Phase acquisition statistics for phase-locked loops

After analyzing the free-acquisition behavior of a phase-locked loop, the influence of noise on phase acquisition is discussed, and it is shown that hung-up is not caused by equivocation. The cumulative probability distribution of acquiring lock within a given time interval is derived, and the results, which are exact for a first-order loop, serve as good approximations to second-order loops with a high damping factor. The performance of the initial phase error estimator is analyzed, and binary quantized and low-pass filtered versions of the (I,Q) phase detector signals are used to estimate the initial phase error quadrant. After deriving the probability of a correct quadrant estimate, the probability of acquiring phase lock within a given time interval for a PLL with acquisition aid is computed. A hang-up-free phase-locked synchronizer is possible for signal-to-noise ratios better than approximately 12 dB.