**1. Introduction**

Condition monitoring of machinery can be defined as the continuous or periodic measurement and interpretation of data in order to indicate the condition of an machine and determine the need for maintenance. Condition monitoring thus is primarily involved with the diagnostics of faults and failures and aims at an accurate and as early as possible fault detection. It is thus oriented towards an unscheduled preventive maintenance plan with continuous monitoring of the machinery as opposed to scheduled periodic maintenance. The possibility of failures of course cannot be diminished, but confident early diagnosis of incipient failures is extremely useful to avoid machinery breakdown and thus ensure a more cost-effective overall operation reducing equipment down-times. Industrial safety is also enhanced as catastrophic events are avoided when a maintenance-for-cause plan is followed.

When faults occur in machines, phenomena like excessive vibration and/or noise, increased temperatures, increased wear rate, etc. are observed. The concept is to monitor, continuously or periodically, these dynamic phenomena utilizing one or more sensors to capture this behavior. One of the earliest approaches was the sound emission monitoring. An expert human ear played the role of the sensor in the early applications, a sophisticated microphone can play the same role today. The most classic approach –widely used until the present- is the vibration monitoring with few or several accelerometers placed upon the machine. The principle is that when damage occurs, the signature of the vibration response changes in the frequency domain, giving a qualitative indication of fault existence. The Acoustic Emission (AE) technique, famous for its sensitivity in the high frequency domain of micro-damage evolution, has found important applications in gearboxes and bearings as Section 4 presents. Other monitoring techniques include oil condition monitoring (oil debris, oil conductivity or humidity etc.), current and voltage transients monitoring in electric motors as well as temperature measurements/thermography. More than 80% of the applications presented in Section 4 involve vibration monitoring, with AE finding more and more applications the last 15 years and current/voltage measurements being always an option in electric machines. Monitoring generally results in a large number of complex signals with valuable diagnostic information hidden under noise or other irrelevant sources. Over the years and the same time with several breakthroughs in the signal processing field, engineers and researchers realized

Utilising the Wavelet Transform in Condition-Based Maintenance: A Review with Applications 275

The current work is organized as follows. Section 2 presents the basic WT versions i.e. DWT, CWT and WPT. Then more recently developed and state-of-the-art wavelet transforms are presented in more detail such as the Dual-Tree Complex Wavelet Transform (DTCWT) as well as Second Generation Wavelet Transforms (SGWT). In section 3 a discussion on the optimum mother wavelet choice issue is conducted and in section 4 a large number of applications -categorized in five application fields- are presented. Section 5 summarizes the

A wavelet is a wave-like oscillation that instead of oscillating forever like harmonic waves drops rather quickly to zero. The continuous wavelet transform breaks up a continuous function *f(t)* into shifted and scaled versions of the mother wavelet *ψ*. It can be defined as the convolution of the input data sequence with a set of functions generated by the mother

� �(�) � �<sup>∗</sup> �

� � *<sup>∞</sup>*

���

���

�� ��� (1)

�� �� (3)

� � ���� (2)

��(�� �) <sup>=</sup> �

scales can be used to pull details from a particular frequency band.

discrete wavelet transform (DWT) which is expressed as:

�|�|

�� �

� � ��(�� �) � � �� � � �

�� �

where *α* represents scale (or pseudo-frequency) and *b* represents time shift of the mother wavelet *ψ*. *ψ\** is the complex conjugate of the mother wavelet *ψ*. The WT's superior timelocalization properties result from the finite support of the mother wavelet: as *b* increases, the analysis wavelet scans the length of the input signal, and *a* increases or decreases in response to changes in the signal's local time and frequency content. Finite support implies that the effect of each term in the wavelet representation is purely localized. This sets the WT apart from the Fourier Transform, where the effects of adding higher frequency sine waves are spread throughout the frequency axis. CWT can be applied with higher resolution to extract information with higher redundancy, that is, a very narrow range of

It turned out quite remarkably that instead of using all possible scales only dyadic scales can be utilized without any information loss. Mathematically this procedure is described by the

��(�� �) <sup>=</sup> √2� � �(�)�∗(2�� � �) ��

where DW(j, k) are the wavelet transforms coefficients given by a two-dimensional matrix, *j*  is the scale that represents the frequency domain aspects of the signal and *k* represents the time shift of the mother wavelet. *f(t)* is the signal that is analyzed and *ψ* the mother wavelet used for the analysis (*ψ\** is the complex conjugate of *ψ*). The inverse discrete wavelet

main conclusions of this work.

**2.1 Continuous Wavelet Transform (CWT)** 

with the inverse transform being expressed as:

**2.2 Discrete Wavelet Transform (DWT)**

transform can be expressed as:

�(�) = � ��

**2. Wavelet transforms**

wavelet:

that the conventional FFT was not suitable to process signals of complex, dynamic nature, often transient and non-stationary, such as the signals from the vibrations of machinery. Among other disadvantages, FFT lacks time localization. To address this problem timefrequency representations were sought and developed. Short-time Fourier Transform (STFT) was introduced as well as non-linear distributions such as the Wigner–Ville distribution (WVD). STFT suffers from the fact that it provides constant resolution for all frequencies since it uses the same window for the analysis of the entire signal. Wigner–Ville distribution and Pseudo-Wigner–Ville distribution are bilinear in nature and artificial cross terms appear in the decomposition results rendering the feature interpretation problematic. Their greatest disadvantage though is that they are generally non-reversible transforms. Wavelet transform (WT) is a relatively recent advancement in the signal processing field. J. Morlet set the first foundations on wavelets back in 1970's but it was not until 1985 when S. Mallat gave wavelets a jump-start through his work in digital signal processing. He discovered some relationships between quadrature mirror filters, pyramid algorithms, and orthonormal wavelet bases. Inspired in part by these results, Y. Meyer constructed the first non-trivial wavelets. A couple of years later, I. Daubechies used Mallat's work to construct a set of wavelet orthonormal basis functions that are perhaps the most elegant, giving a tremendous boost to wavelet applications in numerous scientific fields. The wavelet transform is actually a time-scale method, as it transforms a function from the time domain to the time-scale domain. Scale is indirectly associated with frequency. Furthermore, the wavelet transform is a reversible transform, which makes the reconstruction or evaluation of certain signal components possible, even though the inverse transform may not be orthogonal.

Wavelet transform became very popular in condition monitoring the last 15 years as it is very attractive for the transaction of two major tasks in signals of complex (transient and/or nonstationary) nature: de-noising and feature extraction. De-noising is conducted in order to reduce the fluctuation and pick out hidden or weak diagnostic information. Feature extraction provides usually –though not always- the input to an expert system towards autonomic health degradation monitoring and data-driven prognostics. The generic pattern seen in many studies in the wavelet-based condition monitoring field is summarized in Fig. 1.

Fig. 1. Schematic representation of wavelet-based condition monitoring philosophy

The current work is organized as follows. Section 2 presents the basic WT versions i.e. DWT, CWT and WPT. Then more recently developed and state-of-the-art wavelet transforms are presented in more detail such as the Dual-Tree Complex Wavelet Transform (DTCWT) as well as Second Generation Wavelet Transforms (SGWT). In section 3 a discussion on the optimum mother wavelet choice issue is conducted and in section 4 a large number of applications -categorized in five application fields- are presented. Section 5 summarizes the main conclusions of this work.

### **2. Wavelet transforms**

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

that the conventional FFT was not suitable to process signals of complex, dynamic nature, often transient and non-stationary, such as the signals from the vibrations of machinery. Among other disadvantages, FFT lacks time localization. To address this problem timefrequency representations were sought and developed. Short-time Fourier Transform (STFT) was introduced as well as non-linear distributions such as the Wigner–Ville distribution (WVD). STFT suffers from the fact that it provides constant resolution for all frequencies since it uses the same window for the analysis of the entire signal. Wigner–Ville distribution and Pseudo-Wigner–Ville distribution are bilinear in nature and artificial cross terms appear in the decomposition results rendering the feature interpretation problematic. Their greatest disadvantage though is that they are generally non-reversible transforms. Wavelet transform (WT) is a relatively recent advancement in the signal processing field. J. Morlet set the first foundations on wavelets back in 1970's but it was not until 1985 when S. Mallat gave wavelets a jump-start through his work in digital signal processing. He discovered some relationships between quadrature mirror filters, pyramid algorithms, and orthonormal wavelet bases. Inspired in part by these results, Y. Meyer constructed the first non-trivial wavelets. A couple of years later, I. Daubechies used Mallat's work to construct a set of wavelet orthonormal basis functions that are perhaps the most elegant, giving a tremendous boost to wavelet applications in numerous scientific fields. The wavelet transform is actually a time-scale method, as it transforms a function from the time domain to the time-scale domain. Scale is indirectly associated with frequency. Furthermore, the wavelet transform is a reversible transform, which makes the reconstruction or evaluation of certain signal components possible, even

Wavelet transform became very popular in condition monitoring the last 15 years as it is very attractive for the transaction of two major tasks in signals of complex (transient and/or nonstationary) nature: de-noising and feature extraction. De-noising is conducted in order to reduce the fluctuation and pick out hidden or weak diagnostic information. Feature extraction provides usually –though not always- the input to an expert system towards autonomic health degradation monitoring and data-driven prognostics. The generic pattern seen in many

studies in the wavelet-based condition monitoring field is summarized in Fig. 1.

Fig. 1. Schematic representation of wavelet-based condition monitoring philosophy

though the inverse transform may not be orthogonal.

#### **2.1 Continuous Wavelet Transform (CWT)**

A wavelet is a wave-like oscillation that instead of oscillating forever like harmonic waves drops rather quickly to zero. The continuous wavelet transform breaks up a continuous function *f(t)* into shifted and scaled versions of the mother wavelet *ψ*. It can be defined as the convolution of the input data sequence with a set of functions generated by the mother wavelet:

$$CW(a,b) = \frac{1}{\sqrt{|a|}} \int\_{-\infty}^{x} f(t) \cdot \psi^\* \left(\frac{t-b}{a}\right) dt\tag{1}$$

with the inverse transform being expressed as:

$$f(t) = \frac{1}{c\_{\Psi}} \int\_{-\infty}^{\infty} \int\_{-\infty}^{\infty} \mathcal{C}W(a, b) \cdot \frac{1}{a^2} \cdot \psi\left(\frac{t - b}{a}\right) da db \tag{2}$$

where *α* represents scale (or pseudo-frequency) and *b* represents time shift of the mother wavelet *ψ*. *ψ\** is the complex conjugate of the mother wavelet *ψ*. The WT's superior timelocalization properties result from the finite support of the mother wavelet: as *b* increases, the analysis wavelet scans the length of the input signal, and *a* increases or decreases in response to changes in the signal's local time and frequency content. Finite support implies that the effect of each term in the wavelet representation is purely localized. This sets the WT apart from the Fourier Transform, where the effects of adding higher frequency sine waves are spread throughout the frequency axis. CWT can be applied with higher resolution to extract information with higher redundancy, that is, a very narrow range of scales can be used to pull details from a particular frequency band.

#### **2.2 Discrete Wavelet Transform (DWT)**

It turned out quite remarkably that instead of using all possible scales only dyadic scales can be utilized without any information loss. Mathematically this procedure is described by the discrete wavelet transform (DWT) which is expressed as:

$$DW(j,k) = \sqrt{2^j} \int\_{-\infty}^{+\infty} f(\mathbf{t}) \psi^\*(\mathbf{2}^j \mathbf{t} - k) \, d\mathbf{t} \tag{3}$$

where DW(j, k) are the wavelet transforms coefficients given by a two-dimensional matrix, *j*  is the scale that represents the frequency domain aspects of the signal and *k* represents the time shift of the mother wavelet. *f(t)* is the signal that is analyzed and *ψ* the mother wavelet used for the analysis (*ψ\** is the complex conjugate of *ψ*). The inverse discrete wavelet transform can be expressed as:

$$f(\mathbf{t}) = \mathbf{c} \,\,\Sigma\_l \,\Sigma\_k \, DW(\mathbf{j}, \mathbf{k}) \psi\_{l,k}(\mathbf{t}) \tag{4}$$

$$N = \operatorname{int} \left( \frac{\log \binom{fs}{f}}{\log \text{(2)}} \right) \tag{5}$$

$$W\_{l,k}^{n}(\mathbf{t}) = 2^{j/2} \mathcal{W} \{ 2^j \mathbf{t} - \mathbf{k} \}, \quad j, k \in \mathbf{Z} \tag{6}$$

$$W\_{0,0}^{0}(t) = \varphi(t) = \sqrt{2} \,\Sigma\_k h(k) \varphi(2t - k) \tag{7}$$

$$W\_{0,0}^{1}(t) = \psi(t) = \sqrt{2} \,\Sigma\_k \, g(k) \varphi(2t - k) \tag{8}$$

$$\mathcal{W}\_{0,0}^{2n}(t) = \sqrt{2} \sum\_{k} h(k) \mathcal{W}\_{1,k}^{n}(2t - k) \tag{9}$$

$$\mathcal{W}\_{0,0}^{2n+1}(t) = \sqrt{2} \,\Sigma\_k \, g(k) \mathcal{W}\_{1,k}^n(2t - k) \tag{10}$$

$$\mathcal{W}\_{j,k}^{n} = \langle f(\mathbf{t}), \mathcal{W}\_{j,k}^{n} \rangle = \int f(\mathbf{t}) \, \mathcal{W}\_{j,k}^{n} \, d\mathbf{t} \, . \tag{11}$$

$$
\psi^{\mathbb{C}}(t) = \psi\_h(t) + j\psi\_a(t) \tag{12}
$$

$$d\_l^{Re}(k) = 2^{l/2} \int\_{-\\\\\infty}^{\\\infty} \chi(t) \psi\_h(2^l t - k) dt, \ l = 1, \ldots, l \tag{13}$$

$$c\_{l}^{\text{Re}}(k) = 2^{l/2} \int\_{-\alpha}^{\alpha} \mathbf{x}(\mathbf{t}) \varphi\_{h}(\mathbf{2}^{l}\mathbf{t} - k) d\mathbf{t} \tag{14}$$

$$d\_l^C(k) = d\_l^{Re}(k) + j d\_l^{Im}(k), \ l = 1, \ldots, l \tag{15}$$

$$c\_I^C(k) = c\_I^{Re}(k) + jc\_I^{Im}(k) \tag{16}$$

$$d\_l(\mathbf{t}) = 2^{(l-1)/2} \left[ \sum\_{n} d\_l^{\text{Re}}(k) \psi\_h(\mathcal{Z}^l \mathbf{t} - k) + \sum\_{m} d\_l^{lm}(k) \psi\_g(\mathcal{Z}^l \mathbf{t} - m) \right], \quad l = 1, \dots, l \tag{17}$$

$$c\_{l}(t) = 2^{(l-1)/2} \left[ \sum\_{n} c\_{l}^{Re}(k) \wp\_{h}(2^{l}t - k) + \sum\_{m} c\_{l}^{lm}(k) \wp\_{g}(2^{l}t - m) \right] \tag{18}$$

$$c\_{l+1}^{Re}(k) = \Sigma\_m h\_0(m - 2k) c\_l^{Re}(m) \tag{19}$$

$$d\_{l+1}^{Re}(k) = \Sigma\_m h\_1(m - 2k) c\_l^{Re}(m) \tag{20}$$

$$c\_{l}^{Re}(k) = \Sigma\_{m}\tilde{h}\_{0}(k - 2m)c\_{l+1}^{Re}(m) + \Sigma\_{m}\tilde{h}\_{1}(k - 2m)d\_{l+1}^{Re}(m)\tag{21}$$

$$c\_{l+1}^{lm}(k) = \Sigma\_n g\_0(n - 2k) c\_l^{lm}(n) \tag{22}$$

$$d\_{l+1}^{lm}(k) = \Sigma\_n g\_1(n - 2k) c\_l^{lm}(n) \tag{23}$$

$$c\_{l}^{lm}(k) = \Sigma\_{n} \tilde{g}\_{0}(k - 2n)c\_{l+1}^{lm}(n) + \Sigma\_{n} \tilde{g}\_{1}(k - 2n)d\_{l+1}^{lm}(n) \tag{24}$$

Utilising the Wavelet Transform in Condition-Based Maintenance: A Review with Applications 281

In the DWT decomposition, the highlighted frequencies actually do not exist as the FFT of the original signal confirms. On the contrary artificial peaks do not appear in the DTCWT decomposition as Fig.5 clearly shows proving the reduced frequency aliasing of the DTCWT. A peak highlighted in detail 3 is real though it should appear only in

The classical wavelet techniques (CWT, DWT, WPT) are all dependent on the mother wavelet selection from a library of previously designed wavelet functions, an issue that is discussed in more detail in Section 3. Unfortunately, the standard wavelet functions are independent of a given signal. Towards this direction, the Second Generation Wavelet Transform (SGWT) was developed by (Sweldens, 1998), a new wavelet construction method using the lifting scheme. It is actually an alternative implementation of the classical DWT. The main feature of the SGWT is that it provides an entirely spatial domain interpretation of the transform, as opposed to the traditional frequency domain based constructions. Compared with the classical wavelet transform, the lifting scheme possesses several advantages, including the possibility of adaptive design, in-place calculations, irregular samples and integers-to-integers wavelet transforms. The lifting scheme provides high flexibility, which can be designed according to the properties of the given signal, and thus ensures that the resulting transform is always invertible. It makes good use of similarities between the high and low pass filters to speed up the calculation so that the implementation of the second generation wavelet transform is faster than the first generation wavelet transforms. Additionally, the multi-resolution analysis property is preserved. Consequently, the applications of the SGWT scheme in condition monitoring and fault diagnosis of mechanical equipments have been increasing the last few years (see Section 4). A basic decomposition of the SGWT consists of three main steps (Sweldens, 1998), split, predict, and update. In the split step, an approximate signal *al* at level *l* is split into even samples and

In the prediction step, a prediction operator *P* is designed and applied on *al+1* to predict *dl*+1.

����(�) � ����(�) <sup>−</sup> <sup>∑</sup> ������(� + �) ���

In the update step, a designed update operator *U* is applied on *dl*+1. Adding the result to the

����(�) � ����(�) <sup>+</sup> <sup>∑</sup> ������(� + � − �) ���

where *u j* are the coefficients of *U* and *N* is the length of *u j* . Iteration of the above three steps on the output *a*, generates the detail and approximation coefficients at different levels.

The resultant prediction error *dl*+1 is regarded as the detail coefficients of *al*.

even samples, the resultant *al*+1 is regarded as the approximate coefficients of *al* .

where *pr* the coefficients of *P* and *M* is the length of *pr* .

���� � ��(��), ���� � ��(�� + �) (25)

�������� (26)

�������� (27)

detail 2.

**2.5 Second generation wavelet transforms** 

odd samples (Zhou et al., 2010).

**2.5.1 The Second Generation Wavelet Transform (SGWT)** 

**Normalised frequency**

Fig. 5. 3-level decomposition with DTCWT of x(t)

**0 50 100 150 200**

**samples**

In the DWT decomposition, the highlighted frequencies actually do not exist as the FFT of the original signal confirms. On the contrary artificial peaks do not appear in the DTCWT decomposition as Fig.5 clearly shows proving the reduced frequency aliasing of the DTCWT. A peak highlighted in detail 3 is real though it should appear only in detail 2.

#### **2.5 Second generation wavelet transforms**

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

**original signal**

**detail 1**

**detail 2**

**detail 3**

**approximation**

**spectrum**

**spectrum**

**spectrum**

**spectrum**

**spectrum**

**500**

**100 200**

**200 400**

**100 200**

**50 100**

**500**

**20 40**

**50 100**

> **10 20**

> **10 20**

**original signal**

**detail 1**

**detail 2**

**detail 3**

**spectrum**

**approximation**

**spectrum**

**spectrum**

**spectrum**

**<sup>0</sup> 0.1 0.2 0.3 0.4 0.5 <sup>0</sup>**

**<sup>0</sup> 0.1 0.2 0.3 0.4 0.5 <sup>0</sup>**

**<sup>0</sup> 0.1 0.2 0.3 0.4 0.5 <sup>0</sup>**

**<sup>0</sup> 0.1 0.2 0.3 0.4 0.5 <sup>0</sup>**

**<sup>0</sup> 0.1 0.2 0.3 0.4 0.5 <sup>0</sup>**

**Normalised frequency**

**<sup>0</sup> 0.1 0.2 0.3 0.4 0.5 <sup>0</sup>**

**<sup>0</sup> 0.1 0.2 0.3 0.4 0.5 <sup>0</sup>**

**<sup>0</sup> 0.1 0.2 0.3 0.4 0.5 <sup>0</sup>**

**<sup>0</sup> 0.1 0.2 0.3 0.4 0.5 <sup>0</sup>**

**<sup>0</sup> 0.1 0.2 0.3 0.4 0.5 <sup>0</sup>**

**Normalised frequency**

Fig. 4. 3-level decomposition with DWT of x(t)

**<sup>0</sup> <sup>50</sup> <sup>100</sup> <sup>150</sup> <sup>200</sup> -10**

**<sup>0</sup> <sup>50</sup> <sup>100</sup> <sup>150</sup> <sup>200</sup> -5**

**<sup>0</sup> <sup>50</sup> <sup>100</sup> <sup>150</sup> <sup>200</sup> -5**

**<sup>0</sup> <sup>50</sup> <sup>100</sup> <sup>150</sup> <sup>200</sup> -5**

**<sup>0</sup> <sup>50</sup> <sup>100</sup> <sup>150</sup> <sup>200</sup> -5**

**Samples**

**<sup>0</sup> <sup>50</sup> <sup>100</sup> <sup>150</sup> <sup>200</sup> -10**

**<sup>0</sup> <sup>50</sup> <sup>100</sup> <sup>150</sup> <sup>200</sup> -5**

**<sup>0</sup> <sup>50</sup> <sup>100</sup> <sup>150</sup> <sup>200</sup> -5**

**<sup>0</sup> <sup>50</sup> <sup>100</sup> <sup>150</sup> <sup>200</sup> -10**

**0 10 original signal**

> **0 5**

**0 10 original signal**

> **-2 0 2**

**approximation**

**approximation**

Fig. 5. 3-level decomposition with DTCWT of x(t)

**0 50 100 150 200**

**samples**

#### **2.5.1 The Second Generation Wavelet Transform (SGWT)**

The classical wavelet techniques (CWT, DWT, WPT) are all dependent on the mother wavelet selection from a library of previously designed wavelet functions, an issue that is discussed in more detail in Section 3. Unfortunately, the standard wavelet functions are independent of a given signal. Towards this direction, the Second Generation Wavelet Transform (SGWT) was developed by (Sweldens, 1998), a new wavelet construction method using the lifting scheme. It is actually an alternative implementation of the classical DWT. The main feature of the SGWT is that it provides an entirely spatial domain interpretation of the transform, as opposed to the traditional frequency domain based constructions. Compared with the classical wavelet transform, the lifting scheme possesses several advantages, including the possibility of adaptive design, in-place calculations, irregular samples and integers-to-integers wavelet transforms. The lifting scheme provides high flexibility, which can be designed according to the properties of the given signal, and thus ensures that the resulting transform is always invertible. It makes good use of similarities between the high and low pass filters to speed up the calculation so that the implementation of the second generation wavelet transform is faster than the first generation wavelet transforms. Additionally, the multi-resolution analysis property is preserved. Consequently, the applications of the SGWT scheme in condition monitoring and fault diagnosis of mechanical equipments have been increasing the last few years (see Section 4). A basic decomposition of the SGWT consists of three main steps (Sweldens, 1998), split, predict, and update. In the split step, an approximate signal *al* at level *l* is split into even samples and odd samples (Zhou et al., 2010).

$$a\_{l+1} = a\_l \text{(2l)}, \quad d\_{l+1} = a\_l \text{(2l+1)}\tag{25}$$

In the prediction step, a prediction operator *P* is designed and applied on *al+1* to predict *dl*+1. The resultant prediction error *dl*+1 is regarded as the detail coefficients of *al*.

$$d\_{l+1}(l) = d\_{l+1}(l) - \sum\_{r=-M/2+1}^{M/2} p\_r a\_{l+1}(l+r) \tag{26}$$

where *pr* the coefficients of *P* and *M* is the length of *pr* .

In the update step, a designed update operator *U* is applied on *dl*+1. Adding the result to the even samples, the resultant *al*+1 is regarded as the approximate coefficients of *al* .

$$a\_{l+1}(l) = a\_{l+1}(l) + \sum\_{j=-N/2+1}^{N/2} u\_j d\_{l+1}(l+j-1) \tag{27}$$

where *u j* are the coefficients of *U* and *N* is the length of *u j* . Iteration of the above three steps on the output *a*, generates the detail and approximation coefficients at different levels.

$$a\_{l+1}(i) = a\_{l+1}(i) - \sum\_{j=-N/2+1}^{N/2} u\_j d\_{l+1}(i+j-1) \tag{28}$$

$$d\_{l+1}(l) = d\_{l+1}(l) + \sum\_{r=-M/2+1}^{M/2} p\_r a\_{l+1}(l+r) \tag{29}$$

$$a\_l(2i) = a\_{l+1} \qquad a\_l(2i+1) = d\_{l+1} \tag{30}$$

$$X\_{l\,k\,e} = X\_{l\,k}(2l), \quad X\_{l\,k\,o} = X\_{l\,k}(2l+1) \tag{31}$$

$$X\_{l+1,2} = X\_{l,1o} - P(X\_{l,1e}) \tag{32}$$

$$X\_{l+1,1} = X\_{l,1e} + U(X\_{l+1,2}) \tag{33}$$

$$X\_{l+1,2^{l+1}} = X\_{l,2^lro} - P(X\_{l,2^lr\_ee}) \tag{34}$$

$$X\_{l+1.2^{l+1}-1} = X\_{l.2^l \mathbf{e}} + U(X\_{l+1.2^{l+1}}) \tag{35}$$

$$X\_{l,2^l e} = X\_{l+1,2^{l+1}-1} - U(X\_{l+1,2^{l+1}}) \tag{36}$$

$$X\_{l,2^lo} = X\_{l+1,2^{l+1}} + P(X\_{l,2^le}) \tag{37}$$

$$X\_{l,2^l}(2l) = X\_{l,2^l e} \tag{38}$$

$$X\_{l,2^l}(2l+1) = X\_{l,2^l o} \tag{39}$$

$$X\_{l,1e} = X\_{l+1,1} - U(X\_{l+1,2}) \tag{40}$$

$$X\_{l,1o} = X\_{l+1,2} + P(X\_{l,1e}) \tag{41}$$

$$X\_{l,1}(2l) = X\_{l,1e} \tag{42}$$

$$X\_{l,1}(2l+1) = X\_{l,1o} \tag{43}$$

Utilising the Wavelet Transform in Condition-Based Maintenance: A Review with Applications 285

�(�) <sup>=</sup> <sup>∑</sup> ������ � �

�

*m* is the number of wavelet coefficients and *Cn,i* the *i*th wavelet coefficient at the *n*th scale.

�� = ������

�

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

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��(�) (49)

�(�) = �(�)���������(�) (47)

�� <sup>=</sup> �(�)

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

�(�) = ∑ ������

whereas the Energy to Shannon Entropy ratio is given by:

where the entropy of signal wavelet coefficients is defined as:

��������(�) = − ∑ �� � ������ �

and pi is the energy distribution of the wavelet coefficients,

The total energy is given by:

with ∑ �� �

��� = 1.

Fig. 8. Reconstruction step of SGWPT
