The Laplace transform is a generalization of the Continuous-Time Fourier Transform. It is used because the CTFT does not converge/exist for many important signals, and yet it does for the Laplace-transform (e.g., signals with infinite ${l}_{2}$ norm). It is also used because it is notationally cleaner than the CTFT. However, instead of using complex exponentials of the form ${e}^{i\omega t}$, with purely imaginary parameters, the Laplace transform uses the more general, ${e}^{st}$, where $s=\sigma +i\omega $ is complex, to analyze signals in terms of exponentially weighted sinusoids.

Although Laplace transforms are rarely solved in practice using integration (tables and computers (e.g. Matlab) are much more common), we will provide the bilateral Laplace transform pair here for purposes of discussion and derivation. These define the forward and inverse Laplace transformations. Notice the similarities between the forward and inverse transforms. This will give rise to many of the same symmetries found in Fourier analysis.

Laplace Transform

$$F\left(s\right)={\int}_{-\infty}^{\infty}f\left(t\right){e}^{-(st)}dt$$

1

Inverse Laplace Transform

$$f\left(t\right)=\frac{1}{2\pi i}{\int}_{c-i\infty}^{c+i\infty}F\left(s\right){e}^{st}ds$$

2

Note:

We have defined the bilateral Laplace transform. There is also a unilateral Laplace transform
,
$$F\left(s\right)={\int}_{0}^{\infty}f\left(t\right){e}^{-(st)}dt$$

3

Taking a look at the equations describing the Z-Transform and the Discrete-Time Fourier Transform:

Continuous-Time Fourier Transform

$$\mathcal{F}\left(\Omega \right)={\int}_{-\infty}^{\infty}f\left(t\right){e}^{-(i\Omega t)}dt$$

4

Laplace Transform

$$F\left(s\right)={\int}_{-\infty}^{\infty}f\left(t\right){e}^{-(st)}dt$$

5

$$\mathcal{F}\left(\Omega \right)=F\left(s\right)$$

6

Note:

the CTFT is a complex-valued function of a real-valued variable
$\omega $
(and 2
$\pi $
periodic).
The Z-transform is a complex-valued function of a complex valued variable z.
Plots

With the Fourier transform, we had a complex-valued
function of a **purely imaginary
variable**,
$F\left(i\omega \right)$. This was something we could envision with two
2-dimensional plots (real and imaginary parts or magnitude and
phase). However, with Laplace, we have a complex-valued
function of a **complex variable**.
In order to examine the magnitude and phase or real and
imaginary parts of this function, we must examine
3-dimensional surface plots of each component.

real and imaginary sample plots

magnitude and phase sample plots

While these are legitimate ways of looking at a signal in the Laplace domain, it is quite difficult to draw and/or analyze. For this reason, a simpler method has been developed. Although it will not be discussed in detail here, the method of Poles and Zeros is much easier to understand and is the way both the Laplace transform and its discrete-time counterpart the Z-transform are represented graphically.

Using a computer to find Laplace transforms is relatively
painless. Matlab has two functions,
`laplace`

and
`ilaplace`

, that are both part of the
symbolic toolbox, and will find the Laplace and inverse
Laplace transforms respectively. This method is generally
preferred for more complicated functions. Simpler and more
contrived functions are usually found easily enough by using tables.

Khan Lecture on Laplace

The laplace transform proves a useful, more general form of the Continuous Time Fourier Transform. It applies equally well to describing systems as well as signals using the eigenfunction method, and to describing a larger class of signals better described using the pole-zero method.