Higher Order Convergent Fast Nonlinear Fourier Transform
It is demonstrated in this letter that linear multistep methods for integrating ordinary differential equations can be used to develop a family of fast forward scattering algorithms with higher orders of convergence. Excluding the cost of computing the discrete eigenvalues, the nonlinear Fourier tra...
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| Veröffentlicht in: | IEEE photonics technology letters 2018-04, Vol.30 (8), p.700-703 |
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| creator | Vaibhav, Vishal |
| description | It is demonstrated in this letter that linear multistep methods for integrating ordinary differential equations can be used to develop a family of fast forward scattering algorithms with higher orders of convergence. Excluding the cost of computing the discrete eigenvalues, the nonlinear Fourier transform (NFT) algorithm thus obtained has a complexity of \mathop {O}(KN+C_{p}N\log ^{2}N) such that the error vanishes as \mathop {O}(N^{-p}) where p\in \{1,2,3,4\} and K is the number of eigenvalues. Such an algorithm can be potentially useful for the recently proposed NFT-based modulation methodology for optical fiber communication. The exposition considers the particular case of the backward differentiation formula ( C_{p}=p^{3} ) and the implicit Adams method ( C_{p}=(p-1)^{3},\,p>1 ) of which the latter proves to be the most accurate family of methods for fast NFT. |
| doi_str_mv | 10.1109/LPT.2018.2812808 |
| format | Article |
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Excluding the cost of computing the discrete eigenvalues, the nonlinear Fourier transform (NFT) algorithm thus obtained has a complexity of <inline-formula> <tex-math notation="LaTeX">\mathop {O}(KN+C_{p}N\log ^{2}N) </tex-math></inline-formula> such that the error vanishes as <inline-formula> <tex-math notation="LaTeX">\mathop {O}(N^{-p}) </tex-math></inline-formula> where <inline-formula> <tex-math notation="LaTeX">p\in \{1,2,3,4\} </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">K </tex-math></inline-formula> is the number of eigenvalues. Such an algorithm can be potentially useful for the recently proposed NFT-based modulation methodology for optical fiber communication. The exposition considers the particular case of the backward differentiation formula (<inline-formula> <tex-math notation="LaTeX">C_{p}=p^{3} </tex-math></inline-formula>) and the implicit Adams method (<inline-formula> <tex-math notation="LaTeX">C_{p}=(p-1)^{3},\,p>1 </tex-math></inline-formula>) of which the latter proves to be the most accurate family of methods for fast NFT.]]></description><identifier>ISSN: 1041-1135</identifier><identifier>EISSN: 1941-0174</identifier><identifier>DOI: 10.1109/LPT.2018.2812808</identifier><identifier>CODEN: IPTLEL</identifier><language>eng</language><publisher>IEEE</publisher><subject>Complexity theory ; Convergence ; Eigenvalues and eigenfunctions ; Fourier transforms ; Manganese ; Noise measurement ; Nonlinear Fourier transform ; Scattering ; Zakharov–Shabat scattering problem</subject><ispartof>IEEE photonics technology letters, 2018-04, Vol.30 (8), p.700-703</ispartof><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c263t-59d539a39d11e6b916309c6bb3c7f18d98c52528dbe3ca5aaa1e74c56e1cab183</citedby><cites>FETCH-LOGICAL-c263t-59d539a39d11e6b916309c6bb3c7f18d98c52528dbe3ca5aaa1e74c56e1cab183</cites><orcidid>0000-0002-4800-0590</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://ieeexplore.ieee.org/document/8307084$$EHTML$$P50$$Gieee$$H</linktohtml><link.rule.ids>314,776,780,792,27901,27902,54733</link.rule.ids><linktorsrc>$$Uhttps://ieeexplore.ieee.org/document/8307084$$EView_record_in_IEEE$$FView_record_in_$$GIEEE</linktorsrc></links><search><creatorcontrib>Vaibhav, Vishal</creatorcontrib><title>Higher Order Convergent Fast Nonlinear Fourier Transform</title><title>IEEE photonics technology letters</title><addtitle>LPT</addtitle><description><![CDATA[It is demonstrated in this letter that linear multistep methods for integrating ordinary differential equations can be used to develop a family of fast forward scattering algorithms with higher orders of convergence. Excluding the cost of computing the discrete eigenvalues, the nonlinear Fourier transform (NFT) algorithm thus obtained has a complexity of <inline-formula> <tex-math notation="LaTeX">\mathop {O}(KN+C_{p}N\log ^{2}N) </tex-math></inline-formula> such that the error vanishes as <inline-formula> <tex-math notation="LaTeX">\mathop {O}(N^{-p}) </tex-math></inline-formula> where <inline-formula> <tex-math notation="LaTeX">p\in \{1,2,3,4\} </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">K </tex-math></inline-formula> is the number of eigenvalues. Such an algorithm can be potentially useful for the recently proposed NFT-based modulation methodology for optical fiber communication. The exposition considers the particular case of the backward differentiation formula (<inline-formula> <tex-math notation="LaTeX">C_{p}=p^{3} </tex-math></inline-formula>) and the implicit Adams method (<inline-formula> <tex-math notation="LaTeX">C_{p}=(p-1)^{3},\,p>1 </tex-math></inline-formula>) of which the latter proves to be the most accurate family of methods for fast NFT.]]></description><subject>Complexity theory</subject><subject>Convergence</subject><subject>Eigenvalues and eigenfunctions</subject><subject>Fourier transforms</subject><subject>Manganese</subject><subject>Noise measurement</subject><subject>Nonlinear Fourier transform</subject><subject>Scattering</subject><subject>Zakharov–Shabat scattering problem</subject><issn>1041-1135</issn><issn>1941-0174</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2018</creationdate><recordtype>article</recordtype><sourceid>RIE</sourceid><recordid>eNo9j09Lw0AUxBdRsFbvgpd8gdT39k-ye5RgbCFYD_G8bDYvNdImshsFv70pLV5m5jAz8GPsHmGFCOaxeqtXHFCvuEauQV-wBRqJKWAuL-cMc0YU6prdxPgJgFIJuWB63e8-KCTb0M5ajMMPhR0NU1K6OCWv47DvB3IhKcfv0M-NOrghdmM43LKrzu0j3Z19yd7L57pYp9X2ZVM8VannmZhSZVoljBOmRaSsMZgJMD5rGuHzDnVrtFdccd02JLxTzjmkXHqVEXrXoBZLBqdfH8YYA3X2K_QHF34tgj2S25ncHsntmXyePJwmPRH917WAHLQUf2NGVIU</recordid><startdate>20180415</startdate><enddate>20180415</enddate><creator>Vaibhav, Vishal</creator><general>IEEE</general><scope>97E</scope><scope>RIA</scope><scope>RIE</scope><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0002-4800-0590</orcidid></search><sort><creationdate>20180415</creationdate><title>Higher Order Convergent Fast Nonlinear Fourier Transform</title><author>Vaibhav, Vishal</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c263t-59d539a39d11e6b916309c6bb3c7f18d98c52528dbe3ca5aaa1e74c56e1cab183</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2018</creationdate><topic>Complexity theory</topic><topic>Convergence</topic><topic>Eigenvalues and eigenfunctions</topic><topic>Fourier transforms</topic><topic>Manganese</topic><topic>Noise measurement</topic><topic>Nonlinear Fourier transform</topic><topic>Scattering</topic><topic>Zakharov–Shabat scattering problem</topic><toplevel>online_resources</toplevel><creatorcontrib>Vaibhav, Vishal</creatorcontrib><collection>IEEE All-Society Periodicals Package (ASPP) 2005-present</collection><collection>IEEE All-Society Periodicals Package (ASPP) 1998-Present</collection><collection>IEEE Electronic Library (IEL)</collection><collection>CrossRef</collection><jtitle>IEEE photonics technology letters</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext_linktorsrc</fulltext></delivery><addata><au>Vaibhav, Vishal</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Higher Order Convergent Fast Nonlinear Fourier Transform</atitle><jtitle>IEEE photonics technology letters</jtitle><stitle>LPT</stitle><date>2018-04-15</date><risdate>2018</risdate><volume>30</volume><issue>8</issue><spage>700</spage><epage>703</epage><pages>700-703</pages><issn>1041-1135</issn><eissn>1941-0174</eissn><coden>IPTLEL</coden><abstract><![CDATA[It is demonstrated in this letter that linear multistep methods for integrating ordinary differential equations can be used to develop a family of fast forward scattering algorithms with higher orders of convergence. Excluding the cost of computing the discrete eigenvalues, the nonlinear Fourier transform (NFT) algorithm thus obtained has a complexity of <inline-formula> <tex-math notation="LaTeX">\mathop {O}(KN+C_{p}N\log ^{2}N) </tex-math></inline-formula> such that the error vanishes as <inline-formula> <tex-math notation="LaTeX">\mathop {O}(N^{-p}) </tex-math></inline-formula> where <inline-formula> <tex-math notation="LaTeX">p\in \{1,2,3,4\} </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">K </tex-math></inline-formula> is the number of eigenvalues. Such an algorithm can be potentially useful for the recently proposed NFT-based modulation methodology for optical fiber communication. The exposition considers the particular case of the backward differentiation formula (<inline-formula> <tex-math notation="LaTeX">C_{p}=p^{3} </tex-math></inline-formula>) and the implicit Adams method (<inline-formula> <tex-math notation="LaTeX">C_{p}=(p-1)^{3},\,p>1 </tex-math></inline-formula>) of which the latter proves to be the most accurate family of methods for fast NFT.]]></abstract><pub>IEEE</pub><doi>10.1109/LPT.2018.2812808</doi><tpages>4</tpages><orcidid>https://orcid.org/0000-0002-4800-0590</orcidid><oa>free_for_read</oa></addata></record> |
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| source | IEEE Electronic Library (IEL) |
| subjects | Complexity theory Convergence Eigenvalues and eigenfunctions Fourier transforms Manganese Noise measurement Nonlinear Fourier transform Scattering Zakharov–Shabat scattering problem |
| title | Higher Order Convergent Fast Nonlinear Fourier Transform |
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