# A Combined Noise Reduction Method for Floodgate Vibration Signals Based on Adaptive Singular Value Decomposition and Improved Complete Ensemble Empirical Mode Decomposition with Adaptive Noise

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Basic Principles

#### 2.1. ASVD Noise Reduction

**H**:

**H**is usually constructed as a square matrix or a matrix close to a square matrix. Therefore, the variable r needs to satisfy Equation (2).

**H**is decomposed via SVD:

**0**represents a zero matrix. Then, by setting the singular values representing noise to 0 and reconstructing a new Hankel matrix, the denoised signal can be obtained by the diagonal averaging method.

- (1)
- Each frequency component of the signal relates to a singular value group consisting of two adjacent singular values that have small differences between them.
- (2)
- Due to significant energy differences between different frequency components of the signal, a jump phenomenon is observed between adjacent groups of ESVs.
- (3)
- The energy of white noise is uniformly distributed in the broadband frequency domain, which makes the jump phenomenon of singular value inconspicuous.

**H**according to Equations (1) and (2);

**H**via SVD and record singular values from large to small as $\sigma \left(1\right),\sigma \left(2\right),\cdots ,\sigma \left(r\right)$.

#### 2.2. ICEEMDAN Method

#### 2.3. Combined ASVD–ICEEMDAN Noise Reduction

**X**

_{0}is constructed as a Hankel matrix and then decomposed via SVD to obtain the singular values.

**X**

_{1}can be reconstructed via the ASVD algorithm.

**X**

_{1}via ICEEMDAN to yield several IMF components whose center frequencies are arranged from high to low.

## 3. Numerical Simulation

^{−16}, while that of the ICEEMDAN method is only 1.39 × 10

^{−16}, indicating that the signal decomposed by the ICEEMDAN method has lower residual noise.

## 4. Engineering Example

## 5. Conclusions

- (1)
- An ASVD method is proposed to select effective singular values automatically based on the relationship between singular values and signal components. It avoids the uncertainty of manual selection and improves the efficiency of noise reduction.
- (2)
- The ICEEMDAN algorithm can automatically decompose the signal into several IMF components whose center frequencies are arranged from high to low. Compared to the CEEMDAN algorithm, ICEEMDAN algorithm performs better in terms of suppressing mode mixing and reducing residual noise.
- (3)
- The ASVD–ICEEMDAN method successfully filters out white noise and low-frequency noise in simulated signals, which increases the SNR of the signal (50% noise level) from 4.417 to 16.237 and reduces the RMSE from 2.286 to 0.586. In the engineering case study, the ASVD–ICEEMDAN method effectively filters out the noise and accurately extracts the structural characteristic vibration information, proving it can provide support in operational modal analyses and damage identification in actual structures.
- (4)
- Vibration signals of discharge structures are mainly affected by white noise and low-frequency noise; thus, this method has the potential to be extended to other hydraulic structures under discharge excitation, such as arch dams, gravity dams, and guide walls.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Change in singular values. (Notes: blue lines and numbers represent singular groups, red straight lines represent conspicuous jumping phenomenon, dotted lines represents unconspicuous jumping phenomenon).

**Figure 11.**Pictures of the project and sensor arrangement: (

**a**) top view of the project, (

**b**) measuring point layout of hole 2#, (

**c**) data acquisition instrument, (

**d**) dynamic displacement sensor.

**Figure 12.**Vibration signal of measuring point 10 under condition 1. (Note: Frame (a) and (b) represent the local enlarged drawing windows.)

**Figure 13.**Change in SVDN values of Hankel matrix constructed by vibration signal of measuring point 10 under condition 1.

**Figure 15.**Noise reduction result of measuring point 10 under condition 1. (Note: Frame (a) and (b) represent the local enlarged drawing windows.)

**Figure 16.**Noise reduction result of measuring point 10 under condition 2. (Note: Frame (a) and (b) represent the local enlarged drawing windows.)

Sequence Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

Singular value | 332.520 | 327.151 | 306.594 | 299.108 | 260.133 | 245.444 | 185.043 | 156.517 | 58.924 | 58.764 |

SVDN value | 0.055 | 0.211 | 0.077 | 0.399 | 0.151 | 0.619 | 0.292 | 1.000 | 0.002 | 0.029 |

Sequence number | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |

Singular value | 55.940 | 55.672 | 55.038 | 54.870 | 52.886 | 52.711 | 51.616 | 51.524 | 51.410 | 51.344 |

SVDN value | 0.003 | 0.006 | 0.002 | 0.020 | 0.002 | 0.011 | 0.001 | 0.001 | 0.001 | 0.008 |

IMF Component | $\mathbf{Absolute}\text{}\mathbf{Values}\text{}\mathbf{of}\text{}{\mathit{r}}_{\mathit{s}}$ | IMF Component | $\mathbf{Absolute}\text{}\mathbf{Values}\text{}\mathbf{of}\text{}{\mathit{r}}_{\mathit{s}}$ |
---|---|---|---|

IMF1 | 0.536 | IMF5 | 0.397 |

IMF2 | 0.578 | IMF6 | 0.303 |

IMF3 | 0.498 | IMF7 | 0.094 |

IMF4 | 0.458 | RES | 0.076 |

Noise Reduction Method | Index | |
---|---|---|

SNR/dB | RMSE | |

Original signal | 4.417 | 2.286 |

Moving average | 5.414 | 2.038 |

IIR digital filters | 5.544 | 2.008 |

ASVD | 9.387 | 1.290 |

ICEEMDAN | 5.711 | 1.969 |

ASVD-CEEMDAN | 16.104 | 0.595 |

ASVD–ICEEMDAN | 16.237 | 0.586 |

Sensor Number | Model | Range of Frequency Response | Sensitivity |
---|---|---|---|

1~10 | DPS-0.5-15-H | 0.5~200 Hz | 5 mv/μm |

Working Condition | Order | Suggested Method/Hz | ERA/Hz | Relative Error */% |
---|---|---|---|---|

Condition 1 | 1 | 2.35 | 2.39 | 1.67 |

2 | 3.76 | 3.77 | 0.27 | |

Condition 2 | 1 | 2.47 | 2.43 | 1.65 |

2 | 3.82 | 3.82 | 0 |

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## Share and Cite

**MDPI and ACS Style**

Wang, W.; Zhu, H.; Cheng, Y.; Tang, Y.; Liu, B.; Li, H.; Yang, F.; Zhang, W.; Huang, W.; Zheng, F.
A Combined Noise Reduction Method for Floodgate Vibration Signals Based on Adaptive Singular Value Decomposition and Improved Complete Ensemble Empirical Mode Decomposition with Adaptive Noise. *Water* **2023**, *15*, 4287.
https://doi.org/10.3390/w15244287

**AMA Style**

Wang W, Zhu H, Cheng Y, Tang Y, Liu B, Li H, Yang F, Zhang W, Huang W, Zheng F.
A Combined Noise Reduction Method for Floodgate Vibration Signals Based on Adaptive Singular Value Decomposition and Improved Complete Ensemble Empirical Mode Decomposition with Adaptive Noise. *Water*. 2023; 15(24):4287.
https://doi.org/10.3390/w15244287

**Chicago/Turabian Style**

Wang, Wentao, Huiqi Zhu, Yingxin Cheng, Yiyuan Tang, Bo Liu, Huokun Li, Fan Yang, Wenyuan Zhang, Wei Huang, and Fang Zheng.
2023. "A Combined Noise Reduction Method for Floodgate Vibration Signals Based on Adaptive Singular Value Decomposition and Improved Complete Ensemble Empirical Mode Decomposition with Adaptive Noise" *Water* 15, no. 24: 4287.
https://doi.org/10.3390/w15244287