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Notforsale1SOLUTIONSMANUALtoaccompanyDigitalSignalProcessing:AComputer-BasedApproachFourthEditionSanjitK.MitraPreparedbyChowdaryAdsumilli,JohnBerger,MarcoCarli,Hsin-HanHo,RajeevGandhi,MartinGawecki,ChinKayeKoh,LucaLucchese,MyleneQueirozdeFarias,andTravisSmithCopyright?2011bySanjitK.Mitra.Nopartofthispublicationmaybereproducedordistributedinanyformorbyanymeans,orstoredinadatabaseorretrievalsystem,withoutthepriorwrittenconsentofSanjitK.Mitra,including,butnotlimitedto,inanynetworkorotherelectronicStorageortransmission,orbroadcastfordistancelearning.Notforsale2Chapter99.1WeobtainthesolutionsbyusingEq.(9.3)andEq.(9.4).(a)p=1-10-p/20=1-10-0.24/20=0.0273,s=10-s/20=10-49/20=0.0035.(b)p=1-10-p/20=1-10-0.14/20=0.016,s=10-s/20=10-68/20=0.000398.9.2WeobtainthesolutionsbyusingEqs.(9.3)and(9.4).(a)p=-20log101-p()=-20log10(1-0.04)=0.3546dB,s=-20log10s()=-20log100.08()=21.9382dB.(b)p=-20log101-p()=-20log10(1-0.015)=0.1313dB,s=-20log10s()=-20log100.04()=27.9588dB.9.3G(z)=H2(z),orequivalently,G(ej)=H2(ej)=H(ej)2.LetpandsdenotethepassbandandstopbandripplesofH(ej),respectively.Also,letp,2=2p,ands,2denotethepassbandandstopbandripplesofG(ej),respectively.Thenp,2=1-(1-p)2,ands,2=(s)2.Foracascadeofsections,p,M=1-(1-p)M,ands,M=(s)M.9.4HLP(ej)pspss1+p1p0HHP(ej)s1+p1p-p-ss)p)0Therefore,thepassbandedgeandthestopbandedgeofthehighpassfilteraregivenbyp,HP=-p,ands,HP=-s,respectively.9.5NotethatG(z)isacomplexbandpassfilterwithapassbandintherange0.Itspassbandedgesareatp,BP=op,andstopbandedgesats,BP=os.ArealcoefficientbandpasstransferfunctioncanbegeneratedaccordingtoGBP(z)=HLP(ejoz)+HLP(ejoz)whichwillhaveapassbandintherange0Notforsale3andanotherpassbandintherange0.Howeverbecauseoftheoverlapofthetwospectraasimpleformulaforthebandedgescannotbederived.HLP(ej)pspss1+p1p0G(ej)s1+p1p0oo+so+po-po-s9.6(a)hp(t)=ha(t)?p(t)wherep(t)=(t-nT).n=-Thus,hp(t)=ha(nT)n=-(t-nT)..Wealsohave,g[n]=ha(nT).Now,Ha(s)=ha(t)e-st-dtandHp(s)=hp(t)e-st-dt=ha(nT)(t-nT)e-st-dtn=-=ha(nT)e-snTn=-.ComparingtheaboveexpressionwithG(z)=g[n]z-nn=-=h(nT)z-nn=-,weconcludethatG(z)=Hp(s)s=1Tlnz.WecanalsoshowthataFourierseriesexpansionofp(t)isgivenbyp(t)=1Te-j(2kt/T)k=-.Therefore,hp(t)=1Te-j(2kt/T)k=-ha(t)=1Tha(t)e-j(2kt/T)k=-.Hence,Hp(s)=1THas+j2ktTk=-.Asaresult,wehaveG(z)=1THas+j2ktTk=-s=1Tlnz.(7-1)(b)Thetransformationfromthes-planetoz-planeisgivenbyz=esT.Ifweexpresss=o+jo,thenwecanwritez=rej=eoTejoT.Therefore,Notforsale4z=1,foro1,=1,foro=1,1,foro1.Orinotherwords,apointintheleft-half-planeismappedontoapointinsidetheunitcircleinthez-plane,apointintheright-half-planeismappedontoapointoutsidetheunitcircleinthez-plane,andapointonthej-axisinthes-planeismappedontoapointontheunitcircleinthez-plane.Asaresult,themappinghasthedesirablepropertiesenumeratedinSection9.1.3.(c)However,allpointsinthes-planedefinedby?s=o+joj2kT,k=0,1,2,,,aaremappedontoasinglepointinthez-planeasz=eoTejo2kTT=eoTejoT.Themappingisillustratedinthefigurebelow1-1jzImzRez-plane-planesT3TT3TNotethatthestripofwidth2/Tinthes-planeforvaluesofsintherange-TTismappedintotheentirez-plane,andsoaretheadjacentstripsofwidth2/T.Themappingismany-to-onewithinfinitenumberofsuchstripsofwidth2/T.ItfollowsfromtheabovefigureandalsofromEq.(7-1)thatifthefrequencyresponseHa(j)=0forT,thenG(ej)=1THa(jT)for,andthereisnoaliasing.(d)Forz=ej=ejT,orequivalently,=T.9.7Assumeha(t)iscausal.Now,ha(t)=Ha(s)estds.Hence,g[n]=ha(nT)=Ha(s)esnTds.Therefore,Notforsale5G(z)=g[n]z-nn=0=Ha(s)esnTz-nn=0ds=Ha(s)z-nn=0esnTds=Ha(s)1-esTz-1ds.HenceG(z)=ResiduesHa(s)1-esTz-1allpolesofHa(s).9.8Ha(s)=As+.Thetransferfunctionhasapoleats=-.NowG(z)=Residueats=A(s+)(1-esTz-1)=A1-esTz-1s==A1-e-Tz-1.9.9(a)Has()=2(s+2)(s+3)(s2+4s+5)=-1s+3+0.5-0.5j(s+2-j)+0.5+0.5j(s+2+j)=-1s+3+s+3s+2()2+12=-1s+3+s+2s+2()2+12+1s+2()2+12.UsingEq(9.71),wegetGaz()=-11-e-3Tz-1+1-z-1e-2TcosT()1-2z-1e-2TcosT()+e-4Tz-2+z-1e-2TsinT()1-2z-1e-2TcosT()+e-4Tz-2.SinceT=0.25,wegetGaz()=-11-0.4724z-1+1-0.4376z-11-1.1754z-1+0.3679z-2..(b)Hbs()=2s2+s-1(s+4)(s2+2s+10)=1.5s+4+0.25+0.75j(s+1-3j)+0.25-0.75j(s+1+3j)=1.5s+4+0.5s-8s+1()2+32=-1s+3+0.5s+1s+1()2+32-0.53()3s+1()2+32.UsingEq(9.71),wegetGbz()=1.51-e-4Tz-1+0.51-z-1e-Tcos3T()1-2z-1e-Tcos3T()+e-2Tz-2-1.5z-1e-Tsin3T()1-2z-1e-Tcos3T()+e-2Tz-2.SinceT=0.25,wegetGbz()=1.51-0.3679z-1+0.51-2.1624z-1()1-1.1397z-1+0.6065z-2..(c)Hcs()=-s2+2s+11(s2+2s+5)(s2+s+4)=1.5+j(s+1-2j)+1.5-j(s+1+2j)+-1.5-0.315j(s+0.5-0.515j)+-1.5+0.315j(s+0.5+0.515j)Notforsale6=3s-1/3s+1()2+4-3s-1s+0.5()2+0.515()2=3s+1s+1()2+22+3-2/3()2s+1()2+22-3s+0.5s+0.5()2+0.515()2-3-3/15()0.515s+0.5()2+0.515()2.UsingEq(9.71),wegetGcz()=31-z-1e-Tcos2T()1-2z-1e-Tcos2T()+e-2Tz-2-2z-1e-Tsin2T()1-2z-1e-Tcos2T()+e-2Tz-2.SinceT=0.25,wegetGcz()=31-0.9324z-1()1-1.3669z-1+0.6065z-2-31-0.4629z-1()1-1.5622z-1+0.7788z-2..9.10(a)Gaz()=2zz-e-1.3+5zz-e-2.0=A1zz-e-1T+A2zz-e-2T.SinceT=0.5,1=2.6,2=4,A1=2,A2=5,itfollowsHas()=2s+2.6+5s+4.(b)Gbz()=ze-1.4sin1.6()z2-2ze-1.4cos1.6()+e-2.6=ze-TsinT()z2-2ze-TcosT()+e-2T.SinceT=0.5,=3.2,=2.6,itfollowsHbs()=3.2s+2.6()2+3.22.9.11(a)Has()=Gaz()z=41+s1-s=45s2+18s+9()75s2+154s+91.(b)Hbs()=Gbz()z