Stationarity of Stochastic Processes In The Fractional Fourier Domains

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.