Capacity of MIMO wireless channel with full-duplex amplify-and-forward relay

In this paper, a full-duplex amplify-and-forward (AF) relay is considered that suffers from self-interference in a MIMO wireless channel. We derive the optimal transformation matrix that maximizes the mutual information under average power constraint at the relay output. The capacity of the channel without a direct path is derived and it is shown that the relay maximizes the throughput as if there is no interference from the relay itself. A comparison is made to the channel with a half-duplex relay having the same average transmission power. In particular, a necessary and sufficient condition is derived in terms only of the signal-to-noise ratios (SNRs) for a full-duplex relay to perform better than a half-duplex relay in SISO cases. A simple rule of thumb shows that a full-duplex relay outperforms a half-duplex relay if the sum of the SNR in dB at the relay input and that at the destination input is greater than 0.75. Numerical results show that this result also holds for MIMO channels approximately.