Multivariate Bayesian Cramér-Rao-Type Bound for Stochastic Filtering Involving Periodic States

In many stochastic filtering problems, some of the states have periodic nature, i.e. the observation model is periodic with respect to these states. For estimation of these periodic states, we are interested in the modulo-$T$ error and not in the plain error value. Thus, in this case, the commonly-used Bayesian mean-squared-error (MSE) lower bounds are inappropriate for performance analysis, since the MSE risk is based on the plain error and is inappropriate for periodic state estimation. In contrast, the mean-cyclic-error (MCE) is an appropriate risk for estimation of periodic states. In a mixed periodic and nonperiodic setting, a mixed MCE and MSE lower bound can be useful for performance analysis and design of filters. In this paper, we present the mixed Bayesian Cramér-Rao bound (BCRB) for stochastic filtering. The mixed BCRB is composed of a cyclic part and a noncyclic part for estimation of the periodic and the nonperiodic states, respectively. Direct computation of the mixed BCRB is not practical, since it requires matrix inversion, whose dimensions increase with time. Therefore, we propose a recursive method with low computational complexity for computation of the mixed BCRB at each time step. The mixed BCRB is examined for direction-of-arrival tracking scenarios and compared to the performance of a particle filter. It is shown that in the considered scenarios the mixed BCRB is informative and can be approached by the particle filter. In addition, the inappropriateness of MSE bounds for estimation of periodic states is demonstrated.