## 3 Results and Discussions
### Temporal Dispersion  
To evaluate the time dispersive properties of the UOWC channel, we should first derive the CIR after performing the PTA simulation. Since the statistical results such as the weight and corresponding arrival time of each detected photon have been recorded, we can calculate the summation of the weights within a specific time slot (i.e., 0.2 ns), normalize it by the total emitting photons weights and then plot the CIR curve. Thus, for the \(j\)th time slot, this process is described as [32]:  
\[\rho_{j}=\frac{1}{N_{T}W_{0}}\sum_{i}W_{i}\int_{(j-1)T_{r}}^{|T_{r}}\delta(t-T_{ i})dt \tag{11}\]  
where \(W_{i}\) is the weight of the \(i\)th detected photon corresponding to the arrival time \(T_{i}\), \(T_{r}\) is the time resolution, \(N_{T}\) represents the total number of transmitted photons, and the latter integral is expressed as \(\int_{(j-1)T_{r}}^{|T_{r}}\delta(t-T_{i})dt=\left\{ \begin{array}{cc}1&for\ T_{i}\in[(j-1)T_{r},|T_{r}] \\ 0&for\ T_{i}\
otin[(j-1)T_{r},|T_{r}]\end{array} \right..\)  
Afterwards, the temporal dispersion effect can be quantified by computing the root mean square (RMS) delay spread \(t_{DS}\) given by:  
\[\left\{ \begin{array}{c}t_{DS}=\sqrt{\int_{-\infty}^{\infty}(t-t_{0})^{2}h^{ 2}(t)dt/\int_{-\infty}^{\infty}h^{2}(t)dt}\\ t_{0}=\int_{-\infty}^{\infty}th^{2}(t)dt/\int_{-\infty}^{\infty}h^{2}(t)dt \end{array} \right. \tag{12}\]  
in which \(t_{0}\) is the mean excess delay spread. Figure 5 depicts the CIR for four different water types with different system configurations (i.e., FOV). The curve smoothness is associated with the given time resolution \(T_{r}\) and the total number of simulated photons \(N_{T}\). We consider the Rx with an aperture of 50 cm and Tx with a divergence angle of \(15^{\circ}\). It can be clearly observed that the intensity drops significantly from the peak in 100 m pure sea water and there is almost no \"trailing\" phenomenon on the curve. The trajectory of photons is nearly a straight line to the Rx. As the water turbidity increases, the delay spread tends to be more severe with the fact that the 9 m link in harbor water exhibits a worse channel quality than the longer 29 m coastal link or even the 50 m clear ocean link. This is because a more turbid water medium with larger \(c(\lambda)\) leads to a greater number of scattering events, where \(\Delta s\) is smaller and \(E_{n+1}\) drops below the threshold faster. Therefore, the time delay of arrival photons increases resulting in serious temporal dispersion. At the same time, it can be intuitively found that the delay spread tends to be larger with the increase of FOV. This result is also easy to explain considering the fact that Rx with wider FOV is more likely to capture those photons which have deviated from the optical axis after multiple scattering events. To be more persuasive, the RMS delay spread \(t_{DS}\) is also calculated using (12) to characterize the channel dispersion. We obtain \(t_{DS}\) = (0.74 ns, 1.16 ns, 2.73 ns) and (1.21 ns, 1.44 ns, 2.76 ns) corresponding to FOV = (\(15^{\circ}\), \(35^{\circ}\), \(180^{\circ}\)) for coastal and harbor water, respectively. It is easy to see that the \(t_{DS}\) rises with the increase of FOV and water turbidity, which verifies the aforementioned analysis. Additionally, since the channel dispersion can bring inter-symbol interference (ISI) and limit the system performance, we can also estimate the maximum transmitted data rate through the optical wireless channel without the requirement of equalization by using \(R_{b}\leq 1/10t_{DS}\). Furthermore, our simulation platform provides another option for CIR plots to make the results more intuitive, namely histogram displays. In this way, the time difference between the first and last arrival photons is divided into 100 bins, then the histogram of intensity versus time (Figure 6) can be easily obtained by calculating the normalized weights within each bin similarly as elaborated before.