A Novel Exact Representation of Stationary Colored Gaussian Processes (Fractional Differential Approach)

TL;DR

This paper introduces a new exact method for representing stationary colored Gaussian processes using fractional differential equations, linking spectral density to fractional moments and enabling precise modeling of colored noise.

Contribution

It presents a novel generalized Taylor form for Gaussian processes, providing an exact fractional differential equation representation of colored noise.

Findings

Colored Gaussian noise can be exactly expressed via fractional stochastic differential equations.

The weighting coefficients are explicitly derived and related to fractional moments of the spectral density.

The method offers a precise and analytical approach to modeling stationary colored Gaussian processes.

Abstract

A novel representation of functions, called generalized Taylor form, is applied to the filtering of white noise processes. It is shown that every Gaussian colored noise can be expressed as the output of a set of linear fractional stochastic differential equation whose solution is a weighted sum of fractional Brownian motions. The exact form of the weighting coefficients is given and it is shown that it is related to the fractional moments of the target spectral density of the colored noise.