# Denoised Improved Envelope Spectrum for Fault Diagnosis of Aero-Engine Inter-Shaft Bearing
## 1 Introduction
However, these signal decomposition-based diagnosis methods are still unable to tackle the multi-source interference with a wide-band distribution problem caused by the dual-frequency excitation of an aero-engine. As shown in Figure 1, it displays an inter-shaft bearing vibration signal under dual-frequency excitation. The fast Fourier transform (FFT) spectrum more clearly reveals the rotation frequencies of both the high-pressure and low-pressure rotors. As illustrated in Figure 2, the dual-frequency excitation makes the collected casing signal contain multiple deterministic interferential components. These components include rotation frequency and the harmonics of the HP rotor and LP rotor, as well as numerous combination frequencies [15; 16]. They occupy a very wide frequency band. An alternative and classical approach for identifying bearing faults is the spectral correlation (SC) analysis [17]. By projecting the vibration signal onto a two-dimensional plane of spectral and cyclic frequencies, SC effectively captures the second-order cyclostationary characteristics associated with bearing faults. [PERSON] [18] identified that incipient bearing faults exhibit distinct random cyclostationary characteristics within the vibration signals of the rolling elements. This observation highlighted the importance of cyclic spectral coherence, which is considered a critical tool for the early diagnosis of such faults. This technique not only confirms the presence of fault-related information amidst significant background noise but also serves as an indicator of relative fault severity. Despite its versatility in analyzing cyclostationary signals, the widespread application of cyclic spectral coherence has been limited by its computational intensity. To mitigate this limitation, [PERSON] et al. [19] introduced the Fast-SC algorithm. It is designed to accelerate the computation of spectral coherence. Additionally, they suggested novel methods for calculating the squared envelope spectrum (SES) and the enhanced envelope spectrum (EES) derived from SC. Following these advancements, a variety of fault diagnosis methods have been proposed by leveraging the spectral coherence [20]. Expanding upon this foundation, [PERSON] et al. [21] developed an improved envelope spectrum (IES), achieved by integrating SC across spectral frequency variables. Experimental validation demonstrated that this approach significantly enhances the diagnostic performance of the SES. [PERSON] et al. [22] further compared the performance of two advanced bearing fault diagnosis methodologies: an MED-SK-based approach and a CS-IES-based method. The experimental findings conclusively revealed that the CS-IES-based strategy exhibits superior capability in detecting  
Figure 1: The inter-shaft bearing vibration signal under dual-frequency excitation. Figure 2: The frequency spectrum of inter-shaft bearing vibration signal under dual-frequency excitation. the incipient faults of rotating machinery. [PERSON] et al. [PERSON] et al. (2019) proposed the improved envelope spectrum via the feature optimization-gram (IESFOgram) method, which selects the optimal frequency band and extracts the corresponding improved envelope spectrum (IES) for bearing fault diagnosis. [PERSON] et al. (2019) introduced a method based on blind and targeted features to identify the optimal spectral frequency band of spectral coherence for bearing diagnostics. The results demonstrate that the IES-based methods can accurately discriminate the failure-related spectral frequency bands of spectral coherence, even in the presence of noise and stronger interfering components. Although IES can effectively extract weak faults under high noise conditions, its performance diminishes when addressing the multi-source interference with wide-band distribution problem under dual-frequency excitation ([PERSON] and [PERSON], 2016; [PERSON] et al., 2019).