Document Zbl 1029.94006

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Sinan</a></div> <h2 class="title"> <strong>One-bit sigma-delta quantization with exponential accuracy.</strong> <i>(English)</i> <a class="label nowrap" href="/1029.94006">Zbl 1029.94006</a> </h2> <div class="source"> <a href="/serials/66" title="Journal Profile">Commun. Pure Appl. Math.</a> <a href="/?q=in%3A99898" title="Articles in this Issue">56, No. 11, 1608-1630 (2003)</a>. </div> <div class="abstract">Summary: One-bit quantization is a method of representing band-limited signals by $\pm 1$ sequences that are computed from regularly spaced samples of these signals; as the sampling density $\lambda\to\infty$, convolving these one-bit sequences with appropriately chosen filters produces increasingly close approximations of the original signals. This method is widely used for analog-to-digital and digital-to-analog conversion because it is less expensive and simpler to implement than the more familiar critical sampling followed by fine-resolution quantization. However, unlike fine-resolution quantization, the accuracy of one-bit quantization is not well understood. A natural error lower bound that decreases like $2^{-\lambda}$ can easily be given using information-theoretic arguments. Yet, no one-bit quantization algorithm was known with an error decay estimate even close to exponential decay. In this paper we construct an infinite family of one-bit sigma-delta quantization schemes that achieves this goal. In particular, using this family, we prove that the error signal for $\pi$-band-limited signals is at most $O(2^{-.07\lambda})$.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A1997829">Cited in <strong>31</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A94A14" title="MSC2020">94A14</a> </td> <td class="space"> Modulation and demodulation in information and communication theory </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></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Asigma-delta+modulation">sigma-delta modulation</a>; <a href="/?q=ut%3Asignal+representation">signal representation</a>; <a href="/?q=ut%3Asignal+approximation">signal approximation</a>; <a href="/?q=ut%3Apulse+code+modulation">pulse code modulation</a>; <a href="/?q=ut%3Aband-limited+signals">band-limited signals</a>; <a href="/?q=ut%3Aone-bit+quantization">one-bit quantization</a>; <a href="/?q=ut%3Aone-bit+sigma-delta+quantization">one-bit sigma-delta quantization</a>; <a href="/?q=ut%3Aerror+signal">error signal</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" 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class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1002/cpa.3044" aria-label="DOI for “One-bit sigma-delta quantization with exponential accuracy”" title="10.1002/cpa.3044">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">Anastassiou, Error diffusion coding for A/D conversion, IEEE Trans Circuits and Systems 36 (3) pp 1175– (1989) · <a href="https://doi.org/10.1109/31.34663" class="nowrap">doi:10.1109/31.34663</a></td> </tr><tr> <td>[2]</td> <td class="space">Boas, Entire functions (1954)</td> </tr><tr> <td>[3]</td> <td class="space">Calderbank, The pros and cons of democracy, IEEE Trans Inform Theory 48 (6) pp 1721– (2002) · <a href="/1061.94014" class="nowrap">Zbl 1061.94014</a> · <a href="https://doi.org/10.1109/TIT.2002.1003852" class="nowrap">doi:10.1109/TIT.2002.1003852</a></td> </tr><tr> <td>[4]</td> <td class="space">Chou, Multistage {$\Sigma$}{$\Delta$} modulation, IEEE Trans Inform Theory 35 (4) pp 784– (1989) · <a href="/0684.94007" class="nowrap">Zbl 0684.94007</a> · <a href="https://doi.org/10.1109/18.32155" class="nowrap">doi:10.1109/18.32155</a></td> </tr><tr> <td>[5]</td> <td class="space">Daubechies, Reconstructing a bandlimited function from very coarsely quantized data: A family of stable sigma-delta modulators of arbitrary order, Ann of Math · <a href="/1058.94004" class="nowrap">Zbl 1058.94004</a></td> </tr><tr> <td>[6]</td> <td class="space">de Boor, On calculating with B-splines, J Approximation Theory 6 pp 50– (1972) · <a href="/0239.41006" class="nowrap">Zbl 0239.41006</a></td> </tr><tr> <td>[7]</td> <td class="space">Delta-sigma data converters: Theory, design and simulation (1996)</td> </tr><tr> <td>[8]</td> <td class="space">DeVore, Constructive approximation (1993) · <a href="https://doi.org/10.1007/978-3-662-02888-9" 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Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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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