Spectral-Clustering-Guided Fourier Decomposition Method and Bearing Fault Feature Extraction

Abstract

The Fourier decomposition technique has notable advantages in filtering vibration acceleration signals and enhances the feasibility of frequency-domain mode decomposition. To improve the accuracy of mode extraction, this paper proposed a novel Fourier decomposition technique based on spectral clustering. The methodology comprises three key steps. First, spectral clustering is performed using feature vectors derived from the spectrum envelope, specifically the frequency and amplitude of its maximum value, along with the average amplitude of local spectral peaks. Subsequently, the spectrum is adaptively segmented based on clustering feedback to determine spectral segmentation boundaries. Followed by this, a filter bank is constructed via Fourier decomposition for signal reconstruction. Finally, a harmonic correlation index is computed for all decomposed components to identify fault-sensitive modes exhibiting the highest diagnostic relevance. These selected modes are subsequently subjected to demodulation for fault diagnosis. The effectiveness of the proposed method is validated through both simulated signals and experimental datasets, demonstrating its improved ability to capture critical fault information.

1. Introduction

Rolling bearings are critical components in modern mechanical systems, providing essential rotational support [1,2,3]. Bearing failures frequently generate abnormal vibrations, potentially leading to equipment damage or even catastrophic system failures [4,5]. Within intelligent operation and maintenance frameworks, real-time condition monitoring and proactive diagnosis of wear-prone parts, such as bearings and gears, are essential for enhancing operational safety and reducing economic risks [6,7]. However, the complex operating environments of bearings often introduce environmental noise and transient impulses into the acquired signals, which complicates the accurate assessment of their operational state [8,9]. These challenges underscore the engineering significance of developing advanced fault diagnosis technologies for rolling bearings.

Empirical Mode Decomposition (EMD) [10,11] adaptively decomposes signals into multiple intrinsic mode functions (IMFs) of distinct scales along with a residual component. Nevertheless, EMD notably suffers from theoretical limitations and susceptibility to mode mixing [12]. Despite lacking rigorous mathematical foundations, EMD has demonstrated notable success in practical applications [13]. Zhao et al. applied EMD to monitor vane trailing edge damage and rotor-stator rub in a rotating test rig, demonstrating effective condition monitoring for rotating machinery [14]. Wang et al. integrated EMD with Tunable Q-factor Wavelet Transform (TQWT) for pre-diagnosis of early weak faults in rolling bearings [15,16]. Nevertheless, existing EMD-based approaches fail to completely resolve mode mixing and boundary effects. Building upon the limitations of the Empirical Mode Decomposition (EMD) method, Wu et al. [17] introduced an enhanced approach termed Ensemble Empirical Mode Decomposition (EEMD), which mitigates mode mixing in EMD by leveraging the zero-mean property of white noise. This technique superimposes multiple realizations of Gaussian white noise onto the original signal, performs independent EMD decompositions for each perturbed version, and ultimately averages the resulting Intrinsic Mode Functions (IMFs) to suppress noise-induced artifacts. Variational Mode Decomposition (VMD) [18] is a non-recursive signal decomposition technique developed through a generalization of Wiener Filters. More recently, a successive VMD approach has been introduced by incorporating additional constraints to ensure that modes of interest exhibit minimal overlap with other components in the decomposition results [19]. Consequently, various mode decomposition methodologies in signal have been developed. Qi et al. proposed a novel vibration trend prediction method integrating Adaptive Chirp Mode Decomposition (ACMD) with Long Short-Term Memory (LSTM) neural networks, which was successfully applied to pipeline vibration control [20]. Pan et al. introduced and applied Symplectic Geometry Mode Decomposition (SGMD) to compound fault diagnosis in rotating machinery, enabling efficient signal component reconstruction and noise suppression without requiring user-defined parameters [21,22]. Zhou et al. [23] introduced Empirical Fourier Decomposition (EFD), which synthesizes the frequency-band segmentation technique of Empirical Wavelet Transform (EWT) with the Fourier Decomposition Method (FDM) [24]. This methodology decomposes complex nonlinear non-stationary signals into sums of physically meaningful analytical Fourier intrinsic band functions defined by instantaneous frequencies. For spectral segmentation, EFD adaptively determines spectral boundaries using a locmaxmin-based approach. This strategy overcomes limitations of fixed segmentation modes while enabling adaptive boundary identification and extraction of fault-related components according to signal characteristics. Despite these advantages, dependence of EFD on envelope extrema positions for boundary determination renders it susceptible to strong noise interference, particularly in high-noise environments. Based on cyclostationarity, Shi proposed the Adaptive Cyclostationarity Feature Mode Decomposition (ACFMD) method, which decomposes characteristic mechanical fault features and exhibits enhanced feature extraction performance amidst strong background noise [25,26].

Evaluating equipment operational status requires quantifiable indicators to assess mechanical fault severity. Industrially, simple yet effective time-domain parameters—including peak, average, and root mean square values [27]—are commonly employed. However, given the complexity of actual operating environments, these time-domain indicators often prove insufficient. Consequently, researchers have explored fault signal characteristics in the frequency domain [28]. In 1983, Dwyer introduced spectral kurtosis (SK), demonstrating particular sensitivity to transient impulses. Antoni [29] subsequently provided a rigorous mathematical definition of SK as the energy-normalized fourth-order spectral cumulant, quantifying transient impulse richness within signal components. Nevertheless, random impulses introduce significant interference that disrupts the cyclostationarity of fault-induced transients. Wang [30] established critical theoretical relationships between spectral kurtosis, spectral L2/L1 norm, spectral smoothness, and the spectral Gini index, advancing the fundamental understanding of these indicators. To address these limitations, Zhang [31] developed harmonic spectral kurtosis (HSK), which effectively extracts harmonic features from envelope spectra, quantifies periodic impulses, and suppresses random pulse interference. This study employs a harmonic correlation index (HCI) that combines robustness with high sensitivity to periodic transient impulses, significantly enhancing fault information capture accuracy.

This paper proposes a novel adaptive signal decomposition method termed Spectral Clustering Guided Fourier Decomposition Method (SCGFDM). The methodology employs spectral clustering to classify spectral peaks and evaluates fault components using HCI. Spectral clustering partitions datasets into distinct clusters with high intra-cluster similarity [32,33], enabling adaptive spectral boundary determination. Signal decomposition employs a zero-phase filter bank to prevent mode mixing. Fault intensity quantification utilizes HCI, while a multi-level spectral segmentation framework enhances fault feature extraction. This approach achieves three critical functions: adaptive frequency-band segmentation, optimal demodulation band selection, and fault feature extraction. The core adaptivity of SCGFDM manifests in autonomous spectral boundary determination via frequency-domain analysis and self-selection of optimal demodulation bands eliminating manual screening. Simulation experiments demonstrate a center frequency offset of 0.16% and bandwidth error of 11.50% in extracted optimal bands. Experimental validation confirms effective diagnosis of bearing inner-race and outer-race faults. The paper is structured as follows: Section 2 details the methodological motivation; Section 3 presents the SCGFDM framework; Section 4 validates effectiveness using simulated inner/outer-race compound fault signals; Section 5 demonstrates feasibility through experimental bearing fault diagnosis; Section 6 concludes the study.

2. Principles of VMD and EFD

Konstantin Dragomiretskiy proposed VMD [18], a non-recursive method that concurrently extracts signal modes by optimizing an ensemble with specific center frequencies under a narrow-band prior in the Fourier domain, effectively generalizing the Wiener filter to multiple adaptive bands. The principle of VMD is shown in Figure 1. Its principle ensures theoretically sound and easily understood signal decomposition. Key advantages include significantly enhanced robustness to noise and sampling, computational efficiency via the ADMM optimizer, and superior performance demonstrated on artificial and real data compared to existing recursive techniques like EMD.

In the original FDM, Zhou [23] introduced an EFD approach for frequency-domain processing, leveraging the adaptive decomposition concept of EWT as shown in Figure 2. The signal spectrum is first normalized to the range $\left[0,\pi \right]$. Based on spectral amplitude characteristics, the spectrum is segmented into $N$ contiguous frequency bands. The spectral band boundaries are defined as $S_n=[{\omega }_{n-1},{\omega }_n]$, where ${\omega }_{n-1}$ and ${\omega }_n$ represent the lower and upper frequency limits of the $n$-th band, respectively, while $n\in \left\{\mathrm{1,2},\cdots ,N\right\}$. The boundary conditions are specified as ${\omega }_o=0$ and ${\omega }_N=\pi$, enabling a precise characterization of the spectral band segmentation. Subsequently, a local maximum segmentation technique identifies spectral amplitude maxima. These maxima, combined with amplitudes at frequencies $0$ and $\pi$, form an ordered amplitude sequence. To determine ${\omega }_n$, the first $N-1$ largest extrema of the spectral amplitude are selected. The frequencies corresponding to these extrema are uniquely reindexed in ascending order as ${\Omega }_1,{\Omega }_2,\cdots {\Omega }_{N-1}$ (where ${\Omega }_1<{\Omega }_2{<\cdots <\Omega }_{N-1}$), with ${\Omega }_0=0$ defined additionally. The boundaries ${\omega }_n$ are given by:

$${\omega }_n={\displaystyle \frac{{\Omega }_{n-1}+{\Omega }_n}{2}},n\in \left\{\mathrm{1,2},\dots ,N\right\}$$

(1)

As an alternative to the local maximum technique, the minimum technique can also be employed for adaptive spectrum segmentation. Initially, the same spectrum partitioning and frequency reordering procedures described for the local maximum technique are followed. Subsequently, the minimum value of the spectral amplitude within each frequency interval $[{\Omega }_{n-1},{\Omega }_n]$ is identified. The boundary ${\omega }_n$ is then determined by the following expression:

$${\omega }_n=\arg \mathrm{m}\mathrm{i}\mathrm{n}{X}_n\left(\omega \right)$$

(2)

where $X_n\left(\omega \right)$ denotes the spectral amplitude within the interval $[{\mathsf{\Omega }}_{n-1},{\mathsf{\Omega }}_n]$, and $\mathrm{a}\mathrm{r}\mathrm{g} \mathrm{m}\mathrm{i}\mathrm{n}(·)$ identifies the frequency argument corresponding to the minimum value. This adaptive boundary determination yields precise segmentation results while eliminating trivial residuals in the initial decomposition component. Furthermore, the method resolves inconsistencies between the Low-to-High (LTH) and High-to-Low (HTL) frequency-sweeping approaches in the FDM. Although adaptive, EFD struggles to process signals exhibiting closely spaced modal frequencies or cross-components within the frequency domain. This limitation leads to inaccurate decomposition, resulting in mode mixing and invalid components. Furthermore, the adaptive decomposition method utilized in EWT, based on a wavelet filter bank, may induce mode mixing between components due to the transition band characteristics during signal filtering and reconstruction. While the zero-phase filter bank employed in EFD—essentially a bandpass filter bank—eliminates the transition phase, retaining dominant components within the designated frequency bands while excluding all out-of-band components, it does not fully address the segmentation challenge.

To overcome these limitations, this paper proposes a novel adaptive spectrum segmentation model. This model adaptively segments spectrum boundaries based on inherent signal characteristics, thereby mitigating the influence of external interference on segmentation accuracy. By increasing the iteration count, a multi-level spectrum segmentation framework is constructed to prevent the unreasonable segmentation of diagnostically relevant fault information. Simultaneously, a zero-phase filter bank is implemented to prevent the loss of critical fault information during the decomposition process. Crucially, since EFD decomposition requires manual observation and assessment of each component for fault information—a labor-intensive process—this paper introduces a robustness indicator to automatically identify the optimal demodulation component, enabling efficient fault diagnosis.

3. Spectral-Clustering-Guided Fourier Decomposition Method

Spectral-Clustering-Guided Fourier Decomposition Method (SCGFDM) constructs a sample space by processing spectral peak parameters. This sample space is clustered to adaptively determine spectral segmentation boundaries. Leveraging superior adaptability of spectral clustering to data distributions yields more accurate segmentation than conventional approaches. Inspired by the multi-level spectrum segmentation concept of Fast Kurtogram [34], this method employs spectral clustering to achieve multi-resolution frequency band division. Upon determining spectral boundaries, a zero-phase filter bank filters and reconstructs the signal, enabling adaptive signal decomposition. The HCI detects periodic impulse components to identify the optimal demodulation frequency band for envelope spectrum analysis.

SCGFDM demonstrates fabulous computational efficiency and accuracy in adaptive spectrum segmentation, signal reconstruction and feature extraction. The specific procedure as shown in Figure 3 comprises:

(1)

Acquire vibration signals via sensors;

(2)

Compute the signal spectrum and construct the sample space using appropriate spectral features;

(3)

Cluster the sample space into $k$ groups via spectral clustering, with cluster boundaries defining adaptive frequency band segmentation;

(4)

Construct a zero-phase filter bank based on frequency band boundaries to decompose the signal into intrinsic components;

(5)

Apply HCI to select the optimal demodulation band and perform envelope spectrum analysis for fault diagnosis.

3.1. Theoretical Basis of Spectral Clustering Guide Fourier Decomposition Method

The initial stage of this technique requires effective spectrum segmentation to identify homogeneous signal components while differentiating heterogeneous ones. Spectral clustering, rooted in graph theory, transforms the clustering task into a spectral graph partitioning problem. During this process, each data point corresponds to a graph vertex, with edge weights representing pairwise similarities. Unlike the EFD approach that relies on extreme points for boundary determination, the proposed spectral clustering strategy adaptively defines segmentation boundaries, effectively mitigating exogenous noise interference in boundary positioning. The principal steps as shown in Figure 4 for spectral clustering-based spectrum partitioning are as follows:

(a)

Perform spectral envelope peak detection to retain all spectral maxima.

(b)

Identify local maxima frequencies $\left[f_1,f_2,\cdots ,f_n\right]$ and amplitude $\left[A_1,A_2,\cdots ,A_n\right]$. For adjacent maxima separated by local minima at frequencies $\left[{\dot{f}}_1,{\dot{f}}_2,\cdots ,{\dot{f}}_n\right]$, compute the mean amplitude $A_m$ of the spectral peaks bounded by consecutive minima according to Equation (3).

$$A_m={\displaystyle \frac{{\sum }_{{\dot{f}}_k}^{{\dot{f}}_{k+1}}A\left(\dot{f}\right)}{{\dot{f}}_{k+1}-{\dot{f}}_k}}$$

(3)

(c)

Construct the feature sample set $S=〈f_k,A_k,A_m〉$ using frequency $f_k$, amplitude $A_k$, and average amplitude $A_m$ of each spectral maximum. This three-dimensional feature space minimizes boundary clustering artifacts and reduces noise sensitivity compared to the extremum-based segmentation, thereby improving segmentation accuracy.

(d)

Construct the similarity matrix $A$ from sample set $S$, where each element $A_{ij}$ is computed via Equation (4). Here, $‖s_i-s_j‖$ denotes the Euclidean distance between samples $s_i$ and $s_j$, while $\sigma$ controls the Gaussian kernel bandwidth. For spectral segmentation, $\sigma =0.4$ balances localization (small $\sigma$) and peak sensitivity (large $\sigma$).

$$A_{ij}=\left\{\begin{array}{c}\mathit{exp}\left({\displaystyle \frac{-{‖s_i-s_j‖}^2}{2{\sigma }^2}}\right),i\ne j\\ 0,i=j\end{array}\right.$$

(4)

(e)

Compute the degree matrix $D$ (diagonal matrix where $D_{ii}={\sum }_j{A}_{ij}$) from similarity matrix $A$ as per Equation (5).

$$D=\left(\begin{array}{ccc}{d}_{11}& \cdots & 0\\ ⋮& \ddots & ⋮\\ 0& \cdots & d_{nn}\end{array}\right)$$

(5)

(f)

Calculate the symmetric Laplacian matrix $L_{sym}$ following Equation (6).

$$L_{sym}=D^{-\frac{1}{2}}{AD}^{-\frac{1}{2}}$$

(6)

(g)

Solve $L_{sym}$ to obtain eigenvalues and eigenvectors. Retain the $k$ largest eigenvectors $x_1,x_2,\cdots ,x_k$ to form the $n\times k$ matrix $X=\left[x_1,x_2,\cdots ,x_k\right]$.

(h)

Row-normalize $X$ to matrix $Y$, then apply K-means clustering to the rows of $Y$, where each row vector corresponds to a data point $S_i$ in the original sample set $S$.

First, the number of clusters, denoted as $n$, must be determined. The selection of $n$ significantly influences the clustering outcome. An insufficiently small $n$ may yield overly coarse cluster divisions that fail to accurately capture the inherent data structure. Conversely, an excessively large $n$ may produce overly fragmented clusters, leading to classification inaccuracies. Next, $n$ data points are randomly selected as initial centroids. The Euclidean distance between each remaining data point and every centroid is computed, assigning each point to the cluster corresponding to its nearest centroid. The centroid of each cluster is then recalculated as the mean of all points within that cluster. Subsequently, all data points are reassigned to clusters based on these updated centroids. This iterative process—recalculating centroids and reassigning points—continues until either the centroids exhibit a negligible change between iterations, or a predefined maximum iteration count is reached. This method demonstrates high computational efficiency and delivers satisfactory clustering accuracy.

(i)

The cluster assignment $A_i$ of the $i$-th row in matrix $Y$ corresponds directly to the cluster membership of data point $S_i$ in the original set $S$. The characteristics and effects of clustering can be intuitively perceived from Figure 5. This mapping determines the definitive spectral segmentation boundaries, with a schematic representation provided in Figure 6.

3.2. Spectrum Segmentation Method Based on Spectral Clustering

For signals exhibiting closely spaced modal frequencies, decomposition may induce mode mixing between components. To mitigate this phenomenon, we optimize the filter design to ensure each decomposed component retains its distinctive spectral maxima and fault-related information [35].

Utilizing the enhanced segmentation technique from Section 3.1, frequency band boundaries are identified to partition the spectrum into $N$ bands. A zero-phase filter bank is then constructed for signal reconstruction. Unlike wavelet filters that exhibit transition bands, this zero-phase implementation eliminates boundary artifacts, preserving dominant in-band components while rejecting out-of-band energy. Figure 7 illustrates the filter bank architecture, with construction details as follows:

$${\widehat{\mu }}_n\left(\omega \right)=\left\{\begin{array}{c}1,if{\omega }_{n-1}\le \left|\omega \right|\le {\omega }_n\\ 0,otherwise\end{array}\right.$$

(7)

where $1\le n\le N$ and ${\omega }_n$ denotes the adaptively determined bandwidth parameter obtained from Section 3.1. Following construction of the zero-phase filter bank, frequency-domain filtering is performed by applying the spectral segmentation boundaries defined in Equation (8):

$${\widehat{f}}_n\left(\omega \right)={\widehat{\mu }}_n\left(\omega \right)\widehat{f}\left(\omega \right)$$

(8)

where $\widehat{f}\left(\omega \right)$ denotes the signal spectrum. Multiple frequency-domain components are thereby obtained. Each component undergoes inverse Fourier transformation yielding decomposed time-domain components. Figure 7 illustrates the filtering process schematic.

$$f_n\left(t\right)={\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }{\widehat{f}}_n\left(\omega \right)e^{j\omega t}d\omega$$

(9)

Assuming spectral feature clustering yields upper and lower boundaries of 4100 Hz and 2500 Hz, respectively, frequency-domain filtering is applied to the components of this band, yielding the result shown in Figure 8 below. To validate the proposed filtering technique’s performance, the same frequency band was processed using FIR filters. Comparative analysis demonstrates that the zero-phase filter bank effectively attenuates spectral leakage in the extracted component.

3.3. Hormonic Correlation Index

Fault signals in rotating machinery exhibit distinct regularities. Within electromechanical equipment, the rotational period of uniformly rotating components is typically constant, rendering it predictable. Current diagnostic indicators predominantly explore the second-order cyclostationarity of impulsive components within the signal. However, this approach is susceptible to external disturbances, such as noise or random impulses, potentially leading to inaccurate results. In the time domain, fault signals manifest as periodic transient impulses. Correspondingly, in the frequency domain, they appear as sidebands flanking specific center frequencies. Nevertheless, noise and random impulses often obscure discernible fault characteristics within the spectrum. Applying Hilbert envelope demodulation to the signal yields its envelope spectrum, enabling analysis of the fault characteristic frequency and its harmonic components. While actual fault signals contain extraneous information, a simulated fault model contains solely the impulsive signature. Consequently, leveraging this characteristic for indicator development significantly enhances fault identification accuracy [36].

Calculating the harmonic correlation index (HCI) offers a distinct advantage: it eliminates the need for detailed bearing component parameters. The waveform of the simulated fault model is constructed based solely on the fault characteristic frequency. By comparing the envelope spectrum of the measured fault signal with that of the simulated model and analyzing their harmonic components, the presence of fault information is determined. A high correlation indicates substantial fault-related content within the signal, whereas a low correlation suggests the absence of distinct fault features. This method effectively suppresses interference from random impulses and external noise, thereby improving the accuracy of fault feature recognition.

The Hilbert transform of the fault signal $s\left(t\right)$ is defined as:

$$\widehat{s}\left(t\right)={\displaystyle \frac{1}{\pi }}{\int }_{-\infty }^{+\infty }{\displaystyle \frac{s\left(\tau \right)}{t-\tau }}d\tau$$

(10)

The envelope spectrum of $\widehat{s}\left(t\right)$ is subsequently obtained as:

$$\widehat{v}\left(\omega \right)=\left|{\int }_{-\infty }^{+\infty }\left[\left|\widehat{s}\left(t\right)\right|-\overline{\left|\widehat{s}\left(t\right)\right|}\right]e^{-i\omega t}dt\right|$$

(11)

where $\overline{\left|\widehat{s}\left(t\right)\right|}$ denotes the mean value of $\left|\widehat{s}\left(t\right)\right|$.

The Pearson correlation coefficient quantifies the linear relationship between two similar but distinct signals. To account for differences in their respective positions and scales, the envelope spectrum amplitudes are normalized to the range [0, 1]. Let $U$ denote the normalized measured envelope spectrum of the acquired fault signal and $V$ represent the envelope spectrum of the simulated fault model. The Pearson correlation coefficient between $U$ and $V$ is defined as:

$${\rho }_{U,V}={\displaystyle \frac{cov\left(U,V\right)}{{\sigma }_U·{\sigma }_V}}={\displaystyle \frac{E\left[\left(U-{\mu }_U\right)-\left(V-{\mu }_V\right)\right]}{{\sigma }_U·{\sigma }_V}}$$

(12)

where $E[·]$ denotes the expectation operator, and ${\mu }_U$, ${\mu }_V$, ${\sigma }_U$, and ${\sigma }_V$ represent the means and standard deviations of $U$ and $V$, respectively. The sample Pearson correlation coefficient is given by:

$$r_{UV}={\displaystyle \frac{{\sum }_{i=1}^n\left(U_i-\overline{U}\right)\left(V_i-\stackrel{̿}{V}\right)}{\sqrt{{\sum }_{i=1}^n{\left(U_i-\overline{U}\right)}^2}\sqrt{{\sum }_{i=1}^n{\left(V_i-\overline{V}\right)}^2}}}$$

(13)

where $\overline{U}$ and $\overline{V}$ are the sample means of the normalized fault signal envelope spectrum and the simulated fault model envelope spectrum, respectively.

Figure 9 schematically illustrates this calculation of HCI. The blue profile represents the constructed simulated fault model, while the red profile corresponds to a component of the actual fault signal envelope spectrum. The linear correlation between these profiles determines the fault information content within the actual signal component. As depicted, a high linear correlation with the simulated model identifies this component as the optimal demodulation component.

To demonstrate the superior performance of HCI, four representative signal types—periodic impulse signal, random noise, sinusoidal components, and random pulse—were quantitatively evaluated using harmonic correlation index, industrial-standard kurtosis, and information entropy. Figure 10 and Figure 11 display waveforms of the four signal types alongside their corresponding HCI, Kurtosis, and Information Entropy evaluation indicator (normalized), revealing exceptional sensitivity of HCI to periodic impulse signals while exhibiting robust immunity to other signal components. Whereas Kurtosis and Information Entropy detect fault information yet remain susceptible to interfering components, HCI demonstrates superior robustness in fault detection, thereby offering significant utility in fault diagnosis and feature extraction.

4. Simulation Signal Verification

To validate the effectiveness of SCGFDM and illustrate its calculation procedure, this section employs a simulated bearing signal featuring composite inner and outer race faults. Reflecting the complex operational environment encountered in practice, the simulation incorporates modulation components and additive Gaussian white noise with a low SNR of −2 dB. The sampling frequency was set at 20 kHz, yielding a signal duration of 1.2 s (24,000 samples). The composite fault simulation signal is mathematically represented as:

$$\left\{\begin{array}{c}{s}_o\left(t\right)={\sum }_{j=1}^{100}{A}_o{e}^{2\pi {\xi }_o{f}_{no}t}\times \mathit{sin}\left(2\pi f_{no}\sqrt{1-{\xi }_o^2}\right)u\left(t-{\displaystyle \frac{j}{f_o}}\right)\\ s_i\left(t\right)={\sum }_{k=1}^{100}{A}_i{e}^{2\pi {\xi }_i{f}_{ni}t}\times \mathit{sin}\left(2\pi f_{ni}\sqrt{1-{\xi }_i^2}\right)u\left(t-{\displaystyle \frac{k}{f_i}}\right)\times c\left(t\right)\\ c\left(t\right)={\left[1-\mathit{cos}\left(2\pi f_{r}t\right)\right]}^{{\displaystyle \frac{3}{2}}}\times \mathit{cos}\left(2\pi f_{r}t\right)\\ s\left(t\right)=s_o\left(t\right)+s_i\left(t\right)+c\left(t\right)+n\left(t\right)\end{array}\right.$$

(14)

where $\xi$ denotes the damping coefficient (${\xi }_o={\xi }_i=0.05$), and $f_n$ represents the natural frequency ($f_{no}=2500\mathrm{H}\mathrm{z}$, $f_{ni}=5500\mathrm{H}\mathrm{z}$). The fault characteristic frequencies are set to $f_o=100\mathrm{H}\mathrm{z}$ (outer race) and $f_i=130\mathrm{H}\mathrm{z}$ (inner race) with rotating frequency $f_r=10\mathrm{H}\mathrm{z}$. Amplitudes are $A_o=3\mathrm{m}/{\mathrm{s}}^2$ and $A_i=7\mathrm{m}/{\mathrm{s}}^2$. Here, $s_o\left(t\right)$ represents the outer race fault signal, while $s_i\left(t\right)$ represents the inner race fault signal and $c\left(t\right)$ represents modulation signal. The term $n\left(t\right)$ denotes additive Gaussian white noise at a signal-to-noise ratio (SNR) of −2 dB, where SNR is characterized as $10{\times log}_{10}\left(P_s/P_n\right)$, with $P_s$ and $P_n$ representing the average power of the signal and noise, respectively. Figure 12 presents the outer race fault signal $s_o\left(t\right)$, inner race fault signal $s_i\left(t\right)$, noise $n\left(t\right)$, and composite signal $s\left(t\right)$. As shown in Figure 13, the spectrum reveals resonance bands centered at 5500 Hz (with approximately 900 Hz bandwidth) for the inner race fault and 2500 Hz (with approximately 1400 Hz bandwidth) for the outer race fault. Consequently, particular emphasis should be placed on extracting information within these frequency bands while maintaining fault information fidelity.

Applying SCGFDM to process the signal yields two segmentation results simultaneously, as illustrated in Figure 14. When configuring the frequency band partitioning, dividing the signal into two segments introduces excessive noise, while nine segments produce overly narrow bands. Thus, the initial partitioning was set to three segments. With each incremental decomposition level, the number of frequency bands increases until the spectrum is divided into eight segments, ultimately forming a six-level result. Figure 14a exhibits the inner race fault processing results, and Figure 14c displays the outer race fault results. The SCGFDM demonstrates high precision in identifying bearing inner race faults. The optimal demodulation band marked in the red dotted box in Figure 14a (Level 6: center frequency = 5509 Hz, bandwidth = 885 Hz) aligns closely with the preset inner race resonance band. For the analysis results of outer race fault, the identified optimal resonance band marked in the red dotted box in Figure 14c (Level 6: center frequency = 2013 Hz, bandwidth = 1877 Hz) shows a moderate deviation from the preset center frequency (2500 Hz). Despite bandwidth-induced inclusion of extraneous information, this band retains nearly complete outer race fault characteristics. Due to its high sensitivity to periodic transient impulses, HCI enables comprehensive identification of composite fault information within signal components despite significant noise interference. As evidenced in Figure 14b, the 5000–6000 Hz band containing maximal inner race fault information is fully resolved. The time-domain waveform exhibits distinct periodic impulses, while the envelope spectrum clearly displays the characteristic inner race fault frequency and its harmonics marked in the green dotted box in Figure 14b. Similarly, Figure 14d demonstrates complete detection of the 1000–3000 Hz band housing predominant outer race fault information, with its envelope spectrum revealing explicit outer race fault characteristic frequencies and associated harmonics. Given the near-constant center frequency of fault-induced resonance bands during equipment failure, we employ the center frequency offset rate $r_o$ and bandwidth error rate $r_e$ metrics to quantitatively validate the fault identification accuracy of the proposed SCGFDM.

$$r_o={\displaystyle \frac{f_c-f_n}{f_n}}\times 100\%$$

(15)

$$r_e={\displaystyle \frac{\left|B_w-B_o\right|}{B_o}}\times 100\%$$

(16)

Here, $f_c$ denotes the center frequency of the frequency-domain component, while $f_n$ represents the center frequency of the fault-induced resonance band. A positive $r_o$ value indicates rightward center frequency deviation, whereas a negative value signifies leftward deviation. The terms $B_w$ and $B_o$ correspond to the bandwidths of the frequency-domain component and resonance band, respectively. Computed metrics yield $r_o=0.16\mathrm{\%}$ and $r_e=11.50\mathrm{\%}$ for the inner race fault, and $r_o=-19.48\mathrm{\%}$ with $r_e=6.15\mathrm{\%}$ for the outer race fault. Modulation of the inner race fault frequency via rotational components enriches its signal characteristics, enabling higher spectral clustering accuracy in homogeneous frequency bands. Conversely, despite lower processing precision for spectrally simpler outer race faults ($r_o=-19.48\mathrm{\%}$), the method maintains diagnostically effective outcomes.

The SCGFDM preserves maximal fault information by precisely capturing dominant energy concentrations. Although inner and outer race resonance bands exhibit distinct spectral locations and morphologies in the original signal as shown in Figure 14, the proposed methodology achieves optimal clustering performance. Results confirm that the center frequencies and bandwidths derived by SCGFDM encapsulate critical periodic impulse information, demonstrating significant diagnostic utility for bearing raceway fault detection.

The signal was subsequently processed using EFD with a preset decomposition level of six components. Figure 15 presents the resulting time-domain waveforms and corresponding envelope spectra. Significant noise interference and inner race fault modulation introduce extraneous spectral components, adversely affecting adaptive spectrum segmentation of EFD. This manifests as substantial segmentation deviations. Primary outer race fault components (1st and 3rd) exhibit incomplete fault information due to suboptimal spectral partitioning, and inner race fault information within the 5100–5900 Hz band is fragmented across multiple segments. Components 2, 5, and 6 contain negligible diagnostic content, and critical fault information (e.g., 3300–6000 Hz region) remains largely unretained. This information fragmentation substantially impedes feature extraction. Crucially, no discernible inner or outer race fault features appear in the envelope spectra of reconstructed components, demonstrating how signal complexity compromises maxima-based segmentation of EFD.

To quantitatively evaluate the performance of SCGFDM and EFD, root mean square error (RMSE) analysis was implemented. RMSE values were computed between the six EFD-derived components and the noise-free inner/outer ring bearing signals under compound fault conditions, with results presented in Figure 16. Lower RMSE values indicate superior feature representation accuracy, where RMSE is defined as:

$$RMSE=\sqrt{{\displaystyle \frac{1}{N}}{‖x-\widehat{x}‖}_2^2}$$

(17)

where $x$ denotes the processed component and $\widehat{x}$ represents the noise-free reference signal. SCGFDM achieves inner and outer ring RMSE values of 0.1763 and 0.2397, respectively, against noise-free reference signals, demonstrating its effectiveness in decomposing bearing fault information while improving computational efficiency without compromising feature extraction accuracy.

5. Application

To validate the effectiveness and advantages of the proposed SCGFDM, this study employs the rail transit train bogie transmission system fault dataset released by the National Key Laboratory of Advanced Rail Autonomous Operation at Beijing Jiaotong University (https://drive.google.com/drive/folders/1RlZvFw-v07VvsL2Ni9cS7iFrTPDIhn2r?usp=sharing, accessed on 21 December 2024) [37,38]. The test bench was geometrically scaled at a 1:2 ratio relative to an actual subway bogie. Figure 17 illustrates the test bench configuration and the spatial distribution of the triaxial accelerometers. The reduction gearbox incorporates helical gears with a transmission ratio of 107:16 and HRB 32305 bearings. The fault is depicted in Figure 18 and the fault location is marked by a red dotted circle, while

Table 1. Overview of working conditions.

Serial Number	Speed/Load	Serial Number	Speed/Load
WC1	20 Hz/0 kN	WC6	60 Hz/+10 kN
WC2	40 Hz/0 kN	WC7	20 Hz/−10 kN
WC3	60 Hz/0 kN	WC8	40 Hz/−10 kN
WC4	20 Hz/+10 kN	WC9	60 Hz/−10 kN
WC5	40 Hz/+10 kN

summarizes the nine kinds of operational conditions. Herein, a positive lateral load denotes force application toward the motor side of the transmission chain under test, whereas a negative lateral load indicates loading toward the gearbox side.

5.1. Bearing Inner Ring Fault Data

First, the inner race fault data of the gearbox bearing (Operating Condition WC1) was analyzed. The motor operated at 1198.2 rpm, yielding a rotational frequency of 19.97 Hz. Signals were sampled at 64 kHz over 64,000 data points. The calculated fault characteristic frequency was 158 Hz. Gaussian white noise was introduced to the raw signal at −4 dB SNR. Figure 19a,b displays the waveform and spectrum of the inner race fault signal. While transient impulses are observable in the waveform, no distinct fault signatures are apparent in the time domain. Only a few fault shocks marked by green arrows occur in Figure 19a. The spectrum reveals bearing-related components concentrated within 0–3000 Hz, where amplitudes exceed twice the fundamental shaft rotational frequency.

The proposed SCGFDM was applied to analyze this signal. During clustering, adaptive iteration identified the optimal solution, grouping spectrally similar components to establish segmentation boundaries. As illustrated in Figure 20a, SCGFDM partitions the spectrum into three distinct bands—low-frequency, mid-frequency, and high-frequency—while Figure 20b–d demonstrate that increasing the cluster number $k$ induces substantial variations in both the quantity of spectral components and their corresponding bandwidths.

Figure 21 presents the HCI values for all boundary distributions and their corresponding frequency bands. The maximum HCI indicates the frequency band containing the most significant fault information. Among all components, the third frequency band at Level 6 demonstrates the highest concentration of inner race fault signatures. This component was extracted using a zero-phase filter and subjected to envelope demodulation analysis. Figure 22 displays the spectrum, waveform, and envelope spectrum of the reconstructed component. The waveform exhibits distinct periodic impulses, while the envelope spectrum reveals a clear fault characteristic frequency and its harmonics. Although conventional spectral analysis of the adaptively selected region showed no apparent fault information, spectral clustering-assisted boundary segmentation enables precise identification. This demonstrates that even amidst complex signal interference, the proposed spectral self-demodulation method effectively analyzes information-rich components within the 0–3000 Hz mid-low frequency range. The resulting envelope spectrum exhibits unambiguous fault characteristic frequencies and their multiples without extraneous components. The adaptive segmentation method successfully isolates frequency-domain information of different modes. Consequently, SCGFDM adaptively and accurately identifies bearing inner race faults. In order to study what information component exists in the low and medium frequency components of 0~3000 Hz, the first component at Level 6 shown in Figure 23 is extracted for analysis. Its envelope spectrum shows clear frequency conversion and its multiple frequencies, and no other obvious fault information.

The segmentation technology of EFD relies more on the extreme points in the signal spectrum, but does not consider the impact of noise or interference on the spectrum. Although EFD has adaptive characteristics, it is easily affected by external interference. When there is only an ideal state of single frequency information in the signal, EFD can accurately and adaptively separate them. However, when there is complex modulation information or random pulses in the signal, its modal identification method will be affected by the modulation sidebands, resulting in inaccurate results. Close modulation information will be identified as different modal information by EFD, which is the root cause of the close boundaries in Figure 24. That shows 8 of the boundaries, and the 9th boundary frequency is forced to be set to 5117 Hz, which means that the 8th signal component covers the frequency components of 1550 Hz~5117 Hz, while the frequency components from 5117 Hz to 32,000 Hz are abandoned.

To investigate the information components within the 0–3000 Hz mid–low frequency range, the Level 6 first component was extracted for analysis. Its envelope spectrum exhibits distinct fault characteristic frequencies and their harmonics without extraneous spectral components. While EFD employs spectral extreme points for segmentation, it disregards noise and interference impacts. Although adaptive, EFD remains vulnerable to external disturbances. It achieves accurate separation only for ideal single-frequency signals. However, when processing signals containing complex modulation or random impulses, modal identification of EFD becomes compromised by modulation sidebands, yielding inaccurate results. Closely spaced modulation components are misinterpreted as distinct modes, explaining the overlapping boundaries observed in Figure 25. That displays eight boundaries, with the ninth boundary arbitrarily assigned at 5117 Hz. Consequently, the eighth component spans 1550–5117 Hz, while frequencies from 5117 Hz to the Nyquist frequency (32,000 Hz) are discarded.

5.2. Bearing Outer Ring Fault Data

Subsequently, outer race fault data from the gearbox bearing (Operating Condition WC1) were analyzed. The motor operated at 1998.8 rpm, producing a rotational frequency of 19.98 Hz. Signals were sampled at 64 kHz over 32,000 data points, with a calculated fault characteristic frequency of 102 Hz. Gaussian white noise was introduced at −10 dB SNR. Figure 26a,b displays the waveform and spectrum of the outer race fault signal. Noise contamination obscures periodic impulses in the waveform, while the spectrum exhibits a high-amplitude modulation sideband centered near 21,000 Hz.

The proposed SCGFDM was applied to this signal. Adaptive iterative clustering identified optimal spectral segmentation boundaries, with decomposition-level results presented in Figure 27. Level 2 provides coarse cluster segmentation, while Level 4 achieves finer resolution. Level 5 introduces minimal modifications—notably adding a spectral boundary near 16,000 Hz—without significant structural changes.

Figure 28 displays HCI values for all boundary distributions and corresponding frequency bands. The highest HCI value indicates the frequency band containing the most substantial fault information. Among all components, the fourth frequency band at Level 5 demonstrates the greatest concentration of outer race fault signatures. This component was extracted using a zero-phase filter and subjected to envelope demodulation analysis. Figure 29 presents the spectrum, waveform, and envelope spectrum of the reconstructed component. The waveform exhibits distinct periodic impulses, while the envelope spectrum reveals clear fault characteristic frequencies and their harmonics. This component contains predominant fault information with minimal extraneous components. Despite significant noise interference at −10 dB SNR, SCGFDM successfully clusters spectrally similar components, enabling accurate segmentation boundary determination.

For comparison, EFD was applied with eight decomposition modes. Figure 30 shows EFD-determined segmentation boundaries where the 21,000 Hz band is fragmented into multiple narrow sub-bands, preventing information concentration. This over-segmentation yields physically meaningless modes that hinder fault diagnosis. The −10 dB noise contamination further compromises adaptive segmentation accuracy of EFD. Figure 31 displays the processed waveform and envelope spectrum after component separation. Figure 31 confirms suboptimal spectral segmentation due to noise interference, which critically impairs fault feature extraction. None of the eight decomposed components contain diagnostically useful information. Consequently, EFD performs inadequately when processing complex fault signals—a limitation inherent to its extremum-based segmentation methodology.

6. Conclusions

Leveraging the classification principles of spectral clustering, this study introduces Spectral Clustering Guided Fourier Decomposition (SCGFDM)—a novel adaptive frequency-domain decomposition method for rolling-element bearing fault diagnosis. This approach adaptively determines spectral segmentation boundaries by identifying intrinsic groupings of spectral components, enabling precise fault feature extraction. Building upon frequency-domain decomposition, harmonic correlation index (HCI) facilitate optimal demodulation band selection through interference suppression. Effective spectral segmentation minimizes irrelevant components while maximizing fault information retention. Compared with empirical Fourier decomposition (EFD), SCGFDM achieves superior diagnostic accuracy with reduced computational burden by mitigating interference effects on segmentation. Validation through simulated and experimental signals confirms the efficacy of the proposed rolling bearing fault diagnosis.

This method exhibits certain limitations: both initial boundary count and maximum decomposition level remain heuristically defined. Future research should establish signal-adaptive parameter selection criteria to enhance computational efficiency. For bearing fault signals with severely low SNR or lacking sufficient prior fault information, SCGFDM cannot guarantee highly accurate diagnosis; incorporating alternative spectral features that better characterize frequency-domain information as clustering features represents a highly effective approach to enhance diagnostic performance.