Posterior Cramér-Rao bounds for discrete-time nonlinear filtering with finitely correlated noises

In this paper, a recursive formula of the mean-square error lower bound for the discrete-time nonlinear filtering problem when noises of dynamic systems are temporally correlated is derived based on the Van Trees (posterior) version of the Cramér-Rao inequality. The approximation formula is unified in the sense that it can be applicable to the multi-step correlated process noise, multi-step correlated measurement noise and multi-step cross-correlated process and measurement noise simultaneously. The lower bound is evaluated by two typical target tracking examples respectively. Both of them show that the new lower bound is significantly different from that of the method which ignores correlation of noises. Thus, when they are applied to sensor selection problems, number of selected sensors becomes very different to obtain a desired estimation performance.