SignalandSystemChap33.1Acontinuous-timeperiodicsignalx(t)isrealvalueandhasafundamentalperiodT=8.ThenonzeroFourierseriescoefficientsforx(t)are.Expressx(t)intheformSolution:Fundamentalperiod.3.2Adiscrete-timeperiodicsignalx[n]isrealvaluedandhasafundamentalperiodN=5.ThenonzeroFourierseriescoefficientsforx[n]are,,,Expressx[n]intheformSolution:for,,,,,3.3Forthecontinuous-timeperiodicsignal1SignalandSystemDeterminethefundamentalfrequencyandtheFourierseriescoefficientssuchthat.Solution:fortheperiodofis,theperiodofissotheperiodofis6,i.e.then,,,,3.5Letbeacontinuous-timeperiodicsignalwithfundamentalfrequencyandFouriercoefficients.GiventhatHowisthefundamentalfrequencyofrelatedto?Also,findarelationshipbetweentheFourierseriescoefficientsofandthecoefficientsYoumayusethepropertieslistedinTable3.1.Solution:(1).Because,thenhasthesameperiodas,thatis,(2).2SignalandSystem3.8Supposegiventhefollowinginformationaboutasignalx(t):1.x(t)isrealandodd.2.x(t)isperiodicwithperiodT=2andhasFouriercoefficients.3.for.4.Specifytwodifferentsignalsthatsatisfytheseconditions.Solution:while:isrealandodd,thenispurelyimaginaryandodd，,,.,thenandforsofor3.13Consideracontinuous-timeLTIsystemwhosefrequencyresponseisIftheinputtothissystemisaperiodicsignalWithperiodT=8,determinethecorrespondingsystemoutputy(t).Solution:3SignalandSystemFundamentalperiod.Because另：x(t)为实奇信号，则ak为纯虚奇函数，也可以得到a0为0。So.3.15Consideracontinuous-timeideallowpassfilterSwhosefrequencyresponseisWhentheinputtothisfilterisasignalx(t)withfundamentalperiodandFourierseriescoefficients,itisfoundthat.Forwhatvaluesofkisitguaranteedthat?Solution:for即对于所有的k，for4SignalandSystem也就是说,即12k<100，k<=8，故当k>8时，ak＝0。3.35.Consideracontinuous-timeLTIsystemSwhosefrequencyresponseisWhentheinputtothissystemisasignalx(t)withfundamentalperiodandFourierseriescoefficients,itisfoundthattheoutputy(t)isidenticaltox(t).Forwhatvaluesofkisitguaranteedthat?Solution:T=,.for,thatis,andisinteger,so.Let,,itneeds,for.Chap44.1UsetheFouriertransformanalysisequation(4.9)tocalculatetheFouriertransformsof;(a)(b)SketchandlabelthemagnitudeofeachFouriertransform.Solution:(a).5SignalandSystem(b).4.2UsetheFouriertransformanalysisequation(4.9)tocalculatetheFouriertransformsof:(a)(b)SketchandlabelthemagnitudeofeachFouriertransform.Solution:(a).(b).4.5UsetheFouriertransformsynthesisequation(4.8)todeterminetheinverseFouriertransformof,where6SignalandSystemUseyouranswertodeterminethevaluesoftforwhichx(t)=0.Solution:Ifx(t)=0thenThatis4.6Giventhatx(t)hastheFouriertransform,expresstheFouriertransformsofthesignalslistedbelowinthetermsof.YoumayfindusefultheFouriertransformpropertieslistedinTable4.1.(a)(b)(c)Solution:AccorrdingtothepropertiesoftheFouriertransform,we’llget:(a).7SignalandSystemthen(b).(c).4.11Giventherelationships，AndAndgiventhatx(t)hasFouriertransformandh(t)hasFouriertransform,useFouriertransformpropertiestoshowthatg(t)hastheformDeterminethevaluesofAandB.Solution:forandthen8SignalandSystem4.14Considerasignalx(t)withFouriertransform.Supposewearegiventhefollowingfacts:1.x(t)isrealandnonnegative.2.whereAisindependentoft.3..Determineaclosed-formexpressionforx(t).Solution:From(1),weknowisrealand;From(2),weknow:AndwealsoknowSo=ThatisFrom(3),weknow:ButSo=,thatis,While,Then9
