Signals and Systems - Electrical Engineering

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

ofkinto an infinite number of finite segments of lengthN. We then have

x(nTs)=

N∑− 1

m= 0

∑∞

r=−∞

Xˆ[m+rN]ej

2 π(m+rN)n

N

=

N∑− 1

m= 0

[ ∞

∑

r=−∞

Xˆ[m+rN]

]

ej

2 πmn

N

=

N∑− 1

m= 0

X[m]ej

2 πmn

N

This representation is in terms of complex exponentials with frequencies 2πm/N,m=0,...,

N−1, from 0 to 2π(N− 1 )/N. It is this Fourier series representation that we will develop

next.

Circular Representation of Discrete-Time Periodic Signals

Considering that the periodNof a periodic signalx[n] and the samples in a first periodx 1 [n]

completely characterize a periodic signalx[n], a circular rather than a linear representation would

more efficiently represent the signal. The circular representation is obtained by locating uniformly

around a circle the values of the first period starting withx[0] and putting in a clockwise direc-

tion the remaining termsx[1],...,x[N−1]. Continuing in the clockwise direction are the values

x[N]=x[0],x[N+1]=x[1],...,x[2N−1]=x[N−1], and so on. In general, any valuex[m] where

mis represented as

m=kN+r

for integersk, the exact divisor ofmbyN, and the residue 0≤r<N, equals one of the samples in

the first period—that is,

x[m]=x[kN+r]=x[r]

This representation is calledcircularin contrast to the equivalent linear representation introduced

before:

x[n]=

∑∞

k=−∞

x 1 [n+kN]

which superposes shifted versions of the first period. The circular representation becomes very useful

in the computation of the DFT, as we will see later in this chapter.

Figure 10.8 shows the circular and the linear representations of a periodic signal x[n] of

periodN=4.