## 2 Materials and Methods
### Wavelets  
The wavelet transform has been a well-established concept for a considerable amount of time, originating in 1909. It can be broadly categorized into two main types, the continu  
Figure 4: Amplitude comparison for signals from three different directions: (a) x direction. (b) y direction. (c) z direction. Figure 5: Post-processing diagram. ous wavelet transform (CWT) and the discrete wavelet transform (DWT), each possessing distinct significance in various fields. CWT is a powerful tool that can be used in time-frequency analysis. It can be thought of as an extension of the short-time Fourier transform (STFT). While the STFT uses fixed-length window functions to analyze a signal, CWT employs wavelet functions that are variable in length and can be adapted to the specific features of the signal being analyzed. The definition of CWT is as follows. A function \(\Psi\) with  
\[\int_{\mathbb{R}}\Psi(x)dx=0 \tag{1}\]  
is called a mother wavelet. For every f, \(\Psi\) defines the continuous wavelet transform  
\[W_{\Psi}f(a,b)=\int_{\mathbb{R}}f(x)\overline{\Psi(\frac{x-b}{a})}dx\text{ for all a,b }\in\mathbb{R}_{+}\times\mathbb{R} \tag{2}\]  
where  
\[\Psi_{a,b}=\frac{1}{\sqrt{a}}\Psi(\frac{x-b}{a}) \tag{3}\]  
The function \(\Psi\) is called the wavelet function or the mother wavelet function. It is chosen to be localised at \(x=0\) and at some \(\omega=\omega_{0}>0\) (and/or \(\omega=-\omega_{0}\)). Here \(a\) and \(b\) are named the scale factor and the shift factor. Calculating CWT is computationally expensive. One alternative is to choose a subset of scales and positions to perform the calculations. The selection of scales and positions based on powers of two is a defining feature of discrete wavelet transform (DWT) analysis. Overall, DWT can be viewed as a form of band-pass filtering that separates a signal into different frequency components, allowing for the extraction of useful information and the creation of a detailed representation of the signal. The definition of DWT is shown below:  
\[DWT(a,b)=\frac{1}{\sqrt{a}}\int_{-\infty}^{+\infty}f(x)*\Psi\bigg{(}\frac{x-b }{a}\bigg{)}dx \tag{4}\]  
where  
\[a=2^{j},b=k2^{j},\Big{(}k,j\in\mathbb{Z}^{2}\Big{)} \tag{5}\]  
Here, \(a\) and \(b\) are also the scale factor and the shift factor, respectively. Wavelet denoising involves the use of the time-frequency amplitude matrix produced by the wavelet transform. The process of wavelet denoising uses the discrete wavelet transform (DWT) to break down the original signal into its constituent wavelets and obtain the corresponding time-frequency amplitude matrix. To carry out the denoising process, thresholding is applied to the wavelet coefficients associated with the matrix to remove noise and retain the significant components of the signal. Once the thresholding is complete, the signal is reconstructed using the reverse DWT [43]. The application of wavelet denoising is particularly valuable in processing vibration signals, as it can help to remove unwanted noise and enhance the useful features of the signal. [PERSON] et al. [44] demonstrated the effectiveness of this technique by utilizing a wavelet-denoising algorithm to preprocess vibration signals from wind turbines. [PERSON] et al. [45] introduced a system utilising wavelet denoising in combination with several other signal-processing techniques to improve the accuracy of bearing fault diagnosis. [PERSON] et al. [46] carried out a study about using a wavelet denoising algorithm with multi-level decompositions to monitor the condition of a heavy-haul railway. When implementing wavelet denoising, there are several parameters, methods and thresholding techniques that can be chosen. The parameter maximum level of decomposition in wavelet decomposition depends on both the length of the signal (N) and the choice of the wavelet basis. The maximum number of levels for the decomposition of the obtained signals is 21. The levels of the coefficients are known to influence the kurtosis of the signal [47].