Second-order sigma-delta (\(\Sigma \Delta\)) quantization of finite frame expansions
type: Item
instance of: scholarly article
title: Second-order sigma-delta (\(\Sigma \Delta\)) quantization of finite frame expansions
zbMATH Open document ID: 1088.42018
author: John J. Benedetto
author: Alexander M. Powell
author: Özgür Yılmaz
published in: Applied and Computational Harmonic Analysis
publication date: 2006-03-20T00:00:00Z
review text: The authors consider the quantization problem for finite unit-norm tight frame expansions in \({\mathbb R}^d\). They analyze the performance of a second-order sigma-delta scheme and show that it outperforms the standard quantization techniques. Let \(\{e_n\}_{n=1}^N \subset {\mathbb R}^d\) be a finite unit-norm tight frame of \({\mathbb R}^d\), i.e., that \(\| e_n\| =1\) and that there exists \(A>0\) such that for all \(x \in {\mathbb R}^d\) \[ \sum_{n=1}^N | \langle x,\,e_n\rangle| ^2 = A\,\| x\| ^2. \] Here \(\| \cdot \| \) is the Euclidean norm. Then a given signal \(x\in {\mathbb R}^d\) is decomposed into a frame expansion \(x={d \over N}\,\sum_{n=1}^N x_n\,e_n\). In a quantization step, the coefficients \(x_n\), which are real or complex numbers, are replaced by discrete representations \(q_n\) using a second-order sigma-delta quantization scheme. The quantization error \(\| x - \tilde x\| \) with \({\tilde x} = {d \over N}\, \sum_{n=1}^N q_n\,e_n\) is estimated. It is shown that for special finite frames the quantization error is of order \(N^{-2}\). But in general, the second-order scheme only gives a quantization error of order \(N^{-1}\).
Mathematics Subject Classification ID: 42C15
Mathematics Subject Classification ID: 94A12
Mathematics Subject Classification ID: 94A05
Mathematics Subject Classification ID: 65T99
zbMATH DE Number: 5013515
zbMATH Keywords: quantization
zbMATH Keywords: sigma-delta scheme
zbMATH Keywords: finite frame
zbMATH Keywords: frame expansion
zbMATH Keywords: quantization error
reviewed by: Manfred Tasche
MaRDI profile type: Publication
full work available at URL: j.acha.2005.04.003
OpenAlex ID: W2163296328
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DOI: 10.1016/J.ACHA.2005.04.003
description: scientific article; zbMATH DE number 5013515