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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 responseTransfer function and system impulse responseTransfer function and system frequency responseTransfer function and system stabilityTransfer function and system propertiesTransfer function and system impulse responsemmmmnnnH sb sbsb sbsasa sa()11101110Kszszszsps

3、pspmn1212()()()()()()N sD s()()pole-zero gain formpole-zero plotare zeros12,zzzmare poles12,pppn poles and zeros of H(s)poleszeross-planesjwpole-zero plot2jws3Example 6.23:The pole-zero plot for H(s)is shown in picture,and H()=1.Determine the transfer function H(s).sH ss3()2spH sKsz()()()110Thus K=1

4、 H()1AsSolution:sjw0u(t)etu(t)etu(t)1111s 111ss 111(1)simple poles at saxis Impulse response h(t)corresponding to H(s)Transfer function and system impulse responsesjw011sin(t)etu(t)sin(t)et u(t)sin(t)u(t)ss1(j)(j)ss(1j)(1j)1 ss(1j)(1j)1jj(2)conjugate complex poles1 j1 jTransfer function and system i

5、mpulse response Impulse response h(t)corresponding to H(s)wwHH ss(j)()jFor a stable system,we can obtain frequency response H(jw)The transfer function of a stable system is Its frequency response H(jw)issH ss1()wwwwHH ssj1(j)()jjTransfer function and system frequency response Frequency response H(jw

6、)corresponding to H(s)spH sKsziinjjm()()()11If H(s)is in pole-zero gain formWhen the system is stable,we can determine its frequency responsewwwpHKziinjjm(j)(j)(j)11Transfer function and system frequency response Frequency response H(jw)corresponding to H(s)Necessary and sufficient condition that a

7、LTI system is stable:ROC of transfer function H(s)includes jwaxis in s-plane.stable systemstable systemunstable systemTransfer function and system stabilityoswj0soswjoswjSolution:ssH s21()42Example 6.24:Analyze the stability and causality of the LTI system by ROC.ssH ss(2)(1)()6sRe()2 s1Re()2 sRe()1

8、It includes two simple poles 1,2.0swj220swj10swj1（1）,the system is unstable and causal.sRe()2h tu tu ttt()4e()2e()2（2）,the system is stable and noncausal.s1Re()2 h tutu ttt()4e()2e()2（3),the system is unstable and noncausal.sRe()1 h tututtt()4e()2e()2Solution:Example 6.24:Analyze the stability and c

9、ausality of the LTI system by ROC.ssH s21()42L Lsu tste(),Re()1L Lsutste(),Re()1 For causal LTI systems,if all poles of H(s)are located in the left half of the s-plane,then the LTI system is BIBO stable.For arbitrary LTI systems,if the ROC of H(s)includes jwaxis in s-plane,then the LTI system is BIB

10、O stable.Transfer function and system stabilityExample 6.25:Are these causal LTI systems stable?Solution:ssH sss(1)(2)(1)(),Re()131wsHsss(2)(),Re()02220(1)It has poles at p1=-1 and p2=2,both located in left half of the s-plane.Thus,the causal LTI system is stable.As the systems are causal systems,we

11、 can determine the stability according to the pole locations of H(s)in s-plane.(2)It has poles at p1=-jw0and p2=jw0,not located in left half of the s-plane.Thus,the causal LTI system is unstable.Example 6.26:For a causal LTI system,its zero-state response yzs(t)is,and input signal is x(t)=u(t).Deter

12、mine the transfer function H(s),the system differential equation,impulse response h(t),and its stability.Solution:The unilateral Laplace transform of yzs(t)and x(t):The system transfer function H(s)is ssss ssYss12(1)(2)(),0.511.521zss Re()0sX s(),1s Re()0ttytu tzs2()(0.5e1.5e)()X sssH ssYs()(1)(2)()

13、,21()zs sRe()1X sssH sYss()32(),()212zss Re()1Example 6.26:For a causal LTI system,its zero-state response yzs(t)is,and input signal is x(t)=u(t).Determine the transfer function H(s),the system differential equation,impulse response h(t),and its stability.Solution:ttytu tzs2()(0.5e1.5e)()According t

14、o H(s),the s-domain description of the system isssYssX s2zs(32)()(21)()By inversion of Laplace transform,obtain the differential equation.y ty ty tx tx t()3()2()2()()By partial fraction expansion of H(s),we can obtainBy inversion of Laplace transform,impulse response h(t)isssssH ss3221(),21312s Re()

15、1h tu ttt()(3ee)()2Example 6.26:For a causal LTI system,its zero-state response yzs(t)is,and input signal is x(t)=u(t).Determine the transfer function H(s),the system differential equation,impulse response h(t),and its stability.Solution:ttytu tzs2()(0.5e1.5e)()As the poles are at p1=1 and p2=2,loca

16、ted in the left half of s-plane,the causal LTI system is stable.AcknowledgmentsMaterials 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.Transfer function and system properties