Optimal network beamforming for bi-directional relay networks

We consider a relay network which consists of two transceivers and r relay nodes. We study a half-duplex two-way relaying scheme. First, the two transceivers transmit their information symbols simultaneously and the relays receive a noisy mixture of the two transceiver signals. Then each relay adjusts the phase and the amplitude of its received signal by multiplying it with a complex beamforming coefficient and transmits the so-obtained signal. Aiming at optimally calculating the beamforming weight vector as well as the transceiver transmit powers, we minimize the total transmit power subject to two constraints on the receive signal-to-noise ratios (SNRs) at the two transceivers. We show that the optimal weight vector can be obtained through a simple iterative algorithm which enjoys a linear computational complexity per iteration.