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Received: 5 January 2023 Revised: 5 April 2023 Accepted: 4 May 2023
DOI: 10.1002/eng2.12677
RESEARCH ARTICLE
An improved method for signal de-noising based on
multi-level local mean decomposition
Chao Tang1Heng Chen1Yonghua Jiang1,2 Weidong Jiao2Jianfeng Sun2
Cui Xu2Chen Wang2Haicheng Xia2
1Xingzhi College, Zhejiang Normal
University, Lanxi, China
2Key Laboratory of Intelligent Operation
and Maintenance Technology and
Equipment for Urban Rail Transit of
Zhejiang Province, Zhejiang Normal
University, Jinhua, China
Correspondence
Yonghua Jiang and Weidong Jiao, Key
Laboratory of Intelligent Operation and
Maintenance Technology and Equipment
for Urban Rail Transit of Zhejiang
Province, Zhejiang Normal University,
Jinhua 321004, China.
Email: yonghua_j82@zjnu.cn and
jiaowd1970@zjnu.cn
Funding information
National College Students’ Innovation
and Entrepreneurship Training Program,
Grant/Award Number: 202113276005;
National Natural Science Foundation of
China, Grant/Award Numbers: 51405449,
51575497; Natural Science Foundation of
Zhejiang Province, Grant/Award Number:
LZ22E050001
Abstract
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 meth-
ods is no longer applicable.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 elimi-
nating the pulse components. The proposed method’s superior noise reduction
performanceisdemonstratedthroughtheoreticalanalysisandexperimentalver-
ification. Compared to well-known methods like WT-based and EMD-based
de-noising, the results show that the proposed method has significant compara-
tive advantages in reducing noise in rolling bearing signals.
KEYWORDS
dual-pulse characteristic, empirical mode decomposition, multi-level local mean decomposition,
outlier detection and waveform smoothing, signal de-noising, the superposition and recombination
1INTRODUCTION
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 break-
down, and even casualties.1Therefore, it is essential to have useful and effective fault diagnosis techniques to ensure the
performance and reliability of rotating machinery.2-4
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 back-
ground 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
This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the
original work is properly cited.
© 2023 The Authors. Engineering Reports published by John Wiley & Sons Ltd.
Engineering Reports. 2023;5:e12677. wileyonlinelibrary.com/journal/eng2 1of14
https://doi.org/10.1002/eng2.12677
2of14 TANG .
to be non-stationary, which means their parameters are time-varying.6In order to obtain accurate state features, prepro-
cessing of vibration signals is essential.5Therefore, several traditional methods have been developed for this purpose,
such as the short-time Fourier transform (STFT), demodulated resonance technique (DRT), Wigner-Ville distribution
(WVD), and wavelet transform (WT).7-17 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.18 WVD 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.6,19
Taking these factors into consideration, in 1998, Huang et al.20 proposed the empirical mode decomposition
(EMD), as a self-adaptive signal processing method, which has been widely used in various fields.21-26 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.5However, there are some defects in EMD during the pro-
cess of decomposition, such as end effect, mode mixing, and so forth, which seriously affect its de-noising performance
and restrict its application.4,27-33 On the basis of EMD, Smith proposed the local mean decomposition (LMD) to deal
with non-stationary signals in a self-adaptive way in 2005.34 The 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.35 Furthermore,
more accurate instantaneous frequency could be obtained by LMD.36 The superiority of LMD over EMD has already been
proved.36,37 Nowadays, 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 correc-
tionmethod to process the short-time signals, which couldbe 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 Zhou39 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.23 However, 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 appli-
cable. 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 sig-
nal de-noising to EMD-based de-noising; Section 3introduces 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,wedis-
cuss the simulation analysis and experimental verification results for the proposed method; Finally, the conclusions are
summarized in Section 5.
2LMD-BASED SIGNAL DE-NOISING AND ITS COMPARISONWITH
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.10 After processing with LMD, the original signal x(t) is decomposed into kPFs,
denoted as Pk(t), where k=1, 2, …,K. Thus, the expression for signal reconstruction is as follows:
TANG . 3of14
x(t)= K
k=1
Pk(t)+uk(t),(1)
where, uk(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 impor-
tant 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.34,36
Further 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 com-
ponents 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 decom-
position order according to the logarithm law. The following estimation formula for the amplitude filtering threshold of
the IMF component can be derived23:
Ti=CEi2lnN
Ei=E1
𝛽𝜌−i,i=2,3,4,···,
E1=median(I1)
0.6745 2
,(2)
where, Cis a constant, usually with a typical value of C=0.7. Êiis the noise energy estimate of the ith order IMF com-
ponent. E1is 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
ordertoachieve noise removal. Then, the source signal can be reconstructed together with the low-frequencyPFsbasedon
removingthe trend term, under the assumption that the noiseis 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 sand its virtual noisy observation signal x, where “Pure”
Gaussian white noise uwas added to the simulation signal s, and the signal-to-noise ratio (SNR) of xwas 5dB. The data
points used in the analysis were N=2048. The decomposition result of LMD is shown in Figure 1.ThePFsofs,u,andx
areasshowninFigure1A–C,Figure1D–G,andFigure1H–K, respectively.
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 uis not given
in Figure 1due 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-xof the noisy observation xshowninFigure1H 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-xand PF4-x,
as shown in Figure 1I–K. The energies of PFs of sand xare 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
oftheLMDalgorithmandthedifferent characteristics of PF component signals, and subsequentlyproposenewde-noising
methods.
4of14 TANG .
FIGURE 1 PFs of the “heavy sine” signal s, Gaussian white noise uand virtual noisy observation signal x.
Analyzethe LMD decomposition results of the noise componentuand the noisy observation signal xgiven in Figure 1,
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.
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
−”,asshowninFigure1E–G,J–K. Through a large number of simulation exper-
iments, 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.
3AN IMPROVED DE-NOISING METHOD BASED ON MULTI-LEVEL
LOCAL MEAN DECOMPOSITION
Basedontheabove analysis, an improved method forsignalde-noisingbasedonML-LMD, outlierdetectionandwaveform
smoothing (ML-LMD-OS) is proposed, as shown in Figure 2, which mainly consists of three parts.
1. Multi-levelLMD(ML-LMD)decompositionofthenoisysignal.According to formula (1), x(t)isdecomposedintoKPFs
Pk(i)(t) and a residual component uK(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) P2(i),…,PK(i),uK(i)are linearly superposed to form a restructured signal P(i).
P(i)=K
k=2
P(i)
k+u(i)
K,(3)
where, iis the decomposition level of ML-LMD.
3. Outlier detection and waveform smoothing (OD-WS, OS). After the high-order PFs undergo SR processing, a con-
siderable part of the dual-pulse components are eliminated, but there are still some pulses residues. The data at the
TANG . 5of14
x(t)ML-LMD
SR
ŝ(t)
P2(i)
PK(i)
uK(i)
P1(i)
P(i)
OD-WS
min{ϕ()}
FIGURE 2 Algorithm flow of the proposed method (ML-LMD-OS).
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).41 First,
the mean and variance of the signal P(i)are estimated as follows:
Ebi =M0+n
j=1P(i)
j−M0×1−u2
j2∕n
j=11−u2
j2,(4)
Sbi =
nn
j=1P(i)
j−M02×1−u2
j4∕
n
j=11−u2
j×1−5u2
j,(5)
where, M0is the median of P(i)={P(i)j,j=1, 2, …,n}. Ebi and Sbi are the double-weighted estimates of the mean and
variance of P(i), respectively.
Thedouble-weightestimation is actually a weighted average technique,wheretheweightofsamplepointP(i)jdecrease
as it deviates further away from the distribution center of the data P(i). The weight factor ujcan be calculated by:
uj=P(i)
j−M0∕(cM1),(6)
where, M1is the median of absolute deviation, which is the median of the absolute deviation of the data sample point
Pj(i)relative to the signal median M0. 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 uj>1.0, ujis set to 0 uniformly.
The statistics Ebi and Sbi have strong resistance to outliers, so their estimated values can be used for outlier detection
using testing techniques. Assuming that Pk(i)is identified as an outlier, it can be replaced by the signal median of P(i)to
achieve outlier removal.
P(i)
k=M0=P(i)
n+1∕2
P(i)
n+2+P(i)
n∕2+1
n=1,3,5,···
n=2,4,6,···
.(7)
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.42
For the signal P(1)OD,lettj=j,t={tj,j=1, 2, …,n}. Based on the m-degree polynomial data fitting, the five-point cubic
smoothing method solves the polynomial coefficients a0,a1and amthrough the least-squares criterion below to achieve
WS processing of the signal P(1)OD.
6of14 TANG .
min𝜙(a0,a1,···,am)=min n
j=−n
R2
j=n
j=−nm
i=0
aitj−P(1)
OD j.(8)
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.
4SIMULATION ANALYSIS AND EXPERIMENTAL VERIFICATION
4.1 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 s1(t),
amplitude modulation–frequency modulation (AM–FM) s2(t) and amplitude modulation–phase modulation (AM–PM)
s3(t), are simulated respectively.
s1(t)=sin2𝜋f1t,(9)
s2(t)=sin2𝜋f1tsin2𝜋f2+sin2𝜋f3tt,(10)
s3(t)=sin2𝜋f1tsin2𝜋f2t+sin2𝜋f3t,(11)
where, f1=100Hz, f2=20Hz, f3=60 Hz. Sampling frequency fsis 2048 Hz.
By adding different degrees of Gaussian white noise, the virtual observation signal xi(t), i=1, 2, 3 with different
observation SNR (SNR1) can be obtained, and the observation SNR (SNR1) is 1dB.
Numerous de-noising methods have been recently introduced in various fields. For example, Iqbal43 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 res-
onance 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 s1(t), s2(t)ands3(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
1and SNR2represent 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 𝜶=[𝛼1,𝛼2,𝛼3]=[0.05, 0.05, 0.05], and the number of waveform smoothing cycles is set to N=[N1,N2,
N3]=[10, 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 [M1,IM2] 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 Cis 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.
TANG . 7of14
46-14 -12 -10 -8 -6 -4 -2 0 2
-8
-6
-4
-2
0
2
4
6
8
10
12
LMD-H
LMD-S
ML-LMD-OS
EMD-H
EMD-S
WT-H
WT-S
RNS
2
Bd/
SNR
1
/dB
FIGURE 3 De-noising results of different methods on the standard sine wave signal.
4-14 -12 -1 0 -8 -6 -4 -2 0 2 6
-8
-6
-4
-2
0
4
6
LMD-H
LMD-S
ML-LMD-OS
EMD-H
EMD-S
WT-H
WT-S
4
RNS
2
Bd/
SNR
1
/dB
FIGURE 4 De-noising results of different methods on the AM–FM signal.
64
LMD-H
LMD-S
ML-LMD -OS
EMD-H
EMD-S
WT-H
WT-S
-14 -12 -1 0 -8 -6 -4 -2 0 2
-8
-6
-4
-2
0
2
4
6
8
10
RNS
2
Bd/
SNR
1
/dB
FIGURE 5 De-noising results of different methods on the AM–PM signal.
8of14 TANG .
The comparison results presented in Figures 3–5 indicate that the method based on WT, EMD, LMD and the pro-
posed 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 perfor-
mance for low SNR (SNR1<−7 dB). Additionally, the de-noising method based on EMD shows a significant performance
degradation trend starting from SNR1>3dB. 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 (−7dB<SNR1<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 (SNR1>−7dB).
4.2 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.45 The 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 in 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 1797rpm. The sampling frequency is 12 kHz. The fault data at the fan end
with inner-race fault is selected in this study.
FIGURE 6 The rolling bearing fault simulation experimental device.
TABLE 1 The geometric parameters of the rolling bearing.
Parameter Value
Bearing specs 6205
Body diameter 7.94mm
Pitch diameter 39.04 mm
Outer diameter 52mm
Inner diameter 25 mm
Roller number 9
Contact angle 0◦
TANG . 9of14
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.
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 ana-
lyze 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 8illustrates that the proposed method (ML-LMD-OS) has a good de-noising effect for the noisy vibration sig-
nal 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.
0 200 400 600 800 1000 1200 1400 1600 1800 2000
-1
-0.5
0
0.5
1
Data /N
Amplitude(m/s2)
FIGURE 7 Vibration signal of the fan end bearing with an inner raceway fault.
0 200 400 600 800 1000 1200 1400 1600 1800 2000
-10
0
10
LMD-H
0 200 400 600 800 1000 1200 1400 1600 1800 2000
-20
0
20
LMD-S
0 200 400 600 800 1000 1200 1400 1600 1800 2000
-5
0
5
ML-LMD-OS
0 200 400 600 800 1000 1200 1400 1600 1800 2000
-5
0
5
EMD-H
0 200 400 600 800 1000 1200 1400 1600 1800 2000
-5
0
5
EMD-S
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0
0.5
1
WT-H
0 200 400 600 800 1000 1200 1400 1600 1800 2000
0
0.5
1
WT-S
Data /N
Amplitude(m/s2)
FIGURE 8 De-noised waveform of faulty bearing at the fan end.
10 of 14 TANG .
FIGURE 9 De-noised waveform and FFT of faulty bearing at the fan end.
TANG . 11 of 14
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 signals included 129.2, 493.2, 1362, 1867, 3417, 3804, 4791, and 5425 Hz. From Figure 9B,C,F,G, it can be seen that
the de-noising method based 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,theused
frequency component 1362Hz 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.
FIGURE 10 Correlation coefficient between the de-noised data and the Gaussian white noise reference signal.
12 of 14 TANG .
5CONCLUSIONS
An improved signal de-noising method, ML-LMD-OS, 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) in this study. Simulation analysis and experimental verification results show that the proposed method has
excellent de-noising performance.
It is worth noting that the parameters of each de-noising method are intentionally set to be the same in the simulation
analysis and experimental verification, in order to examine the robustness of performance of each method to the parame-
ter setting in the practical de-noising application. It is found that the high-order PFs of the noisy observation signal has a
significant dual pulse characteristic by the simulation analysis. In experimental verification, it is seen that the de-noising
method of ML-LMD-OS shows good robustness and stability of de-noising effect, while for other methods compared in
this study, whether the de-noising method based EMD, the de-noising method based on WT or the de-noising method
based LMD, in this case, the de-noising results are not satisfactory.
These de-noising methods are essentially amplitude threshold filtering algorithms, and their de-noising performance
depends heavily on the reasonable setting of threshold parameters. It can be predicted that if the threshold parameters of
these methods are reasonably adjusted in further experimental research, the de-noising effect will be improved. However,
this also reveals their limitations in practical applications from the side. In the actual de-noising application, especially in
the fault diagnosis environment, the cut and try method of specifying the parameter cannot be used to improve the per-
formance of signal de-noising because the useful signal (or source signal) is usually unknown in this case. The de-noising
principle of the ML-LMD-OS method is based on the dual pulse characteristics of the PFs extracted by LMD, which
does not depend on the artificial setting of the amplitude filtering threshold. Therefore, the proposed method has strong
adaptability in practical application, which also indicates that it has great application potential.
AUTHOR CONTRIBUTIONS
Chao Tang: Software (lead). Heng Chen: Software (supporting). Yonghua Jiang: Methodology (equal);
writing – original draft (equal). Weidong Jiao: Methodology (equal); writing – original draft (equal). Jianfeng Sun:
Writing – original draft (equal). Cui Xu: Writing – review and editing (equal). Chen Wang: Writing – review and editing
(equal). Haicheng Xia: Writing – review and editing (equal).
ACKNOWLEDGMENTS
This research was funded by the Zhejiang Provincial Natural Science Foundation of China (Grant LZ22E050001),
the National Natural Science Foundation of China (Grant 51405449, Grant 51575497), the National College Students’
Innovation and Entrepreneurship Training Program (Grant 202113276005).
CONFLICT OF INTEREST STATEMENT
The authors declare no conflict of interest.
DATA AVAILABILITY STATEMENT
The data that support the findings of this study are available from Case Western Reserve University Bearing Data Center.
ORCID
Yonghua Jiang https://orcid.org/0000-0003-0890-615X
REFERENCES
1. Liu ZW, He ZJ, Guo W, Tang ZC. A hybrid fault diagnosis method based on second generation wavelet de-noising and local mean
decomposition for rotating machinery. ISA Trans. 2016;61:211-220.
2. Xian GM, Zeng BQ. An intelligent fault diagnosis method based on wavelet packer analysis and hybrid support vector machines. Exp Syst
Appl. 2009;36:12131-12136.
3. Wang HQ, Chen P. Intelligent diagnosis method for rolling element bearing faults using possibility theory and neural network. Comp Ind
Eng. 2011;60:511-518.
4. Jiang YH, Tang C, Zhang XD, Jiao WD, Li G, Huang TT. A novel rolling bearing defect detection method based on bispectrum analysis
and cloud model-improved EEMD. IEEE Access. 2020;8:24323-24333.
TANG . 13 of 14
5. Si L, Wang ZB, Tan C, Liu XH. Vibration-based signal analysis for shearer cutting status recognition based on local mean decomposition
and fuzzy C-means clustering. Appl Sci. 2017;7:1-14.
6. Jiang YH, Tang BP, Qin Y, Liu WY. Feature extraction method of wind turbine based on adaptive Morlet wavelet and SVD. Renew Energy.
2011;36:2146-2153.
7. Lee JH, Kim J, Kim HJ. Development of enhanced Wigner-Ville distribution function. Mech Syst Signal Process. 2001;13:367-398.
8. Xin Y, Li SM. Novel data-driven short-frequency mutual information entropy threshold filtering and its application to bearing fault
diagnosis. Measur Sci Technol. 2019;30:1-13.
9. Lin J, Qu LS. Feature extraction based on Morlet wavelet and its application for mechanical fault diagnosis. J Sound Vib. 2000;234:135-148.
10. Fu YB, Chui CK, Teo CL. Accurate two-dimensional cardiac strain calculation using adaptive windowed Fourier transform and Gabor
wavelet transform. Int J Comput Assist Radiol Surg. 2013;8:135-144.
11. Tang BP, Jiang YH, Yao JB. Fault diagnosis based on reassigned Wigner-Ville distribution spectrogram and SVD. J Vib Measur Diag.
2012;32:301-305.
12. Sanz J, Perera R, Huerta C. Fault diagnosis of rotating machinery based on auto-associative neural networks and wavelet transforms.
J Sound Vib. 2007;302:981-999.
13. Silva RD, Minetto R, Schwartz WR. Adaptive edge-preserving image denoising using wavelet transforms. Pattern Anal Appl.
2013;16:567-580.
14. Chen RX, Huang X, Yang LX. Intelligent fault diagnosis method of planetary gearboxes based on convolution neural network and discrete
wavelet transform. Comp Ind. 2019;106:48-59.
15. Wang F, Ji Z, Peng C. Research on ECG signal denoising based on dual-tree complex wavelet transform. Chin J Sci Instrum.
2013;34:1161-1166.
16. Tang BP, Liu WY, Song T. Wind turbine fault diagnosis based on Morlet wavelet transformation and Wigner-Ville distribution. Renew
Energy. 2010;35:2862-2866.
17. Ren H, Liu WY, Jiang YH, Su XP. A novel wind turbine weak feature extraction method based on cross genetic algorithm optimal MHW.
Measurement. 2017;109:242-246.
18. Cohen L. Time-frequency distributions-a review. Proc IEEE. 1989;77:941-981.
19. Mallat SG. A theory for multiresolution signal decomposition: The wavelet representation. IEEE Trans Pattern Anal Mach Intell.
1989;11:674-693.
20. Huang NE, Shen Z, Long SR, et al. The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time
series analysis. Proc: Math Phys Eng Sci. 1998;454:903-995.
21. Fang L, Sun HC. Study on EEMD-based KICA and its application in fault-feature extraction of rotating machinery. Appl Sci. 2018;8:1-17.
22. Mao Y, Que P. Noise suppression and flaw detection of ultrasonic signals via empirical mode decomposition. Russ J Nondestruct Test.
2007;43:196-203.
23. Kopsinis Y, Mclaughlin S. Development of EMD-based denoising methods inspired by wavelet thresholding. IEEE Trans Signal Process.
2009;57:1351-1362.
24. Kabir MA, Shahnaz C. Denoising of ECG signals based on noise reduction algorithms in EMD and wavelet domains. BiomedSignal Process
Control. 2012;7:481-489.
25. Li Y, Peng JL, Ma HT, Lin HB. Study of the influence of transition IMF on EMD do-noising and the improved algorithm. Chin J Geophys.
2013;56:626-634.
26. Luo YK, Luo ST, Luo FL, Pan MC. Realization and improvement of laser ultrasonic signal denoising based on empirical mode
decomposition. Opt Precis Eng. 2013;21:479-486.
27. Jiang F, Zhu ZC, Li W, Ren Y, Zhou GB, Chang YG. A fusion feature extraction method using EEMD and correlation coefficient analysis
for bearing fault diagnosis. Appl Sci. 2018;8:1-18.
28. Jiao WD, Lin S. Improved empirical mode decomposition based signal de-noising approach using likelihood estimation of residual noise.
Chin J Sci Instrum. 2014;35:2808-2816.
29. Cheng G, Chen XH, Li HY, Li P, Liu HG. Study on planetary gear fault diagnosis based on entropy feature fusion of ensemble empirical
mode decomposition. Measurement. 2016;91:140-154.
30. Ren Y, Suganthan PN, Srikanth N. A novel empirical mode decomposition with support vector regression for wind speed forecasting. IEEE
Trans Neural Networks Learn Syst. 2016;27:1793-1798.
31. Yang YL. Empirical mode decomposition as a time-varying multirate signal processing system. Mech Syst Signal Process.
2016;76-77:759-770.
32. Tang BP, Jiang YH, Zhang XC. Feature extraction method of rolling bearing fault based on singular value decomposition-morphology filter
and empirical mode decomposition. JMechEng. 2010;46:37-42.
33. Jiang YH, Jiao WD, Li RQ, Tang C, Zheng JJ, Cai JC. A study on the method for eliminating mode mixing in B-spline empirical mode
decomposition based on adaptive bandwidth constrained signal. J Vib Shock. 2018;37:83-90.
34. Smith JS. The local mean decomposition and its application to EEG perception data. JRoySocInterface. 2005;2:443-454.
35. DuanYQ, WangCD, Chen Y, LiuPS. Improving the accuracyof fault frequency by meansof local mean decomposition and ratiocorrection
method for rolling bearing failure. Appl Sci. 2019;9:1-15.
36. Wang YX, He ZJ, Zi YY. A comparative study on the local mean decomposition and empirical mode decomposition and their applications
to rotating machinery health diagnosis. J Vibrat Acoust. 2010;132:1-10.
14 of 14 TANG .
37. Cheng JS, Zhang K, Yang Y, Yu DJ. Comparison between the methods of local mean decomposition and empirical mode decomposition.
J Vibrat Shock. 2009;28:13-16.
38. Song MP, Wang JD, Zhao HY, Wang XL. Fault diagnosis method based on AUPLMD and RTSMWPE for a reciprocating compressor valve.
Etropy. 2022;24(10):1480.
39. Yang JZ, Zhou CJ. A fault feature extraction method based on LMD and wavelet packet denoising. Coatings. 2022;12(2):156.
40. Li J, Zhu JH, Xie N, Guo MW. Random error filtering based on local mean decomposition for MEMS gyro. Electron Opt Control.
2011;18:49-51.
41. Lanzante JR. Resistant robust and non-parametric techniques for the analysis of climate data: theory and examples, including applications
to historical radiosonde station data. Int J Climatol. 1998;16:1197-1226.
42. Wu W, Chen B, Wu JF, Huang K. Study on reverse deduction of reservoir-inflow based on cubical smoothing algorithm with five-point
approximation. Water Resour Dev Press. 2013;44:100-102.
43. Iqbal N. DeepSeg: deep segmental denoising neural network for seismic data. IEEE Trans Neural Networks Learn Syst. 2022;1-8.
doi:10.1109/TNNLS.2022.3205421
44. Li C, Zeng ZF, Yi XF, et al. Research on the detection method of MRS Signal initial amplitude based on chaotic detection system. IEEE
Access. 2023;11:7959-7967.
45. Case Western Reserve University Bearing Data Center. Accessed November 12, 2022. https://engineering.case.edu/bearingdatacenter/
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How to cite this article: Tang C, Chen H, Jiang Y, et al. An improved method for signal de-noising based on
multi-level local mean decomposition. Engineering Reports. 2023;5(12):e12677. doi: 10.1002/eng2.12677