  In practical asynchronous bi-directional relaying, symbols transmitted by two
sources cannot arrive at the relay with perfect frame and symbol alignments and
the asynchronous multiple-access channel (MAC) should be seriously considered.
Recently, Lu et al. proposed a Tanner-graph representation of the
symbol-asynchronous MAC with rectangular-pulse shaping and further developed
the message-passing algorithm for optimal decoding of the symbol-asynchronous
physical-layer network coding. In this paper, we present a general channel
model for the asynchronous MAC with arbitrary pulse-shaping. Then, the Bahl,
Cocke, Jelinek, and Raviv (BCJR) algorithm is developed for optimal decoding of
the asynchronous MAC channel. For Low-Density Parity-Check (LDPC)-coded BPSK
signalling over the symbol-asynchronous MAC, we present a formal log-domain
generalized sum-product-algorithm (Log-G-SPA) for efficient decoding.
Furthermore, we propose to use cyclic codes for combating the
frame-asynchronism and the resolution of the relative delay inherent in this
approach can be achieved by employing the simple cyclic-redundancy-check (CRC)
coding technique. Simulation results demonstrate the effectiveness of the
proposed approach.
