# M-Ary Direct Modulation Chirp Spread Spectrum for Spectrally Efficient Communications

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## Abstract

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## 1. Introduction

- Design of a modulation scheme that transmits multiple overlapping up-chirp signals;
- Mathematical formulation of the symbol time and the number of overlaps;
- Deriving the mathematical expression for the bit error probability of the proposed scheme as a function of inter-symbol interference and number of overlaps;
- Performance evaluation using various metrics to demonstrate the effectiveness of our approach.

## 2. Direct Modulation CSS

## 3. M-Ary Phase Shift Keying

## 4. Proposed Model

#### 4.1. Modulation Scheme

#### 4.2. Overlapping Chirps

#### 4.3. Inter-Symbol Interference Due to Overlap

#### 4.4. Zero-ISI Transmission

#### 4.5. Probability of Bit Error

## 5. Performance Analysis and Discussion

#### 5.1. Simulation Environment

#### 5.2. Spectral Efficiency Analysis

#### 5.3. BER Performance Analysis

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AWGN | Additive White Gaussian Noise |

BER | Bit Error Rate |

BOC | Binary Offset Carrier |

BOK | Binary Orthogonal Keying |

CCC | Cross Correlation Coefficients |

CSS | Chirp Spread Spectrum |

DM CSS | Direct Modulation Chirp Spread Spectrum |

ISI | Inter-Symbol Interference |

MPSK | M-ary Phase Shift Keying |

## Appendix A. Expression of the Receiver Output

## Appendix B. Probability of Bit Error

#### Appendix B.1. Euclidean Distance between s_{m} and s_{smi}

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**Figure 3.**Time–frequency representation of the instantaneous frequency of up-chirp signals with normalized time and frequency axes.

**Figure 6.**Theoretical and measured BER as a function of ${E}_{b}/{N}_{0}$ for MPSK CSS with ${B}_{ch}=200$ kHz.

**Figure 7.**Measured BER as a function of ${E}_{b}/{N}_{0}$ for PSK, QAM, and PSK CSS with $\eta \in \{2,3,4\}$.

$\mathit{\alpha}$ | $\frac{\mathit{\pi}}{6}$ | $\frac{\mathit{\pi}}{5}$ | $\frac{\mathit{\pi}}{4}$ | $\frac{3\mathit{\pi}}{10}$ | $\frac{\mathit{\pi}}{3}$ | $\frac{3\mathit{\pi}}{8}$ | $\frac{2\mathit{\pi}}{5}$ | $\frac{5\mathit{\pi}}{12}$ |
---|---|---|---|---|---|---|---|---|

$tan\alpha $ | $\frac{\sqrt{3}}{3}$ | $\sqrt{5-2\sqrt{5}}$ | 1 | $\frac{\sqrt{25+10\sqrt{5}}}{5}$ | $\sqrt{3}$ | $\sqrt{2}+1$ | $\sqrt{5+2\sqrt{5}}$ | $2+\sqrt{3}$ |

$\eta $ | $1.1547$ | $1.4531$ | $2.0$ | $2.7528$ | $3.4641$ | $4.8284$ | $6.1554$ | $7.4641$ |

$\lfloor \eta \rfloor $ | 1 | 1 | 2 | 2 | 3 | 4 | 6 | 7 |

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**MDPI and ACS Style**

Zacko Gbadoubissa, J.E.; Abba Ari, A.A.; Radoi, E.; Gueroui, A.M.
M-Ary Direct Modulation Chirp Spread Spectrum for Spectrally Efficient Communications. *Information* **2023**, *14*, 323.
https://doi.org/10.3390/info14060323

**AMA Style**

Zacko Gbadoubissa JE, Abba Ari AA, Radoi E, Gueroui AM.
M-Ary Direct Modulation Chirp Spread Spectrum for Spectrally Efficient Communications. *Information*. 2023; 14(6):323.
https://doi.org/10.3390/info14060323

**Chicago/Turabian Style**

Zacko Gbadoubissa, Jocelyn Edinio, Ado Adamou Abba Ari, Emanuel Radoi, and Abdelhak Mourad Gueroui.
2023. "M-Ary Direct Modulation Chirp Spread Spectrum for Spectrally Efficient Communications" *Information* 14, no. 6: 323.
https://doi.org/10.3390/info14060323