application of the wavelet transform

Mechanical Systemsand

Signal Processing

www.elsevier.com/locate/jnlabr/ymssp

Mechanical Systems and Signal Processing 18 (2004) 199–221

Review

Application of the wavelet transform in machine conditionmonitoring and fault diagnostics: a review with bibliography

Z.K. Peng, F.L. Chu*

Department of Precision Instruments, Tsinghua University, Beijing 100084, People’s Republic of China

Received 26 November 2002; received in revised form 22 April 2003; accepted 25 April 2003

Abstract

The application of the wavelet transform for machine fault diagnostics has been developed for last 10years at a very rapid rate. A review on all of the literature is certainly not possible. The purpose of thisreview is to present a summary about the application of the wavelet in machine fault diagnostics, includingthe following main aspects: the time–frequency analysis of signals, the fault feature extraction, thesingularity detection for signals, the denoising and extraction of the weak signals, the compression ofvibration signals and the system identification. Some other applications are introduced briefly as well, suchas the wavelet networks, the wavelet-based frequency response function, etc. In addition, some problems inusing the wavelet for machine fault diagnostics are analysed. The prospects of the wavelet analysis insolving non-linear problems are discussed.r 2003 Elsevier Ltd. All rights reserved.

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Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

2. Wavelet transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

3. An application overview of wavelet in fault diagnosis . . . . . . . . . . . . . . . . . . . . . 202

3.1. Time–frequency analysis of signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

3.2. Fault feature extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

3.3. Singularity detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

3.4. Denoising and extraction of the weak signals . . . . . . . . . . . . . . . . . . . . . . 209

3.5. Vibration signal compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

3.6. System and parameter identification . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

3.7. Other applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

*Corresponding author.

E-mail address: [email protected] (F.L. Chu).

0888-3270/04/$ - see front matter r 2003 Elsevier Ltd. All rights reserved.

doi:10.1016/S0888-3270(03)00075-X

1. Introduction

Condition monitoring and fault diagnostics is useful for ensuring the safe running of machines.Signal analysis is one of the most important methods used for condition monitoring and faultdiagnostics, whose aim is to find a simple and effective transform to the original signals.Therefore, the important information contained in the signals can be shown; and then, thedominant features of signals can be extracted for fault diagnostics. Hitherto, many signal analysismethods have been used for fault diagnostics, among which the FFT is one of the most widelyused and well-established methods. Unfortunately, the FFT-based methods are not suitable fornon-stationary signal analysis and are not able to reveal the inherent information of non-stationary signals. However, various kinds of factors, such as the change of the environment andthe faults from the machine itself, often make the output signals of the running machine containnon-stationary components. Usually, these non-stationary components contain abundantinformation about machine faults; therefore, it is important to analyse the non-stationary signals[1]. Because of the disadvantages of the FFT analysis, it is necessary to find supplementarymethods for non-stationary signal analysis. Time–frequency analysis is the most popular methodfor the analysis of non-stationary signals, such as the Wigner–Ville distribution (WVD) [2] and theshort time Fourier transform [3] (STFT). These methods perform a mapping of one-dimensionalsignal xðtÞ to a two-dimensional function of time and frequency TFRðx : t;oÞ; and therefore areable to provide true time–frequency representations for the signal xðtÞ: But each of the time–frequency analysis methods has suffered some problems. It is no doubt that the WVD has goodconcentration in the time–frequency plane. However, even support areas of the signal do notoverlap each other, interference terms will appear on the time–frequency plane. This will misleadthe signal analysis. In order to overcome these disadvantages, many improved methods have beenproposed, such as Choi–Willams distribution (CWD) and cone-shaped distribution (CSD), etc.Without exception, however, elimination of one shortcoming will always lead to the loss of othermerits. For example, the reduction of interference terms will bring the loss of time–frequencyconcentration [4]. The problem with STFT is that it provides constant resolution for allfrequencies since it uses the same window for the analysis of the entire signal. This means that ifwe want to obtain a good frequency resolution using wide windows, which is desired for theanalysis of low-frequency components, we would not be able to obtain good time resolution(narrow window), which is desired for the analysis of high-frequency components. Therefore, theSTFT is suitable for the quasistationary signal analysis (stationary at the scale of the window butnot the real stationary signals). Moreover, there exist no orthogonal bases for STFT, therefore itis difficult to find a fast and effective algorithm to calculate STFT.

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4. Prospects of wavelet in fault diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

5. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

Z.K. Peng, F.L. Chu / Mechanical Systems and Signal Processing 18 (2004) 199–221200

2. Wavelet transform

Over the past 10 years, wavelet theory has become one of the emerging and fast-evolvingmathematical and signal processing tools for its many distinct merits. The novel concept ofwavelet was first put forward definitely by Morlet in 1984. However, at that time, Morlet facedmuch criticism from his colleagues. Later, under the help of Grossman, Morlet formalised thecontinuous wavelet transform (CWT), shown as Eq. (1), and devised the inverse transform

Wxða; b;cÞ ¼ a�1=2Z

xðtÞc�t � b

a

� �dt; ð1Þ

where a is the scale parameter, b is the time parameter, cðtÞ is an analysing wavelet, and the c�ðdÞis the complex conjugate of cðdÞ:In 1985, Meyer constructed a beautiful orthogonal wavelet base with very good time and

frequency localisation properties. In the next year, Meyer and Mallat, a graduate student atUpenn, developed the idea of multi-resolution analysis (MRA) that made it very easy to constructother orthogonal wavelet bases. A more important event was that the MRA led to the famous fastwavelet transform—a simple and recursive filtering algorithm to compute the waveletdecomposition of the signal from its finest scale approximation. Before long, Daubechiesconstructed orthogonal wavelet bases compactly supported in a simple but ingenious way. Inaddition, Daubechies has done many research on wavelet frames that allow more liberty in thechoice of the basis wavelet functions at a little expense of some redundancy, and the ‘‘TenLectures on Wavelets’’ by Daubechies has also been playing an important role for thepopularisation of the wavelet. Daubechies, along with Mallat, is therefore credited with thedevelopment of the wavelet from continuous to discrete signal analysis. In the discrete waveletformalism (DWT), the scale a and the time b are discretised as following:

a ¼ am0 ; b ¼ nam

0 b0; ð2Þ

where m and n are integers. So the continuous wavelet function ca;bðtÞ in Eq. (1) become thediscrete wavelets given by

cm;nðtÞ ¼ a�m=20 cða�m

0 t � nb0Þ: ð3Þ

The discretisation of the scale parameter and time parameter leads to the discrete wavelettransform, defined as

Wxðm; n;cÞ ¼ a�m=20

ZxðtÞc�ða�m

0 t � nb0Þ dt: ð4Þ

In 1992, Coifman, Meyer and Wickerhauser developed the wavelet packet, which is a naturalextension of the MRA.Different from the STFT, the wavelet transform can be used for multi-scale analysis of a signal

through dilation and translation, so it can extract time–frequency features of a signal effectively.Therefore, the wavelet transform is more suitable for the analysis of non-stationary signals [4].Now, the wavelets have obtained great success in machine fault diagnostics for its many distinctadvantages, not only for its ability in the analysis of non-stationary signals.Table 1 gives a comparison of performances of CWT, STFT, WVD, CWD, and CSD.

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Z.K. Peng, F.L. Chu / Mechanical Systems and Signal Processing 18 (2004) 199–221 201

3. An application overview of wavelet in fault diagnosis

3.1. Time–frequency analysis of signals

The wavelet transform is a linear transform, whose physical pattern is to use a series ofoscillating functions with different frequencies as window functions ca;bðtÞ to scan and translatethe signal of xðtÞ; where a is the dilation parameter for changing the oscillating frequency.Although the wavelet transform is similar to the STFT in a certain sense, differences betweenthem exist. Compared with the STFT, whose time–frequency resolution is constant, the time–frequency resolution of the wavelet transform depends on the frequency of the signal. At highfrequencies, the wavelet reaches at a high time resolution but a low frequency resolution, whereas,at low frequencies, high-frequency resolution and low time resolution can be obtained. Suchadaptive ability of time–frequency analysis reinforces the important status of the wavelettransform in the fault diagnostics field. In the physical interpretation, the modulus of the wavelettransform shows how the energy of the signal varies with time and frequency. In engineeringapplications, the square of the modulus of the CWT is often called as scalogram, defined asEq. (5), which has been widely used for fault diagnostics

SGxða; b;cÞ ¼ jWxða; b;cÞj2: ð5Þ

As early as 1990, Leducq [5] had used the wavelet to analyse the hydraulic noise of the centrifugalpump. It was maybe the first paper about the use of the wavelet in diagnostics. In 1993, Wang andMcFadden [6] applied wavelet to analyse the gear vibration signals, and found that the wavelet isable to detect the incipient mechanical failure and to detect different types of faultssimultaneously. Further research was also performed by them [7,8]. In about 1994, Newlandpublished several papers successively [9–12], in which the wavelet transform was introducedsystematically, and the basic theory, the methods, application examples about the use of thewavelet in vibration signal analysis were given. Moreover, he proposed a new wavelet—harmonic

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Table 1

Comparison of the performances of the different methods

Methods Resolution Interference term Speed

CWT Good frequency resolution and low

time resolution for low-frequency

components; low frequency

resolution and good time resolution

for high-frequency components

No Fast

STFT Dependent on window function,

good time or frequency resolution

No Slower than CWT

WVD Good time and frequency

resolution

Severe interference terms Slower than STFT

CWD Good time and frequency

resolution

Less interference terms than

WVD

Very slow

CSD Good time and frequency

resolution

Less interference terms than

CWD

Very slow


wavelet and discussed its properties and applications. In 1999, he used the harmonic wavelet toidentify the ridge and phase of the transient signals successfully [13]. Newland’s work made thewavelet popular in engineering applications, especially for vibration analysis; and later on, thewavelets prevailed in the machine fault diagnostics. Gears, as one kind of the most importantcomponents in machines, were probably the most exploited objects by wavelets, which werepioneered by Wang and McFadden [6]. Boulahbal et al. [14] used the scalogram on the residualvibration signal of gears. Some distinctive features of the cracked tooth were obtained and theprecise location of a crack was detected. Wang et al. [15] experimentally investigated thesensitivity and robustness of the currently well-accepted techniques for gear damage monitoring,including the wavelet transform, and the results show that the wavelet transform is a reliabletechnique for gear health condition monitoring, which is more robust than other methods.Dalpiaz [16] has studied a gear pair affected by a fatigue crack. Yesilyurt and Ball [17] used thewavelet to detect the weakened gear teeth caused by bending fatigue cracks and to assess itsseverity. Many other applications included tooth defects in gear systems [18–20], planetary geartrain [21,22] and spur gear [23], etc. The cracks in rotor systems or in structures were anotherimportant objects for the application of the wavelets. Adewusi and Al-Bedoor [24] analysed thestart-up and steady-state vibration signals of the rotor with a propagating transverse crack byscalograms and space-scale energy distribution graphs. The start-up results showed that the crackreduced the critical speed of the rotor system. The steady-state results showed that thepropagating crack caused changes in vibration amplitudes with the frequencies corresponding to1X, 2X and 4X harmonics. The vibration amplitude with the frequency 1X may increase ordecrease depending on the location of the crack and the side load. However, the amplitude withthe frequency corresponding to 2X increases continuously as the crack propagates. Wavelets werealso used for crack detection. Examples include the edge cracks in cantilever beams [25], cracks inthe rotors [26], cracks in beam structures [27] and in smart structures [28], damages instructures [29,30], cracks in metallic structures [31], cracks in composite plates [32,33], etc.Staszewski [34] made a review on structural and mechanical damage detection using wavelets.Besides gears and cracks, many other objects have been the clients of wavelets. Dalpiaz andRivola [35] assessed and compared the effectiveness and reliability of different vibration analysistechniques for fault detection and diagnostics in cam mechanisms, including wavelets. Tse andPeng [36] compared the effectiveness of the wavelet and the envelope detection (ED) methodusing for rolling element bearing fault diagnosis, and the results showed that both the wavelet andED methods are effective in finding the bearing fault, but the wavelet method is less timeexpensive. Peng et al. [37] analysed three kinds of typical faults: rub-impact, oil whirl andcoupling misalignment, which often occur in rotating machines, by scalograms. Further researchon the rub-impact in the rotor system was carried out with scalograms and wavelet phasespectrums [38].Fig. 1 shows an example of the wavelet scalogram derived from the vibration response in an

industrial machine. Here, a two-dimensional contour plot of the scalogram is used, together withthe classical time and frequency domain representations. It can be seen that the scalogram canbetter exhibit the non-stationary of the analysed signal whose frequency components change withthe time.When the complex wavelet is used, the wavelet transform can provide the amplitude and phase

information of the signals simultaneously. The phase spectrum can be calculated easily from the

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wavelet transform, shown as following:

WPxða; b;cÞ ¼ tg�1Im½Wxða; b;cÞ�Re½Wxða; b;cÞ�

� �: ð6Þ

Compared with the scalograms, the phase spectrum of the wavelet transform is much moredifficult to be interpreted. However, it can also provide useful information in particular aboutsignal discontinuities and impulses. It has a very distinct property: for every discontinuity in thesignal, there will be a taper direct to it accurately, which is formed by the bands with constantphases. It is an inherent property for the wavelet phase spectrum no matter what the chosenwavelet function is. Fig. 2 shows the phase spectrum of the wavelet transform that can be used todetect the signal discontinuities.Staszewski and Tomlinson [39] used the wavelet phase to detect the damaged tooth in a spur

gear. Boulahbal et al. [14] used both the scalograms and the phase maps in conjunction to assessthe condition of an instrumented gear test rig, and the phase map was found to be able to displaydistinctive features in the presence of a cracked tooth. Wong and Chen [40] investigated the non-linear and chaotic behaviour of structural systems by scalograms and phase spectrums.

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Fig. 1. A non-stationary vibration signal (a), its frequency representation (b) and wavelet scalogram (c).

Fig. 2. A simulation signal (a) and its wavelet phase map (b) (at the time of 1 s, the signal’s frequency changed and

started to increase with time).


Good ability in time–frequency analysis makes the wavelet qualify for the transient processanalysis. Chancey and Flowers [41] used the harmonic wavelets to identify transient vibrationcharacteristics and found a relationship between the transient vibration patterns and the absolutevalue of the wavelet coefficients. Kang and Birtwhistle [42] developed a wavelet-based techniqueto characterise the vibration burst signals of the power transformer on-load tap-changer (OLTC).This technique can identify the delays between bursts, the number of bursts and the strengths ofbursts well, all of which are important for the condition assessment of OLTC. Yacamini et al. [43]proposed a method to detect shaft torsional vibrations in AC motors and generators from theirstator currents, and the wavelet was used to deal the transient conditions, under which themeasured stator current was non-stationary. Wang [44] applied the time–frequency-scale waveletmap to detect the transients from different mechanical systems. Al-Khalid et al. [45] used thewavelet transform to detect the fatigue damage in structures, which was modelled as a randomimpulse in the input signal. Gaberson [46] used the wavelet transform to identify the location andmagnitude of the transient events in machinery vibration signals.Without doubt, for the adaptive time–frequency analysis ability, the wavelet is generally able to

perform better than other methods, such as the FFT and STFT, etc. Therefore, the wavelet hasbeen widely used for fault diagnostics, which can be seen through the abundant applications asmentioned above. However, the wavelet also has its shortcomings, but rarely taken into accountin the applications. For example, it will always suffer the effects of the border distortion andenergy leakage, and the phase spectrum of the wavelet is not robust to noise, and therefore once asignal is contaminated by noise, its phase spectrum will change greatly. Moreover, since thedefinition of wavelet transform is essentially based on the convolution, the occurrence of theoverlapping is inevitable. The overlapping will cause undesirable frequency aliasing and bring theinterference terms to the scalograms under certain conditions. It can be seen that the scalogramshown by Fig. 1(c) has many interference terms. The overlapping and interference terms willmislead our analysis of signals. To overcome these problems of the wavelets, Tse and Yang havemade lot of efforts. They had proposed a simple but not very accurate algorithm, which wouldemploy the singularity detection method, to deal with the overlapping problem occurring in theCWT [47]; and also presented a new family of DWTs, which mainly consisted of a series ofButterworth filter banks, to lighten the overlapping problem in the DWT case [48]. The reassignedmethod [37] can reduce the interference terms and improve the readability of the scalograms.However, the computing of the reassigned scalogram is time expensive. The reassigned scalogramof the case in Fig. 1(c) is shown as Fig. 3. It can be seen that there are few interference items in thereassigned scalogram.

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Fig. 3. A reassigned scalogram.


3.2. Fault feature extraction

Apart from the original intention of the wavelet transform for the analysis of non-stationarysignals, another very important and successful application of the wavelet in machine faultdiagnostics is fault feature extraction. Due to the compact support of the basis functions used inthe wavelet transforms, wavelets have good energy concentration properties. Most coefficients cmn

are usually very small, and can be discarded without causing a significant error for signal’spresentation. Therefore, the wavelet transform can present the signal with a limited number ofcoefficients. These coefficients usually can be directly used as the fault features. The key problemis which coefficients should be selected as the fault features and can best describe the fault. Therewere already many solutions to this problem, among which the thresholding method is a typicalone, in which the wavelet coefficients are set to zero according to the threshold function

AðcmnÞ ¼cmn; cmn > y;

0; cmnpy;

(ð7Þ

where y is a threshold.Chen et al. [49,50] decomposed dynamic transient signals by the discrete wavelet transform and

selected the wavelet coefficients by the hard-thresholding method; that is, keeping thosecoefficients that are bigger than the constant thresholding and discarding other smallercoefficients. Then, those coefficients, as being fault features, were inputted into an ART net forfault classification. This method has been applied to a refinery fluid catalytic cracking processsuccessfully. Lin and Qu [51] used the wavelet entropy as a rule to optimise parameters of thewavelet function. Then, the vibration signals from the rolling bearing and the gearbox weredecomposed with the wavelet function. Finally, an improved soft-thresholding method was usedto extract the impulse component as fault features from vibration signals. Yen and Lin [52]decomposed the vibration signals with the wavelet packet transform, and selected the coefficientsas fault features with the aid of a statistics-based criterion. Goumas et al. [53] used discretewavelet transform to analyse the transient signals of the vibration velocity in washing machinesand fault features were extracted from the wavelet coefficient. Then minimum distance Bayesclassifiers were used for classification purposes and such a method was used for product qualitycontrol. Similar investigation on washing machines was carried by Stavrakaki et al. [54] but theyused Karhunen Loeve transform to select features from wavelet coefficients and several classifiers’performance was compared. Lu and Hsu [55] presented a wavelet-based method to detect theexistence and location of structural damage. They found that the changes in the waveletcoefficients of the vibration signals were very sensitive to minor localised damage and themaximum change of the wavelet coefficients was often corresponding to the location of thedamage. Liu et al. [56] proposed a wavelet packet-based method for fault diagnostics. Waveletpacket coefficients were used as features. Ball bearings were studied, and the results showed thatthe coefficients had a high sensitivity to faults. Momoh and Dias [57] used both the FFT and thewavelet transform as feature extractors to diagnose the type and location of faults in the powerdistribution system. They concluded that features extracted from wavelet transforms gave betterresults. Ye and Wu [58] calculated the features with wavelet packet decomposition coefficients ofthe stator current to detect the induction motor rotor bar breakage. Momoh et al. [59] compared

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the performances of feature extractors for DC power system faults, including the FFT, theHartley transform and the wavelet transform; and the conclusions showed that the waveletexhibited a superior performance. Altmann and Mathew [60] presented a novel method, whichwas based on an adaptive network-based fuzzy inference system, to select the wavelet packets ofinterest as fault features automatically. It was proved that the method could enhance the detectionand diagnostics of low-speed rolling-element bearing faults. Akbaryan and Bishnoi [61] used PCAtechnique to reduce the size of the feature space extracted from wavelet coefficients. Mufti andVachtsevano [62] fuzzed the fault features extracted by the wavelet transform; and then, a fuzzyinference was applied. Aminian [63] developed an analog circuit fault diagnostic system based onBayesian neural networks using wavelet transform, normalisation and principal componentanalysis as preprocessors. Essawy et al. [64] presented an automated integrated predictivediagnostics method for the monitoring of the health of complex helicopter gearboxes. In thismethod, the neuro-fuzzy algorithm and the sensor fusion were used, and the wavelets were used toanalyse the vibration data and to prepare them for neural network inputs. The applications ofwavelet coefficients can be found in many other works [65,66].Besides the wavelet coefficients, many other wavelets-based fault features were presented, such

as Xu et al. [67] calculated the singularity exponents of the envelops of vibration signals with thewavelet transform-based method and used those singularity exponents as features to diagnose thebreakers’ fault. Hambaba and Huff [68] decomposed the vibration signals from gears in ahelicopter with discrete wavelet transform, and then approximated the wavelet-transformedsignals at each level. Finally, the probability density functions (PDFs) of the residual errors wereexpanded into Hermite polynomial and the coefficients of this expansion were used as faultfeatures for the detection of the early fatigue cracks in gears. Zheng, Li and Chen used the featureenergy of the time-averaged wavelet spectrum as fault information to detect the gear fault in agear-box [69]. Yen and Lin [70] used the wavelet packet node energy selected by Fisher criterionfunction as fault feature and the network as classifier. Seven types of faults of gearboxes in ahelicopter were investigated by this method. A comparison was made with the Fourier-basedfeatures, and the results showed that the wavelet packet-based method was more robust to thewhite noise. Liu and Ling extended the Mallat and Zhang’s matching pursuit [71] to machinediagnostics. The wavelets were treated as features directly for the detection of diesel enginemalfunctions. The results showed that both the sensitivity and the reliability of this method werevery good [72,73]. Osypiw et al. [74] developed a fast Gaussian wavelet algorithm with verynarrow band-pass filtering technique to extract some main features from the vibration signals,such as the significant frequencies, etc. Ren et al. [75] took the wavelet modulus maximum as thefault features to detect and diagnose the faults in a control system. Chen and Wang used theinstantaneous scale distribution of the wavelet transform for quantifying pattern features, then amulti-layer perceptron pattern classifier was used to identify gearbox faults [76]. Peng et al. [77]used the number of wavelet modulus maxima lines and the singularity exponents as features toidentify the shaft centre orbits of the rotating machines. In Ref. [78], Shibata, Takahash and Shiraused the wavelet transform to analyse the sound signals generated by bearings in the time–frequency domain, and the signal component indicative of a fault was identified. In addition, asymmetrised dot pattern method was also described, which can visualise the sound signals in adiagrammatic representation, and so, it was possible to distinguish differences between normaland faulty bearings.

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In conclusion, based on the wavelet transform, many kinds of fault features can be obtained, allof which can be classified as the wavelet coefficients based, wavelet energy based, singularity basedand wavelet function based, etc. roughly. Since the wavelet coefficients will highlight the changesin signals, which often predicate the occurrence of the fault, the wavelet coefficients-based featuresare relatively suitable for early fault detection. However, because the slight changes in signalsoften have small energy these changes will be easily masked in the wavelet energy-based features.Therefore, the wavelet energy-based features are often not able to detect the early faults. Thesingularity-based features will easily suffer from the influence of noise, even the slight noise willcause the remarkable change in the singularities, and therefore how to lighten the influence ofnoise is worth great research efforts when using the singularity based features.

3.3. Singularity detection

Most information of a signal is often carried by the singularity points, such as the peaks, thediscontinuities, etc. Moreover, at the moment when faults occur, the output signals usuallycontain jump points that often are singularity points. Therefore, singularity detection has playedan important role in fault diagnostics. The polynomial trends in the signals could mask the localweak singularities in signals and this caused some methods failing to detect those singularities. Onthe other hand, the wavelet function can be chosen as the orthogonal to polynomial behaviour ofarbitrarily high order, and therefore can remove the polynomial trends and highlight thesingularity points in signals, thus the singularity points can be detected easily by the wavelet-basedmethods. The wavelet modulus maxima method has been almost a standard method for thedetection of singularity points [79], in which the wavelet modulus maxima lines play an importantrole. The modulus maxima line consists of the points that are local maxima in the time–scaleplane, and whenever the analysed signal xðtÞ has a local H .older exponent hðt0ÞoN (the vanishingmoment of the wavelet function c), there is at least one modulus maxima line pointing towards t0;along which the wavelet coefficients have the scaling behaviours as follows:

jWxða; t0;cÞjBahðt0Þ; ð8Þ

where the hðdÞ are the H .older exponents, which are also often used as fault features. Fig. 4 showsa vibration signal sampled from a rotor with serious rub-impact malfunction and its waveletmodulus maxima.

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Fig. 4. A vibration signal (a) and its wavelet modulus maxima (b).


Now, the wavelet transforms are often used for the detection of the singularity points in outputsignals sampled from the machines, furthermore, for fault diagnostics. Sun and Tang appliedwavelet transform modulus maxima to detect abrupt changes in the vibration signals obtainedfrom operating bearings being monitored [80]. Ruiz, Nougues and Calderon et al used thewavelets to determine the singularities of the transients and to reduce the dimensionality of thedata. Then the processed signals were input into an ANN for the fault classification. This methodwas applied to a batch chemical plant [81]. Tang and Shi [82] combined the dyadic wavelettransform method and the singularity analysis method to separate the weak reflected signals fromthe defects from the noise, and, furthermore, to detect the weak bonding defects occurring insolid-phase welded joints. Dong et al. [83] introduced five kinds of applications of the wavelets inpower system fault signal analysis, including fault location identification through the singularitydetection technique. Jia et al. [84] employed the singularity detection with wavelets to obtain thepolarities and magnitudes of the abrupt change of voltage and current caused by the fault, anddetermined the faulty circuit through the comparison of the polarities and magnitudes. Thismethod was applied for single-phase-to-ground fault analysis. Chen and Lu [85,86] introduced amethod that used the wavelet transform to detect the singular signal and its singularity andextended this method to fault diagnostics for the electro-hydraulic-servo system. Lin et al. [87]used the similar method to analyse the vibration signals of the reciprocating compressor valve.Zhang et al. [88] applied the wavelet-based singularity detection method to detect the position ofthe rub fault occurring in the rotating machines.Undoubtedly, the wavelet transforms are very successful in singularity detection, but when

using the wavelet transforms to detect singularity, regularity could be an important criterion inthe selection of a wavelet function. Usually, the selected wavelet must be sufficiently regular,which implies a long filter impulse response; otherwise some singularities would be overlooked.Additionally, the noise will influence the performance of the wavelet greatly, therefore before thesingularity is detected, the signal preprocessing must be carried out.

3.4. Denoising and extraction of the weak signals

Signal preprocessing is an important step to enhance the data’s reliability and, thereby, toimprove the accuracy of the signal analysis. The core of signal preprocessing is to increase thesignal-to-noise ratio (SNR), that is, to remove the noise and to highlight the signals interested.However, the noise is generally unavoidable, which is usually introduced into signals by variousdisturbances, such as the disturbance from the exotic environment, and from testing instrumentself, etc. Denoising and extraction of the weak signals are very important for fault diagnostics,especially for early fault detection, in which cases features are often very weak and masked by thenoise. The noises are often stochastic signals with broadband, whose frequency band will overlapwith the interested signals’. Therefore, it is difficult to eliminate the noise from the signalseffectively with general filtering methods. In addition, traditional methods require someinformation and assumptions about the signals that one wants to extract from the noise, suchas which class the signal belongs to. With wavelets, it is enough to know that a signal belongs to amuch family, which often includes many more classes, but to know anything more. Just asDonoho [89] said, ‘‘you do as well as someone who makes correct assumptions, and much betterthan someone who makes wrong assumptions.’’ An orthogonal wavelet transform can compress

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the ‘‘energy’’ of the signal in a relatively small number of big coefficients, while the energy of thewhite noise will be dispersed throughout the transform with relatively small coefficients. It gives usmore options to select some sample methods to eliminate the noise. Now, a lot of wavelet-basedmethods for the denoising have been available, for example, the soft-thresholding method [90] byDonoho, and the wavelet shrinkage denoising by Zheng and Li [91].The super merits of the wavelet in the denoising make it to be used widely for signal

preprocessing in the fault diagnostics field. Altmann and Mathew [92] used the discrete waveletpacket analysis-based multiple band-pass filtering to deal with the vibration signals from a low-speed rolling-element bearing and good results were obtained with a significantly improved SNRcompared to its high-pass counterpart. Littler and Morrow [93] applied the discrete wavelettransform for the denoising for power system disturbance signals, and the transient fault signalswere thus enhanced. Yang and Liao [94] proposed a wavelet-based denoising approach, in whichthe threshold of eliminating the noises will be adjusted adaptively according to the backgroundnoises. This method was used in a power quality monitoring system to achieve the purpose of theeasy and correct detection and localisation of the disturbances in the power systems. Menon et al.[95] used the wavelet-based method to eliminate the background operational noises, which weretroublesome in using the acoustic emission technology to detect small fatigue cracks in rotor headcomponents. Shao et al. [96] used the wavelet to preprocess the fault signals, the noises and thespikes were removed, and then the wavelet coefficients obtained were used as the inputs of thenon-linear PCA algorithm for the process performance monitoring. Pineyro et al. [97] comparedthe performances of three methods in the detection of the localised defects in rolling elementbearings, including the second-order power spectral density, the bispectral technique and thewavelet. The wavelet was proved to be useful in the short transient detection for the reason that itcould eliminate the background noise. Lin [98] applied the wavelet-based method to remove thenoise from the machine sound, and, furthermore, to extract the fault features for diagnostics.Watson and Addison [99] employed the wavelet transform modulus maxima filtering to the non-destructive testing signals of the piled foundations. It was proved that the technique allowed forthe effective partitioning of sonic echo signal and noise. Bukkapatnam et al. [100] coupled theneighbourhood method and the wavelet method for signal separation of the chaotic signals. Itscapability was proved by numerical studies and the vibration signal analysis sampled from a weartool in machining. Krishnan and Rangayyan [101] presented a novel wavelet-based denoisingmethod to improve the SNR of the knee joint vibration signals. Hu and Zhou [102] applied thewavelet transform modulus maxima method to eliminate the noise from the residual signal and,therefore, to improve the robustness of the fault detection. This method was applied to thestructure fault detection in the fighter. Liu et al. [103] used the wavelet to preprocess the dieselcylinder vibration signals, mainly focused on the denoising. Duan and Zhang [104] used thewavelet to filter the noise to purify the centre orbit of a rotor.The principle of the wavelet for denoising is different from that of the traditional filter-based

method. In brief, in the filter-based methods, the frequency components outside a certain rangeare often set to zero, which may cause some useful fault information to be lost since some burstfaults often appear as impulses in signals and these impulses always cover a wide frequency range,therefore the filter-based denoising methods will smooth some impulses. On the contrary, thewavelet-based methods are often to set some small wavelet transform coefficients to zero, whichcan retain the impulses in signals well because those impulses are often represented as some big

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wavelet coefficients in the wavelet transform. Especially, the wavelet transform modulus maximamethod can do very well in retaining the useful fault information meanwhile denoising. Therefore,in conclusion, the wavelet-based methods are more suitable for the preprocessing of fault signalsthan the filter-based methods.

3.5. Vibration signal compression

For rotating machines, in order to acquire enough information to assure the diagnosticsaccuracy, a fault diagnostics system must sample signals through many channels simultaneouslywith high sampling speed, and therefore, the data is often massive. Thus, for fault diagnosticssystems, especially for those online systems, it is difficult but necessary to save the real-time dataon hard disks for long time. Additionally, with the development of the internet-based remote faultdiagnostics technique, high-performance data compression algorithms are needed, which will beuseful to solve the bottleneck of the massive data transmission, to reduce the cost of the datatransmission, and further to improve the performance of the remote fault diagnostics system.Data compression maybe the most successful application of the wavelet transform, including one-dimensional signal compression and two-dimensional image compression. Also due to thecompact support of the basis functions used in the wavelet transform, wavelets have good energyconcentration properties. Most wavelet coefficients are therefore very small, and they can bediscarded without causing a significant error in the reconstruction stage, then data compression isachieved. The data compression and decompression algorithms essentially consist of five majorsteps: transform, thresholding, quantisation/encoding, decoder and reconstruction, shown as thefollowing block diagram (Fig. 5).The performance of the compression algorithm can be evaluated by two parameters [105]. The

first one is the compression ratio defined as the ratio of the number of bits of the original data xðtÞto the number of bits of the compressed data #xðtÞ: The second parameter is the normalised mean-square error given as

MSEðxÞ ¼100

Ns2x

XN

i¼1

ðxi � #xiÞ2; ð9Þ

where sx is the standard deviation of the xðtÞ and N is the number of the sample points in theanalysed data.The wavelet transform has been used for vibration signal compression successfully. Staszewski

has studied the performances of the wavelet-based compression methods for different types ofsignals. The results indicated that the wavelet-based methods are more suitable for non-stationaryvibration signal compression. Usually, the more non-stationary the vibration signal is, the betterthe compression performance will be. The criterions for the selection of the wavelet function used

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x(t)

)(ˆ tx

DWT Thresholding EncoderSparse Matrix

Storage

DecoderIDWT

Fig. 5. Block diagram of compressor/decompressor.


for vibration signal compression were given. In general more compactly supported, and thereforeless smooth wavelet functions, are better suitable for non-stationary and irregular signals, such astransient signals. Less compactly supported, and therefore more smooth wavelet functions, arebetter suitable for stationary and regular signals, such as periodic data. Actually, this criterion issuitable for feature extractions as well [105]. Staszewski [106] combined the wavelet with thegenetic algorithm to compress signals. The method was applied to the compression of the gearboxvibration spectra showing a potential for storage, transmission and fault feature selection forcondition monitoring. Makoto et al. [107] tried to use several different Daubechies waveletfunctions to compress the vibration signals sampled from motor bearing rings. Moreover, theyconstructed a requantiser to optimise bit allocation under the given permissible distortion. Thecomparison with discrete cosine transform (DCT)-based methods showed that the wavelet-basedmethod was more effective. Shen and Gao [108] used the wavelet method to compress themechanical vibration signals and the compression ratios of 10–20 were obtained. In Ref. [109], Maand Lu introduced a vibration signal compression method in detail, which was based on thewavelet and the embedded coding of zero tree. The performances of the method weredemonstrated through three kinds of typical vibration signal compression. Xu et al. [110]reviewed the properties of vibration signals in rotating machines and the relation between thewavelet coefficients and the singularities of the signals. On this basis, they proposed a new datacompression algorithm, in which wavelet coefficients were used to present the singularities of thesignals and the frequency components to present the normal characteristics of the signals.Application results showed that this method could achieve high compression ratio with areservation of good local characteristics.Actually, the data compression principles are similar to the denoising, that is, setting some

small coefficients to zero so that we can use a few bits to represent the signal during the encodingstage. Therefore, the use of the wavelet-based method can often obtain high compression ratiomeanwhile retaining the singularities of signals, which often contain most of the fault information.While it is difficult to achieve these purposes for the Fourier transform (FT)-based method, forexample, the DCT-based method, which suits to the regular signal compression but the irregularsignals [107]. In conclusion, compared with the FT-based method, the wavelet transforms haveprominent advantages in vibration signal compression because the wavelet transforms can retainmore fault information during the compression.

3.6. System and parameter identification

Machine faults can be reflected by the changes of the system parameters or modal parameters,such as the natural frequency, damping, stiffness, etc. Therefore, people extended the systemparameter identification to the fault diagnostics field [111]. Different identification methods havebeen developed. These methods can be classified based on the time and the frequency domains.The thought of the time domain techniques is basically to fit the impulse response function of amechanical system. They will be affected by the noise greatly and be with large amount ofcomputation cost. The frequency domain methods are based on the frequency response functions(FRF), which will often give significant errors resulting from the influences of the energy leakageand the spectrum overlap. In order to overcome all these problems, filtering operations areinevitable in system and parameter identifications. The wavelet transform has dominant

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advantages in signal filtering, which, plus other merits such as time–frequency presentation, thecompactly support base, etc., makes the wavelet transform perform well at the parameteridentifications.Staszewski [112] presented a wavelet-based mode decoupling procedure, in which three different

damping estimation procedures for linear systems were presented. One based on the wavelettransform cross-sections, which can be obtained directly from the frequency domain of waveletformula

Wxða; b;cÞ ¼ffiffiffia

p Zxðf Þ #c�a;bðaf Þej2pfb df ; ð10Þ

where #c�ðdÞ is the complex conjugate of #cðdÞ:The second method recovered the impulse response function for a single mode i from the

wavelet transform by using Eq. (11)

xðtÞ ¼1

Cci

Z tþaDtc

t�aDtc

Wxða; b;cÞda

a; ð11Þ

where Dtc and Dfc are the duration and bandwidth of the basic wavelet function, and

Cci ¼Z fiþDfc=a

fi�Dfc=a

j #cðf Þj2

jf jdf : ð12Þ

The third method used the wavelet ridges and skeletons.Staszewski [113] extended the wavelet ridges and skeletons methods for the identification of

non-linear systems and good results were obtained. Further experimental studies can be found inRef. [114]. Akhmetshin and Sendetski [115] introduced the wavelet packet algorithm into free-oscillation testing method for the estimation of fault parameters of structures. Robertson et al.[116,117] presented a wavelet transform-based method to extract the impulse responsecharacteristics from the measured disturbances and response histories of linear structuraldynamic systems. They [118,119] also used the discrete wavelet transforms for the identification ofstructural dynamics models. With these methods, the structural modes, mode shapes and dampingparameters were extracted. Freudinger et al. [120] used the Laplace wavelet to decompose a signalinto impulse responses of single mode subsystems, and, thereby, to identify the model parametersof a flutter. Similar efforts were made by He and Zi [121] for the identification of the naturalfrequency of the hydro-generator axle. Yu et al. [122] discussed the application of the wavelet instructural system identification. Detailed steps for the determination of the impulse response andthe estimation of the response function were given. Their researches also included the modulatedGaussian wavelet for modal parameter identifications [123] and the wavelet for time-varyingmodal parameter identification [124]. Ruzzene et al. [125] used the wavelet transform as a time–frequency representation for system identification purposes, and the results showed that waveletanalysis of the free response of a system allowed the estimation of the natural frequencies andviscous damping ratios. Liu et al. [126] addressed a wavelet-based method for the identification ofdense modal parameters of a flexible space structure. Ma et al. [127] presented a wavelet-basedmethod on identifying the dynamic characteristics of bearings. Experiment results showed thatthis method was well robust to the influence of the measurement error. Many other applications

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of the wavelet in identifications included hydro-generator axles’ viscous damping coefficients [128]and FRF [129].

3.7. Other applications

Compared with the traditional networks, the wavelet neural networks, in which the basisfunctions are drawn from a family of orthonormal wavelets, have better localisationcharacteristics both in the time and frequency domains. These networks allow hierarchical andmulti-resolution learning of input–output maps and therefore have strong ability in the functionapproximation and good resolution when using for pattern recognition. Wavelet neural networkshave been widely used for the fault diagnostics [130–134]. Moyo and Brownjohn applied thewavelet power spectra and the cross-wavelet power spectra to characterise the structural response[135]. Staszewski and Giacomin [136] extended the concept of the FRF to the time–scale domain,and presented the concept of the wavelet-based FRF, which reflected the ratio of output to inputin the time–scale domain and thus fully characterising time-variant physical systems. The methodwas used for the analysis of the vehicle road data. The wavelet has often been used to estimate thePDF to the process monitoring [137–139]. The wavelet can do very well for trend analysis andcondition prediction as well [140,141].

4. Prospects of wavelet in fault diagnostics

With the development for about 10 years, the wavelet transform has been widely used in faultdiagnostics. But, compared with the FT, the applications of the wavelets have still not achieved astandard status. Many reasons have caused the current status of wavelets in fault diagnostics. Forexample, many functions can be used as the wavelet basis, but there is no a standard or a generalmethod to select the wavelet function for different tasks. It is an obstacle for the popularisation ofthe wavelet transform. Some people have paid attention to this problem [105]. Additionally, anignored problem is how to determine the range scales used in the wavelet transform. The solutionto this problem is important. Wavelet transforms with scales out of this range would bring somemeaningless information, which will mislead the signal analysis. Some discussion about thisproblem can be found in Ref. [142]. Unlike the FT, the results of the wavelet transform have nostraightforward physical implications, and therefore it is difficult to obtain useful informationdirectly from the results of the wavelet transform. Furthermore, different wavelet functions mayresult in different analysis results, which will make not only many engineers but also some newlearners of wavelets become confused. Therefore, although the wavelet transform has many meritsin signal processing, almost all engineers are still favorite with the FT. It is, therefore, verynecessary to find an easily understood way to present the results of the wavelet transforms. Thewavelet scalograms, which are similar to the spectrums of the STFTs, can be understood withrelatively easy, so it may be able to remedy the regret of difficulty to understand the results of thewavelet transforms in a certain extent. Solving all those problems mentioned above will promotethe popularisation of the wavelets.Non-linear problem analysis, which is unavoidable in the fault diagnostics field, maybe is

another field in which the wavelet will achieve success. Wong and Chen [38] have investigated the

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non-linear and chaotic behaviours of the structural system by wavelet transforms. Jubran et al.[143] used the modulated Gaussian wavelet to analyse the chaotic behaviour of the flow-inducedvibration of a single cylinder in cross-flow. Pernot and Lamarque [144,145] presented a wavelet-Galerkin procedure to investigate time-periodic systems, including transient vibration andstability analysis. Suh and Chan [146] developed an innovative wavelet-based diagnosticmethodology to perform real-time detection of mechanical chaos occurring in high-speed, high-performance rotor-dynamic systems. Zheng et al. [147] have studied the bifurcation and chaosphenomena with wavelets. It can be expected that the wavelet transform would enjoy greatersuccess in non-linear problem analysis.Hitherto, almost all applications of the wavelet are limited to find some new phenomena or

some new fault features from the sampled signals, based on which some novel and useful faultdetection methods have been presented. But it is a pity that few further researches have beencarried out to discuss the reasons for those new phenomena found by the wavelet-based methods,and, actually, such efforts will be very helpful for the investigation of the fault reasons.Additionally, there is still a pity about the applications of the wavelet, that is, until now, the

wavelets are always used to analyse a single signal and rarely used to analyse two or more signalssimultaneously to find the relationships between them. It is well known that in order to achieveaccurate fault diagnostics, many different signals have to be sampled from machines. There existsome relationships between these sampled signals. Undoubtedly, raveling those relationships willgive more useful information for fault diagnostics. The wavelet cross-scalogram can carry out thecorrelation analysis between two signals on the time–frequency plane and therefore can give moreinformation than the traditional correlation analysis only in the time domain. It can be expectedto be a useful tool for fault diagnostics.

5. Concluding remarks

The applications of the wavelet analysis have covered almost every aspect of the faultdiagnostics. A review on all of them in a couple of pages is certainly not possible. In this review,all applications were divided into several main aspects, including the time–frequency analysis ofsignals, the fault feature extraction, the singularity detection for signals, the denoising andextraction of the weak signals, the compression of vibration signals and the system identification.Some other applications are introduced briefly as well, such as the wavelet networks, the wavelet-based frequency response function, etc. In addition, some problems occurred in the use of thewavelet to fault diagnostics are analysed. The prospect about using the wavelet to solve non-linearproblems is discussed.

Acknowledgements

This research is supported financially by National Key Basic Research Special Fund (No.G1998020309) and Natural Science Foundation of China (Grant No. 50105007).

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