信号与系统信号与系统信号与系统 (13).pdf

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1、BEIJING JIAOTONG UNIVERSITYThe Course Group of Signals and Systems,Beijing Jiaotong University.P.R.CHINA.Copyright 2020Signals and Systems Complex frequency-domain analysis for systemss-domain description for LTI systemsTransfer function and system propertiesImplementation structure for LTI systemss

2、-domain analysis for LTI system responseh(t)H(s)x(t)yzs(t)=x(t)*h(t)X(s)Yzs(s)=X(s)H(s)The relation between H(s)and Yzs(s)s-domain analysis for LTI system responseTime domains-domainresponse Y(s)in s-domainequation in s-domaindifferential equationresponse y(t)solving the equation(convolution operati

3、on)s-domain analysis for LTI system responseunilateralLaplacetransformsolving the equation(algebraic operation)Inversion of unilateralLaplacetransformExample 6.29:The differential equation of a causal LTI system isttty tx ty ty tx tddd56()28()d()d()d()22input x(t)=e-tu(t),initial y(0-)=3 and y(0-)=2

4、,find y(t).by ULT,find the algebraic equation in s-domain solving the algebraic equation,obtain Yzi(s)and Yzs(s)by inversion of ULT,obtain yzi(t)and yzs(t).steps:s-domain analysis for LTI system responses Y ssyy-2()(0)(0)-ssssY sX ssyyys5656()()(0)(0)5(0)2822sX sX s2()8()Yzi(s)Yzs(s)y(t)5y(t)6y(t)x

5、tx t2()8()-sY sy5()(0)Y s6()ttty tx ty ty tx tddd56()28()d()d()d()22s-domain analysis for LTI system responsessYss56()3172zi L L-ytYsu ttt()()(11e8e)()zizi123 zero-input response yzi(t)-ssYssyyy56()(0)(0)5(0)2ziAccording to y(0-)=3，y(0-)=2-ss23118s-domain analysis for LTI system responseRe(s)-2sssYs

6、s561()2812zs-sss123341 zero-state response yzs(t)ssYsX ss56()()282zsAccording to input x(t)=e-tu(t),we can obtain X(s)sX s1()1s-domain analysis for LTI system responseRe(s)-1 L L-ytYsu tttt()()(3e4ee)()zszs123-y tytyttttt()()()3e7e7e,0zizs23-sssYs123()341zs zero-state response yzs(t)s-domain analysi

7、s for LTI system responseRe(s)-1tty tx ty ty tdd712()()d()d()22input x(t)=u(t),y(0-)=1 and y(0-)=2,find y(t).exercise 1answer-ytu ttt1234()(ee)()111zs34-ytu ttt()(6e5e)()zi34-y tttt1234()ee,01171934s-domain analysis for LTI system responsettty tx ty ty tx tddd44()3+2()d()d()d()22input x(t)=4u(t),y(0

8、-)=-2 and y(0-)=3,find y(t).tty ttt-zi22()2ee,0-yttu ttt()(28 e2e)()zs22-y ttttt()24e7 e,022s-domain analysis for LTI system responseexercise 2answerttty tx ty ty tx tddd48()3()d()d()d()22input x(t)=e-tu(t),y(0-)=5 and y(0-)=3,find y(t).-ytttttt2()5ecos(2)esin(2),013zi22-ytttu tttt()0.4e0.4ecos(2)1.

9、7esin(2)()zs22-y ttttttt()0.4e5.4ecos(2)8.2esin(2),022s-domain analysis for LTI system responseexercise 3answer(1)zero-input response yzi(t),zero-state response yzs(t).(2)transfer function H(s),impulse response h(t).(3)the structure diagram of the system H(s).y ty ty tx tx t()7()10()2()3()Input ,and

10、 .-tx tu t()e()-yy(0)1,(0)1Example 6.30:The differential equation of a causal LTI systemDetermine:Solution:By the unilateral Laplace transform(ULT)-s Y ssyysY syY ssX s2()(0)(0)7()7(0)10()(23)()-Y ssyyysssssX s22()(0)(0)7(0)71023710()Yssssss-zi2()87102215By inversion of ULTL L-ytYsttt()()2ee,0zizs12

11、5(1)zero-input yzi(t),zero-state response yzs(t).y ty ty tx tx t()7()10()2()3()According to initial state-yy(0)1,(0)1Re(s)-2-Y ssyyysssssX s22()(0)(0)7(0)71023710()-ssssssYss11710125()231/41/312/72zsL L-ytYsu tttt4312()()(eee)()117zszs125y tytytzizs()()()tttt-2514e73e1912e,0Solution:(1)zero-input yz

12、i(t),zero-state response yzs(t).By inversion of unilateral Laplace transformRe(s)-1Solution:As the system is causal,the transfer function H(s)isH sYsX sssszs2()()()23710ss-1/327/35 LL-h tH su ttt33()()(ee)()71152(2)transfer function H(s),impulse response h(t).ssYsX ss(710)()()232zsBy inversion of un

13、ilateral Laplace transformThe causal LTI system is stable.Re(s)-2Solution:Rewrite the transfer function H(s)as-ssH sss7)1(02311212)s(Y23-)s(X7011-s1-s(3)the structure diagram of the system H(s)ssH ss710()232AcknowledgmentsMaterials 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.s-domain analysis for LTI system response