Posterior matching scheme for Gaussian multiple access channel with feedback

Posterior matching is a method proposed by Ofer Shayevitz and Meir Feder to design capacity achieving coding schemes for general point-to-point memoryless channels with feedback. In this paper, we present a way to extend posterior matching based encoding and variable rate decoding ideas for Gaussian MAC with feedback, referred to as time-varying posterior matching scheme, analyze the achievable rate region and error probabilities of the extended encoding-decoding scheme. The time-varying posterior matching scheme is a generalization of the Shayevitz and Feder´s posterior matching scheme when the posterior distributions of the input messages given output are not fixed over transmission time slots. It turns out that our designed posterior matching obtains the linear-feedback sum-capacity for the symmetric multiuser Gaussian MAC. Besides, the encoding scheme in this paper is designed for the real Gaussian MAC to obtain that performance, which is different from previous approaches where encoding schemes are designed for the complex Gaussian MAC. More importantly, this paper shows potential of posterior matching in designing optimal coding schemes for multiuser channels with feedback.