A Compute-and-Forward Scheme for Gaussian Bi-Directional Relaying with Inter-Symbol Interference

We provide inner and outer bounds on the capacity region for the Gaussian bi-directional relaying over inter-symbol interference channels. The outer bound is obtained by the conventional cut-set argument. For the inner bound, we propose a compute-and-forward coding scheme based on lattice partition chains and study its achievable rate. The coding scheme is a time-domain coding scheme which uses a novel precoding scheme at the transmitter in combination with lattice precoding and a minimum mean squared error receiver to recover linear combinations of lattice codewords. The proposed compute-and-forward coding scheme substantially outperforms decode-and-forward schemes. While it is well known that for the point-to-point communication case, both independent coding along sub-channels and time-domain coding can approach the capacity limit, as a byproduct of the proposed scheme, we show that for the bi-directional relay case, independent coding along sub-channels is not optimal in general and joint coding across sub-channels can improve the capacity for some channel realizations.