Quiet sigma delta quantization, and global convergence for a class of asymmetric piecewise affine maps

TL;DR

This paper introduces a new family of second-order sigma delta quantization schemes that are 'quiet', ensuring zero output at vanishing inputs, and proves their global convergence using Lyapunov methods and asymmetric map structure.

Contribution

The paper presents a novel class of 'quiet' sigma delta quantization schemes and establishes their global convergence for asymmetric piecewise affine maps.

Findings

Quantization output falls to zero at vanishing inputs.

The origin is a globally attractive fixed point.

Convergence is proven using Lyapunov functions and asymmetric map properties.

Abstract

In this paper, we introduce a family of second-order sigma delta quantization schemes for analog-to-digital conversion which are `quiet' : quantization output is guaranteed to fall to zero at the onset of vanishing input. In the process, we prove that the origin is a globally attractive fixed point for the related family of asymmetrically-damped piecewise affine maps. Our proof of convergence is twofold: first, we construct a trapping set using a Lyapunov-type argument; we then take advantage of the asymmetric structure of the maps under consideration to prove convergence to the origin from within this trapping set.