An Optimal Family of Exponentially Accurate One-Bit Sigma-Delta Quantization Schemes

TL;DR

This paper develops an optimized class of one-bit Sigma-Delta quantization schemes that achieve improved exponential accuracy in analog-to-digital conversion by leveraging orthogonal polynomial theory.

Contribution

It introduces a new optimization approach for feedback filters in Sigma-Delta schemes, improving the exponential error decay rate from approximately 0.088 to 0.102.

Findings

Achieved a new exponential decay rate of approximately 0.102.

Connected optimal filters to Chebyshev polynomials of the second kind.

Provided explicit asymptotics for the solutions of the relaxed optimization problem.

Abstract

Sigma-Delta modulation is a popular method for analog-to-digital conversion of bandlimited signals that employs coarse quantization coupled with oversampling. The standard mathematical model for the error analysis of the method measures the performance of a given scheme by the rate at which the associated reconstruction error decays as a function of the oversampling ratio $\lambda$. It was recently shown that exponential accuracy of the form $O(2^{-r\lambda})$ can be achieved by appropriate one-bit Sigma-Delta modulation schemes. By general information-entropy arguments $r$ must be less than 1. The current best known value for $r$ is approximately 0.088. The schemes that were designed to achieve this accuracy employ the "greedy" quantization rule coupled with feedback filters that fall into a class we call "minimally supported". In this paper, we study the minimization problem that corresponds to optimizing the error decay rate for this class of feedback filters. We solve a relaxed version of this problem exactly and provide explicit asymptotics of the solutions. From these relaxed solutions, we find asymptotically optimal solutions of the original problem, which improve the best known exponential error decay rate to $r \approx 0.102$. Our method draws from the theory of orthogonal polynomials; in particular, it relates the optimal filters to the zero sets of Chebyshev polynomials of the second kind.