## 4 Average PER in Malaga-\(\mathcal{M}\) Fading  
We now consider transmission over an OWC link with turbulence. In this environment, the received energy fluctuates as \(\lambda\,h\) with \(h\) denoting the channel state. The average PER is found by integrating Equation (15) over the fading distribution, and we find that  
\[ \begin{split}\overline{P_{e}}=\frac{Q}{2\,(Q-1)}& \sum_{q=1}^{Q-1}\,\binom{Q-1}{q}\,(-1)^{q+1}\sum_{n=0}^{q\,(M-1)}\, \frac{c_{n}^{q}}{(1+q)^{n+M}}\,\sum_{i=0}^{n}\,\frac{(n+M-1)!}{(i+M-1)!\,(n-i)! \,i!}\\ &\times\,\left(\frac{\lambda}{1+q}\right)^{i}\,\int_{0}^{\infty}h ^{i}\,\exp\left(-\frac{q\,\lambda}{1+q}\,h\right)\,f_{h}(h)\,dh\,.\end{split}  \tag{16}\]  
The appearing integrals may be evaluated from mathematical tables or the derivatives of the Laplace transform of the fading, distribution at \(s=\frac{q}{1+q}\,\lambda\). In Malaga-\(\mathcal{M}\) fading \(h\) is a normalized random variable (RV) distributed as [27]  
\[f_{h}(h)=A\,\sum_{j=1}^{\beta}\,a_{j}\,m^{\frac{n+j}{2}}\,h^{\frac{n+j}{2}-1}\, K_{n-j}\left(2\,\sqrt{\frac{\alpha\,\beta\,m}{\gamma\,\beta+\Omega^{\prime}} \,h}\right)\, \tag{17}\]  
where \(K_{v}\left(\cdot\right)\) stands for the modified Bessel function of the second kind ([29], Equation (8.407)). The model parameters are calculated from the following equations [27]  
\[ \begin{split}\Omega^{\prime}&=\Omega\,+\,2\,\rho\,b _{0}+2\,\sqrt{2\,\rho\,b_{0}\,\Omega}\,\cos\left(\phi_{A}-\phi_{B}\right)\,\\ \gamma&=2\,(1-\rho)\,b_{0}\,\\ m&=\gamma+\Omega^{\prime}\,\\ A&=\frac{2\alpha^{\frac{q}{2}}}{\gamma^{1+\frac{q}{2}} \Gamma(\alpha)}\,\,\left(\frac{\gamma\,\beta}{\gamma\,\beta+\Omega^{\prime}} \right)^{\beta+\frac{q}{2}}\,\\ a_{j}&=\binom{\beta-1}{j-1}\,\frac{1}{(j-1)!}\, \left(\frac{\Omega^{\prime}}{\gamma}\right)^{j-1}\,\left(\frac{\alpha}{\beta} \right)^{\frac{j}{2}}\,\left(\gamma\,\beta+\Omega^{\prime}\right)^{1-\frac{j} {2}}\,\end{split}  \tag{18}\]  
and \(\alpha,\beta,\Omega,b_{0},\rho,\phi_{A},\phi_{B}\) are also described in [27]. Figure 2: PPM probabilities of error (PERs) for different modulation orders \(Q\) and noise modes \(M\). The integrals in Equation (16) are evaluated using ([29], Equation (6.631/3)) and ([32], Equation (13.1.33)), and we find that  
\[ \begin{split} I&=\,\int_{0}^{\infty}\,h^{i+\frac{ \alpha+j}{2}-1}\,\exp\left(-\frac{q\,\lambda}{1+q}\,h\right)\,K_{\alpha-j}\left( 2\,\sqrt{\frac{\alpha\,\beta\,m}{\gamma\,\beta+\Omega^{\prime}}\,h}\right)\,dh \\ &=\,\frac{1}{2}\,\frac{b^{\alpha-j}\,m^{\frac{\alpha-j}{2}}}{z_{q }^{\alpha+i}}\,\Gamma(i+\alpha)\,\Gamma(i+j)\,\Psi\left(i+\alpha,1-j+\alpha; \frac{b^{2}\,m}{z_{q}}\right)\.\end{split}  \tag{19}\]  
In the last equation \(\Psi\left(\cdot,\cdot;\cdot\right)\) denotes the Tricomi confluent hypergeometric function ([29], Equation (9.210/2)), and we have introduced \(b=\sqrt{\frac{\alpha\,\beta}{\gamma\,\beta+\Omega^{\prime}}}\) and \(z_{q}=\frac{q}{1+q}\,\lambda\). The PER now becomes  
\[ \begin{split}\overline{P_{e}}=\frac{Q}{2\,(Q-1)}& \sum_{q=1}^{Q-1}\,\binom{Q-1}{q}\,(-1)^{q+1}\,\sum_{n=0}^{q(M-1)}\, \frac{c_{n}^{q}}{(1+q)^{n+M}}\,\sum_{i=0}^{n}\,\binom{n+M-1}{i+M-1}\,\frac{1} {q^{i}}\\ &\times\frac{A}{2}\,\left(\frac{b\,m}{z_{q}}\right)^{\alpha}\, \frac{\Gamma(i+\alpha)}{i!}\,\sum_{j=1}^{\beta}\,\frac{a_{j}\,\Gamma(i+j)}{b ^{j}}\,\Psi\left(i+\alpha,1-j+\alpha;\frac{b^{2}\,m}{z_{q}}\right)\.\end{split}  \tag{20}\]  
Re-arranging the order of summation, we find that  
\[ \begin{split}\overline{P_{e}}=\frac{Q}{2\,(Q-1)}& \sum_{q=1}^{Q-1}\,\binom{Q-1}{q}\,(-1)^{q+1}\,\sum_{n=0}^{q(M-1) }\,\sum_{i=n}^{q(M-1)}\,\binom{i+M-1}{n+M-1}\,\frac{c_{i}^{q}}{(1+q)^{i+M}} \,\frac{1}{q^{n}}\\ &\times\frac{A}{2}\,\left(\frac{b\,m}{z_{q}}\right)^{\alpha}\, \frac{\Gamma(n+\alpha)}{n!}\,\sum_{j=1}^{\beta}\,\frac{a_{j}\,\Gamma(n+j)}{b ^{j}}\,\Psi\left(n+\alpha,1-j+\alpha;\frac{b^{2}\,m}{z_{q}}\right)\,\end{split}