Frequency-Selective KYP Lemma, IIR Filter, and Filter ... - CiteSeerX

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Frequency-Selective KYP Lemma, IIR Filter, and Filter Bank Design Hung Gia Hoang, Hoang Duong Tuan, and Truong Q. Nguyen

Abstract—For a transfer function ( ) of order , Kalman– Yakubovich–Popov (KYP) lemma characterizes a general intractable semi-infinite programming (SIP) condition by a tractable semidefinite programming (SDP) for the entire frequency range. Some recent results generalize this lemma for a certain frequency interval. All these SDP characterizations are given at the expense of the introduced Lyapunov matrix variable of dimension . Consequently, formulation and design of high dimensional problem is challenging. Moreover, existing SDP characterizations for frequency-selective SIP (FS-SIP) do not allow to formulate synthesis problems as SDPs. In this paper, we propose a completely new SDP characterization of general FS-SIP involving SDPs of moderate size and free from Lyapunov variables. Furthermore, a systematic IIR filter and filter bank design is developed in a similar vein, with several simulations provided to validate the effectiveness of our SDP formulation. Index Terms—Filter and filter bank, Kalman–Yakubovich– Popov (KYP) lemma, semidefinite programming.

I. INTRODUCTION HE celebrated Kalman–Yakubovich–Popov (KYP) lemma together with its variations such as positive real lemma and bounded real lemmas (see, e.g., [2], [8], [16], and [24]–[26]) are certainly among the most important results in modern control and signal processing. Essentially, they express computationally intractable semi-infinite programming (SIP) in frequency domain constraints of a transfer function by a computationally tractable semi-definite programming (SDP). The most general KYP lemma (see, e.g., [16]) states that , the SIP given a Hermitian indefinite matrix condition

T

(1) is characterized by a for an -order transfer function SDP involving its state-space realization and a . As a Lyapunov matrix function variable of dimension Manuscript received January 29, 2007; revised April 21, 2008. First published November 07, 2008; current version published February 13, 2009. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Thierry Blu. The work is supported by the Australian Research Council under Grant ARC Discovery Project 0556174. H. G. Hoang and H. D. Tuan are with the School of Electrical Engineering and Telecommunication, University of New South Wales, Sydney, NSW 2052, Australia (e-mail: [email protected]; [email protected]). T. Q. Nguyen is with the Department of Electrical and Computer Engineering, University of California, San Diego, La Jolla CA 92093-0407 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TSP.2008.2009012

of the Lyapunov variable, matter of fact, the dimension scalar variables, increases which is equivalent to dramatically as order increases moderately. Consequently, the resultant SDPs are of large dimension and therefore are hardly solved by existing general-purpose SDP solvers such as [13]. requires a Lyapunov For instance, a function order variable of dimension 100 100, which is equivalent to 5000 scalar variables. This “curse of dimensionality” may not be so visible in control as many physical plants are modeled with low orders, so SDP in tandem with KYP and Lyapunov machineries are the most dominant approach for control problems. However, many signal processing applications require filters of very high order so such kind of SDP is not applicable. Hence, traditional methods such as interpolation (frequency sampling) and window techniques are understandably still popular in spite of lacking flexibility in design [26], [27]. Also, some procedures of iteratively discretizing SIP (1) (see, e.g., [10]) are of interest. Furthermore, many problems in signal processing involve SIPs for a certain frequency interval rather the entire range. For example, one of the main filter design specifications is the peak error between the filter response and the ideal response on and stopband , i.e., SIPs verified for the passband and only. Thus, a generalization for the mentioned KYP lemma, commonly referred to as frequency selective KYP (FS-KYP) lemma, is demanded. Several approaches have been proposed for SDP characterization of FS-KYP lemma as those in [7], [12], which invariably have similar drawback of the original KYP lemma: even more are involved in the SDP formatrix variables of dimension mulation. The formulation of [12] does not allow to formulate a design problem as SDP. More precisely, it leads to a bilinear matrix inequality (BMI) formulation for the problem. In [23], we obtained a new SDP characterization of the frequency selective for finite impulse response inequality . This SDP formulation is of substantially reduced (FIR) order and its dual formulation does not involve any additional variables, hence opens a new way for effective solution of large dimensional digital systems. Our SDP-based methodology can compete well with all other filter designs and is advantageous in term of flexibility. Meanwhile, IIR filter and filter bank design is a fundamental problem in signal processing [26]. For a given specification, an IIR filter requires much lower order as compared to a FIR filter. However, IIR filter and filter bank design is a challenging task since digital IIR filters are mostly derived from their analog counterparts via bilinear transformation or the impulse invariant method [15]. The peak errors as well as cut-off frequencies are not easily controlled. Thus, the aim of this paper is twofold:

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HOANG et al.: FS KYP LEMMA, IIR FILTER, AND FILTER BANK DESIGN

• to develop a new FS-KYP lemma for IIR transfer functions that does not involve any additional variable. The SDP formulation is of reduced order and is flexible for analysis as well as synthesis problems; • to propose a systematic approach for IIR filter and filter bank design in a similar vein. The design problem is formulated as a convex (SDP) optimization problem, while other FS-KYP lemmas based results at best lead to BMI optimization problems. Our solution provides a breakthrough that enables almost all filter and filter bank design problems to be solved efficiently. The structure of the paper is organized as follows. The reduced order formulation is discussed in Section II. Based on the result in Section II, the IIR filter and filter bank design problems are developed in Section III and Section IV, respectively. The decriterion is presented in sign of IIR synthesis bank using Section V. Finally, concluding remarks are given in Section VI. The following notations are used in the paper. Vectors and matrices will be represented by italicized bold lower case and denotes the uppercase letters, respectively. The superscript transpose (without conjugation) whereas the superscript denotes Hermitian transpose. The conventional symbols and are used to denote real and complex spaces. The real and imaginary parts of a complex number are denoted by and , respectively. The standard notation represents a positive semidefinite Hermitian matrix and stands for the inner product of the matrices and , i.e., . Some results of this paper have been announced in [11]. II. REDUCED-ORDER SDP FORMULATION FOR FREQUENCY SELECTIVE KYP LEMMA

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The th moment trigonometric matrix of size is defined as the positive semidefinite matrix [see the equation at the bottom of the page] and accordingly, the matrix is created from by the variable change (2) . It is straightforward to show that

i.e.,

Define also

and accordingly created from

is by the variable change (2), i.e., :

Our main results in this paper are based on the following LMI characterization for the convex hull of the trigonometric curve . Theorem 1: ([23]) The conic hull of the trigonometric curve is fully characterized by LMIs: if and only if it satisfies the LMIs

A. Mathematical Background

(3)

In this paper, we employ the concept of the trigonometric curve and its convex hull which has been introduced in [23]. Let

then the trigonometric curve

and its polar,

is defined as

, is given by

denotes the nearest lower integer to . The where convex hull of is also fully characterized by if and only if it satisfies the LMIs (3) LMIs: with . Note that for even, by definition, is a matrix and accordingly LMIs (3) are unfunction of derstood for some . From the above result, it is straightforward to transform SIP optimization to SDPs. For instance, consider the following general optimization (4)

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where the positive-definite matrix is given and . Then, by Theorem 1, its dual is [see (5) shown at the bottom of the page], which can be rewritten either as a linear conic optimization [13] for efficient computation or more directly, as an SDP:

The most popular particular cases of (9) are as follows. • Frequency-selective bounded realness (FS-BR) with

(10) (11) for

(6)

of (4) is directly retrieved from the The optimal solution optimal solution of (5) by the solution of the following linear system:

Without FS restrictions, it merely states that the -norm of is less than or equal . • Frequency-selective positive realness (FS-PR) with (12) (13)

(7) Thus, the optimal solution of the semi-infinite program (4) can be easily obtained from the program (5) involving just scalar variables. B. Novel FS-KYP Lemma In general, an -order IIR filter is represented by the following transfer function

Without FS restriction, it merely states that the function is positive real. A linear algebra based efficient method has been proposed in -norm of . This method, however, [4] for computing cannot be extended for verifying (11). Also, another heuristic method to solve LMIs in the original KYP lemma has been proposed in [14]. However, both methods do not work for the synsatisfying (11). thesis problem, i.e., one has to design Now, based on the result of Theorem 1, we provide a new look for the FS-SIP (9). Clearly, (9) is equivalent to

(8) (14) By defining

where

With the same Hermitian indefinite matrix frequency selective SIP (FS-SIP) is stated as

of (1), a

FS-SIP (14) is rewritten as

(9)

(15)

(5) for

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HOANG et al.: FS KYP LEMMA, IIR FILTER, AND FILTER BANK DESIGN

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where

essence, the formula (18) transforms the SIP (9) into a trigonometric polynomial that is positive on a finite interval, which is characterized by LMIs via Theorem 1. More important, since and are linear combinations of and , and can be found easily by solving the following simple linear system .. .

.. .

..

.. .

.

From now on, we will use the symbol to denote the maby the variable change (2), i.e., trix created from

where and .

and

are obtained through spectral factorization of

III. IIR FILTER DESIGN .. .

.. .

..

.. .

.

Since the matrix is a linear function in , the FS-SIP (15) holds true if and only if (16) in Theorem Applying the LMI characterization for 1 to (16), we arrive at the following result. Proposition 1: The FS-SIP (9) can be verified by the following SDP: (17) Namely, (9) holds true if and only if the optimal value of SDP (17) is nonnegative. In contrast with all previous results on LMI characterization for SIP (1) or (9) that require a matrix variable of dimension , the above SDP involves merely scalar variables. Observe that SDP formulation (17) is suitable only for analysis problems, i.e., to verify FS-SIP (9) for a given . For a synthesis problem, one has to design to satisfy FS-SIP (9), so and in (8) represent design variables, then (17) is a BMI in . We now reformulate (17) in a form that is more convenient for synthesis purposes. Since is indefinite, by eigenvalue decomposition, there are , and such that

Digital IIR filters are an important class in signal processing. Beside classical methods, optimization-based methods have been widely used in designing IIR filters due to their flexibility and efficiency. The IIR filter design has been formulated as a nonconvex optimization problem, which is then solved by a genetic algorithm in [20]. Consequently, the algorithm convergence is not guaranteed and the local or global optimal solutions cannot be confirmed. In this section, we formulate the IIR filter design problem as a convex SIP in terms of the autocorrelation sequences of the numerator and denominator, which is efficiently solved via SDP (5). Generally, a mask-constrained lowpass IIR filter design problem is formulated as an optimization problem that minimizes the weighted deviation from the ideal magnitude response subject to a set of constraints in the passband and stopband as follows:

(19) subject to: (20a) (20b) (20c) where and are passband and stopband ripples, respectively. Observe that constraints (19) take the same form as (10), hence, like (18) the problem can be expressed in terms of autocorrelation sequences and of and as follows:

[see (10) and (12) for values of and in particular cases]. Then (15) is expressed as (21a) subject to: (21b) (21c) (21d) (21e) (18) where correlation sequences of

and and

where

are the auto. In

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One of the difficulties in optimization-based IIR filter design problem is that no analytic formula is available for the objective function. An alternative way is to reformulate the objective function as

(22) where the weights and are added in the objective function so that the denominator does not become too small and . In our experiwhile minimizing yields good results in most probence, choosing around lems. Hence, the problem is rewritten as (23a)–(23f) shown at and the bottom of the page. For , the optimization problem (23) can be rewritten as the form (4) with

Its solutions are found by solving (5) and (7). To illustrate our result numerically, we adopt one example in [23], where filter design specifications are given in Table I. To fulfill these requirements, for an FIR filter it is necessary that the order is 200 [23]. For Butterworth and Chebyshev filters, the necessary order is 106 and 21, respectively. Using the proposed method, the required order is only 14, which is a significant improvement. The magnitude response of the fourteenth-order IIR filter is presented in Fig. 1 and its poles-zeros diagram is shown in Fig. 2. Observe that all the poles of the designed filter are located inside the unit circle with the maximum pole radius is 0.98219.

Fig. 1. Frequency response of a fourteenth-order IIR filter.

Fig. 2. Poles-zeros diagram of the fourteenth-order IIR filter.

IV. DESIGN OF ORTHOGONAL IIR FILTER BANK A two-channel maximally decimated filter bank, as shown in Fig. 3, is a fundamental block in many signal processing and communications systems [5], [18]. As compared to FIR filter

(23a) subject to:

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(23b) (23c) (23d) (23e) (23f)

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TABLE I SPECIFICATIONS FOR IIR FILTER DESIGN

(30) where is a positive polynomial to be designed such that predefined specifications are met. Assume that the design specifications are given in terms of passband and stopband peak ripples as in the previous section, the design of a IIR filter bank can now be cast as the following SIP:

(31a) subject to: (31b) Fig. 3. QMF bank structure.

banks, the IIR ones have more design freedom and often offer better frequency selectivity. Moreover, if the filter bank is orthogonal, it is guaranteed that the energy of errors due to transmission or quantization will not be amplified, and under certain conditions the filter bank can be used to generate orthogonal wavelet bases [6], [27]. Thus, various approaches have been proposed to the design of orthogonal IIR filter banks [9], [17], [28], which invariably lead to the design of an IIR filter that is parallel sum of two allpass filters. In this section, we present a more general method, in which the subfilters are structure-free. and Consider the filter bank in Fig. 3, assuming that are all IIR filters. It is well known [19], [26] that the filter and bank is alias-free and distortion-free if and only if satisfy the following equations: (24) (25)

(31c) (31d) where

For simplicity, we assume that . Like the IIR filter design case, there is no analytical formula for the objective function. To overcome this problem, we must alternatively find an analytical function which also measures the deviation from the desired response. In this paper, we use the following quadratic function:

and to satisfy (24) There are many ways to choose and (25). A typical approach, which leads to the so-called orthogonal perfect reconstruction filter bank, is to choose and as follows [9], [27]: (26) (27) (28)

(32) Now the IIR filter bank design problem becomes

be the product filter, , the Let design of orthogonal filter bank reduces to the design of a half, i.e., a lowpass filter satisfies halfband condiband filter is derived, is obtained from by tion (28). Once spectral factorization, and the other three filters are derived from according to (26) and (27). As we can always choose as minimum factor of , the stability of all subfilters are guaranteed. It is proved in [9] that every valid halfband filter must admit the following representation:

(33a) subject to:

(29)

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(33b)

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Fig. 4. Magnitude responses of product filters.

Fig. 5. Magnitude responses of analysis filters.

(33c) (33d) where

.. .

.. .

.. .

.. .

..

..

.

.. .

.

.. .

.. .

.. .

Denote by

Fig. 6. Poles-zeros diagram of

H (z).

worth filters, our approach allows direct control of the transition band width. As a result, transition bands of the designed filters are much better than those of Butterworth filters of the same order, as clearly shown in Figs. 4 and 5. The poles-zeros diais presented in Fig. 6 to confirm the gram of the filter stability of the filter. V. NEAR-PERFECT IIR QMF BANK SYNTHESIS

the optimization problem (33) is rewritten compactly in the form (4) (for ). The solution of (33) is found by solving the SDP problem (5) and (7). To illustrate the effectiveness of the proposed method, we design a filter bank comprising of fourth-order IIR filters. In contrast to the maximally flat method in [9], which leads to Butter-

In the previous section, orthogonal perfect reconstruction IIR filter banks have been designed. However, in practice, the distortion-free condition is somewhat too restrictive due to round off errors occurred in the implementation. Moreover, subband signals would usually be processed in many applications. In these situations, it is not necessary that the output signal is a perfect reconstruction of the input. This gives rise to the design of near perfect filter banks, where the aliasing is canceled and distortion is less than a predefined level (see, for example, [1], [3], and [22], and references therein). However, existing approaches often involve state-space representation and KYP lemma, which invariably lead to nonconvex optimization problems of high di-

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mension. In this section, we develop a low-dimensional SDP formulation for the design of synthesis bank of a near-perfect filter bank. in Fig. 3 are given, the Suppose that analysis filters filter bank is guaranteed to be free from aliasing if the synthesis bank is chosen as (34) . In this section, near perfect filter bank is dewith some , signed so as the distortion is guaranteed to be bounded by a pre-specified level , i.e.,

or, equivalently

Fig. 7. Frequency response of a length-8 synthesis filters.

(35) for , . To compensate the unnecessary over-design in which is larger than its minimal achievable one, we optimize the objective

where and are obtained from convolution matrices of and , respectively. Hence, the optimization problem (36), (35) can be rewritten as

(39a) which like the previous section is replaced by

subject to: (39b) (39c) (39d)

(36) where

and

(39e)

are weights. Denote by Now using the augmented variable we can rewrite the SIP (39a) in the form of (4) (for with

it is straightforward to see that of and :

and

are linear combinations

, )

To illustrate the theoretical result, we employ the example in [3] and [21], where the synthesis bank is designed based on the following analysis filters:

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(40)

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Fig. 8. Poles-zeros diagram of

G(z).

and . The magnitude responses of the synthesis filters of order 7 are presented in Fig. 7 with overall distortion error less than 60 dB. The poles-zeros diagram of is shown in Fig. 8, where the maximum poles radius is 0.86642. Note that the magnitude responses of the designed filter of order 7 are better than those of order 14 and 15 in [3] and [22]. This is because our designed filters are globally optimal solution whereas that designed by [3] and [22] are just some suboptimal solutions of the nonconvex optimization formulation. VI. CONCLUSION We presented a totally new FS-KYP lemma that is derived for both analysis and synthesis purposes. The key feature of the proposed FS-KYP lemma is that the resultant SDP is of minimal order, thus enabling high dimensional problems to be solved efficiently using general purpose solvers on a standard computer. A systematic approach for IIR filter and filter bank design problems based on the newly derived FS-KYP lemma has been developed that provides a breakthrough for almost all IIR filter design problems. A number of numerical simulations have been provided confirming the effectiveness of the proposed method in terms of computational complexity and design flexibility. REFERENCES [1] A. Akansu and R. Haddad, Multiresolution Signal Decomposition: Transforms, Subbands and Wavelets. New York: Academic, 2001. [2] B. Anderson and S. Vongpanitlerd, Network Analysis and Synthesis: A Modern System Theory Approach. Englewood Cliffs, NJ: PrenticeHall, 1973. [3] R. N. Banavar and A. A. Kalele, “A mixed norm performance measure for the design of multirate filterbanks,” IEEE Trans. Signal Process., vol. 49, no. 2, pp. 354–359, Feb. 2001. [4] S. Boyd, V. Balakrishnan, and P. Kabamba, “A bisection method for norm of a transfer matrix and related problem,” computing the Math. Control, Signal, Syst., vol. 2, no. 3, pp. 207–219, 1989. [5] C. D. Creusere and S. K. Mitra, “Image coding using wavelets based on perfect reconstruction IIR filter banks,” IEEE Trans. Circuits Syst. Video Technol., vol. 6, no. 5, pp. 447–458, Oct. 1996.

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[6] I. Daubechies, “Orthonormal bases of compactly supported wavelets,” Commun. Pure Appl. Math., vol. 41, pp. 909–996, Nov. 1988. [7] T. Davidson, Z. Luo, and J. F. Sturm, “Linear matrix inequality formulation of spectral mask constraints with applications to FIR filter design,” IEEE Trans. Signal Process., vol. 50, no. 11, pp. 2702–2715, Nov. 2002. [8] B. Dumitrescu, I. Tabus, and P. Stoica, “On the parameterization of positive real sequences and MA parameter estimation,” IEEE Trans. Signal Process., vol. 49, no. 11, pp. 2630–2639, Nov. 2001. [9] C. Herley and M. Vetterli, “Wavelet and recursive filter banks,” IEEE Trans. Signal Process., vol. 41, no. 8, pp. 2536–2556, Aug. 1993. [10] C. Y.-F. Ho, B. W.-K. Ling, Y.-Q. Liu, P. K.-S. Tam, and K.-L. Teo, “Efficient algorithm for solving semi-infinite programming problems and their applications to nonuniform filter bank designs,” IEEE Trans. Signal Process., vol. 54, no. 11, pp. 4223–4232, Nov. 2006. [11] H. G. Hoang, H. D. Tuan, and T. Q. Nguyen, “Frequency selective KYP lemma and its applications to IIR filter bank design,” in Proc. Int. Conf. Acoustic, Speech, Signal Processing (ICASSP), 2007, vol. III, pp. 1457–1460. [12] T. Iwasaki and S. Hara, “Generalized KYP lemma: Unified frequency domain inequalities with design applications,” IEEE Trans. Autom. Control, vol. 50, no. 1, pp. 41–59, Jan. 2005. [13] J. F. Sturm, “Using sedumi 1.02, a matlab toolbox for optimization over symmetric cones,” Optimiz. Methods Softw., pp. 625–653, 1999. [14] P. Parrilo, “On the numerical solution of LMIs derived from KYP lemma,” in Proc. IEEE Conf. Decision Control (CDC), 1998, pp. 2334–2338. [15] J. Proakis and D. Manolakis, Digital Signal Processing: Principles, Algorithms, and Applications, 3rd ed. Englewood Cliffs, NJ: Prentice-Hall, 1996. [16] A. Rantzer, “On the Kalman–Yakubovich–Popov lemma,” Syst. Control Lett., vol. 28, no. 1, pp. 7–10, 1996. [17] I. Selesnick, “Formulas for orthogonal IIR wavelet filters,” IEEE Trans. Signal Process., vol. 46, no. 4, pp. 1138–1141, Apr. 1998. [18] M. J. T. Smith and S. L. Eddins, “Analysis/synthesis techniques for subband image coding,” IEEE Trans. Acoust., Speech, Signal Process., vol. 38, no. 8, pp. 1446–1456, Aug. 1990. [19] G. Strang and T. Nguyen, Wavelets and Filter Banks. Cambridge, MA: Wellesley-Cambridge, 1996. [20] J.-T. Tsai and J.-H. Chou, “Design of optimal digital IIR filters by using an improved immune algorithm,” IEEE Trans. Signal Process., vol. 54, no. 12, pp. 4582–4596, Dec. 2006. [21] H. D. Tuan, L. Nam, H. Tuy, and T. Q. Nguyen, “Multicriterion optimized QMF bank design,” IEEE Trans. Signal Process., vol. 51, no. 10, pp. 2582–2591, Oct. 2003. [22] H. D. Tuan, T. T. Son, P. Apkarian, and T. Q. Nguyen, “Low-order IIR filter bank design,” IEEE Trans. Circuits Syst. I, vol. 52, no. 8, pp. 1673–1683, Aug. 2005. [23] H. D. Tuan, T. T. Son, B. Vo, and T. Q. Nguyen, “LMI characterization for the convex hull of trigonometric curves and applications,” IEEE Trans. Signal Process., to be published. [24] J. Tuqan and P. Vaidyanathan, “The role of the discrete time Kalman–Yakubovich–Popov (KYP) lemma in designing statistically optimum FIR orthonormal filter bank,” in Proc. ISCAS, 1998, vol. 5, pp. 122–125. [25] J. Tuqan and P. Vaidyanathan, “A state-space approach to the design of globally optimal FIR energy compaction filters,” IEEE Trans. Signal Process., vol. 48, pp. 2822–2838, 2000. [26] P. Vaidyanathan, Multirate Systems and Filter Banks. Englewood Cliffs, NJ: Prentice-Hall, 1993. [27] M. Vetterli and J. Kovaˇcevic´a, Wavelets and Subband Coding. Englewood Cliffs, NJ: Prentice-Hall, 1995. [28] X. Zhang, W. Wang, T. Yoshikawa, and Y. Takei, “Design of IIR orthogonal wavelet filter banks using lifting scheme,” IEEE Trans. Signal Process., vol. 54, no. 7, pp. 2616–2624, Jul. 2006.

Hung Gia Hoang received the Bachelor’s degree in electrical engineering from Vietnam National University, Hanoi, in 2000 and the Ph.D. degree from the University of New South Wales, Australia, in 2009. He is working in Vietnam. His research interests include optimization methods and their applications to signal processing and robust control.

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Hoang Duong Tuan was born in Hanoi, Vietnam, in 1964. He received the Diploma and Ph.D. degrees, both in applied mathematics from Odessa State University, Ukraine, in 1987 and 1991, respectively. From 1991 to 1994, he was a Researcher at the Optimization and Systems Division, Vietnam National Center for Science and Technologies. He spent nine academic years in Japan as an Assistant Professor at the Department of Electronic-Mechanical Engineering, Nagoya University, from 1994 to 1999 and then as an Associate Professor at the Department of Electrical and Computer Engineering, Toyota Technological Institute, Nagoya, from 1999 to 2003. Currently, he is an Associate Professor at the School of Electrical Engineering and Telecommunications, the University of New South Wales, Sydney, Australia. His research interests include theoretical developments and applications of optimization based methods in many areas of control, signal processing, communication, and bio-informatics.

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Truong Q. Nguyen 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 a Member of Technical Staff with MIT Lincoln Laboratory from June 1989 to July 1994. Since August 1994 to July 1998, he was with the Electrical and Computer Engineering Department, University of Wisconsin, Madison. He was with Boston University from 1996 to 2001, and he is currently with the Electrical and Computer Engineering Department at the University of California, San Diego. 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 conversion. He is the coauthor (with Prof. G. Strang) of a popular textbook Wavelets & Filter Banks (Wellesley-Cambridge Press, 1997) and the author of several Matlab-based toolboxes on image compression, electrocardiogram compression, and filter bank design. He also hold a patent on an efficient design method for wavelets and filter banks and several patents on wavelet applications, including compression and signal analysis. Prof. Nguyen received the IEEE TRANSACTIONS IN SIGNAL PROCESSING Paper Award (Image and Multidimensional Processing area) for the paper he coauthored with Prof. P. P. Vaidyanathan on linear-phase perfect-reconstruction filter banks (1992). He received the NSF Career Award in 1995 and is currently the Series Editor (Digital Signal Processing) for Academic Press. He served as Associate Editor for the IEEE TRANSACTIONS ON SIGNAL PROCESSING from 1994 to 1996 and for the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS from 1996 to 1997.

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