On the optimal transmission for the MIMO bidirectional broadcast channel

In this work the transmit covariance matrix optimization problem for the MIMO Gaussian bidirectional broadcast channel is studied. A half-duplex relay node establishes bidirectional communication between two nodes using a decode-and-forward protocol. In the initial multiple access phase both nodes transmit their messages to the relay node. In the succeeding phase the relay broadcasts an optimal re-encoded message so that both nodes can decode the other's message using their own message as side information. We show that if the channels are orthogonal then there exist equivalent transmit strategies with different ranks. The study of special cases reveals some of the difficult structure of the optimal solution. In particular a closed form procedure for the case of full rank transmission for invertible channels is derived. Moreover, for parallel channels the optimal solution is completely characterized and discussed.