Kalman Estimation of Single-Tone Parameters and Performance Comparison With MAP Estimator

By incorporating a priori knowledge via the a priori probability density function of the frequency and phase, the Kalman estimator for linear, minimum mean-square error estimation of these single-tone parameters in additive, white, Gaussian noise is presented. First, a linear, two-dimensional state-space model for the frequency and phase of a sinusoid is formulated. Then, two Kalman filters are proposed, one based on Tretter´s phase noise model [S. A. Tretter, ldquoEstimating the Frequency of a Noisy Sinusoid by Linear Regression,rdquo IEEE Transactions on Information Theory, vol. IT-31, no. 6, pp. 832-835, Nov. 1985], and the other based on our newly proposed model in [H. Fu and P. Y. Kam, ldquoMAP/ML Estimation of the Frequency and Phase of a Single Sinusoid in Noise,rdquo IEEE Transactions on Signal Processing, vol. 55, no. 3, Mar. 2007]. Finally, their mean-square error performances are compared with each other and with that of the maximum a posteriori probability (MAP) estimator, using computer simulations. The results show that the MAP estimator performs best and the Kalman filter based on the improved phase noise model has better performance than that based on Tretter´s model, especially at low signal-to-noise ratio.