# LDPC-Coded CAP with Spatial Diversity for UVLC Systems over Generalized-Gamma Fading Channel
## 3 Approximated BER without FEC  
CAP and QAM have the same BER in theory. An approximated conditional bit-error probability (BEP) of \(M\)-QAM is given by [32]  
\[p(\varepsilon|\gamma)=\frac{4\Big{(}\sqrt{M}-1\Big{)}}{\sqrt{M}\log_{2}M}Q\bigg{(} \sqrt{\frac{3}{M-1}}\gamma\bigg{)}+\frac{4\Big{(}\sqrt{M}-2\Big{)}}{\sqrt{M} \log_{2}M}\,Q\bigg{(}3\sqrt{\frac{3}{M-1}}\gamma\bigg{)} \tag{18}\]  
where \(Q(\cdot)\) is the Gaussian \(Q\)-function, and \(\gamma\) is the electrical SNR. The average BER can be obtained by integrating BEP over \(p_{\gamma}(\gamma)\), namely,  
\[P_{b}=\int_{0}^{\infty}p(\varepsilon|\gamma)p_{\gamma}(\gamma)\mathrm{d}\gamma \tag{19}\]  
However, the integration in Equation (19) was complicated, as it was very difficult to replace the multidimensional integral resulting from \(p_{\gamma}(\gamma)\) with an equivalent single-fold integral. So, we adopted the CHF-based approach proposed in [33] to derive analytical expressions for the BER of CAP with the EGC diversity scheme over the GG fading channels. The expressions depend on the single-fold integral, and can be presented in the form of [PERSON]'s H-function or [PERSON]'s G-function, which greatly simplify the calculations. Using the alternative form of \(Q(\cdot)\) as shown as Equation (4.2) in [34], and according to Equations (2), (7) and (9) in [33], the approximated BER can be expressed as follows:  
\[P_{b}=\frac{1}{\pi}\int_{0}^{\infty}\mathrm{Re}\Bigg{\{}\varphi_{b}(w)\phi^{ *}_{\sqrt{\gamma_{\mathrm{EGC}}^{\mathrm{eff}}}}(w)\Bigg{\}}\mathrm{d}w \tag{20}\]  
where \(w\) is the angular frequency, \(\varphi_{b}(w)\) is the CHF of the conditional BEP \(p\Big{(}\varepsilon\bigg{|}\sqrt{\gamma_{\mathrm{EGC}}^{\mathrm{eff}}}\Big{)}\), and \(\phi^{*}_{\sqrt{\gamma_{\mathrm{EGC}}^{\mathrm{eff}}}}(w)\) is the complex conjugate of the CHF of \(\sqrt{\gamma_{\mathrm{EGC}}^{\mathrm{eff}}}\). The real and imaginary parts of \(\varphi_{b}(w)\) are, respectively, given by [35]  
\[ \begin{array}{l}\mathrm{Re}[\varphi_{b}(w)]=\frac{1}{2}\sum\limits_{z=1}^{Z }\int_{0}^{\theta_{z}}a_{z}(\theta)\sqrt{\frac{\pi\sin^{2}\theta}{\delta_{z}} }\exp(-\frac{w^{2}\sin^{2}\theta}{4\delta_{z}})d\theta\\ \mathrm{Im}[\varphi_{b}(w)]=\frac{1}{2}\sum\limits_{z=1}^{Z}\int_{0}^{\theta_ {z}}a_{z}(\theta)\frac{w\sin^{2}\theta}{\delta_{z}}_{1}F_{1}\Big{(}1;\frac{3} {2};-\frac{w^{2}\sin^{2}\theta}{4\delta_{z}}\Big{)}d\theta\end{array}  \tag{21}\]  
where \({}_{1}F_{1}\)\((\cdot)\) is the confluent hypergeometric function of the first kind shown as Equation (9.210/1) in [27], and parameters Z, \(a_{z}\)\(\theta_{z}\), and \(\delta_{z}\) for \(M\)-QAM are given in Table 1.  
Then, when we derive \(\phi_{\sqrt{\gamma_{\mathrm{EGC}}^{\mathrm{eff}}}}(w)\). From Equation (8), we have \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameters** & **Z** & \(\mathbf{a_{z}}\) & \(\mathbf{\vartheta_{z}}\) & \(\mathbf{\theta_{z}}\) \\ \hline \multirow{3}{*}{Value} & \multirow{3}{*}{2} & \(a_{1}=\frac{4}{\pi}\Big{(}\frac{\sqrt{M}-1}{\sqrt{M}\log_{2}M}\Big{)}\) & \(\delta_{1}=\frac{3}{2(M-1)}\) & \multirow{3}{*}{\(\theta_{1}=\theta_{2}=\frac{\pi}{2}\)} \\  & & \(a_{2}=\frac{4}{\pi}\Big{(}\frac{\sqrt{M}-2}{\sqrt{M}\log_{2}M}\Big{)}\) & \(\delta_{2}=\frac{27}{2(M-1)}\) & \multirow{3}{*}{\(\theta_{1}=\theta_{2}=\frac{\pi}{2}\)} \\ \cline{1-1} \cline{5-5}  & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: \(M\)-quadrature-amplitude-modulation (\(M\)-QAM) parameters. \[\sqrt{\gamma_{\text{EGC}}^{\text{eff}}}=\sqrt{\frac{\overline{\gamma}}{N}}\sum_{i= 1}^{N}\alpha_{i} \tag{22}\]  
Assuming the receiver separations are higher than underwater coherence distance, the fading coefficients are independent; hence, the CHF of Equation (22) is given as  
\[\phi_{\sqrt{\gamma_{\text{EGC}}^{\text{eff}}}}(w)=\prod_{i=1}^{N}\phi_{\alpha _{i}}\Bigg{(}\sqrt{\frac{\overline{\gamma}}{N}}w\Bigg{)} \tag{23}\]  
where \(\phi_{\alpha_{i}}(w)\) is the CHF of \(\alpha_{i}\) that can be obtained using the approach detailed in [35].