New nonlinear target tracking algorithms using convolution

Abstract

This dissertation is concerned with producing and updating a probability density function to represent the state of a discrete time stochastic system which is driven by a known non-Gaussian input density function with additive Gaussian and/or non-Gaussian noise. The a posteriori density function of the state conditioned on the measurement data contains all of the available information that can be used in the development of estimation for such systems. In the Bayesian framework of recursive estimation, there exist general recursive expression describing the evolution of the a posteriori density functions in terms of the a priori distributions and the available measurement data. However, the optimal solution to nonlinear recursive estimation has been considered infeasible for real time implementations, since it involves multi-dimensional integrals that lack analytical solutions. Therefore, difficulties in applying this filter to nonlinear and/or non-Gaussian systems have led to the consideration of more general filters and the possibility of approximate solution of the general Bayesian recursion relations. The major contributions of this thesis are the development of a nonlinear filtering algorithm suitable for the application of target tracking in the Bayesian framework, and a numerical approximation to the optimal recursive solution. In this algorithm the probability density functions have been approximated by a grid, which computes a discretized version of the posteriori filter density in a uniform mesh over the interesting region of the state space. The implementation of the grid method consists of a convolution and an element-wise matrix multiplication. The proposed algorithm propagates the entire non-Gaussian conditional probability density functions recursively, but in a computationally efficient manner using either the fast Fourier transform or the discrete wavelet transformation. In the nonlinear filtering algorithm using the discrete wavelet transf...