# Second-Order Statistical Properties of Vehicle-to-Vehicle Rician Fading Channel
## 2 Description of the Theoretical Model for Non-Isotropic V2V Flat Fading Channel  
This section concisely overviews the narrow-band single-user SISO V2V Rician fading channel. The model assumes that both Tx and Rx are mobile and utilize low-height antennas. The propagation occurs in an outdoor environment and is characterized by 2D non-isotropic scattering with a line of sight (LoS) and non-line of sight (NLoS), or a combination of both propagations between the Tx and Rx. Figure 1 illustrates a 2D simplified geometric two-ring model for a V2V Rician fading channel characterized by both LoS and NLoS propagation akin to [10]. The model combines LoS components and diffused components, which is a combination of the single-bounce components at the transmitter (SBT), single-bounce components at the receiver (SBR), and double-bounce components (DB) at the two rings. The SGTR model defines two rings of effective scatterers, one around Tx and the other around Rx, representing the effect of many scatterers with similar spatial locations. Hence, the received complex faded envelope between the Tx-Rx link (\(A_{T}-A_{R}\)) can be expressed as combining the LoS, SBT, SBR, and DB multipath components. \[ \begin{array}{l}h(t)=h^{LOS}(t)+h^{DIF},\\ h^{DIF}=h^{SBT}(t)+h^{SBR}(t)+h^{DB}(t),\end{array}  \tag{1}\]  
The LoS, SBT, SBR, and DB components of the received faded complex envelope are further given as  
Figure 1: SGTR model for MIMO V2V channels with \(M_{T}=M_{R}=2\) antenna elements, showing only double-bounce rays (DB). \[h^{LoS}(t)=\sqrt{\frac{K}{K+1}}\times e^{\beta 2\pi t[f_{Tmax}cos(\pi-a_{R}^{ (LoS)}+\gamma_{T})+f_{Rmax}cos(a_{R}^{(LoS)}-\gamma_{R})]} \tag{2}\] \[h^{SBT}(t)=\sqrt{\frac{\eta_{T}}{K+1}}\underset{M\to\infty}{\lim} \sum_{m}^{M}\frac{a_{m}}{\sqrt{M}}e^{j\phi_{m}}\times e^{j2\pi t[f_{Tmax}cos \left(a_{T}^{(m)}-\gamma_{T}\right)+f_{Rmax}cos\left(a_{R}^{(m)}-\gamma_{R} \right)]}\] (3) \[h^{SBR}(t)=\sqrt{\frac{\eta_{R}}{K+1}}\underset{N\to\infty}{\lim} \sum_{m}^{M}\frac{a_{n}}{\sqrt{N}}e^{j\phi_{n}}\times e^{j2\pi t[f_{Tmax}cos \left(a_{T}^{(n)}-\gamma_{T}\right)+f_{Rmax}cos\left(a_{R}^{(n)}-\gamma_{R} \right)]}\] (4) \[h^{DB}(t)=\sqrt{\frac{\eta_{TR}}{K+1}}\underset{M,N\to\infty}{ \lim}\sum_{m,n=1}^{M,N}\frac{a_{mn}}{\sqrt{MN}}e^{j\phi_{nm}}\times e^{j2\pi t [f_{Tmax}cos\left(a_{T}^{(m)}-\gamma_{T}\right)+f_{Rmax}cos\left(a_{R}^{(n)}- \gamma_{R}\right)]} \tag{5}\]  
In Equations (2)-(5), \(K\) denotes the Rician factor, and the parameters \(\eta_{T}\), \(\eta_{R}\), and \(\eta_{TR}\) specify the relative contributions of the SBT, SBR, and DB to the average power \(E\left[\left|h(t)\right|^{2}\right]\), and satisfy the relationship \(\eta_{T}+\eta_{R}+\eta_{TR}=1\). \(a_{m},a_{n}\), and \(a_{mn}\) represent random path gain and are chosen to satisfy the limiting condition of \(\alpha_{m}^{2}=\alpha_{n}^{2}=\alpha_{mn}^{2}=1\), respectively, ensuring that \(E\left[\left|h(t)\right|^{2}\right]\) is normalized to unity. The Tx and Rx are moving at speeds of \(v_{T}\) and \(v_{R}\) in the directions of \(\gamma_{T}\) and \(\gamma_{R}\), resulting in the maximum Doppler frequencies of \(f_{Tmax}=\frac{v_{T}}{\lambda}\) and \(f_{Rmax}=\frac{v_{R}}{\lambda}\), where \(\lambda\) is the carrier wavelength. The angles of departure (AoDs) are denoted by \(\alpha_{T}^{(m)}\) and \(\alpha_{T}^{(n)}\), and impinge on the Tx, Rx scatterers \(S_{T}^{(m)}\) and \(S_{R}^{(n)}\), while \(\alpha_{R}^{(m)}\) and \(\alpha_{R}^{(n)}\) denote the angles of arrival (AoAs) scattered from the \(S_{T}^{(m)}\) and \(S_{R}^{(n)}\) of Tx and Rx, respectively. The AoD (\(\alpha_{T}^{(m)}\), \(\alpha_{T}^{(n)}\)) and AoA (\(\alpha_{R}^{(m)}\), \(\alpha_{R}^{(n)}\)) are assumed to be a random variable; notably, the phases \(\phi_{m}\), \(\phi_{n}\), and \(\phi_{mn}\) are also assumed to be uniformly distributed random variables from the interval of [-\(\pi,\pi\)] and are independent of (\(\alpha_{T}^{(m)}\), \(\alpha_{T}^{(n)}\)) and (\(\alpha_{R}^{(m)}\), \(\alpha_{R}^{(n)}\)). Also, \(\alpha_{R}^{(LoS)}\) is assumed to be \(\pi\).