Page 366 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 61 FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS 353

6.57. Consider a sequence
x[n] ={l,l,-1,-1,-l,l,l,-1)
Determine the DFT X[k] of x[n] using the decimation-in-time FFT algorithm.
From Figs. 6-3Na) and (61, the phase factors W: and W,k are easily found as follows:
w; = -,
wb)= 1 w~~ - 1 w~~ j
=
=
1 1 1 1
and W:= 1 w,' = - w+ -j w+ ---,-
a -'7T a a

Next, from Eqs. (6.215~) and (6.2156)
f 1.1 =x[2n] = (x[O], x[2], x[4], x[6]) = (1, - 1,- 1,l)
g[n]=x[2n+ 1]= {x[l],x[3],x[S],x[7]} ={I,- 1,1,- 1)

Then, using Eqs. (6.206) and (6.2121, we have

(4 (b)
Fig. 638 Phase factors W: and W,".