Rate regions of a two-way Gaussian relay channel

We study two-way communications over a three-node Gaussian relay channel. Several practical relaying strategies are considered, including bitwise XOR-based decode-and-forward (DF), superposition-based DF, and amplify-and-forward (AF). For each considered two-way relay scheme, we show that its achievable rate pair is governed by a set of linear inequalities. As a result, we obtain explicit, analytical expressions as well as geometrical shapes of the rate regions for each scheme. Our analytical results provide a convenient and efficient way to evaluate the performance of two-way relaying at any channel conditions. Numerical examples are also presented to demonstrate and compare the rate regions and maximum sum-rates of the considered schemes at different relay locations.