Signals and Systems - Electrical Engineering

600 C H A P T E R 10: Fourier Analysis of Discrete-Time Signals and Systems

The Fourier series representation of a periodic signalx[n]of periodNis

x[n]=

k 0 +∑N− 1

k=k 0

X[k]ej

2 π

Nkn (10.27)

where the Fourier series coefficients{X[k]}are obtained from

X[k]=

1

N

n 0 +∑N− 1

n=n 0

x[n]e−j

(^2) Nπkn
(10.28)
The frequencyω 0 = 2 π/Nrad is the fundamental frequency, andk 0 andn 0 in Equations (10.27) and (10.28)
are arbitrary integer values. The Fourier series coefficientsX[k], as functions of frequency 2 πk/N, are periodic
of periodN.
Remarks
n The connection of the above two equations can be verified by using the orthonormality of the{φ[k,n]/

√

N}

functions. In fact, if we multiply x[n]by e−j(^2 π/N)`nand sum these values for n changing over a period,

using Equation (10.27) we get:

∑

n

x[n]e−j^2 πn`/N=

∑

n

∑

k

X[k]ej^2 π(k−`)n/N

=

∑

k

X[k]

∑

n

ej^2 π(k−`)n/N=NX[`]

since

∑

ne

j 2 π(k−`)n/Nis zero, except when k−`= 0 , in which case the sum is equal to N.

n Both x[n]and X[k]are periodic with respect to n and k of the same period N, as can be easily shown

using the periodicity of the functions{φ(k,n)}. Consequently, the sum over k in the Fourier series and the

sum over n in the Fourier coefficients are computed over any period of x[n]and X[k]. Thus, the sum in the

Fourier series can be computed in any period, or from k=k 0 to k 0 +N− 1 for any value of k 0. Likewise,

the summation in the computation of the Fourier coefficients goes from n=n 0 to no+N− 1 , which is

an arbitrary period for any integer value n 0.

n Notice that both x[n]and X[k]can be computed with a computer since the frequency is discrete and only

sums are needed to implement them. We will use these characteristics in the practical computation of the

Fourier transform of discrete-time signals, or DFT.

nExample 10.13

Find the Fourier series of a periodic signal

x[n]= 1 +cos( 2 πn/ 4 )+sin( 2 πn/ 2 ) −∞<n<∞