# A Fusion Frequency Feature Extraction Method for Underwater Acoustic Signal Based on Variational Mode Decomposition, Duffing Chaotic Oscillator and a Kind of Permutation Entropy

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## Abstract

**:**

## 1. Introduction

## 2. Theory

#### 2.1. VMD

#### 2.2. DCO

- (1)
- Put periodic signal $s(t)$ and noise signal $n(t)$ into the system, DCO equation can be expressed in Equation (7).$$\frac{{d}^{2}x}{d{t}^{2}}+k\frac{dx}{dt}-x(t)+{x}^{3}(t)={\gamma}_{d}\mathrm{cos}(\omega t)+s(t)+n(t)$$
- (2)
- Set $k$, $x(0)$ and ${x}^{\prime}(0)$ to 0.5, 0 and 0. The Runge-Kutta of the fourth order is used for a solution of DCO equation.
- (3)
- We can determine whether the angular frequency of the periodic signal $s(t)$ is close to $\omega $ according to the system state. When the system state is the great periodic state, this means that the angular frequency of the periodic signal $s(t)$ is approximated as $\omega $, and vice versa. More detailed explanations about DCO can be found elsewhere [27,28].

#### 2.3. KPE

- (1)
- KPE, as an improved PE, is defined as the distance between the time series and white Gaussian noise. Therefore, KPE and PE have a totally opposite trend. For example, when the time series is white Gaussian noise, PE and KPE are close to 1 and 0 respectively.
- (2)
- The equations of KPE and PE are different. KPE and PE can be expressed as$$\{\begin{array}{l}{H}_{PE}=-{\displaystyle \sum _{j=1}^{K}{P}_{j}\mathrm{ln}{P}_{j}}/\mathrm{ln}(m!)\\ {H}_{KPE}=={\displaystyle \sum _{j=1}^{K}{P}_{j}^{2}}-\frac{1}{m!}\end{array}$$
- (3)
- Compared with PE, KPE has better robustness for time series of different lengths.

## 3. Frequency Feature Extraction Method for Underwater Acoustic Signal

- Step 1:
- Signal decomposition.
- (1)
- Collect underwater acoustic signals by sensors;
- (2)
- Decompose underwater acoustic signals by EMD, M IMFs can be obtained;
- (3)
- Set the decomposition layers of VMD to M;
- (4)
- Decompose underwater acoustic signals by VMD.

- Step 2:
- Feature extraction.
- (1)
- Select the low-frequency IMF for the research, such as the last IMF;
- (2)
- Obtain estimated frequency of selected IMF by VMD;
- (3)
- Detect periodic signal of selected IMF using DCO;
- (4)
- When the phase track of selected IMF is in great periodic, and the KPE of DCO system output reaches the maximum, we can determine the accurate frequency of selected IMF.

- Step 3:
- Classification recognition.
- (1)
- Input frequency characteristics of different kinds of underwater acoustic signals into SVM;
- (2)
- Obtain classification results of different kinds of underwater acoustic signals.

## 4. Frequency Feature Extraction for Simulation Signal

#### 4.1. VMD of Simulation Signal

#### 4.2. Frequency Feature Extraction of IMF Using DCO and KPE

#### 4.2.1. Frequency Feature Extraction of IMF9

#### 4.2.2. Frequency Feature Extraction of IMF8 and IMF7

#### 4.3. Comparison of Different Frequency Feature Extraction Methods

## 5. Application in Underwater Acoustic Signals

#### 5.1. VMD of Ship-Radiated Noise Signal

#### 5.2. Frequency Feature Extraction of Line Spectrum

#### 5.3. Comparison of Different Frequency Feature Extraction Methods

## 6. Conclusions

- (1)
- DCO is first used to detect the frequency of IMF by VMD for underwater acoustic signals in this paper.
- (2)
- KPE is first used to determine the frequency of IMF combined with DCO for underwater acoustic signals in this paper.
- (3)
- VMD-DCO-PE is successfully applied to extract the frequency feature of a simulation signal. Compared with EMD-CF, EEMD-CF and VMD-CF, VMD-DCO-KPE can be more accurate and efficient to extract the frequency feature of a simulation signal.
- (4)
- VMD-DCO-KPE is also applied to extract the frequency feature extraction of line spectrum for underwater acoustic signal. VMD-DCO-KPE has better classification performance than EMD-CF, EEMD-CF and VMD-CF.

## Author Contributions

## Funding

## Conflicts of Interest

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

442.23 Hz | 392.59 Hz | 321.89 Hz | 264.65 Hz | 227.73 Hz | 168.37 Hz | 99.47 Hz | 50.12 Hz | 10.14 Hz |

9.95 Hz | 9.96 Hz | 9.97 Hz | 9.98 Hz | 9.99 Hz | 10.00 Hz | 10.01 Hz |
---|---|---|---|---|---|---|

0.313618 | 0.313618 | 0.313622 | 0.313630 | 0.313622 | 0.313618 | 0.313616 |

49.96 Hz | 49.97 Hz | 49.98 Hz | 49.99 Hz | 50.00 Hz | 50.01 Hz | 50.02 Hz |
---|---|---|---|---|---|---|

0.240762 | 0.240779 | 0.240813 | 0.240961 | 0.240884 | 0.240761 | 0.240753 |

100.00 Hz | 100.01 Hz | 100.02 Hz | 100.03 Hz | 100.04 Hz | 100.05 Hz | 100.06 Hz |
---|---|---|---|---|---|---|

0.163238 | 0.163396 | 0.164079 | 0.164159 | 0.164076 | 0.164027 | 0.163390 |

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

319.2 Hz | 147.13 Hz | 68.94 Hz | 43.66 Hz | 19.54 Hz | 9.68 Hz | 6.32 Hz | 3.02 Hz | 1.97 Hz |

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

338.07 Hz | 149.78 Hz | 73.74 Hz | 44.07 Hz | 16.26 Hz | 9.72 Hz | 4.41 Hz | 2.37 Hz |

Methods | 10 Hz | 50 Hz | 100 Hz |
---|---|---|---|

EMD-CF | 9.68 Hz | 43.66 Hz | 68.94 Hz |

EEMD-CF | 9.72 Hz | 44.07 Hz | 73.74 Hz |

VMD-CF | 10.14 Hz | 50.12 Hz | 99.47 Hz |

VMD-DCO-KPE | 9.98 Hz | 49.99 Hz | 100.03 Hz |

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

15.59 Hz | 66.18 Hz | 26.11 Hz |

**Table 9.**The frequency distribution of IMF8 by VMD-DCO-KPE for three kinds of ship-radiated noise samples.

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

11.82 Hz | 44.29 Hz | 29.85 Hz |

EMD-CF | EEMD-CF | VMD-CF | VMD-DCO-KPE |
---|---|---|---|

67.33% | 74.67% | 80.67% | 100% |

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**MDPI and ACS Style**

Li, Y.; Chen, X.; Yu, J.; Yang, X.
A Fusion Frequency Feature Extraction Method for Underwater Acoustic Signal Based on Variational Mode Decomposition, Duffing Chaotic Oscillator and a Kind of Permutation Entropy. *Electronics* **2019**, *8*, 61.
https://doi.org/10.3390/electronics8010061

**AMA Style**

Li Y, Chen X, Yu J, Yang X.
A Fusion Frequency Feature Extraction Method for Underwater Acoustic Signal Based on Variational Mode Decomposition, Duffing Chaotic Oscillator and a Kind of Permutation Entropy. *Electronics*. 2019; 8(1):61.
https://doi.org/10.3390/electronics8010061

**Chicago/Turabian Style**

Li, Yuxing, Xiao Chen, Jing Yu, and Xiaohui Yang.
2019. "A Fusion Frequency Feature Extraction Method for Underwater Acoustic Signal Based on Variational Mode Decomposition, Duffing Chaotic Oscillator and a Kind of Permutation Entropy" *Electronics* 8, no. 1: 61.
https://doi.org/10.3390/electronics8010061