Wave Analogs of Media Based on Phase Locked Loops

Abstract

Introduction. At present, phase locked loops (PLLs) are widely used: from optimal signal detection and frequency synthesis to automatic control of phase distribution in phased scanned arrays. One of the simplest structures is a multi-stage (chain) PLL, which may contain a specially selected multi-connected control circuit. Such cascaded PLLs have wide application in solving a number of tasks of the theory of optimal estimates, multi-position phase telegraphy, in synchronizing of many tunable generators while preserving specified phase relations between their oscillations, etc. PLLs are actively used in radio physics both in analog and digital versions. One of the promising directions for collective PLLs development is the study of ensembles of neuromorphic networks based on PLL. Aim. To obtain wave analogues characterizing the collective PLL not as a discrete network, but as a continuous (distributed) media. Materials and methods. An unidirectional model (without mutual control circuits) of the cascade structure of the PLL. Results. Wave analogues of cascade-coupled phase synchronization systems that do not contain mutual control circuits were found. A solution of equations of wave analogues was found. A proof of validity of the obtained approximate solution in comparison with the exact one was presented. Conclusion. It was shown that by selecting a filter in a control circuit of each single-circuit circuit with different transmission coefficients, it is possible to obtain various types of continuous media or wave analogues of chain structures based on phase synchronization systems.