Polar Complexity: A New Descriptive Complexity with Applications to Source and Joint Source-Channel Coding

TL;DR

This paper introduces polar complexity as a new measure for binary sequences, develops efficient algorithms to compute it, and applies it to create a lossless, adaptive joint source-channel coding scheme with promising performance and robustness.

Contribution

It defines polar complexity, proposes a novel polar-based source coding scheme, and integrates it with channel coding for adaptive JSCC with optimized complexity-performance tradeoff.

Findings

The proposed scheme is strictly lossless and prefix-free.

Normalized average compression length approaches source entropy asymptotically.

The adaptive JSCC scheme outperforms existing polar-code-based baselines.

Abstract

This paper first presents a new approach to evaluating the descriptive complexity of finite-length binary sequences. Specifically, we investigate the sequence-wise recovery behavior induced by polar compression and successive cancellation decoding (SCD), and define the polar complexity of a sequence as the minimum polar-compression length (PCL) required for its exact reconstruction. To compute the polar complexity efficiently, we further develop both a bisection-search algorithm and a low-complexity estimation method. We then propose a polar-based two-stage source coding scheme, in which each source sequence is represented by its polar complexity followed by the corresponding polar-compressed sequence. The proposed scheme is strictly lossless and prefix-free. In addition, for BMSs, the normalized average compression length of the proposed scheme can asymptotically approach the source entropy under certain conditions. Simulation results further demonstrate that the scheme can operate without prior knowledge of the source statistics and remains robust across different source distributions. Finally, we integrate the proposed polar source coding with polar channel coding to develop an adaptive double-polar joint source-channel coding (JSCC) scheme, where the encoder and decoder share a predefined set of candidate PCLs to balance error performance and decoding complexity. We formulate the design of the candidate-PCL set as an optimization problem and solve it efficiently via dynamic programming. Simulation results show that the proposed adaptive double-polar JSCC scheme provides a flexible performance-complexity tradeoff and outperforms existing polar-code-based JSCC baselines.