Degrees of Freedom for MIMO Two-Way X Relay Channel

We study the degrees of freedom (DOF) of a multiple-input multiple-output (MIMO) two-way X relay channel, where there are two groups of source nodes and one relay node, each equipped with multiple antennas, and each of the two source nodes in one group exchanges independent messages with the two source nodes in the other group via the relay node. It is assumed that every source node is equipped with M antennas while the relay is equipped with N antennas. We first show that the upper bound on the total DOF for this network is 2min {2M,N} and then focus on the case of N ≤ 2M so that the DOF is upper bounded by the number of antennas at the relay. By applying signal alignment for network coding and joint transceiver design for interference cancellation, we show that this upper bound can be achieved when N ≤ ⌊ 8M 5 ⌋. We also show that with signal alignment only but no joint transceiver design, the upper bound is achievable when N ≤ ⌊ 4M 3 ⌋. Simulation results are provided to corroborate the theoretical results and to demonstrate the performance of the proposed scheme in the finite signal-to-noise ratio regime.