9 The Laplace Transform 9. The Laplace Transform 9.1 The Laplace transform (1) Definition X(s)=x(tedt (where s=0+Jo (2) Region of Convergence(ROC) ROC: Range of o for X(s) to converge Representation A. Inequality B Region in S-plane
9 The Laplace Transform 9. The Laplace Transform 9.1 The Laplace Transform (1) Definition + − − X s = x t e dt st ( ) ( ) (where s = + j) (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 m S-plane S-plane a Re I-a Re
9 The Laplace Transform Example for ROC Re Re S-plane S-plane Im Im -a -a
9 The Laplace Transform (3)Relationship between Fourier and Laplace transform X(S) (te dt X(o ∫ x(teyo di X(o=X(sIso or X(s)=X(aloes Example9.19.29.39.5
9 The Laplace Transform (3) Relationship between Fourier and Laplace transform s j j s j t s t X j X s or X s X j X j x t e dt X s x t e dt = = + − − + − − = = = = ( ) ( )| ( ) ( )| ( ) ( ) ( ) ( ) Example 9.1 9.2 9.3 9.5
9 The Laplace Transform 9.2 The Region of Convergence for Laplace transform Property 1: The Roc of X(s) consists of strips parallel to j-axis in the S-plane Property: For rational Laplace transform, the Roc does not contain any poles Property 3: If x(t is of finite duration and is absolutely integrable, then the roc is the entire s plane
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 splane
9 The Laplace Transform Property4: If x(t is right sided, and if the line res=oo is in the roc then all values of s for which Res)>oo will also in the Roc x(t)
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