Image Processing Reference
InDepth 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 antialiasing 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 antialiasing filter.
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