Noise-induced cycle slips in a PLL: computing the average time to first cycle slip

Well-know phase-lock loop (PLL) models exist which include a reference input signal that is embedded in white Gaussian noise. Under stationary phase-locked conditions, these models describe a Markov state vector that remains in a neighborhood of a stable equilibrium point for a random and finite period of time to before suffering a noise-induced cycle slip whereby closed-loop phase increases or decreases by 2π. The average time E[t_a] cannot be computed exactly except for an extremely limited class of PLLs. More generally, E[t_a] must be approximated numerically, and a new method for doing so is outlined. The approximation technique is applied to a simple second-order system, and numerical results are given.