Page 303 - Schaum's Outline of Theory and Problems of Signals and Systems

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

Let n = -m in Eq. (6.13). Then

Letting k = n and m = k in the above expression, we get

Comparing Eq. (6.14) with Eq. (6.91, we see that (l/N,,)x[-k] are the Fourier coefficients
of c[n]. If we adopt the notation
x[n] Bck =c[k] (6.15)
to denote the discrete Fourier series pair, then by Eq. (6.14) we have

1
DFS
~[n]
c--) -x[-k]
No
Equation (6.16) is known as the duality property of the discrete Fourier series.
3. Other Properties:

When x[n] is real, then from Eq. (6.8) or [Eq. (6.10)] and Eq. (6.12) it follows that
*
CPk =CN,,-k = ck (6.1 7)
where * denotes the complex conjugate.
Even and Odd Sequences:
When x[n] is real, let

x[nl =xe[nl +~o[nl
where xe[n] and xo[n] are the even and odd components of x[n], respectively. Let
x[n] Sck

Then
xe[n] Re[ck] (6.18~)
xo[n] 2% j Im[ck] (6.186)

Thus, we see that if x[n] is real and even, then its Fourier coefficients are real, while if
x[n] is real and odd, its Fourier coefficients are imaginary.

E. Parseval's Theorem:
If x[n] is represented by the discrete Fourier series in Eq. (6.9), then it can be shown
that (Prob. 6.10)

Equation (6.19) is called Parseval's identity (or Parseual's theorem) for the discrete
Fourier series.