# Application of L1 Trend Filtering Technology on the Current Time Domain Spectroscopy of Dielectrics

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Basic Theory of L1 Trend Filtering

## 3. Test of the Current Time Domain Spectrum

#### 3.1. The System of Measurement of the Current Time Domain Spectrum

#### 3.2. Test Results of the Current Time Domain Spectrum

## 4. Filtering Results and Analysis

#### 4.1. The Filtering Results with L1 Trend Filtering Algorithms

#### 4.2. Filtering Results with Several Common Filtering Algorithms

#### 4.2.1. Filtering Results with Sliding Mean Filtering

#### 4.2.2. Filtering Results with Savitzky–Golay

#### 4.2.3. Filtering Results with Wavelet Transform

#### 4.3. Analysis and Discussion

## 5. Comprehensive Performance Analysis of L1 Trend Filtering

#### 5.1. Robustness Analysis of L1 Trend Filtering

#### 5.2. Time Complexity Analysis of L1 Trend Filtering

## 6. The Effects of L1 Trend Filtering on the Acquisition of the Trap Distribution

## 7. Conclusions

- Due to the wide range of the polarization currents in the whole test time, the commonly filtering algorithms such as sliding average filtering and Savoitzky–Golay smoothing filtering cannot take into account the filtering effects of the whole time period. If the filtering effects of the initial stage of the currents are good, then the filtering effects of the last stage are poor. If the filtering effects of the last stage of the currents are good, then the initial stage of the filtering curve will be distorted.
- For L1 trend filtering, with the increase of the λ value, the polarization current curve after filtering becomes smoother, but the filtering results will be distorted at the beginning of a period of time when λ exceeds a certain value. For the polarization currents in this paper, when λ is in the range of 1600–24,000, the filtering curve of the polarization currents by L1 trend filtering is smooth and undistorted in the whole test time, and the λ has little influence on the filtering effects.
- For the time series under noises with different SNR, the L1 trend filter can accurately extract the trend items, and the relative error between the given SNR and SNR obtained by L1 trend filtering is about 1%. The execution time obtained by the simulation experiment is also lower than 176.67 s when the number of tested points is no more than 20,000. This result show that L1 trend filtering has great robustness, and L1 trend filter technology can be applied to the filtering of the time domain current spectrum of dielectrics.
- For the effects of L1 trend filtering on the acquisition of the trap distribution, when I(t) is the virtual tested polarization absorption current which is filtered by L1 trend filtering, the I(t)·t ~ log(t) curve obvious overlaps with the I(t)·t ~ log(t) curve when I(t) is the given ideal trend time without any noise. By contrast, due to the noise in the tested polarization current, both the number and position of the peak in the I(t)·t ~ log(t) curve are misjudged, which leads to an incorrect judgment about the trap distribution of dielectrics. The results indicate that the L1 trend filtering of polarization absorption current play an important role in the trap distribution acquisition of dielectrics.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

D=zeros(7200); |

n=7200; |

for i=1:1:n |

D(i,i)=1; |

D(i,i+1)=-2; |

D(i,i+2)=1; |

end |

cvx_begin |

variable x(7202) |

minimize( 0.5*sum((y-x).^2)+10,000*norm(D*x,1)) |

subject to |

x>=0 |

cvx_end |

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**Figure 2.**The test results of 20% SiC/PE composite under electric field strength of 30 kV/mm. (

**a**) The test results of the polarization currents; and (

**b**) the test results of depolarization currents.

**Figure 3.**The diagram of L1 trend filtering result: (

**a**) The filtering results with λ value of 10–100,000; (

**b**) the filtering results with λ value of 1000–30,000; and (

**c**) the filtering results with λ = 10,000.

**Figure 4.**The effect diagram of moving average filtering with different adjacent numbers: (

**a**) the filtering results with m = 10; and (

**b**) the filtering results with m = 1000.

**Figure 5.**The effect diagram Savitzky–Golay smooth filtering with different window widths: (

**a**) The filtering result with m = 40; and (

**b**) the filtering result with m = 400.

**Figure 6.**The effect diagram of wavelet transform filtering with different decomposition levels: (

**a**) Filtering results with J = 3; and (

**b**) filtering results with J = 6.

**Figure 7.**The effect diagram of L1 trend filtering for different degrees of noises with the ideal trend term produced by the extended Debye model: (

**a**) SNR = 10 dB; (

**b**) SNR = 15 dB; (

**c**) SNR = 20 dB; (

**d**) SNR = 25 dB.

**Figure 8.**The effect diagram of L1 trend filtering for different degrees of noises with the ideal trend term produced by the Curie formula: (

**a**) SNR = 10 dB; (

**b**) SNR = 15 dB; (

**c**) SNR = 20 dB; (

**d**) SNR = 25 dB.

**Figure 9.**The diagram the product of the polarization absorption current with time (I × t) with log10 t: (

**a**) The polarization absorption current with noise; (

**b**) the polarization absorption current filtered by L1 trend filtering; and (

**c**) the comparison of polarization absorption current filtered by L1 trend filtering and the given ideal trend time.

Signal-to-Noise Ratio of Given Interference (dB) | 10 | 15 | 20 | 25 |
---|---|---|---|---|

Signal to noise ratio after L1 trend filtering (dB) (λ = 10,000) | 9.99 | 14.88 | 19.81 | 24.64 |

The relative error (%) | 1% | 0.8% | 0.95% | 1.4% |

The Number of Points | 5000 | 10,000 | 20,000 | 30,000 | 40,000 | 50,000 |
---|---|---|---|---|---|---|

The execution time (seconds) (λ = 10,000) | 25.92 | 63.50 | 176.67 | 353.11 | 2512.81 | insufficient memory |

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**MDPI and ACS Style**

Suo, C.; Li, Z.; Sun, Y.; Han, Y.
Application of L1 Trend Filtering Technology on the Current Time Domain Spectroscopy of Dielectrics. *Electronics* **2019**, *8*, 1046.
https://doi.org/10.3390/electronics8091046

**AMA Style**

Suo C, Li Z, Sun Y, Han Y.
Application of L1 Trend Filtering Technology on the Current Time Domain Spectroscopy of Dielectrics. *Electronics*. 2019; 8(9):1046.
https://doi.org/10.3390/electronics8091046

**Chicago/Turabian Style**

Suo, Changyou, Zhonghua Li, Yunlong Sun, and Yongsen Han.
2019. "Application of L1 Trend Filtering Technology on the Current Time Domain Spectroscopy of Dielectrics" *Electronics* 8, no. 9: 1046.
https://doi.org/10.3390/electronics8091046