# Second-Order Statistical Properties of Vehicle-to-Vehicle Rician Fading Channel
## 3 LCR and AFD for the V2V Flat Fading Channel  
In this section, building upon the proposed SGTR framework outlined in (1), we will derive the analytical expressions for the LCR and AFD of the Rician complex faded envelope in a non-isotropic scattering environment. The LCR, denoted as \(L(R_{\zeta})\), represents the average number of times per second that the signal envelope \(\xi(t)=\)\(|h(t)|\) crosses a specified threshold level (\(R\)), considering both positive and negative slope crossings. Leveraging on the established probability density function (PDF) based methodology in [18], the LCR for the Rician flat fading channel can be expressed as  
\[ \begin{array}{c}L_{\zeta}(R)=\frac{2(R_{\zeta})}{\pi^{\frac{2}{ 2}}}\sqrt{B(K+1)}\times e^{-K-K(K+1)R_{\zeta}^{2}}\times\sqrt[n/2]{2}cosh(2 \sqrt{K(K+1)}R_{\zeta}cos\theta)\\ \times\Big{[}e^{-(\phi sin\theta)^{2}}+\sqrt{\pi}\phi\psi sin\theta.erf(\phi \psi sin\theta)\Big{]}d\theta,\end{array}  \tag{6}\]  
where \(cosh(. )\) is the hyperbolic cosine, the error function is denoted by \(erf(. )\), while \(B=\frac{\eta_{b}b_{2}-b_{1}^{2}}{b_{0}^{2}}\), \(\psi=\sqrt{K/2B(K+1)}\), and \(\phi=E+\ b_{1}/b_{0}\), where \(E=2\pi[f_{Tmax}cos(\pi-\alpha_{R_{q}}^{LoS}+\alpha_{R_{q}}^{LoS}+\alpha_{R_{q}}^ {LoS}+\alpha_{R_{q}}^{LoS}+\alpha_{R_{q}}^{LoS}+\alpha_{R_{q}}^{LoS}+\alpha_{R }^{LoS}+\alpha_{R_{q}}^{LoS}+\alpha_{R_{q}}^{LoS}+\alpha_{R_{q}}^{LoS}+\alpha_{R _{q}}^{LoS}+\alpha_{R_{q}}^{LoS}+\alpha_{R_{q}}^{LoS}+\alpha_{R_{q}}^{LoS}+ \alpha\(\gamma_{T}\)) \(+\)\(f_{R_{max}}cos(\alpha_{R_{q}}^{LOS}-\gamma_{R})\)]. Most importantly, the fundamental model parameter \(b_{m}\) is defined as  
\[b_{m}=\frac{d^{m}\rho h^{DIF}(\tau)}{2j^{m}d\tau^{m}}|_{\tau=0}, \tag{7}\]  
with \(\rho h^{DIF}(\tau)=E\big{[}h^{DIF}(\tau)h^{*DIF}(t-\tau)\big{]}/\Omega\), where \(j^{2}=-1\), \(\rho_{h^{DIF}}(\tau)\) is the time ACF of the diffuse component \(h^{DIF}(\tau)\) of the complex fading envelope, \((. )^{*}\) denotes the complex conjugate operation, and \(E[. ]\) designates the statistical expectation operator. The expression in (6) is a general one for LCR in a non-isotropic V2V Rician fading channel and is the same as the LCR models previously presented in ([19], Equation (29)) and ([15], Equation (3)) It can be reduced to expressions presented in ([12], Equation (6)) and ([13], Equation (43)) by setting the parameter \(E=0\), which corresponds to a time-invariant LoS component. In contrast, (6) can be used for a non-isotropic environment with a time-variant LoS component, while the expressions in [12; 13] are only applicable for a non-isotropic environment with a time-invariant LoS component. Based on the model introduced in Section 2, we proceeded to derive the important model parameter \(b_{m}\) which is characterized by the random AoA and AoD of the scatterer distribution. Leveraging the literature on scatterer distribution from prior research, we adopted the von Mises PDF, as it approximates many known distributions and enables close-form solutions for numerous relevant scenarios. The von Mises PDF is defined as \(f(\theta)=exp[(kcos(\theta-\mu)]/2\pi l_{0}(k)\)[20], where \(\theta\in[-\pi,\ \pi)\), \(l_{0}(. )\) denotes the zeroth-order Bessel function of the first kind, \(\mu\in[-\pi,\ \pi)\) is the mean angle of the scatterer distribution, and \(k\) controls the spread of the scatterers around the mean. Hence, substituting (1), (3)-(5) and the von Mises PDF expression for \(b_{m}\), we can derive the closed-form expressions for \(b_{0},\ b_{1}\), and \(b_{2}\).