Digital Signal Processing Reference
In-Depth Information
The result is obtained from the property of the impulse function
b
f(t
0
),
a F t
0
F b
B
f(t)d(t - t
0
)dt =
.
0,
otherwise
L
a
In (12.24), only the impulse functions
occur between
In summary, for a periodic sequence
d(Æ), d(Æ- 2p/N),
Á , d(Æ- 2p[N - 1]/N)
0 FÆ62p.
x[n] = x[n + N],
N
q
2p
2pk
N
2pk
N
¢
≤
¢
≤
[eq(12.23)]
X(Æ) =
X
0
d
Æ-
k=-
q
and
N- 1
1
N
a
2pk
N
¢
≤
e
j2pkn/N
,
[eq(12.24)]
x[n] =
X
0
k= 0
where, from (12.18),
N- 1
x
0
[n]e
-jnÆ
.
X
0
(Æ) =
a
(12.25)
n= 0
There are
N
distinct values of
x[n]
and
N
distinct values of
X
0
(2pk/N).
An example
is now given to illustrate these developments.
Calculation of the DTFT of a periodic sequence
EXAMPLE 12.7
Consider again the periodic signal
x[n]
of Figure 12.4. For this signal,
N = 3, x
0
[0] = 0,
x
0
[1] = 1,
and
x
0
[2] = 1.
From (12.25), the discrete-time Fourier transform of
x
0
[n]
is given
by
N- 1
x
0
[n]e
-jnÆ
= 0e
-j0
+ (1)e
-jÆ
+ (1)e
-j2Æ
X
0
(Æ) =
a
n= 0
= e
-jÆ
+ e
-j2Æ
.
This transform is periodic and the real part of
X
0
(Æ)
is plotted in Figure 12.6. From (12.23), the
discrete-time Fourier transform of
x[n]
is then
3
q
2p
2pk
3
2pk
3
X(Æ) =
X
0
¢
≤
d
¢
Æ-
≤
,
k=-
q
