Image Processing Reference
In-Depth Information
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ω
FIGURE 3.9
Magnitude of DTFT of x
(
n
)
.
S
OLUTION
The DTFT of x(n)is
1
x(n)e
jnv
¼
3e
jv
þ
7e
2jv
3e
3jv
þ
2e
4jv
X( jv
)
¼
2
n¼1
2e
2jv
(e
2jv
þ e
2jv
)
3e
2jv
(e
jv
þ e
jv
)
7e
2jv
¼
þ
¼ e
2jv
[4 cos 2
v
6 cos
v þ
7]
The magnitude of X( jv
)is
jX( jv
)
j¼
7
6 cos
v þ
4 cos 2
v
The magnitude plot is shown in Figure 3.9.
3.6.2 I
NVERSE
DTFT
The inverse DTFT can be computed using the inversion integral used in computing
the inverse z-transform, that is,
þ
1
j2
X
(
z
)
z
n
1
x
(
n
) ¼
d
z
(
3
:
94
)
p
C
Let the integration contour C be the unit circle z
¼
e
j
v
in the complex z-plane. We
can use this closed contour since x
(
n
)
has Fourier transform and its ROC contains the
unit circle. Therefore,
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