**3. Choosing the best wavelet basis**

Utilizing the classical WT (DWT, CWT or WPT) brings on the unresolved issue of mother wavelet selection. Different types of wavelets have different time–frequency structures and thus it is always an issue how to choose the best wavelet function for extracting fault features from a given signal. An "inappropriate" wavelet will reduce the accuracy of the fault detection. There is a plethora of options between various wavelet families (with infinite number of members!) or specific wavelets. Haar, Daubechies (db), Symlets, Coiflets, Gaussian, Morlet, complex Morlet, Mexican hat, biorthogonal wavelets, reverse biorthogonal, Meyer, harmonic wavelets, discrete approximation of Meyer, complex Gaussian, Shannon, and frequency B-spline are among the most well established wavelets. In principle, the wavelet decomposition would achieve a better result if the wavelet basis is ''similar'' to the signal under analysis. The wavelet coefficients reflect the similarity between the signal local and the corresponding wavelet basis. The bigger the coefficient, the more similar the two parts are. Different wavelet basis would lead to quite different results of signal analysis. Currently there are still no generic theoretical guidelines for how to select the optimum wavelet basis, or how to select the corresponding shape parameter and scale level for a particular application. The selection is in many cases done by trial and error. In literature there are some interesting approaches that attempt to address this issue.

(Kankar et al., 2011) presented a methodology for rolling element bearings fault diagnosis using continuous wavelet transform (CWT). Six different base wavelets were considered of which three were real valued and the other three were complex valued. Out of these six wavelets, the base wavelet was selected based on wavelet selection criteria to extract statistical features from wavelet coefficients of raw vibration signals. Two wavelet selection criteria, Maximum Energy to Shannon Entropy ratio and Maximum Relative Wavelet Energy were used and compared to select the appropriate wavelet for feature extraction. The wavelet having Maximum Energy to Shannon Entropy ratio/Maximum Relative Wavelet Energy was considered for fault diagnosis of rolling element bearings. The relative Wavelet Energy is defined as:

$$p\_n = \,^E\langle n \rangle\_{E\_{total}} \tag{44}$$

Where E(n) the energy at each resolution level,

$$E(n) = \Sigma\_{l=1}^{m} \left| \mathcal{C}\_{n,l} \right|^2 \tag{45}$$

*m* is the number of wavelet coefficients and *Cn,i* the *i*th wavelet coefficient at the *n*th scale. The total energy is given by:

$$E(n) = \sum\_{n} \left| \mathcal{C}\_{n,l} \right|^2 \tag{46}$$

whereas the Energy to Shannon Entropy ratio is given by:

$$
\zeta(\mathfrak{n}) = E(\mathfrak{n}) / \mathfrak{S}\_{entropy}(\mathfrak{n}) \tag{47}
$$

where the entropy of signal wavelet coefficients is defined as:

$$S\_{entropy}(n) = -\sum\_{l=1}^{m} p\_l \cdot \log\_2 p\_l \tag{48}$$

and pi is the energy distribution of the wavelet coefficients,

$$p\_l = \left| \mathcal{C}\_{n,l} \right|^2 / E(n) \tag{49}$$

with ∑ �� � ��� = 1.

284 Advances in Wavelet Theory and Their Applications in Engineering, Physics and Technology

Utilizing the classical WT (DWT, CWT or WPT) brings on the unresolved issue of mother wavelet selection. Different types of wavelets have different time–frequency structures and thus it is always an issue how to choose the best wavelet function for extracting fault features from a given signal. An "inappropriate" wavelet will reduce the accuracy of the fault detection. There is a plethora of options between various wavelet families (with infinite number of members!) or specific wavelets. Haar, Daubechies (db), Symlets, Coiflets, Gaussian, Morlet, complex Morlet, Mexican hat, biorthogonal wavelets, reverse biorthogonal, Meyer, harmonic wavelets, discrete approximation of Meyer, complex Gaussian, Shannon, and frequency B-spline are among the most well established wavelets. In principle, the wavelet decomposition would achieve a better result if the wavelet basis is ''similar'' to the signal under analysis. The wavelet coefficients reflect the similarity between the signal local and the corresponding wavelet basis. The bigger the coefficient, the more similar the two parts are. Different wavelet basis would lead to quite different results of signal analysis. Currently there are still no generic theoretical guidelines for how to select the optimum wavelet basis, or how to select the corresponding shape parameter and scale level for a particular application. The selection is in many cases done by trial and error. In literature there are some interesting approaches that attempt to

(Kankar et al., 2011) presented a methodology for rolling element bearings fault diagnosis using continuous wavelet transform (CWT). Six different base wavelets were considered of which three were real valued and the other three were complex valued. Out of these six wavelets, the base wavelet was selected based on wavelet selection criteria to extract statistical features from wavelet coefficients of raw vibration signals. Two wavelet selection criteria, Maximum Energy to Shannon Entropy ratio and Maximum Relative Wavelet Energy were used and compared to select the appropriate wavelet for feature extraction. The wavelet having Maximum Energy to Shannon Entropy ratio/Maximum Relative Wavelet Energy was considered for fault diagnosis of rolling element bearings. The relative

Fig. 8. Reconstruction step of SGWPT

address this issue.

Wavelet Energy is defined as:

**3. Choosing the best wavelet basis**

To find the most suitable mother wavelet, (Rafiee and Tse, 2009), in probably the most thorough study of mother wavelet choice investigation, studied 324 candidate mother wavelet functions from various families including Haar, Daubechies (db), Symlet, Coiflet, Gaussian, Morlet, complex Morlet, Mexican hat, bio-orthogonal, reverse bio-orthogonal, Meyer, discrete approximation of Meyer, complex Gaussian, Shannon, and frequency Bspline. The most similar mother wavelet for analyzing the gear vibration signal was selected based on the following procedure. Raw vibration signals were recorded and synchronized. The feature vector was composed of the variance of CWT coefficients for each of the 24 scales calculated by each of the 50 segmented signals in each gearbox condition. The average of the feature vector in the 50 segmented signals was computed for each gearbox condition. Variances of the mentioned average of the four gearbox conditions were determined for each scale (24 elements). The five highest values of the calculated vector were selected as the feature because the larger the variance, the greater the ability to properly classify faults. The summation of the five elements, called ''SUMVAR'' for simplicity, was compared with those obtained from the other 323 candidate mother wavelets (a total of 324 mother wavelets). The one that had the highest SUMVAR was selected as the most similar function to our vibration signals. In a similar work (Rafiee et al., 2010) following a similar procedure found that "Daubechies 44" ("db44") has the most similar shape across both gear and bearing vibration signals. Results also suggested that although "db44" is the most similar mother wavelet function for the studied vibration signals, it is not the proper function for all wavelet-based processing. The research verified that Morlet wavelet has better similarity to both vibration signals in comparison to many other functions such as Daubechies (1–43), Coiflet, Symlet, complex Morlet, Gaussian, complex Gaussian, and Meyer for both experimental set-ups (i.e. gear testing and bearing testing). Among the studied mother wavelets, results also showed

Utilising the Wavelet Transform in Condition-Based Maintenance: A Review with Applications 287

unpublished study by the authors, an investigation of the optimum parameters for the most effective de-noising with DWT was conducted. The analysis of a representative AE signal from seeded defects in bearings shows how statistical parameters change respectively to the wavelet choice between the 10 first members of the Daubechies family in Fig. 9. Obviously the wavelet that maximizes kurtosis, crest factor and crest value is chosen as optimum,

Fig. 9. Kurtosis, crest value and factor features of de-noised AE signal with various "db"

db1 db2 db3 db4 db5 db6 db7 db8 db9 db10

Crest Value

Kurtosis

Crest Factor

Wavelet based de-noising is a very interesting and important application of wavelets in the processing of signals from condition monitoring. It is very widely adopted in many studies as it is ideal to extract hidden diagnostic information and enhance the impulsive components of complex, non-stationary signals with strong background. Wavelet thresholding is based on the idea that the energy of the signal is concentrated in a few wavelet coefficients, while the energy of noise spreads throughout all the resulted wavelet coefficients. Similarity between the mother wavelet and the signal to be analyzed plays a very important role, making it possible for the signal to concentrate on fewer coefficients and thus its choice is critical in the efficiency of the de-noising task. The first foundations in wavelet-based de-noising were set by (Donoho, 1995). Let *x(t)* be the discrete signal acquired during condition monitoring. The signal series consists of impulses and noise. x(t) can alternatively be expressed as *x(t)=p(t)+n(t),* where *p(t)* indicates the impulses to be determined, whereas *n(t)* indicates equally distributed and independent Gaussian noise with mean zero and standard deviation r. In principle, the wavelet threshold de-noising

**4. Applications overview of wavelets in condition based maintenance** 

"db2" in this case.

wavelets in a DWT de-noising scheme

0

0.2

0.4

0.6

**Normalized Indicators**

0.8

1

1.2

**4.1 Wavelet-based de-noising**

procedure has the following steps:

that db44 is the most similar function across both gear and bearing vibration signals. The drawback of the db44 function is that the high-order db functions take more CPU time than most others. In another work (Rafiee et al., 2009) utilized genetic algorithms (GAs) to optimize the selection of mother wavelet function (among several members of the Daubechies family), the number of the decomposition levels of the wavelet packet transform (WPT) as well as the number of neurons in the ANNs hidden layers used for the fault classification, resulted in a high-speed, effective two-layer ANN with a small-sized structure. "db11", level 4 and 14 neurons have been selected as the best values for Daubechies order, decomposition level, and the number of nodes in hidden layer, respectively. In (Gketsis et al., 2009) the optimum wavelet choice criterion is the maximization of the cross-correlation between the signal of interest and the wavelet. In an application of condition monitoring in electrical machines, they tested several wavelet functions, namely Haar, Daubechies 2, 4, 8, Symlet 2, 3, 4, 8 and Coiflet 3 and concluded to "db2". (Saravanan and Ramachandran, 2009) found that among the 15 members of Daubechies wavelet, "db1" and "db5" gave the maximum classification efficiency of an expert system (Decision Tree) at around 98.7%.

Other researchers prefer more qualitative explanations. (Xu and Li, 2008) support that in the common family of wavelet bases i.e. Morlet, Haar, Shannon, Symmlets, Coiflets and Daubechies wavelets, etc., the most popular is the Daubechies wavelet, as it bears the shortest compactly supported scaling function in all of orthogonal wavelets when given exponent number of vanishing moment. Moreover, it gives the best overall performance in the respect of both mean squared error between reconstruction signal and original signal, and maximizing the SNR improvement. Therefore, the Daubechies wavelet is applied and others are for comparison in this case. (Jazebi et al., 2011) state that one specific mother wavelet is best suited for a particular application. For this purpose, mother wavelet type and decomposition level have been chosen based on experience and trial and error. The research includes detecting and analyzing low amplitude, short duration, fast decaying, and oscillating type of current signals. For this purpose, Daubechies's mother wavelet seems to be an appropriate choice. In comparison with Haar wavelet, Daubechies are best suited for feature extraction due to their low-pass and highpass filters. On the other hand because of its inherent orthogonality, it satisfies Parseval theorem, not like biorthogonal wavelets such as Coiflet and Meyer wavelets . db4 mother wavelet over level d4 has been chosen because the maximum energy localization in details (1–4) was obtained using these parameters.

(Daviu et al., 2007) supports that the Daubechies family is well suited for application of DWT in condition monitoring due to its interesting inherent properties. An important fact they observed when using the Daubechies family, was the overlap between the frequency bands (frequency aliasing) associated with the DWT decomposition of their signals. This is due to the non-ideal filtering process performed by the wavelet signals, a fact that makes that the signal components, included within a certain frequency band and placed in the proximity of its limits, overlap partially with the adjacent band. When using a high-order Daubechies wavelet for signal decomposition, this effect is less intense than when using a low-order one. In other words, high-order wavelets behave as more ideal filters. Maximization of statistical features such as kurtosis or crest factor can be utilized as a criterion for the choice of mother wavelet within a family or among various families. In an

unpublished study by the authors, an investigation of the optimum parameters for the most effective de-noising with DWT was conducted. The analysis of a representative AE signal from seeded defects in bearings shows how statistical parameters change respectively to the wavelet choice between the 10 first members of the Daubechies family in Fig. 9. Obviously the wavelet that maximizes kurtosis, crest factor and crest value is chosen as optimum, "db2" in this case.

Fig. 9. Kurtosis, crest value and factor features of de-noised AE signal with various "db" wavelets in a DWT de-noising scheme
