Phase-Based, Time-Domain Estimation of the Frequency and Phase of a Single Sinusoid in AWGN—The Role and Applications of the Additive Observation Phase Noise Model

Abstract

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.