  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. A second
important issue is how to integrate channel coding with PNC to achieve reliable
communication. This paper investigates these two issues and makes the following
contributions: 1) We propose and investigate a general framework for decoding
at the receiver based on belief propagation (BP). The framework can effectively
deal with symbol and phase asynchronies while incorporating channel coding at
the same time. 2) For unchannel-coded PNC, we show that for BPSK and QPSK
modulations, our BP method can significantly reduce the asynchrony penalties
compared with prior methods. 3) For unchannel-coded PNC, with half symbol
offset between the transmitters, our BP method can drastically reduce the
performance penalty due to phase asynchrony, from more than 6 dB to no more
than 1 dB. 4) For channel-coded PNC, with our BP method, both symbol and phase
asynchronies actually improve the system performance compared with the
perfectly synchronous case. Furthermore, the performance spread due to
different combinations of symbol and phase offsets between the transmitters in
channel-coded PNC is only around 1 dB. The implication of 3) is that if we
could control the symbol arrival times at the receiver, it would be
advantageous to deliberately introduce a half symbol offset in unchannel-coded
PNC. The implication of 4) is that when channel coding is used, symbol and
phase asynchronies are not major performance concerns in PNC.
