A state-space calculus for rational probability density functions and applications to non-Gaussian filtering

We propose what we believe to be a novel approach to performing calculation s for rational density functions using state-space representations of the d ensities. By standard results from realization theory, a rational probabili ty density function is considered to be the transfer function of a linear s ystem with generally complex entries. The stable part of this system is pos itive-real, which we call the density summand. The existence of moments is investigated using the Markov parameters of the density summand. Moreover, explicit formulae are given for the existing moments in terms of these Mark ov parameters. Some of the main contributions of the paper are explicit sta te-space descriptions for products and convolutions of rational densities. As an application which is of interest in its own right, the filtering prob lem is investigated for a linear time-varying system whose noise inputs hav e rational probability density functions. In particular, state-space formul ations are derived for the calculation of the prediction and update equatio ns. The case of Cauchy noise is treated as an illustrative example.