3. Frequency-Domain Analysis of Continuous-Time Si

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-TimeFourierTransform(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.,X(k)iscalledthespectrumofx(t).(3.1)and(3.2)showthatacontinuous-timeperiodicsignalcanbedecomposedintoasetofcontinuous-timeelementarysignals.Anycontinuous-timeelementarysignal,X(k)exp(j2πkt/T),isperiodicandhasthefrequency2πk/TandthecoefficientX(k).,ktT2jexp)k(X)t(xk∑∞-∞=⎟⎠⎞⎜⎝⎛π=.dtktT2jexp)t(xT1)k(XT∫⎟⎠⎞⎜⎝⎛π-=whereX(k)isgivenby(3.1)(3.2)3.1.1.DerivationofContinuous-TimeFourierSeriesAssumethatx(t)canberepresentedby(3.1).WeshowthatX(k)isgivenby(3.2).Substitutingk′forkin(3.1),weobtain.tkT2jexp)k(X)t(xk∑∞-∞=′⎟⎠⎞⎜⎝⎛′π′=Next,(3.3)ismultipliedbyexp(-j2πkt/T),integratedoveroneperiodanddividedbyT.Thatis,(3.3)∫∑∫⎟⎠⎞⎜⎝⎛π-⎥⎦⎤⎢⎣⎡⎟⎠⎞⎜⎝⎛′π′=⎟⎠⎞⎜⎝⎛π-∞-∞=′TkT.dtktT2jexptkT2jexp)k(XT1dtktT2jexp)t(xT1(3.4)Changingtheorderoftheintegrationandthesummationontherightsideof(3.4),weobtain.dtt)kk(T2jexpT1)k(XdtktT2jexp)t(xT1TkT∫∑∫⎥⎦⎤⎢⎣⎡-′π′=⎟⎠⎞⎜⎝⎛π-∞-∞=′(3.5)Since,kk0,kk,1dtt)kk(T2jexpT1T⎩⎨⎧≠′=′=⎥⎦⎤⎢⎣⎡-′π∫(3.6)therightsideof(3.5)equalsX(k)and(3.2)isderived.3.1.2.ConvergenceofContinuous-TimeFourierSeriesTheintegralin(3.2)convergeswhenthefollowingconditionsaresatisfied.(1)Inanyperiod,x(t)isabsolutelyintegrable.Thatis,thereexistsafiniteconstantBsuchthat.Bdt|)t(x|T≤∫(3.7)(2)Inanyperiod,x(t)hasafinitenumberofmaximaandminima.(3)Inanyperiod,x(t)hasafinitenumberofdiscontinuities,andhasboththeleft-sidedlimitandtheright-sidedlimitateachofthesediscontinuities.TheaboveconditionsarecalledtheDirichletconditions.Itshouldbenotedthattheyaresufficientfortheconvergenceoftheintegralin(3.2)butunnecessary.SupposethattheDirichletconditionsaresatisfiedandtheintegralin(3.2)converges.Then,theseriesin(3.1)convergesbutmaynotconvergetox(t)everywhere.Atthecontinuities,itconvergestox(t),butatthediscontinuities,itconvergestotheaverageoftheleft-sidedlimitandtheright-sidedlimitofx(t).Example.DeterminetheFourierseriescoefficientsforeachofthefollowingsignals:(1)x(t)=sin(ω0t).(2)x(t)=1+sin(ω0t)+2cos(ω0t)+cos(2ω0t+π/4).(3)Overperiod[-T/2,T/2),x(t)isdefinedas.2/|t|0,2/|t|,1x(t)⎩⎨⎧ττ≤=3.2.PropertiesofContinuous-TimeFourierSeries3.2.1.LinearitySupposethatx1(t)andx2(t)havethesameperiod,anda1anda2aretwoarbitraryconstants.Ifx1(t)↔X1(k)andx2(t)↔X2(k),then.)nTt()t(x(4)n∑∞-∞=-δ=a1x1(t)+a2x2(t)↔a1X1(k)+a2X2(k).(3.8)3.2.2.DifferentiationIfx(t)↔X(k),then).k(kXT2jdt)t(dxπ↔(3.9)3.2.3.ShiftingIfx(t)↔X(k),then,ktT2jexp)k(X)tt(x00⎟⎠⎞⎜⎝⎛π-↔-(3.10)wheret0isanarbitraryrealnumber.Ifx(t)↔X(k),then),kk(XtkT2jexp)t(x00-↔⎟⎠⎞⎜⎝⎛π(3.11)wherek0isanarbitraryinteger.3.2.4.ScalingIfx(t)↔X(k),thenx(at)↔X[sgn(a)k],(3.12)whereaisanonzerorealnumber.Whena0,(3.12)becomesx(at)↔X(k).(3.13)Thus,onlyX(k)cannotspecifyx(t)uniquely.Tisalsorequired.Lettinga=-1in(3.12),weobtainx(-t)↔X(-k),(3.14)i.e.,thereversalpropertyofthecontinuous-timeFourierseries.From(3.14),thefollowingconclusionscanbedrawn:(1)x(t)even⇔X(k)even.(2)x(t)odd⇔X(k)odd.3.2.5.ConjugationIfx(t)↔X(k),thenx*(t)↔X*(-k).(3.15)From(3.15),thefollowingconclusionscanbedrawn:(1)Im[x(t)]=0⇔X(k)=X*(-k).(2)Re[x(t)]=0⇔X(k)=-X*(-k).(3)Im[X(k)]=0⇔x(t)=x*(-t).(4)Re[X(k)]=0⇔x(t)=-x*(-t).3.2.6.ConvolutionAssumethatx1(t)andx2(t)havethesameperiodT.Ifx1(t)↔X1(k)),k(X)k(TXd)t(x)(x21T21↔ττ-τ∫(3.16),)mk(X)m(X)t(x)t(xm2121∑∞-∞=-↔(3.17)wheretheintegraliscalledtheperiodicconvolutionintegralofx1(t)andx2(t).Ifx1(t)↔X1(k)andx2(t)↔X2(k),thenwherethesumiscalledtheconvolutionsumofX1(k)andX2(k).3.2.7.Parseval’sEquationIfx(t)↔X(k),then.)k(Xdt|)t(x|T1k2T2∑∫∞-∞==(3.18)andx2(t)↔X2(k),then3.3.DefinitionofContinuous-TimeFourierTransformAcontinuous-timesignalx(t)canberepresentedbyacontinuous-timeFourierintegral,i.e.,(),dtjexp)(X21)t(x∫∞∞-ωωωπ=(3.19)(3.20)iscalledthecontinuous-timeFouriertransform,and(3.19)iscalledtheinversecontinuous-timeFouriertransform.X(ω)iscalledthespectrumofx(t).Itcanbeseenfrom(3.19)-(3.20)thatacontinuous-timesignalcanbedecomposedintoasetofcontinuous-timeelementarysignals.Anycontinuous-timeelementarysignal,X(ω)exp(jωt)dω/(2π),isperiodicandhasthefrequencyωandthecoefficientX(ω)dω/(2π).().dttjexp)t(x)(X∫∞∞-ω-=ωwhereX(ω)isgivenby(3.20)3.3.1.DerivationofContinuous-TimeFourierTransformAssumethatx′(t)isx(t)extendedwithperiodT.Then,.ktT2jexpdkT2jexp)(xT1)t(xkT∑∫∞-∞=⎟⎠⎞⎜⎝⎛π⎥⎦⎤⎢⎣⎡τ⎟⎠⎞⎜⎝⎛τπ-τ′=′(3.21)Letting2π/T=Δω,weobtainLettingΔω→0,weobtain()().tjkexpdjkexp)(x21)t(xk2∑∫∞-∞=ωΔπωΔωΔ⎥⎦⎤⎢⎣⎡τωτΔ-τ′π=′(3.22)()().dtjexpdjexp)(x21)t(x∫∫∞∞-∞∞-ωω⎥⎦⎤⎢⎣⎡τωτ-τπ=(3.23)(3.23)showsthatx(t)canbeexpressedas(),dtjexp)(X21)t(x∫∞∞-ωωωπ=(3.24)().dttjexp)t(x)(X∫∞∞-ω-=ωwhere(3.25)3.3.2.ConvergenceofContinuous-TimeFourierTransformWhenthefollowingconditionsaresatisfied,theintegralin(3.20)converges.(1)Over(-∞,∞),x(t)