A Semi-Closed-Form Solution to Optimal Distributed Beamforming for Two-Way Relay Networks

In this correspondence, we present a computationally simple semi-closed-form solution to the problem of designing distributed beamformer for two-way (bi-directional) multi-relay networks. In such a network, the relay nodes use amplify-and-forward relaying protocol to help two transceivers exchange information in a bidirectional manner. We consider a total power minimization approach to optimally find the relay beamforming weights and the transceiver transmit powers. This approach is based on the minimization of the total transmit power, consumed in the whole network, subject to SNR constraints at the two transceivers. We show that as far as the relay beamforming weight vector is concerned, this minimization problem is equivalent to the minimization of the total transmit power for a one-way relay network where the target SNR of the receiving transceiver is equal to the sum of the target SNRs of the two transceivers in the original two-way relay network. Based on this observation, we show that the relay beamforming weight vector can be obtained in a closed from given that an intermediate parameter, namely the transmit power of the transmitter in the equivalent one-way relay network, is available. This intermediate parameter is shown to be the solution to a one-dimensional optimization problem, and thus, it can be obtained using a simple bisection method. Our semi-closed-form solution not only reveals the structure of the optimal beamforming weight vector, but also leads to a one-dimensional search regardless of the number of relays. This provides the computational advantage over the gradient based numerical method of Havary-Nassab

, where the gradient dimension reflects the number of relays.