线性非时变系统的时域描述 (4).pdf

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1、BEIJING JIAOTONG UNIVERSITYThe Course Group of Signals and Systems,Beijing Jiaotong University.P.R.CHINA.Copyright 2020Continuous time system&discrete time systemLinear system&nonlinear systemTime invariant system&time variant systemCausal system&noncausal systemStable system&unstable systemClassifi

2、cation of SystemsContinuous time system&discrete time systemLinear system&nonlinear systemTime invariant system&time variant systemCausal system&noncausal systemStable system&unstable systemClassification of SystemsTime invariant system:if a time delay or time advance of the input signal leads to an

3、 identical time shift in the output signal.The characteristics of the system do not change with time.x ty t()()if x ttytt0zs0()()thenx ky k ifx kny knthenTime invariant system&time variant systemFor the C-T system that is initially at rest.For the D-T system that is initially at rest.0tt01t010t+20tt

4、01t021)0tt(y)t(y)0tt(x)t(xme t sys tna i ravni emiTTime invariant system&time variant systemTime invariant system:if a time delay or time advance of the input signal leads to an identical time shift in the output signal.Otherwise,the system is said to be time variant.=x tittR0()()If the resistance R

5、 is a constantTime invariant=ttxyR)(=vttR0()If the resistance R(t)changes with time.=R t itt()()R0vttR0()Time invariant system&time variant system=titR()(RR)t(Rv)t(RiRthen=Ritt()R0when=x tittR0()()when=ttxyR t)()(thenTime variant=Rittt()()RRA resistor is time invariant,if the resistance R does not c

6、hange with time.=Cy titt()()d1C0=Cit t()d1C0=vttC0()=x tittC0()()=C ty txt()()()d1=C tit t()()d1C0vttC0()Time invariant system&time variant systemC)t(Cv)t(Ci=tiCt)1()d(CCIf the capacitance C is a constantTime invariantthenwhenIf the capacitance C(t)changes with time.whenthenTime variant=x tittC0()()

7、=tC tit()()d1CCA capacitor is time invariant,if the capacitance C does not change with time.=x tittL0()()=L0()d()dy tLittt=vttL0()vttL0()=L0()()d()dy tL titttTime invariant system&time variant system)t(Lv)t(LiL=Lttit()d()dLL=ttLtti()d()dLLIf the inductance L is a constantTime invariantthenwhenIf the

8、 inductance L(t)changes with time.whenthenTime variant=x tittL0()()An inductor is time invariant,if the inductance L does not change with time.All the electric systems consisting of the RLC,which do not change with time,are time invariant.)t(Rv)t(Lv)t(i)t(xRL)t(x)t(Ci)t(Li)t(RiCLR)t(Cv+)t(2RvW2=2R)t

9、(uW1=1RTime invariant systemsTime invariant system&time variant systemExample 1.8:The systems are described by the input-output relationDetermine whether these system are time invariant.(1)y(t)=2x(t)(2)y(t)=cos(t)x(t)(3)yk=k xk k y k xme t sys T-Dnk ynk xme t sys T-DTime invariant system&time varian

10、t system)0tt(y)0tt(xme t sys T-C)t(y)t(xme t sys T-CSolution:(1)y(t)=2x(t)(2)y(t)=cos(t)x(t)(3)yk=k xk(1)=x tty tx tt010()()2()=y tt0()=x tty ttx tt()()cos()()010y tt0()=x kny kk x kn 1y knTime invariant system(2)(3)Generally,if there exist the variable t or its function except x(t)for the C-T syste

11、m,then the system is time variant.Time invariant system&time variant systemSolution:Time variant systemTime variant system=x tx tt10()()=y txtt()()d10An integrator is time invariant.=xt t()d0=y tt0()Example 1.9:Determine whether the integrator and unit delayer are time invariant.)t(x1k x=k y k xDSol

12、ution:ifthenTime invariant system&time variant system()()dty tx=x kx kk10 y kx kk=10 1An unit delayer is time invariant.=y kk0Example 1.9:Determine whether the integrator and unit delayer are time invariant.)t(x1k x=k y k xDSolution:ifthenTime invariant system&time variant system()()dty tx=Continuou

13、s time system&discrete time systemLinear system&nonlinear systemTime invariant system&time variant systemCausal system&noncausal systemStable system&unstable systemClassification of SystemsCausal system&noncausal systemCausal system:if the present value of the output signal depends only on the prese

14、nt or past values of the input signal.In other words,the output signal of the system does not advance the input signal of the system.t01)t(y0tt01)0tt(yt021)t(xme t sys l asuaCt01)t(yt021)t(xme t sySl asuacnoNCausal system&noncausal systemNoncausal system:if the present value of the output signal dep

15、ends on one or more future values of the input signal.In other words,the output signal of the system advances the input signal of the system.The D-T system is described by the input-output relation(the moving-average system).Determine whether it is causal.CausalNoncausalB1 1234y kx kx kx kx k=QuizAC

16、ontinuous time system&discrete time systemLinear system&nonlinear systemTime invariant system&time variant systemCausal system&noncausal systemStable system&unstable systemClassification of SystemsStable system&unstable systemStable system:if and only if every bounded input of the system results in

17、a bounded output(BIBO).tO1)t(xt0)t(yme t syse lba tS1)t(xt0me t sySe lba t snU)t(ytOStable system&unstable systemUnstable system:when the input of the system is bounded,the output of the system may be not bounded.Example 1.10:The system is described by the input-output relationShow that the system i

18、s unstable.y(t)=e2t x(t)Solution:Stable system&unstable systemAssume that the input signal x(t)is bounded|x(t)|Mx We can find that|y(t)|=|e2t x(t)|e2t|MxWhen the input x(t)is bounded,the output y(t)diverges for increasing t.Continuous time system&discrete time systemLinear system&nonlinear systemTim

19、e invariant system&time variant systemCausal system&noncausal systemStable system&unstable systemClassification of SystemsAcknowledgmentsMaterials used here are accumulated by authors for years with helpfrom colleagues,media or other sources,which,unfortunately,cannotbe noted specifically.We gratefully acknowledge those contributors.Classification of Systems