# Preamble-Based Noncoherent Synchronization in Molecular Communication: A Machine Learning Approach
## 3 Proposed Preamble-Based Synchronization Scheme
### Proposed Synchronization Scheme  
To detect the preamble signal at the receiving end, it is essential to estimate the start time of the pulse. Figure 3 illustrates an example of the proposed synchronization scheme, with \(N_{p}=4\). Let us assume that the preamble is transmitted two times. The actual reception point of the preamble is \(t_{r,0}\). Now, at the receiver, the synchronization method utilizes long collection windows (LCWs) and small collection windows (SCWs) to detect the preamble signal. The transmitted preamble concentration is captured in several attempts. In the first attempt, we will compute the concentrations of the pulse in two SCWs, namely \(S_{1}\) and \(S_{2}\), and add them to obtain an output of first LCW, namely, \(Z_{1}\). In the second attempt, the SCWs are moved by the width of one SCW, and the concentrations are again captured in \(S_{2}\) and \(S_{3}\), which results in \(Z_{2}\). In this fashion, by summing up the SCWs equivalent to the  
Figure 2: Pulse-based preamble signal structure [18].  
total length of the preamble, the LCW is derived. The above schemes represents a sliding window approach throughout the SCWs. More generally, we can write it as [18]  
\[Z_{i}=\sum_{k=i}^{i+L-1}S_{k}, \tag{7}\]  
where \(L\) represents the total number of SCWs in a LCW, \(S_{i}\) is the concentration in the \(i\)th SCW, and \(Z_{i}\) is the output of the \(i\)th LCW. The number of SCWs, i.e., \(L\), is defined as [18]  
\[L=\left(\frac{T_{s}}{T_{tx}}\right)N_{int}, \tag{8}\]  
where \(N_{int}\) describes the total number of LCWs (attempts to collect the whole preamble concentration). Thus, each SCW has a width of \(T_{tx}/N_{int}\). Increasing the size of the SCW makes the preamble signal easier to detect, thereby increasing synchronization probability. In cases where errors occur, the actual signal may exhibit a greater time difference, resulting in an increased mean square error. Given this consideration, it is essential to set an appropriate value for the number of SCW.  
After collecting the concentrations in various LCWs, the value is compared and the time instant of the LCW with the maximum concentration is chosen as the estimated synchronization time. In general terms, the estimated synchronization time instant for the \(k\)th LCW is given as [18]  
\[t_{s,k}=t_{s,1}+(k-1)T_{tx}/N_{int}, \tag{9}\]  
where \(t_{s,1}\) is the starting synchronization point for first LCW. The output of the \(k\)th SCW \((S_{k})\) is given by [18]  
\[S_{k}=\sum_{i=t_{s,1}+(k-1)T_{tx}/N_{int}}^{t_{s,1}+kT_{tx}/N_{int}}U(t_{i}), \tag{10}\]  
where \(t_{i}\) represents in the discrete-time points at which \(U(t)\) is sampled. Then, the final estimated synchronization point is given as [18]  
\[\hat{t}_{sym}=t_{s,1}+(i^{*}-1)T_{tx}/N_{int}, \tag{11}\]  
where  
\[i^{*}=\max_{i}(Z_{i}). \tag{12}\]  
Note that the proposed approach can achieve synchronization within the ambiguity of \(T_{tx}/N_{int}\), with a bank of \(N_{int}\) accumulators, as follows [18]:  
\[\hat{t}_{\text{syn}}\in\left[t_{\text{syn}}-\frac{T_{tx}}{2N_{int}},t_{\text{ syn}}+\frac{T_{tx}}{2N_{int}}\right], \tag{13}\]  
where \(t_{\text{syn}}\) denotes the ideal synchronization point. Therefore, we can see that the time accuracy of synchronization depends upon the number of accumulators. In other words, a higher number of accumulators will increase the probability of synchronization.