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In this paper, we investigate the stationarity of stochastic processes in thefractional Fourier domains. We study the stationarity of a stochastic processafter performing fractional Fourier transform (FRFT), and discrete fractionalFourier transform (DFRT) on both continuous and discrete stochastic processes,respectively. Also we investigate the stationarity of the fractional Fourierseries (FRFS) coefficients of a continuous time stochastic process, and thestationarity of the discrete time fractional Fourier transform (DTFRFT) of adiscrete time stochastic process. Closed formulas of the input processautocorrelation function and pseudo-autocorrelation function after performingthe fractional Fourier transform are derived given that the input is astationary stochastic process. We derive a formula for the outputautocorrelation as a function of the $a^{th}$ power spectral density of theinput stochastic process, also we derived a formula for the input fractionalpower spectral density as a function of the fractional Fourier transform of theoutput process autocorrelation function. We proved that, the input stochasticprocess must be zero mean to satisfy a necessary but not a sufficient conditionof stationarity in the fractional domains. Closed formulas of the resultantstatistics are also shown. It is shown that, in case of real input process, theoutput process is stationary if and only if the input process is white. On theother hand, if the input process is a complex process, it should be properwhite process to obtain a stationary output process.

Stationarity of Stochastic Processes In The Fractional Fourier Domains