# Denoised Improved Envelope Spectrum for Fault Diagnosis of Aero-Engine Inter-Shaft Bearing
## 1 Introduction  
An aero-engine is the core component of an aircraft. Its stable operation guarantees the safe flight of the aircraft [1]. The rolling bearing within this system is deemed one of the most critical elements. Its unexpected failure could cause major economic losses. And it might even lead to catastrophic incidents. [2]. Among the rolling bearings of an aero-engine, the inter-shaft bearing is a significant component in the support system ofa dual-rotor aero-engine [3]. A tough working environment is extremely likely to cause various faults, such as wear, scratches, and peeling off of oil film [4]. Once a fault occurs in an inter-shaft bearing, it will weaken the stability of an engine in minor cases and even cause the shutdown of the engine in the air [5; 6]. Therefore, conducting fault diagnosis for the inter-shaft bearing is of great significance. In the current body of research, fault diagnosis of inter-shaft bearings is predominantly performed using vibration signal analysis. After all, the casing signals measured by vibration sensors are able to reveal the health status of an aero-engine's transmission chain. Moreover, the dynamic degradation process of the inter-shaft bearing often leads to variations or abnormalities in vibration signals. Through vibration signal analysis, the spectral characteristics of vibration signals can be obtained. By extracting the fault characteristic frequencies from the spectrum, it is possible to determine whether an inter-shaft bearing has failed and, if it has, what type of fault has occurred. The typical vibration-based diagnosis methods mainly include narrowband-pass filtering, signal decomposition, spectral correlation analysis, and so on. For example, [PERSON] et al. [7] proposed a rotary speed-difference-domain spectrum and envelope spectrum method for fault detection in inter-shaft bearings. The rotary speed difference between the high-pressure (HP) rotor and the low-pressure (LP) rotor is used as a triggering signal. The vibration signals were sampled for the whole cycles of the rotary speed-difference-domain. However, this signal acquisition process is very complex and not suitable for practical conditions. [PERSON] et al. [8] treat the peak factor of a partial envelope spectrum as a new optimization parameter of the frequency band and apply it to resonance demodulation [9]. When a single inner or outer race fault occurs in an inter-shaft bearing, the method proposed by [PERSON] et al. [8] can effectively identify the type of fault. However, the core of this method is still based on the bandpass-filtering principle, which complicates the extraction of weak fault signals from a background of wide frequency distribution noise. As for signal decomposition-based diagnosis methods, they have been widely utilized for fault component extraction. The representative approach includes variational mode decomposition (VMD) [10], wavelet transform (WT) [11], and so on. However, these approaches have a high parameter sensitivity. For example, when the VMD is used for fault component isolation, its two parameters, named mode number and penalty factor, need to be carefully optimized [12]. To effectively extract the fault information of the inter-shaft bearing from vibration signals, [PERSON] et al. [6] proposed an decomposition-reconstruction-based diagnostic method. This method integrates information fusion, WT, singular value decomposition (SVD), Katz fractal dimension, and cross-correlation function (CCF). It is termed as CCF-WT-SVD-Katz. Each component of this algorithm facilitates signal denoising, enhanced reconstruction, and fault feature extraction. Moreover, [PERSON] et al. [13] introduced a method for diagnosing bearing faults by combining intrinsic time-scale decomposition (ITD) [14] and SVD. The results demonstrated that the ITD-SVD method exhibits higher accuracy in noisy conditions compared to ITD and principal component analysis methods.