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This paper presents the theoretical foundation for time-domain, phase-based estimation of the frequency and phase of a single sinusoid in additive white Gaussian noise (AWGN), analogous to the theoretical foundation provided by Rife and Boorstyn for frequency-domain, Fourier-transform-based estimation. It is shown from the maximum a posteriori probability (MAP) and the maximum likelihood (ML) estimation principles that with the additive observation phase noise (AOPN), due to the AWGN, being described by its a posteriori distribution conditioned on the received signal magnitude, the received signal phase is a sufficient statistic for estimating the single-sinusoid angle parameters. Using a geometric approach, the exact statistical model for the AOPN is derived, where the a posteriori probability density function (pdf) and the corresponding a priori pdf are given by explicit, closed-form expressions that are valid for arbitrary signal-to-noise ratios (SNRs). The a posteriori pdf is Tikhonov, and is of particular interest as it establishes the AOPN model for phase-based frequency/phase MAP/ML estimation in the time domain. It is further illustrated that the results derived can yield various AOPN models as special cases, and the underlying physical insights and interconnections that exist among these models are revealed. It is shown that the model derived by Tretter is an ultimate specialization in the high SNR limit of the AOPN models developed here. For high SNR, the a posteriori Tikhonov pdf can be accurately approximated by a Gaussian distribution, which leads to the best linearized AOPN model. The applications of these AOPN models to the design of linear estimators, including the linear minimum mean square error (LMMSE) estimator, the linear minimum variance estimator, and the LMMSE implementation of the weighted phase averager are presented, and their estimation performances are compared through computer simulations, with the Cramer–Rao lower bound (CRLB) and the Bayesian CRLB as the benchmark. To facilitate estimator design, the a priori statistical models of the frequency and phase are proposed from the information-theoretic perspective, and an improved phase unwrapping algorithm over that given by Fu and Kam is presented. It is shown that by incorporating all the information available in the AOPN, the estimation accuracy can be much improved. [ABSTRACT FROM PUBLISHER]

In this paper, a new structured approach for obtaining uniformly best non-Bayesian biased estimators, which attain minimum-mean-square-error performance at any point in the parameter space, is established. We show that if a uniformly best biased (UBB) estimator exists, then it is unique, and it can be directly obtained from any locally best biased (LBB) estimator. A necessary and sufficient condition for the existence of a UBB estimator is derived. It is shown that if there exists an optimal bias, such that this condition is satisfied, then it is unique, and its closed-form expression is obtained. The proposed approach is exemplified in two nonlinear estimation problems, where uniformly minimum-variance-unbiased estimators do not exist. In the considered examples, we show that the UBB estimators outperform the corresponding maximum-likelihood estimators in the MSE sense. [ABSTRACT FROM AUTHOR]