  This work considers the protograph-coded physical network coding (PNC) based
on Alamouti space-time block coding (STBC) over Nakagami-fading two-way relay
channels, in which both the two sources and relay possess two antennas. We
first propose a novel precoding scheme at the two sources so as to implement
the iterative decoder efficiently at the relay. We further address a simplified
updating rule of the log-likelihood-ratio (LLR) in such a decoder. Based on the
simplified LLR-updating rule and Gaussian approximation, we analyze the
theoretical bit-error-rate (BER) of the system, which is shown to be consistent
with the decoding thresholds and simulated results. Moreover, the theoretical
analysis has lower computational complexity than the protograph extrinsic
information transfer (PEXIT) algorithm. Consequently, the analysis not only
provides a simple way to evaluate the error performance but also facilitates
the design of the joint channel-and-PNC (JCNC) in wireless communication
scenarios.
