Algebraic method for optimal beamforming in two-way relay systems with analog network coding

The problem of optimal beamforming for two-way relay (TWR) systems with perfect channel state information (CSI) that use analog network coding (ANC) is reduced to a pair of algebraic equations in two variables that can be solved very inexpensively using the conjugate gradient algorithm. The solution has greatly reduced complexity compared to previous optimal solutions via semidefinite programming (SDP). Together with a previously-designed linearized algorithm for beamforming in the robust (imperfect CSI) case, it provides a high-performance, low-complexity robust beamforming solution for 2-way relays.