Databases Reference
In-Depth Information
and
H
2
(
z
)
=
P
2
(
−
z
)
(57)
Although these filters are perfect reconstruction filters, for applications in data compression
they suffer from one significant drawback. Because these filters may be of unequal bandwidth,
the output of the larger bandwidth filter suffers from severe aliasing. If the output of both bands
is available to the receiver, this is not a problem because the aliasing is canceled out in the
reconstruction process. However, in many compression applications we discard the subband
containing the least amount of energy, which will generally be the output of the filter with
the smaller bandwidth. In this case the reconstruction will contain a large amount of aliasing
distortion. In order to avoid this problem for compression applications, we generally wish to
minimize the amount of aliasing in each subband.
Quadrature mirror filters
(QMF) are a class
of filters that are useful in this situation. We look at these filters in the next section.
14.6.1 Two-Channel PR Quadrature Mirror Filters
Before we introduce quadrature mirror filters, let's rewrite Equation (
48
)as
X
(
z
)
=
T
(
z
)
X
(
z
)
+
S
(
z
)
X
(
−
z
)
(58)
where
1
2
[
H
1
(
(
)
=
)
K
1
(
)
+
H
2
(
)
K
2
(
)
(59)
T
z
z
z
z
z
]
1
2
[
H
1
(
−
S
(
z
)
=
z
)
K
1
(
z
)
+
H
2
(
−
z
)
K
2
(
z
)
]
(60)
In order for the reconstruction of the input sequence
{
x
n
}
to be a delayed, and perhaps scaled,
version of
{
x
n
}
, we need to get rid of the aliasing term
X
(
−
z
)
and have
T
(
z
)
be a pure delay.
To get rid of the aliasing term, we need
S
(
z
)
=
0
,
∀
z
From Equation (
60
), this will happen if
K
1
(
z
)
=
H
2
(
−
z
)
(61)
K
2
(
z
)
=−
H
1
(
−
z
)
(62)
After removing the aliasing distortion, a delayed version of the input will be available at
the output if
cz
−
n
0
T
(
z
)
=
c
is a constant
(63)
Replacing
z
by
e
j
ω
, this means that we want
=
e
j
ω
)
T
(
constant
(64)
e
j
ω
))
=
arg
(
T
(
K
w
K
constant
(65)
