## 3 UWOC System Model
### Digital Signal Evaluation Cycle from Optical to Electronic--A Mathematical Viewpoint  
The digital signals evaluation cycle is based on the illustration in Figure 4. The main components of a typical optical receiver system (Rx), as explained in Section 3.2, include a \begin{table}
\begin{tabular}{c c c} \hline \hline
**Model Name** & **The Equation of the Model** & **Ref.** \\ \hline \multirow{4}{*}{DGF} & The closed-form expression of the double gamma & \\  & functions (DXFs) is given as follows: & \\  & \(\mathrm{h}(t)=c_{1}\Delta te^{-c_{2}\Delta t}+c_{3}\Delta te^{-c_{4}\Delta t}\), & \(t\geq t_{0}\), & [74] \\  & \(\Delta t=(t-t_{0})\) & \(t_{0}=\frac{t}{\sharp}\) & \\ \hline \multirow{4}{*}{WDGF} & The weighted double gamma functions (WDGFs) model is & \\  & given as follows: & \\  & \(\mathrm{h}(t)=c_{1}\Delta t^{\delta}e^{-c_{2}\Delta t}+c_{3}\Delta t^{\beta}e^{ -c_{4}\Delta t}\), & \(t\geq t_{0}\), & [75] \\  & \(\Delta t=(t-t_{0})\) & & \\ \hline \multirow{4}{*}{CEAPE} & A combination of exponential and arbitrary power & \\  & functions (CEAPEs) is given as follows: & \\  & \(\mathrm{h}(t)=\frac{c_{1}\cdot\Delta t^{\alpha}}{(t+c_{2})\cdot\Delta t}\cdot e^{- a\cdot\beta(\Delta t+t_{0})}\) & [76] \\  & \(c_{1}>0,c_{2}>0,\Delta\cdot\infty\ -1\) _and_ \(\beta>0\) & \\ \hline \multirow{3}{*}{BP} & Beta Prime (BP) distribution is given as follows: & \\  & \(\mathrm{h}_{BP}(t)=\frac{\Gamma(\beta_{1}+\beta_{2})}{\Gamma(\beta_{1})\Gamma( \beta_{2})}\cdot\frac{t^{\beta_{1}-1}}{(1+t)^{\beta_{1}+\beta_{2}}}\), & \(t>0\) & [77] \\ \hline \hline \end{tabular} \(C_{1}\), \(C_{2}\), \(C_{3}\), \(C_{4}\), \(\alpha\), \(\beta\), \(\beta_{1}\), and \(\beta_{2}\) are double gamma curve-fitting parameters. \(v\) is the light velocity for the seawater medium under consideration. L is the linkspan distance between the Tx and Rx.  
\end{table} 
Table 1: List of models’ channel impulse response functions. photodetector, preamplifier/amplifier, filter/equaliser, and decision unit (DU). The received optical signal \(r_{opt}(t)\) can be described as follows:  
In a typical optical digital communication system, the transmitted optical signal \(s(t)\) can be represented as follows [5]:  
\[s(t)=\sum_{k=-\infty}^{\infty}a_{k}\ h_{p}(t-kT) \tag{1}\]  
where \(T\) is the signalling period, for a binary (OOK) signal format, if \(\tau\) is the timespan of each bit within a symbol, then T = \(\tau\), so 1/T is the bit rate. \(a_{k}\) is the energy received in the k\({}^{\text{th}}\) symbol; for binary system \(a_{k}\in\{0,\,1\}\), the \(h_{p}(t)\) is the transmitted optical pulse. Typically, the \(s(t)\) experiences temporal and spatial distortions while propagating through a medium channel (air, fibre optic, or water), depending on the profile of the propagation channel impulse response \(h_{c}(t)\). The received optical signal \(r_{opt}(t)\) is the footprint of the convolutional impact of \(h_{c}(t)\) on the \(s(t)\). Hence, \(r_{opt}(t)\) can be expressed as follows:  
\[r_{opt}(t)=s(t)\otimes h_{c}(t) \tag{2a}\] \[r_{opt}(t)=a_{0}h_{p}(t)\otimes h_{c}(t)+\sum_{ \begin{array}{c}k=-\infty\\ k\
eq 0\end{array} }^{\infty}a_{k}\ h_{p}(t-kT)\otimes h_{c}(t) \tag{2b}\]  
where \(\otimes\) denotes the convolution operation. The \(r_{opt}(t)\) is the received optical signal. In this study, we consider a binary direct detection Rx, as depicted in Figure 4. Without losing generality, we will assume that the Rx includes a PIN photodetector with an internal gain (\(g\)) equal to one. The photodetector converts the input photons of \(r_{opt}(t)\) into electronic signal \(r_{sig}(t)=\), which can be expressed as follows:  
\[r_{sig}(t)=\sum_{j=1}^{N(t)}\ h_{d}(t-t_{j})\] (3a) where \[\{t_{j}\}\] denotes the photoelectron emission times.