# PTA-Sync: Packet-Train-Aided Time Synchronization for Underwater Acoustic Applications
## 2 Synchronization Protocol for UANets
### PTA-Sync Protocol  
Assumption and Overview of PTA-Sync  
We make the following assumptions:  
1. UN contains the initial position information and initial propagation delay information. 2. UN acquires the position and speed from sensors, which include an altitude heading reference system (AHRS) and a Doppler velocity log (DVL). Table 1 defines the parameters related to the PTA-Sync results. Figure 2 shows the parameters applied to PTA-Sync of CS and UN, and elucidates the parameters defined in Table 1. A beacon broadcast by CS comprises \(N\) number of sub \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Description** \\ \hline \(N\) & Total number of sub-frame beacons \\ \(B_{i}\) & \(j\)-th sub-frame beacon \\ \(t(j)\) & Time at which CS transmitted \(B_{j}\) \\ \(RT(j)\) & Time at which UN received \(B_{j}\) \\ \(D(j)\) & One-way travel time (OWT) for \(B_{j}\) transmitted by CS to be received by UN \\ \(P_{CS}(j)\) & 3D position vector when CS transmits \(B_{j}\) \\ \(P_{UN}(j)\) & 3D position vector when UN receives \(B_{j}\) \\ \(V_{UN}(j)\) & 3D velocity vector when UN receives \(B_{j}\) \\ \(\Delta D(j)\) & Changes in OWTT of UN when \(B_{j-1}\) and \(B_{j}\) are transmitted \\ \(\tilde{a}(j)\) & Estimated clock skew of UN when \(B_{j-1}\) and \(B_{j}\) sare transmitted \\ \(\overline{\tilde{a}}\) & Mean of \(\tilde{a}\) \\ \(\tilde{b}\) & Estimated clock offset of UN \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter description of PTA-Sync. Figure 1: Target network model. frames, and is transmitted in the form of a packet train. Each sub-frame beacon contains the position information of CS and time stamp transmission time \(t(j)\). Description of PTA-Sync  
The relationship between the reference time of CS and local time of UN can be expressed using the following linear function:  
\[RT(j)=a\cdot(t(j)+D(j))+b, \tag{1}\]  
where \(a\) is the clock skew of UN, and \(b\) is the clock offset of UN. The transmission time difference between \(B_{j-1}\) and \(B_{j}\) in CS, \(\Delta t(j)\), can be expressed as follows:  
\[\Delta t(j)=t(j)-t(j-1). \tag{2}\]  
The reception time difference between \(B_{j-1}\) and \(B_{j}\) in UN, \(\Delta RT(j)\), can be expressed as follows:  
\[\Delta RT(j)=RT(j)-RT(j-1). \tag{3}\]  
Equation (4) can be expressed as follows using Equations (1)-(3):  
\[\Delta RT(j)=a\cdot(\Delta t(j)+\Delta D(j)), \tag{4}\]  
where \(\Delta D(j)\) is defined as follows:  
\[\Delta D(j)=D(j)-D(j-1). \tag{5}\]  
To estimate \(D(j)\), \(\mathbf{P}_{\textit{UN}}(j)\) must be determined. When UN receives \(B_{j}\), the changes in the position of UN are calculated using the speed and direction information at the reception point of \(B_{j-1}\) to calculate the propagation delay changes during \(\Delta t(j)\). \[\mathbf{P}_{\textit{UN}}(j)=\mathbf{P}_{\textit{UN}}(j-1)+\mathbf{V}_{\textit{UN}}(j-1) \cdot\Delta t(j); \tag{6}\]  
note that \(\mathbf{P}_{\textit{UN}}\) and \(\mathbf{V}_{\textit{UN}}\) are three-dimensional vectors. Figure 2: Illustration of parameters for PTA-Sync. UN estimates the propagation delay between CS and UN using the information \(P_{CS}(j)\) obtained by receiving \(P_{UN}(j)\) and \(B_{j}\), and can be expressed as follows:  
\[\hat{D}(j)=|\mathbf{P}_{CS}(j)-\mathbf{P}_{UN}(j)|/c, \tag{7}\]  
where \(c\) is the underwater acoustic speed. Using Equations (4), (5) and (7), the clock skew of UN, \(\hat{a}(j)\), can be estimated while \(B_{j-1}\) and \(B_{j}\) are being transmitted.