# PTA-Sync: Packet-Train-Aided Time Synchronization for Underwater Acoustic Applications
## 3 Simulation and Results
### Simulation Results  
Figure 3 shows the time error according to the elapsed time after synchronization. In this simulation, \(V\_mean\) is 2.5 m/s and 0.5 m/s, \(I\_tx\) is 1.2 s, \(N\_ex\) is 10, \(Nr\) is 15 km, and \(\alpha\) is 0.5. As shown in Figure 3, as \(T\) increases, the time error of all synchronization protocols linearly increases, but the increase rates vary. The time errors of TSMA-IGTS and E2 DTS increase at a higher rate than that of PTS-Sync owing to the OVTT estimation method. TSMA-IGTS and E2 DTS calculate the changes in OVTT by considering the scalar speed rather than vector velocity of UN, thus resulting in errors between the actual OVTT changes and propagation OVTT change estimates from the assumption that moving directions of \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter** & **Value** \\ \hline Mean velocity of UN & [0.5, 5] m/s \\ \hline Number of messages exchanged for & [1; 30] \\ synchronization & \\ \hline Network range & [500, 30,000] m \\ \hline Clock skew & Uniform distribution within the range of 20–50 ppm. \\ \hline Degree of mobility randomness & [0, 1] in the Gauss–Markov model \\ \hline Mean direction of UN & \(\pi\) \\ \hline Data transmission rate & 100 bps \\ \hline Packet size & Varying in the range of [1; 750] bytes \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Simulation parameters. CS and UN are identical. Therefore, TSMA-IGTS and E2 DTS have increasing time errors because propagation delay estimation errors abruptly increase over time in relation to PTA-Sync. Ultimately, PTA-Sync demonstrated the highest synchronization performance in all three protocols as the time elapsed. Figure 4 shows the time error of synchronization protocols with respect to the changes in the mean velocity of UN. As _V_mean_ increases, the time error of PTA-Sync remains relatively consistent, while those of TSMA-IGTS and E2 DTS increase. PTA-Sync is not affected by the increase in velocity because the vector quantity is adequately tracked even when the velocity increases. However, TSMA-IGTS and E2 DTS have larger time errors as propagation delay fluctuation increases when the speed increases due to discrepancy between the vector velocity and scalar speed of UN. Figure 5 shows the synchronization time error with respect to the number of messages sent for synchronization when _V_mean_ = 2.5 m/s, _I_tx_ = 1.2 s, _Nr_ = 15 km, \(T\) = 30 s, and \(x\) = 0.5. The time error of all three synchronization protocols decreases as _N_ex_ increases, but the time error becomes saturated at a specific value of _N_ex_. As the number of messages sent for synchronization increases, the amount of time information also increases, thus increasing synchronization accuracy. However, if _N_ex_ increases beyond a certain point, there is a limitation in performance improvement because synchronization performance is determined mostly by the Gauss probability variables in Equations (11) and (12). PTA-Sync demonstrated 87.83% and 62.43% performance improvement compared to TSMA-IGTS and E2 DTS, respectively, when _N_ex_ = 15, _V_mean_ = 2.5 m/s, and _Nr_ = 15 km. Although the performance improvement in Figure 5b is relatively lower than that of Figure 5a, PTA-Sync still has the best performance in all three protocols. The complexity of PTA-Sync is _O(N_ex)_ which is the same as those of TSMA-IGTS and E2 DTS, because all three protocols employ one-way TS packet transmission in synchronization process. Figure 3: Time error measured at 30 s after synchronization versus \(T\) when _V_mean_ = 2.5 m/s, 0.5 m/s, _I_tx_ = 1.2 s, _N_ex_ = 30, _Nr_ = 15 km, \(x\) = 0.5. Figure 4: Time error measured at 30 s after synchronization versus _V_mean_ when _I_tx_ = 1.2 s, _N_ex_ = 30, _Nr_ = 15 km, \(T\) = 30 s, _a_=0.5. Figure 5: Time error measured at 30 s after synchronization versus _N_ex_ when _I_tx_ = 1.2 s, \(T\) = 30 s, \(a\) = 0.5.