Modeling and identification of reciprocal processes

In this paper we discuss modeling and identification of stationary reciprocal processes defined on a finite interval [1 , N] of the discrete time line. These processes can be seen as a generalization of Markov random fields restricted to one dimension. Non stationary reciprocal processes have been extensively studied in the past especially by Krener, Levy and co-workers. However the specialization of the nonstationary results to the stationary case does not seem to have been pursued in sufficient depth in the literature. Processes (and stochastic models) of this kind are especially useful for describing phenomena which naturally live in a finite region of the ¿time¿ (or space) line, like say stationary images. Identification of these models starting from observed data seems to be a completely open problem which can in principle lead to many interesting applications in signal processing. In this paper we describe two approaches to identification of reciprocal processes; one is based on covariance matching (like in the method of moments) and the other is based on maximum likelihood. While the first leads to the solution of a linear system of the Yule-Walker type, the second approach leads to a circulant band extension problem which is intrisically nonlinear and does not seem to have been discussed in the literature.