Denoise for propeller acoustic signals based on the improved wavelet thresholding algorithm of CEEMDAN

Abstract

The primary objective of this study is to address the issue of noise in underwater propeller signals. The motivation for this research arises from the challenge of distinguishing signal and noise components in underwater propeller signals, particularly in the presence of complex hydrodynamic interactions. So it introduces an improved wavelet threshold algorithm based on CEEMDAN for noise reduction in underwater propeller signals. The algorithm utilizes the energy concentration property of wavelet transform to distinguish signal and noise components. And by setting the noise coefficients based on the Balanced Wavelet Thresholding (BWT) algorithm, effective denoising is achieved. Compared to the traditional wavelet threshold method, the study evaluated three approaches, EMD-CWT, EMD- BWT, and CEEMDAN-BWT for denoising numerical simulation and experimental signals of propeller cavitation wake acoustic pressure generated using CFD software. The findings show that the CEEMDAN-BWT outperforms others in noise reduction. The research results indicate that the CEEMDAN-BWT algorithm performs superiorly in noise reduction. The Signal-to-Noise Ratio (SNR) has been improved by 2.4326 decibels, and the Root Mean Square Error (RMSE) has increased by 0.0893 decibels. The algorithm effectively preserves signal components, enhancing SNR without introducing excessive distortion, demonstrating its potential in significantly reducing propeller signal noise levels and improving signal quality.

Keywords:

• Improving wavelet threshold algorithm
• CEEMDAN
• noise reduction
• propeller signal
• CFD

Reviewing Editor:

• Pham D T, University of Birmingham, United Kingdom of Great Britain and Northern Ireland

SUBJECTS:

• Fluid Mechanics
• Mechanical Engineering
• Electromagnetics & Communication

1. Introduction

Underwater propeller noise, especially cavitation noise, constitutes a significant research focus in the field of underwater acoustics. It plays a vital role in the detection, classification, and identification of underwater targets. Because of the multiple interferences and low signal-to-noise ratio (SNR) experienced during the transmission of underwater propeller noise, effective noise reduction (Erzheng F, Zhihao H and Chengyao G., 2020) is of paramount importance. Currently, the wavelet threshold processing algorithm, as a commonly used denoising method (DONG Xin, LI Guolong, HE Kun, JIA Yachao, XU Kai, LI Biao, 2020), is widely adopted. However, its effectiveness is less than ideal when dealing with non-stationary propeller noise. Therefore, this study is dedicated to enhancing the wavelet threshold algorithm to improve its performance in reducing non-stationary propeller noise. It is specific significant for target recognition based on propeller noise.

To solve the problems of the discontinuity, no asymptotic, and no smoothness of the traditional wavelet threshold function at the threshold, the noise images were denoised using improved methods based on the wavelet domain and the spatial domain and a local adaptive wavelet image method based on the wavelet domain (Parmar & Patil, 2013). An improved threshold function was also proposed for image denoising by Tingpeng and Guicang (2023). Compared with the hard threshold, soft threshold and the existing threshold function, the function by Deng is superior mathematical characteristics. And the superiority of the algorithm is verified by peak signal-to-noise ratio and objective evaluation of mean square value error. In order to solve the defects of the hard and soft threshold function in the existing wavelet threshold denoising method and improve the effect, Fei W, Chenhao M and Kun C (2022) studied the denoising principle of different threshold functions, and proposed an improved threshold function applied to wavelet denoising, which has the advantages of hard and soft threshold processing. An algorithm for noise reduction of bearing vibration signal was proposed by Jian, Taiyong, Zhiguo and Yinan. (2023). First, the wavelet decomposition of the noise-containing signal is performed. Second, as the new threshold function, the generalized cross-validation GCV function was used to process the decomposed wavelet coefficient. On the premise of retaining the signal features, this method removes the noise to a large extent and effectively improves the noise reduction effect. However, the simple wavelet threshold method cannot retain the signal characteristics well, which may lose some important information. Therefore, we can use other methods to achieve a better denoising effect.

EMD is a data decomposition method based on the characteristics of the signal itself, first developed by Huang et al. (1998). The goal of EMD is to decompose the signal into a set of adaptive intrinsic mode functions (Intrinsic Mode Functions, IMFs). CEEMDAN is an adaptive signal analysis method, based on empirical mode decomposition (EMD). It can decompose nonlinear non-stationary signals into intrinsic mode function (IMF), and retain the time-frequency characteristics of the signal (Lei Y, Liu Z, Ouazri J, Lin J.A, 2017). In order to achieve better noise reduction and maximize the signal characteristics, this method is often combined with wavelet threshold. For example, A filtering noise reduction method based on the improved wavelet threshold-empirical mode decomposition (EMD) joint algorithm was proposed to retain the effective feature information in the signal and achieve good noise reduction effect by Bowen and Zimeng (2019). Based on empirical mode decomposition (EMD), a new denoising method (K. Khaldi, M. Turki-Hadj Alouane and A.-O. Boudraa, 2008) is introduced in the criteria based on the energy of IMFs used to select the very noisy IMF that must be filtered. The results show that it greatly improves the noise reduction efficiency but may lose some important information. In the signal noise reduction of the ECG by Xu Y, Luo M, Li T and Song G (2017), an ECG signal correction method based on adaptive noise (CEEMDAN) and wavelet threshold was proposed. The results show that the algorithm can suppress random noise in ECG signals and correct the baseline drift. A hybrid method of β climbing combined with wavelet transform was proposed for ECG signal denoising by Alkareem Alyasseri, A. T. Khader, M. A. Al-Betar and L. M. Abualigah (2017). In the underwater acoustic signal denoising, They (Li, Y., Li, Y., Chen, X., Yu, J., Yang, H., Wang, L., 2018) also demonstrated that the method using CEEMDAN can effectively remove underwater noise while retaining features.In order to address underwater communication challenges in the marine environment, Pauline, Narayanamoorthi and Dhanalakshmi (2022) proposed an optimal cascaded adaptive filtering structure for the highest precision denoising of noisy signals using the Symbolic Data Least Mean Square (SDLMS) algorithm. This structure ultimately offers a cost-effective hardware implementation for Adaptive Noise Canceller (ANC). A fully integrated empirical mode decomposition method based on adaptive noise (CEEMDAN) threshold denoising optimized by non-local mean (NLM) algorithm was also proposed by Zhang, Liu, Hu, Jiang, Xu and Hao (2020). Improving the selection of traditional thresholds may destroy the information, and this algorithm may retain the signal characteristics better.

In this article, we propose an improved wavelet thresholding algorithm based on CEEMDAN to eliminate noise from propeller signals. We decompose the propeller signal using CEEMAN to obtain Intrinsic Mode Function (IMF) components at different scales. Subsequently, it employs the Balanced Wavelet Thresholding (BWT) algorithm to effectively denoise the signal based on the energy distribution of each IMF component. Considering the advantages and disadvantages of the soft and hard threshold algorithm, it suppresses the noise, and maintains the detailed characteristics of the signal (Donoho, 1995). The numerical simulation signal adopts the gas-liquid mixed flow model and the acoustic model to realize the simulated noise of the underwater propeller (Zhu, Zhou and Li, 2017). In the article, the effectiveness and superiority of the algorithm is verified to denoising of propeller numerical and measured data.

2. Model and methods

2.1. Gas–liquid mixture flow model

The propeller cavitation is essentially caused by the phase transition between the gas-liquid phases around the propeller running underwater. In the article, an air–liquid mixture flow model (Zhu, 2014) is used to describe the process of bubble and liquid mixing by a five-blade propeller. The model is based on the principles of conservation of mass, momentum and energy, and the interaction between gas and liquid.

Mass conservation equation describes the mass transport between gaseous phase and liquid phase. (1) $$\frac{\partial ({\alpha }_g{\rho }_g)}{\partial t}+\nabla \cdot ({\alpha }_g{\rho }_g{u}_g)=S_g$$(1) where ${\alpha }_g$ represents the volume fraction of the gas phase, ${\rho }_g$ indicates the gas density, $u_g$ represents the velocity vector of the gas phase, and $S_g$ represents the bubble source term.

Momentum conservation equation describes the momentum transport between the gas and liquid phases. (2) $$\frac{\partial ({\alpha }_g{\rho }_g{u}_g)}{\partial t}+\nabla \cdot ({\alpha }_g{\rho }_g{u}_g{u}_g)=-\nabla P_g+\nabla \cdot {\tau }_g+{\alpha }_g{\rho }_{g}g+\nabla \cdot F_b$$(2) where $P_g$ represents the pressure of the gas phase, ${\tau }_g$ is the viscosity stress tensor of the gas phase, $g$ is the gravity vector, and $F_b$ is the force of the bubble on the gas phase.

Energy conservation equation describes the heat transport between the gas and liquid phases. (3) $$\frac{\partial ({\alpha }_g{\rho }_g{h}_g)}{\partial t}+\nabla \cdot ({\alpha }_g{\rho }_g{h}_g{u}_g)=-\nabla \cdot ({\alpha }_g{\rho }_g{k}_g\nabla T_g)+Q_g$$(3) where $h_g,$ it represents the specific enthalpy of the gas phase, $T_g$ represents the temperature of the gas phase, $k_g$ represents the thermal conductivity of the gas phase, and $Q_g$ represents the heat source term of the gas phase.

2.2. Acoustic model

An acoustic model (Zhou, Zhu and Wang, 2023) is used to simulate the generation and propagation of cavitation noise. Based on the acoustic wave propagation equation, it is used to describe the behavior of sound waves in water. The classical Far-Field Wave Equation (FW-H) is utilized to depict the propagation of sound waves in water. The FW-H equation is expressed in the following form: (4) $$\frac{1}{c^2}\frac{{\partial }^{2}p}{\partial t^2}-{\nabla }^{2}p=S$$(4) where $P$ is the sound pressure (the pressure change of the sound wave), $c$ means the sound speed, ${\nabla }^2$ is the Laplacian operator, and $S$ is the sound source term.

2.3. EMD-CWT of algorithm

Based on the principle of wavelet transform, the wavelet threshold function is a commonly used to reduce signal noise. It represents the signal as wavelet coefficients. noise reduction is performed by thresholding these coefficients (Hongqiang, Xiaocheng & Tong, 2023). The basic steps of the method include the following aspects: (1) Wavelet transforms (Feng & Xinmin, 2005). The original signal is converted into wavelet domain through the wavelet transform to obtain a set of wavelet coefficients. (2) Threshold selection. the appropriate threshold criterion is chosen to determine the wavelet coefficient to be retained or to be discarded. There are hard and soft thresholds in common threshold selection methods. (3) Threshold processing. The wavelet coefficient was thresholded according to the chosen threshold criterion. For a hard threshold, coefficients less than the threshold will be zero; for a soft threshold, the coefficient less than the threshold will be reduced. (4) Wavet inverse transform. The processed wavelet coefficient is transformed back to the time domain through the inverse transform to obtain the noise reduction signal.

Let x be the noise signal, $\phi$ be the wavelet basis, and ${\psi }_{j,k}(t)$ be the wavelet basis function of the k-th coefficient in the j-level wavelet decomposition, then c_j,k may be expressed as: (5) $$c_{j,k}=〈x,{\psi }_{j,k}〉={\int }_{-\infty }^{+\infty }x(t)\cdot {\psi }_{j,k}(t)\mathit{\text{dt}}$$(5)

In the wavelet threshold denoising, the two commonly used threshold methods are soft threshold and hard threshold. The soft threshold is: (6) $$T_s\left(c\right)=\mathit{\text{sign}}\left(c\right)•\text{max}(\left|c\right|-\lambda ,0)$$(6)

The hard threshold is: (7) $$T_h(c)=\{\begin{array}{c}c,\mathit{\text{if}}\left|c\right|>\lambda \\ 0,\mathit{\text{if}}\left|c\right|\le \lambda \end{array}$$(7) where $T_s(c)$ represents the soft threshold, $T_h(c)$ represents the hard threshold, $c$ is the wavelet coefficient, $\lambda$ is the wavelet threshold.

The Empirical Mode Decomposition (EMD) is a data-driven adaptive signal decomposition method to decompose signals into a series of intrinsic mode functions (Intrinsic Mode Functions, IMFs). The basic idea is to decompose the signal into vibrational modes with different time scales. Each vibrational mode is a function of clear frequency characteristics and does not contain excess extreme points. IMFs are constructed by iteratively extracting the maxima and minimum points in the signal, where each IMF represents a different vibrational mode. The signal is first EMD decomposed to obtain IMF, and then the traditional wavelet threshold algorithm is applied to each IMF to remove noise.

For the k-th IMF, the signal $x_k(t)$ is first subjected to wavelet transform to obtain wavelet coefficients $w_k(n).$ The traditional thresholding function is then applied to the wavelet coefficients for thresholding: (8) $${w^{\prime }}_k(n)=T(w_k(n),\lambda )$$(8)

In the above equation, $T$ represents the threshold function, λ is the threshold parameter, and the processed signal $x^{\prime }{}_k(t)$ is obtained by performing the inverse transform on the threshold coefficients. The algorithmic steps taken in this article are as follows:

(1) The EMD decomposition of the noise-containing signal y Conducted by using the EMD function to obtain the IMFs and get the number of decomposed IMF. (2) In order to obtain the optimal wavelet parameters, signal-to-noise ratio (SNR), root mean square error (RMSE), and peak signal-to-noise ratio (PSNR) are used as evaluation metrics. The goal is to maximize SNR, minimize RMSE, and maximize PSNR. Through experimental testing, the final configuration was determined as using the sym4 wavelet, with five decomposition levels, employing a soft thresholding approach, and setting the threshold parameter to 0.001. (3) The wavelet decomposition of IMF is conducted, and the wavelet coefficient and length L are obtained using the function wavedec. According to the set soft and hard threshold type parameter, the soft threshold coefficient is denoised, and the threshold treatment coefficient is obtained using the function wthresh. (4) The processed coefficient and length are reconstructed by the function waverec to obtain the denoised IMF. (5) The denoised IMFs are summed to obtain the reconstructed denoising signal. (6) SNR and RMSE are calculated.

2.4. EMD-BWT of algorithm

Based on the energy concentration property of wavelet transform, the improved wavelet thresholding function utilizes the characteristics that the magnitude of wavelet coefficients corresponds to the signal’s features to distinguish the signal from the noise. The function then sets the noise coefficients to zero or modifies them according to certain rules, aiming to achieve denoising. We introduce a parameter $\mathrm{\lambda }$ to control the threshold level. During the wavelet thresholding process, the wavelet coefficients are compared with the threshold value (denoted as $\mathrm{\lambda }$). For the coefficients with an absolute value greater than $\mathrm{\lambda },$ their signs are preserved, and the absolute value is reduced by $\mathrm{\lambda },$ resulting in new wavelet coefficients. For coefficients with an absolute value less than or equal to $\mathrm{\lambda },$ they are set to zero. This method helps to filter out noise to a certain extent, and preserve the main components of the signal, thus improving the signal quality a signal-to-noise ratio. Due to the local and multiresolution properties of wavelet thresholding, it can accurately identify and remove noise of different scales, and retain the detailed features of the signal.

The formula for the improved wavelet thresholding function can be derived through the following steps:

For wavelet coefficients as $\omega$ with an absolute value less than the threshold value $\lambda ,$ they are preserved as: (9) $${\omega }^{\prime }=\omega ,\mathit{\text{if}}\left|\omega \right|\le \lambda$$(9)

For wavelet coefficients as $\omega$ with an absolute value greater than the threshold value $\lambda ,$ they are processed using a hard thresholding approach. This means subtracting a fixed threshold value and multiplying it by the sign of the coefficient: (10) $${\omega }^{\prime }=(\left|\omega \right|-\lambda )•\mathrm{\text{sgn}}\left(\omega \right),\mathit{\text{if}}\left|\omega \right|>\lambda$$(10)

For the wavelet coefficient $\omega$ between the soft and hard threshold, the soft threshold is used, so that it is multiplied by the coefficient: (11) $${\omega }^{\prime }=\frac{\omega }{1+\beta {\left(\frac{\left|\omega \right|}{\lambda }\right)}^2},\mathit{\text{if}}\left|\omega \right|\le \lambda$$(11)

The coefficient was selected for the following reasons: (1) When $\beta =0,$ the coefficient was 1. There is no soft threshold treatment, and it was degraded to a hard threshold method. (2) When $\beta$ approaching $+\infty$ perhaps $-\infty ,$ the coefficient tends to 0. The effect of the soft threshold method is more and more obvious. (3) When $\left|\omega \right|>\lambda ,$ the coefficient is less than 1. The wavelet coefficient is soft threshold treatment, and the larger $\beta$ the more obvious soft threshold effect.

Thus we obtain the combined threshold formula for soft and hard thresholds. This can be called the equilibrium wavelet threshold formula (Balanced Wavelet Thresholding, BWT). (12) $${\omega }^{\prime }=\{\begin{array}{c}(\left|\omega \right|-\lambda )•\text{sgn}\left(\omega \right),\left|\omega \right|>\lambda \\ \frac{\omega }{1+\beta {\left(\frac{\left|\omega \right|}{\lambda }\right)}^2},\left|\omega \right|\le \lambda \end{array}$$(12)

The horizontal axis in Figure 1 shows the value range of the wavelet coefficient, and the vertical axis shows the value of the treated coefficient. The three curves represent the results of hard, soft and improved threshold processing methods respectively. For hard thresholding treatments, coefficients smaller than the threshold are directly set to 0, and coefficients greater than the threshold are retained, thus yielding a sharp edge. For soft thresholding treatments, coefficients smaller than the threshold were directly set to 0, and coefficients greater than the threshold were retained and subtracted from a fixed value, creating a smooth transition. For the improved threshold approach, the results of the treatment are somewhat between the hard threshold and the soft threshold, and the processed wavelet coefficients maintain better smoothing. Compared with the characteristics of the three curves, the improved thresholding method can better remove the noise, and the treated wavelet coefficient has a better transition effect at the edge. The improved wavelet thresholding function can be integrated with the Empirical Mode Decomposition (EMD) to effectively address noise components within the Intrinsic Mode Functions (IMFs). For the k-th IMF: (13) $${w^{\prime }}_k(n)=T^{\prime }(w_k(n),\lambda )$$(13) where T′ is the improved threshold function and λ is the threshold parameter, and the original signal is inverse transformed to obtain the processed signal $x_k{}^{\prime }(t).$ The algorithmic steps taken here refer to the conventional wavelet thresholding algorithm ones above.

Figure 1. Plot of the hard and soft threshold and the improvement threshold.

2.5. CEEMDAN-BWT of algorithm

CEEMDAN achieves more accurate Intrinsic Mode Function (IMF) components by iteratively decomposing the signal multiple times (Ngoyi, 2023). It introduces noise-assisted functions and an adaptive noise algorithm to better handle non-stationary signals and nonlinear noise. Employing CEEMDAN allows for the decomposition of the signal x(t) into multiple IMFs: (14) $$x(t)=\sum c_i(t)+r(t)$$(14) where $c_i(t)$ represents the i-th IMF, and $r(t)$ is the residual component. The improved wavelet thresholding function is applied to each IMF for threshold processing: (15) $${c^{\prime }}_i(t)=T^{\prime }(c_i(t),\lambda ){c^{\prime }}_i(t)=T^{\prime }(c_i(t),\lambda )$$(15) where T′ is the improved threshold function and λ is the threshold parameter. The reconstructed signal is (16) $$x^{\prime }(t)=\sum {c^{\prime }}_i(t)+r(t)$$(16)

The algorithmic steps taken are as follows:

(1) The noise-containing signal y is generated. (2) CEEMDAN decomposition is conducted, CEEMDAN decomposition related parameters are set, including positive and negative Gaussian white noise standard deviation Nstd, NR of the number of noise added, and the maximum number of iterations MaxIter. The function ceemdan is used with CEEMDAN to decompose the noise-containing signal y to obtain the IMFs and the number of decomposition iterations. the size of IMFs is obtained by decomposition. (3) Using the determination method outlined in Section 2.3.The experiment was set to sym4 wavelet, the number of decomposition layers is 5, the threshold is 0.001, and the equilibrium parameter is 0.005. The improved wavelet threshold noise reduction was applied to each IMF, and the improved wavelet threshold function was used to obtain the noise reduction IMFs. (4) The noise-reduction IMFs are summed to obtain the denoised signal Y. (5) The SNR after being denoised and root mean square error RMSE are calculated.

Figure 2 illustrates the algorithmic flowchart, which involves data preprocessing followed by the application of three different algorithms to generate denoised images. Finally, the image with the best denoising effect is selected.

Figure 2. Flow chart of the noise reduction algorithm.

3. Denoising of numerical simulation signals

The above three noise reduction algorithms are used to compare the noise reduction performance of the propeller simulation signals generated in fluent software, and then the measured propeller signals.

3.1. Simulation signal parameter configuration

First, the flow grid around the propeller is established, and there are 5 million grid cells in the grid. Then the digital model of the propeller is introduced into the meshing software GAMBIT for meshing. The unstructured meshing is used. And then the grid is imported into fluid computing software CFD. Specifically, the fluid reference pressure is a standard atmospheric pressure. The saturated vapor pressure is 2.368 Kpa, The fluid sound velocity is 340 m/s, and a fluid density is 998 kg/m³. The diameter of the propeller is 0.992 m. In this article, we simulate the situation of a five-blade propeller running underwater, whose skew angle is 0. The progress coefficient J = 0.4.5000 data points are collected, and the time step is set to 0.0005s. The propeller rotation speed is 15 r/s. Figure 3, (a) represents the computational domain, and (b) depicts the numerical results of propeller cavitation. The simulation results of the sound pressure signal in the propeller cavitation wake are shown in Figure 4.

Figure 3. Model configuration. (a) Computing domain and (b) cavitation numerical results.

Figure 4. Simulation signal time-domain diagram.

The above three different algorithms are used to reduce the noise of the signal, and the signal-to-noise ratio and root mean square error are calculated to illustrate the noise reduction effect.

3.2. Denoising results of simulation signals

According to the three algorithm steps mentioned above for noise reduction respectively, the wavelet selection and the number of decomposition layers are consistent with the propeller simulation signal experiment. Figure 5 shows the denoised image obtained by simulating the signal using the EMD-CWT, Figure 6 represents the denoised image using the EMD-BWT, and Figure 7 illustrates the denoised image using the CEEMDAN-BWT. From these three images, it is evident that Figure 7 exhibits the best denoising effect, significantly improving the quality of the signal.

Figure 5. EMD-CWT noise reduction diagram.

Figure 6. EMD-BWT noise reduction diagram.

Figure 7. CEEMDAN-BWT noise reduction diagram.

From the noise reduction effect diagram of different algorithms, we can see that the CEEMDAN-improved wavelet threshold algorithm is the best, when we calculate their signal/noise ratio and root mean square error for numerical analysis. The calculation results are shown in Table 1.

Table 1. SNR and RMSE of the different noise reduction algorithms.

Algorithm	EMD-CWT	EMD-BWT	CEEMDAN-BWT
SNR(dB)	11.7874	12.4146	14.2200
RMSE	0.3645	0.3020	0.2752

Through the analysis of the results above, it is found in next aspects. (1) The noise-containing signa is effectively denoised by applying EMD decomposition and he traditional wavelet threshold algorithm. (2) EMD decomposition and improved wavelet threshold algorithm are used to denoise the noise-containing signal, and the further noise reduction effect is clear. The signal to noise ratio (SNR) increased by 0.6242 dB and the root mean square error (RMSE) decreased by 0.0625 units. It can more effectively suppress the noise and improve the clarity and quality of the signal. (3) We apply CEEMDAN decomposition and the improved threshold wavelet algorithm to denoise the noise-containing signals, and observe the best noise reduction effect. The signal to noise ratio (SNR) increased by 2.4326 dB and the root mean square error (RMSE) decreased by 0.0893 units. It shows that it can better capture the local features and frequency components of the signal and effectively suppress the effects of noise.

4. Denoising of measured propeller signals

4.1. Measured signal

The data of the propeller comes from the an online website. We collected the data of three different states of the cargo ship. The basic information includes five-blade propeller, the frequency 32,000 Hz, its 9 r/s the departure state, and 12 r/s-23 r/s the driving state data. Their time domain is shown in Figure 8.

Figure 8. Time domain diagram of different rotational speeds.

To better see the noise reduction effect, we used the most serious data signal, namely 23 r/s noise data, to conduct the experimental noise reduction. Figure 9 shows the original signal plot of a section of the experimentally measured propeller signal for 23r/s.

Figure 9. Spiral noise signal.

4.2. Measured signal noise reduction results

Three different algorithms are used to analyze the propeller noise signal respectively. Gaussian white noise is added before CEEMDAN processing. The standard deviation is set to 0.5. The number of decomposed iterations is 100, and the maximum number of iterations is 20. Separate noise reduction is shown in the following results. Figure 10 represents the denoised signal obtained through the EMD-CWT algorithm for the experimentally measured propeller signal. Figure 11 displays the signal denoised using the EMD-BWT algorithm, and Figure 12 illustrates the signal denoised with the CEEMDAN-BWT algorithm. It can be observed that the denoised signal from CEEMDAN-BWT exhibits the best performance, effectively eliminating noise from the propeller acoustic signal while preserving the maximum signal-to-noise ratio.

Figure 10. EMD-CWT of noise reduction effect diagram.

Figure 11. EMD-BWT of noise reduction effect diagram.

Figure 12. CEEMDAN-BWT of noise reduction effect diagram.

Figure 13 shows the IMF component diagram of the measured propeller signal passing through CEEMDAN. IMF1 to IMF8 are high frequency noise, concentrated at around 1 Khz. IMF9 to IMF14 is low frequency noise, concentrated at around 100 Hz. Then these components are improved by the wavelet threshold algorithm to achieve the noise reduction effect.

Figure 13. IMF component plot.

According to the denoising effect diagrams of three different denoising algorithms, it is evident that the CEEMDAN-improved wavelet threshold algorithm exhibits the best noise reduction performance. It achieves the most effective denoising for the measured propeller signals. However, there are still some residual burr phenomena, indicating that complete denoising is not achieved. In future research, when dealing with more complex and larger noise datasets, further optimization of the CEEMDAN-improved wavelet threshold algorithm parameters, especially the selection of thresholds and adjustments to wavelet parameters, is necessary to enhance denoising effectiveness and reduce residual noise.

5. Conclusion

Compared with results of a traditional wavelet thresholding algorithm in the field of signal denoising, an improved method is conducted to experimental validation in this study. Additionally, an enhanced thresholding wavelet algorithm incorporating CEEMDAN decomposition is applied in both simulated signals and actual measured signals from a propeller. The following are our conclusions:

1. Numerical simulation results demonstrate that the improved wavelet thresholding algorithm outperforms the traditional wavelet thresholding algorithm in terms of denoising effectiveness. By adjusting the threshold and balance parameters, we can better suppress noise and enhance the clarity and quality of the signal.

2. Based on two evaluation metrics, SNR and RMSE, for different denoising algorithms, it is observed that the CEEMDAN + improved thresholding wavelet algorithm exhibits the best performance in denoising. It effectively captures the local features and frequency components of the signal, efficiently suppresses noise interference and achieves optimal denoising results.

3. In the experimental measurements of actual signals, the proposed improved denoising algorithm demonstrates a significant reduction in noise for the measured propeller signals. However, it is important to note that the operating conditions of the measured propeller signals in this study are relatively simple, and the dataset is limited. When faced with more complex and larger datasets, the noise characteristics may vary. Consequently, the parameters in the BWT algorithm, including the selection of wavelet parameters and threshold settings, may need to be adjusted differently from those used in this study.Further research is required to address these aspects, exploring potential adjustments and considerations when dealing with more intricate noise scenarios and larger datasets.

Author contribution

Authors Zhu Zhifeng and CAI BOHUA contributed jointly to the conception and design of the data, as well as rigorous editing of the intellectual content of the article. Yao Yong and Wu Chengkun was responsible for the analysis and interpretation of experimental data and made significant contributions to the drafting of the manuscript. All authors reviewed the final version approved for publication and agreed to be accountable for all aspects of the work. Thanks to all contributors for their efforts and collaboration throughout the research process.

Data derived from public domain resources

The data that support the findings of this study are available. These data were derived from the following resources available in the public domain: [https://github.com/irfankamboh/DeepShip].

Disclosure statement

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this article.

Additional information

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Key Research and Development Projects of Anhui Province [No. 2022107020012].