Bidirectional multi-pair network with a MIMO relay: Beamforming strategies and lack of duality

We address the problem of a multi-user relay network, where multiple single-antenna node pairs want to exchange information by using a multiple antenna relay node. Due to the half-duplex constraint of the relay, the exchange of information takes place in two steps. In the first step, the nodes transmit their data to the relay, while in the second step, the relay is broadcasting the data by using linear and non-linear precoding strategies. We focus on the second step in this paper. We first consider the problem of maximizing the overall rate achievable using linear and dirty-paper type precoding strategies at the relay. Then, we consider minimizing the total power at the relay subject to individual SINR constraints using the same strategies at the relay. We show that the downlink-uplink duality does not hold for the setup considered here, which is a somewhat surprising result. We also show that the beamforming strategy which is optimal in the single-pair case performs very well in the multi-pair case for practically relevant SNR. The results are illustrated by numerical simulations.