Page 302 - Schaum's Outline of Theory and Problems of Signals and Systems

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

B. Discrete Fourier Series Representation:
The discrete Fourier series representation of a periodic sequence x[n] with fundamen-
tal period No is given by

where c, are the Fourier coefficients and are given by (Prob. 6.2)

Because of Eq. (6.5) [or Eq. (6.6)], Eqs. (6.7) and (6.8) can be rewritten as

where C,, denotes that the summation is on k as k varies over a range of No
successive integers. Setting k = 0 in Eq. (6.101, we have

which indicates that co equals the average value of x[n] over a period.
The Fourier coefficients c, are often referred to as the spectral coefficients of x[n].

C. Convergence of Discrete Fourier Series:
Since the discrete Fourier series is a finite series, in contrast to the continuous-time
case, there are no convergence issues with discrete Fourier series.

D. Properties of Discrete Fourier Series:
I. Periodicity of Fourier Coeficients:
From Eqs. (6.5) and (6.7) [or (6.911, we see that

C,+N, = Ck
which indicates that the Fourier series coefficients c, are periodic with fundamental
period No.
2. Duality:

From Eq. (6.12) we see that the Fourier coefficients c, form a periodic sequence with
fundamental period No. Thus, writing c, as c[k], Eq. (6.10) can be rewritten as