Analysis of lock-loss events in discrete-time phase locked loop (PLL)

Lock-loss phenomena were studied extensively for continuous-time phase locked loop (PLL), and closed form analytical expressions for the mean time to lose lock (MTLL) has been derived under the assumptions that the input is a phase modulated sine wave, and that the additive noise associated with the input can be added to the detector output. The analysis of the discrete-time phase detector is more complicated as the associated discrete-time stochastic equation depends on the signal constellation as well as on the detector non-linearity. In this study we present an approach for analyzing lock-loss events and for calculating the MTLL in first-order discrete-time PLL, and prove that the time to lose lock is approximately exponentially distributed. The results can be applied to any linear modulation scheme and, in particular, to any QAM modulation scheme, and can be used for qualitative understanding of the problem as well as for performance assessment and parameter optimization. In that context it is shown that the Leclert-Vandamme phase detector is more robust to the lock-loss problem at the cost of slightly larger residual phase error, compared with the more conventional Simon and Smith phase detector.