Deviations from spectral Dirac comb due to semiperiodic pulses

TL;DR

This paper analyzes how deviations from perfect periodicity, such as jitter and renewal processes, affect the spectral Dirac comb in weakly nonlinear systems, providing new models and insights into spectral broadening.

Contribution

It introduces a stochastic model for semiperiodic fluctuations, deriving closed-form spectral density expressions and exploring the effects of jitter and renewal deviations.

Findings

Deviations from periodicity diminish the Dirac comb, leaving mainly the pulse spectrum.

Jitter modulates higher harmonic amplitudes, causing spectral modulation.

Renewal processes lead to spectral broadening, modeling real-world fluctuations.

Abstract

In the frequency power spectral density, periodic oscillations appear as a Dirac comb at integer multiples of the frequency of the period. In weakly nonlinear systems or systems close to the primary instability threshold, the periodicity may be perturbed, resulting in deviations from the Dirac comb. We review and discuss a stochastic model of such semiperiodic fluctuations, while also providing several new results which widen the applicability of the model. The fluctuations are described as a superposition of pulses with a fixed shape. Closed form expressions are derived for the frequency power spectral density in the case of periodic pulse arrivals and a random distribution of pulse amplitudes. In general, the spectrum is a Dirac comb located at multiples of the inverse periodicity time and modulated by the pulse spectrum. Deviations from strict periodicity in the arrivals are considered in two ways: either as a random offset to each periodic arrival (jitter) or as independently distributed waiting times between arrivals (renewal). In this contribution, we show that both ways of including deviations from periodicity remove the Dirac comb with remarkable efficiency, leaving mainly the spectrum of the pulse function. Where the jitter process modulates the mass of the higher harmonics, the renewal process leads to spectral broadening. We clarify the effects of random pulse amplitudes on the frequency power spectrum and demonstrate the applicability of normally distributed waiting times to modeling. Contrary to the previous literature, we argue that negative waiting times do not pose problems for the theory, broadening the applicability of the normal approximation. Randomness in the pulse arrival times is investigated by numerical realizations of the process, and the model is used to describe time series of kinetic energy of fluctuating motions in two-dimensional thermal convection.