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1 This document is downloaded from DR-NTU, Nanyang Technological University Library, Singapore. Title Optimum masking levels and coefficient sparseness for Hilbert transformers and half-band filters designed using the frequency-response masking technique( Published ) Author(s) Lim, Yong Ching; Yu, Ya Jun; Saramäki, Tapio Citation Lim, Y. C., Yu, Y. J., & Saramäki, T. (2005). Optimum masking levels and coefficient sparseness for Hilbert transformers and half-band filters designed using the frequency-response masking technique. IEEE Transactions on Circuits and Systems-I: Regular Papers, 52(11), Date 2005 URL Rights IEEE Transactions on Circuits and Systems-I: Regular Papers 2005 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.

2 2444 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 52, NO. 11, NOVEMBER 2005 Optimum Masking Levels and Coefficient Sparseness for Hilbert Transformers and Half-Band Filters Designed Using the Frequency-Response Masking Technique Yong Ching Lim, Fellow, IEEE, Ya Jun Yu, Member, IEEE, and Tapio Saramäki, Fellow, IEEE Abstract Hilbert transformers and half-band filters are two very important special classes of finite-impulse response filters often used in signal processing applications. Furthermore, there exists a very close relationship between these two special classes of filters in such a way that a half-band filter can be derived from a Hilbert transformer in a straightforward manner and vice versa. It has been shown that these two classes of filters may be synthesized using the frequency-response masking (FRM) technique resulting in very efficient implementation when the filters are very sharp. While filters synthesized using the FRM technique has been characterized for the general low-pass case, Hilbert transformers and half-band filters synthesized using the FRM technique have not been characterized. The characterization of the two classes of filter is a focus of this paper. In this paper, we re-develop the FRM structure for the synthesis of Hilbert transformer from a new perspective. This new approach uses a frequency response correction term produced by masking the frequency response of a sparse coefficient filter, whose frequency response is periodic, to sharpen the bandedge of a low-order Hilbert transformer. Optimum masking levels and coefficient sparseness for the Hilbert transformers are derived; corresponding quantities for the half-band filters are obtained via the close relationship between these two classes of filters. Index Terms Finite-impulse response (FIR) digital filter, frequency-response masking (FRM), Hilbert transformer, half-band filter, sparse coefficient filter. I. INTRODUCTION THROUGHOUT this paper, the phrase a filter refers to a filter whose -transform transfer function is. We shall also denote the frequency response of by. Furthermore, by the phrase a filter, we refer to the filter whose frequency response is. The conventional frequency-response masking (FRM) structure shown in Fig. 1 was first introduced in [1]. It has been used very successfully for the synthesis of very sharp low-pass, Manuscript received August 16, 2004; revised March 10, This work was supported in part by Nanyang Technological University, Temasek Laboratories@NTU, and the Grant PolyU 5101/02E. This paper was recommended by Associate Editor V. E. DeBrunner. Y. C. Lim is with the School of Electrical and Electronics Engineering, Nanyang Technological University, Singapore ( elelimyc@pmail.ntu.edu.sg). Y. J. Yu is with the Temasek Laboratories, Research TechnoPlaza, Nanyang Technological University, Singapore T. Saramäki is with the Institute of Signal Processing, Tampere University of Technology, FIN Tampere, Finland ( ts@cs.tut.fi). Digital Object Identifier /TCSI Fig. 1. Structure of a filter synthesized using the FRM technique. All the subfilters are assumed to be zero phase. Causality can be restored by introducing appropriate delays into the subfilters. high-pass, and bandpass filters with extremely low complexity. Further improvements on the structure, powerful specialized optimization algorithms, and interesting applications have been developed by many authors for the FRM [2] [24] technique. It was shown in [2] that the FRM technique can also be used to synthesize half-band filters. Since every other coefficient value of a half-band filter is zero, the various estimation formulae derived in [4] are no longer applicable. A Hilbert transformer with odd length may be derived from a low-pass half-band filter by discarding the centre coefficient, multiplying its th coefficient by, and scale up the coefficient values by a factor of two. Making use of this close relationship between a Hilbert transformer and a half-band filter, [3] illustrated the synthesis of a Hilbert transformer using the FRM technique. Although [2] and [3] illustrated the application of the FRM technique for the synthesis of Hilbert transformer and half-band filter, important information on the coefficient sparseness for the bandedge shaping filter and the optimum masking levels were not presented. In this paper, we adopted a new approach toward the synthesis of filters using the FRM technique. In this new approach, the Hilbert transformer is synthesized as a parallel connection of two filters. One of these two filters is a low-order Hilbert transformer. The other one is a series connection of a masking filter and a sparse coefficient bandedge shaping filter producing a correction term that sharpens the bandedge of the low-order Hilbert transformer. A Hilbert transformer with odd length can be derived from one with even length by replacing each in the -transform transfer function of the even length Hilbert transformer by ; this will cause the transition width to shrink by a factor of two. As a result of this straightforward relationship between the odd length and even length Hilbert transformers, the odd length Hilbert transformer will not be discussed in this paper /$ IEEE

3 LIM et al.: OPTIMUM MASKING LEVELS AND COEFFICIENT SPARSENESS 2445 Fig. 2. Structure for the synthesis of a Hilbert transformer using the FRM technique. Our new FRM approach is presented in Section II. In the FRM technique, each delay of a prototype bandedge shaping filter is replaced by delays. The optimum estimate for is presented in Section III together with the estimates for the resulting complexity and effective filter length. The application of our new FRM technique to the design of a Hilbert transformer for suppressing acoustic feedback is presented in Section IV; a factor of 9 reduction in the number of nontrivial coefficients is achieved when compared to the Chebyshev optimum design. Further reduction in the complexity of the filter may be achieved by considering a multi-level masking structure; this is presented in Section V. A two level-masking design is presented in Section VI to meet the requirement of the example of Section IV. The two-level masking approach produces a factor of 17 reduction in the number of nontrivial coefficients comparing to the Chebyshev optimum design. II. HILBERT TRANSFORMER SYNTHESIZED USING FRM TECHNIQUE The -transform transfer function of a Hilbert transformer with length, where is an even integer, is given by In (1), is the th impulse response. Since is even, its group delay, given by, is not an integer and the filter is said to have half-sample delay. Fig. 2 shows the structure for implementing the Hilbert transformer using the FRM technique. The -transform transfer function for the overall filter is given by It is a system of subfilters consisting of a parallel connection of two branches. One of the parallel branches consists of a basic filter. The other branch consists of a cascade of a bandedge sharpening filter and a masking filter. The basic filter provides a low-order approximation (with wide transition width) to the desired specification. The cascade of and provides a transfer function correction term for sharpening the transition band. Let the lengths of, and be, and, respectively. The length of will be. The delay introduced by and that introduced by must be the same; otherwise, pure delays must be introduced into the shorter one to equalize (1) (2) Fig. 3. Frequency responses of the subfilters for even length H (z). Note that =M = is the transition width of the desired transfer function. them. In order to avoid inserting half-sample delay in the implementation, the parity of and that of must be the same. Furthermore, must have anti-symmetrical impulse response. Consider a wide transition band Hilbert transformer as shown in Fig. 3(a) where is even. The computational complexity of is low since its transition band is wide. Now consider a transition band correction filter as shown in Fig. 3(b). When is added to,a sharp transition band Hilbert transformer,, as shown in Fig. 3(c) is obtained. Our objective is to design a very low complexity filter using the FRM technique. Consider a bandedge shaping filter as shown in Fig. 3(d). The complexity of is low because it has wide transition width. Replacing each delay of by delays, a filter as shown in Fig. 3(e) is obtained. A masking filter as shown in Fig. 3(f) is used to mask the unwanted passbands of to produce the frequency response as shown in Fig. 3(b). has low complexity because its frequency response has a wide transition band. Since the length of is even, the length of, i.e.,, must also be even. If is even, must also be even. If is odd, and must have different parities. By considering the gain of in the vicinity of, it is clear that has symmetrical impulse response. Thus, must have anti-symmetrical impulse response to satisfy the condition that must have anti-symmetrical impulse response. The bandedges of and are the same. Let the bandedge of be as shown in Fig. 3. Let the bandedge of be. As can be seen from Fig. 3,.

4 2446 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 52, NO. 11, NOVEMBER 2005 III. OPTIMUM FOR EVEN LENGTH HILBERT TRANSFORMERS Joint simultaneous optimization of, and is a nonlinear optimization problem and can be solved by general purpose or specialized optimization packages. Nevertheless, regardless of the optimization packages used, it is necessary to determine the value of before initiating the optimization process. Let the transition width of the desired Hilbert transformer be where is the sampling frequency and let its passband ripple magnitude be. It is shown in Appendix I that, for, the filter length,, of the Chebyshev optimum design is given by The length of polynomial where (3) may be estimated from the Chebyshev (4a) (4b) (4c) TABLE I VALUES OF 8 (); 8 () AND 8 ()=8 () (10) It can be shown by differentiating (10) with respect to and equating the derivative to zero that, for small, the value of, denoted by, corresponding to the minimum value of is given by (11a) Several values of and for ranging from 0.1 to are tabulated in Table I. As can be seen from Table I, for. This further simplifies (11a) to Equation (4c) maps from 0 to into from to. Let has unity gain at and let the peak stopband ripple magnitude of be. It can be shown that For given by (11a) (11b) (5) (12a) Let.For For given by (11a) and (12b) where Since the desired transition width is of is. Thus, its length is (6a) (6b), the transition width The transition width of is and is given by The transition width of is the same as that of. Thus (7) (8) For, it can be easily shown from (3) and (12) that for. Thus, our new FRM technique will produce a design with a smaller number of nontrivial coefficients than that of the Chebyshev optimum design if the transition width of the desired Hilbert transformer is less than. For an audio signal sampled at 48 khz (44.1 khz), our new FRM technique will produce a saving in complexity comparing to the Chebyshev optimum design if frequency components below 5.2 khz (4.8 khz) must be faithfully transformed. The effective length of the filter designed using our new FRM technique is given by From (3), (7), (9), (11), and (13) (13) The total number of nontrivial coefficients, by (9),is given (14) In (14), the term 0.86 represents the fractional increase in filter length when compared to the Chebyshev optimum design. The fractional increase is less than 10% for.

5 LIM et al.: OPTIMUM MASKING LEVELS AND COEFFICIENT SPARSENESS 2447 TABLE III COEFFICIENT VALUES OF H (z) Fig. 4.Frequency shifter is inserted to suppress oscillation due to acoustic feedback. TABLE II COEFFICIENT VALUES OF H (z) TABLE IV COEFFICIENT VALUES OF H (z ) IV. EXAMPLE A problem often encountered in an acoustic feedback system is oscillation due to feedback. Such oscillation can be suppressed by inserting a frequency shifter into the system as shown in Fig. 4. The frequency shifter shifts the frequency by an adjustable amount ranging from 0 to 5 Hz. A shift of 2 Hz produces no noticeable distortion to most people. The implementation of the shifter requires a sharp Hilbert transformer if low frequency components must also be faithfully shifted. In a particular system, the application required a Hilbert transformer with the following specifications. Sampling rate: 32 khz. Lower-bandedge: 20 Hz. Peak ripple magnitude: The above requirements correspond to. In order to reduce the arithmetic complexity of the Hilbert transformer further, we chose an odd length filter with passband spanning from 20 to Hz. The design was done by first designing an even length Hilbert transformer with. The even length Hilbert transformer was then converted to an odd length Hilbert transformer by replacing the variable of its -transform transfer function by. The transition width shrank from to when was replaced by. The estimated length of a direct form even length Chebyshev optimum design meeting the specification was Using our FRM technique, the value of suggested by (11a) was The values of, and estimated from (7), (8), and (9) for were 106.6, 48.8, and 61.4, respectively. The optimization algorithm reported in [19] was used to optimize the subfilters and a design with, and was obtained. The total number of nontrivial coefficient was.it was only about 10.7% of the 2000 nontrivial coefficients of the Chebyshev optimum design. The effective length of the Chebyshev optimum design (after replacing by ) was The effective length of our FRM design (after replacing by ) was 4107; it was only about 2.7% longer than the Chebyshev optimum design. The coefficient values of the subfilters (after replacing by or which ever is appropriate) of the odd length Hilbert transformer are tabulated in Tables II IV. The passband ripple of the overall filter is shown in Fig. 5 and the frequency responses of the various subfilters are shown in Fig. 6. V. MULTI-LEVEL FRM If the transition width of the required filter is very narrow, the lengths of and may still be very long. In

6 2448 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 52, NO. 11, NOVEMBER 2005 given in, and for even values of and are (16a) (16b) (16c) Fig. 5. Passband ripples of the Hilbert transformer. The total number of nontrivial coefficients, -level masking design is given by of a (17) The optimum values of where are equal to (18) For given in (18), we have (19) Fig. 6. Frequency responses of the individual subfilters. (a) H (e ), (b) H (e ), (c) H (e ), (d) H (e )H (e ), (e) H (e )+ H (e )H (e ). The total number of nontrivial coefficients, -level masking design is given by,of a this case, a multi-level FRM technique may be employed to further reduce the complexity. In the multi-level FRM technique, we define (20a) (15) where is defined as unity. Consider a -level masking structure. The case for is shown in Fig. 7. Let the lengths of, and be, and, respectively. Estimates for (20b) The values of for several values of and are tabulated in Table V. It can be seen from Table V that, for a given value of decreases initially with increasing until it reaches a minimum and then increases with increasing. The value of that will yield the minimum

7 LIM et al.: OPTIMUM MASKING LEVELS AND COEFFICIENT SPARSENESS 2449 TABLE V VALUES OF (K +1)2:32 FOR SEVERAL VALUES OF AND K. RESULTS FOR K = 0CORRESPOND TO THOSE WITHOUT USING OUR PROPOSED FRM TECHNIQUE TABLE VI RANGES OF 1 CORRESPONDING TO VARIOUS VALUES OF K MINIMUM N () FOR Note that both and approach (the base of the natural logarithm) as approaches infinity. Thus, approaches as approaches zero. It is also interesting to note that also approaches as approaches infinity. The effective length of the filter is given by Fig. 7. Three-level masking structure. value of depends on the value of. It can be shown by manipulating (20a) that if where (21a) (24) In (24), the term represents the fractional increase in filter length when compared to the Chebyshev optimum design. For, we have, and applying the result of (18) (25) (21b) The value of that minimizes, denoted by, satisfies the constraint (22) The optimum values of for various transition widths are tabulated in Table VI. If is selected to minimize based on (20), it can be shown by considering (20a) and (18) that is bounded by (23) For. Thus, as for the case where, the FRM technique will produce a Hilbert transformer whose length is about 25% longer than that of the Chebyshev optimum design. VI. TWO-LEVEL MASKING EXAMPLE For the acoustic feedback suppression example in Section IV, the optimum value of as listed in Table VI was 5 (corresponding to ). The value of predicted from (20a) was 90; it is smaller than the Chebyshev optimum design of 2000 by a factor of 22! However, the actual selection of the value of depends on several factors. One of the most important factors influencing the selection of is the availability of user friendly and reliable optimization packages. The difficulty faced in the design process increases

8 2450 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 52, NO. 11, NOVEMBER 2005 TABLE VII COEFFICIENT VALUES OF H (z) TABLE XI COEFFICIENT VALUES OF H (z ) TABLE VIII COEFFICIENT VALUES OF H (z) TABLE IX COEFFICIENT VALUES OF H (z ) Fig. 8. Passband ripples of the Hilbert transformer. The frequency axis is normalized against the sampling frequency. TABLE X COEFFICIENT VALUES OF H (z ) with increasing ; optimization algorithms become less user friendly as increases. Instead of choosing, we choose to illustrate the advantage that can be gained in multi-level masking. For two-level masking, the estimated values were. A design with, and, and was obtained. The total number of nontrivial coefficient was. This is only 6% of the 2000 nontrivial coefficients of the Chebyshev optimum design; the saving was a factor of 17. The effective length of our FRM design (after replacing by ) was 4339; it was only 8.5% longer than the Chebyshev optimum design. The coefficient values of the subfilters (after replacing by or which ever is appropriate) are tabulated in Tables VII XI. The passband ripple of the overall filter is shown in Fig. 8 and the frequency responses of the various subfilters are shown in Fig. 9. VII. HALF-BAND FILTER An odd length low-pass half-band filter may be derived from an even length Hilbert transformer by scaling all the coefficient values of the Hilbert transformer by a factor of half, inserting a zero value coefficient between two coefficients, multiplying its th coefficient by, and replacing the centre coefficient by. The factor of two reduction in transition width due to inserting a zero between every two coefficients is nullified by the factor of two increase in transition width when the centre coefficient is inserted to raise the frequency response of the Hilbert transformer to become that of the half-band filter. Since the coefficient values have been scaled by half, the ripple magnitude of the resultant half-band filter is half that of the original Hilbert transformer. Taking this into consideration, the set of equations for the half-band filter may be obtained from that for the Hilbert transformer by replacing by. Specifically, for a half-band filter, (18) becomes and (22) becomes VIII. WITH SPECIAL PROPERTIES (26) (27) In the examples presented in Sections IV and VI, was optimized to minimize the overall complexity of the Hilbert transformer. However, it should be noted that can be any low-order Hilbert transformer implemented using any structure

9 LIM et al.: OPTIMUM MASKING LEVELS AND COEFFICIENT SPARSENESS 2451 IX. CONCLUSION A set of equations for characterizing the Hilbert transformer based on the implementation structure shown in Fig. 7 have been derived. It is also demonstrated that, in an acoustic feedback oscillation suppression example, the FRM technique produced a Hilbert transformer with reduction in the number of nontrivial coefficients by a factor of 17 when compared to the Chebyshev optimum design. APPENDIX I The expression [28] relating the length, of a low-pass filter to its passband and stopband ripples, and, respectively, and its normalized transition width is A Hilbert transformer may be derived from a low-pass half-band filter by discarding the centre coefficient, multiplying its th coefficient by, discarding trivial coefficients, and scale up the coefficient values by a factor of two. Replacing we have where. Fig. 9. Frequency responses of the individual subfilters and the overall filter. such as those in [25] [27] and can be designed independent on and. The frequency responses of and are than optimized as a correction term to sharpen the frequency response of the overall filter. In this case, the frequency response of the overall filter at the vicinity of is determined mainly by and. If the attenuation of is very high for frequencies far away from, the characteristics of the frequency response of the overall filter will be similar to that of for frequencies far away from. Nevertheless, the overall complexity of the filter will be higher than that where, and are jointly optimized. REFERENCES [1] Y. C. Lim, Frequency-response masking approach for the synthesis of sharp linear phase digital filter, IEEE Trans. Circuits Syst., vol. CAS-33, no. 4, pp , Apr [2] T. Saramäki, Y. C. Lim, and R. Yang, The synthesis of half-band filter using frequency-response masking technique, IEEE Trans Circuits Syst. II, Analog Digit. Signal Process., vol. 42, no. 1, pp , Jan [3] Y. C. Lim and Y. J. Yu, Synthesis of very sharp hilbert transformer using the frequency-response masking technique, IEEE Trans. Signal Process., vol. 53, no. 7, pp , Jul [4] Y. C. Lim and Y. Lian, The optimum design of one- and two-dimensional FIR filters using the frequency response masking technique, IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 40, no. 2, pp , Feb [5], Frequency-response masking approach for digital filter design: Complexity reduction via masking filter factorization, IEEE Trans Circuits Syst. II, Analog Digit. Signal Process., vol. 41, no. 8, pp , Aug

10 2452 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 52, NO. 11, NOVEMBER 2005 [6] M. G. Bellanger, Improved design of long FIR filters using the frequency masking technique, in Proc. IEEE Int. Conf. Acoustics, Speech, Signal Processing, 1996, pp [7] Y. C. Lim and S. H. Low, Frequency-response masking approach for the synthesis of sharp two-dimensional diamond-shaped filters, IEEE Trans Circuits Syst. II, Analog Digit. Signal Process., vol. 45, no. 12, pp , Dec [8] H. Johansson and L. Wanhammar, High-speed recursive digital filters based on the frequency-response masking approach, IEEE Trans. Circuits Syst. II: Analog and Digital Signal Processing, vol. 47, no. 1, pp , Jan [9] W. S. Lu and T. Hinamoto, Optimal design of frequency-responsemasking filters using semidefinite programming, IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 50, no. 4, pp , Apr [10], Optimal design of IIR frequency-response-masking filters using second-order cone programming, IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 50, no. 11, pp , Nov [11] M. B. Furtado, P. S. R. Diniz, and S. L. Netto, Optimized prototype filter based on the FRM approach for cosine-modulated filter banks, Circuits, Syst. Signal Processing, vol. 22, no. 2, pp , Mar./Apr [12] L. C. R. Barcellos, S. L. Netto, and P. S. R. Diniz, Optimization of FRM filters using the WLS-Chebyshev approach, Circuits, Syst. Signal Processing, vol. 22, no. 2, pp , Mar./Apr [13] O. Gustafsson, H. Johansson, and L. Wanhammar, Single filter frequency masking high-speed recursive digital filters, Circuits, Syst. Signal Processing, vol. 22, no. 2, pp , Mar./Apr [14] T. Saramäki and Y. C. Lim, Use of the Remez algorithm for designing FRM based FIR filters, Circuits, Syst. Signal Processing, vol. 22, no. 2, pp , Mar./Apr [15] H. Johansson and T. Saramäki, Two-channel FIR filter banks utilizing the FRM approach, Circuits, Syst. Signal Processing, vol. 22, no. 2, pp , Mar./Apr [16] Y. Lian and C. Z. Yang, Complexity reduction by decoupling the masking filters from bandedge shaping filter in FRM technique, Circuits, Syst. Signal Processing, vol. 22, no. 2, pp , Mar./Apr [17] Y. Lian, Complexity reduction for FRM based FIR filters using the prefilter-equalizer technique, Circuits, Syst. Signal Processing, vol. 22, no. 2, pp , Mar./Apr [18] Y. C. Lim, Y. J. Yu, H. Q. Zheng, and S. W. Foo, FPGA implementation of digital filters synthesized using the FRM technique, Circuits, Syst. Signal Processing, vol. 22, no. 2, pp , Mar./Apr [19] T. Saramäki, J. Yli-Kaakinen, and H. Johansson, Optimization of frequency-response-masking based FIR filters, Circuits, Syst. Signal Processing, vol. 12, no. 5, pp , Oct [20] W. R. Lee, V. Rehbock, and K. L. Teo, Frequency-response masking based FIR filter design with power-of-two coefficients and suboptimum PWR, Circuits, Syst. Signal Processing, vol. 12, no. 5, pp , Oct [21] O. Gustafsson, H. Johansson, and L. Wanhammar, Single filter frequency-response masking FIR filter, Circuits, Syst. Signal Processing, vol. 12, no. 5, pp , Oct [22] S. L. Netto, L. C. R. Barcellos, and P. S. R. Diniz, Efficient design of narrow-band cosine-modulated filter banks using a two-stage frequencyresponse masking approach, Circuits, Syst. Signal Processing, vol. 12, no. 5, pp , Oct [23] Y. Lian, A modified frequency response masking structure for highspeed FPGA implementation of programmable sharp FIR filters, Circuits, Syst. Signal Processing, vol. 12, no. 5, pp , Oct [24] S. W. Foo and W. T. Lee, Application of fast filter bank for transcription of polyphonic signals, Circuits, Syst. Signal Processing, vol. 12, no. 5, pp , Oct [25] M. Z. Komodromos, S. F. Russell, and P. T. P. Tang, Design of FIR Hilbert transformers and differentiators in the complex domain, IEEE Trans. Circuits Systs. I, Fundam. Theory Appl., vol. 45, no. 1, pp , Jan [26] S. Samadi, Y. Igarashi, and H. Iwakura, Design and multiplierless realization of maximally flat FIR digital Hilbert transformers, IEEE Trans. Signal Process., vol. 47, no. 7, pp , Jul [27] S. C. Pei and P. H. Wang, Close-form design of maximally flat FIR Hilbert transformer, differentiators, and fractional delayers by power series expansion, IEEE Trans. Circuits Systs. I, Fundam. Theory Appl., vol. 48, no. 4, pp , Apr [28] O. Herrmann, L. R. Rabiner, and D. S. K. Chan, Practical design rules for optimum finite impulse response low-pass digital filters, Bell Syst. Tech. J., vol. 52, pp , Jul. Aug Yong Ching Lim (S 79 M 82 SM 92 F 00) received the A.C.G.I. and B.Sc. degrees in 1977 and the D.I.C. and Ph.D. degrees in 1980, all in electrical engineering, from Imperial College, University of London, London, U.K. From 1980 to 1982, he was a National Research Council Research Associate in the Naval Postgraduate School, Monterey, CA. From 1982 to 2003, he was with the Department of Electrical Engineering, National University of Singapore, Singapore. Since 2003, he has been with the School of Electrical and Electronics Engineering, Nanyang Technological University, Singapore, where he is currently a Professor. His research interests include digital signal processing and VLSI circuits and systems design. Dr. Lim was selected to receive the 1996 IEEE Circuits and Systems Society s Guillemin Cauer Award, the 1990 IREE (Australia) Norman Hayes Memorial Award, 1977 IEE (UK) Prize and the Siemens Memorial (Imperial College) Award. He served as a lecturer for the IEEE Circuits and Systems Society under the distinguished lecturer program from 2001 to 2002 and as an Associate Editor of the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS from 1991 to 1993 and from 1999 to He has also served as an Associate Editor of Circuits, Systems and Signal Processing from 1993 to He served as the Chairman of the DSP Technical Committee of the IEEE Circuits and Systems Society from 1998 to He served in the Technical Program Committee s DSP Track as the Chairman in IEEE ISCAS 97 and IEEE ISCAS 00 and as a Co-chairman in IEEE ISCAS 99. He is the General Chairman for IEEE APCCAS 06. He is a member of Eta Kappa Nu. Ya Jun Yu (S 99 M 05) received the B.Sc. and M.Eng. degrees in biomedical engineering from Zhejiang University, Hangzhou, China, in 1994 and 1997, respectively, and the Ph.D. degree in electrical and computer engineering from the National University of Singapore, Singapore, in From 1997 to 1998, she was a Teaching Assistant with Zhejiang University. She joined the Department of Electrical and Computer Engineering, National University of Singapore as a Post-Master s Fellow in 1998 and remained in the same department as a Research Engineer until In 2002, she was a Visiting Researcher at the Tampere University of Technology, Tampere, Finland and The Hong Kong Polytechnic University, Hong Kong. Since 2004, she has been with the Temasek Laboratories, Nanyang Technological University, Singapore, as a Research Fellow. Her research interests include digital signal processing and VLSI circuits, and systems design.

11 LIM et al.: OPTIMUM MASKING LEVELS AND COEFFICIENT SPARSENESS 2453 Tapio Saramäki (M 98 SM 01 F 02) was born in Orivesi, Finland, on June 12, He received the Diploma Engineer (with honors) and Doctor of Technology (with honors) degrees in electrical engineering from the Tampere University of Technology (TUT), Tampere, Finland, in 1978 and 1981, respectively. Since 1977, he has held various research and teaching positions at TUT, where he is currently a Professor of Signal Processing and a Docent of Telecommunications (a scientist having valuable knowledge for both the research and education at the corresponding laboratory). He is also a Cofounder and a System-Level Designer of VLSI Solution Oy, Tampere, Finland, originally specializing in VLSI implementations of sigma-delta modulators and analog and digital signal processing algorithms for various applications. He is also the President of Aragit Oy Ltd., Tampere, Finland, which was founded by four TUT professors, specializing in various services for the industry, including the application of information technology to numerous applications. In 1982, 1985, 1986, 1990, and 1998, he was a Visiting Research Fellow (Professor) with the University of California, Santa Barbara, in 1987 with the California Institute of Technology, Pasadena, and in 2001 with the National University of Singapore, Singapore. His research interests are in digital signal processing, especially filter and filter bank design, VLSI implementations, and communications applications, as well as approximation and optimization theories. He has written more than 250 international journal and conference articles, various international book chapters, and holds three worldwide-used patents. Dr. Saramäki received the 1987 Guillemin Cauer Award for the Best Paper of the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS, as well as two other Best Paper awards. In 2004, he was awarded the honorary membership of the A. S. Popov Society for Radio-Engineering, Electronics, and Communications (the highest membership grade in the society and the 80th honorary member since 1945) for great contributions to the development of DSP theory and methods and great contributions to the consolidation of relationships between Russian and Finnish organizations. He is a founding member of the Median-Free Group International. He was an Associate Editor of the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II: ANALOG AND DIGITAL SIGNAL PROCESSING ( ), and is currently an Associate Editor of Circuits, Systems, and Signal Processing ( ). He was also a Distinguished Lecturer of the IEEE Circuits and Systems Society ( ) and the Chairman of the IEEE Circuits and Systems DSP Technical Committee (May 2002 May 2004).