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We present an optimal decoder for physical-layer network coding (PNC) in multipath fading channels. Previous studies on PNC have largely focused on the single-path case. For PNC, multipath introduces not only intersymbol interference (ISI) but cross-symbol interference (Cross-SI) between signals simultaneously transmitted by multiple users as well. To overcome these problems, the decoder at the relay of our PNC design 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 International Telecommunication Union (ITU) channel model can be kept within 0.5 dB. Moreover, it outperforms a multiuser detection (MUD)- xor algorithm by 3 dB (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- and three-path channel models, our algorithm can be extended to cases with more than three paths.

An Optimal Decoding Strategy for Physical-Layer Network Coding Over Multipath Fading Channels