Document Zbl 1088.42018

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Comput. Harmon. Anal.</a> <a href="/?q=in%3A123888" title="Articles in this Issue">20, No. 1, 126-148 (2006)</a>. </div> <div class="abstract">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 $N^{-1}$.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A6720">Manfred Tasche (Rostock)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A5013515">Cited in <strong>1</strong> Review</a></div><div class="clear"><a href="/?q=rf%3A5013515">Cited in <strong>25</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A42C15" title="MSC2020">42C15</a> </td> <td class="space"> General harmonic expansions, frames </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A94A12" title="MSC2020">94A12</a> </td> <td class="space"> Signal theory (characterization, reconstruction, filtering, etc.) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A94A05" title="MSC2020">94A05</a> </td> <td class="space"> Communication theory </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65T99" title="MSC2020">65T99</a> </td> <td class="space"> Numerical methods in Fourier analysis </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aquantization">quantization</a>; <a href="/?q=ut%3Asigma-delta+scheme">sigma-delta scheme</a>; <a href="/?q=ut%3Afinite+frame">finite frame</a>; <a href="/?q=ut%3Aframe+expansion">frame expansion</a>; <a href="/?q=ut%3Aquantization+error">quantization error</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1088.42018" data-ciurl="/ci/05013515" data-biburl="/bibtex/05013515.bib" data-amsurl="/amsrefs/05013515.bib" data-xmlurl="/xml/05013515.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/05013515.pdf" title="Zbl 1088.42018 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/j.acha.2005.04.003" aria-label="DOI for “Second-order sigma-delta ($\Sigma \Delta$) quantization of finite frame expansions”" title="10.1016/j.acha.2005.04.003">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Daubechies, I.; DeVore, R., Approximating a bandlimited function using very coarsely quantized data: A family of stable sigma-delta modulators of arbitrary order, Ann. of Math., 158, 2, 679-710 (2003) · <a href="/1058.94004" class="nowrap">Zbl 1058.94004</a></td> </tr><tr> <td>[2]</td> <td class="space">(Benedetto, J. 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This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </div> </div></article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none"> <span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

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