## 1 Introduction  
The BeiDou navigation satellite system (BDS) provides positioning, navigation, and timing (PNT) service with high continuity, reliability, and stability for global users. The high-precision and stable in-orbit atomic clock plays a key role in global satellite navigation system (GNSS) positioning and navigation. With increasing Beidou-3 satellites equipped with the passive hydrogen maser (PHM) and rubidium atomic frequency standard (RAFS) orbiting in space, the clock performance needs to be explored. Particularly, the clock stability has a significant influence on PNT service. By the end of December 2018, there were 18 medium earth orbit (MEO) satellites and 1 geostationary earth orbit (GEO) satellite in orbit, which form the BDS-3 basic system and mainly serve counties in the China-proposed Belt and Road Initiative. The public signals include B1C, B2a, B2b, and the old BDS-2 signals, B1I and B3I [1]. Table 1 lists the satellite information [2], where the GEO satellite C59 with RAFS is still under testing status. BDS-2 has been studied by many scholars, including the precise orbit determination (POD) [3; 4], modeling of solar radiation pressure [5], attitude mode [6], positioning performance [7; 8], and signal characteristics [9]. In the initial period of BDS-2 providing service, [PERSON] et al. [10] used four estimation strategies to analyze the short-term stability with the best satellite C09 located in inclined geosynchronous orbit (IGSO) being 4 \(\times\) 10\({}^{-13}\) and 1 \(\times\) 10\({}^{-13}\) at 100 s and 1000 s averaging time, respectively. Then, further studies of BDS-2 clock performance were analyzed by many researchers to assess the frequency stability [11; 12; 13; 14; 15; 16]. The clock data were derived mainly by the orbit determination and time synchronization (ODTS) method. [PERSON] et al. [11] gave the initial result, and the stability of BDS-2 RAFS satellites with 1 and 1000 s averaging time was 0.1-7 \(\times\) 10\({}^{-12}\) and 1-3 \(\times\) 10\({}^{-13}\), while for GPS IIF satellites it was 4-6 \(\times\) 10\({}^{-12}\) and 1 \(\times\) 10\({}^{-13}\), respectively. [PERSON] et al. [12] used different POD strategies to assess the performance of BDS-2 clocks, and the stability at the averaging time of 1000 s for GEO and IGSO satellites was 8-12 \(\times\) 10\({}^{-14}\) and 8-17 \(\times\) 10\({}^{-14}\), respectively, which is better than the result of [11]. Then, [PERSON] et al. [13] used the clock product from Wuhan University with 5 min sampling rate to assess the BDS-2 clock performance. The Allan deviation (ADEV) of GEO, IGSO, and MEO clocks was about 3-5 \(\times\) 10\({}^{-13}\), 3-8 \(\times\) 10\({}^{-13}\), and 2-4 \(\times\) 10\({}^{-13}\) at the averaging time of 300 s, which is comparable to the reference [14]. When the averaging time was 10,000 s, the ADEV was 8-11 \(\times\) 10\({}^{-14}\), 6-12 \(\times\) 10\({}^{-14}\), and 4-6\(\times\) 10\({}^{-14}\), correspondingly. Also, the two-way satellite time and frequency transfer (TWSTFT) data was analyzed by [PERSON] et al. [15]. [PERSON] et al. [16] gave the detailed analysis of the long-term in-orbit situation of BDS-2 clocks with a length of about five years. Two IGSO and two MEO experimental satellites of BeiDou-3 were launched between 2015 and 2016. Technologies of the signal system [17], intersatellite links [18], and POD [19] were validated and analyzed. The atomic clock performance was also assessed with the clock data derived by the TWSTFT [20] or ODTS method [21; 22]. [PERSON] et al. [20] analyzed the frequency stability, prediction accuracy, and clock rate variation, and the optimal stability at the averaging time of 86,400 s is 6.5 \(\times\) 10\({}^{-15}\) for the IGSO satellite C32 with the Ka-band measurement noise of 3 \(\times\) 10\({}^{-15}\). [PERSON] et al. [21] gave an initial result with the regional distribution of ground stations. The stability of IGSO satellite clocks at 1000 s averaging time is 2-4 \(\times\) 10\({}^{-14}\), which is 10 times better than that of BDS-2 satellites.