Achievable degrees of freedom of MIMO multiway relaying with pairwise data exchange

We study achievable degrees of freedom (DoF) of a multi-input multi-output (MIMO) multiway relay channel (mRC) where K users, each equipped with M antennas, exchange messages in a pairwise manner with the help of a single TV-antenna relay node. A novel and systematic way of designing beamforming vectors and matrices at the user and at the relay is proposed to realize signal alignment and to implement physical-layer network coding (PNC). It is shown that, for the considered mRC with K = 3 users, the proposed beamforming design achieves the DoF capacity for any (M, N) setups. For the scenarios with K > 3, we show that the proposed scheme can be improved by disabling a portion of relay antennas so as to align signals more efficiently. Our analysis reveals that the obtained achievable DoF is always piecewise linear, and is bounded either by the number of user antennas M or by the number of relay antennas N. Asymptotic DoF as K → ∞ is also derived based on the proposed signal alignment scheme.