An optimal family of exponentially accurate one-bit Sigma-Delta quantization schemes
Article first published online: 14 MAR 2011
Copyright © 2011 Wiley Periodicals, Inc.
Communications on Pure and Applied Mathematics
Volume 64, Issue 7, pages 883–919, July 2011
How to Cite
Deift, P., Krahmer, F. and Güntürk, C. S. (2011), An optimal family of exponentially accurate one-bit Sigma-Delta quantization schemes. Comm. Pure Appl. Math., 64: 883–919. doi: 10.1002/cpa.20367
- Issue published online: 4 APR 2011
- Article first published online: 14 MAR 2011
- Manuscript Received: JAN 2010
- National Science Foundation. Grant Numbers: DMJ-0500923, CCF-0515187
- Alfred P. Sloan Research Fellowship
- Morawetz Fellowship at the Courant Institute
- Charles M. Newman Fellowship at the Courant Institute
- NYU GSAS Dean's Student Travel Grant
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.