Achieving the stationary feedback capacity for Gaussian channels

In this paper, we study a Gaussian channel with memory and with noiseless feedback, for which we present a coding scheme to achieve the stationary feedback capacity (the maximum information rate over all stationary input distributions, conjectured to be the asymptotic feedback capacity). The coding scheme essentially implements the celebrated Kalman filter algorithm; is equivalent to an estimation system over the same channel without feedback; and reveals that the achievable information rate of the feedback communication system can be alternatively given by the decay rate of the Cramer-Rao bound of the associated estimation system. Thus, combined with the control theoretic characterizations of feedback communication (proposed by Elia), this implies that the fundamental limitations in feedback communication, estimation, and control coincide. In addition, the proposed coding scheme simplifies the coding complexity and shortens the coding delay, and its construction amounts to solving a finite-dimensional optimization problem. We also provide a further simplification to the optimal input distribution developed by Yang, Kavcic, and Tatikonda.