信號與系統(tǒng)第二版課后習(xí)題解答(6-7-9)奧本海姆

Chap66.1Consideracontinuous-timeLTIsystemwithfrequencyresponseandrealimpulseresponseh<t>.Supposethatweapplyaninputoutputcanbeshowntobeoftheformtothissystem.TheresultingWhereAisanonnegativerealnumberrepresentinganamplitude-scalingfactorandisatimedelay.<a>ExpressAintermsof.<b>ExpressintermsofSolution:<a>ForSoSo<b>forSo6.3ConsiderthefollowingfrequencyresponseforacausalandstableLTIsystem:<a>Showthat,anddeterminethevaluesofA.<b>Determinewhichofthefollowingstatementsistrueabout,thegroupdelayofthesystem.<Note,whereisexpressedinaformthatdoesnotcontainanydiscontinuities.>1.2.3Solution:<a>forSoA=1(b)forSo6.5Consideracontinuous-timeidealbandpassfilterwhosefrequencyresponseis(a)Ifh<t>istheimpulseresponseofthisfilter,determineafunctiong<t>suchthat<b>Asisincreased,dosetheimpulseresponseofthefiltergetmoreconcentratedorlessconcentratedabouttheorigin?Solution(a)Method1.LetTheyareshowninthefigures,whereSowecangetMethod2.UsingtheinverseFTdefinition,itisobtained(b)moreconcentrated.Chap77.1Areal-valuedsignalx<t>isknowtobeuniquelydeterminedbyitssampleswhenthesamplingfrequencyis.Forwhatvaluesofisguaranteedtobezero?Solution:AccordingtothesamplingtheoremThatisSoif,7.2Acontinuous-timesignalx<t>isobtainedattheoutputofanideallowpassfilterwithcutofffrequency.Ifimpulse-trainsamplingisperformedonx<t>,whichofthefollowingsamplingperiodswouldguaranteethatx<t>canberecoveredfromitssampledversionusinganappropriatelowpassfilter?<a><b><c>Solution:Fromthesamplingtheorem,,thatistheconditions<a>and<c>aresatisfiedwiththesamplingtheorem,<b>isnotsatisfied.7.3Thefrequencywhich,underthesamplingtheorem,mustbeexceededbythesamplingfrequencyiscalledtheNyquistrate.DeterminetheNyquistratecorrespondingtoeachofthefollowingsignals:<a><b><c>Solution:<a>theNyquistrateis<b>theNyquistrateis<c>theNyquistrateis7.4Letx<t>beasignalwithNyquistrate<a>.DeterminetheNyquistrateforeachofthefollowingsignals:<b><c><d>Solution:<a>weletSoSotheNyquistrateofsignal<a>is.<b>weletSoSotheNyquistrateofsignal<b>is.<c>weletSoSotheNyquistrateofsignal<c>is2<d>welet.ForSoSotheNyquistrateofsignal<d>is7.9ConsiderthesignalWhichwewishtosamplewithasamplingfrequencyoftoobtainasignalg<t>withFourierforwhichitisguaranteedthattransform.DeterminethemaximumvalueofWhereistheFouriertransformofx<t>.Solution:Butthefigureaboutbefore-samplingandafter-samplingofisWecanseethatonlywhenfigure.,thebefore-samplingandafter-samplingofhavethesameSoifThemaximumvalueofis.Chap99.2ConsiderthesignalanddenoteitsLaplacetransformbyX<s>.<a>Usingeq.<9.3>,evaluateX<s>andspecifyitsregionofconvergence.<b>DeterminethevaluesofthefinitenumbersAandsuchthattheLaplacetransformG<s>ofhasthesamealgebraicformasX<s>.whatistheregionofconvergencecorrespondingtoG<s>?Solution:<a>.Accordingtoeq.<9.3>,wewillgetROC:Re{s}>-5<b>.If,Re{s}<-5thenit’sobviouslythatA=-1,,Re{s}<-5.9.5ForeachofthefollowingalgebraicexpressionsfortheLaplacetransformofasignal,determinethenumberofzeroslocatedinthefinites-planeandthenumberofzeroslocatedatinfinity:<a><b><c>Solution:<a>.1,1ithasazerointhefinites-plane,thatisAndbecausetheorderofthedenominatorexceedstheorderofthenumeratorby1X<s>has1zeroatinfinity.<b>.0,1ithasnozerointhefinites-plane.becausetheorderofthedenominatorexceedstheorderofthenumeratorby1AndX<s>has1zeroatinfinity.<c>.1,0ithasazerointhefinites-plane,thatisAndbecausetheorderofthedenominatorequalstotheorderofthenumeratorX<s>hasnozeroatinfinity.9.7HowmanysignalshaveaLaplacetransformthatmaybeexpressedasregionofconvergence?initsSolution:Thereare4polesintheexpression,butonly3ofthemhavedifferentrealpart.Thes-planewillbedividedinto4stripswhichparalleltothejw-axisandhavenocut-across.Thereare4signalshavingthesameLaplacetransformexpression.9.8Letx<t>beasignalthathasarationalLaplacetransformwithexactlytwopoleslocatedats=-1ands=-3.If[theFouriertransformofg<t>]converges,determinewhetherx<t>isleftsided,rightsided,ortwosided.Solution:ROC:R<x>+Re{2}Andx<t>havethreepossibleROCstrips:g<t>havethreepossibleROCstrips:IFThentheROCofistwosides.9.9Giventhatis<-1,1>DeterminetheinverseLaplacetransformofSolution:Itisobtainedfromthepartial-fractionalexpansion:,WecangettheinverseLaplacetransformfromgivenformulaandlinearproperty.9.10UsinggeometricevaluationofthemagnitudeoftheFouriertransformfromthecorrespondingpole-zeroplot,determine,foreachofthefollowingLaplacetransforms,whetherthemagnitudeofthecorrespondingFouriertransformisapproximatelylowpass,highpass,orbandpass.<a>:<b>:<c>:Soluti