Amplify-and-Forward Strategies in the Two-Way Relay Channel With Analog Tx–Rx Beamforming

In this paper, we study the multiple-input-multiple-output two-way relay channel (MIMO-TWRC) when the nodes and relay use analog beamforming. Following the amplify-and-forward (AF) strategy, the problem consists of finding the transmit and receive beamformers of the nodes and relay (as well as the power allocated to each one) that achieve the boundary of the optimal rate region. To solve it, we first express the optimal node beamformers in terms of relay beamformers and then show that the optimal rate region can efficiently be characterized using convex optimization techniques. We also extend our study to the multiple-relay scenario when the source nodes are single antenna and propose a distributed algorithm to compute the relay beamforming matrices. The proposed algorithm consists of solving two different subproblems. First, each individual TWRC is optimized independently. Next, a distributed beamforming is applied to make the signals from the relays add up coherently at the source nodes. Numerical examples are provided to illustrate the performance of the proposed techniques and to compare the performance of analog beamforming architectures against conventional MIMO schemes that operate at the baseband.