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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 ⅆ t F ( s ) = ∫ − ∞ ∞ f ( t ) e − s t ⅆ 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 ⅆ s f ( t ) = 1 2 π j ∫ c − j ∞ c + j ∞ F ( s ) e s t ⅆ 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.

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