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

This paper studies the achievable degrees of freedom for multi-user 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: $K$-user MIMO Y channel, multi-pair MIMO two-way relay channel, generalized MIMO two-way X relay channel, and $L$-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. This work aims to investigate the achievability of the DoF cut-set bound $KM$ for the case $N\geq 2M$. 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, 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 i) $\frac{N}{M} \in \big(0, 2+\frac{4}{K(K-1)}\big] \cup \big[K-2, +\infty\big)$ for the $K$-user MIMO Y channel; ii) $\frac{N}{M} \in \big(0, 2+\frac{4}{K}\big] \cup \big[K-2, +\infty\big)$ for the multi-pair MIMO two-way relay channel; iii) $\frac{N}{M} \in \big(0, 2+\frac{8}{K^2}\big] \cup \big[K-2, +\infty\big)$ 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 $KM$.