  We present an optimal decoder for physical-layer network coding (PNC) in a
multipath fading channels. Previous studies on PNC have largely focused on the
single path case. For PNC, multipath not only introduces inter-symbol
interference (ISI), but also cross-symbol interference (Cross-SI) between
signals simultaneously transmitted by multiple users. In this paper, we assume
the transmitters do not have channel state information (CSI). The relay in the
PNC system, however, has CSI. The relay makes use of a belief propagation (BP)
algorithm to decode the multipath-distorted signals received from multiple
users into a network-coded packet. We refer to our multipath decoding algorithm
as MP-PNC. Our simulation results show that, benchmarked against synchronous
PNC over a one-path channel, the bit error rate (BER) performance penalty of
MP-PNC under a two-tap ITU channel model can be kept within 0.5dB. Moreover, it
outperforms a MUD-XOR algorithm by 3dB -- MUD-XOR decodes the individual
information from both users explicitly before performing the XOR network-coding
mapping. Although the framework of fading-channel PNC presented in this paper
is demonstrated based on two-path and three-path channel models, our algorithm
can be easily extended to cases with more than three paths.
