The random Wigner distribution of Gaussian stochastic processes with covariance inS0(?2d)

摘要

The paper treats time-frequency analysis of scalar-valued zero mean Gaussian stochastic processes on?d. We prove that if the covariance function belongs to the Feichtinger algebraS0(?2d)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?2d, (ii) these stochastic processes on?2dare Fourier transform pairs in a certain sense, and (iii) Cohen's class, ie convolution of the Wigner process by a deterministic functionΦ∈C(?2d), gives a finite variance process, and ifΦ∈S0(?2d)thenW?Φcan be expressed multiplicatively in the Fourier domain.