Page 363 - Schaum's Outline of Theory and Problems of Signals and Systems

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

(b) Let Wn+,,,+, denote the entry in the (n + 1)st row and (k + 1)st column of the W4
matrix. Then, from Eq. (6.207)

and we have

6.55. (a) Find the DFT X[k] of x[n] = (0,1,2,3).
(b) Find the IDFT x[n] from X[k] obtained in part (a).

(a) Using Eqs. (6.206) and (6.212), the DFT XIk] of x[n] is given by

(b) Using Eqs. (6.209) and (6.212), the IDFT x[n] of X[k] is given by
1 1 1 1 6
1 - 1
= -
1 - 1 1 - 1 4
1 -j - 1 j - -2-j2

6.56. Let x[n] be a sequence of finite length N such that
x[n] = 0 n<O,n>N

Let the N-point DFT X[k] of x[n] be given by [Eq. (6.9211
N- 1

Suppose N is even and let

The sequences f[n] and g[n] represent the even-numbered and odd-numbered
samples of x[n], respectively.
(a) Show that
N
f [n] = 4.1 = 0 outside 0 sn 5 - - 1
2