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In-Depth Information
phase distortion can be completely eliminated. As discussed earlier, choosing
K
1
(
z
)
=−
H
2
(
−
z
)
K
2
(
z
)
=
H
1
(
−
z
)
(75)
eliminates aliasing. This leaves us with
1
2
[
T
(
z
)
=
H
1
(
−
z
)
H
2
(
z
)
−
H
1
(
z
)
H
2
(
−
z
)
]
In the approach due to Smith and Barnwell [
205
] and Mintzer [
209
], with
N
an odd integer,
we select
z
−
N
H
1
(
−
z
−
1
H
2
(
z
)
=
)
(76)
so that
1
2
z
−
N
z
−
1
z
−
1
T
(
z
)
=
[
H
1
(
z
)
H
1
(
)
+
H
1
(
−
z
)
H
1
(
−
)
]
(77)
Therefore, the perfect reconstruction requirement reduces to finding a prototype low-pass filter
H
(
z
)
=
H
1
(
z
)
such that
z
−
1
z
−
1
Q
(
z
)
=
H
(
z
)
H
(
)
+
H
(
−
z
)
H
(
−
)
=
constant
(78)
Defining
z
−
1
R
(
z
)
=
H
(
z
)
H
(
)
(79)
the perfect reconstruction requirement becomes
Q
(
z
)
=
R
(
z
)
+
R
(
−
z
)
=
constant
(80)
But
R
(
z
)
is simply the Z-transform of the autocorrelation sequence of
h
(
n
)
. The autocor-
relation sequence
ρ(
n
)
is given by
N
ρ(
n
)
=
h
k
h
k
+
n
(81)
k
=
0
The Z-transform of
ρ(
n
)
is given by
N
(
)
=
Z
[
ρ(
)
]=
Z
(82)
R
z
n
h
k
h
k
+
n
k
=
0
We can express the sum
k
=
0
h
k
h
k
+
n
as a convolution:
N
h
n
⊗
h
=
h
k
h
k
+
n
(83)
−
n
k
=
0
Using the fact that the Z-transform of a convolution of two sequences is the product of the
Z-transforms of the individual sequences, we obtain
z
−
1
R
(
z
)
=
Z
[
h
n
]
Z
[
h
−
n
]=
H
(
z
)
H
(
)
(84)
