Cramer-Rao bound on the estimation accuracy of complex-valued homogeneous Gaussian random fields

This paper considers the problem of the achievable accuracy in jointly estimating the parameters of a complex-valued two-dimensional (2-D) Gaussian and homogeneous random field from a single observed realization of it. Based on the 2-D Wold decomposition, the field is modeled as a sum of purely indeterministic, evanescent, and harmonic components. Using this parametric model, we first solve a key problem common to many open problems in parametric estimation of homogeneous random fields: that of expressing the field mean and covariance functions in terms of the model parameters. Employing the parametric representation of the observed field mean and covariance, we derive a closed-form expression for the Fisher information matrix (FIM) of complexvalued homogeneous Gaussian random fields with mixed spectral distribution. Consequently, the Cramér–Rao lower bound on the error variance in jointly estimating the model parameters is evaluated.