An improved method for signal de‐noising based on multi‐level local mean decomposition

The product functions (PFs) extracted by local mean decomposition (LMD) of the noisy signal contain obvious energy‐concentrated pulses. As a result, the conventional amplitude threshold filtering used in wavelet transform (WT)‐based and empirical mode decomposition (EMD)‐based de‐noising methods is no longer applicable. To address this issue, an improved signal de‐noising method is proposed by using the multi‐level local mean decomposition (ML‐LMD), the superposition and recombination (SR) of high‐order PFs, the outlier detection, and waveform smoothing (OD‐WS) to remove noise by eliminating the pulse components. The proposed method's superior noise reduction performance is demonstrated through theoretical analysis and experimental verification. Compared to well‐known methods like WT‐based and EMD‐based de‐noising, the results show that the proposed method has significant comparative advantages in reducing noise in rolling bearing signals.


INTRODUCTION
Nowadays, rotating machinery often experiences faults during operation due to damages to key components, which can lead to abnormal operation.If not detected early, these faults can cause emergency shutdowns, equipment breakdown, and even casualties. 1][4] Vibration signals of rotating machinery are always utilized for fault diagnosis, as they always carry valuable fault information.Therefore, vibration-based signal analysis techniques have been widely used for fault diagnosis of rotating machinery. 5However, the vibration signals collected from rotating machinery in operation are often noisy.If the background noise is too heavy, the useful information would be submerged.Therefore, feasible and effective methods are necessary for feature extraction and fault diagnosis.Moreover, these vibration signals of rotating machines are known to be non-stationary, which means their parameters are time-varying. 6In order to obtain accurate state features, preprocessing of vibration signals is essential. 5][9][10][11][12][13][14][15][16][17] While these methods are useful for fault diagnosis of rotating machinery, they have limitations in terms of adaptability.For example, the time-frequency window size of STFT is fixed, and the best main resonance frequency band of DRT is difficult to determine accurately. 18WVD will cause cross-term interference during signal processing.Although the WT has a variable time-frequency window, the mother wavelet and transform scale must be selected, and the results are fixed-band signals. 6,192][23][24][25][26] The complicated multi-component signal will be decomposed into a sum of intrinsic mode functions (IMFs) by EMD, and each of the IMFs contains the local characteristics of the original signal. 58][29][30][31][32][33] On the basis of EMD, Smith proposed the local mean decomposition (LMD) to deal with non-stationary signals in a self-adaptive way in 2005. 34The complicated multi-component signal will be adaptively decomposed into a set of product functions (PFs) by LMD, and each of the PFs is the product of an amplitude envelope signal and a pure frequency-modulation signal.As a self-adaptive time-frequency signal analysis method, LMD has the capacity of time-frequency analysis and demodulation analysis for signal and is especially suitable for non-stationary signal processing.Compared with EMD, the end effect could be suppressed to a certain extent by LMD. 35Furthermore, more accurate instantaneous frequency could be obtained by LMD. 36The superiority of LMD over EMD has already been proved. 36,37Nowadays, LMD has attracted much attention due to its peculiarity, and has been widely used in the fault diagnosis of rotating machinery.Duan et al. 35 proposed a fault diagnosis method by combining LMD and the ratio correction method to process the short-time signals, which could be utilized in online real-time monitoring technology of rolling bearing failure.Wang et al. 36 compared LMD with EMD, and then applied them to the health diagnosis of two actual industrial rotating machines with rub-impact and steam-excited vibration faults, respectively.The results revealed that LMD is more suitable and performs better than EMD for the incipient fault detection.Song et al. 38 proposed a fault feature extraction method that combines adaptive uniform phase local mean decomposition (AUPLMD) and refined time-shift multiscale weighted permutation entropy (RTSMWPE) to recognize different categories and severities of reciprocating compressor valve faults.Yang and Zhou 39 utilized LMD and wavelet packet transform (WPT) to extract fault features of a diaphragm pump check valve.
As we know, the principle of WT-based de-noising method is similar to EMD-based de-noising method.Both methods are based on the basic assumption of an "energy sparsity distribution" of decomposed signal, which enables de-noising by screening signal decomposition components and processing their amplitudes using a threshold. 23However, there are obvious energy-concentrated pulses in the PFs of noisy signal extracted by LMD.This makes the principle of amplitude threshold filtering, which is commonly used in both WT-based de-noising and EMD-based de-noising, no longer applicable.Therefore, an improved signal de-noising algorithm based on multi-level local mean decomposition (ML-LMD) is introduced in this paper.This algorithm provides an effective way to achieve superior de-noising performance for rotating machinery compared to other de-noising methods.
This study is organized as follows: In Section 2, we review the main steps of the LMD and compare LMD-based signal de-noising to EMD-based de-noising; Section 3 introduces the flowchart and key steps of the proposed improved de-noising method based on ML-LMD, outlier detection and waveform smoothing (ML-LMD-OS); In Section 4, we discuss the simulation analysis and experimental verification results for the proposed method; Finally, the conclusions are summarized in Section 5.

LMD-BASED SIGNAL DE-NOISING AND ITS COMPARISON WITH EMD-BASED DE-NOISING
LMD is an essential process for gradually separating an FM signal from an AM signal, and it includes three basic steps: 1 smoothing the original signal; 2 subtracting the smoothed signal from the original signal; 3 amplitude demodulation processing based on envelope estimation. 10After processing with LMD, the original signal x(t) is decomposed into k PFs, denoted as P k (t), where k = 1, 2, … , K. Thus, the expression for signal reconstruction is as follows: where, u k (t) is the residual component.
To obtain continuous, smooth local mean and envelope functions during the LMD decomposition process, a moving average smoothing process is performed on the local mean and amplitude using time-shift weighting of the continuous extreme value.In EMD processing, cubic spline interpolation is directly applied to the extreme points to obtain the upper and lower envelopes of the signal, and then the mean value is obtained to achieve signal decomposition.This is an important difference between LMD and EMD.Previous research results have shown that LMD can effectively overcome the end effect and mode mixing of EMD.Additionally, its decomposition process is more in line with the natural characteristics of the signal, which enables obtaining more detailed features of the time-frequency distribution of the signal.Consequently, the signal can be interpreted and described in a more physical sense. 34,36urther research has shown that, after EMD processing, the energy of the original noisy observation signal x(t) is only concentrated in a few high-order intrinsic mode function (IMF) components, while the other low-order IMF components are mainly noise.In particular, the first-order IMF component is almost entirely composed of "pure" noise components.In fact, the noise energy contained in the IMF component decreases with the increase of the EMD decomposition order according to the logarithm law.The following estimation formula for the amplitude filtering threshold of the IMF component can be derived 23 : where, C is a constant, usually with a typical value of C = 0.7.Ê i is the noise energy estimate of the ith order IMF component.E 1 is the energy of the first-order "pure" noise IMF component, and the median(⋅) is the median estimation function.
Li et al. 40 suggest that wavelet threshold filtering should be applied to the high-frequency PFs first to reduce noise, in order to achieve noise removal.Then, the source signal can be reconstructed together with the low-frequency PFs based on removing the trend term, under the assumption that the noise is mainly concentrated on the high-frequency PFs extracted by LMD.However, this paper does not provide a specific algorithm implementation.Moreover, serious consequences may occur if the threshold filtering technology is directly applied to the PFs following the existing WT-based or EMD-based de-noising methods in LMD application.The following simulation example illustrates this problem.
The LMD was applied to analyze a "heavy sine" source signal s and its virtual noisy observation signal x, where "Pure" Gaussian white noise u was added to the simulation signal s, and the signal-to-noise ratio (SNR) of x was 5 dB.The data points used in the analysis were N = 2048.The decomposition result of LMD is shown in Figure 1.The PFs of s, u, and x are as shown in Figure 1A-C It can be observed from Figure 1D-G that the noise component u has a dominant pulse component with an obvious energy concentration in its high-order PFs, which is different from the approximately even distribution of energy in each "pure" noise IMF component obtained by EMD (the EMD decomposition result of noise component u is not given in Figure 1 due to the space limitation).This is because of the difference in the signal decomposition algorithm used by LMD and EMD.The energy estimation results of four "pure" noise PFs are [915, 6.150 8 × 105, 3.006 1 × 109, 3.009 2 × 109], which are significantly different from the logarithmic attenuation law based on the EMD study.Moreover, it is noticed that the first-order PF component PF1-x of the noisy observation x shown in Figure 1H is mainly composed of noise components, which is similar to the decomposition result of EMD, but with significant energy concentration pulses.The pulse energy concentration effect becomes more apparent in the higher-order PFs PF2-x, PF3-x and PF4-x, as shown in Figure 1I-K.The energies of PFs of s and x are estimated, and the results are [2548, 336, 24] and [1135, 9.5131 × 108, 2.9385 × 1012, 2.9372 × 1012], which reveal that it is almost impossible to obtain any meaningful change law for determining the amplitude filtering threshold.
Taking into account the analysis above, in the context of LMD-based de-noising, the amplitude threshold filtering principle that current wavelet transform (WT) or empirical mode decomposition (EMD) based de-noising relies on is no longer applicable.Therefore, it is necessary to re-examine new de-noising principles based on the fundamental principles of the LMD algorithm and the different characteristics of PF component signals, and subsequently propose new de-noising methods.According to the above definition, several pairs of dual pulses are found in Figure 1, which are represented by " 1 + and 1 −", " 2 + and 2 −", and " 3 + and 3 −", as shown in Figure 1E-G,J-K.Through a large number of simulation experiments, it is found that the high-order PFs of the noisy observation signal has a significant dual pulse characteristic due to the LMD decomposition characteristic of noise, which lays a theoretical foundation for the study of a new de-noising method based on LMD.

AN IMPROVED DE-NOISING METHOD BASED ON MULTI-LEVEL LOCAL MEAN DECOMPOSITION
Based on the above analysis, an improved method for signal de-noising based on ML-LMD, outlier detection and waveform smoothing (ML-LMD-OS) is proposed, as shown in Figure 2, which mainly consists of three parts.
1. Multi-level LMD (ML-LMD) decomposition of the noisy signal.According to formula (1), x(t) is decomposed into K PFs P k (i)(t) and a residual component u K (i) (t) that describes the trend of x(t) after the ith level LMD, where k = 1, 2, … , K. 2. Superposition and restructuring (SR) of higher-order PFs.The second and higher-order PFs (including residual components) P 2 (i) , … , P K (i) , u K (i) are linearly superposed to form a restructured signal P (i) .
where, i is the decomposition level of ML-LMD.3. Outlier detection and waveform smoothing (OD-WS, OS).After the high-order PFs undergo SR processing, a considerable part of the dual-pulse components are eliminated, but there are still some pulses residues.The data at the PK (i) uK (i) P1 (i) locations of these residual pulses can be considered as outliers that deviate significantly from the center of the overall data distribution.The improved Thompson statistical test method is used to perform outlier detection (OD). 41First, the mean and variance of the signal P (i) are estimated as follows: where, M 0 is the median of P (i) = {P (i) j , j = 1, 2, … , n}.E bi and S bi are the double-weighted estimates of the mean and variance of P (i) , respectively.
The double-weight estimation is actually a weighted average technique, where the weight of sample point P (i) j decrease as it deviates further away from the distribution center of the data P (i) .The weight factor u j can be calculated by: where, M 1 is the median of absolute deviation, which is the median of the absolute deviation of the data sample point P j (i) relative to the signal median M 0 .The parameter c controls the deviation distance of each data point relative to the data distribution center.The value is usually in the range of 6 < c < 9, and a relatively robust choice is c = 7.5.For the case where |u j | > 1.0, u j is set to 0 uniformly.The statistics E bi and S bi have strong resistance to outliers, so their estimated values can be used for outlier detection using testing techniques.Assuming that P k (i) is identified as an outlier, it can be replaced by the signal median of P (i) to achieve outlier removal.
After outlier removal, the residual pulse components in the signal are greatly reduced, but there are some minor pulse residues in local positions.In order to eliminate their adverse effects on signal de-noising, further waveform smoothing (WS) processing is required.There are many algorithms available for data smoothing, among which five-point cubic smoothing method utilizes polynomial least-squares approximation to achieve smooth filtering of sampling points, and the algorithm is simple and effective. 42or the signal P (1) OD , let t j = j, t = {t j , j = 1, 2, … , n}.Based on the m-degree polynomial data fitting, the five-point cubic smoothing method solves the polynomial coefficients a 0 , a 1 and a m through the least-squares criterion below to achieve WS processing of the signal P (1) OD . min After WS processing, the local small pulse residuals in the signal P (1) OD are further reduced, and finally a smoother signal waveform P (1) OD-WS can be obtained.So far, the first-level de-noising based on ML-LMD waveform smoothing is completed for the noisy observation x.If a higher-level de-noising processing is required, steps (1)-( 3) can be repeated.However, the de-noising level cannot be too high, generally not exceeding three levels, as a higher level will degrade the de-noising performance of the algorithm.In addition, in the OD-WS processing step, it is crucial to select the appropriate values for parameters such as the distance control parameter c, the polynomial order m, and the number of smoothing loops.The main goal is effectively remove outliers from the signal processing by superimposed recombination (SR) and to ensure that the resulting waveform is sufficiently smooth.Additionally, the processed signal should contain enough extreme points to facilitate the smooth execution of the next level decomposition of ML-LMD.

Simulation analysis
In the following subsection, the proposed method is employed to analyze three simulation signals to verify that it is able to effectively eliminate the Gaussian white noise.Three types of source simulation signals of standard sine wave s 1 (t), amplitude modulation-frequency modulation (AM-FM) s 2 (t) and amplitude modulation-phase modulation (AM-PM) s 3 (t), are simulated respectively.
where, f 1 = 100 Hz, f 2 = 20 Hz, f 3 = 60 Hz.Sampling frequency f s is 2048 Hz.By adding different degrees of Gaussian white noise, the virtual observation signal x i (t), i = 1, 2, 3 with different observation SNR (SNR1) can be obtained, and the observation SNR (SNR1) is 1 dB.Numerous de-noising methods have been recently introduced in various fields.For example, Iqbal 43 proposed a novel noise reduction framework that employs an intelligent deep convolutional neural network to enhance the signal-to-noise ratio (SNR) of registered seismic signals; Meanwhile, Li et al. 44 introduced a pre-segmentation method for magnetic resonance spectroscopy (MRS) signals, which can reliably extract MRS signals with a signal-to-noise ratio (SNR) of −30 dB, providing technical support for the MRS method to function effectively despite high electromagnetic noise.However, to compare the efficacy of different de-noising methods, the improved ML-LMD-based de-noising method (ML-LMD-OS), as proposed in this study, as well as other methods, including the LMD-based de-noising method (LMD-H and LMD-S), the WT-based translation-invariant threshold denoising method (WT-H and WT-S), 13 and the improved EMD-based de-noising method (EMD-H and EMD-S) 23 are applied on signal s 1 (t), s 2 (t) and s 3 (t), respectively.The results are shown in Figures 3-5.
The suffixes "-H" and "-S" in Figures 3-5 indicate the use of hard threshold and soft threshold, respectively, as explained in Reference 23.SNR 1 and SNR 2 represent the SNR of the noisy observation signal and the de-noising signal, respectively.The number of decomposition levels in the ML-LMD method is set to 3, the critical parameter of outlier detection is set to  = [ 1 ,  2 ,  3 ] = [0.05,0.05, 0.05], and the number of waveform smoothing cycles is set to 35, 2].The decomposition number of EMD in the improved de-noising method based on EMD is fixed at 8. The parameters of threshold calculation and the source signal reconstruction [M 1 , IM 2 ] in the de-noising method based on LMD and improved de-noising method based on EMD are set to [3, 2] and [1, 0] respectively.And the amplitude filter constant C is set to 0.7.The median filtering technique is used to estimate the noise standard deviation in the de-noising method based on WT.Furthermore, the de-noising method based on LMD in this study refers to the filtering threshold method described in Reference 23, which performs threshold filtering directly on the amplitude of the PFs.The comparison results presented in Figures 3-5 indicate that the method based on WT, EMD, LMD and the proposed method (ML-LMD-OS) all achieve relatively stable de-noising results.For the same kind of de-noising method, the hard thresholding (-H) generally outperforms soft thresholding (-S).Compared with the method based on WT and the method based on EMD, the overall performance is similar, but the de-noising method based on WT shows better performance for low SNR (SNR 1 < −7 dB).Additionally, the de-noising method based on EMD shows a significant performance degradation trend starting from SNR 1 > 3 dB.According to the overall performance, the de-noising performance of the proposed method (ML-LMD-OS) is obviously better than that of other methods.Especially in the middle of the observed SNR (−7 dB < SNR 1 < 2 dB), the proposed method performs best.For the AM-PM signal, the proposed method achieves the best de-noising results in the whole middle and high SNR (SNR 1 > −7 dB).

Experimental verification
The rolling bearing data disclosed by the Case Western Reserve University Bearing Data Center are utilized to verify the effectiveness of the proposed method. 45The vibration data of the rolling bearing is collected from the rolling bearing fault simulation experimental device shown in Figure 6.The left end of the experimental device is a 1.47 kW motor, the middle part is a set of torque transducer/encoder, and the right end of the experimental device is a dynamometer.The motor shaft is supported by the test rolling bearings.Single point faults were simulated on the test rolling bearings separately at the inner-race, outer-race and rolling element by electro-discharge machining (EDM).The test bearing is a 6205-type rolling bearing produced by SKF Co.The geometric parameters of the test rolling bearing are listed Table 1.
The vibration data was collected by piezoelectric accelerometers, which were attached to different positions of the experimental device with magnetic bases, including drive shaft end and fan end of the motor housing.The vibration signals were collected by a 16-channel data acquisition instrument, and then sent to Matlab for post-processing.All data files are in *.mat format.The motor speed is 1797 rpm.The sampling frequency is 12 kHz.The fault data at the fan end with inner-race fault is selected in this study.

F I G U R E 6
The rolling bearing fault simulation experimental device.The vibration signal of the fan end bearing with an inner raceway fault shown in Figure 7, where the number is 2048.It is observed that there is a considerable amount of noise interference in the bearing signal, which may be due to the electric noise generated within the data acquisition system, external electromagnetic interference, vibration crosstalk from adjacent operating equipment in the laboratory, power frequency interference, and other factors.Failure to reduce this noise would have a detrimental effect on the rolling bearing fault diagnosis.

TA B L E 1
The proposed method (ML-LMD-OS) is used to process the bearing signal.At the same time, several other methods such as the LMD-based de-noising method (MD-H and LMD-S), the WT-based translation-invariant threshold de-noising method (WT-H and WT-S) and the improved EMD-based de-noising method (EMD-H and EMD-S) are also used to analyze the signal.The parameter settings of each method are the same as those in the simulation analysis in Section 4.1, and the results are shown in Figure 8.
Figure 8 illustrates that the proposed method (ML-LMD-OS) has a good de-noising effect for the noisy vibration signal of the fan end fault bearing.In comparison, the LMD-based de-noising method (LMD-H and LMD-S), the improved EMD-based de-noising method (EMD-H and EMD-S), and the WT-based translation-invariant threshold de-noising method (WT-H and WT-S) performed poorly, showing obvious over de-noising, and the loss of useful information in the signal is serious.In particular, the translation-invariant threshold de-noising method based on WT fails to process, and the useful information in the original noisy signal is eliminated together with the noise interference, which has a serious negative impact on further analysis and extraction of ball bearing fault features.In order to show the details of de-noising, frequency spectrum of the reconstructed signals with different de-noising methods were calculated by FFT, as shown in Figure 9.The main frequency components in frequency spectrum of the original included 129.2, 493.2, 1362, 1867, 3417, 3804, 4791, and 5425 Hz. Figure 9B,C,F,G, it can be that the de-noising method on LMD (LMD-H and LMD-S) and the translation-invariant threshold de-noising method based on WT (WT-H and WT-S) did little to remove high-frequency noise.As can be seen in Figure 9D,E, the used frequency component 1362 Hz was distorted or lost.From Figure 9H, it can be seen that the method maintained the authenticity and high reducibility of the signal while removing noise.In order to compare the performance of different noise reduction methods more clearly, the correlation coefficient between noise reduction data and Gaussian white noise reference signal was calculated, as shown in Figure 10.It can be seen from Figure 10 that the correlation coefficient of method ML-LMD-OS was smaller than other methods, which means better effect of de-noising.

F I G U R E 10
Correlation coefficient between the de-noised data and the Gaussian white noise reference signal.

F
I G U R E 1 PFs of the "heavy sine" signal s, Gaussian white noise u and virtual noisy observation signal x.Analyze the LMD decomposition results of the noise component u and the noisy observation signal x given in Figure1, especially the waveform characteristics of high-order PFs, and define as follows:If a pair of pulses existing in two signals has the same position and opposite directions and can partially or almost completely cancel each other out by superposition, the pair of pulses is called a dual pulse, and the two signals have dual pulse characteristics.

3 4 F I G U R E 5
De-noising results of different methods on the standard sine wave signal.De-noising results of different methods on the AM-FM signal.De-noising results of different methods on the AM-PM signal.

7 2 )F
Vibration signal of the fan end bearing with an inner raceway fault.I G U R E 8 De-noised waveform of faulty bearing at the fan end.F I G U R E 9 De-noised waveform and FFT of faulty bearing at the fan end.
The geometric parameters of the rolling bearing.