Phase and frequency transfer between mutually synchronized oscillators

Accurate phase and frequency transfer between mutually synchronized oscillators is a problem of current interest. This paper addresses a system of two mutually synchronized oscillators with inherently sinusoidal phase detectors. The system is studied in the context of nonlinear phase-locked loop theory. This is accomplished by statistically characterizing the 'time' and 'space and time' increments of the phase processes generated by the two oscillators. The characterization of these increment processes is obtained in the form of steady-state probability density functions (pdf's). The effects of such disturbances as noise, transmission path delays, and oscillator frequency instabilities on the system are studied through their effects on these pdf's. Gaussian approximations for these pdf's (which simplify the extraction of the effects of the disturbances on the system) are also developed. Additionally, the stability of the time intervals and time interval errors generated by the oscillators are studied in the context of cycle slipping phenomena.