# Generalized Hurst Hypothesis: Description of Time-Series in Communication Systems

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## Abstract

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## 1. Introduction

- We propose the generalization of asymptotic Hurst law (Section 2.1).
- The proposed method is not asymptotic, thus it does not impose any limitations on the size of the analyzed TLS.
- We test it on noise data described in Section 2.2.
- We analyze the importance of parameters of the generalized Hurst law (Section 3).
- We discuss the possible applications of the generalized Hurst law (Section 4).

## 2. Materials and Methods

#### 2.1. Description of the Algorithm

- Select L and apply the RTIP-procedure, if necessary.
- Calculate the ratios $R\left(ka\right)/S\left(ka\right)$ up to $K=\lfloor \frac{N}{a}\rfloor $.

#### 2.2. Description of the Data

## 3. Results

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

DFA | detrended fluctuation analysis |

EC | eigen-coordinates method |

LLSM | linear least square method |

LTS | long-time series |

RTIP | reduction to three incident points |

TLS(s) | trendless sequence(s) |

SDR | Software Defined Radio |

USRP | Universal Software Radio Peripheral |

ADC | Analog to Digital Converter |

## Appendix A

#### Appendix A.1

#### Appendix A.2

#### Appendix A.3

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Fit Error | H | A | |
---|---|---|---|

mean | 1.0% | −1.0 | 3.19 |

std | 0.4% | 0.1 | 0.02 |

Fit Error | ${\mathit{B}}_{2}$ | ${\mathit{B}}_{3}$ | Re(${\mathit{H}}_{1}$) | Im(${\mathit{H}}_{1}$) | ${\mathit{H}}_{3}$ | |
---|---|---|---|---|---|---|

mean | 2.0% | 0.048 | −0.05 | −0.16 | −0.04 | −2.4 |

std | 0.5% | 0.5 | 0.59 | 0.25 | 0.4 | 0.5 |

Fit Error | ${\mathit{B}}_{0}$ | Re(${\mathit{H}}_{1,\mathit{compl}}$) | Im(${\mathit{H}}_{1,\mathit{compl}}$) | ${\mathit{H}}_{1,\mathit{real}}$ | ${\mathit{H}}_{2,\mathit{real}}$ | |
---|---|---|---|---|---|---|

mean | 1.0% | 4.17 | −0.05 | 0.55 | 0.59 | −0.52 |

std | 0.3% | 0.4 | 0.07 | 0.15 | 0.43 | 0.38 |

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**MDPI and ACS Style**

Nigmatullin, R.; Dorokhin, S.; Ivchenko, A.
Generalized Hurst Hypothesis: Description of Time-Series in Communication Systems. *Mathematics* **2021**, *9*, 381.
https://doi.org/10.3390/math9040381

**AMA Style**

Nigmatullin R, Dorokhin S, Ivchenko A.
Generalized Hurst Hypothesis: Description of Time-Series in Communication Systems. *Mathematics*. 2021; 9(4):381.
https://doi.org/10.3390/math9040381

**Chicago/Turabian Style**

Nigmatullin, Raoul, Semyon Dorokhin, and Alexander Ivchenko.
2021. "Generalized Hurst Hypothesis: Description of Time-Series in Communication Systems" *Mathematics* 9, no. 4: 381.
https://doi.org/10.3390/math9040381