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This paper investigates coherent detection for physical-layer network coding(PNC) with short packet transmissions in a two-way relay channel (TWRC). PNCturns superimposed EM waves into network-coded messages to improve throughputin a relay system. To achieve this, accurate channel information at the relayis a necessity. Much prior work applies preambles to estimate the channel. Forlong packets, the preamble overhead is low because of the large data payload.For short packets, that is not the case. To avoid excessive overhead, weconsider a set-up in which short packets do not have preambles. A key challengeis how the relay can estimate the channel and detect the network-coded messagesjointly based on the received signals from the two end users. We design acoherent detector that makes use of a belief propagation (BP) algorithm to doso. For concreteness, we focus on frequency-shift-keying (FSK) modulation. Weshow how the BP algorithm can be simplified and made practical withGaussian-mixture passing. In addition, we demonstrate that prior knowledge onthe channel distribution is not needed with our framework. Benchmarked againstthe detector with prior knowledge of the channel distribution, numericalresults show that our detector can have nearly the same performance withoutsuch prior knowledge.

Coherent Detection for Short-Packet Physical-Layer Network Coding with FSK Modulation