[奥本海姆]信号与系统(第二版)课后答案(英文版、非扫描版).docx

Signals&Systems(SecondEdition)LearningInstructions(ExercisesAnswers)DepartmentofComputerEngineering2005.12ContentsChapter 1 ,2Chapter 2 ,17ChiIPter3*35Chapter462ChaPter5*83Chapter6-109ChiIPter7119Chapter 8 -132Chapter 9 -140Chapter 10 i1601 1 1- p = coS 加=2e 22.e T = COSq)+ /sin®= / 泼= "=j>-i2 , v s÷n<=")in 9)Chapter1Answers1.1 ConvertingfrompolartoCartesiancoordinates:e3=G"噂+)*=¢*="jy2e4=j2ei=Ja产=一1.2 convertingfromCartesiantopolarcoordinates:1.3.=Gw,2-2少97&1+/=0勺,K-j)=e(a)EnC,dt>P=°，because48-3j=S-jw21j=e->(不声=e<12(b)W)=J>，M)I=Jherefg足,(矶力=匚力一,PEkJ'g(瞰UmudT吧=1×>2TTT2(C)2(O=cos(t).Therefore,EH=JNX3(矶力=Iu>cos(%=8,(，如丽(加j屋M同百加Po0=O1becauseE<exjin=eX11f=1therefore,EW=PoO=Iim-Ll-V92l=Iim-L"2NLyl2L'c|NqN1/、-n=,'22(f)HM=C。用Therefore,&=独如汽口吟)=至。吟”cos(P30=±(兀)IimJE(+2公)=-奥加T&cos=2加21.4. (a)Thesignalxnisshiftedby3totheright.Theshiftedsignalwillbezeroforn<l,Andn>7.(b) Thesignalxnisshiftedby4totheleft.Theshiftedsignalwillbezeroforn<-6.Andn>0.(c) Thesignalxnisflippedsignalwillbezeroforn<-landn>2.(d) Thesignalx11lisflippedandtheflippedsignalisshiftedby2totheright.ThenewSignalwillbezeroforn<-2andn>4.(e) Thesignalxnisflippedandtheflippedandtheflippedsignalisshiftedby2totheleft.Thisnewsignalwillbezeroforn<-6andn>0.1.5. (a)x(l-t)isobtainedbyflippingx(t)andshiftingtheflippedsignalby1totheright.Therefore,x(l-t)willbezerofort>-2.(b) From(a),weknowthatx(1-t)iszerofort>-2.Similarly,x(2-t)iszerofort>-l,Therefore,x(1-t)+x(2-t)willbezerofort>-2.(c) x(3t)isobtainedbylinearlycompressionx(t)byafactorof3.Therefore,x(3t)willbezerofort<1.(d) x(t3)isobtainedbylinearlycompressionx(t)byafactorof3.Therefore,x(3t)willbezerofort<9.1.6. (a)X(r)isnotperiodicbecauseitiszerofort<0.(b) foralln.Therefore,itisperiodicwithafundamentalperiodof1.(c) X3wisasshownintheFigureS1.6.r111Xn.3I3II?-4014nTherefore,itisperiodicIvithafundamentalP昂Odof4.1.7. (a)(j)=%Jw+-n=(wn-un-4+u-n-u-n-41)Therefore,£(xjl)iszeroforxjl>3(b)Sincex,()isanoddsignal,(x,11)iszeroforallvaluesoft.(C)e,(利川)=如加+X卜叫l-3-QjTherefore,(X3)iszerowhenn<3andwhen同f8.MIr5/(d)()=-()+%4()=/ed-2)-2)ITherefore,(X4)iszeroonlywhenf>.18(a)(9MXP)=_2¾0s(0+)(b) (t11dX2(r)=>"cos()cos(3r+2)=cos(30=e0,cos(3z+)(c) (¾fX3()=ezsin(3+r)=n(3-)(/4)e-2si11(100)=sin(1OOr+)=2lcos(100/+y)乎is供日前Ccomplexexponential.国(r)%(0=9(b) isacomplexexponentialmultipliedbyadecayingexponential.Therefore,j(r)isnotperiodic.n(c) X3nisaperiodicsignal.XJm)=XWisacomplexexponentialwithafundamentalperiodof竺.2=2兀(d)(向isaperiodicsignal.ThefundamentalperiodisgivenbyN=m(,$)YBychoosingm=3.Weobtainthefundamentalperiodtobe10.(e)isnotperiodic,rr1isacomplexexponentialwithwecannottnanyInlegerm4JAsihsuchthatm(2元)isalsoaninteger.Therefore,.isnotperiodic.刈对Wo1.10.x()=2cos(10t+l)-sin(4t-l)PeriodoffirsttermintheRHS=2n.三一105PeriodoffirsttermintheRHS=2.一42Therefore,theoverallsignalisperiodicwithaperiodwhichtheleastcommonmultipleoftheperiodsofthefirstandsecondterms.Thisisequalto.1.lLxnl=1+PeriodoffirsttermintheRHS=1.PeriodofsecondtermintheRHS-(2)=7(whenm=2)1477JPeriodofsecondtermintheRHS-<2(whenm=l)k2xf5)Therefore,theoverallsignalxnisperiodicwithaperiodwhichistheleastcommonMultipleoftheperiodsofthethreetermsinnxn.Thisisequalto35.1.12 .ThesignalxnisasshowninfigureS1.12.xncanbeobtainedbyflippingunandthenShiftingtheflippedsignalby3totheright.Therefore,xn=u-n+3.ThisimpliesthatM=-Iandno=-3.1.13Oj < -2 1,-2 < 2Oj >2Mt)= Lx(T)由=f(+2)-(-WThereforeEg=J=41.14 Thesignalx(t)anditsderivativeg(t)areshowninFigureS1.14.g(t)-1(O=3(r-2)-3(r-2-l)A"-0oThisimpliesthatA1=3,11=0,A2=-3,andt2=1.1.15 (a)Thesignalx2n,whichistheinputtoS2,isthesameasy1n.Therefore,Iyn=X9n-2+Xn-3-221=y1n-2+-y1n-31=2xIn-2+4xIn-3+-(2x1n-3+4xln-4)=2xln-2÷5xln-3+2xln-4Theinput-outputrelationshipforSisyn=2xn-2+5xn-3+2xn-4(b)Theinput-outputrelationshipdoesnotchangeiftheorderinwhichS1andS2areconnectedseriesreversed.WecaneasilyprovethisassumingthatS1followsS2.Inthiscase,thesignalXjnLwhichistheinputtoS1isthesameasy2n.Thereforeylnl=2x1n÷4x1n-11=2y2n+4y2n-l1 1=2(x2n-21+x2n-3)+4(x2(n-31+xn-4)2 ,22=2x2n-2+5x2n-3+2x2n-4Theinput-outputrelationshipforSisonceagainyn=2xn-2+5xn-3+2xn-41.16 (a)Thesystemisnotmemorylessbecauseyndepenonpastvaluesofxn.(b)Theoutputofthesystemwillbey(n=OTl712=0(c)Fromtheresultofpart(b),wemayconcludethatthesystemoutputisalwayszeroforinputsoftheform3一%,kr.Therefore,thesystemisnotinvertible.1.17 (a)Thesystemisnotcausalbecausetheoutputy(t)atsometimemaydependonfuturevaluesofx(t).Forinstance,y(-)=x(0).(b)Considertwoarbitraryinputsx(t)andX2(t).xI(Oy1(t)=x1(sin(t)x2W-÷y2(0=x2(sin(t)1.etX3(t)bealinearcombinationofx(t)andx2(t).Thatis,x(t)=ax(t)+bx(t)3I2Whereaandbarearbitraryscalars.Ifx3(t)istheinputtothegivensystem,thenthecorrespondingoutputy(Oisy(t)=x(sin(t)333=axl(sin(t)+X2(sin(t)=ay1(t)+by2(t)Therefore,thesystemislinear.1.18.(a)Considertwoarbitraryinputsxlnandx2n.x,ny1n=ZXWk=n-%I2fn1Ty2fnl=ZX伙k=n-nft21.etx3nbealinearcombinationofx1nlandx2n.Thatis:x3n=ax1n+bx2nwhereaandbarearbitraryscalars.Ifx3nistheinputtothegivensystem,thenthecorrespondingoutput11+110ynisyn=yxk33=ff-1103n+n+/HlD=网+如灯)=a£芭伙+b2入；伙k=n-nuk=n-t0k=n-11o2=ay1n÷by2nThereforethesystemislinear.(b) Consideranarbitraryinputx1n.Lety 1 nl = XX W=-0 Ibe the corresponding output .Consider a second input x2 n obtained by shifting xln in time:n-n ÷/kn-n -x2 n=x1n-nllThe output corresponding to this input is +Also note thatTherefore,y n= ZX k= ZM伙一nk=n-no 2k=n-nn-ni Yi n-l= x1k=n-nx -iiqy2lnl=y n-njJ=Thisimpliesthatthesystemistime-invariant.yn (2n +1)B.o(c) Ifw<B,thenTherefore,C(2n0+1)B.1.19 (a) (i) Consider two arbitrary inputs x 1 (t) and X 2 (t).Let X 3(t) be a linear combination of x (t) and x 2 (t).That isX(t)y1(t)=t21(t-i)x2(t)y2(t)=t2x2(t-l)x(t)=ax(t)+bx(t)312whereaandbarearbitraryscalars.Ifx3(t)istheinputtothegivensystem,thenthecorrespondingoutputy3(Oisy3(t)=t23(t-l)=t2(axl(t-l)+bx2(t-l)=ay1(l)+by2(l)Therefore,thesystemislinear.(ii)Consideranarbitraryinputsx1(t).Lety(t)=t*"x1(t-l)bethecorrespondingoutput.Considerasecondinputx2(t)obtainedbyshiftingx1(t)intime:x2(t)=x1(t-t0)22Theoutputcorrespondingtothisinputisy(t)=t-x(t-l)=tX(t-1-1)221Alsonotethaty1(t-t0)=(t-t0)2x1(t-1-10)y(t)Thereforethesystemisnottime-invariant.(b)(i)Considertwoarbitraryinputsx1nandx2n.xn>yn=x2111-2x>2ny2n=x22n-2.1.etx3(I)bealinearcombinationofXnandx2n.Thatisx3nl=ax1n+bx2111whereaandbarearbitraryscalars.Ifx3nistheinputtothegivensystem,thenthecorrespondingoutputy11isy3n=x32n-232=(ax,n-2+bxJn-2)=a2x12n-2j+b2x22n-2+2abxln-2X2n-2ay,n+by2nThereforethesystemisnotlinear.(ii)Consideranarbitraryinputx1n.Letynl=x2n-21bethecorrespondingoutput.Considerasecondinputx2nobtainedbyshiftingx1nintime:x2n=x,n-noTheoutputcorrespondingtothisinputisy2n=x22n-2=x12n-2-nJ2Alsonotethatyn-n=xn-2-nIoi0Therefore,y2n=y1n-nJThisimpliesthatthesystemistime-invariant.(c) (i)Considertwoarbitraryinputsxlnandx2n.x,ny1n=x1n+l-xjn-lx2Hy2n=x2n+l-x2n-11.etx3nbealinearcombinationofxlnjandx2n.Thatis:x3n=ax1n+bx2nwhereaandbarearbitraryscalars.Ifx3nistheinputtothegivensystem,thenthecorrespondingoutputy3nisyn=xn+l-xn-l333=ax1n+l+bx2n+l-ax1n-l-bx2n-1=a(xln+1-x1n-1)+b(x2n+l-x2n-l)=ay1n+by,nThereforethesystemislinear.(11) ConsideranarbitraryinputxnJ.Letyn=xn+1-xn-lIIIbethecorrespondingoutput.Considerasecondinputx2nobtainedbyshiftingx1nintime:x2n=x1n-n0lTheoutputcorrespondingtothisinputisy2n=x2tn+U-X?S-l=x,n+l-n0-x1n-l-n0Alsonotethatyn-n=xn+l-n-xn-l-nI11Therefore,y2n-yln-n000Thisimpliesthatthesystemistime-invariant.(d) (i)ConsidertwoarbitraryinputsX(t)andx2(t).X1(t)y1(t)=Odx1(t)X2(0Ty2(0=OJx2(t)1.etx3(t)bealinearcombinationofx(t)andx2(t).Thatisx(t)=ax(t)+bx(t)3I2whereaandbarearbitraryscalars.Ifx3(t)istheinputtothegivensystem,thenthecorrespondingoutputy3(Oisy3=Odx3(t)=Odax(t)+bx2(t)=atx1(t)+bOJx2(t)=ayl(t)+by2(t)Thereforethesystemislinear.(ii)Consideranarbitraryinputsx1(t).Lety1=Odx1.卜。bethecorrespondingoutput.Considerasecondinputx2(t)obtainedbyshiftingx1(t)intime:x2(t)=x1(t-t0)Theoutputcorrespondingtothisinputisyi(I)=Odx2(t)=X2(t)-X2(-r)2_x(t-t)-(t_.-1QICQ2AlsonotethatY1(tt0)=x(t<>)X(TTO)丰y(t)22Thereforethesystemisnottime-invariant.1.20(a)GivenX(f)=e2力y=e口X(f)=e-20一y=6一两Sincethesystemliner九()=l2(e周+e%,)=l2(e/3r+Tkre-J3t)ThereforeX(t)=CoS)Xa)=Cos(3t)(b)WeknowthatXI(t)=cos(2(t-l)=(e-，e2jt÷Ce'")2Usingthelinearityproperty,wemayonceagainwrite1(t)=y(JQ+C,e-2力)_9y(t)=(eje3j,+/)=cos(3t-l)乙_e-3jTherefore,M(t)=cos(2(t-12)y=COS(3tJ)1.2LThesignalsaresketchedinfigured1.21.FigureSI.211.22 ThesignalsaresketchedinfigureS1.221.23 TheevenandoddpartsaresketchedinFigureS1.23x3- nxn-4(b)xnun-3=xn(d)1/21x3nx3n+l(f)xonx3- n2 +(-l)nxn2Figure S 1.24Figure S 1.23171.24 TheevenandoddpartsaresketchedinFigureS1.241.25 (a)periodicperiod=2/(4)=/2(b) periodicperiod=2/(4)=2(c) x(t)=1+cos(4t-2/3)1/2.periodicperiod=2/(4)=/2(d) x(t)=cos(4t)2.periodicperiod=2/(4)=1/2(e) x(t)=sin(4t)u(t)-sin(4t)u(-t)2.Notperiod.(f) Notperiod.1.26 (a)periodic,period=7.(b) Notperiod.(c) periodic,period=8.(d) xn=(l2)cos(3n4+cos(n4).periodic,period=8.(e) periodic,period=16.1.27 (a)Linear,stable(b) Notperiod.(c) 1.inear(d) 1.inear,causal,stable(e) Timeinvariant,linear,causal,stable(f) 1.inear,stable(g) Timeinvariant,linear,causal1.28 (a)Linear,stable(b)Timeinvariant,linear,causal,stable(d) (C)Memoryless,linear,causal(e) 1.inear,stable(f) 1.inear,stableMemoryless,linear,causal,stable1.inear,stable1.29(a)ConsidertwoinputstothesystemsuchthatX1-i>F=况fl2andx2H2F=况*?eNowconsiderathirdinputjyn=)n+Xn.Thecorrespondingsystemoutput用町=火,莒呻,Winbe=况仍四+占四=9U÷9iJtherefore,wemayconcludethatthesystemisadditive1.etusnowassumethatinputstothesystemsuchthatX四P=况andj4Xn.%四，X2p=况叫刎Nowconsiderathirdinput与n=X2n+乃n.ThecorrespondingsystemoutputWillbe%=况/9=COS(九/4)况-sin(w4)I«+cos(z4)况:2!ZK-Sin(九/4)I,£V+cos(九/4)9“04-sin(兀/4)1(»=91ee严XF+况的兀,4X/?2=y小了2网therefore,wemayconcludethatthesystemisadditive(b)(i)Considertwoinputstothesystemsuchthatandx2(z)y2(z)NowconsiderathirdinputXJu=2t+t.ThecorrespondingsystemoutputWillbe111()÷,o2()+()UIdiXlJ(0÷JOjtherefore,wemayconcludethatthesystemisnotadditiveNowconsiderathirdinputx4tj=axt.ThecorrespondingsystemoutputWillbe= y()Therefore,thesystemishomogeneous.n. The corresponding outputs evaluated at n=0 are(ii)Thissystemisnotadditive.Considerthefowlingexample.Let6(n=2n+2+2n+l+2nandn=8n+l+2n+l+y1O=2andy0=322Nowconsiderathirdinputxyn=x2n+xn.=3n+2+4n+l+5nThecorrespondingoutputsevaluatedatn=0is>,30=154.Gnarly,y3O0.Thisx4nx4n-2I94L"Jv.11xoteiselo,4h,(4)j2(t)periodic,periodT;x(t)periodic,periodT/2;1.33(1)Truexnl=xn+11;y(n)=y(n+No)i.e.periodicwithM)=n/2(2)False.ygn=ifNisevenandwithperiod=nifNisodd.nperiodicdoesnoimplyxnisperiodici.e.Letxn=gn+hnwhere1,neven,0,noddandhn=0,neven(1/2)”,oddThenyn=X2nisperiodicbutxnisclearlynotperiodic.(3)True.xn+N=xn;y?n+No=y211whereN=2N(4)True,y2n+N=y2n;y?n+No=y2nwhereNo=N21.34.(a)Consider411=0÷+t=lIfxnisodd,xn+x-n=0.Therefore,the-(b)Letyn=x1nx2n.ThenfUnMnffitionevaluatestozero.y-n=X-nx2-n=-xnx2n=-yn.Thisimpliesthatynisodd.(C)Consider=z£+2Usingtheresultofpart(b),weknowTliatxcnxonisanOdcfsignab.Therefore,usingtheresultofpart(a)wemayconcludethat2支幻川JdM=OTherefore,=÷W=-<C=-OCH=-OO(d)ConsiderEA2力JL+匚点)由+2j：Xy)X。力Again,sincexc(t)xo(t)isodd,r%(r)x0(M=oTherefore,EAMJ"/")%/*"1.35.WewanttofindthesmallestN0suchthatm(2/N)N0=2korN0=kNm,wherekisaninteger,thenNmustbeamultipleofm/kandm/kmustbeaninteger.thisimpliesthatm/kisadivisorofbothmandN.Also,ifwewantthesmallestpossibleN°,thenm/kshouldbetheGCDofmandN.Therefore,No=Ngcd(m,N).1.36.(a)Ifxnisperiodic"+where.。27o.2T"NT=2IknL=c=TiThisimpliesthattarationalnumber.(b)TTo=pqthenxn=is,Thefundamentalperiodisqgcd(p,q)andthefundmentalfrequency22oCQT一gcd(p,q)=LgCd(PM)=-TrgCd(P,q)=0gcd(p,q).qPqPP(c)pgcd(p,q)periodsofx(t)areneeded.1.37.(a)Fromthedefinitionof(f).Wehavex(t+)y()d=y(-t+)x(s)dJo(b)Notefrom=(KK(T)part(a)thatxv()=60(T).Thisimpliesthat(f)even.Therefore,theoddpartofw().iszero.(c)Here,=xt(-)andv(r)=xt(01.38.(a)Weknowthat2(2/)=.z2(f).ThereforeThisimpliesthatislim(2r)=Iim-U->oo2)二9.2(b)TheplotareasshowninFigures3.18.1.39WehaveIimw(r)(0=Iimw(O)(f)=O.ooAlso,Iim(r)=(Z).o2g(t)=fw()(r-)t7=fw()(r-)JJ-OOJOTherefore,0，f>°v(-)=0g")="z<0vw()(r-)=(r-)undefinedfort=O1.40. (a)Ifasystemisadditive,thenalso,ifasystemishomogeneous,then0=M)-XOXO-y(D=00=0jr(r)XOO=O(b)y(t)=x(t)issuchasysterm.(c)No.Forexample,considery(t)y(t)=Jx()dwithx(t)=ut)m(Z-l).Thenx(t)=Ofort>1,buty(t)=1fort>I.1.41. (a)ynl=2xnl.Therefbre,thesystemistimeinvariant.(b) yn=(2n-l)xn.Thisisnottime-invariantbecauseyn-No(2n-1)2xn-N(c) yn=xn1+(-1)n+1+(-1)n,=2xn.Therefore,thesystemistimeinvariant.1.42. (a)ConsidertwosystemSandS?connectedinseries.Assumethatifx(t)andX2(l)aretheinputstoS,.theny(t)andy2(t)aretheoutputs.respectively.Also,assumethatify(t)andy2(t)aretheinputtoS2,thenz(t)andZ2(t)aretheoutputs,respectively.SinceSislinear,wemaywrite必S+b5O()+()whereaandbareconstants.SinceSjslsocaf,wemaywrite孙(/)+加«)3()÷()Wemaythereforeconcludethataztbzt,axSt