Probabilistic Shaping for Finite Blocklengths: Distribution Matching and Sphere Shaping

TL;DR

This paper compares distribution matching and sphere shaping algorithms for short blocklength probabilistic amplitude shaping, highlighting that energy-efficient shaping techniques like MPDM and sphere shaping outperform traditional methods like CCDM in terms of rate loss and power efficiency.

Contribution

It introduces a new perspective on shaping for short blocklengths, emphasizing energy efficiency over distribution matching, and provides a systematic comparison of algorithms like MPDM, ESS, and SpSh.

Findings

MPDM and SpSh have smaller rate losses than CCDM.

SpSh achieves the minimum rate loss among the methods.

Up to 1 dB power efficiency improvement with MPDM and SpSh at blocklengths around 200.

Abstract

In this paper, we provide for the first time a systematic comparison of distribution matching (DM) and sphere shaping (SpSh) algorithms for short blocklength probabilistic amplitude shaping. For asymptotically large blocklengths, constant composition distribution matching (CCDM) is known to generate the target capacity-achieving distribution. As the blocklength decreases, however, the resulting rate loss diminishes the efficiency of CCDM. We claim that for such short blocklengths and over the additive white Gaussian channel (AWGN), the objective of shaping should be reformulated as obtaining the most energy-efficient signal space for a given rate (rather than matching distributions). In light of this interpretation, multiset-partition DM (MPDM), enumerative sphere shaping (ESS) and shell mapping (SM), are reviewed as energy-efficient shaping techniques. Numerical results show that MPDM and SpSh have smaller rate losses than CCDM. SpSh--whose sole objective is to maximize the energy efficiency--is shown to have the minimum rate loss amongst all. We provide simulation results of the end-to-end decoding performance showing that up to 1 dB improvement in power efficiency over uniform signaling can be obtained with MPDM and SpSh at blocklengths around 200. Finally, we present a discussion on the complexity of these algorithms from the perspective of latency, storage and computations.