# PTA-Sync: Packet-Train-Aided Time Synchronization for Underwater Acoustic Applications
## 3 Simulation and Results
### Simulation Results
Figure 6 shows the synchronization performance with respect to the message transmission time interval for synchronization when \(N\_ex=10\), \(V\_mean=2.5\) m/s, \(N\_tr=1.5\) km, \(T=30\) s, and \(\alpha=0.5\). As shown in the figure, the time error decreases as \(I\_tx\) increases and then becomes saturated at a specific value of \(I\_tx\). Propagation delay changes increase as \(I\_tx\) increases, while transmission time difference and reception time difference also increase. \(\Delta D\) is smaller than the increase in \(\Delta t\) and \(\Delta RT\). Consequently, as \(I\_tx\) increases, the synchronization time error becomes saturated when \(\frac{\Delta RT}{\Delta t}\) becomes a dominant factor in Equation (8). In Figure 6a, PTA-Sync improves the synchronization performance by 81.78% and 60.99% on an average in relation to TSMA-IGTS and E2 DTS, respectively. Figure 7 shows the synchronization performance according to the degree of mobility randomness when \(V\_mean=1\) m/s, 2.5 m/s, 5 m/s, \(N\_ex=10\), \(I\_tx=1.2\) s, \(N\_tr=15\) km, and \(T=30\) s. As \(\alpha\) increases, the node has more similarity to the movement pattern of the previous sub-frame and moves more smoothly. Based on the simulation assumptions, the node's movement distortion has a Gaussian distribution, with a mean of 0 and a standard deviation of \(\sqrt{(1-\alpha^{2})}\), and as \(\alpha\) increases, the UN's movement distortion decreases. Accordingly, PTA-Sync shows that as \(\alpha\) increases, the movement distortion decreases, which reduces the propagation delay estimation error, thus reducing the synchronization time error. In contrast, E2 DTS and TSMA-IGTS show that as \(\alpha\) increases, the synchronization time error increases. Because of the mismatch between the UN's vector velocity and scalar speed, as \(\alpha\) increases when the movement direction of UN and CS match, the propagation delay estimation error decreases. However, when the movement directions do not match, although there is a similarity to the movement pattern of the previous frame, the propagation delay estimation error increases. Therefore, when \(0\leq\alpha\leq 0.5\), the similarity of the movement pattern with the previous frame is low, and the influence  
Figure 6: Time error measured at 30 s after synchronization versus \(I\_tx\) when \(N\_ex=10\), \(T=30\) s, \(\alpha=0.5\). of the mobility randomness is dominant. Thus, the propagation delay estimation error is constant, regardless of \(\alpha\), and the synchronization time error has a nearly constant value. When \(0.5\leq\alpha\leq 1\), as the similarity to the previous frame's movement pattern increases, the propagation delay estimation error increases, thus increasing the synchronization time error. Figure 8 shows the time synchronization error according to the network range. The simulation results were obtained when \(Nr=500\) m-30 km, \(I\_tx=1.2\) s, \(N\_ex=10\), \(V\_mean=1\) m/s, \(2.5\) m/s, \(5\) m/s, \(T=30\) s, and \(\alpha=0.5\). As shown in the figure, the time errors of the three synchronization protocols decrease as the network range increases. This is because as the network range increases, the average initial propagation delay with error free value given in the assumption increases, whereas the propagation delay changes with error-prone values are almost constant, regardless of the network range. Therefore, when synchronization is performed, the time error decreases because the influence of uncertain propagation delay changes is reduced compared to the given initial propagation delay. PTA-Sync has the best synchronization performance among the compared protocols, showing an average performance improvement of 79.57% and 45.21% compared to TSMA-IGTS and E2 DTS, respectively, when \(V\_mean=2.5\) m/s. Figure 7: Time error measured at 30 s after synchronization versus \(\alpha\) when \(N\_ex=10\), \(I\_tx=1.2\) s, \(Nr=15\) km, \(T=30\) s.