# Precision Joint RF Measurement of Inter-Satellite Range and Time Difference and Scalable Clock Synchronization for Multi-Microsatellite Formations

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. System Model

#### 2.1. Error Caused by Frequency Deviation

#### 2.2. Motion Error

#### 2.3. Phase Tracking Error

_{0}is the carrier–noise ratio (CNR, Carrier-to-Noise Ratio, typically in units of dBHz); and $D$ is the early-to-late correlation spacing (in chips).

_{1}, t

_{3}, and t

_{5}are denoted as ${\sigma}_{\mathrm{DLL}1}$, ${\sigma}_{\mathrm{DLL}3}$, and ${\sigma}_{\mathrm{DLL}5}$, respectively. The transmitter code-phase jitter noise affects ${T}_{\mathrm{tof}2}$ with the effect denoted as ${\sigma}_{\mathsf{\theta}\mathrm{tof}2}$. ${n}_{\Delta T}$ is the sum of the phase tracking noise errors at times ${T}_{\mathrm{A}}{(t}_{4})$ and ${T}_{\mathrm{B}}{(t}_{3})$, and it can be expressed as:

#### 2.4. Summary of Error Modelling

## 3. Design of Time Difference Reference

## 4. Clock Synchronization Scheme

#### 4.1. Multi-Satellite Clock Synchronization

#### 4.2. Clock Synchronization Performance Analysis

#### 4.3. Simulation of Multi-Satellite Clock Synchronization Scheme

_{13}), Satellites 2 and 3 (noted as CD

_{23}), and Satellites 1 and 2 (noted as CD

_{12}) without phase correction and frequency correction for each satellite. Figure 11b shows the clock deviation with phase correction only. It can be seen that there is still a hundred-ns level gap among the satellite clocks due to the absence of frequency error correction. Figure 11c shows the clock deviation with both phase correction and frequency correction, which remains at only an ns level, demonstrating the best synchronization performance.

## 5. Experimental Verification

#### 5.1. Experimental Platform

#### 5.2. Two-Node Experiment

#### 5.2.1. Joint Measurement Experimental Verification

#### 5.2.2. Clock Synchronization Experimental Verification

#### 5.3. Multi-Node Experiment

#### 5.3.1. Joint Measurement Experimental Verification

#### 5.3.2. Clock Synchronization Experimental Verification

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## Appendix B

## Appendix C

## References

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**Figure 2.**Numerical results of the error caused by frequency deviation: (

**a**) time difference measurement error; (

**b**) range measurement error (normalized to time).

**Figure 11.**Simulation results of inter-satellite clock synchronization: (

**a**) satellite clock deviation without synchronization; (

**b**) satellite clock deviation with only phase correction; and (

**c**) satellite clock deviation with both phase and frequency correction.

**Figure 16.**Experimental clock synchronization results: (

**a**) synchronization error with frequency correction vs. without; (

**b**) detailed view of the synchronization error with frequency correction (standard deviation = 431.60 ps).

**Figure 18.**Multi-node joint measurement experimental results: (

**a**) measurements between node A and B; (

**b**) measurements between node A and C; and (

**c**) measurements between node B and C.

**Figure 19.**Multi-node clock synchronization experimental results: (

**a**) results of node B and C when node A is the master node; (

**b**) results of node A and C when node B is the master node; and (

**c**) results of node A and B when node C is the master node.

Parameters | Value |
---|---|

Double-side front-end bandwidth/MHz | 20 |

Single-side equivalent bandwidth/Hz | 35 |

Carrier-to-noise ratio/dBHz | 70 |

Loop order | 2 |

PN code rate/MHz | 5.115 |

Early-to-late correlation spacing/chip | 1 |

Coherent integration time/$\mathsf{\mu}\mathrm{s}$ | 50 |

Time slot/s | 5 |

Orbital Elements | S0 | S1 | S2 | S3 |
---|---|---|---|---|

$a$/km | 6878.14 | 6878.14 | 6878.14 | 6878.14 |

e | 0 | 0.0087 | 0.0087 | 0.0087 |

$i$/° | 97 | 96.9992 | 97.7495 | 96.2501 |

$\mathsf{\Omega}$/° | 0 | 0.8722 | 359.5632 | 359.5646 |

$w$/° | 0 | 180.1063 | 59.9439 | 299.9498 |

${M}_{0}$/° | 0 | 0.0087 | 0.0087 | 0.0087 |

Simulation Parameters | Value | Simulation Parameters | Value |
---|---|---|---|

${K}_{1}$ | $1+2\times {10}^{-8}$ | ${\mathit{eTD}}_{12}/\mathrm{s}$ | $5\times {10}^{-10}$ |

${K}_{2}$ | $1+5\times {10}^{-8}$ | ${\mathit{eTD}}_{13}/\mathrm{s}$ | $1\times {10}^{-9}$ |

${K}_{3}$ | $1-1\times {10}^{-8}$ | ${\mathit{eTD}}_{21}/\mathrm{s}$ | $-5\times {10}^{-10}$ |

${\mathit{eNCO}}_{1}/\mathrm{Hz}$ | $4.65\times {10}^{-3}$ | ${\mathit{eTD}}_{23}/\mathrm{s}$ | $5\times {10}^{-10}$ |

${\mathit{eNCO}}_{2}/\mathrm{Hz}$ | $-4.65\times {10}^{-3}$ | ${\mathit{eTD}}_{31}/\mathrm{s}$ | $-1\times {10}^{-9}$ |

${\mathit{eNCO}}_{3}/\mathrm{Hz}$ | 0 | ${\mathit{eTD}}_{32}/\mathrm{s}$ | $-5\times {10}^{-10}$ |

Parameters | Value |
---|---|

Nominal frequency of the frequency source/MHz | 40 |

Stability of the frequency source | 1 × 10^{−11} |

Accuracy of the frequency source | 1 × 10^{−8} |

RF frequency/MHz | 2352 |

Transmit power/mW | 400 |

Receiver acquisition sensitivity/dBm | −112 |

Receiver dynamic range/dB | $\ge 60$ |

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## Share and Cite

**MDPI and ACS Style**

Hou, C.; Jin, X.; Zhou, L.; Wang, H.; Yang, X.; Xu, Z.; Jin, Z.
Precision Joint RF Measurement of Inter-Satellite Range and Time Difference and Scalable Clock Synchronization for Multi-Microsatellite Formations. *Sensors* **2023**, *23*, 4109.
https://doi.org/10.3390/s23084109

**AMA Style**

Hou C, Jin X, Zhou L, Wang H, Yang X, Xu Z, Jin Z.
Precision Joint RF Measurement of Inter-Satellite Range and Time Difference and Scalable Clock Synchronization for Multi-Microsatellite Formations. *Sensors*. 2023; 23(8):4109.
https://doi.org/10.3390/s23084109

**Chicago/Turabian Style**

Hou, Cong, Xiaojun Jin, Lishan Zhou, Haoze Wang, Xiaopeng Yang, Zhaobin Xu, and Zhonghe Jin.
2023. "Precision Joint RF Measurement of Inter-Satellite Range and Time Difference and Scalable Clock Synchronization for Multi-Microsatellite Formations" *Sensors* 23, no. 8: 4109.
https://doi.org/10.3390/s23084109