Minimum number of antennas and degrees of freedom of multiple-input–multiple-output multi-user two-way relay X channels

In this study, the minimum number of antennas at each user is derived to obtain 2M² degrees of freedom (DOF) in the M × 1 × M multiple-input-multiple-output (MIMO) multi-user two-way relay X channels. Based on the design of beamforming vectors for the conventional signal space alignment scheme, it is shown that each user needs at least ⌈M²- (M/2)⌉ antennas to obtain DOF 2M² with the relay having M² antennas. As the number of users increases, the number of antennas also increases, which makes the implementation difficult. In an effort to reduce the number of antennas, the authors propose a new beamforming scheme of MIMO multi-user two-way relay X channels using the time extension. Through the numerical analysis, it is confirmed that the proposed scheme with time extension is a good alternative scheme to replace the conventional scheme.