## 4 Performance Analysis
### Coded Transmission Schemes  
The second scenario considered in this section consists of any generic coding scheme for which its closed-form analytical expression for CBER is unknown, although accurately estimated by a curve-fitting method. In this respect, signaling techniques such as OOK-GScc [40,42] and vw-MPPM [41] can be included as representative examples belonging to this group: OOK-GScc is a Markov-chain-based coding scheme that can be studied as a run-length limited (RLL) sequences generator, with the great advantage that the associated decoder can be built by means of a shift register and simple combinational logic, which drastically reduces the decoding complexity if compared to the classical scheme based on the Viterbi algorithm. OOK-GScc provides a native coding rate of 1/4 and, to provide additional coding rates, the procedure consists in introducing a selective number of silence periods behind each coded bit generated by the OOK-GScc scheme. In this paper, effective coding rates of 1/4 and 1/8 are employed in the analysis. On the other hand, the vw-MPPM coding scheme is a nonlinear block coding technique based on translation table between input data codewords and output codewords, with the main goal of increasing the peak-to-average optical power ratio (PAOPR), maintaining the average optical power transmitted to the medium constant. For these coding methods, to the knowledge of the authors, there is no closed-form expression for BER under a simple additive white Gaussian noise (AWGN) channel; thus, they are appropriate for applying the curve-fitting method based on a hyperexponential fitting technique [43], obtaining the following analytical approximate equation:  
\[\text{CBER}(h,\gamma_{0})=P_{b}(e|h)\approx a_{c}\exp\left[-b_{c}\left(\gamma_{ 0}^{2}h^{2}\right)^{c_{c}}\right] \tag{22}\]  
with \(\gamma_{0}\) being the electrical SINR in absence of turbulence defined in (12) and where the hyperexponential fitting parameters are \(a_{c_{c}},b_{c_{c}},c_{c}\in\Re^{+}\). These three parameters depend on the used coding scheme and are calculated by employing a least-squares fitting method over the results of BER accomplished by Monte-Carlo simulations without the effect of the turbulence process. Table 1 shows the resulting fitting parameters for OOK-GScc, with extended coding rates of \(R=1/4\) and \(1/8\), and the vw-MPPM coding schemes calculated in [43]. By applying the same steps as in the previous section to simplify (14), i.e., applying \(P_{b}(e|\infty)=0\) and \(F_{h}(0)=0\), and considering that negative values for the optical irradiance are not allowed, the ABER, \(P_{b}(e)\), is obtained by averaging CBER over the PDF of \(h\) by using the equivalent integration by parts formula:  
\[P_{b}=-\int_{0}^{\infty}\frac{\text{d}}{\text{d}h}\left[P_{b}(e|h)\right]F_{h} (h)\text{d}h \tag{23}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Code** & **Code Rate** & \(a_{c}\) & \(b_{c}\) & \(c_{c}\) \\ \hline \multirow{2}{*}{OOK-GScc} & \(1/4\) & 1.7103 & 30.2515 & 0.8592 \\  & \(1/8\) & 1.2465 & 120.1244 & 0.9386 \\ \hline \multirow{2}{*}{vw-MPPM} & \(9/36\) & 0.7246 & 42.4424 & 0.8600 \\  & \(5/40\) & 1.1890 & 152.1918 & 0.8150 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Hyper-exponential fitting parameters \(a_{c}\), \(b_{c}\) and \(c_{c}\) in the absence of turbulence. where the derivative of (22) with respect to \(h\) is obtained as follows. \[\frac{\mathrm{d}}{\mathrm{d}h}[P_{b}(e|h)]=-2a_{c}b_{c}c_{c}\cdot\gamma_{0}^{2c_{ c}}h^{2c_{c}-1}\exp\Bigl{[}-b_{c}(\gamma_{0}h)^{2c_{c}}\Bigr{]}. \tag{24}\]  
Next, a generalized Gauss-Laguerre quadrature is again applied, as shown in (17), and with the help of (18) and (19).