It is possible to design combinations of analysis and
synthesis filters such that the aliasing from
downsampling/upsampling is completely cancelled. Below we
derive aliasing-cancellation conditions for two-channel
filterbanks. Though the results can be extended to M-channel
filterbanks in a rather straightforward manner, the
two-channel case offers a more lucid explanation of the
principle ideas (see Figure 1).
The aliasing cancellation conditions follow directly from the
input/output equations derived below. Let
i∈01
i
0
1
denote the filterbank branch index. Then
U
i
z=12∑p=01
H
i
z12e(−i)πpXz12e(−i)πp
U
i
z
1
2
p
0
1
H
i
z
1
2
p
X
z
1
2
p
(1)
Yz=∑i=01
G
i
z
U
i
z2=∑i=01
G
i
z12∑p=01
H
i
ze(−i)πpXze(−i)πp=12∑i=01
G
i
z(
H
i
zXz+
H
i
−zX−z)=12(
XzX−z
)(
H
0
z
H
1
z
H
0
−z
H
1
−z
)(
G
0
z
G
1
z
)
Y
z
i
0
1
G
i
z
U
i
z
2
i
0
1
G
i
z
1
2
p
0
1
H
i
z
p
X
z
p
1
2
i
0
1
G
i
z
H
i
z
X
z
H
i
z
X
z
1
2
X
z
X
z
H
0
z
H
1
z
H
0
z
H
1
z
G
0
z
G
1
z
(2)
where
Hz=(
H
0
z
H
1
z
H
0
−z
H
1
−z
)
H
z
H
0
z
H
1
z
H
0
z
H
1
z
.
Hz
H
z
is often called the
aliasing component matrix.
For aliasing cancellation, we need to ensure that
X−z
X
z
does not contribute to the output
Yz
Y
z
.
This requires that
(
H
0
−z
H
1
−z
)(
G
0
z
G
1
z
)=
H
0
−z
G
0
z+
H
1
−z
G
1
z=0
H
0
z
H
1
z
G
0
z
G
1
z
H
0
z
G
0
z
H
1
z
G
1
z
0
which is guaranteed by
G
0
z
G
1
z=−
H
1
−z
H
0
−z
G
0
z
G
1
z
H
1
z
H
0
z
(3)
or by the following pair of conditions for any rational
Cz
Cz
G
0
z=Cz
H
1
−z
G
0
z
C
z
H
1
z
G
1
z=(−Cz)
H
0
−z
G
1
z
C
z
H
0
z
Under these aliasing-cancellation conditions, we get the
input/output relation
Yz=12(
H
0
z
H
1
−z−
H
1
z
H
0
−z)CzXz
Y
z
1
2
H
0
z
H
1
z
H
1
z
H
0
z
C
z
X
z
(4)
where
Tz=12(
H
0
z
H
1
−z−
H
1
z
H
0
−z)Cz
T
z
1
2
H
0
z
H
1
z
H
1
z
H
0
z
C
z
represents the system transfer function. We say
that "perfect reconstruction" results when
yn=xn−l
y
n
x
n
l
for some
l∈N
l
, or equivalently when
Tz=z−l
T
z
z
l
.
The aliasing-cancellation conditions remove one degree of
freedom from our filterbank design; originally, we had the
choice of four transfer functions
H
0
z
H
1
z
G
0
z
G
1
z
H
0
z
H
1
z
G
0
z
G
1
z
, whereas now we can choose three:
H
0
z
H
1
zCz
H
0
z
H
1
z
C
z
.
Why Filterbanks