# Characterizing Periodic Variations of Atomic Frequency Standards via Their Frequency Stability Estimates

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## Abstract

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## 1. Introduction

- Satellite clock bias includes both deterministic and random signals: while the spectra of deterministic signals are estimated from its direct Fourier transform, the spectra of stochastic processes are calculated from the Karhunen–Loeve transform of its covariance [25]. On the other hand, Dong et al. show that the Karhunen–Loeve transform of covariance can distort the spectral response of deterministic signals [26].
- Additionally, Zhou et al. reported that there is no uniform method based on least squares for detecting, fitting, and removing periodic variations in Beidou satellite system (BDS) inclined geosynchronous orbit (IGSO), geostationary earth orbit (GEO), and medium earth orbit (MEO) satellite clock periodic variations [27].

## 2. Methods

#### 2.1. Allan and Hadamard Variances of Periodic Variations

**Theorem**

**1.**

**Theorem**

**2.**

#### 2.2. Characterizing Periodic Variations Using Frequency Stability Estimates

## 3. Results

#### 3.1. Simulated Data

#### 3.2. GPS SVN63 Clock Data

#### 3.3. Other GPS Clock Data

- For more than half of the satellites, the sigma–tau plot of the AVAR computed from clock bias with periodic variations removed using Equation (9) is closer to the standard sigma–tau plot of AVARs than the least squares method.
- “Lumps” of AVARs estimated from PRN01, PRN02, PRN05, PRN07, PRN11, PRN12, PRN13, PRN14 (SVN41), PRN15, PRN16, PRN17, PRN18 (SVN54), PRN20, PRN21, PRN22, PRN23 (SVN76), PRN25, PRN28, PRN30, PRN31, PRN32 (SVN23), and PRN32 (SVN70) clock bias with periodic variations removed using Equation (9) at an averaging time around $4\times {10}^{4}$ s suggest underestimation of 24-hour periodic variations.
- “Lumps” of AVARs estimated from PRN01, PRN02, PRN03, PRN04 (SVN34), PRN04 (SVN74), PRN05, PRN06, PRN09, PRN12, PRN14 (SVN77), PRN18 (SVN75), PRN19, PRN23 (SVN60), PRN23 (SVN76), PRN24, PRN26, PRN27, PRN30, and PRN32 (SVN23) clock bias with periodic variations removed using Equation (9) at an averaging time around $2\times {10}^{4}$ s suggest underestimation of 12-hour periodic variations.
- “Lumps” of AVARs estimated from PRN04 (SVN34), PRN04 (SVN74), PRN05, PRN07, PRN10, PRN11, PRN12, PRN14 (SVN41), PRN17, PRN18 (SVN75), PRN20, PRN21, PRN22, PRN24, PRN25, PRN27, PRN29, and PRN32 (SVN70) clock bias with periodic variations removed using Equation (9) at an averaging time around ${10}^{4}$ s suggest underestimation of 6-hour periodic variations.
- “Lumps” of AVARs estimated from PRN02, PRN04 (SVN34), PRN05, PRN06, PRN07, PRN08, PRN09, PRN11, PRN13, PRN14 (SVN41), PRN15, PRN16, PRN18 (SVN75), PRN19, PRN22, PRN23 (SVN60), PRN25, PRN32 (SVN23), and PRN32 (SVN70) clock bias with periodic variations removed using Equation (9) at an averaging time around $5\times {10}^{3}$ s suggest underestimation of 4-hour periodic variations.
- “Lumps” of AVARs estimated from PRN09, PRN11, PRN13, PRN18 (SVN54), PRN18 (SVN75), PRN22, PRN24, PRN27, PRN29, PRN31, PRN32 (SVN23), and PRN32 (SVN70) clock bias with periodic variations removed using Equation (9) at an averaging time around $3\times {10}^{3}$ suggest underestimation of 3-hour periodic variations.

- The gap between the AVAR estimated from PRN01 clock bias with periodic variations fitted and removed using the least squares method and the AVAR computed from PRN01 clock bias enlarges with increasing averaging interval. Since the tail of AVAR estimated from PRN01 clock bias with periodic variations removed using the least squares method has a similar shape to the AVARs of 12-hour sinusoidal variations, and the AVAR computed from PRN01 clock bias with periodic variations fitted and removed using Equation (9) suggests strong frequency noise, the discrepancies between the AVAR estimated from PRN01 clock bias with periodic variations fitted and removed using the least squares method and the AVAR computed from PRN01 clock bias is caused by overfitting the periodic variations by taking a portion of frequency noises as 12-hour variation.
- The AVAR estimated from PRN22 clock bias with periodic variations fitted and removed using least squares method is greater than the AVAR computed from IGS final combined PRN22 clock bias around averaging time $2\times {10}^{4}$ s. Since AVARs computed from the three PRN22 clock biases increase with the averaging interval for $\tau \ge 2\times {10}^{4}$ s, PRN22 AFS is influenced by strong FM noise processes. It seems that the least squares method overfits the periodic variations of PRN22 by taking a portion of frequency noises as 12-hour variation.

- Overfitting of periodic variations can reduce the clock residuals caused by power–law noise processes. When periodic variations are removed, the interaction between random clock behaviors and periodic variations is suppressed, leading to an increase in clock residuals and prediction RMS.
- Only high-variability estimates (HVAR) are used in solving Equations (6) and (9), which may not capture all the periodic variations present in the data. The maximum averaging time of HVARs estimated from two-day GPS clock bias is $1.44\times {10}^{4}$ s, while the first local minimum of Equation (5) appears at averaging interval $\tau =2\times {10}^{4}$ s. This means that some periodic variations may not be captured by HVAR estimates and could contribute to an increase in the RMS prediction when removed.

## 4. Conclusions and Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

AFS | atomic frequency standards |

BDS | Beidou Satellite System |

BIPM | Bureau International des Poids et Mesures |

CCTF | Consultative Committee for Time and Frequency |

CGPM | General Conference on Weights and Measures |

CLS | Collecte Localisation Satellites |

cpd | cycles per day |

EAL | échelle atomique libre (or Free atomic time scale) |

GNSS | Global Navigation Satellite System |

GPS | US Global Positioning System |

HVAR | Hadamard variance |

IGS | International GNSS Service |

IGST | International GNSS Service Timescale |

JAXA | Japan Aerospace Exploration Agency |

MGEX | Multi-GNSS Experiment |

PRN | Pseudorandom Noise |

RAFS | Rubidium Atomic Frequency Standard |

RMS | root mean square |

PLN | power–law noise |

PSD | power spectral distribution |

SVN | Satellite Vehicle Number |

TAI | International Atomic Time |

TT | Terrestrial Time |

UTC | Coordinated Universal Time |

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**Figure 2.**AVARs of simulated clock data 12-, 6-, 4-, and 3-hour periodic variations fitted and removed by least squares and by Equation (9), respectively.

**Figure 3.**AVARs of GPS SVN63 onboard rubidium clock bias from MJD 56,739.0 to MJD 56,745.9965 12-, 6-, 4-, and 3-hour sinusoidal signals. AVARs of 12-, 6-, 4-, and 3-hour sinusoidal signals are magnified to be tangent to AVARs estimated from SVN63 clock bias at their maxima.

**Figure 5.**AVARs of GPS PRN02∼17 onboard clock bias 12-, 6-, 4-, and 3-hour sinusoidals fitted and removed using least squares method and Equation (9), respectively.

**Figure 6.**AVARs of GPS PRN18∼32 onboard clock bias 12-, 6-, 4-, and 3-hour sinusoidals fitted and removed using least squares method and Equation (9), respectively.

**Table 1.**Amplitudes of simulated periodic variations estimated by least squares method and solving Equation (6).

Frequency (cpa) | Input | Equations (6) and (9) | Least Squares |
---|---|---|---|

$2.0029$ | $9.00\times {10}^{-10}$ | $8.92\times {10}^{-10}$ | $1.62\times {10}^{-9}$ |

$2\times 2.0029$ | $3.30\times {10}^{-10}$ | $3.40\times {10}^{-10}$ | $3.83\times {10}^{-10}$ |

$3\times 2.0029$ | $6.00\times {10}^{-11}$ | $1.35\times {10}^{-10}$ | $3.09\times {10}^{-10}$ |

$4\times 2.0029$ | 0 | 0 | $3.53\times {10}^{-10}$ |

PRN | With Periodics | Equations (6) and (9) | Least Square | Time-Span |
---|---|---|---|---|

G01 | 0.52 | 0.65 | 0.42 | 03-23-14∼12-27-20 |

G02 | 0.72 | 1.64 | 0.72 | 03-23-14∼12-27-20 |

G03 | 0.82 | 0.94 | 0.78 | 03-23-14∼12-27-20 |

G04 | 1.61 | 1.61 | 1.58 | 03-23-14∼12-27-20 |

G05 | 0.71 | 0.80 | 0.62 | 03-23-14∼12-27-20 |

G06 | 0.53 | 0.64 | 0.50 | 03-23-14∼12-27-20 |

G07 | 1.13 | 1.93 | 1.11 | 03-23-14∼12-23-20 |

G08 | 3.39 | 3.61 | 3.36 | 03-23-14∼12-27-20 |

G09 | 0.66 | 0.77 | 0.61 | 03-23-14∼12-27-20 |

G10 | 1.39 | 1.48 | 1.38 | 03-23-14∼12-27-20 |

G11 | 1.34 | 1.85 | 1.31 | 03-23-14∼12-27-20 |

G12 | 0.62 | 5.73 | 0.52 | 03-23-14∼12-27-20 |

G13 | 1.07 | 1.24 | 1.05 | 03-23-14∼12-27-20 |

G14 | 0.73 | 1.00 | 0.73 | 03-23-14∼12-25-20 |

G15 | 0.47 | 0.52 | 0.42 | 03-23-14∼12-27-20 |

G16 | 0.62 | 0.62 | 0.49 | 03-23-14∼12-27-20 |

G17 | 1.56 | 1.70 | 1.53 | 03-23-14∼12-27-20 |

G18 | 0.92 | 1.86 | 0.88 | 03-23-14∼12-27-20 |

G19 | 0.58 | 0.65 | 0.57 | 03-23-14∼12-27-20 |

G20 | 0.62 | 0.95 | 0.61 | 03-23-14∼12-27-20 |

G21 | 0.76 | 1.42 | 0.72 | 03-23-14∼12-26-20 |

G22 | 1.09 | 1.64 | 1.06 | 03-23-14∼12-27-20 |

G23 | 0.53 | 1.04 | 0.53 | 03-23-14∼12-27-20 |

G24 | 4.18 | 4.57 | 4.24 | 03-23-14∼12-27-20 |

G25 | 0.42 | 0.42 | 0.36 | 03-23-14∼12-26-20 |

G26 | 0.63 | 0.76 | 0.59 | 03-23-14∼12-27-20 |

G27 | 0.48 | 0.59 | 0.42 | 03-23-14∼12-27-20 |

G28 | 3.71 | 3.73 | 3.71 | 03-23-14∼12-27-20 |

G29 | 1.26 | 1.47 | 1.22 | 03-23-14∼12-27-20 |

G30 | 0.56 | 0.67 | 0.51 | 03-23-14∼12-27-20 |

G31 | 1.15 | 1.81 | 1.14 | 03-23-14∼12-27-20 |

G32 | 0.87 | 0.94 | 0.77 | 03-23-14∼12-27-20 |

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**MDPI and ACS Style**

Cheng, W.; Nie, G.; Zhu, J.
Characterizing Periodic Variations of Atomic Frequency Standards via Their Frequency Stability Estimates. *Sensors* **2023**, *23*, 5356.
https://doi.org/10.3390/s23115356

**AMA Style**

Cheng W, Nie G, Zhu J.
Characterizing Periodic Variations of Atomic Frequency Standards via Their Frequency Stability Estimates. *Sensors*. 2023; 23(11):5356.
https://doi.org/10.3390/s23115356

**Chicago/Turabian Style**

Cheng, Weiwei, Guigen Nie, and Jian Zhu.
2023. "Characterizing Periodic Variations of Atomic Frequency Standards via Their Frequency Stability Estimates" *Sensors* 23, no. 11: 5356.
https://doi.org/10.3390/s23115356