## 4 _Bt_ Lower Limit
### Lower Limit  
Since the \(\sigma_{\text{t}}\) in (11) is a function of the \(C\)/\(N_{0}\), the resulting \(\sigma_{\text{DPLL}}\) in (14) varies with the given \(C\)/\(N_{0}\). Figure 7 illustrates the \(\sigma_{\text{DPLL}}\) curves for the \(C\)/\(N_{0}\) for various \(B\) and \(T\) values, as an example. Obviously, the measurement error increases as the \(C\)/\(N_{0}\) decreases, and eventually crosses the measurement error threshold line (i.e., horizontal line at 15 deg). As the DPLL cannot operate above the threshold line, the \(C\)/\(N_{0}\) at the crossing point indicates the minimum \(C\)/\(N_{0}\) that the specific DPLL can track, which can be defined as a \(C\)/\(N_{0}\) threshold for the DPLL as:  
\[C/N_{0,\text{thresh}}=\underset{C/N_{0}\in\mathbb{R}}{\arg\min}|\sigma_{\text{ DPLL}}-15| \tag{15}\]  
The \(C\)/\(N_{0}\) thresholds of third-order DPLL for each measurement error parameter are calculated to form the \(C\)/\(N_{0}\) threshold curves presented in Figure 8. As the \(B\) becomes narrower, the \(C\)/\(N_{0}\) threshold declines smoothly, and at some point, it rapidly rises. Such  
Figure 7: Third-order DPLL measurement error curves for the \(C\)/\(N_{0}\) for various \(B\) and \(T\) (jerk stress of 1 g/s and OCXO assumed).  
vertical lines represent the lower limits of the \(B\), which means DPLLs that have \(B\)s smaller than this limit have insufficient gain to work well in that condition. Note that, as can be observed from the figure, the lower limit is dependent on the \(B\), not \(BT\), and \(T\) has a negligible impact on the lower limits. Therefore, the lower limits with respect to the \(B\) are induced first and the \(T\) is multiplied later to form the \(BT\) that is the main interest of this paper. The \(BT\) lower limit is represented as follows:  
\[\begin{array}{ll}BT_{\text{low}}&=T\times B_{\text{min}}\\ &=T\times\underset{B\in(0,\infty)}{\arg\max}\Big{|}\frac{\partial}{\partial B }(C/N_{0,\text{thresh}})\Big{|}\end{array} \tag{16}\]  
where \(B_{\text{min}}\) is the minimum \(B\) [Hz] that the DPLL can operate at specific environmental conditions. In the case of no dynamic stress (0 g/s case), the \(B\) lower limits of TCXO and OCXO deviate from each other because all the phase noise observed as a dynamic component at the receiver is dominated by the oscillator phase noise. Clearly, the \(B\) limit of the OCXO is smaller than TCXO, which verifies that the OCXO has better clock performance than TCXO. One of the reasons why the high-quality oscillator (e.g., OCXO) is preferred for the application of weak signal processing can be inferred from the figure, as the OCXO can provide tracking capability down to approximately 15 dB-Hz of \(C/N_{0}\) even for \(T=20\) ms. When the dynamic stress exists, the two oscillators have similar effects on the lower limit, and as the dynamics get stronger, the limits grow because the required gains increase as well.  
The obtained \(BT\) lower limits of third-order DPLL are arranged in Table 7. Figure 9 shows the \(BT\) lower limits for various jerk dynamic stresses visually. It can be observed that shorter \(T\) is beneficial in the dynamic environment since the \(T=1\) ms have nearly the same \(BT\) limits for all jerk values. Additionally, OCXO has more advantages than TCXO when a longer \(T\) is used, as the difference between the \(BT\) limits of the oscillators becomes larger for the longer \(T\).  
Figure 8: \(C/N_{0}\) threshold curves for \(B\) of third-order DPLL using TCXO and OCXO.