10.3 Fourier Series of Discrete-Time Periodic Signals 599
x[0], x[4],...
x[1], x[5],...
x[2], x[6],...
x[3], x[7],...
··· ···
n
01
2
3
4
5
6
N = 4
x[n]
x[5]
x[2] x[6]
x[4]
x[3]
x[1]
x[0]
(a) (b)
FIGURE 10.8
(a) Circular and (b) linear representation of a periodic discrete-time signalx[n]of periodN= 4. Notice how the
circular representation shows the periodicityx[0]=x[4],...,x[3]=x[7],...for positive as well as negative
integers.
10.3.1 Complex Exponential Discrete Fourier Series
Consider the representation of a discrete-time signalx[n] periodic of periodN, using the orthog-
onal functions{φ[k,n]=ej^2 πkn/N}forn,k=0,...,N−1. Two important characteristics of these
functions are:
n The functions{φ[k,n]}are periodic with respect tokandnwith periodN. In fact,
φ[k+`N,n]=ej
2 π(k+`N)n
N
=ej
2 πkn
N ej^2 π`n
=ej
2 πkn
N
where we used thatej^2 π`n=1. It can be equally shown that the functions{φ(k,n)}are periodic
with respect tonwith a periodN.
n The functions{φ(k,n)}are orthogonal with respect ton—that is,
N∑− 1
n= 0
ej
2 π
Nkn(ej
2 π
N`n)∗=
{
N ifk−`= 0
0 ifk−`6= 0
and can be normalized by dividing them by
√
N. So{φ[k,n]/
√
N}are orthonormal functions.
These two properties will be used in obtaining the Fourier series representation of periodic discrete-
time signals.
