Page 229 - Schaum's Outline of Theory and Problems of Signals and Systems

218 FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS [CHAP. 5

which indicates that the bilateral Laplace transform of x(t) can be interpreted as the
Fourier transform of x( t) e-"'.
Since the Laplace transform may be considered a generalization of the Fourier
transform in which the frequency is generalized from jw to s =a + jo, the complex
variable s is often referred to as the complexfrequency.
Note that since the integral in Eq. (5.39) is denoted by X(s), the integral in Eq. (5.38)
may be denoted as X(jw). Thus, in the remainder of this book both X(o) and X(jw)
mean the same thing whenever we connect the Fourier transform with the Laplace
transform. Because the Fourier transform is the Laplace transform with s = jo, it should
not be assumed automatically that the Fourier transform of a signal ~(r) is the Laplace
transform with s replaced by jw. If x(t) is absolutely integrable, that is, if x(r) satisfies
condition (5.37), the Fourier transform of x(t) can be obtained from the Laplace
transform of x(t) with s = jw. This is not generally true of signals which are not absolutely
integrable. The following examples illustrate the above statements.

EXAMPLE 5.1. Consider the unit impulse function S( t).
From Eq. (3.13) the Laplace transform of S(t) is
J(S(t)} = 1 all s

By definitions (5.31) and (1.20) the Fourier transform of 6(t) is

Thus, the Laplace transform and the Fourier transform of S(t) are the same.

EXAMPLE 5.2. Consider the exponential signal

From Eq. (3.8) the Laplace transform of x(t) is given by

By definition (5.31) the Fourier transform of x(t) is

Thus, comparing Eqs. (5.44) and (5.451, we have

X(w) =X(s)ls-jcu
Note that x(t) is absolutely integrable.

EXAMPLE 5.3. Consider the unit step function u(t ).
From Eq. (3.14) the Laplace transform of u(t) is