title: Self-organized synchronization of digital phase-locked loops with delayed coupling in theory and experiment.
abstract: Self-organized synchronization occurs in a variety of natural and technical systems but has so far only attracted limited attention as an engineering principle. In distributed electronic systems, such as antenna arrays and multi-core processors, a common time reference is key to coordinate signal transmission and processing. Here we show how the self-organized synchronization of mutually coupled digital phase-locked loops (DPLLs) can provide robust clocking in large-scale systems. We develop a nonlinear phase description of individual and coupled DPLLs that takes into account filter impulse responses and delayed signal transmission. Our phase model permits analytical expressions for the collective frequencies of synchronized states, the analysis of stability properties and the time scale of synchronization. In particular, we find that signal filtering introduces stability transitions that are not found in systems without filtering. To test our theoretical predictions, we designed and carried out experiments using networks of off-the-shelf DPLL integrated circuitry. We show that the phase model can quantitatively predict the existence, frequency, and stability of synchronized states. Our results demonstrate that mutually delay-coupled DPLLs can provide robust and self-organized synchronous clocking in electronic systems.
authors.affiliation: Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Str. 38, 01187 Dresden, Germany.|Center for Advancing Electronics Dresden, cfaed, 01062 Dresden, Germany.
authors.forename: Lucas
authors.initials: L
authors.lastname: Wetzel
authors.affiliation: Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Str. 38, 01187 Dresden, Germany.|Center for Advancing Electronics Dresden, cfaed, 01062 Dresden, Germany.
authors.forename: David J
authors.initials: DJ
authors.lastname: Jörg
authors.affiliation: Center for Advancing Electronics Dresden, cfaed, 01062 Dresden, Germany.|Vodafone Chair - Mobile Communications Systems, TU Dresden, 01062 Dresden, Germany.
authors.forename: Alexandros
authors.initials: A
authors.lastname: Pollakis
authors.affiliation: Center for Advancing Electronics Dresden, cfaed, 01062 Dresden, Germany.|Vodafone Chair - Mobile Communications Systems, TU Dresden, 01062 Dresden, Germany.
authors.forename: Wolfgang
authors.initials: W
authors.lastname: Rave
authors.affiliation: Center for Advancing Electronics Dresden, cfaed, 01062 Dresden, Germany.|Vodafone Chair - Mobile Communications Systems, TU Dresden, 01062 Dresden, Germany.
authors.forename: Gerhard
authors.initials: G
authors.lastname: Fettweis
authors.affiliation: Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Str. 38, 01187 Dresden, Germany.|Center for Advancing Electronics Dresden, cfaed, 01062 Dresden, Germany.
authors.forename: Frank
authors.initials: F
authors.lastname: Jülicher