## 3 Derivation of the Lower Capacity Bounds for UWOC with APD
Since the conditional PDF of \(Y_{1}\) on \(x\) is given by  
\[W(Y|x)=\frac{1}{\sqrt{2\pi\sigma^{2}(1+Hx\varsigma^{2})}}e^{-\frac{(y-Hx)^{2 }}{2\sigma^{2}(1+Hx\varsigma^{2})}}. \tag{15}\]0  
and if we choose \(Q_{Exp}(\cdot)\) as the input distribution to our reduced channel \(Y_{1}\), the corresponding output distribution is  
\[W(Y|x)=\frac{1}{\sqrt{2\pi\sigma^{2}(1+Hx\varsigma^{2})}}e^{-\frac{(y-Hx)^{2 }}{2\sigma^{2}(1+Hx\varsigma^{2})}}. \tag{15}\]1  
By using the data processing theorem for relative entropy [15], we obtain the following inequality  
\[W(Y|x)=\frac{1}{\sqrt{2\pi\sigma^{2}(1+Hx\varsigma^{2})}}e^{-\frac{(y-Hx)^{2 }}{2\sigma^{2}(1+Hx\varsigma^{2})}}. \tag{15}\]2where  
\[W(Y|x)=\frac{1}{\sqrt{2\pi\sigma^{2}(1+Hx\varsigma^{2})}}e^{-\frac{(y-Hx)^{2 }}{2\sigma^{2}(1+Hx\varsigma^{2})}}. \tag{15}\]3  
is a function which only relates to parameters \(\varepsilon\) and \(H\), and has noting to do with input \(x\). The detailed derivations of (22) and (23) can be found in Appendix A. Substituting (23), (24), and (17) into (16), we have  
\[W(Y|x)=\frac{1}{\sqrt{2\pi\sigma^{2}(1+Hx\varsigma^{2})}}e^{-\frac{(y-Hx)^{2 }}{2\sigma^{2}(1+Hx\varsigma^{2})}}. \tag{15}\]4  
\[W(Y|x)=\frac{1}{\sqrt{2\pi\sigma^{2}(1+Hx\varsigma^{2})}}e^{-\frac{(y-Hx)^{2 }}{2\sigma^{2}(1+Hx\varsigma^{2})}}. \tag{15}\]5  
Till now, the lower bound of channel capacity about UWOC with APD detector can be calculated by choosing an input distribution \(Q(\cdot)\) which maximizes the entropy \(h(x)\) under the constraints (10), (11), and (12). This can be solved by applying the Lagrange multipliers approach. In addition, we consider the following two different cases of \(\alpha\) value range:  
1. \(\alpha\in\Big{(}0,\frac{1}{3}\Big{)}\). Both the average and peak power constraints are imposed in this case, then the input distribution \(Q(\cdot)\) is expressed as \[Q(x)\triangleq\left\{ \begin{array}{cc}\frac{\sqrt{\mu}}{\sqrt{P\pi xrf( \sqrt{\mu})}}e^{-\frac{\mu}{2}x},&0<x\leq P\\ 0,&else\end{array} \right.\] (26) where \(\mu\) is the solution to equation \[E[x]=\frac{P}{2\mu}-\frac{Pe^{-\mu}}{\sqrt{\mu}\sqrt{\pi}erf\big{(}\sqrt{\mu} \big{)}}=\alpha P,\] (27) in which, \(erf(t)=\frac{2}{\sqrt{\pi}}\int_{0}^{t}e^{-\mu^{2}}du\) denotes the error function. The detailed derivations of (26) and (27) can be found in Appendix B. Please note that, since the function \(f(\mu)=\frac{1}{2\mu}-\frac{e^{-\mu}}{\sqrt{\mu}\sqrt{\pi}erf(\sqrt{\mu})}\) is monotonically decreasing in \((0,\infty)\), i.e., it tends to \(1/3\) for \(\mu\downarrow 0\) and to \(0\) for \(\mu\uparrow\infty\)[15], the average to peak power ratio \(\alpha\) satisfies \(\alpha\in\Big{(}0,\frac{1}{3}\Big{)}\). With (26), we can solve the component terms with respect to the input signal \(x\) in (25) as \[h(x)=-\int_{0}^{P}Q(x)logQ(x)dx=\frac{1}{2}log\frac{P}{\mu}+log\sqrt{\pi}erf( \sqrt{\mu})+\alpha\mu+\frac{1}{2}E_{Q}[logx],\] (28) and \[\frac{1}{2}E_{Q}[log(1+\frac{1}{H\varsigma^{2}x})]\leq\frac{2\sqrt{\mu}}{\sqrt {\pi}erf\big{(}\sqrt{\mu}\big{)}\sqrt{\varsigma^{2}PH}}tan^{-1}\sqrt{\varsigma ^{2}PH}+\]  
\[\frac{\sqrt{\mu}}{\sqrt{\pi}erf\big{(}\sqrt{\mu}\big{)}}log\Big{(}1+\frac{1}{ \varsigma^{2}PH}\Big{)}. \tag{29}\]  
The detailed derivations of (29) can be found in Appendix C. Combining (25) with (24), (28), and (29), the lower capacity bound of UWOC with APD in case I can be obtained as  
\[C\geq C_{Low}=\frac{1}{2}log\frac{P}{\mu}+log\sqrt{\pi}erf(\sqrt{\mu})-\frac {1}{2}log2\pi e\sigma^{2}\varsigma^{2}+a\mu-1+f(H)\, \tag{30}\]  
where  
\[f(H)=\frac{1}{2}log\frac{\varepsilon H+2\sigma^{2}\varsigma^{2}}{\varepsilon} -\frac{\varepsilon H}{\sigma^{2}\varsigma^{2}}+\frac{\sqrt{\varepsilon H( \varepsilon H+2\sigma^{2}\varsigma^{2})}}{\sigma^{2}\varsigma^{2}}-\]\[\frac{2\sqrt{\mu}}{\sqrt{\pi}erf(\sqrt{\mu})\sqrt{\zeta^{2}PH}}tan^{-1}\sqrt{ \zeta^{2}PH}-\frac{\sqrt{\mu}}{\sqrt{\pi}erf(\sqrt{\mu})}log\bigg{(}1+\frac{1}{ \zeta^{2}PH}\bigg{)}. \tag{31}\]
2. \(\alpha\in\left[\frac{1}{3},1\right]\).