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Iterative Algorithms for Envelope Constrained Recursive Filter Design via Laguerre Functions Zhuquan Zang, Member, IEEE, Ba-Ngu Vo, Member, IEEE, Antonio Cantoni, Fellow, IEEE, and Kok Lay Teo, Senior Member, IEEE
Abstract—The envelope constrained (EC) filtering problem is concerned with the design of a time-invariant filter to process a given input pulse such that the output waveform is guaranteed to lie within a prescribed output mask. Using the orthonormal Laguerre functions the EC filtering problem can be posed as a quadratic programming (QP) problem with affine inequality constraints. An iterative algorithm for solving this QP problem is proposed. We also show that for the EC filtering problem, filters based on Laguerre functions offer a more robust and low-order alternative to finite impulse response (FIR) filters. A numerical example, concerned with the design of an equalization filter for a digital transmission channel, is presented to illustrate the effectiveness of the iterative algorithm and the Laguerre filter.
(a)
Index Terms— Digital transmission channel, envelope-constrained filter, equalization filter, IIR filter, iterative algorithm, Laguerre function, quadratic programming, recursive filter.
I. INTRODUCTION
(b)
ONSIDER a time invariant filter with impulse response ( ) used to process a given input which is corrupted by white additive noise, sequence is as shown in Fig. 1(a). The noiseless output waveform required to fit into a pulse shape envelope defined by the upper and , as shown in Fig. 1(b). and lower boundaries The optimal envelope-constrained (EC) filter is defined as the filter which minimizes the output noise power, while satisfying the pulse shape constraints. With the assumption is white with a constant power that the random noise spectrum, it can easily be verified that the output noise power norm of the filter’s impulse is proportional to the squared response. Hence, the (discrete-time) EC filtering problem can be posed as
Fig. 1. EC filtering problem. (a) Block diagram. (b) Pulse shape envelope.
C
subject to (1) where
In standards, the performance of a digital link is often specified in terms of a mask applied to the received signal [2], [3], [14]. Manuscript received March 30, 1998; revised December 16, 1998. This was supported in part by a research grant from the Australian Research Council. This paper was recommended by Associate Editor G. Martinelli. The authors are with the Australian Telecommunications Research Institute, Curtin University of Technology, Perth WA 6001, Australia (e-mail:
[email protected]). Publisher Item Identifier S 1057-7122(99)09265-X.
The EC filter design problem is directly applicable and the would correspond to the test signal specified in the signal standard. Other areas of application include robust antenna and filter design [1], pulse compression in radar and sonar [20], pulse shape design for digital communication systems [21], and seismology [28]. Using an FIR filter structure, the discrete-time EC filtering problem has been formulated as a quadratic programming (QP) problem with affine inequality constraints [6]. Sequential QP via an active set strategy is a popular tool for solving QP problems. However, implementation-wise, this is not suitable for adaptive filters. Iterative methods for solving this problem, which can be implemented adaptively, had been proposed in [7], [25], and [30]. Although FIR filters are attractive, due to their simplicity, they generally require a large number of taps. In this paper, using a filter based on the set of orthonormal Laguerre functions, the discrete-time EC filtering problem is formulated and solved as a QP problem with linear affine inequality constraints. The use of orthonormal functions for signal representation and filter synthesis can be traced back to the 1930’s (see [15] for a summary of this early work). In recent years, there has been a renewed interest in the use of orthogonal functions, particularly Laguerre functions for signal representation, filter synthesis, system modeling and infinitedimensional system approximation, see, e.g., [13], [19], [27], [29], and the references therein. Since filters based on Laguerre
1057–7122/99$10.00 1999 IEEE
ZANG et al.: ITERATIVE ALGORITHM FOR ENVELOPE CONSTRAINED FILTER DESIGN
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functions have an exponential response they are particularly suitable for representing linear time invariant (LTI) systems. In practice, only a few Laguerre functions are often sufficient to yield a very good approximation to an LTI response. In this paper, we will show that for the EC filtering problem, filters based on Laguerre functions offer a more robust and low-order alternative to FIR filters. II. PROBLEM FORMULATION Let
denote the space of all square summable sequence equipped with the norm . In [11] the discrete Laguerre functions of degree are defined as
Fig. 2. Digital Laguerre network.
expressed as (5)
where is the discrete time index and constant. Let
and is a and define
is an orthonormal set, by Parseval’s
Since formula,
(6) . It is known (see [11] and [13]) that the set is an space. orthonormal basis on the denote the Hardy space of all complex-valued Let which are analytic and square integrable outfunctions side the unit disc and equipped with the norm . Let be the transform of ’s, then it can be shown that
, then the EC filterLet ing problem can be expressed as the following constrained optimization problem: subject to
(7)
where (2) .. .
forms an orIt is known that the sequence [27]. This means that thonormal basis in the Hardy space can be expanded as a Laguerre–Fourier series any of the form
.. .
(3) and dewhere notes the complex conjugate of . denote the subspace of genFor a given , let erated by the finite set of orthonormal Laguerre functions . In the following we shall demonstrate that , it by restricting the EC filtering problem to the subspace can be reformulated and solved as a QP problem with affine inequality constraints. Consider the filter shown in Fig. 2. In view of (2), this filter can be interpreted as a finite version of the series in (3). We will refer to the filter in Fig. 2 as a digital Laguerre filter. The transfer function of this discrete time filter is given by a finite sum of a Laguerre–Fourier series
.. .
.. .
.. .
.. .
(8)
(9)
Note that it follows from (2) and (4) that the Laguerre filter has infinite impulse response (IIR) but the response is a linear function of the parameters . It is assumed that the input signal is sufficiently rich (see, is nonzero at points e.g., [10] and [16]) such that (or more). Then, it is easy to see that the columns of the matrix are linearly independent. The problem defined in (7) is a QP problem with affine inequality constraints. In a more standard form, it can be below written as problem subject to
(10)
where
(4) and Let as
denote the -transform of . The output, denoted , of the filter in response to an input can be
(11)
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The set of all feasible filters
is defined as
Problem ( ) involves a convex cost and a convex constraint is not empty, the problem has set. If the feasible region a unique solution [8], [18]. Note that the optimal solution to problem ( ) can be obtained in a straightforward manner using existing softwares such as MATLAB and its related Toolboxes, see [12]. To eliminate trivial envelopes which permit zero response, it is assumed that there exists at least such that . Some one are important geometric properties of the feasible region presented in Lemma 2.1, see [25] for further details. of feasible filters is convex and Lemma 2.1: The set compact. Moreover if the columns of the matrix are linearly independent, then is contained in a closed ball centered at , where is a vector the origin and radius and with components defined by denotes the minimum singular value of the matrix . III. ITERATIVE ALGORITHMS In cases where the underlying signal models are either not known or varying with time, it may be necessary to employ an adaptive filter with parameters that can be adjusted to their optimum value either on a continuous basis or at specific instants of time [10]. This section investigates iterative algorithms for solving the EC filtering problem formulated above. The iterative algorithm serves as a basis for the development of adaptive EC algorithms, just as the steepest descent (SD) algorithm did for the LMS algorithm. A. Approximating the QP Problem In this section, an augmented cost is constructed by compos(referred to as a penalty allocator) with each constraint ing function and then adding this to the original cost. The suboptimum solution is found by minimizing the augmented cost with descent algorithms, such as SD and Newton–Raphson (NR). to ensure global convergence are The criteria for choosing presented. For the NR method with an appropriate line search, a quadratic rate of convergence can be achieved. For each let us define
Fig. 3. Penalty allocator.
Then, for any
, define
as follows:
Clearly, given any , the approximate feasible region is a subset of . Intuitively, by inspecting the above expression, tends toward the one may conclude that as tends to zero actual feasible region . , an augmented cost function is For each defined as (13) The scalar and are called the accuracy and penalty parameters, respectively. The analytic properties of the augmented cost function are given below, which can be established easily and . using the definition of is continuous and Lemma 3.1: The augmented cost strictly convex and, hence, has a unique minimum. Moreover is once continuously differentiable, then 1) If is once continuously differentiable and its gradient is given by
2) If is twice continuously differentiable, then is twice continuously differentiable, with a positive definite Hessian given by
(12) denoting the th row of and the th element of . with define a penalty allocator to be any For any , continuous and convex function identically zero on but increasing otherwise (see Fig. 3). The penalty function is then defined as
Assuming that the feasible set of problem ( ) has a nonempty interior, i.e., int
3) If ,
has a bounded second derivative, i.e., , then also has a bounded Hessian, i.e.,
4) If has a Lipschitz continuous second derivative, i.e., such that there exists a constant , then is also Lipschitz continuous. More precisely
ZANG et al.: ITERATIVE ALGORITHM FOR ENVELOPE CONSTRAINED FILTER DESIGN
These properties allow the augmented cost to be efficiently minimized by descent-direction-based algorithms, such as NR or SD, by choosing an appropriate penalty allocator . Let us demonstrate how the augmented cost can be used to solve can be approximated the EC filtering problem. Problem , defined for by an unconstrained optimization problem as
To justify the approximations in terms of the relationship and problem , several useful between problem results from [25] are adopted. be the solution to problem and Theorem 3.1: Let the solution to problem , then the following holds. for some , then 1) If . such that , if 2) For each for some , then,
3) For any
If some
, let
such that , then,
and
for
From part 1) of the theorem, it is clear that for a fixed but small , a sufficiently large penalty parameter forces the solution of the approximate problem into the feasible region of the EC filtering problem. This is the principle advantage over other penalty methods, where feasibility is only guaranteed as tends to infinity. Provided that for each the penalty is chosen according to 1), 2) shows that the parameter converges to the true solution as approximate solution the accuracy parameter tends to zero. Part 3) of the theorem asserts that if a feasible point is known, then for any given can be calculated error bound , the accuracy parameter and ) (without using any information on the solutions , so that the noise gain deviation, defined as is less than the error bound . B. Descent-Direction-Based Algorithms Having selected appropriate approximation parameters and , next we shall consider a number of iterative algorithms to compute the solution to the approximate problem, . Among i.e., the minimization of the augmented cost the number of algorithms available, the most widely known are descent-direction-based iterative algorithms which permit direct implementation in adaptive EC filters
where is the filter’s coefficient vector at the th iteration, is the step size, and satisfying is the descent direction.
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Assuming that is at least once continuously differentiable, the simplest of the descent directions is the SD direction . Consequently, the SD algorithm can be . The following described as theorem gives the sufficient condition for the SD algorithm to converge. is twice continuously difTheorem 3.2: Suppose that with , . Then, ferentiable for each , the sequence from an arbitrary starting point generated by the SD algorithm with constant step size converges to the solution of if
The SD algorithm is attractive, to a large extent due to its computational simplicity, however, the method has several important practical drawbacks, the most significant of which is that convergence is usually slow. Another popular method is the NR algorithm, which uses second derivative information to calculate the descent direction,
It is assumed that the penalty allocator is chosen so that the Hessian is positive definite (see [25, Lemma 3.1, iii)]). For algorithms using constant step size, a larger step size than the SD algorithm is allowed, as illustrated in the following theorem. is twice continuously difTheorem 3.3: Suppose that with . ferentiable for each Then, from an arbitrary starting point , the sequence generated by the NR algorithm with a constant step size converges to the solution of if
Considerably faster convergence can be achieved with the NR method by performing some line search at each iteration to determine the step size (an iteration in this context refers to a search direction evaluation). For algorithms with a constant stepsize, each search direction evaluation is followed by a single filter update, while algorithms with a line search require at least one update to determine the step size before the next search direction is calculated. A popular form of line search involves the Goldstein condition and the Wolfe–Powell condition (see [8] and [26] for details). The following theorem gives the sufficient conditions for the NR algorithm with a ine search to converge at a quadratic rate. is twice continuously difTheorem 3.4: Suppose that with . If is ferentiable for each Lipschitz continuous, i.e., then, from an arbitrary starting point , the sequence generated by the NR algorithm with step size chosen by the line search procedures of [25], converges to of at a quadratic rate. the solution
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TABLE I LAGUERRE FILTERS WITH
A
FIXED a
TABLE II FIR FILTERS
Fig. 4. Input, output signals, and output mask for an optimal Laguerre EC filter.
IV. SIMULATION RESULTS In this section, the NR algorithm is used to study a design example which involves the equalization of a digital transmission channel, consisting of a coaxial cable operating at the DX3 rate (45 Mb/s) [2], [3]. The coaxial cable has a 30-dB , where is attenuation at a normalized frequency of 10 s). An equalizing filter is the baud interval (22.35 required to shape the impulse response of the cable so that it fits in the envelope given by the DSX3 pulse template (see Fig. 4). The signals for this example are in a continuous-time it had setting, the input is an IIR signal, however, at decayed to a negligible level and thus can be treated as a time limited signal. To apply the NR algorithm presented in the previous section, the signals are discretized by sampling at . regular intervals over Laguerre filters and FIR filters are compared for various sampling periods. The accuracy parameter is chosen to be 10 . For FIR filters, the penalty parameters are obtained from [25, corollary 3.1 ]. With the Laguerre filters, the penalty parameters are obtained via Lemma 2.1 and Theorem 3.1. , This can be done as follows. Given a feasible point . Therefore, according to Lemma 2.1, according to 1) of Theorem 3.1, we can simply choose . The expression for the penalty allocator used in this example is given below, which can be easily verified to be twice continuously differentiable
(14) .
Simulation results for a fixed and II.
are summarized in Tables I
From these simulations several remarks are in order. • The minimum number of taps (or sections) in Laguerre filters are fairly insensitive to the sampling period, as compared to FIR filters. • Laguerre filters have slightly smaller norms. This can be attributed to better approximation of the filter space by the Laguerre basis functions. • Laguerre filters are more tolerant of small variations in the sampling period, as compared to FIR filters, in the sense that an optimal EC filter for a sampling period can also fit input of sampling period into the envelope. Figs. 4 and 5 plot the input, output signals and the output mask , for the two different for the case with sampling period types of filters. Another attractive feature of digital Laguerre filters is that the Laguerre function pole constitutes an extra design parameter which can be manipulated to the designer’s advantage. By varying , optimal EC filters with considerably lower orders than FIR filters can be obtained. This is demonstrated in Table III. Again, this is due to better approximation of the filter space by the Laguerre basis functions. Numerical studies have also shown that Laguerre filters are fairly tolerant of small variations in the pole . It is possible to include the Laguerre function pole as a variable of the optimization problem. However, we have chosen to fix it due to the following considerations. First, for many real systems, the bandwidth of the channel to be equalized is often known at least approximately. In this case, the design parameter in the Laguerre functions can often be chosen offline from a priori information about the channel
ZANG et al.: ITERATIVE ALGORITHM FOR ENVELOPE CONSTRAINED FILTER DESIGN
Fig. 5. Input, output signals, and output mask for an optimal FIR EC filter.
TABLE III LAGUERRE FILTERS WITH DIFFERENT a
impulse response. Roughly speaking, should be chosen to be approximately equal to the dominant pole of the channel which generates the input signal. Second, including in the optimization problem complicates the presentation without providing any additional insight into the proposed iterative method, which is the primary focus of the paper. Moreover, the optimization of has been considered and is the subject of ongoing work [5]. Methods regarding the optimization of the Laguerre function pole for well-behaved input can be found, e.g., in [9], [22], and [23]. Typically, the related optimization problem is multimodal and hence, practically, it may be better to select a fixed location on the basis of a priori information.
V. CONCLUSION We have proposed a filter based on Laguerre functions for the discrete-time EC filtering problem. An iterative algorithm for solving the problem has been presented and a comparison to FIR filters has been made. Numerical studies show that order of optimal EC filters based on Laguerre functions can be much smaller than that of an optimal FIR filter. Furthermore, the order of the Laguerre filter required to achieve good performance is less sensitive to the sampling period than for FIR filters.
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REFERENCES [1] K. M. Ahmed and R. J. Evans, “An adaptive array processor with robustness and broadband capabilities,” IEEE Trans. Antennas Propagat., vol. AP-32, pp. 944–950, Sept. 1984. [2] “DSX-3 isolated pulse template and equations,” Bell Communications, Tech. Ref. TR-TSY-000499, no. 2, pp. 9–17, Dec. 1988. [3] CCITT Recommendation, “Physical/electrical characteristics of hierarchical digital interfaces,” G.703, Fascicle III, 1994. [4] G. J. Clowes, “Choice of the time-scaling factor for linear system approximations using orthonormal Laguerre functions,” IEEE Trans. Automat. Contr., vol. 10, pp. 487–489, Apr. 1965. [5] H. H. Dam, K. L. Teo, Y. Liu, and S. Nordebo, “Optimum pole position for digital Laguerre network with least squares error criterion,” in ICNV’99, submitted for publication. [6] R. J. Evans, T. E. Fortman, and A. Cantoni, “Envelope-constrained filter—Part I: Theory and application,” IEEE Trans. Inform. Theory, vol. IT-23, no. 4, pp. 421–434, 1977. [7] R. J. Evans, A. Cantoni, and T. E. Fortman, “Envelope-constrained filter—Part II: Adaptive structures,” IEEE Trans. Inform. Theory, vol. IT-23, no. 4, pp. 435–444, 1977. [8] R. Fletcher, Practical Methods of Optimizations, 2nd ed. New York: Wiley, 1987. [9] Y. Fu and G.A. Dumont, “An optimum time scale for discrete Laguerre network,” IEEE Trans. Automat. Control, vol. 38, pp. 934–938, June 1993. [10] G. C. Goodwin and K. S. Sin, Adaptive Filtering, Prediction and Control. Englewood Cliffs, NJ: Prentice-Hall, 1984. [11] M. J. Gottlieb, “Polynomials orthogonal on a finite or enumerable set of points,” Amer. J. Math., vol. 60, pp. 453–458, 1938. [12] A. Grace, Optimization Toolbox: for Use with MATLAB. South Natick, MA: Math Works, 1993. [13] R. E. King and P. N. Paraskevopoulos, “Digital Laguerre filters,” Int. J. Circuit Theory Applicat., vol. 5, pp. 81–91, 1977. [14] J. W. Lechleider, “A new interpolation theorem with application to pulse transmission,” IEEE Trans. Commun., vol. 39, pp. 1438–1444, Oct. 1991. [15] Y. W. Lee, Statistical Theory of Communication. New York: Wiley, 1960. [16] L. Ljung, System Identification: Theory for the User. Englewood Cliffs, NJ: Prentice-Hall, 1987. [17] R. Lucky, J. Salz, and E. J. Weldon, Principles of Data Communications. New York: McGraw-Hill, 1968. [18] D. G. Luenberger, Introduction to Linear and Nonlinear Programming, 2nd ed. New York: Addison-Wesley, 1984. [19] P. M. M¨akil¨a, “Laguerre series approximation of infinite dimensional systems,” Automatica, vol. 26, pp. 985–995, 1990. [20] R. J. McAulay and J. R. Johnson, “Optimal mismatched filter design for radar ranging detection and resolution,” IEEE Trans. Inform. Theory, vol. IT-17, no. 6, pp. 696–701, 1971. [21] R. A. Nobakht and M. R. Civanlar, “Optimal pulse shape design for digital communication systems by projections onto convex sets,” IEEE Trans. Commun., vol. 43, pp. 2874–2877, Dec. 1995. [22] T. Oliverira e Silva, “Optimality conditions for truncated Laguerre networks,” IEEE Trans. Signal Processing, vol. 42, pp. 2528–2530, Sept. 1994. , “On the determination of the optimal pole position of Laguerre [23] filters,” IEEE Trans. Signal Processing, vol. 43, pp. 2079–2087, Sept. 1995. [24] K. L. Teo, A. Cantoni, and X. G. Lin, “A new approach to optimization of envelope constrained filters with uncertain input,” IEEE Trans. Signal Processing, vol. 42, pp. 426–429, Feb. 1994. [25] B. Vo, A. Cantoni, and K. L. Teo, “Iterative algorithms for envelope constrained filter design,” in Proc. IEEE Int. Conf. Acoustics, Speech Signal Processing, Detroit, MI, 1995, vol. 2, pp. 1288–1291. [26] B. Vo, A. Cantoni, and K. L. Teo, “Envelope constrained filter with linear interpolator,” IEEE Trans. Signal Processing, vol. 45, pp. 1405–1414, June 1997. [27] B. Wahlberg, “System identification using Laguerre models,” IEEE Trans. Automat. Contr., vol. 36, pp. 551–562, May 1991. [28] L. C. Wood and S. Treitel, “Seismic signal processing,” Proc. IEEE (Special Issue on Digital Signal Processing), vol. 63, no. 4, pp. 649–661, 1975. [29] Z. Zang, A. Cantoni, and K. L. Teo, “Continuous-time envelope constrained filter design via Laguerre filters and optimization methods,” IEEE Trans. Signal Processing, vol. 46, pp. 2601–2610, Oct. 1998. [30] W. Zheng, A. Cantoni, B. Vo, and K. L. Teo, “Recursive procedures for constrained optimization problems and its application in signal processing,” Proc. Inst. Elect. Eng., vol. 142, no. 3, pp. 161–168, 1995.
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Zhuquan Zang (M’93) received the B.S. degree in mathematics from Shandong Normal University, Jinan, China, and the M.S. in mathematics degree from Shandong University, Jinan, China, in 1982 and 1985, respectively, and the Ph.D. degree in engineering from the Australian National University, Canberra, Australia, in 1993. From 1993 to 1994, he was a Research Associate at the University of Western Australia, Nedlands, Australia, as a Research Associate in the areas of optimization, optimal control, and system identification. Since 1994, he has been with the Australian Telecommunications Research Institute, Curtin University of Technology, Perth, Australia, first as a Research Fellow and currently as a Senior Research Fellow. His current research interests include signal processing for communication systems, networking theory, system modeling and identification, and dynamic systems and optimization theory.
Ba-Ngu Vo (M’97) was born in Saigon, Vietnam. He received the B.Sc. and B.E degrees (first class honors) in 1994 from the University of Western Australia and the Ph.D degree in 1997 from the Australian Telecommunications Research Institute, Curtin University of Technology, Perth, Australia. He spent a brief period at ATRI as a Lecturer before spending a year in the Electronic Engineering Department, the Chinese University of Hong Kong. He is currently a Postdoctoral Fellow at the Australian Telecommunications Research Institute. His research interests lie in the application of optimization techniques to signal processing, in particular, filter design with time domain constraints and semiinfinite programming.
Antonio Cantoni (M’74–SM’83–F’98) was born in Soliera, Italy, on October 30, 1946. He received the B.E. (first class honors) and Ph.D. degrees from the University of Western Australia, Nedlands, Australia, in 1968 and 1972, respectively. In 1972, he was a Lecturer in computer science at the Australian National University, Canberra. He joined the Department of Electrical and Electronic Engineering at the University of Newcastle, Shortland, NSW, Australia, in 1973, where he held the Chair of Computer Engineering until 1986. In 1987, he joined QPSX Communications Ltd., Perth, Western Australia, as Director of the Digital and Computer Systems Design Section for the development of the DQDB Metropolitan Area Network. From 1987 to 1990, he was also a Visiting Professor in the Department of Electrical and Electronic Engineering, the University of Western Australia. From 1992 to 1997, he was the Director of the Australian Telecommunications Research Institute (ATRI) at Curtin University of Technology, in Western Australia and also Director of the Cooperative Research Centre for Broadband Telecommunications and Networking. He is currently a Chief Technology Officer at Atmosphere Networks Inc., and a Professor in ATRI. He is interested in adaptive signal processing, electronic system design, and networking and regularly acts as a consultant to industry in these areas. Dr Cantoni was an Associate Editor of the IEEE TRANSACTIONS ON SIGNAL PROCESSING from 1996 to 1998. He is a Fellow of the Australian Academy of Technological Sciences and Engineering.
Kok Lay Teo (S’69–M’74–SM’87) obtained the Ph.D. degree from the Department of Electrical Engineering, the University of Ottawa, Ottawa, ONT, Canada, in 1974. He was with the Department of Mathematics at the University of New South Wales from 1974 to 1985, first as a Lecturer and subsequently a Senior Lecturer. He was an Associate Professor in the Department of Industrial and Systems Engineering at the National University of Singapore from 1985 to 1987. He was an Associate Professor in the Department of Mathematics, the University of Western Australia, Nedlands, Australia, from 1988 to 1996. He joined Curtin University of Technology, Perth, Australia, where he was a Professor of Applied Mathematics from 1997 to 1998. He is now the Chair and Head of the Department of Applied Mathematics, the Hong Kong Polytechnic University. He has delivered four keynote lectures, two fully funded invited lectures, and has published four books and numerous journal and conference papers. His research team has developed a software package, MISER, for solving general constrained optimal control problems. His research interests include both theoretical and practical aspects of optimal control and optimization, in particular, signal processing in telecommunications engineering.
