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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 C-T LTI systemsTransfer function and system propertiesImplementation structure for LTI syst

2、emss-domain analysis for LTI system responseThe differential equation describing a C-T LTI system in time-domaina ytayta y ta y tnnnn()()()()110()(1)b xtbxtb x tb x tmmmm()()()()110()(1)By the differentiation property of bilateral Laplace transform,we can obtain s-domain equation describing the LTI

3、system()a sasa sa Y snnnn1101b sbsbsb X smmmm()1101s-domain description for C-T LTI systemsX sH sY s()()()The system transfer function H(s)is defined asa sasa sab sbsbsbnnnnmmmm11011101a sasa sa Y snnnn()1101b sbsbsb X smmmm()1101Definition of system transfer functionH(s)can describe causal systems

4、and noncausal systemsH(s)is an important function describing the system in s-domain.The relationship between H(s)and h(t)X sH sY s()()()L LL Lth t()=()L Lh t()L Lh tH s()()-1)t(h)t(me t sySITL T-CThey are bilateral Laplace transform pairDefinition of system transfer functionFor a causal LTI system,i

5、ts impulse response h(t)is causal.In this case,the unilateral and bilateral Laplace transform of causal h(t)is identical.X sH sYs()()()zsTherefore,for a causal LTI system,H(s)can be obtained byunilateral Laplace transform of input x(t)and output y(t).L Lh t()If the system is causal,we can use unilat

6、eral Laplace transform to find its H(s).Definition of system transfer function by impulse response h(t):by the differential equation describing the system by input x(t)and output y(t):The main methods of determining H(s)L LL Lx tH sy t()()=()a sasa saH sb sbsbsbnnnnmmmm()11011101H(s)=L L h(t)Definit

7、ion of system transfer functionSolution:(1)The relationship of input and output for the integrator isTherefore,the transfer function H(s)for the integrator is y txxtt()()d=()(1)L LsH sh t()()1The impulse response h(t)is h ttu t()()()(1)Example 6.20:Determine H(s)for the ideal integrator and differen

8、tiator.()d()dy tx ttth tttd()=()d()L LH sh ts()()Solution:(1)The relationship of input and output for the differentiator isTherefore,the transfer function H(s)for the differentiator is The impulse response h(t)is Example 6.20:Determine H(s)for the ideal integrator and differentiator.Another Way?Exam

9、ple 6.21:In the RC circuit,input signal is x(t),output signal is y(t),and initial voltage y(0)is zero.Determine H(s)and h(t)。C)t(x+)t(yR-+The circuit is a causal LTI system,we can use unilateral Laplace transform to determine the transfer function H(s).X sH sYs()()()zsSolution:Cs/1)s(X+)s(YR-+s-doma

10、in circuit with y(0)=0Time-domain circuitBy the inversion of unilateral Laplace transform,we can get h(t)RC th tRCu t(1/)()1e()According to the circuit model in s-domain and Kirchhoffs voltage law,we can obtainX sH sYs()()()zssCRsC11/sRCRC1/1/Example 6.21:In the RC circuit,input signal is x(t),outpu

11、t signal is y(t),and initial voltage y(0)is zero.Determine H(s)and h(t)。Cs/1)s(X+)s(YR-+s-domain circuit with y(0)=0Example 6.22:the differential equation for a causal LTI system is as y(t)+3y(t)+2y(t)=3x(t)+2x(t)Determine its H(s)and h(t)。Solution:As the system is causal,we can apply the unilateral

12、 Laplace transform.The transfer function H(s)for the causal LTI system is By inversion of unilateral Laplace transform,we can determine h(t)ssYssX s2zs(32)()(23)()sssX sH sYs(3)2)()23(2zsss2e(s)11R,11h tu ttt()(ee)()2The largest real part of polesAcknowledgmentsMaterials 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 description for LTI systems