## 3 Analysis Methods
### Main Performance Metrics of the Satellite Atomic Clock  
(1) Fitting residual and periodic characteristic analysis  
The frequency of BDS satellite clocks drift to some extent with frequency changes, especially for RAFS; therefore, their models are usually fitted by means of a quadratic polynomial model [25]:  
\[L(t)=a_{0}+\hat{a}_{1}t+\frac{1}{2}\hat{a}_{2}t^{2} \tag{3}\]  
where \(a_{0}\) represents the initial clock offset, \(\hat{a}_{1}\) is the estimated frequency offset, and \(\hat{a}_{2}\) is the estimated frequency drift. Then, the fit residues can be obtained from Equations (1) and (3):  
\[\Delta x(t)=x(t)-L(t)=(a_{1}-\hat{a}_{1})t+\frac{1}{2}(a_{2}-\hat{a}_{2})t^{2} +\frac{A}{2\pi f_{0}}\sin(2\pi f_{0}+\varphi)|_{0}^{t} \tag{4}\]  
It can be seen from Equation (4) that the fitted residual has the frequency and the clock drift component residual in the low-frequency part, the noise term in the high-frequency part, and the periodic term component in the middle part. Therefore, Fast  
Figure 4: Satellite clock systematics model, including environmental and random effects. Fourier Transform (FFT) is used to identify and analyze satellite atomic clock period terms, with the following formula:  
\[X(k)=\sum\limits_{n=0}^{N-1}\Delta x(n)e^{-j\frac{2\pi}{N}kn} \tag{5}\]  
where \(X(k)\) is the spectral analysis sequence at epoch \(k\), \(\Delta x(n)\) is the clock data residuals fitted by quadratic polynomial, \(N\) is the number of residuals, \(j\) is an imaginary unit, and \(e\) is a mathematical constant. Frequency drift rate is the ratio of the frequency offset before and after the operation of the satellite clock to the two-point sampling time interval, reflecting the frequency variation of the satellite clock within a certain period of time. The drift rate and daily drift rate are mainly used to measure the frequency drift of the atomic clock. The former is solved by short-term fitting of quadratic polynomial, and the latter is calculated by using longer-term data. For example, the frequency of one month is used for fitting, and the calculation formula is [26]:  
\[D=\frac{\sum\limits_{i=1}^{N}\left(y_{i}-\overline{y}\right)(t_{i}-\overline{t })}{\sum\limits_{i=1}^{N}\left(t_{i}-\overline{t}\right)^{2}} \tag{6}\]  
where \(y_{i}\) represents the frequency values at each moment, \(\overline{y}\) is the mean of the relative frequency value, \(t_{i}\) represents the moment of the measurement, and \(\overline{t}\) is equal to \(\frac{1}{N}\sum\limits_{i=1}^{N}t_{i}\). Frequency accuracy is a measure of the deviation between the actual output frequency of a satellite clock and its reference frequency, showing the error of the satellite clock output frequency and the reference ground clock frequency. Usually, a frequency accuracy index can be obtained by fitting the phase data in a linear mode based on the least square principle [27]:  
\[K_{\tau}=\frac{\sum\limits_{i=1}^{N}\left(x_{i}-\overline{x}\right)(t_{i}- \overline{t})}{\sum\limits_{i=1}^{N}\left(t_{i}-\overline{t}\right)^{2}} \tag{7}\]  
where \(x_{i}\) represents the phase values at each moment, \(\overline{x}\) is the mean of the phase value, \(t_{i}\) represents the moment of the measurement, and \(\overline{t}\) is equal to \(\frac{1}{N}\sum\limits_{i=1}^{N}t_{i}\). Frequency stability characterizes the ability of the same frequency source to stably produce a certain frequency over a certain period of time. The commonly used Allan variance belongs to the double-sampling treatment method and is greatly affected by the drift. To this end, the stability of BDS satellite atomic clocks was assessed by the triple-sampled Hadamard variance.