The Cramér-Rao bound (CRB) has been extensively used as a benchmark for estimation performance in both Bayesian and non-Bayesian frameworks. In many practical periodic parameter estimation problems, such as phase, frequency, and direction-of-arrival estimation, the observation model is periodic with respect to the unknown parameters and thus, the appropriate criterion is periodic in the parameter space. Consequently, Bayesian lower bounds on the mean-squared-error (MSE) are not valid bounds for periodic estimation problems. In addition, many Bayesian MSE lower bounds cannot be derived in the periodic case due to their restrictive regularity conditions. For example, the regularity conditions of the Bayesian CRB (BCRB) are not satisfied for parameters with uniform prior distribution. In this letter, we derive a Bayesian Cramér-Rao-type lower bound on the mean-squared-periodic-error (MSPE). The proposed periodic BCRB (PBCRB) is a lower bound on the MSPE of any estimator and has less restrictive regularity conditions than the BCRB. The PBCRB is compared with the MSPE of the minimum MSPE estimator for phase estimation in Gaussian noise and it is shown that the PBCRB is a valid and tight lower bound for this problem.

中文翻译：

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贝叶斯周期性克拉默-饶界


Cramér-Rao 界 (CRB) 已被广泛用作贝叶斯和非贝叶斯框架中估计性能的基准。在许多实际的周期性参数估计问题中，例如相位、频率和到达方向估计，观测模型相对于未知参数是周期性的，因此，适当的准则在参数空间中是周期性的。因此，均方误差 (MSE) 的贝叶斯下界不是周期性估计问题的有效界限。此外，由于其限制性规律性条件，许多贝叶斯 MSE 下界无法在周期性情况下导出。例如，对于具有均匀先验分布的参数，不满足贝叶斯 CRB (BCRB) 的正则性条件。在这封信中，我们推导出均方周期误差 (MSPE) 的贝叶斯 Cramér-Rao 型下界。所提出的周期性 BCRB (PBCRB) 是任何估计器的 MSPE 的下界，并且具有比 BCRB 更少的限制性规则性条件。将 PBCRB 与高斯噪声中相位估计的最小 MSPE 估计器的 MSPE 进行比较，结果表明 PBCRB 是该问题的有效且严格的下界。