Frontiers in Computing and Intelligent Systems 
ISSN: 2832-6024 | Vol. 11, No. 3, 2025 

 

41 

Bearing	Acoustic	Emission	Signal	Processing	based	on	
Improved	SGMD	
Yue Shen *, Yang Yu 

School of Information Science and Engineering, Shenyang University of Technology, Shenyang Liaoning, 110870, China 
* Corresponding author: Yue Shen (Email: 2488119408@qq.com) 

 

Abstract: To solve the problems of poor adaptability to fixed thresholds and obvious endpoint effect in traditional Symplectic 
Geometric Mode Decomposition (SGMD), an improved SGMD (ISGMD) algorithm is proposed to improve the acoustic 
emission signal processing performance in bearing fault diagnosis. Firstly, a dynamic threshold adjustment model was 
constructed by fusing the signal Lyapunov exponent and the fractal dimension to realize the adaptive decomposition of signals 
of different complexity. Secondly, combined with the improved mirror extension and cosine smoothing technology, the endpoint 
effect is suppressed and the waveform distortion is avoided. Furthermore, the composite multi-scale dispersive entropy (CMSD) 
is introduced to screen the effective components, and the multi-scale entropy value is used to quantify the signal characteristics, 
eliminate the noise and reconstruct the components with high information order. Experiments show that ISGMD has excellent 
decomposition effect in noisy simulation signals, and the decomposition does not produce modal aliasing. In the actual bearing 
fault signal analysis, the reconstructed signal retains the outstanding peak characteristics, and the noise component is removed 
to a certain extent. This method significantly improves the robustness of signal decomposition and the ability to extract fault 
features, and provides an effective tool for the diagnosis of AE bearings under complex working conditions. 

Keywords: Symplectic Geometric Mode Decomposition; Acoustic Emission; Signal Reconstruction; Entropy Theory; 
Bearing Fault Diagnosis. 

 

1. Introduction 
As the core component of large-scale rotating machinery, 

the operation status of rolling bearings is directly related to 
the economy and safety of industrial production systems. 
According to statistics, about 30% of rotating machinery 
failures originate from bearing failure, of which local damage 
to rolling elements, inner and outer rings and other 
components accounts for more than 60%. If such faults are 
not diagnosed in time, they can lead to equipment chain 
damage and even safety accidents. Therefore, the 
development of efficient bearing condition monitoring and 
fault diagnosis technology is of great engineering value [1-3]. 

Acoustic Emission (AE) technology [4] can sensitively 
detect early microcracks and local spalling of bearings by 
capturing the transient elastic waves generated by the release 
of stress waves inside the material, and has shown unique 
advantages in the field of bearing fault detection by virtue of 
its non-sensitivity to materials and strong environmental 
adaptability. However, acoustic emission signals are 
susceptible to environmental noise, mechanical friction noise 
and electromagnetic interference, and their nonlinearity and 
non-Gausterity make it difficult for traditional time-frequency 
analysis methods (such as Fourier transform and wavelet 
analysis) to effectively extract fault features. In addition, 
existing adaptive decomposition methods, such as Empirical 
Mode Decomposition (EMD), Local Mean Decomposition 
(LMD) and Variational Mode Decomposition (VMD), have 
problems such as modal aliasing, endpoint effects and 
parameter sensitivity, which limit their application under 
complex working conditions. The Symplectic Geometric 
Mode Decomposition (SGMD) proposed by scholar Pan 
Haiyang solves the eigenvalues of Hamilton matrices through 
the symplectic geometric similarity transformation, which 
effectively suppresses the modal confusion while maintaining 

the intrinsic characteristics of the sequence. Subsequent 
scholars have extended SGMD to the fields of gearbox fault 
current analysis [5], photovoltaic DC denoising [6], fault 
feature clustering extraction [7], and power load prediction 
[8]. However, the existing SGMD methods have obvious 
limitations: the decomposition quality depends on the preset 
similarity threshold, and the fixed parameters are difficult to 
adapt to signals of different complexity. At the same time, 
there is endpoint effect interference, which affects the 
reconstruction accuracy of AE signals. 

In this study, an improved SGMD (ISGMD) method is 
proposed, which improves the adaptability of the algorithm to 
different signals by constructing an adaptive threshold 
mechanism, uses the mirror extension technology to eliminate 
the endpoint effect, and introduces the entropy theory for 
effective component screening and signal reconstruction. 

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2. Theoretical Analysis 

2.1. Symplectic Geometric Modal 
Decomposition Algorithm  

2.1.1. Sub-section Headings 
The core of Symplectic Geometric Mode Decomposition 

(SGMD) is to obtain the trajectory matrix by 
reconstructing the phase space of the acquired signal. Then, 
the symplectic geometric similarity transformation is carried 
out to obtain the eigenvalues and eigenvectors under the 
symplectic geometric framework, and the corresponding 
eigenvectors are used to reconstruct the single-component 
signal. The process of the symplectic geometric modal 



 

42 

decomposition algorithm can be divided into the following 
four stages: 

(1) Phase space reconstruction 
The phase space of the original sequence signal 	x
, ,⋯ ,  (n represents the signal length) is reconstructed 

by the Tankens embedding theorem, and the trajectory matrix 
X of the sequence signal is constructed. 

X ⋮
	
⋮

	

…
⋮
⋯

	
⋮

	
1  

Among them, d is the embedding dimension, and its value 
is determined by the power spectral density (PSD) method: 
the maximum peak frequency of the PSD of the calculated 
signal is , and d=n/3 is taken when the ⁄ <10⁻³ is 
taken, otherwise d=1.2 ⁄ is taken. The delay time τ is 
determined by an autocorrelation function. 

(2) QR feature decomposition 
After the trajectory matrix is constructed, the eigenvalues 

and eigenvectors are further solved by QR feature 
decomposition. Firstly, the covariance matrix A=XᵀX is 
constructed, and the Hamilton matrix is established: 

M 0
0

2  

Convert M to standard by symplectic geometric similarity 
transformation: 

0
0

3  

where W   is the upper triangular matrix, and its 
eigenvaluesλ₁,…,λd satisfy . 

Subsequently, based on the eigenvector  , the initial 
single-component  is calculated: 

4  
In this case, the reconstruction matrix Z is composed of the 

initial single-component reconstruction matrix Z_i group d: 
Z ⋯   that is, Z is a m*d reconstruction 
matrix. 

(3) Diagonal averaging 
After solving the reconstruction matrix Z, it needs to be 

converted into the required one-dimensional time series, so 
the diagonal averaging method is used. For the element Z in 
the matrix   (1≤ i≤m,1≤ j≤d), let ∗ ,  , 
∗ , , m n 1 . If m<d, let ∗ , 

otherwise,	 ∗  . Then, the one-dimensional time series 
, , … , 	is solved by the formula: 

1
, 	
∗ 1 ∗

1
∗ , 		

∗ ∗ ∗

∗

1
1

∗

∗

∗

7 5  

Based on the above formula, the Z_i is converted into a 
one-dimensional time series , , … ,  , which is 
denoted as a one-dimensional initial single-component signal 
(SGC), and the sum _i of these initial single-component 
signals is the original signal. 

(4) Component reorganization 

After the above operation, the initial single-component 
signal of group D is obtained. Some of these single-
component signals have similar periodic characteristics and 
are not completely independent of each other, so it is 
necessary to set a fixed threshold θ for component 
reconstruction of the components of these similar features, 
and extract the noise components separately as several 
components to distinguish them from the active components. 
However, the fixed threshold cannot adapt to the complexity 
of different signals, and the signals acquired under different 
working conditions or at different times have significant 
differences in essential characteristics (such as nonlinear 
characteristics and waveform morphology). Improvements 
are made below. 

2.2. Improve SGMD Algorithm 
(1) Dynamic threshold adaptive mechanism 
The dynamic threshold adjustment strategy using nonlinear 

feature fusion is improved, and the feature evaluation system 
is constructed by Lyapunov exponent and fractal dimension, 
so as to better capture the internal structure of the signal under 
different working conditions, improve the reconstruction 
quality and enhance the follow-up diagnosis effect. The 
specific principles and operations are as follows: 

Lyapunov Index: quantifies the chaotic properties of a 
signal, calculated as: 

λ lim
→

1
ln

0
6  

where δx(t) represents the small perturbation between the 
initial state of the system and the corresponding trajectory. 
The higher the exponential value, the more significant the 
chaotic nature of the system. 

Fractal dimension: The complexity of the signal structure 
is measured based on the box counting method, and the 
calculation formula is as follows: 

D lim
→

ln
ln

7  

where N(ε) denotes the number of boxes of the signal at 
scale ε. The signal complexity is quantified based on the box 
counting method, and the complexity of the signal structure is 
improved with the increase of the dimension value. 

Dynamic Threshold Generation Model: According to the 
Lyapunov exponent and fractal dimension of each signal, the 
threshold θ _d of each signal is dynamically adjusted as 
follows: 

. 1 ∙ ∙ 1 ∙ 8  
where  is the initial threshold. α. β is the adjustment 

coefficient, when the signal presents strong chaos (λ↑) or 
high complexity (D↑ ), the threshold  is automatically 
reduced to enhance the decomposition sensitivity, so as to 
ensure that the signal can obtain the optimal component 
reorganization effect under different working conditions. 

(2) Improve image extension 
The essence of the image extension method is to 

symmetrically extend the original signal into a closed ring 
signal, which fundamentally avoids the endpoint effect [9], 
but it is easy to introduce distortion in asymmetric signals. A 
three-stage optimization scheme is proposed: 

1. Bidirectional extension: bidirectional image expansion 
of the original signal x(t) generates x'(t), forms a ring structure 
to eliminate the endpoints; 

2. Waveform symmetry correction: the time-domain 
flipping process is carried out on the extension section to 



 

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eliminate the sudden trend change; 
3. Cosine smoothing: The ascending cosine function is 

used to weighted the fusion extension section: 

1 9  
As shown in Figure 1, a comparison between the normal 

image extension and the improved image extension, it can be 
seen that the smooth transition at the endpoint connection 
does not produce deformities.  

 
Figure 1. Comparison of image extensions 

Figure 2 shows the original signal decomposed using 
SGMD and the signal processed using improved image 
extension. 

 

 
(a) Improved image extension is not used 

 
(b) Improve image extension 

Figure 2. Break down the contrasts 

 

 It can be seen that the component endpoint effect of the 

decomposition after image extension is significantly 
improved, which improves the robustness of the 
decomposition algorithm, which is helpful to retain the key 
features of the signal in the subsequent actual signal 
decomposition process, and also provides reliability for 
subsequent reconstruction and analysis. It is worth noting that 
the waveform of the two decomposition components of the 
same signal is not synchronized, which is caused by the 
change of the length of the decomposition signal under the 
image extension, and does not change the frequency and other 
characteristics of the signal, which does not affect the results. 

2.3. Signal Reconstruction  
According to the entropy value theory, the Composite 

Multiscale Entropy (CMSD) is used to capture the multi-level 
changes of the signal through the multi-scale entropy value, 
deal with the transient changes or prominent peaks in the 
signal components, and select the effective components. 

As an entropy measurement method based on signal 
multiscale analysis, compound multi-scale dispersion entropy 
aims to measure the complexity and uncertainty of signals at 
different scales. Capturing the multi-level changes of the 
signal from a multi-scale perspective has the advantage of 
interpreting the local and global characteristics of the signal 
at different scales, and is more effective for the non-stationary 
signal of the acoustic emission signal. 

(1) Dispersion entropy calculation: 
The phase space of the component signal x(t) is 

reconstructed to generate an embedding matrix: 
, 1 , … , 1 	 10  

(2) Calculate the window divergence value: 
Divergence measurements are performed on the 

embedding matrix to calculate the divergence within each 
window: 

d
1

‖ ‖ 11  

where a=N-m+1. 
(3) Calculate the entropy value by probability distribution: 

log 12  

where  is the probability of each divergence value. 
(4) Multi-scale fusion: 
The dispersion entropy is calculated in the range of scale 

s=1 to s, and the composite entropy value is weighted to be 
generated: 

1
13  

(5) Threshold screening: 
Calculate the mean entropy of all components Threshold as 

the baseline threshold: 

Threshold
∑

14  

According to the entropy theory [10]: the smaller the 
entropy, the lower the uncertainty of the system and the more 
information it contains, that is, the smaller the entropy value, 
the more "useful" or "orderly" the information in the system 
can better help understand or predict the characteristics of the 
system. Therefore, the benchmark threshold is taken as the 
filter condition, and the components that meet the 

 are retained (low entropy value represents high 
information ordering), and the high entropy noise component 



 

44 

is eliminated. 

3. Experimental Analysis  
In order to verify the effectiveness and robustness of the 

improved SGMD (ISGMD) algorithm, experimental analysis 
of simulated signals and actual bearing signals is carried out. 
The experimental part aims to compare the performance of 
ISGMD with traditional empirical mode decomposition 
(EMD) and variational mode decomposition (VMD) 
algorithms, especially in dealing with noisy signals and 
complex fault modes. 

3.1. Analysis of Noisy Simulated Signals 
The decomposition effect and noise robustness of the 

improved algorithm ISGMD were verified, and the simulated 
signal analysis was carried out, and the experimental part was 
aimed at comparing the performance of ISGMD with the 
traditional empirical mode decomposition (EMD) and 
variational mode decomposition (VMD) algorithms. 

Construct the following simulation signal: 
 

2 sin 60 1 0.5 sin 2
sin 12
0.5 cos 10

15  

Add a noise n(t) with a signal-to-noise ratio of 20dB to x(t), 
and the noise simulation signal is as follows: 

16  
The decomposition was carried out by ISGMD, and the 

EMD decomposition and VMD decomposition were used for 
comparison, and the decomposition results are shown in 
Figure 3, Figure 4, and Figure 5. 

For better observation, only the first three components of 
the decomposition are shown in the SGMD exploded view, 
and the residual components are not shown. In the face of 
noisy signals, SGMD can still be effectively decomposed, and 
the first three components of the separation are basically the 
same as the components of the original signal. Observing the 
frequency domain diagram, it is found that there is no modal 
aliasing for each component. It shows that the SGMD 
decomposition algorithm does not produce modal aliasing 
during the decomposition process, and has good noise 
robustness. 

Observing EMD, it can be seen that the decomposition 
effect is not ideal, and many false components are generated, 
and the decomposed components are not similar to the 
original signal components, and the observation of the 
frequency domain can be seen that the modal aliasing is 
serious. Therefore, the EMD decomposition algorithm is not 
noise robust and cannot suppress modal aliasing. 

Observing the decomposition effect of VMD, it can be seen 
that the first two components of the decomposition are 
roughly similar to the first two components of the original 
signal when the decomposition parameter is set to 4, but there 
are also deviations, while the third component is not 
decomposed. At the same time, the frequency domain was 
observed, and it was found that the first component was 
mixed with the components of the third component, and the 
second component was not mixed with the other components. 
Therefore, the VMD decomposition algorithm is not very 
ideal in the decomposition of simulated signals containing 
noise, and the decomposition effect is not as good as ISGMD, 
and it also does not have the effect of noise robustness and 

suppressing modal aliasing. 
 

 
(a) Time domain 

 
(b) Frequency domain 

Figure 3. ISGMD decomposition results 

3.2. Actual Signal Analysis 
The simulation signal decomposition with noise proves that 

the ISGMD algorithm has the robustness of suppressing 
modal aliasing and noise, and the actual signal decomposition 
analysis is carried out below. As shown in Figure 6, the 
acoustic emission signal in the healthy state of the rolling 
bearing in the actual engineering - the fault signal of the inner 
ring of the acoustic emission can be observed, and the 
characteristic part is not obvious, and there is a large amount 
of noise to cover the real signal.  

Observing the components of the decomposition, it can be 
seen that: 

1. Each component shows different amplitude and 
frequency changes, and the waveform of some components 
has strong periodic or abrupt characteristics, which indicates 
that decomposition can effectively distinguish different signal 
components. 

2. From the third component onwards, the amplitude of the 
subsequent components is relatively small, meaning that they 
contain less energy or are just a fraction of the components in 
the original signal, representing noisy or irrelevant parts. 
Some components, such as SGC1, show large fluctuations 
and represent the main components of the signal. 

 
 



 

45 

 
(a) Time domain 

 
(b) Frequency domain 

Figure 4. EMD decomposition results 

 
(a) Time domain 

 
(b) Frequency domain 

Figure 5. VMD decomposition results 

 
Figure 6. Acoustic emission inner ring fault signal 

 
The signal is broken down below. The components 

decomposed by ISGMD are shown in Figure 7, and only the 
first six components are shown for ease of display. 

 

 
Figure 7. ISGMD decomposition results 

 

3. As can be seen from the figure, although there is a large 
difference between the components, they can show a certain 
regularity in the time domain, which shows that the ISGMD 
decomposition effect is very good, and different features are 
extracted through reasonable decomposition, which 
correspond to the important dynamic changes in the bearing 
signal. 

These characteristics show the effectiveness of the ISGMD 
decomposition algorithm and the advantages of the AE signal: 
it can extract components with different characteristics from 
the AE signal, and these components can correspond to the 
specific failure modes or characteristic frequencies in the 
signal, which can provide help for subsequent signal analysis 
and component reconstruction. 

Then, the composite multi-scale dispersive entropy is used 
to reconstruct the components, and the reconstructed signal is 
shown in Figure 8.  

By observing the reconstructed signals, it can be seen that 
the two types of reconstructed signals retain outstanding peak 
characteristics, and their mean square error and root mean 
square error are very small, indicating that the quality of the 
reconstructed signal is good, and the noise component is 
removed to a certain extent. Among them, the decrease in the 



 

46 

amplitude at some peaks compared to the original signal is 
caused by the amplitude of the removed noise, which is a 
normal situation. 

 

 
Figure 8. Reconstruct the signal 

 

Then, the envelope spectrum transformation of the 
reconstructed signal and the original signal is performed, as 
shown in Figure 9.  

 

 
(a) Raw signal envelope spectrum 

 
(b)Reconstruct the signal envelope spectrum 

Figure 9. Comparison of envelope spectrum 

 

Observing the reconstructed envelope spectrum, it can be 
seen that the fault energy is mainly concentrated in the low 
frequency range, which is consistent with the fault energy 
concentration area of the inner ring fault, and it can be seen 
that the energy is enhanced and highlighted, indicating that 
the reconstruction signal reduces the noise component and 
highlights the fault pulse. 

4. Conclusion  
The symplectic geometric mode decomposition algorithm 

is improved, and the SGMD can be appropriately 
decomposed according to the characteristics of different 
signals through adaptive threshold adjustment. At the same 
time, the improved mirror extension method is used to 
effectively suppress the endpoint effect of the components. 
Experiments have proved that the ISGMD algorithm is 
effective and noise robust. 

According to the entropy value theory, the multi-scale 
dispersion entropy is used to select the effective components 
to reconstruct the signal, and through the observation of 
multiple scales and the selection of the retained signal 
components, the prominent peak characteristics are retained, 
and the noise component is removed to a certain extent. 
Through experiments, the effectiveness of the reconstruction 
method is proved. 

The improved symplectic geometric mode decomposition 
provides a technical basis for real-time monitoring of 
industrial sites, and provides a possibility for the subsequent 
exploration of the deep integration of the algorithm with other 
deep learning models to further improve the generalized 
diagnosis ability of complex failure modes. 

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