Generalized Signal Alignment: On the Achievable DoF for Multi-User MIMO Two-Way Relay Channels

This paper studies the achievable degrees of freedom (DoF) for multi-user multiple-input multiple-output (MIMO) two-way relay channels, where there are K source nodes, each equipped with M antennas, one relay node, equipped with N antennas, and each source node exchanges independent messages with an arbitrary set of other source nodes via the relay. By allowing an arbitrary information exchange pattern, the considered channel model is a unified one. It includes several existing channel models as special cases: 1) K-user MIMO Y channel; 2) multi-pair MIMO two-way relay channel; 3) generalized MIMO two-way X relay channel; and 4) L-cluster MIMO multiway relay channel. Previous studies mainly considered the achievability of the DoF cut-set bound 2N at the antenna configuration N <; 2M by applying signal alignment for network coding. This paper aims to investigate the achievability of the DoF cut-set bound K M for the case N 2M. To this end, we first derive tighter DoF upper bounds for three special cases of the considered channel model. Then, we propose a new transmission framework, generalized signal alignment (GSA), to approach these bounds. The notion of GSA is to form network-coded symbols by aligning every pair of signals to be exchanged in a compressed subspace at the relay. A necessary and sufficient condition to construct the relay compression matrix is given. We show that using GSA, the new DoF upper bound is achievable when: 1) N/m ∈ (0, 2+ 4/{K(K-1)}]∪[K-2, +∞) for the K-user MIMO Y channel; 2) N/M ∈ (0,2 +4/K]∪[K - 2, +∞) for the multi-pair MIMO two-way relay channel; and 3) N/M ∈ (0, 2+ 8/K²]∪[K-2, +∞) for the generalized MIMO two-way X relay channel. We also provide the antenna configuration regions for the general multi-user MIMO two-way relay channel to achieve the total DoF KM.