The QMF design
choices prevented the design of a useful
(i.e., frequency selective)
perfect-reconstruction (PR) FIR filterbank. This motivates us
to re-examine PR filterbank design without the
overly-restrictive QMF conditions. However, we will still
require causal FIR filters with real coefficients.

Recalling that the two-channel
filterbank (Figure 1),

has the input/output relation:

Yz=12(
XzX-z
)(
H
0
z
H
1
z
H
0
−z
H
1
−z
)(
G
0
z
G
1
z
)
Y
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

(1)
we see that the delay-

ll perfect
reconstruction requires

(
2z−l
0
)=(
H
0
z
H
1
z
H
0
−z
H
1
−z
)(
G
0
z
G
1
z
)
2
z
l
0
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
or, equivalently, that

(
G
0
z
G
1
z
)=H -1z(
2z−l
0
)=1detHz(
H
1
−z−
H
1
z
−
H
0
−z
H
0
z
)(
2z−l
0
)=2detHz(
z−l
H
1
(−z)
−(z−l
H
0
−z)
)
G
0
z
G
1
z
H
z
-1
2
z
l
0
1
H
z
H
1
z
H
1
z
H
0
z
H
0
z
2
z
l
0
2
H
z
z
l
H
1
z
z
l
H
0
z

(3)
where

detHz=
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

(4)
For FIR

G
0
z
G
0
z
and

G
1
z
G
1
z
, we require
that

detHz=cz−k
H
z
c
z
k

(5)
for

c∈R
c
and

k∈Z
k
. Under this determinant condition, we find that

G
0
z
G
1
z=2z−(l−k)c
H
1
−z−
H
0
−z
G
0
z
G
1
z
2
z
l
k
c
H
1
z
H
0
z

(6)
Assuming that

H
0
z
H
0
z
and

H
1
z
H
1
z
are causal with non-zero initial coefficient, we
choose

k=l
k
l
to keep

G
0
z
G
0
z
and

G
1
z
G
1
z
causal and free of unnecessary delay.

Summarizing the two-channel FIR-PR conditions:
H
0
z∧
H
1
z= causal real-coefficient FIR
H
0
z
H
1
z
causal real-coefficient FIR
∀
c
,c∈R∧l∈Z:detHz=cz−l
c
c
l
H
z
c
z
l
G
0
z=2c
H
1
−z
G
0
z
2
c
H
1
z
G
1
z=-2c
H
0
−z
G
1
z
-2
c
H
0
z