Page 326 - Schaum's Outline of Theory and Problems of Signals and Systems

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

(a) If x[nl is real, then from Eq. (6.8) we have

Thus,
Cdk =a_,+jb-, = (a, +jbk)* =a,

and we have
a-,=a, and b-,=-bk

(6) If No is even, then from Eq. (6.8)

1 No-'
= - C ( - l)"x[n] = real
NO n=o

If No is odd, then (No - 1) is even and we can write x[n] as

Now, from Eq. (6.17)

Thus,

(No- 1)/2
= c, + 2 z (a, cos kRon - bk sin kRon)
k=l
If No is even, we can write x[n] as