  In this paper, an approximation 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 Cramer-Rao inequality. The formula is unified in the
sense that it can be applied 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.
