A Novel Joint Denoising Method for Hydrophone Signal Based on Improved SGMD and WT

Underwater acoustic technology as an important means of exploring the oceans is receiving more attention. Denoising for underwater acoustic information in complex marine environments has become a hot research topic. In order to realize the hydrophone signal denoising, this paper proposes a joint denoising method based on improved symplectic geometry modal decomposition (ISGMD) and wavelet threshold (WT). Firstly, the energy contribution (EC) is introduced into the SGMD as an iterative termination condition, which efficiently improves the denoising capability of SGMD and generates a reasonable number of symplectic geometry components (SGCs). Then spectral clustering (SC) is used to accurately aggregate SGCs into information clusters mixed-clusters, and noise clusters. Spectrum entropy (SE) is used to distinguish clusters quickly. Finally, the mixed clusters achieve the signal denoising by wavelet threshold. The useful information is reconstructed to achieve the original signal denoising. In the simulation experiment, the denoising effect of different denoising algorithms in the time domain and frequency domain is compared, and SNR and RMSE are used as evaluation indexes. The results show that the proposed algorithm has better performance. In the experiment of hydrophone, the denoising ability of the proposed algorithm is also verified.


Introduction
Thanks to the abundant resources in the ocean, oceanic exploration has received widespread attention [1,2].Underwater acoustic technology is the main means of obtaining information in the ocean due to its advantages in transmission range [3], propagation speed, and energy loss.Since accurate underwater acoustic signals play an important role in the detection and localization, underwater acoustic communication, and data transmission, hydrophones have become a research hotspot as devices for receiving underwater acoustic signals.Influenced by the complex marine environment including waves, marine organisms, and ships, hydrophone signals [4,5] inevitably contain noise, which poses a challenge for acquiring information in the ocean.Thus denoising the hydrophone signals to obtain useful signals has important significance.
The noise in underwater acoustic signals is usually strong random and time-varying in complex and variable marine environments, which brings huge challenges to the extraction of useful signals.Compared with improving hydrophone hardware structure, denoising algorithms are widely used in underwater acoustic signal processing.Some algorithms including Fourier transform, Kalman filter (KF), and wavelet transform (WT) have good denoising effects [6][7][8], but they inevitably have some inherent defects.The Fourier transform can reflect the relationship between the time function and the spectral function but cannot present the specific details in the signal [9], so it is not applicable to the processing and analysis of non-stationary signals.KF filters and predicts signals by matrix combines improved SGMD with wavelet thresholds.Specifically, the contributions in this paper are as follows: 1.
This paper is the first to optimize the termination conditions in SGMD to improve its ability to remove noise signals.

2.
For SGMD generating excessive symplectic geometry components (SGCs), this paper uses spectral clustering to categorize the SGC and efficiently aggregate the noisy signals.

3.
The mixed signals are denoised by WT and the clusters are reconstructed to get the ideal denoised signal.
Section 2 introduces the algorithms and models used in this paper, including improved SGMD, spectral clustering, spectral entropy, and joint denoising algorithms.Section 3 illustrates the advantages of the proposed method in this paper by comparing the experimental results.Finally, Section 4 gives the conclusion.

Algorithms and Models
In signal denoising, a single denoising method makes it difficult to achieve the goal ideally, so the denoising algorithm combining many methods has become the main means.In order to realize the noise reduction for non-stationary signals and ensure the fidelity of the signals, this paper adopts the algorithm combining SGMD and WT to process the signals.For pre-denoising the signal during signal decomposition and obtaining better decomposition results, it introduces the energy contribution (EC) to improve SGMD.In addition, spectral clustering (SC) is used to cluster the decomposed signals according to signal structure features.Based on the idea of classification processing, spectral entropy (SE) is introduced to classify the clustered signals.Figure 1 shows the framework of the algorithm combining many methods.
noising and fault diagnosis of gears.Yu et al. [31] combined SGMD with EMD to accomplish the denoising and debris feature extraction for inductive oil chip sensors.The combination of multiple algorithms to efficiently realize signal denoising is ge ing more attention.
In order to effectively remove the noise signals from the hydrophone signals, this paper conducts some exploratory research.This paper proposes a denoising algorithm that combines improved SGMD with wavelet thresholds.Specifically, the contributions in this paper are as follows: 1.This paper is the first to optimize the termination conditions in SGMD to improve its ability to remove noise signals.2. For SGMD generating excessive symplectic geometry components (SGCs), this paper uses spectral clustering to categorize the SGC and efficiently aggregate the noisy signals.3. The mixed signals are denoised by WT and the clusters are reconstructed to get the ideal denoised signal.
Section 2 introduces the algorithms and models used in this paper, including improved SGMD, spectral clustering, spectral entropy, and joint denoising algorithms.Section 3 illustrates the advantages of the proposed method in this paper by comparing the experimental results.Finally, Section 4 gives the conclusion.

Algorithms and Models
In signal denoising, a single denoising method makes it difficult to achieve the goal ideally, so the denoising algorithm combining many methods has become the main means.In order to realize the noise reduction for non-stationary signals and ensure the fidelity of the signals, this paper adopts the algorithm combining SGMD and WT to process the signals.For pre-denoising the signal during signal decomposition and obtaining be er decomposition results, it introduces the energy contribution (EC) to improve SGMD.In addition, spectral clustering (SC) is used to cluster the decomposed signals according to signal structure features.Based on the idea of classification processing, spectral entropy (SE) is introduced to classify the clustered signals.Figure 1 shows the framework of the algorithm combining many methods.

Improved Symplectic Geometry Model Decomposition
SGMD solves the eigenvalues of the Hamiltonian matrix by the symplectic geometry similarity transform and uses the corresponding eigenvectors to reconstruct the symplectic geometry components (SGCs) [29].SGMD uses a nonlinear transform and is suitable for the analysis of nonlinear signals.It can decompose a nonlinear signal and the decomposed sub-signal is nonlinear.The iteration termination condition in SGMD uses normalized mean square error (NMSE), which may cause overfitting of signal decomposition.Thus, this paper uses EC as the iteration termination condition to improve SGMD.The flowchart of the SGMD and improved SGMD (ISGMD) methods is shown in Figure 2.

Improved Symplectic Geometry Model Decomposition
SGMD solves the eigenvalues of the Hamiltonian matrix by the symplectic geometry similarity transform and uses the corresponding eigenvectors to reconstruct the symplectic geometry components (SGCs) [29].SGMD uses a nonlinear transform and is suitable for the analysis of nonlinear signals.It can decompose a nonlinear signal and the decomposed sub-signal is nonlinear.The iteration termination condition in SGMD uses normalized mean square error (NMSE), which may cause overfi ing of signal decomposition.Thus, this paper uses EC as the iteration termination condition to improve SGMD.The flowchart of the SGMD and improved SGMD (ISGMD) methods is shown in Figure 2. The trajectory matrix  is mapped by Taken's embedding theorem [29].
In Equation (1),  is the embedding dimension and  is the delay time, and  =  − ( − 1).The embedding dimension  and the delay time  directly affect the trajectory matrix .When  = 1, the trajectory matrix  is a Hankel matrix, and the elements of the matrix that are symmetric about the diagonal are the same [32].The trajectory matrix  should remain a Hankel matrix, and the delay time in phase space should be 1.In this paper, the embedding dimension  is determined by using the power spectral density (PSD), and the frequency of the maximum peak  is estimated by the PSD of the initial time series .During the iteration, the embedding dimension  is set to  3 ⁄ if the normalization frequency is less than the given threshold 10 .Otherwise,  is set to  = 1.2 × ( / ) , where  represents the sampling frequency, and  takes an integer value less than  if not an integer value.

Symplectic geometry similarity transformation
The trajectory matrix  is analyzed by autocorrelation to obtain the covariance symmetry matrix  =   , and a Hamilton matrix is constructed [33]:  Phase space reconstruction x = {x 1 , x 2 , ..., x n } denotes the original time series, where n represents the signal length.The trajectory matrix X is mapped by Taken's embedding theorem [29].
In Equation (1), k is the embedding dimension and τ is the delay time, and m = n − (k − 1)τ.The embedding dimension k and the delay time τ directly affect the trajectory matrix X.When τ = 1, the trajectory matrix X is a Hankel matrix, and the elements of the matrix that are symmetric about the diagonal are the same [32].The trajectory matrix X should remain a Hankel matrix, and the delay time in phase space should be 1.In this paper, the embedding dimension k is determined by using the power spectral density (PSD), and the frequency of the maximum peak f max is estimated by the PSD of the initial time series x.During the iteration, the embedding dimension k is set to k/3 if the normalization frequency is less than the given threshold 10 −3 .Otherwise, k is set to k = 1.2 × ( f max / f s ), where f s represents the sampling frequency, and k takes an integer value less than k if not an integer value.

Symplectic geometry similarity transformation
The trajectory matrix X is analyzed by autocorrelation to obtain the covariance symmetry matrix A = X T X, and a Hamilton matrix is constructed [33]: According to Equation (2), set C = B 2 , and both C and B are Hamiltonian matrices by the definition of Hamiltonian matrix.Thus, the symplectic orthogonal matrix Q can be constructed as Equation (3): where the matrix Q is a symplectic matrix containing the orthogonality [34].During the symplectic transformation, due to the structural features of the Hamiltonian matrix being maintained, the transformed matrix is a Hamiltonian matrix.D is the upper triangular matrix, i.e., h ij = 0(i > j + 1).The matrix C can transform to the matrix D by using the Schmidt transform and the eigenvalues of the matrix D can be obtained by the QR algorithm as Equation (4): Assuming that A is a real symmetric matrix, the eigenvalues of A are the same as those of D, and the eigenvalues λ(X) of X are the square roots of σ(D): Sorting the eigenvalues of A by order as Equation ( 5): The matrix Q represents the symplectic eigenvectors of A and ) are the eigenvectors corresponding to the eigenvalues σ i of the matrix A. Let S = Q T X, Z = QS, and Z is the reconstructed trajectory matrix.i-th row in the transformed coefficient matrix S i is denoted as Equation ( 7): The corresponding reconstruction matrix can then be expressed as Equation ( 8): where Z i is the reconstructed single-component matrix.Thus, the reconstructed trajectory matrix in phase space is Equation (9) [29]: 3.

Diagonal averaging
The reconstructed phase space matrix Z can be transformed by diagonal averaging to a sum of k symmetric geometric components with length n.
Define the dimension of the reconstructed trajectory matrix as m × k and reorder the time series of length n using the diagonal averaging technique.The original time series x is decomposed into d time series.Defining matrix [35] as Equation ( 10): According to the diagonal averaging formula, the matrix Z i is converted to a series Y i (y 1 , y 2 , • • • , y k ) with the corresponding length.

Component reconstruction
The initial components are not completely independent, and there may be some relationship between them.Analyzed from a frequency perspective, component signals with similar frequencies can be considered to come from the same original signal.First, the first symplectic geometric component SGC1 is composed by Y 1 and other component signals containing a similar frequency to Y 1 .Then SGC1 is removed from the initial signal and the remaining components are represented by g 1 .However, the original signal usually Sensors 2024, 24, 1340 6 of 19 contains noise signals, which can affect the signal reconstruction, so a termination condition is required for signal reconstruction.The NMSE between the residual signal and the original signal is usually set as the termination condition.The remaining signal will stop decomposition when the NMSE is less than the set threshold h = 10 −2 .The NMSE equation is as Equation (11): where h represents the number of iterations and g h represents the remaining signal.The initial signal is eventually decomposed into the sum of several component signals and the residual component signal g N+1 as Equation ( 12): where N represents the number of decomposed component signals.

Improved Component Reconstruction
The termination conditions in SGMD during the formation reorganization affect the number of SGCs and the ability of signal pre-denoising.NMSE, as the termination condition for SGMD, cannot deal with localized noise in the signal and may overfit the signal during denoising.For SGMD to achieve better signal pre-denoising, the energy contribution is used as the termination condition during the iteration.Energy contribution as the termination condition has many advantages [36,37], including a reasonable number of modes, better pre-denoising effect, taking into account both global and local features of the signal, and more adaptation to the signal structure. The where k represents the signal length.Then the total energy for the signal is calculated [38] as Equation ( 13): Setting the energy contribution as the termination condition, and the remaining signal will stop decomposition when the EC is less than the set threshold Et = 10 −3 .The energy contribution formula is as Equation ( 14): where h represents the number of iterations and g(e) represents the remaining signal.The initial component signal is finally decomposed as SGMD uses NMSE as an iterative termination condition, which focuses more on the similarity with the original signal, and the local features of the information cannot be represented.When the noise fluctuates violently, the decomposed signal also fluctuates.ISGMD uses the EC as an iterative termination condition, which can well extract the local features of the signal and can make the decomposition of the signal more stable.

Spectral Clustering
Spectral clustering can process the datasets and converge to a globally optimal solution by graph theory.A given dataset A = a i a i ∈ R d , i = 1, 2, • • • , k , uses all the samples in graph A to construct a vertex set Y and set it into an undirected graph G = (Y, B, W) [39].In a graph G, any two vertices can generate an edge, and all edges construct a set B. The element w ij in the affinity matrix W is called the affinity factor.The affinity factor indicates the similarity between y i and y j , and the affinity factor w ij > 0. The element w ij in the affinity matrix W can be computed by using the similarity function [39]: In Equation ( 15), d represents the Euclidean distance between two points, and the scale parameter σ > 0.
Set D = diag(d 11 , d 22 , • • • , d kk ) to be the degree matrix and diag represents the diago- nal matrix, where d ii = ∑ k j=1 w ij .L = D − W is the Laplacian matrix, and then computing the normalized graph Laplacian [40] as Equation ( 16): SGMD decomposition produces an excessive number of SGCs, and processing all of them consumes a large amount of computational resources.In order to quickly distinguish the information component from the noise component, the SGCs are processed by using the spectral clustering approach.The similarity metrics and graph structures on which SGMD and spectral clustering are based fit very well, which makes it easier to aggregate SGCs together based on similarity.

Spectral Entropy
The power spectrum of a signal is the cornerstone for spectral entropy [41], and spectral entropy is used to describe the irregularities of the signal spectrum.The spectral entropy utilizes the Fourier transform to calculate the energy distribution in the domain and combines it with the Shannon entropy to obtain the corresponding spectral entropy value [42].The corresponding power spectrum of the signal is obtained by using the discrete Fourier transform [43].At a certain frequency, the power spectrum possesses power w i and the probability of possessing power at this frequency is as Equation ( 17): Let the length of the signal be k.The summation executes from i = 1 to i = k 2 during using the discrete Fourier transform, and the power is normalized.The entropy of the power spectrum is denoted as Equation ( 18): where f l and f h are the lower limit and upper limit, respectively.Normalizing the spectral entropy to get the normalized spectral entropy [43] as Equation ( 19): where Calculating the spectral entropy of a signal will help quickly distinguish the complexity of the signal.The more complex the signal, the greater the value of spectral entropy.By calculating the spectral entropy of a signal, the noise signal, mixed signal, and useful signal are quickly discriminated, which helps to denoise the signal more accurately.

Wavelet Threshold Denoising
The wavelet transform is the basis for wavelet thresholding.The mixed signal is decomposed by wavelet decomposition to produce different wavelet coefficients and the wavelet coefficients of the useful signals are larger than those of the noisy signals [44].According to the theory, the mixed wavelet coefficients are processed by using a suitable threshold, and the wavelet coefficients of the useful signal are reconstructed to obtain the denoised signal.The steps of the wavelet threshold algorithm are as follows [45]:

Wavelet Decomposition
The effect of signal decomposition relies on the choice of wavelet basis function, and a suitable wavelet basis function will give the ideal denoising effect.The characteristics of the signal need to be considered when the signal is processed by the wavelet threshold algorithm.The error in the signal and theoretical result is usually processed by wavelet analysis as a criterion for evaluating the quality of the wavelet basis function.

Threshold for Wavelet Coefficients
The mixed wavelet coefficients are classified by a given threshold.The amplitude of the wavelet coefficients generated by the useful signal is larger than a given threshold, which needs to be reasonably preserved or reduced.The amplitude of the wavelet coefficients generated by the noise signal is smaller than a given threshold, which should be discarded.The choice of wavelet threshold affects the effectiveness of signal denoising.If the wavelet threshold is too large, useful information may be discarded; if the wavelet threshold is too small, noisy signals may be retained due to poor denoising.

Wavelet Reconstruction
The denoised signal can be obtained by inverse wavelet transform on the processed wavelet coefficients.Compared to the noise signal, the useful signal has continuity in the time domain and the amplitude of the wavelet coefficients for the useful signal is greater than the amplitude of the wavelet coefficients for the noise signal in the wavelet domain, so the wavelet transform can separate the noise signal from the useful signal.If the wavelet threshold method is used for denoising, a reasonable threshold should be chosen.Methods of threshold selection include fixed threshold estimation, heuristic threshold estimation, unbiased likelihood estimation, etc. Usually, fixed threshold estimation and heuristic threshold estimation make it easier to remove useful signals from mixed signals.Thus, unbiased likelihood estimation is chosen as the threshold selection method.The steps are as follows [46]: (1) Obtain the absolute value of each element in the signal y(j) and arrange them in ascending order to obtain a new signal sequence s(j) as Equation (20): (2) If the threshold (Th) uses the square root of the elements in the sequence s(j), the corresponding risk accompanying this threshold is risk(j) as Equations ( 21) and ( 22): (3) Find the point on the risk curve where the risky value is minimized and mark it as j min .Thus, the minimum threshold is noted as Equation ( 23): The uniform threshold is based on a Gaussian noise model as Equation ( 24): Sensors 2024, 24, 1340 9 of 19 where N is the signal length and σ is the standard deviation for the noise.The standard deviation for the noise is usually estimated by using Equation ( 25): where v is the wavelet coefficients and median represents the median function.
A threshold is the basis for the threshold function, and a suitable threshold function can ideally filter the mixed wavelet coefficients.Threshold functions are categorized into soft and hard threshold functions.They both can discard wavelet coefficients smaller than a given threshold as noise.When the wavelet coefficients are greater than a given threshold, the hard threshold function maintains the wavelet coefficients, while the soft threshold function subtracts the threshold from the original value.The soft threshold function makes the wavelet coefficients have better continuity, so in this paper the soft threshold function is chosen.
The hard threshold function [47] is denoted as Equation ( 26): The soft threshold function [47] is denoted as Equation ( 27): Using wavelet thresholding for mixed signals further removes the noise signals from the mixed signals, which enhances the extraction of useful signals.

The Proposed Joint Denoising Algorithm (ISGMD-WT)
In the output signal of a hydrophone, the noise signal often swamps the useful signal.Keeping more useful signals while reducing noise is the pursued aim.This paper proposes a novel denoising algorithm combining ISGMD, SC, SE, and WT.The algorithm steps are as follows: Step 1: Optimize the SGMD The SGMD algorithm has a significant advantage for the decomposition of nonstationary signals and pre-noise reduction is accomplished during decomposition.SGMD using NMSE as the termination condition for decomposition hardly obtains an ideal prenoising effect.Thus, SGMD uses energy contribution to optimize the termination condition in this paper.The ISGMD flowchart is shown in Figure 2.
Step 2: Symplectic geometry model decomposition The hydrophone signals were decomposed by ISGMD to obtain a series of SGCs.The SGCs contain both useful and noise signals.The number of SGCs produced by SGMD is usually high, so this paper introduces spectral clustering to categorize SGCs into clusters.
Step 3: Cluster To reduce the number of decomposed signals and quickly distinguish useful SGCs, the spectral clustering algorithm is adopted to cluster the SGCs.Spectral clustering is able to handle nonlinear data structures and aggregate noisy signals into the same cluster.In order to denoise purposely, this paper introduces spectral entropy to classify the aggregated clusters.Step 5: Denoise and Reconstruct Clusters are categorized into noise clusters, mixed clusters, and information clusters by spectral entropy.The noise clusters are directly discarded and the noise-free clusters are obtained by using wavelet thresholding to denoise the mixed clusters.The noise-free clusters and information clusters are reconstructed to obtain the denoised signal.

Simulation and Application
In order to verify the effectiveness and feasibility of the improved SGMD and wavelet threshold (ISGMD-WT) algorithm proposed in denoising hydrophone signals, two analog signals and one hydrophone signal are processed and analyzed in this paper.Signals in the ocean are complex and variable, so the two analog signals simulate a variety of underwater noise environments.The analog signal 1 is to explore the denoising capability of the ISGMD-WT algorithm in an environment containing signals of different frequencies and white noise.Analog signal 2 is to explore the denoising ability of the ISGMD-WT algorithm in an environment with different white noise intensities.To illustrate the advantages of the proposed algorithm in signal denoising, in the experiments, it is compared with the eminent signal denoising algorithms including the wavelet threshold (WT) algorithm, empirical modal decomposition and wavelet threshold (EMD-WT) algorithm, and the variational modal decomposition and wavelet threshold (VMD-WT) algorithm.It is displayed in the experimental results intuitively by using a signal-to-noise ratio (SNR) and root mean square error (RMSE) as evaluation indexes.

Simulation 1
There are many different frequencies of noise in the ocean, and denoising the mixed noise containing different frequencies is an essential operation to extract the target signal.In this simulation experiment, the target signal is mixed with noise signals of other frequencies and Gaussian white noise is added.In the noise, the line spectrum signal of different frequencies represents the ship noise and the underwater acoustic communication signals during operation, and the white noise comes from marine life and wind waves.The mixed signals are as follows: (28) where the target signal f 1 has a frequency of 50 Hz, f 2 , f 3 , f 4 , f 5 , and f 6 are line spectrum signals with different frequencies.The target and mixed signals are displayed in Figure 3.The denoising results of the four denoising algorithms are shown in Figure 4.The denoising evaluation indexes of the four algorithms are recorded in Table 1.
⎩  =  +  +  +  +  +  where the target signal  has a frequency of 50 Hz,  ,  ,  ,  , and  are line spectrum signals with different frequencies.The target and mixed signals are displayed in Figure 3.The denoising results of the four denoising algorithms are shown in Figure 4.The denoising evaluation indexes of the four algorithms are recorded in Table 1.From Figure 4, it can be known that all four algorithms can realize the denoising for the signal.However, the signals obtained by denoising using the WT, EMD-WT, and VMD-WT algorithms still contain significant noise, and the denoised signals possess fluctuations on the whole.Combined with the data in Table 1, the outstanding advantages of the ISGMD-WT in signal denoising can be clearly seen.The ISGMD-WT algorithm ensures the waveform smoothness during signal denoising, and the waveform remains highly  From Figure 4, it can be known that all four algorithms can realize the denoising for the signal.However, the signals obtained by denoising using the WT, EMD-WT, and VMD-WT algorithms still contain significant noise, and the denoised signals possess fluctuations on the whole.Combined with the data in Table 1, the outstanding advantages of the ISGMD-WT in signal denoising can be clearly seen.The ISGMD-WT algorithm ensures the waveform smoothness during signal denoising, and the waveform remains highly consistent with the target signal.The proposed algorithm is significantly superior to the comparison algorithm in terms of SNR and RMSE.

Simulation 2
The complex and diverse noise in the ocean leads to the possibility that hydrophones may receive underwater acoustic signals that contain noise of different intensities.In the noise, the line spectrum signal represents the signal during the vehicle operation, and white signals of different intensities come from marine life and the variable marine environment.To simulate a mixed signal containing variable noise intensity, the simulated signal is set as follows: where f 1 is the target signal and f 2 is the line spectrum signal with a frequency of 150 hz.f 3 is a noise signal, and the A is set to be adjustable in order to satisfy the demand of different noise levels.f is the mixed signal (original signal) received by the hydrophone.In the paper, the noise signals are set with Gaussian white noise of 0 db, 5 db, and 10 db, where the db of white noise is the ratio of the target signal ( f 1 ) and the line spectrum signal ( f 2 ) to the white noise ( f 3 ), while the SNR of the original signal is the ratio of the target signal ( f 1 ) to the white noise ( f 3 ).In Figure 5, Figure 5a shows the mixed signal with different noised decibels in the time domain, and Figure 5b shows the spectrogram of the target signal.Figures 6-8 show the spectrograms of the denoising signals denoised by the four algorithms.
where  is the target signal and  is the line spectrum signal with a frequency of 150 hz. is a noise signal, and the  is set to be adjustable in order to satisfy the demand of different noise levels. is the mixed signal (original signal) received by the hydrophone.
In the paper, the noise signals are set with Gaussian white noise of 0 db, 5 db, and 10 db, where the db of white noise is the ratio of the target signal ( ) and the line spectrum signal ( ) to the white noise ( ), while the SNR of the original signal is the ratio of the target signal ( ) to the white noise ( ).In Figure 5, Figure 5a shows the mixed signal with different noised decibels in the time domain, and Figure 5b shows the spectrogram of the target signal.2, it can be intuitively seen that compared with the other algorithms, the proposed algorithm possesses eminent denoising ability and the denoised signal is highly consistent with the target signal.The algorithm proposed possesses high signal fidelity in the 0 db and 5 db noise decibel environments.In an environment with 10 db noise, the proposed algorithm essentially obtains the target signal from the mixed signal.It is clear that the denoising ability of the proposed algorithm in this paper is be er than other algorithms.

Application in Hydrophone Experiment
In order to verify the effectiveness of the algorithm in real signals, it is applied to the processing of hydrophone signals.The signals in this paper are received from the Olympus-v389-su hydrophone, which has a center frequency of 500 khz and a sampling frequency of 2 Mhz for collecting underwater acoustic signals.The hydrophone is shown in  From Figures 6-8 and Table 2, it can be intuitively seen that compared with the other algorithms, the proposed algorithm possesses eminent denoising ability and the denoised signal is highly consistent with the target signal.The algorithm proposed possesses high signal fidelity in the 0 db and 5 db noise decibel environments.In an environment with 10 db noise, the proposed algorithm essentially obtains the target signal from the mixed signal.It is clear that the denoising ability of the proposed algorithm in this paper is better than other algorithms.

Application in Hydrophone Experiment
In order to verify the effectiveness of the algorithm in real signals, it is applied to the processing of hydrophone signals.The signals in this paper are received from the Olympus-v389-su hydrophone, which has a center frequency of 500 khz and a sampling frequency of 2 Mhz for collecting underwater acoustic signals.The hydrophone is shown in Figure 9a.The experimental platform is shown in Figure 9b.When the pulsed laser is irradiated on the aluminum block, the surface of the aluminum block breaks and emits ultrasonic waves.The ultrasonic waves are picked up by the hydrophone and used as an input signal for the experiment.3. Figure 12 shows the time-domain waveform graph of the four algorithms after denoising the hydrophone signal, and Figure 13 illustrates the spectrograms of the four algorithms after denoising the hydrophone signal.3. Figure 12 shows the time-domain waveform graph of the four algorithms after denoising the hydrophone signal, and Figure 13 illustrates the spectrograms of the four algorithms after denoising the hydrophone signal.3. Figure 12 shows the time-domain waveform graph of the four algorithms after denoising the hydrophone signal, and Figure 13 illustrates the spectrograms of the four algorithms after denoising the hydrophone signal.

Figure 1 .
Figure 1.The framework of the algorithm combining many methods.Figure 1.The framework of the algorithm combining many methods.

Figure 1 .
Figure 1.The framework of the algorithm combining many methods.Figure 1.The framework of the algorithm combining many methods.

Step 4 :
Calculate and classifyIn order to purposely remove the noisy signals, this paper uses spectral entropy to partition the clusters.The spectral entropy measures the randomness and complexity of the signal.By calculating the spectral entropy, clusters can be categorized into noise clusters, mixed clusters, and information clusters.Noisy signals are mainly concentrated in noise clusters, while useful signals are mainly concentrated in information clusters.Mixed clusters contain many noise signals and useful signals.

Figure 3 .
Figure 3. Target and mixed signals, spectrogram of the target signal.(a) Target and mixed signals.(b) Spectrogram of the target signal.

Figure 3 .Figure 4 .
Figure 3. Target and mixed signals, spectrogram of the target signal.(a) Target and mixed signals.(b) Spectrogram of the target signal.Sensors 2023, 23, x FOR PEER REVIEW 11 of 20

Figure 5 .
Figure 5.The original signal and the mixed signals with different noise decibels.

Figure 9a .Figure 9 .
Figure9a.The experimental platform is shown in Figure9b.When the pulsed laser is irradiated on the aluminum block, the surface of the aluminum block breaks and emits ultrasonic waves.The ultrasonic waves are picked up by the hydrophone and used as an input signal for the experiment.

Figure 10 Figure 10 .
Figure 10 shows the time-domain signal received by the hydrophone and the spectrogram of the signal.The hydrophone signal is divided into two main parts: the static phase, and the signal reception phase.The received signal in the static phase consists entirely of noise, and the signal in the signal reception phase consists of the noise and useful signals.In this paper, hydrophone signals are decomposed by the ISGMD algorithm and SGCs are clustered by using spectral clustering.The clusters are computed using spectral entropy, discarding the noise clusters and denoising the mixed clusters by wavelet thresholding to finally obtain the denoised signal.A time-domain diagram of the clusters is presented in Figure 11, which visualizes the waveform and amplitude of each cluster.

Figure 10
Figure 10 shows the time-domain signal received by the hydrophone and the spectrogram of the signal.The hydrophone signal is divided into two main parts: the static phase, and the signal reception phase.The received signal in the static phase consists entirely of noise, and the signal in the signal reception phase consists of the noise and useful signals.In this paper, hydrophone signals are decomposed by the ISGMD algorithm and SGCs are clustered by using spectral clustering.The clusters are computed using spectral entropy, discarding the noise clusters and denoising the mixed clusters by wavelet thresholding to finally obtain the denoised signal.A time-domain diagram of the clusters is presented in Figure 11, which visualizes the waveform and amplitude of each cluster.

Figure 9a .Figure 9 .
Figure9a.The experimental platform is shown in Figure9b.When the pulsed laser is i radiated on the aluminum block, the surface of the aluminum block breaks and emits u trasonic waves.The ultrasonic waves are picked up by the hydrophone and used as a input signal for the experiment.

Figure 10
Figure 10 shows the time-domain signal received by the hydrophone and the spe trogram of the signal.The hydrophone signal is divided into two main parts: the stat phase, and the signal reception phase.The received signal in the static phase consists e tirely of noise, and the signal in the signal reception phase consists of the noise and usef signals.In this paper, hydrophone signals are decomposed by the ISGMD algorithm an SGCs are clustered by using spectral clustering.The clusters are computed using spectr entropy, discarding the noise clusters and denoising the mixed clusters by wavelet thres olding to finally obtain the denoised signal.A time-domain diagram of the clusters is pr sented in Figure 11, which visualizes the waveform and amplitude of each cluster.

Figure 10 .
Figure 10.The time-domain signal received by the hydrophone and the spectrogram of signal.( Time-domain signal.(b) The spectrogram of signal.

Figure 10 .
Figure 10.The time-domain signal received by the hydrophone and the spectrogram of signal.(a) Time-domain signal.(b) The spectrogram of signal.

Figure 11 .
Figure 11.The clusters generated by spectral clustering.The clusters generated by ISGMD and spectral clustering are shown in Figure 9.The ISGMD decomposes the signal into 51 SGCs, which are divided into 20 clusters by spectral clustering, and the clusters are computed using spectral entropy.The noisy signal is mainly in cluster 18, for which the noise-reduction process using wavelet thresholding is used to obtain the noise-reduced cluster.Signal reconstruction obtains the denoised hydrophone signal.The denoising signals generated by the four algorithms are shown in Figures 12 and 13 and the evaluation indexes are recorded in Table3.Figure12shows the time-domain waveform graph of the four algorithms after denoising the hydrophone signal, and Figure13illustrates the spectrograms of the four algorithms after denoising the hydrophone signal.

Figure 11 .
Figure 11.The clusters generated by spectral clustering.The clusters generated by ISGMD and spectral clustering are shown in Figure 9.The ISGMD decomposes the signal into 51 SGCs, which are divided into 20 clusters by spectral clustering, and the clusters are computed using spectral entropy.The noisy signal is mainly in cluster 18, for which the noise-reduction process using wavelet thresholding is used to obtain the noise-reduced cluster.Signal reconstruction obtains the denoised hydrophone signal.The denoising signals generated by the four algorithms are shown in Figures 12 and 13 and the evaluation indexes are recorded in Table3.Figure12shows the time-domain waveform graph of the four algorithms after denoising the hydrophone signal, and Figure13illustrates the spectrograms of the four algorithms after denoising the hydrophone signal.

20 Figure 11 .
Figure 11.The clusters generated by spectral clustering.The clusters generated by ISGMD and spectral clustering are shown in Figure 9.The ISGMD decomposes the signal into 51 SGCs, which are divided into 20 clusters by spectral clustering, and the clusters are computed using spectral entropy.The noisy signal is mainly in cluster 18, for which the noise-reduction process using wavelet thresholding is used to obtain the noise-reduced cluster.Signal reconstruction obtains the denoised hydrophone signal.The denoising signals generated by the four algorithms are shown in Figures 12 and 13 and the evaluation indexes are recorded in Table3.Figure12shows the time-domain waveform graph of the four algorithms after denoising the hydrophone signal, and Figure13illustrates the spectrograms of the four algorithms after denoising the hydrophone signal.

Figure 12 .
Figure 12.Denoised signals generated by the four algorithms.Figure 12. Denoised signals generated by the four algorithms.

Figure 12 .
Figure 12.Denoised signals generated by the four algorithms.Figure 12. Denoised signals generated by the four algorithms.

Table 1 .
Evaluation indexes for different denoising algorithms.

Table 1 .
Evaluation indexes for different denoising algorithms.

Table 2
records the denoising evaluation indexes of the four algorithms under different noise intensities.

Table 2 .
Denoising evaluation indexes of the four algorithms under different noise decibels.

Table 2 .
Denoising evaluation indexes of the four algorithms under different noise decibels.