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On a phase tracking problem: Continuous-discrete filtering approaches

In this paper, we derive and apply numerically and computationally-efficient extended Kalman filter and the proposed higher-order filter for state estimation of a stochastic system. Note that the phase tracking problem can be formalized as a stochastic differential system. The continuous-discrete extended Kalman filter and the proposed higher-order filter are applied to a phase tracking problem. The phase tracking problem is formalized as a non-linear noisy discrete observation equation in which the measurement non-linearity is sinusoid added with additive noise. From the dynamical systems' viewpoint, we state the evolution of the phase angle of the measurement equation as well. As a result of this, we wish to estimate the phase angle from given discrete noisy observations using two non-linear filters: (i) the extended Kalman filter (ii) the proposed higher-order filter. This paper develops two non-linear filters for a phase tracking filtering model. Note that the Ornstein-Uhlenbeck process is the process noise as well as an augmented stochastic state for the phase tracking problem and the Brownian noise process is the observation noise. The filter efficacy is examined by utilizing quite extensive numerical experimentations with one set of data. This paper bridges a gap between non-linear stochastic filtering and phase tracking problem.