信号与系统信号与系统信号与系统 (3).pdf

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

2、able systemClassification of SystemsContinuous time system&discrete time systemLinear system&nonlinear systemTime invariant system&time variant systemCausal system&noncausal systemStable system&unstable systemClassification of SystemsContinuous time system&discrete time system Continuous time system

3、(C-T system)Input signals and output signals of the system are both C-T signals.C-T systemx(t)y(t)D-T systemxkyk Discrete time system(D-T system)Input signals and output signals of the system are both D-T signals.Such as these circuits composed of registers,flip-flops,counters,etc.C-T system均衡器-+)t(

4、Lv)t(u-+RLContinuous time system&discrete time systemD-T systemSuch as these circuits composed of resistors,inductors and capacitors.1G communication system1G mobile phoneContinuous time system&discrete time system4G mobile phone4G communication systemContinuous time system&discrete time systemLinea

5、r system&nonlinear systemTime invariant system&time variant systemCausal system&noncausal systemStable system&unstable systemClassification of SystemsLinear system:with the properties of superposition and homogeneity in term of the system input x(t)and the system output y(t).Homogeneity:Superpositio

6、n:n K x tK y tif x ty tthe)()(),()(ifx ty tx ty t ()()and ()()1122+tty tnx txythe()()()()1212Linear system&nonlinear systemFor the C-T system that is initially at rest.Only if a system satisfies the properties of superposition and homogeneity,the system is defined as a linear system.if x ty tx ty t

7、()()and ()()1122+x tx ty ty t()()()()1212where,are constant factors.For the C-T linear system that is initially at rest.Linear system&nonlinear systemthenme t sys T-Dme t sys T-Dme t sys T-DLinear system&nonlinear systemSimilarly,for the D-T linear system that is initially at rest.ifthen1kx1ky2kx2ky

8、12 x kx k+12 y ky k+Linear system:satisfies both the property of superposition and the property of homogeneity.Nonlinear system:violates either the principle of superposition or the principle of homogeneity.Linear system&nonlinear systemExample 1.6:Consider the C-T systems described by the input-out

9、put relation.For each system,determine whether it is linear.=+=+y tx ty txty tx ttx t2(1)()3()4(2)()4()(3)()2d()d()Solution:x ty tx t=+()()3()4Kx tKx t+()3()4The system violates the principle of homogeneity and is therefore nonlinear.Ky t()(1)Homogeneity?Linear system&nonlinear system=x ty txt2()()4

10、()Kx tKx t()4()2 Ky t()=Kxt224()Example 1.6:Consider the C-T systems described by the input-output relation.For each system,determine whether it is linear.=+=+y tx ty txty tx ttx t2(1)()3()4(2)()4()(3)()2d()d()Solution:（2）Homogeneity?The system violates the principle of homogeneity and is therefore

11、nonlinear.Linear system&nonlinear system=+tx ty tx tx td()()2()d()+Kx tKx ttKx t()2d()d()=+Kx ttx t2d()d()=Ky t()Example 1.6:Consider the C-T systems described by the input-output relation.For each system,determine whether it is linear.=+=+y tx ty txty tx ttx t2(1)()3()4(2)()4()(3)()2d()d()Solution:

12、（3）Homogeneity?Linear system&nonlinear systemThe system satisfies the principle of homogeneity.+x tx tx tx ttx tx t121212()()2d()()d()()=+x ty tx ttx t2222()()2d()d()=+x ty tx ttx t1111()()2d()d()=+x ttx tx ttx t11222d()d()2d()d()=+y ty t12()()=+=+y tx ty txty tx ttx t2(1)()3()4(2)()4()(3)()2d()d()L

13、inear system&nonlinear systemSolution:（3）Superposition?The system satisfies the principle of superposition.The C-T system is described by the input-output relationDetermine whether it is linear.NonlinearLinearB()4()y tx t=QuizAThe C-T system is described by the input-output relationDetermine whether

14、 it is linear.NonlinearLinearBd()()2dx ty tt=QuizANonlinearLinearBd()()2()3()dx ty tx tx tt=+QuizAThe C-T system is described by the input-output relationDetermine whether it is linear.Example 1.7:Determine whether the integrator and unit delayer are linear.)t(x=+x tx tx t12()()()=+-y txxxttd()()d()

15、()121-k x=k y k xDAn integrator system is linear12()d()dxxtt=+-=+y ty t12()()thenLinear system&nonlinear systemSolution:Composite signal:()()dty tx-=+x kx kx k12 =-=-+-ky kx kx kx 1 1112=+y ky k 12Example 1.7:Determine whether the integrator and unit delayer are linear.Linear system&nonlinear system

16、)t(x1-k x=k y k xDSolution:Composite signal:thenAn unit delayer system is linear()()dty tx-=The D-T system is described by the input-output relation(the moving-average system).Determine whether it is linear.NonlinearLinearB1 1234y kx kx kx kx k=+-+-+-QuizA=titR()(RR)t(Rv)t(Ri-+R)t(Lv)t(Li-+LC)t(Cv)t

17、(Ci-+=Lttit()d()dLL=-tiCt)1()d(CCConsider the linearity of a resistor,a capacitor and an inductor.If current i(t)as the input signal,voltage v(t)as the output signal.Linear system&nonlinear system=-y tAxt()()d=ty tAx td()d()=y tAx t()()=+itititRR1R2()()()For the resistor,Then=+ittRittiR()()()RR12RR=

18、+RitRitR1R2()()=+vtvtR1R2()()Therefore a resistor is linear.Linear system&nonlinear system=titR()(RR)t(Rv)t(Ri-+RConsider the linearity of a resistor,a capacitor and an inductor.If current i(t)as the input signal,voltage v(t)as the output signal.Composite signal:=+-tCiCiitt()d)()(d(11CCC1C2=+itititC

19、C1C2()()()C1C2()d()dCiCitt=+-=+vtvtC1C2()()For the capacitor,Then Therefore a capacitor is linear.Linear system&nonlinear systemConsider the linearity of a resistor,a capacitor and an inductor.If current i(t)as the input signal,voltage v(t)as the output signal.Composite signal:C)t(Cv)t(Ci-+=-tiCt)1(

20、)d(CC=+itititLL1L2()()()=+vtvtL1L2()()=+tLttLititti(d()ddd()()L1L2LL=+L1L2d()dd()dLittLittFor the inductor,Then Therefore an inductor is linear.Linear system&nonlinear systemConsider the linearity of a resistor,a capacitor and an inductor.If current i(t)as the input signal,voltage v(t)as the output

21、signal.Composite signal:)t(Lv)t(Li-+L=Lttit()d()dLL)t(Rv)t(Lv-+-+)t(i)t(x-+RL)t(x)t(Ci)t(Li)t(RiCLR)t(Cv+-All the electric systems consisting of resistors,capacitors and inductors are linear.Linear system&nonlinear systemC)t(Rv)t(Lv-+-+)t(i)t(x-+RLSeries RL circuitSeries RLC circuitParallel RLC circuitAcknowledgmentsMaterials 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