Binary Labeling for 2D and 4D constellations

Introduction

The binary labeling of constellations is an important factor to the achievable information rate (AIR) of a constellation. Although normally the mutual information (MI) is used to determine the AIR of a constellation, when the binary labeling is taken into account the GMI is used as described in this publication (https://ieeexplore.ieee.org/document/8240991).

Methodology

• To calculate both the MI and GMI the Gauss-Hermite approximation is used.
• The algorithm used to determine a good labeling for the constellations is the binary switching algorithm (BSA) as described in this paper (https://ieeexplore.ieee.org/document/64657). The cost function used in the BSA is described in the publication (https://ieeexplore.ieee.org/document/6784555).
• The target SNR is found by computing the constellations have an NMI (normalized MI) and NGMI (normalized GMI) of 0.8.
• SNR is defined as the signal power $E[||X||]^2$ over the total noise power $\sigma^2$ (N-dimensional noise power).

Notes

• Most of the constellations for which the BSA was executed were taken from Erik Agrell's database https://codes.se/packings/
• Some of the comments in the last column of the tables are taken from Erik Agrell's database https://codes.se/packings/
• The binary labelings reported might not be optimal. If you know of a better labeling (or can find a better labeling), please let us know so we update the table.

Image

By the NMI (normalized MI) of constellations is 0.8, the required SNR gaps of between Shannon capacity different moduation formats are shown in this figure, which suggests that high dimensional modulation scheme can approach more close to the Shannon limit.

Research Results

The numbers following the constellation have specific meanings in the tables. For example, in 'QAM2_4',the '2' represents the dimensionality of the constellation while the '4' refers to the amount of bits.

2D constellation&Labeling

4D constellation&Labeling

The database is managed by Bin Chen (HFUT, China), Alex Alvarado (TUe, The Netherlands) and Zhiwei Liang (HFUT, China), if you are interested in adding your constellation formats in this database, please contact us (bin.chen@hfut.edu.cn, A.Alvarado@tue.nl).

Please cite the database as
Bin Chen, Zhiwei Liang and Alex Alvarado, “Binary Labeling for 2D and 4D constellations”, https://github.com/TUe-ICTLab/Binary-Labeling-for-2D-and-4D-constellations