The Optimal Kalman Type State Estimator with Multi-Step Correlated Process and Measurement Noises

In this paper, an optimal Kalman type recursive state estimator is presented for the discrete time random dynamic system when the process noise and measurement noise are two-step correlated. Then, we extend it to the more general case of the process noise and measurement noise being n-step correlated. Finally, we verify that the Kalman type filter equation with one-step correlated process noise and measurement noise is globally optimal in the sense that its performance is the same as that of the optimal Mean Square Error state estimation using all observations from initial time up to now.