# A Gyroscope Signal Denoising Method Based on Empirical Mode Decomposition and Signal Reconstruction

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Basic Theory

#### 2.1.1. Empirical Mode Decomposition

#### 2.1.2. Quantifying the Noise in the Gyroscope Sensor through Fractal Gaussian Noise

#### 2.2. Proposed Denoising Method

- Estimate the value of the Hurst parameter (H) and quantify the noise through FGN;
- Implement the EMD algorithm to obtain the IMFs and residual;
- Determine the pure noise IMFs by comparing the significant difference between the actual and theoretical energy values;
- Determine the mixed IMFs using an improved Hausdorff distance (HD) method;
- Threshold the mixed IMFs via a hard interval threshold;
- Reconstruct the optimal signal.

#### 2.2.1. Determining the Value of ${M}_{1}$ and ${M}_{2}$

#### 2.2.2. Thresholding for Mixed IMFs

#### 2.3. Experimental Design

**Scheme 1**: Input signal (the actual gyroscope data);

**Scheme 2**: Partial reconstruction of relevant IMFs based on Euclidean norm (${l}_{2}$-norm) measure, named EMD-${l}_{2}$-norm;

**Scheme 3**: Whole reconstruction of filtered IMFs based on Sliding Average, named EMD-SA;

**Scheme 4**: Proposed method.

## 3. Results and Discussions

#### 3.1. Static Test Signal

#### 3.2. Dynamic Test Signal

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Dell’Olio, F.; Tatoli, T.; Ciminelli, C.; Armenise, M.N. Recent advances in miniaturized optical gyroscopes. J. Eur. Opt. Soc.
**2014**, 9, 1–14. [Google Scholar] [CrossRef] - Wu, B.; Yu, Y.; Xiong, J.; Zhang, X. Silicon Integrated Interferometric Optical Gyroscope. Sci. Rep.
**2018**, 8, 1–7. [Google Scholar] [CrossRef] [PubMed] - Cui, B.; Chen, X. Improved hybrid filter for fiber optic gyroscope signal denoising based on EMD and forward linear prediction. Sensor. Actuat. A.-Phys.
**2015**, 230, 150–155. [Google Scholar] [CrossRef] - Ciminelli, C.; Dell’Olio, F.; Campanella, C.E.; Armenise, M.N. Photonic technologies for angular velocity sensing. Adv. Opt. Photonics
**2010**, 2, 370–404. [Google Scholar] [CrossRef] - Woodman, O.J. An introduction to inertial navigation; Technical Report Number 696; University of Cambridge, Computer Laboratory: Cambridge, UK, 2007. [Google Scholar]
- Yang, X.H.; Ren, J.X.; Zhao, X.M.; Chen, R. MEMS Gyro Signal De-noising Based on Adaptive Stationary Wavelet Threshold. Adv. Mat. Res.
**2012**, 466–467, 986–990. [Google Scholar] [CrossRef] - Zhang, F. Modeling Study on Random Error of Fiber Optic Gyro. Appl. Mech. Mater.
**2013**, 239–240, 167–171. [Google Scholar] [CrossRef] - Ma, J.; Yang, Z.; Shi, Z.; Zhang, X.; Liu, C. Application and Optimization of Wavelet Transform Filter for North-Seeking Gyroscope Sensor Exposed to Vibration. Sensors
**2019**, 19, 3624. [Google Scholar] [CrossRef] - Yang, G.; Liu, Y.; Wang, Y.; Zhu, Z. EMD interval thresholding denoising based on similarity measure to select relevant modes. Signal Process.
**2015**, 109, 95–109. [Google Scholar] [CrossRef] - Gan, Y.; Sui, L.; Wu, J.; Wang, B.; Zhang, Q.; Xiao, G. An EMD threshold de-noising method for inertial sensors. Measurement
**2014**, 49, 34–41. [Google Scholar] [CrossRef] - Antoniadis, A.; Bigot, J.; Sapatinas, T. Wavelet Estimators in Nonparametric Regression: A Comparative Simulation Study. J. Stat. Softw.
**2001**, 6, 1–83. [Google Scholar] [CrossRef] - Cai, T.T.; Silverman, B.W. Incorporating Information on Neighbouring Coefficients into Wavelet Estimation. Sankhyā Indian J. Stat. Ser. B
**2001**, 63, 127–148. [Google Scholar] - Bessous, N.; Zouzou, S.E.; Bentrah, W.; Sbaa, S.; Sahraoui, M. Diagnosis of bearing defects in induction motors using discrete wavelet transform. Int. J. Syst. Assur. Eng. Manag.
**2018**, 9, 335–343. [Google Scholar] [CrossRef] - Donoho, D.L.; Johnstone, I.M.; Kerkyacharian, G.; Picard, D. Density estimation by wavelet thresholding. Ann. Stat.
**1996**, 24, 508–539. [Google Scholar] [CrossRef] - Luo, H.Z.; Lin, X.Y.; Liu, L. Research on GPS/SINS Integrated Navigation System Based on Wavelet Transform. In Proceedings of the 7th International Conference on Wireless Communications, Networking and Mobile Computing (WiCOM 2011), Wuhan, China, 23–25 Septembar 2011; pp. 1–4. [Google Scholar]
- Yan, R.; Gao, R.X.; Chen, X. Wavelets for fault diagnosis of rotary machines: A review with applications. Signal Process.
**2014**, 96, 1–15. [Google Scholar] [CrossRef] - Davari, N.; Gholami, A.; Shabani, M. Performance Enhancement of GPS/INS Integrated Navigation System Using Wavelet Based De-noising method. AUT J. Electr. Eng.
**2016**, 48, 101–111. [Google Scholar] - Zeng, K.; Huang, J.; Dong, M. White Gaussian Noise Energy Estimation and Wavelet Multi-threshold De-noising for Heart Sound Signals. Circ. Syst. Signal Pr.
**2014**, 33, 2987–3002. [Google Scholar] [CrossRef] - Huang, N.E.; Shen, Z.; Long, S.R.; Wu, M.C.; Shih, H.H.; Zheng, Q.; Yen, N.C.; 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. London A Math. Phys. Eng. sci.
**1998**, 454, 903–995. [Google Scholar] [CrossRef] - Yu, D.; Cheng, J.; Yang, Y. Application of EMD method and Hilbert spectrum to the fault diagnosis of roller bearings. Mech. Syst. Signal Pr.
**2005**, 19, 259–270. [Google Scholar] [CrossRef] - Guangfen, W.; Wen, A.; Fajin, G.; Zhenan, T.; Jun, Y. The Hilbert-Huang Transform and Its Application in Processing Dynamic Signals of Gas Sensors. In Proceedings of the 2009 International Conference on Information Engineering and Computer Science, Wuhan, China, 19–20 December 2009; pp. 1–4. [Google Scholar]
- Jun, H.; Zhang, Q.; Sun, G.; Yang, J.C.; Xiong, J. A Vibration Signal Analysis Method based on Enforced De-Noising and Modified EMD. I. J. Signal Process. Image Process. Pattern Recognit.
**2015**, 8, 87–98. [Google Scholar] [CrossRef] - Deng, H.; Liu, J.; Li, H. EMD Based Infrared Image Target Detection Method. J. Infrared Millim. Te.
**2009**, 30, 1205–1215. [Google Scholar] [CrossRef] - Wang, J.; Zhang, J.; Liu, Z. EMD based multi-scale model for high resolution image fusion. GeoSpat. Inf. Sci.
**2008**, 11, 31–37. [Google Scholar] [CrossRef] - Rakshit, M.; Das, S. An efficient ECG denoising methodology using empirical mode decomposition and adaptive switching mean filter. Biomed. Signal Process. Control
**2018**, 40, 140–148. [Google Scholar] [CrossRef] - Dang, S.; Tian, W.; Qian, F. EMD- and LWT-based stochastic noise eliminating method for fiber optic gyro. Measurement
**2011**, 44, 2190–2193. [Google Scholar] [CrossRef] - Flandrin, P.; Rilling, G.; Goncalves, P. Empirical Mode Decomposition as a Filter Bank. IEEE Signal Process. Lett.
**2004**, 11, 112–114. [Google Scholar] [CrossRef] - Wu, Z.; Huang, N.E. A study of the characteristics of white noise using the empirical mode decomposition method. Proc. R. Soc. London A Math. Phys. Eng. Sci.
**2004**, 460, 1597–1611. [Google Scholar] [CrossRef] - Flandrin, P.; Goncalves, P.; Rilling, G. EMD Equivalent Filter Banks, from Interpretation to Applications. In Hilbert-Huang Transform and Its Applications; World Scientific: Singapore, 2004; pp. 57–74. [Google Scholar] [CrossRef]
- Ayenu-Prah, A.Y.; Attoh-Okine, N.O. A criterion for selecting relevant intrinsic mode functions in empirical mode decomposition. Adv.Adapt. Data Anal.
**2010**, 2, 1–24. [Google Scholar] [CrossRef] - Tang, Y.W.; Tai, C.C.; Su, C.C.; Chen, C.Y.; Chen, J.F. A correlated empirical mode decomposition method for partial discharge signal denoising. Meas. Sci. Technol.
**2010**, 21, 085106. [Google Scholar] [CrossRef] - Komaty, A.; Boudraa, A.O.; Augier, B.; Dare-Emzivat, D. EMD-Based Filtering Using Similarity Measure Between Probability Density Functions of IMFs. IEEE T. Instrum. Meas.
**2014**, 63, 27–34. [Google Scholar] [CrossRef] - Darong, H.; Lanyan, K.; Bo, M.; Ling, Z.; Guoxi, S. A New Incipient Fault Diagnosis Method Combining Improved RLS and LMD Algorithm for Rolling Bearings with Strong Background Noise. IEEE Access
**2018**, 6, 26001–26010. [Google Scholar] [CrossRef] - Jiang, C.; Zhang, S.B. A Novel Adaptively-Robust Strategy Based on the Mahalanobis Distance for GPS/INS Integrated Navigation Systems. Sensors
**2018**, 18, 695. [Google Scholar] [CrossRef] - Boudraa, A.O.; Cexus, J.C. Denoising via empirical mode decomposition. Proc. IEEE ISCCSP
**2006**, 2006, 4. [Google Scholar] - Mandelbrot, B.B.; Van Ness, V.J. Fractional Brownian Motions, Fractional Noises and Applications. SIAM Rev.
**1968**, 10, 422–437. [Google Scholar] [CrossRef] - Mandelbrot, B.B.; Wallis, J.R. Computer Experiments with Fractional Gaussian Noises: Part 1, Mathematical Appendix. Water Resour. Res.
**1969**, 5, 260–267. [Google Scholar] [CrossRef] [Green Version] - Rilling, G.; Flandrin, P.; Goncalves, P. Empirical Mode Decomposition, fractional Gaussian noise and Hurst exponent estimation. In Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, Philadelphia, PA, USA, 23–23 March 2005; pp. 489–492. [Google Scholar]
- Huttenlocher, D.P.; Klanderman, G.A.; Rucklidge, W.J. Comparing Images Using the Hausdorff Distance. IEEE Trans. Pettern Anal. Mach. Intell
**1993**, 15, 850–863. [Google Scholar] [CrossRef] - Komaty, A.; Boudraa, A.O.; Dare, D. EMD-based filtering using the Hausdorff distance. In Proceedings of the IEEE International Symposium on Signal Processing and Information Technology (ISSPIT), Ho Chi Minh City, Vietnam, 12–15 December 2012; pp. 292–297. [Google Scholar]
- Kopsinis, Y.; McLaughlin, S. Empirical mode decomposition based soft-thresholding. In Proceedings of the 16th European Signal Processing Conference, Lausanne, Switzerland, 25–29 August 2008. [Google Scholar]
- Xi, X.; Zhang, Y.; Zhao, Y.; She, Q.; Luo, Z. Denoising of surface electromyogram based on complementary ensemble empirical mode decomposition and improved interval thresholding. Rev. Sci. Instrum.
**2019**, 90, 035003. [Google Scholar] [CrossRef] [PubMed] - Prosvirin, A.E.; Islam, M.; Kim, J.; Kim, J.M. Rub-Impact Fault Diagnosis Using an Effective IMF Selection Technique in Ensemble Empirical Mode Decomposition and Hybrid Feature Models. Sensors
**2018**, 18, 2040. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**Flow chart of the detailed screening steps of empirical mode decomposition (EMD). Note: IMF = intrinsic mode function.

**Figure 6.**Gyroscope signal and its denoised results for three denoising methods. The red line is the gyroscope signal; the blue, orange, and green lines are the denoising results of Euclidean norm measure method (EMD-${l}_{2}$-norm), the sliding average filtering method (EMD-SA), and the proposed method, respectively.

**Figure 7.**Amplitude spectrum of the signal. The red line is the gyroscope signal; the blue, orange, and green lines are the denoising results of EMD-${l}_{2}$-norm, EMD-SA, and the proposed method, respectively.

**Figure 8.**Allan variance analysis of the gyroscope signal and denoised signal. The red line is the gyroscope signal; the blue, orange, and green lines are the denoising results of EMD-${l}_{2}$-norm, EMD-SA, and the proposed method, respectively.

**Figure 9.**The gyroscope dynamic signal and its denoised results for three denoising methods in the five dynamic tests. The red line is the gyroscope’s dynamic signal; the blue, orange, and green lines are the denoising results of the EMD-l

_{2}-norm, EMD-SA, and the proposed method, respectively. (

**a**–

**e**) Dynamic gyroscope signals and the denoised results for the three denoising methods, with speeds of 10 °/s, 20 °/s, 30 °/s, 40 °/s, and 50 °/s, respectively.

**Table 1.**Bias drift, angle random walking, quantization noise, and root mean square error (RMSE) for the different schemes.

Scheme | Bias Drift (°/s) | Angle Random Walking (°/$\sqrt{\mathit{h}}$) | Quantization Noise (°) | RMSE (°/s) |
---|---|---|---|---|

1 | 0.0308 | 0.0526 | 0.5130 | 0.0309 |

2 | 0.0075 | 0.0374 | 0.4820 | 0.0079 |

3 | 0.0059 | 0.0273 | 0.3404 | 0.0063 |

4 | 0.0038 | 0.0191 | 0.1622 | 0.0045 |

**Table 2.**Comparison of root mean square error (RMSE) and standard deviation (STD) for the denoising results of dynamic signals with different rotational speeds.

Rotational Speed | Indicator | Scheme 1 | Scheme 2 | Scheme 3 | Scheme 4 |
---|---|---|---|---|---|

10 °/s (H = 0.2472) | RMSE | 0.0951 | 0.0933 | 0.0927 | 0.0892 |

STD | 0.0537 | 0.0505 | 0.0493 | 0.0177 | |

20 °/s (H = 0.3277) | RMSE | 0.1686 | 0.1676 | 0.1672 | 0.1580 |

STD | 0.0685 | 0.0660 | 0.0645 | 0.0276 | |

30 °/s (H = 0.3137) | RMSE | 0.2396 | 0.2389 | 0.2386 | 0.2321 |

STD | 0.0686 | 0.0662 | 0.0652 | 0.0392 | |

40 °/s (H = 0.2472) | RMSE | 0.3144 | 0.3138 | 0.3136 | 0.3069 |

STD | 0.0763 | 0.0741 | 0.0731 | 0.0450 | |

50 °/s (H = 0.3310) | RMSE | 0.3901 | 0.3897 | 0.3896 | 0.3809 |

STD | 0.0860 | 0.0840 | 0.0832 | 0.0500 |

© 2019 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**

Liu, C.; Yang, Z.; Shi, Z.; Ma, J.; Cao, J.
A Gyroscope Signal Denoising Method Based on Empirical Mode Decomposition and Signal Reconstruction. *Sensors* **2019**, *19*, 5064.
https://doi.org/10.3390/s19235064

**AMA Style**

Liu C, Yang Z, Shi Z, Ma J, Cao J.
A Gyroscope Signal Denoising Method Based on Empirical Mode Decomposition and Signal Reconstruction. *Sensors*. 2019; 19(23):5064.
https://doi.org/10.3390/s19235064

**Chicago/Turabian Style**

Liu, Chenchen, Zhiqiang Yang, Zhen Shi, Ji Ma, and Jian Cao.
2019. "A Gyroscope Signal Denoising Method Based on Empirical Mode Decomposition and Signal Reconstruction" *Sensors* 19, no. 23: 5064.
https://doi.org/10.3390/s19235064