The belt conveyor sound denoising method based on MEPSO-VMD and improved wavelet threshold

Abstract

The sound signals of belt conveyors contain rich state information, and accurately extracting fault signals from non-stationary signals such as noise is one of the key technologies for solving belt conveyor fault identification. To address this issue, a denoising method based on Motion -Encoding Particle Swarm Optimization and Variational Mode Decomposition is proposed, combined with Sample Entropy and an improved wavelet threshold for signal screening and filtering. First, with the minimum fitness function value as the target, the optimization algorithm is used to search for the VMD parameters K and α adaptively. Then, the intrinsic mode functions decomposed by VMD are filtered based on their Sample Entropy values. Finally, the IMF components that meet the conditions are decomposed by wavelet threshold for secondary denoising. Since fault signals have weak and nonlinear characteristics, the proposed algorithm effectively extracts weak sound signals from a noisy background. Experimental results show that the proposed algorithm improves the signal-to-noise ratio by approximately 20% compared to traditional denoising methods. In addition, it is further applied to motor bearing fault diagnosis, improving diagnostic efficiency and providing a useful reference for monitoring of rotating machinery, motor systems, and power equipment.

Keywords

Belt conveyor sound denoising variational mode decomposition sample entropy wavelet threshold

Introduction

During the operation of belt conveyor, the equipment is subjected to a special environment characterized by significant pressure fluctuations and high operational loads over extended periods, making it prone to various faults. Common issues include belt tearing, belt misalignment, idler roller failure, and drum failure. In the process of diagnosing equipment faults, the sound signals collected by sensors are inevitably mixed with a large amount of noise,¹ particularly in coastal environments, where noise signals include wind and wave noise, as well as noise generated by other equipment in operation. These noise signals are non-stationary and time-varying, causing serious interference with fault diagnosis of belt conveyor. If these faults are not detected and repaired promptly, they may lead to further damage, resulting in more severe production accidents. Therefore, extracting useful signals from sound signals heavily contaminated with noise is of great importance for the subsequent diagnosis of belt conveyor faults.² The key challenge lies in effectively extracting fault signals from strong background noise, which is crucial for enabling successful fault detection.

Due to the weak and nonlinear characteristics of belt conveyor fault signals, the frequency of the sound signals continuously changes over time. Therefore, appropriate methods are required to reduce signal noise. Empirical Mode Decomposition (EMD) provides a new approach for denoising sound signals.³ Junbai Chen et al. used the EMD method to extract the energy of different frequency bands from the vibration signals of fuel pumps as feature parameters for efficient fault diagnosis of airborne fuel pumps. However, the EMD method has drawbacks such as endpoint effects and modal mixing. With the introduction of ensemble empirical mode decomposition (EEMD) and complementary ensemble empirical mode decomposition (CEEMD) algorithms, these methods effectively reduced the problem of mode mixing, but the residual white noise brought noise interference to the signals.⁴ Wavelet Transform (WT),⁵ introduced in 1980, can decompose signals into multiple scales representing different frequency bands. Gilles combined the advantages of wavelet threshold denoising and the EMD algorithm to propose the Empirical Wavelet Transform (EWT).⁶ Kang Sun et al. applied the EWT algorithm for denoising sound signals from wind turbine gearboxes, demonstrating outstanding performance in suppressing background interference. However, EWT also has some limitations when dealing with non-stationary signals, as it is prone to improper mode segmentation. To overcome these challenges, Dragomiretskiy proposed Variational Mode Decomposition (VMD)^7,8 in 2014. This algorithm demonstrates the ability to adaptively match the optimal center frequency and constrained bandwidth for each Intrinsic Mode Function (IMF), which helps effectively separate IMF components in the frequency domain, resulting in the efficient decomposition of the given signals. It not only solves the inherent frequency mixing issues in traditional EMD and EWT but also shows excellent time-frequency localization characteristics, providing an effective tool for decomposing complex sound signals. However, the decomposition performance of VMD depends on the decomposition parameters K and the penalty factor (alpha).^9,10 A large K value can lead to over-decomposition, while a small value may result in under-decomposition. Similarly, the value of (alpha) ensures the precision of signal decomposition. Therefore, selecting appropriate (K) and (alpha) values is crucial for achieving complete decomposition. To address the challenges in the aforementioned denoising methods, this paper proposes a sound denoising algorithm.

(1) To address the issue of large errors in manually assigned decomposition parameters (K) and penalty factor (α),Motion-Encoding Particle Swarm Optimization (MEPSO)algorithm is proposed to adaptively search for the optimal parameters. Additionally, a motion encoding mechanism is introduced to significantly reduce the complexity of the VMD algorithm, thereby decreasing computation time to meet real-time requirements.

(2) Based on the dimensionless index of signal sequence complexity, the sample entropy values are used to filter suitable IMF components.¹¹ By selecting the noisy signals with higher complexity for subsequent threshold filtering, it ensures that only the noisy signals undergo filtering.¹²

(3) For the IMF components containing noise, a secondary denoising process is carried out using an improved wavelet threshold method. The improved threshold algorithm effectively reduces noise, smoothing the transition of the sound signal, and thereby eliminating certain noise fluctuations. The wavelet threshold allows the filtered signal to be closer to the original, achieving effective denoising of the signal.

Sound collection system and VMD algorithm

Sound collection system

In view of the characteristics of the port belt conveyor and their complex working environments, this paper designs a fault sound collection system. The belt conveyor sound collection system mainly includes a sound collection sensor module, a digital signal processor, and a power module. The sound collection module uses sensors to capture the sound signals generated during the operation of the belt conveyor and transmits this information to the main controller. The main controller processes the data through the A/D conversion module and then transmits it wirelessly to an upper computer for further diagnosis and analysis. As shown in Figure 1(a)–(c), the sound collection sensor CRY2301, the digital signal processor TMS320C6747, and the power module UB18650 are presented. Figure 1(d) shows the signal acquisition platform, and Figure 1(e) shows the site where the sound signals are collected from the port belt conveyor.

Figure 1.

Sound Collection system: (a) The sound collection sensor; (b) The digital signal processor; (c) The power module; (d) Signal Acquisition Platform; (e) Port Belt Conveyor Signal Acquisition Site.

VMD algorithm

The VMD algorithm can more effectively handle background noise with weak signals compared to traditional denoising algorithms. In the Variational Mode Decomposition (VMD) step, by solving the variational problem, each modal component is matched with its specific center frequency and bandwidth constraints, which is crucial for separating weak fault signals from strong noise. Variational Mode Decomposition, was proposed by Dragomiretskiy and colleagues in 2014. Unlike the intrinsic mode function (IMF) concept introduced by Huang, VMD redefines the IMFs as intrinsic mode functions with stricter constraints on bandwidth. These intrinsic mode functions are defined as amplitude-modulated and frequency-modulated components. The mathematical expression for these component mode functions is as follows:

u_{k} (t) = A_{k} (t) \cos (ϕ_{k} (t))

(1)

Where:

A_{k} (t)

represents the envelope amplitude of the signal

u_{k} (t)

, and

ϕ_{k} (t)

is the instantaneous phase. It is important to note that the components represented by this function also satisfy the same constraints as in Empirical Mode Decomposition (EMD). Here, k denotes the number of IMFs, where k = 1,2.

u_{k} (t)

is the k-th IMF, and

A_{k} (t)

is the instantaneous amplitude of the k-th IMF.

VMD solution process is as follows, as shown in Figure 2.

(1) Construction of the Variational Problem: A variational problem refers to the task of finding the extremum of a functional. In VMD, the function represents the constrained variational model, and the extremum to be sought is the minimization of the sum of the bandwidths of the center frequencies for each mode. The constrained variational model in VMD is expressed as follows:

\begin{array}{l} \min_{{u_{k}}, {ω_{k}}} {\sum_{k} {‖ \partial_{t} [(δ (t) + \frac{j}{π t}) * u_{k} (t)] e^{- j ω_{k} t} ‖}_{2}^{2}} \\ s . t . \sum_{k} u_{k} = f \end{array}

(2)

Where:

\partial_{t}

represents the partial derivative with respect to time t;

ω_{k}

corresponds to the center frequency of

u_{k}

\partial_{t}

is the Dirac delta function.

(t) + \frac{j}{π t} * u_{k} (t)

is the Hilbert transform. The equation above represents the process of finding the modal function

u_{k}

and the center frequency

ω_{k}

that minimizes the sum of the bandwidths of each modal component’s center frequency.

(2) The penalty factor (α) and the multiplication operator (λ) are introduced, and the Lagrangian method is used to transform the constrained variational problem into an equivalent unconstrained one. The value of α ensures the accuracy of signal decomposition, while λ guarantees the strictness of the constraints. The augmented Lagrangian function is as follows:

L_{\{u_{k}\}, \{ω_{k}\}, λ} = α \underset{k}{} {‖ \partial_{t} [(δ (t) + \frac{j}{π t}) u_{k} (t)] \exp (- j ω_{k} t) ‖}_{2}^{2} + {‖ f (t) - \underset{k}{} u_{k} (t) ‖}_{2}^{2} + [λ (t), f (t) - \underset{k}{} u_{k} (t)]

(3)

Where:

α

is a penalty factor used to control the importance of the regularization term, in order to avoid overfitting or underfitting;

\exp (- j ω_{k} t)

is used to transform the modal function into the frequency domain;

(3) The Alternating Direction Method of Multipliers (ADMM) is used to solve the equations and obtain the optimal solution to the variational problem:

\begin{array}{l} {\hat{u}}_{k}^{n + 1} (ω) = \frac{\hat{f} (ω) - \sum_{i \neq k} {\hat{u}}_{i} (ω) + \frac{\hat{λ} (ω)}{2}}{1 + 2 α {(ω - ω_{k})}^{2}} \\ ω_{k}^{n + 1} = \frac{\int_{0}^{\infty} ω | {\hat{u}}_{k} (ω) |^{2}}{\int_{0}^{\infty} | {\hat{u}}_{k} (ω) |^{2}} \end{array}

(4)

\begin{array}{l} {\hat{λ}}^{n + 1} (ω) = {\hat{λ}}^{n} (ω) + τ (\hat{f} (ω) - \sum_{k} {\hat{u}}_{k}^{n + 1} (ω)) \\ \sum_{k} {‖ {\hat{u}}_{k}^{n + 1} - {\hat{u}}_{k}^{n} ‖}_{2}^{2} / {‖ {\hat{u}}_{k}^{n} ‖}_{2}^{2} < ε \\ ω \geq 0, u p d a t e λ_{k} \end{array}

(5)

Where:

{\hat{u}}_{k}^{n + 1}

represents the update frequency of theK-th component, and

ε

is the given solution accuracy.

Figure 2.

VMD algorithm flowchart.

MEPSO algorithm for optimizing VMD parameters

The traditional VMD algorithm requires the parameters K and α to be manually assigned, which introduces randomness and uncertainty. The accuracy of parameter selection can affect the overall performance of the denoising algorithm. To improve the stability of the algorithm and meet the real-time denoising requirements of belt conveyors, this paper adopts the Motion-Encoding Particle Swarm Optimization (MEPSO) algorithm to adaptively optimize the key parameters in the VMD algorithm. During the optimization process, the MEPSO algorithm designs specific particle movement rules and objective functions, allowing the two parameters K and α of VMD to be optimized to their optimal values in a short time, effectively reducing the computational complexity and workload of VMD to meet the real-time denoising requirements.

In the MEPSO algorithm, we have designed the following objective function to optimize the parameters K and α of the VMD:

f (K, α) = \sum_{i = 1}^{N} (‖ s_{i} - {\hat{s}}_{i} ‖^{2}) + λ (α \cdot K)

(6)

Where:

f (K, α)

is the objective function, used to measure the error and complexity of the VMD decomposition;

s_{i}

is the modal component (IMF) obtained after decomposition, and

{\hat{s}}_{i}

is the estimated value of the original signal; N is the length of the signal, representing the error evaluation at each time point; λ is the penalty term, used to balance the trade-off between decomposition accuracy and computational complexity.

Subsequently, this paper introduces the motion encoding mechanism, which restricts the movement of particles within their neighborhood, ensuring that there are no excessive or minimal changes during the optimization process. This design helps reduce the complexity of the search space and enhances computational efficiency. The motion encoding mechanism treats each search path as a set of motion segments. Using Uk as the position of each particle, the update equation for MEPSO is as follows:

\begin{array}{l} Δ U_{k + 1} = ω Δ U_{k} + c_{1} r_{1} (L B_{k} - U_{k}) + c_{2} r_{2} (G B_{k} - U_{k}) \\ U_{k + 1} = U_{k} + Δ U_{k + 1} \end{array}

(7)

Where:

ω

、

c_{1}

、

c_{2}

, and

r_{1} {、 r}_{2}

serve the same roles as in the PSO algorithm.

L B_{k}

and

G B_{k}

represent the personal best solution of a particle and the global best position vector set, respectively, at the K-th generation.

During the search process, in order to evaluate the cost associated with the path $U_{k}$ , the path needs to be mapped from motion space to Cartesian space. The mapping process is performed by constraining each motion segment of the UAV path to one of its eight neighbors. Thus, at time t, the motion magnitude $pt$ can be normalized and the motion angle $at$ can be quantized, with the following formula:

\begin{array}{l} ρ_{t}^{*} = 1 \\ α_{t}^{*} = 4 5^{\circ} ⌊ \frac{α_{t}}{4 5^{\circ}} ⌋ \end{array}

(8)

Where: the operator

⌊ \frac{α_{t}}{4 5^{\circ}} ⌋

denotes rounding to the nearest integer.

The Cartesian space coordinates relative to the motion space coordinates can be obtained through equation (7), and the conversion formula is as follows:

\begin{array}{l} u_{k, t}^{*} = | ⌊ \cos α_{t}^{*} ⌋, ⌊ \sin α_{t}^{*} ⌋ | \\ o_{k, t + 1} = o_{k, t} + u_{k, t}^{*} \end{array}

(9)

By introducing the MEPSO algorithm, the parameters K and α of VMD can be adaptively adjusted during the optimization process, reducing the uncertainty caused by manually setting the parameters. Compared to the traditional VMD algorithm, the MEPSO-optimized VMD algorithm ensures high-precision decomposition while effectively reducing computational complexity. Since the MEPSO algorithm can quickly converge to the optimal solution, the optimization process does not consume excessive computational resources, significantly shortening the overall algorithm’s processing time, thereby meeting the real-time denoising requirements of the belt conveyor. The specific process is as follows, and the optimization flow is shown in Figure 3.

Figure 3.

MEPSO optimization process.

Sample entropy and improved wavelet threshold function

Sample entropy

In the VMD decomposition process, components are selected based on entropy values as features. Not all IMF components obtained after modal decomposition contain noise, so a selection step is necessary.¹³ Generally, noise is a random and irregular signal with a higher sample entropy, while feature signals usually have a certain degree of regularity, and their sample entropy is relatively low. By selecting IMF components with high-frequency noise based on the magnitude of the sample entropy,¹⁴ while preserving low-frequency IMF components, subsequent experimental analysis suggests that setting a threshold (TH = 0.15) can effectively filter out the noise components that need to be processed, thus maximizing the retention of feature signals. Sample entropy is a statistic proposed by Richman et al. that does not count its own matches, which is different from approximate entropy.¹⁵ When calculating the sample entropy approximation, it does not include comparison with its own data segment, the calculation error is small, and it does not depend on the data length. Sample entropy has better consistency. Sample entropy is not sensitive to missing data. Even if the data is lost as much as 1/3, it will have little impact on the calculated value of SampEn. The implementation process of the sample entropy function is as follows, and the specific sample entropy analysis process is shown in Figure 4.

(1) Unlike approximate entropy, when traversing all combinations of $X (i)$ and $X (j)$ , the range of $i$ is from 1 to $N ‐ m$ , and cases where $i$ is equal to $j$ are excluded. Therefore, there will be a total of $N ‐ m ‐ 1$ combinations

(2) The ratio of the approximate quantity to the total quantity is denoted as $B_{i}^{m} (r)$ , and the total quantity is “ $N ‐ m ‐ 1$ ”.

(3) $B^{m} (r) = \frac{1}{N - m} \sum_{i = 1}^{N - m} B_{i}^{m} (r)$ , Compared to the steps in approximate entropy, no logarithm is taken here, which avoids the occurrence of ln0.

(4) By increasing the dimension by 1, becoming m+1, and repeating the above steps, $A^{m} (r)$ is obtained.

(5) The sample entropy of the signal sequence is obtained as:

S a m p E n (m, r) = \ln B^{m} (r) - \ln A^{m} (r)

(10)

Figure 4.

Sample entropy analysis process.

Improved wavelet threshold function

After completing the sample entropy analysis, threshold filtering is performed on the screened noise signal. The essence of the wavelet threshold denoising method is to achieve signal denoising by performing threshold processing on wavelet coefficients based on wavelet transform. The wavelet basis function is the foundation of wavelet analysis, and the choice of different wavelet basis functions directly affects the final decomposition and reconstruction results, ultimately determining the quality of the denoised signal.¹⁶ The quality of the wavelet threshold denoising results also depends on the threshold selection; setting the threshold too high can lead to signal distortion, while setting it too low can result in noise remaining in the signal, leading to poor denoising performance.

(1) Selection of Wavelet Basis Function

The choice of wavelet basis function depends on the need for accurately describing the acoustic signal information. The main factors usually considered include whether the function has compact support and whether it possesses sufficient vanishing moments within the specified interval. The Daubechies (db) function is a high-precision wavelet basis function that is not only orthogonal but also satisfies the support condition. Based on the subsequent experimental analysis, this paper selects the db8 wavelet function as the wavelet basis for analyzing acoustic signals, with a decomposition level of 2.¹⁷

(2) Constructing the Wavelet Threshold Function

Traditional hard and soft thresholding functions have some issues in signal denoising. The hard threshold function may cause additional oscillations, while the soft threshold function may introduce a constant bias, which can affect the accuracy and quality of signal reconstruction. To address these issues, the paper proposes a new dual-parameter threshold function, which flexibly adjusts the threshold range through two adjustable parameters (α and β), achieving smoother and more continuous threshold processing.¹⁸ This approach avoids the pseudo-Gibbs phenomenon and reduces signal distortion. The hard threshold and soft threshold functions are as follows:

y_{j, k} = {\begin{cases} ω_{j, k} | ω_{j, k} | > λ \\ 0 | ω_{j, k} | ⩽ λ \end{cases}

(11)

y_{j, k} = {\begin{cases} s i g n (ω_{j, k}) (| ω_{j, k} | - λ) | ω_{j, k} | > λ \\ 0 | ω_{j, k} | ⩽ λ \end{cases}

(12)

The proposed threshold function uses an exponential function as the carrier. For both wavelet coefficients greater than the threshold and those less than the threshold, the threshold function can be determined by continuously adjusting the exponent. The improved threshold function is continuous at the threshold and smooths the wavelet coefficients that are smaller than the threshold, avoiding the fixed bias issue present in the soft threshold function. Additionally, it can automatically adjust parameters according to different noise levels. The improved threshold function, by introducing a transition reduction interval, allows for flexible adjustment under different signal characteristics, effectively enhancing noise suppression capabilities.^19,20 It also preserves the key features of the signal when handling signals in high-noise environments.²¹ The improved wavelet threshold function is shown below, with the transition curves for the improved threshold, hard threshold, and soft threshold shown in Figure 5. The specific denoising process using the improved threshold is shown in Figure 6.

y_{j, k} = {\begin{cases} γ ω_{j, k} + (1 - γ) s g n (ω_{j, k}) (| ω_{j, k} | - λ) | ω_{j, k} | > λ \\ s g n (ω_{j, k}) \times e^{β (| ω_{j, k} | - λ)} \times | ω_{j, k} | | ω_{j, k} | ⩽ λ \end{cases}

(13)

Where:

γ^{2} = e^{- α} \sqrt{ω_{j, k}^{2} - λ^{2}}

α

and

β

are constants.

Figure 5.

Improved threshold function curve.

Figure 6.

Improved threshold denoising process.

The overall structure of the belt conveyor sound denoising method based on MEPSO-VMD and improved wavelet threshold is shown in Figure 7. The specific parameter settings for each part will be introduced in the subsequent experimental content.

Figure 7.

Based on the MEPSO-VMD denoising system.

Experiment analysis of results

Data description and experiment setup

To verify the effectiveness and real-time performance of the denoising algorithm in processing field-measured belt conveyor fault signals, various fault signals such as belt tear, belt deviation, idler failure, and drum failure were collected on-site at Rizhao Port in China. Since belt conveyor fault signals are typically low-frequency signals, the fault signals are easily confused with normal signals. To address this, the number of sensor channels was increased (8 channels), and fault signals were collected using four sensors. The sensors used for the signal collection were CRY2301 unidirectional capacitive acoustic sensors combined with the high-precision data acquisition system PCI8815 as the sound collection system for the field experiment. The parameters of the acoustic sensors include a nominal sensitivity of 80 mV/Pa and a frequency response range of 10 Hz to 20 kHz. Figure 8 shows the signal acquisition site, where sensors were installed under the idler to collect the sound signals. The experiment used four data sets: the idler failure dataset, belt tear dataset, belt deviation dataset, and drum failure dataset. The sampling frequency was 10 kHz, with 4000 sampling points, and each signal frame lasted 25 ms, with 4000 samples for each dataset. The types of fault signals and dataset sizes are shown in Table 1.

Figure 8.

Belt conveyor fault signal collection site.

Table 1.

The port belt conveyor fault signal dataset.

Dataset	Rizhao Port Belt Conveyor Fault Sound Dataset
Sampling frequency (kHz)	48 kHz
Acquisition equipment	The sound collection sensorCRY2301,digital signal processor TMS320C6747, and power module UB18650
Fault type	Idler failure, belt tear, belt deviation, drum failure
Classes	4

To ensure consistency in the experimental environment, all the following experiments were conducted on the same platform (MATLAB2024), with the same configuration (CPU: I9-13900HX, RAM: 16 GB, GTX 4070), using the same fault signals. The specific experimental steps are as follows.

(1) First, in the optimization algorithm phase, the performance of the MEPSO algorithm proposed in this paper is compared with four optimization algorithms and traditional denoising algorithms. The optimization algorithms include PSO (Particle Swarm Optimization), SSA (Sparrow Search Algorithm), COA (Crayfish Optimization Algorithm), and GA (Genetic Algorithm). The traditional algorithms include EMD, VMD, and EWT. Secondly, the complexity and runtime of the MEPSO algorithm are compared with the other three optimization algorithms, and the denoising performance of the MEPSO algorithm is comprehensively analyzed.

(2) In the entropy feature extraction phase, the effects of selecting IMF components using sample entropy, fuzzy entropy (Fuzzy Entropy), permutation entropy (Permutation Entropy), and spectral entropy (Spectral Entropy) are analyzed.

(3) In the secondary denoising phase, the improved threshold function is compared with the hard threshold function and soft threshold function in terms of their Signal-to-Noise Ratio (SNR) when processing different fault signals.

(4) Finally, a comprehensive analysis of the denoising performance of the VMD-MEPSO-Improved Threshold algorithm under different fault types is conducted. By setting two noise environments (SNR = 5 dB and SNR = 10 dB), the denoising effectiveness under different noise conditions is explored.

Analysis of denoising performance of MEPSO algorithm

By setting the optimization algorithm iteration count to 50 generations, the number of optimized parameters to 2 (K and α), and the population size to 2, the optimization of K and α, fitness function value, runtime, Signal-to-Noise Ratio (SNR), Mean Square Error (MSE), and Normalized Cross-Correlation (NCC) are compared between MEPSO and other algorithms to verify the accuracy of the proposed algorithm. To quantitatively analyze the denoising results, this paper uses indicators such as Signal-to-Noise Ratio (SNR), Root Mean Square Error (MSE), and Normalized Cross-Correlation (NCC) to evaluate the denoising quality. The definitions of these indicators are as follows:

\begin{array}{l} S N R = 10 \lg \frac{\sum_{N}^{n = 1} X^{2} (n)}{\sum_{N}^{n = 1} {(X (n) - Y (n))}^{2}} \\ M S E = \sqrt{\frac{1}{n} \sum_{n = 1}^{N} {(X (n) - Y (n))}^{2}} \\ N C C = \frac{\sum_{N}^{n = 1} X (n) Y (n)}{\sqrt{(\sum_{N}^{n = 1} (X^{2} (n))) (\sum_{N}^{n = 1} (Y^{2} (n)))}} \end{array}

(14)

Where

X (n)

represents the original signal, and

Y (n)

represents the denoised signal.

As shown in Table 2, the SNR of MEPSO-VMD (10.2178) indicates its ability to more effectively extract signals and suppress noise. Compared to EMD, VMD, and EWT, the SNR of MEPSO has increased by approximately 21.9%, 7.9%, and 13.0%, respectively. Furthermore, the MEPSO algorithm also demonstrates the smallest reconstruction error and the highest correlation with the original signal. The NCC of MEPSO has increased by approximately 1.26%, 0.19%, and 0.26%, respectively.

Table 2.

Comparison of denoising performance of various traditional algorithms.

Methods	SNR	MSE	NCC
MEPSO-VMD	10.2178	0.02159	0.9936
EMD	7.9728	0.03370	0.9812
VMD	9.4231	0.02243	0.9917
EWT	8.3861	0.02275	0.9910

In Table 3, MEPSO is compared with four other optimization algorithms. The sample length of each fault signal is 4000, and the sample signal time is 0.45 seconds. Both MEPSO and PSO can correctly decompose the number of modes in the simulated signal (K = 4), while the other four algorithms fail to correctly decompose the modes. MEPSO has a faster runtime (21.59% shorter compared to the PSO algorithm, 34.38% shorter compared to the COA algorithm, 33.80% shorter compared to the SSA algorithm, and 41.43% shorter compared to the GA algorithm) and achieves a lower fitness function value, making it capable of real-time processing. As seen from Table 3, the performance of MEPSO is significantly better than the other algorithms, providing a solid foundation for subsequent secondary filtering and decomposition.

Table 3.

Comparison of the performance of various optimization algorithms.

Optimization algorithm	K	a	Optimization time/s	Fitness value
MEPSO	4	12000	0.523	0.2261
PSO	4	13500	0.667	0.2457
COA	7	10477	0.797	0.2972
SSA	9	12877	0.790	0.2709
GA	8	10567	0.893	0.2713

Using the four collected fault signals (idler failure, belt tear, belt deviation, and drum failure) as examples, fitness curves based on different optimization algorithms are plotted, as shown in Figure 9. From Figure 9, it can be observed that the MEPSO algorithm proposed in this paper achieves lower fitness values for all four types of faults. In 50 iterations, the MEPSO algorithm found the best fitness value, which is significantly lower than the traditional PSO algorithm and other algorithms, and it also exhibits more stable fitness values.

Figure 9.

Fitness Curves of Various Algorithms: (a) Fitness curves of various algorithms under idler failure; (b) Fitness curves of various algorithms under belt tear; (c) Fitness curves of various algorithms under belt deviation; (d) Fitness curves of various algorithms under drum failure.

Sample entropy experimental analysis

In actual fault signals, not all IMF components are noise signals; there are also feature signals. Generally, noise is a random and irregular signal, characterized by a higher entropy value, while feature signals usually have certain regularities and relatively lower entropy values.

After completing the MEPSO-VMD decomposition, the sample entropy values of each IMF component will be analyzed to select the IMF components with high sample entropy for filtering to further eliminate noise. The filtering effects of different entropy functions, such as Fuzzy Entropy, Permutation Entropy, Spectral Entropy, and the sample entropy proposed in this paper, are compared. As shown in Figure 10, it is concluded that sample entropy is more accurate in filtering noise signals. In Figure 10(a), the effect of sample entropy is the best, showing a clear stepped pattern. The entropy value of IMF4 is lower than 0.1 (indicating a normal signal with no noise), so it does not need to participate in subsequent threshold decomposition. This also ensures that the normal signal is not interfered with by threshold decomposition. In Figure 10(b), Fuzzy Entropy fails to correctly assess the entropy values of noise, with noise signal entropy being lower than normal signal entropy. Figure 10(c) shows that Permutation Entropy performs second best after sample entropy, but it fails to exclude IMF4, the normal signal. Figure 10(d) illustrates that Spectral Entropy performs the worst, with the entropy values of noise and normal signals mixed together, making it unable to distinguish them correctly. The specific entropy values for each IMF component are displayed in Table 4, where the entropy values of each component are clearly shown.

Figure 10.

Entropy: (a) Sample Entropy; (b) Fuzzy Entropy; (c) Permutation Entropy; (d) Permutation entropy.

Table 4.

Comparison of different types of entropy values.

Optimization algorithm	IMF1	IMF2	IMF3	IMF4
Sample entropy	0.701	0.142	0.159	0.010
Fuzzy entropy	0.242	0.553	0.502	0.197
Permutation entropy	0.451	0.286	0.266	0.187
Spectral entropy	2.794	2.979	2.387	2.293

Improved wavelet threshold experiment analysis

In the final stage, the improved wavelet threshold method will be used for secondary denoising of the IMF components filtered by sample entropy. The denoising performance of the improved threshold algorithm is compared with that of the hard threshold function and soft threshold function. As shown in Figure 11, the denoising effects of the three thresholding methods under different wavelet bases are displayed. After 8-level decomposition with the db8 wavelet base, the best Signal-to-Noise Ratio (SNR) is achieved in the denoising process of different fault types, demonstrating its superiority in denoising. By adjusting the parameters α and β in the improved threshold function formula, the transition curve of the threshold can be modified. Experimental validation reveals that the best results are obtained with α = 2 and β = 0.8. Table 5 shows the denoising SNR for the four fault types under the db8 wavelet base. It can be observed that under the improved threshold, the SNR is significantly improved, and the signal-to-noise ratio is higher than that of the traditional soft and hard thresholding methods. The proposed method demonstrates a higher denoising ability for nonlinear and non-stationary random signals while retaining more signal details.

Figure 11.

Signal-to-Noise Ratios (SNR) of Different Fault Types Under Various Thresholds: (a) SNR under idler failure for various thresholds; (b) SNR under belt tear for various thresholds; (c) SNR under drum failure for various thresholds; (d) SNR under belt deviation for various thresholds.

Table 5.

Signal-to-Noise Ratio (SNR) under Different Thresholds with the db8 Wavelet Basis.

Thresholding method	Idler failure	Belt tear	Drum failure	Belt deviation
Improved threshold	7.68	7.70	7.69	7.63
Soft thresholds	7.53	7.51	7.39	7.40
Hard thresholds	7.51	7.53	7.41	7.52

Comprehensive experimental analysis of belt conveyor fault signals

To verify the effectiveness of the proposed denoising algorithm, this section performs a denoising performance analysis using the proposed algorithm and other algorithms (PSO, SSA, COA, GA, and VMD without optimization) in different noise environments, with noise levels set to 5 dB and 10 dB. To further compare the performance of these denoising methods, 30 real noise signals were generated. On average, the MEPSO-VMD-Improved Threshold provided the best results in terms of SNRout (see Figure 12). In the idler failure denoising experiment, the proposed algorithm demonstrated the best denoising performance, achieving a higher signal-to-noise ratio. The proposed algorithm outperforms the PSO, SSA, COA, GA, and VMD algorithms, particularly when the noise signal’s SNR is higher, yielding better denoising results. The MEPSO-VMD-Improved Threshold also showed a higher denoising success rate. These results indicate that the proposed algorithm is more suitable for signals with jump signals, protruding signals, and high-frequency oscillations at individual positions. However, for low-frequency oscillating signals with low SNR, the performance of the proposed method can be slightly improved by adjusting the search range of the MEPSO algorithm to increase the number of modes (K). Tables 6 and 7 provide detailed information on the average denoising performance of various algorithms in different noise environments.

Figure 12.

Denoising Performance of Various Faults under Different Noise Environments: (a) SNR of various algorithms under idler failure with 5 dB noise; (b) SNR of various algorithms under belt tear with 5 dB noise; (c) SNR of various algorithms under drum failure with 5 dB noise; (d) SNR of various algorithms under belt deviation with 5 dB noise; (e) SNR of various algorithms under idler failure with 10 dB noise; (f) SNR of various algorithms under belt tear with 10 dB noise; (g) SNR of various algorithms under drum failure with 10 dB noise; (h) SNR of various algorithms under belt deviation with 10 dB noise.

Table 6.

Comparison results of performance indicators of different methods (SNR = 5 dB).

Methods	Idler failure	Belt tear	Drum failure	Belt deviation
Ours	16.31	12.32	14.57	13.49
PSO-VMD	13.83	11.91	14.23	9.23
COA-VMD	15.23	10.83	14.21	12.14
SSA-VMD	14.15	10.71	13.91	10.83
GA-VMD	12.94	10.63	11.97	9.76
VMD	12.19	10.91	10.94	8.43

Table 7.

Comparison results of performance indicators of different methods (SNR = 10 dB).

Methods	Idler failure	Belt tear	Drum failure	Belt deviation
Ours	17.82	13.49	14.23	17.39
PSO-VMD	13.12	12.44	12.77	13.21
COA-VMD	16.21	12.32	12.42	14.43
SSA-VMD	16.23	10.77	12.17	14.21
GA-VMD	11.79	10.76	9.63	11.89
VMD	10.49	10.79	8.34	10.12

In addition, the fault detection efficiency of different algorithms was evaluated to further demonstrate the effectiveness of the proposed method in fault signal denoising. The sample length of each type of belt conveyor fault signal was set to 4000, with a signal duration of 0.45 seconds. As shown in Table 8, the proposed algorithm achieves the best performance in terms of real-time denoising capability, essentially meeting the requirements for real-time processing.

Table 8.

Comparison of fault detection efficiency under different denoising methods.

Methods	Original signal	5 dB noise signal	10 dB noise signal
Ours	0.536	0.593	0.633
PSO-VMD	0.667	0.713	0.796
COA-VMD	0.797	0.814	0.875
SSA-VMD	0.790	0.832	0.896
GA-VMD	0.893	0.963	1.001
VMD	0.632	0.697	0.765

Application in motor bearing fault diagnosis

The denoising method proposed in this study is particularly suitable for processing nonlinear, non-stationary fault signals with strong noise interference, such as vibration and impact signals from belt conveyors. These signals are typically characterized by high impulsiveness, complex mode distribution, and significant spectral overlap. Therefore, the proposed algorithm exhibits strong generalization capability and can be applied to signal processing tasks in various industrial scenarios, such as rotating machinery, motor systems, and power equipment monitoring.

In the context of motor fault diagnosis, the PU dataset—a publicly available motor bearing fault diagnosis dataset—was used as the experimental signal source. Two types of fault signals were selected, each with a sample length of 4000 and a signal duration of 0.45 seconds. As shown in Figure 13 and Table 9, the proposed algorithm also demonstrates excellent denoising performance in bearing fault diagnosis. It effectively reduces noise in the original signals, improves the accuracy of feature extraction, and enhances the robustness of the diagnostic model, thereby providing more reliable technical support for motor condition monitoring.

Figure 13.

Denoising Performance of Various Faults under Different Noise Environments: (a) SNR of various algorithms under rolling element fault with 5 dB noise; (b) SNR of various algorithms under inner ring fault with 5 dB noise; (c) SNR of various algorithms under rolling element fault with 10 dB noise; (d) SNR of various algorithms under inner ring fault with 10 dB noise.

Table 9.

Comparison of SNR for motor bearing faults using different algorithms.

Methods	Rolling element fault (5 dB)	Inner ring fault (5 dB)	Rolling element fault (10 dB)	Inner ring fault (10 dB)
Ours	14.92	14.37	15.73	14.28
PSO-VMD	13.62	11.44	13.77	11.21
COA-VMD	13.61	11.52	13.82	13.93
SSA-VMD	13.27	12.74	13.13	13.25
GA-VMD	12.79	11.76	12.63	11.89
VMD	13.69	8.69	5.64	8.92

Conclusion

This paper proposes a denoising method based on the combination of MEPSO-VMD and an improved wavelet threshold, aimed at denoising fault signals of belt conveyors. The proposed method retains more signal details while achieving lower computational complexity and meeting the real-time denoising requirements. Whether through detailed comparative analysis or quantitative analysis using signal-denoising performance metrics, the proposed method performs better in noise removal. Experimental analysis of the VMD-MEPSO method shows that the proposed denoising method increases the Signal-to-Noise Ratio (SNR) by 21% compared to traditional algorithms, while also demonstrating faster response speeds. Experiments simulating different noise environments show that, compared to traditional PSO and other improved algorithms, the proposed denoising method increases the SNR by approximately 20%. Overall, this method not only achieves fault signal denoising for belt conveyors in high-noise environments, but also provides an effective technical approach for motor bearing fault diagnosis, as well as the monitoring of rotating machinery and power equipment.

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study is supported by Natural Science Foundation of Shandong Province, China (ZR2020MF092).

ORCID iD

Liuxiao Wang

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