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An Extension of Enumerative Sphere Shaping for Arbitrary Channel Input Distributions

Frederik Ritter, Andrej Rode, Laurent Schmalen

TL;DR

This work extends Enumerative Sphere Shaping to support arbitrary discrete channel input distributions by introducing distribution-dependent weights in a trellis framework, enabling non-uniform shaping on discrete constellations. By selecting weights via a divergence-optimal approach (w^{(k)} = -log P_A(a^{(k)})), the method minimizes the informational divergence to the target input distribution, improving the mutual information bound. Simulation results on a non-Gaussian optical-channel model show a measurable rate gain over CCDM and the ability to adapt shaping via the trellis' maximum weight level, offering practical rate adaptation benefits. The combination of generalized ESS with probabilistic amplitude shaping demonstrates robust performance improvements and provides an open-source implementation for broader adoption.

Abstract

A non-uniform channel input distribution is key for achieving the capacity of arbitrary channels. However, message bits are generally assumed to follow a uniform distribution which must first be transformed to a non-uniform distribution by using a distribution matching algorithm. One such algorithm is enumerative sphere shaping (ESS). Compared to algorithms such as constant composition distribution matching (CCDM), ESS can utilize more channel input symbol sequences, allowing it to achieve a comparably low rate loss. However, the distribution of channel input symbols produced by ESS is fixed, restricting the utility of ESS to channels with Gaussian-like capacity-achieving input distributions. In this paper, we generalize ESS to produce arbitrary discrete channel input distributions, making it usable on most channels. Crucially, our generalization replaces fixed weights used internally by ESS with weights depending on the desired channel input distribution. We present numerical simulations using generalized ESS with probabilistic amplitude shaping (PAS) to transmit sequences of 256 symbols over a simplified model of an unamplified coherent optical link, a channel with a distinctly non-Gaussian capacity-achieving input distribution. In these simulations, we found that generalized ESS improves the maximum transmission rate by 0.0425 bit/symbol at a frame error rate below 10^{-4} compared to CCDM.

An Extension of Enumerative Sphere Shaping for Arbitrary Channel Input Distributions

TL;DR

This work extends Enumerative Sphere Shaping to support arbitrary discrete channel input distributions by introducing distribution-dependent weights in a trellis framework, enabling non-uniform shaping on discrete constellations. By selecting weights via a divergence-optimal approach (w^{(k)} = -log P_A(a^{(k)})), the method minimizes the informational divergence to the target input distribution, improving the mutual information bound. Simulation results on a non-Gaussian optical-channel model show a measurable rate gain over CCDM and the ability to adapt shaping via the trellis' maximum weight level, offering practical rate adaptation benefits. The combination of generalized ESS with probabilistic amplitude shaping demonstrates robust performance improvements and provides an open-source implementation for broader adoption.

Abstract

A non-uniform channel input distribution is key for achieving the capacity of arbitrary channels. However, message bits are generally assumed to follow a uniform distribution which must first be transformed to a non-uniform distribution by using a distribution matching algorithm. One such algorithm is enumerative sphere shaping (ESS). Compared to algorithms such as constant composition distribution matching (CCDM), ESS can utilize more channel input symbol sequences, allowing it to achieve a comparably low rate loss. However, the distribution of channel input symbols produced by ESS is fixed, restricting the utility of ESS to channels with Gaussian-like capacity-achieving input distributions. In this paper, we generalize ESS to produce arbitrary discrete channel input distributions, making it usable on most channels. Crucially, our generalization replaces fixed weights used internally by ESS with weights depending on the desired channel input distribution. We present numerical simulations using generalized ESS with probabilistic amplitude shaping (PAS) to transmit sequences of 256 symbols over a simplified model of an unamplified coherent optical link, a channel with a distinctly non-Gaussian capacity-achieving input distribution. In these simulations, we found that generalized ESS improves the maximum transmission rate by 0.0425 bit/symbol at a frame error rate below 10^{-4} compared to CCDM.

Paper Structure

This paper contains 9 sections, 8 equations, 3 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Generalization of ESS
  3. Non-unique weights
  4. Choosing a Distribution via Weights
  5. Divergence-Optimal Weights
  6. Implementation Aspects
  7. Open Source Implementation
  8. Simulation Results
  9. Conclusion

Figures (3)

  • Figure 1: Generalized trellis with $N=4$ and non-unique weights $(0, \, 1, \, 1, \, 3)$.
  • Figure 2: Maximum rates achieving a below $10^{-4}$ for a fixed sequence length of $256$$8$- symbols.
  • Figure 3: Average amplitude energy for a sequence length of $224$ symbols and a shaping rate of $1.5$ bit / amplitude. Generalized uses a weight function with factor $f$ computed from a Maxwell-Boltzmann distribution with $\lambda = 0.1$ using \ref{['eq:weight-function']}.