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  1. 1. Sreedevi Gandham, T. Sreenivasulu Reddy / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 6, November- December 2012, pp.1705-1711 Enhanced Signal Denoising Performance by EMD-based Techniques Sreedevi Gandham*, T. Sreenivasulu Reddy Department of Electronics and Communication Engineering Sri Venkateswara University, Tirupati, Andhra Pradesh-517502. India.Abstract Empirical mode decomposition (EMD) is improve the understanding of the way EMDone of the most efficient methods used for non- operates or to enhance its performance, EMD stillparametric signal denoising. In this study wavelet lacks a sound mathematical theory and is essentiallythresholding principle is used in the described by an algorithm. However, partly due todecomposition modes resulting from applying the fact that it is easily and directly applicable andEMD to a signal. The principles of hard and soft partly because it often results in interesting andwavelet thresholding including translation useful decomposition outcomes, it has found a vastinvariant denoising were appropriately modified number of diverse applications such as biomedical,to develop denoising methods suited for watermarking and audio processing to name a few.thresholding EMD modes. We demonstrated In this study, inspired by standard waveletthat, although a direct application of this thresholding and translation invariant thresholding,principle is not feasible in the EMD case, it can a few EMD-based denoising techniques arebe appropriately adapted by exploiting the developed and tested in different signal scenariosspecial characteristics of the EMD decomposition and white Gaussian noise. It is shown that althoughmodes. In the same manner, inspired by the the main principles between wavelet and EMDtranslation invariant wavelet thresholding, a thresholding are the same, in the case of EMD, thesimilar technique adapted to EMD is developed, thresholding operation has to be properly adapted inleading to enhanced denoising performance. order to be consistent with the special characteristicsKeywords: Denoising, EMD, Wavelet of the signal modes resulting from EMD. Thethresholding possibility of adapting the wavelet thresholding principles in thresholding the decomposition modes I Introduction of EMD directly, is explored. Consequently, three Denoising aims to remove the noise and to novel EMD-based hard and soft thresholdingrecover the original signal regardless of the signal’s strategies are presented.frequency content. Traditional denoising schemesare based on linear methods, where the most II EMD: A Brief description andcommon choice is the Wiener filtering and they notationhave their own limitations(Kopsinis and McLauglin, EMD[3] adaptively decomposes a2008).Recently, a new data-driven technique, multicomponent signal [4] x(t) into a number of the so-called IMFs, h (t ),1  i  Lreferred to as empirical mode decomposition (EMD) (i )has been introduced by Huang et al. (1998) for Lanalyzing data from nonstationary and nonlinear x(t )   h(i ) (t )  d (t ) (1)processes. The EMD has received more attention in i 1terms of applications, interpretation, and where d(t) is a remainder which is a non-zero-meanimprovement. Empirical mode decomposition slowly varying function with only few extrema.(EMD) method is an algorithm for the analysis of Each one of the IMFs, say, the (i) th one , ismulticomponent signals that breaks them down into estimated with the aid of an iterative process, calleda number of amplitude and frequency modulated sifting, applied to the residual multicomponent(AM/FM) zero-mean signals, termed intrinsic mode signal.functions (IMFs). In contrast to conventionaldecomposition methods such as wavelets, which   x(t ), i 1 x (i ) (t )   (2)  x(t )   j 1 h (t ), i  2.perform the analysis by projecting the signal under i 1 ( i )consideration onto a number of predefined basis vectors, EMD expresses the signal as an expansion The sifting process used in this paper is theof basis functions that are signal-dependent and are standard one [3]. According to this, during the (n+1)estimated via an iterative procedure called sifting. th sifting iteration, the temporary IMF estimateAlthough many attempts have been made to 1705 | P a g e
  2. 2. Sreedevi Gandham, T. Sreenivasulu Reddy / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 6, November- December 2012, pp.1705-1711hni ) (t ) is improving according to the following ( lower frequencies locally in the time-frequency domain than its preceeding ones.steps.21. Find the local maxima and minima of hni ) (t ) . (2. Interpolate, using natural cubic splines, along thepoints of hni ) (t ) estimated in the first step in order (to form an upper and a lower envelope.3. Compute the mean of two envelopes. (i )4. Obtain the refined estimate hn 1 (t ) of the IMF bysubtracting the mean found in the previous step (i )from the current IMF estimate hn (t ) .5. Proceed from step 1) again unless a stoppingcriterion has been fulfilled. Figure 1.Empirical mode decomposition of the noisy The sifting process is effectively an empirical signal shown in (a).but powerful technique for the estimation of the (i )mean m (t ) of the residual multicomponent signal III Signal Denoising Thresholding is a technique used for signalx(i ) (t ) localy, a quantity that we term total local and image denoising. The discrete waveletmean. Although the notion of the total local mean is transform uses two types of filters: (1) averagingsomewhat vague, especially for multicomponent filters, and (2) detail filters. When we decompose asignals, in the EMD context it means that its signal using the wavelet transform, we are left with (i ) a set of wavelet coefficients that correlates to thesubtraction from x (t ) will lead to a signal, which high frequency subbands. These high frequencyis actually the corresponding IMF, i.e., subbands consist of the details in the data set.h(i ) (t )  x(i ) (t )  m(i ) (t ), that is going to have If these details are small enough, theythe following properties. might be omitted without substantially affecting the 1) Zero mean. main features of the data set. Additionally, these 2) All the maxima and all the minima of small details are often those associated with noise; therefore, by setting these coefficients to zero, we h(i ) (t ) will be positive and negative, are essentially killing the noise. This becomes the respectively. Often, but not always, the basic concept behind thresholding-set all frequency IMFs resemble sinusoids that are both subband coefficients that are less than a particular amplitude and (frequency modulated (AM threshold to zero and use these coefficients in an /FM). inverse wavelet transformation to reconstruct the data set. By construction, the number of say N i  3.1. IMF Thresholding based Denoising An alternative denoising procedure inspired (i )extrema of h (t ) positioned in time instances by wavelet thresholding is proposed. EMD thresholding can exceed the performance achievedr(i ) j  [r , r ,....., rNi)i  ] and the corresponding 1 (i ) 2 (i ) ( by wavelet thresholding only by adapting theIMF points h(i ) (rj(i ) ), j  1,....., N (i) , will thresholding function to the special nature of IMFs. EMD can be interpreted as a subband-like filteringalternate between maxima and minima, i.e,. positive procedure resulting in essentially uncorrelatedand negative values. As a result, in any pair of IMFs. Although the equivalent filter-bank structureextrema rj(i )  [h(i ) (rj(i ) ), h(i ) (rj()1 )] , corresponds i is by no means predetermined and fixed as in wavelet decomposition, one can in principle performto a single zero-crossing z (ji ) . Depending on the thresholding in each IMF in order to locally excludeIMF shape, the number of zero-crossings can be low-energy IMF parts that are expected to be N  i  or N  i  -1. Moreover, each IMF 4 significantly corrupted by noise.either A direct application of waveletlets say the one of the order, I, have fewer extrema thresholding in the EMD case translates tothan all over the lower order of IMFs, j=1, i-1,leading to fewer and fewer oscillations as the IMForder increases. In other words, each IMF occupies 1706 | P a g e
  3. 3. Sreedevi Gandham, T. Sreenivasulu Reddy / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 6, November- December 2012, pp.1705-1711 h(i ) (t ), h(i ) (t )  Tih (i ) (t )    0, h(i ) (t )  Tifor hard thresholding and to sgn(h(i ) (t ))( h(i ) (t )  Ti ),  h(i ) (t )  Tih (i ) (t )     0, h(i ) (t )  Ti Figure 2. EMD Direct Thresholding.The top-rightfor soft thresholding, where, in both thresholding numbers are the SNR values after denoising.cases, indicates the thresholded IMF. The reasonfor adopting different thresholds per mode will The reasoning underlying the significancebecome clearer in the sequel. test procedure above is fairly simple but strong. IfA generalized reconstruction of the denoised signal the energy of the IMFs resulting from theis given by decomposition of a noise-only signal with certain characteristics is known, then in actual cases of M2 L signals comprising both information and noisex(t ) ˆ  h (i ) (t )   k  M1  k  M 2 1 h(i ) (t ) following the specific characteristics, a significant discrepancy between the energy of a noise-only IMF Where the introduction of parameters and the corresponding noisy-signal IMF indicatesgives us flexibility on the exclusion of the noisy the presence of useful information. In a denoisinglow-order IMFs and on the optional thresholding of scenario, this translates to partially reconstructingthe high-order ones, which in white Gaussian noise the signal using only the IMFs that contain usefulconditions contain little noise energy. There are two information and discarding those that carrymajor differences, which are interconnected, primarily noise, i.e., the IMFs that share similarbetween wavelet and direct EMD thresholding amounts of energy with the noise-only case.(EMD-DT) shown above. In practice, the noise-only signal is never First, in contrast to wavelet denoising available in order to apply EMD and estimate thewhere thresholding is applied to the wavelet IMF energies, so the usefulness of the abovecomponents, in the EMD case, thresholding is technique relies on whether or not the energies ofapplied to the samples of each IMF, which are the noise-only IMFs can be estimated directly basedbasically the signal portion contained in each on the actual noisy signal. The latter is usually theadaptive subband. An equivalent procedure in the case due to a striking feature of EMD. Apart fromwavelet method would be to perform thresholding the first noise-only IMF, the power spectra of theon the reconstructed signals after performing the other IMFs exhibit self-similar characteristics akinsynthesis function on each scale separately. to those that appear in any dyadic filter structure. AsSecondly, as a consequence of the first a result, the IMF energies should linearly decreasedifference,the IMF samples are not Gaussian in a semilog diagram of, e.g., with respect to for. Itdistributed with variance equal to the noise variance, also turns out that the first IMF carries the highestas the wavelet components are irrespective of scale. amounts of energy In our study of thresholds, multiples of theIMF-dependent universal threshold, i.e, where is aconstant, are used. Moreover, the IMF energies canbe computed directly based on the variance estimateof the first IMF.3.2 Conventional EMD Denoising The first attempt at using EMD as adenoising tool emerged from the need to knowwhether a specific IMF contains useful information Figure 3. EMD Conventional Denoising. The top-or primarily noise. Thus, significance IMF test right numbers are the SNR values after denoising.procedures were simultaneously developed based onthe statistical analysis of modes resulting from the IV Wavelet based Denoisingdecomposition of signals solely consisting of Employing a chosen orthonormal waveletfractional Gaussian noise and white Gaussian noise, basis, an orthogonal N× N matrix W is appropriatelyrespectively. built. Discrete wavelet transform (DWT) c =Wx where, x=[x(1), x(2), . . . , x(N)] is the vector of the signal samples and c = [c1, c2, . . . , cN] contains the 1707 | P a g e
  4. 4. Sreedevi Gandham, T. Sreenivasulu Reddy / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 6, November- December 2012, pp.1705-1711resultant wavelet coefficients. Under white Gaussian also different loss functions. One of the mainnoise conditions and due to the orthogonality of advantages of our approach is to naturally provide amatrix W, any wavelet coefficient 𝑐 𝑖 follows normal thresholding rule, and consequently, a thresholddistribution with variance the noise variance σ and value adapted to the signal/noise under study.mean the corresponding coefficient value 𝑐 𝑖 of the Moreover, we will also show that the MAPDWT of the noiseless signal x(t). estimation is also closely related to wavelet Provided that the signal under regularization of ill-posed inverse stochasticconsideration is sparse in the wavelet domain the problems with appropriate penalties and lossDWT is expected to distribute the total energy of functions, that parallel Bridge estimation techniquesx(t) in only a few wavelet components lending for nonparametric regression as introduced by thethemselves to high amplitudes. As a result, the sake of simplicity (but see our discussion later), weamplitude of most of the wavelet components is assume in the sequel that the wavelet coefficientsattributed to noise only. The fundamental reasoning of the signal and the noise are two independentof wavelet soft thresholding is to set to zero all the sequences of i.i.d. random variables.components which are lower than a threshold related Hard thresholding zeroes out all values into noise level and appropriately shrink the rest of the the frequency band that is found to be Gaussian. Thecomponents hard thresholding coefficient is There has recently been a great deal ofresearch interest in wavelet thresholding techniquesfor signal and image denoising developed wavelet  0, if the band is Gaussian ch  shrinkage and thresholding methods for  1, if the band is not Gaussian.reconstructing signals from noisy data, where thenoise is assumed to be white and Gaussian. Theyhave also shown that the resulting estimates of theunknown function are nearly minimax over a largeclass of function spaces and for a wide range of lossfunctions. The model generally adopted for theobserved process isy(i)  f (i)   (i), i 1,....., K  2 J  ( J  N  ) Where ξ(i)} is usually assumed to be a Figure 4. Wavelet Translation invariant Hardrandom noise vector with independent and thresholding.identically distributed (i.i.d.) Gaussian componentswith zero mean and variance 𝜎 2 . Note however that Soft Thresholding is obtained bythe assumption of Gaussianity is alleviated in the multiplying values in the specific frequency bandsequel. Estimation of the underlying unknown signal that is found to be Gaussian, by a coefficientf is of interest. We subsequently consider a between 0 and 1. Using a coefficient of 0 is the(periodic) discrete wavelet expansion of the same as hard thresholding, whereas using aobservation signal, leading to the following additive coefficient of 1 is the same as leaving the frequencymodel: band undisturbed. The soft thresholding coefficient is calculated usingWyj ,k  W f j ,k  W j ,k , k {1.......2 j K } 34 ˆ Under the Gaussian noise assumption, c  , 1.5thresholding techniques successfully utilize theunitary transform property of the wavelet Where µ4 g is the bootstrapped kurtosis of thedecomposition to distinguish statistically the signal particular frequency band, and denotes the absolutecomponents from those of the noise. In order to fix value. The bootstrapped kurtosis value µ4 g isterminology, we recall that a thresholding rule sets limited not to exceed 4.5, which will be explainedto zero all coefficients with an absolute value below later. The bootstrapped kurtosis value µ4 g isa certain threshold  j  0 . obtained by using the Bootstrap principle, which We exhibit close connections between calls for resamplings from the data set many timeswavelet thresholding and Maximum A Posteriori with replacement, to obtain N resampled sets of the(MAP) estimation (or, equivalently, wavelet same length as the original data set. Next, theregularization) using exponential power prior kurtosis value for each resample is found. Finally,distributions. Our approach differs from those the Bootstrapped kurtosis µ4 g is defined as thepreviously mentioned by using a different prior and estimated mean obtained from N kurtosis values. It 1708 | P a g e
  5. 5. Sreedevi Gandham, T. Sreenivasulu Reddy / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 6, November- December 2012, pp.1705-1711should be noted that the thresholding coefficient c is 7. Iterate K-1 time between steps 3)-6),where K isa function of the frequency band’s degree of the number of averaging iteration in order toGaussianity. Equation above shows that the obtain k denoised versions ofcoefficient c gets closer to 0 as the bootstrapped    x, ie., x1 , x2 ,....., xk .kurtosis value µ4 g for a specific frequency bandgets closer to the theoretical value 3, and vice versa. 8. Average the resulted denoised signals kTherefore, the closer a frequency band gets to being x(t )  (1/ k ) xk (t ).  Gaussian, the smaller contribution it has after soft k 1thresholding. 4.2. Clear iterative interval thresholding In order to simplify the presentation of our When the noise is relatively low, enhancedresult further, we will drop the dependence on the performance compared to EMD-IIT denoising canresolution level j and the time index k of the be achieved with a variant called clear iterativequantities involved subsequently. interval-thresholding (EMD-CIIT). The need for4.1. Iterative EMD Interval-Thresholding such a modification comes from the fact that the Direct application of translation invariant first IMF, especially when the signal SNR is high, isdenoising to the EMD case will not work. This likely to contain some signal portions as well. If thisarises from the fact that the wavelet components of is the case, then by randomly altering its samplethe circularly shifted versions of the signal positions, the information signal carried on the firstcorrespond to atoms centered on different signal IMF will spread out contaminating the rest of theinstances. In the case of the data-driven EMD signal along its length. In such an unfortunatedecomposition, the major processing components, situation, the denoising performance will decline. Inwhich are the extrema, are signal dependent, leading order to overcome this disadvantage of EMD-IIT itto fixed relative extrema positions with respect to is not the first IMF that is altered directly but thethe signal when the latter is shifted. As a result, the first IMF after having all the parts of the usefulEMD of shifted versions of the noisy signal information signal that it contains removed. Thecorresponds to identical IMFs sifted by the same “extraction” of the information signal from the firstamount. Consequently, noise averaging cannot be IMF can be realized with any thresholding method,achieved in this way. The different denoised either EMD-based or wavelet-based. It is importantversions of the noisy signal in the case of EMD can to note that any useful signal resulting from theonly be constructed from different IMF versions thresholding operation of the first IMF has to beafter being thresholded. Inevitably, this is possible summed with the partial reconstruction of the last 1only by decomposing different noisy versions of the IMFs.signal under consideration itself. Algorithm We know that in white Gaussian noise 1. Perform an EMD expansion of the original noisyconditions, the first IMF is mainly noise, and more signal xspecifically comprises the larger amount of noise 2. Perform a thresholding operation to the first IMFcompared to the rest of the IMFs. By altering in arandom way the positions of the samples of the first of x(t ) to obtain a denoised versionIMF and then adding the resulting noise signal to the  h (1) (t ) of h(1) (t ) .sum of the rest of the IMFs, we can obtain a 3. Compute the actual noise signal that existed indifferent noisy version of the original signal.Algorithm (1)  h(1) (t ), hn (t )  h(1) (t )  h (1) (t ) .1. Perform an EMD expansion of the original 4.Perform a partial reconstruction using the last L-1 noisy signal x. IMFs plus the information signal contained in the2. Perform a partial reconstruction using the last x p (t )  i 2 h(i ) (t )  h (1) (t ).  L L-1 IMFs only,𝑥 𝑝 (t) = 𝑖 𝐿 = 2ℎ 𝑖𝑡 first IMF,3. Randomly alter the sample position of the first 5. Randomly alter the sample positions of the noise- IMF ha (t )  ALTER(h (t )) . only part of the first IMF, (1) (1)4. Construct a different noisy version of the ha (t )  ALTER(hn (t )). (1) (1) original signal xa (t )  x p (t )  ha (t ). (1)5. Perform EMD on the new altered noisy signal xa (t ).6. Perform EMD-IT denoising (12)or(13) on the IMFs of xa (t ) to obtain a denoised version  x1 (t ) of x. 1709 | P a g e
  6. 6. Sreedevi Gandham, T. Sreenivasulu Reddy / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 6, November- December 2012, pp.1705-1711 Figure 5. Result of the EMD-based Denoising with SNR db Clear Iterative Interval Thresholding method using Meth - -5 -2 0 2 5 10 twenty Iterations. Piece ods 10 5. Simulation results and discussion Regul ar EMD 7. 10 14. 15 17. 18 22. signal - 63 .9 27 .9 11 .8 22 4096 CIIT 6 5 8 sampl H(P) es EMD 8. 10 13. 14 17. 19 23. - 26 .6 53 .4 44 .2 06 CIIT 1 2 1 H(c) Figure 6. SNR after denoising with respect to the number of shifting iterations EMD 8. 10 12. 14 17. 18 22. -IIT 28 .1 97 .1 18 .8 81 H(c) 3 9 5 EMD 6. 10 13. 15 16. 18 21. -IIT 78 .6 54 .2 51 .5 93 H(P) 9 2 1 V Conclusions Figure 7. Performance evaluation of the piece- In the present paper, the principles of hard regular signal using EMD-based denoising methods and soft wavelet thresholding, including translation invariant denoising, were appropriately modified to develop denoising methods suited for thresholding EMD modes. The novel techniques presented exhibit an enhanced performance compared to wavelet denoising in the cases where the signal SNR is low and/ or the sampling frequency is high. These preliminary results suggest further efforts for Figure 8. Performance evaluation of EMD-based improvement of EMD-based denoising when hard thresholding techniques. denoising of signals with moderate to high SNR SNR performance and variance of EMD-Based would be appropriate. Denoising methods applied on Doppler and Piece-Regular signal. Table 1 & 2 Refernces [1] Kopsinis and S. McLauugin, “Empirical SNR db mode decomposition based denoising Met - -5 -2 0 2 5 10 techniques,” in Proc. 1st IAPRW WorkshopPiece hods 10 Cong. Inf. ProcessRegul [2] N.E, Huang et al., “The empherical modear EM 7. 10. 14. 15. 17. 18. 22. decomposition and the Hilbert spectrum forsignal D- 63 96 27 95 11 88 22 nonlinear and non-stationary time series4096 CII analysis,” Proc, Ray, Soc. London A, vol, pp.sampl T 903-995, Mar, 1998.es H(P) [3] L. Cohen, Time-Frequency analysis. EM 8. 10. 13. 14. 17. 19. 23. Englewood Cliffs, NJ:Prentice- Hall, 1995. D- 26 61 53 42 44 21 06 [4] S.Mallat, A wavelet Tour of Signal CII Processingm, 2 nd ed. New York: Acedamic, T 1999. H(c) [5] P. Flandrin, G Rilling, and P. Goncalves, 8. 10. 12. 14. 17. 18. 22. “Empirical mode decomposition as a filter EM 28 13 97 19 18 85 81 bank,” IEEE Signal Process. Lett., vol. 11, D- pp. 112-114, Feb. 2004. IIT [6] G. Rilling and P. Flandrin, “One or two H(c) frequencies? The empirical mode EM 6. 10. 13. 15. 16. 18. 21. decomposition answers,” IEEE Trans. Signal D- 78 69 54 22 51 51 93 Process., PP. 85-95, Jan 2008. IIT [6] G. Rilling and P. Flandrin, “One or two H(P) frequencies? The empirical mode 1710 | P a g e
  7. 7. Sreedevi Gandham, T. Sreenivasulu Reddy / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 6, November- December 2012, pp.1705-1711 decomposition answers,” IEEE Trans. Signal Process., PP. 85-95, Jan 2008. [7] Y. Washizawa, T. Tanaka, D.P. Mandic, and A. Cichocki, “A flexible method for envelope estimation in empirical mode decomposition,” in Proc. 10 th Int. Conf. Knowl. –Based Intell. Inf. Eng. Syst. (KES 2006), Bournemouth, U.K., 2006. [8] Y. Kopsinis and S. McLauglin, “Investigation and performance enhancement of the empirical mode decomposition method based on a heuristic search optimization approach, “IEEE Trans. Signal process., pp. 1-13, Jan. 2008. [9] Y. Kopsininsm and S.McLauglin, “ Improvement EMDusing doubly-iterative sifting and high order spiline interpolation, “EURASIP J.Adv. Signal Proecss., vol. 2008, 2008. [10] Z. Wu and N.E. Huang, “A study of the characteristics of white noise using the empirical mode decomposition method, “Proc. Roy. Soc. London A, vol. 460, pp.1597-1611, Jun.2004. [11] N.E. Husng and Z. Wu. “Stastical significance test of intrinsic mode functions, “in Hibert-Huang transform snd its applications, N.E.Huang and S.Shen, Eds., 1 st ed. Singapore: World scientific, 2005. [12] S. Theodordis and K. Koutroumbas, Pattern Recognition, 3 rd ed. New York: Acedamic, 2006. [13] A. O. Boudraa and J. C. Cexus, “Denoising via empirical mode decomposition,” in Proc. ISCCSP 2006. [14] Y. Mao and P. Que, “Noise suppression and flawdetection of ultrasonic signals via empirical mode decomposition,” Rus. J. Nondestruct. Test., vol. 43, pp. 196–203, 2007. [15] T. Jing-Tian, Z. Qing, T. Yan, L. Bin, and Z. Xiao-Kai, “Hilbert-huang ransform for ECG de-noising,” in Proc. 1st Int. Conf. Bioinf. Biomed. Eng. (ICBBE 2007), 20 1711 | P a g e