# X-ray Pulsar Signal Denoising Based on Variational Mode Decomposition

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### Variational Mode Decomposition

- Initialization: ${h}_{k}^{1},{\widehat{\omega}}_{k}^{1},{\widehat{\lambda}}^{1},n\leftarrow 0$;
- $n\leftarrow n+1$;
- Use ${\widehat{\lambda}}^{n+1}\left(\omega \right)={\widehat{\lambda}}^{n}\left(\omega \right)+\tau [\widehat{f}\left(\omega \right)-{\displaystyle \sum _{k}}{\widehat{h}}_{k}^{n+1}\left(\omega \right)]$ and update $\widehat{\lambda}$, $\forall \omega \ge 0$;
- Stop the iteration until $\sum _{k=1}^{K}}{\displaystyle \frac{{\u2225{\widehat{h}}_{k}^{n+1}-{\widehat{h}}_{k}^{n}\u2225}_{2}^{2}}{\u2225{\widehat{h}}_{k}^{n}\u2225}}<\epsilon $ for a chosen criterion $\epsilon $, otherwise return to step 2.

**Remark**

**1.**

## 3. VMD-Based Denoising Design for X-ray Pulsar Signals

#### 3.1. X-ray Pulsar Profile

#### 3.2. Denoise of Pulse Profile Based on VMD

- Calculate the sum$$y\left(k\right)=\sum _{i=1}^{k}u\left(i\right)-k\overline{u},k=1,2,\dots ,N,$$
- Divide the sequence $y\left(k\right)$ into ${N}_{n}=(N/n)$ nonoverlap length-of-n pieces. As for each local trend, one can apply l-order polynomial to fit ${y}_{n}\left(k\right)$. For example, let $l=2$ and define$${y}_{n}\left(k\right)={a}_{n}{k}^{2}+{b}_{n}k+{c}_{n},$$
- Define the root-mean-square (RMS) function by$$F\left(n\right)=\sqrt{\frac{1}{N}\sum _{k=1}^{N}{\left[y\left(k\right)-{y}_{n}\left(k\right)\right]}^{2}}.$$
- Finally, calculate the scaling exponent $\alpha $ by the least square regression approach as follows,$$ln\left(F\right(n\left)\right)=\alpha ln\left(n\right)+C.$$

## 4. Experimental Analysis

#### 4.1. Experiments of Simulation Data

#### 4.2. Experiments of HEASARC-Archived Data

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 8.**The reference and noisy pulsar profile archived pulsar data. (

**a**) Reference pulsar profile obtained via folding all photons; (

**b**) Noisy pulsar profile.

Method | ORI | EP | WT | EMD | VMD |
---|---|---|---|---|---|

SNR | −0.1145 | 22.8210 | 25.9410 | 27.2745 | 29.8821 |

RMSE | 1.0121 | 0.0727 | 0.0508 | 0.0435 | 0.0313 |

PCC | 0.1554 | 0.8619 | 0.9251 | 0.9441 | 0.9668 |

Method | Ori | EP | WT | EMD | VMD |
---|---|---|---|---|---|

SNR | 7.8000 | 25.9117 | 27.9890 | 29.1157 | 30.3373 |

RMSE | 0.4132 | 0.0506 | 0.0391 | 0.0339 | 0.0270 |

PCC | 0.0462 | 0.8748 | 0.9138 | 0.9277 | 0.9480 |

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Chen, Q.; Zhao, Y.; Yan, L.
X-ray Pulsar Signal Denoising Based on Variational Mode Decomposition. *Entropy* **2021**, *23*, 1181.
https://doi.org/10.3390/e23091181

**AMA Style**

Chen Q, Zhao Y, Yan L.
X-ray Pulsar Signal Denoising Based on Variational Mode Decomposition. *Entropy*. 2021; 23(9):1181.
https://doi.org/10.3390/e23091181

**Chicago/Turabian Style**

Chen, Qiang, Yong Zhao, and Lixia Yan.
2021. "X-ray Pulsar Signal Denoising Based on Variational Mode Decomposition" *Entropy* 23, no. 9: 1181.
https://doi.org/10.3390/e23091181