Image Processing Reference
In-Depth Information
H
(
j
ω)
P
ω
π
π
-
P
P
FIGURE 2.32
Frequency response of ideal interpolation filter.
!
1
P
D
x
j
v
2
p
k
G
(
j
v) ¼
(
:
)
F
a
2
108
D
x
P
k
¼1
Therefore, if the signal y
(
n
)
is passed through an ideal LPF with a gain of P and
cutoff frequency of
P
, the output would be the desired upsampled signal g
(
n
)
. This
means that the ideal interpolation LPF has the frequency response given by
jvj <
P
P
H
(
j
v) ¼
(
2
:
109
)
0
otherwise
This frequency response is shown in Figure 2.32.
2.7.3 D
OWNSAMPLING BY
F
ACTOR OF
Q
To downsample a 1-D signal by a factor of Q, we retain every Qth sample. The block
diagram of a downsampler is shown in Figure 2.33.
The output signal is related to the input by
y
(
n
) ¼
f
(
Qn
)
(
:
)
2
110
The DTFT of y
(
n
)
is given by
1
1
1
f
(
k
)
w
(
k
)
e
j
Q
v
y
(
n
)
e
jn
v
¼
f
(
Qn
)
e
jn
v
¼
Y
(v) ¼
(
2
:
111
)
n
¼1
n
¼1
k
¼1
f
(
n
)
y
(
n
)
Q
FIGURE 2.33
Downsampling by a factor of Q.
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