Stationarity of Stochastic Processes In The Fractional Fourier Domains

TL;DR

This paper analyzes how stochastic processes behave in fractional Fourier domains, deriving formulas for autocorrelation and spectral density, and establishing conditions for stationarity in these transformed domains.

Contribution

It provides new formulas linking input and output autocorrelation functions and establishes conditions for stationarity of stochastic processes in fractional Fourier domains.

Findings

Output process is stationary iff input is white for real processes.

Proper white processes are needed for complex inputs to ensure stationarity.

Derived formulas relate autocorrelation and spectral density in fractional Fourier transforms.

Abstract

In this paper, we investigate the stationarity of stochastic processes in the fractional Fourier domains. We study the stationarity of a stochastic process after performing fractional Fourier transform (FRFT), and discrete fractional Fourier transform (DFRT) on both continuous and discrete stochastic processes, respectively. Also we investigate the stationarity of the fractional Fourier series (FRFS) coefficients of a continuous time stochastic process, and the stationarity of the discrete time fractional Fourier transform (DTFRFT) of a discrete time stochastic process. Closed formulas of the input process autocorrelation function and pseudo-autocorrelation function after performing the fractional Fourier transform are derived given that the input is a stationary stochastic process. We derive a formula for the output autocorrelation as a function of the $a^{th}$ power spectral density of the input stochastic process, also we derived a formula for the input fractional power spectral density as a function of the fractional Fourier transform of the output process autocorrelation function. We proved that, the input stochastic process must be zero mean to satisfy a necessary but not a sufficient condition of stationarity in the fractional domains. Closed formulas of the resultant statistics are also shown. It is shown that, in case of real input process, the output process is stationary if and only if the input process is white. On the other hand, if the input process is a complex process, it should be proper white process to obtain a stationary output process.