###### Abstract  
Onboard satellite clocks are the basis of Global Navigation Satellite Systems (GNSS) operation, and their revolution periods are at the level of 2 per day (about 12 h) in the case of the Medium Earth Orbit (MEO) satellites. In this work, the authors analysed the entire BeiDou Navigation Satellite System (BDS) space segment (BDS-2 and BDS-3) in terms of the occurrence of periodic, repetitive signals in the clock products, and checked if they coincide with the orbital periods or their multiples. The Lomb-Scargle (L-S) power spectrum was used as a tool to determine the periods present in the BDS clock products, allowing for analyses based on incomplete input data; in this case, the incomplete data were the phase data with jumps and outliers removed. In addition, continuous wavelet transform (CWT) was used to produce a time--frequency representation showing the more complex behaviour of the satellite clock products. As shown in the case of geostationary and geosynchronous inclined orbit satellites, the main period was 23.935 h, while for the Medium Earth Orbit it was 12.887 h, with the BDS satellite orbital period being 12 h 53 m (12.883 h). Some effects connected with reference clock swapping are also visible in the power spectrum. The conducted analyses showed that the BDS-2 satellite clocks have much higher noise than the BDS-3 satellite clocks, meaning that the number of designated periods is greater, but their reliability is significantly lower. BDS-3 satellites have only been in operation for a very short time, thus this is the first analysis to include this type of data. Moreover, such a wide and complex analysis has not been carried out to date. satellite; GNSS; period; time; clock; BeiDou +
Footnote †: journal: Applied  
Satellite navigation systems are used in many fields and have various applications [3; 4; 5; 6; 7; 8]. Time synchronisation of the clock is the base and the core in GNSS positioning. Insufficient operation or instability of the onboard clocks leads to degradation of the positioning accuracy [9; 10; 11]. The main impact on accuracy in navigation and positioning comes from the onboard satellites' clocks [12]. In recent years, research on the stability of GNSS clocks has been extensively conducted [13; 14; 15]. As clock products process data in 24-h batches, day boundary discontinuities (DBDs) occur between consecutive products. A shift or jump in the clock data is caused by the start of the new data processing package at midnight [16], and its magnitude in the case of clock products might be as high as 1 ns [17]. This kind of phenomenon must be considered when processing involves midnight--the point of the switch to the next file/processing batch. The DBDs can impact either the satellite's clock [18] or orbit [19]. As research shows, this may also affect the positioning accuracy [20; 21]. Research on the analysis of GNSS clock products in terms of periodic signals has already been conducted, but to a very limited extent [22; 23; 24]. The amplitude spectra of the International GNSS Service (IGS) final clock products (after removing 12-h periodic signal and daily rates) for GPS Block IIR satellites show 4 and 6 cycles per day [25], while for the whole GPS space segment, the biggest spectrum amplitudes appear for two and four cycles per day, respectively, according to [PERSON] et al. [26] and [PERSON] et al. [22], also based on the final IGS clock products. Moreover, such phenomena appear not only for the MEO navigational satellites such as GPS or Galileo, and IGSO satellites, e.g., QZSS [27], but also for Low Earth Orbit (LEO), e.g., GRACE [28]. The Lomb-Scargle periodogram method is well-known and is widely used for the spectral analysis of unevenly sampled time series [29]. It allows for the computation of the reliable power spectrum for unevenly sampled data or data with gaps, and has particularly broad applications in astronomy and geodynamics [30]. Usage of the Lomb-Scargle (L-S) and Continuous Wavelet Transform (CWT) methods for GNSS clock analyses has not been conducted so far.