## 6 Conclusions  
In this study, the upper and lower limits of the \(BT\) for the DPLL were deduced theoretically and presented. For the upper limit, the effects of using digital integration methods on the stability of the DPLLs were investigated using the root loci. Stability problems in sampled-data loops occurred when \(BT\) was not sufficiently small, such that the sampled-data loops no longer represented their counterparts in the continuous-time domain, which typically occurs in modern digital GNSS receivers. The computational delay inherent in the discriminator and loop filter was considered in the analysis. All the types of DPLLs become conditionally stable when the unit delay is inserted as the computational delay, while the types vary for the order and integration method when zero computational delays are assumed.  
The lower limits were obtained by analyzing the DPLL measurement error and the corresponding threshold. The thermal noise, oscillator phase noise using Allan deviation, and dynamic stress error were taken into account to constitute the measurement errors. The \(C\)/\(N_{0}\) threshold was defined as the \(C\)/\(N_{0}\) at the crossing point between the measurement error and threshold. By observing the \(C\)/\(N_{0}\) thresholds for varying \(B\), the point that the \(C\)/\(N_{0}\) threshold increases rapidly existed, which is the lower limit. The lower limits are affected by the oscillator quality and dynamic stress because some amount of gain is required for the DPLL to track quickly varying phase and/or phase rate stably.  
Issues related to the actual implementation of the DPLL were suggested with the simulation results. As an example, simulation results using second-order DPLL with SI/SI integration were presented. The phase error oscillates as \(BT\) approaches the deduced upper limits (for both cases with and without the computational delay), which verifies the previous analysis since it behaves as expected. For the numerical analysis of the lower limits, an accurate modeling of the oscillator phase noise on the sampled signal is needed and the dynamic stress, which has a dominant role in determining the lower limits, does not have random characteristics, so it is hard to observe the effect of it using the Monte Carlo simulation. Therefore, such detailed simulation and verification have remained as future works.  
Conceptualization, Y-J.S., T.P. and J.-H.W.; methodology, Y-J.S., T.P. and J.-H.W.; software, Y-J.S. and J.-H.W.; validation, Y-J.S., T.P. and J.-H.W.; formal analysis, Y-J.S. and J.-H.W.; investigation, Y-J.S., T.P. and J.-H.W.; and J.-H.W.; data curation, [PERSON] and [PERSON]; writing--original draft preparation, [PERSON] and [PERSON]; writing--review and editing, [PERSON] and [PERSON]; visualization, [PERSON], [PERSON] and [PERSON]; supervision, [PERSON]; project administration, [PERSON]; funding acquisition, [PERSON]. All authors have read and agreed to the published version of the manuscript.  
This work was supported by a National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (No. NRF-2020R1A2C2013091).  
Not applicable.  
The authors declare no conflict of interest.