Estimating the Frequency and Phase of a Noisy Sinusoid by Kalman Filter

A linear, two-dimensional state-space model involving the instantaneous signal frequency and carrier phase is formulated. This enables Kalman filtering to be used for estimating the frequency and phase. Two Kalman filters are presented here, one based on the old observation model of Tretter (1985) and the other based on our newly proposed model by H. Fu and P.Y. Kam (2006). The Kalman filter for the old observation model requires knowledge of the signal amplitude and the noise variance, while for the new observation model, only knowledge of the noise variance is required. Their mean square estimation error performances are compared using simulations, and it is shown that the filter based on the new observation model performs better, especially at low signal-to-noise ratio. Kalman filtering also allows the incorporation of prior knowledge of the interval of distribution of the frequency to improve the estimation performance.