Digital Signal Processing Reference
In-Depth Information
we write (12.21) as
N
q
2p
2pk
N
2pk
N
¢
≤
¢
≤
X(Æ) =
X
0
d
Æ-
.
(12.23)
k=-
q
Recall from Table 12.2 that a discrete-time Fourier transform is
always
peri-
odic with period
2p;
thus,
X
0
(Æ)
is periodic. Therefore, the Fourier transform of a
sequence
x[n],
which is periodic with period
N
, results in a function
X(Æ)
that is pe-
riodic with period Furthermore, the
N
distinct values of
transform into
N
distinct values of in frequency.
Figure 12.5 gives an example of in (12.23). For convenience, we have as-
sumed that is real and triangular. We also assume that for this exam-
ple. The discrete-time Fourier transform of of (12.23), is then as shown
in Figure 12.5(b). In this figure, the lengths of the arrows denote the weights of the
impulse functions. The period of
2p.
x[n],
0 F n F N - 1,
X
0
(2pk/N), 0 F k F N - 1
X(Æ)
X
0
(Æ)
N = 3
x[n],
X(Æ)
X(Æ)
is
2p,
with an impulse function occurring
every
Æ=2p/3.
Hence, the values
x
[0],
x
[1], and
x
[2] completely describe the peri-
odic function
x[n],
and the values
X(0),
X(2p/3),
and
X(4p/3)
completely describe
the periodic function
Next, we derive the inverse discrete-time Fourier transform of (12.23). From
(12.2),
X(Æ).
2p
1
2p
L
-1
[X(Æ)] = x[n] =
X(Æ)e
jÆn
dÆ
0
2p
N
q
1
2p
L
2p
2pk
N
2pk
N
B
¢
≤
¢
≤R
e
jÆn
dÆ
=
X
0
d
Æ-
(12.24)
k=-
q
0
N- 1
1
N
a
2pk
N
¢
≤
e
j2pkn/N
.
=
X
0
k= 0
X
0
(
)
1
•••
•••
2
0
2
(a)
X
(
)
2
3
•••
3
•••
2
4
3
2
3
0
2
3
4
3
2
8
3
3
Figure 12.5
Discrete-time Fourier
transform of a periodic signal.
(b)
