Phase-locked loop dynamics in the presence of noise by Fokker-Planck techniques

Statistical parameters of the phase-error behavior of a phase-locked loop tracking a constant frequency signal in the presence of additive, stationary, Gaussian noise are obtained by treating the problem as a continuous random walk with a sinusoidal restoring force. The Fokker-Planck or diffusion equation is obtained for a general loop and for the case of frequency-modulated received signals. An exact expression for the steady-state phase-error distribution is available only for the first-order loop, but approximate and asymptotic expressions are derived for the second-order loop. Results are obtained also for the expected time to loss of lock and for the frequency of skipping cycles. Some of the results are extended to tracking loops with nonsinusoidal error functions. Validity thresholds of widely accepted approximate models of the phase-locked loop are obtained by comparison with the exact results available for the first-order loop.