Beyond one-way communication: Degrees of freedom of the 4-user relay MIMO Y channel

We characterize the degrees of freedom (DoF) of a wireless network consisting of 4 user nodes, each with M antennas, and one N-antenna relay node. In this network, each user sends one independent message to each of the other users, and there are no direct links between any two users, i.e., all communication must pass through the relay node. 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,N alternatively. While the information theoretic DoF converse is established for every M,N, the achievability relying on linear signal subspace alignment is established in the spatially-normalized sense in general. The central new insight to emerge from this work is the notion of inter-user signal subspace alignment incorporating the idea of network coding, which is the key to achieve the optimal DoF for multi-way relay interference networks. Moreover, this work also settles the feasibility of linear interference alignment that extends the feasibility framework from one-way to multi-way networks.