# Research on an Optimization Method for a Partially Responsive Continuous Phase Modulated (CPM) Signal Based on an Optimal Generic Function

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## Abstract

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## 1. Introduction

## 2. Establishment of an Optimal Generic Function Model

## 3. Optimization and Solution of Partial Response CPM Symbol Signal

#### 3.1. Optimization of CPM Signals of Different Lengths

#### 3.2. The Effect of the N-Value of the Optimal Generic Function Model on the CPM Function

#### 3.3. Transient and Steady-State Modulation Method for Six-Way Parallel Transmission

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Comparison of normalized energy spectral densities for different length continuous phase modulated (CPM) functions with n = 2 and m = 4.

**Figure 2.**Comparison of normalized energy spectral densities for different length CPM functions with n = 2 and m = 4.

**Figure 3.**Comparison of normalized energy spectral densities for different length CPM functions with n = 6 and m = 4.

**Figure 4.**Minimum shift keying (MSK) energy spectra with CPM function lengths of 2T and 4T and, n = 2, 4 and 6 respectively, and the normalized energy spectral density of the corresponding CPM function.

**Figure 5.**The CPM function trajectory with a CPM function length of 6T and n values of 2, 4, and 6 respectively.

**Figure 6.**MSK energy spectrum with CPM function length of 6T and n = 2, 4 and 6 respectively, and the normalized energy spectral density of the corresponding CPM function.

**Figure 7.**Schematic diagram of a transient steady-state modulation method for six-way parallel transmission of CPM signals with inter-symbol interference.

**Figure 8.**Six-way parallel CPM transmission signals’ phase trajectory and superimposed phase trajectory.

**Figure 9.**The phase modulated signal resulting from the superposition of six parallel CPM transmission signals.

**Table 1.**Fourier series $m=4$, when the symbol length is 2T, 4T and 6T, the coefficient values of the Fourier series obtained by taking 2, 4 and 6 respectively, and the corresponding K values.

Length of Symbol | n | ${\mathit{a}}_{0}$ | ${\mathit{a}}_{1}$ | ${\mathit{a}}_{2}$ | ${\mathit{a}}_{3}$ | ${\mathit{a}}_{4}$ | K |
---|---|---|---|---|---|---|---|

2 | 1.1743 | 0.5569 | −0.0241 | −0.0241 | −0.0015 | 1.3377 | |

2T | 4 | 1.0388 | 0.6630 | 0.1291 | −0.0122 | 0.0023 | 1.5121 |

6 | 0.9479 | 0.6961 | 0.2548 | 0.0284 | −0.00428 | 1.6571 | |

2 | 0.8302 | 0.3937 | −0.0170 | 0.0033 | −0.0010 | 0.9460 | |

4T | 4 | 0.7361 | 0.4698 | 0.0915 | −0.0087 | 0.0016 | 1.0670 |

6 | 0.6590 | 0.4933 | 0.1958 | 0.0317 | −0.0003 | 1.1917 | |

2 | 0.6778 | 0.3215 | −0.0139 | 0.0027 | −0.0009 | 0.7724 | |

6T | 4 | 0.6007 | 0.3834 | 0.0747 | −0.0071 | 0.0013 | 0.8745 |

6 | 0.5481 | 0.4025 | 0.1473 | 0.0164 | −0.0024 | 0.9552 |

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**MDPI and ACS Style**

Qi, J.; Makarov, S.B.; Liu, M.; Li, B.; Xue, W.
Research on an Optimization Method for a Partially Responsive Continuous Phase Modulated (CPM) Signal Based on an Optimal Generic Function. *Symmetry* **2019**, *11*, 1114.
https://doi.org/10.3390/sym11091114

**AMA Style**

Qi J, Makarov SB, Liu M, Li B, Xue W.
Research on an Optimization Method for a Partially Responsive Continuous Phase Modulated (CPM) Signal Based on an Optimal Generic Function. *Symmetry*. 2019; 11(9):1114.
https://doi.org/10.3390/sym11091114

**Chicago/Turabian Style**

Qi, Junwei, Sergey B. Makarov, Mingxin Liu, Beiming Li, and Wei Xue.
2019. "Research on an Optimization Method for a Partially Responsive Continuous Phase Modulated (CPM) Signal Based on an Optimal Generic Function" *Symmetry* 11, no. 9: 1114.
https://doi.org/10.3390/sym11091114