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INTERNATIONAL JOURNAL OF CIRCUIT THEORY AND APPLICATIONS
Int. J. Circ. ¹heor. Appl., 26, 539—546 (1998)
LETTER TO THE EDITOR
AN EFFICIENT DESIGN METHOD OF COSINE-MODULATED
QMF BANKS SATISFYING PR PROPERTY
YING TAN1* AND ZHENYA HE2
1Department of Electronic Engineering and Information Science, University of Science and Technology of China,
Hefei 230027, People+s Republic of China
2Department of Radio Engineering, Southeast University, Nanjing 210096, People+s Republic of China
SUMMARY
This paper deals with the design of cosine-modulated QMF banks satisfying the perfect-reconstruction property (PR).
This problem has been formulated as a novel quadratic-constrained least-squares (QCLS) minimization problem in
which all constrained matrices of the QCLS optimization problem are symmetric and positive definite. In order to
efficiently solve this kind of QCLS optimization problems, we construct a cost function that is a convex function of our
desired prototype filter coefficients. Therefore, a global minimizer of this problem can be easily obtained. Moreover, this
design approach has the ability to obtain solutions that are a compromise between PR conditions and stopband
performance. The results of two design examples are presented to support our derivations and analyses. ( 1998 John
Wiley & Sons, Ltd.
1. INTRODUCTION
Multirate signal processing and filter bank systems have found many applications in areas such as subband
coders for speech signal and image coding during the past 20 years. Recently, the perfect reconstruction (PR)
theory has been established.1~3 Among them, the PR cosine-modulated filter bank (PR, CMFB) has
emerged as an attractive choice of filter bank with respect to implementation cost and design saving. The
impulse responses of the analysis and synthesis filters are cosine-modulated versions of a prototype filter. It is
shown in Reference 3 that the 2M polyphase components of the prototype filter can be grouped into
M power-complementary pairs, where each pair is implemented as a two-channel lossless lattice filter bank.
The lattice coefficients are optimized to minimize the stopband attenuation of the prototype filter, but this is
a highly non-linear optimization problem with respect to lattice coefficients. Consequently, it is difficult to
obtain the PR CMFB with high stopband attenuation. Recently, References 6 and 7 formulated the design
problem as a QCLS problem, and obtained the PR CMFB with high stopband attenuation. In this paper, we
at first derive another form of QCLS problem8 for designing the PR CMFB, where all constrained matrices
are symmetric and positive definite. After that, we solve the QCLS optimization problem by penalty function
method. Finally, the PR cosine-modulated QMF banks with high stopband attenuation can also be easily
obtained as illustrated by two examples.
What follows is a summary of the features of existing design methods and our proposed method.
Drawbacks of existing PR CMFB design methods:
(a) It is necessary to provide an appropriate initial approximate solution for the designed filter since it is
very sensitive to the initial approximations.
* Correspondence to: Y. Tan, Department of Electronic Engineering and Information Science, University of Science and Technology of
China, Hefei 230027, People’s Republic of China. Email: ytan@ipc.ustc.edn.cn
CCC 0098—9886/98/050539—08$17.50
( 1998 by John Wiley & Sons, Ltd.
Received 26 July 1997
Revised 19 March 1998
540
LETTER TO THE EDITOR
(b) All the methods follow an iterative procedure, so they share their calculational complexity.
(c) The optimized solution may be a local minima.
(d) A complex non-linear optimization problem should be converted into a linear optimization problem
by linearizing the processing procedure.
(e) Cost function / is a highly non-linear function with respect to the lattice coefficients. Consequently, PR
filler banks, where the analysis filters have high stopband attenuation, are difficult to obtain.
Advantages of our proposed method:
(a) A global optimal solution can be reached as penalty factors approach infinity simultaneously.
(b) No initialization approximation is needed for the method as it has global Lyapunov stability.
(c) The precision of the solution can be controlled by appropriately choosing values of penalty factors. In
other words, the compromise between PR conditions and stopband attenuation can be set by the
penalty factor.
(d) The coefficients of the prototype filter can be optimized directly.
(e) This method can be easily extended to the multidimensional multichannel QMF bank case the
appropriately rearranging the coefficients of the prototype filter.
The rest of this paper is organized as follows. In Section 2, we present the design problem formulation of
paraunitary CMFB. In Section 3, we describe and analyse our method. In Section 4, two design examples are
presented to illustrate the method. Finally, Section 5 concludes this paper.
Notations. Bold-faced letters represent vectors and matrices. Superscript T denotes transposition, and the
‘tilde’ accent on a function is defined such that A3 (z)"A`(1/z*) where superscript astrisk denotes
the conjugation of coefficients. I and J denote k]k identity and reversal matrices respectively. x
x
y denotes
k
k
the largest integer less than or equal to x.
2. DESIGN OF PARAUNITARY COSINE-MODULATED QMF BANKS
2.1. Background for designing an M-channel maximally decimated QMF bank
Figure 1 shows an M-channel maximally decimated QMF bank. QMF banks are used in a wide range of
speech, image, and other applications, which involve the splitting of an input signal into subbands and,
finally, the reconstruction of the original signal. There are two prevalent approaches to M-channel QMF
filter design. They are the perfect reconstruction QMF banks and the pseudo-QMF banks. The latter have an
efficient design procedure (only the prototype filter is designed), but the former achieves perfect reconstruction of the input i.e. without aliasing, magnitude, or phase distortion).
In a cosine-modulated QMF bank, the transfer functions of the analysis and synthesis filters, denoted by
H (z) and F (z) for k"0, 1,2, M!1, respectively, are obtained by modulating the transfer function P(z) of
k
k
a linear-phase lowpass FIR filter prototype P with bandwidth n/(2M). So, the impulse responses of the
Figure 1. M-channel maximally decimated filter bank
( 1998 John Wiley & Sons, Ltd.
Int. J. Circ. ¹heor. Appl., 26, 539—546 (1998)
541
LETTER TO THE EDITOR
analysis and synthesis filters H and F are given by
k
k
A
A
A
A
B B
B B
n
N!1
n!
#h
h (n)"2p(n) cos (2k#1)
k
k
2M
2
(1a)
n
N!1
n!
!h
f (n)"2p(n) cos (2k#1)
k
k
2M
2
(1b)
where n and k are in the range of 0)n)N!1 and 0)k)M!1, respectively, and p(n) is the impulse
response of the prototype filter P. From equation (1), the analysis and synthesis filters are related by
f (n)"h (N!1!n) or F (z)"z~(N~1)HI (z)
k
k
k
k
As shown in equation 2, in order to cancel significant aliasing terms and assume a relatively flat overall
amplitude response, h must be chosen as
k
n
for 0)k)M!1
h "(!1)k
k
4
Here the lengths of H (z) and F (z) are the same, and are assumed to be multiples of 2M, i.e. N"2mM.
k
k
It has been shown in Reference 3 that the PR conditions for M-channel cosine-modulated QMF bank
when the prototype filter is linear-phase FIR can be written as
1
M
 GI (z)G (z)#GI
(z)GI
(z)"
,
0)k) !1, even M
M`k
M`k
 k k
2M
2

M
1
 GM k (z)Gk (z)#GI M`k (z)GM`k (z)" , 0)k)
!1 odd M
2
2M

1
 2GI
(z)G
(z)"

(M~1)@2
(M~1)@2
2M
G
(2)
where G (z) are the type I polyphase components of prototype filter P(z). Generally, for the case where the
k
prototype filter of cosine-modulated QMF bank has arbitrary-length,7 it turn out that the necessary and
sufficient conditions for perfect reconstruction are the same as equation (2). Thus, for convenience, we only
consider the case where the length of P(z) is an arbitrary multiple of 2M because the other cases are very
similar.
2.2. Quadratic-constraint formulation of PR-CM QMF bank design
A convenient and simple filter design method is directly optimizing the impulse response p(n) of prototype
filter P(z). For this purpose, we must rewrite the PR conditions in equation (2) as some quadratic-constraint
form in p(n). According to the polyphase representation of filter bank8, kth polyphase filter of Type I can be
expressed as
m~1
G (z)" + g (n)z~n
k
k
n/0
(3)
The PR conditions for even M case in equation (2) can be expressed as
1
" d
gTS g #gT S g
M`k r M`k M r
k r k
( 1998 John Wiley & Sons, Ltd.
(4)
Int. J. Circ. ¹heor. Appl., 26, 539—546 (1998)
542
LETTER TO THE EDITOR
where g "[g (0), g (1),2, g (m!1)]T is the impulse response vector of the kth polyphase filter G (z),
k
k
k
k
k
g
"[g
(0), g
(1), , g
(m!1)]T is the coefficient vector of (M#k)-th polyphase filter G
(z),
M`k
M`k
M`k 2 M`k
M`k
0 I
0
0
1, r"0
m~r
#
and d "
S"
is the unit-pulse.
r
r
0 0
I
0
0, rO0
m~r
m]m
m]m
Let gT"pTT , where p"[p(0), p(1),2, p(2mM!1)]T which is an (2mM)-dimensional coefficient vector
k
k
of prototype filter P(z) and T ’s (i, j)-entry
k
1, i"2jM#k
[T ] "
,
k i,j
0, otherwise
C
D
C
D
G
G
and T is 2mM]m matrix.
k
Substituting the related equations of above into equation (4), we can obtain
1
pTA p" d
k,r
M r
(5)
S TT is a group of symmetrical matrices since S is a symmetrical matrix for
where A "T S TT#T
r
M`k r M`k
k,r
k r k
each r(r"0,2, m!1) by its definition. In addition, A is also a group of sparse matrices with each column
k,r
or now only containing two 1 s in our current problem.
By checking equation (5), we have known that equation (5) implies a normalized constraint for the
prototype filter, pTp"l. We combined the above two constraints and then obtained
pTp"1,
a
pT[aA #I] p" dr#1
k,r
M
(6)
where a is a constant which is used to guarantee matrices B "aA #I to be diagonal entry dominant
k,r
k,r
when we choose
2mM
2mM
a(max + [Ak,r]i,j!min + [Ak,r]i,j .
i/l
i/l
iOj
For our problem, each column or row of A has only two 1 s. Consequently, if we choose a)0.5, B must
k,r
k,r
be diagonal entry dominant matrices. On the other hand, we have BT "B by checking equation (6). So,
k,r
k,r
B is positive definite by matrix theory.
k,r
By using the linear phase property of the prototype filter, g can be rewritten as gT"pT [I J]T , where
1@2
k
k
k
p "[p(0), p(1),2, p(mM!1)]T. In terms of p , the above equation can be re-expressed as
1@2
1@2
pT C p "1
1@2 k,r 1@2
(7)
where
CD
1
I
a
C " [I J]B
and q " d #1O0
k,r q
k,r J
r M r
r
Obviously, there exist CT "C , that is to say C is a group of symmetric and positive definite matrices.
k,r
k,r
k,r
Note that k goes from 0 to (M/2)!1 and r is in the range of 0 to m!1, therefore there are mM/2 constrained
conditions in equation (7).
In a similar way, PR conditions for the odd M case in equation (2) are given by
pT C p "1,
1@2 k,r 1@2
p "1
pT C
1@2 ((M~1)@2),r 1@2
(8)
where
C
((M~1)@2),r
( 1998 John Wiley & Sons, Ltd.
CD
1
I
a
" [I J] T
S TT
, q"
d #1,
(M~1)@2
(M~1)@2
r
r
q
J
2M r
r
Int. J. Circ. ¹heor. Appl., 26, 539—546 (1998)
543
LETTER TO THE EDITOR
M
M
and, r is in the same range as equation (7) and k goes from 0 to x
y !1, so there are m x
y constrained
2
2
conditions in equation (8).
In summary, the PR conditions in equation (2) are rewritten as quadratic-constraint conditions on p as
1@2
in equations (7) and (8) with normalized condition in which all the constrained matrices are symmetric and
positive definite.
Since the system in this case is a perfect reconstruction, according to PR theory in equation 2 it is sufficient
to only minimize the stopband energy of the prototype filter under the mM/2 quadratic-constrained
conditions and normalized condition. The cost of the stopband energy /": n
D P (e+u) D 2 du can be
(n@2M)`e 0
further simplified as /"pTQp, where Q is a real, symmetric, and positive-definite matrix with (i, j)th entry
P
n
cos(iu!N/2)cos( ju!N/2) du
(9)
(n@2M)`e
In concise form, the optimization problem of prototype filter design for PR cosine-modulated QMF bank
can be restated as
[Q] "4
i,j





min
p
pT Qp
1@2 1@2
1@2
s.t. pT C p "1, pT p "1/2
1@2 1@2
1@2 k,n 1@2
(10)
where index k and n are chosen to have the same value ranges as that in equation (7) or equation (8).
3. PENALTY FUNCTION METHOD FOR QCLS PROBLEM
A cost function for our QCLS minimization problem can be constructed as
l
J(X, b)"XTQX# + b p (X)
(11)
i i
i/1
where b is a positive constant, p (X) is a penalty function satisfying the following conditions: (a) it is
i
i
a continuous and differentiable function of the unknown variable X, (b) p (X)*0, ∀X, (c) p (X)"0 iff
i
i
XTA X"1 such that
i
p (X)"(XTA X!1)2 and b '0
i
i
i
According to penalty function (PS) theory,11 the minimizer of equation (11) is the exact solution of the
original QCLS optimization problem in equation (10) when penalty factor b approaches infinity. Let Mbk,
i
i
i"1, 2,2, l, k"1, 2,2 be l sequences simultaneously tending to infinity such that for each k, bk*0 and
i
bk`1*bk . Further, let Xk be the minimizer of J. Then, we have the following limit theorem.
i
i
¹heorem 1. ¸imit point of the sequence MXkN is a solution of equation (11).
That is, a maximum objective function value exactly satisfying constraints can be arrived at the limit point
of the sequence MXkN. So, the problem in hand has been transformed to an unconstrained non-linear
optimization problem. The solution of the unconstrained non-linear optimization problem of equation (11) is
just the solution of the constrained non-linear optimization problem of equation (10) as b approaches
i
positive infinity. In the following analysis, we assume that b is a fixed positive value.
i
Firstly, we define the feasible solution space as #"MX D QX"+ l (X)A X, where l (X)"
i
i
i i
2b (1!XTA X)N which consists of all the equilibrium points of J.
i
i
¸emma 1. For ∀X , X 3# and X OX , then XTA X "0, i"1, 2,2, l.
1 2
1
2
1 i 2
( 1998 John Wiley & Sons, Ltd.
Int. J. Circ. ¹heor. Appl., 26, 539—546 (1998)
544
LETTER TO THE EDITOR
The proof of Lemma 1 can be found in Reference 8. Clearly, the statement of Lemma 1 means every two
different vectors in feasible region are mutually orthogonal with respect to each constrained matrix A .
i
¹heorem 2. XM is a stable global minimizer of J in equation (11) if and only if XM satisfies:
E XM E "max
E X E , i"1,2, l, where, E X E stands for the weighted norm of X with matrix M, and # is
Ai
X|#
Ai
M
the feasible solution space.
For the proof of Theorem 2 refer to equation 8, the following four corollaries bring out the significant
features of Theorem 2.
Corollary 1. For a given set of b , every local minimizer of J is also a global minimizer.
i
Since the Hessian matrix H(XM ) at every local minimizer XM of J satisfies
l
XM @H(XM )XM " + 8b E XM E 4 *0
Ai
i
i/1
therefore the result of Corollary 1 follows directly from Theorem 2 by recognizing that the Hessian matrix
H(XM ) at point XM is at least positive semidefinite when XM is a local minimizer of J.
Corollary 2. ¹he minimizer of J is unique (except for the sign).
Corollary 3. ¹he equilibrium points except the stable minimizer correspond to saddle points of J.
This follows from the fact that H(XM ) is indefinite at the equilibrium point XM if it is a vector not satisfying
the condition of Theorem 2.
Corollary 4. ¹he minimum energy value of J is J "+ b (1!E XM E4A ) when a global minimizer arrived.
i
.*/
i i
In the next section, we will present two examples to show the efficiency and flexibility of our method, in
which the penalty functions are chosen in the form p (x)"(XTA X!1)2 and all the penalty parameters in
i
i
equation (11) are chosen to have the same values as in our examples.
4. ILLUSTRATIVE EXAMPLES
In order to verify our analysis and derivations in the preceding sections, we designed an 8-channel PR CMFB
as an example using the proposed method. Let M"8 and m"3; the filter length is N"48. For this case,
there are 13 quadratic-constraints and 24 unknown variables. In our PS method, we choose the penalty
factor b "1)0e#03 and give random initial values. Table I gives the filter coefficients of the designed
i
prototype filter by our PS method. The magnitude responses of the designed prototype filter and analysis
bank H (z) are shown in Figures 2(a) and 2(b). In Figure 2(a), the prototype designed by conventional
k
TC — LAT method in Reference 3 is also shown by dashed-line for comparison. It can be seen that the
stopband attenuation of the designed analysis filter is about !40)32 dB, which is about 3 dB hiher than
those designed using conventional approaches.3 Here the designed filter is only approximate PR CMFB
since the penalty factors have finite positive values. The maximum error of the constraints is less than 10~6
which is very small and can be neglected in some areas.
As the second example, we design a 16-channel PR CMFB. We choose larger penalty factors
b"1)0e#06. The magnitude responses of the designed prototype and analysis bank are shown in Figures
3(a) and 3(b), respectively. The maximum error of quadratic constrained conditions is less than 10~9, which
can be ignored in many practical application fields.
( 1998 John Wiley & Sons, Ltd.
Int. J. Circ. ¹heor. Appl., 26, 539—546 (1998)
545
LETTER TO THE EDITOR
Table I. The coefficients of prototype (N"48) of 8-channel PR CMFB by our PS method for
b"1.e#03
n
p (n)
1@2
n
p (n)
1@2
1
2
3
4
5
6
7
8
9
10
11
12
!3)138047627341292e-003
!3)749265236720407e-003
!4)798317517895776e-003
!7)568400992714401e-003
!1)203752077867772e-002
!1)524549626657524e-002
!1)673603975199828e-002
!1)697064663406244e-002
!1)550869354619807e-002
!1)192078528085052e-002
!5)987753050649529e-003
3)222310033767170e-003
13
14
15
16
17
18
19
20
21
22
23
24
1)688249086906735e-002
3)391606968769700e-002
5)340043348450080e-002
7)495679737521221e-002
9)796611627184632e-002
1)216234343259736e-001
1)447886766314165e-001
1)664158030709835e-001
1)852527850735704e-001
2)002476303839510e-001
2)107604016518628e-001
2)162036408160893e-001
Figure 2. The magnitude responses of the designed prototype
(a) and analysis filter bank (b) for 8-channel PR CM-QMF bank
Figure 3. The magnitude responses of the designed prototype
(a) and analysis filter bank (b) for 16-channel PR CM-QMF
bank
5. CONCLUSIONS AND REMARKS
In this paper, a prototype design of PR CMFB is formulated as a new kind of QCLS optimization problem in
which all constrained matrices are symmetric and positive definite. To solve the QCLS minimization
problem, we use a penalty function approach, which has been verified to have Lyapunov global stability, to
design PR cosine-modulated QMF banks. Design examples show the effectiveness of our method.
( 1998 John Wiley & Sons, Ltd.
Int. J. Circ. ¹heor. Appl., 26, 539—546 (1998)
546
LETTER TO THE EDITOR
ACKNOWLEDGEMENTS
The authors would like to thank Dr X. Q. Gao, University of Southeast, for many helpful discussions. The
authors also thank the anonymous reviewers for their constructive comments on the manuscript. This work
was supported in part by the Climbing Programme—National Key Project for Fundamental Research in
China, Grant NSC92097 and partly by National Natural Foundation of China.
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perfect reconstruction’, IEEE ¹rans. Signal Processing, SP-44(3), 473—483 (1996).
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( 1998 John Wiley & Sons, Ltd.
Int. J. Circ. ¹heor. Appl., 26, 539—546 (1998)
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