## 2 Wavelet Transform for Sounds from Frictional Contacts
### Haar Wavelet Coefficient and PSNR
Wavelet constant \(\alpha\) is always \(1/\sqrt{2}\) because of energy conservation for normalizing:  
\[d_{k}^{2}+c_{k}^{2}=2\alpha^{2}\Big{(}y_{2k}^{2}+y_{2k-1}^{2}\Big{)}. \tag{2}\]  
Figure 3: Sound responses from the metal contact surface: (**a**) transient response and (**b**) STFT result. For the Haar-WT in fine-scale detail and approximation, the consecutive process of the calculation of Equation (1) is performed. In this way, the approximated response in level \(j\) is extracted by accompanying the scaling function \(\phi\) and wavelet function \(\psi\) with detail and coarse coefficients [30]:  
\[y_{j+1}(t)=\sum_{k=0}^{2j-1}\Big{\{}c_{j,k}\phi_{j,k}(t)+d_{j,k}\psi_{jk}(t) \Big{\}}, \tag{3}\]  
where the \(\phi\) and \(\psi\) also represent low-pass filter and high-pass filter components, respectively, which is calculated as  
\[\phi_{j,k}(t)=2^{j/2}\phi\Big{(}2^{j}t-k\Big{)},\phi(t)=\left\{ \begin{array}[] {ll}1&\text{if }t\ \in[0,1]\\ 0&\text{otherwise}\end{array} \right., \tag{4}\]  
\[\psi_{j,k}(t)=2^{j/2}\psi\Big{(}2^{j}t-k\Big{)},\psi(t)=\left\{ \begin{array}[ ]{ll}1&\text{if }t\ \in[0,1/2)\\ -1&\text{if }t\ \in[1/2,1)\\ 0&\text{otherwise}\end{array} \right.. \tag{5}\]  
From Equation (3), detail and coarse coefficients in the level \(j-1\) can be extracted by Euler formulas as  
\[c_{j-1,k}=\int_{2^{-j-1}(k)}^{2^{-j-1}(k+1)}y(t)\phi_{j-1,k}(t)dt, \tag{6}\]  
\[d_{j-1,k}=\int_{2^{-j-1}(k)}^{2^{-j-1}(k+1)}y(t)\psi_{j-1,k}(t)dt. \tag{7}\]  
Here, the wavelet function is denoted as \(\psi(t)=\phi(2t)-\phi(2t-1)\). The wavelet coefficient for coarse and detail components can be briefly extracted by rearranging Equations (6) and (7) with the following relation:  
\[c_{j+1,k}=\alpha\Big{(}c_{j,2k}+c_{j,2k+1}\Big{)},\;d_{j+1,k}=\alpha\Big{(}c_ {j,2k}-c_{j,2k+1}\Big{)}. \tag{8}\]  
We can evaluate the detail and coarse coefficients in level \(j+1\) from the coefficients in level \(j\). By repeating the procedures, we can continuously decompose the coarse coefficient as the level processes. In the last level, the arrangement of the coefficient consists mostly of detail coefficients. Figure 4 shows the progress of the discrete wavelet transform. Each level reflects the frequency band. For \(n\) sampling frequency in level \(j\), coarse and detail coefficients reflect the frequency band from \(0\) Hz to \(n/(j+1)\) Hz and from \(n/(j+1)\) Hz to \(n\) Hz, respectively. The change in the period of the detail coefficients reflects the variation in the high-frequency components that is involuntarily induced by the frictional wear. Regarding the wavelet coefficient in the high-frequency range, it is advantageous for it to be expressed as the single value of change in response to the contact phenomenon between the components. This PSNR is expressed as below [31; 32]  
\[P=10\log_{10}^{(d_{peak}^{2}/E)}, \tag{9}\]  
where \(d_{\text{peak}}\) denotes the detail coefficient peak value, and \(E\) is the mean square error of the coefficient array. The \(E\) is computed as  
\[E=\frac{1}{n}\sum_{i=1}^{n}\Big{(}d_{j}-\hat{d_{j}}\Big{)}^{2}. \tag{10}\]  
Here, \(\hat{d_{j}}\) denotes the predicted value calculated by a regression model based on given wavelet coefficients. The \(E\) represents how far the data are located from the average values. The PSNR statistically represents the fine change in the occurrence of the frictional wear.