# Unified Performance Analysis of Direct Air-to-Underwater RF/UWOC Systems
## 2 System and Channel Models  
As shown in Figure 1, we considered an RF/UWOC system in which airborne unmanned aerial vehicles (UAVs) communicate directly with underwater submarines without a relay connection. Under foggy conditions, signals emitted by UAVs experiencing atmospheric turbulence and pointing errors can lead to channel fading. When passing through the air-water interface, assuming only the influence of water waves is considered and there is no bubble effect on the transmission interface, the signal is directly transmitted to the underwater submarine for communication through the air-water interface. The signal received by the underwater detector is  
\[s(t)=h_{a}h_{f}h_{p}h_{w}It+n \tag{1}\]  
where \(t\) is the transmission signal, \(h_{1}\) is the channel coefficient of atmospheric turbulence, and the randomness of the path gain of the \(h_{2}\) model due to fog conditions. \(h_{p}\) represents the channel gain of the pointing error. \(h_{w}\) simulates the scattering and reflection of water waves at the air-water interface and their effects on the signal. \(I\) represents the path loss of ocean turbulence. \(n\) represents noise with detector \(\sigma^{2}\). Figure 1: System model of RF/UWOC. We have considered providing a unified PDF for the signal-to-noise ratio \(r\) under atmospheric turbulence effects [30]. \[f_{r}(r)=k_{0}H_{p,q}^{m,n}\left[\lambda r\right]_{\left( \begin{subarray}{c}(a_{ r},A_{i})_{i-1}^{p}\\ (b_{i},B_{i})_{i-1}^{q}\end{subarray} \right]} \tag{2}\]  
The parameters in Equation (2) define a specific atmospheric turbulence model, as shown in Table 1.  
The relationship between the signal-to-noise ratio and the channel gain can be defined as  
\[r=\overline{r}h^{2} \tag{3}\]  
where \(\overline{r}\) represents the average signal-to-noise ratio, and \(h\) represents the product of \(h_{a}h_{f}h_{p}h_{w}I\). The PDF of the channel gain under atmospheric turbulence can be expressed as  
\[f_{h_{a}}(h_{a})=2\overline{r}h_{a}H_{p,q}^{m,n}\left[\lambda\overline{r}h_{a} ^{2}\right]_{\left( \begin{subarray}{c}(a_{r},A_{i})_{i-1}^{p}\\ (b_{i},B_{i})_{i-1}^{q}\end{subarray} \right]} \tag{4}\]  
The PDF of the channel gain of \(h_{f}\) under foggy conditions [20] is shown below:  
\[f_{h_{f}}(h_{f})=\frac{v^{k}}{\Gamma(k)}\left[\ln\left(\frac{1}{h_{f}}\right) \right]^{k-1}h_{f}^{p-1} \tag{5}\]  
where \(k\) is the shape parameter \(0<k<1\), \(v=4.343/\mathrm{d}\beta^{\mathrm{fog}}\), \(\beta^{\mathrm{fog}}\) are the scale parameter. If \(k\) and \(v\) are parameters of the fog channel, let \(h_{3}=h_{a}h_{f}\); afterward, Equations (4) and (5) are taken into Equation (6), and the joint PDF under the influence of atmospheric turbulence combined with fog conditions can be expressed as  
\[f_{h_{3}}(h_{3})=\int_{h_{3}}^{\infty}\frac{1}{h_{a}}f_{h_{a}}(h_{a})f_{h_{f}} \left(\frac{h_{3}}{h_{a}}\right)d_{h_{a}} \tag{6}\]  
And expand the H-fox function to obtain the result. \[f_{h_{3}}(h_{3})=2\overline{r}v^{k}h_{3}H_{p+k,q+k}^{m+k,n}\left[\lambda \overline{r}h_{3}^{2}\right]_{\left( \begin{subarray}{c}(a_{r},A_{i})_{i-1}^{ p}(v-1,2)_{i-1}^{k}\\ (b_{i},B_{i})_{i-1}^{q}\end{subarray} \right]} \tag{7}\]  
The expression for PDF with a pointing error of nonzero mean single-sided is as follows [31]. \[ \begin{array}{rl}f(h_{p})=&\frac{e^{2}}{A_{0}^{\infty}}h_{p}^{2}\sqrt{\frac {2\mu}{w_{\mathrm{a}t}}\sqrt{2\ln\frac{\lambda_{0}}{k_{p}}}}\exp(-\frac{2\mu^{ 2}e^{2}}{w_{\mathrm{a}t}^{2}})\\ &\times\mathrm{I}_{-\frac{1}{2}}(\frac{2\mu e^{2}}{w_{\mathrm{a}t}}\sqrt{2\ln \frac{\lambda_{0}}{k_{p}}})\end{array}  \tag{8}\]  
After a series of simplifications, the following results can be obtained. \[ \begin{array}{rl}f(H)=&B\sum\limits_{i=0}^{1}\sum\limits_{k=0}^{\infty}\frac {H(-1)^{k}\mu\mathrm{r}\Gamma(\frac{k+1}{2})}{k(k+1)}\\ &\times H_{p+k+1,q+k+1}^{m+k+1,n}\left[\frac{4\mu H_{p}^{2}}{A_{0}^{2}}\right] _{\mu_{i}(e^{2}-2,1)}\end{array}