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|a This paper considers physical-layer network coding (PNC) with M-ary phase-shift keying (MPSK) modulation in two-way relay channel (TWRC). A low complexity detection technique, termed symbol-based PNC (SPNC), is proposed for the relay. In particular, attributing to the outer product operation imposed on the superposed MPSK signals at the relay, SPNC obtains the network-coded symbol (NCS) straightforwardly without having to detect individual symbols separately. Unlike the optimal multi-user detector (MUD) which searches over the combinations of all users' modulation constellations, SPNC searches over only one modulation constellation, thus simplifies the NCS detection. Despite the reduced complexity, SPNC achieves full diversity in multi-antenna relay as the optimal MUD does. Specifically, antenna selection based SPNC (AS-SPNC) scheme and signal combining based SPNC (SC-SPNC) scheme are proposed. Our analysis of these two schemes not only confirms their full diversity performance, but also implies when SPNC is applied in multi-antenna relay, TWRC can be viewed as an effective single-input multiple-output (SIMO) system, in which AS-PNC and SC-PNC are equivalent to the general AS scheme and the maximal-ratio combining (MRC) scheme. Moreover, an asymptotic analysis of symbol error rate (SER) is provided for SC-PNC considering the case that the number of relay antennas is sufficiently large.
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