  This paper investigates coherent detection for physical-layer network coding
(PNC) with short packet transmissions in a two-way relay channel (TWRC). PNC
turns superimposed EM waves into network-coded messages to improve throughput
in a relay system. To achieve this, accurate channel information at the relay
is a necessity. Much prior work applies preambles to estimate the channel. For
long packets, the preamble overhead is low because of the large data payload.
For short packets, that is not the case. To avoid excessive overhead, we
consider a set-up in which short packets do not have preambles. A key challenge
is how the relay can estimate the channel and detect the network-coded messages
jointly based on the received signals from the two end users. We design a
coherent detector that makes use of a belief propagation (BP) algorithm to do
so. For concreteness, we focus on frequency-shift-keying (FSK) modulation. We
show how the BP algorithm can be simplified and made practical with
Gaussian-mixture passing. In addition, we demonstrate that prior knowledge on
the channel distribution is not needed with our framework. Benchmarked against
the detector with prior knowledge of the channel distribution, numerical
results show that our detector can have nearly the same performance without
such prior knowledge.
