Optimal Design of Complex FIR Filters With Arbitrary ... - IEEE Xplore

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 5, MAY 2006

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Optimal Design of Complex FIR Filters With Arbitrary Magnitude and Group Delay Responses Wei Rong Lee, Lou Caccetta, Kok Lay Teo, Senior Member, IEEE, and Volker Rehbock

Abstract—This paper presents a method for the frequency-domain design of digital finite impulse response filters with arbitrary magnitude and group delay responses. The method can deal with both the equiripple design problem and the peak constrained least squares (PCLS) design problem. Consequently, the method can also be applied to the equiripple passbands and PCLS stopbands design problem as a special case of the PCLS design. Both the equiripple and the PCLS design problems are converted into weighted least squares optimization problems. They are then solved iteratively with appropriately updated error weighting functions. A novel scheme for updating the error weighting function is developed to incorporate the design requirements. Design examples are included in order to compare the performance of the filters designed using the proposed scheme and several other existing methods. Index Terms—Complex finite impulse response (FIR) filters, optimal design, weighted least squares.

I. INTRODUCTION

L

INEAR-phase finite impulse response (FIR) digital filters are frequently used in signal processing applications because of their guaranteed stability and freedom from phase distortion. However, linear-phase filters introduce large signal delays when long filters are required. Moreover, the linear-phase restriction is not needed in the stopbands. Imposing the linear-phase requirements only in the passbands results in complex FIR filter designs. Examples include the equiripple design, the peak constrained least squares (PCLS) design [2], and the equiripple passbands and PCLS stopbands (EPPCLSS) design [3]. The need for such designs arises in various applications. They have been extensively studied [3]–[5]. Several methods have been proposed for the solution of these optimization problems. These include linear programming [6]–[9], the multiple exchange algorithm [4], [10]–[13], Lawson’s algorithm [14]–[17], quadratic programming [18], [19], the implicit multiple exchange algorithms [20]–[22], semidefinite programming [3], and a method based on alternate iterations in the time and frequency domains [23]. While these

Manuscript received November 22, 2004; revised June 15, 2005. The work of K. L. Teo was supported by the Research Grant Council of Hong Kong under Research Grant Ployu5247/04E. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Mariane R. Petraglia W. R. Lee, L. Caccetta, and V. Rehbock are with the Western Australian Centre of Excellence in Industrial Optimization, Department of Mathematics and Statistics, Curtin University of Technology, Perth 6845, Australia. K. L. Teo was with the Department of Applied Mathematics and the Center for Multimedia Signal Processing, Department of Electronics and Information Technology, The Hong Kong Polytechnic University, Kowloon, Hong Kong. He is currently with the Department of Mathematics and Statistics, Curtin University of Technology, Perth 6845, Australia. Digital Object Identifier 10.1109/TSP.2006.872542

methods have generated good solutions, the design criteria used in these paper are not the most appropriate in terms of minimizing the weighted peak ripple (WPR) of the magnitude and the maximum phase deviation from the linear phase in the passbands. This is because the minimum Chebyshev error is not equivalent to the minimum WPR with the maximum phase deviation from the linear phase being less than or equal to the peak ripple of the magnitude in each individual passband. Here, we just compare the values of the phase deviation and the peak ripple of the magnitude where the phase is measured in radians. In this paper, we first reformulate the equiripple design problem. The error criteria used are the WPR of the magnitude and the maximum phase deviation from the linear phase in the passbands. This reformulation leads to a better equiripple design formulation. A new computational method is then developed for solving these complex FIR filter design problems. The method is based on the ripple-weighted approach introduced in [1] and [24]. We show that this method, with slight modifications, can be readily used to solve the PCLS design problem. Consequently, it can also be used to solve the EPPCLSS design problem. All of these problems are effectively converted into corresponding weighted least squares (WLS) problems. They are then solved iteratively. The success of the WLS approach depends on the appropriate error weighting function. In [26], Algazi et al. pointed out a formal relation between WLS design and equiripple design through Lawson’s algorithm [25]. Lim et al. in [27] presented a novel scheme to derive the error weighting function for an equiripple design. Diniz and Netto presented a modified scheme in [28] for solving WLS-Chebyshev FIR digital filters. However, the schemes proposed in these papers only deal with the magnitude of the filter. For the problems addressed in this paper, both the magnitude and the phase deviation from the linear phase should be dealt with. We thus propose a novel scheme to update the error weighting functions for the new formulation and the design requirements for both the equiripple design and the PCLS design. At each iteration, the error weighting function is updated according to the design requirements and the resulting LS problem is solved. It is well known that the LS solution can be obtained analytically. It is also well known that the large number of frequency points required for accurate design of a long filter does not increase the size of the linear system to be solved at each iteration. Consequently, one does not increase the complexity of the problem to be solved. The only computational tasks involved in our method are to update a sequence of error weighting functions and to evaluate a sequence of analytic solutions of LS problems. Furthermore, the error weighting function for the new scheme does not change for most frequencies at each

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iteration, particularly for the PCLS design. This greatly reduces the computational complexity of the proposed approach. This paper is organized as follows. Section II is devoted to the equiripple design. The equiripple design problem is reformulated and the solution method of the problem as well as the error weighting function are developed. Section III is devoted to the PCLS design problem. The solution method is developed and the EPPCLSS design problem is solved as a special case of the PCLS design problem. In Section IV, three complex FIR filters are designed using the proposed method. Section V concludes this paper. II. THE EQUIRIPPLE DESIGN Consider the transfer function of an

-tap FIR filter

and respectively, where . The Chebyshev approximation problem in (2.1) can be reformulated as (2.4) . Here, and are the allowed WPR where is equal to of the magnitude and the allowed maximum phase deviation from the linear phase, respectively. This formulation directly minimizes the WPR and the maximum phase deviation from the linear phase. It provides more freedom than (2.1) in terms of the choice of the allowed WPR and the allowed maximum phase deviation from the linear phase in the passbands. A. Solution Method We can rewrite the frequency response

and denote its frequency response by

as

where and . The error function (2.2) in the magnitude and the error function (2.3) in the phase deviation from the linear phase can be simplified to

where

, , and . Here and throughout this paper, denotes the . Define the complex error function as transpose of

(2.5) and

where and are, respectively, a positive weighting function defined on a frequency set and the desired complex is given by frequency response. Here,

(2.6) , respectively. When the imaginary part, is very close to zero and the real of , of is very close to one, part,

where and are the unions of the passbands and stopbands, respectively, and is the group delay which can take any real 1)/2 and can take different values on the number around ( different region of the passbands. The Chebyshev approximation problem can then be formulated as finding a coefficient vector that solves the optimization problem (2.1)

. Since the terms and

will

be minimized in the subsequent optimization problem, the above assumptions will be fulfilled automatically once this optimization problem has been solved. Therefore, can be approximated by . The design problem can then be formulated as follows. Find a filter coefficient vector that solves the optimization problem

Define the error function in the magnitude and the error function in the phase deviation from the linear phase as (2.7)

(2.2) where

and (2.3)

(2.8)

LEE et al.: COMPLEX FIR FILTERS WITH ARBITRARY MAGNITUDE AND GROUP DELAY RESPONSES

(2.9) (2.10)

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, , and , the solution of the problem For any (2.13) is the LS solution of the following overdetermined linear system: (2.14)

Let

and be dense sets of frequencies uniformly distributed in and , respectively. We then have the following discretized version of the problem (2.7). Find a filter coefficient vector that solves the optimization problem

(2.11) It is well known (see [29]–[31]) that (2.11) is equivalent to the WLS problem with the unknown error weighting functions , , and shown in (2.12) at the bottom of the page. , , , and contribute to the equal ripples of Here, the magnitude in the passbands and the stopbands and the equal ripple of the phase deviation from the linear phase in the passbands, respectively. They will be discussed later. Since

when the term the term

where .. .

.. .

.. .

.. .

.. .

.. .

.. .

.. .

.. .

.. .

.. .

is very close to zero, we replace .. .

in (2.12) with the term and keep the error unchanged. We then have the folweighting function lowing modified version of the WLS optimization problem (2.12):

.. .

.. .

For simplicity, we combine the error weighing functions and the weighting function to form the following combined error weighting function (CEWF):

The diagonal matrix is then a combined error weighting matrix (CEWM). According to [32], the LS solution of the linear system (2.14) is given by (2.13)

(2.15)

(2.12)

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B. The Combined Error Weighting Function The CEWF at the ( lowing formula [27]:

1)th iteration is updated using the fol-

The LS solution (2.15) at the ( 1)th iteration can then be obtained by utilizing the LS solution at the th iteration as indicated by the following equations:

(2.16) where

is an updating function. The choice of the CEWF at the th iteration is crucial for obtaining a fast convergence speed with minimal computational effort. We propose the following algorithm to update the CEWF at each iteration. Initially, the CEWF is chosen to be

is a diagonal matrix with most of its elements on the diagonal equal to zero. Obviously, the evaluations of and require less computation than the evaluations of and , respectively. III. THE PCLS DESIGN The LS design problem can be formulated as finding a coefficient vector that solves the optimization problem

The CEWF at the th iteration will then be updated using (2.16) with if otherwise

(2.17)

where

(3.1) is a positive weighting function defined on the frewhere . decides the ratio of the error energies of quency set themagnitudeandphasedeviationfromthelinearphaseinvarious frequency bands. The discretized version of the problem (3.1), , of frequencies in , is given by using a dense set,

(2.18)

(2.19) (2.20) (2.21) Here, is a constant which should be less than the allowed WPR, , of the magnitude. This updating scheme means that the CEWF only changes around the extremal frequencies, where the filter has a large magnitude or a large phase deviation from the linear phase (larger than , that is). with , where We can replace if otherwise Therefore, the CEWF at the ( using the following formula:

(2.22) 1)th iteration can be updated

(3.2) The peak constraints are given by (3.3) (3.4) where and are as defined in the previous section. The discretized version of the constraints (3.3) and (3.4) can, respectively, be specified as

(2.23)

(3.5)

Let (3.6) and

(3.7)

Using (3.5), (3.7) can be simplified to (3.8) The CEWM at the ( following formula:

1)th iteration can be updated using the

The PCLS design can then be formulated as finding a filter coefficient vector such that (3.2) is minimized subject to the constraints (3.6)–(3.8).


A. Solution Method To solve the PCLS design problem, we form the Lagrangian shown at the bottom of the page. If is the optimal solution of the PCLS problem, then by the Kuhn–Tucker conditions for a local optimum, there exist sets and of nonnegative Lagrange multipliers, as shown in (3.9)–(3.15) at the bottom of the next page. Equations (3.14) and (3.15) can, respectively, be rewritten as (3.16)

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final CEWF should be unchanged for most frequencies, except for the extremal frequencies where the upper bound for the magnitude (or the upper bound for the phase deviation from the linear phase) is reached. The final values of the CEWF at these extremal frequencies should be larger than the initial values. We propose the following algorithm to derive the final CEWF for the solution of the PCLS problem. The initial CEWF is chosen to be The CEWF will then be updated at each iteration using the following updating function: if otherwise

and

(3.17) Here, is the same as in the previous section. It is clear that of (3.9) is the gradient of the WLS with the following CEWF:

(3.18)

(3.19) and

(3.20) Consequently, is the solution of the WLS problem with the CEWF defined by (3.18)–(3.20). We are therefore looking for the WLS solution with the CEWF defined by (3.18)–(3.20) such that (3.10)–(3.17) are satisfied. From (3.18)–(3.20), the initial CEWF should be equal to . This decides the initial ratio of the error energies of the magnitude and phase deviation from the linear phase in various frequency bands. From (3.11), (3.13), and (3.17), we see that the Lagrange multiplier (or ) is positive only when of the the ripple of the magnitude reaches the upper bound WPR (or the ripple of the phase deviation from the linear phase in the passbands reaches the upper bound ). Therefore, the

(3.21)

where are defined in (2.18)–(2.21), i.e., the same four functions we defined for the equiripple design. Effectively, we are taking advantage of the equiripple design formulation for the PCLS design. From (3.21), it can be seen that the CEWF at each iteration will be unchanged for most frequencies except for the frequencies where the magnitudes (or the phase deviations from (or the the linear phase) are larger than the allowed WPR allowed maximum phase deviation , from the linear phase). Therefore, the final CEWF will still be unchanged for most frequencies and will have significant increment at these extremal frequencies. The final CEWF will not satisfy (3.11), (3.13), and (3.17) exactly. Aside from the large increments at these extremal frequencies, there will also be additional increments at the frequencies around these extremal frequencies. These increments provide some additional weights to the constraints around these extremal frequencies so that the maximum values at these points are less than (or ) to help reduce the excessive peak ripples. However, in view of (2.16) and (3.21), we see that the CEWF will change very slowly when the peak ripple of the magnitude and the peak ripple of the phase deviation from the linear phase are very close to the upper bonds and , respectively. Consequently the convergence will be very slow. To overcome this problem, in the updating function (3.21) is reduced by 10% to 20%. B. The EPPCLSS Design The EPPCLSS design is a special case of the PCLS design [1] where the filter reaches minimax criterion in each individual passband. This can be achieved by reducing the values of

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in these passbands. This results in higher error energies in these passbands and, consequently, higher extreme values at the peak ripples in these passbands. The CEWF will be used to deal with these extreme values. Finally, the minimax solution in these passbands will be reached. IV. DESIGN EXAMPLES The design methods described in the previous section are illustrated by the following examples. All computations were performed in Fortran 77 double precision on a Unix Workstation. For the WLS problems, 5001 sample points were used in the frequency range for all the examples. Example 1: This example of a 250-tap filter was considered in [12], [20], and [33]. The design specification is

and .

This filter is designed using the equiripple, the PCLS, and the EPPCLSS designs. For the equiripple design, both the allowed of the magnitude and the allowed maximum phase WPR deviation from the linear phase in the passband are equal to 0.000195. For the error weighting updating function (2.17), . Our method takes 11 iterations to meet we use the minimax criteria. The maximum magnitude errors in the passband and the stopband and the maximum phase deviation from the linear phase in the passband achieved are 0.000185, 0.0000185, and 0.000185, respectively. By comparison, the maximum magnitude errors in the passband and the stopband of the design reported in [20] presented 0.000202 and 0.0000202, respectively; the maximum magnitude errors in the passband and the stopband of the design reported in [12] presented 0.0002015 and 0.00002015, respectively. The complex-valued frequency response of the filter designed using the proposed method is depicted in Fig. 1(a). Its magnitude response and group delay response are depicted in Fig. 1(b)–(d). For the PCLS design, the constraints are chosen to be for both the allowed WPR and the allowed max-

(3.9) (3.10) (3.11) (3.12) (3.13) (3.14) (3.15)


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imum phase deviation from the linear phase in the passband. The initial error energy weighting function is chosen to be (4.1)

.

In (3.21), is replaced by 0.00019, which represents a reduction of about 15%. The algorithm takes 16 iterations to reach the design criteria. The maximum magnitude errors in the passband and the stopband, and the maximum phase deviation from the linear phase in the passband, are 0.000219, 0.0000219, and 0.000218, respectively. By comparison, the WPR for the design reported in [33] presented 0.000283 with the same level of the error energy as the filter designed using the proposed method. The magnitude of the filter designed using the proposed method is depicted in Fig. 1(e) and (f). The final CEWFs in the passband and the stopband are depicted in Fig. 1(g) and (h), respectively. From Fig. 1(g) and (h), it can be seen that the final CEWF for . The incremost frequencies is equal to the initial value ments are concentrated at some extremal frequencies where the upper bound of the WPR is reached, as expected. Finally, this filter is designed using the EPPCLSS design. For the EPPCLSS design, the initial error energy weighting function is chosen to be , . The constraints were chosen to be for both the allowed WPR and the allowed maximum phase deviation from the linear phase in the passband. For (3.21), is replaced by 0.00018, which is a reduction of 10%. The algorithm takes 16 iterations to satisfy the design criteria. The maximum magnitude errors in the passband and the stopband, and the maximum phase deviation from the linear phase in the passband are 0.000199, 0.0000199, 0.000198, respectively. The magnitude of the filter designed using the proposed method is depicted in Fig. 1(i) and (j). Example 2: This 161-tap filter with specification as in [5] is designed using both the equiripple and the PCLS designs. The design specification of the filter is Fig. 1. Frequency response of 250-tap FIR filter in Example 1. (a) Frequency response (3-D) of the filter using the equiripple design. (b) Magnitude response of the filter using the equiripple design. (c) Magnitude response of the filter in the passband using the equiripple design.

and

. For the equiripple design, the allowed WPR of the magnitude and the allowed maximum phase deviations from the linear phase in passband one and passband two are 0.0095, 0.0095,

and 0.00475, respectively. We have to define the weighted peak ripple of the phase deviation from the linear phase in two passbands (similar to the definition of the WPR of the magnitude) with constant weights 1 and 2 for passband one and passband two, respectively. This is because the maximum allowed phase deviations from the linear phase in passband one and passband two are equal to 0.0095 and 0.00475, respectively. . For the error weighting function (2.17), we use Our method takes 17 iterations to meet the minimax design criterion. The maximum magnitude errors in the passbands and the stopbands are 0.00896, 0.00896, 0.000289, 0.00448, and 0.000896, respectively. The maximum phase deviations from the linear phases achieved in passbands one and two

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Fig. 1. (Continued.) Frequency response of 250-tap FIR filter in Example 1. (d) Group delay of the filter using the equiripple design. (e) Magnitude response of the filter using the PCLS design. (f) Magnitude response of the filter in the passband using the PCLS design.

Fig. 1. (Continued.) Frequency response of 250-tap FIR filter in Example 1. (g) The final CEWF in the passband using the PCLS design. (h) The final CEWF in the stopband using the PCLS design. (i) Magnitude response of the filter using the EPPCLSS design.

are 0.00896 and 0.00447, respectively. By comparison, the maximum magnitude errors in the passbands and the stopbands of the design reported in [5] are 0.01, 0.01, 0.000316, 0.05, and 0.001, respectively. The magnitude response and the group delay responses in passband one and passband two of

the filter designed using the proposed method are depicted in Fig. 2(a)–(d). For the PCLS design, the constraints are chosen to be for both the allowed WPR and the allowed weighted peak ripple of the phase deviation from the linear


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Fig. 1. (Continued.) Frequency response of 250-tap FIR filter in Example 1. (j) Magnitude response of the filter in the passband using the EPPCLSS design.

phase in the passbands (as defined in the equiripple design above). The initial error energy weighting function is chosen to be

. In (3.21), is replaced by 0.0085 which represents a reduction of about 11 percent. The algorithm takes 32 iterations to satisfy the design criteria. The maximum magnitude errors in the passbands and the stopbands are 0.00948, 0.00948, 0.0003004, 0.00474, and 0.000948, respectively. By comparison, the maximum magnitude errors in the passbands and the stopbands of the design reported in [5] are 0.01, 0.01, 0.000318, 0.05, and 0.001, respectively, and the filter designed using the proposed method also has a smaller error energy than the one designed in [5]. The magnitude response and the group delay responses in passband one and passband two of the filter designed using the proposed method are depicted in Fig. 2(b), (e) and (f), respectively. Example 3: The final example is a 91-tap filter with specification as in [3]. It is designed using the EPPCLSS design. The design specification of the filter is

and

Fig. 2. Frequency response of 161-tap filter in Example 2. (a) The magnitude response of the filter using the equiripple design. (b) The magnitude response of the filter using the PCLS design. (c) Group delay of the filter in passband 1 using the equiripple design.

The initial error energy weighting function is chosen to be

We chose for both the allowed WPR and the allowed phase deviation from the linear phase in the passband. For (3.21), is reduced to 0.0165, which is about 10% lower than

the allowed peak ripple. The algorithm takes 51 iterations to satisfy the design criteria. The maximum magnitude errors in the passband and the stopband, and the maximum phase deviation from the linear phase in the passband are 0.01799, 0.001799, and 0.01775, respectively. The magnitude of the filter designed using the proposed method is depicted in Fig. 3(a) and (b). By

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Fig. 3. Magnitude response of 91-tap filter using the EPPCLSS design in Example 3. (a) Magnitude response of the filter. (b) Magnitude response in the passband.

V. CONCLUSION

Fig. 2. (Continued.) Frequency response of 161-tap filter in Example 2. (d) Group delay of the filter in passband 2 using the equiripple design. (e) Group delay of the filter in passband 1 using the PCLS design. (f) Group delay of the filter in passband 2 using the PCLS design.

comparing with the filter designed by the method proposed in [3], the filter designed using the proposed method exhibits the . However, comparing same level of peak ripple the magnitude response shown in Fig. 1 of [3] with the magnitude response shown in Fig. 3 of this paper, the proposed method clearly results in a smaller error energy in the stopband.

We have presented an efficient computational method for complex FIR filter design with arbitrarily prescribed magnitude and group delay. We have reformulated the equiripple design problem as a minimax optimization problem with the WPR and the maximum phase deviation from the linear phase as design objectives. This directly minimizes the WPR and the maximum phase deviation from the linear phase in the passbands and provides more flexibility than the conventional formulation for the choice of the design criteria. We have shown that the method for the equiripple design can be easily modified to solve the PCLS design problem. Consequently, it can also be used to solve the EPPCLSS design problem. The method is based on the WLS approach. We have proposed a scheme to update the CEWF appropriately. With this scheme, the CEWF does not change for most frequencies at each subsequent iteration, thus significantly reducing the computational cost, especially for the case of the PCLS design. Numerical examples show that the method is stable and takes about 10 to 50 iterations for convergence with a linear system to be solved at each iteration. Numerical examples also show that the method improves existing solutions by around 10% in both the equiripple design and the PCLS design, due to the new formulations and the efficient solution strategy.


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Wei Rong Lee received the Ph.D. degree in applied mathematics from Curtin University of Technology, Australia, in 1999. From 1999 to 2001, he was a Research Associate with the Department of Mathematics and Statistics, Curtin University of Technology. In 2001, he joined Western Australian Centre of Excellence in Industrial Optimization as a Research Fellow. His main fields of research are theory and applications of optimization and optimal control, in particular, the applications in signal processing, numerical analysis, and stochastic processes with applications to hidden Markov model.

Lou Caccetta received the B.Sc. and Ph.D. degrees from the University of Western Australia. He spent four years in Canada, mostly in the Department of Combinatorics and Optimization, University of Waterloo. He joined Curtin University of Technology, Australia, as a Lecturer in operations research in 1981. He became a Senior Lecturer and Associate Professor there in 1986 and 1990, respectively. He obtained a Personal Chair in 1992 as Professor of mathematics. He was Head of the School of Mathematics and Statistics from 1994 to 2004. He established the Western Australian Centre of Excellence in Industrial Optimization (WACEIO) in 2001, which undertakes industry focused R&D. His research interests include computational algorithms, combinatorial mathematics (graph theory and network optimization), combinatorial optimization (integer programming), industrial modeling, and operations research. His research projects span theory, methods and applications. He has developed, implemented, and tested a number of optimization routines for solving large industrial problems. He is on the Editorial Board of nine journals and is a Reviewer and Referee for a number of journals and funding agencies. He has supervised to completion 20 Ph.D.’s and five Master’s, and is currently supervising 17 Ph.D.’s. He has organized a number of conferences/workshops. He has published more than 185 refereed papers and has received a number of competitive research grants. He has extensive experience in contract R&D and industrial consultancy. Prof. Caccetta is a member of a number of professional affiliations and is currently the National President of the Australian Society for Operations Research. He won the Deans Medallion for Distinguished Research, Faculty of Science, Curtin University of Technology, in 2000 and the Vice-Chancellors Award for Excellence.

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Kok Lay Teo (M’74–SM’87) received the B.Sc. degree in telecommunications engineering from Ngee Ann Technical College, Singapore, and the M.A.Sc. and Ph.D. degrees in electrical engineering from the University of Ottawa, Ottawa, ON, Canada. He was with the Department of Applied Mathematics, University of New South Wales, Australia; the Department of Industrial and Systems Engineering, National University of Singapore; and the Department of Mathematics, University of Western Australia, Australia. In 1996, he joined the Department of Mathematics and Statistics, Curtin University of Technology, Australia, as a Professor. He then became Chair Professor of Applied Mathematics and Head of the Department of Applied Mathematics, Hong Kong Polytechnic University, China, from 1999 to 2004. He is currently a Professor of applied mathematics and Head of the Department of Mathematics and Statistics at Curtin University of Technology. He has published five books and more than 250 journal papers. He has a software package, MISER3.3, for solving general constrained optimal control problems. He is Editor-in-Chief of the Journal of Industrial and Management Optimization. He was an Associate Editor of a number of international journals, including Automatica, Nonlinear Dynamics and Systems Theory, Journal of Global Optimization, Engineering and Optimization, Discrete and Continuous Dynamic Systems (Series A and Series B), and Dynamics of Continuous, Discrete and Impulsive Systems (Series A and Series B). His research interests include both the theoretical and practical aspects of optimal control and optimization and their practical applications such as in signal processing in telecommunications and financial portfolio optimization.


Volker Rehbock received the Ph.D. degree in applied mathematics from the University of Western Australia, Australia, in 1994. Since 1995, he has been with the Department of Mathematics and Statistics, Curtin University of Technology, Australia. His main interests are in the modeling and numerical solution of dynamic optimization problems. He is a coauthor of more than 30 refereed publications in the areas of optimization and optimal control. Currently, he is also Deputy Director of the Western Australian Centre of Excellence in Industrial Optimization.