Digital Signal Processing Reference
In-Depth Information
(a)
(b)
wn
2
()
wn
2
()
xn
2
()
H
2
()
xm
2
()
G
2
()
xn
()
xm
2
()
↑
M
↓
M
f
f
f
f M
/
sM
/
f
f M
s
f
f
s
s
sM
s
s
s
(c)
Xf
()
Xf
2
()
(h)
f
f
f
/2
f
/2
s
s
(d)
(g)
Wf
2
()
Wf
2
()
f
f
f
/2
f
/2
s
s
Xf
2
()
Xf
2
()
(e)
(f)
f
f
f
f M
/(2
)
f
f M
/(2
)
sM
s
sM
s
FIGURE 13.4
Analysis and synthesis stages for channel 2.
components as
W
2
ðzÞ
at the analyzer.
A
fter
w
2
ðnÞ
is filtered by the synthesis bandpass filter,
G
2
ðzÞ
,we
get the reconstructed bandpass signal
x
2
ðnÞ
.
The process in channel 3 is similar to that in channel 1 with the spectral reversal effect and is
illustrated in
Figure 13.5
.
Now let us examine the theory. Without quantization of subband channels, perfect reconstruction of
the filter banks (see
Figure 13.1
)
depends on the analysis and syntheses filter effects. To develop the
perfect reconstruction required of the analysis and synthesis filters, consider a signal in a single
channel flowing up to the synthesis filter in general as depicted in
Figure 13.6
.
sampling rate, that is,
WðzÞ¼HðzÞXðzÞ
(13.1)
x
d
ðmÞ
is the downsampled version of
wðnÞ
while
wðnÞ
is the interpolated version of
wðnÞ
prior to the
synthesis filter and can be expressed as
(
wðnÞ
n ¼
0
; M;
2
M;
.
wðnÞ¼
(13.2)
0
otherwise
Using a delta function
dðnÞ
, that is,
dð
0
Þ¼
1 for
n ¼
0 and
dðnÞ¼
0 for
n
s
0, we can write
wðnÞ
as
"
N
k ¼
0
dðn kMÞ
#
wðnÞ¼
wðnÞ¼iðnÞwðnÞ
(13.3)
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