Space and spectral domain relative entropy for homogeneous random fields

A classical result in spectral estimation establishes that the relative entropy rate between two zero-mean stationary Gaussian processes can be computed explicitly in terms of their spectral densities, hence inducing a pseudo-distance in the cone of positive definite spectra. Relying on the properties of multi-level circulant and multi-level Toeplitz matrices, we establish a similar theory for homogeneous Gaussian random fields, both periodic and non-periodic. As a consequence we derive a natural entropic pseudo-distance on the cone of positive definite multi-dimensional spectra. We also define the spectral relative entropy rate as an entropic divergence index between the spectral measures associated to two homogeneous Gaussian random fields and we show that the relative entropy rate and the spectral relative entropy rate are in fact equal.