Optimal Recursive Non-Linear Filtering: Convergence of Parallel Algorithms with Application to Detection Hypothesis and Phase Tracking Problems

Abstract

This paper sumes up several results obtained for the numerical approximation of optimal non-linear filters. It presents convergence results for numerical algorithms arising from an analysis of non-linear partial differential equations. It extends these results to a class of parallel algorithms describing the evolution of the unnormalised conditional density of probability. Several examples in detection and phase tracking specify the region of application of this type of solution and show the performances and stability that one can expect for such a filter through a statistical analysis of a numerical implementation.Using developments in partial differential equation theory, this paper describes a class of algorithms the implementation of which on a multiprocessor parallel computer is specially easy. Results of convergence guarantee the numerical stability of this solution and a Monte Carlo analysis shows what is in practice the exact meaning of this stability.Two types of application are presented. The first one concerns the detection of hypothesis from a set of equiprobable situations. This leads to performing algorithms for one-line identification of discrete valued parameters in linear and non-linear systems. The second one gives a stable and accurate solution to the phase tracking problem and the associated coding and decoding problem when the noise over signal ratio is too high for classical solution to be efficient. This pair of applications shows what is the area of efficiency of these approximation algorithms and what performances can be a priori expected.