Necessary and sufficient conditions on the parameters of 1-D Gaussian Markov processes

The valid parameter space of infinite- and finite-length (1-D noncausal) Gaussian Markov (GM) processes is investigated. For the infinite-length process, the authors show that the valid parameter space admits a simple description. The procedure the authors present to obtain such a description is readily implementable for any order process. The authors apply this procedure to a nontrivial special class, namely, the second-order 1-D GM processes. The authors obtain a simple and complete description of the valid parameter space for this class, which they observe to be considerably larger than the one implied by the previously known sufficient condition. For the finite-length process, it is shown that the valid parameter space does not admit a simple description. The set of conditions for the infinite-length process, however, serves as a good set of sufficient conditions for the finite-length case. The results readily apply to 2-D Gaussian Markov random fields (GMRFs) with a separable autocorrelation