The Laplace Transform
The Laplace Transform is a generalization of the Fourier Transform. Instead of considering only the imaginary axis, $ j\omega\! $, (as the Fourier Transform does) the Laplace Transform considers all complex values represented by the general complex variable $ s\! $. Take the following simple picture:
Fourier Transform: $ x(t) --> X(\omega)\! $ where $ \omega\! $ is a frequency.
Laplace Transform: $ x(t) --> X(s)\! $ where $ s\! $ is a complex variable.
Mathematically, the Laplace Transform is represented as follows:
Let's consider the case where $ s = j\omega\! $.
$ X(s)|_{s=j\omega} = X(j\omega) = \int_{-\infty}^{\infty} x(t)e^{-j\omega t} dt = F(x(t)) = X (\omega)\! $
So, the Laplace Transform, $ X(s)\! $, evaluated on the imaginary axis, $ X(j\omega)\! $, is equal to the Fourier Transform, $ F(x(t))\! $, at $ \omega\! $. Said another way, the Fourier Transform, $ X(\omega)\! $, is the restriction of the Laplace Transform, $ X(s)\! $, on the imaginary axis, $ s = j\omega\! $.
It is actually possible to obtain the Laplace Transform, $ X(s)\! $, from the Fourier Transform, $ X(\omega)\! $.
$ X(s) = \int_{-\infty}^{\infty} x(t)e^{-st} dt\! $, $ let\! $ $ s = a + j\omega \! $
$ = \int_{-\infty}^{\infty} x(t)e^{-(a+j\omega)t} dt\! $
$ = \int_{-\infty}^{\infty} x(t)e^{-at}e^{-j\omega t} dt\! $
$ = F(x(t)e^{-at})\! $
Therefore, $ X(s) = X(a+j\omega) = F(x(t)e^{-at})\! $
General Example of Laplace Transform
Let $ x(t) = e^{-bt}u(t)\! $
$ X(s) = \int_{-\infty}^{\infty} x(t)e^{-st} dt\! $
$ = \int_{-\infty}^{\infty} e^{-bt}u(t)e^{-st} dt\! $
$ = \int_{0}^{\infty} e^{-bt}e^{-st} dt\! $
$ = \int_{0}^{\infty} e^{-(b+s)t} dt\! $
$ = \int_{0}^{\infty} e^{-(b+a+j\omega)t} dt\! $
if $ b+a\leq 0\! $ or $ Re(s) \leq -b\! $, then the integral diverges.
else, $ Re(s) > -b\! $,
$ = \frac{e^{-(b+a)t}e^{-j\omega t}}{-(b+s)} |_{0}^{\infty}\! $
$ = 0 - \frac{1}{-(b+s)}\! $
$ =\frac{1}{b+s}\! $
$ Re(s) > -b\! $ is called the region of convergence, or ROC, for this problem. Every problem that requires a Laplace transform must include a ROC in its solution. The ROC is those values of $ s\! $ for which the Laplace Transform converges. For this general example the ROC is $ Re(s) > -b\! $. An interesting property of the ROC is that if the ROC excludes the imaginary axis, then the Fourier Transform of that $ x(t)\! $ does not exist. Therefore, if the Fourier Transform of $ x(t)\! $ exists, then the imaginary axis is part of the ROC. This property is useful for checking to see if one's computed Laplace Transform is correct. For additional ROC properties, see my classmates pages in the Homework 10 link.
