# Improving Prediction Accuracy and Extraction Precision of Frequency Shift from Low-SNR Brillouin Gain Spectra in Distributed Structural Health Monitoring

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## Abstract

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## 1. Introduction

- The first group of methods requires retrofitting additional hardware or software-based digital filters to increase the SNR of BGS. For example, in [20,21,22], the low-pass filtering, pump and probe waves intensities modulation and also the modulation of the probe signal wavelength were used. The reports in [23,24] describe the successful use of the wavelet transformations designed for signal filtering. The techniques mentioned above showed their effectiveness, but due to their complexity (the use of atypical and sophisticated algorithms with a significant number of parameters and coefficients) and sensitivity to the spectra shape, the implementation of these methods is limited. The idea of increasing the number of optical pulses for the strain measurement in a single repetition time has also been proposed. Inspired by the pulse coding technique in radar technology, the incorporation of optical pulse coding techniques such as Golay complementary codes has also been proposed for the purpose of improving the SNR [25].
- Then, the most popular in commercial instruments and well-known method is the reconstruction of a Lorentzian function, which is the actual BGS shape. This method is widely used in engineering applications [26]. In recent years the method has seen significant improvements in BFS detection precision [27] as well as in calculation speed increase and works perfectly if the tuning coefficients of the Lorentzian shape are precisely extracted. However, in practice, due to digitization data losses and the noise contributed by various other reasons, the spectrum can be significantly distorted [28,29]. These problems bring new limitations to the BGS reconstructing method’s application.
- The third approach is based on calculating the cross-correlation function of the obtained BGS and some previously generated profiles of the Lorentzian shape [30,31,32,33]. Additionally, instead of this approach, the spectrum can be obtained by inverting the original BGS, which is called backward correlation. This method will be described in this article in more detail later. Practice has shown that correlation methods are quite effective in studying signals with low SNR.
- Another noteworthy method is machine learning [34,35,36,37]. These are artificial intelligence methods used to obtain the correct characteristic feature which is not a direct solution to the problem but is a learning input through applied solutions to many other similar problems. To obtain the desired value of the frequency shift, it is advisable to use learning from precedents, or inductive learning based on identifying empirical patterns in the data. Similar methods, including the generalized linear model (GLM) method, which will also be discussed in detail in this article, have become widespread and have demonstrated their effectiveness not only in increasing the accuracy of determining the BFS but also in predicting it.

## 2. Levenberg–Marquardt (LM) Theory

## 3. Backward Correlation (BWC) Method Theory

## 4. Generalize Liner Model (GLM) Theory

## 5. Experiment Setup

## 6. Digital Noise

## 7. Data Processing Strategy

## 8. Results and Discussion

## 9. Conclusions and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The classical approach of the BFS extraction process (

**a**) in comparison with the proposed study (

**b**).

**Figure 2.**A schematic diagram combined using real data, describing the method operation principle (blue graph—forward spectrum; orange graph—backward spectrum with its shifted copy; dotted line—backward correlation function).

**Figure 7.**The absolute error results obtained after data processing: (

**a**) −2 dB single processing; (

**b**) −2 dB double processing (

**c**) 3 dB single processing; (

**d**) 3 dB double processing; (

**e**) 6 dB single processing (

**f**); 6 dB double processing; (

**g**) 20 dB single processing; (

**h**) 20 dB double processing.

No. | Components/Devices | Make and Model |
---|---|---|

1. | Laser source | Yokogawa AQ4312A |

2. | Signal generator | Hittite HMC-T2220 |

3. | Pulse generator | Agilent 33521A |

4. | Polarization scrambler | General Photonics PCD-104 |

5. | Mach–Zehnder modulator | iXblue MX-LN-20 |

6. | Single side-band modulator | iXblue MPX-LN-20 |

7. | Erbium-doped fiber amplifier | Keopsys CEFA-C-PB-LP |

8. | O/E converter | Tektronix P6703B |

9. | Oscilloscope | Teledyne-LeCroy HDO4054 |

10. | Optical spectrum analyzer | Yokogawa AQ6370B |

11. | Fibre Bragg grating | Generic. FWHM 1nm. Reflectivity >95% |

12. | Fibre under test | Corning SMF-28e+ |

13. | Polarization controller | Newport F-POL-APC |

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## Share and Cite

**MDPI and ACS Style**

Nordin, N.D.; Abdullah, F.; Zan, M.S.D.; A Bakar, A.A.; Krivosheev, A.I.; Barkov, F.L.; Konstantinov, Y.A.
Improving Prediction Accuracy and Extraction Precision of Frequency Shift from Low-SNR Brillouin Gain Spectra in Distributed Structural Health Monitoring. *Sensors* **2022**, *22*, 2677.
https://doi.org/10.3390/s22072677

**AMA Style**

Nordin ND, Abdullah F, Zan MSD, A Bakar AA, Krivosheev AI, Barkov FL, Konstantinov YA.
Improving Prediction Accuracy and Extraction Precision of Frequency Shift from Low-SNR Brillouin Gain Spectra in Distributed Structural Health Monitoring. *Sensors*. 2022; 22(7):2677.
https://doi.org/10.3390/s22072677

**Chicago/Turabian Style**

Nordin, Nur Dalilla, Fairuz Abdullah, Mohd Saiful Dzulkefly Zan, Ahmad Ashrif A Bakar, Anton I. Krivosheev, Fedor L. Barkov, and Yuri A. Konstantinov.
2022. "Improving Prediction Accuracy and Extraction Precision of Frequency Shift from Low-SNR Brillouin Gain Spectra in Distributed Structural Health Monitoring" *Sensors* 22, no. 7: 2677.
https://doi.org/10.3390/s22072677