## Abstract

A new application is given of recently developed singular perturbation methods in the area of mathematical theory of nonlinear filtering. The phenomenon of cycle slipping in a second order phase-locked loop (PLL) which serves as a demodulator for a random FM message is considered. New scaling parameters are introduced into the Ito system of stochastic differential equations of describing the PLL, thus identifying the phenomenon of cycle slipping with Kolmogorov's exit problem. Singular perturbation methods are used to obtain an explicit expression for the mean time between cycle slips. Furthermore, the mechanism of the cycle slips is described, and new parameters are identified which determine the probability of their occurrence.

Original language | English |
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Pages (from-to) | 174-187 |

Number of pages | 14 |

Journal | SIAM Journal on Applied Mathematics |

Volume | 42 |

Issue number | 1 |

DOIs | |

State | Published - 1982 |