Statistical Analyses of Digital Phase-Locked Loops with Time Delay

We analyze the linear stability of the synchronized state in networks of N identical digital phase-locked loops. These are pulse-coupled oscillator arrays in which the frequency (rather than the phase) of each oscillator is updated discontinuously whenever that oscillator reaches a specific phase in its cycle. Three different coupling configurations are studied: one-way rings, two-way rings, and globally coupled arrays. In each case we obtain explicit formulas for the transient time to lock, the critical gain at which the synchronized state loses stability, and the period of the bifurcating solution at the onset of instability. Our results explain the numerical observations of de Sousa Vieira, Lichtenberg, and Lieberman.

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Enhanced Acquisition and Tracking in All Digital Phase-Locked Loops

Circuits Systems and Signal Processing ◽

10.1007/s00034-008-9042-y ◽

2008 ◽

Vol 27 (4) ◽

pp. 537-552 ◽

Cited By ~ 1

Author(s):

Reza Danesfahani ◽

Mohammad Moghaddasi ◽

Mahmood Mahlouji

Keyword(s):

Phase Locked Loops ◽

Digital Phase ◽

Digital Phase Locked Loops

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Advanced digital phase-locked loops

Proceedings of the IEEE 2013 Custom Integrated Circuits Conference ◽

10.1109/cicc.2013.6658505 ◽

2013 ◽

Cited By ~ 4

Author(s):

Salvatore Levantino

Keyword(s):

Phase Locked Loops ◽

Digital Phase ◽

Digital Phase Locked Loops

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NONLINEAR DYNAMICS OF DIGITAL PHASE-LOCKED LOOPS WITH DELAY

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127494000514 ◽

1994 ◽

Vol 04 (03) ◽

pp. 715-726 ◽

Cited By ~ 3

Author(s):

MARIA DE SOUSA VIEIRA ◽

ALLAN J. LICHTENBERG ◽

MICHAEL A. LIEBERMAN

Keyword(s):

Nonlinear Dynamics ◽

Chaotic Behavior ◽

Nonlinear Oscillators ◽

Coupled Systems ◽

Phase Locked Loops ◽

Bifurcation Diagrams ◽

Digital Phase ◽

Coupled Nonlinear Oscillators ◽

Digital Phase Locked Loops ◽

The Stability

We investigate numerically and analytically the nonlinear dynamics of a system consisting of two self-synchronizing pulse-coupled nonlinear oscillators with delay. The particular system considered consists of connected digital phase-locked loops. We find mapping equations that govern the system and determine the synchronization properties. We study the bifurcation diagrams, which show regions of periodic, quasiperiodic and chaotic behavior, with unusual bifurcation diagrams, depending on the delay. We show that depending on the parameter that is varied, the delay will have a synchronizing or desynchronizing effect on the locked state. The stability of the system is studied by determining the Liapunov exponents, indicating marked differences compared to coupled systems without delay.

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