Approximating a bandlimited function using very coarsely quantized data: Improved error estimates in sigmadelta modulation
Author:
C. Sinan Güntürk
Journal:
J. Amer. Math. Soc. 17 (2004), 229242
MSC (2000):
Primary 94A20, 11K06; Secondary 11L07, 41A25
Published electronically:
August 1, 2003
MathSciNet review:
2015335
Fulltext PDF Free Access
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Additional Information
Abstract: Sigmadelta 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 onebit sequences with appropriately chosen kernels produces increasingly close approximations of the original signals. This method is widely used for analogtodigital and digitaltoanalog conversion, because it is less expensive and simpler to implement than the more familiar critical sampling followed by fineresolution quantization. We present examples of how tools from number theory and harmonic analysis are employed in sharpening the error estimates in sigmadelta quantization.
 1.
J. C. Candy and G. C. Temes, Eds.,
Oversampling DeltaSigma 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 ICASSP90, Int. Conf. on Acoustics, Speech and Signal Processing, vol. 3, pp. 17511754, Albuquerque, NM, April 1990.
 3.
I. Daubechies, R. DeVore,
``Approximating a Bandlimited Function Using Very Coarsely Quantized Data: A Family of Stable SigmaDelta Modulators of Arbitrary Order'', to appear in Annals of Mathematics.
 4.
R. M. Gray,
``Spectral Analysis of Quantization Noise in a SingleLoop SigmaDelta Modulator with dc Input,'' IEEE Trans. on Comm., vol. COM37, pp. 588599, June 1989.
 5.
C. S. Güntürk,
``Improved Error Estimates for First Order SigmaDelta Systems,'' Proceedings SampTA99, 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 SigmaDelta Modulation with DC Inputs,'' submitted to IEEE Transactions on Information Theory, in revision.
 7.
C. S. Güntürk,
``OneBit SigmaDelta Quantization with Exponential Accuracy,'' to appear in Communications on Pure and Applied Mathematics.
 8.
L.
Kuipers and H.
Niederreiter, Uniform distribution of sequences,
WileyInterscience [John Wiley & Sons], New YorkLondonSydney, 1974.
Pure and Applied Mathematics. MR 0419394
(54 #7415)
 9.
Charles
K. Chui (ed.), Wavelets, Wavelet Analysis and its
Applications, vol. 2, Academic Press, Inc., Boston, MA, 1992. A
tutorial in theory and applications. MR 1161244
(92k:42001)
 10.
Hugh
L. Montgomery, Ten lectures on the interface between analytic
number theory and harmonic analysis, CBMS Regional Conference Series
in Mathematics, vol. 84, Published for the Conference Board of the
Mathematical Sciences, Washington, DC; by the American Mathematical
Society, Providence, RI, 1994. MR 1297543
(96i:11002)
 11.
S. R. Norsworthy, R. Schreier, and G. C. Temes, Eds.,
DeltaSigma Data Converters: Theory, Design and Simulation, IEEE Press, 1996.
 12.
Elias
M. Stein, Harmonic analysis: realvariable methods, orthogonality,
and oscillatory integrals, Princeton Mathematical Series,
vol. 43, Princeton University Press, Princeton, NJ, 1993. With the
assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
(95c:42002)
 1.
 J. C. Candy and G. C. Temes, Eds.,
Oversampling DeltaSigma 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 ICASSP90, Int. Conf. on Acoustics, Speech and Signal Processing, vol. 3, pp. 17511754, Albuquerque, NM, April 1990.
 3.
 I. Daubechies, R. DeVore,
``Approximating a Bandlimited Function Using Very Coarsely Quantized Data: A Family of Stable SigmaDelta Modulators of Arbitrary Order'', to appear in Annals of Mathematics.
 4.
 R. M. Gray,
``Spectral Analysis of Quantization Noise in a SingleLoop SigmaDelta Modulator with dc Input,'' IEEE Trans. on Comm., vol. COM37, pp. 588599, June 1989.
 5.
 C. S. Güntürk,
``Improved Error Estimates for First Order SigmaDelta Systems,'' Proceedings SampTA99, 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 SigmaDelta Modulation with DC Inputs,'' submitted to IEEE Transactions on Information Theory, in revision.
 7.
 C. S. Güntürk,
``OneBit SigmaDelta 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.,
DeltaSigma Data Converters: Theory, Design and Simulation, IEEE Press, 1996.
 12.
 E. M. Stein,
Harmonic Analysis: RealVariable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, 1993. MR 95c:42002
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Additional Information
C. Sinan Güntürk
Affiliation:
Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, New York 100121185
Email:
gunturk@cims.nyu.edu
DOI:
http://dx.doi.org/10.1090/S0894034703004363
PII:
S 08940347(03)004363
Keywords:
A/D conversion,
sigmadelta 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 9729992 at the Institute for Advanced Study, and the NSF Grant DMS0219072.
Article copyright:
© Copyright 2003
American Mathematical Society
