Generalized Compute-and-Forward Schemes for Multi-Hop Two-Way Relay Channels

Abstract

Wireless network coding for multi-hop two-way relay channels (MH-TRC) has been proven to achieve a significantly improved network throughput than non- network coding (Non-NC) schemes. Our previous work showed that the multi-hop compute-and-forward (MH- CPF) scheme for the MH-TRC with fixed two transmission time intervals was unable to outperform Non-NC in the MH-TRC with an arbitrary number of nodes. In view of this fact, we propose an <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I</i> -time- interval ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I</i> -TI) MH-CPF scheme ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I</i> represents the number of transmission time intervals) for the MH- TRC with arbitrary numbers of nodes and messages. By converting the transmission pattern to a corresponding characteristic matrix, the outage probability of the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I</i> -TI MH-CPF scheme is derived. We investigate the relationships between the numbers of nodes, messages and TIs with the outage probability. It is proven that <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> -1-TI MH-CPF can outperform 2-TI MH-CPF when the number of messages is large, while the MH-CPF scheme with a larger number of TIs has better outage performance than 2-TI MH-CPF in the MH-TRC with a small number of nodes.