The Determination of Frequency in Systems of Mutually Synchronized Oscillators

The synchronization of large systems of geographically separated oscillators is of considerable practical interest for pulse code modulation (PCM) switching. Synchronization could be achieved by establishing a single master oscillator, with every other oscillator slaved either directly to the master or to another oscillator that is slaved directly or indirectly to the master. However, the system would then be vulnerable to failure of a single link or a single oscillator. An alternative called "mutual synchronization" would permit the oscillators to determine the system frequency jointly and to exchange synchronization information over redundant paths. However, the complexity of the system raises questions concerning the factors that determine the system frequency as well as system stability and dynamic response. A broad sufficient condition for the stability of mutually synchronized systems was first established by Bcnes.1 This condition has recently been rederived by a different method, for a slightly more general system, by Gersho and Karafin.2 The model used in both these studies was oversimplified so that it gave a paradoxical result for the system frequency at equilibrium. A model that corrected this oversimplification, by considering the received signal phases observed at each oscillator station at the initial moment when all oscillator controls are put into operation, was first devised by Runyon.3 A corrected model based on the same 1737