Digital Signal Processing Reference
In-Depth Information
Channel
Channel
0
0
xn
0
()
xm
0
()
xm
0
()
1
xn
()
xn
1
()
1
xm
1
()
xm
1
()
xn
()
+
2
2
xn
2
()
xm
2
()
f
xm
2
()
f
s
s
3
3
xn
3
()
xm
3
()
xm
3
()
f
=
f M
/
sM
s
Aanalysis stage Synthesis stage
(a)
(b)
FIGURE 13.1
Filter bank framework with an analyzer and synthesizer.
synthesis filter bank are then combined to reconstruct the original signal
xðnÞ
at the original sampling
rate
f
s
. Each channel essentially generates a bandpass signal. The decimated signal spectrum for
channel 0 can be achieved via a standard downsampling process, while the decimated spectra of
other channels can be obtained using the principle of undersampling of bandpass signals with an
integer band (discussed in Section 12.5), where the inherent frequency aliasing or image properties
of decimation and interpolation are involved. The theoretical development will follow next. With
a proper design of analysis and synthesis filter banks, we are able to achieve perfect reconstruction of
the original signal.
Let us examine the spectral details of each band (subband).
Figure 13.2
depicts the spectral
information of the analysis and synthesis stages, as shown in
Figure 13.2
(
a) and (b).
H
0
ðzÞ
and
G
0
ðzÞ
are the analysis and synthesis filters of channel 0, respectively. At the analyzer (
Figure 13.2
(c) to (e)),
xðnÞ
is bandlimited by a lowpass filter
H
0
ðzÞ
to get
w
0
ðnÞ
and decimated by
M ¼
4 to o
bt
ain
x
0
ðmÞ
.At
the synthesizer (
Figure 13.2
(
f) to (h)),
x
0
ðmÞ
is upsampled by a factor of 4 to obtai
n
w
0
ðnÞ
and then
goes through the anti-aliasing (synthesis) filter
G
0
ðzÞ
to achieve the lowpass signal
x
0
ðnÞ
.
Figure 13.3
depicts the analysis and synthesis stages for channel 1 (see
Figure 13.3
(
a) and (b)).
H
1
ðzÞ
and
G
1
ðzÞ
are the bandpass analysis and synthesis filters, respectively. Similarly, at the analyzer
(
Figure 13.3
(c) to (e)),
xðnÞ
is filtered by a bandpass filter
H
1
ðzÞ
to get
w
1
ðnÞ
and decimated by
M ¼
4
to obtain
x
1
ðmÞ
. Since the lower frequency edge of
W
1
ðzÞ
is
f
c
=B ¼
1
¼
odd number, where
f
c
¼ f
s
=ð
2
MÞ¼B
,
f
c
corresponds to the carrier frequency, and
B
is the baseband bandwidth as
depicted in Section 12.5, the reversed spectrum in the baseband results in
Figure 13.3
(
e). However, this
is not a probl
em
, since at the synthesizer as shown in
Figures 13.3
(f) and (g), the spectral reve
rs
al occurs
again so that
W
1
ðzÞ
will have the same spectral components as
W
1
ðzÞ
at the analyzer. After
w
1
ðnÞ
goes
through the anti-aliasing (synthesis) filter
G
1
ðzÞ
, we achieve the reconstructed bandpass signal
x
1
ðnÞ
.
Figure 13.4
describes the analysis and synthesis stages for channel 2. At the analyzer
(
Figure 13.4
(
c) to (e)),
xðnÞ
is filtered by a bandpass filter
H
2
ðzÞ
to get
w
2
ðnÞ
and decimated by
M ¼
4
to obtain
x
2
ðmÞ
. Similarly, considering the lower frequency edge of
W
2
ðzÞ
as
f
c
¼
2
ðf
s
=ð
2
MÞÞ ¼
2
B
,
f
c
=B ¼
2
¼
even. Therefore, we obtain the nonreversed spectrum in
th
e baseband as shown in
Figure 13.4
(f). At the synthesizer shown in
Figure 13.4
(
g), the spectrum
W
2
ðzÞ
has the same spectral
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