Page 328 - Schaum's Outline of Theory and Problems of Signals and Systems

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

6.9. Let x,[n] and x,[n] be the two periodic signals in Prob. 6.8. Show that

Equation (6.127) is known as Parseval's relation for periodic sequences.
From Eq. (6.126) we have

1 N0-I No- 1
ck = - C x,[n]x2[n] e-jknon = C ddk-m
NO n=o m =O
Setting k = 0 in the above expression, we get

6.10. (a) Verify Parseval's identity [Eq. (6.19)] for the discrete Fourier series, that is,

(6) Using x[n] in Prob. 6.3, verify Parseval's identity [Eq. (6.19)].
(a) Let

and

1 No-' ] No-'
Then d -- C X*[n~e-~k~on= - C X[n]e~kRon (6.128)
- NO n=O NO n=o
Equation (6.128) indicates that if the Fourier coefficients of x[n] are c,, then the Fourier
coefficients of x*[n] are c?,. Setting x,[n] =x[n] and x2[n] =x*[n] in Eq. (6.1271, we
have dk = ck and ek = c?, (or e-, = c:) and we obtain

(b) From Fig. 6-7 and the results from Prob. 6.3, we have

and Parseval's identity is verified.