信号与系统课件教学作者2 Chapter 4 The Frequency-Domain Analysis for Continuous Signals and Systems.ppt

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1、Chapter4TheFrequency-DomainAnalysisforContinuousSignalsandSystemsξ4.1FourierSeriesRepresentationξ4.1.1LinearCombinationsofHarmonicallyRelatedComplexExponentialf(t)=f(t+T)Ω=2π/TIff(t)isrealf(t)=f*(t)Fn=F-n*=

2、Fn

3、ejΨThenEx4.1.1:F0=1/2F1=F-1=1/πF2=F-2=0F3=F-3=-1/(3

4、π)F4=F-4=0F5=F-5=1/(5π)f(t)=?ξ4.1.2DeterminationofFourierSeries=FkTn=0,1,2..f(t):realn=1,2,…Fn=(an-jbn)/2Ex4.1.2:f1(t)=Acosωt,f2(t)=Asinωtf1(t),f2(t)FourierSeries?Ex4.1.4:f(t)=A(1+BcosΩt)cosωtω=5Ωf(t)FourierSeries?Solution:f(t)=Acosωt+½ABcos(ω-Ω)t+½ABcos(ω+Ω)tE

5、x4.1.3:f(t)=1+sin(wt)+2coswt+cos(2wt+π/4)f(t)FourierSeries?Ex4.1.5:f(t)asfigure,f(t)FourierSeries?Solution:Fk=sin(kπ/2)/(kπ)f(t)tT/4T/2-T/4-T/219ξ4.1.3PropertiesofFourierSeries3.TimeReversalf(-t)F-nf1(t)F1nf2(t)F2naf1(t)+bf2(t)Fn=aF1n+bF2n2.TimeShiftingf(t-τ)e-

6、jnΩτFn1.Linearityf(1)(t)jnΩFn4.Differentiation5.Realf(t)realandoddFnpurelyimaginaryandoddf(t)realFn=F*-nf(t)realandevenFnrealandevenξ4.2SpectraofPeriodicSignalξ4.2.1SpectraofPeriodicSignal

7、Fn(ω)

8、MagnitudeSpectraψ(ω)PhaseSpectraξ4.2.2CharacterofSpectraofPeriodic

9、Signala.Discreteω=nΩc.Convergenceb.Harmonicω=0ω=Ωω=2Ωω=nΩξ4.2.3PowerofPeriodicSignalξ4.3TheContinuous-TimeFourierTransformforAperiodicSignalsξ4.3.1FourierTransformω=nΩdω=ΩAfinitenumberofdiscontinuities:SufficientCondition:ξ4.3.2BasicFourierPairsδ(t)112πδ(ω)ejω0

10、t2πδ(ω-ω0)e-atε(t)1/(a+jω)(a>0)ε(t+τ/2)-ε(t-τ/2)τSa(ωτ/2)10ξ4.4TheFourierTransformforperiodicSignalsEx4.4.1:f(t)=cosω0tF(jω)=?ωF(jω)πω0-ω00πξ4.5PropertiesofTheFourierTransform1.Linearity2.TimeShiftingEx4.5.1:f(t)=δ(t+t0)F(jω)=?3.DifferentiationandIntegrationdf(

11、t)/dt→jωF(jω)∫-∞tf(τ)/dτ→1/jωF(jω)+πF(0)δ(ω)Ex4.5.2:f(t)=ε(t)F(jω)=?4.TimeandFrequencyScalingf(at)→1/

12、a

13、*F(jω/a)Ex4.5.3:f(t)asfigureg(t)=f(2t)G(jω)=?f(t)→τSa(ωτ/2)g(t)→1/2τSa(ωτ/4)5.Duality-jtf(t)→dF(jω)/dωf(t)→F(jω)F(jt)→2πf(-ω)f(t-t0)→e-jωt0F(jω)ejω0tf(t)→F(j

14、(ω-ω0))df(t)/dt→jωF(jω)Ex4.5.4:p(t)=gτ(t)·cosω0tP(jω)=?Solution:P(jω)=1/2τSa((ω-ω0)τ/2)+1/2τSa((ω+ω0)τ/2)6.Realf(t)realF(jω)=F*(-jω)f(t)realandevenF(jω)realandevenf(t)realan