## 3 Improvement of Envelope Analysis
### Methods for Overcoming Impulsive Noises  
We investigated the phenomenon where the performance of defect detection through envelope analysis was degraded due to impulsive noise. Figure 9 shows the spectrum at 0.2 s in Figure 7a, which is contaminated by the large impulsive noise, and the spectrum at 0.7 s in Figure 7a, which is sufficiently out of the effect of the impulsive noise. In Figure 9a, the effect of impulsive noise compared to Figure 9b greatly increases the baseline of the spectrum, regardless of whether it is defective. As a result, the relative magnitude of the peak due to the defect is smaller than that in Figure 9b, making it difficult to set a threshold for defect detection. Defect detection through envelope analysis distinguishes between the normal and defect by using the magnitude at the rotational frequency of the rotating machinery, as described above. However, the magnitude of the impulsive noises in a wide frequency range that is irrelevant to rotation also affects the rotational frequency, making it difficult to detect defects. To overcome this disadvantageous phenomenon, we propose using the difference in magnitude at the rotation frequency and at frequencies sufficiently lower than the rotational frequency as a feature of defect detection. This idea is inspired by the frequency characteristics of very short time single rectangular wave given by Equations (12) and (13). \(\tau\) is duration of impulsive noise \(i(t)\) and \(I\) is the amplitude of \(i(t)\). As \(\tau\) gets shorter, \(I(f)\) appears as the horizontal stretched sinc function in frequency domain. Therefore, it can be seen that the magnitude of the relatively lower frequency is larger due to the short \(\tau\). This applies similarly to a single triangle wave. \[i(t)=\left\{ \begin{array}{c}I,-\frac{\tau}{2}<t<\frac{\tau}{2}\\ 0,\text{elsewhere}\end{array} \right. \tag{12}\]  
\[I(f)=\mathcal{F}\{i(t)\}=I\tau\frac{\sin(\tau\pi f)}{\tau\pi f}=I\tau\cdot \text{sinc}(\tau\pi f) \tag{13}\]  
The flowchart is shown in Figure 10. The window length \(T\) is selected considering time resolution for impulsive noise and frequency resolution for rotating frequency. The time step is defined by the degree of overlap, window length \(T\), and total length of the measured signal. The window length \(T\) should be selected as short as possible, but the frequency resolution \(\Delta f\), is reciprocal value of \(T\), sufficient to detect the rotational frequency \(f_{rot}\) must also be guaranteed. Then the magnitude \(M_{n}\) is calculated from first time step to the end of time step \(N\). We can calculate the detection value at the end of the flowchart to define the threshold that can differentiate between normal and faulty. Figure 9: Comparison of envelope spectrum (**a**) without an impulsive noise and (**b**) in the presence of an impulsive noise. The proposed method uses the characteristics of the impulsive rectangular signal, which has a very large instantaneous magnitude in the time domain but has broadly magnitudes from low to high frequencies in the frequency domain. In the spectrums with and without impulsive noise in Figure 9, in the normal signal, there is no difference between the magnitude at the rotational frequency of 18 Hz and the baseline magnitude of the lower frequencies in both. However, the defective signal shows the difference between the peak magnitude at the rotational frequency of 18 Hz and the baseline magnitude of the lower frequencies in both Figure 9a,b. Figure 11 represents the cases of normal and defect depending on the number of impulsive noises contained. In the STFT of Figure 6a without impulsive noises, the difference between the magnitude at the rotational frequency of 18 Hz and the baseline magnitude of the lower frequencies is shown in Figure 11a. As shown in Figure 6b, which compares only the magnitude at the rotation frequency in the STFT, the normal and the defect in Figure 11a are clearly distinguished.