Generalized signal alignment for arbitrary MIMO two-way relay channels

In this paper, we study the achievable degrees of freedom (DoF) for an arbitrary MIMO two-way relay channel, where there are K source nodes, each equipped with M
_i
antennas, for i = 1, 2, ⋯, K, and one relay node, equipped with N antennas. Each source node can exchange independent messages with an arbitrary set of other source nodes assisted by the relay. We extend our newly-proposed transmission scheme, generalized signal alignment (GSA) in [1], to the arbitrary MIMO two-way relay channel with antenna configuration satisfying N ≥ M
_i
+ M
_j
, ∀i ≠ j. The notion of GSA is to form network-coded symbols by aligning every pair of signals to be exchanged in a projected subspace at the relay. 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. Applying the GSA, we show that the DoF upper bound min {Σ
_i=1
^K
M
_{i, 2}
Σ
_i=2
^K
M
_i
, 2N} is tight under the antenna configuration N ≥ max{Σ
_i=1
^K
M
_i
-M
_s
-M
_t
+d
_{s, t}
| ∀
_s
, t}. Here, d
_{s, t}
denotes the DoF of the message exchanged between nodes 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 with GSA is larger than the existing result.