# Research on Spectrum Optimization Technology for a Wireless Communication System

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## Abstract

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## 1. Introduction

## 2. General Function Model with the Minimal Spectral Energy Leakage

## 3. Limitations of the General Function Model

## 4. Solution and Analysis of Signal $\mathit{a}(\mathit{t})$ with High-Quality Spectrum Characteristics

#### 4.1. Numerical Solution of Signal $a(t)$ with High-Quality Spectral Characteristics

#### 4.2. Comparative Analysis of Energy Spectrum under Different Constraints

#### 4.3. Spectral Analysis of Signals with Additional Frequency-Domain Restrictions

#### 4.4. Comparative Analysis of Spectral Characteristics under Different Constraints at n = 2

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Location of frequency-domain constraints selected from normalized energy spectra of different n values.

**Figure 2.**Time-domain plots and normalized energy density plots for n = 2, m = 6 under different constraints.

**Figure 3.**Time-domain plots and normalized energy density plots for n = 4, m = 6 under different constraints.

**Figure 4.**Time-domain plots and normalized energy density plots for n = 6, m = 6 under different constraints.

**Figure 5.**The values of n are 4 and 6. When m is 6, the baseband signal and the empirical signal cosine signal are obtained to make the following normalized energy spectral density diagram.

**Table 1.**The roll-off index n = 2, the coefficients of the Fourier series of the corresponding baseband signal $a(t)$ at different values of the Fourier series m.

m | ${\mathit{a}}_{0}$ | ${\mathit{a}}_{1}$ | ${\mathit{a}}_{2}$ | ${\mathit{a}}_{3}$ | ${\mathit{a}}_{4}$ | ${\mathit{a}}_{5}$ | ${\mathit{a}}_{6}$ | ${\mathit{a}}_{7}$ |
---|---|---|---|---|---|---|---|---|

3 | 1.5478 | 0.8877 | 0.1163 | 0.0025 | - | - | - | - |

4 | 1.5280 | 0.9011 | 0.1438 | 0.0052 | −0.0015 | - | - | - |

5 | 1.5252 | 0.9038 | 0.1413 | −0.0004 | −0.0003 | 0.0002 | - | - |

6 | 1.5077 | 0.9155 | 0.1585 | −0.0029 | 0.0003 | −0.00002 | −0.00001 | - |

7 | 1.4769 | 0.9339 | 0.1931 | −0.0022 | 0.0002 | 0.00001 | 0.00002 | 0.00002 |

**Table 2.**The roll-off index n = 4, the coefficients of the Fourier series of the corresponding baseband signal $a(t)$ at different numbers of terms of the Fourier series m.

m | ${\mathit{a}}_{0}$ | ${\mathit{a}}_{1}$ | ${\mathit{a}}_{2}$ | ${\mathit{a}}_{3}$ | ${\mathit{a}}_{4}$ | ${\mathit{a}}_{5}$ | ${\mathit{a}}_{6}$ | ${\mathit{a}}_{7}$ |
---|---|---|---|---|---|---|---|---|

4 | 1.4479 | 0.9540 | 0.2256 | −0.0027 | 0.0017 | - | - | - |

5 | 1.3873 | 0.9687 | 0.3059 | 0.0319 | −0.0008 | −0.0018 | - | - |

6 | 1.3812 | 0.9727 | 0.3127 | 0.0301 | −0.0006 | −0.00004 | 0.00004 | - |

7 | 1.3597 | 0.9782 | 0.3391 | 0.0393 | −0.0013 | 0.0001 | −0.00001 | −0.000002 |

**Table 3.**The roll-off index n = 6, the coefficients of the Fourier series of the corresponding baseband signal $a(t)$ at different numbers of terms of the Fourier series m.

m | ${\mathit{a}}_{0}$ | ${\mathit{a}}_{1}$ | ${\mathit{a}}_{2}$ | ${\mathit{a}}_{3}$ | ${\mathit{a}}_{4}$ | ${\mathit{a}}_{5}$ | ${\mathit{a}}_{6}$ | ${\mathit{a}}_{7}$ |
---|---|---|---|---|---|---|---|---|

5 | 1.331 | 0.9857 | 0.3844 | 0.0592 | −0.0007 | 0.0003 | - | - |

6 | 1.3279 | 0.9843 | 0.3841 | 0.0598 | −0.0004 | 0.0002 | −0.00007 | - |

7 | 1.3269 | 0.9875 | 0.3842 | 0.0596 | −0.0003 | 0.0002 | −0.00006 | 0.00002 |

n | the Number of Lines | m | ${\mathit{\xi}}_{\mathit{n},\mathit{m}}({\mathit{f}}_{1})$ |
---|---|---|---|

2 | Figure 2, line 1 | 6 | 1.470202 |

Figure 2, line 2 | 1.780325 | ||

4 | Figure 3, line 1 | 1.855372 | |

Figure 3, line 2 | 2.242426 | ||

6 | Figure 4, line 1 | 2.197020 | |

Figure 4, line 2 | 2.309111 |

**Table 5.**When n = 2 and m = 5, 6 and, 7, the frequency-domain principal lobe energy ratio of the code signal in the absence of frequency-domain restriction is compared and analyzed with the spectrum bandwidth calculated by −40 dB and −60 dB.

m | ${\mathit{\xi}}_{2,\mathit{m}}({\mathit{f}}_{1})$ | −40 dB Spectrum Bandwidth | −60 dB Spectrum Bandwidth |
---|---|---|---|

5 | 1.466991 | 3.533 fT | 7.614 fT |

6 | 1.470202 | ||

7 | 1.466096 |

**Table 6.**When n = 2 and m = 5, 6, and 7, the frequency-domain principal lobe energy ratio of the code signal in the frequency-domain is added, and the spectrum bandwidth is calculated with −40 dB and −60 dB.

m | ${\mathit{\xi}}_{2,\mathit{m}}({\mathit{f}}_{1})$ | −40 dB Spectrum bandwidth | −60 dB Spectrum bandwidth |
---|---|---|---|

5 | 1.744317 | 2.57 fT | 4.647 fT |

6 | 1.780325 | 2.61 fT | 3.39 fT |

7 | 1.860039 | 2.722 fT | 3.716 fT |

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## Share and Cite

**MDPI and ACS Style**

Liu, M.; Xue, W.; Jia, P.; Makarov, S.B.; Li, B.
Research on Spectrum Optimization Technology for a Wireless Communication System. *Symmetry* **2020**, *12*, 34.
https://doi.org/10.3390/sym12010034

**AMA Style**

Liu M, Xue W, Jia P, Makarov SB, Li B.
Research on Spectrum Optimization Technology for a Wireless Communication System. *Symmetry*. 2020; 12(1):34.
https://doi.org/10.3390/sym12010034

**Chicago/Turabian Style**

Liu, Mingxin, Wei Xue, Peisong Jia, Sergey B. Makarov, and Beiming Li.
2020. "Research on Spectrum Optimization Technology for a Wireless Communication System" *Symmetry* 12, no. 1: 34.
https://doi.org/10.3390/sym12010034