# Denoised Improved Envelope Spectrum for Fault Diagnosis of Aero-Engine Inter-Shaft Bearing
## 2 Methods
### Improved Adaptive Denoising for Subtracted NIES
These noises typically vary across different sampled signals, and strong noise may hinder the recognition of fault information, as depicted in Figure 3. To mitigate the impact of random and strong noise components effectively, an improved adaptive denoising technique is proposed to accentuate the fault frequencies. The proposed improved adaptive threshold is defined as follows:  
Figure 4: Illustration of spectral subtraction process: (**a**) faulty signal spectrum, (**b**) healthy signal spectrum, (**c**) normalized spectrum after spectral subtraction. The red lines represent the frequencies of faulty components; the black lines represent the frequencies of deterministic interference components. Figure 5: NIES after spectral subtraction using the real healthy and faulty inter-shaft bearing signal spectra. First, the amplitude percentile points (APPs) of all the spectral lines of the \(NIES_{sub}\) are computed and sorted in descending order of their amplitudes:  
\[APPs=\Big{(}A_{i},\frac{N_{i}}{N}\Big{)},i=1,2,\ldots,N \tag{12}\]  
where \(N_{i}\in[0,N-1],N_{i}\in\mathbb{Z}^{+}\). Here, \(N_{i}\) represents the number of spectral lines with amplitudes smaller than \(A_{i}\). As shown in Figure 6, the horizontal axis corresponds to amplitude \(A_{i}\), while the vertical axis represents the percentile \(\frac{N_{i}}{N}\). To determine an appropriate threshold for spectral denoising, the first D% of the APPs are selected for further analysis. As shown in Figure 6, the amplitude percentile point sequence is used to identify the optimal threshold point that distinguishes random noise from meaningful data. This is achieved through a line-based change point detection model. It utilizes a linear approach to detect significant changes in slope. Figure 6 illustrates this model. Two lines are fitted to different segments of the data, and the threshold point is identified where the slope change is most pronounced. Once the adaptive threshold point is determined, the frequencies associated with random noise can be effectively eliminated. As shown in Figure 7a, the positive amplitudes of the \(NIES_{sub}\) are extracted to illustrate the denoising effect of adaptive thresholding. This threshold is represented by a red dotted line. However, although this operation eliminates some random noise with low amplitude, the spectrum still contains high-amplitude noise, as shown in Figure 7b. According to the method's principle, the parameter \(D\) is critical. Based on the principle stated in the study of [PERSON]. [PERSON] et al. [29], the parameter \(D\) was empirically set to 2. Subsequently, the \(NIES_{sub}\) is denoised using a computed adaptive threshold, producing the DIES. The result of DIES is shown in Figure 7c. In summary, the calculation process of the adaptive threshold is as follows:  
\[NIES_{sub}=\frac{NIES_{F}-NIES_{H}}{\|NIES_{F}-NIES_{H}\|_{L2}} \tag{9}\]0 Illustration of line-based change point detection model. \[NIES_{sub}=\frac{NIES_{F}-NIES_{H}}{\|NIES_{F}-NIES_{H}\|_{L2}} \tag{9}\]1 _Cont._  
1. Compute amplitude percentile points of all the spectral lines of the \(NIES_{sub}\);
2. The first D% of the data is selected in descending order;
3. Compute line-based change point using the selected data;
4. Calculate the threshold for denoising.