(2) Region of Convergence ( ROC ) 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 converge Representation: A. Inequality B. Region in S-plane

9 The Laplace Transform Example for ROC Re S-plane Im -a

(3) Relationship between Fourier and Laplace transform 9 The Laplace Transform (3) Relationship between Fourier and Laplace transform Example 9.1 9.2 9.3 9.5

9.2 The Region of Convergence for Laplace Transform 9 The Laplace Transform 9.2 The Region of Convergence for Laplace Transform Property1: 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 poles. Property3: If x(t) is of finite duration and is absolutely integrable, then the ROC is the entire s- plane

Property4: If x(t) is right sided, and if the line Re{s}=0 9 The Laplace Transform Property4: If x(t) is right sided, and if the line Re{s}=0 is in the ROC, then all values of s for which Re{s}>0 will also in the ROC.

Property5: If x(t) is left sided, and if the line Re{s}=0 9 The Laplace Transform Property5: If x(t) is left sided, and if the line Re{s}=0 is in the ROC, then all values of s for which Re{s}<0 will also in the ROC. x(t) T2 t e-0t e-1t

Property6: If x(t) is two sided, and if the line Re{s}=0 9 The Laplace Transform Property6: If x(t) is two sided, and if the line Re{s}=0 is in the ROC, then the ROC will consist of a strip in the s-plane that includes the line Re{s}=0 .

9 The Laplace Transform S-plane Re Im R L

Property7: 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) For rational 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- plane

Property7: 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) 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

Consider a fraction polynomial: 9 The Laplace Transform Appendix Partial Fraction Expansion Consider a fraction polynomial: Discuss two cases of D(s)=0, for distinct root and same root.

9 The Laplace Transform (1) Distinct root: thus

Multiply two sides by (s-1): 9 The Laplace Transform Calculate A1 : Multiply two sides by (s-1): Let s=1, so Generally

9 The Laplace Transform (2) Same root: thus For first order poles:

Multiply two sides by (s-1)r : 9 The Laplace Transform For r-order poles: Multiply two sides by (s-1)r : So

9.3 The Inverse Laplace Transform 9 The Laplace Transform 9.3 The Inverse Laplace Transform So

The calculation for inverse Laplace transform: 9 The Laplace Transform The calculation for inverse Laplace transform: (1) Integration of complex function by equation. (2) Compute by Fraction expansion. General form of X(s): Important transform pair: Example 9.9 9.10 9.11