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Optimal time-division of two-phase decode-and-forward bidirectional relaying

A relay node establishes bidirectional communication between two nodes using a two-phase decode-and-forward protocol. In the first phase nodes 1 and 2 transmit their messages to the relay node, which decodes the messages. In the succeeding phase the relay broadcasts a re-encoded message using the network coding idea so that each node can decode the other's message using its own message as side information. In this work we optimize the time-division between the two phases assuming memoryless Gaussian channels with multiple antennas at the relay node and single antennas at nodes 1 and 2. It shows that the achievable rate region is convex without additional time-sharing between two operating points. The presented equivalent characterization of the achievable rate region simplifies the optimization problem of resource allocation policies.