# Parameter Estimation of LFM Signals Based on FOTD-CFRFT under Impulsive Noise

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## Abstract

**:**

## 1. Introduction

## 2. Impulsive Noise Model

- (1)
- If $X~S(\alpha ,\beta ,\gamma ,a)$, $b$ is a real number, then$$X+b~S(\alpha ,\beta ,\gamma ,a+b)$$
- (2)
- If $X~S(\alpha ,\beta ,\gamma ,a)$, m is a non-zero real number, then$$mX~\{\begin{array}{l}S(\alpha ,\mathrm{sgn}(m)\beta ,\gamma ,ma)\alpha \ne 1\\ S(1,\mathrm{sgn}(m)\beta ,|m|\gamma ,ma-\frac{2}{\mathsf{\pi}}m(\mathrm{ln}\left|m\right|)\beta \gamma )\alpha =1\end{array}$$
- (3)
- Let ${X}_{1}~S(\alpha ,{\beta}_{1},{\gamma}_{1},{a}_{1})$ and ${X}_{2}~S(\alpha ,{\beta}_{2},{\gamma}_{2},{a}_{2})$ be mutually independent α-stable distribution, then$${X}_{1}+{X}_{2}~S(\alpha ,\beta ,\gamma ,a)$$$$\beta =\frac{{\beta}_{1}{\gamma}_{1}^{\alpha}+{\beta}_{2}{\gamma}_{2}^{\alpha}}{{\gamma}_{1}^{\alpha}+{\gamma}_{2}^{\alpha}},\gamma =({\gamma}_{1}^{\alpha}+{\gamma}_{2}^{\alpha}{)}^{1/\alpha},a={a}_{1}+{a}_{2}$$

## 3. Parameter Estimation Method

#### 3.1. Fractional-Order Tracking Differentiator

#### 3.2. Parameter Estimation Model

## 4. Simulation Experiment

#### 4.1. FOTD Analysis

#### 4.2. Comparisons

#### 4.2.1. Standard SαS Distribution Noise

#### 4.2.2. Non-Standard SαS Distribution Noise

- (1)
- When $a\ne 0,\beta =0$, the correction formula is$$X-a~S(\alpha ,0,\gamma ,0)$$
- (2)
- When $a=0,\beta \ne 0$, let ${X}^{\prime}~S(\alpha ,-\beta ,\gamma ,0)$, the correction formula is$$X+{X}^{\prime}~S(\alpha ,0,{2}^{1/\alpha}\gamma ,0)$$
- (3)
- When $a\ne 0,\beta \ne 0$, let ${X}^{\prime}~S(\alpha ,-\beta ,\gamma ,0)$, the correction formula is$$X+{X}^{\prime}-a~S(\alpha ,0,{2}^{1/\alpha}\gamma ,0)$$

#### 4.3. Measured Noise Experiment

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 5.**The tracking performance of FOTD: (

**a**) time-domain waveform of the noisy signal; (

**b**) time-domain waveform of the tracked signal; (

**c**) fractional spectrum of the noisy signal; (

**d**) fractional spectrum of the tracked signal.

**Figure 6.**The parameter analysis of FOTD: (

**a**) tracking factor r; (

**b**) filtering factor ${h}_{0}$; (

**c**) fractional derivative order $\theta $.

**Figure 7.**The fractional spectrogram of the noisy signal (GSNR = 0 dB): (

**a**) FTD-FRFT; (

**b**) Sigmoid-FPSD; (

**c**) PANT-LVD; (

**d**) FOTD-CFRFT.

**Figure 8.**The fractional spectrogram of the noisy signal ($\mathrm{GSNR}=-3\mathrm{dB}$) and its projection: (

**a**) FTD-FRFT; (

**b**) Sigmoid-FPSD; (

**c**) PANT-LVD; (

**d**) FOTD-CFRFT.

**Figure 9.**RMSE of estimated parameter for four methods: (

**a**) center frequency; (

**b**) chirp rate. (

**c**) center frequency (with the outlier detection algorithm); (

**d**) chirp rate (with the outlier detection algorithm).

**Figure 10.**The scatter diagram of estimated parameters in 100 Monte Carlo experiments ($\mathrm{GSNR}=-6\mathrm{dB}$): (

**a**) FTD-FRFT; (

**b**) Sigmoid-FPSD; (

**c**) PANT-LVD; (

**d**) FOTD-CFRFT.

**Figure 11.**RMSE of estimated parameter for different $\alpha $: (

**a**) center frequency; (

**b**) chirp rate.

**Figure 12.**RMSE of center frequency and chirp rate for four methods at different $\beta $: (

**a**) center frequency; (

**b**) chirp rate.

**Figure 13.**The fractional spectrum of the noisy signal when $\beta =0.8$. (

**a**) FTD-FRFT; (

**b**) Sigmoid-FPSD; (

**c**) PANT-LVD; (

**d**) FOTD-CFRFT.

**Figure 14.**RMSE of center frequency and modulation frequency for four methods at different $a$: (

**a**) center frequency; (

**b**) chirp rate.

**Figure 15.**The fractional spectrum of the noisy signal when $a=3$. (

**a**) FTD-FRFT; (

**b**) Sigmoid-FPSD; (

**c**) PANT-LVD; (

**d**) FOTD-CFRFT.

**Figure 17.**The fractional spectrum of LFM signal and its projections under the weak measured impulsive noise: (

**a**) FTD-FRFT; (

**b**) Sigmoid-FPSD; (

**c**) PANT-LVD; (

**d**) FOTD-CFRFT.

**Figure 18.**The fractional spectrum of noisy signal and its projections under the strong measured impulsive noise: (

**a**) FTD-FRFT; (

**b**) Sigmoid-FPSD; (

**c**) PANT-LVD; (

**d**) FOTD-CFRFT.

−6 dB | −5 dB | −4 dB | −3 dB | −2 dB | −1 dB | 0 dB | 1 dB | 2 dB | 3 dB | ||
---|---|---|---|---|---|---|---|---|---|---|---|

f_{0} | FTD-FRFT | 1.2102 | 2.2815 | 5.5838 | 7.6531 | 13.4799 | 18.5058 | 20.0341 | 20.0968 | 20.0952 | 20.0950 |

Sigmoid-FPSD | −4.7484 | 11.7004 | 10.2649 | 20.2852 | 20.2182 | 20.1562 | 20.1863 | 20.1862 | 20.1058 | 20.1158 | |

PANT-LVD | 16.0291 | 18.4695 | 19.6947 | 20.9199 | 20.9199 | 20.9199 | 20.9199 | 20.9199 | 20.9199 | 20.9199 | |

FOTD-CFRFT | 20.2796 | 20.2574 | 20.2608 | 20.2617 | 20.2639 | 20.2652 | 20.2605 | 20.2603 | 20.2649 | 20.2646 | |

k | FTD-FRFT | −3.5269 | 0.9849 | 1.3671 | 8.1601 | 25.9438 | 40.8245 | 49.4254 | 50.1678 | 50.0054 | 49.9729 |

Sigmoid-FPSD | 45.4743 | 54.6101 | 50.1517 | 50.1191 | 50.1191 | 50.1678 | 50.1840 | 50.1677 | 50.1516 | 50.1353 | |

PANT-LVD | 37.9866 | 43.9949 | 47.0189 | 50.1228 | 50.0730 | 50.1329 | 50.2327 | 50.2826 | 50.3425 | 50.3924 | |

FOTD-CFRFT | 50.0982 | 50.1338 | 50.0889 | 50.1279 | 50.1832 | 50.1652 | 50.1783 | 50.1738 | 50.2006 | 50.1938 |

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**MDPI and ACS Style**

Wang, H.; Guo, Y.; Yang, L.
Parameter Estimation of LFM Signals Based on FOTD-CFRFT under Impulsive Noise. *Fractal Fract.* **2023**, *7*, 822.
https://doi.org/10.3390/fractalfract7110822

**AMA Style**

Wang H, Guo Y, Yang L.
Parameter Estimation of LFM Signals Based on FOTD-CFRFT under Impulsive Noise. *Fractal and Fractional*. 2023; 7(11):822.
https://doi.org/10.3390/fractalfract7110822

**Chicago/Turabian Style**

Wang, Houyou, Yong Guo, and Lidong Yang.
2023. "Parameter Estimation of LFM Signals Based on FOTD-CFRFT under Impulsive Noise" *Fractal and Fractional* 7, no. 11: 822.
https://doi.org/10.3390/fractalfract7110822