Page 228 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 51 FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS

If x(t) is a real signal, then from Eq. (5.31) we get

Then it follows that

and Ix(-o)l = Ix(o)l 4(-4 = -$(@) (5.36b)
Hence, as in the case of periodic signals, the amplitude spectrum IX(o)( is an even
function and the phase spectrum 4(o) is an odd function of o.

D. Convergence of Fourier Transforms:
Just as in the case of periodic signals, the sufficient conditions for the convergence of
X(o) are the following (again referred to as the Dirichlet conditions):
1. x(l) is absolutely integrable, that is,

2. x(t) has a finite number of maxima and minima within any finite interval.
3. x(t) has a finite number of discontinuities within any finite interval, and each of these
discontinuities is finite.
Although the above Dirichlet conditions guarantee the existence of the Fourier transform for
a signal, if impulse functions are permitted in the transform, signals which do not satisfy
these conditions can have Fourier transforms (Prob. 5.23).

E. Connection between the Fourier Transform and the Laplace Transform:
Equation (5.31) defines the Fourier transform of x(r as

The bilateral Laplace transform of x(t), as defined in Eq. (4.31, is given by

Comparing Eqs. (5.38) and (5.39), we see that the Fourier transform is a special case of
the Laplace transform in which s = jo, that is,

Setting s = u + jo in Eq. (5.39), we have

m m
e-("+~")'dt = 1 [x(t) e-"1 e-jW'dt
-m
X(u + jw) = Y(x(t) e-"'1