FOURIER TRANSFORMS OF DISCRETE-TIME SIGNALS - Signals, Systems, and Transforms - page 618

an analog signal and which produces a finite set of discrete-frequency spectrum val-
ues. This Fourier transform approximation is well suited for calculation by a digital
computer. It is called the discrete Fourier transform (DFT).
To use a digital computer to calculate the Fourier transform of a continuous-
time signal, we must use sampled values of in the form of a discrete-time signal
x [ n ]. From Section 12.1, we have the discrete-time Fourier transform (DTFT):

x(t)

(x[n]) = q
n=- q

x[n]e -jnÆ . [eq(12.1)] X(Æ) =

The DTFT is computed from discrete-time samples, but is a continuous
function of the frequency variable and cannot be represented exactly in a digital
computer. However, using digital computations, we can approximate

X(Æ)

X(Æ) by cal-

culating discrete-frequency samples of the continuous-frequency function.
To generate the discrete-frequency samples, we must, for practical reasons,
limit ourselves to a finite set of discrete-time samples. For the purpose of this de-
velopment, we will let the symbol N represent the number of samples chosen to
represent the discrete-time signal. We choose the value of N sufficiently large so
that our set of samples adequately represents all of x[n]. We can select our finite

set of samples by multiplying the infinite set x[n] by a rectangular windowing

function [5]:

1, n = 0, 1, 2, Á , N - 1
0,

b

w R [n] = .

otherwise

Then the set of samples used to calculate the frequency spectrum is

x[n], n = 0, 1, 2, Á , N - 1
0,

b

x N [n] = x[n]w R [n] = . (12.26)

otherwise

The frequency spectrum of the signal x N [n] is given by (12.1):

q

N- 1

x N [n]e -jnÆ = a x[n]e -jnÆ .

X N (Æ) = (x N [n]) = a (12.27)

n=- q n= 0

For the remainder of the development, we consider the set of N samples of
to be the complete signal. We will drop the subscript on and refer to the
finite discrete-time sequence simply as This is justified by the assumption that
we have chosen the finite set of samples so that they adequately represent the entire
signal.
We now select N samples of to represent the frequency spectrum. We
could choose more than N samples, but we must choose at least N to avoid creating

x N [n] x N [n]

x[n].

X N (Æ)