Page 225 - Schaum's Outline of Theory and Problems of Signals and Systems

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

A plot of lckl versus the angular frequency w is called the amplitude spectrum of the
periodic signal x(t), and a plot of 4, versus w is called the phase spectrum of x(t 1. Since
the index k assumes only integers, the amplitude and phase spectra are not continuous
curves but appear only at the discrete frequencies ko,,. They are therefore referred to as
discrete frequency spectra or line spectra.
For a real periodic signal x(t) we have c-, = c:. Thus,

Hence, the amplitude spectrum is an even function of w, and the phase spectrum is an odd
function of o for a real periodic signal.

C. Power Content of a Periodic Signal:

In Chap. 1 (Prob. 1.18) we introduced the average power of a periodic signal x(t) over
any period as

If x(t) is represented by the complex exponential Fourier series in Eq. (5.4), then it can be
shown that (Prob. 5.14)

Equation (5.21) is called Parserlal's identity (or Parse~lal's theorem) for the Fourier series.

5.3 THE FOURIER TRANSFORM
A. From Fourier Series to Fourier Transform:

Let X( t ) be a nonperiodic signal of finite duration, that is,

x(t) = 0 Itl> TI
Such a signal is shown in Fig. 5-l(a). Let x,,)(t) be a periodic signal formed by repeating
x(r) with fundamental period T,, as shown in"~i~. 5-l( b). If we let To --+ m, we have

lim xT,lt) =x(t)
TO+=
The complex exponential Fourier series of xril(t) is given

m

where

Since ~,,~(t ) = x(r) for It 1 < T,,/2 and also since x( t) = 0 outside this interval, Eq. (5.24a)