Bobrovsky-Zakai-Type Bound for Periodic Stochastic Filtering

Mean-squared-error (MSE) lower bounds are commonly used for performance analysis and system design. Recursive algorithms have been derived for computation of Bayesian bounds in stochastic filtering problems. In this letter, we consider stochastic filtering with a mixture of periodic and nonperiodic states. For periodic states, the modulo-$T$ estimation error is of interest and the MSE lower bounds are inappropriate. Therefore, in this case, the mean-cyclic error and the MSE risks are used for estimation of the periodic and nonperiodic states, respectively. We derive a Bobrovsky-Zakai-type bound for mixed periodic and nonperiodic stochastic filtering. Then, we derive a recursive computation method for this bound in order to allow its computation in dynamic settings. The proposed recursively-computed mixed Bobrovsky-Zakai bound is useful for the design and performance analysis of filters in stochastic filtering problems with both periodic and nonperiodic states. This bound is evaluated for a target tracking example and is shown to be a valid and informative bound for particle filtering performance.