Signals and Systems - Electrical Engineering

10.3 Fourier Series of Discrete-Time Periodic Signals 597

representation for discrete-time periodic signals that is discrete and periodic in frequency so it can be
implemented in a computer. This is the basis of the so-called discrete Fourier transform (DFT), which
is fundamental in digital signal processing (see Section 10.4). An algorithm called the Fast Fourier
Transform (FFT) implements it very efficiently (see Chapter 12).

Before finding the representation of periodic discrete-time signals recall that:

n A discrete-time signalx[n] is periodic if there is a positive integerNsuch thatx[n+kN]=x[n]
for any integerk. This valueNis the smallest positive integer that satisfies this condition and it is
called the period ofx[n]. For the periodicity to hold we need thatx[n] be of infinite support (i.e.,
x[n] must be defined in−∞<n<∞).
n According to the eigenfunction property of discrete-time LTI systems, whenever the input to such
systems is a complex exponentialAej(ω^0 n+θ)the corresponding output in the steady state is

y[n]=Aej(ω^0 n+θ)H(ejω^0 )

whereH(ejω^0 )is the frequency response of the system at the input frequencyω 0. The advantage

of this, as demonstrated in the continuous-time domain, is that if we are able to express the input

signal as a linear combination of complex exponentials, then superposition gives us a linear

combination of the responses to each exponential. Thus, if the input signal is of the form

x[n]=

∑

k

A[k]ejωkn

then the output will be

y[n]=

∑

k

A[k]ejωknH(ejωk)

This property is valid whether the frequency components of the input signal are harmonically
related (whenx[n] is periodic) or not.
n We showed before that a signalx(t)that is periodic of periodT 0 can be represented by its Fourier
series

x(t)=

∑∞

k=−∞

Xˆ[k]ej

2 πkt

T (^0) (10.26)
If we samplex(t)using a sampling periodTs=T 0 /NwhereNis a positive integer, we get
x(nTs)=

∑∞

k=−∞

Xˆ[k]ej

2 πknTs

T (^0) =

∑∞

k=−∞

Xˆ[k]ej^2 πNkn

The last summation repeats the frequencies between 0 to 2π. To avoid these repetitions, we let

k=m+rNwhere 0≤m≤N−1 andr=0,±1,±2,...—that is, we divide the infinite support