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

Research output: Contribution to journalArticle

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.

Original languageEnglish (US)
Pages (from-to)883-919
Number of pages37
JournalCommunications on Pure and Applied Mathematics
Volume64
Issue number7
DOIs
StatePublished - Jul 2011

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Delta sigma modulation
Oversampling
Quantization
Decay Rate
Modulation
Polynomials
Filter
Feedback
Optimal Filter
Information Entropy
Zero set
Analog to digital conversion
Chebyshev Polynomials
Asymptotically Optimal
Error Analysis
Orthogonal Polynomials
Minimization Problem
Error analysis
Standard Model
Entropy

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

An optimal family of exponentially accurate one-bit Sigma-Delta quantization schemes. / Deift, Percy; Krahmer, Felix; Gunturk, C. Sinan.

In: Communications on Pure and Applied Mathematics, Vol. 64, No. 7, 07.2011, p. 883-919.

Research output: Contribution to journalArticle

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