On Symbol Mapping for Binary Physical-Layer Network Coding with PSK Modulation

In a system employing physical-layer network coding, the superimposed constellation at the relay may have ambiguity where one received signal is associated with more than one possible combination of users´ data. We find the necessary and sufficient condition on a user bit mapping which removes this ambiguity for phase shift keying modulation. Further, we introduce the concept of semi-Gray mapping which improves the system BER performance and achievable rate. The necessary and sufficient condition of having a semi-Gray mapping is also found. Interestingly, the widely used binary reflected Gray mapping does not satisfy the semi-Gray mapping condition resulting in a poorer performance compared to our proposed semi-Gray scheme.