On the sum capacity of the Gaussian multiple access channel with feedback

This paper studies the sum capacity C(P) of the N-sender additive white Gaussian noise (AWGN) multiple access channel (MAC), under equal power constraint P, when noiseless output feedback is available to all the N senders. The multi-letter characterization of the sum capacity, in terms of directed information, is considered as an optimization problem. The main result of this paper is to solve this problem when it is restricted to Gaussian causally conditional input distributions. Also, a dependence balance bound in terms of directed information is introduced, which for the case of memoryless channels is the same as the bound introduced by Kramer and Gastpar. This bound is used to capture the causality, however, since it is in general "non-convex" makes the problem technically hard. A general upper bound is obtained by forming the Lagrange dual problem and it is then shown that this upper bound coincides with the sum-rate achieved by Kramer's Fourier-MEC scheme. This result generalizes earlier work by Kramer and Gastpar on the achievable sum rate under a "per-symbol" power constraint to the one under the standard "block" power constraint.