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Geometrically-Shaped Constellation for Visible Light Communications at Short Blocklength

Jia-Ning Guo, Ru-Han Chen, Jian Zhang, Longguang Li, Xu Yang, Jing Zhou

TL;DR

This work addresses short-packet visible light communication under simultaneous peak and average intensity constraints by developing a second-order, large-deviation-based analysis of the optimal shaping region and proposing a Construction B lattice framework that combines coarse shaping with fine coding. The approach yields near-maximum shaping gains and significantly larger coding gains by leveraging dense lattice cosets, culminating in an energy-efficient 24-dimensional Leech lattice constellation that outperforms conventional schemes in simulations. The results provide both theoretical finite-blocklength insights and a practical constellation-construction methodology with fast encoding/decoding suitable for low-latency VLC scenarios. Overall, the paper offers a principled path to close the shaping-gap in short-blocklength VLC and demonstrates tangible OSNR gains for real-world indoor settings.

Abstract

In this paper, we present a general framework of designing geometrically shaped constellations for short-packet visible light communications with a peak- and an average-intensity constraints. By leveraging tools from large deviation theory, we first characterize the second-order asymptotics of the optimal constellation shaping region under aforementioned intensity constraints, which serves as a good performance measure for the best geometric shaping in finite blocklength. To further incorporate a sufficiently large coding gain and a nearly-maximum shaping gain, we construct multidimensional constellations by the nested structure of Construction B lattices, where the constellation shaping is implemented by controlling the boundary of the embedded sublattice, i.e., a strategy called coarsely shaping and finely coding. Fast algorithms for constellation mapping and demodulation are presented as well. As an illustrative example, we present an energy-efficient $24$-dimensional constellation design based on the Leech lattice, whose superiority over existing constellation designs is verified by numerical results.

Geometrically-Shaped Constellation for Visible Light Communications at Short Blocklength

TL;DR

This work addresses short-packet visible light communication under simultaneous peak and average intensity constraints by developing a second-order, large-deviation-based analysis of the optimal shaping region and proposing a Construction B lattice framework that combines coarse shaping with fine coding. The approach yields near-maximum shaping gains and significantly larger coding gains by leveraging dense lattice cosets, culminating in an energy-efficient 24-dimensional Leech lattice constellation that outperforms conventional schemes in simulations. The results provide both theoretical finite-blocklength insights and a practical constellation-construction methodology with fast encoding/decoding suitable for low-latency VLC scenarios. Overall, the paper offers a principled path to close the shaping-gap in short-blocklength VLC and demonstrates tangible OSNR gains for real-world indoor settings.

Abstract

In this paper, we present a general framework of designing geometrically shaped constellations for short-packet visible light communications with a peak- and an average-intensity constraints. By leveraging tools from large deviation theory, we first characterize the second-order asymptotics of the optimal constellation shaping region under aforementioned intensity constraints, which serves as a good performance measure for the best geometric shaping in finite blocklength. To further incorporate a sufficiently large coding gain and a nearly-maximum shaping gain, we construct multidimensional constellations by the nested structure of Construction B lattices, where the constellation shaping is implemented by controlling the boundary of the embedded sublattice, i.e., a strategy called coarsely shaping and finely coding. Fast algorithms for constellation mapping and demodulation are presented as well. As an illustrative example, we present an energy-efficient -dimensional constellation design based on the Leech lattice, whose superiority over existing constellation designs is verified by numerical results.
Paper Structure (35 sections, 4 theorems, 61 equations, 8 figures, 3 tables)

This paper contains 35 sections, 4 theorems, 61 equations, 8 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Channel Model
  3. Preliminaries
  4. Construction B Lattice
  5. Truncated Cubes
  6. Geometric Shaping
  7. Shaping Gain
  8. Optimal Shaping Region
  9. $D_n$ Lattice Points in Truncated Cubes
  10. Optimal Shaping in Finite-Blocklength Regimes
  11. Finite-Blocklength Analysis of Optimal Shaping Parameter
  12. Finite-Blocklength Analysis of Maximum Shaping Gain
  13. Extension to APP-Limited Quadrature Gaussian Channel
  14. Optimally-Shaped Constellation Based on Construction B Lattices
  15. Basic Idea of Constructing Geometrically-Shaped Constellations
  16. ...and 20 more sections

Key Result

Theorem 1

(Optimal Shaping Chen2020FFT) The optimal solution to the problem prob:opt_shaping is $\mathscr{T}_n \left(t_n^{\star}\right)$ as defined in Eq. eq:truncated_cube, where the parameter $t_n^{\star}$ is determined by and $\textnormal{ P}_n(\cdot)$ is given by Eq. eq.moment. Accordingly, the maximum shaping gain in Euclidean-$n$ space is given by

Figures (8)

  • Figure 1: The approximation error $t_n^{\star}-n\alpha-{1}/{\mu^*}$ versus various blocklengths $n$ and constraint parameters $\alpha$.
  • Figure 2: True shaping gain and the second-order approximation with different constraint parameters.
  • Figure 3: The relationship between the maximum shaping gain $\overline{\mathtt{SG}}_{\textnormal{VLC}}(n;\alpha)$ and the constraint parameter $\alpha$ with different dimensions.
  • Figure 4: Coarsely shaping and finely coding for selecting $D_2$ points from a truncated cube.
  • Figure 5: Constellation mapping for the OSLC.
  • ...and 3 more figures

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 1