# A Non-WSSUS Channel Simulator for V2X Communication Systems
## 6 Simulation Examples
### Channel Estimation Techniques for the IEEE 802.11p Standard
#### 6.1.2 Spectral Temporal Averaging Technique  
The STA technique presented in [32] takes as the initial point the LS estimator in (15). This estimation scheme is intended for time-varying channels, as it tracks the channel variations according to:  
\[\langle\mathbf{\hat{Y}}_{zz}^{\mathrm{d}}\rangle_{m,k}=\frac{\langle\mathbf{ Y}^{\mathrm{d}}\rangle_{m,k}}{\langle\mathbf{\hat{H}}_{zz}^{\mathrm{d}} \rangle_{m,k-1}} \tag{17}\]  
for \(m=1,2,\ldots,N_{p}\) and \(k=2,3,\ldots,F\), where \(\mathbf{\hat{H}}_{zz}^{\mathrm{d}}\) and \(\mathbf{\hat{Y}}_{zz}^{\mathrm{d}}\) are the estimated channel and the equalized data matrix, respectively, when the STA technique is employed. In the STA technique, the \(k\)-th equalized symbol represented by \(\langle\mathbf{\hat{Y}}_{zz}^{\mathrm{d}}\rangle_{m,k}\) depends on the estimate \(\langle\mathbf{\hat{H}}_{zz}^{\mathrm{d}}\rangle_{m,k-1}\) computed from the previous data symbol. The first estimate \(\langle\mathbf{\hat{H}}_{zz}^{\mathrm{d}}\rangle_{m,1}=\langle\mathbf{\hat{H} }_{zz}^{\mathrm{d}}\rangle_{m,1}\) corresponds to that obtained with the LS scheme from (15).  
Then, the equalized symbol \(\langle\tilde{\mathbf{Y}}^{\text{d}}_{zz}\rangle_{m,k}\) shown in (17) is demodulated in order to obtain an estimate \(\langle\tilde{\mathbf{X}}^{\text{d}}\rangle_{m,k}\) of the transmitted OFDM data symbols, where \(\tilde{\mathbf{X}}^{\text{d}}\) represents the estimation of the transmitted data matrix. Such estimates are used to obtain dynamic channel estimations based on:  
\[\langle\tilde{\mathbf{H}}^{\text{d}}_{sc}\rangle_{m,k}=\frac{\langle\mathbf{Y}^ {\text{d}}\rangle_{m,k}}{\langle\tilde{\mathbf{X}}^{\text{d}}\rangle_{m,k}} \tag{18}\]  
for \(m=1,2,\ldots,N_{p}\) and \(k=2,3,\ldots,F\) where \(\tilde{\mathbf{H}}^{\text{d}}_{sc}\) denotes an estimate of the channel matrix.  
The estimates \(\langle\tilde{\mathbf{H}}^{\text{d}}_{sc}\rangle_{m,k}\) given in (18) are affected by AWGN. For this reason, an averaging operation is computed for each subcarrier as follows:  
\[\langle\tilde{\mathbf{H}}^{\text{d}}\rangle_{m,k}=\frac{1}{2\beta+1}\sum_{n=- \beta}^{n=+\beta}\langle\tilde{\mathbf{H}}^{\text{d}}_{sc}\rangle_{m,k+n} \tag{19}\]  
for \(m=1,2,\ldots,N_{p}\) and \(k=2,3,\ldots,F\), where \(\tilde{\mathbf{H}}^{\text{d}}\) is a matrix that contains all the averages in the frequency domain and \(\beta\) is an integer variable that controls the number of subcarriers in the averaging process. In general, when we have a frequency selective channel, the value of \(\beta\) must be small. On the other hand, for non-selective channels \(\beta\) must be higher [32]. Then, the estimated channel is given by:  
\[\langle\tilde{\mathbf{H}}^{\text{d}}_{zz}\rangle_{m,k}=\left(1-\frac{1}{\alpha }\right)\langle\tilde{\mathbf{H}}^{\text{d}}_{zz}\rangle_{m,k-1}+\frac{1}{ \alpha}\langle\mathbf{H}^{\text{d}}\rangle_{m,k} \tag{20}\]  
for \(m=1,2,\ldots,N_{p}\) and \(k=2,3,\ldots,F\), where \(\alpha\) is an updating and weighting factor restricted to \(\alpha>1\) and \((1-\,1/\alpha)>0\). A large value of \(\alpha\) is used in slow-fading channels, while a small value is preferred for fast-fading channels [32]. Finally, the channel estimation shown in Equation (20) is used in (17) to equalize the next OFDM data symbol and repeat the whole process until the end of the frame.