Page 367 - Schaum's Outline of Theory and Problems of Signals and Systems

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

and by Eqs. (6.217~) and (6.21 76) we obtain

Noting that since x[n] is real and using Eq. (6.204),. X[7], X[6], and X[5] can be easily
obtained by taking the conjugates of X[l], X[2], and X[3], respectively.

6.58. Let x[n] be a sequence of finite length N such that
x[n] = 0 n <0, n rN
Let the N-point DFT X[k] of x[n] be given by [Eq. (6.92)]
N-1
X[k] = x[n] w,kn w N- - e-~(277/N) k=0,1, ..., N-1 (6.224)
n=O
Suppose N is even and let

(a) Show that the N-point DFT X[k] of x[n] can be expressed as

N
X[2k + 11 = Q[k] k=0,1, ..., -- 1 (6.2266)
2
(N/2) - 1 N
where P[kl= C p[nlW,k;z k =0,1, ..., -- 1 (6.227~)
n=O 2
(N/2)- 1 N
- -
Q[k] = C 4[nlW,k;2 k =0,1, ..., 1 (6.2276)
n=O 2
(6) Draw a flow graph to illustrate the evaluation of X[k] from Eqs. (6.226~) and
(6.2266) with N = 8.
(a) We rewrite Eq. (6.224) as

Changing the variable n = m + N/2 in the second term of Eq. (6.228), we have