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In-Depth Information
2
H
10
(
z
)
G
10
(
z
)
2
z
-1
z
-1
2
H
11
(
z
)
G
11
(
z
)
2
2
H
20
(
z
)
G
20
(
z
)
2
z
-1
z
-1
2
H
21
(
z
)
G
21
(
z
)
2
F I GU R E 14 . 27
Polyphase representation of the two-band subband coder.
If we impose the mirror condition
H
2
(
z
)
=
H
1
(
−
z
),
T
(
z
)
becomes
H
1
(
1
2
H
1
(
−
(
)
=
)
−
)
(107)
T
z
z
z
The polyphase decomposition of
H
1
(
z
)
is
z
2
z
−
1
H
11
(
z
2
H
1
(
z
)
=
H
10
(
)
+
)
Substituting this into Equation (
107
)for
H
1
(
z
)
and
z
2
z
−
1
H
11
(
z
2
H
1
(
−
z
)
=
H
10
(
)
−
)
for
H
1
(
−
z
)
, we obtain
2
z
−
1
H
10
(
z
2
z
2
T
(
z
)
=
)
H
11
(
)
(108)
can have the form
cz
−
n
0
Clearly, the only way
T
(
z
)
is if both
H
10
(
z
)
and
H
11
(
z
)
are simple
delays; that is,
h
0
z
−
k
0
H
10
(
z
)
=
(109)
h
1
z
−
k
1
H
11
(
z
)
=
(110)
This results in
2
h
0
h
1
z
−
(
2
k
0
+
2
k
1
+
1
)
T
(
z
)
=
(111)
which is of the form
cz
−
n
0
as desired. The resulting filters have the transfer functions
h
0
z
−
2
k
0
h
1
z
−
(
2
k
1
+
1
)
H
1
(
z
)
=
+
(112)
h
0
z
−
2
k
0
h
1
z
−
(
2
k
1
+
1
)
H
2
(
z
)
=
−
(113)
