Algorithm for Taming Rubidium Atomic Clocks Based on Longwave (Loran-C) Timing Signals

Abstract

This paper explores effective methods for taming rubidium atomic clocks with longwave timing signals. In an in-depth analysis of the time-difference data between the 1PPS timing signal output from the ground-wave signal received by a long-wave receiver and the 1PPS signal from UTC, we observe that the time-difference data has significant short-term jitter and long-term periodicity effects. To meet this challenge, we adopt several innovative strategies. First, we use the Fourier transform algorithm to analyse the time-frequency characteristics of the time-difference data in detail and accordingly propose a de-jittering correction algorithm for the long-wave timing data, which is aimed at improving the stability of the long-wave timing signals. Secondly, the time difference model of the rubidium clock is innovatively modified, and a quadratic polynomial superimposed with a periodic fluctuation term is constructed, which can accurately solve and eliminate the periodic components and obtain smoother time difference data. Finally, the parameters of the rubidium clock are accurately estimated by the least-squares method using the corrected smoother time difference data, and the output frequency of the rubidium clock is adjusted accordingly so that the rubidium clock is tamed effectively by the long-wave timing signal successfully. The experimental results show that the long-term stability of the tamed rubidium clock is significantly improved to 3.52 × 10⁻¹³/100,000 s; meanwhile, the phase deviation of the output 1PPS from the UTC of the tamed rubidium clock after entering the stabilisation period is kept within 25 ns.

1. Introduction

The importance of the time-frequency system as the country’s key infrastructure is self-evident [1,2]. In this system, atomic clocks play a pivotal role, which is the core equipment to maintain the time-frequency accuracy [3,4]. Currently, the time-frequency standard mainly relies on the generation of caesium and hydrogen clocks; however, the high cost, large size, and demanding operating environment of these two types of clocks limit their wide application in both military and civilian fields. In comparison, rubidium atomic clocks exhibit higher practical value in the civilian field due to their significant advantages of being able to achieve a smaller size and lower cost. The model of the rubidium clock selected in this article is PRS10, and this model has good adaptability in numerous civilian scenarios. However, whether it is a caesium atomic clock, a hydrogen atomic clock, or a rubidium atomic clock, as long as they are in a free state, they all face the problems of frequency shift and ageing. After running for a long time, atomic clocks will generate cumulative phase errors and frequency errors, making it difficult to meet the stringent requirements of high-precision time-frequency signals [5].

Correcting the rubidium clock output signal with the help of an external time source is necessary to effectively improve the accuracy and stability of the rubidium clock output signal. Among many external time sources, the timing signal output from the GNSS (Global Navigation Satellite System) receiver stands out. It not only possesses high accuracy and stability [6] but also has the excellent ability to work continuously around the clock and thus logically becomes the key reference source to tame the system. With the continuous progress and innovation of technology, the field of GNSS receivers has made remarkable developments. On the one hand, their cost has been gradually reduced, making them more affordable for a broader range of applications; on the other hand, their operation has become easier, lowering the threshold for use. These advantages have led to the fact that most taming systems nowadays adopt the 1PPS timing signal output from GNSS receivers to precisely tame the high-stability crystals or rubidium atomic clocks [7,8,9]. In this application process, the research on taming crystals by utilising the long-term stability of GNSS timing signals has received extensive attention and in-depth exploration [10]. According to the research status of frequency scale taming technology at home and abroad, the related techniques and algorithms have become increasingly mature after years of accumulation and development. The current mainstream methods include least squares [11,12], Kalman filtering [13,14], and PID control algorithms. Each of these methods has its characteristics, which provide diversified technical support for realising the high accuracy and stability of frequency marker taming.

Although GNSS has been widely used in many fields due to its high accuracy, there are obvious shortcomings in signal reception, which is susceptible to interference and has poor anti-jamming ability, and satellite rejection occurs occasionally [15,16,17,18,19]. To compensate for this shortcoming, the technology of taming rubidium atomic clocks based on longwave timing signals has received increasing attention. The long-wave timing system (LORAN-C regime: the long-range navigation system) has become an indispensable complement and alternative to satellite navigation systems due to its excellent anti-interference capability and stable signal performance [20,21,22,23,24]. Studying longwave taming rubidium atomic clock algorithms is especially critical in complex and changing application scenarios, especially when GNSS signals are subject to interference and become unstable or fail. This research focuses on maintaining time and frequency accuracy, which is not only to improve the accuracy and stability of time synchronisation to ensure the continuous and stable operation of various systems relying on precise time but also to expand the engineering application of land-based longwave timing systems and to promote its in-depth application and innovative development in more fields.

However, long-wave signals in the ground wave propagation process, their propagation delay will be affected by atmospheric conditions, propagation path changes, and many other complex factors, and then show very significant time-varying characteristics [25,26,27] (given the time-varying characteristics of long-wave signals propagating along the ground, this paper deals with long-wave receivers focusing on the reception of ground-wave signals). This time-varying characteristic undoubtedly poses a significant challenge to taming rubidium atomic clocks using longwave timing signals. Specifically, due to the instability of the time delay, the time difference between the 1PPS signal output from the longwave receiver and that from the rubidium clock not only fluctuates wildly but also often exhibits recognisable periodic characteristics. Therefore, although the long-wave timing system has a series of significant advantages, such as strong anti-jamming ability, signal stability, etc., it is almost difficult to find reports in the open literature, both in academia and industry, on the use of long-wave timing signals to tame the rubidium atomic clocks directly. Therefore, how to efficiently process the time difference data, which is interfered with by multiple complex factors, to realise the precise timing of rubidium clocks has become the core problem to be overcome in the long-wave taming of rubidium frequency markers. The solution to this problem is vital to promote the practical application of longwave timing technology and to enhance the precision and stability of the time and frequency systems.

In view of this, this paper will deeply investigate the characteristics of the 1PPS timing signal output from a longwave receiver and is dedicated to finding a more accurate correction algorithm for the longwave timing signal. On the basis of successfully obtaining the correction algorithm, this paper will further analyse the time-frequency characteristics of the time difference data between the longwave 1PPS signal and the rubidium 1PPS signal in depth and detail. Through rigorous and innovative methods, we strive to transform the original time-difference data with strong fluctuations and significant jitter into a smooth and reliable time-difference data sequence. After completing the above key steps, this paper will precisely estimate the key parameters of the rubidium atomic clock with the help of the classical least squares algorithm. In this way, the high-precision taming of rubidium clocks is finally realised, and the accuracy and stability of rubidium clock output signals are significantly improved, providing solid technical support and guarantee for the research and application in related fields.

2. Long-Wave Timing Signal Characterization

To deeply analyse the characteristics of the 1PPS (1 Pulse Per Second) signal output from long-wave receivers, the group has built a set of long-wave ground-wave propagation time delay measurement systems at the National Time Service Centre (NTSC). The system receives the long-wave time signal from the Pucheng transmitter in Shaanxi Province. It measures the time difference with the high-precision standard 1PPS time signal output from the NTSC to accurately determine the characteristics of the 1PPS signal output from the long-wave timing receivers, including, but not limited to, the signal’s accuracy, stability, and possible fluctuation patterns. It should be particularly noted that the 1PPS output of China’s Coordinated Universal Time UTC (NTSC) maintained by the National Time Service Centre of the Chinese Academy of Sciences demonstrates extremely high stability: the stability within 5 days can reach 8.3 × 10⁻¹⁶, and the stability within 30 days can reach 4.7 × 10⁻¹⁶ Based on this remarkable stability when measuring the time difference data in this paper, the 1PPS output of UTC (NTSC) is selected as the sole reference source.

In addition, to provide a comprehensive and effective comparison and evaluation basis for the subsequent experiments, the group has also set up two additional signal measurements. One channel focuses on the 1PPS signal output from the GNSS receiver to provide a reference standard for the analysis of the long-wave signal characteristics by utilising the high-precision characteristics of the GNSS signal; the other channel focuses on the 1PPS signal output from the rubidium clock in the free state to understand the signal characteristics of the rubidium clock and provide essential data for the study of the taming effect of the long-wave signal on the rubidium clock. The basic structure of the experimental measurement system is shown in Figure 1, which provides an intuitive and essential reference for the subsequent research work.

2.1. Long-Wave Propagation Delay Characteristics

Figure 2 clearly and intuitively shows a comparison of the data collected by the measurement system over 10 consecutive days. After an in-depth statistical analysis of the data, we obtained the following key information:

• From the perspective of data quantisation, the standard deviation of the longwave timing data is 64.3812 ns, while the standard deviation of the GNSS timing data is only 11.3402 ns. This contrasting data strongly indicates that compared with the high-precision and long-term stable timing signals output by GNSS, the long-wave timing results are a little less accurate, and the fluctuation degree of the signals is also relatively more significant. In the process of long-wave signal propagation, it is very easy to be interfered with by many complicated factors, among which the frequent changes in atmospheric conditions are the most prominent. These factors may cause the signal delay to be unstable, which will lead to the fluctuation of long-wave timing signals.
• Long-wave timing signals show an extremely significant feature—the apparent phenomenon of sky cycles. The generation of this periodicity is most likely closely related to various geophysical factors. The rotation of the Earth makes the relative position of long-wave signals change in the propagation process; the temperature difference between day and night leads to periodic changes in the physical properties of the atmospheric medium. The combined effect of these geophysical phenomena exerts a periodic influence on the propagation path and speed of the long-wave ground waves, resulting in regular fluctuations in the time delay of the received long-wave signals.

Figure 3 and

Table 1. Stability comparison of longwave raw, GNSS, and rubidium clock-free states.

Data Source	Allan’s Var—Time Interval
	1 s	10 s	100 s	1000 s	10,000 s	100,000 s
GNSS	9.17 × 10−9	1.04 × 10−9	1.12 × 10−10	1.33 × 10−11	1.68 × 10−12	1.89 × 10−13
Long-wave Origin	1.19 × 10−8	1.47 × 10−9	1.50 × 10−10	1.79 × 10−11	3.89 × 10−12	6.08 × 10−13
Rubidium clock-free states	1.18 × 10−10	2.98 × 10−11	7.68 × 10−12	2.34 × 10−13	7.70 × 10−13	1.78 × 10−12

present a comprehensive stability analysis of the longwave, GNSS, and Rubidium clock timing data in the free state.

The results are shown in Figure 3 and Table 1. It can be clearly found from the above-detailed analysis that GNSS has better performance in terms of long-term stability compared with the rubidium clock in the free state. Therefore, GNSS signals are often used to discipline atomic clocks. However, the long-wave time service system is limited by the characteristics of signal propagation, and there is a certain gap in long-term stability compared with GNSS. Therefore, it is difficult to use it directly to discipline the rubidium clock, which points out the direction for the subsequent optimisation and improvement of relevant technologies.

2.2. Frequency Domain Characterisation of Longwave Time-Difference Data

Based on the obvious day-periodic phenomenon in the time domain of the longwave time-difference data in the previous section, we select one day of the original longwave time-difference data to analyse the cyclic characteristics of the longwave data more deeply. To carry out the study, we convert the time-domain data to the frequency domain with the help of the Fourier Transform (FT) [28,29].

The Fourier Transform will help us to reveal the hidden periodicity or explore other potential periodic disturbances in the long wave raw time difference data [30]. The conversion formula for the discrete Fourier transform is described below:

$$\mathrm{F}\left(\mathsf{\omega }\right)={\sum }_{\mathrm{n}=-\mathrm{\infty }}^{\mathrm{\infty }}\mathrm{f}\left(\mathrm{n}\right){\mathrm{e}}^{-\mathrm{i}\mathsf{\omega }\mathrm{n}}$$

(1)

The results of the FT analysis of the longwave raw time-difference data are shown in Figure 4. Taking point A in the figure as an example, the frequency $f=1.15741\times {10}^{-5} \mathrm{H}\mathrm{z}$, corresponding to the period $T=1/f/3600\approx 24$ h. If the vertical coordinate amplitude of 10 is used as the threshold, the period terms are as follows: 24 h, 12 h, and 8 h corresponding to points A, B, and C in the figure, respectively. It can be seen that the most obvious is the 24-h day period, which corresponds to point A in the figure, and also conforms to the “day period” phenomenon of long wave propagation delay.

2.3. Long-Wave Time-Difference Data Correction Algorithm

Based on the above-detailed analysis of the characteristics of longwave timing signals, we have come to an important conclusion: longwave timing signals have an extremely pronounced periodic fluctuation component. Ideally, that is, if there is no external interference and no accumulation of errors, the delay measured by a longwave receiver at a fixed location should be maintained at a constant value.

Therefore, we are able to construct a mathematical model to describe the periodic fluctuations of longwave timing signals. The core objective of this mathematical model is to accurately capture and quantify the periodic fluctuations in longwave timing signals, thus laying a solid foundation for more in-depth analysis and efficient processing of longwave timing signals:

$${x1}_{Loran}\left(t\right)=C+{\sum }_{i=1}^n{k}_i\mathit{sin}\left({\omega }_{i}t+{\phi }_i\right)+{\epsilon }_L\left(t\right)$$

(2)

Among them,

• where ${x1}_{Loran}\left(t\right)$ represents the longwave timing result at time t.
• $C$ is the constant of timing in ideal cases.
• where $k_i$ is the amplitude of the $i$-th periodic fluctuation component.
• where ${\omega }_i$ is the frequency of the $i$-th periodic fluctuation component, which is typically related to periods of natural phenomena such as the Earth’s rotation and atmospheric changes.
• where ${\phi }_i$ is the phase of the $i$-th periodic fluctuation component, which represents the starting moment or phase shift of the fluctuation.
• ${\epsilon }_L\left(t\right)$ is a random error term used to represent random fluctuations or noise that cannot be fully captured by the periodic fluctuation model.

Equation (2) can be expressed in matrix form as follows:

$$X=AZ$$

(3)

In this context, $X={\left[\begin{array}{ccccc}{x}_{L1}& x_{L2}& x_{L3}& ...& x_{Lm}\end{array}\right]}^T$ represents the measured values, specifically the longwave time difference dataset ${x1}_{Loran}$, and $m$ denotes the number of measured values. The longwave time difference data of m points is: ${\displaystyle {\sum }_{t=1}^{m}x{1}_{Loran}\left(t\right)}={\displaystyle {\sum }_{t=1}^m\left[C+{\displaystyle {\sum }_{i=1}^n{K}_i\sin \left({\omega }_{i}t+{\phi }_i\right)}\right]}$. Therefore,

$$\begin{array}{l}\left[\begin{array}{c}{x}_{L1}\\ x_{L2}\\ ⋮\\ x_{Lm}\end{array}\right]=\left[\begin{array}{c}C+{\displaystyle {\sum }_{i=1}^n{k}_i\sin \left({\omega }_i{t}_1+{\phi }_i\right)}\\ C+{\displaystyle {\sum }_{i=1}^n{k}_i\sin \left({\omega }_i{t}_2+{\phi }_i\right)}\\ ⋮\\ C+{\displaystyle {\sum }_{i=1}^n{k}_i\sin \left({\omega }_i{t}_m+{\phi }_i\right)}\end{array}\right]=\left[\begin{array}{c}C+{\displaystyle {\sum }_{i=1}^n{k}_i\left[\sin \left({\omega }_i{t}_1\right)\cos \left({\phi }_i\right)+\cos \left({\omega }_i{t}_1\right)\sin \left({\phi }_i\right)\right]}\\ C+{\displaystyle {\sum }_{i=1}^n{k}_i\left[\sin \left({\omega }_i{t}_2\right)\cos \left({\phi }_i\right)+\cos \left({\omega }_i{t}_2\right)\sin \left({\phi }_i\right)\right]}\\ ⋮\\ C+{\displaystyle {\sum }_{i=1}^n{k}_i\left[\sin \left({\omega }_i{t}_m\right)\cos \left({\phi }_i\right)+\cos \left({\omega }_i{t}_m\right)\sin \left({\phi }_i\right)\right]}\end{array}\right]\\ =\left[\begin{array}{c}C+k_1\left[\sin \left({\omega }_1{t}_1\right)\cos \left({\phi }_1\right)+\cos \left({\omega }_1{t}_1\right)\sin \left({\phi }_1\right)\right]+k_2\left[\sin \left({\omega }_2{t}_1\right)\cos \left({\phi }_2\right)+\cos \left({\omega }_2{t}_1\right)\sin \left({\phi }_2\right)\right]+\cdots +k_n\left[\sin \left({\omega }_n{t}_1\right)\cos \left({\phi }_n\right)+\cos \left({\omega }_n{t}_1\right)\sin \left({\phi }_n\right)\right]\\ C+k_1\left[\sin \left({\omega }_1{t}_2\right)\cos \left({\phi }_1\right)+\cos \left({\omega }_1{t}_2\right)\sin \left({\phi }_1\right)\right]+k_2\left[\sin \left({\omega }_2{t}_2\right)\cos \left({\phi }_2\right)+\cos \left({\omega }_2{t}_2\right)\sin \left({\phi }_2\right)\right]+\cdots +k_n\left[\sin \left({\omega }_n{t}_2\right)\cos \left({\phi }_n\right)+\cos \left({\omega }_n{t}_2\right)\sin \left({\phi }_n\right)\right]\\ ⋮\\ C+k_1\left[\sin \left({\omega }_1{t}_m\right)\cos \left({\phi }_1\right)+\cos \left({\omega }_1{t}_m\right)\sin \left({\phi }_1\right)\right]+k_2\left[\sin \left({\omega }_2{t}_m\right)\cos \left({\phi }_2\right)+\cos \left({\omega }_2{t}_m\right)\sin \left({\phi }_2\right)\right]+\cdots +k_n\left[\sin \left({\omega }_n{t}_m\right)\cos \left({\phi }_n\right)+\cos \left({\omega }_n{t}_m\right)\sin \left({\phi }_n\right)\right]\end{array}\right]\end{array}$$

In this context, $Z={\left[\begin{array}{cccccc}C& k_1\mathit{cos}\left({\phi }_1\right)& k_1\mathit{sin}\left({\phi }_1\right)& \cdots & k_n\mathit{cos}\left({\phi }_n\right)& k_n\mathit{sin}\left({\phi }_n\right)\end{array}\right]}^T$ represents the parameter to be estimated, where $C$ denotes the constant term, $k_n$ represents the coefficient of the $n$-th periodic term, and ${\phi }_n$ represents the phase of the $n$-th periodic term.

$$A=\left[\begin{array}{cccccccc}1& \mathit{sin}\left({\omega }_1{t}_1\right)& \mathit{cos}\left({\omega }_1{t}_1\right)& \mathit{sin}\left({\omega }_2{t}_1\right)& \mathit{cos}\left({\omega }_2{t}_1\right)& \cdots & \mathit{sin}\left({\omega }_n{t}_1\right)& \mathit{cos}\left({\omega }_n{t}_1\right)\\ 1& \mathit{sin}\left({\omega }_1{t}_2\right)& \mathit{cos}\left({\omega }_1{t}_2\right)& \mathit{sin}\left({\omega }_2{t}_2\right)& \mathit{cos}\left({\omega }_2{t}_2\right)& \cdots & \mathit{sin}\left({\omega }_n{t}_2\right)& \mathit{cos}\left({\omega }_n{t}_2\right)\\ 1& \mathit{sin}\left({\omega }_1{t}_3\right)& \mathit{cos}\left({\omega }_1{t}_3\right)& \mathit{sin}\left({\omega }_2{t}_3\right)& \mathit{cos}\left({\omega }_2{t}_3\right)& \cdots & \mathit{sin}\left({\omega }_n{t}_3\right)& \mathit{cos}\left({\omega }_n{t}_3\right)\\ 1& \mathit{sin}\left({\omega }_1{t}_4\right)& \mathit{cos}\left({\omega }_1{t}_4\right)& \mathit{sin}\left({\omega }_2{t}_4\right)& \mathit{cos}\left({\omega }_2{t}_4\right)& \cdots & \mathit{sin}\left({\omega }_n{t}_4\right)& \mathit{cos}\left({\omega }_n{t}_4\right)\\ 1& ⋮& ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 1& \mathit{sin}\left({\omega }_1{t}_m\right)& \mathit{cos}\left({\omega }_1{t}_m\right)& \mathit{sin}\left({\omega }_2{t}_m\right)& \mathit{cos}\left({\omega }_2{t}_m\right)& \cdots & \mathit{sin}\left({\omega }_n{t}_m\right)& \mathit{cos}\left({\omega }_n{t}_m\right)\end{array}\right]$$

represents the coefficient matrix, where ${\omega }_n$ denotes the frequency of the $n$-th periodic term. To estimate the parameter $Z$, we can transform the matrix Equation (3) into Equation (4).

$$Z={\left(A^{T}A\right)}^{-1}{A}^{T}X$$

(4)

Equation (4) can be solved via the least squares method.

On the basis of the establishment of the longwave timing mathematical model, we focus only on the periodic term, denoted as $\sum _{i=1}^n{k}_i\mathit{sin}\left({\omega }_{i}t+{\phi }_i\right)$. If we remove this periodic term, we can achieve the correction of longwave timing time difference data, as shown in Formula (5):

$${x2}_{Loran}\left(t\right)={x1}_{Loran}\left(t\right)-{\sum }_{i=1}^n{k}_i\mathit{sin}\left({\omega }_{i}t+{\phi }_i\right)$$

(5)

Using the parameters to be estimated, $Z$, obtained from (2) to (4) above, and substituting them into Equation (5), the result after removing the period term in the original long-wave time-difference data is shown in Figure 5.

Through the detailed statistical analysis of the data in Figure 5, the standard deviation of the long-wave timing data before correction is as high as 64.38 ns, and after correction, this value drops to 11.87 ns. In comparison, the standard deviation of BeiDou’s timing data is 11.34 ns. From the comparison of these data, it can be clearly seen that the accuracy of long-wave timing data has been significantly improved with the help of the established long-wave timing correction model, and the accuracy of the corrected longwave timing data is very close to that of BeiDou. This result fully shows that the long-wave timing correction model has excellent results in optimising the accuracy of long-wave timing data, which greatly reduces the gap between long-wave timing and BeiDou timing in terms of accuracy.

The stability of the longwave timing data before and after correction, the GNSS timing data, and the free-state rubidium clock timing data are further analysed, and the results are shown in Figure 6 and

Table 2. Comparison of the stability of the free state of the GNSS and rubidium clocks before and after long-wave corrections.

Data Source	Allan’s Var—Time Interval
	1 s	10 s	100 s	1000 s	10,000 s	100,000 s
GNSS	9.17 × 10−9	1.04 × 10−9	1.12 × 10−10	1.33 × 10−11	1.68 × 10−12	1.89 × 10−13
Long-wave Origin	1.19 × 10−8	1.47 × 10−9	1.50 × 10−10	1.79 × 10−11	3.89 × 10−12	6.08 × 10−13
Long-wave Corrected	1.19 × 10−8	1.45 × 10−9	1.46 × 10−10	1.69 × 10−11	2.15 × 10−12	2.06 × 10−13
Rubidium clock-free states	1.18 × 10−10	2.98 × 10−11	7.68 × 10−12	2.34 × 10−13	7.70 × 10−13	1.78 × 10−12

. The results are shown in Figure 6 and Table 2. After comparison, it can be clearly found that the stability of the long-wave timing difference data is significantly improved after the correction, especially the long-term stability, which is also very close to the long-term stability of GNSS.

The above results clearly show that the long-wave timing signal has achieved significant improvement in both accuracy and stability and has the potential to tame the rubidium atomic clock, which lays a solid foundation for the subsequent work and is expected to promote the in-depth application and development of long-wave timing technology in the field of time and frequency.

3. Long-Wave Tame Rubidium Atomic Clock Control System

The long-wave tamed rubidium atomic clock control system selects the 1PPS (one pulse per second) signal output from the long-wave timing receiver as the reference signal, through which the frequency of the rubidium atomic clock is precisely adjusted. In this system, a multi-channel time interval counter is responsible for accurately measuring the time difference between the 1PPS signal from the longwave timing receiver and the 1PPS signal from the rubidium atomic clock. Immediately after that, the computer will collect these time difference measurements and use them to estimate the adjustment amount of the rubidium atomic clock according to the atomic clock taming algorithm. Ultimately, the rubidium atomic clock will be precisely controlled based on the estimation results.

In addition, in order to provide more comprehensive and accurate comparison and evaluation data for subsequent experiments, the group also added a measurement path, which is specifically designed to measure the time difference between the 1PPS signal output from the UTC (Coordinated Universal Time) and the rubidium atomic clock. The basic block diagram of the Rubidium clock control system is shown in Figure 7.

3.1. Atomic Clock Time Difference Model

The time difference between a rubidium atomic clock and Coordinated Universal Time (UTC) exhibits a specific model that has been elaborated in a related study. Specifically, this model can be precisely described by a mathematical expression [31]:

$$Y_R\left(t\right)=a+bt+{\displaystyle \frac{1}{2}}c{t}^2+{\epsilon }_R\left(t\right)$$

(6)

where, the constant term $a$ is the phase difference between the rubidium atomic clock and the standard time (UTC) at the initial moment. This phase difference may originate from a variety of factors, including, but not limited to, the startup moment of the rubidium atomic clock and the accuracy of the synchronisation of the standard time.

$b$ is the deviation between the frequency of the rubidium atomic clock at the initial moment and the standard frequency, which again may be influenced by a variety of factors, including the physical parameter settings within the rubidium atomic clock, fluctuations in external environmental conditions (e.g., temperature, pressure, etc.), and so on.

$c$ t is frequency drift, also known as ageing, which is a phenomenon in which a Rubidium atomic clock deviates linearly in frequency relative to the standard frequency due to the ageing of its own parameters, etc., during long-term operation. This deviation tends to be quadratic in time, i.e., the time difference between the rubidium atomic clock and the UTC accumulates and increases over time.

${\epsilon }_R\left(t\right)$ is noise, a random variable that represents the effect of various uncertainties on the time accuracy of the rubidium atomic clock. These uncertainties may include electromagnetic interference, mechanical vibration, temperature variations, and many other factors.

In controlling a rubidium atomic clock, we have two main methods: adjusting the phase or adjusting the frequency. However, making adjustments to the phase may lead to temporal discontinuities, i.e., the phenomenon of time jumps. Therefore, in this study, we prefer the method of frequency adjustment to control the rubidium atomic clock. By adjusting the frequency, we can make sure that the phase of the rubidium clock output 1PPS signal is continuous and can compensate for the phase of the rubidium clock output by frequency adjustment. In this way, even if there are effects of frequency drift and noise, we can still maintain the time synchronisation accuracy between the Rubidium atomic clock and the UTC through regular frequency adjustments.

3.2. Improved Rubidium Clock Time Difference Modelling

Through the in-depth analysis of the 1PPS signal from the longwave receiver in Section 2, we have come to the clear conclusion that the signal is characterised by extremely significant periodic fluctuations. This discovery enables us to further deduce that when we analyse the time difference between the 1PPS signal from the Rubidium clock and the 1PPS signal from the longwave timing receiver (i.e., the Rubidium—Loran time difference data), the results do not only reflect the characteristics of the rubidium clock itself but also incorporate the characteristics inherent in the longwave signal. In other words, the Rubidium—Loran time difference data is actually a product of the interplay between the characteristics of Rubidium clocks and the characteristics of longwave signals.

In view of the above conclusion, we decided to optimise the original atomic clock time-difference model in order to achieve a more accurate description and prediction of the Rubidium–Loran time-difference data. Specifically, we innovatively introduce a new model of atomic clock time difference, which skilfully combines quadratic polynomials and periodic fluctuation terms. In order to accurately capture the periodic fluctuations of long-wave signals, we use trigonometric functions to construct the periodic fluctuation terms. In this way, the model is able to portray the complex characteristics of the long-wave signal and its interaction with the rubidium atomic clock signal in a more detailed way. The improved rubidium atomic clock time difference model is expressed as follows:

$$y_{RL\_original}\left(t\right)=a+bt+{\displaystyle \frac{1}{2}}c{t}^2+{\displaystyle {\sum }_{i=1}^n{K}_i\sin \left({\omega }_{i}t+{\phi }_i\right)+{\epsilon }_{RL}\left(t\right)}$$

(7)

3.3. Correcting Atomic Clock Time Difference Data

The significant fluctuation of the periodic term in the Rubidium–Loran time-difference data greatly increases the difficulty of accurately estimating the rubidium frequency adjustment, which makes it difficult to realise direct and accurate estimation. Therefore, in order to effectively solve this problem, we will correct the original longwave and rubidium clock time difference data based on the constructed improved rubidium clock time difference model, and the specific implementation methods are as follows.

In the correction process, we will continue to follow the correction method described in Section 2.3. The subtlety of this method is that it can skilfully transform the complex and abstract mathematical model into a more intuitive and easy-to-handle matrix form. The first step is to reduce the model presented in Equation (7) to the following matrix form:

$$Y_{RL}=AX$$

(8)

where $Y_{RL}={\left[\begin{array}{cccc}{y}_1& y_2& \dots & y_m\end{array}\right]}^T$ denotes Rubidium–Loran time difference data.

$$A=\left[\begin{array}{cccccccccc}1& t_1& {\displaystyle \frac{1}{2}}{t}_1^2& \sin \left({\omega }_1{t}_1\right)& \cos \left({\omega }_1{t}_1\right)& \sin \left({\omega }_2{t}_1\right)& \cos \left({\omega }_2{t}_1\right)& \cdots & \sin \left({\omega }_n{t}_1\right)& \cos \left({\omega }_n{t}_1\right)\\ 1& t_2& {\displaystyle \frac{1}{2}}{t}_2^2& \sin \left({\omega }_1{t}_2\right)& \cos \left({\omega }_1{t}_2\right)& \sin \left({\omega }_2{t}_2\right)& \cos \left({\omega }_2{t}_2\right)& \cdots & \sin \left({\omega }_n{t}_2\right)& \cos \left({\omega }_n{t}_2\right)\\ 1& t_3& {\displaystyle \frac{1}{2}}{t}_3^2& \sin \left({\omega }_1{t}_3\right)& \cos \left({\omega }_1{t}_3\right)& \sin \left({\omega }_2{t}_3\right)& \cos \left({\omega }_2{t}_3\right)& \cdots & \sin \left({\omega }_n{t}_3\right)& \cos \left({\omega }_n{t}_3\right)\\ 1& t_4& {\displaystyle \frac{1}{2}}{t}_4^2& \sin \left({\omega }_1{t}_4\right)& \cos \left({\omega }_1{t}_4\right)& \sin \left({\omega }_2{t}_4\right)& \cos \left({\omega }_2{t}_4\right)& \cdots & \sin \left({\omega }_n{t}_4\right)& \cos \left({\omega }_n{t}_4\right)\\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \ddots & \cdots & \cdots \\ 1& t_m& {\displaystyle \frac{1}{2}}{t}_m^2& \sin \left({\omega }_1{t}_m\right)& \cos \left({\omega }_1{t}_m\right)& \sin \left({\omega }_2{t}_m\right)& \cos \left({\omega }_2{t}_m\right)& \cdots & \sin \left({\omega }_n{t}_m\right)& \cos \left({\omega }_n{t}_m\right)\end{array}\right]$$

denotes the coefficient matrix.

$$X={\left[\begin{array}{cccccccccc}a& b& c& k_1\cos \left({\phi }_1\right)& k_1\sin \left({\phi }_1\right)& k_2\cos \left({\phi }_2\right)& k_2\sin \left({\phi }_2\right)& \cdots & k_n\cos \left({\phi }_n\right)& k_n\sin \left({\phi }_n\right)\end{array}\right]}^T$$

denotes the parameters to be estimated.

Substituting the raw time difference data into the matrix system of equations, each parameter to be estimated is obtained by solving the matrix system of equations, as shown in Equation (9):

$$X={\left(A^{T}A\right)}^{-1}{A}^T{Y}_{RL}$$

(9)

Then, subtracting the period term fit from the raw data gives the smooth longwave and rubidium atomic clock time difference data:

$$y_{RL\_corrected}\left(t\right)=y_{RL\_original}\left(t\right)-{\displaystyle {\sum }_{i=1}^n{K}_i\sin \left({\omega }_{i}t+{\phi }_i\right)}$$

(10)

The data before and after the processing of the time difference between the longwave and the rubidium clock are clearly shown in Figure 8. By scrutinizing the data in Figure 8, we can intuitively notice that the corrected time difference data between the longwave and rubidium clocks is much smoother and more stable than before the processing.

In addition, in order to realise a more accurate quantitative analysis, we adopt the following method: Generally speaking, the 1PPS signal output from UTC is regarded as noise-free, and thus the clock difference data between UTC and rubidium clocks can most realistically reflect the performance characteristics of rubidium clocks.

Therefore, in order to further confirm that the corrected longwave and rubidium clock difference data have a significant advantage in estimating the rubidium clock adjustment, we carry out the following comparison operation: the difference between the longwave pre-corrected and post-corrected clock difference data $y_{RL\_original}$ and $y_{RL\_corrected}$ and the clock difference data $y_{RUtc}$ between UTC and rubidium clocks, respectively, are calculated. A randomly selected set of data is calculated, and the final results are shown in Figure 9. The standard deviation of the longwave time difference data before correction is 41.87 ns, while after correction, the value is reduced to 10.42 ns. This change clearly shows that the corrected longwave and rubidium clock time difference data are much more stable and accurate and can more accurately serve the estimation of rubidium clock adjustment.

After 10 consecutive sets of experiments, the statistical results are shown in

Table 3. Comparison of the standard deviation of the long-wave time difference correction before and after making the difference with UTC, respectively (number of groups of 10).

	Standard Deviation (ns)
	1	2	3	4	5	6	7	8	9	10
Before correction	41.87	63.59	57.81	49.08	44.47	41.07	48.79	53.13	54.69	55.26
After correction	10.42	10.40	10.88	12.41	11.74	12.92	12.51	11.83	12.03	11.38

. From the data in the table, it can be clearly seen that there is a significant difference between the long-wave time difference data before and after the correction, and this difference once again strongly proves that the corrected long-wave time difference data is closer to the real characteristics of the rubidium clock and can more accurately reflect the actual situation of the rubidium clock.

The results of 10 consecutive sets of Rb clock frequency deviation estimates using the least squares method described in Section 4.1 are visualised in Figure 10 and

Table 4. Comparison of frequency deviation before long-wave correction, after long-wave correction, and UTC clock difference estimation.

	1	2	3	4	5	6	7	8	9	10
Before correction	62.34	57.90	51.77	52.71	56.06	54.50	51.22	45.88	51.825	51.37
After correction	56.50	56.29	54.64	54.00	51.88	53.20	52.11	51.20	49.41	48.82
UTC clock difference	56.48	55.77	54.60	53.85	53.03	53.03	51.82	51.09	50.17	49.59

.

A comparison of the data clearly shows that the frequency deviation of the Rb clocks estimated from the corrected longwave time-difference data is much closer to the frequency deviation estimated from the UTC-measured clock differentials than to the uncorrected data. This significant difference proves that the corrected longwave time-difference data are more accurate in estimating rubidium clock adjustments and can provide a more reliable basis for accurate rubidium atomic clock tuning.

4. Rubidium Atomic Clock Control Principle

Among the many algorithms for estimating the parameters of rubidium clocks, the methods for developing estimates based on clock difference data are extremely well-developed and diverse. In this study, we have chosen the least squares method as the central research tool.

The least squares method has a number of significant advantages. First and foremost, it can accurately and efficiently estimate the optimal value of an unknown parameter without any a priori knowledge, which makes it effective in complex Rb clock parameter estimation scenarios where no a priori information is available. In addition, the least squares method is excellent in terms of computational complexity, and its relatively low computational complexity allows it to handle large-scale datasets with ease. Whether it is a large amount of clock difference data or a large number of operations that need to be completed in a short period of time, the least squares method is able to produce reliable results quickly with efficient and stable operations, which provides a strong support for the accurate estimation of rubidium atomic clock parameters.

4.1. Least Squares

The least squares method is used to model the estimation of the performance parameters of the rubidium atomic clock, expressed in matrix form as follows:

$$Z=BW$$

(11)

where $Z={\left[\begin{array}{cccc}{z}_1& z_2& \dots & z_m\end{array}\right]}^T$ denotes the corrected long-wave time-difference data.

$B=\left[\begin{array}{ccc}1& t_1& {\displaystyle \frac{1}{2}}{t}_1^2\\ 1& t_2& {\displaystyle \frac{1}{2}}{t}_2^2\\ 1& t_3& {\displaystyle \frac{1}{2}}{t}_3^2\\ \dots & \dots & \dots \\ 1& t_n& {\displaystyle \frac{1}{2}}{t}_n^2\end{array}\right]$ denotes the coefficient matrix. $W={\left[\begin{array}{ccc}a& b& c\end{array}\right]}^T$ denotes the rubidium clock performance parameters to be solved.

According to Equation (7), the performance parameters $a$, $b$, and $c$ of the rubidium atomic clock can be obtained by solving the system of matrix equations.

4.2. Taming Process and Experimental Results

After estimating the performance parameters of the rubidium clocks by the least squares method, the rubidium clocks were adjusted. In the initial state, the clock difference between the longwave and the rubidium clock is extremely significant, so we can carry out a coarse synchronisation with the initial phase difference $a$ and the initial frequency deviation $b$ obtained from the first estimation. After the completion of the coarse synchronisation, in order to ensure the phase continuity of the 1PPS signal output from the rubidium clock, the phase of the rubidium clock output is compensated by adjusting the frequency. The adjustment strategy used in this process is based on the ping-pong algorithm. The ping-pong algorithm, by virtue of its unique operation mechanism, can accurately control the amplitude and rhythm of the frequency adjustment so as to realise efficient and accurate compensation of the phase of the rubidium atomic clock output and ensure the stability and reliability of the rubidium clock output signal.

Assuming that the threshold of the clock difference is set to $\pm a_{Th}$ and the threshold of the frequency deviation is set to $\pm b_{Th}$ after coarse synchronisation, the frequency adjustment amount is calculated as follows:

• When the clock differential prediction, $x>a_{Th}$, and the frequency deviation, $y>0$, indicate that the rubidium clock output is positively oriented away from the reference frequency standard, the frequency adjustment, $\Delta f=x/\tau +y$, is adjusted downward, where $\tau$ refers to the calibration interval.
• When the clock difference predicted value $x<-a_{Th}$, frequency deviation $y<0$, indicates that the rubidium clock output signal reverse direction away from the reference frequency standard, frequency adjustment amount $\Delta f=x/\tau +y$, upward adjustment.
• When the clock difference prediction value $x>a_{Th}$, frequency deviation $y<0$, or $x<-a_{Th},y>0$, indicates that the output signal of the rubidium clock is gradually synchronized to the reference frequency standard, and no adjustment is made at this time.
• When the clock difference predicted value $-a_{Th}<x<a_{Th}$, but the frequency deviation $y<-b_{Th}$, or $y>b_{Th}$, indicating that the frequency deviation has exceeded the control range, frequency adjustment amount $\Delta f=y$, when $y>b_{Th}$, downward adjustment, when $y<-b_{Th}$, downward adjustment.
• When the clock difference prediction value $-a_{Th}<x<a_{Th}$, frequency deviation $-b_{Th}<y<b_{Th}$, indicating that the clock difference and frequency deviation are within the controllable range, do not make adjustments.

The actual clock deviation during the training process and after stabilisation when the longwave timing signal is used to tame the rubidium clock is shown in Figure 11. The standard deviation of the tamed rubidium clock is 15.47 ns, and its fluctuation range is controlled within 25 ns. A further comparison of the stability of the rubidium clock after taming and in the free state is presented in Figure 12 and

Table 5. Stability comparison of rubidium clocks in free state and after taming.

Data Source	Allan’s Var—Time Interval
	1 s	10 s	100 s	1000 s	10,000 s	100,000 s
Rubidium clock free states	1.18 × 10−10	2.98 × 10−11	7.68 × 10−12	2.34 × 10−13	7.70 × 10−13	1.78 × 10−12
Rubidium clock after taming	1.06 × 10−10	3.35 × 10−11	1.06 × 10−12	3.36 × 10−13	1.37 × 10−13	3.52 × 10−13

. By analysing the graphical data, we can clearly find that the long-term stability of the tamed rubidium clock in 100,000 s is increased from 1.78 × 10⁻¹² to 3.52 × 10⁻¹³, which fully demonstrates the positive role of the long-wave timing signal in taming the rubidium clock and effectively improves the performance of the rubidium clock.

In summary, the taming of rubidium atomic clocks by utilising modified long-wave timing signals has achieved remarkable results in two aspects. On the one hand, on the basis of ensuring that the short-term stability of rubidium clocks is not jeopardised, the long-term stability of rubidium clocks is significantly improved, and their performance is more reliable on a long time scale. On the other hand, the tamed rubidium clocks output 1PPS signals that are better synchronised with UTC (NTSC). This not only provides strong support for high-precision time-frequency applications but also further expands the potential of long-wave timing signals in related fields.

5. Discussion

5.1. Interpretation and Comparison of Results

This research innovatively proposes an algorithm to tame rubidium atomic clocks for long-wave timing signals and makes breakthroughs in several key indexes. In terms of longwave timing signal processing, the correction algorithm established by deeply analysing the characteristics of longwave groundwave transmission delay is amazingly effective. The standard deviation of timing is sharply reduced from the initial 64.38 ns to 11.87 ns, which is extremely close to that of BeiDou’s 11.34 ns, and the significant change of this data intuitively reflects the excellent performance of the algorithm in improving signal accuracy. Meanwhile, the corrected long-wave timing signal surpasses the free-state rubidium atomic clock in long-term stability, which lays a solid foundation for the application of long-wave in the field of atomic clock taming. Compared with the traditional way of relying on a single atomic clock or other unstable reference sources, the introduction of long-wave signals opens up a new way of constructing a more stable and accurate time reference.

In the rubidium atomic clock time difference model improvement, the model constructed with a quadratic polynomial superimposed on the periodic fluctuation term is unique. The periodic jitter induced by long-wave characteristics is successfully revealed and effectively removed from the clock difference data, which greatly improves the accuracy and stability of the time difference data and provides reliable data support for accurately estimating the rubidium clock adjustment. Compared with the previous method of simply processing the clock difference data, this strategy of mining the data characteristics and optimising them in a targeted manner greatly improves the accuracy of rubidium clock control.

The experimental verification session strongly proves the reliability of the algorithm. With the help of the least-squares and ping-pong algorithms, the rubidium clock is successfully tamed so that the trained rubidium clock maintains its short-stability characteristics while achieving an excellent long-stability of 3.52 × 10⁻¹³/100,000 s. The tamed rubidium clock’s output signal is stabilised within 25 ns of UTC (NTSC). Both the long stability performance and the clock difference control accuracy show obvious advantages, which emphasise the strong competitiveness of this algorithm in practical applications.

5.2. Impact and Significance of the Research Results

From the theoretical level, this study fills the gap in the application of long waves in the field of atomic clock taming and enriches the theoretical system of atomic clock control. The in-depth study of long-wave transmission characteristics and the exploration of the interaction mechanism with atomic clocks provide an important theoretical basis for subsequent interdisciplinary research. This innovative algorithmic idea, which organically combines long-wave communication technology and atomic clock technology, injects new vitality into the research in the field of time and frequency and is expected to promote the cross-fertilisation and development of related disciplines.

In practical applications, the algorithm of long-wave timing signal taming rubidium atomic clocks has a wide range of application prospects. In timekeeping systems, it provides a reliable backup guarantee for maintaining high-precision time and frequency in complex scenarios. Especially in extreme situations, such as GNSS being rejected or failing, the advantages of long-wave technology are highlighted, and it becomes a key technical means in the field of time synchronisation. This is of great significance for military defenxe, aerospace, communications, electric power, and many other fields that rely on precise time. For example, in military operations, precise time synchronisation is the key to achieving coordinated operations and precision strikes; in the aerospace field, high-precision time reference is crucial for spacecraft orbit control and communication navigation. The results of this research will greatly enhance the stability and reliability of operations in these fields under complex environments and provide strong technical support for national security and development.

6. Conclusions

In this paper, we innovatively propose an algorithm to tame rubidium atomic clocks by using longwave timing signals, and this breakthrough research effectively fills the gap in the application of longwave in the field of atomic clock taming. Firstly, we analyse the delay characteristics of long-wave ground wave transmission and carefully establish a correction algorithm for the 1PPS timing signal output from a long-wave receiver. Subsequently, simulation verification is carried out with the help of measured data, and the results clearly show that the corrected longwave timing signal is better than the free-state rubidium clock in terms of long-term stability, which establishes the feasibility of using it as a reliable reference source for taming rubidium clocks and builds up a solid foundation for the subsequent research.

In further exploration, we use this as a basis for boldly improving the time difference model of rubidium atomic clocks. By constructing a time-difference model with quadratic polynomials superimposed on a periodic fluctuation term, we have successfully revealed the periodic jitter problem caused by long-wave characteristics in the clock difference data. Based on this, we skilfully remove the periodic terms to effectively overcome this jitter problem and significantly improve the accuracy and stability of the clock difference data, whose timing standard deviation is reduced from 64.38 ns to 11.87 ns, which is close to that of Beidou’s 11.34 ns, and which can be more accurately used for estimating the adjustments of rubidium clocks.

In the experimental validation session, we use the least squares method for precision estimation of rubidium clock parameters, as well as the ping-pong algorithm to successfully complete the rubidium clock taming process. The experimental results show that the trained rubidium clock retains the short-stability characteristics well while achieving a good long-stability of 3.52 × 10⁻¹³/100,000 s. Meanwhile, the clock difference between the 1PPS output of the tamed rubidium clock and the UTC (NTSC) is always kept within a very small range of 25 ns after the clock is stabilised. These experimental results fully verify the effectiveness and practicality of the algorithm proposed in this paper.

Looking ahead, the application of long-wave timing signals in atomic clock taming is promising. It not only provides a reliable backup guarantee for the timekeeping system to maintain high-precision time and frequency in complex scenarios but also becomes an indispensable key technology in the field of time synchronization, especially in extreme situations such as the rejection or failure of GNSS, which protects many fields relying on precise time.