On the Exchange Rate for Bi-Directional Relaying over Inter-Symbol Interference Channels

We propose two compute-and-forward coding schemes for the bi-directional relay channel with inter- symbol interference (ISI) based on lattice codes and study their achievable rates. The first coding scheme is similar in spirit to coded orthogonal frequency division multiplexing (OFDM) with independent coding across sub-carriers and uses nested-lattice code with a power allocation strategy that can exploit the group property of lattices. The second 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 schemes substantially outperform decode-and-forward schemes. While it is well known that for the point-to-point communication case, both the coded OFDM approach and the time-domain coding scheme can approach the capacity limit, we show that for the bi-directional relaying case, the performance of the two coding schemes are different. Particularly, we show that independent coding across sub-channels is not optimal and joint coding across sub-channels can improve the exchange capacity for some channel realizations.