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An optimal family of exponentially accurate one‐bit Sigma‐Delta quantization schemes
Author(s) -
Deift Percy,
Krahmer Felix,
Güntürk C. Sınan
Publication year - 2011
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20367
Subject(s) - oversampling , mathematics , quantization (signal processing) , bandlimiting , sigma , exponential function , chebyshev filter , asymptotically optimal algorithm , delta sigma modulation , algorithm , discrete mathematics , computer science , mathematical analysis , fourier transform , bandwidth (computing) , computer network , physics , quantum mechanics
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 − r λ ) 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.