3. Frequency-Domain Analysis of Continuous-Time Signals and Systems .pdf

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1、3.Frequency-DomainAnalysisofContinuous-TimeSignalsandSystems3.1.DefinitionofContinuous-TimeFourierSeries(3.3-3.4)3.2.PropertiesofContinuous-TimeFourierSeries(3.5)3.3.DefinitionofContinuous-TimeFourierTransform(4.0-4.2)3.4.PropertiesofContinuous-TimeFour

2、ierTransform(4.3-4.6)3.5.FrequencyResponse(3.2,3.8,4.4)3.6.LinearConstant-CoefficientDifferentialEquations(4.7)3.1.DefinitionofContinuous-TimeFourierSeriesAcontinuous-timesignalx(t)withperiodTcanberepresentedbyacontinuous-timeFourierseries,i.e.,2x(t

3、)X(k)expjkt,(3.1)kTwhereX(k)isgivenby12X(k)x(t)expjktdt.(3.2)TTTX(k)iscalledthespectrumofx(t).(3.1)and(3.2)showthatacontinuous-timeperiodicsignalcanbedecomposedintoasetofcontinuous-timeelementarysignals.Anycontinuous-timeelementarysig

4、nal,X(k)exp(j2kt/T),isperiodicandhasthefrequency2k/TandthecoefficientX(k).3.1.1.DerivationofContinuous-TimeFourierSeriesAssumethatx(t)canberepresentedby(3.1).WeshowthatX(k)isgivenby(3.2).Substitutingkforkin(3.1),weobtain2x(t)X(k)expjkt.(3.3

5、)kTNext,(3.3)ismultipliedbyexp(j2kt/T),integratedoveroneperiodanddividedbyT.Thatis,12x(t)expjktdtTTT122X(k)expjktexpjktdt.(3.4)TTkTTChangingtheorderoftheintegrationandthesummationontherightsideof(3.4),weob

6、tain12x(t)expjktdtTTT12X(k)expj(kk)tdt.(3.5)TTTkSince121,kkTexpj(kk)tdt,(3.6)TT0,kktherightsideof(3.5)equalsX(k)and(3.2)isderived.3.1.2.ConvergenceofContinuous-TimeFourierSeriesTheintegralin(3.2)convergeswhen

7、thefollowingconditionsaresatisfied.(1)Inanyperiod,x(t)isabsolutelyintegrable.Thatis,thereexistsafiniteconstantBsuchthat

8、x(t)

9、dtB.(3.7)T(2)Inanyperiod,x(t)hasafinitenumberofmaximaandminima.(3)Inanyperiod,x(t)hasafinitenumberofdiscontinuities,andhasboth

10、theleft-sidedlimitandtheright-sidedlimitateachofthesediscontinuities.TheaboveconditionsarecalledtheDirichletconditions.Itshouldbenotedthattheyaresufficientfortheconvergenceoftheintegralin(3.2)butunnecessary.SupposethattheDirichle