TY - INPR A1 - Kreitmeier, Wolfgang A1 - Linder, Tamas T1 - Entropy Density and Mismatch in High-Rate Scalar Quantization with Rényi Entropy Constraint N2 - 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 $\alpha$ KW - Maßtheorie KW - Quantisierung KW - Entropie KW - Asymptotic quantization theory KW - distortion density KW - entropy density KW - quantizer mismatch KW - Rényi-entropy Y1 - 2011 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:739-opus-26132 N1 - 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 ER - TY - INPR A1 - Kreitmeier, Wolfgang T1 - Asymptotic optimality of scalar Gersho quantizers N2 - In his famous paper Gersho stressed that the codecells of optimal quantizers asymptotically make an equal contribution to the distortion of the quantizer. Motivated by this fact, we investigate in this paper quantizers in the scalar case, where each codecell contributes with exactly the same portion to the quantization error. We show that such quantizers of Gersho type - or Gersho quantizers for short - exist for non-atomic scalar distributions. As a main result we prove that Gersho quantizers are asymptotically optimal. KW - Maßtheorie KW - Informationstheorie KW - Signaltheorie KW - Approximation KW - Kodierung KW - Quantisierung KW - Asymptotically optimal quantization KW - Quantization error KW - Scalar quantization KW - Gersho quantizer KW - High rate quantization Y1 - 2012 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:739-opus-27080 N1 - This is a preprint of an article accepted for publication in Constructive Approximation, ISSN: 0176-4276 (print version) ISSN: 1432-0940 (electronic version) Copyright (c) by Springer. The final publication is available at link.springer.com URL: http://dx.doi.org/10.1007/s00365-013-9214-2 ER -