A General Formulation of Modulated Filter Banks

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986 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 47, NO. 4, APRIL 1999

A General Formulation of Modulated Filter BanksPeter Niels Heller, Member, IEEE, Tanja Karp, and Truong Q. Nguyen, Senior Member, IEEE

Abstract This paper presents a general framework for max-

imally decimated modulated lter banks. The theory covers theknown classes of cosine modulation and relates them to complex-modulated lter banks. The prototype lters have arbitrarylengths, and the overall delay of the lter bank is arbitrary,within fundamental limits. Necessary and sufcient conditionsfor perfect reconstruction (PR) are derived using the polyphaserepresentation. It is shown that these PR conditions are identicalfor all types of modulationmodulation based on the discrete co-sine transform (DCT), both DCT-III/DCT-IV and DCT-I/DCT-II,and modulation based on the modied discrete Fourier transform(MDFT). A quadratic-constrained design method for prototypelters yielding PR with arbitrary length and system delay isderived, and design examples are presented to illustrate thetradeoff between overall system delay and stopband attenuation(subchannelization).

Index Terms Cosine-modulated lter bank, DCT, lter bank,MDCT, modulated lter bank.

I. INTRODUCTION

M ULTIRATE lter banks are used in many applications,ranging from data compression (speech, audio, image,and video) to multicarrier modulation (xDSL) and featuredetection. Fig. 1 shows an -channel maximally decimatedlter bank with the corresponding nearly ideal lter responses.The input signal is bandpass ltered and downsampled to yieldcritical sampling of the subband signals. The subbandsignals can be processed independently to extract essential

information, and they can be combined to yield a reconstructedsignal, which may be similar to or very different from theoriginal signal. A lter bank where the output is a delayedcopy of the input is a perfect reconstruction (PR) system.The theory and design of such PR lter banks have beenextensively studied [13], [31].

An important subclass are the modulated lter banksthosefor which the full -channel lter bank is obtained bymodulation of a single lowpass prototype lter. The rstexamples were the pseudo-QMF banks [14], [22], [25], whichoffered approximate but not true PR. The rst PR lter bank where the analysis and synthesis lters were limited to alength of 2 was presented in [24]. Later work extended

the idea to longer lters and other modulation schemes basedManuscript received April 6, 1994; revised April 17, 1997. This work was

supported in part by Deutsche Forschungsgemeinschaft under Grant Fli 116/10and the National Science Foundation under Grant MIP 9501589. The associateeditor coordinating the review of this paper and approving it for publicationwas Prof. Roberto H. Bamberger.

P. N. Heller is with Aware Inc., Bedford, MA 01730-1432 USA (e-mail:[email protected]).

T. Karp is with Mannheim University, Mannheim, Germany (e-mail:[email protected]).

T. Q. Nguyen is with the Department of Electrical and Computer Engineer-ing, Boston University, Boston, MA 02215 USA (e-mail: [email protected]).

Publisher Item Identier S 1053-587X(99)02139-X.

on the Type IV discrete cosine transform or DCT-IV [8], [10],

[12], [15], [24] and the modied discrete Fourier transform(MDFT) [5], [9]. These lter banks offer the advantage of high computational efciency (employing fast DCT or DFTalgorithms) and high-performance lter designs [17], [21]).

The rst fundamental constructions of modulated lterbanks were based on a linear-phase prototype lter of length2 with reconstruction delay 2 1. Later work ex-tended the PR constructions to arbitrary length prototypes[19] and DCT-II modulation [7], [11]. Systems with linear-phase prototype lters face a tradeoff between latency andlter performanceshorter lters will decrease the systemlatency at the cost of poorer subchannelization (as measuredby stopband attenuation). Several authors [17], [18], [20],[30] have introduced more general values 2 1 for thereconstruction delay, which force the modulated lter bank tobe biorthogonalthe synthesis lters will no longer be time-reversed versions of the analysis lters. The authors of [29] and[34] describe the construction of pseudo-QMF (near-perfect-reconstruction) and PR modulated lter banks, respectively,with fully general reconstruction delay. Such biorthogonalmodulated lter banks ameliorate but do not eliminate thetradeoff between system latency and subchannelization. Eachof these extensions offers advantagesDCT-II modulationyields modulated lter banks with linear-phase lters, whichis useful in image compression, while reduced reconstruction

delay is important for applications to speech, audio [28], andtelecommunications.This paper combines and extends the cited results into a

single framework that encompasses PR with arbitrary recon-struction delay and all known modulation schemes (DCT-IV,DCT-II, and MDFT). In particular, the following results aredemonstrated:

necessary and sufcient conditions for prototype lterswith arbitrary length and reconstruction delay to yieldPR biorthogonal modulated lter banks;

variable overall system delay (within thefundamental limits imposed by the lter lengths);

arbitrary modulation with DCT-IV, DCT-II, or MDFTschemes;

equivalence of the various modulationsthe same con-ditions for PR apply to all the modulation schemes; aprototype that satises these conditions will yield PR inany scheme;

a quadratic-constrained least-squares method for high-performance arbitrary-delay PR modulated lter bank designs;

an application to subband coding of audio signals.Outline of the Paper: The paper begins with cosine-

1053587X/99$10.00 1999 IEEE


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HELLER et al: GENERAL FORMULATION OF MODULATED FILTER BANKS 991

We use the convention that and are nonzero forand zero otherwise. The polyphase

matrices of the -band lter banks can be described withthe aid of several xed transform matrices. and are

matrices with entries

and are matrices with entries

We will also need the matrices and dened by

and

as well as

and

with , , , , as before. Then, the analysisand synthesis polyphase matrices are

(40)

Thus, the overall polyphase matrix of size is

For PR to hold with a system delay of ,must satisfy

(41)

As shown in Appendix A, when

(42)

whereas if

(43)

As in the DCT III/IV modulation, splits into a sum of two matrices with

and

The nonzero entries of are again on the antidiagonaland can be written

(44)

while those of are at the same positions as for in(41). For

(45)

(46)


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Fig. 2. Type-1 MDFT FB. The total number of analysis and synthesis lters is . Depending on the subband, the real part is taken alternately from thezeroth or rst subband polyphase component. The imaginary part is taken from the remaining subband polyphase component.

, , and have the same form as stated in (41)and (44)(48), respectively, resulting in the same conditionsfor PR. Since and from (49) and (50) are not integers,there is no need to consider them, and and donot have any common nonzero entries.

We can summarize the results derived in this section in thefollowing theorem.

Theorem 2: Given a -channel lter bank constructedby the DCT-I modulation (35)(38) or DCT-II modulation(51)(54) of the prototype lters and with overalldelay , then the nec-essary and sufcient conditions on the analysis and synthesispolyphase components and ,for PR are the same as those derived for the DCT-III/IVmodulation in Theorem 1.

IV. MDFT F ILTER BANKS

In this section, we present two types of modied DFT(MDFT) lter banks, derive the PR conditions for the pro-totype lters, and relate them to the ones already derived forcosine-modulated lter banks. Historically, the rst modulatedlter banks known to offer an efcient implementation werethe complex-modulated FBs. They are also called DFT FBssince they can be realized in a highly efcient form bymeans of polyphase lters and an IDFT transform [1]. Theanalysis lters and synthesis lters of a -channel DFT lterbank are obtained by complex modulation of the prototypelters and . However, it is well known that thecritically subsampled -channel DFT FB with FIR analysis

and synthesis lters satises PR only in the case where theprototype lters are rectangular windows of size [31].This case corresponds to a pure block transform.

The MDFT lter bank [5], [6] is a critically sampledcomplex-modulated lter bank derived from a -channelDFT FB with oversampling factor 2 by introducing severalmodications in the subbands.

a two-step decimation of the subband signals : After dec-imating by the factor (corresponding to an over-sampling factor of 2), each of the subbands isdecomposed into two (even and odd) polyphase compo-nents that are complex valued.

Critical subsampling is obtained by taking the real partof one polyphase component and the imaginary part of the other polyphase component in each subband andalternating from one subband to the next.

On the synthesis side, the subband signals are rst recon-structed from these 2-ary polyphase components and thenprocessed by the oversampled DFT synthesis FB.

Depending on the complex modulation of the analysis and syn-thesis lters, we get two different types of MDFT lter banksthat differ in the phase of the modulation and the way onetakes the real and imaginary part of the subbands polyphasecomponents. In the following, we assume a complex-valuedinput signal. The case of real input signals will be discussedat the end of the section.

A. Type-1 MDFT Filter Banks

In this section, we generalize the results shown for the pa-raunitary case in [9]. The complex analysis and synthesis ltersare obtained from the prototypes and , respectively,by

where . After ltering with the analysislters , each subband signal is broken into even and oddpolyphase components. For even-indexed subbands, we takethe real part of the even subband polyphase component andthe imaginary part of the odd polyphase component and viceversa for odd-indexed subbands, as in Fig. 2.

Theorem 3: Given an analysis prototype lter satisfy-ing the PR conditions (30) and (31) or (32) and (33) as well as(34) for an -channel lter bank with DCT-III/IV modulationand an overall delay of samples, then the channelMDFT lter bank will also be PR, and the overall delay of the lter bank will besamples.


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Proof of Theorem 3: From [9], we know that the lterbank output signal can be written as 1

(55)

where denotes the transform of the conjugated complexinput signal. The + sign is valid for odd and the sign for even. When expressing the analysis and synthesislters by the prototype lters and , respectively,the second part of the sum in (55) cancels (see Appendix B)so that the output signal is given by

(56)

Thus, we have shown that all odd aliasing terms cancel withinthe MDFT lter bank whenever the analysis and synthesisprototype lters satisfy (34). Note that (56) describes thetransfer function of a -channel DFT FB with analysislters , synthesis lters , andan oversampling factor of 2 in the subbands. PR solutions forDFT FBs oversampled by 2 (see, e.g., [2][4]) are thereforealso valid for MDFT lter banks if the prototypes polyphasecomponents satisfy (34).

The analysis polyphase matrix of the oversampledDFT lter bank is given by

where denotes the DFT-modulation matrixwith entries . The synthesis polyphasematrix containing the delay of (56) is

with so thatthe size overall polyphase matrix can be written

An argument similar to that of Appendix B shows that. Choosing the overall delay of

the lter bank to besamples, must satisfy

1 For , 0 must be interpreted as 0

.

Fig. 3. Type-2 MDFT FB. The real part is always taken from the ze-roth subband polyphase component and the imaginary part from the rstcomponent.

(57)

Substituting for and expressing this in terms of thepolyphase lters , we obtain the solution already givenby (30) and (31) for , (32) and (33) for

, and (34).

B. Type-2 MDFT Filter Banks

Another way to obtain MDFT lter banks is by modulatingthe analysis and synthesis lters as

(58)

for and always taking the real part in theundelayed branch of the subbands and the imaginary part inthe delayed branch, as in Fig. 3. The result is called a type-2MDFT lter bank.

Theorem 4: The PR conditions for channel type-2MDFT lter banks are the same as for channel type-1 MDFT lter banks and are thus identical to those forchannel CMFBs with DCT-IV or DCT-II modulation. Thedelay introduced by the lter bank is .

Proof of Theorem 4: It can be shown that the input signaland the reconstructed signal for type-2 MDFT lter banks arerelated by (56) if we use the analysis and synthesis lters (58).Thus, the overall polyphase matrix satises

where


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(a) (b)

Fig. 6. , prototype lters with delay 3, 5, and 7. (a) Impulse responses. (b) Frequency responses.

(a) (b) (c)

Fig. 7. Magnitude frequency responses of lter banks obtained by DCT-IV and DCT-II modulations of the same , , prototype lter.

are plotted in Fig. 4(b). We see that the system withhas the highest stopband attenuation, whereas the system with

has the poorest stopband attenuation.b) A second set of design examples demonstrate the effect

of keeping the delay parameter constant while varying thelter length. Fig. 5 displays the magnitude responses of twolters with , reconstruction delay of 63 samples (and ), and lengths 64 and 96. The longer prototypehas superior stopband attenuation (43 dB rst sidelobe heightversus 34 dB for the shorter prototype).

c) A third set of design examples shows the effect of changing while keeping and xed. Here, and

, whereas and takes the values 3, 5, and 7. The

impulse responses of the three lters are shown in Fig. 6(a)and the magnitude frequency responses in Fig. 6(b). As before,decreasing delay leads to lower stopband attenuation (lower-performance lters). The prototype for has two zerocoefcients near its tail, whereas the prototype hasone zero coefcient, and the has none. Thesezeros of the impulse response, forced by the PR conditions(30)(33), affect the lter performance, as described in [19].

d) The nal example explores the use of a given prototypefor different types of modulation. First, consider a prototypelter with , , and . Since and

have different parity, we can modulate the prototype using

either a DCT-IV or a DCT-II. Fig. 7(a) shows the magnitudefrequency response of the lter bank generated using DCT-IVmodulation. Notice that each subband is covered once, and allthe subband lters have the same amplitude. The lter bank has eight bands and reconstruction delay 95. Fig. 7(b) and(c) show the magnitude frequency responses of the DCT-IIlter bank generated from the same prototype; plot (b) showsthe lters generated using (51), and plot (c) shows thelters generated by (52). Notice that the subbands centered atDC and are covered once, whereas the remaining subbandsare covered twice by lters with amplitudes that have beenscaled down by . This DCT-II lter bank has 16 bands andreconstruction delay 103.

Now, consider the example of a prototype lter with ,, and . Here, and have the same parity so

that we will use a DCT-III or DCT-I to modulate the prototype.Fig. 8(a) shows the magnitude frequency response of the lterbank generated from the prototype using DCT-III modulation(1). Again, each subband is covered once, and all the subbandlters have the same amplitude. The lter bank haschannels and reconstruction delay . Fig. 8(b) and (c)show the magnitude frequency responses of the DCT-I lterbank generated from the same prototypeplot (b) shows the

lters generated using (35), and plot (c) shows thelters generated using (36). This 16-band lter bank


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Fig. 10. Frequency response of length 512 prototype lters with delays 447and 511.

(a)

(b)

(c)

Fig. 11. Original audio signal and reconstructed signals coded at 128 kb/susing the MPEG-1 prototype lter and the low-delay prototype lter.

VII. CONCLUSIONS

In this paper, the theory, structure, and design of generalbiorthogonal modulated lter banks (cosine-modulated andmodied DFT) have been developed, using the polyphaserepresentation. Working with the design parameters of thenumber of channels , the prototype lter length , and the

system delay , we have developed necessary and sufcientconditions on the polyphase components of the prototype lterfor the modulated lter bank to satisfy perfect reconstruction.Furthermore, the relationships among modulated lter bankswith different modulation schemes (DCT-III/IV, DCT-I/II,and MDFT) have been explored. A prototype lter satisfyingperfect reconstruction for one of these modulation schemeswill yield PR with any of the other types of modulation, andin fact, MDFT modulation has been shown to be equivalentto DCT-II modulation.

A quadratic-constrained least-squares technique has beenpresented for the design of prototype lters for PR modulatedlter banks, with full generality in setting the parameters ,

, and . Implementation of the technique shows that the bestsubchannelization (greatest stopband attenuation) is obtainedfor . Systems with have lower stopbandattenuation (as do systems with ), leading to atradeoff between subchannelization and system delay for agiven lter length.

The application of biorthogonal modulated lter banks to

audio compression has been investigated. A comparison of pre-echo noise in MPEG-type subband audio coders withand was presented, and the system

with shorter delay showed a decrease in pre-echo noise power.

APPENDIX A

We prove here several identities involving the matrix prod-ucts and .

Proof 1: In the case of DCT-III/IV modulation, we willprove that has the form given in (3). Consider theelement of the matrix

(67)


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First, examine the upper sum in (67) and replaceby

Substituting , we obtain

otherwise.

The value must be chosen so that are in the range, yielding

with solutions and . Thus, the upper sumin (67) becomes

For the second sum in (67), we proceed similarly. Thephase term can be expressed as

, yielding

Thus, the th matrix entry is

and the matrix as a whole has the form of (3).Proof 2: In the case of DCT-I/II modulation, we prove

that and have the form given in (42) and (43).First, consider the th element of

(68)

The lower sum in (68) reduces to ,contributing a matrix to . The upper sumin (68) reduces to

where , and . When , thisbecomes


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whereas when , the upper sum equals

Thus, the overall matrix satises (42) and (43). Theproof for is similar.

APPENDIX B

We show here that the second part of (55) is equal to zero.This is equivalent to the cancellation of all odd alias spectra.

Substituting yields

Note that we do not have to change the summation indexwhen substituting for since we sum over a whole periodof and . Expressing the prototype lters bytheir polyphase components and taking (34) into considerationyields that the whole expression is equal to zero.

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[1] M. G. Bellanger and J. L. Daguet, TDM-FDM transmultiplexer: Digitalpolyphase and FFT, IEEE Trans. Commun., vol. COMM-22, pp.11991204, Sept. 1974.

[2] H. Bolcskei, F. Hlawatsch, and H. G. Feichtinger, Oversampled FIRand IIR DFT lter banks and Weyl-Heisenberg-frames, in Proc. IEEE ICASSP, Atlanta, GA, 1996.

[3] R. E. Crochiere and L. R. Rabiner, Multirate Digital Signal Processing.Englewwod Cliffs, NJ: Prentice-Hall, 1983.

[4] Z. Cvetkovi c and M. Vetterli, Oversampled modulated lter banks andtight Gabor frames in (Z), in Proc. IEEE ICASSP, Detroit, MI, 1995.

[5] N. J. Fliege, Computational efciency of MDFT lter banks, in Proc.27th Asilomar Conf. Signals, Syst., Comput., Asilomar, CA, 1993.

[6] , Modied DFT polyphase SBC lter banks with almost perfectreconstruction, in Proc. IEEE ICASSP, Adelaide, Australia, 1994, pp.III149III152.

[7] R. A. Gopinath, Modulated lter banks and waveletsA generalunied theory, in Proc. IEEE ICASSP, Atlanta, GA, 1996.

[8] R. Gopinath and C. Burrus, Theory of modulated lter banks andmodulated wavelet tight frames, in Proc. IEEE ICASSP, Minneapolis,MN, 1993, pp. III169III172.

[9] T. Karp and N. Fliege, MDFT Filter banks with perfect reconstruction,in Proc. IEEE ISCAS, Seattle, WA, 1995.[10] R. D. Koilpillai and P. P. Vaidyanathan, Cosine modulated FIR lter

banks satisfying perfect reconstruction, IEEE Trans. Signal Processing,vol. 40, pp. 770783, Apr. 1992.

[11] Y.-P. Lin and P. P. Vaidyanathan, Linear-phase cosine modulatedmaximally decimated lter banks with perfect reconstruction, IEEE Trans. Signal Processing, vol. 43, pp. 25252539, Nov. 1995.

[12] H. Malvar, Extended lapped transforms: Fast algorithms and applica-tions, IEEE Trans. Signal Processing, Nov. 1992.

[13] H. Malvar, Signal Processing with Lapped Transforms. Norwood, MA:Artech House, 1992.

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[17] K. Nayebi, T. P. Barnwell, III, and M. J. T. Smith, Time-domain lterbank analysis: A new design theory, IEEE Trans. Signal Processing,vol. 40, pp. 14121429, 1992.

[18] , Low delay FIR lter banks: Design and evaluation, IEEE Trans. Signal Processing, vol. 42, pp. 2431, Jan. 1994.

[19] T. Q. Nguyen and R. D. Koilpillai, The theory and design of arbitrary-length cosine-modulated lter banks and wavelets, satisfying perfectreconstruction, IEEE Trans. Signal Processing, vol. 44, pp. 473483,Mar. 1996.

[20] T. Q. Nguyen, A class of generalized cosine modulated lter bank, inProc. IEEE ISCAS, 1992, pp. 943946.

[21] , Digital lter bank design quadratic constrained formulation, IEEE Trans. Signal Processing, vol. 43, pp. 21032108, Sept. 1995.

[22] J. Nussbaumer and M. Vetterli. Computationally efcient QMF lterbanks, in Proc. IEEE ICASSP, San Diego, CA, 1984.

[23] J. Princen and A. Bradley, Analysis/synthesis lter bank design basedon time domain alias cancellation, IEEE Trans. Acoust., Speech, SignalProcessing, vol. ASSP-34, pp. 476492, 1986.

[24] T. A. Ramstad and J. P. Tanem, Cosine modulated analysis synthesislter bank with critical sampling and perfect reconstruction, in Proc. IEEE ICASSP, Toronto, Ont., Canada, 1991, pp. 17891792.

[25] J. H. Rothweiler, Polyphase quadrature ltersA new subband codingtechnique, in Proc. IEEE ICASSP, 1983, pp. 12801283.

[26] K. Schittkowski, NLPQL: A FORTRAN subroutine solving constrainednonlinear programming problems, Ann. Oper. Res., vol. 5, pp. 485500,1986.

[27] M. R. Schroeder, Number Theory in Science and Communication. NewYork: Springer-Verlag, 1986.

[28] G. Schuller, A low delay lter bank for audio coding with reducedpre-echoes, in Proc. 99th AES Conv., New York, NY, Oct. 1995.

[29] , A new factorization and structure for cosine modulated lterbanks with variable system delay, in Proc. 30th Asilomar Conf. Signals,Syst., Comput., Pacic Grove, CA, Nov. 1996.

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Peter Niels Heller (M91) received the B.S. degreein mathematics with highest honors from the Uni-versity of North Carolina, Chapel Hill, in 1981 andthe M.A. and Ph.D. degrees in mathematics fromPrinceton University, Princeton, NJ, in 1983 and1986, respectively.

He is presently a Scientist at Aware, Inc., Bed-ford, MA. From 1985 to 1986, he was a C. L.E. Moore Instructor of Mathematics at the Massa-chusetts Institute of Technology (MIT), Cambridge,

and he returned to MIT as a Visiting Lecturerin Mathematics in 1989. During 1990, he developed geometric modelingsoftware at Parametric Technology Corp., Waltham, MA. in December 1990,he joined Aware. His research interests include wavelets and multiratesystems and their application to data compression and telecommunications.He is currently directing the development of a wavelet-based broadbandwireline modem chip. In 1994, he copresented the Nordic postgraduate courseon wavelets, lter banks, and applications at the Helsinki University of Technology, Helsinki, Finland.

Dr. Heller held National Science Foundation and Sloan Foundation fellow-ships while a graduate student at Princeton.

Tanja Karp was born in Germany in 1969. She re-ceived the Dipl.-Ing. degree in electrical engineeringand the Dr.-Ing. degree from Hamburg University of Technology, Hamburg, Germany, in 1993 and 1997,respectively.

Since 1997, she has been with Mannheim Uni-versity, Mannheim, Germany, as a Research andTeaching Assistant. Her research interests includemultirate signal processing, lter banks, source andchannel coding, and signal processing for commu-nications.

Truong Q. Nguyen (S85M90SM95) received the B.S., M.S., and Ph.D.degrees in electrical engineering from the California Institute of Technology,Pasadena, in 1985, 1986, and 1989, respectively.

He was with the Lincoln Laboratory, Massachusetts Institute of Technology(MIT), Cambridge, from June 1989 to July 1994, as a Member of the TechnicalStaff. From 1993 to 1994, he was a Visiting Lecturer at MIT and an AdjunctProfessor at Northeastern University, Boston, MA. From August 1994 to July1996, he was an Assistant Professor at the University of Wisconsin, Madison.He is now with the Boston University. His research interests are in digitaland image signal processing, multirate systems, wavelets and applications,

and biomedical signal processing.Prof. Nguyen was a recipient of a fellowship from Aerojet Dynamics for

advanced studies. He received the IEEE T RANSACTIONS ON SIGNAL PROCESSINGAward (Image and Multidimensional Processing area) for the paper hecowrote, with Prof. P. P. Vaidyanathan, on linear-phase perfect-reconstructionlter banks in 1992. He received the NSF Career Award in 1995 and is thecoauthor (with Prof. G. Strang) of a textbook on Wavelets and Filter Banks(Wellesley, MA: Wellesley-Cambridge). He served as Associate Editor for theIEEE TRANSACTIONS ON SIGNAL PROCESSING and for the IEEE T RANSACTIONSON CIRCUITS AND SYSTEMS II. He is a member of Tau Beta Pi and Eta KappaNu.

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A General Formulation of Modulated Filter Banks