Page 305 - Schaum's Outline of Theory and Problems of Signals and Systems

292 FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS [CHAP. 6

N,
Since xN,,[n] = x[n] for In1 I and also since x[n] = 0 outside this interval, Eq. (6.22~)
can be rewritten as

Let us define X(R) as

Then, from Eq. (6.22b) the Fourier coefficients c, can be expressed as

Substituting Eq. (6.24) into Eq. (6.21), we have

From Eq. (6.231, X(R) is periodic with period 27r and so is eJRn. Thus, the product
X(R)e*'" will also be periodic with period 27r. As shown in Fig. 6-2, each term in the
summation in Eq. (6.25) represents the area of a rectangle of height ~(kR,)e'~~11" and
width R,. As No + m, 0, = 27r/N0 becomes infinitesimal (R, + 0) and Eq. (6.25) passes
to an integral. Furthermore, since the summation in Eq. (6.25) is over N,, consecutive
intervals of width 0, = 27r/N,,, the total interval of integration will always have a width
27r. Thus, as NO + a: and in view of Eq. (6.20), Eq. (6.25) becomes
1 /
x[n] = - ~(0) ejRn dR (6.26)
27r 2v
Since X(R)e 'On is periodic with period 27r, the interval of integration in Eq. (6.26) can be
taken as any interval of length 27r.

Fig. 6-2 Graphical interpretation of Eq. (6.25).