Analysis of the statistics and threshold of the phase-lock loop

The mean-square value of the phase error in the phase-lock loop is determined from the linear differential equation. Conditions are then derived, for its minimisation, with an integral plus proportional filter in the loop, for an input consisting of an f.m. f.d.m. carrier plus thermal noise. The statistics of the phase error are then deduced using the Fokker-Planck technique for an arbitrary loop filter. From the statistics, the loss of lock rate is derived and each loss of lock is assumed to produce an inpulse in the baseband. The well known result of Rice is used to determine the output signal/noise ratio, and the point of threshold arising from the loss of lock is shown.