A Two-Way Network Beamforming Approach Based on Total Power Minimization With Symmetric Relay Beamforming Matrices

We study the total transmit power minimization problem for a two-way relay network under two constraints on the transceivers' received signal-to-noise-ratios. The network considered herein consists of multiple multi-antenna relay nodes and two single-antenna transceivers. Each relay transforms the vector of its received signals, by multiplying this vector with a complex beamforming matrix, thereby obtaining a new vector whose entries are transmitted over different antennas of that relay. Assuming the relay beamforming matrices and the transceivers' transmit powers as the design parameters, we first study the total power minimization problem under the assumption that the relay beamforming matrices are symmetric. Under such an assumption, we show that the total power minimization problem is amenable to a semi-closed-form solution, and thus, it can be solved efficiently. We then consider the case, where the relay beamforming matrices may not be symmetric and show that in this case, the total power minimization problem can be solved using a computationally prohibitive algorithm which involves a 2-D search over a grid in the space of the transceivers' transmit powers and semi-definite programming at each vertex of this grid. Our numerical results show that the symmetric assumption on the relay beamforming matrices incurs only insignificant loss, while this assumption allows us to significantly reduce the computational burden of solving the total power minimization problem.