A New Model for Applying Extended Kalman Filtering to Extract Harmonic Signals from Noisy Measurements

The extended Kalman filter has been used to estimate a harmonic signal from noisy measurements. Most algorithms are based on the Cartesian model, which is a discretization in time of the continuous state space model associated with the differential equation that is satisfied by a sinusoidal signal with constant amplitude and frequency. In order to handle the more realistic case where both amplitude and frequency are changing, this basic model is modified by including ad hoc extensions. This paper starts by deriving a differential equation that explicitly includes time varying amplitude and frequency, and it is shown that this can be reduced to a Bessel's equation of order 1/2 that has a closed form solution. This is used to derive an explicit expression for a discrete-time model, which forms the basis of an extended Kalman filter. Simulation results show that this algorithm outperforms other approaches, particularly for harmonic signals where the frequency is changing rapidly.