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Asymptotic quantization errors for unbounded quantizers

Author: M. Shykula
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 75 (2006).
Journal: Theor. Probability and Math. Statist. 75 (2007), 189-199
MSC (2000): Primary 60G99; Secondary 94A29, 94A34
Published electronically: January 25, 2008
MathSciNet review: 2321192
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Abstract | References | Similar Articles | Additional Information


We consider non-uniform scalar quantization for a wide class of unbounded random variables (or values of a random process sampled in time). Asymptotic stochastic structures for quantization errors are derived for two types of quantizers when the number of quantization levels tends to infinity. The corresponding results for bounded random variables are generalized. Some numerical examples illustrate the rate of convergence.

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  • 1. W. R. Bennett, Spectrum of quantized signal, Bell. Syst. Tech. J. 27 (1948), 446-472. MR 0026287 (10:133f)
  • 2. J. A. Bucklew and G. L. Wise, Multidimensional asymptotic quantization theory with $ r$th power distortion measures, IEEE Trans. Inform. Theory 28 (1982), 239-247. MR 651819 (83b:94026)
  • 3. S. Cambanis and N. Gerr, A simple class of asymptotically optimal quantizers, IEEE Trans. Inform. Theory 29 (1983), 664-676. MR 730904 (85a:94001)
  • 4. S. Graf and H. Luschgy, Foundations of Quantization for Probability Distributions, Springer-Verlag, New York, 2000. MR 1764176 (2001m:60043)
  • 5. R. M. Gray and T. Linder, Mismatch in high rate entropy constrained vector quantization, IEEE Trans. Inform. Theory 49 (2003), 1204-1217. MR 1984821 (2004e:94026)
  • 6. R. M. Gray and D. L. Neuhoff, Quantization, IEEE Trans. Inform. Theory 44 (1998), 2325-2383. MR 1658787 (99i:94029)
  • 7. D. H. Lee and D. L. Neuhoff, Asymptotic distribution of the errors in scalar and vector quantizers, IEEE Trans. Inform. Theory 42 (1996), 446-460.
  • 8. J. Li, N. Chaddha, and R. M. Gray, Asymptotic performance of vector quantizers with a perceptual distortion measure, IEEE Trans. Inform. Theory 45 (1999), 1082-1091. MR 1686244 (2000b:94008)
  • 9. T. Linder, On asymptotically optimal companding quantization, Probl. Contr. Inform. Theory 20 (1991), 475-484.
  • 10. H. Luschgy and G. Pagès, Functional quantization of Gaussian processes, Jour. Func. Annal. 196 (2002), 486-531. MR 1943099 (2003i:60006)
  • 11. O. Seleznjev, Spline approximation of random processes and design problems, Jour. Stat. Plan. Infer. 84 (2000), 249-262. MR 1747507 (2001e:62082)
  • 12. M. Shykula and O. Seleznjev, Uniform quantization of random processes, Univ. Umeå Research Report 1 (2004), 1-16.
  • 13. M. Shykula and O. Seleznjev, Stochastic structure of asymptotic quantization errors, Stat. Prob. Letters 76 (2006), 453-464. MR 2266598
  • 14. P. L. Zador, Asymptotic quantization error of continuous signals and the quantization dimensions, IEEE Trans. Inform. Theory 28 (1982), 139-148. MR 651809 (83b:94014)

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Additional Information

M. Shykula
Affiliation: Department of Mathematics and Mathematical Statistics, Umeå University, SE-901 87 Umeå, Sweden

Keywords: Non-uniform scalar quantization, random process, stochastic structure
Received by editor(s): July 24, 2005
Published electronically: January 25, 2008
Article copyright: © Copyright 2008 American Mathematical Society

American Mathematical Society