Atomic clock is the core component of navigation satellite payload, playing a decisive role in the realization of positioning function. So the monitoring for anomalies of the satellite atomic clock is very important. In this paper, a complete autonomous monitoring method for the satellite clock is put forward, which is, respectively, based on PhaseLocked Loop (PLL) and statistical principle. Our methods focus on anomalies in satellite clock such as phase and frequency jumping, instantaneous deterioration, stability deterioration, and frequency driftrate anomaly. Now, method based on PLL has been used successfully in China’s newest BeiDou navigation satellite.
The most important function of navigation satellite is to support its users to acquire their position through the satellite signal, during which satellite time is one of the most important factors. Because of the changes in temperature, humidity, radiation, and the aging of the satellite clock, the physical and electric part of clock may both have problems, which will bring anomaly in clock signal, resulting in large error in the prediction of satellite time or even unpredictability, which may lead to disastrous consequence. So the anomaly monitoring of satellite clock is very important.
So far, researchers have proposed schemes to monitor anomalies of clock, such as Interferometric Detection Method [
Under normal circumstance, ground station can evaluate the health condition and performance of the clock by continuously tracking satellite signals. But when the satellite flies beyond the ground station’s sight, or owing to some reasons, the satellite cannot contact with the ground station in a few hours or even days; the satellite needs to judge the status of the clock all by itself. SelfMonitoring for clock anomaly, which is in the absence of ground station, is that the satellite monitors its clock by itself to make a judgment on the satellite clock running state.
The common anomalies of the satellite clock are signal loss, phase jumping, frequency jumping, instantaneous deterioration, stability, and frequency driftrate deterioration.
The contributions of this paper can be summarized as follows.
In this paper, a set of SelfMonitoring algorithms is proposed to improve the reliability of satellite. Two methods are put forward to monitor satellite clock anomalies. The first method is based on PLL, and it can detect signal loss and phase and frequency jumping. Based on the measurement data from intercomparison among three clocks, Modified DAVAR is used to detect phase and frequency jumping and instantaneous deterioration; we use windowed overlapping Hadamard variance to evaluate clock stability in real time and the threestate Kalman filter to detect large drift rate.
The method based on PLL has been proved effective and used in newest BeiDou satellite. And the other research on SelfMonitoring Method in this paper can be used in next generation navigation satellites.
Generally speaking, there are two methods to evaluate atomic clocks: (
Because there is no standard reference in the satellite and the performance of satellite clocks is similar, we make use of the second method to realize SelfMonitoring for anomalies. The schematic diagram is shown in Figure
Global schematic diagram of SelfMonitoring Method.
Firstly, we define that
In this paper,
The detailed structure of SelfMonitoring Module based on PLL in Figure
Schematic diagram of SelfMonitoring Method based on PLL.
Figure
Once phase or frequency jumping occurs, the output of Phase Detector in PLL will follow. In this section, the response of Phase Detector to these two anomalies will be derived.
According to [
In the following, the tracking property of Phase Detector for phase and frequency jumping will be deduced.
Supposing that the phase jumping can be written as
Through factorization, (
Considering (
From (
Response of Phase Detector to phase jumping during simulation.
Assuming that frequency jumping is
After factorization,
As can be seen from (
Response of Phase Detector to frequency jumping during simulation.
As shown in Figure
Response of Phase Detector to signal loss during simulation.
It can be seen that, from Figures
In practice, Probability of False Alarm (PFA) and Detection Probability (PD) are usually used to evaluate the detection method. The basic principle in setting parameters (loop parameters and detection threshold) is to improve PD and minimize PFA at the same time.
The loop parameters and detection threshold are mainly determined by clock noise level and required resolution. We usually use Allan variance (
During the following simulations, we simulate 10000 realizations.
Detection performance for phase jumping.
Loop parameter  Threshold  PD  PFA  Delay (m) 



1.0000 

1 

1.0000 

1  

0.9999 

1  

0.9978 

1  

0.9833 

1 
Detection performance for frequency jumping.
Loop parameter  Threshold  PD  PFA  Delay (m) 



1.0000 

234 

1.0000 

296  

1.0000 

337  

0.9933 

380  

0.9873 

429 
Detection performance of phase jumping for different clock stability.
Clock stability  Resolution  Threshold  PD  PFA  Delay (m) 




0.9978 

1 



0.9972 

1 



0.9955 

1 
Detection performance of frequency jumping for different clock stability.
Clock stability  Resolution  Threshold  PD  PFA  Delay (m) 




0.9933 

380 



0.9947 

384 



0.9978 

381 
The noise level of atomic clock directly determines the detection resolution, which we can see from Tables
The method based on PLL can realize SelfMonitoring for phase jumping, frequency jumping, and signal loss. The computation complexity is low and costs little time to detect anomaly. But if we want to enhance its weak anomaly detection performance, we need to lower the working frequency of Phase Detector, which will lead to longer detection delay. In practice, we pay more attention to large frequency jumping in satellite atomic clock, which will obviously affect the positioning accuracy and our PLL method is designed for it.
We usually use Allan variance [
Powerlaw spectrum is used to analyze noise property in frequency domain:
The slope of Allan variance gives us a knowledge of noise distribution in different averaging time.
After obtaining a sufficient number of measurement data, Allan variance can be used to calculate the stability of different averaging time. But when anomaly occurs, the results given by Allan variance may lose practical significance. As shown in Figure
Relative frequency deviation and its Allan deviation when anomaly occurs.
Measured value of relative frequency deviation
Allan deviation
As can be seen from Figure
When we calculate DAVAR, a sliding window is used to cut the data. The window length is
From expression (
Expression (
In this section, we will firstly analyze and compare the detection performances of DAVAR and Modified DAVAR when facing phase and frequency jumping and then show that Modified DAVAR is also effective in detecting instantaneous stability deterioration.
The monitoring method for phase and frequency jumping is based on statistics.
In Figure
Phase and frequency jumping.
Phase jumping
Frequency jumping
We simulated 1000 sampling points and phase and frequency jumping occurred at the 500th point. We simulated one realization and saved the response data of DAVAR and Modified DAVAR to jumping at every time instant; then we repeated 10000 realizations in the same way. Of course the 1000 sampling points’ data is different in every realization. Then we got the average response at every time instant that is shown in Figures
Comparison of two variances for phase jumping.
Average response of DAVAR to phase jumping
Average response of Modified DAVAR to phase jumping
Comparison of two variances for frequency jumping.
Average response of DAVAR to frequency jumping
Average response of Modified DAVAR to frequency jumping
Because the peak value of response of DAVAR and Modified DAVAR to jumping determines if the jumping could be detected, we focus on the peak value in each realization. Assuming that, in one realization, the maximum value of response of DAVAR to frequency jumping is
To make the statistical result more clear, we list the statistical characteristic of
Statistical characteristic of
Statistical item  DAVAR  Modified DAVAR 

Minimum value 


Maximum value 


Mean value 


Standard deviation 


In Tables
Detection performance for phase jumping.
Detection item  DAVAR  Modified DAVAR 

Resolution 


Threshold 


PD 


PFA 


Detection delay (points)  1  1 
Detection performance for phase jumping.
Detection item  DAVAR  Modified DAVAR 

Resolution  12  4 
Threshold 


PD 


PFA 


Detection delay (points)  1  5 
Detection performance of phase jumping for different clock stability.

Resolution  Threshold  PD  PFA  Delay (points) 


12 

0.9978 

1 

12 

0.9961 

1 

12 

0.9977 

1 
Detection performance of frequency jumping for different clock stability.

Resolution  Threshold  PD  PFA  Delay (points) 


4 

0.9927 

5 

4 

0.9923 

5 

4 

0.9934 

5 
What should be pointed out is that the unit of resolution in phase and frequency jumping in Tables
During the simulation, we simulate 10000 realizations and use same threshold for both DAVAR and Modified DAVAR. The window length
Table
From Tables
In addition, the window length
Figure
Instantaneous deterioration of clock.
Instantaneous deterioration
Modified DAVAR
Modified DAVAR can be considered as a statistical tool; it is effective to detect phase and frequency jumping. Compared with PLL method, we need to measure the time error data firstly and then calculate the statistical characteristics of clock. Modified DAVAR can monitor weaker frequency jumping compared to PLL method, but PLL method is independent of a second standard reference and timecomparison device, which will give us more flexibility. Taking their respective characteristics into account, cooperation between them may be a good choice to improve the SelfMonitoring reliability.
In this section, we will introduce two existing methods, which are called LS method and Kalman filter method.
Figures
LS method in detecting frequency jumping.
Kalman filter method in detecting frequency jumping.
After we have done numerical simulations, we give Table
Detection performance of frequency jumping for different clock stability.
Detection method  Resolution  PD  PFA  Delay 

PLL method 

0.9933 

0.4 s 
DAVAR method 

0.9874 

1 s 
Modified DAVAR method 

0.9927 

5 s 
LS method 

0.9944 

1 s 
Kalman filter method 

0.9926 

1 s 
Different observation quantity is needed for different methods. In our opinion, DAVAR, Modified DAVAR, LS, and Kalman filter are all effective in detecting weak frequency jumping. They need a second standard reference and timecomparison device to get the time error measurement
In this section, we use the wellknown “threecornered hat” approach [
According to the references, we can get
Because Allan variance is convergent for the five kinds of noises at different averaging time, it is often used to evaluate the stability of clock. However Allan variance cannot rule out frequency drift. Especially when the drift is almost equal to Allan variance for certain averaging time, if we use Allan variance to calculate
In order to make full use of the measurement data and also track slow change of satellite clock timely, we use windowed overlapping Hadamard variance, as shown in (
Expression (
As opposed to the Allan variance, which makes use of a second difference, the Hadamard variance employs a third difference that leads to reduction in the degrees of freedom by one. The Hadamard variance requires more data to produce a single stability calculation, as compared to the Allan variance, given equal averaging time
According to [
Two points:
Two groups of points:
LS:
Three points:
Kalman filter:
To compare these six estimators, we generate simulation data by wellknown Stable 32 software. The parameters of phase data are shown in Table
Parameters of generated phase data.



Drift rate/day  Frequency deviation 







From Table
Statistical property of the estimation result of six estimators.
Estimator 





















Comparison of different driftrate estimators.
Driftrate estimation of different estimators
Partial enlarged view of (a)
The driftrate estimation of Kalman filter.
In fact, no matter which method we choose to evaluate the drift rate, we must know the time error data
When the satellite is within the sight of ground station, we can calculate the drift rate with certain estimator by comparing the satellite time with the timescale on the ground. But when the satellite cannot contact with the station, the only available data is the intercomparison data
In order to evaluate the drift rate, there may be two methods. (
In the following, we will compare these two methods. The threestate Kalman filter method is called method 1, and the combination of twostate Kalman filter with
The basic Kalman fiter equations are as follows.
System equation is
Observation equation is
For the twostate Kalman filter,
For the threestate Kalman filter,
To compare these two methods, we make use of Stable 32 to generate simulation data. The parameters of the phase data are shown in Table
Parameter setting.
Parameters 



Drift rate/day  Frequency deviation 

Clock 1 





Clock 2 





Clock 3 





Firstly, we will show the prediction performance of both twostate and threestate Kalman filter for
Prediction performance of Kalman filter for
For method 2, after we have got the prediction of
Figure
Figures
Driftrate estimation of method 1.
Drift rate of clock 1
Drift rate of clock 2
Drift rate of clock 3
Driftrate estimation of method 2.
Drift rate of clock 1
Drift rate of clock 2
Drift rate of clock 3
This paper puts forward a set of SelfMonitoring Methods for common anomalies. We use PLL to realize SelfMonitoring for signal loss and phase and frequency jumping. Based on the measurement data from intercomparison among three clocks, Modified DAVAR is used to detect phase and frequency jumping and instantaneous deterioration; we use windowed overlapping Hadamard variance to evaluate clock stability in real time and the threestate Kalman filter for large driftrate anomaly.
The method based on PLL has been proved effective and used in newest BeiDou satellite. And the other research on SelfMonitoring Method in this paper can be used in next generation navigation satellites after year 2019.
The authors declare that they have no competing interests.