#
Advances in Crest Factor Minimization for Wide-Bandwidth Multi-Sine Signals with Non-Flat Amplitude Spectra^{ †}

^{*}

^{†}

## Abstract

**:**

^{1}Hz to 10

^{7}Hz. The crest factor (CF) determines the information density of a multi-sine signal. Minimizing the CF yields higher information density and is the goal of the presented work. Four algorithms and a combination of two of them are presented. The first two algorithms implement different iterative optimizations of the amplitude and phase angle values of the signal. The combined algorithm alternates between the first and second optimization algorithms. Additionally, a simulated annealing approach and a genetic algorithm optimizing the CF were implemented.

## 1. Introduction

^{2}Hz to 10

^{7}Hz, resulting in a need for excitation signals with more than 20 frequencies distributed over the measurement bandwidth to provide sufficient spectral resolution.

#### 1.1. Multi-Sine

_{n}, amplitude a

_{n}, and phase φ

_{n}according to the following equation:

#### 1.2. Crest Factor

## 2. State of the Art

## 3. Optimization Approaches

#### 3.1. Iterative–Stochastic Optimization

#### 3.2. Simulated Annealing

#### 3.3. Genetic Algorithm

#### 3.4. Experiments

^{6}Hz as this was a reasonable tradeoff between CF reduction and calculation effort. The frequency distribution was fixed as was the number of iterations.

## 4. Results and Discussion

#### 4.1. CF Minimization

#### 4.2. Time per CF Reduction

## 5. Conclusions and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Appendix A

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Parameter | Value |
---|---|

Frequency bandwidth | ${10}^{2}-{10}^{6}$ |

Number of frequencies | 10, 20, 50, 100 |

Frequency distribution | Linear |

Amplitude distribution | Uniform, linear, exponential |

${n}_{CF}$ ^{1} | 40,000 |

^{1}Number of CF calculations.

Configuration | Time per Iteration (s) | ||||
---|---|---|---|---|---|

Frequency Components | Amplitude Distribution | SA | GA | Mixed | Clip |

10 | Uniform | 0.328 | 0.328 | 0.322 | 0.923 |

10 | Linear | 0.327 | 0.329 | 0.321 | 0.919 |

10 | Exponential | 0.326 | 0.328 | 0.321 | 0.922 |

20 | Uniform | 0.560 | 0.572 | 0.560 | 1.145 |

20 | Linear | 0.559 | 0.567 | 0.533 | 1.151 |

20 | Exponential | 0.560 | 0.566 | 0.530 | 1.148 |

50 | Uniform | 1.081 | 0.959 | 1.034 | 1.704 |

50 | Linear | 0.978 | 0.855 | 0.900 | 1.576 |

50 | Exponential | 0.790 | 0.717 | 0.646 | 1.350 |

100 | Uniform | 1.967 | 2.002 | 1.980 | 2.404 |

100 | Linear | 1.967 | 2.000 | 1.979 | 2.406 |

100 | Exponential | 1.058 | 1.054 | 1.059 | 1.093 |

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**MDPI and ACS Style**

Althoff, H.; Eberhardt, M.; Geinitz, S.; Linder, C.
Advances in Crest Factor Minimization for Wide-Bandwidth Multi-Sine Signals with Non-Flat Amplitude Spectra. *Comput. Sci. Math. Forum* **2022**, *2*, 11.
https://doi.org/10.3390/IOCA2021-10908

**AMA Style**

Althoff H, Eberhardt M, Geinitz S, Linder C.
Advances in Crest Factor Minimization for Wide-Bandwidth Multi-Sine Signals with Non-Flat Amplitude Spectra. *Computer Sciences & Mathematics Forum*. 2022; 2(1):11.
https://doi.org/10.3390/IOCA2021-10908

**Chicago/Turabian Style**

Althoff, Helena, Maximilian Eberhardt, Steffen Geinitz, and Christian Linder.
2022. "Advances in Crest Factor Minimization for Wide-Bandwidth Multi-Sine Signals with Non-Flat Amplitude Spectra" *Computer Sciences & Mathematics Forum* 2, no. 1: 11.
https://doi.org/10.3390/IOCA2021-10908