Cyclic Communication and the Inseparability of MIMO Multi-Way Relay Channels

The K-user multiple-input multiple-output (MIMO) multi-way relay channel (Y-channel) consisting of K users with M antennas each and a common relay node with N antennas is studied in this paper. Each user wants to exchange messages with all the other users through the relay. A transmission strategy is proposed for this channel. The proposed strategy is based on two steps: 1) channel diagonalization and 2) cyclic communication. The channel diagonalization is applied by using zero-forcing beam-forming. After the channel diagonalization, the channel is decomposed into parallel sub-channels. Cyclic communication is then applied, where signal-space alignment for network-coding is used over each sub-channel. The proposed strategy achieves the optimal degree-of-freedom (DoF) region of the channel if N ≤ M. To prove this, a new DoFs outer bound is derived. As a by-product, we conclude that the MIMO Y-channel is not separable, i.e., independent coding on separate sub-channels is not enough, and one has to code jointly over several sub-channels.