Journal
APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS
Volume 32, Issue 1, Pages 131-138Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.acha.2011.06.003
Keywords
Dynamical systems in applications; Rate of convergence; Degree of approximation; Best constants; Approximation by arbitrary nonlinear expressions; Widths and entropy; Harmonic analysis on Euclidean spaces - probabilistic methods; Probability theory - combinatorial probability; Information and communication; Circuits
Categories
Funding
- National Science Foundation
- Courant Institute
- Division Of Mathematical Sciences
- Direct For Mathematical & Physical Scien [0902720] Funding Source: National Science Foundation
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Several analog-to-digital conversion methods for bandlimited signals used in applications, such as Sigma Delta quantization schemes, employ coarse quantization coupled with oversampling. The standard mathematical model for the error accrued from such methods 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(-alpha lambda)) can be achieved by appropriate one-bit Sigma-Delta modulation schemes. However, the best known achievable rate constants alpha in this setting differ significantly from the general information theoretic lower bound. In this paper, we provide the first lower bound specific to coarse quantization, thus narrowing the gap between existing upper and lower bounds. In particular, our results imply a quantitative correspondence between the maximal signal amplitude and the best possible error decay rate. Our method draws from the theory of large deviations. (C) 2011 Elsevier Inc. All rights reserved.
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