信号与系统奥本海姆英文版课后答案chapter7

137Chapter7Answers7.1FromtheNyquistsamplingtheorem,weknowthatonlyifX(jw)=0for|w|ws/2willbesignalberecoverablefromitssamples.Therefore,X(jw)5000л.7.2FromtheNyquisttheorem,weknowthatthesamplingfrequencyinthiscasemustbeatleastws=2000п.Inotherwords,thesamplingperiodshouldbeatmostT=2п/(ws)=1＊10-3.Clearly,only(a)and(e)satisfythiscondition.7.3(a)WecaneasilyshowthatX(jw)=0for|w|4000п.Therefore,theNyquistrateforthissignaliswN=2(4000п)=8000п.(b)FromtheTables4.1and4.2weknowthatX(jw)isarectangularpulseforwhichX(jw)=0for|w|4000п.Therefore,theNyquistrateforthissignaliswN=2(4000п)=800п.(c)FromtheTables4.1and4.2weknowthatX(jw)istheconvolutionoftworectangularpulseseachofwhichiszerofor|w|4000п.Therefore,X(jw)=0for|w|8000пandtheNyquistrateforthissignaliswN=2(8000п)=16000п.7.4Ifthesignalx(t)hasaNyquistrateofwo,thenitsFouriertransformX(jw)=0for|w|wo/2.(a)Fromchapter4,y(t)=x(t)+x(t-1)FTY(jw)=X(jw)+e-jwtX(jw).Clearly,wecanonlyguaranteethatY(jw)=0for|w|wo/2.Therefore,theNyquistratefory(t)isalsowo.(b)Fromchapter4,y(t)=dttdx)(FTY(jw)=jwX(jw).Clearly,wecanonlyguaranteethatY(jw)=0for|w|wo/2.Therefore,theNyquistratefory(t)isalsowo.(c)Fromchapter4,y(t)=x2(t)FTY(jw)=(1/2п)[X(jw)*X(jw)]Clearly,wecanonlyguaranteethatY(jw)=0for|w|wo.Therefore,theNyquistratefory(t)isalso2wo.(d)Fromchapter4,y(t)=x(t)cos(wot)FTY(jw)=(1/2)X(j(w-wo))+(1/2)X(j(w+wo)).Clearly,wecanguaranteethatY(jw)=0for|w|wo+wo/2.Therefore,theNyquistratefory(t)is3wo.7.5UsingTable4.2,p(t)FTT2KTK)/2(FromTable4.1p(t-1)FTT2e-jwTjkkeTk2)2(.Sincey(t)=x(t)p(t-1),wehaveY(jw)=(1/2п)[X(jw)*FT{P(t-1)}]=(1/T)TjkKeTkjX2))2((Therefore,Y(j)consistsofreplicatesofX(j)shiftedbyk2/Tandaddedtoearthother(seeFigureS7.5).Inordertorecoverx(t)fromy(t).weneedtobeabletoisolateonereplicaofX(j)fromY(j).FigureS7.5Fromthefigure,itisclearthatthisispossibleifwemultiplyY(j)withotherwiseTjHc,0||,)(Where(2/0)c(2/T)-(2/0).7.6Considerthesignalw(t)=x1(t)x2(t).TheFouriertransformW(j)ofw(t)isgivenbyW(j)=21)(*)(21jXjX.Since0)(1jXfor||≥1andX2(j)=0for||≥2,wemayconcludethatW(j)=0for||≥1+2.Consequently，theNyquistrateforw(t)iss=2(1+2).Therefore,themaximumsampling-2/02/00X(j)TA2020Y(j)TjeTA/2)(TjeTA/2)(T2T2138periodwhichwouldstillalloww(t)toberecoveredisT=2/(s)=/(1+2).7.7Wenotethatx1(t)=h1(t)*{nnTtnTx)()(}FormFigure7.7inthebook,weknowthattheoutputofthezero-orderholdmaybewrittenasx0(t)=h0(t)*{nnTtnTx)()(}whereh0(t)isasshowninFigureS7.7BytakingtheFouriertransformofthetwoaboveequations,wehaveX1(j)=H1(j)Xp(j)X0(j)=H0(j)Xp(j)WenowneedtodetermineafrequencyresponseHd(j)forafilterwhichproducesx1(t)atitsoutputwhenx0(t)isitsinput.Therefore,weneedX0(j)Hd(j)=X1(j)Thetriangularfunctionh1(t)maybeobtainedbyconvolvingtworectangularpulsesasshowninFigureS7.7FigureS7.7Therefore,h1(t)={(1/T)h0(t+T/2)}*{(1/T)h0(t+T/2)}TakingtheFouriertransformofbothsidesoftheaboveequation,H1(j)=T1eTjH0(j)H0(j)ThereforeX1(j)=H1(j)Xp(j)=T1eTjH0(j)H0(j)Xp(j)=T1eTjH0(j)X0(j)ThereforeHd(j)=T1eTjH0(j)=e2/jwTTT)2/sin(27.8(a)Yes,aliasingdoesoccurinthiscase.Thismaybeeasilyshownbyconsideringthesinusoidaltermofx(t)fork=5.Thistermisasignaloftheformy(t)=(1/2)5sin(5t).Ifx(t)issampledasT=0.2,thenwewillalwaysbesamplingy(t)atexactlyitszero-crossings(ThisissimilartotheideapresentedinFigure7.17ofyourtextbook).Therefore,thesignaly(t)appearstobeidenticaltothesignal(1/2)5sin(0t)forfrequency5isaliasedintoasinusoidoffrequency0inthesampledsignal.(b)Thelowpassfilterperformsbandlimitedinterpolationonthesignalx(t).Butsincealiasinghasalreadyresultedinthelossofthesinusoid(1/2)5sin(5t),theoutputwillbeoftheformx(t)=kk)21(40sin(kt)TheFourierseriesrepresentationofthissignalisoftheform0h0(t)1Tt0tT-Th1(t)01/TT/2-T/2t0T/2-T/21/Tt=*139x(t)=44kkae)/(tkj0Whereak=-j(1/2)1kj(1/2)1k7.9TheFouriertransformX(j)ofx(t)isasshowninFigureS7.9FigureS7.9Weknowfromtheresultsonimpulse-trainsamplingthatG(jw)=kjXT((1s)),WhereT=2/s=1/75.therefore,G(jw)isasshowninFigureS7.9.Clearly,G(jw)=(1/T)X(j)=75X(j)for||50.7.10(a)Weknowthatx(t)isnotaband-limitedsignal.Therefore,itcannotundergoimpulse-trainsamplingwithoutaliasing.(b)FormthegivenX(j)itisclearthatthesignalx(t)whichisbandlimited.Thatis,X(j)=0for||0.Therefore,itmustbepossibletoperformimpulse-trainsamplingonthissignalwithoutexperiencingaliasing.Theminimumsamplingraterequiredwouldbes=20,ThisimpliesthatthesamplingperiodcanatmostbeT=2/s=/0(c)Whenx(t)undergoesimpulsetrainsamplingwithT=2/0,wewouldobtainthesignalg(t)withFouriertransformG(jw)=T1kTkjX))/2((ThisisasshownintheFigureS7.10FigureS7.10Itisclearfromthefigurethatnoaliasingoccurs,andthatX(jw)canberecoveredbyusingafilterwithfrequencyresponseT00H(jw)=0otherwiseTherefore,thegivenstatementistrue.7.11WeknowfromSection7.4thatXd(je)=T1kcTkjX))/2(((a)SinceXd(je)isjustformedbyshiftingandsummingreplicasofX(jw),wemayarguethatifXd(je)isreal,thenX(jw)mustalsobereal(b)Xd(je)consistsofreplicasofX(jw)whichares