## 4 Capacity Modelling of the Proposed System  
Turbulence-induced scintillation is the major factor that can impact the propagation of a spatial mode multiplexed signal across the FSO link [23]. These scintillation distortions are produced due to local fluctuations in pressure and temperature in the atmosphere. This random change in the atmospheric refractive index results in the leakage of signal power from one mode to adjacent modes. This phenomenon results in modes overlapping with power penalties.  
The scintillation theory states that the intensity fluctuations of the received OAM-OFDM-SMM multiplexed signal \(I=x\times y\) is the product of large (\(\upalpha\)) and small (\(\beta\)) scale turbulent eddies in GG turbulence conditions of the atmospheric channel. These intensity fluctuations are represented by the conditional PDF corresponding to \(\upalpha\) and \(\beta\) parameters at the system receiver end by Equations (20) and (21) respectively [24]. Here the Rytovconstant is given by \(\sigma^{2}=1.23C_{n}^{2}k^{7/6}z^{11/6}\) and \(C_{n}^{2}\) is the refractive structure parameter whereas \(\Gamma(.)\) denote the Gamma function.  
\[f_{x}(x)=\frac{\alpha(\alpha x)^{\alpha-1}}{\Gamma\alpha}e^{-\alpha x}\text{ for }x,\ \alpha\ >\ 0 \tag{20}\]  
\[f_{y}(y)=\frac{\beta(\beta y)^{\beta-1}}{\Gamma\beta}e^{-\beta y}\text{ for }y,\ \beta\ >\ 0 \tag{21}\]  
\[\alpha=\left(\exp\left[\frac{0.49\sigma^{2}}{\left(1+1.11\sigma^{2}4\right)^{ 7/6}}\right]-1\right)^{-1}>0 \tag{22}\]  
\[\beta=\left(\exp\left[\frac{0.51\sigma^{2}}{\left(1+0.691\sigma^{2}4\right)^{ 5/6}}\right]-1\right)^{-1}>0 \tag{23}\]  
The conditional PDF of intensity fluctuation \(I\) for the fixed values of \(x\) is further expressed by Equation (24), whereas the unconditional PDF can be further derived by taking the average of this Equation (24) over the Gamma distribution using Equation (20). Finally, the PDF for the GG distribution function can be derived for the modified Bessel function \(K(.)\) of second kind and written by Equation (25).  
\[f_{y}(I/x)=\frac{\beta(\beta I/x)^{\beta-1}}{x\Gamma\beta}e^{-(\beta I/y)} \tag{24}\]  
\[f(I)=\frac{2(\alpha\beta)^{(\alpha+\beta)/2}}{\Gamma\alpha\Gamma\beta}I^{( \alpha+\beta)/2-1}K_{\alpha-\beta}\sqrt{2\alpha\beta I} \tag{25}\]  
Considering Equation (25), and the bit error rate (BER) equation for the orthogonal modulation scheme [22], the BER of the OAM-OFDM signal, for the transmitted and received OAM state, \(l_{q}\) and \(l\) and \(m_{th}\) subcarrier frequency, is given by  
\[\langle ber_{l,m}\rangle=\int\limits_{0}^{\infty}0.5f_{l}(I)erfc\sqrt{\frac{l^ {2}\gamma_{l,m}}{2}}dI \tag{26}\]  
In the above Equation (26), \(f_{l}(I)\), represents the PDF of the intensity fluctuation of the OFDM signal at the receiver end due to atmospheric turbulence, whereas, \(\gamma_{l,m}=\frac{P_{l,m}(l_{2})}{\sum\limits_{l_{q}\in l}^{l_{q}}P_{l,m}(l_ {q};z)+\frac{N_{0}}{P_{l,m}}}\) denotes the signal-to-noise ratio (SNR) of the OAM-OFDM channel for the transmitted power \(P_{tx}\) and, \(erfc(.)\) is the complementary error function [25].  
In this proposed FSO system, a binary symmetric channel (BSC) is adopted, whose transmission channel capacity is derived in Refs. [7,26]. Hence the capacity of the OAM-OFDM FSO system can be expressed by Equation (27)  
\[C(\langle ber_{l,m}\rangle)=1+\langle ber_{l,m}\rangle\log_{2}\langle ber_{l,m} \rangle+(1-\langle ber_{l,m}\rangle)\log_{2}(1-\langle ber_{l,m}\rangle) \tag{27}\]  
hence, the total capacity of OAM-OFDM-SMM multiplexed FSO transmission systems can be expressed as  
\[C=\sum\limits_{l=-K}^{\pm K}\sum\limits_{m=0}^{M-1}C(\langle ber_{l,m}\rangle) \tag{28}\]