  This paper addresses the problem of joint channel estimation and channel
decoding in physical-layer network coding (PNC) systems. In PNC, multiple users
transmit to a relay simultaneously. PNC channel decoding is different from
conventional multi-user channel decoding: specifically, the PNC relay aims to
decode a network-coded message rather than the individual messages of the
users. Although prior work has shown that PNC can significantly improve the
throughput of a relay network, the improvement is predicated on the
availability of accurate channel estimates. Channel estimation in PNC, however,
can be particularly challenging because of 1) the overlapped signals of
multiple users; 2) the correlations among data symbols induced by channel
coding; and 3) time-varying channels. We combine the expectation-maximization
(EM) algorithm and belief propagation (BP) algorithm on a unified factor-graph
framework to tackle these challenges. In this framework, channel estimation is
performed by an EM subgraph, and channel decoding is performed by a BP subgraph
that models a virtual encoder matched to the target of PNC channel decoding.
Iterative message passing between these two subgraphs allow the optimal
solutions for both to be approached progressively. We present extensive
simulation results demonstrating the superiority of our PNC receivers over
other PNC receivers.
