## 2 Background
### Single Ring Geometrical Scattering Channel Model  
To test the channel estimation techniques, we need to define how to represent the communication channel. This representation needs to take most critical physical effects of the vehicular environment into consideration, and it also needs to be as close to reality as possible in order to properly evaluate the system performance. Considering that the vehicular channel does not fulfill the WSSUS condition, the channel representation proposed in [14; 15; 16] is used in this work. The channel introduced in [14; 15; 16] is a statistical geometrical model for mobile-to-mobile systems. The scenario has  
Figure 4: IEEE 802.11p packet preamble structure. Figure 3: IEEE 802.11p subcarrier distribution. a mobile transmitter with variable velocity and acceleration, as well as a receptor moving at a variable velocity and acceleration. The receiver is surrounded by interfering objects positioned in a circle whose center is the receptor itself. Therefore, the model depicts a realistic representation of a mobile to mobile system scenario. Furthermore, the model proposed in [14; 15; 16] is physically complete but mathematically simple in comparison with other non-WSSUS geometry-based statistical channel models. In the model presented in Figure 5, \(V_{T}\) corresponds to the vector describing the velocity of the transmitter, \(V_{R}\) is the vector describing the velocity of the receptor, \(a_{T}\) represents the vector describing the acceleration of the transmitter, \(a_{R}\) denotes the vector describing the acceleration of the receptor, \(d\) comes to be the radius of the ring, and \(l\)-\(th>IO\) corresponds to the \(l\)-\(th\) interfering object. As a result, the channel transfer function is given as follows [14; 15; 16],  
\[h(\tau,t)=\sum_{l=1}^{\mathcal{L}}g_{l}exp\{j[\theta_{l}-2\pi f_{c}\tau_{l}(t)] \}\delta[t-\tau_{l}(t)]\Omega_{T_{0}}(t-t_{0}), \tag{2}\]  
where \(f_{c}\) corresponds to the central frequency of the OFDM signal, whereas \(g_{l}\), \(\theta_{l}\), and \(\alpha_{l}^{P}\) are random variables. Specifically, \(g_{l}\) is a complex attenuation factor and follows a uniform distribution between \([0,\sqrt{\frac{1}{2\mathcal{L}}}]\), \(\theta_{l}\) is the phase shift that follows a uniform distribution between \([0,2\pi]\), and \(\alpha_{l}^{P}\) is the angle between the \(l\)-\(th>IO\) and the transmitter or receiver and follows a uniform distribution between \([0,2\pi]\). However, \(\Omega_{T_{0}}(t-t_{0})\) is a windowing function equal to \(1\) if \(t\in[t_{0},t_{0}+T_{0}]\) and \(0\) otherwise, where \(t_{0}\) is the reference point where the observation begins and \(T_{0}\) is the length of the observation window. This windowing function is used in order to ignore the large scale effects. The time-varying propagation delays of the multipath components are given as [14; 15; 16],  
\[\tau_{l}(t)=\frac{d_{l}^{T}+d_{l}^{R}}{c}-\frac{t}{f_{c}}\left[f_{l}^{S}+\frac {f_{l}^{S}(t)}{2}\right], \tag{3}\]  
where \(f_{l}^{S}\) and \(\dot{f}_{l}^{S}\) are the Doppler frequency shift caused by the vehicles' velocity and acceleration, respectively, and are defined in the following form,  
\[f_{l}^{S}=f_{max}^{T}cos(\alpha_{l}^{T}-\gamma_{T})+f_{max}^{R}cos(\alpha_{l}^ {R}-\gamma_{R}), \tag{4}\]  
\[f_{l}^{S}(t)=f_{max}^{T}(t)cos(\alpha_{l}^{T}-\beta_{T})+f_{max}^{R}(t)cos( \alpha_{l}^{R}-\beta_{R}), \tag{5}\]  
where \(d_{l}^{T},d_{l}^{R}\) are the distances of the transmitter and the receiver to the \(l-th\)\(IO\), \(c\) is the speed of light, whereas \(\gamma_{T}\) and \(\gamma_{R}\) are the velocity vector angles of the transmitter and receiver velocity vector, respectively. At the same time, \(\beta_{T}\) and \(\beta_{R}\) are the acceleration vector angles of the transmitter and receiver, respectively.