Constant Composition Distribution Matching (1503.05133v1)

Abstract: Distribution matching transforms independent and Bernoulli(1/2) distributed input bits into a sequence of output symbols with a desired distribution. Fixed-to-fixed length, invertible, and low complexity encoders and decoders based on constant composition and arithmetic coding are presented. Asymptotically in the blocklength, the encoder achieves the maximum rate, namely the entropy of the desired distribution. Furthermore, the normalized divergence of the encoder output and the desired distribution goes to zero in the blocklength.

• The paper introduces a fixed-to-fixed length distribution matching method that converts Bernoulli input bits into symbols with a predetermined distribution using constant composition codes.
• It demonstrates asymptotic optimality by showing that the encoder’s rate converges to the entropy of the desired distribution while reducing normalized divergence with increasing block length.
• It employs arithmetic coding for efficient, online codebook construction, significantly reducing memory overhead compared to traditional variable-length approaches.

Constant Composition Distribution Matching: An Overview

Patrick Schulte and Georg Bocherer's paper, "Constant Composition Distribution Matching," presents an advanced approach to distribution matching through a fixed-to-fixed length (f2f) framework. This work illustrates the effective transformation of Bernoulli-distributed input bits into output symbols with a predetermined distribution, emphasizing the efficiency and complexity reductions achieved by leveraging constant composition codes and arithmetic coding.

Key Contributions

The paper addresses several pivotal aspects in the domain of distribution matching:

1. Fixed-to-Fixed Length Encoding: Unlike prior approaches predominantly focused on variable-length encoding, this research emphasizes f2f encoding, mitigating synchronization issues and variable rate problems that previously afflicted transmission systems.
2. Asymptotic Optimality: The proposed constant composition distribution matcher (ccdm) approaches optimality in the limit of large block lengths. Specifically, as block length increases, the rate achieved by the encoder aligns with the entropy of the desired distribution, and the divergence between the encoder output and the desired distribution diminishes to zero.
3. Feasibility and Efficiency: The use of arithmetic coding enables efficient online codebook construction, avoiding the impractical storage requirements of offline codebook generation for large inputs.

Theoretical Implications

The paper meticulously derives the asymptotic behavior of the ccdm. The researchers demonstrate that the achievable rate $R$ satisfies the condition $R \leq H(A)$, where $H(A)$ is the entropy of the desired distribution. Furthermore, through rigorous analysis, the authors establish that the normalized informational divergence is minimized as block length increases, confirming the asymptotic equivalence of the ccdm’s output and the target distribution.

Practical Implications

The practical significance of this research extends into robust communication systems. By maintaining a constant composition in the matcher output, the system effectively emulates a discrete memoryless source, beneficial for systems requiring precise rate adaptation or operating close to channel capacity, such as additive white Gaussian noise channels.

Numerical Results

The numerical evaluation presented in the paper underscores that the ccdm requires substantially fewer resources compared to previously optimal solutions. For instance, while achieving a normalized divergence of 0.06 bits/symbol, the ccdm requires merely four times the block length compared to the optimal variable-length schemes, yet operates with significantly less memory overhead.

Future Work

Looking forward, the paper suggests future exploration into finite block length performance improvements for f2f codes. Such developments could further cement distribution matching's role in efficient and reliable data transmission across various communication paradigms.

Conclusion

Schulte and Bocherer’s work on constant composition distribution matching offers a comprehensive framework for f2f length encoding, blending theoretical precision with practical feasibility. Their approach mitigates common concerns in earlier methods, like large buffer sizes and error propagation, and sets a foundation for future advancements in distribution matching with applications in modern communication systems.