# A New Underwater Acoustic Signal Denoising Technique Based on CEEMDAN, Mutual Information, Permutation Entropy, and Wavelet Threshold Denoising

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. CEEMDAN

- (1)
- Add white noise ${n}_{i}\widehat{(}t)$ to the target signal $x(t)$:$${x}_{i}\widehat{(}t)=x(t)+{n}_{i}\widehat{(}t),\begin{array}{cc}& i=1,\text{}2,\text{}\cdots ,N\end{array}$$
- (2)
- Decompose ${x}_{i}\widehat{(}t)$ by EMD to obtain the first IMF ${c}_{i1}\widehat{(}t)$ and residual mode ${r}_{i}(t)$:$$\left(\begin{array}{c}{x}_{1}\widehat{(}t)\\ {x}_{2}\widehat{(}t)\\ \cdots \\ {x}_{i}\widehat{(}t)\\ \cdots \\ {x}_{N}\widehat{(}t)\end{array}\right)\stackrel{\mathrm{EMD}}{\to}\left(\begin{array}{cc}{c}_{11}\widehat{(}t)& {r}_{1}(t)\\ {c}_{21}\widehat{(}t)& {r}_{2}(t)\\ \cdots & \cdots \\ {c}_{i1}\widehat{(}t)& {r}_{i}(t)\\ \cdots & \cdots \\ {c}_{N1}\widehat{(}t)& {r}_{N}(t)\end{array}\right)$$
- (3)
- Obtain the first IMF of CEEMDAN by calculating the mean of ${c}_{i}{}_{1}\widehat{(}t)$:$${c}_{1}\widehat{(}t)=\frac{1}{N}{\displaystyle \sum _{i=1}^{N}{c}_{i}{}_{1}\widehat{(}t)}$$
- (4)
- Obtain the residual mode of ${c}_{1}\widehat{(}t)$:$${r}_{1}\widehat{(}t)=x(t)-{c}_{1}\widehat{(}t)$$
- (5)
- Decompose white noise ${n}_{i}\widehat{(}t)$ by EMD:$$\left(\begin{array}{c}{n}_{1}\widehat{(}t)\\ {n}_{2}\widehat{(}t)\\ \cdots \\ {n}_{i}\widehat{(}t)\\ \cdots \\ {n}_{N}\widehat{(}t)\end{array}\right)\stackrel{\mathrm{EMD}}{\to}\left(\begin{array}{ccccc}{c}_{{n}_{1}1}\widehat{(}t)& {c}_{{n}_{1}2}\widehat{(}t)& \cdots & {c}_{{n}_{1}j}\widehat{(}t)& {r}_{{n}_{1}}\widehat{(}t)\\ {c}_{{n}_{2}1}\widehat{(}t)& {c}_{{n}_{2}2}\widehat{(}t)& \cdots & {c}_{{n}_{2}j}\widehat{(}t)& {r}_{{n}_{2}}\widehat{(}t)\\ \cdots & \cdots & \cdots & \cdots & \cdots \\ {c}_{{n}_{i}1}\widehat{(}t)& {c}_{{n}_{i}2}\widehat{(}t)& \cdots & {c}_{{n}_{i}j}\widehat{(}t)& {r}_{{n}_{i}}\widehat{(}t)\\ \cdots & \cdots & \cdots & \cdots & \cdots \\ {c}_{{n}_{N}1}\widehat{(}t)& {c}_{{n}_{N}2}\widehat{(}t)& \cdots & {c}_{{n}_{N}j}\widehat{(}t)& {r}_{{n}_{N}}\widehat{(}t)\end{array}\right)$$$${E}_{1}({n}_{i}\widehat{(}t))={\left(\begin{array}{cccccc}{c}_{{n}_{1}1}\widehat{(}t)& {c}_{{n}_{2}1}\widehat{(}t)& \cdots & {c}_{{n}_{i}1}\widehat{(}t)& \cdots & {c}_{{n}_{N}}\widehat{(}t)\end{array}\right)}^{T}$$
- (6)
- Construct signal $xne{w}_{1}(t)$ and decompose it by EMD (only decompose the first IMF):$$xne{w}_{1}(t)={r}_{1}\widehat{(}t)+{E}_{1}({n}_{i}\widehat{(}t))$$$$xne{w}_{1}(t)={r}_{1}\widehat{(}t)+\left(\begin{array}{c}{c}_{{n}_{1}1}\widehat{(}t)\\ {c}_{{n}_{2}1}\widehat{(}t)\\ \cdots \\ {c}_{{n}_{i}1}\widehat{(}t)\\ \cdots \\ {c}_{{n}_{N}1}\widehat{(}t)\end{array}\right)\stackrel{\mathrm{EMD}}{\to}\left(\begin{array}{c}{c}_{{r}_{1}{n}_{1}1}\widehat{(}t)\\ {c}_{{r}_{1}{n}_{2}1}\widehat{(}t)\\ \cdots \\ {c}_{{r}_{1}{n}_{i}1}\widehat{(}t)\\ \cdots \\ {c}_{{r}_{1}{n}_{N}1}\widehat{(}t)\end{array}\right)$$
- (7)
- Obtain the second IMF ${c}_{2}\widehat{(}t)$ and residual mode ${r}_{2}\widehat{(}t)$ of CEEMDAN:$${c}_{2}\widehat{(}t)=\frac{1}{N}{\displaystyle \sum _{i=1}^{N}{c}_{{r}_{1}{n}_{i}}{}_{1}\widehat{(}t)}$$$${r}_{2}\widehat{(}t)={r}_{1}\widehat{(}t)-{c}_{2}\widehat{(}t)$$
- (8)
- Obtain $xne{w}_{j-1}(t)$ and repeat step (6) and (7), ${c}_{j}\widehat{(}t)$ and ${r}_{j}\widehat{(}t)$ are expressed as:$$xne{w}_{j-1}(t)={r}_{j-1}\widehat{(}t)+{E}_{j-1}({n}_{i}\widehat{(}t))$$$${c}_{j}\widehat{(}t)=\frac{1}{N}{\displaystyle \sum _{i=1}^{N}{c}_{{r}_{j-1}{n}_{i}}{}_{1}\widehat{(}t)}$$$${r}_{j}\widehat{(}t)={r}_{j-1}\widehat{(}t)-{c}_{j}\widehat{(}t)$$
- (9)
- $x(t)$ is expressed as:$$x(t)={\displaystyle \sum _{j=1}^{L}{c}_{j}\widehat{(}t)}+r(t)$$

- (1)
- CEEMDAN has better decomposition effect and lower computational cost than EEMD and CEEMD.
- (2)
- CEEMDAN is suitable for analyzing non-linear, non-stationary and non-Gaussian signals, in theory, it can decompose all signals.
- (3)
- CEEMDAN is self-adaptive and based on characteristic time scale of the data itself without basis function.

#### 2.2. MI

_{i}represents the MI of IMF

_{i}and IMF

_{i}

_{+1}. As shown in Table 1 and Table 2, the center frequency decreases with the increase of IMF, the first three MI of IMFs are obviously less than the other ones of IMFs. According to the prior information of ship signal, its main frequency range is less than 5000Hz, the first three IMFs are noise IMFs, which is consistent with the judgment of MI. Therefore, we can use MI to identify noise IMFs in this paper, when the MI of IMF

_{i}and IMF

_{i}

_{+1}increases, obviously more than the former MIs, the former i − 1 IMFs are considered as noise IMFs.

#### 2.3. PE

- (1)
- Reconstruct time series $X=\{{x}_{1},{x}_{2},\cdots ,{x}_{N}\}$:$$\{\begin{array}{l}\{x(1),x(1+\tau ),\cdots ,x(1+(m-1)\tau )\}\\ \begin{array}{ccc}& \vdots & \end{array}\\ \{x(j),x(j+\tau ),\cdots ,x(j+(m-1)\tau )\}\\ \begin{array}{ccc}& \vdots & \end{array}\\ \{x(K),x(K+\tau ),\cdots ,x(K+(m-1)\tau )\}\text{}(K=n-(m-1)\tau )\end{array}$$
- (2)
- Rearrange each row vectorin ascending order:$$x(i+(j1-1)\tau )\le x(i+(j2-1)\tau )\le \cdots \le x(i+(jm-1)\tau )$$
- (3)
- Obtain a symbol-sequence for each row vector as:$$S(g)=(j1,j2,\cdots ,jm)\text{}(g=1,2,\cdots ,l\text{}\mathrm{and}\text{}l\le m!)$$
- (4)
- Define PE as:$${H}_{P}(m)=-{\displaystyle \sum _{j=1}^{l}{P}_{j}\mathrm{ln}}{P}_{j}$$
- (5)
- Define normalized PE as:$${H}_{P}={H}_{P}(m)/\mathrm{ln}(m!)$$

#### 2.4. Wavelet Threshold Denoising

- (1)
- A proper wavelet basis function and decomposition level are selected to perform wavelet decomposition on the noisy signal.
- (2)
- Threshold is performed by selecting an appropriate threshold method for high frequency coefficients at different decomposition scales.
- (3)
- The low frequency coefficient of wavelet decomposition and the thresholdhigh frequency coefficient of different scales are used to reconstruct.

## 3. Denoising Algorithm for Underwater Acoustic Signal

- (1)
- The underwater signal is decomposed by CEEMDAN, we can obtain a lot of IMFs, which contain noise IMFs, noise-dominant IMFs, and real IMFs.
- (2)
- Calculate MIs of two neighboring IMFs in ascending order.
- (3)
- Identify noise IMF according to MIs. If the MI of the $K$-th IMF and $(K+1)$-th IMF increases obviously than the former MIs, the former $K-1$ IMFs are considered as noise IMFs.
- (4)
- Screen out noise IMFs and calculate the PEs of the other IMFs.
- (5)
- Identify noise-dominant IMF according to PEs. If the PE of IMF is more than 0.5, weconsider it as noise-dominant IMF, otherwise real IMF.
- (6)
- Denoise noise-dominant IMFs by wavelet threshold denoising (WTD). We use the wavelet soft threshold denoising for noise-dominant IMFs, wavelet basis function, and decomposition level are db4 and 4, respectively.
- (7)
- The denoised signal can be obtain by reconstructing denoised noise-dominant IMFs and real IMFs.

## 4. Denoising for Simulation Signal

#### 4.1. CEEMDAN for Simulation Signal

#### 4.2. Identifying Noise IMFs

#### 4.3. Identifying Noise-Dominant IMFs

#### 4.4. Denoising for Noise-Dominant IMFs and Reconstruction

#### 4.5. Comparison of Different Denoising Methods

#### 4.5.1. Wavelet Denoising

#### 4.5.2. Comparison of Denoising Effect

## 5. Denoising for Chaotic Signal

## 6. Denoising for Underwater Acoustic Signal

## 7. Conclusions

- (1)
- CEEMDAN, as an adaptive decomposition algorithm, is introduced for underwater acoustic signal denoising.
- (2)
- Compared with existing denoising methods, IMFs by CEEMDAN are divided into three parts (noise IMFs, noise-dominant IMFs, and real IMFs) for the first time.
- (3)
- Four kinds of signals (Blocks, Bumps, Doppler, and Heavysine) with different SNRs are denoised by EMD-MI, EEMD-MI, CEEMDAN-MI, CEEMDAN-MI-PE, and WSTD, the proposed denoising method has lower RMSE and higher SNR, which has a better performance.
- (4)
- For chaotic signals with different SNR and underwater acoustic signals, the CEEMDAN-MI-PE is also an effective denoising method, which is beneficial to the subsequent processing of underwater acoustic signals.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Li, Y.X.; Li, Y.A.; Chen, Z.; Chen, X. Feature extraction of ship-radiated noise based on permutation entropy of the intrinsic mode function with the highest energy. Entropy
**2016**, 18, 393. [Google Scholar] [CrossRef] - Tucker, J.D.; Azimi-Sadjadi, M.R. Coherence-based underwater target detection from multiple disparatesonar platforms. IEEE J. Ocean Eng.
**2011**, 36, 37–51. [Google Scholar] [CrossRef] - Li, Y.; Li, Y.; Chen, X.; Yu, J. A novel feature extraction method for ship-radiated noise based on variational mode decomposition and multi-scale permutation entropy. Entropy
**2017**, 19, 342. [Google Scholar] - Wang, S.G.; Zeng, X.Y. Robust underwater noise targets classification using auditory inspired time-frequency analysis. Appl. Acoust.
**2014**, 78, 68–76. [Google Scholar] [CrossRef] - Huang, N.E.; Shen, Z.; Long, S.R.; Wu, M.C.; Shi, H.H.; Zheng, Q.A.; Yen, N.; Tung, C.C.; Liu, H.H. The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc. R. Soc. Lond.
**1998**, 454, 903–995. [Google Scholar] [CrossRef] - Wu, Z.; Huang, N.E. Ensemble empirical mode decomposition: A noise-assisted data analysis method. Adv. Adapt. Data Anal.
**2009**, 1, 1–41. [Google Scholar] [CrossRef] - Yeh, J.R.; Shieh, J.S.; Huang, N.E. Complementary ensemble empirical mode decomposition: A novel noise enhanced data analysis method. Adv. Adapt. Data Anal.
**2010**, 2, 135–156. [Google Scholar] [CrossRef] - Torres, M.E.; Colominas, M.A.; Schlotthauer, G.; Flandrin, P. A complete ensemble empirical mode decomposition with adaptive noise. In Proceedings of the 2011 IEEE International Conference on Acoustics, Speech and Signal (ICASSP), Prague, Czech Republic, 22–27 May 2011; pp. 4144–4147. [Google Scholar]
- Gao, B.; Woo, W.L.; Dlay, S.S. Single channel blind source separation using EMD-subband variable regularized sparse features. IEEE Trans. Audio Speech Lang. Process.
**2011**, 19, 961–976. [Google Scholar] [CrossRef] - Bi, F.; Li, L.; Zhang, J.; Ma, T. Source identification of gasoline engine noise based on continuous wavelet transform and EEMD–Robust ICA. Appl. Acoust.
**2015**, 100, 34–42. [Google Scholar] [CrossRef] - Li, N.; Yang, J.; Zhou, R.; Liang, C. Determination of knock characteristics in spark ignition engines: An approach based on ensemble empirical mode decomposition. Meas. Sci. Technol.
**2016**, 27, 045109. [Google Scholar] [CrossRef] - Lee, D.H.; Ahn, J.H.; Koh, B.H. Fault detection of bearing systems through EEMD and optimization algorithm. Sensors
**2017**, 17, 2477. [Google Scholar] [CrossRef] [PubMed] - Lv, Y.; Yuan, R.; Wang, T.; Li, H.; Song, G. Health degradation monitoring and early fault diagnosis of a rolling bearing based on CEEMDAN and improved MMSE. Materials
**2018**, 11, 1009. [Google Scholar] [CrossRef] [PubMed] - Kuai, M.; Cheng, G.; Pang, Y.; Li, Y. Research of planetary gear fault diagnosis based on permutation entropy of CEEMDAN and ANFIS. Sensors
**2018**, 18, 782. [Google Scholar] [CrossRef] [PubMed] - Queyam, A.B.; Pahuja, S.K.; Singh, D. Quantification of feto-maternal heart rate from abdominal ECG signal using empirical mode decomposition for heart rate variability analysis. Technologies
**2017**, 5, 68. [Google Scholar] [CrossRef] - Sharma, R.; Pachori, R.B.; Acharya, U.R. Application of entropy measures on intrinsic mode functions for the automated identification of focal electroencephalogram signals. Entropy
**2015**, 17, 669–691. [Google Scholar] [CrossRef] - Shih, M.T.; Doctor, F.; Fan, S.Z.; Jen, K.K.; Shieh, J.S. Instantaneous 3D EEG signal analysis based on empirical mode decomposition and the hilbert–huang transform applied to depth of anaesthesia. Entropy
**2015**, 17, 928–949. [Google Scholar] [CrossRef] - Li, Y.; Li, Y. Feature extraction of underwater acoustic signal using mode decomposition and measuring complexity. In Proceedings of the 2018 15th International Bhurban Conference on Applied Sciences and Technology (IBCAST), Islamabad, Pakistan, 9–13 January 2018; pp. 757–763. [Google Scholar]
- An, X.; Yang, J. Denoising of hydropower unit vibration signal based on variational mode decomposition and approximate entropy. Trans. Inst. Meas. Control
**2016**, 38, 282–292. [Google Scholar] [CrossRef] - Figlus, T.; Gnap, J.; Skrúcaný, T.; Šarkan, B.; Stoklosa, J. The use of denoising and analysis of the acoustic signal entropy in diagnosing engine valve clearance. Entropy
**2016**, 18, 253. [Google Scholar] [CrossRef] - Bai, L.; Han, Z.; Li, Y.; Ning, S. A hybrid de-noising algorithm for the gear transmission system based on CEEMDAN-PE-TFPF. Entropy
**2018**, 20, 361. [Google Scholar] [CrossRef] - Xu, Y.; Luo, M.; Li, T.; Song, G. ECG signal de-noising and baseline wander correction based on CEEMDAN and wavelet threshold. Sensors
**2017**, 17, 2754. [Google Scholar] [CrossRef] [PubMed] - Zhan, L.; Li, C. A comparative study of empirical mode decomposition-based filtering for impact signal. Entropy
**2017**, 19, 13. [Google Scholar] [CrossRef] - Li, C.; Zhan, L.; Shen, L. Friction signal denoising using complete ensemble EMD with adaptive noise and mutual information. Entropy
**2015**, 17, 5965–5979. [Google Scholar] [CrossRef] - Li, Y.; Li, Y.; Chen, X.; Yu, J. Denoising and feature extraction algorithms using npe combined with vmd and their applications in ship-radiated noise. Symmetry
**2017**, 9, 256. [Google Scholar] [CrossRef] - Li, Y.; Li, Y.; Chen, X.; Yu, J. Research on ship-radiated noise denoising using secondary variational mode decomposition and correlation coefficient. Sensors
**2018**, 18, 48. [Google Scholar] - Kvålseth, T.O. On normalized mutual information: measure derivations and properties. Entropy
**2017**, 19, 631. [Google Scholar] [CrossRef] - Zanin, M.; Gómez-Andrés, D.; Pulido-Valdeolivas, I.; Martín-Gonzalo, J.A.; López-López, J.; Pascual-Pascual, S.I.; Rausell, E. Characterizing normal and pathological gait through permutation entropy. Entropy
**2018**, 20, 77. [Google Scholar] [CrossRef] - Gao, Y.; Villecco, F.; Li, M.; Song, W. Multi-Scale permutation entropy based on improved LMD and HMM for rolling bearing diagnosis. Entropy
**2017**, 19, 176. [Google Scholar] [CrossRef] - Bandt, C.; Pompe, B. Permutation entropy: A natural complexity measure for time series. Phys. Rev. Lett.
**2002**, 88, 174102. [Google Scholar] [CrossRef] [PubMed] - Wang, X.; Xu, J.; Zhao, Y. Wavelet based denoising for the estimation of the state of charge for lithium-ion batteries. Energies
**2018**, 11, 1144. [Google Scholar] [CrossRef] - Bandt, C. A new kind of permutation entropy used to classify sleep stages from invisible EEG microstructure. Entropy
**2017**, 19, 197. [Google Scholar] [CrossRef]

**Figure 1.**The flow chart of complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN).

**Figure 5.**The time-domain waveforms for simulation signals. (

**a**) Blocks,

**(b**) Bumps, (

**c**) Doppler, and (

**d**) Heavysine.

**Figure 7.**The decomposition result of the noisy Blocks signal with 0 dB. (

**a**) Empirical mode decomposition (EMD), (

**b**) Ensemble EMD (EEMD), and (

**c**) CEEMDAN.

**Figure 8.**The six kinds of reconstructed signals by different decomposition methods. (

**a**) EMD, (

**b**) EEMD, and (

**c**) CEEMDAN.

**Figure 9.**The denoising results for different methods. (

**a**) EMD-MI, (

**b**) EEMD-MI, (

**c**) CEEMDAN-MI, and (

**d**) CEEMDAN-MI-PE.

**Figure 10.**Lorenz noisy and denoised signals with different SNRs and their attractor trajectories. (

**a**) Lorenz signal, (

**b**) Lorenz attractor trajectory, (

**c**) Lorenz noisy signal with 0 dB, (

**d**) Noisy attractor trajectory with 0 dB, (

**e**) Lorenz noisy signal with 10 dB, (

**f**) noisy attractor trajectory with 10 dB, (

**g**) denoised Lorenz signal with 0 dB, (

**h**) denoised attractor trajectory (0 dB), (

**i**) denoised Lorenz signal with 10 dB, and(

**j**) denoised attractor trajectory (10 dB).

**Figure 11.**Ship-1 and denoised Ship-1 signals and their attractor trajectories. (

**a**) Ship-1, (

**b**) attractor trajectory for ship-1, (

**c**) denoised Ship-1, and (

**d**) attractor trajectory for denoised ship-1.

**Figure 12.**Ship-2 and denoised Ship-2 signals and their attractor trajectories. (

**a**) Ship-2, (

**b**) attractor trajectory for ship-2, (

**c**) denoised Ship-2, (

**d**) attractor trajectory for denoised ship-2.

**Figure 13.**Ship-3 and denoised Ship-3 signals and their attractor trajectories. (

**a**) Ship-3, (

**b**) attractor trajectory for ship-3, (

**c**) denoised Ship-3, (

**d**) attractor trajectory for denoised ship-3.

IMF1 | IMF2 | IMF3 | IMF4 | IMF5 | IMF6 | IMF7 | IMF8 | IMF9 | IMF10 |
---|---|---|---|---|---|---|---|---|---|

12333 | 9068.1 | 6296.3 | 3065.5 | 1595.4 | 902.08 | 446.55 | 340.21 | 127.43 | 67.039 |

MI_{1} | MI_{2} | MI_{3} | MI_{4} | MI_{5} | MI_{6} | MI_{7} | MI_{8} | MI_{9} |
---|---|---|---|---|---|---|---|---|

0.0676 | 0.0501 | 0.0511 | 0.1279 | 0.1751 | 0.3437 | 0.8394 | 0.9998 | 1.6231 |

Methods | ${\mathit{M}}_{1}$ | ${\mathit{M}}_{2}$ | ${\mathit{M}}_{3}$ | ${\mathit{M}}_{4}$ | ${\mathit{M}}_{5}$ | ${\mathit{M}}_{6}$ | ${\mathit{M}}_{7}$ | ${\mathit{M}}_{8}$ | ${\mathit{M}}_{9}$ |
---|---|---|---|---|---|---|---|---|---|

EMD | 0.0034 | 0.0113 | 0.0633 | 0.1805 | 0.4233 | 0.6419 | 1.332 | 2.1514 | 3.0829 |

EEMD | 0.0014 | 0.0023 | 0.0475 | 0.1969 | 0.5045 | 0.8873 | 1.8319 | 1.5387 | 3.0034 |

CEEMDAN | 0.0143 | 0.0419 | 0.0647 | 0.0803 | 0.2169 | 0.5376 | 0.8044 | 1.5476 | 2.3663 |

IMF5 | IMF6 | IMF7 | IMF8 | IMF9 | IMF10 |
---|---|---|---|---|---|

0.5869 | 0.4945 | 0.4547 | 0.4263 | 0.4038 | 0.3767 |

Parameter | EMD-MI | EEMD-MI | CEEMDAN-MI | CEEMDAN-MI-PE |
---|---|---|---|---|

SNR/dB | 7.1052 | 8.6122 | 9.0433 | 9.3663 |

RMSE | 0.8031 | 0.7496 | 0.7189 | 0.7078 |

**Table 6.**(

**a**) wavelet soft-threshold denoising (WSTD) results for Blocks signal. (

**b**) WSTD results for Bumps signal. (

**c**) WSTD results for Doppler signal. (

**d**) WSTD results for Heavysine signal.

SNR | Parameter | Decomposition Level | |||||
---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | ||

(a) | |||||||

−10 dB | SNR/db | −7.2271 | −4.8104 | −2.3898 | 0.2149 | 1.6290 | 1.0946 |

RMSE | 3.6839 | 3.5798 | 3.5307 | 3.5614 | 3.3392 | 3.4362 | |

−5 dB | SNR/db | −2.3435 | 0.3025 | 3.1728 | 5.7484 | 6.3767 | 3.8336 |

RMSE | 1.4457 | 1.5207 | 1.4748 | 1.6308 | 1.5565 | 1.2164 | |

0 dB | SNR/db | 3.2235 | 6.2477 | 8.1067 | 8.4866 | 7.7998 | 5.8663 |

RMSE | 0.8504 | 0.7895 | 0.8176 | 0.8341 | 0.7473 | 0.6417 | |

5 dB | SNR/db | 7.6526 | 10.1584 | 11.4382 | 10.2663 | 9.6887 | 8.4297 |

RMSE | 0.3915 | 0.4133 | 0.4045 | 0.4114 | 0.4347 | 0.4945 | |

(b) | |||||||

−10 dB | SNR/db | −7.4756 | −4.5317 | −2.0528 | 0.3106 | 0.8154 | 0.5481 |

RMSE | 1.6983 | 1.6584 | 1.7329 | 1.6642 | 1.3528 | 1.5681 | |

−5 dB | SNR/db | −3.2325 | −0.1944 | 2.3835 | 4.5283 | 4.3014 | 3.5033 |

RMSE | 1.5826 | 1.5639 | 1.5729 | 1.4888 | 1.5554 | 1.9642 | |

0 dB | SNR/db | 2.6556 | 5.1528 | 7.8573 | 8.3710 | 6.9682 | 6.3942 |

RMSE | 0.3337 | 0.3153 | 0.3046 | 0.2942 | 0.3022 | 0.3025 | |

5 dB | SNR/db | 7.4436 | 10.3617 | 11.2783 | 10.7887 | 9.1181 | 9.2302 |

RMSE | 0.2211 | 0.2385 | 0.2360 | 0.2486 | 0.2182 | 0.2427 | |

(c) | |||||||

−10 dB | SNR/db | −6.6660 | −4.3199 | −0.7130 | 1.4574 | 3.4092 | 3.2951 |

RMSE | 0.9655 | 0.9220 | 0.9301 | 0.8570 | 0.9781 | 1.2149 | |

−5 dB | SNR/db | −1.8736 | 0.9657 | 4.1477 | 6.8774 | 7.0568 | 6.4521 |

RMSE | 0.0502 | 0.0538 | 0.0511 | 0.0491 | 0.0342 | 0.0254 | |

0 dB | SNR/db | 2.4724 | 4.5147 | 8.0985 | 8.8992 | 9.1371 | 8.8475 |

RMSE | 0.0694 | 0.0756 | 0.0545 | 0.0445 | 0.0272 | 0.0284 | |

5 dB | SNR/db | 8.1516 | 10.7101 | 11.0123 | 11.2306 | 11.5458 | 10.0998 |

RMSE | 0.0324 | 0.0333 | 0.0338 | 0.0343 | 0.0285 | 0.0127 | |

(d) | |||||||

−10 dB | SNR/db | −6.1624 | −3.3838 | −0.9295 | 1.7670 | 5.8097 | 4.1349 |

RMSE | 2.8465 | 2.6462 | 2.5926 | 2.4760 | 2.6292 | 4.2984 | |

−5 dB | SNR/db | −2.3572 | 0.2426 | 3.1175 | 6.0666 | 7.3376 | 6.857 |

RMSE | 0.8090 | 0.8554 | 0.8105 | 0.7434 | 0.7203 | 0.7268 | |

0 dB | SNR/db | 2.9317 | 5.4369 | 8.9585 | 11.8276 | 14.4169 | 13.8013 |

RMSE | 0.2792 | 0.1840 | 0.1973 | 0.1924 | 0.1682 | 0.3920 | |

5 dB | SNR/db | 8.1963 | 10.8592 | 13.4744 | 15.5710 | 17.9746 | 17.6310 |

RMSE | 0.2759 | 0.2346 | 0.2099 | 0.2277 | 0.1028 | 0.1503 |

**Table 7.**(

**a**) Denoising results of different methods for Blocks signal. (

**b**) Denoising results of different methods for Bumps signal. (

**c**) Denoising results of different methods for Doppler signal. (

**d**) Denoising results of different methods for Heavysine signal.

SNR | Parameter | Denoising Method | ||||
---|---|---|---|---|---|---|

EMD-MI | EEMD-MI | CEEMDAN-MI | CEEMDAN-MI-PE | WSTD | ||

(a) | ||||||

−10 dB | SNR/db | 1.8632 | 2.0988 | 2.2803 | 2.5588 | 1.6290 |

RMSE | 5.0621 | 3.3228 | 2.5753 | 2.4237 | 3.3392 | |

−5 dB | SNR/db | 4.8004 | 6.4972 | 6.6097 | 6.8239 | 6.3767 |

RMSE | 1.6806 | 1.4843 | 1.3438 | 1.3401 | 1.5565 | |

0 dB | SNR/db | 6.2426 | 8.4579 | 9.2502 | 9.8326 | 7.7998 |

RMSE | 0.8261 | 0.7834 | 0.7651 | 0.7051 | 0.7473 | |

5 dB | SNR/db | 11.3699 | 11.5903 | 11.7158 | 11.8733 | 11.4382 |

RMSE | 0.6600 | 0.3917 | 0.4086 | 0.3489 | 0.4045 | |

(b) | ||||||

−10 dB | SNR/db | −0.1258 | 0.4728 | 0.9903 | 1.2130 | 0.8154 |

RMSE | 1.7652 | 1.4865 | 1.0045 | 1.0041 | 1.3528 | |

−5 dB | SNR/db | 3.4745 | 4.4220 | 4.6204 | 4.7355 | 4.5283 |

RMSE | 1.8357 | 1.5325 | 1.5554 | 1.4859 | 1.5888 | |

0 dB | SNR/db | 6.8187 | 7.7571 | 8.7208 | 9.0641 | 8.3710 |

RMSE | 0.4253 | 0.3461 | 0.1969 | 0.1950 | 0.2942 | |

5 dB | SNR/db | 9.5890 | 10.5161 | 11.4614 | 11.5623 | 11.2783 |

RMSE | 0.3158 | 0.2058 | 0.1609 | 0.1513 | 0.2360 | |

(c) | ||||||

−10 dB | SNR/db | 3.0361 | 3.3580 | 3.5508 | 4.1789 | 3.4092 |

RMSE | 1.2158 | 0.5124 | 0.4747 | 0.4597 | 0.9781 | |

−5 dB | SNR/db | 5.7752 | 6.3432 | 7.2250 | 7.2939 | 7.0568 |

RMSE | 0.0604 | 0.0463 | 0.0263 | 0.0213 | 0.0342 | |

0 dB | SNR/db | 8.3418 | 8.6526 | 8.8555 | 9.5866 | 9.1371 |

RMSE | 0.0235 | 0.0190 | 0.0182 | 0.0165 | 0.0272 | |

5 dB | SNR/db | 11.2457 | 11.7545 | 11.8473 | 12.1583 | 11.5458 |

RMSE | 0.0298 | 0.0025 | 0.0015 | 0.0013 | 0.0285 | |

(d) | ||||||

−10 dB | SNR/db | 6.0781 | 6.236 | 6.4919 | 6.6696 | 5.8097 |

RMSE | 1.8252 | 1.6397 | 1.5395 | 1.4666 | 2.6292 | |

−5 dB | SNR/db | 7.1830 | 8.1239 | 8.2463 | 8.3975 | 7.3376 |

RMSE | 0.7325 | 0.6431 | 0.6324 | 0.6216 | 0.7203 | |

0 dB | SNR/db | 14.896 | 15.128 | 15.2882 | 15.4476 | 14.4169 |

RMSE | 0.1224 | 0.1158 | 0.1142 | 0.1139 | 0.1682 | |

5 dB | SNR/db | 17.8843 | 19.5125 | 19.7096 | 19.7125 | 17.9746 |

RMSE | 0.1052 | 0.0931 | 0.0925 | 0.0921 | 0.1028 |

SNR | Parameter | CEEMDAN-MI-PE |
---|---|---|

0 dB | SNR/db | 13.254 |

RMSE | 1.8762 | |

10 dB | SNR/db | 20.146 |

RMSE | 0.3993 |

Parameter | Ship-1 | Ship-2 | Ship-3 | |
---|---|---|---|---|

Before denoising | PE | 0.8094 | 0.9231 | 0.8856 |

NPE | 0.1227 | 0.0495 | 0.0739 | |

After denoising | PE | 0.5537 | 0.5381 | 0.5148 |

NPE | 0.2680 | 0.2765 | 0.2861 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Li, Y.; Li, Y.; Chen, X.; Yu, J.; Yang, H.; Wang, L.
A New Underwater Acoustic Signal Denoising Technique Based on CEEMDAN, Mutual Information, Permutation Entropy, and Wavelet Threshold Denoising. *Entropy* **2018**, *20*, 563.
https://doi.org/10.3390/e20080563

**AMA Style**

Li Y, Li Y, Chen X, Yu J, Yang H, Wang L.
A New Underwater Acoustic Signal Denoising Technique Based on CEEMDAN, Mutual Information, Permutation Entropy, and Wavelet Threshold Denoising. *Entropy*. 2018; 20(8):563.
https://doi.org/10.3390/e20080563

**Chicago/Turabian Style**

Li, Yuxing, Yaan Li, Xiao Chen, Jing Yu, Hong Yang, and Long Wang.
2018. "A New Underwater Acoustic Signal Denoising Technique Based on CEEMDAN, Mutual Information, Permutation Entropy, and Wavelet Threshold Denoising" *Entropy* 20, no. 8: 563.
https://doi.org/10.3390/e20080563