We consider the full-duplex (FD) two-way amplify-and-forward relay system with imperfect cancelation of loopback self-interference (SI) and investigate joint design of relay and receive beamforming for minimizing the mean square error under a relay transmit power constraint. Due to loopback channel estimation error and limitation of analog-to-digital converter, the loopback SI cannot be completely canceled. Multiple antennas at the relay can help loopback SI suppression but beamforming is required to balance between the residual SI suppression and the desired signal transmission. Moreover, the relay beamforming matrix should be updated every time slot because the residual SI in the previous time slot is amplified by the current beamforming matrix and added to the received signals from the two sources in the current time slot. We derive the optimally balanced relay beamforming and receive beamforming matrices in closed form based on minimum mean square error, taking into account the propagation of the residual loopback SI from the first to the current time slot. We also propose beamforming design using only the channels of the m latest time slots, not from the first time slot. Based on our numerical results, we also identify when FD is beneficial and propose selection between FD and half-duplex according to signal-to-noise ratio and interference-to-noise ratio.

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