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Generalized Signal Alignment For Arbitrary MIMO Two-Way Relay Channels

In this paper, we consider the arbitrary MIMO two-way relay channels, where there are $K$ source nodes, each equipped with $M_i$ antennas, for $i=1,2,\cdots,K$, and one relay node, equipped with $N$ antennas. Each source node can exchange independent messages with arbitrary other source nodes assisted by the relay. We extend our newly-proposed transmission scheme, generalized signal alignment (GSA) in [1], to arbitrary MIMO two-way relay channels when $N>M_i+M_j$, $\forall i \neq j$. The key idea of GSA is to cancel the interference for each data pair in its specific subspace by two steps. This is realized by jointly designing the precoding matrices at all source nodes and the processing matrix at the relay node. Moreover, the aligned subspaces are orthogonal to each other. By applying the GSA, we show that a necessary condition on the antenna configuration to achieve the DoF upper bound $\min \{\sum_{i=1}^K M_i, 2\sum_{i=2}^K M_i,2N\}$ is $N \geq \max\{\sum_{i=1}^K M_i-M_s-M_t+d_{s,t}\mid \forall s,t\}$. Here, $d_{s,t}$ denotes the DoF of the message exchanged between source node $s$ and $t$. In the special case when the arbitrary MIMO two-way relay channel reduces to the $K$-user MIMO Y channel, we show that our achievable region of DoF upper bound is larger than the previous work.