A Singular Perturbation Method for the Computation of the Mean First Passage Time in a Nonlinear Filter

We give a new application of recently developed singular perturbation methods in the area of mathematical theory of nonlinear filtering. We consider the phenomenon of cycle slipping in a second order phase-locked loop (PLL) which serves as demodulator for a random FM message. We introduce new scaling parameters into the Ito system of stochastic differential equations describing the PLL, thus identifying the phenomenon of cycle slipping with Kolmogorov’s exit problem. We use singular perturbation methods to obtain an explicit expression for the mean time between cycle slips. Furthermore, we describe the mechanism of cycle slips and identify new parameters which determine the probability of their occurrence.