The Laplace transform
The Laplace transform of signal x(t) is defined as
$ X(s) = \int_{-\infty}^{\infty}x(t)e^{-st} $
The ROC for Laplace transforms
There are 8 properties that we should know for ROC(region of convergence)
1.The ROC of X(s) consists of strips parallel to the jw-axis in the s-plane.
2.for rational Laplace transform, the ROC does not contain any poles.
3. If x(t) is of finite duration and is absolutely integrable, then the ROC is the entire s -plane.
4.If x(t) is right sided, and if the line Re{s} = $ \sigma_{0} $ is in the ROC, then all values of s for which Re{s} > $ \sigma_{0} $ will also be in the ROC.
5.If x(t) is left sided, and if the line Re{s} = $ \sigma_{0} $ is in the ROC, then all values of s for which Re{s} < $ \sigma_{0} $ will also be in the ROC.
6.If x(t) is two sided, and if the line Re{s} = $ \sigma_{0} $ is in the ROC, then the ROC will consist of a strip in the s-plane that includes the line Re{s} = $ \sigma_{0} $.
7.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.
8.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.
