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

Preview (16 of 926 Pages)

100%

Solution Manual For Discrete-Time Signal Processing, 3rd Edition - Page 1 preview image

Loading page ...

|"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

Solution Manual For Discrete-Time Signal Processing, 3rd Edition - Page 2 preview image

Loading page ...

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

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

Loading page ...

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&

Solution Manual For Discrete-Time Signal Processing, 3rd Edition - Page 4 preview image

Loading page ...

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)

Solution Manual For Discrete-Time Signal Processing, 3rd Edition - Page 5 preview image

Loading page ...

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)

Solution Manual For Discrete-Time Signal Processing, 3rd Edition - Page 6 preview image

Loading page ...

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)

Solution Manual For Discrete-Time Signal Processing, 3rd Edition - Page 7 preview image

Loading page ...

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&

Solution Manual For Discrete-Time Signal Processing, 3rd Edition - Page 8 preview image

Loading page ...

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)

Solution Manual For Discrete-Time Signal Processing, 3rd Edition - Page 9 preview image

Loading page ...

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

Solution Manual For Discrete-Time Signal Processing, 3rd Edition - Page 10 preview image

Loading page ...

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&

Solution Manual For Discrete-Time Signal Processing, 3rd Edition - Page 11 preview image

Loading page ...

|"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&

Solution Manual For Discrete-Time Signal Processing, 3rd Edition - Page 12 preview image

Loading page ...

|"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)

Solution Manual For Discrete-Time Signal Processing, 3rd Edition - Page 13 preview image

Loading page ...

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

Solution Manual For Discrete-Time Signal Processing, 3rd Edition - Page 14 preview image

Loading page ...

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

Solution Manual For Discrete-Time Signal Processing, 3rd Edition - Page 15 preview image

Loading page ...

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)

Solution Manual For Discrete-Time Signal Processing, 3rd Edition - Page 16 preview image

Loading page ...

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)

Preview Mode

This document has 926 pages. Sign in to access the full document!

Report

Study Now!

Document Details

Related Documents

ENGR 111 Spring 2025 ScheduleFeb 19

MyOakton Instructions Change Term View Placement I

Computer Assembling Plant Design

Reliability Analysis and Maintenance Strategies for Complex Systems and Components

Analysis and Calculations of Resistances, Wire Sizes, and Magnetomotive Force in DC Machine Field Windings

ECET105 � Digital Fundamentals Homework Assignment #5

INDE 2333 Engineering Statistics I Practice Exam 2

Test Bank for Automotive Service Management, 3rd edition

Solution Manual for Sensors and Actuators: Engineering System Instrumentation, 2nd Edition

Solution Manual For Refrigeration and Air Conditioning Technology, 8th Edition

Company

Explore

Study Tools