## 5 Diversity Reception in Malaga-\(\mathcal{M}\) Fading  
In diversity reception, \(L\) pre-amplified receiving branches are utilized and their outputs are added in an EGC to produce the total signal prior to soft-decision decoding. To simplify our analysis, we assume that the amplifiers are identical and that perfect synchronization is achieved among the branches, despite the fact that the spatial separation of apertures introduces delay. In a practical system, the spatial coherence length of the atmospheric channel only measures a few centimetres, and it follows that the apertures only need to be separated by a few centimetres to achieve uncorrelated reception. This results in a relative delay among branches of several picoseconds, which will only impact the system's performance for comparable PPM slot durations that reach the bandwidth limits of the receiver electronics. It follows from the previous discussion that the received signals are independent and identically distributed \(\chi^{2}\) RVs, while their instantaneous (within a single PPM symbol) values equal \(\frac{\lambda}{L}\,h_{\ell}\), with \(h_{\ell}\) denoting the channel state for the corresponding receiver. It is also assumed that the transmitted symbol energy is reduced as factor of \(L\) so that the total received optical energy at the combiner remains equal to \(\lambda\) (on average) and independent of the diversity order \(L\). Given that the sum of independent and identically distributed \(\chi^{2}\) RVs is also a \(\chi^{2}\) RV with \(N=L\,M\) degrees of freedom and \(\lambda_{egc}\,=\frac{\lambda}{L}\sum_{\ell=1}^{L}\,h_{\ell}\), the instantaneous PER is calculated as  
\[ \begin{split} P_{e}&=\frac{Q}{2\,(Q-1)}\,\sum_{q=1 }^{Q-1}\,\binom{Q-1}{q}\,(-1)^{q+1}\,\exp\left(-\frac{q}{1+q}\,\frac{\lambda} {L}\,\sum_{\ell=1}^{L}\,h_{\ell}\right)\,^{q}\sum_{n=0}^{(N-1)}\,\frac{c_{n}^ {q}}{(1+q)^{n+N}}\\ &\qquad\qquad\times\,\sum_{i=0}^{n}\,\frac{(n+N-1)!}{(i+N-1)!\,( n-i)!\,i!}\,\left(\frac{\lambda}{L}\,\frac{\sum_{\ell=1}^{L}\,h_{\ell}}{1+q} \right)^{i}\\ &=\frac{Q}{2\,(Q-1)}\,\sum_{q=1}^{Q-1}\,\binom{Q-1}{q}\,(-1)^{q+ 1}\,\sum_{n=0}^{q(N-1)}\,\frac{c_{n}^{q}}{(1+q)^{n+N}}\,\sum_{i=0}^{n}\, \binom{n+N-1}{i+N-1}\,\,\left(\frac{\lambda}{1+q}\right)^{i}\\ &\qquad\qquad\times\,\sum_{i_{1}+\ldots+i_{L}=i}\,\prod_{\ell=1} ^{L}\,\frac{h_{\ell}^{i_{\ell}}}{i_{\ell}!}\,\exp\left(-\frac{q}{1+q}\,\frac{ \lambda}{L}\,h_{\ell}\right).\end{split}  \tag{22}\]  
Figure 3: PPM average PER in Malaga-\(\mathcal{M}\) fading. The plots correspond to no fading (solid), weak fading (dash), moderate fading (dash-dot) and strong fading (dash-dot). The noise modes are equal to \(M=2\) and \(M=200\) in (**a**) and (**b**), respectively. The average probability now equals  
\[ \begin{split}\overline{P_{e}}=\frac{Q}{2\left(Q-1\right)}& \sum_{q=1}^{Q-1}\ \binom{Q-1}{q}\ (-1)^{q+1}\sum_{n=0}^{q\ (N-1)}\ \frac{c_{n}^{q}}{(1+q)^{n+N}}\ \sum_{i=0}^{n}\ \binom{n+N-1}{i+N-1}\ \left(\frac{ \frac{\lambda}{L}}{1+q}\right)^{i}\\ &\times\sum_{i_{1}+\ldots+i_{L}=i}\ \prod_{\ell=1}^{L}\ \frac{1}{i_{\ell}!}\int_{h_{\ell}=0}^{\infty}h_{\ell}^{i_{\ell}}\ \exp\left(-\frac{q}{1+q}\ \frac{\lambda}{L}\ h_{\ell}\right)\ f_{h_{\ell}}(h_{\ell})\ dh_{\ell}\,, \end{split}  \tag{23}\]  
and using Equation (19) we find that  
\[ \begin{split}\overline{P_{e}}&=\frac{Q}{2\left(Q-1 \right)}\ \sum_{q=1}^{Q-1}\ (\ \binom{Q-1}{q}\ (-1)^{q+1}\sum_{n=0}^{q\ (N-1)}\ \frac{c_{n}^{q}}{(1+q)^{n+N}}\ \sum_{i=0}^{n}\ \binom{n+N-1}{i+N-1}\ \frac{1}{q}\\ &\times\sum_{i_{1}+\ldots+i_{L}=i}\ \prod_{\ell=1}^{L}\ \frac{ \lambda}{2}\ \left(\frac{b\,m}{z_{q}}\right)^{\alpha}\ \frac{\Gamma(i_{\ell}+\alpha)}{i_{\ell}!}\ \sum_{j=1}^{\beta}\ \frac{a_{j}\,\Gamma(i_{\ell}+ \alpha)}{y!}\ \Psi\left(i_{\ell}+\alpha,1-j+\alpha;\frac{b^{2}\,m}{z_{q}}\right)\,, \end{split}  \tag{24}\]  
where \(z_{q}=\frac{q}{1+q}\ \frac{\lambda}{L}\). Despite the apparent complexity of the last equation, the appearing terms are calculated recursively using the theory of extended binomial coefficients.