Suppose that we are interested in "real world", "practical" dynamical systems, that is, "stochastic" systems, whose evolution can only be known up to their probability distributions. The usual problems here will be filtering, smoothing, detection, parameter estimation and stochastic control. Each of these problems concerns the probability distributions as the central mathematical objects to deal with. Often these functions are difficult to deal with and often they are infinite dimensional. One exception is the ubiquitous Gaussian density, which is completely described by its two sufficient statistics, viz, the mean and variance. Furthermore, the Gaussian density functions enjoy many useful properties.The problem of dealing with non-Gaussian models arises quite naturally in many settings. For example there might be non-Gaussian initial conditions, (see Bensoussan and Makowski), and models with non-Gaussian noise. Approximating probability densities using Gaussian mixtures has appeared in the literature in several forms. Two examples of interest are 1) approximating a non-Gaussian noise model; and 2) approximating a filter density. Alternative approximations to densities via Gram-Charlier and Edgeworth expansions have been investigated, but only with limited success. In particular, truncated Edgeworth expansions can be negative and so do not approximate probability densities. In contrast, one can show a Gaussian mixture can approximate any probability density and, appropriately scaled, is itself always a probability density.

In this seminar we introduce finite Gaussian mixtures and state a fundamental Theorem to justify their application. A first example is given concerning a Markov chain observed in arbitrary, non-Gaussian noise. Filters and smoothers will be described for this model. We conclude with a state estimation example of a Jump Markov System extensively used in tracking. Here, a Gaussian mixture approximates the filter density. Further, our scheme is easily extended to a Jump Markov system with non-Gaussian observation noise. Our schemes do not suffer the non-trivial deficiencies of particle filters in this particular scenario. Further, they provide a well based formulation from which questions of analysis can be investigated.

The research reported here is joint work with Prof R J Elliott, University of Calgary, Canada, and Prof F Dufour, Universite de Bordeaux, France.

Abstract 2: We provide scaling results for the sum capacity of the multi-access, uplink channel in a flat fading environment, when there is interference from other cells. We consider a scaling regime where the number of antennas per user remains fixed but the number of antennas at the base station and the number of users in each cell grow large together. We characterize the asymptotic behaviour of the spectral efficiencies in each cell, in three scenarios 1) single cell processing with full frequency reuse 2) single cell processing, with frequency re-use partitioning of adjacent cells and 3) base station cooperative decoding (macrodiversity). It is shown that base station cooperation provides very significant gains in spectral efficiency over single cell processing.