Improved Ensemble Empirical Mode Decomposition.zip_EEMD_ensemble

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src="https://static.pudn.com/prod/directory_preview_static/62604a4b090cbf2c4ec3c3b3/bg1.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0"> </div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls0 ws0"> </div><div class="t m0 x3 h3 y3 ff2 fs1 fc0 sc1 ls0 ws0">I<span class="_ _0"></span>m<span class="_ _0"></span>p<span class="_ _0"></span>r<span class="_ _0"></span>o<span class="_ _0"></span>v<span class="_ _0"></span>e<span class="_ _0"></span>d<span class="_ _0"></span> <span class="_ _0"></span>E<span class="_ _0"></span>n<span class="_ _0"></span>s<span class="_ _0"></span>e<span class="_ _0"></span>m<span class="_ _0"></span>b<span class="_ _0"></span>l<span class="_ _0"></span>e<span class="_ _0"></span> <span class="_ _0"></span>E<span class="_ _0"></span>m<span class="_ _0"></span>p<span class="_ _0"></span>i<span class="_ _0"></span>r<span class="_ _0"></span>i<span class="_ _0"></span>c<span class="_ _0"></span>a<span class="_ _0"></span>l<span class="_ _0"></span> <span class="_ _0"></span>M<span class="_ _0"></span>o<span class="_ _0"></span>d<span class="_ _0"></span>e<span class="_ _0"></span> <span class="_ _0"></span>D<span class="_ _1"></span>e<span class="_ _0"></span>c<span class="_ _1"></span>o<span class="_ _0"></span>m<span class="_ _2"></span>p<span class="_ _1"></span>o<span class="_ _0"></span>s<span class="_ _0"></span>i<span class="_ _0"></span>t<span class="_ _0"></span>i<span class="_ _0"></span>o<span class="_ _0"></span>n<span class="_ _0"></span> <span class="_ _0"></span>a<span class="_ _0"></span>n<span class="_ _0"></span>d<span class="_ _0"></span> <span class="_ _0"></span>i<span class="_ _0"></span>t<span class="_ _0"></span>s<span class="_ _0"></span> </div><div class="t m0 x4 h3 y4 ff2 fs1 fc0 sc1 ls0 ws0">A<span class="_ _0"></span>p<span class="_ _0"></span>p<span class="_ _1"></span>l<span class="_ _0"></span>i<span class="_ _0"></span>c<span class="_ _0"></span>a<span class="_ _0"></span>t<span class="_ _0"></span>i<span class="_ _0"></span>o<span class="_ _0"></span>n<span class="_ _0"></span>s<span class="_ _0"></span> <span class="_ _0"></span>t<span class="_ _0"></span>o<span class="_ _0"></span> <span class="_ _0"></span>G<span class="_ _0"></span>e<span class="_ _0"></span>a<span class="_ _0"></span>r<span class="_ _0"></span>b<span class="_ _0"></span>o<span class="_ _0"></span>x<span class="_ _0"></span> <span class="_ _0"></span>F<span class="_ _0"></span>a<span class="_ _0"></span>u<span class="_ _0"></span>l<span class="_ _0"></span>t<span class="_ _0"></span> <span class="_ _0"></span>S<span class="_ _0"></span>i<span class="_ _0"></span>g<span class="_ _0"></span>n<span class="_ _0"></span>a<span class="_ _0"></span>l<span class="_ _0"></span> <span class="_ _1"></span>P<span class="_ _0"></span>r<span class="_ _0"></span>o<span class="_ _0"></span>c<span class="_ _0"></span>e<span class="_ _0"></span>s<span class="_ _0"></span>s<span class="_ _0"></span>i<span class="_ _0"></span>n<span class="_ _0"></span>g<span class="_ _0"></span> </div><div class="c x5 y5 w2 h4"><div class="t m0 x6 h5 y6 ff3 fs2 fc0 sc0 ls0 ws0">Jinshan Lin </div><div class="t m0 x7 h5 y7 ff3 fs2 fc0 sc0 ls0 ws0"> </div><div class="t m0 x8 h5 y8 ff3 fs2 fc0 sc0 ls0 ws0"> School of Mechatronics and Vehic<span class="ls1">le</span> <span class="_ _3"></span>Engineering<span class="ls2">, </span>Weifang University </div><div class="t m0 x9 h5 y9 ff3 fs2 fc0 sc0 ls0 ws0">Weifang<span class="ls2">, </span>261061<span class="ls2">, C</span>hina </div><div class="t m0 x7 h6 ya ff4 fs0 fc0 sc0 ls0 ws0"> </div></div><div class="t m0 x1 h7 yb ff3 fs0 fc0 sc0 ls0 ws0"> </div><div class="t m0 xa h7 yc ff3 fs0 fc0 sc0 ls0 ws0">Abstract </div><div class="t m0 x1 h8 yd ff1 fs2 fc0 sc0 ls0 ws0">Ensemble <span class="_"> </span>empirical <span class="_"> </span>mode <span class="_ _4"> </span>decomposition <span class="_"> </span>(EEMD) <span class="_"> </span>is <span class="_ _4"> </span>a <span class="_"> </span>noise-</div><div class="t m0 x1 h8 ye ff1 fs2 fc0 sc0 ls0 ws0">assisted <span class="_ _3"></span>method <span class="_ _5"></span>and <span class="_ _3"></span>also <span class="_ _5"></span>a <span class="_ _3"></span>significant <span class="_ _5"></span>improvement <span class="_ _3"></span>on <span class="_ _5"></span>empirical<span class="_ _3"></span> </div><div class="t m0 x1 h8 yf ff1 fs2 fc0 sc0 ls0 ws0">mode <span class="_ _3"></span>decomposition <span class="_ _3"></span>(EMD). <span class="_ _3"></span>However,<span class="_ _3"></span> <span class="_ _3"></span>the <span class="_ _3"></span>EE<span class="_ _3"></span>MD <span class="_ _3"></span>method <span class="_ _3"></span>la<span class="_ _3"></span>cks </div><div class="t m0 x1 h8 y10 ff1 fs2 fc0 sc0 ls0 ws0">a guide <span class="_ _3"></span>to <span class="_ _3"></span>choosing the <span class="_ _3"></span>appropriate <span class="_ _3"></span>amplitude <span class="_ _3"></span>of added <span class="_ _3"></span>noise and </div><div class="t m0 x1 h8 y11 ff1 fs2 fc0 sc0 ls0 ws0">its <span class="_ _6"> </span>computation <span class="_ _6"> </span>efficiency<span class="_ _3"></span> <span class="_ _6"> </span>is <span class="_ _6"> </span>fairly <span class="_ _6"> </span>low. <span class="_ _6"> </span>To <span class="_ _6"> </span>alleviate <span class="_ _6"> </span>the </div><div class="t m0 x1 h8 y12 ff1 fs2 fc0 sc0 ls0 ws0">problems <span class="_ _4"> </span>of <span class="_ _7"> </span>the <span class="_ _7"> </span>EE<span class="_ _3"></span>MD <span class="_ _7"> </span>method, <span class="_ _4"> </span>the <span class="_ _7"> </span>improved <span class="_ _4"> </span>complementary </div><div class="t m0 x1 h8 y13 ff1 fs2 fc0 sc0 ls0 ws0">EEMD <span class="_ _8"> </span>method <span class="_ _8"> </span>(ICEEMD) <span class="_ _8"> </span><span class="ls3">wa</span>s <span class="_ _8"> </span>proposed. <span class="_ _8"> </span>Furthermore, <span class="_ _8"> </span>the </div><div class="t m0 x1 h8 y14 ff1 fs2 fc0 sc0 ls0 ws0">ICEEMD <span class="_ _7"> </span>method <span class="_ _7"> </span>was <span class="_ _7"> </span>used <span class="_ _7"> </span>to <span class="_ _7"> </span>ana<span class="_ _3"></span>lyze <span class="_ _7"> </span>realisti<span class="_ _3"></span>c <span class="_ _7"> </span>gearbox <span class="_ _7"> </span>faulty<span class="_ _3"></span> </div><div class="t m0 x1 h8 y15 ff1 fs2 fc0 sc0 ls0 ws0">signals. <span class="_ _5"></span>The <span class="_ _9"></span>results <span class="_ _5"></span>i<span class="_ _3"></span>ndicate <span class="_ _9"></span>that <span class="_ _5"></span>the <span class="_ _9"></span>ICEEMD <span class="_ _9"></span>method <span class="_ _5"></span>has <span class="_ _9"></span>some </div><div class="t m0 x1 h8 y16 ff1 fs2 fc0 sc0 ls0 ws0">advantages <span class="_ _a"> </span>over <span class="_ _a"> </span>the <span class="_ _a"> </span>EEMD <span class="_ _a"> </span>method <span class="_ _b"> </span>in <span class="_ _b"> </span>al<span class="_ _3"></span>leviating <span class="_ _a"> </span>the <span class="_ _b"> </span>mode </div><div class="t m0 x1 h8 y17 ff1 fs2 fc0 sc0 ls0 ws0">mixing <span class="_ _c"></span>and <span class="_ _c"></span>splitting <span class="_ _7"></span>as <span class="_ _9"></span>well <span class="_ _c"></span>as<span class="_ _3"></span> <span class="_ _c"></span>reducing <span class="_ _c"></span>the<span class="_ _3"></span> <span class="_ _c"></span>time <span class="_ _c"></span>cost <span class="_ _7"></span>and <span class="_ _9"></span>also </div><div class="t m0 x1 h8 y18 ff1 fs2 fc0 sc0 ls0 ws0">outperforms <span class="_ _3"></span>the <span class="_ _5"></span>CEEMD <span class="_ _5"></span>method <span class="_ _3"></span>in <span class="_ _9"></span>alleviating <span class="_ _5"></span>the <span class="_ _3"></span>mode <span class="_ _5"></span>mixing </div><div class="t m0 x1 h8 y19 ff1 fs2 fc0 sc0 ls0 ws0">and <span class="_ _5"></span>splitting. <span class="_ _5"></span>The <span class="_ _5"></span>paper <span class="_ _5"></span>also <span class="_ _5"></span>indi<span class="_ _3"></span>cates <span class="_ _5"></span>that <span class="_ _5"></span>the <span class="_ _9"></span>ICEEMD <span class="_ _5"></span>method </div><div class="t m0 x1 h8 y1a ff1 fs2 fc0 sc0 ls0 ws0">seems <span class="_"> </span>to <span class="_"> </span>be <span class="_"> </span>an <span class="_"> </span>effective <span class="_"> </span>and <span class="_"> </span>efficient <span class="_"> </span>method <span class="_"> </span>for <span class="_ _4"> </span>processing </div><div class="t m0 x1 h8 y1b ff1 fs2 fc0 sc0 ls0 ws0">gearbox fault signals. </div><div class="t m0 x1 h9 y1c ff5 fs0 fc0 sc0 ls0 ws0">Keywords:<span class="ff6 fs2"> <span class="_ _d"> </span>Complementary <span class="_ _d"> </span>Ensemble <span class="_ _d"> </span>Empirical <span class="_ _d"> </span>Mode </span></div><div class="t m0 x1 ha y1d ff6 fs2 fc0 sc0 ls0 ws0">Decomposition(CEEMD)<span class="ls2">, <span class="_ _a"> </span></span>Improved <span class="_ _a"> </span>Complementary <span class="_ _e"> </span>Ensemble </div><div class="t m0 x1 ha y1e ff6 fs2 fc0 sc0 ls0 ws0">Empirical <span class="_ _6"> </span>Mode <span class="_ _6"> </span>Decomposition(ICEEMD)<span class="ls2">, <span class="_ _6"> </span></span>Gearbox<span class="ls2">, <span class="_ _f"> </span></span>Signal </div><div class="t m0 x1 h9 y1f ff6 fs2 fc0 sc0 ls0 ws0">Processing.<span class="ff5 fs0"> </span></div><div class="t m0 x1 hb y20 ff3 fs3 fc0 sc0 ls0 ws0">1. Introduction </div><div class="t m0 x1 h2 y21 ff1 fs0 fc0 sc0 ls0 ws0">It <span class="_ _8"> </span>is <span class="_ _8"> </span>a <span class="_ _8"> </span>challenging <span class="_ _8"> </span>task <span class="_ _8"> </span>to <span class="_ _8"> </span>develop <span class="_ _8"> </span>signal <span class="_ _e"> </span>pr<span class="_ _3"></span>ocessing </div><div class="t m0 x1 h2 y22 ff1 fs0 fc0 sc0 ls0 ws0">techniques <span class="_ _3"></span>for <span class="_ _3"></span>non<span class="_ _3"></span>-stationary <span class="_ _3"></span>and <span class="_ _3"></span>no<span class="_ _3"></span>isy <span class="_ _3"></span>signals, <span class="_ _3"></span>which <span class="_ _5"></span>has </div><div class="t m0 x1 h2 y23 ff1 fs0 fc0 sc0 ls0 ws0">attracted <span class="_ _10"> </span>considerable <span class="_ _10"> </span>attentions <span class="_ _10"> </span>recently <span class="_ _10"> </span>[1]. <span class="_ _10"> </span>Many </div><div class="t m0 x1 h2 y24 ff1 fs0 fc0 sc0 ls0 ws0">methods, <span class="_ _c"></span>such <span class="_ _7"></span>as <span class="_ _c"></span>short <span class="_ _7"></span>tim<span class="_ _2"></span>e <span class="_ _7"></span>frequency <span class="_ _c"></span>transform <span class="_ _7"> </span>[2] <span class="_ _11"></span>and </div><div class="t m0 x1 h2 y25 ff1 fs0 fc0 sc0 ls0 ws0">wavelet transform <span class="_ _3"></span>[3] , <span class="_ _3"></span>have <span class="_ _3"></span>been <span class="_ _3"></span>proposed <span class="_ _3"></span>for <span class="_ _3"></span>solving <span class="_ _3"></span>the </div><div class="t m0 x1 h2 y26 ff1 fs0 fc0 sc0 ls0 ws0">problem and<span class="_ _3"></span> <span class="_ _3"></span>proved <span class="_ _3"></span>us<span class="_ _3"></span>eful <span class="_ _3"></span>in <span class="_ _3"></span>some <span class="_ _3"></span>applications.<span class="_ _3"></span> <span class="_ _3"></span>Ho<span class="_ _3"></span>wever, </div><div class="t m0 x1 h2 y27 ff1 fs0 fc0 sc0 ls0 ws0">because <span class="_ _4"> </span>these <span class="_ _4"> </span>methods <span class="_ _4"> </span>usually <span class="_ _4"> </span>need <span class="_ _4"> </span>a <span class="_ _4"> </span>priori <span class="_"> </span>knowledge </div><div class="t m0 x1 h2 y28 ff1 fs0 fc0 sc0 ls0 ws0">about <span class="_ _c"></span>the <span class="_ _c"></span>researched <span class="_ _11"></span>signals, <span class="_ _c"></span>they <span class="_ _c"></span>naturally <span class="_ _11"></span>lack <span class="_ _c"></span>the <span class="_ _c"></span>self<span class="_ _3"></span>-</div><div class="t m0 x1 h2 y29 ff1 fs0 fc0 sc0 ls0 ws0">adaption <span class="_ _e"> </span>for <span class="_ _8"> </span>the <span class="_ _e"> </span>researched <span class="_ _8"> </span>signals. <span class="_ _e"> </span>The <span class="_ _e"> </span>W<span class="_ _3"></span>igner-Ville </div><div class="t m0 x1 h2 y2a ff1 fs0 fc0 sc0 ls0 ws0">distribution <span class="_ _a"> </span>has <span class="_ _a"> </span>high <span class="_ _e"> </span>time-frequency <span class="_ _a"> </span>resolution, <span class="_ _a"> </span>but <span class="_ _a"> </span>its </div><div class="t m0 x1 h2 y2b ff1 fs0 fc0 sc0 ls0 ws0">cross <span class="_ _d"> </span>terms <span class="_ _12"> </span>is <span class="_ _12"> </span>unbear<span class="_ _3"></span>able <span class="_ _d"> </span>[4]. <span class="_ _12"> </span>Empirical <span class="_ _12"> </span>mode </div><div class="t m0 x1 h2 y2c ff1 fs0 fc0 sc0 ls0 ws0">decomposition <span class="_ _f"> </span>(EMD) <span class="_ _6"> </span>is <span class="_ _6"> </span>a <span class="_ _f"> </span>self<span class="_ _3"></span>-adaptive <span class="_ _f"> </span>method <span class="_ _6"> </span>and </div><div class="t m0 x1 h2 y2d ff1 fs0 fc0 sc0 ls0 ws0">suitable <span class="_ _e"> </span>to <span class="_ _8"> </span>analyzing <span class="_ _e"> </span>the <span class="_ _8"> </span>non-stationary <span class="_ _e"> </span>and <span class="_ _8"> </span>nonlinear </div><div class="t m0 x1 h2 y2e ff1 fs0 fc0 sc0 ls0 ws0">signals <span class="_ _3"></span>[4], <span class="_ _3"></span>which <span class="_ _3"></span>has <span class="_ _3"></span>been <span class="_ _3"></span>successfully <span class="_ _3"></span>applied <span class="_ _3"></span>to <span class="_ _3"></span>various </div><div class="t m0 x1 h2 y2f ff1 fs0 fc0 sc0 ls0 ws0">fields <span class="_ _5"></span>[4]. <span class="_ _9"></span>Nevertheless, <span class="_ _9"></span>when <span class="_ _9"></span>the <span class="_ _5"></span>EMD <span class="_ _9"></span>algor<span class="_ _3"></span>ithm <span class="_ _5"></span>is <span class="_ _5"></span>us<span class="_ _3"></span>ed </div><div class="t m0 x1 h2 y30 ff1 fs0 fc0 sc0 ls0 ws0">to <span class="_ _9"></span>deal <span class="_ _5"></span>with <span class="_ _9"></span>a <span class="_ _9"></span>signal <span class="_ _9"></span>with <span class="_ _9"></span>intermittency, <span class="_ _9"></span>the <span class="_ _9"></span>mode <span class="_ _5"></span>mixing </div><div class="t m0 x1 h2 y31 ff1 fs0 fc0 sc0 ls0 ws0">often emerges as <span class="_ _3"></span>an annoying<span class="_ _3"></span> problem <span class="_ _3"></span>[5-7]. <span class="_ _3"></span>To ov<span class="_ _3"></span>ercome </div><div class="t m0 x1 h2 y32 ff1 fs0 fc0 sc0 ls0 ws0">the <span class="_ _f"> </span>mode <span class="_ _f"> </span>mixing <span class="_ _f"> </span>problem, <span class="_ _f"> </span>ensemble <span class="_ _f"> </span>empirical <span class="_ _f"> </span>mode </div><div class="t m0 x1 h2 y33 ff1 fs0 fc0 sc0 ls0 ws0">decomposition <span class="_ _3"></span>(EEMD) <span class="_ _5"></span>is <span class="_ _3"></span>pr<span class="_ _3"></span>esented <span class="_ _5"></span>in <span class="_ _3"></span>place <span class="_ _5"></span>of <span class="_ _5"></span>EMD <span class="_ _9"></span>[8]<span class="ls4">. </span></div><div class="t m0 x1 h2 y34 ff1 fs0 fc0 sc0 ls0 ws0">The <span class="_ _7"> </span>EEMD <span class="_ _4"> </span>met<span class="_ _2"></span>ho<span class="_ _3"></span>d <span class="_ _7"> </span>adds <span class="_ _4"> </span>some <span class="_ _11"></span>wh<span class="_ _3"></span>ite <span class="_ _7"> </span>noise <span class="_ _7"> </span>with <span class="_ _7"> </span>limited </div><div class="t m0 x1 h2 y35 ff1 fs0 fc0 sc0 ls0 ws0">amplitude <span class="_ _b"> </span>to <span class="_ _e"> </span>the <span class="_ _b"> </span>research<span class="_ _3"></span>ed <span class="_ _a"> </span>signals, <span class="_ _a"> </span>sufficiently <span class="_ _a"> </span>taking </div><div class="t m0 xb h2 y36 ff1 fs0 fc0 sc0 ls0 ws0">advantage <span class="_ _7"> </span>of <span class="_ _7"> </span>the <span class="_ _7"> </span>statistical <span class="_ _7"></span>characteristics <span class="_ _11"></span>of <span class="_ _7"> </span>white <span class="_ _7"> </span>noise </div><div class="t m0 xb h2 y37 ff1 fs0 fc0 sc0 ls0 ws0">whose <span class="_ _c"></span>energy <span class="_ _9"></span>den<span class="_ _3"></span>sity <span class="_ _9"></span>is <span class="_ _c"></span>uniformly <span class="_ _9"></span>distributed <span class="_ _c"></span>throug<span class="_ _3"></span>hout </div><div class="t m0 xb h2 y38 ff1 fs0 fc0 sc0 ls0 ws0">the frequency domain, then <span class="_ _3"></span>projects <span class="_ _3"></span>the signal components </div><div class="t m0 xb h2 y39 ff1 fs0 fc0 sc0 ls0 ws0">onto <span class="_ _4"> </span>the <span class="_ _7"> </span>prop<span class="_ _3"></span>er <span class="_ _4"> </span>frequency <span class="_ _7"> </span>bands<span class="_ _3"></span> <span class="_ _4"> </span>and, <span class="_ _7"> </span>finally, <span class="_ _4"> </span>the <span class="_ _7"> </span>added </div><div class="t m0 xb h2 y3a ff1 fs0 fc0 sc0 ls0 ws0">white <span class="_ _11"></span>noise <span class="_ _11"></span>can <span class="_ _11"></span>been <span class="_ _7"> </span>counteracted <span class="_ _11"></span>by <span class="_ _7"> </span>ensemble <span class="_ _11"></span>mean <span class="_ _11"></span><span class="ls2">of </span></div><div class="t m0 xb h2 y3b ff1 fs0 fc0 sc0 ls0 ws0">enough <span class="_ _f"> </span>corresponding <span class="_ _f"> </span>components <span class="_ _f"> </span>[8]. <span class="_ _8"> </span>Therefore, <span class="_ _f"> </span>the </div><div class="t m0 xb h2 y3c ff1 fs0 fc0 sc0 ls0 ws0">EEMD <span class="_ _3"></span>method <span class="_ _3"></span>is <span class="_ _3"></span>considered <span class="_ _3"></span>as <span class="_ _5"></span>a <span class="_ _3"></span>significant <span class="_ _5"></span>improvement </div><div class="t m0 xb h2 y3d ff1 fs0 fc0 sc0 ls0 ws0">on <span class="_ _5"></span>the <span class="_ _5"></span>EMD <span class="_ _9"></span>method <span class="_ _5"></span>and <span class="_ _5"></span>recommended <span class="_ _5"></span>as <span class="_ _5"></span>a <span class="_ _9"></span>substitute <span class="_ _5"></span>for </div><div class="t m0 xb h2 y3e ff1 fs0 fc0 sc0 ls0 ws0">the <span class="_ _a"> </span>EMD <span class="_ _a"> </span>method <span class="_ _a"> </span>[8]. <span class="_ _a"> </span>Indeed<span class="_ _3"></span>, <span class="_ _a"> </span>the <span class="_ _a"> </span>EEMD <span class="_ _a"> </span>method <span class="_ _a"> </span>has </div><div class="t m0 xb h2 y3f ff1 fs0 fc0 sc0 ls0 ws0">shown <span class="_ _e"> </span>its <span class="_ _e"> </span>superiority <span class="_ _e"> </span>over <span class="_ _e"> </span>the <span class="_ _e"> </span>EMD <span class="_ _e"> </span>method <span class="_ _a"> </span>in <span class="_ _e"> </span>some </div><div class="t m0 xb h2 y40 ff1 fs0 fc0 sc0 ls0 ws0">applications [9]. </div><div class="t m0 xb h2 y41 ff1 fs0 fc0 sc0 ls0 ws0"> </div><div class="t m0 xb h2 y42 ff1 fs0 fc0 sc0 ls5 ws0">How<span class="ls0">ever, <span class="_ _b"> </span>the <span class="_ _b"> </span>EEMD <span class="_ _b"> </span>method <span class="_ _b"> </span>lacks <span class="_ _b"> </span>a <span class="_ _b"> </span>guide <span class="_ _b"> </span>to <span class="_ _b"> </span>how <span class="_ _a"> </span>to </span></div><div class="t m0 xb h2 y43 ff1 fs0 fc0 sc0 ls0 ws0">choose <span class="_ _11"></span>the <span class="_ _11"></span>appropriate <span class="_ _11"></span>amplitude <span class="_ _c"></span>of <span class="_ _11"></span>the <span class="_ _11"></span>added <span class="_ _11"></span>noise <span class="_ _11"></span>and </div><div class="t m0 xb h2 y44 ff1 fs0 fc0 sc0 ls0 ws0">its <span class="_ _7"> </span>computation <span class="_ _7"> </span>eff<span class="_ _3"></span>iciency <span class="_ _7"> </span>is <span class="_ _4"> </span>fairly <span class="_ _7"> </span>low. <span class="_ _4"> </span>As <span class="_ _7"> </span>a <span class="_ _4"> </span>result, <span class="_ _7"> </span>the </div><div class="t m0 xb h2 y45 ff1 fs0 fc0 sc0 ls0 ws0">inappropriate <span class="_ _11"></span>amplitude <span class="_ _11"></span>of <span class="_ _11"></span>the <span class="_ _11"></span>added <span class="_ _11"></span>wh<span class="_ _3"></span>ite <span class="_ _11"></span>noise <span class="_ _11"></span>for <span class="_ _7"> </span>the </div><div class="t m0 xb h2 y46 ff1 fs0 fc0 sc0 ls0 ws0">EEMD <span class="_ _4"> </span>method <span class="_ _4"> </span>is <span class="_ _4"> </span>go<span class="_ _3"></span>ing <span class="_ _4"> </span>to <span class="_"> </span>cause <span class="_ _4"> </span>the <span class="_ _4"> </span>mode <span class="_ _4"> </span>mixing <span class="_ _4"> </span>and </div><div class="t m0 xb h2 y47 ff1 fs0 fc0 sc0 ls0 ws0">splitting <span class="_ _4"> </span>that <span class="_ _4"> </span>often <span class="_"> </span>exists <span class="_ _7"> </span>in <span class="_"> </span>the <span class="_ _4"> </span>EMD <span class="_ _4"> </span>method <span class="_ _4"> </span>[<span class="ls2">10<span class="ls4">, <span class="_ _4"> </span></span>11</span>]<span class="ls4">. </span></div><div class="t m0 xb h2 y48 ff1 fs0 fc0 sc0 ls0 ws0">Although <span class="_ _3"></span>the <span class="_ _3"></span>reference <span class="_ _3"></span>[<span class="_ _3"></span>8] <span class="_ _3"></span>suggested <span class="_ _3"></span>that <span class="_ _3"></span>the <span class="_ _5"></span>amplitude of </div><div class="t m0 xb h2 y49 ff1 fs0 fc0 sc0 ls0 ws0">the <span class="_ _7"> </span>add<span class="_ _3"></span>ed <span class="_ _4"> </span>white <span class="_ _7"> </span>noise <span class="_ _4"> </span>should <span class="_ _4"> </span>be <span class="_ _7"> </span>a<span class="_ _3"></span>bout <span class="_ _4"> </span>0.2 <span class="_ _4"> </span>times <span class="_ _7"> </span>of <span class="_ _4"> </span>the </div><div class="t m0 xb h2 y4a ff1 fs0 fc0 sc0 ls0 ws0">standard deviation <span class="_ _3"></span>of the <span class="_ _3"></span>investigated sign<span class="_ _3"></span>al, unfo<span class="_ _3"></span>rtunately, </div><div class="t m0 xb h2 y4b ff1 fs0 fc0 sc0 ls0 ws0">with <span class="_ _c"></span>the <span class="_ _11"></span>suggested <span class="_ _c"></span>value, <span class="_ _11"></span>the <span class="_ _c"></span>decomposition <span class="_ _11"></span>results <span class="_ _c"></span>from </div><div class="t m0 xb h2 y4c ff1 fs0 fc0 sc0 ls0 ws0">the <span class="_ _13"> </span>EEMD <span class="_ _13"> </span>method <span class="_ _13"> </span>often <span class="_ _13"> </span>deviate <span class="_ _14"> </span>from <span class="_ _14"> </span>the <span class="_ _13"> </span>realistic </div><div class="t m0 xb h2 y4d ff1 fs0 fc0 sc0 ls0 ws0">contents <span class="_ _3"></span>of <span class="_ _5"></span>the <span class="_ _5"></span>signals <span class="_ _5"></span>[<span class="ls2">11</span>]. <span class="_ _3"></span>In<span class="_ _3"></span> <span class="_ _3"></span>add<span class="_ _3"></span>ition, <span class="_ _5"></span>to <span class="_ _3"></span>fu<span class="_ _3"></span>rther <span class="_ _5"></span>remove </div><div class="t m0 xb h2 y4e ff1 fs0 fc0 sc0 ls0 ws0">the residual of the added white noise and red<span class="_ _3"></span>uce a waste of </div><div class="t m0 xb h2 y4f ff1 fs0 fc0 sc0 ls0 ws0">time <span class="_ _3"></span>fo<span class="_ _3"></span>r <span class="_ _5"></span>the <span class="_ _9"></span>EEMD <span class="_ _5"></span>method, <span class="_ _9"></span>the <span class="_ _5"></span>complementary <span class="_ _5"></span>ensemble </div><div class="t m0 xb h2 y50 ff1 fs0 fc0 sc0 ls0 ws0">empirical <span class="_ _10"> </span>mode <span class="_ _10"> </span>deco<span class="_ _3"></span>mposition <span class="_ _10"> </span>(CEEMD) <span class="_ _15"> </span>has <span class="_ _10"> </span>been </div><div class="t m0 xb h2 y51 ff1 fs0 fc0 sc0 ls0 ws0">addressed <span class="_ _a"> </span>to <span class="_ _b"> </span>replace <span class="_ _a"> </span>the <span class="_ _a"> </span>EEMD <span class="_ _b"> </span>method <span class="_ _a"> </span>as <span class="_ _a"> </span>a <span class="_ _b"> </span>standard </div><div class="t m0 xb h2 y52 ff1 fs0 fc0 sc0 ls0 ws0">version <span class="_"> </span>of <span class="_ _4"> </span>the <span class="_"> </span>EMD <span class="_ _4"> </span>method <span class="_"> </span>[<span class="ls2">10</span>]. <span class="_"> </span>Notwithsta<span class="_ _2"></span>nd<span class="_ _3"></span>ing, <span class="_"> </span>the </div><div class="t m0 xb h2 y53 ff1 fs0 fc0 sc0 ls0 ws0">CEEMD <span class="_"> </span>method <span class="_ _a"> </span>only <span class="_"> </span>partly <span class="_ _a"> </span>enhances <span class="_"> </span>the <span class="_ _a"> </span>computation </div><div class="t m0 xb h2 y54 ff1 fs0 fc0 sc0 ls0 ws0">efficiency <span class="_ _e"> </span>of <span class="_ _8"> </span>the <span class="_ _e"> </span>EEMD <span class="_ _8"> </span>met<span class="_ _2"></span>ho<span class="_ _3"></span>d, <span class="_ _8"> </span>and <span class="_ _e"> </span>the <span class="_ _e"> </span>abo<span class="_ _3"></span>ve <span class="_ _e"> </span>first </div><div class="t m0 xb h2 y55 ff1 fs0 fc0 sc0 ls0 ws0">problem <span class="_ _13"> </span>reg<span class="_ _3"></span>arding <span class="_ _16"> </span>the <span class="_ _16"> </span>EEMD <span class="_ _13"> </span>method <span class="_ _16"> </span>still <span class="_ _13"> </span>remains </div><div class="t m0 xb h2 y56 ff1 fs0 fc0 sc0 ls0 ws0">untouched. Ad<span class="_ _3"></span>ditionally, if the <span class="_ _3"></span>researched <span class="_ _3"></span>signal is <span class="_ _3"></span>a <span class="_ _3"></span>noisy </div><div class="t m0 xb h2 y57 ff1 fs0 fc0 sc0 ls0 ws0">signal <span class="_ _7"> </span>in <span class="_ _7"> </span>itself, <span class="_ _7"> </span>its <span class="_ _7"> </span>intrinsic <span class="_ _4"> </span>noise <span class="_ _7"> </span>will <span class="_ _7"> </span>inevitably <span class="_ _7"> </span>interact </div><div class="t m0 xb h2 y58 ff1 fs0 fc0 sc0 ls0 ws0">with <span class="_ _11"></span>the <span class="_ _7"> </span>noise <span class="_ _7"> </span>added <span class="_ _7"> </span>through <span class="_ _7"> </span>the <span class="_ _7"> </span>EEMD <span class="_ _7"> </span>method, <span class="_ _11"></span>wh<span class="_ _3"></span>ich </div><div class="t m0 xb h2 y59 ff1 fs0 fc0 sc0 ls0 ws0">may <span class="_ _9"></span>further <span class="_ _9"></span>complicate <span class="_ _9"></span>the <span class="_ _c"></span>above <span class="_ _c"></span>first <span class="_ _9"></span>prob<span class="_ _3"></span>lem <span class="_ _9"></span>regarding </div><div class="t m0 xb h2 y5a ff1 fs0 fc0 sc0 ls0 ws0">the <span class="_ _b"> </span>EEMD <span class="_ _a"> </span>method. <span class="_ _b"> </span>In<span class="_ _3"></span> <span class="_ _b"> </span>particular, <span class="_ _a"> </span>when <span class="_ _a"> </span>the <span class="_ _b"> </span>researched </div><div class="t m0 xb h2 y5b ff1 fs0 fc0 sc0 ls0 ws0">signals <span class="_ _6"> </span>become <span class="_ _6"> </span>ver<span class="_ _3"></span>y <span class="_ _6"> </span>noisy, <span class="_ _6"> </span>the <span class="_ _14"> </span>above <span class="_ _6"> </span>first <span class="_ _14"> </span>problem </div><div class="t m0 xb h2 y5c ff1 fs0 fc0 sc0 ls0 ws0">regarding the EEMD method leaves a gap. </div><div class="t m0 xb h2 y5d ff1 fs0 fc0 sc0 ls0 ws0"> </div><div class="t m0 xb h2 y5e ff1 fs0 fc0 sc0 ls0 ws0">This <span class="_ _7"> </span>paper <span class="_ _4"> </span>explores <span class="_ _7"> </span>the <span class="_ _7"> </span>above <span class="_ _7"> </span>two <span class="_ _4"> </span>problems <span class="_ _11"></span>con<span class="_ _3"></span>cerning </div><div class="t m0 xb h2 y5f ff1 fs0 fc0 sc0 ls0 ws0">the <span class="_ _17"> </span>EEMD <span class="_ _17"> </span>method. <span class="_ _15"> </span>Then<span class="_ _3"></span>, <span class="_ _17"> </span>the <span class="_ _15"> </span>improved <span class="_ _17"> </span>CEEMD </div><div class="t m0 xb h2 y60 ff1 fs0 fc0 sc0 ls0 ws0">(ICEEMD) method was pr<span class="_ _3"></span>oposed. Ap<span class="_ _3"></span>plications to analysis </div><div class="c xc y61 w3 hc"><div class="t m0 x0 hd y62 ff7 fs4 fc1 sc0 ls0 ws0">IJCSI International Journal of Computer Science Issues, Vol. 9, Issue 6, No 2, November 2012<span class="fc2 sc0"> </span></div><div class="t m0 x0 hd y63 ff7 fs4 fc1 sc0 ls0 ws0">ISSN (Online): 1694-0814 </div><div class="t m0 x0 hd y64 ff7 fs4 fc1 sc0 ls0 ws0">www.IJCSI.org</div></div><div class="c xd y65 w4 he"><div class="t m0 x0 hd y64 ff7 fs4 fc1 sc0 ls0 ws0">194</div></div><div class="c xe y66 w5 he"><div class="t m0 x0 hd y64 ff7 fs4 fc1 sc0 ls0 ws0">Copyright (c) 2012 International Journal of Computer Science Issues. 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<div id="pf2" class="pf w0 h0" data-page-no="2"><div class="pc pc2 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/62604a4b090cbf2c4ec3c3b3/bg2.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0"> </div><div class="t m0 xf h2 y67 ff1 fs0 fc0 sc0 ls0 ws0"> </div><div class="t m0 x1 h2 y68 ff1 fs0 fc0 sc0 ls0 ws0">of <span class="_ _9"></span>defective <span class="_ _9"></span>gearbox<span class="_ _3"></span> <span class="_ _9"></span>signals <span class="_ _c"></span><span class="ls2">prove</span>d <span class="_ _9"></span>the <span class="_ _9"></span>superiority <span class="_ _c"></span>of <span class="_ _9"></span>the </div><div class="t m0 x1 h2 y69 ff1 fs0 fc0 sc0 ls0 ws0">ICEEMD method over the EEMD method. </div><div class="t m0 x1 hb y6a ff3 fs3 fc0 sc0 ls0 ws0">2. The EMD and its Several Variations </div><div class="t m0 x1 hf y6b ff1 fs5 fc0 sc0 ls0 ws0">2.1 The EMD method </div><div class="t m0 x1 h2 y6c ff1 fs0 fc0 sc0 ls0 ws0">The <span class="_ _4"> </span>EMD <span class="_ _4"> </span>method <span class="_ _4"> </span>can <span class="_ _4"> </span>self-adaptively <span class="_ _4"> </span>decompose <span class="_ _4"> </span>any <span class="_ _4"> </span>a </div><div class="t m0 x1 h2 y6d ff1 fs0 fc0 sc0 ls2 ws0">non<span class="ls0">-stationary <span class="_ _c"></span>and <span class="_ _11"></span>nonlinear <span class="_ _11"></span>signal <span class="_ _c"></span>into <span class="_ _11"></span>a <span class="_ _11"></span>set <span class="_ _c"></span>of <span class="_ _11"></span>intrinsic </span></div><div class="t m0 x1 h2 y6e ff1 fs0 fc0 sc0 ls0 ws0">mode <span class="_ _f"> </span>functions <span class="_ _6"> </span>(IMFs) <span class="_ _6"> </span>from <span class="_ _6"> </span>high <span class="_ _f"> </span>fr<span class="_ _3"></span>equency <span class="_ _f"> </span>to <span class="_ _6"> </span>low </div><div class="t m0 x1 h2 y6f ff1 fs0 fc0 sc0 ls0 ws0">frequency, which may be written as <span class="_ _18"> </span> </div><div class="c x10 y70 w6 h10"><div class="t m2 x11 h11 y71 ff1 fs6 fc0 sc0 ls0 ws0">1</div><div class="t m2 x12 h12 y72 ff1 fs7 fc0 sc0 ls0 ws0">(<span class="_ _e"> </span>)<span class="_ _19"> </span>(<span class="_ _e"> </span>)<span class="_ _1a"> </span>(<span class="_ _e"> </span>)</div><div class="t m2 x13 h13 y73 ff6 fs6 fc0 sc0 ls0 ws0">N</div><div class="t m2 x14 h13 y74 ff6 fs6 fc0 sc0 ls6 ws0">iN</div><div class="t m2 x15 h13 y71 ff6 fs6 fc0 sc0 ls0 ws0">i</div><div class="t m2 x16 h14 y72 ff6 fs7 fc0 sc0 ls0 ws0">x<span class="_ _e"> </span>t<span class="_ _1b"> </span>c<span class="_ _10"> </span>t<span class="_ _1c"> </span>r<span class="_ _1d"> </span>t</div><div class="t m2 x13 h15 y71 ff8 fs6 fc0 sc0 ls0 ws0"></div><div class="t m2 x17 h16 y72 ff8 fs7 fc0 sc0 ls7 ws0"></div><div class="t m2 x18 h17 y75 ff8 fs8 fc0 sc0 ls0 ws0"></div></div><div class="t m0 x19 h2 y76 ff1 fs0 fc0 sc0 ls0 ws0"> <span class="_ _3"></span> <span class="_ _3"></span> <span class="_ _3"></span> <span class="_ _3"></span> <span class="_ _3"></span> <span class="_ _3"></span> <span class="_ _3"></span> <span class="_ _3"></span> <span class="_ _3"></span> <span class="_ _3"></span> <span class="_ _3"></span> <span class="_ _5"></span>(1)</div><div class="t m0 x1 h2 y77 ff1 fs0 fc0 sc0 ls0 ws0">where <span class="_ _3"></span><span class="ff6">c</span></div><div class="t m0 x1a h18 y78 ff6 fs9 fc0 sc0 ls0 ws0">i</div><div class="t m0 x1b h2 y77 ff1 fs0 fc0 sc0 ls0 ws0">(<span class="ff6">t</span>) <span class="_ _3"></span>indicates <span class="_ _3"></span>the <span class="_ _5"></span><span class="ff6">i</span>th <span class="_ _3"></span>IMF<span class="ff6"> <span class="_ _5"></span></span>and <span class="_ _3"></span><span class="ff6">r</span></div><div class="t m0 x1c h18 y78 ff6 fs9 fc0 sc0 ls0 ws0">N</div><div class="t m0 x1d h2 y77 ff1 fs0 fc0 sc0 ls0 ws0">(<span class="ff6">t</span>) <span class="_ _3"></span>represents <span class="_ _5"></span>the <span class="_ _5"></span>r</div><div class="t m0 x1 h2 y79 ff1 fs0 fc0 sc0 ls0 ws0">esidual of the signal <span class="_ _3"></span><span class="ff6">x</span>(<span class="ff6">t</span>). An <span class="_ _3"></span>IMF is a <span class="_ _3"></span>function <span class="_ _3"></span>which mus</div><div class="t m0 x1 h2 y7a ff1 fs0 fc0 sc0 ls0 ws0">t satisfy the <span class="_ _3"></span>following two <span class="_ _3"></span>conditions: (1<span class="_ _3"></span>) the <span class="_ _3"></span>number <span class="_ _3"></span>of <span class="_ _3"></span>e</div><div class="t m0 x1 h2 y7b ff1 fs0 fc0 sc0 ls0 ws0">xtrema <span class="_ _3"></span>and <span class="_ _3"></span>the <span class="_ _5"></span>number <span class="_ _3"></span>of <span class="_ _3"></span>zero <span class="_ _5"></span>crossings <span class="_ _5"></span>either <span class="_ _3"></span>equal <span class="_ _5"></span>to <span class="_ _3"></span>e</div><div class="t m0 x1 h2 y7c ff1 fs0 fc0 sc0 ls0 ws0">ach <span class="_ _3"></span>other <span class="_ _3"></span>or <span class="_ _3"></span>differ <span class="_ _3"></span>at <span class="_ _3"></span>most by <span class="_ _3"></span>one, <span class="_ _3"></span>and<span class="_ _3"></span> <span class="_ _3"></span>(2) <span class="_ _3"></span>at <span class="_ _3"></span>any <span class="_ _3"></span>point, <span class="_ _3"></span>th</div><div class="t m0 x1 h2 y7d ff1 fs0 fc0 sc0 ls0 ws0">e local <span class="_ _3"></span>avera<span class="_ _3"></span>ge <span class="_ _3"></span>of <span class="_ _3"></span>the <span class="_ _3"></span>upper <span class="_ _3"></span>envelope <span class="_ _3"></span>and <span class="_ _3"></span>the <span class="_ _3"></span>lower <span class="_ _5"></span>envel</div><div class="t m0 x1 h2 y7e ff1 fs0 fc0 sc0 ls0 ws0">ope is <span class="_ _3"></span>zero <span class="_ _3"></span>[8]. <span class="_ _3"></span>The <span class="_ _3"></span>residual <span class="_ _3"></span><span class="ff6">r</span></div><div class="t m0 x1e h18 y7f ff6 fs9 fc0 sc0 ls0 ws0">N</div><div class="t m0 x1f h2 y7e ff1 fs0 fc0 sc0 ls0 ws0">(<span class="ff6">t</span>) usually <span class="_ _3"></span>is <span class="_ _3"></span>a <span class="_ _3"></span>monotonic <span class="_ _3"></span>f</div><div class="t m0 x1 h2 y80 ff1 fs0 fc0 sc0 ls0 ws0">unction or a constant. </div><div class="t m0 x1 hf y81 ff1 fs5 fc0 sc0 ls2 ws0">2.<span class="ls0">2 The Ensemble <span class="ls8">EMD</span> </span></div><div class="t m0 x1 h2 y82 ff1 fs0 fc0 sc0 ls0 ws0">An <span class="_ _3"></span>ann<span class="_ _3"></span>oying <span class="_ _3"></span>prob<span class="_ _3"></span>lem <span class="_ _3"></span>associated <span class="_ _3"></span>with <span class="_ _5"></span>the <span class="_ _5"></span>EMD <span class="_ _5"></span>method <span class="_ _5"></span>is </div><div class="t m0 x1 h2 y83 ff1 fs0 fc0 sc0 ls0 ws0">the <span class="_ _9"></span>mode <span class="_ _9"></span>mixing <span class="_ _5"></span>du<span class="_ _3"></span>e <span class="_ _9"></span>to <span class="_ _9"></span>intermittency, <span class="_ _5"></span>defined <span class="_ _c"></span>as <span class="_ _9"></span>either <span class="_ _9"></span>a </div><div class="t m0 x1 h2 y84 ff1 fs0 fc0 sc0 ls0 ws0">single <span class="_ _a"> </span>IMF <span class="_ _a"> </span>consisting <span class="_ _b"> </span>of<span class="_ _3"></span> <span class="_ _a"> </span>widely <span class="_ _b"> </span>disparate <span class="_ _a"> </span>scales, <span class="_ _a"> </span>or <span class="_ _b"> </span>a </div><div class="t m0 x1 h2 y85 ff1 fs0 fc0 sc0 ls0 ws0">signal <span class="_ _9"></span>residing <span class="_ _c"></span>in <span class="_ _9"></span>diff<span class="_ _3"></span>erent <span class="_ _9"></span>IMF <span class="_ _c"></span>components. <span class="_ _c"></span>To <span class="_ _c"></span>alleviate </div><div class="t m0 x1 h2 y86 ff1 fs0 fc0 sc0 ls0 ws0">the <span class="_ _5"></span>imperfection <span class="_ _5"></span>of<span class="_ _3"></span> <span class="_ _5"></span>the <span class="_ _9"></span>EMD <span class="_ _9"></span>method, <span class="_ _5"></span>the <span class="_ _9"></span>ensemble <span class="_ _5"></span>EMD </div><div class="t m0 x1 h2 y87 ff1 fs0 fc0 sc0 ls0 ws0">(EEMD), <span class="_ _14"> </span>a <span class="_ _13"> </span>noise-assisted <span class="_ _14"> </span>method, <span class="_ _14"> </span>is <span class="_ _13"> </span>proposed. <span class="_ _14"> </span>The </div><div class="t m0 x1 h2 y88 ff1 fs0 fc0 sc0 ls0 ws0">EEMD method can be stated as follows: </div><div class="c x20 y89 w7 h19"><div class="t m3 x21 h1a y8a ff1 fsa fc0 sc0 ls0 ws0">(<span class="_ _a"> </span>)<span class="_ _1e"> </span>(<span class="_ _a"> </span>)<span class="_ _1f"> </span>(<span class="_ _a"> </span>),<span class="_ _20"> </span>1<span class="_ _1"></span>,<span class="_ _9"></span>2<span class="_ _2"></span>,<span class="_ _21"> </span><span class="fc2 sc0">,</span></div><div class="t m3 x12 h1b y8b ff6 fsb fc0 sc0 ls9 ws0">mm</div><div class="t m3 x16 h1c y8a ff6 fsa fc0 sc0 ls0 ws0">x<span class="_ _1d"> </span>t<span class="_ _22"> </span>x<span class="_ _a"> </span>t<span class="_ _1c"> </span>w<span class="_ _18"> </span>t<span class="_ _23"> </span>m<span class="_ _24"> </span><span class="fc2 sc0">N</span><span class="_ _25"></span><span class="ff8"><span class="_ _26"> </span><span class="_ _27"> </span></span></div></div><div class="t m0 x22 h1d y8c ff1 fsc fc0 sc0 ls4 ws0"> <span class="ls0">(2)</span></div><div class="t m0 x1 h2 y8d ff1 fs0 fc0 sc0 ls0 ws0"> <span class="_ _28"> </span> </div><div class="c x23 y8e w8 h1e"><div class="t m4 x1 h1f y8f ff1 fsd fc0 sc0 lsa ws0">,,</div><div class="t m4 x24 h1f y90 ff1 fsd fc0 sc0 ls0 ws0">1</div><div class="t m4 x21 h20 y91 ff1 fse fc0 sc0 ls0 ws0">(<span class="_ _a"> </span>)<span class="_ _29"> </span>(<span class="_ _a"> </span>)<span class="_ _2a"> </span>(<span class="_ _a"> </span>),<span class="_ _1b"> </span>1<span class="_ _2b"></span>,<span class="_ _c"></span>2,<span class="_ _21"> </span><span class="fc2 sc0">,</span></div><div class="t m4 xc h21 y92 ff6 fsd fc0 sc0 ls0 ws0">L</div><div class="t m4 x12 h21 y8f ff6 fsd fc0 sc0 ls0 ws0">m<span class="_ _24"> </span>m<span class="_ _4"> </span>i<span class="_ _2a"> </span>m<span class="_ _2c"> </span>L</div><div class="t m4 x11 h21 y90 ff6 fsd fc0 sc0 ls0 ws0">i</div><div class="t m4 x16 h22 y91 ff6 fse fc0 sc0 ls0 ws0">x<span class="_ _12"> </span>t<span class="_ _2d"> </span>c<span class="_ _2e"> </span>t<span class="_ _1c"> </span>r<span class="_ _2f"> </span>t<span class="_ _30"> </span>m<span class="_ _31"> </span><span class="fc2 sc0">N</span></div><div class="t m4 xc h23 y90 ff8 fsd fc0 sc0 ls0 ws0"></div><div class="t m4 x25 h24 y91 ff8 fse fc0 sc0 ls0 ws0"><span class="_ _32"> </span><span class="_ _33"> </span></div><div class="t m4 x26 h25 y93 ff8 fsf fc0 sc0 ls0 ws0"></div></div><div class="t m0 x27 h1d y94 ff1 fsc fc0 sc0 ls0 ws0"> <span class="_ _3"></span> <span class="_ _3"></span> <span class="_ _3"></span> <span class="_ _3"></span> <span class="_ _3"></span> <span class="_ _3"></span> <span class="_ _3"></span> <span class="_ _3"></span>(3)</div><div class="c x23 y95 w8 h1e"><div class="t m4 x1 h1f y8f ff1 fsd fc0 sc0 lsa ws0">,,</div><div class="t m4 x24 h1f y90 ff1 fsd fc0 sc0 ls0 ws0">1</div><div class="t m4 x21 h20 y91 ff1 fse fc0 sc0 ls0 ws0">(<span class="_ _a"> </span>)<span class="_ _29"> </span>(<span class="_ _a"> </span>)<span class="_ _2a"> </span>(<span class="_ _a"> </span>),<span class="_ _2d"> </span>1<span class="_ _1"></span>,<span class="_ _c"></span>2,<span class="_ _21"> </span><span class="fc2 sc0">,</span></div><div class="t m4 xc h21 y92 ff6 fsd fc0 sc0 ls0 ws0">L</div><div class="t m4 x12 h21 y8f ff6 fsd fc0 sc0 ls0 ws0">m<span class="_ _24"> </span>m<span class="_ _4"> </span>i<span class="_ _2a"> </span>m<span class="_ _2c"> </span>L</div><div class="t m4 x11 h21 y90 ff6 fsd fc0 sc0 ls0 ws0">i</div><div class="t m4 x16 h22 y91 ff6 fse fc0 sc0 ls0 ws0">x<span class="_ _12"> </span>t<span class="_ _2d"> </span>c<span class="_ _2e"> </span>t<span class="_ _1c"> </span>r<span class="_ _2f"> </span>t<span class="_ _30"> </span>m<span class="_ _31"> </span><span class="fc2 sc0">N</span></div><div class="t m4 xc h23 y90 ff8 fsd fc0 sc0 ls0 ws0"></div><div class="t m4 x25 h24 y91 ff8 fse fc0 sc0 ls0 ws0"><span class="_ _32"> </span><span class="_ _33"> </span></div><div class="t m4 x26 h25 y93 ff8 fsf fc0 sc0 ls0 ws0"></div></div><div class="t m0 x27 h1d y96 ff1 fsc fc0 sc0 ls0 ws0"> <span class="_ _3"></span> <span class="_ _3"></span> <span class="_ _5"></span> <span class="_ _3"></span> <span class="_ _3"></span> <span class="_ _3"></span> <span class="_ _3"></span> <span class="_ _5"></span> <span class="_ _3"></span>(4) </div><div class="t m0 x1 h2 y97 ff1 fs0 fc0 sc0 ls0 ws0">where <span class="_ _4"> </span><span class="ff6">x</span>(<span class="ff6">t</span>) <span class="_ _7"> </span>is <span class="_ _7"> </span>the <span class="_ _4"> </span>original <span class="_ _7"> </span>signal, <span class="_ _4"> </span><span class="ff6">w</span></div><div class="t m0 x28 h26 y98 ff1 fs9 fc0 sc0 ls0 ws0">m</div><div class="t m0 x29 h2 y97 ff1 fs0 fc0 sc0 ls0 ws0">(<span class="ff6">t</span>) <span class="_ _7"> </span>is <span class="_ _7"> </span>the <span class="_ _4"> </span><span class="ff6">m</span>th <span class="_ _7"> </span>added </div><div class="t m0 x1 h2 y99 ff1 fs0 fc0 sc0 ls0 ws0">white <span class="_ _3"></span>noise,<span class="ff6"> <span class="_ _3"></span>x</span></div><div class="t m0 x2a h18 y9a ff6 fs9 fc0 sc0 ls0 ws0">m</div><div class="t m0 x2b h2 y99 ff1 fs0 fc0 sc0 ls0 ws0">(<span class="ff6">t</span>) <span class="_ _3"></span>is <span class="_ _3"></span>the <span class="_ _3"></span>no<span class="_ _3"></span>isy <span class="_ _3"></span>signal <span class="_ _3"></span>of <span class="_ _5"></span>the <span class="_ _5"></span><span class="ff6">m</span>th <span class="_ _3"></span>trial, <span class="_ _3"></span><span class="ff6">c</span></div><div class="t m0 x2c h18 y9a ff6 fs9 fc0 sc0 lsb ws0">m,i</div><div class="t m0 x2d h2 y99 ff1 fs0 fc0 sc0 ls0 ws0">(<span class="ff6">t</span><span class="lsc">) </span></div><div class="t m0 x1 h2 y9b ff1 fs0 fc0 sc0 ls0 ws0">is <span class="_ _c"></span>the <span class="_ _11"></span><span class="ff6">i</span>th <span class="_ _11"></span>IMF <span class="_ _11"></span>of <span class="_ _11"></span>the <span class="_ _11"></span><span class="ff6">m</span>th <span class="_ _11"></span>trial,<span class="ff6"> <span class="_ _11"></span>L</span> <span class="_ _11"></span>is <span class="_ _11"></span>the <span class="_ _11"></span>number <span class="_ _11"></span>of <span class="_ _11"></span>IMFs </div><div class="t m0 x1 h2 y9c ff1 fs0 fc0 sc0 ls0 ws0">from <span class="_ _9"></span>the <span class="_ _9"></span>EMD <span class="_ _9"></span>method, <span class="_ _9"></span>and <span class="_ _9"></span><span class="ff6">N</span> <span class="_ _9"></span>is <span class="_ _9"></span>the <span class="_ _c"></span>ensemble <span class="_ _5"></span>nu<span class="_ _3"></span>mber <span class="_ _9"></span>of </div><div class="t m0 x1 h2 y9d ff1 fs0 fc0 sc0 ls0 ws0">the EEMD method. </div><div class="t m0 x1 h2 y9e ff1 fs0 fc0 sc0 ls0 ws0">The <span class="_ _8"> </span>EEMD <span class="_ _8"> </span>m<span class="_ _2"></span>ethod <span class="_ _8"> </span>adds <span class="_ _8"> </span>white <span class="_ _8"> </span>noise <span class="_ _8"> </span>with <span class="_ _8"> </span>the <span class="_ _8"> </span>finite </div><div class="t m0 x1 h2 y9f ff1 fs0 fc0 sc0 ls0 ws0">amplitude <span class="_ _7"> </span>to <span class="_ _4"> </span>the <span class="_ _7"> </span>sign<span class="_ _3"></span>al, <span class="_ _7"> </span>su<span class="_ _3"></span>fficiently <span class="_ _7"> </span>taking <span class="_ _4"> </span>advantage <span class="_ _4"> </span>of </div><div class="t m0 x1 h2 ya0 ff1 fs0 fc0 sc0 ls0 ws0">the <span class="_ _7"> </span>uniform <span class="_ _11"></span>statistic <span class="_ _7"> </span>characteristics <span class="_ _11"></span>of <span class="_ _7"> </span>white <span class="_ _7"> </span>noise <span class="_ _7"> </span>in <span class="_ _7"> </span>the </div><div class="t m0 x1 h2 ya1 ff1 fs0 fc0 sc0 ls0 ws0">frequency <span class="_ _c"></span>domain, <span class="_ _9"></span>pr<span class="_ _3"></span>ojects <span class="_ _c"></span>the <span class="_ _c"></span>different <span class="_ _c"></span>frequency <span class="_ _c"></span>signal </div><div class="t m0 x1 h2 ya2 ff1 fs0 fc0 sc0 ls0 ws0">components <span class="_ _9"></span>onto <span class="_ _9"></span>the <span class="_ _9"></span>correspon<span class="_ _3"></span>ding <span class="_ _9"></span>frequency <span class="_ _9"></span>banks<span class="_ _3"></span> <span class="_ _9"></span>and, </div><div class="t m0 x1 h2 ya3 ff1 fs0 fc0 sc0 ls0 ws0">as <span class="_ _5"></span>a <span class="_ _5"></span>result, <span class="_ _5"></span>eff<span class="_ _3"></span>ectively <span class="_ _5"></span>overcomes <span class="_ _5"></span>the <span class="_ _5"></span>mode <span class="_ _5"></span>mixing <span class="_ _5"></span>due <span class="_ _5"></span>to </div><div class="t m0 x1 h2 ya4 ff1 fs0 fc0 sc0 ls0 ws0">the <span class="_ _c"></span>existence <span class="_ _11"></span>of <span class="_ _11"></span>intermittency <span class="_ _c"></span>[8]. <span class="_ _11"></span>Nonetheless, <span class="_ _11"></span>to <span class="_ _11"></span>totally </div><div class="t m0 x1 h2 ya5 ff1 fs0 fc0 sc0 ls0 ws0">clear <span class="_ _5"></span>the <span class="_ _9"></span>residual <span class="_ _5"></span>of <span class="_ _9"></span>the <span class="_ _5"></span>added <span class="_ _9"></span>white <span class="_ _5"></span>no<span class="_ _3"></span>ise <span class="_ _5"></span>from <span class="_ _9"></span>the <span class="_ _9"></span>IMFs </div><div class="t m0 xb h2 y68 ff1 fs0 fc0 sc0 ls0 ws0">of <span class="_ _3"></span>the <span class="_ _5"></span>EEMD <span class="_ _5"></span>method, <span class="_ _5"></span>a <span class="_ _3"></span>larg<span class="_ _3"></span>e <span class="_ _3"></span>ensemble <span class="_ _5"></span>number <span class="_ _5"></span>is <span class="_ _3"></span>usually </div><div class="t m0 xb h2 y69 ff1 fs0 fc0 sc0 ls0 ws0">demanded, which will cause a tremendous waste of time. </div><div class="t m0 xb hf ya6 ff1 fs5 fc0 sc0 ls0 ws0">2.3 The Complementary EEMD method </div><div class="t m0 xb h2 ya7 ff1 fs0 fc0 sc0 ls0 ws0">To <span class="_ _e"> </span>better <span class="_ _a"> </span>eliminate <span class="_ _a"> </span>the <span class="_ _e"> </span>residual <span class="_ _e"> </span>of <span class="_ _e"> </span>added <span class="_ _e"> </span>white <span class="_ _a"> </span>noise </div><div class="t m0 xb h2 ya8 ff1 fs0 fc0 sc0 ls0 ws0">persisting <span class="_ _5"></span>in <span class="_ _5"></span>the <span class="_ _9"></span>IMFs <span class="_ _5"></span>of <span class="_ _5"></span>the <span class="_ _9"></span>EEMD <span class="_ _5"></span>method <span class="_ _5"></span>and<span class="_ _3"></span> <span class="_ _5"></span>raise <span class="_ _9"></span>the </div><div class="t m0 xb h2 ya9 ff1 fs0 fc0 sc0 ls0 ws0">computation <span class="_ _10"> </span>eff<span class="_ _3"></span>iciency <span class="_ _10"> </span>of<span class="_ _3"></span> <span class="_ _15"> </span>the <span class="_ _10"> </span>EEMD <span class="_ _15"> </span>method, <span class="_ _15"> </span>the </div><div class="t m0 xb h2 yaa ff1 fs0 fc0 sc0 ls0 ws0">complementary <span class="_ _7"> </span>ensemble <span class="_ _4"> </span>EMD <span class="_ _4"> </span>(CEEMD) <span class="_ _4"> </span>[<span class="ls2">10</span>], <span class="_ _4"> </span>a <span class="_ _4"> </span>novel </div><div class="t m0 xb h2 yab ff1 fs0 fc0 sc0 ls0 ws0">noise-enhanced <span class="_ _15"> </span>method, <span class="_ _10"> </span>is <span class="_ _10"> </span>pr<span class="_ _3"></span>esented. <span class="_ _15"> </span>The <span class="_ _10"> </span>CEEMD </div><div class="t m0 xb h2 yac ff1 fs0 fc0 sc0 ls0 ws0">method <span class="_ _7"> </span>adds <span class="_ _4"> </span>white <span class="_ _4"> </span>noise <span class="_ _7"> </span>in <span class="_ _4"> </span>pairs <span class="_ _7"> </span>with <span class="_ _4"> </span>one <span class="_ _4"> </span>positive <span class="_ _7"> </span>and </div><div class="t m0 xb h2 yad ff1 fs0 fc0 sc0 ls0 ws0">another <span class="_ _c"></span>negative <span class="_ _c"></span>to <span class="_ _11"></span>the <span class="_ _c"></span>original <span class="_ _c"></span>signal <span class="_ _11"></span>and <span class="_ _c"></span>then <span class="_ _11"></span>produces </div><div class="t m0 xb h2 yae ff1 fs0 fc0 sc0 ls0 ws0">two <span class="_ _10"> </span>sets <span class="_ _16"> </span>of <span class="_ _10"> </span>ensemble <span class="_ _10"> </span>IMFs. <span class="_ _16"> </span>Hence, <span class="_ _10"> </span>two <span class="_ _34"> </span>different </div><div class="t m0 xb h2 yaf ff1 fs0 fc0 sc0 ls0 ws0">combinations <span class="_ _7"> </span>of <span class="_ _4"> </span>the <span class="_ _7"> </span>original <span class="_ _7"> </span>signal <span class="_ _4"> </span>and <span class="_ _7"> </span>the <span class="_ _7"> </span>added <span class="_ _4"> </span>white </div><div class="t m0 xb h2 yb0 ff1 fs0 fc0 sc0 ls0 ws0">noise can be obtained, i.e. </div><div class="c x2e yb1 w9 h27"><div class="t m5 x20 h28 yb2 ff1 fs10 fc0 sc0 lsd ws0">()</div><div class="t m5 x2f h28 yb3 ff1 fs10 fc0 sc0 lse ws0">11</div><div class="t m5 x30 h28 yb4 ff1 fs10 fc0 sc0 lsd ws0">()</div><div class="t m5 x2a h28 yb5 ff1 fs10 fc0 sc0 ls0 ws0">,<span class="_ _2d"> </span>1<span class="_ _1"></span>,<span class="_ _c"></span>2,<span class="_ _21"> </span><span class="fc2 sc0">,</span></div><div class="t m5 x31 h28 yb6 ff1 fs10 fc0 sc0 lsd ws0">()</div><div class="t m5 x2f h28 yb7 ff1 fs10 fc0 sc0 lsf ws0">11</div><div class="t m5 x30 h28 yb8 ff1 fs10 fc0 sc0 lsd ws0">()</div><div class="t m5 x32 h29 yb9 ff6 fs11 fc0 sc0 ls0 ws0">m</div><div class="t m5 x33 h29 yba ff6 fs11 fc0 sc0 ls0 ws0">m</div><div class="t m5 x32 h29 ybb ff6 fs11 fc0 sc0 ls0 ws0">m</div><div class="t m5 x1a h2a yb2 ff6 fs10 fc0 sc0 ls10 ws0">xt</div><div class="t m5 x12 h2a yb4 ff6 fs10 fc0 sc0 ls11 ws0">xt</div><div class="t m5 x34 h2a yb5 ff6 fs10 fc0 sc0 ls12 ws0">m<span class="fc2 sc0">N</span></div><div class="t m5 x35 h2a yb6 ff6 fs10 fc0 sc0 ls13 ws0">wt</div><div class="t m5 x12 h2a yb8 ff6 fs10 fc0 sc0 ls14 ws0">xt</div><div class="t m5 x36 h2b ybc ff8 fs11 fc0 sc0 ls0 ws0"></div><div class="t m5 x36 h2b ybd ff8 fs11 fc0 sc0 ls0 ws0"></div><div class="t m5 x16 h2c ybe ff8 fs10 fc0 sc0 ls15 ws0"></div><div class="t m5 x37 h2c ybf ff8 fs10 fc0 sc0 ls16 ws0"></div><div class="t m5 x38 h2c yc0 ff8 fs10 fc0 sc0 ls17 ws0"></div><div class="t m5 x39 h2c yc1 ff8 fs10 fc0 sc0 ls18 ws0"></div><div class="t m5 x16 h2c yc2 ff8 fs10 fc0 sc0 ls15 ws0"></div><div class="t m5 x37 h2c yc3 ff8 fs10 fc0 sc0 ls16 ws0"></div><div class="t m5 x38 h2c yc4 ff8 fs10 fc0 sc0 ls17 ws0"></div><div class="t m5 x3a h2c yb7 ff8 fs10 fc0 sc0 ls0 ws0"></div><div class="t m5 x38 h2c yc5 ff8 fs10 fc0 sc0 ls17 ws0"></div><div class="t m5 x37 h2c yc6 ff8 fs10 fc0 sc0 ls16 ws0"></div><div class="t m5 x16 h2c yc7 ff8 fs10 fc0 sc0 ls15 ws0"></div></div><div class="t m0 x3b h1d yc8 ff1 fsc fc0 sc0 ls4 ws0"> <span class="ls0">(5) </span></div><div class="t m0 xb h2 yc9 ff1 fs0 fc0 sc0 ls0 ws0">where <span class="_ _4"> </span><span class="ff6">x</span>(<span class="ff6">t</span>) <span class="_ _7"> </span>is <span class="_ _7"> </span>the <span class="_ _4"> </span>original <span class="_ _7"> </span>sign<span class="_ _3"></span>al, <span class="_ _4"> </span><span class="ff6">w</span></div><div class="t m0 x3c h18 yca ff6 fs9 fc0 sc0 ls0 ws0">m</div><div class="t m0 x3d h2 yc9 ff1 fs0 fc0 sc0 ls0 ws0">(<span class="ff6">t</span>) <span class="_ _7"> </span>is <span class="_ _4"> </span>the <span class="_ _7"> </span><span class="ff6">m</span>th <span class="_ _4"> </span>added </div><div class="t m0 xb h2 ycb ff1 fs0 fc0 sc0 ls0 ws0">white <span class="_ _c"></span>noise, <span class="_ _c"></span><span class="ff6">x</span></div><div class="t m0 x3e h18 ycc ff6 fs9 fc0 sc0 ls0 ws0">m</div><div class="t m0 x3f h26 ycd ff1 fs9 fc0 sc0 ls0 ws0">+</div><div class="t m0 x40 h2 ycb ff1 fs0 fc0 sc0 ls0 ws0">(<span class="ff6">t</span>) <span class="_ _c"></span>is <span class="_ _c"></span>the <span class="_ _c"></span>sum <span class="_ _c"></span>of<span class="_ _3"></span> <span class="_ _c"></span>the <span class="_ _11"></span>original <span class="_ _11"></span><span class="ff6">x</span>(<span class="ff6">t</span>) <span class="_ _c"></span>and <span class="_ _11"></span>the </div><div class="t m0 xb h2 yce ff6 fs0 fc0 sc0 ls0 ws0">m<span class="ff1">th <span class="_ _3"></span>added <span class="_ _3"></span>white <span class="_ _3"></span>noise <span class="_ _3"></span></span>w</div><div class="t m0 x41 h18 ycf ff6 fs9 fc0 sc0 ls0 ws0">m</div><div class="t m0 x42 h2 yce ff1 fs0 fc0 sc0 ls0 ws0">(<span class="ff6">t</span>), <span class="_ _3"></span>and <span class="_ _3"></span><span class="ff6">x</span></div><div class="t m0 x43 h18 ycf ff6 fs9 fc0 sc0 ls0 ws0">m</div><div class="t m0 x44 h26 yd0 ff1 fs9 fc0 sc0 ls0 ws0">-</div><div class="t m0 x3c h2 yce ff1 fs0 fc0 sc0 ls0 ws0">(<span class="ff6">t</span>) <span class="_ _3"></span>is <span class="_ _3"></span>the <span class="_ _3"></span>difference <span class="_ _3"></span>of </div><div class="t m0 xb h2 yd1 ff1 fs0 fc0 sc0 ls0 ws0">the <span class="_ _c"></span>or<span class="_ _3"></span>iginal <span class="_ _11"></span><span class="ff6">x</span>(<span class="ff6">t</span>) <span class="_ _11"></span>and <span class="_ _11"></span>the <span class="_ _11"></span><span class="ff6">m</span>th <span class="_ _11"></span>added <span class="_ _11"></span>white <span class="_ _11"></span>noise <span class="_ _11"></span><span class="ff6">w</span></div><div class="t m0 x45 h18 yd2 ff6 fs9 fc0 sc0 ls0 ws0">m</div><div class="t m0 xd h2 yd1 ff1 fs0 fc0 sc0 ls0 ws0">(<span class="ff6">t</span>). <span class="_ _11"></span>In </div><div class="t m0 xb h2 yd3 ff1 fs0 fc0 sc0 ls0 ws0">light <span class="_ _8"> </span>of <span class="_ _f"> </span>(3) <span class="_ _8"> </span>and<span class="_ _3"></span> <span class="_ _8"> </span>(4),<span class="_ _3"></span> <span class="_ _8"> </span>the <span class="_ _f"> </span>original <span class="_ _8"> </span>signal <span class="_ _f"> </span><span class="ff6">x</span>(<span class="ff6">t</span>) <span class="_ _f"> </span>can <span class="_ _8"> </span>be </div><div class="t m0 xb h2 yd4 ff1 fs0 fc0 sc0 ls0 ws0">expressed as </div><div class="c x46 yd5 wa h2d"><div class="t m6 x47 h2e yd6 ff8 fs12 fc0 sc0 ls0 ws0"><span class="_ _35"> </span></div><div class="t m6 x48 h2e yd7 ff8 fs12 fc0 sc0 ls0 ws0"><span class="_ _36"> </span></div><div class="t m7 x49 h2f yd8 ff1 fs13 fc0 sc0 ls19 ws0">,,</div><div class="t m7 x5 h2f yd9 ff1 fs13 fc0 sc0 ls1a ws0">11</div><div class="t m7 x4a h2f yda ff1 fs13 fc0 sc0 ls1b ws0">,,</div><div class="t m7 x2f h2f ydb ff1 fs13 fc0 sc0 ls0 ws0">1</div><div class="t m7 x26 h30 ydc ff1 fs14 fc0 sc0 ls0 ws0">1</div><div class="t m7 x12 h30 ydd ff1 fs14 fc0 sc0 ls0 ws0">(<span class="_ _e"> </span>)<span class="_ _37"> </span>(<span class="_ _e"> </span>)<span class="_ _38"> </span>(<span class="_ _e"> </span>)</div><div class="t m7 x15 h30 yde ff1 fs14 fc0 sc0 ls0 ws0">2</div><div class="t m7 x15 h30 ydf ff1 fs14 fc0 sc0 ls0 ws0">1</div><div class="t m7 x4b h30 ye0 ff1 fs14 fc0 sc0 ls0 ws0">(<span class="_ _e"> </span>)<span class="_ _39"> </span>(<span class="_ _e"> </span>)</div><div class="t m7 x25 h30 ye1 ff1 fs14 fc0 sc0 ls0 ws0">2</div><div class="t m7 x4c h31 ye2 ff6 fs13 fc0 sc0 ls1c ws0">LN</div><div class="t m7 x23 h31 yd8 ff6 fs13 fc0 sc0 ls0 ws0">m<span class="_ _4"> </span>i<span class="_ _19"> </span>m<span class="_ _4"> </span>i</div><div class="t m7 x38 h31 yd9 ff6 fs13 fc0 sc0 ls1d ws0">im</div><div class="t m7 x14 h31 ye3 ff6 fs13 fc0 sc0 ls0 ws0">N</div><div class="t m7 x4d h31 yda ff6 fs13 fc0 sc0 ls0 ws0">m<span class="_ _4"> </span>L<span class="_ _3a"> </span>m<span class="_ _4"> </span>L</div><div class="t m7 x4e h31 ydb ff6 fs13 fc0 sc0 ls0 ws0">m</div><div class="t m7 x16 h32 ydd ff6 fs14 fc0 sc0 ls0 ws0">x<span class="_ _e"> </span>t<span class="_ _3b"> </span>c<span class="_ _3c"> </span>t<span class="_ _1c"> </span>c<span class="_ _3c"> </span>t</div><div class="t m7 x11 h32 yde ff6 fs14 fc0 sc0 ls0 ws0">N</div><div class="t m7 x4f h32 ye0 ff6 fs14 fc0 sc0 ls0 ws0">r<span class="_ _3d"> </span>t<span class="_ _2f"> </span>r<span class="_ _3d"> </span>t</div><div class="t m7 x13 h32 ye1 ff6 fs14 fc0 sc0 ls0 ws0">N</div><div class="t m7 x4b h33 ye4 ff8 fs13 fc0 sc0 ls1e ws0"></div><div class="t m7 x4c h33 yd9 ff8 fs13 fc0 sc0 ls1f ws0"></div><div class="t m7 x50 h33 ye5 ff8 fs13 fc0 sc0 ls20 ws0"></div><div class="t m7 x38 h33 ydb ff8 fs13 fc0 sc0 ls0 ws0"></div><div class="t m7 x17 h34 ydd ff8 fs14 fc0 sc0 ls0 ws0"><span class="_ _3e"> </span><span class="_ _29"> </span></div><div class="t m7 x1b h34 ye0 ff8 fs14 fc0 sc0 ls0 ws0"></div><div class="t m7 x14 h35 ye6 ff8 fs15 fc0 sc0 ls21 ws0"></div><div class="t m7 x4e h35 ye7 ff8 fs15 fc0 sc0 ls0 ws0"></div></div><div class="t m0 x51 h1d ye8 ff1 fsc fc0 sc0 ls4 ws0"> <span class="ls0">(6) </span></div><div class="t m0 xb h2 ye9 ff1 fs0 fc0 sc0 ls0 ws0">where <span class="_ _9"></span><span class="ff6">c</span></div><div class="t m0 x52 h18 yea ff6 fs9 fc0 sc0 ls0 ws0">+</div><div class="t m0 x53 h18 yeb ff6 fs9 fc0 sc0 lsb ws0">m,i</div><div class="t m0 x54 h2 ye9 ff1 fs0 fc0 sc0 ls0 ws0">(<span class="ff6">t</span>) <span class="_ _9"></span>is <span class="_ _5"></span>the <span class="_ _9"></span><span class="ff6">i</span>th <span class="_ _9"></span>IMF <span class="_ _9"></span>of <span class="_ _9"></span><span class="ff6">x</span></div><div class="t m0 x55 h18 yeb ff6 fs9 fc0 sc0 ls0 ws0">m</div><div class="t m0 x56 h26 yea ff1 fs9 fc0 sc0 ls0 ws0">+</div><div class="t m0 x57 h2 ye9 ff1 fs0 fc0 sc0 ls0 ws0">(<span class="ff6">t</span>) <span class="_ _9"></span>and <span class="_ _9"></span><span class="ff6">c</span></div><div class="t m0 x58 h18 yea ff6 fs9 fc0 sc0 ls0 ws0">-</div><div class="t m0 x59 h18 yeb ff6 fs9 fc0 sc0 lsb ws0">m,i</div><div class="t m0 x5a h2 ye9 ff1 fs0 fc0 sc0 ls0 ws0">(<span class="ff6">t</span>) <span class="_ _9"></span>is <span class="_ _9"></span>the <span class="_ _9"></span><span class="ff6">i</span>th </div><div class="t m0 xb h2 yec ff1 fs0 fc0 sc0 ls0 ws0">IMF of <span class="_ _3"></span><span class="ff6">x</span></div><div class="t m0 x5b h18 yed ff6 fs9 fc0 sc0 ls0 ws0">m</div><div class="t m0 x5c h26 yee ff1 fs9 fc0 sc0 ls0 ws0">-</div><div class="t m0 x5d h2 yec ff1 fs0 fc0 sc0 ls0 ws0">(<span class="ff6">t</span>). Althoug<span class="_ _3"></span>h the <span class="_ _3"></span>CEEMD <span class="_ _3"></span>method appears <span class="_ _3"></span>to <span class="_ _3"></span>be </div><div class="t m0 xb h2 yef ff1 fs0 fc0 sc0 ls0 ws0">able <span class="_ _3"></span>to <span class="_ _5"></span>completely <span class="_ _3"></span>remove <span class="_ _5"></span>the <span class="_ _5"></span>residual <span class="_ _5"></span>of <span class="_ _5"></span>the <span class="_ _5"></span>added <span class="_ _9"></span>white </div><div class="t m0 xb h2 yf0 ff1 fs0 fc0 sc0 ls0 ws0">noise <span class="_ _5"></span>persisting <span class="_ _9"></span>in <span class="_ _9"></span>the <span class="_ _9"></span>IMFs <span class="_ _9"></span>of <span class="_ _9"></span>the <span class="_ _9"></span>EEMD <span class="_ _9"></span>method, <span class="_ _9"></span>it <span class="_ _5"></span>still </div><div class="t m0 xb h2 yf1 ff1 fs0 fc0 sc0 ls0 ws0">remains unsolved how to <span class="_ _3"></span>choose the appro<span class="_ _3"></span>priate amplitude </div><div class="t m0 xb h2 yf2 ff1 fs0 fc0 sc0 ls0 ws0">of the added white noise for the EEMD method. <span class="_ _3"></span> </div><div class="t m0 xb hb yf3 ff3 fs3 fc0 sc0 ls0 ws0">3. The Improved CEEMD Method </div><div class="t m0 xb hf yf4 ff1 fs5 fc0 sc0 ls0 ws0">3.1 The Choice of Amplitude of th<span class="_ _3"></span>e Added Noise </div><div class="t m0 xb h2 yf5 ff1 fs0 fc0 sc0 ls0 ws0">The <span class="_ _3"></span>amplitude of <span class="_ _3"></span>the <span class="_ _3"></span>added <span class="_ _3"></span>wh<span class="_ _3"></span>ite noise <span class="_ _3"></span>is <span class="_ _3"></span>a <span class="_ _3"></span>key <span class="_ _5"></span>parameter </div><div class="t m0 xb h2 yf6 ff1 fs0 fc0 sc0 ls0 ws0">of <span class="_ _9"></span>the <span class="_ _9"></span>EEMD <span class="_ _c"></span>method, <span class="_ _9"></span>which <span class="_ _9"></span>will <span class="_ _c"></span>exert <span class="_ _9"></span>a <span class="_ _9"></span>decisive <span class="_ _c"></span>impact </div><div class="t m0 xb h2 yf7 ff1 fs0 fc0 sc0 ls0 ws0">on <span class="_ _8"> </span>whether <span class="_ _8"> </span>or <span class="_ _8"> </span>not <span class="_ _8"> </span>the <span class="_ _8"> </span>EEMD <span class="_ _8"> </span>method <span class="_ _8"> </span>can <span class="_ _8"> </span>yiel<span class="_ _2"></span>d <span class="_ _8"> </span>the </div><div class="t m0 xb h2 yf8 ff1 fs0 fc0 sc0 ls0 ws0">reasonable <span class="_ _3"></span>decomposition <span class="_ _3"></span>results. <span class="_ _3"></span>If <span class="_ _3"></span>the <span class="_ _5"></span>added <span class="_ _3"></span>noise <span class="_ _3"></span>is <span class="_ _5"></span>too </div><div class="t m0 xb h2 yf9 ff1 fs0 fc0 sc0 ls0 ws0">weak to bring the <span class="_ _3"></span>changes of<span class="_ _3"></span> extrema of the or<span class="_ _3"></span>iginal signal, </div><div class="t m0 xb h2 yfa ff1 fs0 fc0 sc0 ls0 ws0">the <span class="_ _5"></span>EEMD <span class="_ _5"></span>method <span class="_ _5"></span>will <span class="_ _3"></span>deg<span class="_ _3"></span>enerate <span class="_ _5"></span>into <span class="_ _5"></span>the <span class="_ _5"></span>EMD <span class="_ _9"></span>method. </div><div class="t m0 xb h2 yfb ff1 fs0 fc0 sc0 ls0 ws0">Conversely, <span class="_ _c"></span>if <span class="_ _c"></span>the <span class="_ _c"></span>added <span class="_ _c"></span>noise <span class="_ _c"></span>is <span class="_ _c"></span>too <span class="_ _c"></span>strong <span class="_ _c"></span>to <span class="_ _c"></span>reveal <span class="_ _c"></span>the </div><div class="t m0 xb h2 yfc ff1 fs0 fc0 sc0 ls0 ws0">original signal, <span class="_ _3"></span>the EEMD <span class="_ _3"></span>method will derive <span class="_ _3"></span>meaningless </div><div class="t m0 xb h2 yfd ff1 fs0 fc0 sc0 ls0 ws0">results which are <span class="_ _3"></span>mainly controlled by <span class="_ _3"></span>the added <span class="_ _3"></span>noise <span class="_ _3"></span>and </div><div class="t m0 xb h2 yfe ff1 fs0 fc0 sc0 ls0 ws0">scarcely <span class="_ _c"></span>associated <span class="_ _c"></span>with <span class="_ _11"></span>the <span class="_ _c"></span>original <span class="_ _11"></span>signal <span class="_ _11"></span>[<span class="ls2">11</span>], <span class="_ _c"></span>wh<span class="_ _3"></span>ether </div><div class="t m0 xb h2 yff ff1 fs0 fc0 sc0 ls0 ws0">or not the <span class="_ _3"></span>ensemble number is large enough.<span class="_ _3"></span> The referen<span class="_ _3"></span>ce </div><div class="t m0 xb h2 y100 ff1 fs0 fc0 sc0 ls0 ws0">[<span class="ls2">11</span>] <span class="_ _4"> </span>demonstrated <span class="_"> </span>that <span class="_ _4"> </span>the <span class="_"> </span>decom<span class="_ _2"></span>po<span class="_ _3"></span>sition <span class="_ _4"> </span>resu<span class="_ _3"></span>lts <span class="_ _4"> </span>of <span class="_"> </span>the </div><div class="t m0 xb h2 y101 ff1 fs0 fc0 sc0 ls0 ws0">EEMD <span class="_ _9"></span>method <span class="_ _9"></span>var<span class="_ _3"></span>ied <span class="_ _9"></span>with <span class="_ _c"></span>the <span class="_ _c"></span>different <span class="_ _c"></span>amplitude <span class="_ _9"></span>of <span class="_ _c"></span>the </div><div class="c xc y61 w3 hc"><div class="t m0 x0 hd y62 ff7 fs4 fc1 sc0 ls0 ws0">IJCSI International Journal of Computer Science Issues, Vol. 9, Issue 6, No 2, November 2012<span class="fc2 sc0"> </span></div><div class="t m0 x0 hd y63 ff7 fs4 fc1 sc0 ls0 ws0">ISSN (Online): 1694-0814 </div><div class="t m0 x0 hd y64 ff7 fs4 fc1 sc0 ls0 ws0">www.IJCSI.org</div></div><div class="c xd y65 w4 he"><div class="t m0 x0 hd y64 ff7 fs4 fc1 sc0 ls0 ws0">195</div></div><div class="c xe y66 w5 he"><div class="t m0 x0 hd y64 ff7 fs4 fc1 sc0 ls0 ws0">Copyright (c) 2012 International Journal of Computer Science Issues. All Rights Reserved.</div></div><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a></div><div class="pi" data-data='{"ctm":[1.568627,0.000000,0.000000,1.568627,0.000000,0.000000]}'></div></div>