## 3 Optimal Bandwidth Table
\begin{bmatrix}\frac{1}{3}\left(\arctan\frac{\omega_{0}}{\omega_{0}}\ -\arctan\frac{\omega_{0}}{\omega_{0}}\right)\ +\frac{1}{4\sqrt{3}}\ln\frac{\left(\omega_{0}^{2}-\omega_{0}\omega_{2}\sqrt{3 }+\omega_{2}^{2}\right)\left(\omega_{0}^{2}+\omega_{0}\omega_{1}\sqrt{3}+ \omega_{2}^{2}\right)}{\left(\omega_{0}^{2}+\omega_{0}\omega_{2}\sqrt{3}+ \omega_{2}^{2}\right)\left(\omega_{0}^{2}-\omega_{0}\omega_{1}\sqrt{3}+ \omega_{1}^{2}\right)}\\ +\frac{1}{6}\left(\arctan\frac{-\sqrt{3}\omega_{0}+2\omega_{1}}{\omega_{0}}+ \arctan\frac{\sqrt{3}\omega_{0}+2\omega_{1}}{\omega_{0}}-\arctan\frac{-\sqrt {3}\omega_{0}+2\omega_{1}}{\omega_{0}}-\arctan\frac{\sqrt{3}\omega_{0}+2\omega _{1}}{\omega_{0}}\right)}\end{bmatrix} }\mathrm{[deg]} \tag{4}\]  
where \(k_{g}\) is the oscillator's g-sensitivity [parts/g], \(G_{g}\) is the single-sided vibration spectral density [g\({}^{2}\)/Hz], and \(\omega_{1}\) and \(\omega_{2}\) are the lower and upper limits [rad/s] of the constant vibration PSD, respectively. Additionally, the parameter values for the vibration-induced oscillator phase noise calculation are selected based on the assumption that the receiver uses a high-quality oscillator and suffers a moderate amount of vibration. These parameter values are presented in Table 2.  
The LOS dynamic stress error of the third-order DPLL is modeled as follows [1]:  
\[\theta_{e}=\frac{R^{\prime\prime\prime}}{\omega_{0}^{3}}\ [\deg] \tag{5}\]  
where \(R^{\prime\prime\prime}\) is the LOS jerk dynamic stress [deg/s\({}^{3}\)]. Since the third-order DPLL can track up to the acceleration, the dynamic stress error of the third-order DPLL is induced by only \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \(h_{0}\) & \(2.51\times 10^{-26}\) [s] \\ \(h_{-1}\) & \(2.51\times 10^{-23}\) [s/s] \\ \(h_{-2}\) & \(2.51\times 10^{-22}\) [s/s\({}^{2}\)] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Clock parameters for oven-controlled crystal oscillator [26]. the jerk value. Notably, the dynamic stress error is inversely proportional to the natural frequency of the loop filter, so the noise bandwidth (which is proportional to the natural frequency) should be widened to reduce the dynamic stress error. The overall root mean square error (RMSE) of the DPLL measurement error presented in (1) has a trade-off with respect to the noise bandwidth. In other words, a simple selection of the noise bandwidth to a low or high value can increase the measurement error, and as a result, the performance of the DPLL may be degraded. Therefore, the optimal bandwidth is selected as a point that minimizes the RMSE as follows:  
\[B_{opt}=\underset{B_{n}\in(0,\infty)}{\text{arg min}}\sigma_{DPLL} \tag{6}\]  
Although the optimal bandwidth can be found, the resulting RMSE can exceed the rule-of-thumb tracking threshold (i.e., 30 degrees for the pilot DPLL as shown in (1)) because the operating condition can be too harsh (e.g., very low SNR, very high dynamics, low oscillator quality, etc.). In that case, the calculated bandwidth is not saved to the optimal bandwidth table because the stable operation of the DPLL cannot be ensured theoretically, and the usage of the calculated value may yield the divergence of the loop filter. The minimum RMSE determined by the calculated optimal bandwidth is expressed as follows:  
\[\sigma_{min}=\underset{B_{n}\in(0,\infty)}{\text{min}}\sigma_{DPLL}=\left. \sigma_{DPLL}\right|_{B_{n}=B_{opt}} \tag{7}\]  
Examples of the overall measurement error calculation results for the extreme cases, namely low SNR and high dynamics conditions, are illustrated in Figure 5. Each condition is selected in consideration of the analysis result obtained from Section 2 where the maximum value of the C/N\({}_{0}\) and the jerk are selected to be similar to the analysis result.