Signal Denoising of MEMS vector hydrophone based on optimized VMD,

compressed sensing, and Wavelet threshold

HONGPING HU, NANA ZOU, YANPING BAI

School of Science, North University of China,

Taiyuan, Shanxi, 030051

CHINA

Abstract: With the noise in underwater acoustic signal extracted from ocean background, the denoising algorithm

based on the Variational Mode Decomposition (VMD) optimized by improved Grasshopper Optimization Algorithm

(IGOA), the compressed sensing (CS) and wavelet threshold (WT) is proposed in this paper, named by

IGOA-VMD-CS-WT, where VMD optimized by IGOA is utilized to perform sign composition and the obtained

Intrinsic Mode Functions (IMF) are divided into effective components and noise components using cross-correlation

coefficient of each IMF. CS is performed on sparse representation of noise components and the obtained sparse

coefficients are processed with WT for the filters. The effective components and the denoised components are

reconstructed to the denoised signal by the Orthogonal Matching Pursuit. The experiments show that

IGOA-VMD-CS-WT has the highest signal-to-noise ratios and the least root mean square errors under different noise

levels and has the better denoising effect on the denoising of the actual data.

Keywords: Grasshopper Optimisation Algorithm, Variational Mode Decomposition, Compressed Sensing, Wavelet

Threshold, Signal Denoising

Received: May 13, 2021. Revised: July 23, 2022. Accepted: August 17, 2022. Published: September 19, 2022.

1. Introduction

N underwater acoustics, vector hydrophone with its

convenience for submarine detection has succeeded in

attracting wide attention. According to the principle of the fish

lateral organs and the acoustic theory of cylinder, MEMS

vector hydrophone performs the detection of underwater signal

through the stimulation perception of varistor [1-2]. With small

size, high sensitivity, low-frequency detection, and other

excellent performance, there exist many kinds of hydrophones

and their application [3-4]. However, owing to the complicated

underwater acoustic environment in the ocean or lake, there

exists inevitably noise and distortion in the signal collected by

MEMS vector hydrophone, which affects the subsequent signal

detection, directional positioning, classification, and

recognition. Therefore, it is essential to adopt the denoising

algorithms for performing signal denoising to facilitate the

smooth development of the follow-up work.

There are rich denoising algorithms, such as Fourier

transform(FT) [5], wavelet transform (WT)[6], singular

spectrum decomposition(SSD)[7], Empirical Mode

Decomposition (EMD)[8] and their improvements, Variational

Mode Decomposition(VMD)[9] and the joint denoising

algorithms. Generally speaking, there exist certain denoised

effects obtained by these algorithms, but these algorithms have

shortcomings, which leads to the careful consideration. FT can

show the relationship between the time domain and frequency

domain and can be applied to the analysis and processing of

stationary signals, but it cannot reflect the characteristics of

specific time signals. WT is more effective and practical, and

for the signal denoising and the image denoising, WT has the

better-denoised results and is more suitable for the unstable

signal denoising. The principle components in SSD need to be

selected. Self-adaptability is one of the advantages of EMD

and the decomposed modal functions obtained by EMD are

screened out by their properties. And EMD is applied to

non-linear and non-stationary signals and achieves a better

denoising effect. However, there are shortcomings of EMD,

such as the lack of a strictly mathematical basis, low efficiency,

and mode aliasing, which lead to the limitations to a certain

extent in its applications and development. VMD proposed in

2013 is a self-adaptive, non-recursive signal decomposition

algorithm, which performs partitioning the signal in the

frequency domain to achieve effective separation. Different

from EMD, VMD has a very strong mathematical foundation,

better noise robustness, and higher operational efficiency. But

the parameters of VMD are set up in advance. At present, a

large number of experimental researches show that a hybrid

denoising algorithm has better performance than a single

denoising algorithm. For example, VMD is combined with the

nonlinear wavelet threshold (NWT) to establish the joint

denoising method VMD-NWT for performing denoising and

baseline drift removal [10].

Recently, there are more and more metaheuristic

algorithms which are divided into two types[11,12]: single

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solution-based and population-based, where there is only one

solution during the optimization phase in the former type and

there are N solutions in every iteration during the optimization

phase in the latter type. Especially, in the population-based

metaheuristic algorithms, Harris Hawks Optimizer[12] is

obtained from the cooperative behavior and chasing style of

Harris’ haws in nature called surprise pounce; the genetic

algorithm [13,14] mimics the Darwinian theory of evolution; the

particle swarm optimization(PSO)[15] mimics the birds flocking

behaviors; Grasshopper Optimization Algorithm(GOA)[16]

mimics the behavior of grasshopper swarms in nature for

solving optimization problems.

Generally, the parameters of VMD are the number of

Intrinsic Mode Functions (IMFs) and the penalty factor. The

suitable parameters make the denoising results more effective.

The births of metaheuristic algorithms support the

opportunities for the parameters of VMD. In the reference [17],

the parameters of VMD are optimized by the whale

optimization algorithm (WOA), the power spectrum entropy

(PSE) is taken to be the fitness function of WOA, and thus the

whale-optimized VMD and correlation coefficient (CC) are

combined to propose a denoising and baseline drift removal

algorithm. In the reference [18], a hybrid algorithm of

Multi-Verse Optimizer and PSO is proposed to optimize the

parameters of VMD and then is combined with WT denoising

and CC judgment to perform the signal denoising of MEMS

vector hydrophone.

The definition of Compressed sensing (CS) is that if a

high-dimensional signal is compressible or sparse in a certain

transform domain, it is mapped into a low dimensional space

by a measurement matrix unrelated to the transform basis, and

then the original signal is reconstructed from the small number

of projection measurements with high probability by solving an

optimization problem. The combinations of CS and other

algorithms have been widely used in the medical, earthquake,

image optimization, and other fields, such as the combination

of EMD, CS, and WT for denoising microseismic signals [19],

and a method of noise attenuation of the weak seismic signal

based on CS and CEEMD [20], the hybrid algorithm based on

improved SSA and CS for lidar signal denoising [21].

In this paper, the improved GOA (IGOA) is proposed to

be applied to optimize the parameters of VMD for performing

sign composition and the obtained IMFs are divided into the

effective components and noise components according to the

cross-correlation coefficient of each IMF. And CS is used to

perform sparse representation of the noise components and the

obtained sparse coefficients are processed by being combined

with WT for the filters. Then the effective components and the

denoised components are reconstructed to the denoised signal.

Thus the denoising algorithm based on VMD optimized by

IGOA, CS, and WT is proposed in this paper, named

IGOA-VMD-CS-WT. The simulation experiments show that

the proposed denoising algorithm IGOA-VMD-CS-WT in this

paper has the highest signal-to-noise ratios and the least root

mean square errors for the simulated signals under different

noise levels, superior to the other compared denoising

algorithms. Finally, the proposed denoising algorithm

IGOA-VMD-CS-WT is applied to perform the denoising of the

actual data.

The remaining paper is organized as follows. Section 1 is the

introduction. Section 2 gives a brief description of the relevant

principles and methods, and the proposed method IGOA-

VMD-CS-WT in Section 3 is described in detail. Experiment

results for denoising are given in Section 4 for the simulated

signal and the Fenji lake trial data obtained from the North

University of China. Section 5 is the conclusion.

2. Basic Principles

2.1 Grasshopper Optimization Algorithm

Because VMD cannot be decomposed adaptively, it needs to

set parameters in advance, so this paper optimizes its

parameters by GOA.

GOA[16] proposed in 2017 simulates the characteristics of

small-scale movement during the locust larvae period and

random large-scale movement during the adult period, which

constitutes the local development and global exploration

process of the GOA algorithm. Without considering the gravity

factor and assuming that the wind direction always points to the

target location, the behavior of the locust swarm can be

represented by the following mathematical model:

 

ˆ,

Nji

d d d

i j i d

j j i ij

ub lb

X c c s x x T





  



(1)

where

represents the position of the ith grasshopper of the

grasshopper swarm in the

th dimension,

ub lb

are the

upper bound and the lower bound of the

th dimension in the

ith grasshopper,

ˆd

is the dth dimension of the optimal

grasshopper.

ij j i

d x x

denotes the distance between the

ith grasshopper and the jth grasshopper in the

th dimension.



is a unit vector from the ith grasshopper to the jth

grasshopper.

max min

max ,

c c t T





(2)

where

max min

,cc

are the maximum value and the minimum value

, respectively,

denotes the current iteration, and

is the

maximum number of iterations. In GOA,

max min

1, 0.000001cc

are selected. The coefficient

is used

for global exploration and local development of the balance

algorithm. The coefficient

decreases the search range of the

individual as the number of iterations increases, and is used to

control the exploration and development of the algorithm; the

coefficient

on the inner side is used to control the attraction

zone, comfort zone, and repulsion zone among locusts. And

()

s r fe e





. (3)

Equation (3) is the function of the interaction force between

locusts and other grasshoppers. When

( ) 0sr 

, the value range

is called the attraction zone. At this time, the locusts attract

each other. Citation; when

( ) 0sr 

, the value range of

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called the repelling zone, and the locusts repel each other;

when

( ) 0sr 

, it is neither attractive nor repulsive, and the

value of

is called the comfort zone. In addition,

and

are

the attracting strength parameter and the attracting scale

parameter, value

0, 1.5fl

From the above analysis, it can be seen that the basic

implementation steps of the GOA algorithm are as follows:

Step1: Initialize the population size

, the parameters

max min

,cc

and the maximum number of iterations

max

, initialize

the population position, and calculate the fitness value of each

grasshopper, and select the optimal value as the target position.

Step2: Enter the main loop, and update the parameter

according to formula (2).

Step3: According to formula (1) update the individual

position of the grasshopper and calculate each grasshopper.

Step4: Judge whether the conditions for stopping the loop

are satisfied. If it is satisfied, the algorithm will jump out of the

loop and return to the target value; otherwise, the algorithm

will repeat Step 2 and Step 3.

This article updates the GOA algorithm and uses the updated

GOA algorithm to optimize the parameters of VMD.

2.2 Variational Mode Decomposition

VMD proposed in 2014 is an adaptive modal decomposition

algorithm, which can make the signal decomposition problem

be transformed into a variational problem[9]. Thus the

variational problem is obtained as follows:

  

min ( ) * ( )

. . ( )

Kjwkt

t u t e

st u x t







































. (4)

In equation (4), the original vibration signal

()xt

is the

accumulation of each IMF obtained by decomposition;

   

, , ,

u u u u

refers to the modal components obtained

after the original signal is decomposed;

   

, , ,

w w w w

represents the center frequency corresponding to each IMF

obtained by decomposition;



is the partial derivative of

;

()t



is the impulse function.

To obtain the optimal solution to the variational problem, the

Lagrange operator

()t



and the quadratic penalty factor



are introduced to make the constrained variational problem be

transformed into an unconstrained variational problem, as

follows:

   

 

, , ( ) * ( )

( ) ( ) ( ), ( ) ( ) ,

Kjwkt

k k t k

L u w t u t e

x t u t t x t u t

   













  









   





(5)

where

()t



is the Lagrangian multiplier;



is the secondary

penalty factor.

ˆ ˆ

, , ( )

u w t





are iteratively updated by

the alternating direction multiplier algorithm. Then the "saddle

point" of equation (5) can be obtained, that is, the optimal

solution of equation (4). In the following, we will use the

improved GOA algorithm to optimize the parameters of VMD

and find the best decomposition result to facilitate subsequent

operations.

2.3 Compressed Sensing

CS is a process of downsampling and decoding the known

signal with sparsity[22]. The effective signal is sparse in the

sparse domain, but the noise signal is not sparse. Therefore, CS

can be utilized to separate the effective signal from the noise.

The basic idea of CS is the following: If the sparsity of an

unknown signal is

or is transformed into

spares by a

sparse basis, then based on

spares, combined with

non-linear transformation, the original signal can be accurately

reconstructed. The signal

can be sparsely expressed as a

linear combination of the standard orthogonal basis

 

, , , N

  



as follows:







, (6)

where



is the sparse vector of signal

, and the sparsity is

The smaller the

is, the higher the sparsity is. The signal can

be reconstructed accurately until the observation matrix is used

to make the signal with the sparse expression be reduced the

dimension linearly for projection. The sparseness of the signal

is determined by the sparse coefficients



, so a

MN

measurement matrix is selected to perform a linear

transformation on the signal to obtain

yx

. The signal

measured and then the measured value

yR

is obtained.

Thus

y x T



    

, (7)

where

T 

is the sensor matrix and



is the sparse

coefficient vector to be solved.



is the orthogonal basis

matrix. If the matrix

meets the Restricted Isometry Property

(RIP) criterion,

sparse can obtain an optimal solution from

the

measured value

. The problem of solving



by the

observation matrix

can be transformed into an optimization

problem in the

norm, as follows:

ˆarg min , . .st T y

  



, (8)

where



is the

norm of



. The convex relaxation

algorithm, the greedy algorithm and the combinatorial

optimization algorithm are usually employed to solve Eq.(8).

The classical sparse transform methods include discrete cosine

transform [23], discrete Fourier transform [24], discrete

wavelet transform [25], and other transforms.

In this paper, we choose CS to perform downsampling on the

IMF components with the small correlation coefficients

obtained by VMD, and the sparse vector θ is obtained. There

are two main ways to obtain the measurement matrix: one is to

construct it according to the signal characteristics, and the

other is to select an existing matrix, such as a random Gaussian

matrix. Discrete cosine transform and fast Fourier transform

are used to be the sparse basis, Gaussian random matrix and

partial Hadamard matrix are chosen to be the measurement

matrix, and the orthogonal matching pursuit algorithm is used

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to perform the reconstruction. In addition, recent research have

shown that the amplified noise signal in the process of random

measurement leads to the reduction of the signal-to-noise ratio

and the noise folding emerged. Based on this, WT is used to

filter the sparse vector θ for reducing the noise folding.

2.4 Wavelet Threshold filtering method

WT filtering method is that the threshold function is used to

quantize the decomposed coefficients and then the denoised

signal is obtained by the signal reconstruction. There are the

common threshold functions, such as hard threshold, soft

threshold, Stein Unbiased Risk threshold, heuristic threshold,

and soft hard threshold tradeoff.

The data-driven threshold proposed by Donoho [26] widely

used at present is defined as follows:

 

2 log , ( 1, 2, , )

0.6745

median

T N i K





, (9)

where



is the ith IMF component, the median is the median

function,

is the threshold value of the ith reconstruction

coefficient



which is the ith sparse coefficient of the sparse

vector



obtained by the CS algorithm. The wavelet obtained

by soft threshold estimation has good sparse continuity and can

not produce additional impact. Therefore soft threshold

denoising is adopted in this paper, whose expression is as

follows:

 

sgn( ) , ( 1, 2, )

i i i

i new

  



  











(10)

3. The Proposed Denoising Algorithm

3.1 Improved Grasshopper Optimization

Algorithm

In the GOA, the position of the grasshoppers is updated by

the optimal grasshopper, the current grasshopper, and the other

grasshoppers. And the convergence speed of GOA is slow and

easy to fall into the local optimal grasshopper. To overcome

this disadvantages of GOA, Levy flight and Nonlinear Weight

are employed to improve GOA in this paper to obtain the

improved GOA, written as IGOA. Levy flight makes

individuals conducive to jumping out of the local optimum and

finding out the global optimum. Nonlinear weight makes GOA

converge to the current local optimum quickly. Levy flight [29]

is a method to provide random factors, which are distributed as

follows:

~ ,1 3.Levy u t





  

. (11)

The nonlinear weight

is taken to be the sigmoid function

[30] as follows:

10 5

10.1

1 1.5

Max iter

e





, (12)

where

is a random number between 0 and 1, and

is the

current number of iterations, which makes the algorithm

decrease slowly in the early stage and converge rapidly in the

later stage. The updated position of the grasshopper is as

follows:

 

ˆ(dim),

Nji

d d d

i j i d

j j i ij

ub lb

X c c s x x T r levy





    



(13

)

where

, , ( ),

d d ij d

ub lb s d T

are similar to the representation in

Subsection 2.1.

Different from GOA, the parameter

in IGOA is used to be

equation (12) and the updated position of the grasshopper in

IGOA is used to be equation (13). And the number of the terms

in equation (13) is more than that in equation (1) and this term

is related to Levy fight. The pseudo-code of the IGOA

algorithm is shown in Figure 1.

Initialize cmax,cmin, and the population size N, spatial dimension dim

Initialize maximum number of iterations and the swarm Xi(i=1,2,…,N)

Calculate the fitness of each search agent

T=the best search algent

while (l<maximum number of iterations)

Update c using Equation (12)

For each search agent

Normalize the distances between grasshoppers

Update the position of the current search agent by Eq.(13)

Bring the current search agent back if it goes outside the

boundaries

end for

Update T if there is a better search algent

l=l+1

end while

Return T

Figure 1. Pseudo codes of the IGOA algorithm.

3.2 The joint denoising algorithm

IGOA-VMD-CS-WT

Two parameters of VMD (the number K and the penalty

factor



of IMFs) need to be set up in advance, which causes

the signal cannot be decomposed adaptively by VMD. Given

the noise folding phenomenon of CS in the case of a low

signal-to-noise ratio, WT is used to process the sparse

coefficients after sparse representation of CS to make them

more sparse, which reduces the noise overlap phenomenon.

Based on the above analysis, a joint denoising method based

on the VMD optimized by IGOA, CS, and WT is proposed in

this paper, named by IGOA-VMD-CS-WT, where PSE [27-28] is

adopted to be the fitness function.

The concrete steps of IGOA-VMD-CS-WT are as follows:

Step1. Obtain the main frequency

of the noisy signal by

Fourier transform. The parameters of VMD are optimized by

IGOA to adaptively select the number

of IMFs and the

penalty factor



Step2. According to the number

of IMFs and the penalty

factor



obtained from Step1, the original noisy signal is

decomposed by VMD and the

IMF components are

retained.

Step3. Calculate the cross-correlation coefficients and

center frequency between each IMF component and the

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original noisy signal. Based on the cross-correlation value of

each IMF component, and the absolute deviation value

between the dominant frequency and the center frequency, the

bound component is found to distinguish the effective

components from the noise components. The effective IMF

component is retained directly.

Step4. The noise IMF components processed by

downsampling are sparsely expressed. Thus the sparse

coefficients

 

, , , p

   



are obtained, where

represents the number of noise IMF components, and then WT

is used to filter the sparse coefficient to get the new sparse

coefficients

 

1, 2, ,

, , ,

new new new p new

   



. Finally, the

Orthogonal Matching Pursuit algorithm is used to reconstruct

the sparse coefficients to get a denoised noise component.

Step5. The effective components and the denoised noise

components are reconstructed to obtain the denoised signal of

the original noisy signal.

The flowchart of IGOA-VMD-CS-WT is shown in Figure 2.

Figure 2. Flowchart of IGOA-VMD-CS-WT

In figure 2, the calculation of fitness value of every agent in the part

of VMD optimized by IGOA is as follows:

Every agent in the population is composed of two parameters: the

number K and the penalty factor



of IMFs. For every agent,

the reconstructed signal of the input noisy signal is obtained by

VMD. PSE is calcuated to be fitness value of every agent by

the reconstructed signal and the input noisy signal.

4. Simulation Experiments

4.1 Simulation signal

In this paper, the simulation signal is obtained from two

sinusoidal signals for performing the denoising performance of

the IGOA-VMD-CS-WT. By considering the dynamic changes

in the marine environment, the known Gaussian white noise

under the different decibels is added to the simulation signal to

simulate the noisy signal received by the MEMS vector

hydrophone.

The simulation signals are as follows:

 

( ) sin 2 400 ( )

( ) sin 2 200 ( )

( ) ( ) ( )

y t t n t

y t y t y t



    



    







(14)

where 200 and 400 represent the frequencies of two sinusoidal

signals, respectively, and is the Gaussian white noise with

the different decibels.

In this paper, we take the signal-to-noise ratio (SNR)

 

()

10log ( ) '( )

SNR x n x n















(15)

and the root mean square error (RMSE)

 

1( ) '( )

RMSE x n x n

N





(16)

to be the performance indicators of denoising, where

is the

number of snapshots, and

)(' nx

and

)(nx

are the denoised

signal and the original signal, respectively.

4.2 Denoising of the simulation signal by

IGOA-VMD-CS-WT

Due to the limited space, we take the noisy signal with the

noisy level at 10dB processed by IGOA-VMD-CS-WT for

example in this paper. Figure 3(a) shows the noisy signal with

the noisy level at 10dB. Then Fourier transform is performed

on the noisy signal and the corresponding frequency spectrum

is obtained, shown in Figure 3(b) and its main frequency

obtained further.

The IGOA is used to optimize two parameters

and



VMD and the selected

and



are adaptive. Here, the noisy

simulated signal is decomposed into 7 IMF components by the

optimized VMD. These IMF components and their

corresponding spectra are shown in Figure 4. Table 1 shows the

cross-correlation coefficients between every IMF component

and the noisy simulation signal.

From Figure 4, it can be observed that the energies of IMF1

and IMF2 are the highest, which are the closest to 200Hz and

400Hz, respectively. From Table 1, we can see that the

correlation coefficients of IMF1 and IMF2 are the highest.

Therefore, IMF1 and IMF2 are regarded to be effective signal

components. The other components IMF3-IMF7 with the low

correlation coefficients are regarded to be the noise signal

components.

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Figure 3. Noisy signal and its corresponding frequency

spectrum.

Figure 4. The IMF components and their corresponding spectra

obtained by the optimized VMD

For the noise IMF components, CS is utilized to perform

denoising. Firstly, 10% of every noise IMF component is

selected to perform downsampling. Secondly, the orthogonal

basis matrix and measurement matrix are constructed, and the

sparse coefficients

 

, , , p

   



, where

is the number

of IMF noise components, are obtained by the sparse

representation of the downsampled signal. Thirdly, the WT

proposed in this paper is used to process the sparse

coefficients, and the new sparse coefficients

 

1, 2, ,

, , ,

new new new p new

   



are

obtained. Finally, the orthogonal pursuit matching algorithm is

used to reconstruct

 

1, 2, ,

, , ,

new new new p new

   



, and then the

effective IMF components and the denoised noise components

are reconstructed to obtain the denoised signal of the original

noisy signal. Table 2 shows the signal sparsity and the best

sparsity by CS.

4.3 Experimental results

 Comparison of Denoised Effects

In this paper, the joint denoising algorithm IGOA-VMD

-CS-WT proposed is the combination of VMD optimized by

IGOA, CS, and WT. To verify the effectiveness of the

IGOA-VMD-CS-WT, we employ VMD, VMD-CS,

VMD-WT, VMD-CS-WT, GOA-VMD-CS, GOA-VMD-WT

and IGOA -VMD-WT for comparison with

IGOA-VMD-CS-WT on the same noisy signal.

Figure 5 shows the comparisons between the original

signal without signal and the denoised signals of the noisy

signal with 10 dB Gaussian white noise obtained by these 8

compared algorithms, respectively. Observed in Figure 5, the

denoised effects of these 8 compared algorithms are all better,

the sharp burrs of the signal are all eliminated, the waveforms

are allsmoother as a whole, there are almost no distortion

phenomenons, and the edges of the denoisied signal by these 8

compared algorithms have the worse denoised effects except

Figure 4(h). The compared results show that the comparison,

IGOA-VMD-CS-WT can be used to perform the denoising of

the noisy signal, especially to improve the denoising of the

edge signal of the noisy signal.

Figure 6 shows the comparisons between the original signal

without noise and the denoised signals of the noisy signal with

-5 dB Gaussian white noise obtained by these 8 compared

algorithms, respectively.

Observed in Figure 6, the denoised signals by VMD-CS and

GOA-VMD-CS are not smooth as a whole and there is noise

folding, while the waveforms of the denoised signed by VMD,

VMD-WT, VMD-CS-WT, GOA-VMD-WT,

IGOA-VMD-WT, and IGOA-VMD-CS-WT are all relatively

smoother. And owing to WT introduced, the noise folding

phenomena are improved. Especially, the denoised signal by

IGOA-VMD-CS-WT has the best fitting effect with the

original signal without noise.

Based on the above analyses, the compared results show

that the denoised effect of IGOA-VMD-CS-WT is better than

other methods.

 Comparison of Denoised Performance Indicators

The noisy signals with five different decibel Gaussian

noises, 15dB, 10dB, 5dB, 0dB, and -5dB, are used to perform

denoising by the 8 algorithms: VMD, VMD-CS, VMD-WT,

VMD-CS-WT, GOA-VMD-CS, GOA-VMD-WT,

IGOA-VMD-WT, and IGOA-VMD-CS-WT, respectively.

The SNRs and RMSEs obtained by these 8 algorithms are

shown in

Table 3, where the first column represents the decibel of

Gaussian white noise decibels added to the original signal.

Observed from Table 3, the SNRs obtained by

IGOA-VMD-CS-WT are all the maximum among these 8

algorithms, which are 23.6105 for 15dB, 19.7220 for 10dB,

15.9566 for 5dB, 12.5489 for 0dB, and 9.2686 for -5dB,

respectively, and the RMSEs obtained by IGOA-VMD

-CS-WT are all the minimum among these 8 algorithms,

Figure 5. Denoised Effects of the noisy signal with 10dB

Gaussian white noise

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Figure 6. Denoised Effects of the noisy signal with -5dB Gaussian white noise

Table 1. Correlation Coefficients between each IMF Component and the noisy simulation signal.

Table 2. Signal Sparsity and the Optimal Sparsity

Noise IMF Component

IMF3

IMF4

IMF5

IMF6

IMF7

Signal Sparsity

Optimal Sparsity

Table 3. The SNRs and RMSEs obtained by these 8 algorithms

Performance indicator

VMD

VMD-CS

VMD-WT

VMD-CS-WT

GOA-VMD-CS

GOA-VMD-WT

IGOA-VMD-WT

IGOA-VMD-CS-WT

15dB

SNR

16.1697

17.7802

21.6527

22.3275

21.0024

22.1576

22.9151

23.6105

RMSE

0.1554

0.1291

0.0827

0.0766

0.0904

0.0780

0.0710

0.0660

10dB

SNR

15.7538

16.1507

18.4869

18.6940

17.2304

18.6400

19.1025

19.7220

RMSE

0.1630

0.1558

0.1191

0.1164

0.1376

0.1172

0.1110

0.1041

5dB

SNR

13.8800

13.2914

14.6015

15.7434

13.4869

15.8678

15.9022

15.9566

RMSE

0.2023

0.2437

0.1862

0.1641

0.2117

0.1611

0.1603

0.1593

0dB

SNR

10.2194

9.0332

11.1042

11.6070

9.1175

12.0430

12.3616

12.5489

RMSE

0.3038

0.3554

0.2785

0.2628

0.3501

0.2504

0.2410

0.2359

-5dB

SNR

6.8692

4.4314

6.9463

7.2939

4.6269

7.3714

7.6339

9.2686

RMSE

0.4535

0.6006

0.4495

0.4321

0.5870

0.4280

0.4154

0.3440

which are 0.0660 for 15dB, 0.1041 for 10dB, 0.1593 for 5dB,

0.2359 for 0dB and 0.3440 for -5dB. Thus the denoised

effect of IGOA-VMD-CS-WT is significantly better than

those of the other compared algorithms under the different

decibel noises. That is, IGOA-VMD-CS-WT is superior to

VMD, VMD-CS, VMD-WT, VMD-CS-WT, GOA-VMD

-CS, GOA-VMD-WT, IGOA-VMD-WT. In addition, the

denoised effect of IGOA-VMD-WT is slightly better than

GOA-VMD-WT, which indicates that the improved GOA

improves the SNR and RMSE and further that IGOA is

better than GOA.

Based on the comparisons of denoised results and the

performance indicators, when the noise level is low, the

simple CS is used to perform denoising, and then there exist

noise folding phenomena, but once the WT denoising is

introduced, the noise folding phenomena are improved and

the denoised signal is better, the SNRs are higher. All of

these further demonstrate that the proposed

IGOA-VMD-WT algorithm has a better-denoised effect on

the noisy signal with different decibel noise and the baseline

drift removal, outperforms VMD, VMD-CS, VMD-WT,

VMD-CS-WT, GOA-VMD-CS, GOA-VMD-WT,

IGOA-VMD-WT.

5. Lake Trial Experiments

The measured data used in this paper are derived from fenji

experiments conducted by the North University of China in

2011 and 2014 in fenhe, respectively.

5.1 Experiment 1: The measured data from the

fenji331Hz data packet tested in 2011.

In this experiment, the vector hydrophone with 5-element

linear array was fixed on the shore and the transducer was

placed on the tugboat. The anchor was dropped at the

different positions, and then the measured data was collected

by the transducers. The sound source with a transmitting

signal frequency of 331Hz is 6 meters far away from the

vector hydrophone with a sampling frequency of 10kHz.

The data with the snapshots1000 arbitrarily intercepted from

the measured data of five road array signals are regarded to

IMF component

IMF1

IMF2

IMF3

IMF4

IMF5

IMF6

IMF7

Correlation Coefficients

0.6884

0.6737

0.1211

0.1017

0.0981

0.0961

0.0973

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be the noisy data for performing the signal denoising.

Figure 7(a)(b)-Figure11(a)(b) show the signals of the

1-Road array to the 5-Road array and their corresponding

frequency spectra, respectively. And Figure7(c)(d)

-Figure11(c)(d) show the denoised signals of 1-Road array to

5-Road array processed by IGOA-VMD-WT and their

corresponding frequency spectra.

Observed in Figure 7(a)(b)-Figure11(a)(b), there exists a

small amount of noise in 1-Road and 2-Road array signals,

while there exist a large amount of noise in 3-Road, 4-Road

and 5-Road array signals. Observed in Figure7(c)

-Figure11(c), the sharp burrs of the signals have been

effectively eliminated, the signal becomes smooth and tidy

and the distorted part of the signal is effectively improved.

Observed in Figure7(d)-Figure11(d), the energies of the

signals are hardly lost and the noises have been eliminated

effectively.

5.2 Experiment 2: The measured data from the

fenji measured data in 2014.

In this experiment, the MEMS vector hydrophone with

5-element arrays of interval distance of 0.5 meters was

placed on the shore, and the transducer was placed on the

tugboat. The MEMS vector hydrophone was placed under

the water 2 meters and was kept horizontal. Thus the MEMS

vector hydrophone could continuously output the sound

pressure and the circuit signals. The signal signals with

315Hz, 500Hz, 630Hz, 800Hz, and 1000Hz are obtained by

adjusting the transducer, respectively. The data with the

snapshots 1000 are arbitrarily intercepted from the different

frequency packets to be regarded as noisy signals.

Figure 12(a)(b)-Figure 16(a)(b) show the signals with 315Hz,

500Hz, 630Hz, 800Hz, and 1000Hz and their corresponding

frequency spectra, respectively. And Figure7(c)(d)-

Figure11(c)(d) show the denoised signals with 315Hz, 500Hz,

630Hz, 800Hz and 1000Hz processed by IGOA-VMD-WT and

their corresponding frequency spectra.

Observed in Figure 12 (a)(b)-Figure 14(a)(b), there exists the

low-frequency noise in the upper part of the signal and the

high-frequency noise in the lower part of the signal, and there

exist the serious distortions in the signals. Observed in

Figure15(a)(b)-Figure16(a)(b), there exists a slight noise and no

serious distortion in the signals, but the waveforms of the signals

are not smooth. And the baseline drift removals are improved.

Observed in Figure12(c)-Figure16(c), the noises of the noisy

measured signals are all eliminated and the waveforms of the

denoised measured signals are all well recovered. And there are

no distortion phenomena and the basic characteristics of the

original signals are all well preserved. Observed in

Figure12(b)(d) -Figure16(b)(d), the energies of the signals are

also well preserved with no loss.

5.3 Experimental results

Based on the above two experiments, the IGOA-VMD-CS -WT

algorithm proposed in this paper can effectively eliminate the

noise, better recover the basic characteristics of the signal, and

improve the baseline drift removal. Therefore, the

IGOA-VMD-WT algorithm proposed in this paper is effective

and suitable for signal denoising. At the same time, the

IGOA-VMD-WT algorithm proposed in this paper gives

support for the next location, classification, and recognition of

the signal.

6. Conclusion

In this paper, IGOA is used to optimize the parameters of

VMD to establish the hybrid algorithm IGOA-VMD. Based on

the combination of IGOA-VMD, CS, and WT, the joint signal

denoising algorithm IGOA-VMD-CS-WT of MEMS vector

hydrophone is proposed. The simulation experiments show that

the proposed IGOA-VMD-CS-WT can effectively eliminate the

noises from the noisy signals, and has the minimum RMSEs and

the maximum SNRs. And the compared results show that the

proposed IGOA-VMD-CS-WT outperforms the other

compared algorithms and weakens the noise folding. Further,

the proposed IGOA-VMD-CS-WT is applied to perform the

denoising of the measured signals derived from fenji

experiments conducted by the North University of China in

2011 and 2014 in fenhe. And the denoised results show that the

proposed IGOA-VMD-CS-WT can effectively eliminate the

noise of the measured signals and improve the baseline drift

removal.

In addition, the IGOA is obtained from GOA by introducing

the Levy flight and the nonlinear weight. But the selections of

different nonlinear weight lead to the different IGOA and the

combination of GOA and the other one or two warm intelligence

algorithms generates different IGOA. These improved GOAs

are combined with VMD, CS, and WT to establish the new joint

denoising algorithms. The denoising performance of these

algorithms will be verified in the future

and be conducive to the subsequent signal location,

classification, and recognition of signals.

Acknowledgment

The authors would like to thank the editors and anonymous

reviewers for their valuable comments and suggestions, which

greatly improved the presentation of the paper. This work was

supported in part by the Fundamental Research Program of

Shanxi Province (Grant No. 20210302123019,

202103021224195, 202103021224212, 202103021223189),

and by the Shanxi Scholarship Council of China (Grant No.

2020-104,2021-108).

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Figure 7. The signal and the denoised signal of 1-Road array

signal and their corresponding frequency spectra

Figure 8. The signal and the denoised signal of 2-Road array

signal and their corresponding frequency spectra

Figure 9. The signal and the denoised signal of 3-Road array

signal and their corresponding frequency spectra

Figure 10. The signal and the denoised signal of 4-Road array

signal and their corresponding frequency spectra

Figure 11. The signal and the denoised signal of 5-Road array

signal and their corresponding frequency spectra

Figure 12. The signal and the denoised signal with 315Hz and

their corresponding frequency spectra

Figure 13. The signal and the denoised signal with 500Hz and

their corresponding frequency spectra

Figure 14. The signal and the denoised signal with 630Hz and

their corresponding frequency spectra

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Figure 15. The signal and the denoised signal with 800Hz and

their corresponding frequency spectra

Figure 16. The signal and the denoised signal with 1000Hz and

their corresponding frequency spectra

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HONGPING HU received a Ph.D. degree from

the North University of China, Shanxi, China, in 2009. She is

currently a Professor and a master’s tutor in the School of

Science, North University of China, Shanxi, China. Her

research interests include combinatorial mathematics, artificial

intelligence, signal processing, and image processing.

Contribution of Individual Authors to the Creation of

a Scientific Article (Ghostwriting Policy)

Hongping Hu and Nana Zou have implemented the

proposed denoising algorithm of Section III. Nana Zou

carried out the simulation and the experiments and wrote

the manuscript. And Hongping Hu and Yanping Bai were

responsible for revising and giving suggestions.

Creative Commons Attribution License 4.0

(Attribution 4.0 International , CC BY 4.0)

This article is published under the terms of the Creative

Commons Attribution License 4.0

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