Page 294 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 51 FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS

Next, taking the Fourier transform of Eq. (5.182), we have

Substituting Eq. (5.187) into Eq. (5.186), we obtain

Taking the inverse Fourier transform of Eq. (5.1881, we get

a sin wM( t - kT,)
= C x(kT,)
k= -m WM(~ -kTs)
From Probs. 5.58 and 5.59 we conclude that a band-limited signal which has no frequency
components higher that fM hertz can be recovered completely from a set of samples taken at
the rate of f, (1 2 fM) samples per second. This is known as the uniform sampling theorem for
low-pass signals. We refer to T, = X/W, = 1 /2 fM (oM = 27r fM as the Nyquist sampling
interval and f, = 1/T, = 2 fM as the Nyquist sampling rate.

5.60. Consider the system shown in Fig. 5-35(a). The frequency response H(w) of the ideal
low-pass filter is given by [Fig. 5-35(b)]

Show that if w, = 0J2, then for any choice of T,,

Fig. 5-35