# A Novel GBSM for Non-Stationary V2V Channels Allowing 3D Velocity Variations
## 3 Proposed V2V Channel Model  
The basic GBSM is suitable for V2V channels, but it is worth mentioning that all channel parameters need to be calculated in real time. To make this model more applicable to real-world V2V communications, the effects of the velocity or trajectory need to be considered. In this paper, we approximated the velocity with the form of the momentum as:  
\[\mathbf{v}^{i,Ci_{n,m}^{i}}(t)=\mathbf{v}^{i,Ci_{n,m}^{i}}(0)+\int_{0}^{t} \mathbf{a}^{i,Ci_{n,m}^{i}}(t^{\prime})dt^{\prime} \tag{5}\]  
where \(\mathbf{v}^{i,Ci_{n,m}^{i}}(0)\) denotes the initial relative speed vector and \(\mathbf{a}^{i,Ci_{n,m}^{i}}(t)\) denotes the corresponding speed acceleration vector, respectively. For the 3D time-variant speed acceleration, we can obtain:  
\[ \begin{split}&\left[\mathbf{v}_{\mathbf{x}}^{i,Ci_{n,m}^{i}}(t) \quad\mathbf{v}_{y}^{i,Ci_{n,m}^{i}}(t)\quad\mathbf{v}_{z}^{i,Ci_{n,m}^{i}}(t) \right]\\ &=\left[\mathbf{v}_{\mathbf{x}}^{i,Ci_{n,m}^{i}}(0)\quad\mathbf{ v}_{y}^{i,Ci_{n,m}^{i}}(0)\quad\mathbf{v}_{z}^{i,Ci_{n,m}^{i}}(0)\right]+\int_{0}^{t} \left[\mathbf{a}_{\mathbf{x}}^{i,Ci_{n,m}^{i}}(t^{\prime})\quad\mathbf{a}_{ \mathbf{y}}^{i,Ci_{n,m}^{i}}(t^{\prime})\quad\mathbf{a}_{\mathbf{z}}^{i,Ci_{ n,m}^{i}}(t^{\prime})\right]dt^{\prime}.\end{split}  \tag{6}\]  
By taking into account the 3D relative velocity speed and acceleration, both the relative speed and acceleration correspond to the xyz axis and can be rewritten as:  
\[\|\mathbf{v}\|^{i,Ci_{n,m}^{i}}(t)=\|\mathbf{v}\|^{i,Ci_{n,m}^{i}}(0)+\int_{0} ^{t}\|\mathbf{a}\|^{i,Ci_{n,m}^{i}}(t^{\prime})dt^{\prime} \tag{7}\]  
\[\mathbf{q}_{\mathbf{v}}^{i,Ci_{n,m}^{i}}(t)=\mathbf{q}_{\mathbf{v}}^{i,Ci_{n, m}^{i}}(0)+\int_{0}^{t}\mathbf{b}^{i,Ci_{n,m}^{i}}(t^{\prime})dt^{\prime} \tag{8}\]  
where \(\mathbf{q}_{\mathbf{v}}^{i,Ci_{n}^{i}}(0)\) and \(\mathbf{b}^{i,Ci_{n}^{i}}(t)\) denote the initial relative movement direction and the acceleration of the direction, respectively. Under the consideration of moving scatterers, the Doppler frequency can be described as [27]:  
\[f_{n,m}(t)=f_{n,m}^{Tx}(t)+f_{n,m}^{Tx}(t)+\cdots+f_{n,m}^{Tx}(t)+f_{n,m}^{Rx} (t) \tag{9}\]  
where \(f_{n,m}^{Tx}(t)\), \(f_{n,m}^{Rx}(t)\), \(f_{n,m}^{Tx}(t)\), \(f_{n,m}^{Tx}(t)\) mean the Doppler frequencies due the movements of terminals and scatterers, respectively. Since the scatterers of the virtual link are unknown and have a random movement, it is reasonable to assume that the mean value of the Doppler frequency is zero, which is caused by the virtual scatterers. Moreover, (9) can be simplified as:  
\[f_{n,m}(t)=\frac{1}{\lambda}\bigg{(}\|\mathbf{v}\|^{Tx,Ci_{n,m}^{Tx}}(t) \cdot\mathbf{q}_{\mathbf{v}}^{Tx,Ci_{n,m}^{Tx}}(t)\cdot\mathbf{s}_{n,m}^{Tx }(t)+\|\mathbf{v}\|^{Ci_{n,m}^{Rx},Rx}(t)\cdot\mathbf{q}_{\mathbf{v}}^{Cx_{n,m}^{Rx}}(t)\cdot\mathbf{s}_{n,m}^{Rx}(t)\bigg{)} \tag{10}\]where \(\|\textbf{v}\|^{Tx_{c}C_{u,m}^{Tx}}(t)\) indicates the correlative velocity of the Tx and cluster \(C_{n,m}^{Tx}\) and \(\|\textbf{v}\|^{C_{u,m}^{Rx}}_{c,m}(t)\) defines the speed of the Rx and cluster \(C_{n,m}^{Rx}\), respectively. For the velocities that are changing, the movement direction vectors can be defined as:  
\[\textbf{q}_{v}^{Tx_{c}C_{u,m}^{Tx}}(t)=\textbf{q}_{v}^{Tx_{c}C_{u,m}^{Tx}}(0)+ \int_{0}^{t}\textbf{b}^{Tx_{c}C_{u,m}^{Tx}}(t^{\prime})dt^{\prime}= \begin{bmatrix} \cos\alpha_{v}^{Tx_{c}C_{u,m}^{Tx}}(t)\cos\varphi_{v}^{Tx_{c}C_{u,m}^{Tx}}(t) \\ \cos\alpha_{v}^{Tx_{c}C_{u,m}^{Tx}}(t)\sin\varphi_{v}^{Tx_{c}C_{u,m}^{Tx}}(t) \\ \sin\alpha_{v}^{Tx_{c}C_{u,m}^{Tx}}(t)\end{bmatrix} \]  
\[\textbf{q}_{v}^{C_{u,m}^{Tx},Rx}(t)=\textbf{q}_{v}^{C_{u,m}^{Rx},Rx}(0)+\int_{ 0}^{t}\textbf{b}^{Cx_{u,m}^{Rx},Rx}(t^{\prime})dt^{\prime}= \begin{bmatrix} \cos\alpha_{v}^{Rx_{c}C_{u,m}^{Rx}}(t)\cos\varphi_{v}^{Rx_{c}Rx}(t)\\ \cos\alpha_{v}^{C_{u,m}^{Rx},Rx}(t)\sin\varphi_{v}^{C_{u,m}^{Rx},Rx}(t)\\ \sin\alpha_{v}^{Rx_{c}C_{u,m}^{Rx}}(t)\end{bmatrix} .