Acquisition Behavior of a First-Order Digital Phase-Locked Loop

A general "phase-plane" technique for investigating the stability of nonlinear difference equation is described and applied to determine the stability criterion of a first-order digital phase-locked loop (DPLL). If the loop is stable, it is shown that acquisition behavior can be modeled as a first passage time problem. Using this model, one can evaluate the acquisition probability and the mean time to acquire, from the appropriate Chapman-Kolmogorov (C-K) equation describing the transition of the phase error sequence. This approach is verified by direct simulation.