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

The $K$-user 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 via the relay. A transmission strategy is proposed for this channel. The proposed strategy is based on two steps: channel diagonalization and cyclic communication. The channel diagonalization is applied by using zero-forcing beam-forming. After 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 DoF region of the channel if $N\leq M$. To prove this, a new degrees-of-freedom 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.