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.

谱估计中的经典结果表明，可以根据它们的谱密度显式计算两个零均值平稳高斯过程之间的相对熵率，从而在正定谱锥中产生伪距离。依靠多级循环和多级Toeplitz矩阵的性质，我们为周期和非周期的齐次高斯随机场建立了相似的理论。结果，我们在正定多维光谱的圆锥上得出了自然的熵伪距。我们还定义了频谱相对熵率 作为与两个均匀高斯随机场相关的频谱度量之间的熵发散指数，我们表明相对熵率和频谱相对熵率实际上是相等的。