# Interference Analysis for Vehicle-to-Vehicle Communications at 28 GHz
## 4 Ergodic Capacity  
We now provide the ergodic capacity expression for the above mentioned scenario. The ergodic capacity can be defined as  
\[C_{erg}\equiv\int_{0}^{\infty}\int_{0}^{\infty}\log\left(\frac{\lambda_{c}}{ \lambda_{i}}\right)f_{\lambda_{c}\lambda_{i}}\left(\lambda_{c},\lambda_{i} \right)d\lambda_{c}d\lambda_{i}=F_{1}+F_{2}-F_{3}, \tag{16}\]  
where \(F_{1}=\int_{0}^{\infty}\log\left(\lambda_{c}\right)\times f_{\lambda_{c}} \left(\lambda_{c}\right)\times F_{\lambda_{i}}\left(\lambda_{c}\right)d \lambda_{c}\), \(F_{2}=\int_{0}^{\infty}\log\left(\lambda_{i}\right)\times f_{\lambda_{i}} \left(\lambda_{i}\right)F_{\lambda_{c}}\left(\lambda_{i}\right)d\lambda_{i}\), and \(F_{3}=\int_{0}^{\infty}\log\left(\lambda_{i}\right)f_{\lambda_{i}}\left( \lambda_{i}\right)d\lambda_{i}\). Using the identities of Meijer G function ([37] (8.4.3), (8.4.6)) and then applying ([37] (8.3.1)), we obtain  
\[F_{1}=\frac{2^{-\left(\mu_{c}+\mu_{i}+2\right)}\alpha_{c}\left( \bar{\lambda}_{c}\right)^{\left(-\frac{\alpha_{c}\mu_{c}}{2}\right)}}{\Gamma \left(\mu_{c}\right)\Gamma\left(\mu_{i}+1\right)\left(\bar{\lambda}_{i} \right)^{\left(\frac{\alpha_{i}\mu_{i}}{2}\right)}}\times\sum_{k=0}^{\infty} \frac{klc_{k,B}}{\left(\mu_{c}\right)_{k}}\times\sum_{n=0}^{\infty}\frac{n!m_ {n,i}}{\left(\mu_{i}+1\right)_{n}}\] \[\times\sum_{r=0}^{k}\left(-1\right)^{r}2^{r}\left( \begin{matrix} k+\mu_{c}-1\\ k-r\end{matrix} \right)\sum_{s=0}^{n}\frac{\left(-1\right)^{s}2^{s}\left(\mu_{i}+ 1\right)^{s}\left( \begin{matrix}n+\mu_{i}\\ n-s\end{matrix} \right)}{s!\mu_{i}^{s}\left(\bar{\lambda}_{i}\right)^{\left( \frac{\alpha_{i}\mu_{i}}{2}\right)}}\] \[\times H_{1,0,0,1,2,2}^{0.1;1;0.1;2}\left( \begin{matrix}1-\left( \mu_{c}+r+\frac{\bar{\alpha}_{i}\left(\mu_{i}+s\right)}{\alpha_{c}}\right) \Big{|}_{\begin{matrix}0,1\end{matrix}\right)\left(\begin{matrix}1,1\end{matrix} \right)\left(\begin{matrix}1,1\end{matrix}\right)}\\ \left(0,1\right)\left(\begin{matrix}1,1\end{matrix}\right)\left(\begin{matrix} 0,1\end{matrix}\right)\end{matrix} \xi_{3},\xi_{4}\right)\] \[\times\left(2\left(\bar{\lambda}_{c}\right)^{\frac{\alpha_{c}}{2} }\right)^{\left(\frac{\bar{\alpha_{i}}\left(\mu_{i}+s\right)}{\alpha_{c}}+\mu_ {c}+r\right)}, \tag{17}\]  
where \(\xi_{3}=\frac{\left(2\left(\bar{\lambda}_{c}\right)^{\frac{\alpha_{c}}{2}} \right)^{\frac{\alpha_{c}}{2}}}{2\left(\lambda_{i}\right)^{\frac{\alpha_{c}}{2} }}\), \(\xi_{4}=2^{\frac{2}{\alpha_{c}}}\bar{\lambda}_{c}\), and \(H_{a,bc,d,e,f}^{g,h;j,k,l}(. )\) is the EGBFHF function [38]. The EGBFHF function can be easily implemented in standard computation softwares like Mathematica. Due to limitation of space, we omit the computation of \(F_{2}\) which can obtained by following the steps of \(F_{1}\). For \(F_{2}\), we get the expression like \(F_{1}\), however, the values of \(\alpha_{c},\mu_{c}\) and \(\bar{\lambda}_{c}\) in \(F_{1}\) are replaced by \(\alpha_{i},\mu_{i}\) and \(\bar{\lambda}_{i}\) and vice versa. In order to get \(F_{3}\), we use ([37] (8.4.3), (8.4.6)) along with identity ([36] (8.970)) which finally yields  
\[F_{3}=\sum_{k=0}^{\infty}\sum_{l=0}^{k}\frac{\left(-1\right)^{l} 2^{\left(l-\mu_{i}-0.5\right)}klc_{k,i}\left( \begin{matrix}k+\mu_{i}-1\\ k-l\end{matrix} \right)}{\left(2\pi\right)^{\left(\alpha_{i}-0.5\right)}l!\Gamma \left(\mu_{i}\right)\left(\mu_{i}\right)k_{\bar{\lambda}}\left(\bar{\lambda}_{ i}\right)^{\left(\frac{\alpha_{i}\left(\mu_{i}+1\right)}{2}\right)}}\] \[\times G_{2u_{i},2u_{i}}^{2+2\alpha_{i}\left(\frac{1}{16\left( \bar{\lambda}_{i}\right)^{\alpha_{i}}}\right|}\frac{\[PERSON]\left(\alpha_{i},\xi_{5} \right),\[PERSON]\left(\alpha_{i},\xi_{6}\right)}{\[PERSON]\left(2,0\right),\[PERSON]\left(\alpha _{i},\xi_{7}\right),\[PERSON]\left(\alpha_{i},\xi_{8}\right)}\Bigg{)}, \tag{18}\]where \(\xi_{5}=-\frac{\alpha_{i}(\mu_{i}+l)}{2}\), \(\xi_{6}=1-\frac{\alpha_{i}(\mu_{i}+l)}{2}\), \(\xi_{7}=-\frac{\alpha_{i}(\mu_{i}+l)}{2}\), and \(\xi_{8}=1-\frac{\alpha_{i}(\mu_{i}+l)}{2}\).