The-local-mean-decomposition-and-its-application-

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The local mean decomposition and its application to EEG perception data
The-local-mean-decomposition-and-its-application-
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内容介绍
<html xmlns="http://www.w3.org/1999/xhtml"> <head> <meta charset="utf-8"> <meta name="generator" content="pdf2htmlEX"> <meta http-equiv="X-UA-Compatible" content="IE=edge,chrome=1"> <link rel="stylesheet" href="https://static.pudn.com/base/css/base.min.css"> <link rel="stylesheet" href="https://static.pudn.com/base/css/fancy.min.css"> <link rel="stylesheet" href="https://static.pudn.com/prod/directory_preview_static/625aa3d7be9ad24cfaec4b49/raw.css"> <script src="https://static.pudn.com/base/js/compatibility.min.js"></script> <script src="https://static.pudn.com/base/js/pdf2htmlEX.min.js"></script> <script> try{ pdf2htmlEX.defaultViewer = new pdf2htmlEX.Viewer({}); }catch(e){} </script> <title></title> </head> <body> <div id="sidebar" style="display: none"> <div id="outline"> </div> </div> <div id="pf1" class="pf w0 h0" data-page-no="1"><div class="pc pc1 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/625aa3d7be9ad24cfaec4b49/bg1.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">The<span class="_ _0"> </span>local<span class="_ _0"> </span>mean<span class="_ _0"> </span>decomposition<span class="_ _0"> </span>and<span class="_ _0"> </span>its</div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls1 ws0">application<span class="_ _0"> </span>to<span class="_ _0"> </span>EEG<span class="_ _0"> </span>perception<span class="_ _0"> </span>data</div><div class="t m0 x3 h3 y3 ff1 fs1 fc0 sc0 ls1 ws0">Jonathan<span class="_ _1"> </span>S.<span class="_ _1"> </span>Smith</div><div class="t m1 x4 h4 y4 ff1 fs2 fc0 sc0 ls1 ws0">&#8224;</div><div class="t m0 x5 h5 y5 ff2 fs3 fc0 sc0 ls1 ws0">Neurotechno<span class="_ _2"> </span>Ltd,<span class="_ _2"> </span>Marlow,<span class="_ _2"> </span>Buckinghamshire<span class="_ _2"> </span>SL7<span class="_ _2"> </span>1SJ,<span class="_ _2"> </span>UK</div><div class="t m0 x6 h6 y6 ff3 fs3 fc0 sc0 ls1 ws0">This<span class="_ _3"> </span>paper<span class="_ _3"> </span>describes<span class="_ _3"> </span>the<span class="_ _3"> </span>local<span class="_ _3"> </span>mean<span class="_ _3"> </span>decomp<span class="_ _4"></span>osition<span class="_ _3"> </span>(LMD),<span class="_ _3"> </span>a<span class="_ _3"> </span>new<span class="_ _3"> </span>iterative<span class="_ _3"> </span>approach<span class="_ _3"> </span>to</div><div class="t m0 x6 h6 y7 ff3 fs3 fc0 sc0 ls1 ws0">demodulating<span class="_ _3"> </span>amplitude<span class="_ _3"> </span>and<span class="_ _3"> </span>frequency<span class="_ _3"> </span>modulated<span class="_ _3"> </span>signals.<span class="_ _3"> </span>The<span class="_ _3"> </span>new<span class="_ _3"> </span>method<span class="_ _3"> </span>decomposes</div><div class="t m0 x6 h6 y8 ff3 fs3 fc0 sc0 ls1 ws0">such<span class="_ _2"> </span>signals<span class="_ _5"> </span>into<span class="_ _5"> </span>a<span class="_ _2"> </span>set<span class="_ _5"> </span>of<span class="_ _6"> </span>functions,<span class="_ _5"> </span>each<span class="_ _2"> </span>of<span class="_ _5"> </span>which<span class="_ _2"> </span>is<span class="_ _5"> </span>the<span class="_ _5"> </span>produ<span class="_ _4"></span>ct<span class="_ _5"> </span>of<span class="_ _5"> </span>an<span class="_ _2"> </span>envelope<span class="_ _5"> </span>signal<span class="_ _2"> </span>and<span class="_ _5"> </span>a</div><div class="t m0 x6 h6 y9 ff3 fs3 fc0 sc0 ls1 ws0">frequency<span class="_ _7"> </span>modulated<span class="_ _7"> </span>signal<span class="_ _7"> </span>from<span class="_ _7"> </span>which<span class="_ _7"> </span>a<span class="_ _8"> </span>time-varying<span class="_ _7"> </span>instantaneous<span class="_ _7"> </span>frequency<span class="_ _7"> </span>can<span class="_ _8"> </span>be</div><div class="t m0 x6 h6 ya ff3 fs3 fc0 sc0 ls1 ws0">derived.<span class="_ _2"> </span>The<span class="_ _2"> </span>LMD<span class="_ _2"> </span>method<span class="_ _2"> </span>can<span class="_ _2"> </span>be<span class="_ _2"> </span>used<span class="_ _2"> </span>to<span class="_ _2"> </span>analyse<span class="_ _2"> </span>a<span class="_ _5"> </span>wi<span class="_ _4"></span>de<span class="_ _2"> </span>variety<span class="_ _2"> </span>of<span class="_ _5"> </span>natural<span class="_ _2"> </span>si<span class="_ _4"></span>gnals<span class="_ _5"> </span>such<span class="_ _2"> </span>as</div><div class="t m0 x6 h6 yb ff3 fs3 fc0 sc0 ls1 ws0">electrocardiograms,<span class="_ _5"> </span>function<span class="_"> </span>al<span class="_ _5"> </span>magnetic<span class="_ _5"> </span>resonan<span class="_"> </span>ce<span class="_ _5"> </span>imaging<span class="_ _5"> </span>data,<span class="_ _5"> </span>and<span class="_ _5"> </span>earthquake<span class="_ _5"> </span>data.<span class="_ _6"> </span>The</div><div class="t m0 x6 h6 yc ff3 fs3 fc0 sc0 ls1 ws0">paper<span class="_ _1"> </span>presents<span class="_ _1"> </span>the<span class="_ _2"> </span>results<span class="_ _1"> </span>of<span class="_ _1"> </span>applying<span class="_ _1"> </span>LMD<span class="_ _2"> </span>to<span class="_ _1"> </span>a<span class="_ _1"> </span>set<span class="_ _2"> </span>of<span class="_ _1"> </span>scalp<span class="_ _1"> </span>electroencephalogram<span class="_ _1"> </span>(EEG)</div><div class="t m0 x6 h6 yd ff3 fs3 fc0 sc0 ls1 ws0">visual<span class="_ _6"> </span>percepti<span class="_ _4"></span>on<span class="_ _6"> </span>data.<span class="_ _5"> </span>The<span class="_ _5"> </span>LMD<span class="_ _6"> </span>instan<span class="_ _4"></span>taneous<span class="_ _5"> </span>frequency<span class="_ _6"> </span>and<span class="_ _5"> </span>energy<span class="_ _5"> </span>structure<span class="_ _5"> </span>of<span class="_ _6"> </span>the<span class="_ _5"> </span>EEG</div><div class="t m0 x6 h6 ye ff3 fs3 fc0 sc0 ls1 ws0">is<span class="_ _9"> </span>examined,<span class="_ _9"> </span>and<span class="_ _9"> </span>compared<span class="_ _9"> </span>with<span class="_ _9"> </span>results<span class="_ _9"> </span>obtained<span class="_ _9"> </span>using<span class="_ _9"> </span>the<span class="_ _9"> </span>spectrogram.<span class="_ _6"> </span>The<span class="_ _9"> </span>nature<span class="_ _9"> </span>of<span class="_ _9"> </span>visual</div><div class="t m0 x6 h6 yf ff3 fs3 fc0 sc0 ls2 ws0">perception<span class="_ _0"> </span>is<span class="_ _a"> </span>investigated<span class="_ _a"> </span>by<span class="_ _0"> </span>measuring<span class="_ _a"> </span>the<span class="_ _0"> </span>degree<span class="_ _a"> </span>of<span class="_ _0"> </span>EEG<span class="_ _a"> </span>instantaneous<span class="_ _0"> </span>phase</div><div class="t m0 x6 h6 y10 ff3 fs3 fc0 sc0 ls1 ws0">concentration<span class="_ _0"> </span>that<span class="_ _b"> </span>occurs<span class="_ _b"> </span>following<span class="_ _0"> </span>stimulus<span class="_ _b"> </span>onset<span class="_ _b"> </span>ove<span class="_ _4"></span>r<span class="_ _b"> </span>multiple<span class="_ _b"> </span>trials.<span class="_ _0"> </span>The<span class="_ _b"> </span>analysis</div><div class="t m0 x6 h6 y11 ff3 fs3 fc0 sc0 ls1 ws0">sugg<span class="_ _4"></span>ests<span class="_ _a"> </span>that<span class="_ _a"> </span>there<span class="_ _a"> </span>is<span class="_ _0"> </span>a<span class="_ _a"> </span>statis<span class="_ _4"></span>tica<span class="_ _4"></span>lly<span class="_ _a"> </span>signi&#64257;<span class="_ _4"></span>cant<span class="_ _a"> </span>dif<span class="_ _c"></span>fere<span class="_ _4"></span>nce<span class="_ _a"> </span>betwee<span class="_ _4"></span>n<span class="_ _0"> </span>t<span class="_ _4"></span>he<span class="_ _0"> </span>t<span class="_ _4"></span>heta<span class="_ _a"> </span>phase</div><div class="t m0 x6 h6 y12 ff3 fs3 fc0 sc0 ls1 ws0">concentrations<span class="_ _2"> </span>of<span class="_ _2"> </span>the<span class="_ _5"> </span>perception<span class="_ _2"> </span>and<span class="_ _2"> </span>no<span class="_ _5"> </span>pe<span class="_ _4"></span>rception<span class="_ _2"> </span>EEG<span class="_ _5"> </span>data.</div><div class="t m0 x7 h7 y13 ff1 fs3 fc0 sc0 ls1 ws0">Keywords:<span class="_ _2"> </span>demodulated<span class="_ _2"> </span>signal<span class="_ _2"> </span>time&#8211;frequency<span class="_ _2"> </span>representation;<span class="_ _2"> </span>electroencephalogram;</div><div class="t m0 x6 h7 y14 ff1 fs3 fc0 sc0 ls1 ws0">instantaneous<span class="_ _5"> </span>frequency;<span class="_ _5"> </span>local<span class="_ _5"> </span>mean<span class="_ _5"> </span>decomposition;<span class="_ _5"> </span>phase<span class="_ _5"> </span>concentra<span class="_ _4"></span>tion;<span class="_ _5"> </span>visual<span class="_ _5"> </span>perception</div><div class="t m0 x8 h7 y15 ff1 fs3 fc0 sc0 ls1 ws0">1.<span class="_ _5"> </span>INTRO<span class="_ _4"></span>DUCTION</div><div class="t m0 x8 h6 y16 ff3 fs3 fc0 sc0 ls1 ws0">The<span class="_ _6"> </span>nature<span class="_ _6"> </span>of<span class="_ _6"> </span>information<span class="_ _5"> </span>is<span class="_ _6"> </span>one<span class="_ _6"> </span>of<span class="_ _6"> </span>the<span class="_ _5"> </span>central<span class="_ _6"> </span>issues<span class="_ _6"> </span>in</div><div class="t m0 x8 h6 y17 ff3 fs3 fc0 sc0 ls1 ws0">modern<span class="_ _6"> </span>science.<span class="_ _6"> </span>Given<span class="_ _6"> </span>a<span class="_ _6"> </span>signal,<span class="_ _6"> </span>or<span class="_ _6"> </span>a<span class="_ _6"> </span>set<span class="_ _6"> </span>of<span class="_ _6"> </span>data,<span class="_ _6"> </span>how<span class="_ _6"> </span>do</div><div class="t m0 x8 h6 y18 ff3 fs3 fc0 sc0 ls1 ws0">we<span class="_ _2"> </span>ext<span class="_ _4"></span>ract<span class="_ _2"> </span>informat<span class="_ _4"></span>ion<span class="_ _2"> </span>from<span class="_ _1"> </span>it,<span class="_ _2"> </span>or<span class="_ _1"> </span>give<span class="_ _2"> </span>meaning<span class="_ _1"> </span>to<span class="_ _2"> </span>it?</div><div class="t m0 x8 h6 y19 ff3 fs3 fc0 sc0 ls1 ws0">Fourier<span class="_ _3"> </span>analysis<span class="_ _d"> </span>is<span class="_ _d"> </span>a<span class="_ _d"> </span>particul<span class="_ _4"></span>arly<span class="_ _d"> </span>powerful<span class="_ _3"> </span>method<span class="_ _d"> </span>of</div><div class="t m0 x8 h6 y1a ff3 fs3 fc0 sc0 ls3 ws0">describing<span class="_ _a"> </span>signals<span class="_ _a"> </span>in<span class="_ _0"> </span>terms<span class="_ _a"> </span>of<span class="_ _a"> </span>their<span class="_ _a"> </span>energy<span class="_ _0"> </span>and</div><div class="t m0 x8 h6 y1b ff3 fs3 fc0 sc0 ls1 ws0">frequency.<span class="_ _2"> </span>Representing<span class="_ _2"> </span>a<span class="_ _2"> </span>signal<span class="_ _2"> </span>as<span class="_ _2"> </span>being<span class="_ _2"> </span>the<span class="_ _2"> </span>sum<span class="_ _2"> </span>of<span class="_ _5"> </span>a</div><div class="t m0 x8 h6 y1c ff3 fs3 fc0 sc0 ls1 ws0">set<span class="_ _6"> </span>of<span class="_ _5"> </span>plane<span class="_ _5"> </span>waves,<span class="_ _5"> </span>each<span class="_ _5"> </span>of<span class="_ _6"> </span>whi<span class="_ _4"></span>ch<span class="_ _6"> </span>has<span class="_ _5"> </span>a<span class="_ _5"> </span>constant<span class="_ _5"> </span>phase,</div><div class="t m0 x8 h6 y1d ff3 fs3 fc0 sc0 ls1 ws0">frequency,<span class="_ _d"> </span>and<span class="_ _1"> </span>envelope<span class="_ _4"></span>,<span class="_ _1"> </span>can<span class="_ _d"> </span>often<span class="_ _d"> </span>be<span class="_ _1"> </span>very<span class="_ _d"> </span>revealing.</div><div class="t m0 x8 h6 y1e ff3 fs3 fc0 sc0 ls4 ws0">For<span class="_ _0"> </span>example,<span class="_ _a"> </span>Fourier<span class="_ _0"> </span>analysis<span class="_ _a"> </span>can<span class="_ _0"> </span>show<span class="_ _a"> </span>that<span class="_ _0"> </span>an</div><div class="t m0 x8 h6 y1f ff3 fs3 fc0 sc0 ls1 ws0">apparently<span class="_ _1"> </span>very<span class="_ _1"> </span>complicated<span class="_ _1"> </span>quasiperiodic<span class="_ _1"> </span>signal<span class="_ _1"> </span>con-</div><div class="t m0 x8 h6 y20 ff3 fs3 fc0 sc0 ls1 ws0">sists<span class="_ _5"> </span>merely<span class="_ _5"> </span>of<span class="_ _5"> </span>the<span class="_ _5"> </span>sum<span class="_ _5"> </span>o<span class="_ _4"></span>f<span class="_ _5"> </span>two<span class="_ _5"> </span>tones<span class="_ _5"> </span>with<span class="_ _5"> </span>incomm<span class="_"> </span>ensu-</div><div class="t m0 x8 h6 y21 ff3 fs3 fc0 sc0 ls1 ws0">rate<span class="_ _2"> </span>frequencies.<span class="_ _2"> </span>However,<span class="_ _2"> </span>it<span class="_ _1"> </span>is<span class="_ _5"> </span>generally<span class="_ _2"> </span>the<span class="_ _2"> </span>case<span class="_ _2"> </span>tha<span class="_"> </span>t</div><div class="t m0 x8 h6 y22 ff3 fs3 fc0 sc0 ls1 ws0">complicated<span class="_ _3"> </span>amplitude<span class="_ _3"> </span>and<span class="_ _d"> </span>frequ<span class="_ _4"></span>ency<span class="_ _d"> </span>modu<span class="_ _4"></span>lated<span class="_ _d"> </span>sig-</div><div class="t m0 x8 h6 y23 ff3 fs3 fc0 sc0 ls1 ws0">nals,<span class="_ _2"> </span>such<span class="_ _1"> </span>as<span class="_ _5"> </span>mos<span class="_ _4"></span>t<span class="_ _2"> </span>biological<span class="_ _1"> </span>signals,<span class="_ _2"> </span>have<span class="_ _2"> </span>correspond<span class="_ _4"></span>-</div><div class="t m0 x8 h6 y24 ff3 fs3 fc0 sc0 ls1 ws0">ingly<span class="_ _9"> </span>complicated<span class="_ _9"> </span>Fourier<span class="_ _9"> </span>spectra.<span class="_ _9"> </span>Such<span class="_ _e"> </span>signals<span class="_ _e"> </span>can<span class="_ _9"> </span>also</div><div class="t m0 x8 h6 y25 ff3 fs3 fc0 sc0 ls1 ws0">be<span class="_ _7"> </span>represented<span class="_ _f"> </span>as<span class="_ _7"> </span>being<span class="_ _7"> </span>the<span class="_ _f"> </span>product<span class="_ _7"> </span>of<span class="_ _7"> </span>an<span class="_ _f"> </span>envelope</div><div class="t m0 x8 h6 y26 ff3 fs3 fc0 sc0 ls1 ws0">signal<span class="_ _6"> </span>and<span class="_ _5"> </span>a<span class="_ _5"> </span>frequency-modulated<span class="_ _5"> </span>signal,<span class="_ _5"> </span>or<span class="_ _6"> </span>as<span class="_ _5"> </span>the<span class="_ _5"> </span>sum</div><div class="t m0 x8 h6 y27 ff3 fs3 fc0 sc0 ls1 ws0">of<span class="_ _9"> </span>a<span class="_ _6"> </span>&#64257;nite<span class="_ _9"> </span>set<span class="_ _6"> </span>of<span class="_ _9"> </span>such<span class="_ _6"> </span>product<span class="_ _6"> </span>functions.<span class="_ _6"> </span>A<span class="_ _9"> </span>well-behaved</div><div class="t m0 x8 h6 y28 ff3 fs3 fc0 sc0 ls1 ws0">and<span class="_ _2"> </span>physically<span class="_ _1"> </span>meaningful<span class="_ _10"> </span>time-varying<span class="_ _10"> </span>instantaneous</div><div class="t m0 x8 h6 y29 ff3 fs3 fc0 sc0 ls1 ws0">frequency<span class="_ _3"> </span>can<span class="_ _3"> </span>then<span class="_ _3"> </span>be<span class="_ _3"> </span>calculated<span class="_ _3"> </span>from<span class="_ _3"> </span>the<span class="_ _3"> </span>frequency</div><div class="t m0 x8 h6 y2a ff3 fs3 fc0 sc0 ls1 ws0">modulated<span class="_ _2"> </span>signal.</div><div class="t m0 x9 h6 y2b ff3 fs3 fc0 sc0 ls5 ws0">The<span class="_ _11"> </span>local<span class="_ _11"> </span>mean<span class="_ _11"> </span>decomposition<span class="_ _11"> </span>(LMD)<span class="_ _11"> </span>was</div><div class="t m0 x8 h6 y2c ff3 fs3 fc0 sc0 ls6 ws0">developed<span class="_ _0"> </span>recently<span class="_ _a"> </span>by<span class="_ _0"> </span>the<span class="_ _a"> </span>author<span class="_ _0"> </span>to<span class="_ _a"> </span>decompose</div><div class="t m0 x8 h6 y2d ff3 fs3 fc0 sc0 ls1 ws0">amplitude<span class="_ _0"> </span>and<span class="_ _0"> </span>frequency<span class="_ _0"> </span>modulated<span class="_ _0"> </span>signals<span class="_ _0"> </span>into<span class="_ _0"> </span>a</div><div class="t m0 x8 h6 y2e ff3 fs3 fc0 sc0 ls1 ws0">small<span class="_ _8"> </span>set<span class="_ _7"> </span>of<span class="_ _8"> </span>product<span class="_ _7"> </span>functions,<span class="_ _7"> </span>each<span class="_ _8"> </span>of<span class="_ _7"> </span>which<span class="_ _7"> </span>is<span class="_ _8"> </span>the</div><div class="t m0 x8 h6 y2f ff3 fs3 fc0 sc0 ls7 ws0">product<span class="_ _a"> </span>of<span class="_ _a"> </span>an<span class="_ _12"> </span>envelope<span class="_ _a"> </span>signal<span class="_ _12"> </span>and<span class="_ _a"> </span>a<span class="_ _a"> </span>frequency</div><div class="t m0 x8 h6 y30 ff3 fs3 fc0 sc0 ls1 ws0">modulated<span class="_ _8"> </span>signal<span class="_ _7"> </span>from<span class="_ _7"> </span>which<span class="_ _8"> </span>a<span class="_ _7"> </span>time-varying<span class="_ _7"> </span>instan-</div><div class="t m0 x8 h6 y31 ff3 fs3 fc0 sc0 ls1 ws0">taneous<span class="_ _b"> </span>phase<span class="_ _b"> </span>and<span class="_ _b"> </span>inst<span class="_ _4"></span>antaneous<span class="_ _b"> </span>frequency<span class="_ _0"> </span>can<span class="_ _b"> </span>be</div><div class="t m0 xa h6 y32 ff3 fs3 fc0 sc0 ls1 ws0">derived.<span class="_ _b"> </span>The<span class="_ _b"> </span>instantaneous<span class="_ _b"> </span>frequency<span class="_ _b"> </span>and<span class="_ _f"> </span>enve<span class="_ _4"></span>lope</div><div class="t m0 xa h6 y33 ff3 fs3 fc0 sc0 ls1 ws0">values<span class="_ _d"> </span>can<span class="_ _3"> </span>then<span class="_ _3"> </span>be<span class="_ _3"> </span>plotted<span class="_ _3"> </span>together<span class="_ _3"> </span>in<span class="_ _3"> </span>the<span class="_ _3"> </span>form<span class="_ _3"> </span>of<span class="_ _d"> </span>a</div><div class="t m0 xa h6 y34 ff3 fs3 fc0 sc0 ls8 ws0">demodulated<span class="_ _0"> </span>signal<span class="_ _a"> </span>time&#8211;f<span class="ls9">requency<span class="_ _0"> </span>representation</span></div><div class="t m0 xa h6 y35 ff3 fs3 fc0 sc0 ls1 ws0">(tfr).<span class="_ _10"> </span>Although<span class="_ _10"> </span>a<span class="_ _1"> </span>form<span class="_ _10"> </span>of<span class="_ _10"> </span>instan<span class="_ _4"></span>taneous<span class="_ _10"> </span>frequency<span class="_ _10"> </span>can</div><div class="t m0 xa h6 y36 ff3 fs3 fc0 sc0 ls1 ws0">be<span class="_ _8"> </span>calculated<span class="_ _7"> </span>from<span class="_ _8"> </span>the<span class="_ _7"> </span>spectrogr<span class="_ _4"></span>am<span class="_ _8"> </span>(the<span class="_ _8"> </span>con<span class="_ _4"></span>ditional</div><div class="t m0 xa h6 y37 ff3 fs3 fc0 sc0 ls1 ws0">mean<span class="_ _1"> </span>frequency<span class="_ _4"></span>,<span class="_ _1"> </span>investigated<span class="_ _d"> </span>by<span class="_ _1"> </span><span class="fc1">Loughlin<span class="_ _d"> </span>1999</span>),<span class="_ _1"> </span>it<span class="_ _1"> </span>is</div><div class="t m0 xa h6 y38 ff3 fs3 fc0 sc0 ls1 ws0">window-dependent.<span class="_ _5"> </span>The<span class="_ _5"> </span>conventional<span class="_ _5"> </span>method<span class="_ _5"> </span>of<span class="_ _5"> </span>deriv-</div><div class="t m0 xa h6 y39 ff3 fs3 fc0 sc0 ls1 ws0">ing<span class="_ _6"> </span>a<span class="_ _5"> </span>time-varying<span class="_ _5"> </span>frequency<span class="_ _5"> </span>from<span class="_ _6"> </span>a<span class="_ _5"> </span>signal<span class="_ _6"> </span>is<span class="_ _5"> </span>to<span class="_ _6"> </span>use<span class="_ _5"> </span>the</div><div class="t m0 xa h6 y3a ff3 fs3 fc0 sc0 ls1 ws0">so-called<span class="_ _8"> </span>analytic<span class="_ _8"> </span>signal<span class="_ _8"> </span>developed<span class="_ _8"> </span>by<span class="_ _3"> </span><span class="fc1">Gabor<span class="_ _8"> </span>(1946)</span>.</div><div class="t m0 xa h6 y3b ff3 fs3 fc0 sc0 ls1 ws0">The<span class="_ _8"> </span>instantaneous<span class="_ _7"> </span>frequency<span class="_ _8"> </span>is<span class="_ _7"> </span>de&#64257;ned<span class="_ _8"> </span>as<span class="_ _7"> </span>being<span class="_ _8"> </span>the</div><div class="t m0 xa h6 y3c ff3 fs3 fc0 sc0 ls1 ws0">derivative<span class="_ _b"> </span>of<span class="_ _f"> </span>the<span class="_ _b"> </span>phase<span class="_ _13"> </span>of<span class="_ _13"> </span>the<span class="_ _13"> </span>analytic<span class="_ _13"> </span>signal.<span class="_ _b"> </span>The</div><div class="t m0 xa h6 y3d ff3 fs3 fc0 sc0 ls1 ws0">problem<span class="_ _7"> </span>with<span class="_ _7"> </span>this<span class="_ _7"> </span>approach<span class="_ _7"> </span>is<span class="_ _7"> </span>that,<span class="_ _7"> </span>particul<span class="_ _4"></span>arly<span class="_ _7"> </span>for</div><div class="t m0 xa h6 y3e ff3 fs3 fc0 sc0 lsa ws0">amplitude<span class="_ _a"> </span>and<span class="_ _0"> </span>frequenc<span class="_"> </span>y<span class="_ _a"> </span>modulated<span class="_ _0"> </span>signals,<span class="_ _a"> </span>the</div><div class="t m0 xa h6 y3f ff3 fs3 fc0 sc0 ls1 ws0">resulting<span class="_ _9"> </span>instantane<span class="_ _4"></span>ous<span class="_ _9"> </span>frequency<span class="_ _6"> </span>can<span class="_ _9"> </span>be<span class="_ _9"> </span>wildly<span class="_ _6"> </span>erratic,</div><div class="t m0 xa h6 y40 ff3 fs3 fc0 sc0 ls1 ws0">even<span class="_ _3"> </span>negati<span class="_ _4"></span>ve<span class="_ _3"> </span>at<span class="_ _8"> </span>times<span class="_ _3"> </span>(<span class="fc1">Mand<span class="_ _4"></span>el<span class="_ _3"> </span>1974</span>;<span class="_ _8"> </span><span class="fc1">Melville<span class="_ _3"> </span>1983<span class="_ _4"></span></span>;</div><div class="t m0 xa h6 y41 ff3 fs3 fc1 sc0 ls1 ws0">Boashash<span class="_ _8"> </span>1992<span class="fc0">).<span class="_ _8"> </span>Such<span class="_ _7"> </span>an<span class="_ _8"> </span>instantaneous<span class="_ _7"> </span>frequency<span class="_ _8"> </span>is</span></div><div class="t m0 xa h6 y42 ff3 fs3 fc0 sc0 lsb ws0">extremely<span class="_ _0"> </span>hard<span class="_ _a"> </span>to<span class="_ _0"> </span>interpret,<span class="_ _a"> </span>and<span class="_ _0"> </span>is<span class="_ _a"> </span>of<span class="_ _0"> </span>debatable</div><div class="t m0 xa h6 y43 ff3 fs3 fc0 sc0 ls1 ws0">physical<span class="_ _6"> </span>signi&#64257;ca<span class="_ _4"></span>nce.<span class="_ _6"> </span>Conseq<span class="_ _4"></span>uently,<span class="_ _6"> </span>the<span class="_ _5"> </span>analytic<span class="_ _6"> </span>si<span class="_ _4"></span>gnal</div><div class="t m0 xa h6 y44 ff3 fs3 fc0 sc0 ls1 ws0">instantaneous<span class="_ _d"> </span>frequency<span class="_ _d"> </span>has<span class="_ _1"> </span>been<span class="_ _d"> </span>of<span class="_ _d"> </span>limited<span class="_ _d"> </span>practical</div><div class="t m0 xa h6 y45 ff3 fs3 fc0 sc0 ls1 ws0">utility.<span class="_ _2"> </span>By<span class="_ _2"> </span>contrast,<span class="_ _2"> </span>the<span class="_ _10"> </span>new<span class="_ _2"> </span>LMD<span class="_ _2"> </span>scheme<span class="_ _2"> </span>is<span class="_ _2"> </span>a<span class="_ _2"> </span>robust<span class="_ _4"></span>,</div><div class="t m0 xa h6 y46 ff3 fs3 fc0 sc0 ls1 ws0">and<span class="_ _2"> </span>conceptually<span class="_ _10"> </span>simple<span class="_ _2"> </span>way<span class="_ _2"> </span>of<span class="_ _10"> </span>analysing<span class="_ _2"> </span>complicated</div><div class="t m0 xa h6 y47 ff3 fs3 fc0 sc0 ls1 ws0">data<span class="_ _7"> </span>in<span class="_ _7"> </span>terms<span class="_ _f"> </span>of<span class="_ _7"> </span>time-varying<span class="_ _7"> </span>frequ<span class="_ _4"></span>ency,<span class="_ _7"> </span>phase<span class="_ _7"> </span>and</div><div class="t m0 xa h6 y48 ff3 fs3 fc0 sc0 ls1 ws0">energy.<span class="_ _b"> </span>Most<span class="_ _b"> </span>importantly,<span class="_ _b"> </span>the<span class="_ _b"> </span>results<span class="_ _b"> </span>of<span class="_ _b"> </span>the<span class="_ _b"> </span>LMD</div><div class="t m0 xa h6 y49 ff3 fs3 fc0 sc0 ls1 ws0">analysis<span class="_ _b"> </span>are<span class="_ _0"> </span>physically<span class="_ _b"> </span>plaus<span class="_ _4"></span>ible;<span class="_ _b"> </span>the<span class="_ _0"> </span>instantaneous</div><div class="t m0 xa h6 y4a ff3 fs3 fc0 sc0 lsc ws0">frequency<span class="_ _0"> </span>generally<span class="_ _0"> </span>cor<span class="_"> </span>responds<span class="_ _a"> </span>to<span class="_ _0"> </span>the<span class="_ _0"> </span>oscillatory</div><div class="t m0 xa h6 y4b ff3 fs3 fc0 sc0 ls1 ws0">frequency<span class="_ _2"> </span>we<span class="_ _5"> </span>can<span class="_ _2"> </span>actually<span class="_ _2"> </span>see<span class="_ _2"> </span>in<span class="_ _5"> </span>the<span class="_ _2"> </span>signal.</div><div class="t m0 xb h6 y4c ff3 fs3 fc0 sc0 ls1 ws0">The<span class="_ _2"> </span>LMD<span class="_ _2"> </span>method<span class="_ _2"> </span>can<span class="_ _2"> </span>be<span class="_ _10"> </span>used<span class="_ _2"> </span>to<span class="_ _2"> </span>analyse<span class="_ _2"> </span>any<span class="_ _2"> </span>data,</div><div class="t m0 xa h6 y4d ff3 fs3 fc0 sc0 ls1 ws0">but<span class="_ _e"> </span>is<span class="_ _9"> </span>of<span class="_ _9"> </span>particular<span class="_ _9"> </span>relevance<span class="_ _9"> </span>with<span class="_ _9"> </span>regard<span class="_ _9"> </span>to<span class="_ _e"> </span>the<span class="_ _9"> </span>analysis</div><div class="t m0 xa h6 y4e ff3 fs3 fc0 sc0 lsd ws0">of<span class="_ _0"> </span>amplitude<span class="_ _a"> </span>and<span class="_ _0"> </span>frequency<span class="_ _a"> </span>modulated<span class="_ _a"> </span>biological</div><div class="t m0 xa h6 y4f ff3 fs3 fc0 sc0 ls1 ws0">signals.<span class="_ _0"> </span>It<span class="_ _0"> </span>could<span class="_ _0"> </span>be<span class="_ _0"> </span>used,<span class="_ _0"> </span>for<span class="_ _0"> </span>example,<span class="_ _0"> </span>to<span class="_ _0"> </span>analyse</div><div class="t m0 xa h6 y50 ff3 fs3 fc0 sc0 ls1 ws0">changes<span class="_ _8"> </span>in<span class="_ _3"> </span>blood<span class="_ _8"> </span>pressure,<span class="_ _8"> </span>or<span class="_ _3"> </span>to<span class="_ _8"> </span>detect<span class="_ _8"> </span>small<span class="_ _3"> </span>&#64258;uctu-</div><div class="t m0 xa h6 y51 ff3 fs3 fc0 sc0 lse ws0">ations<span class="_ _0"> </span>in<span class="_ _a"> </span>the<span class="_ _a"> </span>frequency<span class="_ _0"> </span>of<span class="_ _0"> </span>the<span class="_ _a"> </span>electrocardiogram</div><div class="t m0 xc h8 y52 ff2 fs4 fc0 sc0 ls1 ws0">J.<span class="_ _6"> </span>R.<span class="_ _5"> </span>Soc.<span class="_ _5"> </span>Interface<span class="_ _6"> </span><span class="ff3">(2005)<span class="_ _5"> </span><span class="ff1">2</span>,<span class="_ _6"> </span>443&#8211;454</span></div><div class="t m0 xd h8 y53 ff3 fs4 fc0 sc0 ls1 ws0">doi:10.1098/rsif.2005.0058</div><div class="t m0 xe h8 y54 ff2 fs4 fc0 sc0 ls1 ws0">Published<span class="_ _6"> </span>online<span class="_ _5"> </span><span class="ff3">28<span class="_ _6"> </span>July<span class="_ _5"> </span>2005</span></div><div class="t m2 x8 h9 y55 ff3 fs5 fc0 sc0 ls1 ws0">&#8224;</div><div class="t m0 xf ha y56 ff3 fs6 fc0 sc0 ls1 ws0">jsrsmith@neurotechno.co.uk.</div><div class="t m0 x8 ha y57 ff2 fs6 fc0 sc0 ls1 ws0">Received<span class="_ _9"> </span><span class="ff3">17<span class="_ _9"> </span>May<span class="_ _6"> </span>2005</span></div><div class="t m0 x8 ha y58 ff2 fs6 fc0 sc0 ls1 ws0">Accepted<span class="_ _9"> </span><span class="ff3">23<span class="_ _9"> </span>June<span class="_ _6"> </span>2005</span></div><div class="t m0 x10 h6 y59 ff3 fs3 fc0 sc0 lsf ws0">443<span class="_ _14"> </span><span class="ff4 fs6 ls1">q<span class="_ _9"> </span><span class="ff3">2005<span class="_ _9"> </span>The<span class="_ _6"> </span>Royal<span class="_ _6"> </span>Society</span></span></div><a class="l" rel='nofollow' onclick='return false;'><div class="d m3"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m3"></div></a><a class="l" rel='nofollow' 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