Physical Layer Network Coding for Two-Way Relaying with QAM

TL;DR

This paper explores physical layer network coding with QAM modulation in two-way relaying, demonstrating that QAM offers fewer singular fade states and proposing Latin Square-based mappings to improve performance under fading conditions.

Contribution

It extends network coding design to M-QAM signal sets, analyzes singular fade states, and introduces Latin Square mappings to mitigate fading effects, surpassing PSK-based methods.

Findings

QAM has fewer singular fade states than PSK.

Latin Square mappings can eliminate certain fade states.

QAM-based network coding outperforms PSK in fading scenarios.

Abstract

The design of modulation schemes for the physical layer network-coded two way relaying scenario was studied in [1], [3], [4] and [5]. In [7] it was shown that every network coding map that satisfies the exclusive law is representable by a Latin Square and conversely, and this relationship can be used to get the network coding maps satisfying the exclusive law. But, only the scenario in which the end nodes use $M$-PSK signal sets is addressed in [7] and [8]. In this paper, we address the case in which the end nodes use $M$-QAM signal sets. In a fading scenario, for certain channel conditions $\gamma e^{j \theta}$, termed singular fade states, the MA phase performance is greatly reduced. By formulating a procedure for finding the exact number of singular fade states for QAM, we show that square QAM signal sets give lesser number of singular fade states compared to PSK signal sets. This results in superior performance of $M$-QAM over $M$-PSK. It is shown that the criterion for partitioning the complex plane, for the purpose of using a particular network code for a particular fade state, is different from that used for $M$-PSK. Using a modified criterion, we describe a procedure to analytically partition the complex plane representing the channel condition. We show that when $M$-QAM ($M >4$) signal set is used, the conventional XOR network mapping fails to remove the ill effects of $\gamma e^{j \theta}=1$, which is a singular fade state for all signal sets of arbitrary size. We show that a doubly block circulant Latin Square removes this singular fade state for $M$-QAM.