An optimal family of exponentially accurate one-bit Sigma-Delta quantization schemes

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 λ. It was recently shown that exponential accuracy of the form O(2^−rλ) 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 discrete 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 ≈ 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. © 2011 Wiley Periodicals, Inc.