Exact Phase Noise Model and Its Application to Linear Minimum Variance Estimation of Frequency and Phase of a Noisy Sinusoid

The exact statistical models for the measurement phase noise in estimating the frequency and phase of a single sinusoid over the additive white Gaussian noise channel are derived. The a posteriori Probability Density Function (PDF) and the a priori PDF of the phase noise derived are given by explicit, closed-form expressions that are valid for arbitrary signal-to-noise ratios. It is shown that as far as estimating the single sinusoid angle parameters is concerned, the phase of the received signal samples alone is a sufficient statistic, provided that the phase noise is modeled by the a posteriori PDF, which has a Tikhonov distribution.