Lower bounds for the error decay incurred by coarse quantization schemes

TL;DR

This paper establishes fundamental lower bounds on the error decay rates for coarse quantization schemes in analog-to-digital conversion, narrowing the gap between theoretical limits and existing methods.

Contribution

It provides the first specific lower bounds for coarse quantization error decay, linking maximal signal amplitude to decay rate, using large deviations theory.

Findings

Lower bounds on error decay rates are derived.

The bounds relate signal amplitude to achievable accuracy.

Results improve understanding of fundamental limits in coarse quantization.

Abstract

Several analog-to-digital conversion methods for bandlimited signals used in applications, such as Sigma Delta quantization schemes, employ coarse quantization coupled with oversampling. The standard mathematical model for the error accrued from such methods measures the performance of a given scheme by the rate at which the associated reconstruction error decays as a function of the oversampling ratio L > 1. It was recently shown that exponential accuracy of the form O(2(-r L)) can be achieved by appropriate one-bit Sigma Delta modulation schemes. However, the best known achievable rate constants r in this setting differ significantly from the general information theoretic lower bound. In this paper, we provide the first lower bound specific to coarse quantization, thus narrowing the gap between existing upper and lower bounds. In particular, our results imply a quantitative correspondence between the maximal signal amplitude and the best possible error decay rate. Our method draws from the theory of large deviations.