Design of orthogonal pr-fir filterbanks via halfband
spectral factorization
Recall that analysis-filter design for orthogonal PR-FIR
filterbanks reduces to the design of a real-coefficient causalFIR prototype filter
that satisfies the power-symmetry condition
Power-symmetric filters are closely related to "halfband"
filters. A zero-phase halfband filter is a zero-phase filter
with the property
When, in addition,
has real-valued coefficients, its DTFT is
"amplitude-symmetric":
The amplitude-symmetry property is illustrated in
:
If, in addition to being real-valued,
Recall that zero-phase filters have real-valued DTFTs.
is non-negative, then
constitutes a valid power response. If we can find
such that
,
then this
will satisfy the desired power-symmetry property
.
First, realize
is easily modified to ensure non-negativity: construct
for sufficiently large
, which will raise
by
uniformly over
(see
).
The resulting
is non-negative and satisfies the amplitude-symmetry condition
.
We will make up for the additional gain later. The procedureby which
can be calculated from the raised halfband
, known as
spectral factorization , is
described next.
Since
is conjugate-symmetric around the origin, the roots of
come in pairs
. This can be seen by writing
in the factored form below, which clearly
corresponds to a polynomial with coefficientsconjugate-symmetric around the
-order coefficient.
where
. Note that the complex numbers
are symmetric across the unit circle in the
z-plane .
Thus, for ever root of
inside the unit-circle, there exists a root outside of theunit circle (see
).
Let us assume, without loss of generality, that
. If we form
from the roots of
with magnitude less than one:
then it is apparent that
. This
is the so-called
minimum-phase spectral factor of
.
Actually, in order to make
, we are not required to choose all roots inside the
unit circle; it is enough to choose one root from everyunit-circle-symmetric pair. However, we do want to ensure
that
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 (
).
Because
has real-valued coefficients, we know that its roots satisfythis conjugate-symmetry property. Then forming
from the roots of
that are strictly inside (or strictly outside) the unitcircle, we ensure that
has real-valued coefficients.
Finally, we say a few words about the design of the halfband
filter
. The
window design method is one technique that
could be used in this application. The window design methodstarts with an ideal lowpass filter, and windows its
doubly-infinite impulse response using a window function withfinite time-support. The ideal real-valued zero-phase halfband
filter has impulse response (where
):
which has the important property that all even-indexedcoefficients except
equal zero. It can be seen that this latter property is
implied by the halfband definition
since, due to odd-coefficient cancellation, we find
Note that windowing the ideal halfband does not alter the
property
, thus the window design
is guaranteed to be halfband feature. Furthermore,
a real-valued window with origin-symmetry preserves thereal-valued zero-phase property of
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
radians and all passband/stopband specifications
are symmetric with respect to
.
Design procedure summary
We now summarize the design procedure for a
length-
analysis lowpass
filter for an orthogonal perfect-reconstruction FIRfilterbank:
Design a zero-phase real-coefficient halfband lowpass
filter
where
is a
positive even integer (via,
e.g. ,
window designs, LS, or equiripple).
Calculate
,
the maximum negative value of
.
(Recall that
is real-valued for all
because it has a
zero-phase response.) Then create "raised halfband"
via
, ensuring that
, forall
.
Compute the roots of
, which should come in unit-circle-symmetric
pairs
. Then collect the roots with magnitude less than
one into filter
.
is the desired prototype filter except for a
scale factor. Recall that we desire
Using
Parseval's Theorem , we see that
should be scaled to give
for which
.
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