Precoding designs based on minimum distance for two-way relaying mimo systems with physical network coding

In this paper, we propose new precoding methods for two-way multiple input multiple output physical network coding (PNC) systems which employ the modulo operation. In our work, the transmit and receive filters are determined to maximize the minimum distance of the received constellations assuming global channel state information. The precoding operations are separately optimized for the multiple access (MA) and the broadcast stages, and the optimal precoding is obtained by applying a semidefinite relaxation method. Especially, we prove that for the system with linear detection the modulo operation for the PNC achieves optimality with the derived precoding for the MA stage in terms of the minimum distance. Also, we present a closed-form solution for the optimal filter designs for two special cases. For computing solutions, we transform our max min problem into a simple maximization problem by imposing additional constraints. Also, we propose a suboptimal non-iterative precoding scheme whose performance is within 1 dB at a bit error rate (BER) of 10^{-4} compared to the optimum iterative method with much reduced complexity. Finally, the simulation results show that the proposed systems achieve 2-3 dB gains at a BER of 10^{-4} compared to the optimal amplify-and-forward systems.