# Enumerative Sphere Shaping for Wireless Communications with Short Packets

## TL;DR

This paper introduces Enumerative Sphere Shaping (ESS) as an efficient alternative to CCDM for probabilistic amplitude shaping in wireless communications, especially effective at short block lengths, improving energy efficiency over AWGN and frequency-selective channels.

## Contribution

The paper proposes ESS to reduce rate loss at short block lengths, analyzes its computational complexity, and demonstrates its practical benefits in wireless communication systems.

## Key findings

- ESS outperforms CCDM in rate loss at short block lengths.
- ESS is computationally more efficient than shell mapping.
- Simulations show significant energy efficiency gains with ESS.

## Abstract

Probabilistic amplitude shaping (PAS) combines an outer shaping layer with an inner, systematic forward error correction (FEC) layer to close the shaping gap. Proposed for PAS, constant composition distribution matching (CCDM) produces amplitude sequences with a fixed empirical distribution. We show that CCDM suffers from high rate losses for small block lengths, and we propose to use Enumerative Sphere Shaping (ESS) instead. ESS minimizes the rate loss at any block length. Furthermore, we discuss the computational complexity of ESS and demonstrate that it is significantly smaller than shell mapping (SM), which is another method to perform sphere shaping. We then study the choice of design parameters for PAS. Following Wachsmann et al., we show that for a given constellation and target rate, there is an optimum balance between the FEC code rate and the entropy of the Maxwell-Boltzmann distribution that minimizes the gap-to-capacity. Moreover, we demonstrate how to utilize the non-systematic convolutional code from IEEE 802.11 in PAS. Simulations over the additive white Gaussian noise (AWGN) and frequency-selective channels exhibit that ESS is up to 1.6 and 0.7 dB more energy-efficient than uniform signaling at block lengths as small as 96 symbols, respectively, with convolutional and low-density parity-check (LDPC) codes.