Image Processing Reference
In-Depth Information
where w ( k )
is de
ned as
1
1
k ¼
0,
Q,
2Q,
...
w ( k ) ¼
¼
d( k nQ )
(
2
:
112
)
0
otherwise
n ¼1
Since w ( k )
is periodic with a period of Q, it can be expanded using discrete Fourier
series (DFS) as
X
Q 1
1
Q
e j 2 Q nk
w ( k ) ¼
(
2
:
113
)
n ¼
0
Substituting Equation 2.113 into 2.111 yields
1
1
X
X
1
Q 1
Q 1
1
Q
1
Q
f ( k ) w ( k ) e j Q v ¼
e j 2 p n Q e j Q v ¼
f ( k ) e j v 2 p Q k
Y (v) ¼
f ( k )
k ¼1
k ¼1
n ¼ 0
n ¼ 0
k ¼1
Q
X
1
F v
p n
1
Q
2
¼
(
2
:
114
)
Q
n ¼ 0
As evident from Equation 2.114, there are Q terms in the expansion of Y (v)
. This
expansion shows that there will be aliasing if the signal bandwidth is more than Q .To
avoid aliasing, we need to LPF the signal before downsampling. The frequency
response of the ideal LPF is
jvj < Q
1
H ( j v) ¼
(
2
:
115
)
0
otherwise
The frequency response of the ideal anti-aliasing LPF is shown in Figure 2.34.
The overall block diagram of a downsampler is shown in Figure 2.35.
H ( j ω)
1
ω
π
π
-
Q
Q
FIGURE 2.34
Frequency response of anti-aliasing filter.
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