Digital Signal Processing Reference
In-Depth Information
12.1
DISCRETE-TIME FOURIER TRANSFORM
In Section 6.4, we saw that the Fourier transform
X
s
(v) =
q
n=-
q
x(nT)e
-jnvT
[eq(6.17)]
appears in modeling the sampling process. If, in this equation, we make the usual
substitution
x(nT) = x[n]
and make the change of variables
vT =Æ,
we have the
defining equation of the
discrete-time Fourier transform
:
q
x[n]e
-jnÆ
.
X(Æ) =
(x[n]) =
a
(12.1)
n=-
q
(
#
)
In this equation, denotes the discrete-time Fourier transform and is the
discrete-frequency variable
. We see then that the discrete-time Fourier transform is
inherent in the Fourier-transform model of the sampling operation in Figure 6.29.
The inverse discrete-time Fourier transform is defined as
Æ
Æ
1
+ 2p
1
2p
L
1
2p
L
2p
-1
[X(Æ)] =
X(Æ)e
jnÆ
dÆ=
X(Æ)e
jnÆ
dÆ,
x[n] =
(12.2)
Æ
1
where is arbitrary. This is denoted by placing the value directly underneath
the integral symbol. We show later that the integrand is periodic with period
This inversion integral can be derived directly from that of the Fourier transform,
(5.2) [3]. We denote a discrete-time Fourier transform pair by
Æ
1
2p
2p.
x[n]
Î
"
X(Æ).
(12.3)
It is important to note that is a function of the
discrete
variable
n
, while
the transform in (12.1) is a function of the
continuous
variable Hence,
is a continuous function of frequency, while is a discrete function of time.
In general, we obtain the discrete sequence by sampling a continuous-
time signal or the sequence is interpreted as being the samples of a continuous-
time signal. For this case, the discrete-frequency variable is related to the
real-frequency variable by the equation Hence, we see that discrete
frequency is a
scaled
version of real frequency Recall that this scaling also ap-
peared in the study of discrete-time signals and systems in Chapters 9 and 10. The
principal application of the discrete-time Fourier transform is in the analysis of
sampled signals.
Next, we consider two examples of discrete-time Fourier transforms.
x[n]
X(Æ)
Æ.
X(Æ)
x[n]
x[n]
x(t),
Æ
v
Æ=vT.
Æ
v.
