthetechniques,includingimpulseinvarianceandbilineartransformwhichwillbediscussedinthissection.S平面到Z平面虛軸到 圓HS平面到Z平面虛軸到 圓Hc(s)H(z)左半平面的極點(diǎn)映射到 圓內(nèi)filterdesignbyimpulseinvariance（沖擊不變）principle：filterdesignbyimpulseinvariance（沖擊不變）principle：Theimpulseresponseofdiscrete-timefilterischosenproportionaltoequallyspacedsamplesoftheimpulseresponseofcontinuous-timefilter:Tdhc(t)TdSowecangetthesystemfunctionofthediscrete-timefilterbythefollowingsteps:H(s)ht)[n]dc(nd)[n]H(z)cc2.TransformingEquationItiseasytocarryoutasatransformationonthesystemfunction.Letusconsiderthesystemfunctionofthecontinuous-timefilterexpressedintermsofapartialfractionexpansion:N1AkH (s)cssk0kThesystem2.TransformingEquationItiseasytocarryoutasatransformationonthesystemfunction.Letusconsiderthesystemfunctionofthecontinuous-timefilterexpressedintermsofapartialfractionexpansion:N1AkH (s)cssk0kThesystemfunctionofthediscrete-timefiltercanbeexpressedas:H(z)Hc(s)1Td1eskTdz1sskss11eskdz1k0k01eskTdz1kssk要求記住證明見課堂筆記3.Relationshipbetweenthefrequencyresponseofthediscrete-timefilterandthecontinuous-timefilterH3.Relationshipbetweenthefrequencyresponseofthediscrete-timefilterandthecontinuous-timefilterH(ei)H(jj2k)cTTkdd混迭小的情況下，近似認(rèn)為頻響是線性關(guān)系。Figure7.6Ifthecontinuous-timefilterisbandlimited,sothatHc(j)0thenH(ej)H(jd)cTrelationbetweenfrequenciesofthediscrete-timefilterandthecontinuous-timefilter:Td,relationbetweenfrequenciesofthediscrete-timefilterandthecontinuous-timefilter:Td,,It’snotone-to-onemapping.Thiseffectsthealiasofthefrequencyresponse.虛軸到圓planezplane是一對(duì)多，頻率軸每繞一圈頻響累加一次，所以頻響有混迭。3π/Tdπ/Td-π/TdFigure7.7-3π/Td4.Stability(Relationshipbetweenpoles)TdAkH(z)H(s)c14.Stability(Relationshipbetweenpoles)TdAkH(z)H(s)c1Td1eskTdz11Td1eskTdz1ss1eskdk0k0ssksskkPoles:ekd e(kjk)dekTdejkTdk 0,szkkisthn|zkd1k注意：只有極點(diǎn)滿足此關(guān)系5.designstepsppeffT,sseffTepp/5.designstepsppeffT,sseffTepp/d,ss/prototype（原型）continuous-timespecificationsHc(s)(2)designN1k0N1k0(3)H(z)H(s)c11z1ss1ekdssksskk注意：如果給出的等效模擬指標(biāo)，則必須首先按照比例T線性轉(zhuǎn)換成數(shù)字指標(biāo)。6.AboutTdIthasnorelationshipwiththesamplingperiodT6.AboutTdIthasnorelationshipwiththesamplingperiodTassociatedwiththeC/DandD/Cconversioninfigure7.1;Becausewebeginthedesignproblemwiththediscrete-timefilterspecifications,theparameterTdhasnoroleincontrollingaliasingandintheresultingdiscrete-timefilter.wecanchoosearbitrarilythevalueof,generallywechoose1asaresult.TdHowever,weshouldkeepthesamevalueofThe若從離散時(shí)間指標(biāo)開始，則Td的取值大小不影響混迭程度和離散時(shí)間系統(tǒng)。若從模擬原型的系統(tǒng)函數(shù)開始，則Td的取值影響數(shù)字系統(tǒng)的指標(biāo)。根據(jù)設(shè)計(jì)步驟（1）/T,/Tppdssd以Butterworth為例21pN11010plogc1s101p2Nps1010110j根據(jù)設(shè)計(jì)步驟（1）/T,/Tppdssd以Butterworth為例21pN11010plogc1s101p2Nps1010110j2k11Thepolesare:2se,k0N12NkcN1Hc(s)k0 c s1(ssk)skk0N1k0N1k0(3)H(z)H(s)c11z1ss1ekdssksskk7.CharacterAdvantage:linearmappingof7.CharacterAdvantage:linearmappingoffrequency;Disadvantage:aliasingoffrequencyresponse;Restrictionofapplication:cannotbeappliestohighpassandbandstopfilterswhicharenotbandlimited.EXAMPLEdesign a BW digital lowpasss filter,needEXAMPLEdesign a BW digital lowpasss filter,need:p0.2,s0.4,p1dB,s12dB,wp=0.2*pi; ws=0.4*piap=1; as=12Td=1; Wp=wp/Td; [N,Wc]=buttord(Wp,Ws,ap,as,'s')[Bs,As]=butter(N,Wc,'s')[Bz,Az]=impinvar(Bs,As,1/Td)[H,W]=freqs(Bs,As);plot(W/pi,20*(log10(abs(H))),'r*')holdon[H,w]=freqz(Bz,Az);plot(w/pi,20*(log10(abs(H))))axis([0.2,0.4,-20,0])gridOUTPUT：Bs=0As=1.0000001.60310.51501.28490.5150Bz=00.14530.08550Az=1.0000-1.47820.9106-0.2013bOUTPUT：Bs=0As=1.0000001.60310.51501.28490.5150Bz=00.14530.08550Az=1.0000-1.47820.9106-0.2013bbz1bz2bz3bz4H(z) 0 1 2 3 4 aaz1az2az3az4012340.2rad/s,1rad/s(1)T ps手算：dpsTTddp210101(2)Nlog2.92083ps11010ps 0.7870c1p2N100.2rad/s,1rad/s(1)T ps手算：dpsTTddp210101(2)Nlog2.92083ps11010ps 0.7870c1p2N10101j2k110.39350.7870k0,2s e2N 2j0.6816kck1NN1 1 H(s) c N1cs(ss)k01ksk0k0.787030.3934j0.22710.3934j0.22710.7865(s20.7870s0.6194)(s0.7870) s(0.3935j0.6816) s(0.7870) j0.6861)(3)H(z)Hc(s)0.3934j0.22710.7865 0.3934j0.2271111eskTz11e0.3935j0.6861z1 1e0.7870z1 1e0.3935j0.6816z1sskdigital highpasssdesignaBWfilterdigital highpasssdesignaBWfilter,need:p0.6,s0.5,p1dB,s12dB,[N,Wc]=buttord(0.6*pi,0.5*pi,1,12,'s')[Bs,As]=butter(N,Wc,'high','s')[Bz,Az]=impinvar(Bs,As,1)[H,W]=freqs(Bs,As);plot(W/pi,20*(log10(abs(H))),'r')figureplot(w/pi,20*(log10(abs(H))))其他類型的濾波器設(shè)計(jì)區(qū)別只在模擬濾波器設(shè)計(jì)部分，A/D都一樣EXAMPLE思考本例的問題出在哪里。思考本例的問題出在哪里。7.1.2filter7.1.2filterdesignbybilineartransform（雙線性變換）Thetechniquediscussedinthissectionavoidstheproblemofaliasingbyusingthebilineartransformation,analgebraictransformationbetweenthevariablessandzthatmapstheentirej-axisinthes-planetoonerevolutionoftheunitcircleinthez-plane.Thetransformationbetweenthecontinuous-timeanddiscrete-timefrequencyvariablesmustbenonlinear.Therefore,theuseofthistechniqueisrestrictedtosituationsinwhichthecorrespondingwarpingofthefrequencyaxisisacceptable.1.PrinciplezSjΩ1平面Im(z)Ω1σ1Re(z)Ω=（-∞，∞）Ω1=1.PrinciplezSjΩ1平面Im(z)Ω1σ1Re(z)Ω=（-∞，∞）Ω1=dT）ω=（-π，π）+π/Td2.TransformingEquationdenotingthecontinuous-timesystemWithHc(s)functionandH(z)thediscrete-timesystemfunction,2.TransformingEquationdenotingthecontinuous-timesystemWithHc(s)functionandH(z)thediscrete-timesystemfunction,thebilineartransformationcorrespondstoreplacingsby2 1z1s (z1)1dThatis,2 1z1H(z)Hc(1)1z要求記住3.stableandcausalz1(Td/2)swe3.stableandcausalz1(Td/2)swesolveforztoobtain1(Td/2)ssjand,substitutingz1Td/2jTd/21Td/2jTd/2If<0,then,itfollowsthat|z|<1foranyvalueof.Thatis,ifapoleofisintheleft-halfs-plane,itsHc(s)imageinthez-planewillbeinsidetheunitcircle.Therefore,causalstablecontinuous-timefiltersmapintocausalstablediscrete-timefilters.左半平面所有點(diǎn)到圓內(nèi)4.relationbetweenfrequenciesIfthen0z1jTd/21jTd/2|z|=1.Thatis,the-axismapsontotheunitcirclej4.relationbetweenfrequenciesIfthen0z1jTd/21jTd/2|z|=1.Thatis,the-axismapsontotheunitcirclej(one-to-onemapping).虛軸到圓上So,substitutezei,Getrelationbetweenfrequencies21ejej/2ej/2ej/2ej/222jtan(/2)j1ejTTTdddtan2T 22arctandT2d也可直接一步推導(dǎo)頻率間的關(guān)系：substitue sj,zejinto21z1sT1z也可直接一步推導(dǎo)頻率間的關(guān)系：substitue sj,zejinto21z1sT1z1d21ejej/2ej/2ej/2ej/22j1ejTTdd2jtan(/2)tan2Td2T2arctanT2arctand2relationbetweenfrequencyresponseH(ejrelationbetweenfrequencyresponseH(ej)Hc(j)2Tdtan2頻率軸的是單調(diào)增，所以頻響特性得以保持。So,thespecificationforthecontinuous-timefiltershouldbecalculatedbypre-warp（預(yù)畸變）:p2tanpT2So,thespecificationforthecontinuous-timefiltershouldbecalculatedbypre-warp（預(yù)畸變）:p2tanpT2dtan2ssT2d5.designstepsppeffT,s1)discrete-timespecificationsseffT225.designstepsppeffT,s1)discrete-timespecificationsseffT22 tan( /2), tan(/2)ppSsddprototypecontinuous-timespecifications2)Hc(s)3)H(z)Hc(s)|21z1sz1Td注意：如果給出的等效模擬指標(biāo)，則必須首先用比例T線性轉(zhuǎn)換成數(shù)字指標(biāo)。6.AboutTd6.AboutTdconsequenceinthedesignprocedure,sinceweassumethatthedesignproblemalwaysbeginswithWhenthesespecificationsaremappedtocontinuous-timespecificationsandthecontinuous-timefilteristhenmappedbacktoadiscrete-timefilter,theeffectofTdwillcancel.根據(jù)設(shè)計(jì)步驟（1）22tan(/2),/2)ppSs以Butterworth為例21pN11010plogc1s101p2Nps10根據(jù)設(shè)計(jì)步驟（1）22tan(/2),/2)ppSs以Butterworth為例21pN11010plogc1s101p2Nps1010110j2k11Thepolesare:2se,k0N12NkcN1Hc(s)k0 c s1(ssk)skk03)H(z)Hc(s)|21z1sz1Td7.CharactersAdvantage：avoidstheproblemofaliasing7.CharactersAdvantage：avoidstheproblemofaliasingencounteredwiththeuseofimpulseinvariance,becauseitmapstheentireimaginaryaxisofthes-planeontotheunitcircleinthez-plane；Disadvantage:thenonlinearcompressionofthefrequencyaxisdiscussedbefore.Application:thedesignofdiscrete-timefiltersusingthebilineartransformationisusefulonlywhenthiscompressioncanbetoleratedorcompensatedfor.例如模擬微分器（幅度響應(yīng)線性）不能通過雙線性變換法得到數(shù)字微分器。designalowpassfilter,needp0.2613designalowpassfilter,needp0.2613,s0.4018,p0.75dB,s20dB,wp=0.2613*pi;Td=1;ws=0.4018*pi;ap=0.75;Ws=2/Td*tan(ws/2);Wp=2/Td*tan(wp/2)[N,Wc]=buttord(Wp,Ws,ap,as,’s’)[Bs,As]=butter(N,Wc,‘s’)[Bz,Az]=bilinear(Bs,As,1/Td)[H,W]=freqs(Bs,As);plot(W/pi,20*(log10(abs(H))),’Rx‘)holdon[H,w]=freqz(Bz,Az);plot(w/pi,20*(log10(abs(H))))ylabel(‘虛線：模擬濾波器幅度[dB]實(shí)線：數(shù)字濾波器幅度[dB]’)xlabel(‘虛線：模擬角頻率[*π弧度/秒]實(shí)線:數(shù)字角頻率[*π弧度]’)axis([0.25,0.5,-20,-0.45])gridEXAMPLEORwp=0.2613*pi;ws=0.4018*pi;ap=0.75;as=20[N,wc]=buttord(wp/pi,wsORwp=0.2613*pi;ws=0.4018*pi;ap=0.75;as=20[N,wc]=buttord(wp/pi,ws/pi,ap,as)[Bz,Az]=butter(N,wc)[H,w]=freqz(Bz,Az);plot(w/pi,20*(log10(abs(H))))直接得到數(shù)字系統(tǒng)函數(shù)，不加’s’，PI去掉，采用的是雙線性變換法。designahighpassfilter,needpdesignahighpassfilter,needp0.6,s0.5,p1dB,s12dB,[N,Wc]=buttord(0.6,0.5,1,12);[b,a]=butter(N,Wc,'high');[H,w]=freqz(b,a);gridEXAMPLEDesignabandpassDesignabandpassfilter00.5rad,plsl0.45rad,pu0.55rad:3dB0.4rad,su0.6rad:10dBEXAMPLE[N,wc]=buttord([0.450.55],[0.40.6],3,10雙線性變換法[B,A]=butter(N,wc)[H,w]=freqz(B,A);plot(w/pi,20*(log10(abs(H))))[N,wc]=buttord([0.450.55],[0.40.6],3,10雙線性變換法[B,A]=butter(N,wc)[H,w]=freqz(B,A);plot(w/pi,20*(log10(abs(H))))ylabel(‘20log|H(ejω)|[dB]’)xlabel(‘?dāng)?shù)字角頻率[*π弧度]’)axis([0.4,0.6,-10,0]);gridonOutput:N=wc=B=A=20.44100.02711.00000.5590-0.05411.483800000.02710.59207.11.designsteps T7.11.designsteps T, T等效的模擬系統(tǒng)指標(biāo)p peff s seff 數(shù)字指標(biāo)pp/Td,SS/Td2 2 原型模擬濾波器的指標(biāo)pT tan(p/2),ST tan(s/2)d d1 Td)H(s) ssk ekdz1 z)c 2 1z1sT1z1d2.沖擊響應(yīng)不變法：頻率軸線性多對(duì)一 ,頻響有混迭,不適用于高通等.雙線性變換法：頻率軸有畸變一對(duì)一 ,頻響無混迭,不適用于微分器.7.1節(jié)思考題（1）一個(gè)3dB截止頻率為0.5弧度的高通數(shù)字IIR濾波器，用在圖7.1中對(duì)連續(xù)時(shí)間信號(hào)作濾波，圖中T=0.005s現(xiàn)采用模擬濾波器法來設(shè)計(jì)該數(shù)字濾波器，取Td=1，則原型模擬濾波器的3dB截止頻率可以選擇(A)2tan(0.5/2)7.1節(jié)思考題（1）一個(gè)3dB截止頻率為0.5弧度的高通數(shù)字IIR濾波器，用在圖7.1中對(duì)連續(xù)時(shí)間信號(hào)作濾波，圖中T=0.005s現(xiàn)采用模擬濾波器法來設(shè)計(jì)該數(shù)字濾波器，取Td=1，則原型模擬濾波器的3dB截止頻率可以選擇(A)2tan(0.5/2)(B)0.5(D)0.5/0.0052tan(0.5/2)(C)0.005（2）圖7.1中的數(shù)字系統(tǒng)是采線性變換法設(shè)計(jì)的截止頻率為弧度的低通濾波器，整個(gè)等效連續(xù)時(shí)間系統(tǒng)的截止頻率是（B）(A)0.3T(C)2tan0.3T1(D)2tanT22（3）關(guān)于數(shù)字（3）關(guān)于數(shù)字IIR濾波器的模擬濾波器法設(shè)計(jì)正確的說法是（B）沖擊響應(yīng)不變法不能用來設(shè)計(jì)帶通濾波器。雙線性變換法能將最小相位模擬濾波器 成最小相位數(shù)字濾波器。必須將S平面左半平面全部 到Z平面的 圓內(nèi)，才能保證將因果穩(wěn)定的模擬濾波器轉(zhuǎn)換成因果穩(wěn)定的數(shù)字濾波器。（D）脈沖響應(yīng)不變法將連續(xù)時(shí)間濾波器的零點(diǎn)sk 間濾波器的零點(diǎn)eskd7.1.3補(bǔ)充頻率變換模擬域頻率變換：低通：ssc高通：s7.1.3補(bǔ)充頻率變換模擬域頻率變換：低通：ssc高通：sc/s帶通：s(s2)/[s()]c1 c2c2c1帶阻：ss()]/(s2)c2c1c1 c2c2:上截止頻率，c1：下截止頻率不用背數(shù)字域頻率變換：1低通低通：z1低通高通：z1數(shù)字域頻率變換：1低通低通：z1低通高通：z12kk1z2 k1 k1低通帶通：z1k1z22k1k1k12kk1z2z1 k1 k1低通帶阻：z1k1z22k1k1k1,k與轉(zhuǎn)換前后濾波器的截止頻率有關(guān)7.1節(jié)作業(yè)37.1節(jié)作業(yè)37.2Designof7.2DesignofFIRfiltersbywindowingFIRfiltersarealmostentirelyrestrictedtodiscrete-timeimplementations.Consequently,thedesigntechniquesforFIRfiltersarebasedondirectlyapproximatingthedesiredfrequencyresponseofthediscrete-timesystems.Furthermore,mosttechniquesforapproximatingthemagnituderesponseofanFIRsystemassumealinearphaseconstraint.ThesimplestmethodofFIRfilterdesigniscalledthewindowmethod（窗函數(shù)法）.designideaspropertiesofcommonlyusedwindowseffectofthesewindowstofrequencyresponsedesignstep7.2.1designideas（1）Thewindowmethodgenerally7.2.1designideas（1）Thewindowmethodgenerallybeginswithanidealdesiredfrequencyresponse,it’sinversetransformis12H(ej)ejndh[n]ddManyidealizedsystemsaredefinedbypiecewise-constantorpiecewise-functionalfrequencyresponseswithdiscontinuitiesattheboundariesbetweenbands.Asaresults,thesesystemshaveimpulseresponsesthatarenon-causalandinfinitelylong.（2）Themoststraightforwardapproach（2）ThemoststraightforwardapproachtoobtainingacausalFIRapproximationtosuchsystemsistotruncatetheidealresponse:where,thewindoww[n]istherectangularwindoworothers.Thecorrespondingeffectinthefrequencydomainis12H(ej)H(ej)W(ej())dd加上不同形狀和長(zhǎng)度的窗，形成對(duì)理想頻響的不同程度的逼近廣義相位=0W(ej0)M1H（ej)h(n廣義相位=0W(ej0)M1H（ej)h(n)IFT(H(ej))dddM13(d)W(ej)sin[(M1)/2]jw[n]/2)2(M1)12(f)H(ej)H(ej)W(ej)dGibbs（吉布斯）phenomenon(1)Thewidth(1)Thewidthoftheresultingtransitionband（過渡帶）widthofthemainlobe（主瓣）oftheFouriertransformofthewindow.(2)Thepassbandandstopbandripples（波動(dòng)）aretransformofthewindow.Theripplesinthepassbandandthestopbandareapproximatelythesame,andarenotdependentonMandcanbechangedonlybychangingtheshapeofthewindow.濾波器的過渡帶寬窗頻譜的主瓣寬窗長(zhǎng)和窗形狀。濾波器的阻帶衰減窗頻譜的旁瓣相對(duì)幅度窗形狀。采用時(shí)域緩變的窗形狀（其高頻成分少，低頻多，即旁瓣相對(duì)幅度小，主瓣寬）能增加阻帶率減。7.2.2propertiesofcommonlyusedwindows)rectanguar:[n]0nM0，other2n，M0nM2(27.2.2propertiesofcommonlyusedwindows)rectanguar:[n]0nM0，other2n，M0nM2(2)bartlett(triangular):w[n]22nMnMother，M20，2n(3)hanning:w[n]0.50.5cos0nMM，other一階升余弦窗00.540.46cos2n0nM(4)hamming:w[n]， Mother00.420.5cos2n0.08cos4n0nM(5)blackman:w[n]， MMother0不用背布萊克曼窗族（除了三角形窗）Figure布萊克曼窗族（除了三角形窗）Figure7.21窗族FT窗族FT的對(duì)數(shù)幅度（最大幅度歸一化成0dB）：1MM(1[(n]2)2I1MM(1[(n]2)2I)/022(6)kaiser:w[n]0nMI0()數(shù)可取任意實(shí)數(shù)Figure7.24(a)凱澤窗族FT的對(duì)數(shù)幅度（最大幅度歸一化成0dB）隨著FT的對(duì)數(shù)幅度（最大幅度歸一化成0dB）隨著B增：主瓣寬增M=20B=6隨著N增：主瓣寬減Figure7.24FouriertransformofKaiserfamilyTable7.1加KAISER窗度帶寬Table7.1加KAISER窗度帶寬窗頻譜的旁瓣衰減濾波器的阻帶衰減窗頻譜的主瓣寬加凱窗，濾波器的阻帶衰減、窗形狀、過度帶寬主瓣寬濾波器的過度帶寬（略小于主瓣寬）、阻帶衰減。7.2.3effectofthesewindowstofrequencyresponse定性：窗譜的主瓣寬定量：影響濾波器的阻帶衰減;影響濾波器的過渡帶寬.布窗查表7.2.3effectofthesewindowstofrequencyresponse定性：窗譜的主瓣寬定量：影響濾波器的阻帶衰減;影響濾波器的過渡帶寬.布窗查表7.1：根據(jù)大量實(shí)驗(yàn)結(jié)果，濾波器的阻帶衰減可較精確給出，但過度帶寬只能大致估計(jì)正比于窗的主瓣寬。凱窗采用如下公式較精確地計(jì)算：A5021A50A21.584(A21)0.400A8 過度帶寬A20log阻帶衰減102.285M同樣的阻帶衰減，采用凱澤窗比布窗過度帶寬略小。7.2.4designstep將4個(gè)指標(biāo)轉(zhuǎn)換成：理想截止頻率，阻帶衰減和過度帶寬。如果給出的是等效模擬濾波器的指標(biāo)需先轉(zhuǎn)換成數(shù)字指標(biāo)。1.obtaintheidealimpulseresponsec7.2.4designstep將4個(gè)指標(biāo)轉(zhuǎn)換成：理想截止頻率，阻帶衰減和過度帶寬。如果給出的是等效模擬濾波器的指標(biāo)需先轉(zhuǎn)換成數(shù)字指標(biāo)。1.obtaintheidealimpulseresponsec(ps)/2jMe，passbandband2H(ej)H()ej()dd1H(ej)ejndh[n]dd2.Determinethewindowshapeaccordingtopassbandandstopbandripples:min(p,2.Determinethewindowshapeaccordingtopassbandandstopbandripples:min(p,s)A20log10refertotables7.1ifBlackmanwindow;refertothefollowingformulationifKaiserwindow:0.1102(A8.7)A5021A50A210.584(A21)0.403.Determinethewindowlengthaccordingthetransitionwidth(Misevenforhigh-passandband-stop3.Determinethewindowlengthaccordingthetransitionwidth(Misevenforhigh-passandband-stopfilter）|s |pD(1)refertotables7.1ifBlackmanwindow:M2A8(2)refertotheformulationifKaiserwindow:M4.truncatetheidealimpulseresponseh[n]hd[n]w[n]2.2855.verifyH(ej)adjustc,M,andspecifications.(until)H(ej)satisfytheBlackmanwindowdesignaFIRhighpassfilter,need:p0.7,s0.54,p3dB,s40BlackmanwindowdesignaFIRhighpassfilter,need:p0.7,s0.54,p3dB,s40dBc(s)/20.62(1)p1H(ej)ejndh[n]ddsin[(nM/2)]sin[0.62(nM/2)],n (nM/2)(nM/2)(2)p/20/20hammingmin(p,s)min(110)0.01,A40dBs(3)M(4)h[n]sin[(n13)]sin[0.62(n13)]0.540.46cos(2n)R[n](n13)(n13)2726Example1（5）verifyh=fir1(26,0.62,'high（5）verifyh=fir1(26,0.62,'high',hamming(27))H=fft(h,512);%或者freqz(h,1)axis([0.5,0.7,-50,0]);grido