The Laplace transform can be thought of as a generalization of the Fourier
transform in order to include a larger class of functions, to allow the use of
complex variable theory, to solve initial value differential equations, and to
give a tool for input-output description of linear systems. Its use in system
and signal analysis became popular in the 1950's and remains as the central
tool for much of continuous time system theory. The question of convergence
becomes still more complicated and depends on values of

**Definition of the Laplace Transform**

The definition of the Laplace transform (LT) of a real valued function defined
over all positive time

Notice that the Laplace transform becomes the Fourier transform on the
imaginary axis, for

There is a considerable literature on the Laplace transform and its use in continuous-time system theory. We will develop most of these ideas for the discrete-time system in terms of the z-transform later in this chapter and will only briefly consider only the more important properties here.

The unilateral Laplace transform cannot be used if useful parts of the signal exists for negative time. It does not reduce to the Fourier transform for signals that exist for negative time, but if the negative time part of a signal can be neglected, the unilateral transform will converge for a much larger class of function that the bilateral transform will. It also makes the solution of initial condition differential equations much easier.

Examples can be found in [1][2] and are similar to those of the z-transform presented later in these notes. Indeed, note the parallals and differences in the Fourier series, Fourier transform, and Z-transform.