## 1 Introduction  
The Beidou satellite navigation system has gone through the development stages of the experimental system (BDS-1), the regional system (BDS-2), and the global system (BDS-3) [1]. The service area of Beidou-1 was China, and Beidou-2 provided positioning, navigation, and timing (PNT) services for the Asia-Pacific region; there are still more than 10 Beidou-2 satellites in operation. From 2017 to 2020, the number of Beidou satellites increased rapidly, and finally, the main system of Beidou third-generation network (BDS-3) was officially commissioned in July 2020 and began to provide global services, marking the expansion of the scope of BDS services from the region to the world [2, 3, 4]. In terms of the working principle, BDS is a high-precision time synchronization system which requires stable frequency standards and a high-precision time scale to realize positioning and time service [5, 6]. The satellite atomic clock, which could provide a stable on-satellite time, is an important payload for BDS satellites. The performance and working condition of satellite atomic clocks are directly related to the research on precision clock offset estimation and prediction technology, precision single-point positioning services, and so on. Therefore, it is of great significance to comprehensively evaluate the performance of BDS satellite atomic clocks and analyze their long-term on-orbit characteristics. In addition,high-precision satellite atomic clocks are also key to precise point positioning (PPP) and are important in applications such as PPP-RTK [7; 8; 9]. With the development of GNSS, some analysis and discussion on the satellite atomic clocks of GPS, GLONASS, and Galileo systems have been carried out [10; 11]. The on-orbit working characteristics of satellite atomic clocks has been obtained, including frequency stability level, periodic characteristics, and so on, which provide important reference for the operation of GNSS [12; 13]. Overall, the frequency accuracy of GPS and Galileo satellite atomic clocks is in the range of \(10^{-11}\), the daily frequency drift rate is in the range of \(10^{-14}\), and the daily frequency stability is about 2-5 \(\times\)\(10^{-15}\)[14]. In terms of BDS, the existing research generally focuses on BDS-2 or some kinds of BDS-3 satellites. [PERSON] et al. [15] and [PERSON] et al. [16] studied the on-orbit performance characterization of BDS-2 and made a detailed analysis of the early BDS satellite atomic clock. [PERSON] et al. [17; 18; 19] analyzed some parts of BDS-3 satellite atomic clocks from different angles, including precise orbit and clock determination for BDS-3 experimental satellites and performance of the BDS-3 experimental satellite PHM, mostly based on data over several months. However, a complete overview of the long-term and dynamic characteristics for the satellite atomic clock is lacking after the official commissioning of BDS-3 [2; 20]. Since all on-orbit satellites provide PNT services for users, their atomic clocks should be studied and analyzed in detail, so as to analyze the overall working status of the system. In order to accurately reflect the long-term operation characteristics of a BDS on-orbit satellite atomic clock and comprehensively evaluate its performance, this study was carried out on the basis of the precise clock offset data after the official commissioning of BDS-3. Firstly, in this paper, the characteristics of the original phase and average frequency are analyzed, and the MAD-based outlier detection method is used to provide clean experimental data for subsequent analysis. On this basis, the parameters and fitting residuals, long-term frequency drift rate, frequency accuracy, and frequency stability of the satellite atomic clocks are calculated; further to this, these indicators are discussed according to the system and atomic clock type. The fitting residual is processed by fast Fourier transform (FFT); the main periodic terms due to orbital operation are thus obtained.