An SNR balancing approach to two-way relaying

We consider a relay network which consists of two transceivers and r relay nodes. Assuming that the transceivers and the relays are all equipped with single antennas, we devise a two-way amplify-and-phase-adjust relaying scheme. In this scheme, each relay multiplies its received signal by a complex weight and transmits the so-obtained signal thereby participating in a distributed beamforming process. We deploy an SNR balancing technique where the smallest 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 a unique solution which can be obtained through an iterative procedure with a linear computational complexity per iteration. We also prove that for any channel realization, 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. We further devise a distributed implementation of our proposed scheme which requires a minimal cooperation among the two transceivers and the relays. In fact, we show that our technique 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.