MIMO Multipair Two-Way Relaying With Distributed Relays: Joint Signal Alignment and Interference Neutralization

We study the degrees of freedom (DoFs) of the multiple input multiple output (MIMO) multipair two-way distributed relay channel (mTDRC), where
$K$
pairs of users, each equipped with
$M$
antennas, exchange messages in a pairwise manner with the help of two geographically separated relay nodes, each with
$N$
antennas. We establish a general framework for the DoF analysis by combining the ideas of signal space alignment and interference neutralization. The proposed framework involves a joint design of user transmit beamformers, relay precoders, and user receive beamformers. Novel signal alignment techniques are proposed to reduce the number of linearly independent constraints for interference neutralization. Based on that, the original joint transceiver and relay design problem boils down to a problem solely on the design of the relay precoders. This problem is then solved utilizing recent development on the solvability of linear matrix equations. As a result, we derive an achievable DoF for the MIMO mTDRC with an arbitrary configuration of
$(K,M,N)$
. For
$K=2$
, the optimal DoF of the considered network is derived for
$({M}/{N})\in (0, {1}/{2}) \cup (1,\infty )$
by showing that the obtained achievable DoF meets a DoF upper bound. This result beats the state of the art by establishing the optimal DoF of the considered network in an extra range of
$({M}/{N})\in (1,2)$
. In addition, the achievable DoF obtained for
$K=2$
is much higher than the existing result in the range of
$({M}/{N})\in ({11}/{16},1)$
. Furthermore, we show that the optimal DoF of the considered network with
$K\geq 3$
is derived for
$({M}/{N})\in (0, ({2K+\sqrt {2K}})/({4K^{2}-2K})) \cup ({3}/{2},\infty )$
.