Fractional Sampling Theorem for $\alpha$ -Bandlimited Random Signals and Its Relation to the von Neumann Ergodic Theorem

Considering that fractional correlation function and the fractional power spectral density, for α-stationary random signals, form a fractional Fourier transform pair. We present an interpolation formula to estimate a random signal from a temporal random series, based on the fractional sampling theorem for α-bandlimited random signals. Furthermore, by establishing the relationship between the sampling theorem and the von Neumann ergodic theorem, the estimation of the power spectral density of a random signal from one sample signal becomes a suitable approach. Thus, the validity of the sampling theorem for random signals is closely linked to an ergodic hypothesis in the mean sense.