A semi-closed-form solution to optimal decentralized beamforming for two-way relay networks

In this paper, we investigate the design of a distributed beamforming technique for two-way relay networks consisting of two transceivers and multiple relay nodes. We consider the problem of the total transmit power minimization subject to two constraints that maintain the transceivers´ SNRs above given thresholds ¿₁ and ¿₂. For this problem, we obtain a semi-closed-form solution for the beamforming weight vector. Our solution is based on the observation that beamforming weight vector depends on the sum of SNR thresholds ¿₁ + ¿₂. As long as this sum does not change, the beamforming weight vector will not change. Hence, a two-way relay vector can be equivalently modeled as a one-way relay network where ¿₁ and ¿₂ are replaced with 0 and ¿₁ + ¿₂, respectively. Based on this equivalent one-way relaying scheme, the beamforming weight vector is obtained in a closed form if a certain intermediate parameter is given. This parameter is the transmit power of the source in the equivalent one-way relaying scheme and can be obtained efficiently using a simple bisection method. Once this parameter is obtained, the beamforming weight vector can be obtained in a closed form.