A THRESHOLD DENOISING METHOD BASED ON EMD

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1 Joural of Theoretical ad Applied Iformatio Techology 1 th Jauary 13. Vol. 47 No.1-13 JATIT & LLS. All rights reserved. ISSN: E-ISSN: A THRESHOLD DENOISING METHOD BASED ON EMD JIANZHAO HUANG, JIAN XIE, FENG LI, LIANG LI Xi a High-tech Research Istitute, Xi a 71, Shaxi, Chia ABSTRACT The deoisig method based o empirical mode decompositio (EMD) ca be broadly divided ito: IMF extractio method ad IMF threshold approach. Aimig to the problems of how to select IMFs i extractio method ad the processig of the selected IMFs, a threshold deoisig method based o EMD is put forward. I this method, the stadard of IMF selectio i eergy viewpoit is offered, ad the IMFs uppig to the stadard are selected firstly, the, through comparig the average eergy of all uselected IMFs with the eergy of each selected IMF, the sigular selected IMFs are cofirmed, ad deoised by threshold. Fially, the deoised sigal is obtaied by summig up all selected IMFs. The curret method combies the soft threshold deoisig method with the IMF selectio together, compared with other deoisig methods, the effectiveess ad superiority of the method is validated. The result provides support for improvig the deoisig effect i egieerig. Keywords: EMD, Threshold Deoisig, IMF Selectio, Sigular IMF 1. INTRODUCTION The Empirical Mode Decompositio (EMD) has bee proposed as a adaptive time-frequecy data aalysis method [1]. The major advatage of the EMD is that the basis fuctios are derived from the sigal itself. It has bee proved quite versatile i a broad rage of applicatios for extractig sigals from data geerated ioisy oliear ad ostatioary processes. By studyig the filterig properties of the EMD, it is foud that the EMD has the similar biary filter characteristics as the wavelet trasform []. As i wavelet aalysis, the eergy will ofte be cocetrated o the high frequecy temporal modes ad decreases towards the coarser oes [3]. Accordig to this idea, there will be a mode after which the eergy distributio of the sigal overcomes that of the oise. This particular mode, allows us to separate sigal from oise. Modes coarser tha this particular mode are domiated by the sigal, while fier modes are oise domiated. Based o the priciple metioed above, the deoisg method based o EMD ca be broadly divided ito two directios: (1) IMF extractio method. IMFs are selected without ay processig, ad summed up to get the deoised sigal. I article [4], the mii-eighborig root mea square error is used as the stadard to select IMFs, the deosig result is compared with the average, media ad wavelet filterig methods. Based o the EMD decompositio characteristics of white oise [], the product of the eergy desity ad the average period is calculated, the trip poit of the product is cosidered as the stadard to select IMFs [6], but the quatitative idicator of the trip poit is ot give i the paper. () The threshold approach. IMFs are dealt with the threshold fuctio. For oise reductio of the speech sigal [7], all decomposed IMFs are dealt with hard threshold fuctio, ad the method does better tha the wavelet deosig method. I article [8], EMD ad soft threshold deoisig method are combied together, ad all IMFs are dealt with the soft threshold fuctio. A mode cell is defied as the sigal betwee the two adjacet zero-crossigs amog a IMF [9], the deoisig process is to make the cell s choice, ad a sigle data is replaced by a oscillatig uit i the article. The problem i this method is whe the uit is rejected, the useful sigal i the cell will be loss at the same time. Based o the work metioed above, a threshold deoisig method based o EMD is preseted i this paper. Firstly, the algorithm based o the eergy to determie the trip poit is desiged for IMF selectio, the, by comparig the eergy of the selected IMFs with excluded IMFs, sigular selected IMFs are dealt with soft threshold fuctio, ad fially the deoised sigal is obtaied by summig up the selected IMFs. Compared with other deoisig methods uder differet oise itesity, it is proved that the best IMFs ca be summed up ad properly deoised by the proposed method. 419

2 Joural of Theoretical ad Applied Iformatio Techology 1 th Jauary 13. Vol. 47 No.1-13 JATIT & LLS. All rights reserved. ISSN: E-ISSN: THEORY OVERVIEW.1 EMD Basic The EMD decomposes a give sigal xt () ito a series of IMFs through a iterative process called siftig; each oe with a distict time scale [1]. By defiitio, a IMF satisfies two coditios: (1) the umber of extrema ad the umber of zeros crossigs may differ by o more tha oe; () the average value of the evelope defied by the local maxima, ad the evelope defied by the local miima, is zero. Give a sigal, the effective algorithm of EMD ca be summarized as follows [1]. 1) Idetify all extrema of xt (). ) Iterpolate betwee miima (resp. maxima), edig up with some evelope e mi () t (resp. e () t ). max 3) Compute the average values mt (), mt () = ( emi () t + emax ())/ t. 4) Extract the detail dt () = xt () mt (). ) Iterate o the residual mt (). At last, EMD eds up with a represetatio of the form: K xt () = m() t + d () t (1) k k = 1 Where mk () t stads for a residual tred ad the modes { dk ( t), k = 1, K} are costraied to be zero-mea amplitude modulatio frequecy modulatio waveforms.. IMF SELECTION The statistic characteristics of white oise decomposed by EMD are summed up as follows []: the IMFs are all ormally distributed, ad the Fourier spectra of the IMFs are all idetical ad cover the same area o a semi-logarithmic period scale, ad the product of the eergy desity of IMF ad its correspodig averaged period is a costat, ad that the eergy-desity fuctio is chi-squared distributed. The eergy desity of IMF ad its correspodig averaged period are defied as follows: N 1 E = ( c ( i)) () N i = 1 max k N T = (3) N ET = cost (4) Where c is the -th IMF, E is the eergy desity, ad N is the legth of the data; T is the average period, N max is the maximum umbers of the c. The product of the eergy desity ad the correspodig average period is a costat. The E, T ad ET of each IMF are calculated i accordace with equatios ()-(4). Because the ET of white oise is a costat ad the highfrequecy IMF is usually the oise. So there will be a trip poit i the curve of ET. Excludig all IMFs before the trip poit, the summatio of left IMFs is the deoised sigal [6]. 3. THRESHOLD DENOISING METHOD 3.1 The Problem To study the characteristics of the eergy-based IMF extractio method, a simulatio sigal xt () is used to do aalysis. The simulatio sigal is superimposed by three siusoidal sigals correspodig to the period of 1s, 6s ad s, the samplig iterval is 1s, ad the samplig poits are take as 4, poits. Addig differet itesity of Gaussia white oise with the simulatio sigal xt (). The oise variaces σ are from. to 4, arithmetic icreased by.. The ET of each IMF is calculated uder differet oise itesity accordig to the equatios ()-(4). The simulatio results show that: the total umbers of IMFs are differet uder differet oise itesity, the variace greater the more. For the radom of white oise geerated by Matlab software, the results will be differeces for each ruig, but this radomess does ot affect the statistical properties of white oise. The average ET of top eight IMFs for 1 experimets is show i Table 1. As ca be see from the Table 1, the averages ET of IMFs uder differet oise itesity are differece. View from the portrait, the average ET of IMFs icreases totally with the icreased oise itesity, ad there are idividual circumstaces, such as IMF7 colum; View from the ladscape, the average ET values do t obey the law of gradual icrease with the decompositio umbers, there are sigular IMFs, such as IMF6 colum for each row. The judgmet stadard of trip poit is ot give, ad there is o discussio of the sigular IMFs i article [6]. So there are two problems eed to be solved i practical applicatios: (1) the quatitative idicators of the trip poit uder differet oise itesity; () the processig method of the sigular IMFs. 4

3 Joural of Theoretical ad Applied Iformatio Techology 1 th Jauary 13. Vol. 47 No.1-13 JATIT & LLS. All rights reserved. ISSN: E-ISSN: IMFs Noise Itesity Table 1: The Average ET Of Top Eight Imfs For 1 Experimets e e e e e e Optimizd Threshold Deoisig Method Aimig to the two problems metioed above, a optimized threshold deoisig method based o EMD is put forward. The specific steps of the method are as follows: 1) Obtai the IMFs by EMD. ) Calculate the eergy desity, the average period ad the product for each IMF by usig the equatios ()-(4). 3) Determie the trip poit. Defiite Q =E+1T+1 / ET ( = 1,, 3, N 1) () Where N is the total umber of IMFs. After large simulatio experimets, the first IMF satisfyig the coditio Q > is cosidered as the trip poit. 4) Calculate the average value of all IMFs before the trip poit: ET = mea( E T ) (6) ave i= 1 ) Defiite the IMF compoet meetig the coditio EmTm < * ETave ( m = + 1, +,, N 1) as the sigular IMF. If there is a sigular IMF existig, do the soft threshold fuctio [1]. The threshold is estimated by the followig formula: ( media( abs( IMFj )) /.674) l M thr j = l( j + 1) (7) Where M is the legth of the sigal, j meas the j th IMF. 6) After steps 1) ~ ), summig up all IMFs after the th IMF ad the tred compoet. 4. DENOISING EXPERIMENTS To validate the feasibility ad effectiveess of the proposed method, compared with the literature [4], [6] ad [7], simulatio experimets are used to do test. The trip poit selectig method is ot provided i the, so the proposed method i the article is employed for i the programmig. The sigal to oise ratio (SNR), root mea square error (RMSE) ad correlatio coefficiet (R) are served as the deoisig evaluatio idex. 4.1 Low Frequecy Experimet The simulatio sigal x(t) is superimposed by two siusoidal sigals with the period of 1s ad 6s, ad each siusoidal sigal s amplitude value is 1. The samplig iterval is 1s, ad the samplig poits are take as 4, poits. Addig differet itesity of Gaussia white oise with the simulatio sigal x(t). The oise variaces σ are from. to 4, arithmetic icreased by.. Due to limited space, oly the deoised sigals uder two differet oise variaces coditios are show i Fig.1 ad Fig.. The SNR, RMSE ad R of the four differet methods correspodig to differet oise itesity are show i Table. 41

4 Joural of Theoretical ad Applied Iformatio Techology 1 th Jauary 13. Vol. 47 No.1-13 JATIT & LLS. All rights reserved. ISSN: E-ISSN: Table : Evaluatio Idexes Of Four Methods Uder Differet Noise Itesity Experimets evaluatio oise variaces method idex SNR RMSE R origial sigal oisy sigal sample poits origial sigal oisy sigal sample poits Figure 1: Deoisig Uder Noise Desity 1. Figure : Deoisig Uder Noise Desity 3. It ca be see from Table, uder the same oise itesity, ad the article method have the same idicators, ad they do better tha ad [7]. The reaso why ad the article method have the same idicators is that: through calculatig the ET of all IMFs uder differet oise itesity, it is foud that there is oe sigular IMF existig. That meas the step ) metioed i optimized threshold de-oisig method does t work, ad the selectig methods of the trip poit are the same i ad the article method. So the deoisig idicators are o differece. 4. High Frequecy Experimet The simulatio sigal x(t) is superimposed by three siusoidal sigals with the period of 1s, 6s ad s, ad each siusoidal sigal s amplitude value is 1. The samplig iterval is 1s, ad the samplig poits are take as 4, poits. Compared with the sigal i experimet oe, the high frequecy compoet is added. Addig differet itesity of Gaussia white oise with the simulatio sigal x(t). The oise variaces σ are from. to 4, arithmetic icreased by.. The deoised sigals uder the oise variaces 1. ad 3. are show i Fig.3 ad Fig.4. The SNR, RMSE ad R of the four differet methods correspodig to differet oise itesity are show i Table 3. It ca be see from Table 3, the article method does better tha the other three methods i all idex. Compared with the low frequecy experimet, the article method does better tha, this demostrates that the stadard for sigular IMF selectio is proper ad the soft threshold plays the role i the process. Ad this article method does better tha other methods i SNR, RMSE ad R. 4

5 Joural of Theoretical ad Applied Iformatio Techology 1 th Jauary 13. Vol. 47 No.1-13 JATIT & LLS. All rights reserved. ISSN: E-ISSN: Table 3 : Evaluatio Idexes Of Four Methods Uder Differet Noise Itesity Experimets Evaluatio oise variaces method idex SNR RMSE R origial sigal oisy sigal sample poits origial sigal oisy sigal sample poits Figure 3: Deoisig Uder Noise Desity 1. Figure 4: De-oisig Uder Noise Desity 3.. CONCLUSION Based o the aalysis of the deoisig method based o EMD, aimig to the problems of the stadard for the trip poit ad the processig of sigular IMFs, a threshold deoisig method based o EMD is preseted. By takig sigal to oise ratio, root mea square error ad correlatio coefficiet as the evaluatio idex, method is applied to do deosig experimet for simulatio sigals with differet oise itesity, ad compared with other deoisig methods. It is proved that method ca optimize determie the locatio of the trip poit, ad do the threshold o sigular IMFs. The method of this articlce does better tha other three methods i deoisig. REFERENCES: [1] N. E. Huag, Z. She, S. R. Log, M. L.Wu, H. H. Shih, Q. Zheg, N. C. Ye, C. C. Tug, H. H. Liu, The empirical mode decompositio ad Hilbert spectrum for oliear ad ostatioary time series aalysis, Proc. R. Soc. Lodo A, Vol. 44, No. 1971, 1998, pp [] P. Fladri, G. Rillig, P. Gocalves, Empirical mode decompositio as a filter bak, IEEE Sig. Proc. Lett, Vol. 11, No., 4, pp [3] A. O. Boudraa, J. C. Cexus, ad Z. Saidi, EMDbased sigal oise reductio, It. J. Sigal Process., Vol. 1, No. 1, 4, pp [4] A. O. Boudraa, J. C. Cexus, EMD-based sigal filterig, IEEE Tras. Istru. Meas. Vol. 6, No. 6, 7, pp [] Z. H. Wu, N. E. Huag, A study of the characteristics of white oise usig the empirical mode decompositio method, Proc. R. Soc. Lod.A, Vol. 46, No. 46, 4, pp [6] Z. Q. Li, P. Cao, N. Y. Wag, Z. J. Liu, J. R. Zhag, FBG demodulatio system based o EMD deoise, ACTA PHOTONICA SINICA, Vol. 39, No. 8, 1, pp [7] K. khaldi, A. O. Boudraa, A. Bouchikhi, M. Turki-hadj Aliuae, E. S. Diop, Speech sigal oise reductio by EMD, Proceedigs of the 3rd Iteratioal Symposium o 43

6 Joural of Theoretical ad Applied Iformatio Techology 1 th Jauary 13. Vol. 47 No.1-13 JATIT & LLS. All rights reserved. ISSN: E-ISSN: Commuicatios Cotrol ad Sigal Processig, IEEE Coferece Publishig Services, March 1-14, 8, pp [8] A. O. Boudraa, J. C. Cexus, Deoisig via empirical mode decompositio, Proceedigs of the IEEE Iteratioal Symposium o Cotrol Commuicatios ad Sigal Processig, IEEE Coferece Publishig Services, March 13-1, 6, pp [9] C. S. Qu, Y. Z. Lu, Y. Ta, A modified empirical mode decompositio method with applicatios to sigal de-oisig, ACTA AUTOMATICA SINICA, Vol. 36, No. 1, 1, pp [1] D. L. Dooho, De-oisig by soft-thresholdig, IEEE Tras. o Iform. Theory, Vol. 41, No. 3, 199, pp

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