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Quantization of Random Processes and Related Statistical Problems
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
2006 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis we study a scalar uniform and non-uniform quantization of random processes (or signals) in average case setting. Quantization (or discretization) of a signal is a standard task in all nalog/digital devices (e.g., digital recorders, remote sensors etc.). We evaluate the necessary memory capacity (or quantization rate) needed for quantized process realizations by exploiting the correlation structure of the model random process. The thesis consists of an introductory survey of the subject and related theory followed by four included papers (A-D).

In Paper A we develop a quantization coding method when quantization levels crossings by a process realization are used for its coding. Asymptotical behavior of mean quantization rate is investigated in terms of the correlation structure of the original process. For uniform and non-uniform quantization, we assume that the quantization cellwidth tends to zero and the number of quantization levels tends to infinity, respectively.

In Papers B and C we focus on an additive noise model for a quantized random process. Stochastic structures of asymptotic quantization errors are derived for some bounded and unbounded non-uniform quantizers when the number of quantization levels tends to infinity. The obtained results can be applied, for instance, to some optimization design problems for quantization levels.

Random signals are quantized at sampling points with further compression. In Paper D the concern is statistical inference for run-length encoding (RLE) method, one of the compression techniques, applied to quantized stationary Gaussian sequences. This compression method is widely used, for instance, in digital signal and image processing. First, we deal with mean RLE quantization rates for various probabilistic models. For a time series with unknown stochastic structure, we investigate asymptotic properties (e.g., asymptotic normality) of two estimates for the mean RLE quantization rate based on an observed sample when the sample size tends to infinity.

These results can be used in communication theory, signal processing, coding, and compression applications. Some examples and numerical experiments demonstrating applications of the obtained results for synthetic and real data are presented.

Place, publisher, year, edition, pages
Umeå: Matematik och matematisk statistik , 2006. , 107 p.
Keyword [en]
scalar quantization, random process, rate, distortion, additive noise model, run-length encoding, compression, sample estimate, asymptotical normality
National Category
Probability Theory and Statistics
Identifiers
URN: urn:nbn:se:umu:diva-883ISBN: 91-7264-183-5 (print)OAI: oai:DiVA.org:umu-883DiVA: diva2:144878
Public defence
2006-10-27, MA 121, MIT-hus, Umeå University, SE-901 87, Umeå, Sweden, 10:15
Opponent
Supervisors
Available from: 2006-10-05 Created: 2006-10-05 Last updated: 2013-01-15Bibliographically approved
List of papers
1. Uniform and non-uniform quantization of Gaussian processes
Open this publication in new window or tab >>Uniform and non-uniform quantization of Gaussian processes
2012 (English)In: Mathematical Communications, ISSN 1331-0623, E-ISSN 1848-8013, Vol. 17, no 2, 447-460 p.Article in journal (Refereed) Published
Abstract [en]

Quantization of a continuous-value signal into a discrete form (or discretization of amplitude) is a standard task in all analog/digital devices. We consider quantization of a signal (or random process) in a probabilistic framework. The quantization method presented in this paper can be applied to signal coding and storage capacity problems. In order to demonstrate a general approach, both uniform and non-uniform quantization of a Gaussian process are studied in more detail and compared with a conventional piecewise constant approximation. We investigate asymptotic properties of some accuracy characteristics, such as a random quantization rate, in terms of the correlation structure of the original random process when quantization cellwidth tends to zero. Some examples and numerical experiments are presented.

Place, publisher, year, edition, pages
University of Osijek, 2012
Keyword
quantization; rate; Gaussian process; level crossings
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:umu:diva-61615 (URN)000311954500008 ()
Available from: 2012-11-20 Created: 2012-11-20 Last updated: 2017-12-07Bibliographically approved
2. Stochastic structure of asymptotic quantization errors
Open this publication in new window or tab >>Stochastic structure of asymptotic quantization errors
2006 (English)In: Statistics and Probability Letters, Vol. 76, no 5, 453-464 p.Article in journal (Refereed) Published
Identifiers
urn:nbn:se:umu:diva-5377 (URN)
Available from: 2006-10-05 Created: 2006-10-05Bibliographically approved
3. Asymptotic quantization errors for unbounded quantizers
Open this publication in new window or tab >>Asymptotic quantization errors for unbounded quantizers
In: Theory of Probability and Mathematical Statistics, Vol. 75Article in journal (Refereed) Accepted
Identifiers
urn:nbn:se:umu:diva-5378 (URN)
Available from: 2006-10-05 Created: 2006-10-05Bibliographically approved
4. Quantization of stationary random sequences and related statistical problems
Open this publication in new window or tab >>Quantization of stationary random sequences and related statistical problems
In: Scandinavian Journal of StatisticsArticle in journal (Refereed) Submitted
Identifiers
urn:nbn:se:umu:diva-5379 (URN)
Available from: 2006-10-05 Created: 2006-10-05Bibliographically approved

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Citation style
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