Alternative dual frames for digital-to-analog conversion in sigma–delta quantization
We design alternative dual frames for linearly reconstructing signals from sigma–delta (ΣΔ) quantized finite frame coefficients. In the setting of sampling expansions for bandlimited functions, it is known that a stable r th order sigma–delta quantizer produces approximations where the approximation...
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Veröffentlicht in: | Advances in computational mathematics 2010, Vol.32 (1), p.73-102 |
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creator | Lammers, Mark Powell, Alexander M. Yılmaz, Özgür |
description | We design alternative dual frames for linearly reconstructing signals from sigma–delta (ΣΔ) quantized finite frame coefficients. In the setting of sampling expansions for bandlimited functions, it is known that a stable
r
th order sigma–delta quantizer produces approximations where the approximation error is at most of order 1 /
λ
r
, and
λ
> 1 is the oversampling ratio. We show that the counterpart of this result is not true for several families of redundant finite frames for
when the canonical dual frame is used in linear reconstruction. As a remedy, we construct alternative dual frame sequences which enable an
r
th order sigma–delta quantizer to achieve approximation error of order 1/
N
r
for certain sequences of frames where
N
is the frame size. We also present several numerical examples regarding the constructions. |
doi_str_mv | 10.1007/s10444-008-9088-1 |
format | Article |
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r
th order sigma–delta quantizer produces approximations where the approximation error is at most of order 1 /
λ
r
, and
λ
> 1 is the oversampling ratio. We show that the counterpart of this result is not true for several families of redundant finite frames for
when the canonical dual frame is used in linear reconstruction. As a remedy, we construct alternative dual frame sequences which enable an
r
th order sigma–delta quantizer to achieve approximation error of order 1/
N
r
for certain sequences of frames where
N
is the frame size. We also present several numerical examples regarding the constructions.</description><identifier>ISSN: 1019-7168</identifier><identifier>EISSN: 1572-9044</identifier><identifier>DOI: 10.1007/s10444-008-9088-1</identifier><language>eng</language><publisher>Boston: Springer US</publisher><subject>Approximation ; Computational Mathematics and Numerical Analysis ; Computational Science and Engineering ; Construction ; Conversion ; Counters ; Errors ; Frames ; Mathematical analysis ; Mathematical and Computational Biology ; Mathematical Modeling and Industrial Mathematics ; Mathematical models ; Mathematics ; Mathematics and Statistics ; Visualization</subject><ispartof>Advances in computational mathematics, 2010, Vol.32 (1), p.73-102</ispartof><rights>Springer Science+Business Media, LLC. 2008</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-p160t-b6f3f6434f050461c3c97d7666d648781fd8620806ae4e0fd1e4334cd9e5efc63</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s10444-008-9088-1$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s10444-008-9088-1$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,776,780,27901,27902,41464,42533,51294</link.rule.ids></links><search><creatorcontrib>Lammers, Mark</creatorcontrib><creatorcontrib>Powell, Alexander M.</creatorcontrib><creatorcontrib>Yılmaz, Özgür</creatorcontrib><title>Alternative dual frames for digital-to-analog conversion in sigma–delta quantization</title><title>Advances in computational mathematics</title><addtitle>Adv Comput Math</addtitle><description>We design alternative dual frames for linearly reconstructing signals from sigma–delta (ΣΔ) quantized finite frame coefficients. In the setting of sampling expansions for bandlimited functions, it is known that a stable
r
th order sigma–delta quantizer produces approximations where the approximation error is at most of order 1 /
λ
r
, and
λ
> 1 is the oversampling ratio. We show that the counterpart of this result is not true for several families of redundant finite frames for
when the canonical dual frame is used in linear reconstruction. As a remedy, we construct alternative dual frame sequences which enable an
r
th order sigma–delta quantizer to achieve approximation error of order 1/
N
r
for certain sequences of frames where
N
is the frame size. We also present several numerical examples regarding the constructions.</description><subject>Approximation</subject><subject>Computational Mathematics and Numerical Analysis</subject><subject>Computational Science and Engineering</subject><subject>Construction</subject><subject>Conversion</subject><subject>Counters</subject><subject>Errors</subject><subject>Frames</subject><subject>Mathematical analysis</subject><subject>Mathematical and Computational Biology</subject><subject>Mathematical Modeling and Industrial Mathematics</subject><subject>Mathematical models</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Visualization</subject><issn>1019-7168</issn><issn>1572-9044</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2010</creationdate><recordtype>article</recordtype><recordid>eNotkM1OAyEcxInRxFp9AG_cPKF_CgvssWn8Sky8qFeCC2xoKLQL24Mn38E39Encpp5mJplMMj-ErincUgB5VyhwzgmAIi0oRegJmtFGLqbE-enkgbZEUqHO0UUpawBohWxm6GMZqxuSqWHvsB1NxH4wG1ewzwO2oQ_VRFIzMcnE3OMup70bSsgJh4RL6Dfm9_vHulgN3o0m1fA1TeV0ic68icVd_escvT_cv62eyMvr4_Nq-UK2VEAln8IzLzjjHhrggnasa6WVQggruJKKeqvEAhQI47gDb6njjPHOtq5xvhNsjm6Ou9sh70ZXqt6E0rkYTXJ5LFo1jZCt4Ifm4tgs2yGk3g16ncfpeCyagj4w1EeGemKoDww1ZX9F12af</recordid><startdate>2010</startdate><enddate>2010</enddate><creator>Lammers, Mark</creator><creator>Powell, Alexander M.</creator><creator>Yılmaz, Özgür</creator><general>Springer US</general><scope>7SC</scope><scope>8FD</scope><scope>JQ2</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope></search><sort><creationdate>2010</creationdate><title>Alternative dual frames for digital-to-analog conversion in sigma–delta quantization</title><author>Lammers, Mark ; Powell, Alexander M. ; Yılmaz, Özgür</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-p160t-b6f3f6434f050461c3c97d7666d648781fd8620806ae4e0fd1e4334cd9e5efc63</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2010</creationdate><topic>Approximation</topic><topic>Computational Mathematics and Numerical Analysis</topic><topic>Computational Science and Engineering</topic><topic>Construction</topic><topic>Conversion</topic><topic>Counters</topic><topic>Errors</topic><topic>Frames</topic><topic>Mathematical analysis</topic><topic>Mathematical and Computational Biology</topic><topic>Mathematical Modeling and Industrial Mathematics</topic><topic>Mathematical models</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Visualization</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Lammers, Mark</creatorcontrib><creatorcontrib>Powell, Alexander M.</creatorcontrib><creatorcontrib>Yılmaz, Özgür</creatorcontrib><collection>Computer and Information Systems Abstracts</collection><collection>Technology Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><jtitle>Advances in computational mathematics</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Lammers, Mark</au><au>Powell, Alexander M.</au><au>Yılmaz, Özgür</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Alternative dual frames for digital-to-analog conversion in sigma–delta quantization</atitle><jtitle>Advances in computational mathematics</jtitle><stitle>Adv Comput Math</stitle><date>2010</date><risdate>2010</risdate><volume>32</volume><issue>1</issue><spage>73</spage><epage>102</epage><pages>73-102</pages><issn>1019-7168</issn><eissn>1572-9044</eissn><abstract>We design alternative dual frames for linearly reconstructing signals from sigma–delta (ΣΔ) quantized finite frame coefficients. In the setting of sampling expansions for bandlimited functions, it is known that a stable
r
th order sigma–delta quantizer produces approximations where the approximation error is at most of order 1 /
λ
r
, and
λ
> 1 is the oversampling ratio. We show that the counterpart of this result is not true for several families of redundant finite frames for
when the canonical dual frame is used in linear reconstruction. As a remedy, we construct alternative dual frame sequences which enable an
r
th order sigma–delta quantizer to achieve approximation error of order 1/
N
r
for certain sequences of frames where
N
is the frame size. We also present several numerical examples regarding the constructions.</abstract><cop>Boston</cop><pub>Springer US</pub><doi>10.1007/s10444-008-9088-1</doi><tpages>30</tpages></addata></record> |
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subjects | Approximation Computational Mathematics and Numerical Analysis Computational Science and Engineering Construction Conversion Counters Errors Frames Mathematical analysis Mathematical and Computational Biology Mathematical Modeling and Industrial Mathematics Mathematical models Mathematics Mathematics and Statistics Visualization |
title | Alternative dual frames for digital-to-analog conversion in sigma–delta quantization |
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