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
9 The Laplace Transform Property5: If X( is left sided, and if the line Res=oc is in the roc then all values of s for which Res]<oo will also in the ROc e-o e- o 2
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 -0 t e -1 t
9 The Laplace Transform Property6: If x(t) is two sided, and if the line Re(s]=oo is in the roc then the roc will consist of a strip in the s-plane that includes the line Res]=oo x(t) (b)
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 R Re m OL Re R OL Re
9 The Laplace Transform S-plane Re Re Re Im Im Im L R L R
9 The Laplace Transform Property7: If the Laplace transform X(s)of x(tis rational, then its Roc is bounded by poles or extends to infinity. In addition, no poles of X(s)are contained in the Roc Property 8: If the Laplace transform X(s)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 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 splane
9 The Laplace Transform 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 Property 8: If the Laplace transform X(s)of x(tis 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 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