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.