Beyond One-Way Communication: Degrees of Freedom of Multi-Way Relay MIMO Interference Networks

We characterize the degrees of freedom (DoFs) of a multiway relay wireless network consisting of four user nodes, each with M antennas, and one N-antenna relay node. Each user node sends one independent message to each of the other users, and there are no direct links between any two user nodes. For this network, we show that the symmetric DoF value per message is given by max(min((M/3), (N/7)), min(((2M)/7), (N/6))) normalized by space dimensions, i.e., piecewise linear depending on M and N alternatively. While the information theoretic DoF upper bound is established for every M and N antennas, the achievability relying on linear signal subspace alignment is established in the spatially normalized sense in general. In addition, by deactivating four messages to form a twoway relay MIMO X channel, we also present the DoF result max(min((M/2), (N/5)), min(((2M)/5), (N/4))) in a similar piecewise linear type. The key ingredient to these new results is to first establish the principle of beamforming design via linear dimension counting perspective, which is essentially the fundamental element for establishing both the DoF converse and the DoF achievability. Moreover, we also settle the feasibility of linear interference alignment that extends the feasibility framework from one-way one-hop to multiway relay interference networks.