# Cooperative Terrestrial-Underwater FSO System: Design and Performance Analysis
## 4 Performance Analysis  
In this section, the closed analytical expression of the average BER (ABER) for the FSO-TU cooperative system is developed. To this end, we first obtain the conditional BER (CBER) expression associated with an ideal scenario in absence of channel fluctuations, considering the OOK modulation scheme with IM/DD and AWGN. In this scenario, from [16], and under the assumption that channel state information (CSI) is available at the receiver, the CBER is given by  
\[Pb(e/h)=\frac{1}{2}\text{erfc}\left(\frac{i_{s}h}{2\sqrt{2(\sigma_{n}^{2}+ \sigma_{s}^{2})}}\right), \tag{25}\]  
where \(i_{s}\) is the ideal signal current without considering channel degradation effects, which in this case is obtained as \(i_{s}=P_{1}R_{1}R_{2}g\), where \(P_{1}\) represents the average of transmitted optical power, \(R_{1}\) and \(R_{2}\) are the responsiveness corresponding to the detectors of both links, \(g\) is the repeater gain, \(h=h_{1}h_{2}\) represents the random irradiance fluctuation and \(\sigma_{n}^{2}\) and \(\sigma_{s}^{2}\) are the noise and ISI variances. In practical terms, irradiance fluctuation can be estimatedusing training sequences or pilot symbols [37]. Thereafter, pilot-based channel estimation methods, namely least squares and minimum mean-square error, can be employed. Following [38; 39], we assume in (25) that the ISI interference is Gaussian distributed. Now, we define the signal-to-interference-plus-noise ratio (SINR) without considering the effects of optical channel degradation as follows:  
\[\gamma_{0}=\frac{i_{s}}{\sqrt{\sigma_{n}^{2}+\sigma_{s}^{2}}}. \tag{26}\]  
In order to solve the integral involving ABER, calculated by averaging \(Pb(e|h)\) over the PDF of the irradiance, \(f_{h}(h)\), the SINR can be approximated by averaging the noises and the inter-symbol interference over oceanic turbulence in the way proposed in [24; 38]. \[\gamma_{0}\simeq\frac{i_{s}}{\sqrt{<\sigma_{n}^{2}>+<\sigma_{s}^{2}>}}. \tag{27}\]  
Now, to be consistent with the notation used in (24), we can identify \(\gamma=\gamma_{0}h\). In this way, the probability of error \(P_{b}(e)\) is calculated, following the development presented in [24], from the average of \(P_{b}(e|h)\) and the probability density function of the irradiance, \(f_{h}(h)\). Hence, the BER is derived as follows:  
\[P_{b}=\int_{0}^{\infty}\frac{1}{2}\text{erfc}\bigg{(}\frac{\gamma_{0}h}{2\sqrt {2}}\bigg{)}f_{h}(h)\text{d}h. \tag{28}\]  
Applying integration by parts to Equation (28), we can obtain the \(P_{b}\) from the CDF given in (24), using the following formula:  
\[P_{b}=\bigg{(}P_{b}(e|h)F_{h}(h)\bigg{)}\bigg{|}_{0}^{\infty}-\int_{0}^{\infty }\frac{\text{d}}{\text{d}h}\bigg{[}P_{b}(e|h)\bigg{]}F_{h}(h)\text{d}h. \tag{29}\]  
Since \(P_{b}(e|\infty)=0\) and \(F_{h}(0)=0\) (note that negative values are not allowed for the optical irradiance), then the above expression can be reduced to the following expression:  
\[P_{b}=-\int_{0}^{\infty}\frac{\text{d}}{\text{d}h}\bigg{[}P_{b}(e|h)\bigg{]}F _{h}(h)\text{d}h=-\int_{0}^{\infty}\frac{\text{d}}{\text{d}h}\bigg{[}\frac{1} {2}\text{erfc}\bigg{(}\frac{\gamma_{0}h}{2\sqrt{2}}\bigg{)}\bigg{]}F_{h}(h) \text{d}h, \tag{30}\]  
where \(F_{h}(h)\) is obtained directly from (24). Next, we apply Equation (06.27.13.0005.01) [36] to derive an expression for the derivative of \(P_{b}(e|h)\) with respect to \(h\):  
\[\frac{\text{d}}{\text{d}h}[P_{b}(e|h)]=-\frac{\gamma_{0}}{2\sqrt{2\pi}}\exp \bigg{[}-\bigg{(}\frac{\gamma_{0}h}{2\sqrt{2}}\bigg{)}^{2}\bigg{]}. \tag{31}\]  
Next, (31) is introduced in (30) to solve the resulting integral. To this end, a generalized Gauss-Laguerre quadrature is proposed, (31), and defined as  
\[\int_{0}^{\infty}x^{v}e^{-x}f(x)\text{d}x=\sum_{i=1}^{n}H_{i}f(x_{i})+E_{n}, \tag{32}\]  
where \(v\) is a constant, \(x_{i}\) represents the ith zero of the Laguerre polynomial, \(L_{n}^{v}(x)\), \(H_{i}\) is the corresponding weight coefficients associated with the Gauss-Laguerre quadrature and \(E_{n}\) denotes the truncation error.