信号与系统奥本海姆英文版课后答案chapter8

150Chapter8Answers8.1UsingTable4.1,taketheinverseFouriertransformof(())cYj.Thisgives()2()cjwtytxte.Therefore,()2cjwtmte.8.2(a)TheFouriertransformY(jw)ofy(t)isgivenby()(())cYjwXtj.Clearly,()Yjwisjustashiftedversionof()Xjw.Therefore,x(t)mayberecoveredfromy(t)simplybymultiplyingy(t)bycjwte.Thereisnoconstraintthatneedstobeplacedonctoensurethat()xtisrecoverablefrom()yt.(b)Weknowthat1()Re()()cos().cytytxttTheFouriertransform1()Yjwof1()ytisasshowninFigureS8.2111()(())(())22ccYjwXjXjIfwewanttopreventthetwoshiftedreplicasof()Yjwfrommultipliedbycos(2000)t,theoutputwillbe11()()cos(2000)()sin(2000)cos(2000)()sin(4000)2xtgttxtttxttTheFouriertransformofthissignalis111()((4000))((4000))44XjwXjXjjj.Thisimpliesthat1()Xjwiszerofor||2000.When()ytispassedthroughalowpassfiterwithcutofffrequency2000,theoutputwillclearlybezero.Therefore()yt=0.8.4Considerthesignal3()()sin(400)2sin(400)ytgttt23sin(200)sin(400)2sin(400)sin(200)1cos(800)/22sin(400)1cos(800)/2(1/2)sin(200)(1/4)sin(1000)sin(600)sin(400)(1/2)sin(1200)sin(400)tttttttttttttIfthissignalispassedthroughalowpassfilterwithcutofffrequency400,thentheoutputwillbe1sin(200)yt.8.5Thesignal()xtisasshowninFigureS8.5ttEnvelopof()tA()xt0t1000c1000c0cc(())cXj(())cXj()YjFigureS8.2151FigureS8.5Theenvelopeofthesignal()tisasshowninFigureS8.5.Clearly,iswewanttouseasynchronousdemodulationtorecoverthesignal()xt,weneedtoensurethatAisgreaterthantheheighthofthehighestsidelobe(seeFigureS8.5).Letusnowdeterminetheheightofthehighestsidelobe.Thefirstzero-crossingofthesignal()xtoccursattimeotsuchthat001000,1/1000.ttSimilarly,thesecondzero–crossinghappensattime1tsuchthat1110002,2/1000.ttThehighestsidelobeoccursattime01()/2tt.thatis,attime23/2000t.Atthistime,theamplitudeof()xtis2sin(3/2)2000()3/20003xtTherefore,Ashouldatleastbe20003.ThemodulationindexcorrespondingtothesmallestpermissiblevalueofAis..MaxvaluemMinpossibleofvalue()xtof1A=100032000/328.6LetusdenotetheFouriertransformofsin()/()ctby()Hjw.Thiswillberectangularpulsewhichisnonzeroonlyintherange||c.TakingTheFouriertransformofthefirstequationgivenintheproblem,wehave()()cos()()cos()()()cos()1()(1/2)(())(())1()cccccGjFTxttFTxttHjwFTxttHjwXjXjHjw()GjisasshowninFigureS8.6TheFouriertransformof()cos()cgttisalsoshowninFigureS8.6.Clearly,ifwewanttorecover()xtfrom()cos()cgtt,thenwehavetopass()cos()cgttthroughanideallowpassfilterwithgain4andcutofffrequencyM.Therefore,A=4.8.7InFigureS8.7,weshow()Xj,()Gj,and()Qj.WealsoshowapoltofTheFouriertransformof()cos()cgtt,thenweneedtoensurethat(1)02c,and(2)anideallowpassfilterwithpassbandgainof2andacutofffrequencyofcisusedtofilter0()cos()gtt.c1/2()Qj01/2cc0cM01/21/2M()Xj1()Gj2cFigureS8.6Mc2cc()coscFTgtt1/41/41/4Mc0c()Xj()Gj11/21/20M01528.8(a)FromFigureS8.8,itisclearthat()Yjisconjugate-symmetric.Therefore,()ytisreal.(b)ThispartoftheproblemexploresthedemodulationofSSBsignalsthroughsynchronousdemodulation.Thisideaisexploredinmoredetailinproblem8.29.LetusassumethatweusethesynchronousdemodulationsystemshownintheFigureS8.8.TheFourierTransform1()Yjofthesignal1()ytisshownintheFigureS8.8.Clearly,ifweuseanideallowpassfilterwithcutofffrequencycandpasshandgainof2,weshouldrecovertheoriginalsignal()xt.Therefore,2sin()()sin()*cctxtyttt.8.9Letthesignal1()xtand2()xthaveFourierTransform1()Xjand2()XjasshownintheFigureS8.9.WhenSSBmodulationisperformedonthesignals1()xtand2()xt,wewouldobtainthesignal1()ytand2()yt,respectively.TheFourierTransform1()Yjand2()YjofthesesignalswouldbeasshownintheFigureS8.9(seeSection8.4fordetails).(a)Fromthefigure,itisclearthatthesignal12()()()ytytytwouldhaveaFourierTransform()YjwhichisasshownintheFigureS8.9.FigureS8.8-2Wc2WcMM-Aj-WcWc-Aj/2Aj/2-WcAj/2Aj/2c-WcMM01/20M0M()XjA()()XjHj0WcAjA/2FT{[x(t)*h(t)].coswct}X(jw)FT{y(t)sinwct}FigureS7.20()cosFTgtt00c0c00c00cFigureS8.7153Fromthisfigure,itisobviousthat()Yjiszerofor||2c.（b）Inordertoobtain1()xtfrom()yt,wehavetofirstremoveanycontributionin()ytfrom2()xt.Fromthepreviouslydrawfigures,itisclearthatwecanremoveallcontributionto()ytfrom2()xtbyfirstlowpassfiltering()ytusingalowpassfilterwithcutofffrequencyc.Wemaythenfollowthisbyasynchronousdemodulationsystem.ThisideaisillustratedintheFigureS8.9.Therefore.10sinsin()()*cos*2cctAtxtytttInordertodeterminethevalueofthegainA,wefirstplottheFourierTransformof10sin()()*cosctxtyttt.FromthisitisclearthatA=4.8.10(a)FromSection8.5,weknowthatinordertoavoidaliasing,2/2MT,whereMistheMaximumfrequencyintheoriginalsignalandTistheperiodof()ct.Inthiscase,310T.Therefore1000M.Therefore,()0Xjfor1000.(b)ForFigure8.24,weknowthattheFourierTransform()Yjofthesignal()ytconsistsofshiftedreplicasof()Xj.Thereplicaof()Xjcenteredaround0isscaledby/T,whereisthewidthofeachpulseof()ct.Byusingalowpassfilter,wemayrecover()Xjfrom()Yj.Thel