Page 322 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 61 FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS 309

6.2. Using the orthogonality condition Eq. (6.119), derive Eq. (6.8) for the Fourier
coefficients.
Replacing the summation variable k by m in Eq. (6.71, we have

Using Eq. (6.115) with N = NO, Eq. (6.120) can be rewritten as

Multiplying both sides of Eq. (6.121) by qt[n] and summing over n = 0 to (No - l), we obtain

Interchanging the order of the summation and using Eq. (6.1191, we get

Thus,

6.3. Determine the Fourier coefficients for the periodic sequence x[n] shown in Fig. 6-7.
From Fig. 6-7 we see that x[n] is the periodic extension of (0,1,2,3) with fundamental
period No = 4. Thus,

By Eq. (6.8) the discrete-time Fourier coefficients c, are

Note that c, = c,-, = cT [Eq. (6.17)].