## 2 Wavelet Transform for Sounds from Frictional Contacts
### Haar Wavelet Coefficient and PSNR  
The instantaneous motion caused by friction between two components induces plastic deformation of the surface and generates acoustic energy and heat. Noise from the frictional wear generates as the local response rather than the modal vibration responses. The contact pressure and sliding velocity have significant influence on the response. The frequency spectrum variation at regular intervals requires monitoring for the inspection of the rapid time transition section of the noise caused by friction. Because of the trade-off relationship between time and frequency intervals in short-time Fourier transform (STFT), it is not straight forward to extract components from the response due to wear in a short time. Figure 1: Mechanisms of two different wears: (**a**) adhesive wear and (**b**) abrasive wear. Figure 2: Experimental setting for scratching on the metal for adhesive and abrasive friction: (**a**) schematic section view of scratching experiments using linear motor; (**b**) metal friction using linear motor; (**c**) iron powder on metal to realize abrasive wear. The experiments were performed at the same relative speed of 50 mm/s. During the experiments, care was given to replace the mating surfaces to minimize the influence of wear by the previous measurements. By adjusting the height of the jig, the contact pressure between the two metals was varied. Measurements were conducted for three different contact pressures of 5, 11, and 23 kPa. This contact pressure was measured using the load cell (Model KMR-20 kN, HBM Inc., Darmstadt, Germany), which is set on the scratching metal block. All cases were performed under dry friction. In the experiment, the sound radiation was acquired with a sampling rate of 131,072 Hz by using a data acquisition device (Model 3560-B-030, Bruel & Kjaer Inc., Narum, Denmark). System response was acquired by using a microphone sensor (Model 378B02, PCB Piezotronics Inc., New York, NY, United States), and it was installed and measured at the same height 10 cm away from the scratch. To confirm the reproducibility of the experiment, the same signal processing procedure described above was applied to a friction signal of 1 s at different data points. Wavelet transform (WT) and PSNR calculations were performed on the acquired time data using MATLAB. Figure 3a shows the sound response measured by the microphone for the two metals, as represented in Figure 2b. Since the friction randomly varied in magnitude, it was difficult to capture consistent statistical characteristics. Figure 3b shows the STFT results. The frequency distribution of the acoustic energy changed continuously and irregularly with time. Despite this, there is a section that showed a very high response of up to 2500 Hz; this frequency response does not contain the contribution from friction components. It was generated from the vibrations of two metals where scratches occurred. This makes it difficult to derive the friction components. This variability is a major obstacle in extracting the inherent characteristics of friction when monitoring a metal system. The process of decomposing the signal for each band was efficient. WT complements the shortcomings of this STFT. In particular, Haar-WT is a convenient method for analyzing acoustic signals expressed in time. In addition, it shows good performance for the decomposition of the signal with the average change of magnitudes [26; 27; 28]. Haar-WT distinguishes detail and coarse coefficients, which is calculated from the original response, \(y\), as represented by [29]:  
\[d_{k}=\alpha(y_{2k}-y_{2k-1}),\ c_{k}=\alpha(y_{2k}+y_{2k-1}), \tag{1}\]  
where subscription \(k\) represents the order of acoustic response samples. The \(c\) and \(d\) denote the coarse and detail components, respectively.