  Stationary reciprocal processes defined on a finite interval of the integer
line can be seen as a special class of Markov random fields restricted to one
dimension. Non stationary reciprocal processes have been extensively studied in
the past especially by Jamison, Krener, Levy and co-workers. The specialization
of the non-stationary theory to the stationary case, however, does not seem to
have been pursued in sufficient depth in the literature. Stationary reciprocal
processes (and reciprocal stochastic models) are potentially useful for
describing signals which naturally live in a finite region of the time (or
space) line. Estimation or identification of these models starting from
observed data seems still to be an open problem which can lead to many
interesting applications in signal and image processing. In this paper, we
discuss a class of reciprocal processes which is the acausal analog of
auto-regressive (AR) processes, familiar in control and signal processing. We
show that maximum likelihood identification of these processes leads to a
covariance extension problem for block-circulant covariance matrices. This
generalizes the famous covariance band extension problem for stationary
processes on the integer line. As in the usual stationary setting on the
integer line, the covariance extension problem turns out to be a basic
conceptual and practical step in solving the identification problem. We show
that the maximum entropy principle leads to a complete solution of the problem.
