Cyclic fractional Gaussian noise: time and frequency domain properties

TL;DR

This paper introduces cyclic fractional Gaussian noise (cfGn), a new stochastic model combining cyclostationarity and long-range dependence, with detailed theoretical analysis and potential applications in signal monitoring.

Contribution

It extends amplitude-modulated stationary models using 2d fractional Brownian motion increments to capture both periodicity and long-memory effects.

Findings

cfGn maintains periodic autocovariance while exhibiting long-memory traits.

The model's properties are rigorously derived, including autocovariance and cyclic spectrum.

Monte Carlo simulations confirm theoretical asymptotic behaviors.

Abstract

This article introduces cyclic fractional Gaussian noise (cfGn), a stochastic model that integrates second-order cyclostationarity with long-range dependence property. While classical cyclostationary processes are widely discussed in the literature, they often lack the capacity to account for the persistent, slow-decaying correlations found in complex empirical data. To bridge this gap, we extend the amplitude-modulated stationary framework by utilizing increments of two-dimensional fractional Brownian motion (2d fBm) as the underlying driving process. The proposed cfGn model is constructed by summing two components, which include periodic deterministic functions modulating the univariate coordinates of 2d fGn. We provide a rigorous derivation of the considered model's properties, specifically the autocovariance function (ACVF) and frequency-domain characteristics, including the cyclic spectrum. Through theoretical considerations of asymptotic properties and Monte Carlo simulations, we demonstrate that cfGn preserves periodic behavior of ACVF while inheriting long-memory traits which is manifested in time and frequency domains. This framework offers a robust foundation, for instance, in signal-based condition monitoring in systems where periodic fault components coexist with long-range dependent background noise.