Noise and Transmission Delays in Mutual Synchronization Systems

This paper considers the effects of noise and transmission delays in a phase-averaging mutual synchronization system. The basic analytical ideas are developed with respect to a two-station system which contains first-or second-order analog phase-lock loops. From the equation describing the phase error dynamics of each system, a generalized Fokker-Planck equation is derived. It is used to obtain a good approximate analytic expression for the phase error probability density function (PDF), which is confirmed by simulation. An important quantity for such mutual synchronization systems, under random disturbances, is the variance of the frequency deviation. This may be obtained directly from a knowledge of the phase error PDF. The above development is then extended to a mutual synchronization system of arbitrary size.