Cramér-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 complex-valued 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.