# Denoised Improved Envelope Spectrum for Fault Diagnosis of Aero-Engine Inter-Shaft Bearing
## 2 Methods
### IES Algorithm  
With the help of the Fast-SC algorithm, the spectral correlation and a normalized spectral correlation can be computed efficiently. It provides a powerful tool for analyzing cyclic frequency and spectral frequency. The IES is determined using spectral coherence (SCoh) [18] and a normalized form of SC [27]. The SC is mathematically expressed as the double discrete Fourier transform of the instantaneous autocorrelation function [19]. By integrating the SC over a spectral frequency axis within a specific spectral band containing diagnostic information, the IES is derived to amplify fault-related spectral components, providing the basis for solving the multi-source interference with wide-band distribution problem under dual-frequency excitation. Consider a discrete vibration signal represented as \(x(t_{N})\), where \(t_{N}=N/F_{s}\), N = 0, 1,..., \(L-1\) and \(F_{s}\) denotes sampling frequency. The variable \(L\) denotes the length of the vibration signal. In cyclostationary theory, the instantaneous autocorrelation function \(R_{x}(t_{N},\ \tau_{M})\) is essential for calculating the SCoh of cyclostationary processes. This function, related to the bearing vibration signal, includes period \(T\), which is defined as follows:  
\[R_{x}(t_{N},\ \tau_{M})=E\big{(}x(t_{N})x(t_{N}-\tau_{M})^{*}\big{)}=R_{x}(t_{N}+T,\ \tau_{M}) \tag{1}\]  
where \(E\) denotes the ensemble average operator, \({}^{*}\) represents the complex conjugate, \(\tau_{M}=M/F_{s}\) is the time delay, and \(M\) corresponds to the discrete time points. The spectral correlation \(SC_{x}(\alpha,f)\) is then defined as the double discrete Fourier transform of Equation (1). Its mathematical expression is provided as follows:  
\[SC_{x}(\alpha,f)=\lim_{N\to\infty}\frac{1}{(2N+1)F_{s}}\sum_{n=-N}^{N}\sum_{m =-\infty}^{\infty}R_{x}(t_{n},\tau_{m})e^{-2\pi n}\frac{\alpha}{F_{s}}e^{-2 \pi m}\frac{f}{F_{s}} \tag{2}\]  
The function \(SC_{x}(\alpha,f)\) measures the correlation between two frequency components of a signal located at \(f\) and \(f+\alpha\). It can be interpreted as the decomposition of the analyzed signal with respect to the \"modulation frequency\" \(\alpha\) and \"carrier frequency\" \(f\). SCoh\(\gamma_{x}(\alpha,f)\) is the normalized form of SC, where the amplitude is scaled between 0 and 1:  
\[\gamma_{x}(\alpha,f)=\frac{SC_{x}(\alpha,f)}{\sqrt{SC_{x}(0,f)SC_{x}(0,f-\alpha )}} \tag{3}\]  
where \(SC_{x}(0,f)\) represents the classic power spectral density. Based on the \(\text{Scoh}\gamma_{x}(\alpha,f)\), the IES method [28] utilizes the local extrema of each cyclic frequency spectrum slice (CFSS) to identify candidate fault frequencies (CFFs). A CFSS is defined as the \(\text{SCoh}\gamma_{x}(\alpha,f)\), in which the cyclic frequency \(\alpha\) exhibits non-zero values along lines parallel to the spectral frequency axis in the \((\alpha,f)\) plane. Certain vibration signalsdemonstrate second-order cyclostationarity. Their spectral correlation structure shows clear local maxima, which are distributed along CFSSs. These CFSSs are oriented parallel to the circle frequency \(\alpha\) axis. Consequently, CFSSs with a higher number of local maxima are more likely to correspond to cyclic frequencies caused by localized defects in rolling element bearings. Assume that the cyclic frequencies of \(\gamma_{x}(\alpha,f)\) are equidistantly discretized as \(\{\alpha_{n}|\alpha_{1},\alpha_{2},\ldots,\alpha_{N}=\alpha_{\max}\}\), where \(N\) denotes the number of the CFSSs. The key information can be captured by defining \(\eta=[\eta_{(1)},\eta_{(2)},\ldots,\eta_{(N)}]^{T}\) to qualify the number of the local maxima in each CFSS. The \(n\)th element \(\eta_{(n)}\) represents the number of local maxima in the \(n\)th \(CFSS\gamma_{x}(\alpha,f)\). By sorting \(\eta_{(n)}\) in descending order, \(\widetilde{\eta}=[\widetilde{\eta}_{(1)},\widetilde{\eta}_{(2)},\ldots, \widetilde{\eta}_{(N)}]^{T}\) is obtained [28].