  This paper deals with M$^2$-signals, namely multivariate (or vector-valued)
signals defined over a multidimensional domain. In particular, we propose an
optimization technique to solve the covariance extension problem for stationary
random vector fields. The multidimensional Itakura-Saito distance is employed
as an optimization criterion to select the solution among the spectra
satisfying a finite number of moment constraints. In order to avoid
technicalities that may happen on the boundary of the feasible set, we deal
with the discrete version of the problem where the multidimensional integrals
are approximated by Riemann sums. The spectrum solution is also discrete, which
occurs naturally when the underlying random field is periodic. We show that a
solution to the discrete problem exists, is unique and depends smoothly on the
problem data. Therefore, we have a well-posed problem whose solution can be
tuned in a smooth manner. Finally, we have applied our theory to the target
parameter estimation problem in an integrated system of automotive modules.
Simulation results show that our spectral estimator has promising performance.
