Computation of the exact cramer-rao lower bound for 2-D ARMA parameter Estimation-I: the quarter-plane case

A closed-form expression for computing the exact Cramer-Rao lower bound (CRLB) on unbiased estimates of the parameters of a two-dimensional (2-D) autoregressive moving average (ARMA) model is developed. The formulation is based on a matrix representation of 2-D homogeneous Gaussian random process that is generated uniformly from the related 2-D ARMA model. The formulas derived for the exact Fisher information matrix (FIM) are an explicit function of the 2-D ARMA parameters and are valid for real-valued homogeneous quarter-plane (QP) 2-D ARMA random fields, where data are propagated using only the past values. It is noteworthy that our approach is practical especially for quantifying the accuracy of 2-D ARMA parameter estimates realized with short data records. Computer simulations display the behavior of the derived CRLB expression for some QP causal 2-D ARMA processes, as a function of the number of data points. The extension of this algorithm for the nonsymmetric half-plane (NSHP) case will be presented in a subsequent paper