# Coloured Noise Analysis of a Phase-Locked Loop System: Beyond Itô and Stratonovich Stochastic Calculi

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## Abstract

The phase-locked loop (PLL) system is an appealing electronic circuit that synchronizes oscillator output signal with the reference signal. In theoretical studies, it is revealed that the PLL system fails to achieve the locking condition in the presence of small noise influences. For this reason, it is reasonable to accomplish noise analysis of the PLL system. In this paper, we accomplish the PLL noise analysis in coloured noise framework that is more general as well as confirms the real noise statistics in contrast to the white noise. The extended phase space and stochastic differential rules for the random state vector satisfying vector stochastic differential equation are exploited to accomplish the PLL coloured noise analysis. It is worthwhile to mention that the three concepts, non-linearity, stochasticity, multi-dimensionality, received attentions in applied mathematics literature. However, applications of these three mathematical concepts to engineering problems are relatively very scarce. More precisely, this paper accounts for non-linearity, non-Markovian stochasticity as well as multi-dimensionality in the noise analysis of the PLL dynamics. Since the non-linearity, multi-dimensionality are ubiquitous in real physical systems as well as stochasticity will set future research directions in applied mathematics, systems and control, this paper will have lasting influence on future stochastic problems.

## Keywords

Extended state space Non-linearity Non-Markovian stochasticity Ornstein–Uhlenbeck (OU) process Second-order PLL systems Stochastic differential equations (SDEs) Stochastic differential rules## Notes

### Acknowledgments

The Authors express gratefulness to anonymous Reviewers for their finer comments on the manuscript. That helped improve the content of the present version of the paper.

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