TY - JOUR

T1 - A state-space calculus for rational probability density functions and applications to non-Gaussian filtering

AU - Hanzon, B.

AU - Ober, R.J.

PY - 2001

Y1 - 2001

N2 - We propose what we believe to be a novel approach to performing calculations for rational density functions using state-space representations of the densities. By standard results from realization theory, a rational probability density function is considered to be the transfer function of a linear system with generally complex entries. The stable part of this system is positive-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 Markov parameters. Some of the main contributions of the paper are explicit state-space descriptions for products and convolutions of rational densities. As an application which is of interest in its own right, the filtering problem is investigated for a linear time-varying system whose noise inputs have rational probability density functions. In particular, state-space formulations are derived for the calculation of the prediction and update equations. The case of Cauchy noise is treated as an illustrative example.

AB - We propose what we believe to be a novel approach to performing calculations for rational density functions using state-space representations of the densities. By standard results from realization theory, a rational probability density function is considered to be the transfer function of a linear system with generally complex entries. The stable part of this system is positive-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 Markov parameters. Some of the main contributions of the paper are explicit state-space descriptions for products and convolutions of rational densities. As an application which is of interest in its own right, the filtering problem is investigated for a linear time-varying system whose noise inputs have rational probability density functions. In particular, state-space formulations are derived for the calculation of the prediction and update equations. The case of Cauchy noise is treated as an illustrative example.

U2 - 10.1137/S036301299731610X

DO - 10.1137/S036301299731610X

M3 - Article

SN - 0363-0129

VL - 40

SP - 724

EP - 740

JO - SIAM Journal on Control and Optimization

JF - SIAM Journal on Control and Optimization

ER -