NONLINEAR WAVELET PACKET DENOISING OF IMPULSIVE VIBRATION SIGNALS NIKOLAOS G. NIKOLAOU, IOANNIS A. ANTONIADIS Departent of Mechanical Engineering, Machine Design and Control Systes Section National Technical University of Athens P.O. Box 6478, Athens 57 GREECE Abstract: In this paper an application is presented of the wavelet packet ethod for denoising of ipulsive vibration signals. Vibration response of achines often includes signals with periodic excitation of resonances. The ai is to extract inforation regarding the physical echanis which generates the ipulsive characteristics of the signals. The signals are transfored using the wavelet packet ethod, and the resulting coefficients are nonlinearly odified. The reconstructed tie wavefor of the signal using the odified coefficients ay reveal the ipulsive characteristics, resulting in a safer identification of the source of the ipacts. The approach and its paraeters are evaluated on industrial signals resulting fro defective bearings. Key-Words: - wavelet packet, vibration, ipulsive signal,denoising Introduction The Fourier analysis is widely used for the detection of periodicities of vibration signals. It is based on the assuptions of stationary signals, however, any signals contain nuerous transient characteristics. For exaple, the vibration response of rotating achines often includes ipact generated transient signals, typical cases being the vibration response of achines subjected to defects or wear of certain achine parts. In those cases the the ipact periodicity is usually the interesting inforation which characterizes the ipact source, while the overall frequency content of the signals is not significant. A coon approach to processing of vibration signals is to easure the vibration level and generate the Fast Fourier transfor of the easured signals. Often, frequency spectru peaks characterizing the ipact source are not easily observed, or ay have ultiple interpretations. In order to overcoe this proble a nuber of tie doain and frequency doain ethods have been proposed [-4]. The ai is to develop signal processing ethods that are able to extract patterns that relate to the source of the ipact echanis. Joint tie and frequency doain ethods, such as the Short Tie Fourier Transfor, the Wigner-Ville Distribution and the Wavelet Transfor, have been widely used in any signal processing areas. Wavelets have been established as a widespread tool, due to their flexibility and to their efficient coputational ipleentation [5-8]. They have been introduced in vibrations [9] and there are specific case studies for bearing fault detection and for other achine coponents []. In any cases the application of wavelets has been cobined and enriched by using additional features, such as Gaussian/exponential-enveloped functions [], and de-noising ethods []. The Wavelet Packet Transfor is a generalization of the wavelet transfor and has been used in signal processing for denoising or copression of signals [3-4]. Applications in achining process have also been proposed [5]. In this paper a wavelet packet transfor is used as a tool for the denoising of vibration signals with ipulsive characteristics. The ai is to extract the ipulsive inforation and reject the unwanted inforation due to other factors. The unwanted inforation is assued to exist as a part of each signal coponent. Thus, a nonlinear odification of the all the wavelet packet coefficients is applied and the signal is reconstructed. In chapter, a brief review of the basics of the wavelet transfors is presented. In section 3, the wavelet based denoising and the paraeters of the ipleentation are discussed. In section 4 the proposed approach is evaluated on industrial easureents for two types of bearing faults.
Wavelet Transfors. The Continuous Wavelet Transfor The continuous wavelet transfor (CWT) of a finite energy signal x(t) with the analyzing wavelet ø(t) is the convolution of x(t) with a translated and scaled wavelet : + W( α, b) = xt () ψ () tdt () ab, The wavelet coefficient W(á,b) easures the siilarity between the signal x(t) and the analyzing wavelet ø(t) at different scales as defined by the paraeter a, and different tie positions as defined by the paraeter b. ψα, b () t = a t b ψ a () Sall values of a give a contracted version of the basic wavelet and allows the analysis of high frequency coponents, while large values strech the basic wavelet and allows analysis of low frequency coponents of the signal. The factor á / is used for energy preservation. Equations () and () indicate that the wavelet analysis is a tie-frequency analysis, or, ore properly tered, a tie-scale analysis. The wavelet transfor can be also considered as a special filtering operation. At successively larger scales the frequency resolution iproves and the tie resolution decreases.. The Discrete Wavelet Transfor The discrete wavelet transfor is perfored by choosing fixed values a = a b= nb a (3) where,n are integers. The discrete wavelet analysis can be ipleented: + ψα Wn (, ) = a xt () ( t nb ) dt (4) An orthonoral basis can be constructed for a = and b = /, ψ ψn = ( t nb ) dt (5) A fast algorith can be ipleented by using the scaling filter which is a lowpass filter L related to the scaling function ö(t), and the wavelet filter, which is a highpass filter H, related to the wavelet function ø(t). The coputation of these filters and their properties have been widely analyzed in [5, 6]. s ca cd L H (a) Decoposition fl fl (b) Reconstruction LR HR ca cd Figure. Basic steps of discrete wavelet transfor (a) Decoposition, (b) Reconstruction The fast wavelet algorith can be ipleented in two opposite directions, decoposition and reconstruction. In the decoposition step in Fig. (a), the discrete signal s is convolved with a low-pass filter L and a high-pass filter H, resulting in two vectors ca and cd. The eleents of the vector ca are called approxiation coefficients and the eleents of the vector cd are called detail coefficients. The sybol denotes downsapling i.e. oitting the odd indexed eleents of the filtered signal, so the nuber of the coefficients produced by the basic step is approxiately the sae as the nuber of eleents of the discrete signal s. In the reconstruction step in Fig (b) a pair of filters LR and HR are convolved with the vectors ca and cd respectively. Two signals are produced resulting in a reconstruction signal A called Approxiation, and a reconstruction signal D called Detail. The sybol denotes upsapling e.g. inserting zeros between the eleents of the vectors ca and cd. An iportant property of this step is A D s s = A + D (6) The procedure of the basic step is repeated on the approxiation vector ca and successively on every new approxiation vector ca j. This idea is presented by eans of a wavelet tree with J levels, where J is the nuber of iterations of the basic step.
The nonlinear denoising assues that the noise coponents exist in each coefficient vector and involves a tesholding approach in order to reove those coponents. sgn( y) y t, y > t y =, y < t or in a ore generalized for [] (8) Figure. An exaple of tee level wavelet packet decoposition tree.3 The Wavelet Packet Transfor (WPT) The wavelet packet transfor is a generalization of the wavelet transfor. Let us define two functions W (t)=ö(t), W (t)=ø(t) where ö(t) and ø(t) are the scaling and wavelet functions respectively. Then in an orthogonal case we can write functions W (t), =,,,, as + N- W ()= t hnw ( ) ( t-n) n= N- W ()= t gnw ( ) ( t-n) (7) n= W t W t n j/ -j j,,n() = ( ) where j is a scale paraeter and n is a tie localization paraeter. The analyzing functions W j,,n are called wavelet packet atos. In practice a fast algorith is applied by using the basic step of Fig.. The difference is that both details and approxiations are split into finer coponents, resulting in a wavelet packet tree. In Fig. an exaple of a wavelet packet decoposition tree of tee levels is presented. 3. Denoising 3. Wavelet Based Denoising The wavelet decoposition allows searching an optial decoposition aong L trees if a signal has been decoposed at L levels. Wavelet decoposition involves the selection of an optial decoposition tree aong L. Several criteria have been proposed for the optiization of the decoposition as the Shannon entropy. An application of the wavelet analysis is to reove undesired coponents fro the signal tough a denoising approach. The linear denoising approach assues that the udesired coponents (noise) are located in certain scales and the signal is reconstructed without those coponents. sgn( y)( y qt), y > t y =, y < t where <q< and when q= hard tesholding is applied, when q= soft tesholding is applied. There are several criteria for the selection of teshold [3]: Steins unbiased risk estiate (SURE) is an adapted teshold selection rule. (9) t = log ( nlog ()) n () where n is the nuber of saples of the decoposition level. Fixed teshold approach FIXTHRESH t e = log( n) () calculates the teshold with respect to the length of the signal. The HEURISTIC SURE approach a variant of the SURE approach The MINIMAXI procedure t =.3936+.89log( n) () These odels assue noise distributed with zero ean and variance of and have to be rescaled when dealing with unscaled noise. A ethod proposed in [] is based on the continuous wavelet transfor using the Morlet wavelet. A siple inverse transfor which requires only one integration is used [7] and in a discrete for if A is the doain of a. xt C ψ Wab a 3 () = (, ) da (3) 3 xk ( ) = W( a,k) a (4) C ψ 3. Paraeter selection The purpose of this application is to isolate the ipulsive coponents of the signal and reject the Á
rest of the signal. These characteristics are assued to exist in the largest coefficient values of each coefficient vector. The signal is decoposed using the wavelet packet transfor. It is decoposed at a specific depth L, for exaple L=3 results in 3 coefficient vectors. This decoposition has the advantage that the wavelet coefficients of different coefficient vectors, which belong to the sae level L, are equivalent in ters of signal energy. By selecting decoposition which produces coefficient vectors which belong to the sae level, the sae teshold value for all coefficient vectors can be selected. The teshold is selected as a portion of the axiu of the absolute value of the set of all the wavelet packet coefficients t ax( cof ) = (5) k Hard tesholding is applied because it was found to be ore effective for the purpose of isolation of ipact generated peaks of signals. A variation of this ethod is also proposed. According to this variation the coefficients y are odified according to the forula: th k y = y sign( y) (6) od where k integer. This nonlinear approach odifies the coefficients in a way that coefficients with larger absolute values contribute ore in the reconstruction of the final signal. The RMS level of the signal is odified because of Eq. 6.This is a qualitative process aiing at isolating of ipulsive characteristics. 4 Results and evaluation The ethod is tested on ipulsive vibration signals resulting fro defective rolling eleent bearings. Defects or wear cause ipacts at frequencies governed by the operating speed of the unit and the geoetry of the bearings, which in turn excite various achine natural frequencies. For exaple the characteristic defect frequency f o of a bearing with an outer race fault is d fo =.5 z( cos a) f (7) E where d is the roller diaeter, E is the pitch diaeter, z is the nuber of rolling eleents, a is the angle of contact. The general assuption with rolling eleent bearing faults is that a easured signal contains a low-frequency phenoenon that acts as the odulator to a high-frequency carrier signal. In bearing failure analysis, the low-frequency phenoenon is the ipact caused by a defect of a bearing; the high-frequency carrier is a cobination of the natural frequencies of the associated rolling eleent or even of the achine. The goal of denoising is to suppress the oscillation caused by each ipact and isolate an ipulse sequence. Two characteristic industrial cases are presented. Both signals were supplied by Alouinu of Greece S.A sapled at 8.33 khz. The bearing exained in Case A is of type 8cck/w33 anufactured by SKF. In case B the type of the bearing is8c3. In both cases the signals were wavelet packet decoposed up to the level L=3, the Daubechies wavelet db4 was used. k th =.5 and k =5 were selected. In case A, an extended outer race fault (fluting) was created on the outer race by electric arc caused by electric welding in the background of the bearing. In Fig. 3 the easured tie wavefor is illustrated. The signal is denoised by applying Eq.(5). The resulting wavefor is illustrated in Fig. 4. Peaks spaced at the characteristic outer race defect period (approxiately 5. s) are observed and the shaft rotation frequency odulation becoes clearer. In case B the defect was on the outer race, but it has a localized shape. The tie doain shape in Fig. 5 does not reveal clear ipulsive characteristics. In Fig. 6 the denoised signal applying Eq. (5) is presented. Peaks spaced at the characteristic outer race defect period (approxiately 6.4 s) are observed. In Fig. 7 the denoised signal applying Eq. (6) is presented. Peaks spaced at the characteristic outer race defect period are also observed. Spacing of the ipacts is approxiate due to speed variation and sliding effects. 5 Discussion - Conclusion Denoising of vibration signals by odifying wavelet packet coefficients was presented. It offers a better visual inspection of the ipulsive content of the tie doain signal. The wavelet packet denoising can be helpful when used in cobination with the traditional frequency doain ethods. It akes diagnosis of faults safer, since the interpretation of peaks in the frequency spectru ay have ultiple interpretations. Hard tesholding was applied. Daubechies wavelets were used.
3 A p l i t u d e ( g ) - - -3..4.6.8. Ti e ( s ) Figure 3. A vibration signal easured on a bearing with an extended outer race fault (case A) 3 A p l i t u d e ( g ) 5 5 5-5 - -5 - f o f o f R -5..4.6.8. Ti e ( s ) Figure 4. Denoised signal of Fig. 3 by tesholding WP coefficients. Aplitude (g) 5-5 - -5..4.6.8. Tie (s) Figure 5. Vibration signal fro an outer race localized defect (case B) 8 6 Aplitude 4 - -4-6 -8..4.6.8. Tie (s) Figure 6. Denoised signal of Fig. 5 by thesholding WP coefficients
Aplitude x 4 8 6 4 - -4-6 f o -8..4.6.8. Tie (s) Figure 7. Denoised signal of Fig. 5 by odifying WP coefficients using Eq.(6) It was observed that the use of lower order wavelets results in ore poor representation of the frequency content of each ipact response, but results in a ore clear detection of the presence of ipacts. The wavelet packet ethod is sipler than the Morlet denoising ethod a characteristic of which is redundancy. The Morlet based denoising ethod sees to be ore effective than the wavelet packet denoising ethod in extracting the frequency content of each ipulse response. However, in the tested cases the interesting diagnostic inforation is the periodicity and the intensity of the ipacts rather than the frequency content of each ipact response. In several cases, hard tesholding sees to be ore effective than soft tesholding in ters of intensity of the ipacts. The selection of tesholding level was selected as a fixed portion of the axiu absolute value of the coefficients aking this choice sipler. However, the selection of teshold affects directly the resulting tie wavefor. The choice of teshold reains an open atter and should be the subject of future work. References: [] R.B. Randall, Frequency Analysis, 3 rd Ed, Bruel & Kjaer, 987 [] N. Tandon, A. Choudhury, A review of vibration and acoustic easureent ethods for the detection of defects in rolling eleent bearings, Tribology International, Vol.3, 999, pp. 469-48 [3] P.W. Tse, Y. H. Peng, and R. Ya, Wavelet analysis and envelope detection for rolling eleent bearing fault diagnosis-their effectiveness and flexibilities, ASME Journal of Vibration and Acoustics, Vol.3, No 3,, pp. 33-3. [4] Dialpaz G., Rivola A, Rubini R., Effectiveness and sensitivity of vibration processing techniques for local fault detection of gears, Mechanical systes and signal processing, Vol.4(3),,, pp. 387-4. [5] I. Daubechies, Ten lectures on wavelets, SIAM, 99. [6] G. Strang, T. Nguyen, Wavelets and Filter Banks, Wellesley-Cabridge Press, 996. [7] M. J. Shensa, Discrete inverses for nonorthogonal wavelet transfors, IEEE transactions on signal processing, Vol.44. No 4, 996, pp. 798-87. [8] D. Donoho, De-noising by soft-tesholding, IEEE Transactions Infor Theory, Vol.4, 995, pp. 6-67. [9] D.E. Newland, An Introduction to Rando Vibration, Spectral and Wavelet Analysis, Harlow, Longan, 993 [] J.C. Li, J. Ma, Wavelet decoposition of vibrations for detection of bearing-localized defects, NDT&E International, Vol.3, No.3, 997, pp.43-49 [] P. D McFadden, J. G. Cook and L. M. Forster, Decoposition of gear vibration signals by the generalized S transfor, Mechanical Systes and Signal Processing, Vol.3, No.5, 999, pp. 69-77. [] J. Lin., L. Qu, Feature extraction based on Morlet wavelet and its application for echanical fault diagnosis, Journal of Sound and Vibration, Vol.34 No.,, pp.35-48. [3] P. E Tikkanen, Nonlinear wavelet and wavelet packet denoising of electrocardiogra signal, Biological Cybernetics,Vol. 8,999, pp.59-67. [4] J. Altan, Mathew J., Multiple band pass autoregressive deodulation for rolling eleent bearing fault diagnosis,, Mechanical Systes and Signal Processing, Vol.5, No 5,, pp. 963-997. [5] Y. Wu, R. Du, Feature extraction and assessent using wavelet packets for onitoring of achining processes, Mechanical Systes and Signal Processing, Vol., No., 996, pp.9-53.