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

DOI:
https://doi.org/10.1090/S0894-0347-03-00436-3

Published electronically:
August 1, 2003

MathSciNet review:
2015335

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Sigma-delta quantization is a method of representing bandlimited signals by sequences that are computed from regularly spaced samples of these signals; as the sampling density , 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.

**1.**J. C. Candy and G. C. Temes, Eds.,*Oversampling Delta-Sigma Data Converters: Theory, Design and Simulation*,

IEEE Press, 1992.**2.**W. Chou, T. H. Meng, and R. M. Gray,

``Time Domain Analysis of Sigma Delta Modulation,''*Proceedings ICASSP-90*, Int. Conf. on Acoustics, Speech and Signal Processing, vol. 3, pp. 1751-1754, Albuquerque, NM, April 1990.**3.**I. Daubechies, R. DeVore,

``Approximating a Bandlimited Function Using Very Coarsely Quantized Data: A Family of Stable Sigma-Delta Modulators of Arbitrary Order'', to appear in Annals of Mathematics.**4.**R. M. Gray,

``Spectral Analysis of Quantization Noise in a Single-Loop Sigma-Delta Modulator with dc Input,''*IEEE Trans. on Comm.*, vol. COM-37, pp. 588-599, June 1989.**5.**C. S. Güntürk,

``Improved Error Estimates for First Order Sigma-Delta Systems,''*Proceedings SampTA-99*, Int. Workshop on Sampling Theory and Applications, Loen, Norway, August 1999.**6.**C. S. Güntürk and N. T. Thao,

``Refined Analysis of MSE in Second Order Sigma-Delta Modulation with DC Inputs,'' submitted to IEEE Transactions on Information Theory, in revision.**7.**C. S. Güntürk,

``One-Bit Sigma-Delta Quantization with Exponential Accuracy,'' to appear in Communications on Pure and Applied Mathematics.**8.**L. Kuipers and H. Niederreiter,*Uniform Distribution of Sequences*,

Wiley, 1974. MR**54:7415****9.**Y. Meyer,*Wavelets and Operators*,

Cambridge University Press, 1992. MR**92k:42001****10.**H. L. Montgomery,*Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis*,

AMS, 1994. MR**96i:11002****11.**S. R. Norsworthy, R. Schreier, and G. C. Temes, Eds.,*Delta-Sigma Data Converters: Theory, Design and Simulation*,

IEEE Press, 1996.**12.**E. M. Stein,*Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals*,

Princeton University Press, 1993. MR**95c:42002**

Retrieve articles in *Journal of the American Mathematical Society*
with MSC (2000):
94A20,
11K06,
11L07,
41A25

Retrieve articles in all journals with MSC (2000): 94A20, 11K06, 11L07, 41A25

Additional Information

**C. Sinan Güntürk**

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

Email:
gunturk@cims.nyu.edu

DOI:
https://doi.org/10.1090/S0894-0347-03-00436-3

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