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Fractional Fourier Analysis of Random Signals and the Notion of /spl alpha/-Stationarity of the Wigner - Ville Distribution

In this paper, a generalized notion of wide-sense α-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 α -stationary random signals, the fractional correlation and the fractional power spectral density functions form a fractional Fourier transform pair. Thus, the concept of α -stationarity plays an important role in the analysis of random signals through the fractional Fourier transform for signals nonstationary in the standard formulation, but α -stationary. Furthermore, we define the α-Wigner-Ville distribution in terms of the fractional correlation function, in which the standard Fourier analysis is the particular case for α = π/2 , and it leads to the Wiener-Khinchin theorem.