信号与系统 课件 英文版6

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, nopoles of X(s) are contained in the ROC.Property8: If the Laplace transform X(s) For ratio

6、nal Laplace transform, the ROC does not contain any poles.Property3: If x(t) is of finite duration and is absolutely integrable, then the ROC is the entire s- plane9 The Laplace TransformProperty7: If the Laplace transform X(s) of x(t) is rational, then its ROC is bounded by poles or extends to infi

7、nity. In addition, no poles of X(s) are contained in the ROC.Property8: If the Laplace transform X(s) of x(t) 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

8、the leftmost pole.Example 9.7 9.8 9 The Laplace Transform Appendix Partial Fraction ExpansionConsider a fraction 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 TransformMultiply 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) Integ