rcada_eemd--快速eemd算法_算法_其他

<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/627edac49b6e2b6d55553e29/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/627edac49b6e2b6d55553e29/bg1.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">Physica A 400 (2014) 159–167</div><div class="t m0 x2 h3 y2 ff2 fs1 fc1 sc0 ls0 ws0">Contents lists available at ScienceDirect</div><div class="t m0 x3 h4 y3 ff1 fs2 fc1 sc0 ls0 ws0">Physica A</div><div class="t m0 x4 h3 y4 ff2 fs1 fc1 sc0 ls0 ws0">journal homepage: www.elsevier.com/locate/physa</div><div class="t m0 x5 h5 y5 ff1 fs3 fc1 sc0 ls0 ws0">On the computational complexity of the empirical mode</div><div class="t m0 x5 h5 y6 ff1 fs3 fc1 sc0 ls0 ws0">decomposition algorithm</div><div class="t m0 x5 h6 y7 ff1 fs4 fc1 sc0 ls0 ws0">Yung-Hung Wang</div><div class="t m0 x6 h7 y8 ff1 fs5 fc0 sc0 ls0 ws0">a</div><div class="t m0 x7 h6 y7 ff1 fs4 fc1 sc0 ls0 ws0">, Chien-Hung Yeh</div><div class="t m0 x8 h7 y8 ff1 fs5 fc0 sc0 ls0 ws0">a,b</div><div class="t m0 x9 h6 y7 ff1 fs4 fc1 sc0 ls0 ws0">, Hsu-Wen Vincent Young</div><div class="t m0 xa h7 y8 ff1 fs5 fc0 sc0 ls0 ws0">a</div><div class="t m0 xb h6 y7 ff1 fs4 fc1 sc0 ls0 ws0">, Kun Hu</div><div class="t m0 xc h7 y8 ff1 fs5 fc0 sc0 ls0 ws0">c</div><div class="t m0 xd h6 y7 ff1 fs4 fc1 sc0 ls0 ws0">,</div><div class="t m0 x5 h6 y9 ff1 fs4 fc1 sc0 ls0 ws0">Men-Tzung Lo</div><div class="t m0 xe h7 ya ff1 fs5 fc0 sc0 ls0 ws0">a,∗</div><div class="t m0 x5 h8 yb ff1 fs7 fc1 sc0 ls0 ws0">a</div><div class="t m0 xf h9 yc ff4 fs0 fc1 sc0 ls0 ws0">Research Center for Adaptive Data Analysis, National Central University, Chungli, Taiwan, ROC</div><div class="t m0 x5 h8 yd ff1 fs7 fc1 sc0 ls0 ws0">b</div><div class="t m0 xf h9 ye ff4 fs0 fc1 sc0 ls0 ws0">Department of Electrical Engineering, National Central University, Chungli, Taiwan, ROC</div><div class="t m0 x5 h8 yf ff1 fs7 fc1 sc0 ls0 ws0">c</div><div class="t m0 xf h9 y10 ff4 fs0 fc1 sc0 ls0 ws0">Medical Biodynamics Program, Division of Sleep Medicine, Brigham and Women’s Hospital, Harvard Medical School,</div><div class="t m0 x5 h9 y11 ff4 fs0 fc1 sc0 ls0 ws0">Boston, MA 02115, United States</div><div class="t m0 x5 ha y12 ff5 fs8 fc1 sc0 ls0 ws0">h i g h l i g h t s</div><div class="t m0 x5 hb y13 ff3 fs9 fc1 sc0 ls0 ws0">• The order of the computational complexity of the EMD is equivalent to FFT.</div><div class="t m0 x5 hb y14 ff3 fs9 fc1 sc0 ls0 ws0">• Optimized program is proposed to speed up the computation of EMD up to 1000 times.</div><div class="t m0 x5 hb y15 ff3 fs9 fc1 sc0 ls0 ws0">• Fast HHT with optimized EMD/EEMD algorithm can operate in real-time.</div><div class="t m0 x5 hc y16 ff5 fsb fc1 sc0 ls0 ws0">a r t i c l e i n f o</div><div class="t m0 x5 h9 y17 ff4 fs0 fc1 sc0 ls0 ws0">Article history:</div><div class="t m0 x5 h2 y18 ff1 fs0 fc1 sc0 ls0 ws0">Received 30 October 2013</div><div class="t m0 x5 h2 y19 ff1 fs0 fc1 sc0 ls0 ws0">Received in revised form 30 December 2013</div><div class="t m0 x5 h2 y1a ff1 fs0 fc1 sc0 ls0 ws0">Available online 21 January 2014</div><div class="t m0 x5 h9 y1b ff4 fs0 fc1 sc0 ls0 ws0">Keywords:</div><div class="t m0 x5 h2 y1c ff1 fs0 fc1 sc0 ls0 ws0">EMD</div><div class="t m0 x5 h2 y1d ff1 fs0 fc1 sc0 ls0 ws0">EEMD</div><div class="t m0 x5 h2 y1e ff1 fs0 fc1 sc0 ls0 ws0">Time</div><div class="t m0 x5 h2 y1f ff1 fs0 fc1 sc0 ls0 ws0">Space</div><div class="t m0 x5 h2 y20 ff1 fs0 fc1 sc0 ls0 ws0">Complexity</div><div class="t m0 x10 ha y21 ff5 fs8 fc1 sc0 ls0 ws0">a b s t r a c t</div><div class="t m0 x2 hb y22 ff1 fsa fc1 sc0 ls0 ws0">It has been claimed that the empirical mode decomposition (EMD) and its improved version</div><div class="t m0 x2 hb y23 ff1 fsa fc1 sc0 ls0 ws0">the ensemble EMD (EEMD) are computation intensive. In this study we will prove that</div><div class="t m0 x2 hb y24 ff1 fsa fc1 sc0 ls0 ws0">the time complexity of the EMD/EEMD, which has never been analyzed before, is actually</div><div class="t m0 x2 hb y25 ff1 fsa fc1 sc0 ls0 ws0">equivalent to that of the Fourier Transform. Numerical examples are presented to verify</div><div class="t m0 x2 hb y26 ff1 fsa fc1 sc0 ls0 ws0">that EMD/EEMD is, in fact, a computationally efficient method.</div><div class="t m0 x11 hb y27 ff6 fsc fc1 sc0 ls0 ws0">© 2014 Elsevier B.V. All rights reserved.</div><div class="t m0 x5 hd y28 ff7 fsd fc1 sc0 ls0 ws0">1. Introduction</div><div class="t m0 x12 he y29 ff1 fsd fc1 sc0 ls0 ws0">EMD [1] is a nonlinear and nonstationary time domain decomposition method. It is an adaptive, data-driven algorithm</div><div class="t m0 x5 he y2a ff1 fsd fc1 sc0 ls0 ws0">that decomposes a time series into multiple empirical modes, known as intrinsic mode functions (IMFs). Each IMF</div><div class="t m0 x5 he y2b ff1 fsd fc1 sc0 ls0 ws0">represents a narrow band frequency–amplitude modulation that is often related to a specific physical process. For signals</div><div class="t m0 x5 he y2c ff1 fsd fc1 sc0 ls0 ws0">with intermittent oscillations, one intrinsic mode can comprise oscillations with a variety of wavelengths at different</div><div class="t m0 x5 he y2d ff1 fsd fc1 sc0 ls0 ws0">temporal locations. Collectively, the simultaneous exhibition of these disparate oscillations is known as the mode mixing</div><div class="t m0 x5 he y2e ff1 fsd fc1 sc0 ls0 ws0">phenomenon, which can complicate analyses and obscure physical meanings. To overcome this problem, the EEMD</div><div class="t m0 x5 he y2f ff1 fsd fc1 sc0 ls0 ws0">algorithm [2] and the noise-assisted MEMD [3] have been proposed. In particular, the EEMD has drawn a lot of attention</div><div class="t m0 x12 h2 y30 ff4 fs0 fc1 sc0 ls0 ws0">Abbreviations: EMD, empirical mode decomposition; EEMD, ensemble empirical mode decomposition; IMFs, intrinsic mode functions; MEMD,</div><div class="t m0 x5 h2 y31 ff1 fs0 fc1 sc0 ls0 ws0">multivariate empirical mode decomposition; ADD, addition; MUL, multiplication; DIV, division; COMP, comparison; BFV, blood flow velocity signal.</div><div class="t m0 xf hf y32 ff3 fse fc1 sc0 ls0 ws0">∗</div><div class="t m0 x12 h2 y33 ff1 fs0 fc1 sc0 ls0 ws0">Correspondence to: Research Center for Adaptive Data Analysis/National Central University, No. 300, Jhongda Rd., Jhongli City, Taoyuan County 32001,</div><div class="t m0 x5 h2 y34 ff1 fs0 fc1 sc0 ls0 ws0">Taiwan, ROC. Tel.: +886 3 426 9734; fax: +886 3 426 9736.</div><div class="t m0 x12 h2 y35 ff4 fs0 fc1 sc0 ls0 ws0">E-mail address: mzlo@ncu.edu.tw (M.-T. Lo).</div><div class="t m0 x5 h2 y36 ff1 fs0 fc1 sc0 ls0 ws0">0378-4371/$ – see front matter © 2014 Elsevier B.V. All rights reserved.</div><div class="t m0 x5 h2 y37 ff1 fs0 fc0 sc0 ls0 ws0">http://dx.doi.org/10.1016/j.physa.2014.01.020</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><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><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><a class="l" rel='nofollow' onclick='return false;'><div class="d m1"></div></a></div><div class="pi" data-data='{"ctm":[1.763889,0.000000,0.000000,1.763889,0.000000,0.000000]}'></div></div> </body> </html>

<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/627edac49b6e2b6d55553e29/bg2.jpg"><div class="t m0 xf h2 y1 ff1 fs0 fc1 sc0 ls0 ws0">160 Y.-H. Wang et al. / Physica A 400 (2014) 159–167</div><div class="t m0 x13 h9 y38 ff7 fs0 fc1 sc0 ls0 ws0">Table 1</div><div class="t m0 x13 h2 y39 ff1 fs0 fc1 sc0 ls0 ws0">EMD.</div><div class="t m0 x14 h10 y3a ff3 fs10 fc1 sc0 ls0 ws0">[c</div><div class="t m0 x15 h11 y3b ff4 fs7 fc1 sc0 ls0 ws0">m</div><div class="t m0 x16 h10 y3a ff3 fs10 fc1 sc0 ls0 ws0">] = Main (NS, n</div><div class="t m0 x17 h11 y3b ff4 fs7 fc1 sc0 ls0 ws0">m</div><div class="t m0 x18 h9 y3a ff8 fs10 fc1 sc0 ls0 ws0">, y</div><div class="t m0 x19 h8 y3b ff1 fs7 fc1 sc0 ls0 ws0">0</div><div class="t m0 x3 h12 y3a ff8 fs10 fc1 sc0 ls0 ws0">)</div><div class="t m0 x1a h10 y3c ff4 fs0 fc1 sc0 ls0 ws0">r(t) = y</div><div class="t m0 x1b h8 y3d ff1 fs7 fc1 sc0 ls0 ws0">0</div><div class="t m0 x1c h9 y3c ff8 fs10 fc1 sc0 ls0 ws0">(t)</div><div class="t m0 x1a h2 y3e ff4 fs0 fc1 sc0 ls0 ws0">for (m = 1 : n</div><div class="t m0 x1d h11 y3f ff4 fs7 fc1 sc0 ls0 ws0">m</div><div class="t m0 x1e h2 y3e ff1 fs0 fc1 sc0 ls0 ws0">) { % extract each IMF</div><div class="t m0 x1f h2 y40 ff4 fs0 fc1 sc0 ls0 ws0">x(t) = r(t) % x(t) is proto-IMF</div><div class="t m0 x1f h2 y41 ff4 fs0 fc1 sc0 ls0 ws0">for(s = 1 : NS) { % sifting iteration:</div><div class="t m0 x20 h10 y42 ff3 fs10 fc1 sc0 ls0 ws0">[t</div><div class="t m0 x21 h8 y43 ff1 fs7 fc1 sc0 ls0 ws0">max</div><div class="t m0 x22 h9 y42 ff8 fs10 fc1 sc0 ls0 ws0">, x</div><div class="t m0 x23 h8 y43 ff1 fs7 fc1 sc0 ls0 ws0">max</div><div class="t m0 x24 h9 y42 ff8 fs10 fc1 sc0 ls0 ws0">, t</div><div class="t m0 x25 h8 y43 ff1 fs7 fc1 sc0 ls0 ws0">min</div><div class="t m0 x26 h9 y42 ff8 fs10 fc1 sc0 ls0 ws0">, x</div><div class="t m0 x27 h8 y43 ff1 fs7 fc1 sc0 ls0 ws0">min</div><div class="t m0 x28 h2 y42 ff3 fs10 fc1 sc0 ls0 ws0">] = identify_extrema(x);</div><div class="t m0 x20 h10 y44 ff4 fs0 fc1 sc0 ls0 ws0">U(t) = spline(t</div><div class="t m0 x26 h8 y45 ff1 fs7 fc1 sc0 ls0 ws0">max</div><div class="t m0 x29 h9 y44 ff8 fs10 fc1 sc0 ls0 ws0">, x</div><div class="t m0 x28 h8 y45 ff1 fs7 fc1 sc0 ls0 ws0">max</div><div class="t m0 x2a h2 y44 ff8 fs10 fc1 sc0 ls0 ws0">); % Upper envelope</div><div class="t m0 x20 h10 y46 ff4 fs0 fc1 sc0 ls0 ws0">L(t) = spline(t</div><div class="t m0 x2b h8 y47 ff1 fs7 fc1 sc0 ls0 ws0">min</div><div class="t m0 x27 h9 y46 ff8 fs10 fc1 sc0 ls0 ws0">, x</div><div class="t m0 x2c h8 y47 ff1 fs7 fc1 sc0 ls0 ws0">min</div><div class="t m0 x2d h2 y46 ff8 fs10 fc1 sc0 ls0 ws0">); % Lower envelope</div><div class="t m0 x20 h2 y48 ff4 fs0 fc1 sc0 ls0 ws0">x(t) = x(t) − (U (t) + L(t))/2; % detail extraction (1)</div><div class="t m0 x1f h2 y49 ff1 fs0 fc1 sc0 ls0 ws0">}</div><div class="t m0 x1f h9 y4a ff4 fs0 fc1 sc0 ls0 ws0">c</div><div class="t m0 x2e h11 y4b ff4 fs7 fc1 sc0 ls0 ws0">m</div><div class="t m0 x2 h2 y4a ff8 fs10 fc1 sc0 ls0 ws0">(t) = x(t) (2)</div><div class="t m0 x1f h2 y4c ff4 fs0 fc1 sc0 ls0 ws0">r(t) = r(t) − x(t) (3)</div><div class="t m0 x1a h2 y4d ff1 fs0 fc1 sc0 ls0 ws0">}</div><div class="t m0 xf he y4e ff1 fsd fc1 sc0 ls0 ws0">during the past few years. During the last decade, the EMD/EEMD was shown to be more effective than the traditional Fourier</div><div class="t m0 xf he y4f ff1 fsd fc1 sc0 ls0 ws0">method in many problems from various fields such as physics [4], biomedicine [5–8], mechanical health diagnosis [9], image</div><div class="t m0 xf he y50 ff1 fsd fc1 sc0 ls0 ws0">analysis [10] and others.</div><div class="t m0 x2f he y51 ff1 fsd fc1 sc0 ls0 ws0">It seems a common belief that a major drawback of the EMD/EEMD is that they require a long computation time. For</div><div class="t m0 xf he y52 ff1 fsd fc1 sc0 ls0 ws0">example, in Ref. [11], the authors report that it takes more than an hour to perform EMD on a signal with 30,000 data points</div><div class="t m0 xf he y53 ff1 fsd fc1 sc0 ls0 ws0">using a modern personal computer. The enormous amount of computation time hampers the applications of EMD/EEMD in</div><div class="t m0 xf he y54 ff1 fsd fc1 sc0 ls0 ws0">analyzing long data and pseudo real-time hardware applications. For this reason, parallel computing such as Cuda [12], Open</div><div class="t m0 xf he y55 ff1 fsd fc1 sc0 ls0 ws0">MP [13] and hardware implementations like VLSI [14] and FPGA [15] have been recently adopted to improve the execution</div><div class="t m0 xf he y56 ff1 fsd fc1 sc0 ls0 ws0">time.</div><div class="t m0 x2f he y57 ff1 fsd fc1 sc0 ls0 ws0">In this study, we proved that the EMD/EEMD is actually not computation intensive as has been claimed in the past. We</div><div class="t m0 xf he y58 ff1 fsd fc1 sc0 ls0 ws0">first analyze the time and storage complexity for EMD/EEMD and their variants in different applications under an appropriate</div><div class="t m0 xf he y59 ff1 fsd fc1 sc0 ls0 ws0">implementation, which is based on single core, sequential implementation without any approximations. Finally, numerical</div><div class="t m0 xf he y5a ff1 fsd fc1 sc0 ls0 ws0">examples are presented to verify the performance of EMD/EEMD.</div><div class="t m0 x2f he y5b ff1 fsd fc1 sc0 ls0 ws0">The remainder of this article is divided into the following sections: in Section 2, we briefly review the EMD method. In</div><div class="t m0 xf he y5c ff1 fsd fc1 sc0 ls0 ws0">Section 3 we present an implementation of the EMD/EEMD method and then analyze their time and storage complexities.</div><div class="t m0 xf he y5d ff1 fsd fc1 sc0 ls0 ws0">Section 4 illustrates the efficacy of the proposed method with experimental results and we conclude our study in Section 5.</div><div class="t m0 xf hd y5e ff7 fsd fc1 sc0 ls0 ws0">2. The EMD algorithm</div><div class="t m0 x2f he y5f ff1 fsd fc1 sc0 ls0 ws0">Given an input signal y</div><div class="t m0 x30 h13 y60 ff1 fs11 fc1 sc0 ls0 ws0">0</div><div class="t m0 x31 hd y5f ff8 fs12 fc1 sc0 ls0 ws0">(t), y</div><div class="t m0 x14 h13 y60 ff1 fs11 fc1 sc0 ls0 ws0">0</div><div class="t m0 x4 he y5f ff3 fs12 fc1 sc0 ls0 ws0">∈ R, t ∈ Z and t = [1 : n], the EMD decomposes a signal y</div><div class="t m0 x32 h13 y60 ff1 fs11 fc1 sc0 ls0 ws0">0</div><div class="t m0 x33 he y5f ff8 fs12 fc1 sc0 ls0 ws0">(t) into a series of intrinsic</div><div class="t m0 xf he y61 ff1 fsd fc1 sc0 ls0 ws0">mode functions (IMFs) which are extracted via an iterative sifting process. First, the local maxima and minima of the signal</div><div class="t m0 xf he y62 ff1 fsd fc1 sc0 ls0 ws0">are connected, respectively, by cubic splines to form the upper/lower envelopes. The average of the two envelopes is then</div><div class="t m0 xf he y63 ff1 fsd fc1 sc0 ls0 ws0">subtracted from the original signal. This sifting process is then repeated several times (define the number of siftings as NS,</div><div class="t m0 xf he y64 ff1 fsd fc1 sc0 ls0 ws0">usually 10) to obtain the first IMF that oscillates at relatively higher frequencies than does the residual signal that is obtained</div><div class="t m0 xf he y65 ff1 fsd fc1 sc0 ls0 ws0">by subtracting the IMF from the original signal. Then, the residual is deemed as the input for a new round of iterations. In</div><div class="t m0 xf he y66 ff1 fsd fc1 sc0 ls0 ws0">turn, subsequent IMFs with lower oscillation frequencies are derived using the same process and the newly obtained residue.</div><div class="t m0 xf he y67 ff1 fsd fc1 sc0 ls0 ws0">The result of the EMD is a decomposition of the signal y</div><div class="t m0 x34 h13 y68 ff1 fs11 fc1 sc0 ls0 ws0">0</div><div class="t m0 x35 he y67 ff8 fs12 fc1 sc0 ls0 ws0">(t) into the sum of the IMFs and a residue r(t). That is,</div><div class="t m0 x36 hd y69 ff4 fsd fc1 sc0 ls0 ws0">y</div><div class="t m0 x37 h13 y6a ff1 fs11 fc1 sc0 ls0 ws0">0</div><div class="t m0 x38 h14 y69 ff8 fs12 fc1 sc0 ls0 ws0">(t) =</div><div class="t m0 x39 h15 y6b ff4 fs11 fc1 sc0 ls0 ws0">n</div><div class="t m0 x3a h16 y6c ff4 fs13 fc1 sc0 ls0 ws0">m</div><div class="t m0 x3b h17 y6d ff9 fs12 fc1 sc0 ls0 ws0"></div><div class="t m0 x3b h13 y6e ff4 fs11 fc1 sc0 ls0 ws0">m=1</div><div class="t m0 x3c hd y69 ff4 fsd fc1 sc0 ls0 ws0">c</div><div class="t m0 x3d h15 y6a ff4 fs11 fc1 sc0 ls0 ws0">m</div><div class="t m0 x3e h14 y69 ff8 fs12 fc1 sc0 ls0 ws0">(t) + r(t)</div><div class="t m0 xf he y6f ff1 fsd fc1 sc0 ls0 ws0">where n</div><div class="t m0 x3f h15 y70 ff4 fs11 fc1 sc0 ls0 ws0">m</div><div class="t m0 x40 he y6f ff1 fsd fc1 sc0 ls0 ws0">is the number of IMFs. The EMD algorithm is listed in Table 1.</div><div class="t m0 xf hd y71 ff7 fsd fc1 sc0 ls0 ws0">3. Time and space complexity</div><div class="t m0 xf hd y72 ff4 fsd fc1 sc0 ls0 ws0">3.1. EMD</div><div class="t m0 x2f he y73 ff1 fsd fc1 sc0 ls0 ws0">In this section, we propose an implementation of the EMD and analyze its time and space complexity. The arithmetic</div><div class="t m0 xf he y74 ff1 fsd fc1 sc0 ls0 ws0">operations involved include addition (ADD), multiplication (MUL), division (DIV ) and comparison (COMP). Fixed-point</div><div class="t m0 xf he y75 ff1 fsd fc1 sc0 ls0 ws0">arithmetic operations are negligible compared with the floating-point operations. To facilitate the analysis, the EMD</div><div class="t m0 xf he y76 ff1 fsd fc1 sc0 ls0 ws0">procedure is divided into the main, extrema identification and cubic spline procedures. Capital T and M denote the time</div><div class="t m0 xf he y77 ff1 fsd fc1 sc0 ls0 ws0">and space (storage) complexity, respectively. The EMD algorithm is listed in Table 1 and is explained in detail below.</div><div class="t m0 xf hd y78 ff4 fsd fc1 sc0 ls0 ws0">3.1.1. Main procedure</div><div class="t m0 x2f he y79 ff1 fsd fc1 sc0 ls0 ws0">Each of the input, residue r(t), proto-IMF x(t) and the upper/lower envelope U(t)/L(t) is of length n and requires</div><div class="t m0 xf he y37 ff4 fsd fc1 sc0 ls0 ws0">n floating point storage. The output signal consists of n</div><div class="t m0 x41 h15 y7a ff4 fs11 fc1 sc0 ls0 ws0">m</div><div class="t m0 x2a he y37 ff1 fsd fc1 sc0 ls0 ws0">IMFs each with length of n, hence it requires n</div><div class="t m0 x42 h15 y7a ff4 fs11 fc1 sc0 ls0 ws0">m</div><div class="t m0 x43 he y37 ff3 fs12 fc1 sc0 ls0 ws0">· n floating-</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><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><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":[1.763889,0.000000,0.000000,1.763889,0.000000,0.000000]}'></div></div>