How can we choose the two filters
h
0
n
h
0
n
,
h
1
n
h
1
n
so that the two-channel analysis filter bank represents an
orthonormal transformation of
xn
x
n
(so that
QTQ=I
Q
Q
I
)? In other words, how do we choose the filters so as
to ensure
yn=xn
yn
xn
?
Look at
Yz
Y
z
in terms of
Xz
X
z
. Using multirate identities, we get:
Yz=12
H
0
1z
H
0
zXz+
H
0
-zX-z+12
H
1
1z
H
1
zXz+
H
1
-zX-z
Y
z
1
2
H
0
1
z
H
0
z
X
z
H
0
z
X
z
1
2
H
1
1
z
H
1
z
X
z
H
1
z
X
z
(1)
or
Yz=12
H
0
z
H
0
1z+
H
1
z
H
1
1z+12
H
0
-z
H
0
1z+
H
1
-z
H
1
1z
Y
z
1
2
H
0
z
H
0
1
z
H
1
z
H
1
1
z
1
2
H
0
z
H
0
1
z
H
1
z
H
1
1
z
(2)
If
yn=xn
yn
xn
, then
QTQ=I
Q
Q
I
and the analysis filter bank represents an orthonormal
transform.
We have
yn=xn
yn
xn
only if
Yz=Xz
Yz
Xz
so we must have:
H
0
z
H
0
1z+
H
1
z
H
1
1z=2
H
0
z
H
0
1
z
H
1
z
H
1
1
z
2
(3)
H
0
-z
H
0
1z+
H
1
-z
H
1
1z=0
H
0
z
H
0
1
z
H
1
z
H
1
1
z
0
(4)
The problem is to find
H
0
z
H
0
z
and
H
1
z
H
1
z
that satisfy these two
perfect reconstruction
conditions.
Suppose we set
H
1
z=z-L
H
0
-1z
H
1
z
z
L
H
0
1
z
(5)
with
LL odd, or, equivalently,
h
1
n=-1L-n
h
0
L-n
h
1
n
1
L
n
h
0
L
n
Then the left hand side of
Equation 4
becomes
H
0
-z
H
0
1z+-z-L
H
0
1zzL
H
0
-z
H
0
z
H
0
1
z
z
L
H
0
1
z
z
L
H
0
z
or (because
LL is odd)
H
0
-z
H
0
1z-
H
0
1z
H
0
-z
H
0
z
H
0
1
z
H
0
1
z
H
0
z
which is zero. So with the choice for
H
1
z
H
1
z
if
Equation 5 the condition
Equation 4 is automatically satisfied.
Equation 3 becomes
H
0
z
H
0
1z+z-L
H
0
-1zzL
H
0
-z=2
H
0
z
H
0
1
z
z
L
H
0
1
z
z
L
H
0
z
2
or
H
0
z
H
0
1z+
H
0
-z
H
0
-1z=2
H
0
z
H
0
1
z
H
0
z
H
0
1
z
2
(6)
So we need to choose
H
0
z
H
0
z
so as to satisfy this equation. For convenience,
define
r
0
n
r
0
n
to be the autocorrelation of
h
0
n
h
0
n
,
r
0
n≔
h
0
n*
h
0
-n
≔
r
0
n
h
0
n
h
0
n
or equivalently
R
0
z≔
H
0
z
H
0
1z
≔
R
0
z
H
0
z
H
0
1
z
(7)
Then
Equation 6 becomes
R
0
z+
R
0
-z=2
R
0
z
R
0
z
2
or equivalently
r
0
n+-1n
r
0
n=2δn
r
0
n
1
n
r
0
n
2
δ
n
Note that
r
0
n+-1n
r
0
n=2δn⇔
r
0
2n=δn
⇔
r
0
n
1
n
r
0
n
2
δ
n
r
0
2
n
δ
n
That means, the autocorrelation of the filter
h
0
n
h
0
n
must be a halfband filter,
r
0
2n=δn
r
0
2
n
δ
n
(8)
We can therefore obtain
h
0
n
h
0
n
by performing spectral factorization of
r
0
n
r
0
n
. Given
r
0
n
r
0
n
, we can find
h
0
n
h
0
n
so as to satisfy
Equation 7
provided that
Rⅇⅈω≥0
R
ω
0
for all
ωω and
provided that
r
0
n
r
0
n
is symmetric
(
r
0
n=
r
0
-n
r
0
n
r
0
n
).
In conclusion, the two-channel analysis filter bank can be an
orthogonal transform if the filters
H
0
z
H
0
z
and
H
1
z
H
1
z
satisfy the appropriate conditions (the perfect reconstruction
condition Equation 6).
