Bias-Free Phase Tracking with Linear and Nonlinear Systems

The arctan function is a well-known Maximum Likelihood (ML) estimator of the phase angle α ∈ [-π,π) of a complex signal in additive white Gaussian noise. In this paper we revisit the arctan-based ML phase estimator and identify the bias problem for phase tracking. We show that the posteriori probability density function of α becomes a bi-modal distribution for small values of signal to noise ratio ρ and larger values of α. In such cases the mean and the mode differ from each other, and as a result when such ML phase estimates are used as an input to a linear system (LS), example for phase tracking, the resulting output (which is essentially the mean value of the phase) differs from its true value which is the mode. In such situations there exist a mean (tracking) error at the output of the LS from its true value α, and in (non-Bayesian) statistical terms there exist a bias in the estimates. In this paper, we provide some statistical analysis to explain the above problem, and also provide solutions for bias correction when a LS is used for tracking phase. Furthermore, we also provide two nonlinear phase tracking systems, 1) a Monte-Carlo based sequential phase tracking technique and 2) a second-order digital-phase locked loop based method, for bias-free phase tracking which eliminate the bias problem that occurs in the case of linear phase tracking with ML estimates.