Solution Manual For Discrete-Time Signal Processing, 3rd Edition

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|"aqxd9/21/093:59AMPage1&Nw©2010byOppenheim.PublishedbyPearsonPrenticeHall,PearsonEducation,Inc.,UpperSaddleRiver,NJ.Allrightsreserved.Thismaterialisprotectedunderallcopyrightlawsastheycurrentlyexist.Noportionofthismaterialmaybereproduced,inanyformorbyanymeans,withoutpermissioninwritingfromthepublisher.2.1.(a)T(aln))=glnlaln]«Stable:Let|2[n]|<Mthen[T{z[n]|<|g[n]|M.So,itisstableif|g[n]|isbounded.«Causal:yin]=g[njzi[n]andyo[n]=gln]za[n],soifzy[n]=za[n]foralln<no,theny1[n]=2[n]foralln<no,andthesystemiscausal.©Linear:Tani[n]+bzafn])=gfnl(azan]+baafn]=ag[njai[n]+bg[njza[n]=aT(niln])+bT(zafn))Sothisislinear.oNottime-invariant:T(z[n—no])=glnjzn—no)# yln=no]=gln—noje(n—no]whichisnotTL*Memoryless:y[n]=T(z[n])dependsonlyonthen‘*valueofz,soitismemoryless.(b)T(zfn])=Xion,2lk]NotStable:|z[n]|<M=|T(z[n])|<Tj,|z[k]|<|n—no|M.Asn=co,T—oo,sonotstable.*NotCausal:T'(z[n])dependsonthefuturevaluesofzn]whenn<ny,sothisisnotcausal.®Linear:nT(azi[n)+baz)=)azi[k]+balk]k=nonn|=a}w]+b)mpm|>k=ngk=noTan=aT(z:[n])+bT(z2[n])Thesystemislinear.eNotTL:nT(zfnnol)=lknok=non-no=Yakk=0n—no#yn—nol=)alk]k=noThesystemisnotTIL.«NotMemoryless:Valuesofy[n]dependonpastvaluesforn>nq,sothisisnotmemoryless.(©)T(zln))RIN,zlK]oStable:[T(zfnl)]<Tm,2{k]]<Site,2lklM<[2no+1Mfor[ala]<M,soitisstable.«NotCausal:T'(z[r])dependsonfuturevaluesofz[n],soitisnotcausal.eoLinear:ntnoT(azy[n]+bz2[n])=>az[k]+bza[k]k=n—non+ngning=aYall+bYz[k=aT(zi[n])+bT(z2[n])k=n—nok=n-no(+1&Nw

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DownloadedfromStudyXY.com®+StudyXYSdYe.o>\|iF’prE\3SStudyAnythingThisContentHasbeenPostedOnStudyXY.comassupplementarylearningmaterial.StudyXYdoesnotendroseanyuniversity,collegeorpublisher.Allmaterialspostedareundertheliabilityofthecontributors.wv8)www.studyxy.com

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ch02.qxd9/21/093:59AMPage2&NP©2010byOppenheim.PublishedbyPearsonPrenticeHall,PearsonEducation,Inc.,UpperSaddleRiver,NJ.Allrightsreserved.Thismaterialisprotectedunderallcopyrightlawsastheycurrentlyexist.Noportionofthismaterialmaybereproduced,inanyformorbyanymeans,withoutpermissioninwritingfromthepublisher.Thisislinear.oTL:ninoT(eln—no]=3zlk—nok=n-non=3=Hk=n-ng=yln—noThisisTL«Notmemoryless:Thevaluesofy[n]dependon2ngothervaluesofz,notmemoryless.(d)T(afn])=z[n—no)«Stable:|T(z[n])|=[z[n—no]|<Mif|z[n]<M,sostable.eoCausality:Ifng>0,thisiscausal,otherwiseitisnotcausal.®Linear:T(azi[n]+bzz[n])=azi[n—no]+bzafn—ng]=aT(z[n])+bT(z2[n])Thisislinear.oTE:T(z[n—ng]=z[n—no—na)=y[n—ng).ThisisTL*Notmemoryless:Unlessng=0,thisisnotmemoryless.(e)T(z[n))=el"oStable:|z[n]|<M,|T(z[n])|=|e*I")]<ell<eMthisisstable.«Causal:Itdoesn'tusefuturevaluesofz{n],soitcausal.*Notlinear:T(azyn]+baofn])=essiiniomaln]a—eomilnlgbaaln]aN%N%#aT(z1[n])+bT(z2[n])Thisisnotlinear.oTLT(zfn—no])=e*"~"l=y[n—ng),sothisisTL*Memoryless:y[n]dependsonthen**valueofzonly,soitismemoryless.()T(z[n])=azln]+b«Stable:|T'(z{n])|=|az{n]+b]<a|M|+|b],whichisstableforfiniteaandb.«Causal:Thisdoesn’tusefuturevaluesofz[n],soitiscausal.*Notlinear:T(cz1[n]+dz2[n])=aczi[n]+adza[n]+b#cI(zi[n])+dT(z2[n])Thisisnotlinear.2&

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ch02.gqxd9/21/093:59AMPage3AIS5:©2010byOppenheim.PublishedbyPearsonPrenticeHall,PearsonEducation,Inc.,UpperSaddleRiver,NJ.Allrightsreserved.Thismaterialisprotectedunderallcopyrightlawsastheycurrentlyexist.Noportionofthismaterialmaybereproduced,inanyformorbyanymeans,withoutpermissioninwritingfromthepublisher.oTLT(z[n—no)=az[n—no]+b=y[n—nl.ItisTL©Memoryless:y[n]dependsonthen**valueofz[n]only,soitismemoryless.(8)T(z[n])=z[-n]«Stable:|T(z[n])]<|z[~n]|<M,soitisstable.«Notcausal:Forn<0,itdependsonthefuturevalueofz|n],soitisnotcausal.eLinear:T(azy[n]+bz2[n])=azi[-n]+bzaz[-n]=aT(z:[n])+bT(z2[n])Thisislinear.eNotTI:T(a[n—no])=az[-n—nd#yln—no)=z[-n+ng]ThisisnotTIL.«Notmemoryless:Forn#0,itdependsonavalueofzotherthanthen**value,soitisnotmemoryless.(h)T(z[n])=z[n]+un+1]Stable:|T'(z[n])|<M+3forn>—1and|T(z[n])|<Mforn<—1,soitisstable.oCausal:Sinceitdoesn’tusefuturevaluesofz[n],itiscausal.*Notlinear:T(azy[n]+basfn])=aza[n]+baaln]+Buln+1]#aT(z1[n))+bT(z2[n])ENThisisnotlinear.AYYeNotTI:AvT(z[n—ng]=z[n—ng]+3uln+1]=yln—no=z[n—no]+3ufn—no+1]ThisisnotTI.«Memoryless:y[n]dependsonthen*valueofzonly,sothisismemoryless.3&NP)

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ch02.qxd9/21/093:59AMPage4&NP)©2010byOppenheim.PublishedbyPearsonPrenticeHall,PearsonEducation,Inc.,UpperSaddleRiver,NJ.Allrightsreserved.Thismaterialisprotectedunderallcopyrightlawsastheycurrentlyexist.Noportionofthismaterialmaybereproduced,inanyformorbyanymeans,withoutpermissioninwritingfromthepublisher.2.2.ForanLTIsystem,theoutputisobtainedfromtheconvolutionoftheinputwiththeimpulseresponseofthesystem:ooyln)=>h[k)z[n—Kk]k=—co(a)Sincehlk]#0,for(No<n<Ny),Nyvin)=3hiklzln—K]k=NoTheinput,z[n]#0,for(N2<n<N3),soz[n—kK]#0,forNa<(n—k)<NgNotethattheminimumvalueof(n—k)isN;.Thus,thelowerboundonn,whichoccursfork=NoisNy=No+Na.Usingasimilarargument,Ns=Ny+Ns.‘Therefore,theoutputisnonzerofor(No+Nz)<n<(Ny+Na).(b)Iffn]#0,forsomen,<n<(no+N—1),andhln]#0,forsomeny<n<(ny+M—1),theresultsofpart(a)implythattheoutputisnonzerofor:(mo+m)<n<(no+m+M+N-2)ASotheoutputsequenceisM+N—1sampleslong.ThisisanimportantqualityoftheconvolutionAAvforfinitelengthsequencesasweshallseeinChapter8.Av4&NP)

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ch02.qxd9/21/093:59AMPage5anIS5:©2010byOppenheim.PublishedbyPearsonPrenticeHall,PearsonEducation,Inc.,UpperSaddleRiver,NJ.Allrightsreserved.Thismaterialisprotectedunderallcopyrightlawsastheycurrentlyexist.Noportionofthismaterialmaybereproduced,inanyformorbyanymeans,withoutpermissioninwritingfromthepublisher.2.3.Wedesirethestepresponsetoasystemwhoseimpulseresponseish[n]=a"u[-n],for0<a<1Theconvolutionsum:-ln]=3hlklzln—&]k=—o0Thestepresponseresultswhentheinputistheunitstep:l=={bfrn20=“10,forn<0Substitutionintotheconvolutionsumyields©yinl=Ya*u[-Kuln-kk=—coForn<0:©yn)=Yatk=—oo£3-Fak=-n=ao"T1-aForn>0:[3]yin]=)otJank=—coJanN%~N%=Ya=o_1CT1-a5P55!NP)

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ch02.qxd9/21/093:59AMPage6&NP)©2010byOppenheim.PublishedbyPearsonPrenticeHall,PearsonEducation,Inc.,UpperSaddleRiver,NJ.Allrightsreserved.Thismaterialisprotectedunderallcopyrightlawsastheycurrentlyexist.Noportionofthismaterialmaybereproduced,inanyformorbyanymeans,withoutpermissioninwritingfromthepublisher.2.4.Thedifferenceequation:.1yln]=vin—11+guln—21=22[n—1]‘Tosolve,wetaketheFouriertransformofbothsides.Y(e)—Sy(eye+FHe)e=2.X(e)emiThesystemfunctionisgivenby:iY(ev)wy==1H(eY)X(e%)2e~IvTT-fevgfeTheimpulseresponse(forz[n]=6[n])istheinverseFouriertransformofH(ei).Hey=—3+8_1+gev 1-—ZevThus,11Hin]==8(3)"uln]+8(3)"ul.VanVanNYNY6an&

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ch02.qxd9/21/093:59AMPage7&NP)©2010byOppenheim.PublishedbyPearsonPrenticeHall,PearsonEducation,Inc.,UpperSaddleRiver,NJ.Allrightsreserved.Thismaterialisprotectedunderallcopyrightlawsastheycurrentlyexist.Noportionofthismaterialmaybereproduced,inanyformorbyanymeans,withoutpermissioninwritingfromthepublisher.2.5.(a)Thehomogeneousdifferenceequation:yln]—5yln—1]+6y[n—2]=0TakingtheZ-transform,1-5:71+6272=0(1-211-32"Y)=0.Thehomogeneoussolutionisoftheformyaln]=A1(2)"+42(3)".(b)Wetakethez-transformofbothsides:Y(z)[1-52z71+6272)=2271X(2)Thus,thesystemfunctionisY(z)26)=x6)_227!TT1-5z71+6z2=2_,_2To1-27171-31wheretheregionofconvergenceisoutsidetheoutermostpole,becausethesystemiscausal.HencetheROCis|z|>3.Takingtheinversez-transform,theimpulseresponseishin]=—2(2)"u[n]+2(3)"u[n].(c)Letz[n]=u[n](unitstep),then111>X(@)==>andY(z)=X(2)H(z)_2:71ToA-zH-22"0)(1-327)Partialfractionexpansionyields143Ye=i=TfTheinversetransformyields:y[n]=uln]—4(2)"u[n]+3(3)"u[n].7&NP)

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ch02.qxd9/21/093:59AMPage8&NP©2010byOppenheim.PublishedbyPearsonPrenticeHall,PearsonEducation,Inc.,UpperSaddleRiver,NJ.Allrightsreserved.Thismaterialisprotectedunderallcopyrightlawsastheycurrentlyexist.Noportionofthismaterialmaybereproduced,inanyformorbyanymeans,withoutpermissioninwritingfromthepublisher.2.6.(a)Thedifferenceequation:1yn]—Fin—1]=z[n]+2z[n—1]+zn-2]TakingtheFouriertransformofbothsides,Y(e*)1—367)=X(&)[1+2679+e792],Hence,thefrequencyresponseiswv—Y(e)He?)=X(ef)_142e7iv4emi1-Lew(b)Asystemwithfrequencyresponse:1—le—jw4g=i%wHv)=—2-"72"1+ke7v+Fei_Ye)TX(ev)crossmultiplying,..1,VE)+3e+Je)=X(lL-ge+eI),andtheinversetransformgives131yn]+vin—1]+Fyn—2]=z[n]—33m—1]+z[n-3].aaN%N%8PIE

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ch02.qxd9/21/093:59AMPage9&NP)©2009byOppenheim.PublishedbyPearsonPrenticeHall,PearsonEducation,Inc.,UpperSaddleRiver,NJ.Allrightsreserved.Thismaterialisprotectedunderallcopyrightlawsastheycurrentlyexist.Noportionofthismaterialmaybereproduced,inanyformorbyanymeans,withoutpermissioninwritingfromthepublisher.2.7.z[n]isperiodicwithperiodNifz[n]=z[n+NV]forsomeintegerN.(a)a[n]isperiodicwithperiod12:(En)=GI(F)n+N)_pil(Fn+2ak)=ork=94forintegersk,NMakingk=1andN=12showsthatz[n]hasperiod12.(b)z[n]isperiodicwithperiod8:EN)=GCE)M+N)_i(3Ent2nk)35=>2k=“LN,forintegersk,N=pN=Shforintegersk,NThesmallestkforwhichbothkandNareintegersareis3,resultingintheperiodNbeing8.(¢)z[n]=[sin(wn/5)]/(wn)isnotperiodicbecausethedenominatortermislinearinn.(d)Wewillshowthatz[n]isnotperiodic.Supposethatz[n]isperiodicforsomeperiodN:ECL=(FH)n+N)=Jpntank)3=>2wk=—=N,forintegersk,N7teg=>N=2v/2k,forsomeintegersk,N'ThereisnointegerkforwhichNisaninteger.Hencez[n]isnotperiodic.VanVanNYNY9an&

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|"qxd9/21/093:59AMPage10PIE©2010byOppenheim.PublishedbyPearsonPrenticeHall,PearsonEducation,Inc.,UpperSaddleRiver,NJ.Allrightsreserved.Thismaterialisprotectedunderallcopyrightlawsastheycurrentlyexist.Noportionofthismaterialmaybereproduced,inanyformorbyanymeans,withoutpermissioninwritingfromthepublisher.2.8.WetaketheFouriertransformofbothh[n]andzn],andthenusethefactthatconvolutioninthetimedomainisthesameasmultiplicationinthefrequencydomain.i5HE®)=—1+ze=vY(e®)=H(e™)X(e)_51Toltleiw1-le_3,2oldieT1-lee1n,1n,yin]=2(3)"uln]+3(~3)"uln]PanPanAvAv10A&

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|"aqxd9/21/093:59AMPage11&NP)©2009byOppenheim.PublishedbyPearsonPrenticeHall,PearsonEducation,Inc.,UpperSaddleRiver,NJ.Allrightsreserved.Thismaterialisprotectedunderallcopyrightlawsastheycurrentlyexist.Noportionofthismaterialmaybereproduced,inanyformorbyanymeans,withoutpermissioninwritingfromthepublisher.2.9.(a)Pirstthefrequencyresponse:Y(e)—Seven)+FEY)=FHKE)Y(ev)wy=HY)=X@%)_LetTo1-fedv+LetteNowwetaketheinverseFouriertransformtofindtheimpulseresponse:.-22wy=—_—HE")=oleivtioTeiw11hn]==2(3)"uln]+2(3)"uln]Forthestepresponses[n]:0sln]=>hkun—k]k=—con=>h[k]k=—co__pl=a/t1-1/2)=-21-1/3ufn]+21-1/3uln]=LinLin=+E"-2p)(b)Thehomogeneoussolutiony(n]solvesthedifferenceequationwhenz[n]=0.ItisintheformTatyn[n]=3A(c)",wherethec'ssolvethequadraticequationIa5IF1I-§+s=0Sofor¢=1/2andc¢=1/3,thegeneralformforthehomogeneoussolutionis:11,ln]=Ai(3)"+42(3)(c)Thetotalsolutionisthesumofthehomogeneousandparticularsolutions,withtheparticularsolutionbeingtheimpulseresponsefoundinpart(a):yin]=waln]+ypln]_LinLin_olym,Li.=MG"+AE)"+=23)un]+25)ula)Nowweusetheconstrainty[0]=y[1]=1tosolveforA;andAp:yl)=A+42-2+2=1yl]=A/2+A42/3-2/3+1=1A+4=1A/2+42/3=2/3WithA;=2andA;=—1solvingthesimultaneousequations,wefindthattheimpulseresponseiunl=23)"~3)"+~2(3)"uln]+2(3)"uln]233211&NP)

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ch02.qxd9/21/093:59AMPage12&NP©2010byOppenheim.PublishedbyPearsonPrenticeHall,PearsonEducation,Inc.,UpperSaddleRiver,NJ.Allrightsreserved.Thismaterialisprotectedunderallcopyrightlawsastheycurrentlyexist.Noportionofthismaterialmaybereproduced,inanyformorbyanymens,withoutpermissioninwritingfromthepublisher.2.10.(a)y[n]=hin]xzn]=Ydu[-k-1un-kk=—oc3a,n<-1_|)a1>a,n>-1koan2a<-_Jimens?-Yen>-11-1/a’(b)First,letusdefinev[n]=2"u[-n—1].Then,frompart(a),weknowthat2"p<1win]=ufn]*fn]={ohatNow,yn]=uln—4]xvln]=wn-4273n<3=1,n>3(c)Giventhesamedefinitionsforv[n]andw[n]frompart(b),weusethefactthath[n]=2"u[—(n—1)—1]=v[n—1]toreduceourwork:[Ssy[n]=z[n]*hfn][Ss=zfn]sufn-1]=wn-1]_[2°n<o0-1,n>0(d)Again,weusev[n]andw[n]tohelpus.vin]=z[n]+hln]=(ufn]—un—10)»v[n]=wn]—w[n—10]=(2u[—(n+1)]+u[r])—(2"°u[-(n—9)]+un—10)2(n+l)_9(n=9)p<2=1-2(n-9),-1<n<8§0,n>912&NP

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ch02.qxd9/21/093:59AMPage13&Nw©2009byOppenheim.PublishedbyPearsonPrenticeHall,PearsonEducation,Inc.,UpperSaddleRiver,NJ.Allrightsreserved.Thismaterialisprotectedunderallcopyrightlawsastheycurrentlyexist.Noportionofthismaterialmaybereproduced,inanyformorbyanymeans,withoutpermissioninwritingfromthepublisher.2.11.Firstwere-writez[n]asasumofcomplexexponentials:meIxn/4_g—inn/4ln]=sin(F)=————SincecomplexexponentialsareeigenfunctionsofLTIsystems,H(ei*/4)eim/4_f(e=in/4)e=imn/4oyHeDeHeTeJEvaluatingthefrequencyresponseatw=+7/4:.1—eI%/2I)=—/———=201-j)=—in/4H(e'T)Yr2(1-j)=2v2e1-2.if)=—==x/4H(e™i%)Tri2(1+j)=2v/2e"Weget:2y/Be—in/Aeimn/4_9,/Feix/4g=inn/4ih=HE7=2V2sin(rn/4—7/4).aaNw)Nw)13&Nw

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ch02.qxd9/21/093:59AMPage14&NP)©2010byOppenheim.PublishedbyPearsonPrenticeHall,PearsonEducation,Inc.,UpperSaddleRiver,NJ.Allrightsreserved.Thismaterialisprotectedunderallcopyrightlawsastheycurrentlyexist.Noportionofthismaterialmaybereproduced,inanyformorbyanymeans,withoutpermissioninwritingfromthepublisher.2.12.Thedifferenceequation:yin]=nyfn—1]+z[n]Sincethesystemiscausalandsatisfiesinitial-restconditions,wemayrecursivelyfindtheresponsetoanyinput.(a)Supposez[n]=dfn]:yin]=0,forn<0yf]=1yi=1v2]=2yd=6vid]=24y[n]=h[n]=nlu[n](b)Todetermineifthesystemislinear,considertheinput:z{n]=ad[n]+bs[n]performingtherecursion,yln]=0,forn<0y[0]=a+byll]=a+bv[2]=2(a+b)4[3]=6(a+b)JanJanNF;y[4]=24(a+b)NF;Becausetheoutputofthesuperpositionoftwoinputsignalsisequivalenttothesuperpositionoftheindividualoutputs,thesystemisLINEAR.(c)Todetermineifthesystemistime-invariant,considertheinput:z{n]=b[n—1)therecursionyieldsy[n]=0,forn<0[0]=0yi]=1v2=2y3]=6vid=24Usinghln]frompart(a),hn—1]=(n—1)'ufn—1]#yln]lzpnj=sin—1)Conclude:NOTTIMEINVARIANT.14&NP)

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ch02.qxd9/21/093:59AMPage15anIS5:©2009byOppenheim.PublishedbyPearsonPrenticeHall,PearsonEducation,Inc.,UpperSaddleRiver,NJ.Allrightsreserved.Thismaterialisprotectedunderallcopyrightlawsastheycurrentlyexist.Noportionofthismaterialmaybereproduced,inanyformorbyanymeans,withoutpermissioninwritingfromthepublisher.2.13.EigenfunctionsofLTIsystemsareoftheforma”,sofunctions(a),(b),and(e)areeigenfunctions.Noticethatpart(d),cos(won)=.5(e7“°™+e~7“0")isasumoftwoa”functions,andisthereforenotaneigenfunctionitself.aJayAvAv15P55!NP)

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