Variational Mode Decomposition.rar

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vmd仿真程序在matlab中的应用,可用于数字信号处理方面
Variational Mode Decomposition.rar
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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/625d3f64be9ad24cfa7c5415/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/625d3f64be9ad24cfa7c5415/bg1.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">Copyright (c) 2013 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing pubs-permissions@ieee.org.</div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls0 ws0">This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI</div><div class="t m0 x3 h2 y3 ff1 fs0 fc0 sc0 ls0 ws0">10.1109/TSP.2013.2288675, IEEE Transactions on Signal Processing</div><div class="t m0 x4 h3 y4 ff2 fs1 fc0 sc0 ls0 ws0">IEEE<span class="_ _0"> </span>TRANSA<span class="_ _1"></span>CTIONS<span class="_ _0"> </span>ON<span class="_ _0"> </span>SIGNAL<span class="_ _0"> </span>PR<span class="_ _1"></span>OCESSING,<span class="_ _0"> </span>V<span class="_ _1"></span>OL.<span class="_ _0"> </span>X,<span class="_ _0"> </span>NO.<span class="_ _0"> </span>Y<span class="_ _2"></span>,<span class="_ _0"> </span>ZZZ<span class="_ _0"> </span>20WW<span class="_ _3"> </span>1</div><div class="t m0 x5 h4 y5 ff2 fs2 fc0 sc0 ls0 ws0">V<span class="_ _4"></span>ariational<span class="_ _5"> </span>Mode<span class="_ _5"> </span>Decomposition</div><div class="t m0 x6 h5 y6 ff2 fs3 fc0 sc0 ls0 ws0">K<span class="_ _1"></span>onstantin<span class="_ _6"> </span>Dragomiretskiy<span class="_ _6"> </span>and<span class="_ _6"> </span>Dominique<span class="_ _7"> </span>Zosso</div><div class="t m0 x7 h6 y7 ff3 fs4 fc0 sc0 ls0 ws0">&#8727;</div><div class="t m0 x8 h5 y6 ff2 fs3 fc0 sc0 ls0 ws0">,<span class="_ _6"> </span><span class="ff4">Member<span class="_ _2"></span>,<span class="_ _6"> </span>IEEE</span></div><div class="t m0 x9 h7 y8 ff5 fs5 fc0 sc0 ls0 ws0">Abstract<span class="ff6">&#8212;In<span class="_ _0"> </span>the<span class="_ _0"> </span>late<span class="_ _8"> </span>nineties,<span class="_ _0"> </span>Huang<span class="_ _0"> </span>introduced<span class="_ _0"> </span>the<span class="_ _8"> </span>algorithm</span></div><div class="t m0 x4 h8 y9 ff6 fs5 fc0 sc0 ls0 ws0">called<span class="_ _9"> </span>Empirical<span class="_ _9"> </span>Mode<span class="_ _9"> </span>Decomposition,<span class="_ _9"> </span>which<span class="_ _9"> </span>is<span class="_ _9"> </span>widely<span class="_ _9"> </span>used</div><div class="t m0 x4 h8 ya ff6 fs5 fc0 sc0 ls0 ws0">today<span class="_ _a"> </span>to<span class="_ _a"> </span>recursi<span class="_ _1"></span>vely<span class="_ _a"> </span>decompose<span class="_ _a"> </span>a<span class="_ _a"> </span>signal<span class="_ _a"> </span>into<span class="_ _a"> </span>different<span class="_ _a"> </span>modes</div><div class="t m0 x4 h8 yb ff6 fs5 fc0 sc0 ls0 ws0">of<span class="_ _8"> </span>unknown<span class="_ _0"> </span>but<span class="_ _8"> </span>separate<span class="_ _8"> </span>spectral<span class="_ _8"> </span>bands.<span class="_ _8"> </span>EMD<span class="_ _8"> </span>is<span class="_ _8"> </span>known<span class="_ _8"> </span>f<span class="_ _1"></span>or<span class="_ _8"> </span>lim-</div><div class="t m0 x4 h8 yc ff6 fs5 fc0 sc0 ls0 ws0">itations<span class="_ _6"> </span>like<span class="_ _b"> </span>sensitivity<span class="_ _b"> </span>to<span class="_ _6"> </span>noise<span class="_ _b"> </span>and<span class="_ _6"> </span>sampling,<span class="_ _6"> </span>These<span class="_ _b"> </span>limitations</div><div class="t m0 x4 h8 yd ff6 fs5 fc0 sc0 ls0 ws0">could<span class="_"> </span>only<span class="_"> </span>partially<span class="_ _0"> </span>be<span class="_"> </span>addressed<span class="_"> </span>by<span class="_ _0"> </span>more<span class="_"> </span>mathematical<span class="_"> </span>attempts</div><div class="t m0 x4 h8 ye ff6 fs5 fc0 sc0 ls0 ws0">to<span class="_ _b"> </span>this<span class="_ _b"> </span>decomposition<span class="_ _b"> </span>pr<span class="_ _1"></span>oblem,<span class="_ _b"> </span>lik<span class="_ _1"></span>e<span class="_ _b"> </span>synchrosqueezing,<span class="_ _c"> </span>empirical</div><div class="t m0 x4 h8 yf ff6 fs5 fc0 sc0 ls0 ws0">wav<span class="_ _1"></span>elets<span class="_ _c"> </span>or<span class="_ _b"> </span>recursive<span class="_ _c"> </span>variational<span class="_ _b"> </span>decomposition.</div><div class="t m0 x9 h8 y10 ff6 fs5 fc0 sc0 ls0 ws0">Here,<span class="_ _b"> </span>we<span class="_ _6"> </span>propose<span class="_ _b"> </span>an<span class="_ _6"> </span>entirely<span class="_ _b"> </span>non-recursive<span class="_ _b"> </span>variational<span class="_ _6"> </span>mode</div><div class="t m0 x4 h8 y11 ff6 fs5 fc0 sc0 ls0 ws0">decomposition<span class="_ _a"> </span>model,<span class="_ _a"> </span>wher<span class="_ _1"></span>e<span class="_ _a"> </span>the<span class="_ _d"> </span>modes<span class="_ _d"> </span>are<span class="_ _d"> </span>extracted<span class="_ _a"> </span>concur-</div><div class="t m0 x4 h8 y12 ff6 fs5 fc0 sc0 ls0 ws0">rently<span class="_ _e"></span>.<span class="_ _d"> </span>The<span class="_ _a"> </span>model<span class="_ _d"> </span>looks<span class="_ _a"> </span>for<span class="_ _d"> </span>an<span class="_ _a"> </span>ensemble<span class="_ _d"> </span>of<span class="_ _a"> </span>modes<span class="_ _a"> </span>and<span class="_ _d"> </span>their</div><div class="t m0 x4 h8 y13 ff6 fs5 fc0 sc0 ls0 ws0">respecti<span class="_ _1"></span>ve<span class="_ _d"> </span>center<span class="_ _d"> </span>fr<span class="_ _1"></span>equencies,<span class="_ _d"> </span>such<span class="_ _d"> </span>that<span class="_ _d"> </span>the<span class="_ _d"> </span>modes<span class="_ _d"> </span>collectiv<span class="_ _1"></span>ely</div><div class="t m0 x4 h8 y14 ff6 fs5 fc0 sc0 ls0 ws0">repr<span class="_ _1"></span>oduce<span class="_ _f"> </span>the<span class="_ _f"> </span>input<span class="_ _f"> </span>signal,<span class="_ _f"> </span>while<span class="_ _f"> </span>each<span class="_ _f"> </span>being<span class="_ _f"> </span>smooth<span class="_ _10"> </span>after</div><div class="t m0 x4 h8 y15 ff6 fs5 fc0 sc0 ls0 ws0">demodulation<span class="_"> </span>into<span class="_"> </span>baseband.<span class="_"> </span>In<span class="_"> </span>F<span class="_ _1"></span>ourier<span class="_"> </span>domain,<span class="_ _11"> </span>this<span class="_"> </span>corresponds</div><div class="t m0 x4 h8 y16 ff6 fs5 fc0 sc0 ls0 ws0">to<span class="_ _c"> </span>a<span class="_ _c"> </span>narrow-band<span class="_ _8"> </span>prior<span class="_ _e"></span>.<span class="_ _c"> </span>W<span class="_ _e"></span>e<span class="_ _c"> </span>show<span class="_ _c"> </span>important<span class="_ _c"> </span>relations<span class="_ _c"> </span>to<span class="_ _c"> </span>Wiener</div><div class="t m0 x4 h8 y17 ff6 fs5 fc0 sc0 ls0 ws0">&#64257;lter<span class="_ _c"> </span>denoising.<span class="_ _8"> </span>Indeed,<span class="_ _c"> </span>the<span class="_ _c"> </span>proposed<span class="_ _c"> </span>method<span class="_ _c"> </span>is<span class="_ _c"> </span>a<span class="_ _c"> </span>generalization</div><div class="t m0 x4 h8 y18 ff6 fs5 fc0 sc0 ls0 ws0">of<span class="_ _7"> </span>the<span class="_ _12"> </span>classic<span class="_ _7"> </span>Wiener<span class="_ _7"> </span>&#64257;lter<span class="_ _12"> </span>into<span class="_ _7"> </span>multiple,<span class="_ _12"> </span>adaptive<span class="_ _7"> </span>bands.<span class="_ _7"> </span>Our</div><div class="t m0 x4 h8 y19 ff6 fs5 fc0 sc0 ls0 ws0">model<span class="_ _d"> </span>provides<span class="_ _d"> </span>a<span class="_ _d"> </span>solution<span class="_ _a"> </span>to<span class="_ _d"> </span>the<span class="_ _a"> </span>decomposition<span class="_ _d"> </span>problem<span class="_ _d"> </span>that</div><div class="t m0 x4 h8 y1a ff6 fs5 fc0 sc0 ls0 ws0">is<span class="_ _7"> </span>theoretically<span class="_ _7"> </span>well<span class="_ _7"> </span>founded<span class="_ _7"> </span>and<span class="_ _7"> </span>still<span class="_ _12"> </span>easy<span class="_ _7"> </span>to<span class="_ _12"> </span>understand.<span class="_ _7"> </span>The</div><div class="t m0 x4 h8 y1b ff6 fs5 fc0 sc0 ls0 ws0">variational<span class="_ _12"> </span>model<span class="_ _a"> </span>is<span class="_ _d"> </span>ef&#64257;ciently<span class="_ _d"> </span>optimized<span class="_ _d"> </span>using<span class="_ _a"> </span>an<span class="_ _d"> </span>alternating</div><div class="t m0 x4 h8 y1c ff6 fs5 fc0 sc0 ls0 ws0">direction<span class="_ _d"> </span>method<span class="_ _a"> </span>of<span class="_ _d"> </span>multipliers<span class="_ _a"> </span>approach.<span class="_ _d"> </span>Preliminary<span class="_ _a"> </span>results</div><div class="t m0 x4 h8 y1d ff6 fs5 fc0 sc0 ls0 ws0">show<span class="_ _f"> </span>attractive<span class="_ _f"> </span>perf<span class="_ _1"></span>ormance<span class="_ _f"> </span>with<span class="_ _f"> </span>respect<span class="_ _f"> </span>to<span class="_ _f"> </span>existing<span class="_ _10"> </span>mode</div><div class="t m0 x4 h8 y1e ff6 fs5 fc0 sc0 ls0 ws0">decomposition<span class="_"> </span>models.<span class="_"> </span>In<span class="_"> </span>particular<span class="_ _e"></span>,<span class="_"> </span>our<span class="_ _0"> </span>proposed<span class="_"> </span>model<span class="_"> </span>is<span class="_"> </span>much</div><div class="t m0 x4 h8 y1f ff6 fs5 fc0 sc0 ls0 ws0">more<span class="_ _c"> </span>robust<span class="_ _c"> </span>to<span class="_ _6"> </span>sampling<span class="_ _b"> </span>and<span class="_ _b"> </span>noise.<span class="_ _b"> </span>Finally<span class="_ _1"></span>,<span class="_ _b"> </span>we<span class="_ _b"> </span>show<span class="_ _b"> </span>promising</div><div class="t m0 x4 h8 y20 ff6 fs5 fc0 sc0 ls0 ws0">practical<span class="_ _b"> </span>decomposition<span class="_ _6"> </span>results<span class="_ _b"> </span>on<span class="_ _6"> </span>a<span class="_ _b"> </span>series<span class="_ _6"> </span>of<span class="_ _b"> </span>arti&#64257;cial<span class="_ _6"> </span>and<span class="_ _b"> </span>real</div><div class="t m0 x4 h8 y21 ff6 fs5 fc0 sc0 ls0 ws0">data.</div><div class="t m0 x9 h7 y22 ff5 fs5 fc0 sc0 ls0 ws0">Index<span class="_ _13"> </span>T<span class="_ _e"></span>erms<span class="ff6">&#8212;Mode<span class="_ _13"> </span>decomposition,<span class="_ _13"> </span>variational<span class="_ _13"> </span>pr<span class="_ _1"></span>oblem,</span></div><div class="t m0 x4 h8 y23 ff6 fs5 fc0 sc0 ls0 ws0">Wiener<span class="_ _a"> </span>&#64257;lter<span class="_ _e"></span>,<span class="_ _a"> </span>AM-FM,<span class="_ _14"> </span>spectral<span class="_ _a"> </span>decomposition,<span class="_ _14"> </span>Hilbert<span class="_ _a"> </span>trans-</div><div class="t m0 x4 h8 y24 ff6 fs5 fc0 sc0 ls0 ws0">form,<span class="_ _c"> </span>Fourier<span class="_ _c"> </span>transform,<span class="_ _c"> </span>augmented<span class="_ _b"> </span>Lagrangian.</div><div class="t m0 xa h9 y25 ff2 fs6 fc0 sc0 ls0 ws0">I<span class="_ _15"></span>.<span class="_ _9"> </span>I<span class="_ _15"></span><span class="fs4">N<span class="_ _15"></span>T<span class="_ _15"></span>RO<span class="_ _15"></span>D<span class="_ _15"></span>U<span class="_ _15"></span>C<span class="_ _15"></span>T<span class="_ _15"></span>I<span class="_ _15"></span>O<span class="_ _15"></span>N</span></div><div class="t m0 x4 ha y26 ff6 fs7 fc0 sc0 ls0 ws0">E</div><div class="t m0 xb h9 y27 ff2 fs6 fc0 sc0 ls0 ws0">MPIRICAL<span class="_"> </span>MODE<span class="_"> </span>DECOMPOSITON<span class="_ _0"> </span>(EMD)<span class="_"> </span>proposed</div><div class="t m0 xb h9 y28 ff2 fs6 fc0 sc0 ls0 ws0">by<span class="_ _7"> </span>Huang<span class="_ _7"> </span>et<span class="_ _12"> </span>al.<span class="_ _7"> </span>[1]<span class="_ _7"> </span>is<span class="_ _12"> </span>an<span class="_ _7"> </span>algorithmic<span class="_ _12"> </span>method<span class="_ _7"> </span>to<span class="_ _7"> </span>detect</div><div class="t m0 x4 h9 y29 ff2 fs6 fc0 sc0 ls0 ws0">and<span class="_"> </span>decompose<span class="_ _0"> </span>a<span class="_ _11"> </span>signal<span class="_"> </span>into<span class="_"> </span>principal<span class="_"> </span>&#8220;modes&#8221;.<span class="_ _11"> </span>This<span class="_"> </span>algorithm</div><div class="t m0 x4 h9 y2a ff2 fs6 fc0 sc0 ls0 ws0">recursiv<span class="_ _1"></span>ely<span class="_"> </span>detects<span class="_ _8"> </span>local<span class="_ _8"> </span>minima/maxima<span class="_ _8"> </span>in<span class="_ _8"> </span>a<span class="_ _8"> </span>signal,<span class="_ _8"> </span>estimates</div><div class="t m0 x4 h9 y2b ff2 fs6 fc0 sc0 ls0 ws0">lower/upper<span class="_ _6"> </span>en<span class="_ _1"></span>velopes<span class="_ _6"> </span>by<span class="_ _12"> </span>interpolation<span class="_ _7"> </span>of<span class="_ _7"> </span>these<span class="_ _7"> </span>extrema,<span class="_ _7"> </span>re-</div><div class="t m0 x4 h9 y2c ff2 fs6 fc0 sc0 ls0 ws0">mov<span class="_ _1"></span>es<span class="_ _b"> </span>the<span class="_ _6"> </span>av<span class="_ _1"></span>erage<span class="_ _b"> </span>of<span class="_ _6"> </span>the<span class="_ _b"> </span>env<span class="_ _1"></span>elopes<span class="_ _b"> </span>as<span class="_ _6"> </span>&#8220;low-pass&#8221;<span class="_ _b"> </span>centerline,</div><div class="t m0 x4 h9 y2d ff2 fs6 fc0 sc0 ls0 ws0">thus<span class="_ _a"> </span>isolating<span class="_ _a"> </span>the<span class="_ _a"> </span>high-frequency<span class="_ _a"> </span>oscillations<span class="_ _a"> </span>as<span class="_ _a"> </span>&#8220;mode&#8221;<span class="_ _a"> </span>of</div><div class="t m0 x4 h9 y2e ff2 fs6 fc0 sc0 ls0 ws0">a<span class="_ _a"> </span>signal,<span class="_ _a"> </span>and<span class="_ _a"> </span>continues<span class="_ _a"> </span>recursiv<span class="_ _1"></span>ely<span class="_ _d"> </span>on<span class="_ _a"> </span>the<span class="_ _a"> </span>remaining<span class="_ _a"> </span>&#8220;low-</div><div class="t m0 x4 h9 y2f ff2 fs6 fc0 sc0 ls0 ws0">pass&#8221;<span class="_ _d"> </span>centerline.<span class="_ _a"> </span>In<span class="_ _d"> </span>some<span class="_ _a"> </span>cases,<span class="_ _a"> </span>this<span class="_ _d"> </span>sifting<span class="_ _a"> </span>algorithm<span class="_ _d"> </span>does</div><div class="t m0 x4 h9 y30 ff2 fs6 fc0 sc0 ls0 ws0">indeed<span class="_ _9"> </span>decompose<span class="_ _9"> </span>a<span class="_ _9"> </span>signal<span class="_ _9"> </span>into<span class="_ _9"> </span>principal<span class="_ _9"> </span>modes,<span class="_ _16"> </span>howe<span class="_ _1"></span>v<span class="_ _1"></span>er</div><div class="t m0 x4 h9 y31 ff2 fs6 fc0 sc0 ls0 ws0">the<span class="_ _7"> </span>resulting<span class="_ _7"> </span>decomposition<span class="_ _7"> </span>is<span class="_ _7"> </span>highly<span class="_ _7"> </span>dependent<span class="_ _7"> </span>on<span class="_ _7"> </span>methods</div><div class="t m0 x4 h9 y32 ff2 fs6 fc0 sc0 ls0 ws0">of<span class="_ _9"> </span>extremal<span class="_ _9"> </span>point<span class="_ _9"> </span>&#64257;nding,<span class="_ _9"> </span>interpolation<span class="_ _9"> </span>of<span class="_ _9"> </span>extremal<span class="_ _9"> </span>points</div><div class="t m0 x4 h9 y33 ff2 fs6 fc0 sc0 ls0 ws0">into<span class="_ _b"> </span>carrier<span class="_ _b"> </span>en<span class="_ _1"></span>v<span class="_ _1"></span>elopes,<span class="_ _b"> </span>and<span class="_ _b"> </span>the<span class="_ _c"> </span>stopping<span class="_ _b"> </span>criteria<span class="_ _b"> </span>imposed.<span class="_ _b"> </span>The</div><div class="t m0 x4 h9 y34 ff2 fs6 fc0 sc0 ls0 ws0">lack<span class="_ _8"> </span>of<span class="_ _c"> </span>mathematical<span class="_ _c"> </span>theory<span class="_ _8"> </span>and<span class="_ _c"> </span>the<span class="_ _c"> </span>aformentioned<span class="_ _8"> </span>degrees<span class="_ _8"> </span>of</div><div class="t m0 xc hb y35 ff2 fs4 fc0 sc0 ls0 ws0">Copyright</div><div class="t m0 xd hb y36 ff2 fs4 fc0 sc0 ls0 ws0">c</div><div class="t m0 xe h6 y35 ff3 fs4 fc0 sc0 ls0 ws0">&#58893;<span class="_ _d"> </span><span class="ff2">2013<span class="_ _d"> </span>IEEE.<span class="_ _a"> </span>Personal<span class="_ _d"> </span>use<span class="_ _a"> </span>of<span class="_ _d"> </span>this<span class="_ _d"> </span>material<span class="_ _a"> </span>is<span class="_ _d"> </span>permitted.</span></div><div class="t m0 x4 hb y37 ff2 fs4 fc0 sc0 ls0 ws0">Howe<span class="_ _1"></span>ver<span class="_ _e"></span>,<span class="_ _7"> </span>permission<span class="_ _6"> </span>to<span class="_ _7"> </span>use<span class="_ _7"> </span>this<span class="_ _6"> </span>material<span class="_ _7"> </span>for<span class="_ _6"> </span>any<span class="_ _6"> </span>other<span class="_ _7"> </span>purposes<span class="_ _6"> </span>must<span class="_ _7"> </span>be</div><div class="t m0 x4 hb y38 ff2 fs4 fc0 sc0 ls0 ws0">obtained<span class="_ _0"> </span>from<span class="_ _0"> </span>the<span class="_ _0"> </span>IEEE<span class="_ _0"> </span>by<span class="_ _0"> </span>sending<span class="_ _0"> </span>a<span class="_ _0"> </span>request<span class="_ _0"> </span>to<span class="_ _0"> </span>pubs-persmissions@ieee.org.</div><div class="t m0 x4 hc y39 ff7 fs8 fc0 sc0 ls0 ws0">&#8727;</div><div class="t m0 xf hb y3a ff2 fs4 fc0 sc0 ls0 ws0">T<span class="_ _e"></span>o<span class="_ _8"> </span>whom<span class="_ _8"> </span>correspondence<span class="_ _8"> </span>should<span class="_ _8"> </span>be<span class="_ _8"> </span>addressed.</div><div class="t m0 x4 hb y3b ff2 fs4 fc0 sc0 ls0 ws0">This<span class="_"> </span>work<span class="_"> </span>is<span class="_ _17"> </span>supported<span class="_"> </span>by<span class="_"> </span>the<span class="_"> </span>Swiss<span class="_ _17"> </span>National<span class="_"> </span>Science<span class="_"> </span>F<span class="_ _1"></span>oundation<span class="_"> </span>(SNF)<span class="_"> </span>under</div><div class="t m0 x4 hb y3c ff2 fs4 fc0 sc0 ls0 ws0">grants<span class="_ _b"> </span>PBELP2<span class="_ _7"> </span>137727<span class="_ _b"> </span>and<span class="_ _b"> </span>P300P2<span class="_ _6"> </span>147778,<span class="_ _6"> </span>the<span class="_ _c"> </span>W<span class="_ _1"></span>.<span class="_ _c"> </span>M.<span class="_ _b"> </span>Keck<span class="_ _c"> </span>Foundation,</div><div class="t m0 x4 hb y3d ff2 fs4 fc0 sc0 ls0 ws0">ONR<span class="_ _6"> </span>grants<span class="_ _7"> </span>N000141210838<span class="_ _7"> </span>and<span class="_ _7"> </span>N000141210040,<span class="_ _6"> </span>and<span class="_ _7"> </span>NSF<span class="_ _7"> </span>grant<span class="_ _7"> </span>DMS-</div><div class="t m0 x4 hb y3e ff2 fs4 fc0 sc0 ls0 ws0">1118971.</div><div class="t m0 x4 hb y3f ff2 fs4 fc0 sc0 ls0 ws0">The<span class="_"> </span>authors<span class="_"> </span>are<span class="_"> </span>with<span class="_"> </span>the<span class="_"> </span>Department<span class="_"> </span>of<span class="_"> </span>Mathematics,<span class="_"> </span>Uni<span class="_ _1"></span>versity<span class="_"> </span>of<span class="_"> </span>California,</div><div class="t m0 x4 hb y40 ff2 fs4 fc0 sc0 ls0 ws0">Los<span class="_ _6"> </span>Angeles<span class="_ _7"> </span>(UCLA),<span class="_ _6"> </span>Box<span class="_ _7"> </span>951555,<span class="_ _7"> </span>Los<span class="_ _6"> </span>Angeles,<span class="_ _7"> </span>CA<span class="_ _6"> </span>90095-1555,<span class="_ _7"> </span>USA,</div><div class="t m0 x4 h6 y41 ff3 fs4 fc0 sc0 ls0 ws0">{<span class="ff2">konstantin,zosso</span>}<span class="ff2">@math.ucla.edu.</span></div><div class="t m0 x10 hd y42 ff8 fs6 fc0 sc0 ls0 ws0">0<span class="_ _14"> </span>0<span class="ff9">.</span>2<span class="_ _18"> </span>0<span class="ff9">.</span>4<span class="_ _18"> </span>0<span class="ff9">.</span>6<span class="_ _18"> </span>0<span class="ff9">.</span>8<span class="_ _14"> </span>1<span class="_ _13"> </span>0<span class="_ _14"> </span>0<span class="ff9">.</span>2<span class="_ _18"> </span>0<span class="ff9">.</span>4<span class="_ _18"> </span>0<span class="ff9">.</span>6<span class="_ _18"> </span>0<span class="ff9">.</span>8<span class="_ _14"> </span>1<span class="_ _5"> </span>0<span class="_ _14"> </span>0<span class="ff9">.</span>2<span class="_ _18"> </span>0<span class="ff9">.</span>4<span class="_ _18"> </span>0<span class="ff9">.</span>6<span class="_ _18"> </span>0<span class="ff9">.</span>8<span class="_ _14"> </span>1</div><div class="t m0 x11 hd y43 ff2 fs6 fc0 sc0 ls0 ws0">(a)<span class="_ _b"> </span><span class="ff9">f</span></div><div class="t m0 x12 h3 y44 ff2 fs1 fc0 sc0 ls0 ws0">saw</div><div class="t m0 x13 hd y43 ff8 fs6 fc0 sc0 ls0 ws0">(<span class="ff9">t</span>)<span class="_ _19"> </span><span class="ff2">(b)<span class="_ _b"> </span><span class="ff9">&#966;</span></span>(<span class="ff9">t</span>)<span class="_ _1a"> </span><span class="ff2">(c)<span class="_ _b"> </span><span class="ff9">&#966;</span></span></div><div class="t m0 x14 he y45 ffa fs1 fc0 sc0 ls0 ws0">0</div><div class="t m0 x15 hd y43 ff8 fs6 fc0 sc0 ls0 ws0">(<span class="ff9">t</span>)</div><div class="t m0 x11 hb y46 ff2 fs4 fc0 sc0 ls0 ws0">Fig.<span class="_ _8"> </span>1.<span class="_ _16"> </span>Sawtooth<span class="_ _8"> </span>signal<span class="_ _8"> </span>and<span class="_ _c"> </span>its<span class="_ _8"> </span>phase/frequency<span class="_ _e"></span>.<span class="_ _c"> </span>(a)<span class="_ _8"> </span>The<span class="_ _8"> </span>original<span class="_ _c"> </span>signal.<span class="_ _8"> </span>(b)</div><div class="t m0 x11 hb y47 ff2 fs4 fc0 sc0 ls0 ws0">The<span class="_ _0"> </span>estimated<span class="_ _8"> </span>phase<span class="_ _8"> </span>after<span class="_ _0"> </span>Hilbert<span class="_ _8"> </span>transform.<span class="_ _8"> </span>The<span class="_ _0"> </span>phase<span class="_ _8"> </span>exhibits<span class="_ _0"> </span>jumps<span class="_ _8"> </span>at<span class="_ _0"> </span>the</div><div class="t m0 x11 hb y48 ff2 fs4 fc0 sc0 ls0 ws0">discontinuous<span class="_ _8"> </span>saw-peaks.<span class="_ _8"> </span>(c)<span class="_ _c"> </span>The<span class="_ _8"> </span>instantaneous<span class="_ _8"> </span>frequency<span class="_ _c"> </span>is<span class="_ _8"> </span>obtained<span class="_ _8"> </span>as<span class="_ _c"> </span>the</div><div class="t m0 x11 hb y49 ff2 fs4 fc0 sc0 ls0 ws0">time-deriv<span class="_ _1"></span>ati<span class="_ _1"></span>ve<span class="_ _8"> </span>of<span class="_ _c"> </span>the<span class="_ _c"> </span>phase.<span class="_ _c"> </span>At<span class="_ _c"> </span>the<span class="_ _c"> </span>discontinuous<span class="_ _b"> </span>saw-peaks,<span class="_ _8"> </span>this<span class="_ _c"> </span>is<span class="_ _c"> </span>clearly</div><div class="t m0 x11 hb y4a ff2 fs4 fc0 sc0 ls0 ws0">meaningless.<span class="_ _c"> </span>Such<span class="_ _c"> </span>a<span class="_ _c"> </span>signal<span class="_ _c"> </span>is<span class="_ _c"> </span>an<span class="_ _c"> </span>admissible<span class="_ _b"> </span>mode<span class="_ _c"> </span>according<span class="_ _c"> </span>to<span class="_ _c"> </span>the<span class="_ _c"> </span>original</div><div class="t m0 x11 hb y4b ff2 fs4 fc0 sc0 ls0 ws0">IMF<span class="_ _b"> </span>de&#64257;nition<span class="_ _b"> </span>based<span class="_ _b"> </span>on<span class="_ _b"> </span>zero-crossings<span class="_ _6"> </span>and<span class="_ _c"> </span>local<span class="_ _6"> </span>extrema<span class="_ _c"> </span>[1],<span class="_ _b"> </span>but<span class="_ _b"> </span>not<span class="_ _b"> </span>part</div><div class="t m0 x11 hb y4c ff2 fs4 fc0 sc0 ls0 ws0">of<span class="_ _0"> </span>the<span class="_ _0"> </span>newer<span class="_ _0"> </span>de&#64257;nition<span class="_ _0"> </span>based<span class="_ _0"> </span>on<span class="_ _0"> </span>amplitude-<span class="_ _0"> </span>and<span class="_ _8"> </span>frequency-modulated<span class="_ _0"> </span>signals</div><div class="t m0 x11 hb y4d ff2 fs4 fc0 sc0 ls0 ws0">(1)<span class="_ _8"> </span>which<span class="_ _8"> </span>is<span class="_ _8"> </span>also<span class="_ _8"> </span>used<span class="_ _8"> </span>here<span class="_ _8"> </span>[13].</div><div class="t m0 x11 h9 y4e ff2 fs6 fc0 sc0 ls0 ws0">freedom<span class="_"> </span>reducing<span class="_"> </span>the<span class="_"> </span>algorithm&#8217;<span class="_ _1"></span>s<span class="_"> </span>rob<span class="_ _1"></span>ustness<span class="_"> </span>all<span class="_"> </span>leav<span class="_ _1"></span>e<span class="_"> </span>room<span class="_"> </span>for</div><div class="t m0 x11 h9 y4f ff2 fs6 fc0 sc0 ls0 ws0">theoretical<span class="_ _12"> </span>dev<span class="_ _1"></span>elopment<span class="_ _12"> </span>and<span class="_ _d"> </span>improv<span class="_ _1"></span>ement<span class="_ _12"> </span>on<span class="_ _12"> </span>the<span class="_ _d"> </span>robustness</div><div class="t m0 x11 h9 y50 ff2 fs6 fc0 sc0 ls0 ws0">of<span class="_ _c"> </span>the<span class="_ _b"> </span>decomposition<span class="_ _c"> </span>[2],<span class="_ _c"> </span>[3],<span class="_ _b"> </span>[4],<span class="_ _c"> </span>[5].<span class="_ _b"> </span>In<span class="_ _c"> </span>some<span class="_ _c"> </span>experiments<span class="_ _c"> </span>it</div><div class="t m0 x11 h9 y51 ff2 fs6 fc0 sc0 ls0 ws0">has<span class="_ _6"> </span>been<span class="_ _6"> </span>shown<span class="_ _6"> </span>that<span class="_ _6"> </span>EMD<span class="_ _6"> </span>shares<span class="_ _6"> </span>important<span class="_ _7"> </span>similarities<span class="_ _6"> </span>with</div><div class="t m0 x11 h9 y52 ff2 fs6 fc0 sc0 ls0 ws0">wa<span class="_ _1"></span>velets<span class="_ _c"> </span>and<span class="_ _6"> </span>(adapti<span class="_ _1"></span>ve)<span class="_ _b"> </span>&#64257;lter<span class="_ _b"> </span>banks<span class="_ _b"> </span>[6].</div><div class="t m0 x16 h9 y53 ff2 fs6 fc0 sc0 ls0 ws0">Despite<span class="_ _7"> </span>the<span class="_ _7"> </span>limited<span class="_ _7"> </span>mathematical<span class="_ _12"> </span>understanding<span class="_ _7"> </span>and<span class="_ _7"> </span>some</div><div class="t m0 x11 h9 y54 ff2 fs6 fc0 sc0 ls0 ws0">obvious<span class="_ _6"> </span>shortcomings,<span class="_ _7"> </span>the<span class="_ _6"> </span>EMD<span class="_ _7"> </span>method,<span class="_ _7"> </span>has<span class="_ _6"> </span>had<span class="_ _7"> </span>signi&#64257;cant</div><div class="t m0 x11 h9 y55 ff2 fs6 fc0 sc0 ls0 ws0">impact<span class="_"> </span>and<span class="_ _8"> </span>is<span class="_"> </span>widely<span class="_ _8"> </span>used<span class="_"> </span>in<span class="_"> </span>a<span class="_ _8"> </span>broad<span class="_"> </span>variety<span class="_"> </span>of<span class="_"> </span>time-frequency</div><div class="t m0 x11 h9 y56 ff2 fs6 fc0 sc0 ls0 ws0">analysis<span class="_ _7"> </span>applications.<span class="_ _12"> </span>These<span class="_ _7"> </span>inv<span class="_ _1"></span>olv<span class="_ _1"></span>e<span class="_ _7"> </span>signal<span class="_ _12"> </span>decomposition<span class="_ _7"> </span>in</div><div class="t m0 x11 h9 y57 ff2 fs6 fc0 sc0 ls0 ws0">audio<span class="_ _6"> </span>engineering<span class="_ _6"> </span>[7],<span class="_ _7"> </span>climate<span class="_ _6"> </span>analysis<span class="_ _7"> </span>[8],<span class="_ _6"> </span>and<span class="_ _6"> </span>various<span class="_ _6"> </span>&#64258;ux,</div><div class="t m0 x11 h9 y58 ff2 fs6 fc0 sc0 ls0 ws0">respiratory<span class="_ _e"></span>,<span class="_ _b"> </span>and<span class="_ _c"> </span>neuromuscular<span class="_ _b"> </span>signals<span class="_ _b"> </span>found<span class="_ _c"> </span>in<span class="_ _b"> </span>medicine<span class="_ _b"> </span>and</div><div class="t m0 x11 h9 y59 ff2 fs6 fc0 sc0 ls0 ws0">biology<span class="_ _b"> </span>[9],<span class="_ _b"> </span>[10],<span class="_ _6"> </span>[11],<span class="_ _b"> </span>[12],<span class="_ _b"> </span>to<span class="_ _6"> </span>name<span class="_ _b"> </span>just<span class="_ _b"> </span>a<span class="_ _6"> </span>fe<span class="_ _1"></span>w<span class="_ _b"> </span>examples.</div><div class="t m0 x11 hf y5a ff4 fs6 fc0 sc0 ls0 ws0">A.<span class="_ _a"> </span>What<span class="_ _b"> </span>is<span class="_ _6"> </span>a<span class="_ _b"> </span>mode?</div><div class="t m0 x16 h9 y5b ff2 fs6 fc0 sc0 ls0 ws0">In<span class="_ _9"> </span>the<span class="_ _9"> </span>original<span class="_ _16"> </span>EMD<span class="_ _9"> </span>description,<span class="_ _9"> </span>a<span class="_ _16"> </span>mode<span class="_ _9"> </span>is<span class="_ _16"> </span>de&#64257;ned<span class="_ _9"> </span>as</div><div class="t m0 x11 h9 y5c ff2 fs6 fc0 sc0 ls0 ws0">a<span class="_ _12"> </span>signal<span class="_ _12"> </span>whose<span class="_ _d"> </span>number<span class="_ _12"> </span>of<span class="_ _12"> </span>local<span class="_ _d"> </span>extrema<span class="_ _12"> </span>and<span class="_ _12"> </span>zero-crossings</div><div class="t m0 x11 h9 y5d ff2 fs6 fc0 sc0 ls0 ws0">differ<span class="_ _12"> </span>at<span class="_ _a"> </span>most<span class="_ _a"> </span>by<span class="_ _a"> </span>one<span class="_ _d"> </span>[1].<span class="_ _a"> </span>In<span class="_ _a"> </span>most<span class="_ _a"> </span>later<span class="_ _a"> </span>related<span class="_ _d"> </span>works,<span class="_ _a"> </span>the</div><div class="t m0 x11 h9 y5e ff2 fs6 fc0 sc0 ls0 ws0">de&#64257;nition<span class="_ _a"> </span>is<span class="_ _14"> </span>slightly<span class="_ _a"> </span>changed<span class="_ _14"> </span>into<span class="_ _a"> </span>so-called<span class="_ _14"> </span><span class="ff4">Intrinsic<span class="_ _14"> </span>Mode</span></div><div class="t m0 x11 h9 y5f ff4 fs6 fc0 sc0 ls0 ws0">Functions<span class="_ _b"> </span><span class="ff2">(IMF),<span class="_ _b"> </span>based<span class="_ _6"> </span>on<span class="_ _b"> </span>modulation<span class="_ _b"> </span>criteria:</span></div><div class="t m0 x16 h10 y60 ff4 fs6 fc0 sc0 ls0 ws0">De&#64257;nition<span class="_ _b"> </span>1:<span class="_ _b"> </span><span class="ff6">(Intrinsic<span class="_ _6"> </span>Mode<span class="_ _6"> </span>Function)</span></div><div class="t m0 x11 h9 y61 ff2 fs6 fc0 sc0 ls0 ws0">Intrinsic<span class="_ _8"> </span>Mode<span class="_ _8"> </span>Functions<span class="_ _c"> </span>are<span class="_ _8"> </span>amplitude-modulated-frequency-</div><div class="t m0 x11 h9 y62 ff2 fs6 fc0 sc0 ls0 ws0">modulated<span class="_ _b"> </span>(AM-FM)<span class="_ _b"> </span>signals,<span class="_ _6"> </span>written<span class="_ _b"> </span>as:</div><div class="t m0 x8 hd y63 ff9 fs6 fc0 sc0 ls0 ws0">u</div><div class="t m0 x17 h11 y64 ffb fs1 fc0 sc0 ls0 ws0">k</div><div class="t m0 x18 hd y63 ff8 fs6 fc0 sc0 ls0 ws0">(<span class="ff9">t</span>)<span class="_"> </span>=<span class="_"> </span><span class="ff9">A</span></div><div class="t m0 x19 h11 y64 ffb fs1 fc0 sc0 ls0 ws0">k</div><div class="t m0 x1a hd y63 ff8 fs6 fc0 sc0 ls0 ws0">(<span class="ff9">t</span>)<span class="_ _17"> </span>cos(<span class="ff9">&#966;</span></div><div class="t m0 x1b h11 y64 ffb fs1 fc0 sc0 ls0 ws0">k</div><div class="t m0 x1c hd y63 ff8 fs6 fc0 sc0 ls0 ws0">(<span class="ff9">t</span>))<span class="ff9">,<span class="_ _1b"> </span><span class="ff2">(1)</span></span></div><div class="t m0 x11 hd y65 ff2 fs6 fc0 sc0 ls0 ws0">where<span class="_"> </span>the<span class="_"> </span>phase<span class="_ _11"> </span><span class="ff9">&#966;</span></div><div class="t m0 x8 h11 y66 ffb fs1 fc0 sc0 ls0 ws0">k</div><div class="t m0 x1d hd y65 ff8 fs6 fc0 sc0 ls0 ws0">(<span class="ff9">t</span>)<span class="_ _0"> </span><span class="ff2">is<span class="_"> </span>a<span class="_ _0"> </span>non-decreasing<span class="_ _11"> </span>function,<span class="_"> </span><span class="ff9">&#966;</span></span></div><div class="t m0 x1e he y67 ffa fs1 fc0 sc0 ls0 ws0">0</div><div class="t m0 x1e h11 y68 ffb fs1 fc0 sc0 ls0 ws0">k</div><div class="t m0 x1f h12 y65 ff8 fs6 fc0 sc0 ls0 ws0">(<span class="ff9">t</span>)<span class="_"> </span><span class="ffc">&#8805;<span class="_ _8"> </span></span>0<span class="ff2">,</span></div><div class="t m0 x11 hd y69 ff2 fs6 fc0 sc0 ls0 ws0">the<span class="_"> </span>en<span class="_ _1"></span>velope<span class="_"> </span>is<span class="_"> </span>non-negati<span class="_ _1"></span>v<span class="_ _1"></span>e<span class="_"> </span><span class="ff9">A</span></div><div class="t m0 x20 h11 y6a ffb fs1 fc0 sc0 ls0 ws0">k</div><div class="t m0 x21 h12 y69 ff8 fs6 fc0 sc0 ls0 ws0">(<span class="ff9">t</span>)<span class="_"> </span><span class="ffc">&#8805;<span class="_ _8"> </span></span>0<span class="ff2">,<span class="_"> </span>and,<span class="_"> </span>very<span class="_"> </span>importantly<span class="_ _1"></span>,</span></div><div class="t m0 x11 hd y6b ff2 fs6 fc0 sc0 ls0 ws0">both<span class="_ _16"> </span>the<span class="_ _f"> </span>en<span class="_ _1"></span>v<span class="_ _1"></span>elope<span class="_ _16"> </span><span class="ff9">A</span></div><div class="t m0 x22 h11 y6c ffb fs1 fc0 sc0 ls0 ws0">k</div><div class="t m0 x23 hd y6b ff8 fs6 fc0 sc0 ls0 ws0">(<span class="ff9">t</span>)<span class="_ _16"> </span><span class="ff2">and<span class="_ _f"> </span>the<span class="_ _16"> </span>instantaneous<span class="_ _f"> </span>frequency</span></div><div class="t m0 x11 hd y6d ff9 fs6 fc0 sc0 ls0 ws0">&#969;</div><div class="t m0 x24 h11 y6e ffb fs1 fc0 sc0 ls0 ws0">k</div><div class="t m0 x25 hd y6d ff8 fs6 fc0 sc0 ls0 ws0">(<span class="ff9">t</span>)<span class="_ _c"> </span>:=<span class="_ _b"> </span><span class="ff9">&#966;</span></div><div class="t m0 x26 he y6f ffa fs1 fc0 sc0 ls0 ws0">0</div><div class="t m0 x26 h11 y70 ffb fs1 fc0 sc0 ls0 ws0">k</div><div class="t m0 x27 hd y6d ff8 fs6 fc0 sc0 ls0 ws0">(<span class="ff9">t</span>)<span class="_ _6"> </span><span class="ff2">vary<span class="_ _b"> </span>much<span class="_ _6"> </span>slower<span class="_ _6"> </span>than<span class="_ _6"> </span>the<span class="_ _6"> </span>phase<span class="_ _6"> </span><span class="ff9">&#966;</span></span></div><div class="t m0 x28 h11 y6e ffb fs1 fc0 sc0 ls0 ws0">k</div><div class="t m0 x29 hd y6d ff8 fs6 fc0 sc0 ls0 ws0">(<span class="ff9">t</span>)<span class="_ _6"> </span><span class="ff2">[13],</span></div><div class="t m0 x11 h9 y71 ff2 fs6 fc0 sc0 ls0 ws0">[14].</div><div class="t m0 x16 h12 y72 ff2 fs6 fc0 sc0 ls0 ws0">In<span class="_ _b"> </span>other<span class="_ _c"> </span>words,<span class="_ _b"> </span>on<span class="_ _b"> </span>a<span class="_ _b"> </span>suf&#64257;ciently<span class="_ _c"> </span>long<span class="_ _b"> </span>interv<span class="_ _1"></span>al<span class="_ _b"> </span><span class="ff8">[<span class="ff9">t<span class="_ _17"> </span><span class="ffc">&#8722;<span class="_ _11"> </span></span>&#948;,<span class="_ _17"> </span>t<span class="_ _11"> </span></span>+<span class="_ _11"> </span><span class="ff9">&#948;<span class="_ _15"></span></span>]</span>,</div><div class="t m0 x11 h12 y73 ff9 fs6 fc0 sc0 ls0 ws0">&#948;<span class="_ _9"> </span><span class="ffc">&#8776;<span class="_ _14"> </span><span class="ff8">2</span></span>&#960;<span class="_ _15"></span>/&#966;</div><div class="t m0 x26 he y74 ffa fs1 fc0 sc0 ls0 ws0">0</div><div class="t m0 x26 h11 y75 ffb fs1 fc0 sc0 ls0 ws0">k</div><div class="t m0 x27 hd y73 ff8 fs6 fc0 sc0 ls0 ws0">(<span class="ff9">t</span>)<span class="ff2">,<span class="_ _d"> </span>the<span class="_ _a"> </span>mode<span class="_ _a"> </span><span class="ff9">u</span></span></div><div class="t m0 x2a h11 y76 ffb fs1 fc0 sc0 ls0 ws0">k</div><div class="t m0 x20 hd y73 ff8 fs6 fc0 sc0 ls0 ws0">(<span class="ff9">t</span>)<span class="_ _d"> </span><span class="ff2">can<span class="_ _a"> </span>be<span class="_ _a"> </span>considered<span class="_ _a"> </span>to<span class="_ _a"> </span>be<span class="_ _d"> </span>a</span></div><div class="t m0 x11 hd y77 ff2 fs6 fc0 sc0 ls0 ws0">pure<span class="_ _c"> </span>harmonic<span class="_ _c"> </span>signal<span class="_ _c"> </span>with<span class="_ _b"> </span>amplitude<span class="_ _c"> </span><span class="ff9">A</span></div><div class="t m0 x2b h11 y78 ffb fs1 fc0 sc0 ls0 ws0">k</div><div class="t m0 x2c hd y77 ff8 fs6 fc0 sc0 ls0 ws0">(<span class="ff9">t</span>)<span class="_ _c"> </span><span class="ff2">and<span class="_ _c"> </span>instantaneous</span></div><div class="t m0 x11 hd y79 ff2 fs6 fc0 sc0 ls0 ws0">frequency<span class="_ _c"> </span><span class="ff9">&#966;</span></div><div class="t m0 x2d he y7a ffa fs1 fc0 sc0 ls0 ws0">0</div><div class="t m0 x2d h11 y7b ffb fs1 fc0 sc0 ls0 ws0">k</div><div class="t m0 x2e hd y79 ff8 fs6 fc0 sc0 ls0 ws0">(<span class="ff9">t</span>)<span class="_ _c"> </span><span class="ff2">[13].<span class="_ _b"> </span>Note<span class="_ _c"> </span>that<span class="_ _b"> </span>the<span class="_ _c"> </span>newer<span class="_ _c"> </span>de&#64257;nition<span class="_ _b"> </span>of<span class="_ _c"> </span>signal</span></div><div class="t m0 x11 h9 y7c ff2 fs6 fc0 sc0 ls0 ws0">components<span class="_ _b"> </span>is<span class="_ _6"> </span>slightly<span class="_ _6"> </span>more<span class="_ _6"> </span>restricti<span class="_ _1"></span>ve<span class="_ _b"> </span>than<span class="_ _6"> </span>the<span class="_ _b"> </span>original<span class="_ _6"> </span>one:</div><div class="t m0 x11 h9 y41 ff2 fs6 fc0 sc0 ls0 ws0">while<span class="_ _b"> </span>all<span class="_ _b"> </span>signals<span class="_ _b"> </span>adhering<span class="_ _6"> </span>to<span class="_ _b"> </span>the<span class="_ _b"> </span>above<span class="_ _c"> </span>IMF<span class="_ _b"> </span>de&#64257;nition<span class="_ _6"> </span>satisfy</div></div><div class="pi" data-data='{"ctm":[1.568627,0.000000,0.000000,1.568627,0.000000,0.000000]}'></div></div> </body> </html>
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