## 3 The \(M\)-Distribution with Nonzero Boresight Pointing Errors  
In this section, the \(M\) distributed turbulence model with NZB PE is investigated. In [8], the probability density function (PDF) of the combined effects for a P2P link has been extracted as:  
\[ \begin{split} f_{comb,I_{m}}(I_{m})=\frac{\xi_{mod,m}^{2}\pi^{( \mathbb{R}\text{ }\sigma\text{ }\mathbb{R})}}{2}& I_{m}^{-1}\sum\limits_{j=1}^{( \mathbb{R}\text{ }\sigma\text{ }\mathbb{R})}\mathbf{a}_{j}^{(\mathbb{R}\text{ }\sigma\text{ }\mathbb{R})}\left(B^{(\mathbb{R}\text{ }\sigma\text{ }\mathbb{R})}\right)^{-\frac{\alpha_{m}+j}{2}}\times\\ &\times G_{1,3}^{3,0}\left(B^{(\mathbb{R}\text{ }\sigma\text{ }\mathbb{R})}\frac{I_{m}}{\Lambda_{mod,m}}\left|\begin{array}{c}\xi_{mod,m}^ {2}+1\\ \xi_{mod,m}^{2},\alpha_{m},\ \beta_{m}\end{array}\right.\right)\end{split}  \tag{9}\]  
where \(G_{p,d}^{m,m}[\cdot]\) stands for the Meijer G-function [21] (Equation (9.301)). The \(M\)-distribution is divided in two categories, which are determined by the value of the \(\beta_{m}\) parameter [6]. In this  
Figure 2: Block diagram of the CDMA RoFSO system with receivers’ spatial diversity. The ABER for each user, \(P_{b,k,MRC,AV}\), of the CDMA RoFSO link with diversity reception technique over \(M\) turbulence channels and NZB PE, is estimated by solving the following multiple integrals, [14; 15]:  
\[P_{b,k,AV}=\int_{\overline{I}}f_{comb,\overline{I}}\left(\stackrel{{ \rightarrow}}{{I}}\right)P_{b,k,QAM,MRC}\left(\stackrel{{ \rightarrow}}{{I}}\right)d\stackrel{{\rightarrow}}{{I}} \tag{14}\]  
where the vector \(\stackrel{{\rightarrow}}{{I}}=(I_{1},I_{2},\ldots,I_{M})\) consists of all the irradiance values for each link of the receivers diversity scheme [17]. Thus, taking into account the following approximation for the \(Q\)-function [23]:  
\[Q(x)\approx\frac{1}{12}\left[\exp\left(-\frac{x^{2}}{2}\right)+2\exp\left(-2x ^{2}\right)+2\exp\left(-\frac{2x^{2}}{3}\right)\right] \tag{15}\]  
and by using the Equation (13), we conclude to the following integrals:  
\[ \begin{array}{l}P_{b,k,AV}\approx\frac{1-L^{-1/2}}{3\log_{2}(L)}\left[\int_ {0}^{\infty}f_{comb,I_{m}}(I_{m})\exp\left(\frac{3\sum\limits_{m=1}^{M}CNIR_{ k,m}(I_{m})}{2(1-L)}\right)dI_{m}+\\ +2\int_{0}^{\infty}f_{comb,I_{m}}(I_{m})\exp\left(\frac{6\sum\limits_{m=1}^{M} CNIR_{k,m}(I_{m})}{1-L}\right)dI_{m}+2\int_{0}^{\infty}f_{comb,I_{m}}(I_{m})\exp \left(\frac{2\sum\limits_{m=1}^{M}CNIR_{k,m}(I_{m})}{1-L}\right)dI_{m}\right] \end{array}  \tag{16}\]  
The irradiance values, \(I_{m}\), are statistically independent and identically distributed (i.i.d.) random variables. Due to the independency of the \(I_{m}\) vectors, the \(M\)-fold integral in (16) is partitioned into a product of \(M\) one-dimensional integrals, [24]. As a result, we obtain the following expression for the estimation of the ABER:  
\[ \begin{array}{l}P_{b,k,AV}\approx\frac{1-L^{-1/2}}{3\log_{2}(L)}\left[\prod \limits_{m=1}^{M}\int_{0}^{\infty}f_{comb,I_{m}}(I_{m})\exp\left(-\frac{3 CNIR _{k,m}(I_{m})}{2(L-1)}\right)dI_{m}+\right.\\ \left.+2\prod\limits_{m=1}^{M}\int_{0}^{\infty}f_{comb,I_{m}}(I_{m})\exp \left(-\frac{6 CNIR_{k,m}(I_{m})}{L-1}\right)dI_{m}+\right.\\ \left.+2\prod\limits_{m=1}^{M}\int_{0}^{\infty}f_{comb,I_{m}}(I_{m})\exp \left(-\frac{2 CNIR_{k,m}(I_{m})}{L-1}\right)dI_{m}\right]\end{array}  \tag{17}\]  
Next, we replace the exponential terms with the equivalent Meijer-G expressions for the exponential function, i.e., \(\exp(z)=G_{0,1}^{1,0}(-z|0)\), and we get:  
\[