Recall that analysis-filter design for orthogonal PR-FIR
filterbanks reduces to the design of a real-coefficient causal
FIR prototype filter
H
0
z
H
0
z
that satisfies the power-symmetry condition
|
H
0
ⅇⅈω|2+|
H
0
ⅇⅈπ−ω|2=1
H
0
ω
2
H
0
ω
2
1
(1)
Power-symmetric filters are closely related to "halfband"
filters. A zero-phase halfband filter is a zero-phase filter
Fz
F
z
with the property
Fz+F-z=1
F
z
F
-z
1
(2)
When, in addition,
Fz
F
z
has real-valued coefficients, its DTFT is
"amplitude-symmetric":
Fⅇⅈω+Fⅇⅈπ−ω=1
F
ω
F
ω
1
(3)
The amplitude-symmetry property is illustrated in
Figure 1:
If, in addition to being real-valued,
Fⅇⅈω
F
ω
is non-negative, then
Fⅇⅈω
F
ω
constitutes a valid power response. If we can find
H
0
z
H
0
z
such that
|
H
0
ⅇⅈω|2=Fⅇⅈω
H
0
ω
2
F
ω
,
then this
H
0
z
H
0
z
will satisfy the desired power-symmetry property
|
H
0
ⅇⅈω|2+|
H
0
ⅇⅈπ−ω|2=1
H
0
ω
2
H
0
ω
2
1
.
First, realize
Fⅇⅈω
F
ω
is easily modified to ensure non-negativity: construct
qn=fn+εδn
q
n
f
n
ε
δ
n
for sufficiently large
ɛɛ, which will raise
Fⅇⅈω
F
ω
by εε uniformly over
ωω (see Figure 2).
The resulting
Qz
Q
z
is non-negative and satisfies the amplitude-symmetry condition
Qⅇⅈω+Qⅇⅈπ−ω=1+2ε
Q
ω
Q
ω
1
2
ε
.
We will make up for the additional gain later. The procedure
by which
H
0
z
H
0
z
can be calculated from the raised halfband
Qz
Q
z
, known as spectral factorization, is
described next.
Since
qn
q
n
is conjugate-symmetric around the origin, the roots of
Qz
Q
z
come in pairs
a
i
1
a
i
¯
a
i
1
a
i
. This can be seen by writing
Qz
Q
z
in the factored form below, which clearly
corresponds to a polynomial with coefficients
conjugate-symmetric around the
0
th
0
th
-order coefficient.
Qz=∑n=-(N−1)N−1qnz-n=A∏i=1N−11−
a
i
z-11−
a
i
¯z
Q
z
n
N
1
N
1
q
n
z
n
A
i
1
N
1
1
a
i
z
1
a
i
z
(4)
where
A∈
ℝ
+
A
ℝ
+
. Note that the complex numbers
a
i
1
a
i
¯
a
i
1
a
i
are symmetric across the unit circle in the
z-plane.
Thus, for ever root of
Qz
Q
z
inside the unit-circle, there exists a root outside of the
unit circle (see
Figure 3).
Let us assume, without loss of generality, that
|
a
i
|<1
a
i
1
. If we form
H
0
z
H
0
z
from the roots of
Qz
Q
z
with magnitude less than one:
H
0
z=A∏i=1N−11−
a
i
z-1
H
0
z
A
i
1
N
1
1
a
i
z
(5)
then it is apparent that
|
H
0
ⅇⅈω|2=Qⅇⅈω
H
0
ω
2
Q
ω
. This
H
0
z
H
0
z
is the so-called
minimum-phase spectral factor of
Qz
Q
z
.
Actually, in order to make
|
H
0
ⅇⅈω|2=Qⅇⅈω
H
0
ω
2
Q
ω
, we are not required to choose all roots inside the
unit circle; it is enough to choose one root from every
unit-circle-symmetric pair. However, we do want to ensure
that
H
0
z
H
0
z
has real-valued coefficients. For this, we must ensure that
roots come in conjugate-symmetric pairs,
i.e., pairs having symmetry with respect to
the real axis in the complex plane (Figure 4).
Because
Qz
Q
z
has real-valued coefficients, we know that its roots satisfy
this conjugate-symmetry property. Then forming
H
0
z
H
0
z
from the roots of
Qz
Q
z
that are strictly inside (or strictly outside) the unit
circle, we ensure that
H
0
z
H
0
z
has real-valued coefficients.
Finally, we say a few words about the design of the halfband
filter
Fz
F
z
. The window design method is one technique that
could be used in this application. The window design method
starts with an ideal lowpass filter, and windows its
doubly-infinite impulse response using a window function with
finite time-support. The ideal real-valued zero-phase halfband
filter has impulse response (where
n∈ℤ
n
):
f
¯
n=sinπ2nπn
f
¯
n
2
n
n
(6)
which has the important property that all even-indexed
coefficients except
f
¯
0
f
¯
0
equal zero. It can be seen that this latter property is
implied by the halfband definition
F
¯
z+
F
¯
z-1=1
F
¯
z
F
¯
z
1
since, due to odd-coefficient cancellation, we find
1=
F
¯
z+
F
¯
z-1=2∑m=-∞∞
f
¯
2mz-2m⇔
f
¯
2m=12δm
1
F
¯
z
F
¯
z
⇔
2
m
f
¯
2
m
z
2
m
f
¯
2
m
1
2
δ
m
(7)
Note that windowing the ideal halfband does not alter the
property
f
¯
2m=12δm
f
¯
2
m
1
2
δ
m
, thus the window design
Fz
F
z
is guaranteed to be halfband feature. Furthermore,
a real-valued window with origin-symmetry preserves the
real-valued zero-phase property of
f
¯
n
f
¯
n
above. It turns out that many of the other popular
design methods (
e.g., LS and equiripple)
also produce halfband filters when the cutoff is specified at
π2
2
radians and all passband/stopband specifications
are symmetric with respect to
ω=π2
ω
2
.
We now summarize the design procedure for a
length-NN analysis lowpass
filter for an orthogonal perfect-reconstruction FIR
filterbank:
-
Design a zero-phase real-coefficient halfband lowpass
filter
Fz=∑n=-(N−1)N−1fnz-n
F
z
n
N
1
N
1
f
n
z
n
where NN is a
positive even integer (via, e.g.,
window designs, LS, or equiripple).
-
Calculate
ε
ε,
the maximum negative value of
Fⅇⅈω
F
ω
.
(Recall that
Fⅇⅈω
F
ω
is real-valued for all
ωω because it has a
zero-phase response.) Then create "raised halfband"
Qz
Q
z
via
qn=fn+εδn
q
n
f
n
ε
δ
n
, ensuring that
Qⅇⅈω≥0
Q
ω
0
, forall ωω.
-
Compute the roots of
Qz
Q
z
, which should come in unit-circle-symmetric
pairs
a
i
1
a
i
¯
a
i
1
a
i
. Then collect the roots with magnitude less than
one into filter
H
^
0
z
H
^
0
z
.
-
H
^
0
z
H
^
0
z
is the desired prototype filter except for a
scale factor. Recall that we desire
|
H
0
ⅇⅈω|2+|
H
0
ⅇⅈπ−ω|2=1
H
0
ω
2
H
0
ω
2
1
Using Parseval's Theorem, we see that
h
^
0
n
h
^
0
n
should be scaled to give
h
0
n
h
0
n
for which
∑n=0N−1
h
0
2n=12
n
0
N
1
h
0
n
2
1
2
.
