Cyclic Bayesian Cramér-Rao bound for filtering in circular state space

Mean-squared-error (MSE) lower bounds are widely used for performance analysis in stochastic filtering problems. In many problems of this type, the nature of part of the unknown state parameters is circular or periodic. In this case, we are interested in the modulo-T estimation errors and not in the plain error values. Thus, the MSE risk and conventional MSE bounds are inappropriate for periodic stochastic filtering problems. A commonly used risk for periodic parameter estimation is the mean-cyclic-error (MCE). In this paper, we derive a cyclic version of the Bayesian Cramér-Rao bound (BCRB) on the MCE of any recursive filter. The performance of the cyclic BCRB is evaluated for phase tracking and compared to the MCEs of existing filters.