  A key issue in physical-layer network coding (PNC) is how to deal with the
asynchrony between signals transmitted by multiple transmitters. That is,
symbols transmitted by different transmitters could arrive at the receiver with
symbol misalignment as well as relative carrier-phase offset. In this paper, 1)
we propose and investigate a general framework based on belief propagation (BP)
that can effectively deal with symbol and phase asynchronies; 2) we show that
for BPSK and QPSK modulations, our BP method can significantly reduce the SNR
penalty due to asynchrony compared with prior methods; 3) we find that symbol
misalignment makes the system performance less sensitive and more robust
against carrier-phase offset. Observation 3) has the following practical
implication. It is relatively easier to control symbol timing than
carrier-phase offset. Our results indicate that if we could control the symbol
offset in PNC, it would actually be advantageous to deliberately introduce
symbol misalignment to desensitize the system to phase offset.
