Skip to content Skip to navigation

OpenStax_CNX

You are here: Home » Content » m07 - The Laplace Transform

Navigation

Recently Viewed

This feature requires Javascript to be enabled.
 

m07 - The Laplace Transform

Module by: C. Sidney Burrus. E-mail the author

Summary: The Laplace Transform can be viewed as a generalization of the Fourier Transform giving a function of a complex variable. The Laplace Transform converts a linear, constant coefficient differential equation into an algebraic equation

The Laplace Transform

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 s s used in the inverse transform which must be in a ``region of convergence" (ROC).

Definition of the Laplace Transform

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

F ( s ) = f ( t ) e s t t F ( s ) = f ( t ) e s t t
(1)
and the inverse transform (ILT) is given by the complex contour integral
f ( t ) = 1 2 π j c j c + j F ( s ) e s t s f ( t ) = 1 2 π j c j c + j F ( s ) e s t s
(2)
where s = σ + j ω s = σ + j ω is a complex variable and the path of integration for the ILT must be in the region of the s s plane where the Laplace transform integral converges. This definition is often called the bilateral Laplace transform to distinguish it from the unilateral transform (ULT) which is defined with zero as the lower limit of the forward transform integral (Equation 1). Unless stated otherwise, we will be using the bilateral transform.

Notice that the Laplace transform becomes the Fourier transform on the imaginary axis, for s = j ω s = j ω . If the ROC includes the j ω j ω axis, the Fourier transform exists but if it does not, only the Laplace transform of the function exists.

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.

References

  1. A. Papoulis. (1962). The Fourier Integral and Its Applications. McGraw-Hill.
  2. R. N. Bracewell. (1985). The Fourier Transform and Its Applications. (Third). New York: McGraw-Hill.

Content actions

Download module as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Add module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks