Page 327 - Schaum's Outline of Theory and Problems of Signals and Systems

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

(Nu-2)/2
+
Then x[n] = c0 + ( - I)~C,~/, x 2Re(ckejkRun)
k=l
(Nu-2)/2
= c, + ( - l)nc,v/2 + 2 x (a, cos kfl,,n - b, sin kRon)
k = 1

6.8. Let x,[n] and x,[n] be periodic sequences with fundamental period No and their
discrete Fourier series given by

Show that the sequence x[n] =x,[n]x,[n] is periodic with the same fundamental
period No and can be expressed as

where ck is given by

Now note that

Thus, x[n] is periodic with fundamental period No. Let

1 Nu-' 1 No-'
Then Ck = - C X[n]e-ikfM = - z xl[n]x2[n] e-',"on
No .=o No n=o

since

and the term in parentheses is equal to ek-,.