# MEMS Hydrophone Signal Denoising and Baseline Drift Removal Algorithm Based on Parameter-Optimized Variational Mode Decomposition and Correlation Coefficient

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Basis

#### 2.1. Variational Mode Decomposition (VMD)

- Step 1:
- Initialize the parameters, set $\{{\widehat{u}}_{k}^{1}\}$, $\{{\omega}_{k}^{1}\}$, ${\widehat{\lambda}}^{1}$ and $n$ to 0;
- Step 2:
- Update the values of $\{{\widehat{u}}_{k}^{\mathrm{n}+1}\}$, $\{{\omega}_{k}^{\mathrm{n}+1}\}$, and ${\widehat{\lambda}}^{\mathrm{n}+1}$ according to Formulas (4)–(6);
- Step 3:
- Determine whether the termination condition (7) is satisfied, and repeat step 2 until Formula (7) is satisfied.

#### 2.2. Whale-Optimization Algorithm (WOA)

#### 2.3. Power Spectrum Entropy (PSE)

- Step 1:
- Calculation formula of the power spectrum of signal $x(t)$:$$s(f)=\frac{1}{2\pi L}{\left|x(w)\right|}^{2}$$
- Step 2:
- Obtain the probability density function of the spectrum of all frequency components by normalization:$${P}_{i}=\frac{s({f}_{i})}{{\displaystyle {\sum}_{k=1}^{N}s({f}_{k})}}(i=1,2,3,\cdots ,N)$$
- Step 3:
- The PSE value is defined as:$$H=-{\displaystyle {\sum}_{k=1}^{N}{P}_{i}\mathrm{log}{P}_{i}}$$

#### 2.4. Correlation Coefficient (CC)

## 3. The Method Proposed in This Paper (WOA–VMD–CC)

## 4. Simulation

#### 4.1. Simulation Experiment 1

#### 4.2. Simulation Experiment 2

## 5. Application in the Experiments of MEMS Hydrophone

#### 5.1. MEMS Vector Hydrophone and Signal Acquisition

#### 5.2. Denoising and Baseline Drift Removal Experiments for MEMS Vector Hydrophones

## 6. Conclusions

- (1)
- In many literatures, the parameters of VMD are selected by empirical method. In this paper, the whale-optimization algorithm is used to optimize the parameters $(K,\alpha )$ of the VMD. While taking into account the mutual influence between the two parameters, it is easier to find the global optimal solution, which provides an idea for adaptively searching for the parameters of the VMD.
- (2)
- Power spectrum entropy (PSE) can reflect the variation characteristics of the frequency in the signal. In the whale-optimization algorithm, PSE is used as the fitness function, and it is easier to find the optimal parameters $(K,\alpha )$, which can improve the accuracy of signal decomposition. There is no modal aliasing when decomposing the signal using the algorithm proposed in this paper.
- (3)
- This paper calculates the CCs of the IMFs and the original signal, and denoises the signal by setting the CC threshold. The correlation between the denoised signal and the original signal is fully considered, so that the denoised signal retains more important information of the original signal.
- (4)
- Conventional digital signal-processing methods tend to lose useful information when removing baseline drift. The algorithm in this paper has a good performance for baseline drift removal of signals with different characteristics. At the same time, more useful information is retained.
- (5)
- Compared with conventional digital signal-processing methods and other related algorithms proposed recently, the denoising effect of this algorithm is better.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 4.**The denoising results of different algorithms. (

**a**) variational mode decomposition-new permutation entropy (VMD-NPE); (

**b**) 2VMD-correlation coefficient (CC); (

**c**) empirical mode decomposition and wavelet analysis (EMD-WT); (

**d**) WOA-VMD-CC; (

**e**) least squares fitting with wavelet soft threshold (LSF-WST); (

**f**) Savitzky–Golay smoothing filter (SGFT)-WST.

**Figure 5.**VMD time-frequency diagram with different parameters. (

**a**) K is equal to the decomposition level by EMD, α is the empirical value of 2000. (

**b**) Parameters obtained by WOA, K is 7, α is 6240.

**Figure 8.**Denoising results of different algorithms under different decibel noises. (

**a**) −10 db noise; (

**b**) −5 db noise; (

**c**) 0 db noise; (

**d**) 5 db noise.

**Figure 9.**Denoising and baseline drift removal results of traditional digital signal processing methods in different noise intensity environments. (

**a**) LSF-WST; (

**b**) SGSF-WST.

**Figure 12.**Processing results of different measured signals. (

**a**) sound source frequency 315 Hz; (

**b**) sound source frequency 500 Hz; (

**c**) sound source frequency 800 Hz.

Index | Input Signal | VMD-NPE | 2VMD-CC | EMD-WT | WOA-VMD-CC | LSF-WST | SGSF-WST |
---|---|---|---|---|---|---|---|

SNR | 5.8810 | 9.3281 | 9.4184 | 7.8212 | 18.5693 | 8.5445 | 8.5935 |

RMSE | 1.6560 | 1.5796 | 0.4782 | 1.6382 | 0.1667 | 0.5288 | 0.5030 |

Index | VMD-NPE | WOA-VMD-NPE | 2VMD-CC | WOA-2VMD-CC | WOA-VMD-CC |
---|---|---|---|---|---|

SNR | 9.3281 | 10.7579 | 9.4184 | 17.0467 | 18.5693 |

RMSE | 1.5796 | 1.5480 | 0.4782 | 0.1987 | 0.1667 |

**Table 3.**Different denoising evaluation indexes of different algorithms under different decibel noises.

Noise | Index | Input Signal | VMD-NPE | 2VMD-CC | EMD-WT | WOA-VMD-CC | LSF-WST | SGSF-WST |
---|---|---|---|---|---|---|---|---|

−10 db | SNR | −12.9540 | −5.4163 | −8.3212 | −6.1719 | 1.3264 | −7.1079 | −7.9754 |

RMSE | 13.6503 | 6.0592 | 8.3504 | 6.7044 | 2.5752 | 6.8001 | 7.5148 | |

−5 db | SNR | −7.9970 | −1.6381 | −2.5925 | −1.9593 | 5.8786 | −1.4992 | −2.6494 |

RMSE | 8.0206 | 4.5489 | 4.0434 | 4.7587 | 1.5247 | 3.5635 | 4.0707 | |

0 db | SNR | −2.9637 | 3.2030 | 1.9962 | 1.6200 | 12.3969 | 2.0547 | 2.0923 |

RMSE | 5.1673 | 3.6328 | 2.3840 | 3.8343 | 0.7199 | 2.3680 | 2.3585 | |

5 db | SNR | 2.0469 | 8.7231 | 11.0416 | 4.9440 | 15.1702 | 6.1727 | 6.4195 |

RMSE | 3.8618 | 3.2409 | 0.8415 | 3.4430 | 0.5231 | 1.4740 | 1.4328 |

Noise | Index | Input Signal | LSF-WOA-VMD-CC | SGSF-WOA-VMD-CC | WOA-VMD-CC |
---|---|---|---|---|---|

5db | SNR | 2.0469 | 11.5326 | 13.3392 | 15.1702 |

RMSE | 3.8618 | 0.7952 | 0.6459 | 0.5231 |

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**MDPI and ACS Style**

Yan, H.; Xu, T.; Wang, P.; Zhang, L.; Hu, H.; Bai, Y.
MEMS Hydrophone Signal Denoising and Baseline Drift Removal Algorithm Based on Parameter-Optimized Variational Mode Decomposition and Correlation Coefficient. *Sensors* **2019**, *19*, 4622.
https://doi.org/10.3390/s19214622

**AMA Style**

Yan H, Xu T, Wang P, Zhang L, Hu H, Bai Y.
MEMS Hydrophone Signal Denoising and Baseline Drift Removal Algorithm Based on Parameter-Optimized Variational Mode Decomposition and Correlation Coefficient. *Sensors*. 2019; 19(21):4622.
https://doi.org/10.3390/s19214622

**Chicago/Turabian Style**

Yan, Huichao, Ting Xu, Peng Wang, Linmei Zhang, Hongping Hu, and Yanping Bai.
2019. "MEMS Hydrophone Signal Denoising and Baseline Drift Removal Algorithm Based on Parameter-Optimized Variational Mode Decomposition and Correlation Coefficient" *Sensors* 19, no. 21: 4622.
https://doi.org/10.3390/s19214622