In this paper, we consider a relay network which consists of two single-antenna transceivers and n <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sub> single-antenna relay nodes. Considering a two time slot two-way relaying scheme, each relay adjusts the phase and the amplitude of the mixture signal it receives from the two transceivers during the first time slot, by multiplying it with a complex beamforming coefficient. Then each relay transmits the so-obtained signal in the second time slot. Aiming at optimally calculating the beamforming coefficients as well as the transceiver transmit powers, we study two different approaches. In the first approach, we minimize the total transmit power (dissipated in the whole network) subject to two constraints on the transceivers' received signal-to-noise ratios (SNRs). We prove that such a power minimization technique has a unique solution. We also show that the optimal weight vector can be obtained through a simple iterative algorithm which enjoys a linear computational complexity per iteration. We also prove that for symmetric relaying schemes (where the two constraints on the transceiver SNRs are the same), half of the minimum total transmit power will be allocated to the two transceivers and the remaining half will be shared among the relaying nodes. In the second approach, we will study an SNR balancing technique. In this technique, the smaller of the two transceiver SNRs is maximized while the total transmit power is kept below a certain power budget. We show that this problem has also a unique solution which can be obtained through an iterative procedure with a linear computational complexity per iteration. We also prove that this approach leads to a power allocation scheme, where half of the maximum power budget is allocated to the two transceivers and the remaining half will be shared among all the relay nodes. For both approaches, we devise distributed schemes which require a minimal cooperation among the two transceivers and the relays. In fact, we show that both techniques can be implemented such that the bandwidth, required to obtain the beamforming weights in a distributed manner, remains constant as the size of the network grows.

Optimal Distributed Beamforming for Two-Way Relay Networks