An estimation algorithm of a second-order phase-locked loop system

The stochastic Phase Locked Loop (PLL) system is an appealing and non-trivial case in stochastic system theory. The equation describing the stochastic PLL system is a second-order non-linear fluctuation equation. In this paper, the stochastic version of the Phase Locked Loop system, which accounts the noise contribution at the output of the multiplier, is analysed from estimation-theoretic viewpoint. The estimation theory of the second-order PLL system is accomplished using the Kolmogorov forward and backward equations. As a result of this, a set of two estimation evolution equations for the stochastic problem of concern here is obtained. Approximate evolution equations that can be regarded as a convenient structure from the numerical simulation view point are derived as well.