Fractional Fourier analysis of random signals and the notion of α -Stationarity of the Wigner-Ville distribution

Abstract

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=pi2, and it leads to the Wiener-Khinchin theorem. © 1991-2012 IEEE.