SignalandSystemChap66.1Consideracontinuous-timeLTIsystemwithfrequencyresponseandrealimpulseresponseh(t).Supposethatweapplyaninputtothissystem.TheresultingoutputcanbeshowntobeoftheformWhereAisanonnegativerealnumberrepresentinganamplitude-scalingfactorandisatimedelay.(a)ExpressAintermsof.(b)ExpressintermsofSolution:(a)ForSoSo(b)forSo6.3ConsiderthefollowingfrequencyresponseforacausalandstableLTIsystem:(a)Showthat,anddeterminethevaluesofA.(b)Determinewhichofthefollowingstatementsistrueabout,thegroupdelayofthesystem.(Note,whereisexpressedinaformthatdoesnotcontainany1SignalandSystemdiscontinuities.)1.2.3Solution:(a)forSoA=1(b)forSo6.5Consideracontinuous-timeidealbandpassfilterwhosefrequencyresponseis(a)Ifh(t)istheimpulseresponseofthisfilter,determineafunctiong(t)suchthat(b)Asisincreased,dosetheimpulseresponseofthefiltergetmoreconcentratedorlessconcentratedabouttheorigin?Solution(a)Method1.LetTheyareshowninthefigures,whereSowecanget2SignalandSystemMethod2.UsingtheinverseFTdefinition,itisobtained(b)moreconcentrated.3SignalandSystemChap77.1Areal-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)4SignalandSystemSolution:(a)theNyquistrateis(b)theNyquistrateis(c)theNyquistrateis7.4Letx(t)beasignalwithNyquistrate.DeterminetheNyquistrateforeachofthefollowingsignals:(a)(b)(c)(d)Solution:(a)weletSoSotheNyquistrateofsignal(a)is.(b)weletSo5SignalandSystemSotheNyquistrateofsignal(b)is.(c)weletSoSotheNyquistrateofsignal(c)is2.(d)weletForSoSotheNyquistrateofsignal(d)is7.9ConsiderthesignalWhichwewishtosamplewithasamplingfrequencyoftoobtainasignalg(t)withFouriertransform.DeterminethemaximumvalueofforwhichitisguaranteedthatWhereistheFouriertransformofx(t).Solution:Butthefigureaboutbefore-samplingandafter-samplingofis6SignalandSystemWecanseethatonlywhen,thebefore-samplingandafter-samplingofhavethesamefigure.SoifThemaximumvalueofis.Chap99.2ConsiderthesignalanddenoteitsLaplacetransformbyX(s).7SignalandSystem(a)Usingeq.(9.3),evaluateX(s)andspecifyitsregionofconvergence.(b)DeterminethevaluesofthefinitenumbersAandsuchthattheLaplacetransformG(s)ofhasthesamealgebraicformasX(s).whatistheregionofconvergencecorrespondingtoG(s)?Solution:(a).Accordingtoeq.(9.3),wewillgetROC:Re{s}>-5(b).,Re{s}<-5Ifthenit’sobviouslythatA=-1,,Re{s}<-5.9.5ForeachofthefollowingalgebraicexpressionsfortheLaplacetransformofasignal,determinethenumberofzeroslocatedinthefinites-planeandthenumberofzeroslocatedatinfinity:(a)(b)(c)Solution:(a).1,1ithasazerointhefinites-plane,thatisAndbecausetheorderofthedenominatorexceedstheorderofthenumeratorby1X(s)has1zeroatinfinity.(b).0,18SignalandSystemithasnozerointhefinites-plane.Andbecausetheorderofthedenominatorexceedstheorderofthenumeratorby1X(s)has1zeroatinfinity.(c).1,0ithasazerointhefinites-plane,thatisAndbecausetheorderofthedenominatorequalstotheorderofthenumeratorX(s)hasnozeroatinfinity.9.7HowmanysignalshaveaLaplacetransformthatmaybeexpressedasinitsregionofconvergence?Solution: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:IFThentheROCofis(-1,1)istwosides.9SignalandSystem9.9GiventhatDeterminetheinverseLaplacetransformofSolution:Itisobtainedfromthepartial-fractionalexpansion:,WecangettheinverseLaplacetransformfromgivenformulaandlinearproperty.9.10UsinggeometricevaluationofthemagnitudeoftheFouriertransformfromthecorrespondingpole-zeroplot,determine,foreachofthefollowingLaplacetransforms,whetherthemagnitudeofthecorrespondingFouriertransformisapproximatelylowpass,highpass,orbandpass.(a):(b):(c):Solution:(a).It’slowpass.(b).It’sbandpass.10SignalandSystem(c).It’shighpass.9.13Let,Where.AndtheLaplacetransformofg(t)is.DeterminethevaluesoftheconstantsandSolution:,andTheLaplacetransform:and,FromthescalepropertyofLaplacetransform,,So,Fromgiven,Wecandetermine:11
