A Network Linear Block Coding Approach to Selective Detect-and-Forward Multi-Way Relaying With Differential Modulation

Abstract

This paper introduces a network linear block coding framework for multi-way relaying with differential MPSK modulation. We consider a system with $K$ user terminals and $L$ relays employing a selective detect-and-forward protocol. We show that such a system can be represented as a systematic $(K+L,K)$ linear block code. This framework allows the use of linear systematic block codes as network codes yielding power saving gains at user terminals. We analyse the theoretical performance of our scheme with optimal decoding, showing that our system can achieve a diversity order that equals to the minimum Hamming distance of the associated code. Then we consider a sub-optimal decoder based on log-domain belief propagation, and present numerical results for three 4-ary linear block codes and two binary LDPC codes as examples. Theoretical and simulation results demonstrate a significant performance gain of our scheme over uncoded transmission, even though some relays may be silent occasionally. We present a technique based on hard thresholding the received samples at the terminals, that makes the performance closer to the case when no relay is silent, without any significant increase in complexity. Furthermore, we also present guidelines for choosing suitable codes, and show that systematic block codes with sparse generator matrices are in particular desirable for our system.