One-Bit Sigma-Delta modulation on the circle

TL;DR

This paper investigates Sigma-Delta modulation for analog-to-digital conversion of signals on a circle, proposing updates to improve initialization errors and reconstruction accuracy in manifold-based signal processing.

Contribution

It introduces novel update strategies for first and second order Sigma-Delta schemes tailored for circular manifold models, enhancing error performance.

Findings

Improved reconstruction error with the proposed updates.

Effective handling of initialization mismatches.

Enhanced accuracy in A/D conversion on the circle.

Abstract

Manifold models in data analysis and signal processing have become more prominent in recent years. In this paper, we will look at one of the main tasks of modern signal processing, namely, at analog-to-digital (A/D) conversion in connection with a simple manifold model - the circle. We will focus on Sigma-Delta modulation which is a popular method for A/D conversion of bandlimited signals that employs coarse quantization coupled with oversampling. Classical Sigma-Delta schemes provide mismatches and large errors at the initialization point if the signal to be converted is defined on the circle. In this paper, our goal is to get around these problems for Sigma-Delta schemes. Our results show how to design an update for the first and the second order schemes based on the reconstruction error analysis such that for the updated scheme the reconstruction error is improved.