Jointly optimal distributed beamforming and power control in asynchronous two-way relay networks

In this paper, we consider an asynchronous two-way relay network, where multiple single-antenna relay nodes enable bi-directional communication between two single-antenna transceivers using amplify- and-forward (AF) signaling in a multiple access broadcast channel (MABC) protocol. We assume that each relaying path, which originates from one transceiver, goes through one of the relays, and ends at the other transceiver, causes a delay which can be significantly different from the delays caused by other relaying paths. Such a two-way relay channel can produce inter-symbol-interference at the two transceivers. Assuming a block transmission scheme, we use cyclic prefix insertion to eliminate inter-block-interference. Aiming to optimally obtain the relay beamforming weights and the transceivers´ transmit powers, we minimize the total consumed power in the network, subject to two constraints on the transceivers´ data rates. We rigorously prove that at the optimum, the end-to-end channel impulse response (CIR) must have only one non-zero tap, and hence, only those relays which contribute to that non-zero tap are switched on. We propose a simple search algorithm to optimally determine which tap of the end-to-end CIR is non-zero. Finally, we present a semi-closed-form solution for the optimal the relays´ beamforming weights and the transceivers´ transmit powers.