# Auto-correlation function and frequency spectrum due to a super-position of uncorrelated exponential pulses

## TL;DR

This paper analyzes the auto-correlation and frequency spectrum of a super-position of uncorrelated exponential pulses, revealing their independence from pulse overlap and intermittency, with specific spectral shapes depending on pulse symmetry and shape.

## Contribution

It provides a detailed theoretical analysis of spectral properties of superimposed exponential pulses, including effects of pulse shape, duration distribution, and noise, aligning with experimental plasma data.

## Key findings

- Power spectral density is Lorentzian for constant pulse duration and shape.
- Symmetric pulses produce a squared Lorentzian spectrum.
- Random pulse durations do not affect the high-frequency power law tail.

## Abstract

The auto-correlation function and the frequency power spectral density due to a super-position of uncorrelated exponential pulses are considered. These are shown to be independent of the degree of pulse overlap and thereby the intermittency of the stochastic process. For constant pulse duration and a one-sided exponential pulse shape, the power spectral density has a Lorentzian shape which is flat for low frequencies and a power law at high frequencies. The algebraic tail is demonstrated to result from the discontinuity in the pulse function. For a strongly asymmetric two-sided exponential pulse shape, the frequency spectrum is a broken power law with two scaling regions. In the case of a symmetric pulse shape, the power spectral density is the square of a Lorentzian function. The steep algebraic tail at high frequencies in these cases is demonstrated to follow from the discontinuity in the derivative of the pulse function. A random distribution of pulse durations is shown to result in apparently longer correlation times but has no influence on the asymptotic power law tail of the frequency spectrum. The effect of additional random noise is also discussed, leading to a flat spectrum for high frequencies. The probability density function for the fluctuations is shown to be independent of the distribution of pulse durations. The predictions of this model describe the variety of auto-correlation functions and power spectral densities reported from experimental measurements in the scrape-off layer of magnetically confined plasmas.