On the Capacity Region of the Gaussian Multiple Access Channel with Noisy Feedback

We provide a new outer bound on the capacity region of the two-user Gaussian multiple access channel (MAC) with AWGN-corrupted feedback. Our outer bound is based on the idea of dependence balance due to Hekstra and Willems. Evaluating our outer bound is non-trivial as it involves taking a union over joint densities of three random variables, one of which is an auxiliary random variable. We resolve this difficulty by proving that it is sufficient to consider jointly Gaussian random variables when evaluating our outer bound. As the feedback noise variances become large, our outer bound collapses to the capacity region of the Gaussian MAC without feedback, thereby yielding the first non-trivial result for a Gaussian MAC with noisy feedback. Furthermore, as the feedback noise variances tend to zero, our outer bound collapses to the capacity region of the Gaussian MAC with noiseless feedback, which was established by Ozarow. For all non-zero, finite values of the feedback noise variances, our outer bound strictly improves upon the cutset outer bound.