Cycle slipping in nonlinear circuits under periodic nonlinearities and time delays

TL;DR

This paper provides analytical estimates for cycle slipping in nonlinear PLL-based circuits with delays, capturing complex dynamics and the effects of small perturbations, which are crucial for reliable telecommunications.

Contribution

It introduces a novel analytical approach to estimate cycle slipping in systems with delays and high-order dynamics, extending prior stochastic-focused results.

Findings

Derived bounds for the number of cycle slips in delayed systems

Analyzed the impact of small parameter perturbations on cycle slipping

Captured effects of high-order dynamics and delays on phase error behavior

Abstract

Phase-locked loops (PLL), Costas loops and other synchronizing circuits are featured by the presence of a nonlinear phase detector, described by a periodic nonlinearity. In general, nonlinearities can cause complex behavior of the system such multi-stability and chaos. However, even phase locking may be guaranteed under any initial conditions, the transient behavior of the circuit can be unsatisfactory due to the cycle slipping. Growth of the phase error caused by cycle slipping is undesirable, leading e.g. to demodulation and decoding errors. This makes the problem of estimating the phase error oscillations and number of slipped cycles in nonlinear PLL-based circuits extremely important for modern telecommunications. Most mathematical results in this direction, available in the literature, examine the probability density of the phase error and expected number of slipped cycles under stochastic noise in the signal. At the same time, cycle slipping occurs also in deterministic systems with periodic nonlinearities, depending on the initial conditions, properties of the linear part and the periodic nonlinearity and other factors such as delays in the loop. In the present paper we give analytic estimates for the number of slipped cycles in PLL-based systems, governed by integro-differential equations, allowing to capture effects of high-order dynamics, discrete and distributed delays. We also consider the effects of singular small parameter perturbations on the cycle slipping behavior.