Publications

Note

This page currently reflects Manton's research activities prior to his secondment, in 2006, to the Australian Research Council. It will be updated in the near future, now that Manton has returned to academia in June 2008.

Research Interests (Prior to 2006)

• Introduction of Fundamental Areas of Mathematics into Engineering: Traditionally, signal processing and wireless communication problems are tackled using linear algebra. However, great advantage can be gained by using more appropriate branches of mathematics when tackling non-linear problems. The rationale is that to capture the essence of an inherently non-linear problem in general requires the use of an appropriate non-linear mathematical formulation of the problem. If this mathematical formulation is elegant, for example by containing rich structure, then it is likely that more efficient algorithms can be developed for solving the original engineering problem than if a more traditional approach had been taken. Relevant areas of mathematics include differential geometry, algebraic geometry and algebra.


• Numerical Computation: In wireless communications and signal processing, it is commonly required to implement an algorithm to compute certain eigenvectors of a matrix or to solve a polynomial equation. Often, the global performance of such algorithms cannot be ensured, that is, it cannot be guaranteed that for any input, the algorithm will terminate in a pre-specified time and return an answer which is within a pre-specified accuracy of the true answer. I'm interested in trying to design new algorithms with guaranteed performance for solving commonly encountered problems in signal processing and wireless communications.


• Application Driven: Areas which interest me in wireless communications include channel identification and equalization, code design (including affine precoder design), OFDM systems, systems with guard intervals and super-imposed training. In signal processing, my interests include filtering, estimation theory and subspace tracking.

Selected Publications (Prior to 2006)

Optimization on Manifolds (Differential Geometry in Signal Processing)

• Manton, J.H. (2002). Optimization Algorithms Exploiting Unitary Constraints. IEEE Transactions on Signal Processing, 50(3), pp. 635-650.

  Prior to this work, the only standard method for numerically optimising a cost function on a manifold was first to endow the manifold with a Riemannian structure and then apply a Riemannian algorithm (obtained in the natural way from its Euclidean counterpart). We argue that the Riemannian structure is artificial; rather, what is important is the family of cost functions for which the algorithm is to work well for. Unless this family is somehow related to the introduced Riemannian geometry (such as if the cost function involves Riemannian distances), there is no reason for using a Riemannian method. This paper proposes a significantly more general framework for optimising cost functions on manifolds, which includes the Riemannian approach as a special case. In addition, it applies this framework to several commonly occurring problems in signal processing, demonstrating the benefits over the traditional Riemannian approach.

• Mahony, R. and Manton, J. H. (2002). The Geometry of the Newton Method on Non-Compact Lie Groups. Journal of Global Optimization, 23(3) pp. 309-327. (Invited paper.)

• Manton, J. H. (2002). An Improved Least Squares Blind Channel Identification Algorithm for Linearly and Affinely Precoded Communication Systems. IEEE Signal Processing Letters, 9(9) pp. 282-285.

  This paper demonstrates that a singularity in the original cost function preventing standard optimization techniques from working, can be removed by reformulating the problem on a suitable manifold, in this case complex projective space.

• Manton, J. H., Mahony, R. and Hua, Y. (2003). The Geometry of Weighted Low Rank Approximations. IEEE Transactions on Signal Processing, 51(2) pp. 500-514.

• Manton, J. H. (2004). On the Various Generalisations of Optimisation Algorithms to Manifolds. Sixteenth International Symposium on Mathematical Theory of Networks and Systems, July, Katholieke Universiteit Leuven, Belgium. (Invited paper.)

• Manton, J. H. (2004). A Globally Convergent Numerical Algorithm for Computing the Centre of Mass on Compact Lie Groups. Eighth International Conference on Control, Automation, Robotics and Vision, December, Kunming, China.

• Manton, J. H. (2005). On the Role of Differential Geometry in Signal Processing. International Conference on Acoustics, Speech and Signal Processing, March, Philadelphia, USA, vol. 5, pp. 1021-1024. (Invited paper.)

Channel Identifiability in Wireless Communications (Algebraic Geometry in Signal Processing)

• Manton, J. H. (2002). Finite Alphabet Source Recovery in Polynomial Systems. Systems and Control Letters, 47(4) pp. 279-286.

  One of the results proved in this paper is that if a transmitted block of data symbols coming from a finite alphabet has leading and trailing zeros (such as in a block transmission system using guard intervals) then it is theoretically possible to determine the channel having received only one block of data. The following paper shows that if the finite alphabet assumption is dropped, then two received blocks suffice in order to be able to identify the channel.

• Manton, J. H. and Neumann, W. D. (2003). Totally Blind Channel Identification By Exploiting Guard Intervals. Systems and Control Letters, 48(2) pp. 113-119.

  Given two received blocks, if it is known that there is a sequence of zeros separating each block at the transmitter, then this paper proves this is sufficient knowledge for the receiver to be able to identify the channel. We are currently writing up a paper which presents a practical algorithm for identifying the channel in this case.

• Manton, J. H., Neumann, W. D. and Norbury, P. T. (2005). On the Algebraic Identifiability of Finite Impulse Response Channels Driven by Linearly Precoded Signals. Systems and Control Letters, 54(2) pp. 125-134.

  Abstract: It is common in wireless communications to perform some form of linear precoding operation on the source signal prior to transmission over a channel. Although the traditional reason for precoding is to improve the performance of the communication system, this paper draws attention to the fact that the receiver can identify the impulse response of the channel without any prior knowledge of the transmitted signal simply by solving a system of polynomial equations. Since different precoders lead to different systems of equations, this paper addresses the fundamental issue of determining which classes of linear precoders lead to a system of equations having a unique solution. In doing so, basic properties of polynomial equations which are useful for studying other identifiability issues commonly encountered in engineering and the applied sciences are presented.

Affine Precoders and Super-Imposed Training in Wireless Communications

• Manton, J. H., Mareels, I. M. Y. and Hua, Y. (2000). Affine Precoders for Reliable Communications. IEEE International Conference on Acoustics, Speech and Signal Processing, June, Istanbul, Turkey. vol. 5, pp. 2749-2752.

  Super-imposed training is a current hot topic in wireless communications. In fact, super-imposed training is a special case of an affine precoder, and this paper is believed to be the first to introduce the idea of affine precoders.

• Manton, J. H. (2003). Design and Analysis of Linear Precoders Under A Mean Square Error Criterion, Part I: Foundations and Worst Case Designs. Systems and Control Letters, 49(2) pp. 121-130.

• Manton, J. H. (2003). Design and Analysis of Linear Precoders Under A Mean Square Error Criterion, Part II: MMSE Designs and Conclusions. Systems and Control Letters, 49(2) pp. 131-140.

OFDM Systems and Systems with Guard Intervals in Wireless Communications

• Manton, J. H. (2001). Optimal Training Sequences and Pilot Tones for OFDM Systems. IEEE Communications Letters, 5(4) pp. 151-153.

• Manton, J. H. (2001). Dissecting OFDM: The Independent Roles of the Cyclic Prefix and the IDFT Operation. IEEE Communications Letters, 5(12) pp. 474-476.

• Manton, J. H. (2002). An OFDM Interpretation of Zero Padded Block Transmissions. Systems and Control Letters, 47(5) pp. 393-399.

  Abstract: Orthogonal frequency division multiplex (OFDM) systems are particularly straightforward to understand in the frequency domain. The purpose of this paper is to draw attention to the fact that zero padded block transmissions also have a straightforward frequency domain interpretation. The advantages of this interpretation are illustrated by three additional results which further the understanding of zero padded block transmission systems; it is shown that the zero padding spreads the spectrum of the source symbols uniformly, it is explained why zero padded block transmission systems do not always outperform OFDM systems, and it is shown how pilot tones can be incorporated into zero padded block transmissions.

Filtering of Stochastic Processes

• Manton, J.H., Krishnamurthy, V. and Poor, H.V. (1998). James-Stein State Filtering Algorithms. IEEE Transactions on Signal Processing, 46(9) pp. 2431-2447.

• Manton, J. H., Krishnamurthy, V. and Elliott, R. J. (1999). Discrete Time Filters for Doubly Stochastic Poisson Processes and Other Exponential Noise Models. International Journal of Adaptive Control and Signal Processing, 13(5) pp. 393-416.

  Abstract: The well-known Kalman filter is the optimal filter for a linear Gaussian state-space model. Furthermore, the Kalman filter is one of the few known finite-dimensional filters. In search of other discrete-time finite dimensional filters, this paper derives filters for general linear exponential state-space models, of which the Kalman filter is a special case. One particularly interesting model for which a finite-dimensional filter is found to exist is a doubly stochastic discrete-time Poisson process whose rate evolves as the square of the state of a linear Gaussian dynamical system. Such a model has wide applications in communications systems and queueing theory. Another filter, also with applications in communications systems, is derived for estimating the arrival times of a Poisson process based on negative exponentially delayed observations.

Other Ideas Waiting to be Taken Further

• Manton, J. H. and Vo, B.-N. (2001). Fisher Information Decision Directed Quantisation: A Fast Sub-Optimal Solution to Discrete Optimisation Problems with Applications in Wireless Communications. International Conference on Optimization Theory and Applications, December, Hong Kong. vol. 1, pp. 316-323. (Invited paper.)

  The order in which symbols are quantised can affect the accuracy of the result. This paper proposes the idea of using the Fisher Information Matrix to determine which symbol is likely to be the most accurate and hence which symbol to quantise first. The remaining symbols are quantised likewise by applying this idea recursively. We believe this idea to have broad applicability but have not taken it further as yet.

• Manton, J. H. (2002). The Convex Geometry of Sub-Channel Attenuation Coefficients in Linearly Precoded OFDM Systems. IEEE Transactions on Information Theory, 48(5) pp. 1203-1206.

  This paper shows that OFDM systems naturally have a convex geometry associated with them. In this geometry, channels with spectral nulls are geometrically significant. Although we have not investigated further, we believe that practical algorithms should be able to exploit this convex geometry.

• Manton, J. H., Helmke, U. and Mareels, I. M. Y. (2005). A Dual Purpose Principal and Minor Component Flow. Systems and Control Letters, 54, pp. 759-769.

  This paper presents the first flow on Euclidean space capable of computing both the principal and minor components of a matrix simply by changing the sign of the matrix. Prior to this work, it was believed that minor component computation was harder than principal component computation. We are currently in the process of writing a paper which presents practical numerical algorithms based on this flow.

• An, S., Liu, W. and Manton, J. H. (2005). Extreme Point Results for Robust Stability of Nested Polynomial Families with Real Coefficients. International Journal of Statistics and Systems. Accepted for publication (12 July 2005).

  Abstract: A family of polynomials is robustly stable if each member of the family is stable, meaning all its zeros lie in the open left half plane. Consider a one dimensional affine family of polynomials, visualised as a line segment. For some such families, it is known the family is robustly stable if and only if the two end points are stable. Indeed, a sufficient condition is for the line segment to be parallel to what is called a convex direction. This paper extends this result to nested polynomial families. Specifically, a set of directions is found such that, if the line segment is parallel to one of these directions, then any nested family obtained by composing a fixed polynomial with the line segment is robustly stable if and only if its end points are stable. Moreover, it is proved this set of directions is the largest possible. An extension to higher dimensional families is also derived.

• Manton, J.H. (2005). A Centroid (Karcher Mean) Approach to the Joint Approximate Diagonalisation Problem: The Real Symmetric Case. Digital Signal Processing. Accepted for publication (23 June 2005).

  Abstract: A real symmetric matrix is diagonalisable by a suitable orthonormal change of basis. The joint approximate diagonalisation problem is to find an orthonormal change of basis which simultaneously diagonalises, or approximately diagonalises, two or more real symmetric matrices. A number of important signal processing problems require the computation of a joint approximate diagonaliser. However, no algorithm to date is guaranteed to find the optimal diagonaliser. This paper reformulates the diagonalisation problem as a convex optimisation problem on a Riemannian manifold and is thus able to guarantee global convergence to the optimal diagonaliser.

Expository Material

• Manton, J. H., Groves, J. R. J. and Hua, Y. (2000). On Properties of the Solutions of Systems of Polynomial Equations. Third Asian Control Conference, July, Shanghai, China.

  Whereas most if not all of the results derived in this paper are known to experts in algebraic geometry, the results are not easy to come by in the literature. Moreover, straightforward proofs from first principles are given.

• Moran, W. and Manton, J.H. (2003). Group Theory in Radar and Signal Processing. Computational Noncommutative Algebra and Applications. Kluwer. (Invited book chapter.)

  This book chapter explains how several theoretical questions in radar theory are best answered by appealing to group representation theory.