Page 359 - Schaum's Outline of Theory and Problems of Signals and Systems

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

6.50. Consider two sequences x[n] and h[n] of length 4 given by

(a> Calculate y[n] =x[n] 8 h[n] by doing the circular convolution directly.
(b) Calculate y[n] by DFT.
(a) The sequences x[n] and h[n] can be expressed as
I
1
1
x[n] = {l,O,- 1,O) and h[n] = (l,~,~,~)
By Eq. (6.108)

The sequences x[i] and h[n - iImod4 for n = 0,1,2,3 are plotted in Fig. 6-36(a). Thus, by
Eq. (6.108) we get

which is plotted in Fig. 6-36(b).
(b) By Eq. (6.92)

Then by Eq. (6.107) the DFT of y[n] is
Y[k] = X[k]H[k] = (1 - w,Zk)(l + i~qk + ;wqZk + twak)

- -
- 1 + iwk - lw2k- 1~3k Lw4k - Lw5k
2 4 4 4 8 4 4 4 $ 4
Since W:k = (w:)~ lk and wdjk = W(4+')k = wqk, we obtain
4
=
y[k]=$+$~qk-f~:~-iw~~~ k=0,1,2,3
Thus, by the definition of DFT [Eq. (6.9211 we get