On the DoF Region for the Asymmetric MIMO Two-Way X Relay Channel

In this paper, we study the degrees of freedom (DoF) region for the multiple-input multiple-output two-way X relay channel with asymmetric antenna setting. In this channel model, there are two groups of source nodes, each group contains two source nodes, every source node in one group exchanges independent messages with every source node in another group via a common relay node, and each node has a different number of antennas. First, we derive an outer bound of the DoF region by using the cut-set theorem and the genie-message approach. Then, we obtain an inner bound of the DoF region by proposing a new transmission scheme that collectively utilizes antenna deactivation, pairwise signal alignment, cyclic signal alignment, and generalized signal alignment techniques. In the case of symmetric data exchange, our inner bound coincides with the outer bound, and thus, our proposed transmission strategy is optimal. We also obtain the optimal sum DoF for the special case, where the source nodes within a same group are equipped with the same number of antennas. This paper provides new insights for the study of more complicated relay networks.