# Performance Analysis of a Vertical FSO Link with Energy Harvesting Strategy
## 3 System Performance
### BER Performance  
Now, we derive closed-form expressions of the BER for the OOK scheme used for SLIPT. Hence, the received BER can be written as  
\[\text{BER}=p_{0}P_{e}(1|0)+p_{1}P_{e}(0|1), \tag{20}\]  
where \(P_{e}(1|0)\) and \(P_{e}(0|1)\) represent the probability of false alarm and the probability of missed detection, respectively. In our analysis, we assume equally-likely symbols, \(p_{0}=p_{1}=0.5\) in many different realistic scenarios. First, for an ideal scenario without neither turbulence nor pointing error (in (3), \(h\) is reduced to \(h=A_{o}h_{l}\)), the terms \(P_{e}(1|0)\) and \(P_{e}(0|1)\) can be expressed as [50]:  
\[P_{e}(1|0)=\frac{1}{2}\text{erfc}\Bigg{(}\frac{i_{t}-i_{s_{0}}} {\sqrt{2\sigma_{n}^{2}}}\Bigg{)} =\frac{1}{2}\text{erfc}\Bigg{(}\frac{i_{t}-i_{s_{1}}\zeta}{\sqrt{ 2\sigma_{n}^{2}}}\Bigg{)} \tag{21}\] \[P_{e}(0|1)=\frac{1}{2}\text{erfc}\Bigg{(}\frac{i_{s_{1}}-i_{t}} {\sqrt{2\sigma_{n}^{2}}}\Bigg{)}; \tag{22}\]  
with \(i_{s0}\) and \(i_{s1}\) denoting the received photocurrent corresponding to the transmission of a logical '0' and a logical '1', respectively; where \(i_{t}\) is the optimal detection threshold and \(\sigma_{n}^{2}\) is the noise variance given in Equation (2). On another note, and from (13), the photocurrents \(i_{s0}\) and \(i_{s1}\) can be expressed as \(i_{s0}=Rh_{s}^{2}P_{m}\) and \(i_{s1}=RhP_{m}\), respectively. In addition, \(i_{t}\) can be written as a function of \(\zeta\) as  
\[i_{t}=\frac{i_{s_{1}}+i_{s_{0}}}{2}=\frac{i_{s_{1}}(1+\zeta)}{2}. \tag{23}\]  
Recall that, for this ideal AWGN channel, \(h\) has a deterministic behavior since neither turbulence-induced nor misalignment fadings were considered yet. Accordingly, its associated BER is obtained by substituting (21) and (22) into (20), and it is given by  
\[\text{BER}=\text{erfc}\left(\frac{RhP_{m}(\zeta-1)/2}{\sqrt{2\sigma_{n}^{2}}} \right). \tag{24}\]  
Second, we consider a more realistic scenario where atmospheric turbulence is now considered, although misalignment fading is still not included Therefore, the channel attenuation coefficient, \(h\), becomes a RV since the turbulence-induced scintillation now incorporated into \(h\) follows the Gamma-Gamma pdf shown in (5). Thus (24) must be averaged with the corresponding pdf describing the behavior of \(h\), and (48) (Eqs. (07.34.21.0013.01) and (07.34.21.0085.01)) are, again, used to solve the resulting integral:  
\[\text{BER}=\frac{2^{(\alpha+\beta)}}{8\pi\sqrt{\pi}\Gamma(\alpha)\Gamma(\beta )}G_{5,2}^{2,4}\left(\frac{8(RA_{o}h_{t}P_{m}(\zeta-1)/2)^{2}}{(\sigma_{n} \alpha\beta)^{2}}\right| \begin{array}{c}\frac{1-\alpha}{2},\frac{2-\alpha}{ 2},\frac{1-\beta}{2},\frac{2-\beta}{2},1\\ 0,\frac{1}{2}\end{array} \right) \tag{25}\]  
Finally, when pointing errors are also taken into account, (21) and (22) must be now averaged with respect to (12). Hence,  
\[\text{BER}=\frac{2^{(\alpha+\beta)}\gamma^{2}}{16\pi\sqrt{\pi} \Gamma(\alpha)\Gamma(\beta)}\\ G_{7,4}^{2,6}\left(\frac{8(RA_{o}h_{t}P_{m}(\zeta-1)/2)^{2}}{(\sigma_{n} \alpha\beta)^{2}}\right| \begin{array}{c}\frac{1-\gamma^{2}}{2},\frac{2- \gamma^{2}}{2},\frac{1-\alpha}{2},\frac{2-\alpha}{2},\frac{1-\beta}{2},\frac{2 -\beta}{2},1\\ 0,\frac{1}{2},\frac{-\gamma^{2}}{2},\frac{1-\gamma^{2}}{2}\end{array} \right), \tag{26}\]  
after having used (48) (Eqs. (07.34.21.0013.01) and (07.34.21.0085.01)). As a previous step, either the complementary error function, \((\text{erfc}(\cdot))\), and the Bessel's K function were expressed in terms of Meijer's G functions from (48) (Eqs. (03.04.26.0006.01) y (06.27.26.0006.01)). As can be seen from these latter equations, the exact expression for the error probability is given in terms of Meijer's G-function, which may be difficult to facilitate further analytical studies.