Twelfth International Congress on Sound and Vibration OPTIMIZATION OF MORLET WAVELET FOR MECHANICAL FAULT DIAGNOSIS Jiří Vass* and Cristina Cristalli** * Dept. of Circuit Theory, Faculty of Electrical Engineering, Czech Technical University Technická, 166 7 Prague, Czech Republic vassj@fel.cvut.cz ** The Loccioni Group, AEA s.r.l. Via Fiume 16, 600 30 Angeli di Rosora, Italy Abstract Condition monitoring of rotating machines is commonly based on analysis of machine vibrations. In the presence of a mechanical fault, vibration signals comprise periodic impulses with a characteristic frequency corresponding to a particular defect. However, due to a heavy noise in the industrial applications, vibration signals have very low signal-to-noise ratio, thus requiring the extraction of impulses by an appropriate technique. Therefore, a novel denoising method based on the Morlet wavelet with adaptive time-frequency resolution has recently been proposed. However, this adaptation may under certain conditions violate the admissibility condition on mother wavelets, resulting in severe distortion of the wavelet scalogram. This paper presents a modification of the Morlet wavelet, restricting the adaptive parameter to a specific allowable range. As a result, undesired scalogram distortion is avoided, thus guaranteeing stable performance of the de-noising algorithm. Consequently, the improved method provides more reliable fault diagnosis, as well as higher computational efficiency, suitable for industrial production applications. INTRODUCTION Rolling bearings are essential mechanical parts of various rotating machines, such as universal motors for electric household appliances. Therefore, detection and identification of bearing faults has been the subject of extensive research [1,], in
order to guarantee high standards of quality control on production lines. Vibration analysis is the most common tool for diagnosis of manufacturing defects, although vibration signals have a low signal-to-noise ratio (SNR), caused by large noise from electric brushes and windings of motors [3]. For this reason, it is desirable to suppress the noise by an appropriate pre-processing, prior to classification of faults. Since it is well known that periodic impulses correspond to the repetition frequency of bearing defects [1], a novel wavelet-based denoising method has been proposed [4,5], employing the similarity between the impulse signal and the Morlet wavelet. Furthermore, the extraction of impulses with variable decay is provided by an adaptive parameter β controlling the time-frequency resolution of the Morlet wavelet. However, our study reveals that the parameter β must be adapted only within a limited range in order to meet mathematical requirements on wavelet functions. This paper hence proposes a modification of the Morlet wavelet, defining the solution to avoid possible adaptation problems. THEORY OF THE WAVELET TRANSFORM The continuous wavelet transform (CWT) of a finite energy signal x(t) with an analyzing wavelet ψ(t) is the convolution of x(t) with a scaled and conjugated mother wavelet: W x (a,b) = + x(t)ψa,b(t)dt = 1 + a x(t)ψ ( t b a ) dt, (1) where the asterisk denotes complex conjugation and ψ a,b (t) is a family of daughter wavelets, defined by the dilation parameter a and the translation parameter b. CWT is generally a non-local transform, since the value of the wavelet coefficient W x (a,b) at a point (a 0,b 0 ) depends on x(t) for all t, as given in Eq. (1). Therefore, the decay of ψ(t) must be sufficiently fast in order to ensure the localized time-frequency analysis using CWT. Moreover, the wavelet function must satisfy the admissibility condition [8]: C ψ = + 0 Ψ(ω) dω <, () ω where Ψ(ω) is the Fourier transform of ψ(t) and C ψ is the admissibility constant depending on the chosen wavelet [9]. The weak condition for existence of the integral in Eq. () requires the DC spectral component (the mean value) to be zero: Ψ(0) = µ ψ = + ψ(t)dt = 0. (3)
The Morlet wavelet The Morlet wavelet was proposed in [8], although numerous variations can be found in literature [9]. In this paper, we use the real-part of the complex Morlet wavelet: ( ) β ψ(t) = e β t / cos(ω c t). (4) π The term in brackets represents a Gaussian window derived from the probability density function of the Gaussian distribution N(µ, σ), putting µ = 0 and substituting 1/σ=β. The second term is the modulating harmonic signal with the center frequency ω c determining the number of significant oscillations of the Morlet wavelet [9]. As implies from Eq. (4), the bandwidth parameter β controls the decay rate of the exponential envelope in time, and hence regulates the time resolution of ψ(t). Simultaneously, β corresponds to the frequency bandwidth of the Gauss filter Ψ(ω), and thus determines the resolution in the frequency domain. For this reason, it is possible to find the optimal value β opt for a given vibration signal, adjusting the time-frequency resolution of CWT to the decay rate of impulses to be extracted. PRINCIPLE OF THE DENOISING METHOD [5] Since wavelet coefficients measure the similarity between the signal x(t) and the wavelet family ψ a,b (t), fundamental task in application of CWT is always a proper selection of the mother wavelet. Lin [4] ingeniously proposed the use of the Morlet wavelet, since its shape is similar to an impulse, resulting in high correlation between x(t) and ψ(t). As a consequence, impulses are visually magnified in the wavelet scalogram W x (a,b), and correspond to a small number of large coefficients. On the other hand, noise is uncorrelated with the Morlet wavelet, which leads to a representation by a large number of small wavelet coefficients. As a result, noise can effectively be cancelled in the wavelet domain using generalized soft thresholding [5] or thresholding based on maximum likehood estimation (MLE) [7]. Finally, the denoised vibration signal is reconstructed using the inverse CWT. As the success of the method depends on adaptation of the parameter β, Lin and Qu [5] designed a criterion for finding the optimal β opt using Shannon entropy of wavelet coefficients. The criterion is referred to as the minimum wavelet entropy criterion and operates as follows. First of all, each of the candidate values β k is utilized to compute the CWT of the noisy signal. Then, wavelet coefficients are normalized, and used for calculation of a single value of Shannon entropy E k. Finally, the value of β corresponding to the minimum of wavelet entropy is chosen as optimal, and applied in denoising of the vibration signal. However, the interval of allowable values β k was imprecisely selected in [5] and [6], resulting in unstable performance of the algorithm. Therefore, our research
was concentrated on solving this problem, yielding redefinition of the Morlet wavelet, summarized in Table 1 and discussed in the following sections. FULFILMENT OF THE WAVELET CONDITIONS Figure 1 presents the influence of the parameter β on the time-domain shape of the Morlet wavelet with fixed center frequency ω c = π rad/s. As can be observed, there are three typical cases of the mother wavelet ψ(t). First, the original selection β = 1 by Goupillaud et al. [8] yields the only correct shape satisfying the wavelet conditions. Second, the shape for β = 0.3 represents the case of an excessively small value, resulting in unfulfillment of the fast decay condition. Indeed, it is obvious that the wavelet function is nonzero outside the support interval ( 4, 4) due to insufficient decay of the Gaussian window with small β. Finally, the shape for β = 3 illustrates the tendency of the Morlet wavelet to become a Dirac impulse as β increases to infinity. The mean value of ψ(t) is clearly nonzero, which results in violation of the zero-mean condition (3) and consequent inadmissibility of the Morlet wavelet. Let us therefore examine the mean value function µ ψ = f (β,ω c ) depicted in Fig. 3. As can be noticed, µ ψ rapidly increases with increasing β and constant ω c = π rad/s. This property is in accordance with the previous observation that the Morlet wavelet gradually approaches the Dirac impulse for β (Fig.1), hence demanding the limitation of β to a certain maximal value. In addition, since µ ψ oscillates around zero with varying center frequency, certain values of ω c appear to be highly suitable for construction of the Morlet wavelet, significantly contributing to µ ψ 0. Therefore, we intend to select a profitable value of ω c in order to achieve fulfilment of the zero-mean condition, as illustrated in Fig.. This graph can be considered as a view from above on the function µ ψ after application of binary thresholding, which divides the graph into two distinctive areas. The white area corresponds to approximate fulfilment of the zero-mean condition: µ ψ 0.005, whereas the black area denotes the complementary zone. The white area comprise the desirable center frequencies repeated with the period π/4, which can readily be explained using the DFT spectrum of the Morlet wavelet. Leakage in DFT of the Morlet wavelet It is well-known that frequency spectra may exhibit unwanted leakage caused by unfulfillment of the coherent sampling condition [10], which can be rewritten for a wavelet function as: N f w = (M+ α) π ω c (5) where f w is the sampling frequency of the wavelet, M is the number of complete periods, N = 104 is the number of DFT points, and α < 0.5 is the displacement term.
1. 0.8 β = 0.3 β = 1 β = 3 π 7/4π ψ(t) 0.4 ω c [rad/s] 6/4π 0 5/4π 0.4 4 0 t [ ] 4 Figure1 Influenceofthe parameter β ontheshapeofthemorletwavelet π 0.5 1 1.5.5 β [ ] Figure Approximatefulfilment of thezero-meancondition µ ψ [ ] 0.08 0.06 0.04 0.0 0 0.0 π 7/4π ω c [rad/s] 6/4π 5/4π π 0 0.5 1 1.5 β [ ].5 Figure 3 Mean value of the Morlet wavelet as a function of principal parameters
scales a scales a 0 18 16 14 1 10 8 6 4 (a) 0 100 00 300 time b [Sa] 0 18 16 14 1 10 8 6 4 (c) 0 100 00 300 time b [Sa] scales a scales a 0 18 16 14 1 10 8 6 4 (b) 0 100 00 300 time b [Sa] 0 18 16 14 1 10 8 6 4 (d) 0 100 00 300 time b [Sa] Figure 4 Scalograms of simulated signals using the Morlet wavelet with fixed β = 1.83 and variable ω c [rad/s]: (a) clean signal, ω c = π, (b) clean signal, ω c = 7 4π, (c) noisy signal, ω c = π, (d) noisy signal, ω c = 7 4 π Parameter Original value Modified value normalization factor 1 β/ π center frequency, ω c [rad/s] π 7 4 π allowable range for β 0.1,4 0.5,1.83 number of tested values of β 40 6 Table 1 Modification of the Morlet wavelet parameters
When this condition is not met (i.e. α 0), leakage significantly modifies various spectral components, including the DC offset Ψ(0) = µ ψ. It is thus clear that the zeromean condition can conveniently be satisfied, selecting the center frequency in such a manner that cos (ω c t) in Eq. (4) is sampled at the integer number of periods M. Moreover, properly chosen ω c not only avoids spectral leakage, but also enables extension of allowable range for β, as depicted in Fig.. SCALOGRAM DISTORTION This section is concerned with scalogram distortion caused by inappropriate handling of the Morlet wavelet. In order to demonstrate importance of the problem, we have simulated a clean vibration signal using amplitude modulation between a harmonic signal and an impulse train with an exponential decay [11]. The clean signal consists only of a single impulse located at sample 100, whereas its noisy version was generated by adding white Gaussian noise (SNR = -5 db). Figure 4(a) shows scalogram of the clean signal, evidently blurred due to inadmissibility of the Morlet wavelet with the original center frequency and β belonging to the original interval. In contrast, Figure 4(b) depicts the improved scalogram with modified ω c, in which the wavelet coefficients corresponding to the impulse are clearly visible. Furthermore, the impulse remains sufficiently distinguishable even in the noisy coefficients displayed in Fig. 4(c). On the other hand, noisy scalogram with the original parameters provides little information about the impulse feature, as shown in Fig. 4(d). In fact, Figure 4(a) and (d) cannot be regarded as correct CWT scalograms, since properties of the wavelet transform have been violated, resulting in impossible reconstruction of the time-domain signal. SUMMARY In this paper, we have extended the research of Jing Lin et al. [5,6] proving that the adaptive Morlet wavelet cannot be varied arbitrarily in order to avoid scalogram distortion. Table 1 provides the modified parameters of the wavelet in comparison with the original values. Specifically, normalization factor was added to Eq. (4), center frequency was increased, and allowable range for adaptive parameter β was strictly limited in order to satisfy fundamental conditions on mother wavelets. Moreover, the number of examined values of β within the range was reduced, requiring only 15% of the original computational time. Primarily, advantages of Morlet wavelet adaptation have been preserved, while ensuring stable results of the denoising algorithm [5], necessary for successful feature extraction.
ACKNOWLEDGEMENTS This work has been supported by the GA ČR grant No. 10/03/H085 Biological and Speech Signal Modelling, the research program MSM6840770014 Research in the Area of the Prospective Information and Navigation Technologies, and the CTU internal grant 13016D/05/A. This research is supervised by Prof. Ing. Pavel Sovka, CSc., Department of Circuit Theory, FEE CTU and Ing. Radislav Šmíd, Ph.D., Department of Measurement, FEE CTU. The help of Mgr. Zuzana Lenochová, Eng. Barbara Torcianti and Ing. Petr Prášek, Ph.D. is much acknowledged. REFERENCES [1] Tandon N., Choudhury A., A review of vibration and acoustic measurement methods for the detection of defects in rolling element bearings, Tribology International, 3 (8), 469-480 (1999) [] Peng Z.K., Chu F.L., Application of the wavelet transform in machine condition monitoring and fault diagnostics: a review with bibliography, Mechanical Systems and Signal Processing, 18 (), 199-1 (004) [3] Rodriguez R.M., Cristalli C., Paone N., Comparative study between laser vibrometer and accelerometer measurements for mechanical fault detection of electric motors, SPIE Proceedings, 487, 51-59 (00) [4] Lin J., Feature extraction of machine sound using wavelet and its application in fault diagnostics, NDT&E International, 34, 5-30 (001) [5] Lin J., Qu L., Feature extraction based on Morlet wavelet and its application for mechanical fault diagnosis, Journal of Sound and Vibration, 34 (1), 135-148 (000) [6] Lin J., Zuo M.J., Gearbox fault diagnosis using adaptive wavelet filter, Mechanical Systems and Signal Processing, 17 (6), 159-169 (003) [7] Lin J., Zuo M.J., Fyfe K.R., Mechanical fault detection based on the wavelet de-noising technique, Journal of Vibration and Acoustics, 16 (1), 9-16 (004) [8] Goupillaud P., Grossmann A., Morlet J., Cycle-octave and related transforms in seismic signal analysis, Geoexploration, 3, 85-10 (1984) [9] Addison P. S., Watson J. N., Feng T., Low-oscillation complex wavelets, Journal of Sound and Vibration, 54 (4), 733-76 (00) [10] Sedláček M., Titěra M., Interpolations in frequency and time domains used in FFT spectrum analysis, Measurement, 3, 185-193 (1998) [11] Sheen Y.-T., A complex filter for vibration signal demodulation in bearing defect diagnosis, Journal of Sound and Vibration, 76 (1-), 105-119 (004)