The random Wigner distribution of Gaussian stochastic processes with covariance in S0(ℝ2d)

Abstract

The paper treats time‐frequency analysis of scalar‐valued zero mean Gaussian stochastic processes on ℝd. We prove that if the covariance function belongs to the Feichtinger algebra S0(ℝ2d) then: (i) the Wigner distribution and the ambiguity function of the process exist as finite variance stochastic Riemann integrals, each of which defines a stochastic process on ℝ2d, (ii) these stochastic processes on ℝ2d are Fourier transform pairs in a certain sense, and (iii) Cohen′s class, ie convolution of the Wigner process by a deterministic function Φ ∈ C(ℝ2d), gives a finite variance process, and if Φ ∈ S0(ℝ2d) then W∗Φ can be expressed multiplicatively in the Fourier domain.