Digital Signal Processing Reference
In-Depth Information
(a)
(b)
wn
3
()
wn
3
()
xn
3
()
H
3
()
xm
3
()
G
2
()
xn
()
xm
3
()
↑
M
↓
M
f
f
f
f M
/
sM
/
f
f M
s
f
f
s
s
sM
s
s
s
(c)
Xf
()
Xf
3
()
(h)
f
f
f
/2
f
/2
s
s
(d)
(g)
Wf
3
()
Wf
3
()
f
f
f
/2
f
/2
s
s
Xf
3
()
Xf
3
()
(e)
(f)
f
f
f
f M
/(2
)
f
f M
/(2
)
sM
s
sM
s
FIGURE 13.5
Analysis and synthesis stages for channel 3.
where
iðnÞ
is defined as
iðnÞ¼
P
k ¼
0
dðn kMÞ¼dðnÞþdðn MÞþdðn
2
MÞþ
/
Clearly,
iðnÞ
is a periodic function (impulse train with a period of
M
samples) as shown in
We can determine the discrete Fourier transform of the impulse train with a period of
M
samples as
M
1
n¼
0
iðnÞe
j
M
1
n¼
0
dðnÞe
j
2
pkn
M
2
pkn
M
IðkÞ¼
¼
¼
1
(13.4)
wn
()
wn
()
d
xm
()
H
()
xn
()
↓
M
↑
M
f
f
f
f
f M
/
s
s
s
sM
s
d
xm
()
wn
()
wn
()
n
m
n
FIGURE 13.6
Signal flow in one channel.
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