Deviations from spectral Dirac comb due to semiperiodic pulses

TL;DR

The paper develops a stochastic shot-noise framework for semiperiodic oscillations, deriving how a Dirac comb in the PSD—expected for strictly periodic pulse trains—is modified when arrivals are renewal or jittered. Pulses of fixed shape with random amplitudes and durations are superposed at times governed by a stationary point process, with the PSD given by a pulse-spectrum factor modulated by arrival statistics; the analysis covers periodic, Poisson, and Gamma waiting times, and extends to degenerate duration distributions. A key finding is that even modest deviations from strict periodicity efficiently erase higher harmonics and leave mainly the pulse spectrum, though zero-frequency delta terms may persist depending on the waiting-time statistics. The framework is applied to turbulent bursting in 2D thermal convection, where the model reproduces both time-series features and spectral properties, and supports a normal-waiting-time approximation that remains effective across a wide range of distributions. Overall, the work provides a rigorous PSD-based method to infer waiting-time and pulse-shape characteristics in semiperiodic systems and highlights why Dirac combs are seldom observed in weakly nonlinear or chaotic dynamics.

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 and durations.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.

Figures (5)

• Figure 1: Power spectral density (left) and autocorrelation function (right) for a sum of Lorentzian pulses with periodic arrival times and Laplace distributed amplitudes with various asymmetry parameters $\lambda$ in the case $\mathopen{}\mathclose{\left< w \right> = 5 \tau_\text{d}$ and $A_\text{rms}=1$. The analytical expressions are given by the black and gray dashed lines, respectively. The filled circles indicate the peak of the numerical PSD, while the black crosses indicate the mass of the delta function. For the numerical realizations, $T = 10^5\tau_\text{d}$ and the sampling time is $10^{-2}}\tau_\text{d}$.
• Figure 2: Power spectral density (left) and autocorrelation function (right) for a sum of Lorentzian pulses with normal waiting times according to a renewal process with mean value $\mathopen{}\mathclose{\left< w \right> = 5 \tau_\text{d}$, and various values of $w_\text{rms}$. Left, the analytical expression in Eq. \ref{['eq:jitter-gauss']} is presented by the black dashed line. Right, the black dotted line denotes the pulse autocorrelation.
• Figure 3: The contribution to the power spectral density due to periodic arrivals with jitter for various values of $w_\text{rms}/\mathopen{}\mathclose{\left< w \right>$. The legend in (b) is valid for all plot panels. 'an.' and 'num.' refer respectively to the analytic result, Eq. \ref{['eq:renewal-gauss']}, and the numerical result described in the main text. Note the logarithmic frequency scale in panel (c).
• Figure 4: Time series and power spectral density of the kinetic energy of the fluctuating motions $\mathcal{E}$ in turbulent thermal convection with $\mu = \kappa = 1.6\times10^{-3}$ (left) and $\mu = \kappa = 1.0\times10^{-4}$ (right). The data is fitted with a superposition of pulses shown in orange and green.
• Figure 5: Estimated pulse properties of the kinetic energy of the fluctuating motions $\mathcal{E}$ turbulent thermal convection. Top: Average burst shape and fitted exponential pulse for $\mu = \kappa = 1.6\times10^{-3}$ (left) and $\mu = \kappa = 1.0\times10^{-4}$ (right). Bottom: Amplitude and waiting time distributions of the pulses using Gaussian kernel density estimation.