[Paper Review] Partition-Based Distribution Matching

Tobias Fehenberger, David S. Millar+3 more|arXiv (Cornell University)|Jan 25, 2018

Advanced Data Compression Techniques3 citations

TL;DR

This paper proposes partition-based distribution matching (PBDM), a novel fixed-length invertible mapping that partitions a target distribution into equal-size subsets to reduce rate loss compared to conventional constant-composition distribution matching (CCDM). By using a variable-length prefix to select composition and arithmetic coding for the remainder, PBDM achieves up to a fourfold block length reduction and a 2.5–5× gain in block length efficiency over CCDM at medium to high SNRs.

ABSTRACT

Distribution matching is a fixed-length invertible mapping from a uniformly distributed bit sequence to shaped amplitudes and as such plays an important role in the probabilistic amplitude shaping framework. With conventional constantcomposition distribution matching (CCDM), all output sequences have identical composition. In this paper, we propose partitionbased distribution matching (PBDM) where the composition is constant over all output sequences. When considering the desired distribution as a multiset, PBDM corresponds to partitioning this multiset into equal-size subsets. We show that PBDM allows to address more output sequences and thus has lower rate loss than CCDM in all nontrivial cases. By imposing some constraints on the partitioning, a constructive PBDM algorithm is proposed which comprises two parts. A variable-length prefix of the binary data word determines the composition to be used, and the remainder of the input word is mapped with a conventional CCDM algorithm, such as arithmetic coding, according to the chosen composition. For a specific target distribution and a fixed rate loss, we numerically find that PBDM gives a four-fold reduction in block length in comparison to CCDM. Simulations of 64-ary quadrature amplitude modulation over the additive white Gaussian noise channel demonstrate that the block-length saving of PBDM over CCDM for a fixed gap to capacity varies with the signal-to-noise ratio (SNR) and is approximately a factor of 2.5 to 5 at medium and high SNRs, respectively.

Motivation & Objective

• To reduce rate loss in distribution matching for probabilistic amplitude shaping by allowing variable composition across output sequences.
• To address the limitation of CCDM, which enforces identical composition on all output sequences, leading to higher rate loss.
• To design a constructive algorithm that enables practical implementation of distribution matching with lower block lengths.
• To demonstrate significant gains in block length efficiency over CCDM under realistic modulation and SNR conditions.

Proposed method

• PBDM partitions the target distribution into equal-size subsets, each with a fixed composition, enabling variable composition across output sequences.
• A variable-length prefix of the input determines which composition subset to use, based on the desired distribution.
• The remaining input bits are mapped using a conventional CCDM algorithm, such as arithmetic coding, according to the selected composition.
• The method ensures invertibility and fixed-length output by combining composition selection with standard CCDM mapping.
• Constraints on partitioning are imposed to ensure constructive and efficient implementation.
• The approach maintains the invertibility and fixed-length properties required for practical probabilistic shaping systems.

Experimental results

Research questions

• RQ1 Can a distribution matching scheme be designed that reduces rate loss by allowing variable composition across output sequences compared to CCDM?
• RQ2 How much block length reduction can be achieved with PBDM compared to CCDM for a given rate loss and target distribution?
• RQ3 What is the achievable gain in block length efficiency of PBDM over CCDM in practical systems like 64-ary QAM over AWGN?
• RQ4 How does the performance gain of PBDM vary with signal-to-noise ratio (SNR) in practical communication systems?
• RQ5 Can a constructive and efficient PBDM algorithm be designed that enables practical implementation?

Key findings

• PBDM achieves a four-fold reduction in block length compared to CCDM for a fixed rate loss and specific target distribution.
• Simulations show PBDM reduces block length by a factor of approximately 2.5 to 5 compared to CCDM at medium to high SNRs.
• PBDM consistently reduces rate loss in all nontrivial cases compared to CCDM.
• The block-length gain of PBDM increases with higher SNR, reaching up to a factor of 5 in high-SNR regimes.
• The proposed PBDM algorithm is constructive and practical, using a variable-length prefix to select composition and standard CCDM for the remainder.
• PBDM enables more output sequences than CCDM under the same rate loss, improving spectral efficiency.