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polyphase representation. Thus,
z
2
z
−
1
G
11
(
z
2
G
1
(
z
)
=
G
10
(
)
+
)
(102)
z
2
z
−
1
G
21
(
z
2
G
2
(
z
)
=
G
20
(
)
+
)
(103)
Consider the output of the synthesis filter
G
1
(
z
)
given an input
Y
1
(
z
)
. From Equation (
41
),
the output of the upsampler is
z
2
U
1
(
z
)
=
Y
1
(
)
(104)
and the output of
G
1
(
z
)
is
z
2
V
1
(
z
)
=
Y
1
(
)
G
1
(
z
)
(105)
z
2
z
2
z
−
1
Y
1
(
z
2
z
2
=
Y
1
(
)
G
10
(
)
+
)
G
11
(
)
(106)
The first term in the equation above is the output of an upsampler that
follows
a filter
with transfer function
G
10
(
z
2
z
2
z
)
with input
Y
(
z
)
. Similarly,
Y
1
(
)
G
11
(
)
is the output of an
upsampler that follows a filter with transfer function
G
11
(
z
)
with input
Y
(
z
)
. Thus, this system
can be represented as shown in Figure
14.26
.
Putting the polyphase representations of the analysis and synthesis portions together, we
get the system shown in Figure
14.27
. Looking at the portion in the dashed box, we can see
that this is a completely linear time-invariant system.
The polyphase representation can be a very useful tool for the design and analysis of filters.
While many of its uses are beyond the scope of this chapter, we can use this representation to
prove our statement about the two-band perfect reconstruction quadrature mirror filters.
Recall that we want
1
2
[
cz
−
n
0
T
(
z
)
=
H
1
(
z
)
H
2
(
−
z
)
−
H
1
(
−
z
)
H
2
(
z
)
]=
G
10
(
z
)
2
z
-1
G
11
(
z
)
2
G
20
(
z
)
2
z
-1
G
21
(
z
)
2
F I GU R E 14 . 26
Polyphase representation of the synthesis portion of a two-band
subband coder.
