  In the present paper we consider the problem of estimating the
multidimensional power spectral density which describes a second-order
stationary random field from a finite number of covariance and generalized
cepstral coefficients. The latter can be framed as an optimization problem
subject to multidimensional moment constraints, i.e., to search a spectral
density maximizing an entropic index and matching the moments. In connection
with systems and control, such a problem can also be posed as finding a
multidimensional shaping filter (i.e., a linear time-invariant system) which
can output a random field that has identical moments with the given data when
fed with a white noise, a fundamental problem in system identification. In
particular, we consider the case where the dimension of the random field is
greater than two for which a satisfying theory is still missing. We propose a
multidimensional moment problem which takes into account a generalized
definition of the cepstral moments, together with a consistent definition of
the entropy. We show that it is always possible to find a rational power
spectral density matching exactly the covariances and approximately the
generalized cepstral coefficients, from which a shaping filter can be
constructed via spectral factorization. In plain words, our theory allows to
construct a well-posed spectral estimator for any finite dimension.
