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Approximating a bandlimited function using very coarsely quantized data: Improved error estimates in sigma-delta modulation

Author: C. Sinan Güntürk
Journal: J. Amer. Math. Soc. 17 (2004), 229-242
MSC (2000): Primary 94A20, 11K06; Secondary 11L07, 41A25
Published electronically: August 1, 2003
MathSciNet review: 2015335
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Abstract: Sigma-delta quantization is a method of representing bandlimited signals by $0{-}1$ sequences that are computed from regularly spaced samples of these signals; as the sampling density $\lambda \to \infty$, convolving these one-bit sequences with appropriately chosen kernels produces increasingly close approximations of the original signals. This method is widely used for analog-to-digital and digital-to-analog conversion, because it is less expensive and simpler to implement than the more familiar critical sampling followed by fine-resolution quantization. We present examples of how tools from number theory and harmonic analysis are employed in sharpening the error estimates in sigma-delta quantization.

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Additional Information

C. Sinan Güntürk
Affiliation: Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, New York 10012-1185

Keywords: A/D conversion, sigma-delta modulation, sampling, quantization, uniform distribution, discrepancy, exponential sums.
Received by editor(s): April 11, 2003
Published electronically: August 1, 2003
Additional Notes: The author’s research was supported in part by the Francis Robbins Upton honorific fellowship from Princeton University, the NSF Grant 97-29992 at the Institute for Advanced Study, and the NSF Grant DMS-0219072.
Article copyright: © Copyright 2003 American Mathematical Society