## 2 System Model and Channel Model
### FSO System Model  
Since the atmospheric turbulence is generally a medium to strong turbulence case, the Gamma-Gamma model is more consistent with the actual test data in the turbulence case. The FSO channel between S and R is modeled as Gamma-Gamma distribution with pointing error. The statistical behavior of the received optical irradiance is characterized by means of the Gamma-Gamma turbulence model, which has been widely used to model the FSO channel in the recent literature depending on its doubly stochastic scintillation model [31]. The probability distribution function (PDF) of channel coefficient \(h_{FSO}\) is given as [32].  
\[f_{h_{FSO}}(h_{FSO})=\frac{\xi^{2}}{h_{FSO}\Gamma(\alpha)\Gamma(\beta)}G_{1,3}^{ 3,0}\Big{(}\alpha\beta\frac{h_{FSO}}{I_{FSO}M_{o}}|\xi_{\mathcal{C}^{2}-1, \mathcal{R}-1,\mathcal{R}-1}^{2}\Big{)} \tag{1}\]  
where \(\Gamma(.)\) is the well-known Gamma function [33] and \(G_{P,\mathcal{R}}^{m,\mathcal{R}}(x_{\mathcal{R}_{1},\mathcal{R}_{2}}^{|\alpha _{1}-\mathcal{R}_{2}|})\) is the Meijer's G-function defined in [34]. The atmospheric turbulence induced fading channel gains of FSO links, denoted as \(I_{FSO}\) is modeled with pointing errors and atmospheric attenuation, where \(\xi=\alpha_{e}/2\sigma_{s}\) is the ratio of equivalent beam radius and zero boresight pointing error displacement standard deviation at the photo-detector (PD). The constant term \(M_{o}\) is the power fraction that the detector receives when there is no pointing error [35], and \(1/\alpha\) and  
Figure 1: Block diagram of ultrasonic and RIS-assisted for Dual-Hop Hybrid FSO-mmWave massive MIMO of V2X system.  
Figure 2: DF workflow diagram.  
\(1/\beta\) are the variances of the large and small turbulence eddies, respectively. Parameters \(\alpha\) and \(\beta\) are the distance-dependent fading variables which correspond to atmospheric turbulence conditions, as described in [35].  
\[\alpha=\left\{\exp\left[\frac{0.49\sigma_{R}^{2}}{\left(1+1.11\sigma_{R}^{\frac{ 12}{2}}\right)^{\frac{5}{6}}}\right]-1\right\}^{-1} \tag{2}\]  
\[\beta=\left\{\exp\left[\frac{0.51\sigma_{R}^{2}}{\left(1+0.69\sigma_{R}^{\frac {12}{2}}\right)^{\frac{5}{6}}}\right]-1\right\}^{-1} \tag{3}\]  
where \(\sigma_{R}^{2}=1.23C_{n}^{2}v^{7/6}d_{SR}^{11/6}\) denotes the Rytov variance; \(C_{n}^{2}\), \(v\), and \(d_{SR}\) represent the refractive index structure constant, wave number, and the link length, respectively. \(C_{n}^{2}\) usually takes values in the range \(10^{-17}-10^{-13}\) from weak to strong turbulence conditions.  
In order to evaluate the performance of the FSO link, we need to derive formulas for the BER and outage probability of the FSO link. For the FSO link with IM/DD detection, the instantaneous SNR is given as [36]  
\[\gamma_{F}=\frac{2P_{A}^{2}R^{2}}{\sigma_{n}^{2}}h_{FSO}^{2} \tag{4}\]  
where \(P_{A}\) and \(R\) represent the average transmitted power and the detector responsivity, respectively; \(\sigma_{n}^{2}\) is the variance of additive white Gaussian noise with zero mean. The average SNR is defined in [36] as  
\[\overline{\gamma_{F}}=\frac{2P_{A}^{2}R^{2}}{\sigma_{n}^{2}}E\Big{[}h_{FSO}^{ 2}\Big{]}=\frac{2P_{A}^{2}R^{2}}{\sigma_{n}^{2}}(\varsigma M_{o}I_{FSO})^{2} \tag{5}\]  
According to [36], (1) can be rewritten as a function of \(\gamma_{F}\) instead of \(h_{FSO}\), as  
\[f_{\gamma_{F}}(\gamma_{F})=\frac{\varsigma^{2}}{\gamma_{F}\Gamma(\alpha) \Gamma(\beta)}G_{1,3}^{3,0}\bigg{(}\alpha\beta\xi\sqrt{\frac{\gamma_{F}}{ \overline{\gamma_{F}}}}|\varsigma_{2,\alpha,\beta}^{2+1}\bigg{)} \tag{6}\]