This note demonstrates that the behavior of two mutually synchronized phase-locked oscillators using delay compensation is equivalent to that of a first-order loop with the loop parameters degraded by the residual uncompensated delays of the system. In particular, the probability density of phase differences between the two oscillators is derived and the numerical results are also presented. The cycle slipping rate is easily obtained from this density.

This paper presents a unified mathematical model for studying the behavior of various network synchronization techniques, e.g., plesiochronous or independent clocks, hierarchical master-slave, delay-compensated and uncompensated network models are presented in the form of a set of nonlinear space-time matrix equations in which network interconnection matrices are manifested. Each equation in the set characterizes the diffusion of phase (time) and frequency generated at each network node. The matrices of this model define the topological structure of the network. The time-frequency model is used to characterize the timefrequency stability of the network clock ensemble. Expressions for the steady-state network frequency and the time differences between nodal clocks are derived and compared for plesiochronous, delay compensated and uncompensated networks. A network frequency stability measure is introduced, evaluated and performance comparisons are made for specific network configurations. Finally, we illustrate the steady-state frequency stability achievable whenMsubnetworks are connected to form a larger network.

This paper investigates a new method of compensating for the transmission delays encountered in communications networks employing mutual synchronization. The compensation technique suggested belongs to the category of the single-ended methods in the sense that all the phase measurements affecting the control signal supplied to the local oscillator are carried out at that particular node; however, each oscillator does get signals which reflect, to some extent, the phase relationships between the signals received and the local oscillator at all other nodes in the network. The steady-state frequency of theNnodal network is found for general phase detector characteristics in order to demonstrate the profitable properties of the returnable timing system. The effects of mismatched phase detectors and imperfect delay compensation on the network steady-state frequency is given.

Mutual synchronization between two geographically separated phase-locked oscillators is investigated for two different interconnection methods. This includes the so-called single-ended (uncompensated delay) and the double-ended (compensated delay) methods. Explicit formulas for determining the steady-state frequency and phase errors are developed for both methods. Necessary conditions for the existance of a synchronous state are derived in terms of basic system parameters. Finally, a comparison Of the methods of oscillator interconnection is made as a function of the uncompensated channel delays. Graphical results are presented which are useful in determining steadystate system performance.