# A Novel GBSM for Non-Stationary V2V Channels Allowing 3D Velocity Variations
## 5 Statistical Properties for the Proposed Model
### Time-Variant ACF  
For non-stationary channels, the time-variant ACF at different time instants turns into a function of the time delay \(\tau\) and time \(t\). It can be measured between random variables, which increases the predictability of random events. For continuous time-signals, the ACF characterizes the time correlation of a stochastic process. The ACF under non-stationary scattering scenarios is a time-variant function of time delay \(\tau\) and time \(t\). The time-variant ACF can be specified as:  
\[r_{hh}(t,\tau)=E\{h^{*}(t)h(t\ +\ \tau)\} \tag{29}\]  
where \((\cdot)^{*}\) defines the complex conjugation operation. The computation of (29) defines that the ACF shifts through the time delay. The time-variant ACF can be specified as:  
\[r_{hh}(t,\tau)=\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\int_{-\pi} ^{\pi}\begin{array}{l}\sqrt{p(\varphi_{n}^{Tx}(t+\tau),\alpha_{n}^{Tx}(t+ \tau))}\sqrt{p(\varphi_{n}^{Rx}(t),\alpha_{n}^{Rx}(t))}\\ \sqrt{p(\varphi_{n}^{Rx}(t+\tau),\alpha_{n}^{Rx}(t+\tau))}\sqrt{p(\varphi_{n} ^{Rx}(t),\alpha_{n}^{Rx}(t))}\\ e^{-2\pi\int_{t}^{t+\tau}f_{t,m}(\tau)d\tau}d\varphi_{n}^{Tx}d\alpha_{n}^{Tx} d\varphi_{n}^{Rx}d\alpha_{n}^{Rx}\end{array} \tag{30}\]  
\[r_{hh}(t,\tau)=\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\sqrt{p(\varphi_{n}^{i}(t+ \tau),\alpha_{n}^{i}(t+\tau))}\sqrt{p(\varphi_{n}^{i}(t),\alpha_{n}^{i}(t))}e^ {-2\pi\int_{t}^{t+\tau}f_{t,m}(\tau)d\tau}d\varphi_{n}^{i}d\alpha_{n}^{i} \tag{31}\]  
where \(i\in\{Tx,Rx\}\) when the clusters \(C_{n}^{Tx}\) and \(C_{n}^{Rx}\) are independent of each other, and (31) is equal to the product of the ACFs at Tx and Rx. For simplicity, the antenna array is positioned at the root of the coordinate system. With the help of (10), we can simplify (31) as:  
\[\tilde{r}_{hh}(t,\tau)=\frac{1}{(4\pi)^{m}I_{m/2-1}^{2}(\kappa)( \kappa^{1-m})} \tag{32}\] \[\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\left(\begin{array}{l}\sqrt{ \exp\kappa^{i}\Bigg{(}\cos\alpha\Bigg{(}\cos(\varphi^{i}(t+\tau)-\tilde{ \varphi}_{n}^{i}(t+\tau))\cos\alpha^{i}(t+\tau)\Bigg{)}\Bigg{)}}\\ \cdot\sqrt{\exp\kappa^{i}\Bigg{(}\cos\alpha\Bigg{(}\cos(\varphi^{i}(t)-\varphi _{n}^{i}(t))\cos\alpha^{i}(t)\Bigg{)}\Bigg{)}}\\ \cdot e^{-2\pi\int_{t}^{t+\tau}|\psi^{i}C_{n}^{i}(t)\cdot\varphi_{n}^{i}C_{n} ^{i}(t)\cdot\varphi_{n}^{i},m(t)}\end{array}\right)\]In (32), the form of the distribution remains fixed, and the relative angle offset is nearly fixed at a very short time interval when \(\kappa^{i}\) is constant or nearly constant over a short time interval, for a few milliseconds. Here, (32) can be described as:  
\[\begin{split}&\bar{r}_{hh}(t,\tau)=\frac{1}{(4\pi)^{\mu}l_{m/2-1} ^{2}(\kappa)^{(\kappa^{1-m})}}\\ &\int_{-\pi}^{\pi}\int_{-\pi}^{\pi}\left(\begin{aligned} \sqrt{\exp\kappa^{i}\left(\cos\alpha\left(\frac{\cos (\varphi^{i}(t+\tau)-\varphi_{n}^{i}(t+\tau))\cos\alpha^{i}(t+\tau)}{\cos\tilde {x}_{n}^{i}(t+\tau)}\right)}\\ \cdot\sqrt{\exp\kappa^{i}\left(\cos\alpha\left(\frac{\cos(\varphi^{i }(t)-\varphi_{n}^{i}(t))\cos\alpha^{i}(t)}{\cos\tilde{x}_{n}^{i}(t)+\sin \alpha^{i}(t)\sin\tilde{x}_{n}^{i}(t)}\right)}\right)\cdot e^{-j2\pi\int_{t}^{ t+\tau}A\cdot B}\end{aligned}\right)d\varphi_{n}^{i}da_{n}^{i}\end{split} \tag{33}\]  
where:  
\[\begin{split}& A=\left(\|\mathbf{v}\|^{i,C_{n}^{i}}(0)+\int_{0}^ {t}\|\mathbf{a}\|^{i,C_{n}^{i}}(t^{\prime})dt^{\prime}\right)\\ & B=\left(\cos\left(\varphi_{n}^{i}(t)-\varphi^{i,C_{n}^{i}}(t) \right)\cos\alpha_{n}^{i}(t)\cos\alpha_{n}^{i,C_{n}^{i}}(t)+\sin\alpha_{n}^{i }(t)\sin\alpha_{n}^{i,C_{n}^{i}}(t)\right).\end{split} \tag{34}\]  
The ACFs mentioned in (33) are more common and even appropriate for several propagation scenarios. Finally, by submitting the values of (34) into (33), we can obtain the approximate expression of this proposed model.