IJCSI Interntionl Journl of Computer Science Issues, Vol. 9, Issue, No, Novemer ISSN (Online): 9- www.ijcsi.org 9 Improved Ensemle Empiricl Mode Decomposition nd its Applictions to Gerox Fult Signl Processing Jinshn in School of Mechtronics nd Vehicle Engineering, Weifng University Weifng,, Chin Astrct Ensemle empiricl mode decomposition (EEMD) is noisessisted method nd lso significnt improvement on empiricl mode decomposition (EMD). However, the EEMD method lcks guide to choosing the pproprite mplitude of dded noise nd its computtion efficiency is firly low. To llevite the prolems of the EEMD method, the improved complementry EEMD method (ICEEMD) ws proposed. Furthermore, the ICEEMD method ws used to nlyze relistic gerox fulty signls. The results indicte tht the ICEEMD method hs some dvntges over the EEMD method in lleviting the mode mixing nd splitting s well s reducing the time cost nd lso outperforms the CEEMD method in lleviting the mode mixing nd splitting. The pper lso indictes tht the ICEEMD method seems to e n effective nd efficient method for processing gerox fult signls. Keywords: Complementry Ensemle Empiricl Mode Decomposition(CEEMD), Improved Complementry Ensemle Empiricl Mode Decomposition(ICEEMD), Gerox, Signl Processing.. Introduction It is chllenging tsk to develop signl processing techniques for non-sttionry nd noisy signls, which hs ttrcted considerle ttentions recently []. Mny methods, such s short time frequency trnsform [] nd wvelet trnsform [3], hve een proposed for solving the prolem nd proved useful in some pplictions. However, ecuse these methods usully need priori knowledge out the reserched signls, they nturlly lck the selfdption for the reserched signls. The Wigner-Ville distriution hs high time-frequency resolution, ut its cross terms is unerle []. Empiricl mode decomposition (EMD) is self-dptive method nd suitle to nlyzing the non-sttionry nd nonliner signls [], which hs een successfully pplied to vrious fields []. Nevertheless, when the EMD lgorithm is used to del with signl with intermittency, the mode mixing often emerges s n nnoying prolem [5-7]. To overcome the mode mixing prolem, ensemle empiricl mode decomposition (EEMD) is presented in plce of EMD []. The EEMD method dds some white noise with limited mplitude to the reserched signls, sufficiently tking dvntge of the sttisticl chrcteristics of white noise whose energy density is uniformly distriuted throughout the frequency domin, then projects the signl components onto the proper frequency nds nd, finlly, the dded white noise cn een countercted y ensemle men of enough corresponding components []. Therefore, the EEMD method is considered s significnt improvement on the EMD method nd recommended s sustitute for the EMD method []. Indeed, the EEMD method hs shown its superiority over the EMD method in some pplictions [9]. However, the EEMD method lcks guide to how to choose the pproprite mplitude of the dded noise nd its computtion efficiency is firly low. As result, the inpproprite mplitude of the dded white noise for the EEMD method is going to cuse the mode mixing nd splitting tht often exists in the EMD method [, ]. Although the reference [] suggested tht the mplitude of the dded white noise should e out. times of the stndrd devition of the investigted signl, unfortuntely, with the suggested vlue, the decomposition results from the EEMD method often devite from the relistic contents of the signls []. In ddition, to further remove the residul of the dded white noise nd reduce wste of time for the EEMD method, the complementry ensemle empiricl mode decomposition (CEEMD) hs een ddressed to replce the EEMD method s stndrd version of the EMD method []. Notwithstnding, the CEEMD method only prtly enhnces the computtion efficiency of the EEMD method, nd the ove first prolem regrding the EEMD method still remins untouched. Additionlly, if the reserched signl is noisy signl in itself, its intrinsic noise will inevitly interct with the noise dded through the EEMD method, which my further complicte the ove first prolem regrding the EEMD method. In prticulr, when the reserched signls ecome very noisy, the ove first prolem regrding the EEMD method leves gp. This pper explores the ove two prolems concerning the EEMD method. Then, the improved CEEMD (ICEEMD) method ws proposed. Applictions to nlysis Copyright (c) Interntionl Journl of Computer Science Issues. All Rights Reserved.
IJCSI Interntionl Journl of Computer Science Issues, Vol. 9, Issue, No, Novemer ISSN (Online): 9- www.ijcsi.org 95 of defective gerox signls proved the superiority of the ICEEMD method over the EEMD method.. The EMD nd its Severl Vritions. The EMD method The EMD method cn self-dptively decompose ny non-sttionry nd nonliner signl into set of intrinsic mode functions (IMFs) from high frequency to low frequency, which my e written s N x( t) ci( t) rn( t) () i where c i (t) indictes the ith IMF nd r N (t) represents the r esidul of the signl x(t). An IMF is function which mus t stisfy the following two conditions: () the numer of e xtrem nd the numer of zero crossings either equl to e ch other or differ t most y one, nd () t ny point, th e locl verge of the upper envelope nd the lower envel ope is zero []. The residul r N (t) usully is monotonic f unction or constnt.. The Ensemle EMD An nnoying prolem ssocited with the EMD method is the mode mixing due to intermittency, defined s either single IMF consisting of widely disprte scles, or signl residing in different IMF components. To llevite the imperfection of the EMD method, the ensemle EMD (EEMD), noise-ssisted method, is proposed. The EEMD method cn e stted s follows: x ( t) x( t) w ( t), m,,, N () m m x ( t) c ( t) r ( t), m,,, N (3) m m, i m, i x ( t) c ( t) r ( t), m,,, N () m m, i m, i where x(t) is the originl signl, w m (t) is the mth dded white noise, x m (t) is the noisy signl of the mth tril, c m,i (t) is the ith IMF of the mth tril, is the numer of IMFs from the EMD method, nd N is the ensemle numer of the EEMD method. The EEMD method dds white noise with the finite mplitude to the signl, sufficiently tking dvntge of the uniform sttistic chrcteristics of white noise in the frequency domin, projects the different frequency signl components onto the corresponding frequency nks nd, s result, effectively overcomes the mode mixing due to the existence of intermittency []. Nonetheless, to totlly cler the residul of the dded white noise from the IMFs of the EEMD method, lrge ensemle numer is usully demnded, which will cuse tremendous wste of time..3 The Complementry EEMD method To etter eliminte the residul of dded white noise persisting in the IMFs of the EEMD method nd rise the computtion efficiency of the EEMD method, the complementry ensemle EMD (CEEMD) [], novel noise-enhnced method, is presented. The CEEMD method dds white noise in pirs with one positive nd nother negtive to the originl signl nd then produces two sets of ensemle IMFs. Hence, two different comintions of the originl signl nd the dded white noise cn e otined, i.e. x () () m t xt, m,,, N x () wm () t (5) m t where x(t) is the originl signl, w m (t) is the mth dded white noise, x + m (t) is the sum of the originl x(t) nd the mth dded white noise w m (t), nd x - m (t) is the difference of the originl x(t) nd the mth dded white noise w m (t). In light of (3) nd (), the originl signl x(t) cn e expressed s N x( t) c m, i ( t) c m, i ( t) N i m () N r m, ( t) r m, ( t) N m where c + m,i(t) is the ith IMF of x + m (t) nd c - m,i(t) is the ith IMF of x - m (t). Although the CEEMD method ppers to e le to completely remove the residul of the dded white noise persisting in the IMFs of the EEMD method, it still remins unsolved how to choose the pproprite mplitude of the dded white noise for the EEMD method. 3. The Improved CEEMD Method 3. The Choice of Amplitude of the Added Noise The mplitude of the dded white noise is key prmeter of the EEMD method, which will exert decisive impct on whether or not the EEMD method cn yield the resonle decomposition results. If the dded noise is too wek to ring the chnges of extrem of the originl signl, the EEMD method will degenerte into the EMD method. Conversely, if the dded noise is too strong to revel the originl signl, the EEMD method will derive meningless results which re minly controlled y the dded noise nd scrcely ssocited with the originl signl [], whether or not the ensemle numer is lrge enough. The reference [] demonstrted tht the decomposition results of the EEMD method vried with the different mplitude of the Copyright (c) Interntionl Journl of Computer Science Issues. All Rights Reserved.
IJCSI Interntionl Journl of Computer Science Issues, Vol. 9, Issue, No, Novemer ISSN (Online): 9- www.ijcsi.org 9 dded noise nd considered it pproprite for the EEMD method to set SNR in the rnge of 5- db. In fct, for the simple simulted exmple y which the conclusions were drwn in [], when the mplitude of the dded noise is set s., which is considered n optiml choice y [], the SNR etween the originl signl nd the dded noise is pproximtely 37 db which is outside the rnge of 5- db. Accordingly, the conclusions given y [], relting to the choice of the mplitude of the dded white noise of the EEMD method, is not entirely dependle. Then, the prolem is further investigted in this pper using two simulted signls. Thus, the Person s correltion coefficient (PCC) is used s prmeter to mesure the performnce of the EEMD method with different mplitude of the dded white noise. Here, the ensemle numer of the EEMD method is set s. First, simple noiseless simulted signl ws used to exmine the choice of the mplitude of the dded white noise for the EEMD method. The signl is comintion of low-frequency sinusoid component x (t) nd highfrequency dmped trnsient component x (t), shown in Fig., nd its formul cn refer to []. The reltionship etween the PCCs nd the mplitude of the dded white noise is illustrted in Fig.. As shown in Fig., when the mplitude of the dded white noise is.3, the two PCCs lmost simultneously rech their mximum vlues. Tle exhiits the verge powers of two components of the signl nd the squre roots of the verge powers. As seen in Tle, the vlue of.3 just equls to the squre root of the verge power of the wek trnsient component x (t). As result, the squre root of the verge power of the wek trnsient component pprently pproximtes the optiml mplitude of the dded white noise. Amplitude -.5 -.5 -.5..5..5 Fig. A simple noiseless simulted signl nd two components. signl x(t) x(t) Person's correltion coefficient (PCC)......3 PCC corresponding to x PCC corresponding to x..... The mplitude of the dded white noise Fig. The reltionship etween the Person s correltion coefficient (PCC) nd the mplitude of the dded white noise for the simple signl. Tle : The verge powers of two components of the simple noiseless simulted signl nd the squre roots of the verge powers Prmeter The two components of the simple signl x () t x () t The men power. -5.5 The squre root of the men power.3.77 Susequently, complex noiseless simulted signl ws utilized to further verify the conclusion. The signl consisting of four components imittes relistic virtion signls of rolling ering, shown in Fig. 3, nd its formul cn refer to []. The reltionship etween the PCCs nd the mplitude of the dded white noise is illustrted in Fig.. As shown in Fig., when the mplitude of the dded white noise lies in the rnge of.5-.3, the four PCCs lmost simultneously rech their mximum vlues. Tle depicts the verge powers of four components of the signl nd the squre roots of the verge powers. As shown in Tle, the vlue of.5 is just equl to the squre root of the verge power of the wek sinusoid component x (t) nd the vlue of.3 is just equl to the squre root of the verge power of the wek trnsient component x (t). More generlly, Fig. indictes tht n optiml internl of the mplitude of the dded white noise for the EEMD method my lie etween the squre root of the verge power of the wek sinusoid component nd the squre root of the verge power of the wek trnsient component, which is in ccordnce with the conclusions drwn from the ove simple exmple. Copyright (c) Interntionl Journl of Computer Science Issues. All Rights Reserved.
IJCSI Interntionl Journl of Computer Science Issues, Vol. 9, Issue, No, Novemer ISSN (Online): 9- www.ijcsi.org 97 Amplitude. -. signl. -. x(t).5 -.5 x(t).5 -.5 x3(t). -. x(t)...3..5 Fig. 3 A complexly-simulted signl nd its four components. should e out. times of the stndrd devition of the investigted signl, the conclusion is pretty rough nd inpproprite in mny cses []. To llevite the prolem existing in the EEMD method for nlyzing noisy signl, the improved CEEMD (ICEEMD) is proposed. First, the noisy signl is roughly decomposed using the CEEMD lgorithm. Then, oth the wek trnsient component nd the wek sinusoid component re otined. As result, the optiml intervl of the mplitude of the dded white noise cn e determined. In the end, with the mplitude of the dded white noise lying in the optiml intervl otined in the previous step, the CEEMD method is gin performed. Person's correltion coefficient (PCC).... PCC corresponding to x PCC corresponding to x..3 PCC corresponding to x3 PCC corresponding to x.5..... The mplitufe of the dded white noise Fig. The reltionship etween the Person s correltion coefficient (PCC) nd the mplitude of the dded white noise in complex noiseless simulted signl. Tle : The verge powers of four components of the complex noiseless simulted signl nd the squre roots of the verge powers Prmeter The four components of the complex signl x () t x () t x () t x () t 3 The men power.93.9377 5.5.7 The squre root of the men power 3. The improved CEEMD.3.39.33.5 On the premise tht the noise is neglected, the ove section gives n optiml intervl of the mplitude of the dded white noise for the EEMD method. Actully, relistic signl is usully contminted y strong or wek noise. When the EEMD method is pplied to noisy signl, the intrinsic noise will inevitly interct with the dded noise, which cn mke gret impct on how much extrinsic noise should e dded. Although the reference [] hs suggested tht the mplitude of the dded white noise. Experiment verifiction To further ssess its performnce, the ICEEMD method ws exploited to exmine the gerox virtion dt provided y Kyvn J. Rfiee []. The gerox ws running y the driving ger meshing with the driven one. The rottion speed of the input shft is.5hz, the rottion speed of the output shft is 9.Hz nd the meshing frequency is 97.5Hz []. The signls were mesured from the driving ger. The norml gerox signl nd the roken-tooth gerox signl re shown in Fig. 5. Susequently, the EEMD method, the CEEMD method nd the ICEEMD method were utilized to explore the two signls, nd the corresponding HHT spectr re shown in Fig., Fig. 7 nd Fig., respectively. As shown in Fig. (), Fig. 7() nd Fig. (), there re no ovious periodic chrcteristics in the HHT spectr of the norml gerox signl; conversely, s shown in Fig. (), Fig. 7() nd Fig. (), there re ovious periodic chrcteristics in the HHT spectr of the roken-tooth gerox signl. However, s seen in Fig. () nd Fig. 7(), none of other explicit informtion cn e found, in ddition to the frequency nds scttered nerly throughout the frequency rnge only t the instnt when the shocks hppen, which implies tht there occurs the mode mixing or splitting in the two methods ecuse of the inpproprite mplitude of the dded noise. Insted, s seen in Fig. (), in ddition to the frequency nds scttered nerly throughout the frequency rnge only t the instnt when the shocks hppen, there is nother n instntneous frequency curve similr to cosine curve (highlighted with the red curve) with the modultion frequency of Hz nd the crrier frequency of out Hz, where the frequency Hz lmost equls to the rottion speed of the input shft nd the frequency Hz pproches the twelve times of the meshing frequency, which denotes tht there occurs frequency modultion phenomenon in the roken-tooth signl. Consequently, the comprisons etween the three HHT spectr from Fig. (), Fig. 7() nd Fig. () prove tht the ICEEMD method gretly llevites the mode mixing nd splitting of the EEMD/CEEMD method nd Copyright (c) Interntionl Journl of Computer Science Issues. All Rights Reserved.
IJCSI Interntionl Journl of Computer Science Issues, Vol. 9, Issue, No, Novemer ISSN (Online): 9- www.ijcsi.org 9 cn extrct more nd useful informtion from the fulty signls, which is essentil for fult dignosis of geroxes. In ddition, Fig. 9 presents tht ICEEMD, comprle to CEEMD, cn reduce wste of time y % compred with the EEMD method. This indictes tht the ICEEMD method is seemingly n effective nd efficient method for gerox fult signl processing. Amplitude(m.s - ) Amplitude(m.s - ) - - - -.5..5..5 -.5..5..5 Fig. 5 The two gerox virtion signls: () The norml gerox signl; () The roken-tooth gerox signl..5..5..5.5..5..5 5 5 5 3.5..5..5.5..5..5 5 5 5 3 Fig. 7 The HHT spectr of the two virtion signls using the CEEMD method: () The norml gerox signl; () The roken-tooth gerox signl..5..5..5.5..5..5 5 5 5 3 Fig. The HHT spectr of the two virtion signls using the ICEEMD method (The red cosine curve is dded to highlight the instntneous frequency curve.): () The norml gerox signl; () The roken-tooth gerox signl. Fig. The HHT spectr of the two virtion signls using the EEMD method: () The norml gerox signl; () The roken-tooth gerox signl. Copyright (c) Interntionl Journl of Computer Science Issues. All Rights Reserved.
IJCSI Interntionl Journl of Computer Science Issues, Vol. 9, Issue, No, Novemer ISSN (Online): 9- www.ijcsi.org 99 Computing time(s) 5 3 EEMD CEEMD ICEEMD Algorithm Fig. 9 Comprisons of computing time etween the three different methods for the roken-tooth signl. 5. Conclusions The pper ims to provide guidnce on choosing the pproprite mplitude of the dded white noise for the EEMD method nd reduce the tremendous time wste occurring in the EEMD method. To solve those prolems, sed on the CEEMD method, the ICEEMD method is ddressed in this pper. Besides, the numericl exmples nd the experimentl exmples hve tested the ility of the ICEEMD method. The comprisons with the EEMD nd CEEMD methods show tht the ICEEMD method outperforms the EEMD method in lleviting the mode mixing nd splitting s well s reducing the time wste nd lso outperforms the CEEMD method in lleviting the mode mixing nd splitting. This pper indictes tht the ICEEMD method is seemingly n effective nd efficient method for gerox fult signl processing. In ddition, comined with some other methods[3, ], the ICEEMD method my chieve etter results in nlyzing gerox fulty signls. Acknowledgments The uthor thnks Kyvn J. Rfiee for his generous providing free gerox virtion dtset on his personl officil wepge. The work is supported y Nturl Science Fund of Shndong Province (ZREE7), Development Progrm of Science nd Technology of Weifng City (5, 3) nd Young Science Fund of Weifng University (Z). References [] W. Brtelmus, R. Zimroz, A new feture for monitoring the condition of geroxes in non-sttionry operting conditions, Mechnicl Systems nd Signl Processing, Vol. 3, No. 5, 9, pp. 5-53. []. Stish, Short-time Fourier nd wvelet trnsforms for fult detection in power trnsformers during impulse tests, Proceedings of the Institute of Electricl Engneering Science, Mesurement nd Technology, Vol. 5, No., pp. 77-. [3] X. Jing, S. Mhdevn, Wvelet spectrum nlysis pproch to model vlidtion of dynmic systems, Mechnicl Systems nd Signl Processing, Vol. 5, No,, pp. 575-59. [] N.E. Hung, Z. Shen, S.R. ong, et l, The empiricl mode decomposition nd the Hilert spectrum for nonliner nd non-sttionry time series nlysis, Proceedings of the Royl Society of ondon Series A - Mthemticl Physicl nd Engineering Sciences, Vol. 5, No. 97, 99, pp. 93-995. [5] R. Ricci, P. Penncchi, Dignostics of ger fults sed on EMD nd utomtic selection of intrinsic mode functions, Mechnicl Systems nd Signl Processing, Vol. 5, No. 3,, pp. -3. [] J. Cheng, D. Yu, J. Tng, Y. Yng, Appliction of frequency fmily seprtion method sed upon EMD nd locl Hilert energy spectrum method to ger fult dignosis, Mechnism nd Mchine Theory, Vol. 3, No.,, pp. 7-73. [7]. in, J. Honging, Signl feture extrction sed on n improved EMD method, Mesurement, Vol., No. 5, 9, pp. 79-3. [] Z.H. Wu, N.E. Hung, Ensemle empiricl mode decomposition: A noise-ssisted dt nlysis method, Advnces in Adptive Dt Anlysis, Vol., No., 9, pp. -. [9] Y. ei, Z. He, Y. Zi, Appliction of the EEMD method to rotor fult dignosis of rotting mchinery, Mechnicl Systems nd Signl Processing, Vol. 3, No., 9, pp. 37-33. [] J.I.A.R. Yeh, J.S. Shieh, N.E. Hung, et l, Complementry ensemle empiricl mode decomposition: A novel noise enhnced dt nlysis method, Advnces in Adptive Dt Anlysis, Vol., No.,, pp. 35-5. [] J. Zhng, R. Yn, R.X. Go, Z. Feng, Performnce enhncement of ensemle empiricl mode decomposition, Mechnicl Systems nd Signl Processing, Vol., No. 7,, pp. -3. [] J. Rfiee, P. Tse, Use of utocorreltion of wvelet coefficients for fult dignosis, Mechnicl Systems nd Signl Processing, Vol. 3, No. 5, 9, pp. 55-57. [3] iyong M, Nizhng Feng, Qi Wng, Non-Negtive Mtrix Fctoriztion nd Support Vector Dt Description Bsed One Clss Clssifiction, Interntionl Journl of Computer Science Issues, Vol. 9, No. 5,, pp. 3-. [] Ho Zho, Appliction of Wvelet De-noising in Virtion Torque Mesurement, Interntionl Journl of Computer Science Issues, Vol. 9, No. 5,, pp. 9-33. Jinshn in received his BE degree from Shndong University in Mechnicl Engineering nd Automtion in. Next, he received his ME degree from University of Jinn in Mechnicl nd Electronic Engineering in. Since, he hve tught t Weifng University. Currently, his reserch interests include fult detection, pttern clssifiction nd signl processing. Copyright (c) Interntionl Journl of Computer Science Issues. All Rights Reserved.