Daughter Coloured Noises: The Legacy of Their Mother White Noises Drawn from Different Probability Distributions

White noise is fundamentally linked to many processes; it has a flat power spectral density and a delta-correlated autocorrelation. Operators acting on white noise can result in coloured noise, whether they operate in the time domain, like fractional calculus, or in the frequency domain, like spectral processing. We investigate whether any of the white noise properties remain in the coloured noises produced by the action of an operator. For a coloured noise, which drives a physical system, we provide evidence to pinpoint the mother process from which it came. We demonstrate the existence of two indices, that is, kurtosis and codifference, whose values can categorise coloured noises according to their mother process. Four different mother processes are used in this study: Gaussian, Laplace, Cauchy, and Uniform white noise distributions. The mother process determines the kurtosis value of the coloured noises that are produced. It maintains its value for Gaussian, never converges for Cauchy, and takes values for Laplace and Uniform that are within a range of its white noise value. In addition, the codifference function maintains its value for zero lag-time essentially constant around the value of the corresponding white noise.

Automated summary

White noise, with flat power spectral density and delta-correlated autocorrelation, can be transformed into colored noise through operator manipulation in either time or frequency domain. This study investigates whether any white noise properties remain in colored noises generated by operators and provides evidence to infer the mother process from which a colored noise originated. The study demonstrates kurtosis and codifference as two indices that categorize colored noises based on their mother processes, such as Gaussian, Laplace, Cauchy, and Uniform white noise distributions. The results show that different mother processes determine the kurtosis values of colored noises, while codifference function remains constant around the corresponding white noise value.