Symmetric beamforming for multi-antenna two-way relay networks

This paper addresses the problem of total power minimization of a two-way relay network, where two single-antenna nodes exchange information through multiple multi-antenna relays with symmetric beamforming matrices. More specifically, each relay multiplies the vector of its received signals with a symmetric beamforming matrix to obtain the vector of its transmit signals and broadcasts the elements of the so-obtained transmit signal vector on its different antennas. Considering the two-time-slot multiple access broadcast (MABC) scheme, we aim to minimize the total transmit power subject to signal-to-noise- ratio (SNR) requirements by optimally determining the transceivers' transmit powers and the relay beamforming matrices. We prove that this power minimization problem has a semi-closed-form solution. That is, the symmetric beamforming matrices can be obtained in closed-forms given a certain intermediate parameter. This parameter can also be obtained using Newton-Raphson method or a bisection technique. Our simulation results show that the average total power consumed in the network is twice the average total relay power, which is in average twice the average power of each of the transceivers. Our numerical examples also show that concentrating all the antennas in a few relays reduces the total power consumption as apposed to having a large number of relays with very few antennas.