Error Performance of Physical-Layer Network Coding in Multiple-Antenna TWRC

In this paper, the error performance of physical-layer network coding (PNC) is studied in multiple-antenna two-way relay channels (TWRCs), where two users exchange their messages with the help of a relay. We assume that the TWRC system uses binary phase-shift keying (BPSK) modulation and that all channels are Rayleigh faded. In particular, we consider a general system model where two users may have different numbers of antennas and different average signal-to-noise ratios (SNRs) to the relay. Considering a TWRC with multiple antennas at both users and the relay [referred to as multiple-input multiple-output (MIMO)-TWRC], we derive the upper and lower bounds on the uplink error probability of PNC based on maximum-likelihood (ML) detection. In this scheme, a transmit antenna selection (TAS) strategy is applied at two users, where each user selects the strongest transmit antenna for transmission. Furthermore, we also present the upper and lower bounds for two special cases, namely, the single-input multiple-output (SIMO)-TWRC with single-antenna users and the multiple-antenna relay and the multiple-input single-output (MISO) TWRC with multiple-antenna users and the single-antenna relay. Then, the diversity order is investigated for the schemes. We show that the PNC with TAS outperforms the PNC with space-time block codes (STBCs) significantly. The numerical results also demonstrate that our bounds are very tight to the Monte Carlo simulation.