# BDS Satellite Clock Prediction Considering Periodic Variations
## 2 Materials and Methods
### Analysis of Periodic Variations in BDS Satellite Clock Offsets
The spectral peak in the 90-day and 270-day amplitude spectrums for C25 is approximately 3.72 cpd, while 1.87 cpd in the 180-day amplitude spectrum. Distinct amplitude spectrum results of different FFT lengths are probably led by periodic variations. Performing FFT on clock offset series gives the global spectrum analysis result for the entire time series. However, it cannot give the variations of frequency components in the time domain. The satellite atomic clocks in space are affected by space environment changes, temperature variations, and various disturbances, which make the periodic noise time-varying. Therefore, the main periodic term obtained by FFT may be inaccurate. The STFT has a unique advantage in processing non-stationary signals. To perform STFT, long-term clock offset series are broken up into chunks, and to reduce artifacts at chunk boundary, the chunks overlap each other. Each chunk is Fourier transformed, and the magnitude for each point in time and frequency domain is recorded. Therefore, the STFT can effectively detect the variations of amplitude spectrum. In this section, the STFT is used to analyze the periodic variations of BDS satellite clock offsets. \begin{table}
\begin{tabular}{c c} \hline \hline
**Items** & **Models** \\ \hline Satellite attitude & BDS-2 (GEO: orbit-normal; IGSO/MEO: yaw-steering and orbit-normal); BDS-3: continuous yaw-steering \\ Satellite antenna PCO/PCV & PCO according to igs14.ab; ignoring PCV \\ Reference frame & IGb14 \\ Geopotential & 12 \(\times\) 12 EGM2008 model [38] \\ N-body & Jet Propulsion Laboratory (JPL) DE405 \\ ocean tides & FES2004 [39] \\ Solid earth tides & IERS conventions 2010 [40] \\ pole tides & IERS conventions 2010 [40] \\ Relativity & IERS conventions 2010 [40] \\ Solar radiation & The 5-parameter ECOM1 model [35; 41] \\ Antenna thrust & BDS-2 (GEO: none; IGSO: 185 W; MEO: 130 W [42]); \\  & BDS-3 (MEO: 130 W) \\ Earth Rotation Parameters & Static Earth’s Albedo model [43] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Processing strategies of WHU BDS precise orbit determination [35; 37]. Although the constellation deployment of BDS-3 has been completed, some satellites of BDS-3 are in orbit only for a short time, and there are no sufficient precise clock products. Therefore, the discussion in this paper is limited to 15 satellites of BDS-2 and 15 satellites of BDS-3. The BDS-3 satellites are manufactured by China Academy of Space Technology (CAST) and Shanghai Engineering Center for Microsatellites (SECM). The satellites from the two manufacturers have different body sizes and attitude modes, and employ two different satellite platforms. Distinct attitude control models employed by CAST satellites and the SECM satellites affect the accuracy of orbit determination and clock offsets [19]. Therefore, it is necessary to classify the satellite clock offsets that from the CAST or the SECM. The detailed information of the BDS satellites is listed in Table 2.  
To fully analyze the periodic variations, we investigate the periodic noise of the BDS satellite clock offsets in frequency and the time-frequency domain. First, we collected the raw clock products (DOY 001 to 360, 2019), and calculated the fitting residuals of the QPM after preprocessing. Then, we perform a spectrum analysis based on FFT and a time-frequency analysis based on STFT for the residuals. Limited by paper length, one satellite from each constellation is randomly selected to show the results. Figure 3 gives \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**System** & **Satellite Type** & **Clock Type** & **Manuf** & **PRN** \\ \hline \multirow{3}{*}{BDS-2} & GEO & Rb & CAST & C01, C02, C03, C04, C05 \\  & IGSO & Rb & CAST & C06, C07, C08, C09, C10, C13, C16 \\  & MEO & Rb & CAST & C11, C12, C14 \\ \hline \multirow{3}{*}{BDS-3} & MEO & Rb & CAST & C19, C20, C21, C22, C23, C24, C32, C33 \\  & & H & SECM & C25, C26, C27, C28, C29, C30, C34 \\ \hline \end{tabular}
\end{table}