1. Introduction
As one of the critical moving components in mechanical equipment, rolling bearings find widespread application within various industries involving rotating machinery. Nevertheless, due to prolonged operational periods and the impact of working environments, rolling bearings often succumb to a range of faults, such as fatigue fractures, inadequate lubrication, and localized damage. If these faults are not promptly detected and diagnosed, they can lead to equipment downtime, damage, or even accidents, significantly hampering production efficiency and safety. Consequently, the investigation of rolling bearing fault diagnosis has consistently remained an essential and burgeoning topic within the field of mechanical engineering. The precise and rapid identification of fault features from rolling bearings holds substantial significance for predicting equipment lifespan, devising maintenance schedules, and enhancing equipment reliability [
1,
2].
In recent years, with the continuous advancement of signal processing and fault diagnosis techniques, an increasing number of methods have been proposed for extracting fault features from rolling bearings. Among these, time–frequency analysis methods play a crucial role in rolling bearing fault diagnosis [
3]. The development of time–frequency analysis methods can be traced back to the 1940s [
4] when Fourier analysis emerged as a principal tool in signal processing [
5,
6]. Fourier analysis involves decomposing a signal into a series of superimposed sine and cosine functions, facilitating frequency–domain analysis of the signal. However, traditional Fourier analysis methods have difficulties in dealing with non-stationary and nonlinear signals effectively, whereas rolling bearing signals typically exhibit non-stationary and nonlinear characteristics. As a result, adaptive time–frequency analysis methods have emerged as suitable alternatives, offering more flexibility and accuracy in handling non-stationary and nonlinear signals. In recent years, numerous adaptive time–frequency analysis methods have been proposed and applied to rolling bearing fault feature extraction. Among these, methods such as empirical mode decomposition (EMD) [
7], local mean decomposition (LMD) [
8], and ensemble empirical mode decomposition (EEMD) [
9] have achieved significant research progress in the realm of time–frequency analysis. Notably, EMD is a data-driven adaptive time–frequency analysis method that decomposes a signal into a series of intrinsic mode functions (IMFs), thereby extracting time-varying characteristics from the signal. LMD, on the other hand, decomposes a signal into local mean components and detail components, facilitating a time–frequency representation of the signal. EEMD, based on EMD, introduces random sampling to reduce mode-mixing effects and improve the precision of time–frequency analysis through multiple reconstructions. These methods have found widespread application in rolling bearing fault feature extraction. However, these methods often encounter challenges related to mode mixing and endpoint effects when dealing with non-stationary signals.
In recent years, the Fourier decomposition method (FDM) has emerged as a distinctive approach to handling non-stationary signals [
10]. The FDM technique employs the Fourier transform to jointly analyze signals in both the frequency and time domains, decomposing them into fundamental frequency components and modulation functions. This combined analysis offers a more accurate depiction of signal time-varying characteristics, effectively avoiding mode mixing issues. Presently, this method has found applications across various domains. Karan Singh Parmar et al. [
11] developed an automated diagnostic system for hypertension detection using electrocardiogram (ECG) signals. In their work, they applied the FDM to decompose ECG signals, followed by an assessment of signal transfer entropy and logarithmic energy entropy features. These features were subsequently used for classification in a classifier, yielding promising recognition outcomes. C. Dou et al. [
12] substantiated through numerical analysis that the FDM overcomes the performance limitations of EMD in the adaptive separation of closely spaced frequency signal components. Moreover, by comparing the proposed method with conventional time–frequency analysis techniques, their results revealed its superior performance in characterizing vibration signals from turbine gearbox systems. Minqiang Deng et al. [
13] introduced a fault diagnosis method for Gearboxes based on Resonance Bandwidth Fourier Decomposition (RBBFD). By comparison with existing techniques, they validated the superior performance of their method in gearbox fault diagnosis. Binish Fatimah et al. [
14] proposed a recognition method for hand movements using surface electromyographic signals. Their approach involved the use of FDM to decompose signals, followed by the calculation of entropy, kurtosis, and norms of each signal, which were then utilized for training a classification model. Through testing on the UCI dataset and NinaPro DB5, their proposed method exhibited promising recognition results.
However, real-world mechanical systems often experience interference from strong background noise, posing challenges in accurate feature extraction and analysis of bearing faults. Traditional fault diagnosis methods perform poorly in such conditions, necessitating the need for more precise and effective technical solutions. Singular Value Decomposition (SVD), as a matrix decomposition technique [
15,
16], has been extensively applied in signal processing and data dimensionality reduction. SVD can decompose a signal into distinct singular values and corresponding singular vectors, thereby effectively removing noise components associated with smaller singular values and achieving signal denoising. Presently, the SVD method has been employed to extract useful features from signals in various domains, enhancing signal clarity and accuracy. Ren Y et al. [
17] addressed the difficulty in feature extraction from pressure fluctuation signals in tailwater pipes under a strong noise background by proposing a denoising method based on adaptive local iterative filtering (ALIF) and SVD. Evaluation through simulation experiments demonstrated that the ALIF-SVD denoising approach effectively eliminated noise while preserving useful signal information. Zhong C et al. [
18] presented a rolling bearing radial basis neural network fault diagnosis method based on improved Ensemble Empirical Mode Decomposition-SVD (EEMD-SVD), validating its effectiveness through simulated bearing failure data. To suppress noise in atmospheric lidar echo signals, Cheng X et al. [
19] proposed an Enhanced EEMD-SVD-Lifting Wavelet Transform (LWT) denoising algorithm. Comparative simulation experiments highlighted the superior denoising performance of the EEMD-SVD-LWT algorithm. Additionally, the CYCBD method, introduced by BUZZONI et al. [
20] in 2018, as a second-order cyclic stationary blind deconvolution technique, has demonstrated exceptional performance in complex background noise scenarios. It has been applied to signal filtering and envelope demodulation analysis in fault diagnosis [
21,
22], enhancing the effectiveness of fault feature extraction.
Based on these considerations, this paper proposes a novel approach for fault feature extraction by combining improved FDM-SVD and CYCBD. First, the FDM technique is utilized to decompose bearing fault signals into a series of signal components. Subsequently, a fusion of kurtosis, skewness, and permutation entropy-based criteria is employed to select and reconstruct useful signal components. Next, Singular Value Decomposition (SVD) is applied to denoise the reconstructed signals. Finally, the CYCBD method is employed to filter the denoised signals, followed by envelope demodulation analysis. By synergistically leveraging the strengths of these methods, our aim is to overcome the interference of strong background noise in bearing fault signal analysis, thereby achieving more accurate and reliable fault feature extraction.
The main contributions of this study are as follows:
- (1)
This paper introduces the application of FDM in decomposing the original signals. FDM not only divides the signals into different frequency components but also effectively eliminates adverse effects such as mode mixing. Through signal decomposition, we can gain a deeper understanding of the behavioral characteristics of bearings across various frequencies, thereby enabling a more accurate diagnosis of bearing faults. Furthermore, this study improves the selection mechanism for sensitive signal components by establishing a comprehensive weighted screening criterion that balances the strengths and weaknesses of multiple indicators. This criterion avoids potential blindness in selecting and discarding signal components. By objectively evaluating each signal component obtained through time–frequency analysis, we select the optimal signal components, which can address potential issues of insufficient extraction of bearing fault feature information and enhance the accuracy of fault diagnosis.
- (2)
In this study, the Maximum Second-Order Cyclostationary Blind Convolution (CYCBD) method is employed to filter the signals after noise reduction through Singular Value Decomposition (SVD). This approach effectively highlights the periodic impact components within the signals, enabling us to extract the characteristic frequencies of bearing faults more clearly. This processing technique provides novel insights into the field of signal processing and enriches the theoretical and technological framework of the discipline.
The structure of this paper is as follows:
Section 2 introduces the fundamental principles of improved FDM, SVD, and CYCBD algorithms.
Section 3 delineates the specific implementation steps of the proposed methodology.
Section 4 validates the effectiveness of the proposed approach through simulation experiments.
Section 5 verifies the effectiveness of the proposed approach through actual bearing operation data.
Section 6 presents the discussion and conclusion.
4. Simulation Verification
In order to verify the rationality of the method proposed in this study, simulation signals simulating the vibration of rolling bearings were introduced for experiments. The expression of this signal is as follows:
where the displacement constant
y0 = 5; the carrier frequency
fn = 3000 Hz; the damping ratio
= 0.1;
fs = 20 kHz;
t represents the sampling time, with a period
T = 0.01 s; the number of sampling points is
N = 4096; and the fault frequency
f0 = 100 Hz.
To simulate a bearing fault in accordance with real-world scenarios, noise exhibiting a signal-to-noise ratio of −10 dB was introduced into the aforementioned simulated signal. The simulation and analysis were conducted using MATLAB software (R2009a). The temporal domain representation of the generated simulation signal is depicted in
Figure 2a. When the signal-to-noise ratio (SNR) is −10 dB, as shown in
Figure 2b, it can be observed that due to the influence of strong background noise, the periodic characteristics of the temporal signal waveform have been masked. Consequently, valuable fault information cannot be identified from the temporal domain representation. Thus, further signal analysis is imperative.
The subsequent step involves the FDM analysis of the simulated signals after the introduction of noise. Following signal decomposition, a total of 35 signal components and the ultimate residual component are obtained. The decomposition results of the FDM are illustrated in
Figure 3. Due to constraints on the article’s length, only the decomposition outcomes of the first 20 signal components are presented. From the illustration, it is evident that this method effectively suppresses modal aliasing phenomena and substantially mitigates endpoint effects. To underscore the superiority of the fault diagnosis approach proposed in this paper, a comparative analysis is conducted against traditional methods. Specifically, the simulated noisy signals are subjected to EMD, EEMD, ITD, and VMD decompositions. The decomposition results of EMD, EEMD, ITD, and VMD are shown in
Figure 4a–d, respectively. The analysis of the decomposition results of EMD and EEMD reveals that after EMD decomposition, 12 intrinsic mode functions (IMFs) and 1 residual component were obtained. Through EEMD decomposition, 11 IMFs and 1 residual component were derived. Notably, both methods exhibited instances of mode mixing in their decomposition results. Additionally, the decomposition using the ITD method yielded five PRC components and one residual component, while the VMD method produced six IMF components. After obtaining the signal components through the time–frequency analysis method, further calculations were performed to determine the kurtosis, skewness, and permutation entropy of each signal component. Subsequently, the filtering indices for these signal components were derived.
Table 1,
Table 2,
Table 3,
Table 4 and
Table 5 present the calculated comprehensive screening indices for all signal components obtained using FDM, EMD, EEMD, ITD, and VMD methods, respectively. A comparative analysis of the indices for the 35 signal components obtained through FDM in
Table 1 reveals that the indices for the 20th, 21st, and 23rd signal components are significantly higher than those of the other components, making them suitable for signal reconstruction. Similarly, a comparison of the indices for the 12 signal components derived from EMD decomposition in
Table 2 demonstrates that the index value for the second signal component is notably higher than the rest, thus selecting it for signal reconstruction. Furthermore, an examination of the indices for the 11 signal components obtained through EEMD decomposition in
Table 3 indicates that the index value for the second signal component exceeds those of the other components, leading to its selection for signal reconstruction.
Table 4 showcases the various PRC components derived from ITD decomposition, with the second signal component exhibiting the most significant index value among them, justifying its selection for signal reconstruction. Lastly,
Table 5 presents the IMF components obtained through VMD decomposition. Among these, the fourth signal component stands out with the most prominent index value, thus chosen for signal reconstruction.
Subsequently, the component signals obtained through signal decomposition based on the FDM algorithm and constructed screening rules were reconstructed. Subsequently, the Hankel matrix was further constructed and subjected to SVD decomposition. The resulting singular value spectrum and singular value energy differential spectrum are presented in
Figure 5 and
Figure 6, respectively.
Figure 5a–e depicts the singular value distribution curves obtained through the FDM, EMD, EEMD, ITD, and VMD methods, respectively. It can be observed from the figures that, within the range of 1 to 100, the singular values continuously decrease and tend to stabilize as the sequence number increases.
Figure 6a–e present the singular value energy differential spectrum obtained using the FDM, EMD, EEMD, ITD, and VMD methods, respectively. As evident from
Figure 6a, the energy of the 16th peak in the signal is relatively high, while subsequent peaks exhibit lower energy levels. Consequently, the reconstruction order corresponding to this peak is selected. Similarly,
Figure 6b reveals that the energy of the 28th peak is significantly higher than the following peaks, prompting the selection of the reconstruction order associated with this peak. Likewise,
Figure 6c indicates that the energy of the 22nd peak in the signal is relatively larger compared to subsequent peaks, leading to the selection of the corresponding reconstruction order.
Figure 6d demonstrates that the energy of the 28th peak is relatively greater than subsequent peaks, justifying the choice of the corresponding reconstruction order. Finally,
Figure 6e shows that the energy of the 18th peak stands out among subsequent peaks, thus justifying the selection of the reconstruction order corresponding to this peak.
Figure 7,
Figure 8,
Figure 9,
Figure 10 and
Figure 11 present the time-domain waveforms of signals denoised using five different methods: FDM, EMD, EEMD, ITD, and VMD. From these figures, we can clearly observe that each method effectively reduced varying degrees of noise interference after applying the SVD denoising technique. To accurately evaluate and compare the denoising effects of the above methods, this study employed quantitative analysis using assessment metrics such as the signal-to-noise ratio (SNR), cross-correlation coefficient, kurtosis, and Root Mean Square Error (RMSE). The relevant calculation results have been compiled in
Table 6.
From
Table 6, we can observe that after denoising the signal using the FDM-SVD method, the resulting signal-to-noise ratio (SNR) is 2.241, the cross-correlation coefficient is 0.664, the kurtosis value is 4.869, and the RMSE is 0.179. In contrast, for the EMD-SVD method, the corresponding values are 1.772, 0.599, 4.511, and 0.189, respectively. The EEMD-SVD method yields an SNR of 1.856, a cross-correlation coefficient of 0.560, a kurtosis value of 4.518, and a RMSE of 0.187. The corresponding values for the ITD-SVD method are 0.248, 0.269, 2.846, and 0.226. Lastly, the VMD-SVD method exhibits metric values of 1.623, 0.574, 4.005, and 0.193. Overall, the results obtained using the FDM-SVD method are relatively superior. To conduct a more comprehensive comparative analysis, this study proceeds to perform envelope spectrum analysis on the denoised signals obtained through these methods. The results are presented in
Figure 12,
Figure 13,
Figure 14,
Figure 15 and
Figure 16.
The envelope spectrum obtained using five methods, namely FDM-SVD, EMD-SVD, EEMD-SVD, ITD-SVD, and VMD-SVD, are presented in
Figure 12,
Figure 13,
Figure 14,
Figure 15 and
Figure 16, respectively. A detailed analysis of these figures reveals that, after the process of signal time–frequency decomposition and noise reduction, all these methods can extract fault characteristic frequencies to varying degrees, albeit with the presence of irrelevant frequency interference. Additionally, the amplitudes of the extracted fault frequency peaks by other methods are relatively low. However, compared to the other methods, the FDM-SVD method exhibits superior performance in extracting fault characteristic frequencies and their multiples, with relatively higher peak values and reduced irrelevant interference near the fault frequencies.
To enhance the periodic impulse features in the signal, the noise-reduced signals obtained through FDM-SVD, EMD-SVD, EEMD-SVD, ITD-SVD, and VMD-SVD methods will be filtered using the CYCBD method. Subsequently, the Hilbert envelope spectra are generated.
Figure 17,
Figure 18,
Figure 19,
Figure 20 and
Figure 21 depict the envelope spectra obtained using the FDM-SVD-CYCBD, EMD-SVD-CYCBD, EEMD-SVD-CYCBD ITD-SVD-CYCBD, and VMD-SVD-CYCBD methods, respectively. Analysis of these five methods reveals that after CYCBD filtering, all five methods can accurately extract the fault characteristic frequency and its multiples, ranging from double to septuple frequencies. Notably, the amplitude of the fault characteristic frequency and its multiples extracted by the FDM-SVD-CYCBD method is the highest. Furthermore, the FDM-SVD-CYCBD method significantly reduces the influence of irrelevant frequency interference in the signal.
The vibration signal of a bearing can essentially be regarded as the convolution result of the source signal and channel characteristics. In signal processing, deconvolution, as a method to solve the inverse convolution filter, aims to restore the original source signal and remove the influence of the channel on the signal. Therefore, methods based on blind deconvolution theory have inherent advantages in noise reduction processing for bearing vibration signals. Other common methods include Minimum Entropy Deconvolution (MED), Maximum Correlation Kurtosis Deconvolution (MCKD), and Optimal Minimum Entropy Deconvolution Adjusted (OMEDA) [
33,
34,
35]. To comprehensively and deeply compare the effectiveness of different deconvolution methods in signal filtering, this study combined FDM-SVD for signal preprocessing. Based on this, we applied MED, OMEDA, and MCKD for filtering. Subsequently, Hilbert envelope spectra were generated and presented in
Figure 22,
Figure 23 and
Figure 24.
Observing these three figures, it can be seen that although the envelope spectra obtained after filtering with the aforementioned deconvolution methods can extract fault characteristic frequencies, these frequencies often have low amplitudes and are susceptible to interference from other unrelated frequency components. This interference not only blurs the fault characteristics but also significantly hinders accurate fault diagnosis. To more objectively assess the performance of various methods, we listed the envelope spectrum sparsity values of the signals filtered using the above deconvolution techniques in
Table 7. In the case of mechanical faults in bearings, discrete peaks in the vibration signal become particularly prominent, thereby significantly enhancing sparsity. The appearance of discrete peaks at specific frequencies in the envelope spectrum further increases sparsity, explicitly indicating the presence of different fault pulses in the signal.
Through comparative analysis of these results, it is evident that the envelope spectrum sparsity values after filtering with the CYCBD method are significantly higher than those of other methods, fully demonstrating the superior performance of CYCBD in signal filtering. It not only effectively extracts fault features but also significantly reduces interference from unrelated frequency components, providing more accurate and reliable data support for subsequent fault diagnosis. It is worth mentioning that the CYCBD algorithm, as a novel blind deconvolution method based on maximum second-order cyclic indices, exhibits unique advantages in the field of signal processing. As a robust and sensitive indicator, the cyclic stationarity index plays a crucial role in early fault detection and identification. Compared to traditional methods such as MED, MCKD, and OMEDA, CYCBD demonstrates stronger capability in fault feature extraction. Especially in scenarios with significant pulse noise, the deconvolution effect of CYCBD is more outstanding, effectively filtering out noise and highlighting fault features.