Exact phase noise model for single-tone frequency estimation in noise

The a posteriori and the a priori statistical models for the measurement phase noise (MPN) in estimating the frequency and phase of a single tone over the additive white Gaussian noise channel are derived. Both models are described by exact, explicit, closed-form probability density functions (pdf´s) that are valid for arbitrary signal-to-noise ratio. It is shown that as far as estimating the single-tone angle parameters is concerned, the received signal phase is a sufficient statistic, provided that the MPN is modelled by the a posteriori pdf. It is illustrated that the results derived can yield various MPN models as special cases, and the underlying physical insights and interconnections that exist among these models are revealed. The application of these models to the design of a linear minimum variance estimator is presented.