Multicriterion optimized QMF bank design

2582 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 51, NO. 10, OCTOBER 2003

Multicriterion Optimized QMF Bank DesignHoang Duong Tuan, Member, IEEE, Le Hai Nam, Hoang Tuy, and Truong Q. Nguyen, Senior Member, IEEE

Abstract—The quadrature mirror filter (QMF) bank with mul-ticriterion constraints such as minimal aliasing and/or minimalerror coding is among the most important problems in filterbankdesign, for solving which linear algebra-based methods are stillheuristic and do not always work, especially for large filter length.It is shown in this paper that this problem can be reduced either toconvex linear matrix inequality (LMI) optimization (when filtersare of nonlinear phase) or to semi-infinite linear (SIP) program-ming (when filters are of linear phase), which can be very efficientlysolved either by the standard LMI solvers or our previously devel-oped SIP solver. The proposed computationally tractable optimiza-tion formulations are confirmed by several simulations.

Index Terms—Global optimization, linear matrix inequality(LMI), QMF bank.

I. INTRODUCTION

QUADRATURE mirror filter (QMF) bank is a very im-portant class in digital filterbank because of its poten-tial applications in subband coding [16], [23], [24] and

spread spectrum signal processing for intercept communication[14]. The starting point is the two-channel QMF bank designproblem, which can be formulated as follows (see Fig. 1).

Given two analysis filters and , design two (syn-thesis) filters and such that

for some

(1)

(2)

It is well known that (1) and (2) deal with the perfect reconstruc-tion of the filterbank in the absence of the noises : (1) isthe distortion-free condition, while (2) is the aliasing-free one.To accomplish this common performance measure, the standarddesign procedure (see e.g., [1, p. 138]) is first to make a partic-ular choice , for the syn-thesis filters to satisfy the aliasing-free condition (2), and then,

are designed to complement the perfect reconstruction con-dition (1). With this procedure, there may not be much freedomleft for the designed analysis filters to satisfy other perfor-mance measures to make QMF bank reliable for practical ap-plications. The above procedure offers much more freedom for

Manuscript received January 29, 2002; revised March 5, 2003. The associateeditor coordinating the review of this paper and approving it for publication wasProf. Arnab K. Shaw.

H. D. Tuan and L. H. Nam are with the Department of Electrical andComputer Engineering, Toyota Technological Institute, Nagoya 468-8511,Japan (e-mail: [email protected]; [email protected]).

H. Tuy is with the Institute of Mathematics, Nghia Do, Tu Liem, Hanoi,Vietnam (e-mail: [email protected]).

T. Q. Nguyen is with the Department of Electrical and Computer Engineering,University of California at San Diego, La Jolla, CA 92093-0407 USA (e-mail:[email protected]).

Digital Object Identifier 10.1109/TSP.2003.816877

Fig. 1. QMF filter bank with two analysis and two synthesis channels.

the design, and as will be shown later, it allows us to imposeother typical constraints for our filterbank to give it the desiredcharacteristics.

By changing to , , 1, if necessary, without lossof generality, we can set . Thus, a natural optimizationformulation for handling constraints (1) and (2) is as follows:

(3)where in (11) denotes the standard -norm (see its def-inition in the Appendix), i.e., we try to minimize the distortioneffect while preserving the aliasing-free condition.

Note that (2) implies that

(4)

with some . Substituting this value into (3), one gets theequivalent optimization formulation of (3)

(5)

with

(6)

When is assumed stable proper rational (i.e.,), (5) is just a model-matching or Nehari problem [7],

which has been well studied in the classical control, andthus, the optimal solution of (3) can be easily found [3].However, the order of this optimal solution is too high,and generally, the resulting synthesis filters are requiredto be of infinite impulse response (IIR) even when the analysisfilters are of finite impulse response (FIR). The classical fre-quency-domain based techniques of control are unable toaddress the case of optimal fixed order of. In fact, nowadays,the main unsolved obstacle for making control theory morereliable tool for practical purpose is the high order of its op-timal solutions. Therefore, for computational implementation,one has to use the heuristic balanced truncation technique (seee.g., [2]) to reduce the order of solution [8]. Moreover, it isimpossible within this approach to impose additional constraintsfor multiperformances.

1053-587X/03$17.00 © 2003 IEEE

TUAN et al.: MULTICRITERION OPTIMIZED QMF BANK DESIGN 2583

In this paper, we propose a most natural approach to problem(5) by considering it just as a filtering problem, as it actuallyis. The rationale is that the filtering problem is simpler thanthe general control problem and, therefore, can be moreproperly and efficiently handled than the latter [18]. Actually,our purpose is twofold:

• to show that all the above-mentioned difficulties can beovercome in the filtering approach;

• more importantly, to show that a variety of additional con-straints to give the designed filterbank desired character-istics such as no distortion, aliasing energy minimization,constrained nonlinearity minimization and coding errorminimization, which cannot be handled by the above men-tioned existing approaches, can be dealt with by our ap-proach relatively easily.

Because of space limitations, in what follows, we will restrictour consideration to the case of FIR analysis and synthesis fil-ters. It should be noted that for this case, the optimization for-mulation in (5) may lead to a very difficult problem because

in (4) generally is of IIR, even when are of FIR.For more on IIR filter design, see [20].

The structure of the paper is as follows. Section II gives dif-ferent optimization formulations for the QMF bank design prob-lems. The semi-definite programming-based solution methodfor nonlinear phase filters is presented in Section III, whereaslinear programming solution methods for linear-phase filters aretreated in Section IV. Section V is specially devoted to the de-velopment of an optimization algorithm for solving the errorcoding minimization of QMF. Some simulation examples de-scribed in Section VI confirm the viability of our theoreticaldevelopment in the previous sections. For convenience, someformulas and proofs are given in the Appendix.

II. OPTIMIZATION-BASED DESIGN FORMULATIONS

First, let us mention that the optimal solution ,of optimization problem (3) is not unique, and actually, there isa set of such optimal solutions. Thus, assuming that the optimalvalue of (3) is , its optimal solution can be fully characterizedby constraint (2) and the inequality

(7)

Under these constraints, we can tackle more challenging QMFbank design problems such as the following:

• Aliasing energy minimization [1, pp. 295]:

subject to (8)

where is the autocorrelation function of the inputsignal , which is the commonly used AR(1) processwith intersample autocorrelation coefficient so that

, , and is thefrequency domain representation of .

• Constrained nonlinearity minimization [1, pp. 296]:

subject to (2) and (7) (9)

where are the coefficients of , , 1,which can be easily seen as linear functions in the coef-ficients of the synthesis filters (see (43) and (45) inAppendix A).

It is clear that the objectives in (8) and (9) are convex quadraticin the coefficients of synthesis filters , , whereas insubsequent sections, (2) and (7) will be shown to be convex inthese coefficients as well. Consequently, both (8) and (9) areconvex optimization problems, which can be solved efficientlyby existing optimization codes. We now formulate another im-portant problem of coding error minimization, which is non-convex but nevertheless can be very efficiently solved by themethod to be developed in Section V.

Let and in Fig. 1 be the noise source by quanti-zation, which are assumed wide sense stationary (WSS), white,uncorrelated, and uniformly distributed with zero means. Letdenote the number of bits per sample of the-subband signal

with variance .Then, the variance of is given by withsome constant[9]. Thus, the distortion caused by quantizationis

where is the standard -norm (see the definition in theAppendix B). Now, under the fixed bit rate assumption

const, the coding error minimization problem can beformulated as

(2), (7)

(2), (7)

(10)

Note that a similar objective also arises with the coding gainmaximization problem as well [10]. Now, the objective func-tion (10) is nonconvex, so the optimization problem (10) seemsto be a very difficult one. Nevertheless, as it will be shown inSection V, (10) belongs to the realm of concave and monotonicoptimization [21], [22] and based on this fact, we can developan extremely efficient algorithm that requires just a few convexor linear programs to be solved.


III. N ONLINEAR PHASE FIR FILTER DESIGN

Suppose that the FIR analysis filters and are of-order, i.e.,

(11)

Then, using (44) in Appendix A, FIR filter defined by (6)is expressed as

(12)

where

(13)

Accordingly, the synthesis filters and are designedamong the classes of FIR with some order:

(14)

First, by (44) in Appendix A

(15)

Hence, (2) is equivalent to the following linear constraints on:

(16)

On the other hand, using (43) in Appendix A, we can write

(17)where are affine functions on and are defined by

(18)

Now, with the restriction , the following state spacerepresentation is well known:

(19)

with matrices , , and of dimensions, , given by

(20)

(21)

Note that , are independent of the design filter coefficient, whereas is affine dependent on.The following result is well known in control theory (see e.g.,

[4]).Lemma 1: One has

if and only if the following linear matrix in-equality (LMI) is feasible in the symmetric matrix variableof size and the vector coefficient variable

(22)

Combining this with (49) defining the -norm of FIR filtersleads to the following LMI optimization formulation for the op-timization problem (3), (8), (9).

Theorem 1: Suppose that , , and are defined by(20) and (21). Then, (3) is equivalent to the following LMI op-timization problem:

subject to (18), (21), (16), (22) (23)


Analogously, the optimization problems (8) and (9) are equiva-lent to the following LMI optimization problems:

subject to (18),(21),(16),(22)

(24)

subject to (18),(21),(16),(22)

(25)

Before closing this section, let us mention some efficient LMIcodes in Matlab such as [5] and [17], which will be used in oursimulation.

IV. L INEAR-PHASE FIR FILTER DESIGN

One may note that with the state-space representation (19),LMI characterization (22) is obtained at the expense of the ad-ditional Lyapunov variable , whose dimension grows up veryquickly and become huge even with moderate sizes of anal-ysis and synthesis filters. As a result, LMI optimization prob-lems (23)–(25) may not be efficiently solved by the best ex-isting LMI codes [5], [17], which are able to handle LMIs upto 1000 to 2000 scalar variables. This fact has motivated us [19]to develop an efficient solution method for semi-infinite pro-gramming (SIP) and then apply it to solutions of peak-error con-strained linear-phase FIR filter design (an extension to nonlinearphase FIR filter is under our study). Let us now discuss how SIParises with our filterbank design and how it can be solved.

Suppose that and are linear-phase filters with evenorders 2 and 2 , under the assumption

, i.e., they can be re-expressed as

(26)

Then, using (46) in Appendix A, the constraint (2) is written as(27), shown at the bottom of the page. Now, using (45) to writeout

(28)

where are linear in

(29)

(27)


Thus, the following result is an immediate consequence of SIPformulation of -norm for linear-phase FIR filters given inAppendix B.

Theorem 2: The -norm optimization problem (3) can beexpressed as the following semi-infinite linear program:

(27) (30)

(31)

(32)

Analogously, the optimization problem (8) is equivalent to thefollowing problem of minimizing a convex quadratic objectivefunction under a semi-infinite linear constraint

(27), (31), (32) (33)

For the linear program-based solution method for problem(30)–(32), see [11] and [19]. The obvious advantage of theoptimization formulation (30)–(33) over (23) and (25) is thatthe variable number of the former is always moderate becausethere is no additional variable involved. Moreover, the linearprograms codes are much more flexible than LMI codes, andso, (30)–(32) can be solved without difficulty.

V. GLOBAL OPTIMIZATION FOR CODING ERRORMINIMIZATION

Back to the coding error minimization problem (10), we ob-serve that its objective function can be equiva-lently replaced by , i.e., (10) is equivalent tofollowing optimization problem:

(2), (7) (34)

(35)

Note that all constraints (2), (7), and (35) are convex LMI orconvex quadratic, whereas the objective function

in (34) is nonconvex but separable concave [21]and monotonic in [22]. These useful structures will beexploited later to develop an efficient algorithm for solving (34).

Denote by the projection of the feasible set onto the-space:

s.t. (2), (7), (35) (36)

This is easily seen to be a convex set, and the problem now isto minimize the concave functionover this convex set.

We will solve this problem by a branch and bound (BB) al-gorithm, which requires two basic operations [21].

1) Bounding: Given any rectangle , com-pute a number (lower bound) such that

(2), (7), (35) (37)

(bounding rule).

2) Branching: Given any rectangle divide it into subrect-angles (subdivision rule).

Once these two basic operations have been defined, the algo-rithm starts from an initial partition consisting of the single rec-tangle At any iteration , each rectangle inthe current partition is assigned a lower bound computedaccording to the bounding rule. In addition, a feasible solutionthat is currently the best solution so far obtained is known (thevalue of the objective function associated with this current bestsolution is the current best value). Obviously, allwithlarger than this currently best solution can be deleted from fur-ther consideration since there will be no better solution inside

. The partition member with smallest lower bound amongthe current partition is then selected and further subdivided ac-cording to the subdivision rule, before the algorithm goes to thenext iteration. The algorithm stops when differs fromthe current best value less than a prescribed tolerance: Thecurrent best solution is then an approximate optimal solutionwithin tolerance ( -optimal solution). Clearly, the algorithmcan stop after a finite number of iterations only under the fol-lowing consistency condition:

(38)

This condition also shows that the convergence speed of an al-gorithm depends on how in (37) is tight compared with

. The branching rule also plays a crucial role in speedingup the convergence of (38) as well.

A. Bounding

The bounding operation for (34) proceeds as follows. Givenany rectangle in the ( )-space, by theformula [21], an affine minorant of over is

with

Therefore, a lower bound defined by (37) is provided by theoptimal value of the program

(2), (7), (35) (39)

which is either the LMI or SIP, depending on which among LMIcharacterization (22) and SIP characterization (31) and (32) isused for the -norm constraint (7).

Note that the function matches withat each vertex of the rectangle because matcheswith at the endpoints of [ ], and matches with

at the endpoints of [ ]. Therefore, if the optimalsolution ( ) of coincides with a vertexof the rectangle , then yields the exact minimum of

over all feasible solutions in .

B. Branching

Define

(40)


Fig. 2. RectangleM 2 P and subdivision rule.

(41)

and the rectangle From the strict con-cavity and monotonicity of , it easily follows that anoptimal solution of (34) must be attained on the arc ,where denotes the boundary of. Therefore, the search foran optimal solution can be restricted to the rectangle.

Now, denote by the family of rectanglessuch that the diagonal [ ] is the chord of

an arc of that is entirely contained in . Clearly, .Starting from , we use the following branching rule.

Consider any rectangle . Denote by ( )an optimal solution of (39). We therefore have the followinglemma.

Lemma 2: It must be true that .Proof: See Appendix C.

From Lemma 2, the point ( ) divides thearc into two subarcs contained in the subrectangles

and (with diag-onals [ ] and [ ],respectively). Since the minimum of over mustbe found in either of these subrectangles—except that they areretained for further consideration—the two other subrectanglesare discarded. Thus, after dividing via ( )into four subrectangles, we retain only the upper left andlower right subrectangles , and

(see Fig. 2).

C. Rectangular Algorithm

The above development leads to the following branch andbound procedure for solving (34).

Initialization: Start with the above constructed rec-tangle and an optimal solution( ) of . Define , and set

Set .

Step 1) For each , solve to obtain theoptimal value and an optimal solution( ). Define the current best solution( ) by comparing with

, .

TABLE IDATA OF h

Step 2) In , delete all such that. Let be the set of remaining rectangles. If, terminate: ( ) is an -optimal solution of

the problem.Step 3) Choose , and

divide it into four subrectangles via the point( ). Denote by the collectionof the upper left and lower right subrectangles, andlet Set ,and go back to Step 1.

Proposition 1: The above rectangular algorithm terminatesafter finitely many iterations yielding an-optimal solution.

Proof: See Appendix C.

VI. SIMULATION

In this section, the simulation is set up for demonstrating theviability of results developed in the previous Sections.

A. Nonlinear Phase Filterbank

The analysis filters are the same as in [8, Ex. 1], i.e.,isa 19th-order lowpass filter designed using the Remez algorithmwith transition band , and , i.e.,

, . The data of is pro-vided in Table I.

Some details of simulation for this example are the following:

• -norm optimization:The order of the synthesis filtersis , and the delay is chosen like in

[8]. Solving using Matlab LMI toolbox gives around 2%reconstruction error over all frequency range (the optimalvalue of the corresponding optimization problem (23) is

), which is as good as the resultreported in [8], although our synthesis filters are of FIR,whereas their counterparts are of IIR with the same orderin [8]. This is really a surprising result. The resulting fre-quency responses for these filters are plotted in Fig. 3. Fre-quency responses of are also shown in the same figure.

• Nonlinearity optimization:The same data is used for sim-ulation of direct case with nonlinearity minimization con-


Fig. 3. Frequency response of nonlinear phase filters inH -normoptimization.

Fig. 4. Frequency response of nonlinear phase filters in nonlinearityminimization.

straint by solving problem (25). The reconstruction erroris set to a fixed value at . Then, the nonlin-earity is improving 5.1% compared with the no-constraintcase of -norm only optimization (23). In particular, thevalues of in (9)are 0.924 199 43 and 0.973 671 40 for (24) and (25), re-spectively. Magnitude frequency responses forandare shown in Fig. 4.

B. Linear-Phase Filterbank

For the linear phase case, the analysis filters are taken from[12, Ex. 4.1], which are summarized in Table II.

Again, some simulation results for this example follow:

• -norm optimization:With the delay equal ,the orders of synthesis filters and are 57 and 55,respectively. Now, solving (30)–(32) by the algorithm of[19] gives almost zero construction error (the optimalvalue is ). Frequency responses for this caseare shown in Figs. 5 and 6.

TABLE IIANALYSIS FILTERS

Fig. 5. Frequency response of nonlinear-phase filters in coding errorminimization.

Fig. 6. Frequency response of linear-phase filters inH -norm optimization.


Fig. 7. Frequency response of linear-phase filters in coding errorminimization.

• Error coding optimization:The same example as aboveis used here, except with the construction error set toa fixed value at in (33). The ”coding er-rors” for the nonoptimized and optimizedcases are 0.5147 and 0.5047, respectively, representingabout 2% improvement. Note that the rectangularalgorithm in Section V-C starts with the initial box

and terminatesafter just two iterations, giving the optimal solution. Thefrequency responses for and are presented in Fig. 7.

VII. CONCLUDING REMARKS

In this paper, we have shown that several multicriterion QMFbank design problems can be formulated as computationallytractable optimization problems and, thus, are effectively solvedby the existing softwares in PC. Because of space limitations,we have presented only results related to some of them, suchas aliasing energy, constrained nonlinearity, and error codingminimization. Presently, the extension of our solution approachto other important problems of filterbank design, such as max-imally flat filters, Smith-Barnwell PR-CQF, and the LeGall-Tabatabai PR filterbank [1], is under way.

APPENDIX A

• For

(42)

one has

(43)

(44)

• For linear-phase filters

one has (45) and (46), shown at the top of the next page.Note that (43)–(46) have been used earlier in [6].

APPENDIX BNORMS OFFILTERS/TRANSFERFUNCTIONS

Let be a filter/transfer function with real coefficients.Then, and norms of are defined as

(47)

(48)

Thus, when is a FIR filter

then

(49)

(50)

where is autocorrelation of defined by

(51)

When is linear phase, for instance, with even length

(52)

then is defined by

(53)

APPENDIX CCONVERGENCE OF THERECTANGULAR ALGORITHM

Proof of Lemma 2

Note that the objective function of problemis still monotonic. Hence, if

, then there are and such thatand .

Then, and, contradicting

the optimality of ( ) for .For the proof of Proposition 1, we need the following auxil-

iary result.Lemma 3: Let ,

be any infinite nested sequence such that every is a


(45)

(46)

child of in the above subdivision process. Then, thereexists a subsequence such that the sequence

tends to a vertex of the rectangle

Proof: Let . For every since, we must have either

or . If for infinitely many ,then is a cluster point of ; otherwise, is a clusterpoint of . In any case, there exists a subsequence

such that .Likewise, there exists a subsequence suchthat Then,

, whereis a vertex of the rectangle .

A. Proof of Proposition 1

If the algorithm is infinite, then an infinite nested sequenceof rectangles , can be found, as inLemma 3. By this Lemma, we have, for some subsequence

such that as ,, where ( ) is a vertex of the rectangle .

Then, . Sincefor every , it follows

that , i.e., ( )is an optimal solution of the problem. On the other hand, since

, by taking a cluster point ( )of the sequence , we have ; hence, forlarge enough , contradictingthe fact that .

ACKNOWLEDGMENT

The authors would like to thank J. Sturm for allowing them toaccess to his SeDuMi code [17] and P. Apkarian for his programhelping interaction between codes [5] and [17].

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Hoang Duong Tuan(M’92) was born in Hanoi, Vietnam, in 1964. He receivedthe diploma and the Ph.D. degree, both in applied mathematics, from OdessaState University, Odessa, Ukraine, in 1987 and 1991, respectively.

From 1991 to 1994, he was a Researcher with the Optimization and SystemsDivision, Vietnam National Center for Science and Technologies, Hanoi.From 1994 to 1999, he was an Assistant Professor with the Departmentof Electronic-Mechanical Engineering, Nagoya University, Nagoya, Japan.He joined the Toyota Technological Institute, Nagoya, in 1999, where heis an Associate Professor with the Department of Electrical and ComputerEngineering. His research interests include theoretical developments andapplications of optimization-based methods in broad areas of control, signalprocessing, and communication.

Le Hai Nam received B.E. in degree in telecommunications from Hanoi Uni-versity of Technology, Hanoi, Vietnam in 1987 and the M.E. in telecommunica-tions from the Asian Institute of Technology, Bangkok, Thailand, in 1993. Hereceived the D.E. degree in electronic and electrical engineering from TokyoInstitute of Technology, Tokyo, Japan, in 1997.

From October 1997 to September 1998, he was with NTT Multimedia Net-works Laboratories at NTT Yokosuka R&D Center, Yokosuka, Japan, as a post-doctoral fellow. From October 1998 to June 2000, he was a JSPS post-doc-toral fellow at Tokyo Institute of Technology. Since July 2000, he has been apost-doctoral fellow at Toyota Technological Institute, Nagoya, Japan. His re-search interests are in the areas of antifading techniques for mobile comminica-tions, OFDM-CDMA, and signal processing for communications.

Hoang Tuy received the Ph.D. degree in mathematics from Moscow State Uni-versity, Moscow, U.S.S.R., in 1959.

He was Dean of the Faculty of Physics/Mathematics from 1960 to 1962 andthen Dean of the Faculty of Mathematics of Hanoi University, Hanoi, Vietnam,from 1962 to 1968. Since 1968, he has been with the Institute of Mathematics,National Center for Science and Technology, Hanoi. From 1968 to 1980, heserved as Chief of the Department of Optimization and, from 1980 to 1990, asDirector of the Institute of Mathematics. He has visited many universities aroundthe world, including University Paris IX, Paris, France; Graz Technical Univer-sity, Graz, Austria; Montreal University, Montreal, QC, Canada; and LinkopingUniversity, Linkoping, Sweden. His current research interest is global optimiza-tion and applications. He is author of three monographs:Global Optimization(Berlin, Germany: Springer, first ed. 1990, second ed. 1993, third ed. 1996)(coauthored with R. Horst),Optimization on Low Rank Nonconvex Structures(Boston, MA: Kluwer, 1997) (coauthored with H. Konno and P. T. Thach), andConvex Analysis and Global Optimization(Boston, MA: Kluwer, 1998).

Truong Q. Nguyen (SM’95) received the B.S., M.S., and Ph.D. degrees inelectrical engineering from the California Institute of Technology, Pasadena,in 1985, 1986, and 1989, respectively.

He was with the Lincoln Laboratory, Massachusetts Institute of Technology,Cambridge, from June 1989 to July 1994, as a member of technical staff. FromAugust 1994 to July 1998, he was with the Department of Electrical and Com-puetr Engineering, University of Wisconsin, Madison. He was with Boston Uni-versity, Boston, MA, from 1996 to 2001, and he is currently with the Depart-ment of Electrical and Computer Engineering, University of California at SanDiego, La Jolla. His research interests are in the theory of wavelets and filter-banks and applications in image and video compression, telecommunications,bioinformatics, medical imaging and enhancement, and analog/digital conver-sion. He is the coauthor (with Prof. G. Strang) of a popular textbookWaveletsand Filter Banks(Wellesley, MA: Wellesley-Cambridge Press, 1997) and theauthor of several matlab-based toolboxes on image compression, electrocardio-gram compression, and filterbank design. He also holds a patent on an efficientdesign method for wavelets and filterbanks and several patents on wavelet ap-plications including compression and signal analysis.

Prof. Nguyen received the IEEE TRANSACTIONS ON SIGNAL PROCESSING

Paper Award (Image and Multidimensional Processing area) for the paper heco-wrote with Prof. P. P. Vaidyanathan on linear-phase perfect-reconstructionfilterbanks in 1992. He received the NSF Career Award in 1995 and is currentlythe Series Editor (Digital Signal Processing) for Academic Press. He served asAssociate Editor for the IEEE TRANSACTIONS ON SIGNAL PROCESSINGfrom1994 to 1996 and for the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS

from 1996 to 1997.

Multicriterion optimized QMF bank design

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