  In this paper we study the problem of characterizing and computing the
nonanticipative rate distortion function (NRDF) for partially observable
multivariate Gauss-Markov processes with hard mean squared error (MSE)
distortion constraints. For the finite time horizon case, we first derive the
complete characterization of this problem and its corresponding optimal
realization which is shown to be a linear functional of the current time
sufficient statistic of the past and current observations signals. We show that
when the problem is strictly feasible, it can be computed via semidefinite
programming (SDP) algorithm. For time-varying scalar processes with average
total MSE distortion we derive an optimal closed form expression by means of a
dynamic reverse-waterfilling solution that we also implement via an iterative
scheme that convergences linearly in finite time, and a closed-form solution
under pointwise MSE distortion constraint. For the infinite time horizon, we
give necessary and sufficient conditions to ensure that asymptotically the
sufficient statistic process of the observation signals achieves a steady-state
solution for the corresponding covariance matrices and impose conditions that
allow existence of a time-invariant solution. Then, we show that when a finite
solution exists in the asymptotic limit, it can be computed via SDP algorithm.
We also give strong structural properties on the characterization of the
problem in the asymptotic limit that allow for an optimal solution via a
reverse-waterfilling algorithm that we implement via an iterative scheme that
converges linearly under a finite number of spatial components. Our results are
corroborated with various simulation studies and are also compared with
existing results in the literature.
