Symbol and Bit Mapping Optimization for Physical-Layer Network Coding with Pulse Amplitude Modulation

In this paper, we consider a two-way relay network in which two users exchange messages through a single relay using a physical-layer network coding (PNC) based protocol. The protocol comprises two phases of communication. In the multiple access (MA) phase, two users transmit their modulated signals concurrently to the relay, and in the broadcast (BC) phase, the relay broadcasts a network-coded (denoised) signal to both users. Nonbinary and binary network codes are considered for uniform and nonuniform pulse amplitude modulation (PAM) adopted in the MA phase, respectively. We examine the effect of different choices of symbol mapping (i.e., mapping from the denoised signal to the modulation symbols at the relay) and bit mapping (i.e., mapping from the modulation symbols to the source bits at the user) on the system error-rate performance. A general optimization framework is proposed to determine the optimal symbol/bit mappings with joint consideration of noisy transmissions in both communication phases. Complexity-reduction techniques are developed for solving the optimization problems. It is shown that the optimal symbol/bit mappings depend on the signal-to-noise ratio (SNR) of the channel and the modulation scheme. A general strategy for choosing good symbol/bit mappings is also presented based on a high-SNR analysis, which suggests using a symbol mapping that aligns the error patterns in both communication phases and Gray and binary bit mappings for uniform and nonuniform PAM, respectively.