On the BCJR Algorithm for Asynchronous Physical-layer Network Coding
  In practical asynchronous bi-directional relaying, symbols transmitted by two source nodes cannot arrive at the relay with perfect symbol alignment and the symbol-asynchronous multiple-access channel (MAC) should be seriously considered. Recently, Lu et al. proposed a Tanner-graph representation of symbol-asynchronous MAC with rectangular-pulse shaping and further developed the message-passing algorithm for optimal decoding of the asynchronous physical-layer network coding. In this paper, we present a general channel model for the asynchronous multiple-access channel with arbitrary pulse-shaping. Then, the Bahl, Cocke, Jelinek, and Raviv (BCJR) algorithm is developed for optimal decoding of asynchronous MAC channel. This formulation can be well employed to develop various low-complexity algorithms, such as Log-MAP algorithm, Max-Log-MAP algorithm, which are favorable in practice. 
Xiaofu Wu, Chunming Zhao, and Xiaohu You
cs.IT math.IT