Mutual Synchronization of Geographically Separated Oscillators

Consider a network of N geographically separated stations that are connected by directed communication links. A local clock, or oscillator, is situated at each station. The problem of synchronizing the frequencies of the oscillators is of considerable practical interest for continental pulse code modulation (PCM) systems. The local oscillators have frequencies which may be altered in proportion to a control signal. In the absence of external control, each oscillator operates at a different frequency. The network is "connected" in the sense that from any station to any other station there is either a direct transmission link or an indirect path via one or more intermediate stations. A fixed time delay is associated with each transmission link. In an important but unpublished paper, V. E. Benes 1 has examined a linear control scheme in which each station receives the phases of 1689 1694 T H E B E L L SYSTEM TECHNICAL JOURNAL, D E C E M B E R 15(66 neighboring stations, i.e., those stations connected to it by direct transmission links. The phases are averaged and compared with the local phase; the error is filtered and applied as a correction to the frequency of the oscillator. Similar schemes were also proposed by Runyon. 2 Benes has proved that under suitable conditions the system is stable, i.e., the oscillators asymptotically settle to a common frequency and the phase differences have finite asymptotic values. He also finds explicit formulae for the final frequency and asymptotic phase differences.