A Rolling Bearing Vibration Signal Noise Reduction Processing Algorithm Using the Fusion HPO-VMD and Improved Wavelet Threshold

Abstract

In order to solve the problem of random noise in rolling bearing vibration signals under complex working conditions, this paper use a symmetry VMD theory to set up a rolling bearing vibration signal noise reduction processing algorithm using the fusion HPO-VMD and improved wavelet threshold. Based on the theory of variational mode decomposition (VMD), we introduce the hunter–prey optimization (HPO) algorithm to optimize the core parameters of VMD with the minimum envelope entropy as the objective function and obtain the optimal decomposition modes that contain the rolling bearing vibration signal. And then, we propose to use an improved wavelet threshold processing method to denoise the decomposed rolling bearing vibration signal to improve the recognition effect. Through the acquisition and test of the rolling bearing vibration signal, the proposed algorithm is verified; the results show that the method can reduce random noise and avoid the information loss caused by excessive noise reduction and improve the signal-to-noise ratio.

1. Introduction

Rolling bearings are widely used in all kinds of rotating machinery and equipment, and they are also one of the most easily damaged parts in the equipment. The bearing vibration signal has the characteristics of impact, nonlinearity, instability, weak signal, and strong modulation. Moreover, due to the complex operating conditions of rotating machinery and equipment and serious background noise pollution, the fault signal of rolling bearings is often submerged in noise [1,2,3]. Therefore, the noise reduction of the rolling bearing signal is of great significance for subsequent fault signal analysis and fault diagnosis.

At present, the commonly used noise reduction methods include wavelet transform (WT), empirical mode decomposition (EMD), aggregate empirical mode decomposition (EEMD), variational mode decomposition (WMD), etc. [4,5,6,7,8,9]. The disadvantage of WT in signal denoising processing is that the selection of the wavelet basis function and the wavelet threshold function is not unique, and it is difficult to ensure the optimal denoising effect. In order to overcome the shortcomings of WT, Xiong et al. [10] proposed a new improved wavelet threshold function for signal noise reduction by combining fast spectral kurtosis, bandpass filtering, and Hilbert envelope analysis. In the process of signal decomposition, the EMD method has some defects, such as mode aliasing, end effects, and false components, which make the separation degree of characteristic frequency and noise in the signal insufficient. Zhang et al. [11] used the improved EEMD to separate the inherent noise hidden in the original signal and proposed a new method for the weak fault feature extraction of rolling bearings based on improved EEMD and adaptive threshold denoising. The method can suppress the mode aliasing phenomenon to a certain extent, but it also has the disadvantages of large computation and unsatisfactory spectrum splitting effects. As a non-recursive signal decomposition method, VMD overcomes the shortcomings of EMD and other algorithms, such as mode aliasing and frequency effects, and has good noise robustness and noise reduction effects, with it being widely used. However, VMD decomposition results are largely affected by the penalty factor and the number of mode decomposition layers, and these parameters need to be artificially preset if they are not properly set. This results in inadequate decomposition, which affects the final result. The optimization parameters of the penalty factor and the number of mode decomposition layers are mainly obtained in the following ways: First, they are selected according to prior knowledge or the center frequency observation method, For example, Yang et al. [12] determined the penalty factor according to the center frequency of each IMF component during VMD decomposition of the original signal. This method has poor adaptation and cannot guarantee the accuracy of VMD decomposition. Second, it is selected using evaluation indicators, such as a penalty factor determined by the energy and correlation coefficient. The penalty factor selected using this method is not applicable to all signals. Chen et al. [13] used the WOA algorithm to optimize the number of modal decomposition and penalty factor of VMD, effectively extracted the bearing fault signal hidden in the interference of strong background noise, and avoided the over-decomposition and under-decomposition problems caused by improper parameter setting. This method has achieved good results, but particle swarm optimization is prone to falling into local optima in the search process.

To enhance the robustness of signal detection systems and improve the signal-to-noise ratio of sensors, some experts have adopted different combinations of signal noise reduction processing methods, aiming to improve the recognition ability of rolling vibration signals or target signals in complex environments. For example, the research team led by Li et al. [14,15,16] proposed and studied a projectile signal recognition algorithm using variational mode decomposition and wavelet mode maximum fusion in a sky screen sensor. They developed a measurement and matching recognition algorithm of multiple projectile dispersion positions using a linear array CCD and sky screen fusion; based on the research results of the signal noise reduction algorithms, their team also applied their research results to damage testing, forming a target damage algorithm under the mechanism of high signal-to-noise ratio parameters. Zhang et al. [17] developed a prediction technique for the remaining service life of rolling bearings based on the hunter–prey optimization algorithm and improved particle filtering. In order to solve the problem that there is a lot of noise in the vibration signals collected by the rotor system, which makes the axis locus chaotic and difficult to extract fault features, Li et al. [18] proposed a combination method of the Sequential Variational Mode Decomposition (SVMD) algorithm and rotor axis trajectory purification method based on SVMD and SVD. Although these methods have achieved good results in the actual application of vibration signals, due to the existence of some random uncertainties in the vibration detection environment, the current algorithms have certain limitations.

In order to process vibration signals more effectively and improve the signal-to-noise ratio of the signals, this paper proposes a method for reducing the noise of rolling bearing vibration signals by combining HPO-VMD and the improved wavelet threshold. The main innovations and contributions of this paper are as follows:

(1)

Taking rolling bearing vibration signals as the research object, we use a symmetry VMD theory to set up a rolling bearing vibration signal noise reduction processing algorithm using the fusion HPO-VMD and improved wavelet threshold.

(2)

Based on the theory of variational mode decomposition (VMD), we introduce the hunter–prey optimization (HPO) algorithm to optimize the core parameters of VMD with the minimum envelope entropy as the objective function and obtain the optimal decomposition modes that contain the rolling bearing vibration signal.

(3)

We propose to use an improved wavelet threshold processing method to denoise the decomposed rolling bearing vibration signal to improve the recognition effect. Through the acquisition and test of the rolling bearing vibration signal, the proposed algorithm is verified.

The paper is structured as follows: Section 2 introduces the noise reduction method based on parameter optimization, HPO-VMD, and improved wavelet threshold. Section 3 introduces the simulation signal analysis. Section 4 introduces the experimental results and analysis. Finally, the conclusions are drawn in Section 5.

2. The Noise Reduction Method Based on Parameter Optimization HPO-VMD and Improved Wavelet Threshold

2.1. VMD Decomposition Mechanism

Variational mode decomposition (VMD) is a signal processing technique used to decompose complex signals into a series of intrinsic mode functions; this function is called IMF for short. In the signal decomposition process of VMD, VMD adopts a symmetric decomposition, which can form a symmetrical reconstruction of the signal and effectively improve the processing effect of the signal.

The core of VMD is to treat signal decomposition as a variational problem, which is realized by solving an optimization problem. Each IMF has a fixed central frequency, which is determined adaptively during decomposition [19,20]. The key to VMD is the use of variational methods to minimize the bandwidth of each mode and thus achieve efficient signal decomposition.

It is assumed that each “mode” has a limited bandwidth of different center frequencies. The main process of this method is to use a Wiener filter to denoise and then to obtain K modes by initializing the finite bandwidth parameter and the center angular frequency [21]. The algorithm includes the construction of the variational problem and the solution of the variational problem. The algorithm has two basic criteria: First, the sum of the bandwidth of the center frequency of each modal component is minimum; second, the sum of all modal components is equal to the original signal. The modal component can be defined as the component mode function of amplitude modulation and frequency modulation, which can be expressed by Formula (1).

$$y_k(t)=A_k(t)\cos [{\phi }_k(t)]$$

(1)

where $A_k(t)$ is the envelope amplitude function of the rolling bearing vibration signal, which can represent the peak value of the vibration signal component; ${\phi }_k(t)$ is the instantaneous phase of the vibration signal $y_k(t)$; and $k$ is the total number of modals and $k=1,2,\dots ,K$.

For each modal $y_k(t)$, its analytic signal can be calculated by means of Hilbert transformation, and the analytic signal corresponding to each modal is obtained using Formula (2).

$$f_k(t)=\left(\delta (t)+{\displaystyle \frac{j}{\pi t}}\right)\ast y_k(t)$$

(2)

where $f_k(t)$ is the decomposed signal under each modal and $\delta (t)$ is a unit pulse time function.

Multiply Formula (2) with a pure exponential function for estimating the center frequency to obtain the bandwidth $F_i(t)$ of the modal function, which can be expressed by Formula (3).

$$F_k(t)=f_k(t)e^{-j{\omega }_{k}t}=\left[\left(\delta (t)+{\displaystyle \frac{j}{\pi t}}\right)\ast y_k(t)\right]e^{-j{\omega }_{k}t}$$

(3)

In order to calculate each demodulation signal, the Gaussian smoothness gradient norm is used as the bandwidth estimation, and the sum of each order modal is summed, and the optimization goal is to make the least squares problem of the sum of the estimated bandwidths; it can be expressed by Formula (4).

$$\left\{\begin{array}{l}{\min }_{\left\{y_k,{\omega }_k\right\}}\left\{{\displaystyle \sum _k{‖{\mathsf{\partial }}_t\left[\left(\delta (t)+\frac{j}{\pi t}\right)\ast y_k(t)\right]e^{-j{\omega }_{k}t}‖}_2^2}\right\}\\ s.t.{\displaystyle \sum _k{y}_k(t)=f(t)}\end{array}\right.$$

(4)

where ${\mathsf{\partial }}_t$ is the gradient operation and ${\displaystyle \sum _k{y}_k(t)=f(t)}$ is constraint conditions that the sum of each modal signal is equal to the original signal.

By using the quadratic penalty factor $\alpha$ and the Lagrange multiplication operator $\lambda \left(t\right)$, the constrained variational problem is transformed into a non-constrained variational problem. The factor $\alpha$ guarantees the reconstruction accuracy of the signal and $\lambda \left(t\right)$ guarantees the strictest constraints; the extended Lagrange expression can be obtained using Formula (5).

$$\begin{array}{ll}L\left(\left\{y_k\right\},\left\{{\omega }_k\right\},\lambda \right)=& \alpha {\displaystyle \sum _k{‖{\mathsf{\partial }}_t\left[\left(\delta \left(t\right)+\frac{j}{\pi t}\right)·y_k\left(t\right)\right]e^{-j{\omega }_{k}t}‖}_2^2}+{‖f\left(t\right)-{\displaystyle \sum _k{y}_k\left(t\right)}‖}_2^2\\ \hspace{1em}& +〈\lambda \left(t\right),f\left(t\right)-{\displaystyle \sum _k{y}_k\left(t\right)}〉\end{array}$$

(5)

The saddle points corresponding to Formulas (4) and (5) are obtained by using the alternating direction multiplier method (ADMM). First, the total number of modal $k$ is specified in advance; the frequency–domain expression of ${\widehat{y}}_k^1$, the corresponding central frequency ${\omega }_k^1$, and the Lagrange multiplier ${\widehat{\lambda }}^1$ are initialized. Then, modal ${\widehat{y}}_k$ and center frequency ${\omega }_k$ are updated by Formulas (6) and (7), respectively.

$${\widehat{y}}_k^{n+1}\left(\omega \right)\leftarrow {\displaystyle \frac{\widehat{f}\left(\omega \right){\displaystyle {\sum }_{i<k}{\widehat{y}}_i^{n+1}\left(\omega \right)+{\displaystyle {\sum }_{i>k}{\widehat{y}}_i^n\left(\omega \right)+\frac{{\widehat{\lambda }}^n\left(\omega \right)}{2}}}}{1+2\alpha {\left(\omega -{\omega }_k^n\right)}^2}}$$

(6)

$${\omega }_k^{n+1}\leftarrow {\displaystyle \frac{{\displaystyle {\int }_0^{\infty }\omega {\left|{\widehat{y}}_k^{n+1}\left(\omega \right)\right|}^{2}d\omega }}{{\displaystyle {\int }_0^{\infty }{\left|{\widehat{y}}_k^{n+1}\left(\omega \right)\right|}^{2}d\omega }}}$$

(7)

In summary, there are four parameters that need to be specified in advance. They are the total number of modal $k$, the quadratic penalty factor $\alpha$, the noise tolerance $\tau$ and the convergence criterion tolerance $\epsilon$. Compared with the two parameters $k$ and $\alpha$, $\tau$ and $\epsilon$ have less impact on the decomposition result, so the default values in the original VMD algorithm are usually adopted. Because the total number of modal $k$ is specified without any prior knowledge of the analysis signal, it is difficult to guarantee the appropriateness of $k$ and the accuracy and efficiency of signal decomposition [22,23]. In addition, the secondary penalty factor $\alpha$ is related to the performance of suppressing noise interference, so it should be chosen carefully. Therefore, finding the optimal parameter combination that matches the analysis signal is the key to the VMD method.

2.2. Improved Wavelet Threshold Function

The core of wavelet transform (wT) lies in the design of the threshold function and the selection of the threshold. The traditional wavelet denoising method usually adopts a hard threshold or soft threshold function, but these two methods have some limitations. Among them, the hard threshold function will exhibit discontinuity at the threshold, which easily leads to an unsmooth reconstruction signal. Although the soft threshold function is continuous, it may cause signal distortion due to the introduction of a fixed offset. More importantly, the threshold value in the traditional method is usually set to a constant value, which fails to fully consider the law of noise coefficient change under different decomposition scales, thus affecting the effect of denoising.

The core of wavelet transform (WT) is the design of the threshold function and the selection of a threshold. Traditional wavelet denoising methods usually use the hard threshold or soft threshold function, but these two methods have certain limitations [24,25]. Among them, the soft threshold function is used to achieve signal denoising by retaining important signal coefficients and scaling and smoothing smaller coefficients. However, when the absolute value of the number of signals is greater than the selected threshold, soft threshold processing will produce a constant deviation, which will affect the degree of approximation between the reconstructed signal and the real signal. The hard threshold function compares the signal value with the set threshold. The part below the threshold is set to zero, while the part above the threshold remains unchanged. There is poor continuity, which may cause additional oscillation of the signal, and the hard threshold will be less than or equal to the threshold. The signal strength is completely eliminated or reduced to zero, which may lead to the loss of signal features. More importantly, the threshold value in the traditional method is usually set to a constant value, which fails to fully consider the law of noise coefficient change under different decomposition scales, thus affecting the denoising effect.

Compared with the traditional wavelet filtering method, the adaptive threshold method based on the decomposition scale can dynamically adjust the threshold size, more effectively suppress the noise at different scales, and retain more detailed information of the original signal. In the process of wavelet decomposition, with the increase in decomposition scale, the noise coefficient will gradually decrease. The noise wavelet coefficient of the j-th layer is about $\sqrt{2}$ times that of the j+1-th layer. According to this rule, this section adopts a new threshold method based on adaptive adjustment of decomposition scale to improve denoising performance. The decomposition level $J$ of the signal is dynamically calculated by the signal length $J={\log }_2(N)-3$. The threshold value of this method is calculated using Formula (8).

$$\left\{\begin{array}{l}{\lambda }_1={\sigma }^{\prime }\sqrt{2{\log }_{10}N}\\ {\lambda }_j={\displaystyle \frac{\sqrt{2}}{2}}{\lambda }_{j-1}\end{array}\right.$$

(8)

where ${\lambda }_j$ is the threshold of the j-th layer, and $j=2,3,\cdots ,L$; ${\sigma }^{\prime }$ is the standard deviation of noise; and $N$ is the sequence length [26].

The threshold method based on adaptive adjustment of decomposition scale can make the threshold change dynamically with the scale: the high-frequency scale threshold is slightly higher, and more effective signal details are retained; the low-frequency scale threshold is reduced, and the noise is more thoroughly suppressed. By adapting the multi-scale distribution of noise, the purity and detail retention of the denoised signal are greatly improved. Through this improvement, the reconstruction effect of wavelet denoising can be significantly improved, and the problem that the difference between the reconstructed signal and the original signal is too large due to the constant threshold in the traditional wavelet filtering method is solved.

2.3. Rolling Bearing Vibration Signal Noise Reduction Processing Algorithm Using HPO-VMD and Improved Wavelet Threshold

The hunter–prey optimization (HPO) algorithm is a mathematical model that simulates the continuously optimized area and direction when prey is hunted. It is widely used in the field of finding the optimal solution or a solution close to the optimal solution [27,28]. In this algorithm, the hunter obtains the best hunting position by adjusting its position, and the prey moves to the safe position to avoid the attack of the hunter and the safest position of the prey is the optimal solution to the problem to be optimized.

For the complex and varied non-stationary vibration signals generated in bearing working conditions, it is difficult to separate the interference signals effectively by simply relying on human experience to select parameters. Therefore, this paper selects the improved hunter–prey optimization algorithm to realize the adaptive optimization of k and α and then realizes the effective decomposition and reconstruction of signals.

The position of the prey population needs to be initialized, where the i-th prey position $x_i$ of the prey population is initialized to a random number within the upper and lower limits [l, u]; the position can be obtained using Formula (9).

$$x_i=rand(1,d)\ast (u-l)+l$$

(9)

where d is the dimension of the variable. After the position of the prey population is initialized, the fitness should be determined according to the objective signal [29].

The hunter search mechanism can be expressed by Formula (10), where $x_{i,j}(t)$ represents the current position of the hunter.

$$x_{i,j}\left(t+1\right)=x_{i,j}\left(t\right)+0.5\left[\left(2CZ{P}_{pos}-x_{i,j}\left(t\right)\right)+\left(2\left(1-C\right)Z\overline{x_{i,j}}-x_{i,j}\left(t\right)\right)\right]$$

(10)

where $x_{i,j}(t)$ represents the next position of the hunter and $P_{pos}$ is the position of the prey.

Define C as the balance parameter, which can be obtained using Formula (11), among which $Iter$ is the current iteration number of the algorithm and $MaxIter$ is the maximum iteration number of the algorithm.

$$C=1-{\displaystyle \frac{0.98\times Iter}{MaxIter}}$$

(11)

Define Z as the adaptive parameter; it can be obtained using Formula (12), $\overrightarrow{R_1}$ and $\overrightarrow{R_3}$ are random vectors in [0, 1], and $R_2$ is random numbers in [0, 1]; P is the index value of $\overrightarrow{R_1}<C$ and IDX is the index value of $\overrightarrow{R_1}$ that satisfies condition $(P==0)$.

$$P=\overrightarrow{R_1}<C;IDX=(P==0);Z=R_2\otimes IDX+\overrightarrow{R_3}\otimes (\sim IDX)$$

(12)

The hunter selects the prey farthest from the average position of the population as the hunting target and needs to calculate the Euclidean distance of each member. Define $D_{euc}(i)$ as the Euclidean distance of each member; it can be obtained using Formula (13);

$$D_{euc}(i)=\sqrt{{\displaystyle \sum _{j=1}^d{(x_{i,j}-{\overline{x}}_{i,j})}^2}}$$

(13)

Consider that after the prey is captured, the hunter will continue to move to the new prey position and need to add a decreasing mechanism; the decreasing mechanism can be obtained using Formula (14).

$$\overline{k}=round(C\times \overline{N})$$

(14)

where $\overline{N}$ is the number of search agents;

As the number of iterations changes, the prey position $\overline{P_{pos}}$ is constantly updated, and $\overline{P_{pos}}$ can be expressed by Formula (15).

$$\overline{P_{pos}}={\overline{x}}_l|iisorted{D}_{euc}(\overline{k})$$

(15)

The prey position update can be obtained using Formula (16).

$$x_{i,j}\left(t+1\right)=T_{pos}+CZ\cos (2\pi R_4)\times \left(T_{pos}-x_{i,j}\left(t\right)\right)$$

(16)

where $R_4$ is the random number in [0, 1], and $T_{pos}$ is the globally optimal position. In the process of searching for the global optimal solution, the hunter and the prey are selected according to the parameter $R_5$ and the size of the adjusting parameter $\beta$; $R_5$ is the random number in [0, 1]. If $R_5<\beta$, the search agent is the hunter, and the Formula (10) is selected to update the position; otherwise, the search agent is the prey, and the position is updated by Formula (16).

The specific steps of the HPO-VMD algorithm are as follows:

(1)

Initialize population parameters, such as, population number N, maximum iteration number Max_iteration, and search upper and lower limits $[l_b,u_b]$.

(2)

Establish the fitness function. The envelope entropy can better reflect the sparsity of the signal. A smaller envelope entropy means that the modal component has stronger sparsity, and the decomposition effect is better. Therefore, the minimum envelope entropy can be obtained using Formula (17).

$$\left\{\begin{array}{l}Fitness=\min (f(i))\\ f(i)=-{\displaystyle \sum _{i=1}^N}p(i)\times \log 10(p(i))\\ p(i)=a(i)/{\displaystyle \sum _{i=1}^N}a(i)\end{array}\right.$$

(17)

where $N$ is the number of IMF components; $f(i)$ is the envelope entropy adjusted by Hilbert; $p(i)$ is the standardized form; and $a(i)$ is the envelope signal [30].

(3)

Update the adaptive parameter Z, balance parameter C, and update the position of hunter and prey.

(4)

Calculate the hunter fitness, screen the global optimal fitness value and the optimal individual position, and iterate until the end of the output optimal parameter combination [k, α].

(5)

Decompose the original signal into k components according to the optimized parameters.

(6)

Because each IMF component is derived from the decomposition of the original signal, the correlation coefficient between each component and the original signal can represent its importance. The larger the correlation coefficient is, the higher the similarity degree is, the larger the ontology data contained, and the correlation coefficient can be expressed by Formula (18).

$${\rho }_{k^{\prime }}={\displaystyle \frac{{\displaystyle \sum _{i=1}^n}\left(x_{i,k^{\prime }}-{\overline{x}}_{k^{\prime }}\right)\left(y_i-\overline{y}\right)}{\sqrt{{\displaystyle \sum _{i=1}^n}{\left(x_{i,k^{\prime }}-{\overline{x}}_{k^{\prime }}\right)}^2}\sqrt{{\displaystyle \sum _{i=1}^n}{\left(y_i-\overline{y}\right)}^2}}}$$

(18)

where x is the k-th IMF component; y is the original signal; and n is the number of sampling points [31].

(7)

By setting threshold $\beta \prime$ based on Formula (19), the effective IMF component is selected for reorganization, and irrelevant components are removed to achieve noise reduction.

$${\beta }^{\prime }=\sqrt{\sum _{i=1}^n{\left({\rho }_i-\overline{\rho }\right)}^2/k^{\prime }}$$

(19)

where $\overline{\rho }$ is the mean value of the correlation coefficient [32].

(8)

According to the parameter model, the signal of rolling bearing noise is decomposed by means of VMD. The correlation coefficients between each component and the original signal are calculated respectively, and the effective components are selected according to the principle of correlation coefficient.

(9)

The layered threshold denoising and reconstruction based on wavelet transform are performed on the effective components to obtain the rolling bearing signal only after noise reduction.

Following the steps above, the flow chart of the denoising method using HPO-VMD and improved wavelet threshold is shown in Figure 1.

3. Simulation Verification

In order to verify the effect of combined HPO-VMD and improved wavelet threshold noise reduction, the simulation signal is applied, Gaussian white noise of −2 dB, 5 dB, and 10 dB is added to the signal, and three different simulation signals containing noise are obtained using Formula (20).

$$\left\{\begin{array}{l}x\left(t\right)={\displaystyle \sum _{i=1}^N}{A}_{i}h\left(t-T_{\mathrm{i}}\right)+\eta \\ h\left(t\right)=\exp \left(-C_{2}t\right)\cos \left(2\pi f_{n}t\right)\end{array}\right.$$

(20)

where $\eta$ is Gaussian noise of different proportions, $A_i$ is the i-th impact amplitude [32], the attenuation coefficient $C_2=1100$, the sampling frequency is 12 KHz, and the resonance frequency $f_n=1.5 \mathrm{KHz}$. Taking 5 dB as an example, the comparison before and after adding noise to the simulation signal is shown in Figure 2.

To verify the denoising method of the HPO-VMD and improved wavelet threshold, taking the 5 dB Gaussian white noise simulation signal as an example, the combined HPO algorithm is used to optimize VMD and find the optimal parameter combination. The range of modal decomposition is [2, 15], the range of penalty factor is [200, 5000], the number of populations used for searching is 20, and the maximum number of iterations is 15. Due to the strong randomness of the iteration process, the parameter combination of decomposition and mode aliasing caused by parameter setting is eliminated after multiple iteration optimization. The optimal parameter combination is obtained as [6, 3000], and its fitness function curve is shown in Figure 3.

The simulation signal of Figure 2 is decomposed into 6 IMF components; the IMF1-IMF6 components are shown in Figure 4. At the same time, a set of default parameters [5, 2000] were set to obtain their decomposition diagram. Compared with the decomposed image obtained by default parameters, the optimized image has more IMF4 components; From the spectrum diagram of the IMF4 component, it can be seen that this component is a part of the original signal feature but is omitted in the decomposition diagram obtained by default parameters. This shows that the IMF component optimized by the HPO algorithm can more accurately express the original signal.

The optimized parameter combination was substituted into VMD decomposition to obtain IMF6 components, and the correlation coefficients of IMF components were calculated. The pure component and the noise-containing component are divided according to the correlation coefficient, and the noise-containing component is denoised by using the improved wavelet threshold. The time–domain diagram after denoising is shown in Figure 5.

To verify the noise reduction effect of the proposed method, we conducted a comparative verification. Signal denoising follows the two criteria: one is the filter out the noise in the original signal as much as possible, and the other is to ensure that the signal after noise removal is as undistorted as possible. At the same time, the original signal feature information can be retained. We evaluated the noise reduction quality by means of signal-to-noise ratio (SNR) and root mean square error (RMSE). SNR represents the ratio of actual signal to noise. In general, when the SNR is larger, the restoration of the original signal is greater, and the denoising effect is better. RMSE represents the difference between the denoised signal and the original signal; when the RMSE is smaller, the difference is smaller, and the denoising effect is better. The details can be expressed by Formulas (21) and (22).

$$SNR=10\lg \left[{\displaystyle \frac{{\displaystyle \sum _{i=1}^{n^{\prime }}}s{(i)}^2}{{\displaystyle \sum _{i=1}^{n^{\prime }}}{[s(i)-\widehat{s}(i)]}^2}}\right]$$

(21)

$$RMSE=\sqrt{{\displaystyle \frac{1}{n^{\prime }}}{\displaystyle \sum _{i=1}^{n^{\prime }}}{[s(i)-\widehat{s}(i)]}^2}$$

(22)

where $s(i)$ represents the original signal; $\widehat{s}(i)$ represents the signal after denoising; and $n^{\prime }$ represents the number of sampling points.

In order to further verify the noise reduction effect, EMD noise reduction, VMD noise reduction, wavelet soft threshold noise reduction, wavelet hard threshold noise reduction, SSA-VMD [12], CCEMD-SVD [24], and combined HPO-VMD and improved wavelet threshold noise reduction are used to process signals, respectively. According to Formulas (6) and (7), the signal-to-noise ratio and mean square error of the four noise reduction methods are calculated, respectively, and the effects of the different noise reduction methods are analyzed.

Table 1. The comparison results under different noise reduction methods with different noise sizes.

Noise Reduction Method	Noise Size −2 dB		Noise Size 5 dB		Noise Size 10 dB
	SNR/dB	MSE	SNR/dB	MSE	SNR/dB	MSE
EMD	2.95	2.15	3.10	2.05	3.40	1.95
VMD	4.10	1.85	4.80	1.65	5.90	1.40
Wavelet soft threshold	8.30	1.35	9.50	1.25	10.80	1.15
Wavelet hard threshold	7.74	1.57	9.02	1.38	10.37	1.26
Reference [12]	11.07	1.37	12.37	1.19	13.46	1.12
Reference [24]	11.98	1.29	12.84	1.08	14.58	1.03
Method of this paper	14.00	1.05	15.20	0.95	17.10	0.90

is the comparison results under different noise reduction methods with different noise sizes.

From the noise reduction indexes of different noise reduction methods in Table 1, it can be seen that compared with other methods, the noise reduction method of HPO-VMD combined with improved wavelet threshold performs best at all noise levels, with the highest SNR and the lowest mean square error, indicating that this method is more ideal for noise suppression and signal feature retention. Taking 5 dB noise as an example, compared with wavelet soft threshold noise reduction, the signal-to-noise ratio of HPO-VMD-wavelet noise reduction is increased by 60%, and the mean square error is reduced by 24%. The results show that the optimized method can not only remove noise more effectively but also maintain high signal fidelity in a low SNR environment. At the same time, compared with 10 dB noise, the SNR is 17.1 dB and the MSE is 0.90 under HPO-VMD-wavelet denoising. Compared with the SSA-VMD algorithm, the SNR is improved by 22% and the MSE is reduced by 20%. Compared with the CEEMDAN-SVD algorithm, the SNR is increased by 15% and the MSE is reduced by 14%, indicating that the method still has significant advantages in high SNR environments. It can not only effectively remove the residual noise but also reduce the error that may be introduced in the denoising process, so that the signal quality is further improved.

4. Experimental Results and Analysis

To further verify the effectiveness of the proposed method, the experimental data of the rolling bearing simulation platform published by Case Western Reserve University were selected for analysis. In this paper, the bearing model is SKF6205-2RS, and the approximate motor speed is 1797 r/min. The structural parameters of the bearing are shown in

Table 2. The main parameters of the test bearing.

Bearing Pitch Diameter/mm	Rolling Diameter/mm	Number of Rolling Elements	Contact Angle/Degree
39.04	7.94	9	0

.

According to the structural parameters, the characteristic frequency of the bearing inner ring is about 159.93 Hz, the rotation frequency is about 29.95 Hz, the sampling frequency of the vibration signal is 48 KHz, and the sampling number is 5000. According to the collected actual data, the vibration signal and waveform diagram of the bearing are obtained, as shown in Figure 6.

In order to verify the parameter optimization effect, the gray wolf algorithm (GWO), whale algorithm (WOA), sparrow search algorithm (SSA) and HPO algorithm were used to compare the parameter optimization effect.

We input the same measured signal and used the MATLAB 2022 simulation environment to optimize the VMD parameters several times. The SSA and WOA convergence speed is fast, but it is easy to fall into local optima; for the GWO and HPO algorithms, it is not easy to fall into local optima, but HPO convergence speed is faster, so the HPO optimization effect is better. The comparison of the results of the single fitness function curve is shown in Figure 7.

The bearing signal is optimized by the HPO algorithm. First, the parameters of the HPO algorithm are initialized, where the decomposition number K is set to 2–15, the penalty factor α is set to 200–5000, and the maximum number of iterations is set to 30. The corresponding decomposition layer number K is 4, and the penalty factor is 3200. The optimized parameter combination was substituted into VMD decomposition to obtain the IMF component, and the correlation coefficient of the IMF component was calculated. The pure component and the noise-containing component are divided according to the correlation coefficient, and the noise-containing component is denoised by using the improved wavelet threshold. The comparison diagram after denoising is shown in Figure 8. Among them, blue is the original signal; it can be seen that there is a strong interference signal in the original bearing signal. Red is the signal after noise reduction using this method.

In order to verify the resonance noise reduction effect of the proposed algorithm, three different types of noise with a signal-to-noise ratio (SNR) of −5 dB were added to the original bearing signal; Figure 9 is the result of processing under three different types of noise. Among them, Figure 9a shows the addition of Gaussian white noise; Figure 9b shows the addition of pink noise, and Figure 9c shows the addition of brown noise. Gaussian white noise contains more high-interference signals, such as impact signals; the pink noise and brown noise contain more non-stationary signals, which fully characterize the influence of different types of interference on the signal. In order to evaluate the effect of noise reduction, the proposed algorithm and the unoptimized VMD algorithm are respectively adopted for noise reduction processing on the noisy signals. The first chart in each group shows the signal after adding noise, and the second and third charts show the signal denoising by using unoptimized VMD and combining HPO-VMD and improved wavelet threshold, respectively.

For signals with Gaussian white noise added, HPO-VMD and the improved wavelet threshold noise reduction algorithm successfully weaken evenly distributed noise components and effectively improve signal clarity. In the case of pink noise added, the proposed method can effectively suppress strong low-frequency noise and recover the true contour of the signal more accurately. The proposed algorithm can not only reduce the noise level significantly but also maintain the original trend of the signal, which shows its superiority in dealing with persistent low-frequency noise. Compared with traditional VMD and EMD denoising, HPO-VMD and improved wavelet threshold denoising methods have a stronger ability to improve signal quality and retain original waveform features and, to a certain extent, solve the phenomenon of excessive denoising and signal loss, so as to better prove their effectiveness in the denoising processing of bearing fault data.

At the same time, in order to comprehensively evaluate the effect of noise reduction, we introduce the following evaluation indicators to quantitatively analyze the noise reduction results, and the statistical results of the signal after noise reduction are respectively calculated, as shown in

Table 3. Statistical table of noise reduction effect under different noise types.

Noise Type	Method	SNR	RSME	The Correlation Coefficient
White	Method of this paper	2.85	0.6501	0.8123
	VMD	−0.43	1.0502	0.4614
Pink	Method of this paper	3.30	0.4503	0.7358
	VMD	1.23	0.8676	0.5757
Brown	Method of this paper	1.78	1.0054	0.5654
	VMD	−6.72	2.1667	0.2312

.

The statistical results from the table show that by comparing different noise reduction methods, the proposed method is superior to the traditional VMD noise reduction algorithm in SNR, MSE, correlation coefficient, and other indicators, verifying the superiority of the combined HPO-VMD and improved wavelet threshold noise reduction algorithm in the noise reduction of rolling bearing fault signals.

From the perspective of the frequency domain, taking the environment of white noise and pink noise as an example, Figure 10 shows the unprocessed signal after adding Gaussian white noise to the original signal and the signal processed by the denoising algorithm proposed in this paper. Among them, the blue curve represents the envelope spectrum obtained after denoising the bearing signal with Gaussian white noise added by using the proposed denoising processing algorithm, and the black curve represents the bearing signal without denoising processing. Figure 11 shows the unprocessed signal after adding pink noise to the original signal and the signal processed by the denoising algorithm proposed in this paper. Among them, the orange curve represents the envelope spectrum obtained after denoising the bearing signal with pink noise added by using the proposed denoising processing algorithm, and the black curve represents the bearing signal without denoising processing.

By comparing the amplitude of the noise base (such as high-frequency white noise and low-frequency pink noise), the ability of this algorithm to suppress all kinds of noise is shown directly. For example, the noise base of the high frequency band in the white noise environment is significantly reduced, indicating that the method can effectively filter out broadband interference. In addition, at the bearing fault characteristic frequency, its ratio to the noise base is significantly increased, which is more conducive to fault detection. Under different noise spectrum characteristics (white noise full-frequency interference and pink noise low-frequency interference dominated), the proposed method can stably extract the fault frequency, and the harmonic components can be retained intact, which verifies its ability to adapt to complex industrial scenes.

In order to further verify the applicability and robustness of the proposed method at different sampling frequencies, we selected the same set of bearing fault signals and conducted experiments at three sampling frequencies of 12 kHz (original sampling rate), 24 kHz, and 48 kHz, respectively. Under the premise of maintaining the same algorithm parameters, the signals at different sampling rates are processed separately. A variety of typical noise reduction methods were selected in the experiment, including EMD noise reduction, VMD noise reduction, wavelet soft threshold noise reduction, wavelet hard threshold noise reduction, the SSA-VMD algorithm proposed in Reference [12], the improved SSA-VMD algorithm proposed in Reference [21], and the joint HPO-VMD and improved wavelet threshold method proposed in this paper. According to Formulas (6) and (7), the signal-to-noise ratio (SNR) and mean square error (MSE) of the signals processed using each method were calculated, respectively, to comprehensively evaluate the performance of different noise reduction methods at different sampling frequencies. The noise reduction effect of each method at different sampling rates is shown in

Table 4. Noise reduction results of different methods.

Noise Reduction Method	12 kHz		24 kHz		48 kHz
	SNR/dB	MSE	SNR/dB	MSE	SNR/dB	MSE
EMD	3.97	2.13	3.29	2.03	3.95	1.91
VMD	5.10	1.88	5.35	1.76	6.02	1.68
Wavelet soft threshold	8.30	1.59	9.50	1.37	10.80	1.15
Wavelet hard threshold	7.73	1.62	8.59	1.58	9.98	1.46
Reference [12]	10.03	1.45	11.07	1.39	13.72	1.23
Reference [21]	10.74	1.36	11.85	1.27	14.93	1.13
Method of this paper	14.57	1.25	15.35	0.96	16.59	0.89

.

According to the experimental results shown in Table 4, the proposed joint HPO-VMD and the improved wavelet threshold method show superior noise reduction performance at different sampling frequencies. Both signal-to-noise ratio (SNR) and mean square error (MSE) indicators are significantly better than other comparison methods. Specifically, as the sampling frequency increases from 12 kHz to 48 kHz, the SNR of the proposed method increases from 14.57 dB to 16.59 dB, and the MSE decreases from 1.25 to 0.89, showing good scale adaptability and stability. At the same time, compared with the soft threshold, the wavelet hard threshold, and the advanced methods in the literature [12,24], the proposed method always maintains the highest SNR and the lowest MSE at each sampling rate, indicating that it achieves a better balance between retaining signal characteristics and suppressing noise. In contrast, methods such as EMD and VMD have a large gap in SNR and MSE, and the performance fluctuation is obvious when the sampling rate changes, and the robustness is relatively poor. In summary, the proposed method not only has good adaptability at different sampling frequencies but also outperforms the existing mainstream methods in terms of noise reduction accuracy and signal fidelity, which verifies its wide applicability and stability in practical applications when facing sampling rate changes.

In summary, the combination of HPO-VMD and the improved wavelet threshold noise reduction algorithm can significantly reduce the full-band noise energy, improve the signal-to-noise ratio, retain the original signal characteristics, and adapt to various noise types well, to a certain extent, solve the phenomenon of excessive noise reduction and signal loss, and better retain the bearing fault characteristics while reducing noise. Therefore, the combination of HPO-VMD and the improved wavelet threshold denoising algorithm has strong feasibility and advantages in practical industrial applications.

In order to further illustrate the influence of the key parameters of HPO on noise reduction performance under the background of pink noise, as shown in

Table 5. Key HPO optimization parameters.

Population Size	Iteration Times	SNR (dB)	Characteristic Correlation Coefficient	Time-Consuming/(s)
10	100	2.27	0.79	0.6
20	100	3.30	0.84	0.8
30	100	3.12	0.81	1.1

, the number of fixed iterations is 100, and the population size is 10, 20, and 30, respectively. The experimental results verify the robustness of HPO and the basis of parameter selection:

It can be seen from Table 5 that when the number of iterations is the same and the number of populations is different, the populations size is 20, and the signal-to-noise ratio (SNR) of the HPO-VMD combined with the improved wavelet threshold denoising method proposed in this paper is up to 3.30 dB, which is 45.4% higher than that when the number of populations is 10. At the same time, the characteristic correlation coefficient is up to 0.84. When the population size increases to 30, the SNR decreases to 3.12 dB instead, indicating that over-scale causes excessive exploration and reduces convergence efficiency. The calculation time increases linearly with the population size, but when the population size is 20, the time only increases by 33.3%, the SNR increases by 45.4%, and the cost performance is the best.

5. Conclusions

Aiming to solve the problem of serious random noise interference in rolling bearing signals under complex working conditions, this paper proposes a signal denoising method combining HPO-VMD and an improved wavelet threshold. By introducing the historical optimal particle swarm optimization algorithm (HPO), this method realizes the adaptive optimization of the number of VMD mode decomposition and the penalty factor, which effectively avoids the loss of signal characteristics caused by the artificial setting of parameters. Based on the optimized IMF component, combined with the wavelet threshold function with the adaptive threshold and compromise attenuation strategy, the balance between noise suppression and feature retention is effectively improved. The verification results on the bearing data set of Case Western Reserve University show that compared with the traditional VMD method, the signal-to-noise ratio (SNR) of this method is increased by 4.62 dB on average, the mean square error (MSE) is reduced by 66% on average, and the correlation coefficient is increased by 28% on average, which shows good noise reduction performance and robustness. In addition, due to the clear overall structure of the algorithm, the parameter optimization overhead is concentrated in the initialization stage and does not rely on deep learning or GPU acceleration; it has strong lightweight characteristics and engineering portability. Especially after reasonable algorithm clipping and code optimization, it can achieve quasi-real-time operation on typical edge devices such as Raspberry Pi and Jetson Nano. In the future, deployment tests will be further carried out on the embedded platform to verify its practicality and promote its application in industrial scenarios.