The FIR
perfect-reconstruction (PR) conditions leave some
freedom in the choice of
H
0
z
H
0
z
and
H
1
z
H
1
z
. Orthogonal PR filterbanks are defined
by causal real-coefficient
even-length-NN analysis filters
that satisfy the following two equations:
1=
H
0
z
H
0
z-1+
H
0
-z
H
0
-z-1
1
H
0
z
H
0
z
H
0
-z
H
0
z
(1)
H
1
z=±z-N-1
H
0
-z-1
H
1
z
±
z
N
1
H
0
z
(2)
To verify that these design choices satisfy the FIR-PR
requirements for
H
0
z
H
0
z
and
H
1
z
H
1
z
, we evaluate
detHz
H
z
under the second condition above. This yields
detHz=±
H
0
z
H
1
z-1-
H
0
-z
H
1
z=-z-N-1
H
0
z
H
0
z-1+
H
0
-z
H
0
-z-1=z-N-1
H
z
±
H
0
z
H
1
z
H
0
-z
H
1
z
z
N
1
H
0
z
H
0
z
H
0
-z
H
0
z
z
N
1
(3)
which corresponds to
c=-1
c
-1
and
l=N-1
l
N
1
in the FIR-PR determinant condition
detHz=cz-l
H
z
c
z
l
. The remaining FIR-PR conditions then imply that
the synthesis filters are given by
G
0
z=-2
H
1
z-1=2z-N-1
H
0
z-1
G
0
z
-2
H
1
z
2
z
N
1
H
0
z
(4)
G
1
z=2
H
0
-z=2z-N-1
H
1
z-1
G
1
z
2
H
0
z
2
z
N
1
H
1
z
(5)
The orthogonal PR design rules imply that
H
0
ⅇⅈω
H
0
ω
is "power symmetric" and that
H
0
ⅇⅈω
H
1
ⅇⅈω
H
0
ω
H
1
ω
form a "power complementary" pair. To see the power
symmetry, we rewrite the first design rule using
z=ⅇⅈω
z
ω
and
-1=ⅇ±ⅈπ
-1
±
π
, which gives
1=
H
0
ⅇⅈω
H
0
ⅇ-ⅈω+
H
0
ⅇⅈω-π
H
0
ⅇ-ⅈω-π=|
H
0
ⅇⅈω|2+|
H
0
ⅇⅈω-π|2=|
H
0
ⅇⅈω|2+|
H
0
ⅇⅈπ-ω|2
1
H
0
ω
H
0
ω
H
0
ω
H
0
ω
H
0
ω
2
H
0
ω
2
H
0
ω
2
H
0
ω
2
(6)
The last two steps leveraged the fact that the
DTFT of a
real-coefficient filter is conjugate-symmetric. The
power-symmetry property is illustrated in
Figure 1:
Power complementarity follows from the second orthogonal PR
design rule, which implies
|
H
1
ⅇⅈω|=|
H
0
ⅇⅈπ-ω|
H
1
ω
H
0
ω
. Plugging this into the previous equation, we find
1=|
H
0
ⅇⅈω|2+|
H
1
ⅇⅈω|2
1
H
0
ω
2
H
1
ω
2
(7)
The power-complimentary property is illustrated in
Figure 2:
FIR Perfect-Reconstruction Conditions
Why Filterbanks
DTFT
