# Denoised Improved Envelope Spectrum for Fault Diagnosis of Aero-Engine Inter-Shaft Bearing
## 2 Methods
### Improved Adaptive Denoising for Subtracted NIES  
The IES is able to efficiently extract the weak fault components of a bearing under high background noise. However, when the IES encounters both strong noise and a dual-frequency excitation problem, its performance diminishes. Specifically, the spectrum obtained by the use of IES retains interfering frequency components and random noise, complicating the identification of FCFs and their harmonics. For instance, Figure 3 illustrates the IES of an inter-shaft bearing vibration signal with an inner race fault. In the figure, the frequency response at BPFI (ball pass frequency inner race) and its \(i\) harmonics (\(i=\) {1 st, 2 nd, 3 rd, 4 th, 5 th,...}) are considered as the frequencies related to the inner race fault of the bearing. Furthermore, it can be observed that the deterministic interference and noise components obscure the fault components. Notably, some interfering components exhibit higher amplitudes than the fault-related frequencies. To address these limitations, an improved adaptive denoising method is proposed. The core principle of this method lies in that the deterministic interfering componentsare generally considered to exist in both healthy and faulty signals within the frequency domain. Then, to mitigate their impact on fault diagnosis, an improved noise subtraction (INS) method is introduced to eliminate these interfering components. Initially, the normalized IES (NIES) for both faulty and healthy signals is computed. A subtraction operation is further performed between the two NIES to derive a subtracted NIES. For an IES comprising \(N_{IES}\) spectral lines with amplitudes \([A_{1},A_{2},\ldots,A_{N}]\), the NIES is defined as follows:  
\[NIES=\frac{IES}{\|IES\|_{L1}} \tag{8}\]  
where \(\|IES\|_{L1}=\sum_{i=1}^{N}\lvert A_{i}\rvert\) for \(i=1,2,\ldots,N\). Using this formula, the NIES for both faulty and healthy signals are calculated. The subtraction operation is then applied, yielding the analytic solution:  
\[NIES_{sub}=\frac{NIES_{F}-NIES_{H}}{\|NIES_{F}-NIES_{H}\|_{L2}} \tag{9}\]  
where \(NIES_{F}\) and \(NIES_{H}\) represent the normalized IES of the faulty and healthy signals, respectively. Let  
\[(NIES_{F}-NIES_{H})=[A_{1}^{*},\ldots A_{2}^{*},\ldots A_{N}^{*}]^{T} \tag{10}\]  
where \(A_{n}^{*}\) denotes the spectral line of the subtraction result. The \(L_{2}\) norm is computed as:  
\[\left\lVert NIES_{F}-NIES_{H}\right\rVert_{L2}=\sqrt{\sum_{i=1}^{N}\left(A_{i }^{*}\right)^{2}} \tag{11}\]  
Following spectral subtraction, the \(NIES_{sub}\) method facilitates the identification of fault-related components, random noise, and deterministic interference components. The latter may also appear in the spectrum of healthy signals. An illustration of the spectral subtraction process is shown in Figure 4. The red lines represent the frequencies of faulty components, while the black lines represent the frequencies of deterministic interference components. Since the spectral energy ratio of deterministic interference components in the fault spectrum is smaller than that in the healthy spectrum, spectral components can be distinguished by symbols after the normalization and subtraction process. As illustrated in Figure 5, the \(NIES_{sub}\) is obtained through spectral subtraction using the real healthy and faulty inter-shaft bearing signal spectra. In the figure, the frequency response at BPFO (ball pass frequency outer race) and its \(i\) harmonics (\(i\) = {1 st, 2 nd, 3 rd, 4 th, 5 th,...}) are considered as the frequencies related to the outer race fault of the bearing. In the \(NIES_{sub}\), frequencies with positive amplitudes correspond to fault-related frequencies. Those with negative amplitudes reflect deterministic interference components. Figure 3: IES of an inter-shaft bearing vibration signal with inner race fault. However, in addition to deterministic interferential components, the subtracted spectrum still contains much random noise.