Algorithmic complexity of $β$-expansions and application to A/D conversion

TL;DR

The paper analyzes how the Kolmogorov complexity of prefixes of binary expansions compares with $\beta$-expansions, revealing that $\beta$-expansions can exhibit higher complexity and affect compressibility in $\beta$-based A/D converters. By formalizing computable multivalued mappings between $\beta$-prefixes and binary prefixes, it derives lower and upper bounds on relative complexity, and introduces the operator $M_\beta$ to select canonical $\beta$-expansions. It provides refined results for almost all bases and for algebraic bases, with Pisot numbers giving zero relative complexity and enabling a linear-time algorithm to control complexity. A practical denoising algorithm for A/D conversion is proposed, leveraging Pisot bases (notably the golden ratio) to maintain binary-expansion-like complexity while preserving robustness. Overall, the work offers a rigorous framework and actionable methods for managing algorithmic complexity in $\beta$-expansions relevant to high-precision, noise-tolerant quantization systems.

Abstract

We establish diverse relationships between the algorithmic (Kolmogorov) complexity of the prefixes of any binary expansion and $β$-expansions. These relationships allow to develop intuitions on the complexity behavior of $β$-expansions, and raise problems related to compressibility of binary sequences generated in the context of A/D conversion relying on $β$-expansions. Our last contribution is to solve these problems.

Figures (1)

• Figure 1: Pipeline of A/D conversion with controlled algorithmic complexity. Here $s \in [0,1]$ and $\xf$ is a $\beta$-expansion of $s$. $M_\beta \xf$ displays the same algorithmic complexity as the greedy binary expansion of $s$.

Theorems & Definitions (91)

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