Polar Coded Quantization for Distributed Source Coding

Abstract: Scalar quantization and probabilistic shaping are applied to the distributed source coding of Gaussian sources, with mean-square error distortion. A coding scheme with a modulo interval, dithering, and truncated Gaussian shaping is shown to achieve the corner points of the Berger-Tung region. The theory is illustrated by designing short-block-length multilevel 5G polar codes for Wyner-Ziv (WZ) polar coded quantization (PCQ). WZ-PCQ substantially reduces the total distortion compared to separate PCQ of the source blocks.

• The paper demonstrates that a Wyner-Ziv construction with modulo operations, dithering, and truncated Gaussian shaping achieves the corner points of the Berger-Tung region for quadratic Gaussian sources.
• It introduces a multilevel, short-block-length polar code structure compatible with 5G systems, significantly reducing distortion compared to standard quantization methods.
• Empirical results validate that the proposed PCQ-mod system outperforms traditional schemes under asymmetric rate allocations and strong source correlations, bridging theory and practical implementation.

Polar Coded Quantization for Distributed Source Coding

Overview and Motivation

The paper addresses the construction of practical coding schemes for distributed source coding (DSC) of correlated Gaussian sources under mean-square error (MSE) distortion. The focus is on leveraging polar codes with probabilistic shaping and scalar quantization, both with and without a modulo operation and dithering. The primary theoretical achievement is a demonstration that a Wyner-Ziv (WZ) construction, incorporating modulo intervals, dithering, and truncated Gaussian shaping, achieves the corner points of the Berger-Tung region for quadratic Gaussian sources. The work is carried through to short-block-length multilevel 5G polar code implementations, providing empirical evidence of the distortion-rate gains.

Theoretical Contributions

Precise Characterization of the Berger-Tung Region with Practical Codes

The Berger-Tung region describes the achievable rate-distortion tuples for multi-terminal source coding; it is especially important in quadratic Gaussian problems, where the rate region is explicitly characterized, and joint decoding with optimal estimators is feasible. Previous works have shown the optimality of nested lattice codes and WZ constructions for these problems, but practical realizations at short to moderate block lengths remained elusive.

This paper's core theoretical result is the rigorous proof that, when scalar quantization is augmented by a modulo operation, dithering, and optimal (truncated Gaussian) probabilistic shaping, the resulting WZ coding architecture can achieve the corner points of the Berger-Tung achievable region for distributed Gaussian source coding. This is particularly notable because such an architecture is compatible with polar-coded quantization (PCQ), opening the path to low-complexity, high-performance implementations.

Analysis of Successive Decoding and Construction of the Coding Scheme

The authors detail a two-encoder, one-decoder model, where each encoder processes its own source without access to the other. The decoder receives both messages and reconstructs the sources subject to MSE constraints, following the Berger-Tung model. The innovation lies in showing that (i) one encoder uses a standard Shannon-Rate-Distortion code, and (ii) the other employs a WZ construction with side information, implemented via PCQ with a modulo interval and dithering (PCQ-mod).

Through analysis of mutual information expressions and the use of Markov properties of the Gaussian source, the authors derive encoding, inflation, and shaping parameters that align with the theoretical region. They provide explicit formulas for the code parameters, modulo intervals, dithers, constellation sizes, and shaping variances.

Practical Design and Empirical Validation

Short-Block-Length Multilevel Polar Codes

The work devises explicit 5G-compatible multilevel polar codes for PCQ and PCQ-mod, targeting block length $n=256$ and levels matching standard ASK constellations (e.g., 8-ASK and 16-ASK). The codes are designed using successive cancellation list (SCL) encoding and decoding with list passing across levels, aligning with practical constraints in future wireless systems.

Strong Numerical Results

Empirical results confirm that for a bivariate Gaussian source (covariance matrix $Q_{\bm X}$ with off-diagonal correlation), the PCQ-mod/WZ approach dramatically outperforms standard PCQ. At $(R_1, R_2) = (1, 2)$ bits/sample:

• Case 1 (WZ-PCQ-mod / PCQ): $E[\Delta_1] = 0.372$, $E[\Delta_2] = 0.198$, $E[\Delta_1] + E[\Delta_2] = 0.570$
• Case 2 (PCQ/PCQ): $E[\Delta_1] = 0.531$, $E[\Delta_2] = 0.205$, $E[\Delta_1] + E[\Delta_2] = 0.736$

This demonstrates a substantial reduction in total distortion by employing WZ-coded quantization with side information, particularly under conditions of strong source correlation and asymmetric rates.

Implications and Comparison to Prior Art

Advancement over Lattice and Classical Code Designs

By mapping the WZ scenario to a practical, multilevel polar code construction with modulo and dithering, the scheme provides advantages previously exclusive to high-dimensional lattice approaches, now available with much lower complexity and shorter block lengths. The construction naturally integrates with shaping techniques (discrete or truncated Gaussian) and is amenable to practical system constraints relevant to 5G and beyond.

Offset from Traditional Techniques

While binary-source polar WZ codes previously existed, they lacked multilevel capability and thus were not suited for moderate to high effective rates. The PCQ-mod architecture generalizes the approach and supports efficient distributed quantization, significantly improving achievable distortion, especially when side information is strong.

Future Directions

The methodology invites several extensions:

• Extension to higher-order sources, multidimensional settings, and more general source models (beyond Gaussian).
• Application to other multiterminal source coding scenarios, e.g., CEO problems with more than two agents and various noise models.
• Further reduction of block length and complexity for ultra-low-delay, ultra-reliable applications.
• Integration with real-time shaping and adaptive parameter adjustment for nonstationary sources.
• Investigation of information-theoretic secrecy benefits induced by dithering.

Conclusion

The paper presents both a theoretical and practical advance in distributed source coding for correlated Gaussian sources. By incorporating a modulo operation, dithering, and probabilistic shaping into a WZ/PCQ framework using multilevel polar codes, the authors achieve performance at the Berger-Tung region boundary, even for short block lengths relevant to current communication standards. The numerical results establish clear superiority of WZ-PCQ-mod over separate quantization in terms of total distortion, particularly under unbalanced rate allocations or strong source correlation. This research closes the gap between information-theoretic optimality and practical code implementation for distributed lossy coding of continuous sources.

Reference: "Polar Coded Quantization for Distributed Source Coding" (2604.18335)