Design criterion of linear physical-layer network coding for fading two-way relay channels

We investigate a linear physical-layer network coding (LPNC) scheme for fading two-way relay channels, where the transmitters cannot track the channel. In this scheme, the relay computes and forwards integer combinations of the two users´ messages. We develop a design criterion for adaptively choosing the integer coefficients at the relay, which minimizes the high-SNR error probability of the LPNC scheme. We derive an explicit expression for the optimized integer coefficients, and show that the crucial factor is the ratio between the two users´ fading channel coefficients. We also extend the LPNC scheme to the complex-valued model and derive the optimized integer coefficient vectors. Numerical results show that our designed LPNC scheme outperforms existing schemes by more than 5 dB.