Colored Noise

Abstract

Consider the linear stochastic system with the following state-space description:

$$\left\{ {\begin{array}{*{20}{c}} {{{{\text{x}}}_{{k + 1}}} = {{A}_{k}}{{{\text{x}}}_{k}} + {{\Gamma }_{k}}{{{\underline \xi }}_{k}}} \\ {{{{\text{v}}}_{k}} = {{C}_{k}}{{{\text{x}}}_{k}} + {{{\underline \eta }}_{k}},} \\ \end{array} } \right.$$

(5.1)

where A _k , Γ _k , and C _k are known n × n, n × p, and q ×n constant matrices, respectively, with 1 ≤ p, q ≤ n. The problem is to give a linear unbiased minimum variance estimate of x _k with initial quantities E(x₀) and Var(x₀) under the assumption that

• (i) ${\underline \xi _k} = {M_k} - 1{\underline \xi _{k - 1}} + {\underline \beta _k}$

• (ii) ${\underline \eta _k} = {N_k} - 1{\underline \eta _{k - 1}} + {\underline \gamma _k}$

where ${\underline \xi _{ - 1}} = {\underline \eta _{ - 1}} = 0$, $\left\{ {{{\underline \beta }_k}} \right\}$ and $\left\{ {{{\underline \gamma }_k}} \right\}$ are uncorrelated zero-mean Gaussian white noise sequences satisfying

$$E\left( {{{\underline \beta }_k}\lambda _\ell ^{\rm T}} \right) = 0,\;E\left( {{{\underline \beta }_k}\beta _\ell ^{\rm T}} \right) = Q{k^\delta }k\ell ,\;E\left( {{{\underline \gamma }_k}\lambda _\ell ^{\rm T}} \right) = {R_k}\delta k\ell ,$$

and M _{k −1} and N _{k −1} are known p × p and q × q constant matrices. The noise sequences $\left\{ {{{\underline \xi }_k}} \right\}$ and $\left\{ {{{\underline \eta }_k}} \right\}$ satisfying (i) and (ii) will be called colored noise processes. This chapter is devoted to the study of Kalman filtering with this assumption on the noise sequences.