电子科大信号系统英文版课件Chapter9 LaplaceTransform-B

1、9 The Laplace Transform 9. The Laplace Transform 9.1 The Laplace Transform (1) Definition(2) Region of Convergence ( ROC )ROC: Range of for X(s) to convergeRepresentation: A. Inequality B. Region in S-plane9 The Laplace TransformExample for ROCReReS-planeS-planeImIm-a-a9 The Laplace Transform (3) Re

2、lationship between Fourier and Laplace transform Example 9.1 9.2 9.3 9.5 9 The Laplace Transform 9.2 The Region of Convergence for Laplace TransformProperty1: The ROC of X(s) consists of strips parallel to j-axis in the s-plane.Property2: For rational Laplace transform, the ROC does not contain any

3、poles.Property3: If x(t) is of finite duration and is absolutely integrable, then the ROC is the entire s- plane9 The Laplace TransformProperty4: If x(t) is right sided, and if the line Res=0 is in the ROC, then all values of s for which Res0 will also in the ROC. 9 The Laplace TransformProperty5: I

4、f x(t) is left sided, and if the line Res=0 is in the ROC, then all values of s for which Res0 will also in the ROC. x(t)T2te-0te-1t9 The Laplace TransformProperty6: If x(t) is two sided, and if the line Res=0 is in the ROC, then the ROC will consist of a strip in the s-plane that includes the line

5、Res=0 . 9 The Laplace TransformS-planeReReReImImImRLLR9 The Laplace TransformProperty7: If the Laplace transform X(s) of x(t) is rational, then its ROC is bounded by poles or extends to infinity. In addition, no poles of X(s) are contained in the ROC.Property8: If the Laplace transform X(s) of x(t)

6、is rational, then if x(t) is right sided, the ROC is the region in the s-plane to the right of the rightmost pole. If x(t) is left sided, the ROC is the region in the s-plane to the left of the leftmost pole.Example 9.7 9.8 9 The Laplace Transform Appendix Partial Fraction ExpansionConsider a fracti

7、on polynomial:Discuss two cases of D(s)=0, for distinct root and same root.9 The Laplace Transform(1) Distinct root:thus9 The Laplace TransformCalculate A1 : Multiply two sides by (s-1):Let s=1, so Generally9 The Laplace Transform(2) Same root:thusFor first order poles:9 The Laplace TransformMultipl

8、y two sides by (s-1)r : For r-order poles:So 9 The Laplace Transform 9.3 The Inverse Laplace TransformSo9 The Laplace TransformThe calculation for inverse Laplace transform:(1) Integration of complex function by equation.(2) Compute by Fraction expansion. General form of X(s):Important transform pai

9、r:Example 9.9 9.10 9.11 9 The Laplace Transform 9.4 Geometric Evaluation of the Fourier Transform from the Pole-Zero PlotGeneral form of X(s):Relation between Fourier and Laplace transform:9 The Laplace Transformor9 The Laplace Transform 9.4.1 First-order SystemPole-zero plot:System function of firs

10、t-order system:9 The Laplace Transform9 The Laplace Transform 9.4.2 All-Pass SystemPole-zero plot:System function :9 The Laplace TransformFrequency response :9 The Laplace Transform 9.5 Properties of the Laplace Transform 9.5.1 Linearity of the Laplace TransformExample 9.13 9 The Laplace Transform 9

11、.5.2 Time Shifting9 The Laplace Transform 9.5.3 Shifting in the s-Domainr1r2r2+Re(s0)r1+Re(s0)9 The Laplace Transform9 The Laplace Transform 9.5.4 Time ScalingEspeciallyr1r2r1/ar2/ar2/ar1/aS-plane9 The Laplace Transform 9.5.5 ConjugationWhen x(t) is real, X(s)=X*(s*) 9.5.6 Convolution Property9 The

12、Laplace Transform 9.5.7 Differentiation in the Time Domain9 The Laplace Transform 9.5.8 Differentiation in the s-DomainExample 9.14 9.159 The Laplace Transform 9.5.9 Integration in the Time-DomainUnder the specific constrains that x(t)=0 for t0 contains no impulses or highter order singularities at

13、the origin,Initial-value theorem:Final-value theorem:9 The Laplace Transform 9.5.10 The Initial- and Final-Value TheoremsExample 9.16Page 691: Table 9.19 The Laplace Transform 9.5.11 Table of PropertiesPage 692: Table 9.2 9.6 Some Laplace Transform Pairs9 The Laplace Transform9.7 Analysis and Charac

14、terization of LTI Systems Using the Laplace TransformSystem output: Y(s)=H(s)X(s) LTI system x(t)y(t)H(s) - System function ( Transfer/transition function )9 The Laplace Transform9.7.1 Causality(1) The ROC associated with the system function for a causal system is a right-half plane.(2) For a system

15、 with a rational system function, causality of the system is equivalent to the ROC being the right-half plane to the right of the rightmost pole.Causal LTI system: h(t)=0 for t0 .Example 9.17 9.18 9.199 The Laplace Transform9.7.2 Stability(1) An LTI system is stable if and only if the ROC of its sys

16、tem function H(s) includes the j-axis.(2) A causal system with rational system function H(s) is stable if only if all of the poles of H(s) lie in the left-half of the s-plane - I.e., all of the poles have negative real parts.Example 9.20 9.219 The Laplace Transform9.7.3 LTI Systems Characterized by

17、Linear Constant-Coefficient Differential EquationsDifferential equation:Example 9.24so9 The Laplace Transform9.7.4 Examples Relating System Behavior to the System FunctionExample 9.25 9.26 9 The Laplace Transform9.8 System Function Algebra and Block Diagram Representation9.8.1 System Function for In

18、terconnections of LTI Systems9 The Laplace Transform(1) Parallel interconnectionFor overall system: h(t)=h1(t)+h2(t) and H(s)=H1(s)+H2(s)9 The Laplace Transform(2) Series interconnectionFor overall system: h(t)=h1(t)*h2(t) and H(s)=H1(s)H2(s)9 The Laplace Transform(3) Feedback interconnectionFor ove

19、rall system:9 The Laplace Transform9.8.2 Block Diagram Representations for Causal LTI Systems Described by Differential Equations and Rational System FunctionsBasic elements: (1) Integrator (2) Amplifier (3) Adder1sk9 The Laplace TransformBlock Diagram construction: (1) Direct form (2) Parallel form : H(s) = H1(s) + H2(s) (3) Series form : H(s) =