1042
IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING,
VOL. 37.
NO.
7,
JULY
1989
Improved Technique for Design of Perfect
Reconstruction
FIR
QMF Banks with
Lossless Polyphase Matrices
P.
P.
VAIDYANATHAN,_SENIOR
MEMBER,
IEEE,
TRUONG Q. NGUYEN,
STUDENT
MEMBER,
IEEE,
ZINNUR DOGANATA, STUDENT MEMBER,
IEEE,
AND TAP10 SARAMAKI
Abstract-This paper develops an improved technique for the design
of analysis filters in an
M
channel maximally decimated FIR perfect
reconstruction
QMF
bank, having a lossless polyphase-component ma-
trix
E(z).
As
in earlier work, the aim is to optimize the parameters
characterizing
E(z)
until the sum of the stopband energies
of
the anal-
ysis filters is minimized. There are four new ingredients in the proce-
dure reported here. The first is a technique for efficient initialization
of one of the
M
analysis filters, as a spectral factor of an Mth band
filter. This factorization itself is done without root-finding techniques,
in an efficient manner using the eigenjlters approach. The second com-
ponent is the initialization of the internal parameters which character-
ize
E(z),
based on the above spectral factor. In earlier work, the pa-
rameters characterizing the lossless
E(
z)
were rotation angles, which
resulted in slow convergence of the above-mentioned optimization. The
third ingredient of the improved approach is a modified characteriza-
tion, mostly free from rotation angles, of the lossless FIR
E(z).
The
fourth component of improvement is the incorporation of symmetry
among the analysis filters,
so
as to minimize the number of unknown
parameters being optimized. The resulting design procedure always
gives better filter responses than earlier ones
(for
a given filter length),
and converges much faster.
I. INTRODUCTION
N this paper we consider the maximally decimated
I
M-channel analysis/synthesis system shown in Fig. 1.
The basic operational principles of this analysis/synthesis
system and its applications are discussed in a number of
references
[
11-[ 131. The analysis filters
Hk
(z)
split the
signal
x
(n)
into
M
subband signals which are then deci-
mated by
M
and encoded prior to transmission. At the
synthesis end, the
M
subband signals are decoded, inter-
polated, and recombined using the synthesis filters
Fk
(2).
Ignoring the nonlinear coding/decoding error and quan-
tization errors in the filter implementations, the signal
i(n)
suffers from three errors [5], viz., aliasing, ampli-
tude distortion, and phase distortion. Several techniques
have been discussed in the past for eliminating (partially
or completely) some or all of these errors (see [1]-[13]).
In this paper we shall restrict
our
attention to one such
technique, and incorporate several improvements.
Manuscript received May 30, 1988; revised November
25,
1988. This
work was supported
in
part by the National Science Foundation under
Grants
DCI
8552579 and
MIP
8604456.
P.
P.
Vaidyanathan, T. Q. Nguyen, and
Z.
Doganata are with the De-
partment of Electrical Engineering, California Institute of Technology,
Pasadena,
CA
91 125.
T.
Saramaki is with the Department of Electrical Engineering, Tampere
University of Technology, SF-33
101
Tampere, Finland.
IEEE
Log Number 8928127.
Fig.
1.
The M-channel maximally decimated QMF bank
In the literature, two-channel versions of the system of
Fig. 1 have been referred to as the
quadrature mirrorfil-
ter
(QMF) banks
[
11-[4]. This is because the magnitude
responses
I
HI(
e'")
1
and
I
Ho(
ej")
I
are
images
of
each
other with respect to the frequency a/2 which is a
quarter
of the sampling frequency 2a. For the case of general
M,
the structure of Fig. 1 should not actually be called the
QMF bank because the traditional 2-channel meaning does
not hold. However, as the word QMF has been used in
the past by other authors, we shall retain the same jargon.
The technique we shall discuss here is the one described
in [12] (also see [13]). Here, all three distortions men-
tioned above are eliminated,
so
that
i
(n)
is a delayed ver-
sion of
x(n),
i.e.,
i(n)
=
cx(n
-
no),
c
#
0.
This is
called the perfect reconstruction (abbreviated PR) prop-
erty, and the QMF bank of Fig.
1
is then said to be a PR
system.
The method in [12] is based on the following observa-
tion. Each analysis filter
Hk
(
z)
can be written in the form
Hk(z)
=
Cki'
z-/Ekl(zM),
and each synthesis filter
Fk(z)
can be written in the
form
Fk(z)
=
EM-'
/=0
ZF-
-'h
rk(zM).
The quantities
Ekl(z),
0
I
1
I
M
-
1 are the
M
polyphase components [14], [2] of the kth
analysis filter
Hk(z).
Similarly,
Rlk(z)
are the
M
poly-
phase components of
Fk(z).
These components can be
used to define two
M
X
M
matrices
E(z)
=
[
Ekl(z)] and
R
(z)
=
[
Rlk
(z)
1, called the polyphase component matri-
ces for the analysis bank and synthesis bank, respectively.
With these definitions, the QMF bank of Fig. 1 can al-
ways be redrawn as in Fig. 2. The method described in
[12] constrains the matrix
E(z)
to be FIR and lossless
(i.e.,
E(eJw)
to be unitary for all
U).
Under this condi-
tion, if the matrix
R(z)
is chosen as'
R(z)
=
I?(,),
then
'See the end
of
this section for meanings of A',
A,
etc
0096-35 18/89/0700- 1042$01
.OO
0
1989 IEEE
VAIDYANATHAN
er
ul.:
DESIGN OF PERFECT RECONSTRUCTION
FIR
QMF
BANKS
____
1043
z'
Fig.
2.
Redrawing
of
Fig.
1
in
terms
of
polyphase matrices
E(z)
and
R(z).
the system of Fig. 1 is forced to be a PR system. With
lossless
E(z),
this choice of
R(z)
is equivalent to choos-
ing the synthesis filters as
so
that
fk(
n)
are time-reversed (and conjugated) versions
of
h,(n).
Thus, once the analysis filters
hk(n),
0
5
k
5
M
-
1
are known, we can findfk(n),
0
5
k
5
M
-
1,
of the PR system easily, without matrix inversions.
No-
tice that, with this setup
{
Fk(z)
}
are also FIR filters of
the same lengths as
{
Hk
(z)
}
's.
The central problem in such a design of a PR QMF
bank is therefore the design of analysis filters
Hk
(z)
under
the constraint that
E(z)
be lossless. With
E(z)
con-
strained to be lossless, the aim is to minimize the sum of
stopband energies
M5'
k=O
j
stopband
IHk(elw)12
dw
(2)
by optimizing the parameters characterizing
E(
z).
The
passbands of
Hk
(z)
automatically come out to be good for
reasons mentioned in
[12].
A.
Summary
of
Past Designs
In the above optimization, the objective function (2) is
highly nonlinear with respect to the parameters of
E(z).
Moreover, no convenient
initialization point
for the pa-
rameters of
E(z)
was available in the past. Finally, the
parameters in
E(z)
were rotation angles 1121, which re-
quired the computation of several cosines and sines in or-
der to evaluate (2) for a given parameter set. These facts
resulted in very slow convergence of the optimization
process. Because of the possibility of multiple minima, it
was also necessary to perform optimization with several
random initial points. For these reasons, the designs in
[
121 were restricted to low-order analysis filters. For ex-
ample, the three channel design in
[
121 has analysis filters
of length 15, each filter having a stopband attenuation of
only about 20 dB.
B.
Some Recent Improvements
It was soon observed 1151 that, if the analysis filters
were further constrained to have painvise image property
(3)
(H,(ei")(
=
IHM-I-,
(
e;(w
-
"')
1
this reduces the number
of
degrees of freedom [i.e., the
number of parameters in
E(z)]
for a given filter length,
resulting in faster optimization and better filters. Such a
design example for
M
=
3,
with analysis filter lengths
equal to 62, can be found in
[
151, providing stopband at-
tenuations of about
35
dB. The initialization of parame-
ters in [15], however, continued to be random.
Subsequent to this, it was realized that the initialization
of parameters can actually be done more judiciously
[
161.
This is based on several facts. First, if
E(z)
is lossless,
then each analysis filter
Hk(z)
is a spectral factor of an
Mth band filter (i.e.,
Hk(z)
Hk(z) are Mth band filters).
Second, if any one of the analysis filters, say,
Ho(z),
is
fixed at a certain value, then the lossless constraint of
E(z)
takes away most of the freedom available for the
choice of
Hk(z),
1
I
k
I
M
-
1. Finally, if
Ho(z)
is
somehow initialized, there exists a
synthesis procedure
to
find (i.e., initialize) the majority of parameters of
E(z).
This leads to a substantial improvement in the design
speed, and in the filter performance upon convergence.
The design example for
M
=
3,
reported in
[
161, is based
on this scheme, coupled with the imposition of the sym-
metry property
(3).
The analysis filters have length 56 and
stopband attenuations exceeding 70 dB.
The final phase of improvement is based on a renewed
characterization of an FIR lossless matrix
E(z).
The ear-
lier characterizations
[
171,
[
181 were in the form of a cas-
cade of constant unitary-matrix building blocks, separated
by delay elements. The unitary matrices are themselves
characterized by rotation angles 1171. Even though this
characterization is completely general
[
181 (in the sense
that every lossless FIR matrix can be realized in this form)
and canonic (i.e., has the minimum number of delay ele-
ments and parameters for a given filter length), the pres-
ence of angles makes it necessary to compute several co-
sines and sines, during each computation of
(2).
On most
general purpose computers, the computation of a cosine
(or sine) is about 20 times slower than a multiplication
operation. This is, therefore, a major computational over-
head, while carrying
out
the optimization on most ma-
chines.
A
second characterization of FIR lossless matri-
ces was recently outlined
[
191, which again is general and
canonic, but is mostly free from rotation angles. This,
coupled with the initialization techniques, and the impo-
sition of symmetry
(3),
has now emerged into a more ef-
ficient algorithm for the design of analysis filters
Hk
(z)
of an FIR PR system.
C.
Outline
of
the Paper
The purpose of this paper is to give a comprehensive
presentation of these results.
As
many of these improve-
ments have been reported only in conference proceedings
[
161,
[
191,
1201
(which are incomplete due to space con-
straints), our presentation here will be self-contained (with
the exception that the results in the recent TRANSACTIONS
paper [15] will be freely used). Section I1 describes our
technique to initialize an analysis filter of the PR QMF
bank. Section I11 develops a new minimal characterization
for
M
X
M
FIR lossless systems and for
M
X
1
FIR loss-
less vectors. Section IV derives the number of degrees of
freedom available in the design of an
M
x
M
FIR lossless
system, if one row is fixed (corresponding to fixing or
initializing
an analysis filter). Section V ties up these im-
1044
IEEE
TRANSACTIONS ON ACOUSTICS,
SPEECH,
AND SIGNAL PROCESSING,
VOL. 31,
NO.
7.
JULY
1989
provements with the imposition of pairwise symmetry of
analysis filters
[15]. A design example is presented in
Section VI.
Notations:
The downgoing and upgoing arrows in Fig.
1
represent decimators and interpolators as defined in any
one of
[2],
[I
11-[13].
Boldfaced italic letters denote ma-
trices. The symbol
Z
denotes the identity matrix, whose
dimensions will be clear from the context (if not, a sub-
script will be used to indicate it). The notations
AT, A*,
and
At
denote transposition, conjugation, and transposed-
conjugation, respectively. For an
M
X
1
vector
U,
the
norm is denoted as
(1
U
11.
A
p
X
M
matrix
A
(
p
1
M)
is said to be orthogonal if
ATA
=
CZ,
and unitary if
AtA
=
CZ,
where
c
#
0
is scalar. Subscript
*
denotes
conjugation of coefficients only. Thus, if
E(z)
=
eo
+
e,z-',
then
E,(z)
=
e:
+
efz-'.
Finally,
A(z)
stands
for
Hi(z-I).
For real-coefficient systems,
A(z)
=
H*(z-I).
In any case, on the unit circle, we have
A(z)
=
Ht(z).
A
p
x
M
transfer matrix
E(z)
is said to be
lossless if a) its entries
&(z)
are stable and b)
E(eJ")
is
unitary for all
W.
For
p
=
M,
this property is a discrete-
time version of the well-known property of scattering ma-
trices of
LC multiports in electrical network theory [29],
[30]. (A lossless matrix with real coefficients is said to be
LBR.) Such a system automatically satisfies the property
E(z)
E(z)
=
cz
for all
z,
(4)
where
c
#
0
is a constant. For
E(z)
to be lossless, we
require
p
I
M.
If
p
=
M,
note that (4) is equivalent to
E(z)
E(z)
=
cZ. Note that product of lossless matrices
is lossless. The
M
components of an
M
X
1 lossless vec-
tor are said to be power complementary (since the mag-
nitude-squares add up to a constant for all
U).
Note that
every column of a lossless system is lossless (hence power
complementary).
The degree of a
p
X
M
system (also called McMillan
degree
[22])
E(
z)
is equal to the number of scalar delays
(i.e.,
I-'
building blocks) required to implement it. Even
though it is difficult to find the degree by inspection, it is
simple to find in some cases. For example, the degree of
an
M
x
1
FIR
transfer matrix
~(z)
G
c:=~
h(n)z-"
with
h
(K)
#
0
is equal to
K.
The degree of an
M
X
M
system of the form
zP1A
is equal to the rank
r
of
A
be-
cause we can write
z-
'A
=
B[z-~Z,]
C
where B is
M
X
r and
C
is r
x
M.
11. INITIALIZATION OF
AN
ANALYSIS FILTER
The set of
M
analysis filters
Hk(z),
expressed in terms
of the polyphase components
Ekl
(z),
can be represented
as in Fig. 3(a). Since
Hk(z)
are FIR, the entries of
E(z)
are FIR. In our method, we constrain
E(z)
to be lossless.
The discussions in this section assume
&(z)
[hence
E(z)] to have real coefficients.
A
structural representa-
tion for
E(z),
developed in
[17]
and
[IS],
is shown in
Fig. 3(b). This is a cascade of
N
(real) orthogonal matri-
ces
Rk,
0
I
k
5
N
-
1
[where
N
-
1
is the degree of
E(z)
(b)
Fig.
3.
Implementation
of
FIR lossless
E(z)
as a cascade
of
unitary ma-
trices separated
by
delays.
(b)
Fig.
4.
Details
of
the building
blocks
in Fig.
3(b).
In Fig.
4(b),
Tk
is a
(k
+
1)
X
(k
+
1)
matrix with appearance as in Fig. 4(a). The matrix
Tk
has
k
criss-crosses.
E(z)]
separated by diagonal matrices of the form
0
**-
0
1
.
***
.
0
.].
...
. ..
1
0
...
Recall (Appendix
D)
that an arbitrary
M
x
M
real or-
thogonal matrix requires
(:)
real parameters (rotation an-
gles) for complete characterization. The
N
-
1
matrices
Rk,
0
I
k
5
N
-
2 are special types of orthogonal ma-
trices with
M
-
l planar rotations, as shown in Fig. 4(a).
In this figure, each criss-cross represents a planar rotation
operation of the form
cos
0
sin
0
[
sin
0
-cos
0
1.
The rightmost matrix
RN-
on the other hand, is a general
orthogonal matrix characterized by
( :)
planar rotations
(see Appendix
D) as in Fig. 4(b). It is shown in
1181
that
VAIDYANATHAN
E[
NI.:
DESIGN OF PERFECT RECONSTRUCTION FIR QMF
BANKS
every
real-coefficient FIR lossless system
E(
z)
of degree
N
-
1
can be represented as in Fig. 3(b) with matrices as
in Fig.
4.
Conversely, the structure of Fig. 3(b) with ma-
trices as in Fig. 4 necessarily represents a real-coeffi-
cient FIR lossless system. Thus, the angles in Fig. 4 form
a complete set for characterizing real-coefficient FIR loss-
less
E(z).
Moreover, the representation in Fig. 3(b) is
canonic (i.e., minimal in number of delays, and in num-
ber of planar rotation parameters).
A. The Parameter Space and the Number
of
Degrees
of
Freedom
The total number of planar rotation angles in Fig. 3(b)
is
In order to minimize the objective function (2), it is nec-
essary to optimize these
Np
parameters. From Fig. 3(b),
the maximum length of an analysis filter with this setup
is
L'
=
M(N
-
1)
+
M.
(7)
Clearly, the number of parameters
N,,
grows linearly with
respect to
L'
and quadratically with respect to
M.
For ex-
ample, with
M
=
3 and analysis filters of length
L'
=
57,
we have
Np
=
39. We thus have a large parameter space,
and a very nonlinear objective function. With no clue for
initialization of angles in Fig. 3(b), the optimization task
is formidable indeed.
In Section 111 we shall show that if we can make an
initial guess of one of the length
L'
transfer functions, say
HO(z),
then almost all the
Np
parameters can be initialized
based only on
Ho(z).
To be more specific, once
HO(z)
is
fixed, only
Nf=
(r)
-
(M
-
1)
parameters are still undetermined in Fig. 3(b). Equation
(8)
therefore measures the number of degrees of freedom
available for the design of
Hk(z),
1
I
k
I
M
-
1,
once
HO(z)
has been fixed. This
is
a consequence of the fact
that losslessness of
E
(z
)
puts a constraint on
Hk
(z),
k
#
0,
once
HO(z)
is fixed.
For example, with
M
=
2, we have
Nf
=
0
which shows
that we have no freedom of choice of
H,(z)
once
Ho(z)
is fixed. This is consistent with the earlier observations
[5], [12] that, for the
M
=
2 case, losslessness of
E(z)
completely constrains
H,(z)
to be
H,(z)
=
zp'""HO(
-zpi).
For the
M
=
3 case,
Nf
=
1
so
that,
once
HO(z)
is fixed, only one degree of freedom can be
exercised in choosing
Hl(z)
and
H2(z).
It is important to
realize that
Nf
is
independent
of
the filter length
L'.
Con-
sequently, if
HO(z)
is known, we have to optimize only
the remaining
Nf
parameters, regardless of filter length,
to obtain
Hk(z),
1
I
k
I
M
-
1. (In practice, we have
the option of reoptimizing all the parameters, after ini-
tialization of a subset of parameters based on the initial
choice of
Ho(
z
).
)
In order to exploit this to our advantage, it
is
first nec-
essary to find an appropriate initialization for
Ho(z),
which is the purpose of this section. We can mathemati-
cally express Fig. 3(a) as
(9)
where
and
(11)
(qz)
=
[I
z-l
.
. .
z-(M-I)]q
From the fact that
E(z)
is lossless, it can be proved (see
Appendix A) that the function
Gk(Z)
fik(Z)
Hk(z)
(12)
is an Mth band filter [23], [32], i.e., it satisfies
M-
1
c
Gk(zW')
=
c
/=O
where
W
=
e-J2a/M
and
c
is a nonzero constant. Con-
versely, given an arbitrary FIR filter
Ho(
z)
such that
GO(
z)
is an Mth band filter, there will exist FIR filters
Hk(z),
1
I
k
I
M
-
1,
such that the resulting
E(z)
is
lossless
(as we shall see in Section
IV).
Thus, one way to initialize
HO(z)
would be to first de-
sign a low-pass Mth band filter
GO(z)
and then compute a
spectral factor
Ho(z)
of
Go(z).
Of course, it is necessary
to ensure that
GO(eJw)
is nonnegative for all
U,
so
that a
spectral factor
HO(z)
exists. Such an Mth band filter can
be obtained as follows: first design an Mth band equirip-
ple zero-phase FIR filter
G(z)
as in [23], by using the
McClellan-Parks program [24]. Then define
Go(
z)
=
G(z)
+
6
where
6
is the peak stopband ripple of
G(eJw),
so
that
G0(e1@)
has all the desired properties. Finally,
compute the coefficients of
Ho(z)
such that
GO(z)
=
HO(z-l)
Ho(z)
(this is the spectral factorization step).
If
Ho(z)
is required to have a stopband attenuation of
70
dB (as an arbitrary example), then
GO(
z)
has stopband
attenuation 140 dB. This implies that
G(z)
has large or-
der, and has several zeros in the stopband. Under such
conditions, the spectral factorization
is
an inaccurate pro-
cess, and gives rise to numerical difficulties even when
clever techniques such as [25] are employed. In this sec-
tion we show how
Ho(
z)
with the above properties can be
directly
designed, thereby eliminating the spectral facto-
rization step. According to our experience, this direct ap-
proach always gives rise to more accurate and faster de-
signs for the filter
HO(
z).
B.
The Eigenjlter Approach for Initialization
of
Ho(
z)
Any FIR transfer function
Ho(z)
can be written as
HO(Z)
=
f&(z)
HOI(Z)
(14)
I046
IEEE TRANSACTIONS
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ACOUSTICS, SPEECH. AND SIGNAL PROCESSING,
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37.
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7.
JULY
1Y89
where
Hol(z)
has all zeros on the unit circle, and
Hoo(z)
has none on the unit circle. If
Ho(
z)
is a spectral factor of
Go(z),
then we can write
Go(z)
=
Gdz)
(15)
where
Go0(z)
has no zeros on the unit circle. The form
(15)
reflects the fact that zeros of
Go(z)
on the unit circle
must be
double
so that
Go(ej")
is nonnegative for all W.
Since
Go(z)
has zero phase,
Goo(z)
has zeros occurring
in reciprocal pairs.
Our aim is
to
design
Ho(z)
such that its stopband energy
is minimized under the constraint that
HO(z-l)
Ho(z)
is
an Mth band filter. This is equivalent to minimizing the
quantity
E
=
lT
"S
IGm(eJ")l
IHol(eJ")12
dw
(17)
under the constraint that
Go(
z)
be an Mth band filter. The
advantage of writing
Go(z)
in the form
(15)
and opti-
mizing (17) is that the resulting design will automatically
reveal the coefficients of
Hol(z)
and
Goo(z).
If we take
Hoo(z)
to be a spectral factor of
Goo(z),
the design of
Ho(z)
is complete. This involves finding only the spectral
factor of
Goo(
z)
which is a much lower order polynomial
than
Go(z).
In addition,
Goo(z)
has
no
zeros
on the unit
circle. It is therefore easy to find a spectral factor
Hoo(z)
In what follows, we shall denote the orders of
Hoo(z)
and
Ho,(
z)
by
lo
and
11,
respectively. Accordingly,
Goo(
z)
has order 210 and
Go(z)
has order
2(
lo
+
lI).
Notice that
in the above setup
Goo(z)
and
Hol(z)
are linear-phase
transfer functions. Assume, for a moment, that the coef-
ficients of
Go&)
are known. Given the function
Goo(z),
this task of finding the linear-phase polynomial
Ho,(
z)
to
minimize
(
17) is precisely the
weighted eigenjilter design
problem described in
[28]
(to be elaborated below). While
minimizing (17) we shall impose a constraint that avoids
the trivial solution
Hol(z)
=
0.
Now assume that, for a given
Goo(z),
we have found
the coefficients of
Hal(
z)
using the eigenfilters approach.
With
Hol(z)
fixed at this value, we can recompute the
coefficients of
Goo(z)
such that the quantity
Go(z)
in
(15)
is indeed an Mth band filter. With this new
Goo(z),
we
can again solve for
Hol(z)
to minimize
(17).
A few rep-
etitions (typically four or five) result in an Mth band filter
Go(
z)
with excellent stopband attenuation. Once
Goo(
z)
and
Hol(z)
are found in this manner, it is only necessary
to find a spectral factor
Hoo(z)
of
Goo(z).
The analysis
filter
Ho(z)
is then obtained as in
(14).
Summarizing, the
steps in the design of
Ho(z)
are as follows.
of
Goo(
z
1.
1) Initialize
Goo(
el")
to be unity for all
w.
2) For this
Goo(z),
find the linear phase transfer func-
tion
Ho,(z)
of order
I,
such that (17) is minimized (under
appropriate constraints to be elaborated).
3) With
HO,(z)
so
fixed, find the linear phase transfer
function
Goo(z)
of order 210 such that the product
(15)
is
an Mth band filter.
4) If the result
Go(z)
is not satisfactory,
go
to step
2.
Even though there is no formal proof that this approach
converges, excellent designs for
Go(
z)
could be obtained
in every case attempted. At most, five iterations between
steps
2
and 3 were necessary in all cases. The details of
steps
2
and 3 are described next.
The Eigenjilter Approach
(281
for Step
2:
Let
Ho,(z)
be a real-coefficient linear-phase FIR filter [26]
of
the form
/I
&I(z)
=
C
hol(n)z-"
(18)
h01(n)
=
hOl(,l
-
.I.
(19)
I1
=
0
with
Assume first that
lI
is even and let
MI
=
11/2.
We can
express the frequency response as
[26], [27, p. 721
MI
HoI(e'")
=
e-JMIw
C
b,,
cos
(con).
(20)
The coefficients
b,,
are given by
bo
=
h(M,)
and
b,,
=
2h(M1
-
n),
n
#
0.
If we define the vectors
,1=0
c(w)
=
[l
cos
(U)
* *
-
cos
(M,w)]'
(21)
we can express (20) as
HoI(e.j")
=
e-/'"""b'c(w).
Now
consider the quantity
where
W(eJw)
is a nonnegative real function of
W.
Sup-
pose we wish to minimize
(22)
under the constraint
b'b
=
1.
(23
1
We can rewrite
(22)
as
G
=
b'[
jT
W(e/")
c(w)
c'(w)
dw
b.
(24)
The integral in
(24)
is an
(MI
+
1)
x
(MI
+
I)
matrix
P.
We can rewrite (24) as
"Y
1
E
=
b'Pb.
(25)
Clearly, the matrix
P,
which is real and symmetric, is
positive definite (except under trivial situations such a5
ws
=
T,
or
W(eJ")
I
Hol(
eJw)
1'
=
0
for all
U).
The vector
b
minimizing
(25)
under the constraint (23) is the eigen-
vector of
P
corresponding to its minimum eigenvalue.
Once
b
is found in this way, the coefficients of
Hol(z)
which minimize
(22)
can be obtained. For obvious rea-
sons,
Hol(z),
so found, will be called an eigenfilter. It is
clear that with
W(
e'")
taken as
I
Goo(
el")
1,
we can find
the optimal linear-phase
Hol(
z)
using this eigenfilter ap-
proach.