## 3 Performance Analysis of UOWC Links under Air Bubble-Induced Fading
### Bit Error Rate Analysis
Similarly, the average BER when considering large air bubbles is obtained by averaging over the PDF of large air bubbles defined in Equation (3) as follows:  
\[P_{b_{\text{large}}}=\int_{0}^{\infty}Q\Big{(}\sqrt{2\gamma}\cdot L\cdot h_{b }\Big{)}\cdot f_{h_{\text{large}}}(h_{b})\,dh_{b}. \tag{7}\]  
From Equations (3) and (7) can be formulated as a weighted sum of integrals as follows:  
\[P_{b_{\text{large}}}=\int_{0}^{\infty}Q\Big{(}\sqrt{2\gamma}\cdot L\cdot h_{b }\Big{)}\cdot w\cdot f_{h_{\text{small}}}(h_{b};a_{1},d_{1},p_{1})\,dh_{b}+ \int_{0}^{\infty}Q\Big{(}\sqrt{2\gamma}\cdot L\cdot h_{b}\Big{)}\cdot(1-w) \cdot f_{h_{\text{small}}}(h_{b};a_{2},d_{2},p_{2})\,dh_{b}. \tag{8}\]  
Thus, \(P_{b_{\text{large}}}\) can be easily derived from Equation (5) as a weighted sum of \(P_{b_{\text{small}}}\) as follows:  
\[P_{b_{\text{large}}}(L,a_{1},d_{1},p_{1},a_{2},d_{2},p_{2},w)=w\cdot P_{b_{ \text{small}}}(L,a_{1},d_{1},p_{1})+(1-w)\cdot P_{b_{\text{small}}}(L,a_{2}, d_{2},p_{2}). \tag{9}\]  
Since the obtained exact closed-form expressions could obscure the impact of PDF channel parameters on the UOWC system performance, more mathematically tractable expressions are obtained based on the asymptotic behavior at high SNR. According to ([32], Proposition 1), the asymptotic BER at high SNR can be derived with the behavior of the PDF of the considered random variable (\(h_{b}\)) near the origin. Hence, the asymptotic BER at high SNR tends to \(P_{b}\doteq(G_{c}\gamma)^{-G_{d}}\), where \(G_{c}\) is the coding gain, \(G_{d}\) is the diversity order, and \(\doteq\) is the asymptotic equality sign [32]. On a log-log scale \(G_{c}\) and \(G_{d}\) specify a relative horizontal shift and the slope of the average BER and outage curves in the asymptotic regime, respectively. Both PDFs expressed in Equations (2) and (3) can be approximated by a single polynomial term for \(h_{b}\to 0\), i.e., near the origin, as follows  
\[f_{h_{\text{small}}}(h_{b})\doteq\frac{p}{a^{d}\Gamma(d/p)}\;h_{b}^{d-1}, \tag{10}\]  
and  
\[f_{h_{\text{large}}}(h_{b})\doteq w\frac{p_{1}}{a_{1}^{d_{1}}\Gamma(d_{1}/p_{1 })}\;h_{b}^{d_{1}-1}+(1-w)\frac{p_{2}}{a_{2}^{d_{2}}\Gamma(d_{2}/p_{2})}\;h_{b }^{d_{2}-1}. \tag{11}\]  
Then, by substituting Equation (10) into Equation (5), and making use of ([30], Equation (07.34.21.0009.01)), the asymptotic closed-form expression for the average BER in the presence of small air bubbles can be solved as follows:  
\[P_{b_{\text{small}}}\doteq\frac{p\,\Gamma\Big{(}\frac{1+d}{2}\Big{)}}{2\, \sqrt{\pi}\,d(L\,a)^{d}\,\Gamma\Big{(}\frac{d}{p}\Big{)}}\gamma^{-\frac{d}{2}}. \tag{12}\]Since the \(P_{\text{h}_{\text{h}_{\text{m}}}}\) is a weighted sum of \(P_{\text{h}_{\text{m}}\text{m}}\), the asymptotic closed-form expression for the average BER can be solved as follows:  
\[P_{\text{h}_{\text{h}_{\text{m}}}}\doteq\frac{w}{2\sqrt{\pi}}\frac{p_{1}\Gamma \left(\frac{1+d_{1}}{2}\right)}{(La_{1})^{d_{1}}d_{1}\Gamma\left(\frac{d_{1}}{ p_{1}}\right)}\gamma^{-\frac{d_{1}}{2}}+\frac{1-w}{2\sqrt{\pi}}\frac{p_{2} \Gamma\left(\frac{1+d_{2}}{2}\right)}{(La_{2})^{d_{2}}d_{2}\Gamma\left(\frac{d_ {2}}{p_{2}}\right)}\gamma^{-\frac{d_{2}}{2}}. \tag{13}\]