## 4 Comparison between Theoretical and Simulation Results of \(C_{2\varphi}\)
### Lock Probability Analysis of FLD Output  
The output distribution of FLD is mainly related with the \(C/N_{0}\) of the received signal. With a given \(C/N_{0}=\lambda\), the locked threshold of the tracking loop can be determined by the false alarm probability \(P_{FA}\) and the lock probability \(P_{D}\). \[P_{FA}=P\{C_{2\varphi}>Th|noise\ input\ only\ \}=\int\limits_{Th}^{1}\frac{1}{ \pi\sqrt{(1+c_{2\varphi})(1-c_{2\varphi})}}dc_{2\varphi} \tag{25}\]  
\[ \begin{array}{ll}P_{D}&=P\{C_{2\varphi}>Th|C/N_{0}=\lambda\}\\ &=P\{Y=\frac{M(1-C_{2\varphi})}{2\varphi^{2}}<\frac{M(1-Th)}{2\varphi^{2}}|C/N_ {0}=\lambda\}\\ &=F_{Y}(\frac{M(1-Th)}{2\varphi^{2}}|C/N_{0}=\lambda)\end{array}  \tag{26}\]  
Figure 1: PDF of theoretical and simulation under high \(C/N_{0}\) (\(T=1\) ms, \(M=20\)). Figure 2: Probability density function (PDF) of theoretical and simulation under high \(C/N_{0}\) (\(T=1\) ms, \(M=1\)). to indicate the average time required that tracking loop detect state changed from false alarm to correct detection under no signal condition. MTLL (\(E\{T_{D}\}\)) will imply that how long tracking loop change the signal detection state to missing detection state when the signal presented. Over a total time TT, MTLL (\(E\{T_{F}\}\)) under false alarm \(P_{FA}\) can be obtained [14] as:  
\[ \begin{array}{ll}E\{T_{F}\}|_{TT->\infty}&=\sum\limits_{N=1}^{TT}NP_{FA}^{N}( 1-P_{FA})|_{TT->\infty}\\ &=\frac{1}{1-P_{FA}}\end{array}  \tag{27}\]  
where the false alarm probability \(P_{FA}\) with a certain threshold (\(M=1\)) is shown in (25), \(N\) is time slot. MTLL (\(E\{T_{F}\}\)) indicates average time that frequency locked loop used to detect no signal state. Based on (25) and (27), we can get the relationship between MTLL (\(E\{T_{F}\}\)) and lock threshold. As shown in Figure 7, it indicates that relation between MTLL (\(E\{T_{F}\}\)) and threshold under no signal situation. The continuous tracking time is gradually decreased when increasing \(Th\). MTLL (\(E\{T_{F}\}\)) is almost same when \(Th\) above zero. The MTLL (\(E\{T_{F}\}\)) of false alarm in FLD shown in Figure 7 is similar with that in PLD in [14]. Figure 5: Loop lock probability under different \(C/N_{0}\) with \(M=1\). Figure 6: Relation between FLD false alarm probability and threshold (\(Th\)). When the signals present, the MTLL (\(E\{T_{D}\}\)) can be obtained [14] as:  
\[E\{T_{D}\}|_{TT->\infty}=\frac{1}{1-P_{D}} \tag{28}\]  
where the lock probability \(P_{D}\) with a certain threshold \(Th\) is shown in (26). MTLL (\(E\{T_{D}\}\)) indicates the average time that the frequency locked loop takes to maintain signal detection state. Based on (26) and (28), we can get the relationship between MTLL (\(E\{T_{D}\}\)) and \(C/N_{0}\) with a certain lock threshold. The relationship curves are shown in Figures 8 and 9 with \(M=1\) and \(M=20\), the continuous tracking time MTLL (\(E\{T_{D}\}\)) gradually increases when increasing \(C/N_{0}\) or coherent integration time. At the same time, the verification of MTLL is conducted with the help of Monte Carlo simulation in Figures 10 and 11. The theoretical results show that the MTLL (\(E\{T_{F}\}\)) is only 2 s when the signal does not present and the threshold \(Th\) is equal to 0 in Figure 7. Oversized noise can cause false alarm that leads to error lock in frequency locked loop. We can judge the error lock state in is 2 s. The MTLL (\(E\{T_{D}\}\)) is about 40 s when the integration time is 1 ms under signal with 40 dB-Hz \(C/N_{0}\), and 10 ms for the signal with 30 dB-Hz \(C/N_{0}\) that threshold \(Th\) is equal to 0 (\(M=1\)) in Figure 8 when the signal present. The comparison with PLD [14] is shown in Figure 12. The threshold setting of PLD at 0.7 has a similar curve to the threshold setting of FLD at 0.4, while PLD at 0.6 is similar to the FLD at 0.2. It means  
Figure 8: Relation between mean-time-to-lose-lock (MTLL) (\(E\{T_{D}\}\)) and \(C/N_{0}\) (\(M=1\)) under signal presented situation. Figure 7: Relation between MTLL (\(E\{T_{F}\}\)) and threshold \(Th\) under no signal presented situation.