Two-way lossy communication of correlated Gaussian sources with amplify-and-forward relaying

We consider the problem of two-way lossy communication in an amplify-and-forward (AF) relay channel. Particularly, we study the problem in the case of communication between users with correlated Gaussian sources. First, we prove necessary conditions for achievability of a distortion tuple under a two-phase communication protocol. Then we study achievable schemes thorough separate source-channel coding and uncoded transmission. We derive the achievable distortion region and show that the separation scheme is optimal. In contrast to point-to-point communication where both schemes are optimal, here we further show that uncoded transmission fails to be optimal for correlated sources. However, for the special case of independent sources, uncoded transmission is also optimal. Eventually, we establish similar results for achievability of a distortion tuple in a three-phase communication protocol.