## 4 Performance Analysis
### Uncoded OOK Scheme  
Since the most basic form of pulsed modulation in digital communications is OOK, we propose its analysis as a typical example of a coding technique for which its CBER can be analytically derived. The firstly calculated CBER for a given electrical signal-to-noise ratio in absence of optical turbulence, \(\gamma_{0}\), represents the previous step for obtaining the final ABER featuring the system. From [22], the CBER of IM/DD with AWGN channel using OOK is expressed as follows:  
\[P_{b}(e|h)=\frac{1}{2}\text{erfc}\bigg{(}\frac{i_{S_{0}}h}{2\sqrt{2(\sigma_{n} ^{2}+\sigma_{s}^{2})}}\bigg{)}, \tag{10}\]  
with \(i_{S0}=attRP_{t}\) denoting the signal current in the absence of turbulence-induced fading, with \(att\), \(R\) and \(P_{t}\) representing the attenuation coefficient associated with the medium (not considered in this paper), the responsivity and the average of transmitted optical power, respectively. In (10), \(\sigma_{s}^{2}\) represents the variance associated with the ISI affecting the system in the optical underwater channel, as commented in Section 2. There, it was described how the ISI term is affected by \(h_{2}\), \(\sigma_{s}^{2}(h_{2})\), since it results from multipath interference and its inherent waveform time dispersion. Following [63; 64], we have assumed in (10) that the ISI interference is Gaussian distributed. Next, we define the signal-to-interference-plus-noise ratio (SINR) as follows:  
\[\gamma_{0}=\frac{i_{S0}}{\sqrt{\sigma_{n}^{2}+\sigma_{s}^{2}(h_{2})}}, \tag{11}\]  
where \(\sigma_{s}^{2}(h2)\) represents the dependency of \(\sigma_{s}^{2}\) on \(h_{2}\). Now, to solve the integral involving ABER, calculated by averaging \(P_{b}(e|h)\) over the PDF of the irradiance, \(f_{h}(h)\), the SINR in (11) can be approximated by averaging the noises and the inter-symbol interference over oceanic turbulence in the way proposed in [64]. \[\gamma_{0}\approx\frac{i_{S0}}{\sqrt{<\sigma_{n}^{2}>+<\sigma_{s}^{2}(h_{2})> }}. \tag{12}\]In (12), \(<\cdot>\) denotes the average over turbulence. Thus, and for the sake of consistency with notation in Equation (9), we can identify \(\gamma=\gamma_{0}h\), with \(h\) representing the random normalized irradiance fluctuation. Therefore, the ABER, \(P_{b}(e)\), is calculated by averaging \(P_{b}(e|h)\) over the PDF of the irradiance, \(f_{h}(h)\). Hence, we have the following. \[P_{b}=\int_{0}^{\infty}\frac{1}{2}\text{erfc}\left(\frac{\gamma_{0}h}{2\sqrt{2 }}\right)f_{h}(h)\text{d}h. \tag{13}\]  
In (13), the PDF of the optical irradiance is defined according to the combined Malaga-Weibull model, as detailed in the previous section. Following [65], the ABER, \(P_{b}(e)\), is obtained by averaging CBER over the CDF of \(h\), \(F_{h}(h)\), by using the integration by parts with the following formula. \[P_{b}=\left(P_{b}(e|h)F_{h}(h)\right)\Bigg{|}_{0}^{\infty}-\int_{0}^{\infty} \frac{\text{d}}{\text{d}h}\left[P_{b}(e|h)\right]F_{h}(h)\text{d}h. \tag{14}\]  
Since \(P_{b}(e|\infty)=0\) and \(F_{h}(0)=0\) (note that negative values for the optical irradiance are not allowed), then the latter expression can be reduced to the following:  
\[P_{b}=-\int_{0}^{\infty}\frac{\text{d}}{\text{d}h}\left[P_{b}(e|h)\right]F_{ h}(h)\text{d}h=-\int_{0}^{\infty}\frac{\text{d}}{\text{d}h}\left[\frac{1}{2} \text{erfc}\left(\frac{\gamma_{0}\cdot h}{2\sqrt{2}}\right)\right]F_{h}(h) \text{d}h, \tag{15}\]  
where \(F_{h}(h)\) is directly obtained in (9). Now, we apply [66] (Eq. (06.27.13.0005.01)) to derive an expression for the derivative of \(P_{b}(e|h)\) with respect to \(h\). \[\frac{\text{d}}{\text{d}h}[P_{b}(e|h)]=-\frac{\gamma_{0}}{2\sqrt{2\pi}}\exp \left[-\left(\frac{\gamma_{0}h}{2\sqrt{2}}\right)^{2}\right]. \tag{16}\]  
Next, we introduce (16) in (15) in order to solve the resulting integral.