## Entropy Density and Mismatch in High-Rate Scalar Quantization with Rényi Entropy Constraint

• Properties of scalar quantization with $r$th power distortion and constrained R\'enyi entropy of order $\alpha\in (0,1)$ are investigated. For an asymptotically (high-rate) optimal sequence of quantizers, the contribution to the R\'enyi entropy due to source values in a fixed interval is identified in terms of the "entropy density" of the quantizer sequence. This extends results related to the well-known point density concept in optimal fixed-rate quantization. A dual of the entropy density result quantifies the distortion contribution of a given interval to the overall distortion. The distortion loss resulting from a mismatch of source densities in the design of an asymptotically optimal sequence of quantizers is also determined. This extends Bucklew's fixed-rate ($\alpha=0$) and Gray \emph{et al.}'s variable-rate ($\alpha=1$)mismatch results to general values of the entropy order parameter $\alpha$

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Author: Wolfgang Kreitmeier, Tamas Linder urn:nbn:de:bvb:739-opus-26132 Preprint German 2011 2012/03/27 Universität Passau Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, Canada 2012/03/27 Asymptotic quantization theory; Rényi-entropy; distortion density; entropy density; quantizer mismatch Entropie; Maßtheorie; Quantisierung This is a preprint of an article accepted for publication in the IEEE Transactions on Information Theory Journal, ISSN: 0018-9448. The original publication is available at http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=18  IEEE Transactions on Information Theory Journal, ISSN: 0018-9448 Fakultät für Informatik und Mathematik / Mitarbeiter Lehrstuhl/Einrichtung der Fakultät für Informatik und Mathematik 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik 28-XX MEASURE AND INTEGRATION (For analysis on manifolds, see 58-XX) / 28Dxx Measure-theoretic ergodic theory [See also 11K50, 11K55, 22D40, 37Axx, 47A35, 54H20, 60Fxx, 60G10] / 28D20 Entropy and other invariants 41-XX APPROXIMATIONS AND EXPANSIONS (For all approximation theory in the complex domain, see 30E05 and 30E10; for all trigonometric approximation and interpolation, see 42A10 and 42A15; for numerical approximation, see 65Dxx) / 41Axx Approximations and expansions / 41A46 Approximation by arbitrary nonlinear expressions; widths and entropy 62-XX STATISTICS / 62Hxx Multivariate analysis [See also 60Exx] / 62H30 Classification and discrimination; cluster analysis [See also 68T10] 94-XX INFORMATION AND COMMUNICATION, CIRCUITS / 94Axx Communication, information / 94A17 Measures of information, entropy 94-XX INFORMATION AND COMMUNICATION, CIRCUITS / 94Axx Communication, information / 94A29 Source coding [See also 68P30] open_access Standardbedingung laut Einverständniserklärung

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