  This paper investigates noncoherent detection in a two-way relay channel
operated with physical layer network coding (PNC), assuming FSK modulation and
short-packet transmissions. For noncoherent detection, the detector has access
to the magnitude but not the phase of the received signal. For conventional
communication in which a receiver receives the signal from a transmitter only,
the phase does not affect the magnitude, hence the performance of the
noncoherent detector is independent of the phase. PNC, however, is a multiuser
system in which a receiver receives signals from multiple transmitters
simultaneously. The relative phase of the signals from different transmitters
affects the received signal magnitude through constructive-destructive
interference. In particular, for good performance, the noncoherent detector in
PNC must take into account the influence of the relative phase on the signal
magnitude. Building on this observation, this paper delves into the
fundamentals of PNC noncoherent detector design. To avoid excessive overhead,
we do away from preambles. We show how the relative phase can be deduced
directly from the magnitudes of the received data symbols. Numerical results
show that our detector performs nearly as well as a "fictitious" optimal
detector that has perfect knowledge of the channel gains and relative phase.
