A New Class of Bayesian Cyclic Bounds for Periodic Parameter Estimation

Many practical signal processing applications involve estimation of parameters with periodic nature, such as phase, frequency and direction-of-arrival estimation. The commonly used mean-squared-error (MSE) risk does not take periodicity into account and thus, is inappropriate for periodic parameter estimation in which one is interested in the modulo- T error rather than the plain error value. As a result, MSE lower bounds are not valid for periodic estimation. In addition, conventional Bayesian MSE lower bounds, such as the Bayesian Cramér-Rao bound (BCRB) and the Bobrovsky-Zakai bound (BZB) require restrictive regularity conditions and usually do not exist in periodic settings. An alternative risk, which is commonly used in periodic parameter estimation problems, is the mean-cyclic-error (MCE). In this paper, we establish a new class of Bayesian lower bounds on the MCE of any estimator. The new class includes cyclic versions of the BCRB and the BZB that have less restrictive regularity conditions than those of the conventional BCRB and BZB, respectively. The tightest bound in the proposed class is derived and its tightness is discussed. In addition, the proposed class is extended to mixed vector parameter estimation with both periodic and nonperiodic parameters. The new cyclic lower bounds are compared with the MCE performance of the minimum MCE and maximum a-posteriori probability estimators for von-Mises parameter estimation and for frequency estimation.