# Application of Compressed Sensing Based on Adaptive Dynamic Mode Decomposition in Signal Transmission and Fault Extraction of Bearing Signal

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## Abstract

**:**

## 1. Introduction

## 2. Methodologies

#### 2.1. Compressed Sensing

#### 2.1.1. Compression Process

#### 2.1.2. Reconstruction Process

#### 2.2. Dynamic Mode Decomposition

_{.}

#### 2.3. The Improved Particle Swarm Optimization Algorithm

- (1).
- Initialize the position and velocity of each particle in the population randomly.
- (2).
- Evaluate the fitness of each particle. The position and fitness of all particles will be stored in the individual extreme value (${p}_{best}$), and the best position and fitness value in all ${p}_{best}$ will be put into the global extreme value ${g}_{best}$.
- (3).
- Update the particle displacement and velocity in Equations (20) and (21).
- (4).
- Update the weight in Equation (22).
- (5).
- Compare the fitness value of each particle with its best position and take the current fitness value as the best position of the particles if they are close. Compare current ${p}_{best}$ and ${g}_{best}$ to update ${g}_{best}$.
- (6).
- When the termination requirement is met, the search will stop, and results will output. Otherwise, it will go back to step (3), and go on.

#### 2.4. Mode Selection and Dictionary Construction

#### 2.5. Error Measurement

## 3. Application of ADMD-CS in the Simulation Experiment

## 4. Application of Algorithm in Experimental Data

## 5. Conclusions

- (1).
- A new fitness function of the IPSO algorithm is defined, namely a synthetic envelope kurtosis characteristic energy difference ratio. Additionally, a better decomposition effect can be achieved by using optimized target parameters.
- (2).
- A nonlinear dynamic inertia weight is used to optimize the traditional PSO algorithm, which is out of local search, and adaptively selects the truncated rank and threshold value to obtain a better decomposition effect and accurately extract fault features.
- (3).
- Combined ADMD with CS to form a new method, namely ADMD-CS. It compresses and reconstructs the signal to achieve the goal of reducing storage space and improving transmission efficiency.
- (4).
- Enhance the power of the mode in the frequency domain and use OMP to suppress the noise to obtain a better reconstructed signal.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**The schematic diagram of compressed sensing, based on adaptive dynamic mode decomposition.

**Figure 6.**The sparse coefficient of each dictionary. (

**a**) FFT coefficients; (

**b**) Haar coefficients; (

**c**) DCT coefficients; (

**d**) DMD coefficients.

**Figure 7.**Reconstructed time and quality of different algorithms. (

**a**) Reconstructed time; (

**b**) reconstructed quality.

**Figure 8.**Simulation and reconstructed signals of each dictionary. (

**a**) Simulation signal; (

**b**) FFT; (

**c**) Haar; (

**d**) DCT; (

**e**) DMD; and (

**f**) noise-free signal.

**Figure 9.**Comparison of various noise reduction methods. (

**a**) Noise-free signal; (

**b**) SVD; (

**c**) EMD; (

**d**) ADMD-CS.

**Figure 12.**Frequency domains comparison of various algorithms. (

**a**) Original signal; (

**b**) SVD; (

**c**) EMD; (

**d**) ADMD-CS.

a_{k} | γ | φ_{A} | τ_{k} | c_{A} | B | f_{i} | f_{o} | f_{b} | f_{c} |
---|---|---|---|---|---|---|---|---|---|

4 | 0 | 0 | 0.02 | 1 | 800 | 150 | 180 | 170 | 25 |

L1 | OMP | IT | |
---|---|---|---|

Time/s | 0.122 | 0.013 | 0.005 |

RMSE | 0.241 | 0.241 | 0.243 |

DICTIONARY | FFT | HAAR | DCT | DMD | |
---|---|---|---|---|---|

Noisy signal | CORR | 0.3719 | 0.1184 | 0.4277 | 0.8109 |

RMSE | 0.1360 | 0.1857 | 0.1262 | 0.0659 | |

Noise-free signal | CORR | 0.4154 | 0.1358 | 0.5024 | 0.9278 |

RMSE | 0.1231 | 0.1747 | 0.1102 | 0.0351 |

**Table 4.**SNR of recovered and original signals, after processing the simulation signal with different methods.

Methods | SNR |
---|---|

SVD | −0.3188 |

EMD | −1.6664 |

ADMD-CS | 0.3017 |

Methods | Original Signal | SVD | EMD | ADMD-CS |
---|---|---|---|---|

Storage space | 99 kb | 99 kb | 99 kb | 33.5 kb |

**Table 6.**SNR of the recovered and original signals, after processing experimental signal by different methods.

Methods | SNR |
---|---|

SVD | 0.6193 |

EMD | 0.2920 |

ADMD-CS | 0.8407 |

Methods | Original Signal | SVD | EMD | ADMD-CS |
---|---|---|---|---|

Storage space | 50.1 kb | 50.1 kb | 50.1 kb | 20.5 kb |

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**MDPI and ACS Style**

Cai, Z.; Dang, Z.; Wen, M.; Lv, Y.; Duan, H.
Application of Compressed Sensing Based on Adaptive Dynamic Mode Decomposition in Signal Transmission and Fault Extraction of Bearing Signal. *Machines* **2022**, *10*, 353.
https://doi.org/10.3390/machines10050353

**AMA Style**

Cai Z, Dang Z, Wen M, Lv Y, Duan H.
Application of Compressed Sensing Based on Adaptive Dynamic Mode Decomposition in Signal Transmission and Fault Extraction of Bearing Signal. *Machines*. 2022; 10(5):353.
https://doi.org/10.3390/machines10050353

**Chicago/Turabian Style**

Cai, Zhixin, Zhang Dang, Ming Wen, Yong Lv, and Haochun Duan.
2022. "Application of Compressed Sensing Based on Adaptive Dynamic Mode Decomposition in Signal Transmission and Fault Extraction of Bearing Signal" *Machines* 10, no. 5: 353.
https://doi.org/10.3390/machines10050353