Fractional Fourier Analysis of Random Signals and the Notion of α-Stationarity of the Wigner-Ville Distribution

In this paper, a generalized notion of wide-sense alpha-stationarity for random signals is presented. The notion of stationarity is fundamental in the Fourier analysis of random signals. For this purpose, a definition of the fractional correlation between two random variables is introduced. It is shown that for wide-sense alpha-stationary random signals, the fractional correlation and the fractional power spectral density functions form a fractional Fourier transform pair. Thus, the concept of alpha-stationarity plays an important role in the analysis of random signals through the fractional Fourier transform for signals nonstationary in the standard formulation, but alpha-stationary. Furthermore, we define the alpha-Wigner-Ville distribution in terms of the fractional correlation function, in which the standard Fourier analysis is the particular case for alpha = pi/2 and it leads to the Wiener-Khinchin theorem.