信号与系统奥本海姆英文版课后答案chapter9

158Chapter9Answers9.1（a）Thegivenintegralmaybewrittenas(5)0tjteedtIf-5,thenthefunction(5)tegrowstowardswithincreasingtandthegivenintegraldoesnotconverge.butif-5,thentheintegraldoesconverge(b)Thegivenintegralmaybewrittenas0(5)tjteedtIf-5,thenthefunction(5)tegrowstowardsastdecreasestowards-andthegivenintegraldoesnotconverge.butif-5,thentheintegraldoesconverge(c)Thegivenintegralmaybewrittenas5(5)5tjteedtClearlythisintegralhasafinitevalueforallfinitevaluesof.(d)Thegivenintegralmaybewrittenas(5)tjteedtIf-5,thenthefunction(5)tegrowstowardsastdecreasestowards-andthegivenintegraldoesnotconvergeIf-5,,thenfunction(5)tegrowstowardswithincreasingtandthegivenintegraldoesnotconvergeIf=5,thentheintegralstilldoesnothaveafinitevalue.therefore,theintegraldoesnotconvergeforanyvalueof.(e)Thegivenintegralmaybewrittenas0(5)tjteedt+(5)0tjteedtThefirstintegralconvergesfor-5,thesecondinternalconvergesif-5,therefore,thegiveninternalconvergeswhen5.(f)Thegivenintegralmaybewrittenas0(5)tjteedtIf5,thenthefunction(5)tegrowstowardsastdecreasetowards-andthegivenintegraldoesnotconverge.butif5,thentheintegraldoesconverge.9.2(a)X(s)=5(1)tdteutedt=(5)0stedt=(5)5sesAsshowninExample9.1theROCwillbeRes-5.(b)Byusingeg.(9.3),wecaneasilyshowthatg(t)=A5teu(-t-0t)hastheLaplacetransformG(s)=0(5)5stAesTheROCisspecifiedasRes-5.Therefore,A=1and0t=-19.3UsingananalysissimilartothatusedinExample9.3weknownthatgivensignalhasaLaplacetransformoftheformX(s)115ssThecorrespondingROCisResmax(-5,Re{}).SincewearegiventhattheROCisRe{s}-3,weknowthatRe{}=3.therearenoconstraintsontheimaginarypartof.9.4WeknowformTable9.2that111()sin(2)()()()LtxtetutXsXs,Re{s}-1WealsoknowformTable9.1thatx(t)=1()LxtX(s)=1()Xs159TheROCofX(s)issuchthatif0swasintheROCof1()Xs,then-0swillbeintheROCofX(s).Puttingthetwoaboveequationstogether,wehavex(t)=1x(-t)=sin(2)()tetutLX(s)=1()Xs=-222(1)2s,Res1thedenominatoroftheform2s-2s+5.Therefore,thepolesofX(s)are1+2jand1-2j.9.5(a)thegivenLaplacetransformmaybewrittenas()Xs=24(1)(3)sss.Clearly,X(s)hasazeroats=-2.sinceinX(s)theorderofthedenominatorpolynomialexceedstheorderofthenumeratorpolynomialby1,X(s)hasazeroat.Therefore,X(s)hasonezeroinfinites-planeandonezeroatinfinity.(b)ThegivenLaplancetransformmaybewrittenasX(s)=1(1)(1)sss=11sClearly,X(s)hasnozerointhefinites-plane.SinceinX(s)theofthedenominatorpolynomialexceedstheorderthenumeratorpolynomialby1,X(s)hasazeroat.thereforeX(s)hasnozerointhefinites-planeandonezeroatinfinity.(c)ThegivenLaplacetransformmaybewrittenas22(1)(1)()1(1)sssXssssClearly，X（s）hasazeroats=1.sinceinX(s)theorderofthenumeratorpolynomialexceedstheorderofthedenominatorpolynomialby1,X(s)haszerosat.therefore,X(s)hasonezerointhes-planeandnozeroatinfinity.9.6(a)No.Fromproperty3inSection9.2weknowthatforafinite-lengthsignal.theROCistheentires-plane.therefore.therecanbenopolesinthefinites-planeforafinitelengthsignal.Clearlyinthisproblemthisnotthecase.(b)Yes.Sincethesignalisabsolutelyintegrable,TheROCmustinclude,thej-axis.Furthermore,X(s)hasapoleats=2.therefore,onevalidROCforthesignalwouldbeRe{s}2.Fromproperty5insection9.2weknowthatthiswouldcorrespondtoaleft-sidedsignal(C)No.Sincethesignalisabsolutelyintegrable,TheROCmustinclude,thej-axis.Furthermore,X(s)hasapoleats=2.therefore,wecanneverhaveanROCoftheformRe{s}.Fromproperty5insection9.2weknewthatx(t)cannotbearight-sidesignal(d)Yes.Sincethesignalisabsolutelyintegrable,TheROCmustinclude,thej-axis.Furthermore,X(s)hasapoleats=2.therefore,onevalidROCforthesignalcouldbeRe{s}2suchthat0.Fromproperty6insection9.2,weknowthatthiswouldcorrespondtoatwosidesignal9.7WemayfinddifferentsignalwiththegivenLaplacetransformbychoosingdifferentregionsofconvergence,thepolesofthegivenLaplacetransformare02s13s21322sj31322sjBasedonthelocationsofthelocationsofthesepoles,wemychooseformthefollowingregionsofconvergence:(i)Re{s}-12(ii)-2Re{s}-12(iii)-3Re{s}-2(iv)Re{s}-3Therefore,wemayfindfourdifferentsignalsthegivenLaplacetransform.9.8FromTable9.1,weknowthatG(t)=2()()(2)LtextGsXs.TheROCofG(s)istheROCofX(s)shiftedtotherightby2WearealsogiventhatX(s)hasexactly2polesats=-1ands=-3.sinceG(s)=X(s-2),G(s)alsohasexactlytwopoles,locatedats=-1+2=1ands=-3+2=-1sincewearegivenG(j)exists,wemayinferthatj-axisliesintheROCofG(s).Giventhisfactandthelocationsofthepoles,wemayconcludethatg(t)isatwosidesequence.Obviouslyx(t)=2teg(t)willalsobetwosided1609.9UsingpartialfractionexpansionX(s)=4243ssTakingtheinverseLaplacetransform,X(t)=443()2()tteuteut9.10Thepole-zeroplotsforeachofthethreeLaplacetransformsisasshowninFigureS9.10(a)formSection9.4weknewthatthemagnitudeoftheFouriertransformmaybeexpressedaswesethattheright-handsideoftheaboveexpressionismaximumfor=0anddecreasesasbecomesincreasingmorepositiveormorenegative.Therefore1()Hjisapproximatelylowpass(b)FromSection9.4weknowthatthemagnitudeoftheFouriertransformmaybeexpressas(lengthofvectorfromto0)1