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变分模态分解算法,可分解非平稳多分量信号至多个单分量。
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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/6254b2b147503a0a93bc3845/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/6254b2b147503a0a93bc3845/bg1.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">IEEE<span class="_ _0"> </span>TRANSACTIONS<span class="_ _0"> </span>ON<span class="_ _0"> </span>SIGNAL<span class="_ _0"> </span>PROCESSING,<span class="_ _0"> </span>VOL.<span class="_ _0"> </span>62,<span class="_ _0"> </span>NO.<span class="_ _0"> </span>3,<span class="_ _0"> </span>FEBRUAR<span class="_ _1"></span>Y<span class="_ _0"> </span>1,<span class="_ _0"> </span>2014<span class="_ _2"> </span><span class="ls1">531</span></div><div class="t m0 x2 h3 y2 ff1 fs1 fc0 sc0 ls2 ws0">V<span class="_ _3"></span>ariational<span class="_ _4"> </span>Mode<span class="_ _4"> </span>Decomposition</div><div class="t m0 x3 h4 y3 ff1 fs2 fc0 sc0 ls3 ws0">Konstantin<span class="_ _5"> </span>Dragomiretskiy<span class="_ _6"> </span>and<span class="_ _7"> </span>Dominique<span class="_ _5"> </span>Zosso<span class="ff2 ls4">,<span class="_ _5"> </span>Member<span class="_ _8"></span>,<span class="_ _5"> </span>IEEE</span></div><div class="t m0 x4 h5 y4 ff3 fs3 fc0 sc0 ls5 ws0">Abstract&#8212;<span class="ff4 ls6">During<span class="_ _9"> </span>the<span class="_ _9"> </span>late<span class="_ _6"> </span>1990s,<span class="_ _9"> </span>Huang<span class="_ _9"> </span>introduced<span class="_ _9"> </span>the<span class="_ _6"> </span>algo-</span></div><div class="t m0 x1 h5 y5 ff4 fs3 fc0 sc0 ls7 ws0">rithm<span class="_ _7"> </span>cal<span class="_ _1"></span>led<span class="_ _7"> </span>Empirical<span class="_ _7"> </span>Mode<span class="_ _a"> </span>Decomposition,<span class="_ _a"> </span>w<span class="_"> </span>hi</div><div class="t m0 x5 h5 y6 ff4 fs3 fc0 sc0 ls8 ws0">ch<span class="_ _7"> </span>is<span class="_ _a"> </span>widely</div><div class="t m0 x1 h5 y7 ff4 fs3 fc0 sc0 ls9 ws0">used<span class="_ _b"> </span>today<span class="_ _b"> </span>to<span class="_ _b"> </span>recursively<span class="_ _b"> </span>decomp<span class="lsa">ose<span class="_ _b"> </span>a<span class="_ _b"> </span>signal<span class="_ _b"> </span>into<span class="_ _c"> </span>different<span class="_ _b"> </span>modes</span></div><div class="t m0 x1 h5 y8 ff4 fs3 fc0 sc0 lsb ws0">of<span class="_ _a"> </span>u<span class="_"> </span>nkn<span class="_ _d"></span>own<span class="_ _7"> </span>but<span class="_ _a"> </span>separate<span class="_ _7"> </span>spectral<span class="_ _a"> </span>bands.<span class="_ _7"> </span>EMD<span class="_ _a"> </span>is<span class="_ _7"> </span>known<span class="_ _a"> </span>for</div><div class="t m0 x1 h5 y9 ff4 fs3 fc0 sc0 lsc ws0">limitations<span class="_ _0"> </span>like<span class="_ _b"> </span>sensitivity<span class="_ _0"> </span>to<span class="_ _b"> </span>noise<span class="_ _0"> </span>and<span class="_ _b"> </span>sa</div><div class="t m0 x6 h5 ya ff4 fs3 fc0 sc0 lsd ws0">mpling.<span class="_ _0"> </span>These<span class="_ _b"> </span>limitations</div><div class="t m0 x1 h5 yb ff4 fs3 fc0 sc0 lse ws0">could<span class="_ _b"> </span>only<span class="_ _b"> </span>partially<span class="_ _c"> </span>be<span class="_ _b"> </span>addresse<span class="lsf">d<span class="_ _b"> </span>by<span class="_ _c"> </span>more<span class="_ _b"> </span>mathematical<span class="_ _b"> </span>attempts</span></div><div class="t m0 x1 h5 yc ff4 fs3 fc0 sc0 lsf ws0">to<span class="_ _e"> </span>this<span class="_ _e"> </span>decomposition<span class="_ _e"> </span>problem,<span class="_ _5"> </span>lik<span class="ls10">e<span class="_ _e"> </span>synchrosqueez<span class="_ _1"></span>ing,<span class="_ _e"> </span>empirical</span></div><div class="t m0 x1 h5 yd ff4 fs3 fc0 sc0 lsf ws0">wavelets<span class="_ _0"> </span>or<span class="_ _b"> </span>recursive<span class="_ _0"> </span>variational<span class="_ _0"> </span>de</div><div class="t m0 x7 h5 ye ff4 fs3 fc0 sc0 ls11 ws0">co<span class="_ _d"></span>mp<span class="_ _d"></span>os<span class="_ _d"></span>it<span class="_ _d"></span>i<span class="_ _d"></span>on<span class="_ _d"></span>.<span class="_ _b"> </span>He<span class="_ _d"></span>re,<span class="_ _b"> </span>we<span class="_ _b"> </span>pro<span class="_ _d"></span>p<span class="_ _d"></span>os<span class="_ _d"></span>e</div><div class="t m0 x1 h5 yf ff4 fs3 fc0 sc0 ls12 ws0">an<span class="_ _c"> </span>entirely<span class="_ _c"> </span>non-recursive<span class="_ _b"> </span>variati<span class="lsf">onal<span class="_ _5"> </span>mode<span class="_ _b"> </span>decomposition<span class="_ _5"> </span>model,</span></div><div class="t m0 x1 h5 y10 ff4 fs3 fc0 sc0 ls13 ws0">where<span class="_ _6"> </span>the<span class="_ _6"> </span>modes<span class="_ _a"> </span>ar<span class="_ _1"></span>e<span class="_ _6"> </span>extracted<span class="_ _a"> </span>concurr<span class="_ _1"></span>ently.<span class="_ _6"> </span>The<span class="_ _a"> </span>model<span class="_ _6"> </span>looks</div><div class="t m0 x1 h5 y11 ff4 fs3 fc0 sc0 ls14 ws0">for<span class="_ _9"> </span>an<span class="_ _6"> </span>ensemble<span class="_ _6"> </span>of<span class="_ _6"> </span>modes<span class="_ _6"> </span>and<span class="_ _9"> </span>their<span class="_ _6"> </span>re</div><div class="t m0 x8 h5 y12 ff4 fs3 fc0 sc0 ls15 ws0">spective<span class="_ _6"> </span>center<span class="_ _6"> </span>frequen-</div><div class="t m0 x1 h5 y13 ff4 fs3 fc0 sc0 ls16 ws0">cies,<span class="_ _5"> </span>such<span class="_ _5"> </span>that<span class="_ _5"> </span>the<span class="_ _5"> </span>modes<span class="_ _5"> </span>collectively<span class="_ _e"> </span>repr<span class="_ _1"></span>oduce<span class="_ _5"> </span>th<span class="_ _d"></span>e<span class="_ _5"> </span>input<span class="_ _5"> </span>signal,</div><div class="t m0 x1 h5 y14 ff4 fs3 fc0 sc0 ls13 ws0">while<span class="_ _9"> </span>each<span class="_ _6"> </span>being<span class="_ _9"> </span>smooth<span class="_ _6"> </span>after<span class="_ _9"> </span>demodulation<span class="_ _6"> </span>into<span class="_ _9"> </span>baseband.<span class="_ _6"> </span>In</div><div class="t m0 x1 h5 y15 ff4 fs3 fc0 sc0 ls14 ws0">Fourier<span class="_ _6"> </span>domain,<span class="_ _6"> </span>this<span class="_ _a"> </span>corr<span class="_ _1"></span>espond</div><div class="t m0 x9 h5 y16 ff4 fs3 fc0 sc0 ls17 ws0">s<span class="_ _6"> </span>to<span class="_ _6"> </span>a<span class="_ _6"> </span>narrow-band<span class="_ _6"> </span>prior.<span class="_ _6"> </span>W<span class="_ _1"></span>e</div><div class="t m0 x1 h5 y17 ff4 fs3 fc0 sc0 ls6 ws0">show<span class="_ _e"> </span>important<span class="_ _e"> </span>relations<span class="_ _5"> </span>to<span class="_ _e"> </span>Wiener<span class="_ _5"> </span><span class="ff5 ls1">&#64257;<span class="_"> </span></span><span class="ls18">lt<span class="_ _d"></span>er<span class="_ _e"> </span>deno<span class="_ _d"></span>ising.<span class="_ _e"> </span>In<span class="_ _d"></span>deed,<span class="_ _e"> </span>the</span></div><div class="t m0 x1 h5 y18 ff4 fs3 fc0 sc0 ls19 ws0">proposed<span class="_ _e"> </span>method<span class="_ _9"> </span>is<span class="_ _e"> </span>a<span class="_ _e"> </span>generali<span class="ls1a">zation<span class="_ _9"> </span>of<span class="_ _e"> </span>the<span class="_ _9"> </span>classic<span class="_ _9"> </span>W<span class="_ _1"></span>iener<span class="_ _9"> </span><span class="ff5 ls1">&#64257;</span><span class="ls1b">lter</span></span></div><div class="t m0 x1 h5 y19 ff4 fs3 fc0 sc0 ls1c ws0">into<span class="_ _7"> </span>multiple,<span class="_ _7"> </span>adaptive<span class="_ _a"> </span>b<span class="_ _d"></span>a</div><div class="t m0 xa h5 y1a ff4 fs3 fc0 sc0 lse ws0">nds.<span class="_ _a"> </span>Our<span class="_ _7"> </span>model<span class="_ _a"> </span>provides<span class="_ _a"> </span>a<span class="_ _a"> </span>solution</div><div class="t m0 x1 h5 y1b ff4 fs3 fc0 sc0 lsc ws0">to<span class="_ _e"> </span>the<span class="_ _9"> </span>decomposition<span class="_ _9"> </span>problem<span class="_ _e"> </span>that<span class="_ _9"> </span><span class="ls1d">is<span class="_ _e"> </span>theoretically<span class="_ _e"> </span>well<span class="_ _9"> </span>founded</span></div><div class="t m0 x1 h5 y1c ff4 fs3 fc0 sc0 lsd ws0">and<span class="_ _5"> </span>still<span class="_ _e"> </span>easy<span class="_ _e"> </span>to<span class="_ _e"> </span>understand.<span class="_ _5"> </span>The<span class="_ _e"> </span>variational<span class="_ _e"> </span>model<span class="_ _5"> </span>is<span class="_ _e"> </span>ef<span class="ff5 ls1">&#64257;</span><span class="ls1e">ciently</span></div><div class="t m0 x1 h5 y1d ff4 fs3 fc0 sc0 ls17 ws0">optimized<span class="_ _6"> </span>using<span class="_ _6"> </span>an<span class="_ _6"> </span>alter</div><div class="t m0 xb h5 y1e ff4 fs3 fc0 sc0 ls8 ws0">nating<span class="_ _6"> </span>direction<span class="_ _6"> </span>method<span class="_ _6"> </span>of<span class="_ _6"> </span>multipliers</div><div class="t m0 x1 h5 y1f ff4 fs3 fc0 sc0 ls6 ws0">approach.<span class="_ _5"> </span>Preli<span class="_ _1"></span>minary<span class="_ _5"> </span>results<span class="_ _5"> </span>show<span class="_ _e"> </span>attractive<span class="_ _5"> </span>performance<span class="_ _5"> </span>with</div><div class="t m0 x1 h5 y20 ff4 fs3 fc0 sc0 ls6 ws0">respect<span class="_ _a"> </span>to<span class="_ _a"> </span>existing<span class="_ _7"> </span>mode<span class="_ _a"> </span>decomposition<span class="_ _7"> </span>models.<span class="_ _a"> </span>In<span class="_ _a"> </span>particular<span class="_ _f"></span>,</div><div class="t m0 x1 h5 y21 ff4 fs3 fc0 sc0 ls19 ws0">our<span class="_ _5"> </span>pr<span class="_ _f"></span>oposed<span class="_ _5"> </span>model<span class="_ _5"> </span>is<span class="_ _c"> </span>m</div><div class="t m0 x2 h5 y22 ff4 fs3 fc0 sc0 lsc ws0">uch<span class="_ _5"> </span>more<span class="_ _c"> </span>robust<span class="_ _c"> </span>to<span class="_ _5"> </span>sampling<span class="_ _5"> </span>and<span class="_ _c"> </span>noise.</div><div class="t m0 x1 h5 y23 ff4 fs3 fc0 sc0 ls12 ws0">Finally<span class="_ _f"></span>,<span class="_ _5"> </span>we<span class="_ _e"> </span>show<span class="_ _e"> </span>promising<span class="_ _5"> </span>practical<span class="_ _5"> </span>decomposition<span class="_ _5"> </span>results<span class="_ _5"> </span>on<span class="_ _e"> </span>a</div><div class="t m0 x1 h5 y24 ff4 fs3 fc0 sc0 ls1f ws0">series<span class="_ _b"> </span>of<span class="_ _c"> </span>arti<span class="ff5 ls1">&#64257;</span><span class="lsc">cial<span class="_ _c"> </span>and<span class="_ _c"> </span>real<span class="_ _b"> </span>data.</span></div><div class="t m0 x4 h5 y25 ff3 fs3 fc0 sc0 ls20 ws0">Index<span class="_ _c"> </span>T<span class="_ _f"></span>er<span class="_ _1"></span>ms&#8212;<span class="ff4 ls1a">AM-FM,<span class="_ _c"> </span>augmented<span class="_ _5"> </span>Lagran<span class="ls21">gian,<span class="_ _c"> </span>Fourier<span class="_ _c"> </span>trans-</span></span></div><div class="t m0 x1 h5 y26 ff4 fs3 fc0 sc0 ls9 ws0">form,<span class="_ _b"> </span>Hilbert<span class="_ _b"> </span>transform,<span class="_ _b"> </span>mode<span class="_ _c"> </span>de<span class="ls21">composition,<span class="_ _b"> </span>spectral<span class="_ _c"> </span>decompo-</span></div><div class="t m0 x1 h5 y27 ff4 fs3 fc0 sc0 lsd ws0">sition,<span class="_ _b"> </span>variational<span class="_ _c"> </span>problem,<span class="_ _b"> </span>Wiener<span class="_ _b"> </span><span class="ff5 ls1">&#64257;</span><span class="ls1a">lter.</span></div><div class="t m0 xc h6 y28 ff1 fs4 fc0 sc0 ls22 ws0">I.<span class="_ _10"> </span>I<span class="fs5 ls23">NTRODUCTION</span></div><div class="t m0 x1 h7 y29 ff4 fs6 fc0 sc0 ls1 ws0">E</div><div class="t m0 xd h6 y2a ff1 fs4 fc0 sc0 ls24 ws0">MPIRICAL<span class="_ _6"> </span>MODE<span class="_ _9"> </span>decompo<span class="_"> </span>sito<span class="_"> </span>n<span class="_ _6"> </span>(EMD)<span class="_ _9"> </span>proposed<span class="_ _6"> </span>by</div><div class="t m0 xd h6 y2b ff1 fs4 fc0 sc0 ls25 ws0">Huang<span class="_ _0"> </span><span class="ff2 ls26">et<span class="_ _b"> </span>al.<span class="_ _0"> </span></span><span class="ls27">[1]<span class="_ _0"> </span>is<span class="_ _0"> </span>an<span class="_ _b"> </span>algorithmic<span class="_ _b"> </span>method<span class="_ _0"> </span>to<span class="_ _b"> </span>detect<span class="_ _0"> </span>and<span class="_ _0"> </span>de-</span></div><div class="t m0 x1 h6 y2c ff1 fs4 fc0 sc0 ls28 ws0">compose<span class="_ _b"> </span>a<span class="_ _c"> </span>signal<span class="_ _b"> </span>into<span class="_ _b"> </span>principal<span class="_ _c"> </span>&#8220;modes&#8221;.<span class="_ _b"> </span>This<span class="_ _b"> </span>algorithm<span class="_ _c"> </span>recur-</div><div class="t m0 x1 h6 y2d ff1 fs4 fc0 sc0 ls29 ws0">sively<span class="_ _0"> </span>detects<span class="_ _0"> </span>local<span class="_ _0"> </span>minima/max<span class="ls2a">ima<span class="_ _0"> </span>in<span class="_ _0"> </span>a<span class="_ _0"> </span>signal,<span class="_ _0"> </span>estimates<span class="_ _0"> </span>lower/</span></div><div class="t m0 x1 h6 y2e ff1 fs4 fc0 sc0 ls2b ws0">upper<span class="_ _b"> </span>envelopes<span class="_ _c"> </span>by<span class="_ _b"> </span>interpolation<span class="_ _b"> </span>of<span class="_ _c"> </span>these<span class="_ _b"> </span>extrema,<span class="_ _b"> </span>removes<span class="_ _b"> </span>the</div><div class="t m0 x1 h6 y2f ff1 fs4 fc0 sc0 ls2c ws0">average<span class="_ _0"> </span>of<span class="_ _0"> </span>the<span class="_ _0"> </span>envel<span class="_"> </span>opes<span class="_ _b"> </span>as<span class="_ _0"> </span>&#8220;low-pass&#8221;<span class="_ _0"> </span>centerlin<span class="_"> </span>e,<span class="_ _0"> </span>thu<span class="_ _d"></span>s<span class="_ _0"> </span>isolating</div><div class="t m0 x1 h6 y30 ff1 fs4 fc0 sc0 ls2d ws0">the<span class="_ _c"> </span>high-frequency<span class="_ _c"> </span>oscillations<span class="_ _c"> </span>as<span class="_ _c"> </span>&#8220;mode&#8221;<span class="_ _c"> </span>of<span class="_ _c"> </span>a<span class="_ _b"> </span>s<span class="_"> </span>ign<span class="_"> </span>a<span class="_"> </span>l,<span class="_ _c"> </span>and<span class="_ _c"> </span>con-</div><div class="t m0 x1 h6 y31 ff1 fs4 fc0 sc0 ls2d ws0">tinues<span class="_ _9"> </span>recursively<span class="_ _6"> </span>on<span class="_ _9"> </span>the<span class="_ _9"> </span>remain<span class="_"> </span>ing<span class="_ _6"> </span>&#8220;low-pass&#8221;<span class="_ _9"> </span>centerline.<span class="_ _9"> </span>I<span class="_"> </span>n</div><div class="t m0 x1 h6 y32 ff1 fs4 fc0 sc0 ls2e ws0">some<span class="_ _a"> </span>cases<span class="_ _1"></span>,<span class="_ _a"> </span>this<span class="_ _6"> </span>sifting<span class="_ _6"> </span>algorithm<span class="_ _a"> </span>does<span class="_ _6"> </span>indeed<span class="_ _a"> </span>decompose<span class="_ _6"> </span>a</div><div class="t m0 x1 h6 y33 ff1 fs4 fc0 sc0 ls27 ws0">signal<span class="_ _5"> </span>into<span class="_ _5"> </span>principal<span class="_ _5"> </span>modes,<span class="_ _c"> </span>how<span class="_"> </span><span class="ls2f">ever<span class="_ _5"> </span>the<span class="_ _5"> </span>resulting<span class="_ _c"> </span>decomposi-</span></div><div class="t m0 xe h8 y34 ff1 fs5 fc0 sc0 ls30 ws0">Manuscript<span class="_ _9"> </span>received<span class="_ _9"> </span>April<span class="_ _9"> </span>10,<span class="_ _e"> </span>201<span class="ls31">3;<span class="_ _9"> </span>revised<span class="_ _9"> </span>August<span class="_ _9"> </span>09,<span class="_ _9"> </span>2013;<span class="_ _e"> </span>acce<span class="_"> </span>pted</span></div><div class="t m0 x1 h8 y35 ff1 fs5 fc0 sc0 ls32 ws0">October<span class="_ _5"> </span>10,<span class="_ _e"> </span>2013.<span class="_ _5"> </span>Date<span class="_ _5"> </span>of<span class="_ _e"> </span>publication<span class="_ _5"> </span>November<span class="_ _e"> </span>05,<span class="_ _5"> </span>2013;<span class="_ _e"> </span>date<span class="_ _5"> </span>of<span class="_ _e"> </span>current</div><div class="t m0 x1 h8 y36 ff1 fs5 fc0 sc0 ls33 ws0">version<span class="_ _b"> </span>January<span class="_ _b"> </span>13,<span class="_ _b"> </span>2014.<span class="_ _b"> </span>The<span class="_ _b"> </span>associate<span class="_ _0"> </span>e<span class="_"> </span>ditor<span class="_ _b"> </span>coordinating<span class="_ _b"> </span>the<span class="_ _b"> </span>review<span class="_ _b"> </span>of<span class="_ _b"> </span>this</div><div class="t m0 x1 h8 y37 ff1 fs5 fc0 sc0 ls34 ws0">manuscript<span class="_ _b"> </span>and<span class="_ _0"> </span>approving<span class="_ _b"> </span>it<span class="_ _b"> </span>for<span class="_ _0"> </span>publication<span class="_ _b"> </span>was<span class="_ _b"> </span>Dr<span class="_ _1"></span>.<span class="_ _b"> </span>Akbar<span class="_ _b"> </span>Sayeed.<span class="_ _b"> </span>This<span class="_ _0"> </span>work</div><div class="t m0 x1 h8 y38 ff1 fs5 fc0 sc0 ls35 ws0">was<span class="_ _e"> </span>supported<span class="_ _5"> </span>by<span class="_ _e"> </span>the<span class="_ _e"> </span>Swiss<span class="_ _e"> </span>National<span class="_ _e"> </span>Science<span class="_ _e"> </span>Foundation<span class="_ _e"> </span>(SNF)<span class="_ _e"> </span>by<span class="_ _e"> </span>Grants</div><div class="t m0 x1 h8 y39 ff1 fs5 fc0 sc0 ls33 ws0">PBELP2_137727<span class="_ _e"> </span>and<span class="_ _9"> </span>P300P2_147778,<span class="_ _e"> </span>the<span class="_ _9"> </span>W.<span class="_ _9"> </span>M.<span class="_ _e"> </span>Keck<span class="_ _9"> </span>Foundation,<span class="_ _e"> </span>ONR</div><div class="t m0 x1 h8 y3a ff1 fs5 fc0 sc0 ls33 ws0">Grants<span class="_ _0"> </span>N000141210838<span class="_ _0"> </span>and<span class="_ _b"> </span>N00014121<span class="_ _1"></span>0040,<span class="_ _0"> </span>and<span class="_ _b"> </span>NSF<span class="_ _0"> </span>Grant<span class="_ _b"> </span>DMS-1<span class="_ _f"></span>118971.</div><div class="t m0 xe h8 y3b ff1 fs5 fc0 sc0 ls36 ws0">The<span class="_ _7"> </span>auth<span class="_"> </span>or<span class="_"> </span>s<span class="_ _11"> </span>are<span class="_ _7"> </span>with<span class="_ _11"> </span>the<span class="_ _7"> </span>Departm<span class="_ _d"></span>ent<span class="_ _7"> </span>of<span class="_ _11"> </span>Mathematics,<span class="_ _7"> </span>Univ<span class="_ _d"></span>ersity<span class="_ _7"> </span>of</div><div class="t m0 x1 h8 y3c ff1 fs5 fc0 sc0 ls37 ws0">California,<span class="_ _6"> </span>Los<span class="_ _6"> </span>Angeles<span class="_ _6"> </span>(UCLA),<span class="_ _a"> </span>Los<span class="_ _6"> </span>Angeles,<span class="_ _6"> </span>CA<span class="_ _6"> </span>90095<span class="_ _6"> </span>USA<span class="_ _a"> </span>(e-mail<span class="_ _1"></span>:</div><div class="t m0 x1 h8 y3d ff1 fs5 fc0 sc0 ls38 ws0">konstantin@math<span class="_ _1"></span>.ucla.edu;<span class="_ _0"> </span>zosso@math.ucla.edu).</div><div class="t m0 xe h8 y3e ff1 fs5 fc0 sc0 ls39 ws0">Color<span class="_ _0"> </span>versions<span class="_ _0"> </span>of<span class="_ _0"> </span>one<span class="_ _0"> </span>or<span class="_ _0"> </span>mor<span class="_ _1"></span>e<span class="_ _0"> </span>of<span class="_ _0"> </span>the<span class="_ _0"> </span><span class="ff6 ls1">&#64257;</span><span class="ls34">gures<span class="_ _0"> </span>in<span class="_ _0"> </span>this<span class="_ _0"> </span>paper<span class="_ _0"> </span>are<span class="_ _0"> </span>availab<span class="_ _1"></span>le<span class="_ _0"> </span>online</span></div><div class="t m0 x1 h8 y3f ff1 fs5 fc0 sc0 ls3a ws0">at<span class="_ _b"> </span>http://ieeexplore.ieee.org.</div><div class="t m0 xe h8 y40 ff1 fs5 fc0 sc0 ls3b ws0">Digital<span class="_ _b"> </span>Object<span class="_ _b"> </span>Identi<span class="ff6 ls1">&#64257;</span><span class="ls38">er<span class="_ _b"> </span>10.1<span class="_ _f"></span>109/TSP.2013.2288675</span></div><div class="t m0 xf h6 y4 ff1 fs4 fc0 sc0 ls3c ws0">tion<span class="_ _c"> </span>is<span class="_ _5"> </span>highly<span class="_ _c"> </span>dependent<span class="_ _5"> </span>on<span class="_ _c"> </span>methods<span class="_ _5"> </span>of<span class="_ _c"> </span>extremal<span class="_ _5"> </span>point<span class="_ _c"> </span><span class="ff6 ls1">&#64257;</span><span class="ls2a">nding,</span></div><div class="t m0 xf h6 y41 ff1 fs4 fc0 sc0 ls3d ws0">interpolation<span class="_ _5"> </span>of<span class="_ _5"> </span>extremal<span class="_ _c"> </span>p<span class="_"> </span>oin<span class="_ _d"></span>t<span class="ls3e">s<span class="_ _c"> </span>into<span class="_ _5"> </span>carrier<span class="_ _c"> </span>envelopes,<span class="_ _5"> </span>and<span class="_ _c"> </span>the</span></div><div class="t m0 xf h6 y42 ff1 fs4 fc0 sc0 ls2b ws0">stopping<span class="_ _b"> </span>criteria<span class="_ _b"> </span>impo<span class="_"> </span>sed.<span class="_ _c"> </span>The<span class="_ _b"> </span>l<span class="ls3f">ack<span class="_ _b"> </span>of<span class="_ _b"> </span>mathem<span class="_"> </span>atical<span class="_ _b"> </span>theory<span class="_ _b"> </span>and</span></div><div class="t m0 xf h6 y43 ff1 fs4 fc0 sc0 ls40 ws0">the<span class="_ _b"> </span>aformention<span class="_"> </span>ed<span class="_ _c"> </span>degrees<span class="_ _b"> </span>of<span class="_ _c"> </span>freedom<span class="_ _b"> </span>reducing<span class="_ _c"> </span>the<span class="_ _b"> </span>algorithm&#8217;<span class="_ _f"></span>s</div><div class="t m0 xf h6 y44 ff1 fs4 fc0 sc0 ls40 ws0">robustness<span class="_ _5"> </span>all<span class="_ _5"> </span>leave<span class="_ _5"> </span>room<span class="_ _5"> </span>for<span class="_ _5"> </span>theoretical<span class="_ _c"> </span>d<span class="_"> </span>evelo<span class="_ _d"></span>pment<span class="_ _5"> </span>and<span class="_ _5"> </span>im-</div><div class="t m0 xf h6 y45 ff1 fs4 fc0 sc0 ls41 ws0">provement<span class="_ _e"> </span>on<span class="_ _e"> </span>the<span class="_ _e"> </span>robustness<span class="_ _e"> </span>of<span class="_ _e"> </span>the<span class="_ _e"> </span>decomposition<span class="_ _e"> </span>[2]&#8211;<span class="_"> </span>[5<span class="_ _d"></span>].<span class="_ _5"> </span>In</div><div class="t m0 xf h6 y46 ff1 fs4 fc0 sc0 ls42 ws0">some<span class="_ _5"> </span>experim<span class="_"> </span>ents<span class="_ _e"> </span>it<span class="_ _e"> </span>has<span class="_ _5"> </span>been<span class="_ _e"> </span>shown<span class="_ _e"> </span>that<span class="_ _e"> </span>EMD<span class="_ _e"> </span>shares<span class="_ _5"> </span>im<span class="_"> </span>por-</div><div class="t m0 xf h6 y47 ff1 fs4 fc0 sc0 ls42 ws0">tant<span class="_ _b"> </span>sim<span class="_"> </span>ilarities<span class="_ _c"> </span>with<span class="_ _c"> </span>wavelets<span class="_ _c"> </span>and<span class="_ _c"> </span>(adaptive)<span class="_ _c"> </span><span class="ff6 ls1">&#64257;</span><span class="ls43">lter<span class="_ _c"> </span>banks<span class="_ _c"> </span>[6].</span></div><div class="t m0 x10 h6 y48 ff1 fs4 fc0 sc0 ls44 ws0">Despite<span class="_ _6"> </span>the<span class="_ _6"> </span>limited<span class="_ _6"> </span>mathema<span class="ls3e">tical<span class="_ _6"> </span>understanding<span class="_ _6"> </span>and<span class="_ _6"> </span>some</span></div><div class="t m0 xf h6 y49 ff1 fs4 fc0 sc0 ls26 ws0">obvious<span class="_ _b"> </span>shortcomings,<span class="_ _b"> </span>the<span class="_ _0"> </span>EMD<span class="_ _b"> </span>method<span class="_ _b"> </span>has<span class="_ _0"> </span>had<span class="_ _b"> </span>signi<span class="ff6 ls1">&#64257;</span><span class="ls45">cant<span class="_ _0"> </span>im-</span></div><div class="t m0 xf h6 y4a ff1 fs4 fc0 sc0 ls2e ws0">pact<span class="_ _9"> </span>and<span class="_ _9"> </span>is<span class="_ _9"> </span>widely<span class="_ _9"> </span>used<span class="_ _e"> </span>in<span class="_ _9"> </span>a<span class="_ _9"> </span>broad<span class="_ _9"> </span>variety<span class="_ _9"> </span>of<span class="_ _9"> </span>time-frequency</div><div class="t m0 xf h6 y4b ff1 fs4 fc0 sc0 ls2e ws0">analysis<span class="_ _6"> </span>applications.<span class="_ _6"> </span>These<span class="_ _6"> </span><span class="ls2d">involve<span class="_ _a"> </span>signal<span class="_ _9"> </span>deco<span class="_"> </span>mp<span class="_ _d"></span>osition<span class="_ _6"> </span>in</span></div><div class="t m0 xf h6 y4c ff1 fs4 fc0 sc0 ls3e ws0">audio<span class="_ _9"> </span>engineering<span class="_ _6"> </span>[7],<span class="_ _9"> </span>clima<span class="ls37">te<span class="_ _9"> </span>analysis<span class="_ _9"> </span>[8],<span class="_ _6"> </span>and<span class="_ _9"> </span>various<span class="_ _9"> </span><span class="ff6 ls1">&#64258;<span class="_ _d"></span></span><span class="ls2c">ux,</span></span></div><div class="t m0 xf h6 y4d ff1 fs4 fc0 sc0 ls37 ws0">respiratory<span class="_ _f"></span>,<span class="_ _5"> </span>and<span class="_ _e"> </span>neuromuscular<span class="_ _e"> </span><span class="ls46">signals<span class="_ _5"> </span>found<span class="_ _5"> </span>in<span class="_ _e"> </span>medicine<span class="_ _5"> </span>and</span></div><div class="t m0 xf h6 y4e ff1 fs4 fc0 sc0 ls40 ws0">biology<span class="_ _c"> </span>[9]&#8211;[12],<span class="_ _5"> </span>to<span class="_ _c"> </span>name<span class="_ _c"> </span>just<span class="_ _c"> </span>a<span class="_ _5"> </span>few<span class="_ _c"> </span>examples.</div><div class="t m0 xf h9 y4f ff2 fs4 fc0 sc0 ls47 ws0">A.<span class="_ _10"> </span>What<span class="_ _5"> </span>is<span class="_ _c"> </span>a<span class="_ _5"> </span>Mode?</div><div class="t m0 x10 h6 y50 ff1 fs4 fc0 sc0 ls26 ws0">In<span class="_ _b"> </span>the<span class="_ _0"> </span>original<span class="_ _b"> </span>EMD<span class="_ _0"> </span>description,<span class="_ _b"> </span>a<span class="_ _0"> </span>mode<span class="_ _b"> </span>is<span class="_ _0"> </span>de<span class="ff6 ls1">&#64257;</span><span class="ls2b">ned<span class="_ _b"> </span>as<span class="_ _0"> </span>a<span class="_ _b"> </span>signal</span></div><div class="t m0 xf h6 y51 ff1 fs4 fc0 sc0 ls2a ws0">whose<span class="_ _0"> </span>number<span class="_ _0"> </span>of<span class="_ _0"> </span>local<span class="_ _12"> </span>extrema<span class="_ _0"> </span>and<span class="_ _12"> </span>zero-crossin<span class="_"> </span>gs<span class="_ _0"> </span>differ<span class="_ _12"> </span>at<span class="_ _0"> </span>most</div><div class="t m0 xf h6 y52 ff1 fs4 fc0 sc0 ls48 ws0">by<span class="_ _b"> </span>on<span class="_"> </span>e<span class="_ _c"> </span>[1].<span class="_ _b"> </span>In<span class="_ _c"> </span>most<span class="_ _b"> </span>later<span class="_ _c"> </span>related<span class="_ _b"> </span>works,<span class="_ _c"> </span>the<span class="_ _b"> </span>de<span class="ff6 ls1">&#64257;<span class="_ _d"></span></span><span class="ls2d">nition<span class="_ _b"> </span>is<span class="_ _c"> </span>slightly</span></div><div class="t m0 xf h6 y53 ff1 fs4 fc0 sc0 ls24 ws0">changed<span class="_ _5"> </span>into<span class="_ _5"> </span>so-called<span class="_ _5"> </span><span class="ff2 ls43">Intrinsic<span class="_ _5"> </span>Mode<span class="_ _5"> </span>Functions<span class="_ _5"> </span></span><span class="ls49">(IMF),<span class="_ _5"> </span>based</span></div><div class="t m0 xf h6 y54 ff1 fs4 fc0 sc0 ls41 ws0">on<span class="_ _5"> </span>modulation<span class="_ _5"> </span>criteria:</div><div class="t m0 x10 h6 y55 ff2 fs4 fc0 sc0 ls4a ws0">De<span class="ff7 ls1">&#64257;</span><span class="ls2c">nition<span class="_ _a"> </span>1:<span class="_ _6"> </span>(<span class="ff4 ls37">Intrinsic<span class="_ _a"> </span>Mode<span class="_ _6"> </span>Function</span><span class="ls22">):<span class="_ _4"> </span><span class="ff1 ls4b">Intrinsic<span class="_ _6"> </span>Mo<span class="_ _d"></span>de</span></span></span></div><div class="t m0 xf h6 y56 ff1 fs4 fc0 sc0 ls41 ws0">Functions<span class="_ _13"> </span>are<span class="_ _13"> </span>amplitude-modulated-frequen<span class="_"> </span>cy-m<span class="_"> </span>od<span class="_ _d"></span>ulated</div><div class="t m0 xf h6 y57 ff1 fs4 fc0 sc0 ls4c ws0">(AM-FM)<span class="_ _5"> </span>signals,<span class="_ _5"> </span>written<span class="_ _5"> </span>as:</div><div class="t m0 x11 h6 y58 ff1 fs4 fc0 sc0 ls4d ws0">(1)</div><div class="t m0 xf h6 y59 ff1 fs4 fc0 sc0 ls2c ws0">where<span class="_ _e"> </span>the<span class="_ _9"> </span>phase</div><div class="t m0 x12 h6 y5a ff1 fs4 fc0 sc0 ls29 ws0">is<span class="_ _e"> </span>a<span class="_ _9"> </span>non-decreasing<span class="_ _e"> </span>function,</div><div class="t m0 x13 h6 y5b ff1 fs4 fc0 sc0 ls29 ws0">,<span class="_ _e"> </span>the<span class="_ _e"> </span>envelope<span class="_ _e"> </span>is<span class="_ _e"> </span>n<span class="_"> </span>on-neg<span class="_ _d"></span>ative<span class="_ _14"> </span><span class="ls27">,<span class="_ _e"> </span>and,<span class="_ _e"> </span>very<span class="_ _9"> </span>impor-</span></div><div class="t m0 xf h6 y5c ff1 fs4 fc0 sc0 ls26 ws0">tantly<span class="_ _f"></span>,<span class="_ _b"> </span>both<span class="_ _0"> </span>the<span class="_ _b"> </span>envelope</div><div class="t m0 x14 h6 y5d ff1 fs4 fc0 sc0 ls46 ws0">and<span class="_ _0"> </span>the<span class="_ _b"> </span>instant<span class="_ _1"></span>aneous<span class="_ _0"> </span>frequency</div><div class="t m0 x15 h6 y5e ff1 fs4 fc0 sc0 ls42 ws0">vary<span class="_ _e"> </span>much<span class="_ _e"> </span>slower<span class="_ _e"> </span>than<span class="_ _9"> </span>the<span class="_ _e"> </span>phase<span class="_ _15"> </span><span class="ls4e">[13],</span></div><div class="t m0 xf h6 y5f ff1 fs4 fc0 sc0 ls28 ws0">[14].</div><div class="t m0 x10 h6 y60 ff1 fs4 fc0 sc0 ls4f ws0">In<span class="_ _0"> </span>other<span class="_ _12"> </span>words,<span class="_ _0"> </span>on<span class="_ _0"> </span>a<span class="_ _0"> </span>suf<span class="_ _1"></span><span class="ff6 ls1">&#64257;<span class="ff1 ls50">ciently<span class="_ _b"> </span>long<span class="_ _0"> </span>interval</span></span></div><div class="t m0 x16 h6 y61 ff1 fs4 fc0 sc0 ls1 ws0">,</div><div class="t m0 x17 h6 y62 ff1 fs4 fc0 sc0 ls26 ws0">,<span class="_ _5"> </span>the<span class="_ _5"> </span>mode<span class="_ _16"> </span><span class="ls2f">can<span class="_ _5"> </span>be<span class="_ _5"> </span>considered<span class="_ _c"> </span>to<span class="_ _5"> </span>be<span class="_ _5"> </span>a<span class="_ _5"> </span>pure<span class="_ _5"> </span>har-</span></div><div class="t m0 xf h6 y63 ff1 fs4 fc0 sc0 ls51 ws0">monic<span class="_ _0"> </span>sign<span class="_"> </span>a<span class="_"> </span>l<span class="_ _0"> </span>w<span class="_ _d"></span>ith<span class="_ _0"> </span>amplitud<span class="_ _d"></span>e</div><div class="t m0 x18 h6 y64 ff1 fs4 fc0 sc0 ls52 ws0">and<span class="_ _0"> </span>instantaneous<span class="_ _0"> </span>frequency</div><div class="t m0 x19 h6 y65 ff1 fs4 fc0 sc0 ls2a ws0">[13].<span class="_ _b"> </span>Note<span class="_ _b"> </span>that<span class="_ _b"> </span>the<span class="_ _b"> </span>newer<span class="_ _b"> </span>d<span class="_"> </span>e<span class="ff6 ls1">&#64257;</span><span class="ls53">nition<span class="_ _b"> </span>of<span class="_ _b"> </span>signal<span class="_ _b"> </span>compone<span class="_ _1"></span>nts</span></div><div class="t m0 xf h6 y66 ff1 fs4 fc0 sc0 ls43 ws0">is<span class="_ _5"> </span>slightly<span class="_ _5"> </span>more<span class="_ _5"> </span>restrict<span class="_"> </span>ive<span class="_ _5"> </span>th<span class="_"> </span>a<span class="_"> </span>n<span class="_ _5"> </span>the<span class="_ _5"> </span>orig<span class="_ _d"></span>inal<span class="_ _5"> </span>one:<span class="_ _5"> </span>while<span class="_ _5"> </span>all<span class="_ _5"> </span>sig-</div><div class="t m0 xf h6 y67 ff1 fs4 fc0 sc0 ls54 ws0">nals<span class="_ _e"> </span>adhering<span class="_ _e"> </span>to<span class="_ _9"> </span>the<span class="_ _e"> </span>above<span class="_ _9"> </span>IMF<span class="_ _e"> </span>de<span class="ff6 ls1">&#64257;</span><span class="ls55">nition<span class="_ _9"> </span>satisfy<span class="_ _9"> </span>the<span class="_ _e"> </span>ori<span class="_ _d"></span>ginal</span></div><div class="t m0 xf h6 y68 ff1 fs4 fc0 sc0 ls52 ws0">EMD<span class="_ _b"> </span>mode<span class="_ _0"> </span>prop<span class="_"> </span>erties,<span class="_ _b"> </span>the<span class="_ _b"> </span>convers<span class="ls2c">e<span class="_ _0"> </span>is<span class="_ _b"> </span>not<span class="_ _b"> </span>necessarily<span class="_ _0"> </span>true<span class="_ _b"> </span>[13].</span></div><div class="t m0 xf h6 y69 ff1 fs4 fc0 sc0 ls52 ws0">The<span class="_ _b"> </span>immediate<span class="_ _c"> </span>consequence<span class="_ _b"> </span>of<span class="_ _c"> </span>the<span class="_ _b"> </span>newer<span class="_ _c"> </span>IMF<span class="_ _b"> </span>de<span class="ff6 ls1">&#64257;</span><span class="ls2c">nition,<span class="_ _c"> </span>how-</span></div><div class="t m0 xf h6 y6a ff1 fs4 fc0 sc0 ls3e ws0">ever<span class="_ _f"></span>,<span class="_ _5"> </span>is<span class="_ _5"> </span>limited<span class="_ _c"> </span>bandwidth,<span class="_ _5"> </span>as<span class="_ _c"> </span>we<span class="_ _5"> </span>show<span class="_ _5"> </span>in<span class="_ _c"> </span>the<span class="_ _5"> </span>next<span class="_ _c"> </span>paragraphs,</div><div class="t m0 xf h6 y6b ff1 fs4 fc0 sc0 ls2e ws0">and<span class="_ _e"> </span>which<span class="_ _e"> </span>is<span class="_ _5"> </span>the<span class="_ _e"> </span>central<span class="_ _e"> </span>assumption<span class="_ _e"> </span>that<span class="_ _e"> </span>allows<span class="_ _5"> </span>m<span class="_"> </span>ode<span class="_ _e"> </span>separa-</div><div class="t m0 xf h6 y6c ff1 fs4 fc0 sc0 ls26 ws0">tion<span class="_ _6"> </span>in<span class="_ _6"> </span>the<span class="_ _6"> </span>proposed<span class="_ _6"> </span>v<span class="_"> </span>ariation<span class="_"> </span>al<span class="_ _a"> </span>mode<span class="_ _9"> </span>decom<span class="_"> </span>po<span class="_ _d"></span>sition<span class="_ _6"> </span>model.</div><div class="t m0 xf h6 y6d ff1 fs4 fc0 sc0 ls2c ws0">Our<span class="_ _e"> </span>restricted<span class="_ _9"> </span>class<span class="_ _e"> </span>of<span class="_ _e"> </span>admissible<span class="_ _9"> </span>mode<span class="_ _e"> </span>functions<span class="_ _9"> </span>is<span class="_ _e"> </span>also<span class="_ _e"> </span>mo-</div><div class="t m0 xf h6 y6e ff1 fs4 fc0 sc0 ls29 ws0">tivated<span class="_ _9"> </span>by<span class="_ _9"> </span>analysis<span class="_ _9"> </span>in<span class="_ _9"> </span>[3],<span class="_ _9"> </span>for<span class="_ _9"> </span>they<span class="_ _9"> </span>are<span class="_ _9"> </span>well-behaved<span class="_ _9"> </span>with<span class="_ _9"> </span>re-</div><div class="t m0 xf h6 y6f ff1 fs4 fc0 sc0 ls2e ws0">spect<span class="_ _6"> </span>to<span class="_ _9"> </span>tim<span class="_"> </span>e-frequency<span class="_ _6"> </span>analysis<span class="_ _6"> </span>of<span class="_ _6"> </span>extracted<span class="_ _9"> </span>m<span class="_ _d"></span>odes,<span class="_ _9"> </span>as<span class="_ _6"> </span>rou-</div><div class="t m0 xf h6 y70 ff1 fs4 fc0 sc0 ls56 ws0">tinely<span class="_ _e"> </span>pe<span class="_"> </span>rf<span class="_"> </span>or<span class="_"> </span>med<span class="_ _9"> </span>in<span class="_ _e"> </span>the<span class="_ _e"> </span>Hi<span class="_"> </span>lber<span class="_"> </span>t-<span class="_ _d"></span><span class="ls48">Huang-Transform<span class="_ _5"> </span>[1].<span class="_ _e"> </span>In<span class="_ _9"> </span>con-</span></div><div class="t m0 xf h6 y71 ff1 fs4 fc0 sc0 ls57 ws0">trast,<span class="_ _e"> </span>the<span class="_ _e"> </span>original<span class="_ _9"> </span>IMF<span class="_ _5"> </span>d<span class="_ _d"></span>e<span class="ff6 ls1">&#64257;</span><span class="ls41">nition<span class="_ _e"> </span>also<span class="_ _e"> </span>admits<span class="_ _9"> </span>non-</span></div><div class="t m0 x1a h6 y72 ff1 fs4 fc0 sc0 ls41 ws0">and<span class="_ _e"> </span>dis-</div><div class="t m0 xf h6 y73 ff1 fs4 fc0 sc0 ls2b ws0">continuous<span class="_ _b"> </span>s<span class="_ _1"></span>ignals<span class="_ _0"> </span>like<span class="_ _b"> </span>the<span class="_ _12"> </span>sawtooth,<span class="_ _b"> </span>for<span class="_ _12"> </span>wh<span class="_"> </span>ich<span class="_ _b"> </span>the<span class="_ _12"> </span>estimated<span class="_ _b"> </span>in-</div><div class="t m0 xf h6 y74 ff1 fs4 fc0 sc0 ls41 ws0">stantaneous<span class="_ _c"> </span>frequency<span class="_ _c"> </span>is<span class="_ _c"> </span>not<span class="_ _b"> </span>ph<span class="_"> </span>ysically<span class="_ _c"> </span>m<span class="_"> </span>eanin<span class="_"> </span>gfu<span class="_"> </span>l,<span class="_ _c"> </span>see<span class="_ _c"> </span>Fig.<span class="_ _c"> </span>1.</div><div class="t m0 x1b h8 y75 ff1 fs5 fc0 sc0 ls58 ws0">1053-587X<span class="_ _b"> </span>&#169;<span class="_ _b"> </span>2013<span class="_ _b"> </span>IEEE.<span class="_ _b"> </span>Personal<span class="_ _b"> </span>use<span class="_ _b"> </span>is<span class="_ _0"> </span>p<span class="_"> </span>erm<span class="_ _d"></span>itted,<span class="_ _b"> </span>but<span class="_ _b"> </span>republication/redistribution<span class="_ _b"> </span>requires<span class="_ _b"> </span>IEEE<span class="_ _b"> </span>permission.</div><div class="t m0 x1c h8 y76 ff1 fs5 fc0 sc0 ls59 ws0">See<span class="_ _b"> </span>http://www.ieee.org/publications_standards/publi<span class="ls5a">cations/rights/index.ht<span class="ls31">ml<span class="_ _b"> </span>for<span class="_ _b"> </span>more<span class="_ _b"> </span>information.</span></span></div><div class="t m0 x4 ha y77 ff8 fs7 fc0 sc0 ls1 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