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Chapter4Frequency-domainApproachtoLTISystemsMaintopics:•FrequencyresponseofLTIsystems•Propertiesoffrequencyresponsesandbodeplots•FrequencyresponseofLTIsystemscharacterizedbyLCCDEs•Frequencydomainapproachtosystemoutputs•SometypicalLTIsystemsKey:ToanalyzetheeffectsofLTIsystemsoninputsfromfrequency-domain!14.2FrequencyresponseofLTIsystemsRecall:foranLTIx→yonehasy[n]=h[n]∗x[n]jφ0Now,letA=ρewithρ≥0,Ω0andφ0allreal-valuedconstants,consider:+∞x[n]=AejΩ0n→y[n]=h[m]x[n−m]m=X−∞+∞=h[m]AejΩ0(n−m)m=X−∞+∞=[Ah[m]e−jΩ0m]ejΩ0nm=X−∞△=BejΩ0nWhatdoesthistell?Theoutputisalsoasinusoidwiththesamefrequencyandandifferentam-plitudethatisgivenby2+∞B=Ah[m]e−jΩ0m(1)m=X−∞Notethatthe2ndfactorontherightisequaltoH(ejΩ0),where+∞△H(ejΩ)=h(m)e−jΩm=|H(ejΩ)|ejφh(Ω)(2)m=X−∞usuallyreferredtoasthefrequencyresponseofthe(LTI)systemwith•|H(ejΩ)|themagnituderesponseand•andφh(Ω)thephaseresponseofthesystem.Therefore,x[n]=AejΩ0n→y[n]=AH(ejΩ0)ejΩ0n(3)BjΩ0Whatdoesitsignify?A=H(e)–amplitudegain!3Now,considerx[n]=ρxcos(Ω0n+φx).Aswellknown,jΩ1njΩ2n△x[n]=A1e+A2e=x1[n]+x2[n]ρxjφxρx−jφxwhereA1=2e,Ω1=Ω0andA2=2e,Ω2=−Ω0.Accordingto(3),wehavejΩknjΩkjΩknxk[n]=Ake→yk[n]=AkH(e)e,k=1,2(4)jΩ1jΩ1njΩ2jΩ2nlinearity⇒y[n]=y1[n]+y2[n]=A1H(e)e+A2H(e)eρxjφx∗j(−Ω)∗jΩNotingthatA1=2e=A2,Ω1=Ω0=−Ω2andH(e)=H(e)=|H(ejΩ)|e−jφ(Ω)(ash[n]isassumedreal-valued),onefinallyreachx[n]=ρxcos(Ω0n+φx)→y[n]=ρycos(Ω0n+φy)(5)wherejΩ0ρy=ρ|H(e)|,φy=φx+φh(Ω0)4Measuringfrequencyresponse:ρjΩ0y|H(e)|=,φh(Ω0)=φy−φx(6)ρxSweepingΩ0from0toπ,wecanthenfindthecompletefrequencyresponseofthesystem.ThisiswhatwepracticallydoindeterminingthefrequencyresponseforanLTIsystem.Now,letusconsidertwoexamplesthatwillhelpushaveabetterunderstand-ingoffrequencyresponse.Example4.1:WehaveanLTIsystem11x[n]=κ+(−1)n→y[n]=x[n]+x[n−1]22whereκconstantbutunknown.SeeFigure1(a).5x[n]32.521.510.50n051015(a)y[n]32.521.510.50n051015(b)Figure1:Time-domainwaveformsforExample4.1.(a)x[n]withκ=2;(b)y[n].Withtheinputsignalx[n]giveninFigure1(a),theoutputcanbeobtaineddirectlyfromandisshowninFigure1(b).Itseemsthattheunknownconstantκis2.Whyisthat?Theanswercanbeobtainedfromtheconceptoffrequencyresponsewiththeanalysisbelow.611Firstofall,h[n]=2δ[n]+2δ[n−1]leadstoh[n]=0,∀n6=0,1and1h[0]=h[1]=2.So,11H(ejΩ)=+e−jΩ=cos(Ω/2)e−jΩ/222Therefore,themagnituderesponseis|H(ejΩ)|=|cos(Ω/2)|andthephaseresponseisofformφh(Ω)=−Ω/2,|Ω|≤π.Analysis:Withx[n]=κcos(0n+0)+cos(πn+0),applying(5)andlinearityyieldsj0jπy[n]=κ×|H(e)|cos[0n+0+φh(0)]+1×|H(e)|cos[πn+0+φh(π)]ThenH(ej0)=1,H(ejπ)=0⇒y[n]=κ⇒κ=2whichisconfirmedbytheactualcomputationresultinFigure1(b).7SeeFigure2for−π≤ω≤π.1.2|H(ejΩ)|10.80.60.40.2Ω0ππ−π−202π(a)φh(Ω)1.510.50−0.5−1−1.5Ωππ−π−202π(b)jΩFigure2:FrequencyresponseforExample4.1,wherethex-axisdenotesangularfrequency[−π,π].(a)|H(e)|-themagnituderesponse;(b)φh(Ω)-thephaseresponsenThissystemblocksthehighfrequencycomponentx2[n]=(−1)butletsx1[n]=κpass-atypicallow-passfilteringoperation.8Example4.2:AcausalLTIsystemisgivenbythefollowing11y[n]=x[n]−x[n−1]22ComputeH(ejΩ)anddeterminey[n]tothesamex[n]usedinExample4.1.Solution:Applyingthesameprocedure,wehaveh[n]=0,∀n6=0,1and11jπ/2h[0]=2,h[1]=−2.Notingj=e,wehave11ΩπH(ejΩ)=−e−jΩ=jsin(Ω/2)e−jΩ/2=sin(Ω/2)e−j(2−2)22Withx[n]=K+(−1)n,theoutputisinthesameformy[n]=K×|H(ej0)|cos[0n+0+φ(0)]+1×|H(ejπ)|cos[πn+0+φ(π)]butthistime,H(ej0)=0,H(ejπ)=1,whichleadstony[n]=(−1)=x2[n]9nThissystemblocksx1[n]=Kandletsx2[n]=(−1)pass-atypicalhigh-passfilteringoperation.10BothmagnitudeandphaseresponsesareplottedinFigure3for−π≤Ω≤π.sin(Ω/2)10.50−0.5−1Ωππ−π−202π(a)|H(ejΩ)|10.5Ω0ππ−π−202π(b)φh(Ω)10−1Ωππ−π−202π(c)Figure3:FrequencyresponseforExample4.2,wherethex-axisdenotesangularfrequency[−π,π],and(a)sin(Ω/2);(b)|H(ejΩ)|-themagnituderesponse;(c)φh(Ω)-thephaseresponse11ForanCTLTIsystemofh(t),thefrequencyresponseisdefinedas+∞−jωτjφh(ω)H(jω)=Z−∞h(τ)edτ=|H(jω)|e(7)1where|H(jω)|andφh(ω)arethemagnitudeandphaseresponses,respectively.Itcanbeshownthatjωxtjωxtx(t)=Ae↔y(t)=AH(jωx)e(8)andfurthermore,x(t)=ρxcos(ωxt+φx)→y(t)=ρycos(ωxt+φy)(9)with△△φy=φx+φh(ωx),ρy=ρx|H(jωx)|aslongastheLTIsystemisreal-valued.1Whendiscussingfrequencyresponseofacontinuous-timeandsystemh(t)andadiscrete-timesystemh[n]simultaneously,weoftenuseH(jω)andH(ejΩ)todenotethefrequencyresponse,respectively.ItshouldbepointedoutthatthereisnoconnectionbetweenthetwofunctionsbotharedenotedwithH()unlessh(t)andh[n]arerelated.Thisargumentisparticularlytrueforthephaseresponsesφh(ω)andφh(Ω).12dx(t)Example4.3:Thedifferentiatory(t)=dtisacausalLTIsystem.Deter-mineitsfrequencyresponse.δ(t)Solution:Ash(t)=dthasanFTjω,thefrequencyresponseisπH(jω)=jω=ωej23.5|H(jω)|32.521.510.5ω0ππ−π−202π(a)φh(ω)1.510.50−0.5−1−1.5ωππ−π−202π(b)Figure4:(a)Magnituderesponse;(b)Phaseresponse.13Clearly,thissystemblacksthelowfrequencycomponentsandamplifiesthehighones.Suchasystemcanbeusedforedgedetection.AnimageisrepresentedbyagreyscalefunctionG=f(x,y)insuchawaythatG=0meansthatthepixel(x,y)isblack,whilealargevalueofGcorrespondstoawhitepixel.v△u∂f(x,y)∂f(x,y)˜u22G(x,y)→G(x,y)=u{}+{}uut∂x∂y(a)(b)Figure5:Effectsofdifferentiatorsonanimagekids.jpg.144.3PropertiesoffrequencyresponsesandbodeplotsFordiscrete-timeLTIsystems,wehavejΩjΩjφ(Ω)jΩ•H(e)=|H(e)|ehandhenceboth|H(e)|andφh(Ω)areallperiodicinΩwithaperiodof2π.jΩ•Forreal-valuedh[n],|H(e)|isanevenfunction,whileφh(Ω)isanoddfunction.Itisduetothesepropertiesthatthemagnitudeandphaseresponses|H(ejΩ)|jΩ(or20log10|H(e)|)andφh(Ω)areusuallyplottedwithΩjustfor0≤Ω<π15Forareal-valuedCTLTIsystem,isgivenforω≥0because•|H(jω)|isanevenfunction,whileφh(ω)isanoddfunction.•Veryoftenthebodeplotisused,inwhichboth20log10|H(jω)|andφh(ω)arepresentedwithalogarithmicscaledfrequencylog10ωforω>0.−1520log10|H(jω)|−20−25−30−35−40−45ω012310101010(a)φh(ω)10.80.60.40.20ω012310101010(b)1+jω/ω1121010Figure6:BodeplotforH(jω)=κ1+jω/ω2withκ=0.01,ω=10andω=100.(a)20log|H(jω)|;(b)φh(ω),wherethex-axisislogω.16Theuseoflogarithmicscaleallowsdetailstobedisplayedoverawiderdynamicrange.E.g.,thedetailedvariationsaroundavalueof10−5andavalueof105onthesamegraph,thelogarithmicscalingprovesapowerfultool.Thesameargumentappliestothelogarithmicfrequencyscaleusedinabodeplot,wherethefrequencyvariesfrom0Hztoinfinity,whileitisnotusedinfrequencyresponseplotofdiscrete-timesystemsasthefrequencyrangeisjustfrom0toπ.17Now,letusconsiderthefrequencyresponseofthefollowingsystem1+jω/ωH(jω)=κ1(10)1+jω/ω2withκ,ω1,ω2allconstant.So,20log10|H(jω)|=20log10|κ|+20log10|1+jω/ω1|−20log10|1+jω/ω2|Thestraight-lineapproximationofBodemagnitudeplotisthegraphobtainedwiththefollowingapproximationrule:0,0≤ω<|ωk|20log10|1+jω/ωk|≈(11)20logω−20log|ω|,ω≥|ω|1010kkThestraight-lineapproximationoftheBodemagnitudeplotfortheH(jω)givenby(10)isshowninFigure7.18−1520log10|H(jω)|−20−25−30−35−40−45ω012310101010(a)−3520log10|H(jω)|−40−45−50−55−60−65ω012310101010(b)1+jω/ω1101012Figure7:Thestraight-lineapproximation(solid-line)for20log|H(jω)|=20log|κ1+jω/ω2|(dotted-line)withκ=0.01.(a)ω=10andω=100;(b)ω1=100andω2=10.WhenwerefertoH(jω),itindicatesthatthesystemisstable.2Otherwise,itdoesnotexist.Also,sinceboth|1+jω|and|1−jω|havethesamestraight-lineωpωpapproximation,astraight-linecorrespondstotwopossiblefrequencyresponses.2Asdiscussedbefore,theDTFTofasequence,sayh[n],existsifitsabsolutelysummable.WethenrealizethattheconceptoffrequencyresponseappliestostableLTIsystemsonly.ThesameappliestotheFT.19Example4.4:Astraight-lineapproximationofacausalLTIsystemisgivenbyFigure8.Determinethefrequencyresponseofthissystem.6520log10|H(jω)|605550454035ω012341010101010Figure8:Straight-lineapproximationforExample4.4.Solution:Denoteω1=10,ω2=100andω3=1000.Observingtheplotgiven,weknowthatthefrequencyresponseofthesystemisofform1ω21H(jω)=κω(1±j)ω1±jω21±jω1ω3where|κ|=1060/20=103.20Since11ωe−ωptu(t)↔,−ωeωptu(−t)↔p1+jωp1−jωωpωpandthesystemiscausal,wehave(jω±ω)2H(jω)=±1032(jω+ω1)(jω+ω3)whichyieldsfourpossiblefrequencyresponses.214.4FrequencyresponseofLTIsystemscharacterizedbyLCCDEsConsidertheclassofLTIsystemsthatareconstrainedwithNMx[n]→y[n]:y[n]+aky[n−k]=bmx[n−m](12)kX=1mX=0TheexistenceofsuchLTIsystemshasbeenprovedinChapter2.WhatistheH(ejΩ)forsuchasystem?Itcanbeshown(seethetextbook)thatM−jΩmbmejΩPm=0H(e)=N−jΩk(13)1+akePk=1Withsuchanexpression,theresponseresponsecanbeevaluatedmucheasilyoncethecoefficientsak,bmaregiven.22Example4.5:DeterminethefrequencyresponseoftheLTIsystemgivenby1y[n]−y[n−2]=2x[n]4Solution:Basedon(13),thefrequencyresponseisgivendirectlyjΩ2H(e)=1−j2Ω1−4eThisprocedureavoidscomputingtheunitimpulseresponseofthesystemthoughthelattercanbeobtainedbytheIDTFTofH(ejω)withjΩ11H(e)=1−jΩ+1−jΩ1−2e1+2eClearly,h[n]=0.5nu[n]+(−0.5)nu[n]Clearly,thesystemiscausalandstabledueton|h[n]|<+∞.P23ForanCTLTIsystemconstrainedwithdNy(t)NdN−ky(t)MdM−kx(t)+αk=βk(14)dtNkX=1dtN−kkX=0dtM−kifitsfrequencyresponseexists,thenMM−kβk(jω)Pk=0H(jω)=NNN−k(15)(jω)+αk(jω)Pk=1Example4.6:ConsidertherectifiershowninFigure1.3,whereR>>randx(t)=cos(2πF0t),F0=1Hz.ThedesignproblemistochooseRandCsuchthattheoutputy(t)isclosetoaconstant.Figures9(b)and9(c)showtheoutputy(t)forRC=0.01andC=10,respectively.Trytoexplainwhythedifferenceissobigbyevaluatingthevoltagey(t)acrossthecapacitorC.24p(t)10.50t−2.5−2−1.5−1−0.500.511.52(a)y001(t).10.50t−2.5−2−1.5−1−0.500.511.52(b)y10(t)10.50t−2.5−2−1.5−1−0.500.511.52(c)Figure9:SignalsforExample4.6.(a)p(t);(b)y(t)withRC=0.01;(c)y(t)withRC=10.Solution:Ideally,p(t)showninFigure1.3isaperiodicwithT0=1/F0=1second.SeeFigure9(a).Denote△x0(t)=x(t)wT0/2(t)⇒p(t)=x0(t−kT0)Xk25jω0mtAsp(t)isperiodic,p(t)=mc[m]e,whereP1sin((m−1)π/2)sin((m+1)π/2)c[m]=X(j2πm)=[+]04(m−1)π/2(m+1)π/2Particularly,11c[0]=,c[1]=π4SincetheRCcircuitisanLTIsystemconstrainedwithdy(t)1y(t)+RC=p(t)⇔H(jω)=dt1+jωRCTherefore,thelinearitysuggestsc[m]jmω0tjmω0ty(t)=Xc[m]H(jmω0)e=Xemm1+jmω0RC△1=d[m]ejmω0t=+d[m]ejmω0tXmπmX6=0260.40.4c[m]c[m]0.30.30.20.20.10.10m0m010203040500102030405011|H(jω)||H(jω)|0.80.60.50.40.2ω0ω01020304050010203040500.40.4d[m]d[m]0.30.30.20.20.10.10m0m0102030405001020304050(a)(b)Figure10:SpectralrelationshipforExample4.6.(a)RC=0.01;(b)RC=10.WithRC=10,(sayC=10µFandR=100kΩ)themissionofgeneratingadccanbeaccomplished.274.5FrequencydomainapproachtosystemoutputsTheimportantconclusions:+∞jΩjΩjΩy[n]=h[m]x[n−m]Y(e)=X(e)H(e)Pm=−∞⇔(16)+∞y(t)=h(τ)x(t−τ)dτY(jω)=X(jω)H(jω)−∞RHere,justprovidetheproofforCTcase.Firstofall,wenote+∞y(t)=Z−∞x(τ)h(t−τ)dτand1h(t−τ)=+∞H(jω)ejω(t−τ)dω2πZ−∞28Substitutingthelatterintotheformer,weobtain1y(t)=+∞x(τ)[+∞H(jω)ejω(t−τ)dω]dτZ−∞2πZ−∞1=+∞H(jω)[+∞x(τ)e−jωτdτ]ejωtdω2πZ−∞Z−∞1=+∞H(jω)X(jω)ejωtdω2πZ−∞ThisimpliesthattheFTofy(t)isH(jω)X(jω)andhencecompletestheproof.Theoutputy(intime-domain)canthenbeevaluatedbytheinversetransform,namely1πjΩjΩjnΩy[n]=H(e)X(e)edΩ2πR−π(17)1+∞jωty(t)=X(jω)H(jω)edω2π−∞R29Example4.8:ConsideracausalLTIsystemdescribedwithd2y(t)dy(t)dx(t)+4+3y(t)=+2x(t)dt2dtdtDeriveaclosed-formexpressionfortheoutputy(t)inresponsetox(t)=e−tu(t).Solution:ForthisLTIsystem,wehavejω+2H(jω)=(jω)2+4jω+3Timedomainapproach:h(t)isrequired:jω+2ABH(jω)==+(jω+1)(jω+3)jω+1jω+3where,bycomparingthecoefficientsofthenumerator,A=B=1/2,thatthe1−t−3tunitimpulseresponseish(t)=[2e+e]u(t)andhencetheoutputcanbeevaluatedwithy(t)=x(t)∗h(t).301Frequencydomainapproach:NoteX(jω)=1+jω.Thenjω+21Y(jω)=H(jω)X(jω)=(jω+1)(jω+3)1+jωCCC=1+2+31+jω(1+jω)23+jωBycomparingthecoefficients,onehasC1=1/4,C2=1/2,C3=−1/4Finally,1y(t)=[e−t+2te−t−e−t]u(t)4Thisapproachseemssimplerthandirectlycomputingtheconvolution.31Themostimportantadvantageofthefrequency-domainapproachoverthetime-domainoneisduetothefactthatthedesignofsystemscanbemademucheasierinfrequency-domain.40|X(ejΩ)|3020100ππΩ−π−202π(a)|H(ejΩ)|10.5ππΩ−π−202π(b)40jΩ|X0(e)|3020100ππΩ−π−202π(c)Figure11:Spectraofx[n]=x0[n]+e[n],h[n]andy[n].Aprioriinformation:Ω0<Ωe32LookatthefilterH(ejΩ)=71e−jΩk,usedinChapter2forprocessingx[n]Pk=08showninFigurer2.3.1.2|H(ejΩ)|10.80.60.40.20Ω−π0πφ(Ω)h3210−1−2−3Ω−π0πjΩ81−jΩkFigure12:FrequencyresponseofH(e)=k=09e:(a)Magnituderesponse;(b)Phaseresponse.PAsobserved,|H(ejΩ)|≈=1aroundΩ=0,muchlargerthanthatforthehigherfrequencies.334.6SometypicalLTIsystemsTointroduceseveralclassesofwell-knownLTIsystems.ForLTIsystems,|Y(·)|=|H(·)X(·)|Y(·)=H(·)X(·)⇒(18)φ(·)=φ(·)+φ(·)yhxGenerallyspeaking,•both|H(·)X(·)|andφh(·)affectthespectrumoftheinputsignalthoughindifferentmanners.•Besides,assuggestedbyParsevalTheorem,φh(·)hasnoteffectonthesignalenergydistributionoftheinputsignal.34All-passsystemsAsystemissaidtobeall-passif|H(·)|=c(say,c=1)forallfrequencies.Continuous-timecase:E.g.,H(jω)=e−jαω:|H(jω)|=1,φ(ω)=−αω,yieldingy(t)=x(t−α).Discrete-timecase:jΩe−jΩ−βe−jΩ(1−βejΩ)E.g.,H(e)=1−βe−jΩ=1−βe−jΩwithβreal-valuedisall-pass.jΩ−jΩ−jpΩIngeneral,letA(e)=1+a1e+···+apewith{ak}areal-valuedset.Thenthefollowingisanall-passfilter:e−jpΩA(e−jΩ)H(ejΩ)=A(ejΩ)Willanall-passLTIsystemaffecttheenergydensityoftheinputsignal?35LinearphaseresponsesystemsAdiscrete-timesystemissaidoflinearphaseresponseifitsphaseresponseφh(Ω)islinearinfrequencyvariableΩ.Thegroupdelayisameasureusedinstudyofthistopic:dφ(Ω)g(Ω)=△−h(19)dΩSo,anLTIsystemh[n]oflinearphaseresponseactuallyimpliesthatthegroupdelayg(Ω)isconstant.Alltheseapplytocontinuous-timeLTIsystems.36Example4.9:Lety[n]betheoutputofthesystemH(ejΩ)=e−jαΩinresponsetox[n],whereαisnotnecessarilyaninteger.Whatisthetime-domainrelationshipbetweenx[n]andy[n]?Solution:Firstofall,11y[n]=πY(ejΩ)ejnΩdΩ=πX(ejΩ)ej(n−α)ΩdΩ2πZ−π2πZ−π11=πx[m]e−jmΩej(n−α)ΩdΩ=x[m]πej(n−α−m)ΩdΩ2πZ−πXmXm2πZ−πsin[(n−α−m)π]=x[m]Xmπ(n−α−m)Astobeseeninnextchapter,ifx[n]isobtainedbysamplingx(t)withsamplingperiodTsmallenough,theny[n]=x(nT−αT).Isthissystemcausal?37IdealchannelsLetx(t)beasignalwhosespectrumisjustspreadwithin(ω1,ω2).Theidealchannelfortransmittingthissignalshouldhaveafrequencyresponse−jτωρe,ω1≤ω≤ω2CI(jω)=(20)notinterested,otherwisewithρ>0,τconstantasthereceivedsignaly(t)isgivenbyy(t)=ρx(t−τ)andhencecontainsexactlythesameinformationasx(t)does.Inpractice,duetothemulti-patheffectincommunicationssystemsthesignalreceivedisr(t)=x(t)+βx(t−τ).ThefrequencyresponseofthischannelisC(jω)=1+βe−jωτSeeFigure13forβ=0.25,τ=0.005.381.5|C(jω)|10.50ω−1500−1000−500050010001500(a)0.5φC(ω)0−0.5ω−1500−1000−500050010001500(b)Figure13:(a)Magnituderesponse|C(jω)|;(b)Phaseresponse.Onewaytorecoverbackthetransmittedsignalx(t)istofeedthereceivedsignalr(t)intoawelldesignedsystemE(jω),calledchannelequalizersuchthatr(t)→x(t−ξ).Morediscussionscanbefoundfromthetextbooksoncommunications.39TypicalfrequencyresponsesofidealfiltersIdeallow-pass:|C(ω)|=wωl(ω−ωl/2),ω≥0|D(Ω)|=wΩl(Ω−Ωl/2),0≤Ω≤πIdealhigh-pass:|C(ω)|=u(ω−ωh),ω≥0|D(Ω)|=u(Ω−Ωh)−u(Ω−π),0≤Ω≤πIdealband-pass:|C(ω)|=u(ω−ωl)−u(ω−ωh),ω≥0|D(Ω)|=u(Ω−Ωl)−u(Ω−Ωh),0≤Ω≤π40Idealband-stop:|C(ω)|=1−[u(ω−ωl)−u(ω−ωh)],ω≥0|D(Ω)|=1−[u(Ω−Ωl)−u(Ω−Ωh)],0≤Ω≤πFigure14showsgraphicallythemagnituderesponsesforfourtypesofdigitalfilters.jΩjΩHlp(e)Hhp(e)11πΩπΩ0Ωl0Ωh(a)(b)jΩjΩHbp(e)Hbs(e)11πΩπΩ0ΩlΩh0ΩlΩh(c)(d)Figure14:Fourtypesofidealdigitalfilterswith0<Ωl<Ωh<π.(a)Low-pass;(b)High-pass;(c)Band-pass;(d)Band-stop.41StudyExample4.10byyourself-atypicalexampletobeconsiderinthecourseCommunicationsPrinciples!Thoughhavingdifferentfrequencycharacteristics,theseidealfiltershaveonethingincommon-allnon-causalandhencepracticallynotimplementable.Filterdesign:min||H(ejΩ)−D(Ω)||Hwhere−jΩ−jNΩjΩb0+b1e+···+bNeH(e)=−jΩ−jNΩ1+a1e+···+aNeisacausalLTIsystem.42jΩ33−jnΩForexample,thecausalH(e)=bnePn=020log|H(ejΩ)|0−50−100Ω−π0πFigure15:Magnituderesponse(indB)ofacausallow-passdigitalfilterwithΩp=0.4πandΩs=0.6π.WhenΩl=Ωh,thestop-bandfilteriscalledanidealnotchfilterandcanbeapproximatedwithy[n]+a1y[n−1]+a2y[n−2]=b0x[n]+b1x[n−1]+b2x[n−2]π2wherea1=−2ρcos4,a2=ρ,b0=b2=1,b1=−2cosθ0.431.2|H(ejΩ)|10.80.60.40.2Ω0π3π02π22π(a)3φh(Ω)210−1−2−3Ωπ3π02π22π(b)Figure16:Frequencyresponseofa2ndorderIIRnotchsystemwithρ=0.99,θ0=π/4.(a)Magnituderesponse(b)Phaseresponse.44