Posterior cramer-rao bounds for adaptive discrete-time system identification

A mean square error lower bound for the discrete-time nonlinear filtering problem is derived based on the Van Trees' (posterior) version of the Cramer-Rao inequality. This lower bound is applicable to multidimensional nonlinear, possibly non-Gaussian, dynamical systems and is more general than the previous bounds in the literature. The case of singular conditional distribution of the one-step-ahead state vector, given the present state, is considered. The bound is evaluated for three important examples: the recursive estimation of slowly varying parameters of an autoregre-sive process; tracking a slowly varying frequency of a single cisoid in noise; tracking parameters of a sinusoidal frequency with sinusoidal phase modulation.