Analysis of identified 2-D noncausal models

There are two approaches to the identification of noncausal autoregressive systems in two dimensions differing in the assumed noise model. For both approaches, the maximum likelihood estimator formulated in the frequency domain is presented. The Fisher information matrix is evaluated and found to be the sum of a block-Toeplitz and a block-Hankel matrix. The variance of the parameters, however, cannot be used for comparison of the two approaches, so the variance in the frequency domain is evaluated, assuming that the true system in each case can be described by a model of that type, possibly high-order. In particular, the variance of the spectrum estimate is derived. If the number of parameters tends to infinity, it is shown that the two approaches give the same spectrum estimate variance. The question of which set of true spectra can be described by the respective approaches is discussed