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(a)
H
L
(
z
)
2
H
L
(
z
)
2
H
L
(
z
)
2
(b)
A
(
z
)
8
F I GU R E 14 . 21
Equivalent structures for recursive filtering using a two-band split.
This constitutes the analysis bank. The subband signals
y
k
(
are encoded and transmitted.
At the synthesis stage the subband signals are then decoded, upsampled by a factor of
L
by interlacing adjacent samples with
L
n
)
1 zeros, and then passed through the synthesis or
interpolation filters. The output of all these synthesis filters is added together to obtain the
reconstructed signal. This constitutes the synthesis filter bank. Thus, the analysis and synthesis
filter banks together take an input signal
x
−
(
n
)
and produce an output signal
x
ˆ
(
n
)
. These filters
could be any combination of FIR and IIR filters.
Depending on whether
M
is less than, equal to, or greater than
L
, the filter bank is called
an
underdecimated, critically (maximally) decimated
,or
overdecimated
filter bank. For most
practical applications, maximal decimation or “critical subsampling” is used.
A detailed study of
M
-band filters is beyond the scope of this chapter. Suffice it to say that
in broad outline much of what we said about two-band filters can be generalized to
M
-band
filters. (For more on this subject, see [
206
].)
14.8 The Polyphase Decomposition
A major problem with representing the combination of filters and downsamplers is the time-
varying nature of the up- and downsamplers. An elegant way of solving this problem is with
the use of
polyphase decomposition
. In order to demonstrate this concept, let us first consider
the simple case of two-band splitting. We will first consider the analysis portion of the system
shown in Figure
14.22
. Suppose the analysis filter
H
1
(
z
)
is given by
h
1
z
−
1
h
2
z
−
2
h
3
z
−
3
H
1
(
z
)
=
h
0
+
+
+
+···
(93)
H
1
(
z
)
2
H
2
(
z
)
2
F I GU R E 14 . 22
Analysis portion of a two-band subband coder.
