Optimal transmit strategy for the 2×1 MISO bidirectional broadcast channel

We consider the broadcast phase of a three-node network, where a relay node with multiple antennas establishes a bidirectional communication between the two other nodes each equipped with a single antenna using a spectrally efficient two phase protocol. In the first phase, both nodes transmit a message to the relay node, which decodes the messages. In the second phase the relay broadcasts a re-encoded composition of them. In this work, we show that for two transmit antennas at the relay node transmitting a single beam is always an optimal transmit strategy. We show that if a single beam is optimal the beamforming vector is a linear combination of the two single-user optimal strategies. Further, we analyze the dependency of the capacity region on the correlation between the two channels.