Optimal filtering for polynomial measurement nonlinearities with additive non-Gaussian noise

We consider the problem of estimating the n-dimensional state of a dynamic system based on m-dimensional discrete-time measurements. The measurements depend nonlinearly on the state and are corrupted by white non-Gaussian noise. The problem is solved by recursively calculating the complete posterior density of the state given the measurements. For that purpose, a new exponential type density is introduced, the so called pseudo Gaussian density, which is used to represent the complicated non-Gaussian posterior densities resulting from the recursion. For polynomial measurement nonlinearities and pseudo Gaussian noise densities, it is shown that the result of the optimal Bayesian measurement update is exactly obtained by a Kalman filter operating in a higher dimensional space. The resulting filtering algorithms are easy to implement and always guarantee valid posterior densities.