## 3 Theoretical Background
### Radiofrequency Impairment: CFO
\[\epsilon=\frac{\Delta f}{f_{1}}=\Delta f\frac{N_{FFT}}{F_{sampling}}\] \[y(n)=\sum_{\Gamma=0}^{\tau-1}x(n-\Gamma)\times h(\Gamma)\times e^{\frac{2\pi n \epsilon}{N_{FFT}}}+W(n) \tag{5}\]0  
As previously presented, the main impact of the CFO on the OFDM subcarrier is the presence on Intercarrier Interference (ICI) between the main subcarriers and the adjacent subcarriers. Figures 6 and 7 show a comparison without and with the CFO, respectively, after the Fast Fourier Transform operation performed at the receiver side [34]. The greater the CFO value is, the more intense the ICI between subcarriers. Then, the ICI increases with the number of subcarriers used per OFDM symbol transmission, as the subcarrier spacing value diminishes when the number of subcarriers increases. * Doppler effect case study In the case of the receiver mobility, a Doppler shift affects the signal frequency and then the OFDM subcarriers. According to the moving direction and the various speeds of the receiver, the values of the Doppler shift vary, inducing a transmitter frequency decrease or increase [36]. Figure 8 presents an illustration of the Doppler effect when the receiver is moving. Let us use the following notations:
* \(\Delta f_{D}\) is the Doppler shift, which is also the frequency difference between the transmitter frequency and the receiver frequency, previously called \(\Delta f\). * \(v\) is the vehicle speed, which varies \((m/s)\). * \(f_{0}\) is the transmitter frequency \((Hz)\). * \(c\) is the speed of light (electromagnetic wave) \((3\times 10^{8}m/s)\). * \(\varphi\) is the angle between the direction of motion and the line of sight to the transmitter \((0<\varphi<\pi)\). * \(\Delta f_{D}=v\times\frac{f_{0}}{c}\times\cos(\varphi)\) There are three specific cases of Doppler shift:
* Case 1: the vehicle is moving towards the transmitter, so \(0^{\circ}<\varphi<90^{\circ}\), i.e., \(1>\cos(\varphi)>0\), and finally \(\Delta f_{D_{max}}>\Delta f_{D}>0\). * Case 2: the vehicle is moving away from the transmitter, so \(90^{\circ}<\varphi<180^{\circ}\), i.e., \(0>\cos(\varphi)>-1\) (\(\cos(\varphi)\) is negative), and finally \(0>-\Delta f_{D}>-\Delta f_{D_{max}}\). * Case 3: the vehicle is driving around the transmitter in circles; \(\varphi=90^{\circ}\), \(\cos\varphi=0\), and the transmit frequency remains unchanged. \[y(n)=\sum_{\Gamma=0}^{\tau-1}x(n-\Gamma)\times h(\Gamma)\times e^{\frac{2\pi n \epsilon}{N_{FFT}}}+W(n) \tag{5}\]1 OFDM subcarriers without CFO—adapted from [34]. To conclude, the Doppler shift is directly proportional to the carrier frequency \(f_{0}\) of the transmitter. Considering the same OFDM mode (number of subcarriers), the lower the frequency \(f_{0}\), the smaller the Doppler effect. Hence, the OFDM subcarrier spacing is weakly affected. Conversely, the higher the frequency \(f_{0}\), the larger the Doppler effects. Hence, the OFDM subcarrier spacing is highly affected. From all the above, the normalized CFO expression is defined, assuming the DVB-T2 transmission parameters in Equation (13). \[y(n)=\sum_{\Gamma=0}^{\tau-1}x(n-\Gamma)\times h(\Gamma)\times e^{\frac{2\pi n \epsilon}{N_{FFT}}}+W(n) \tag{5}\]2  
The maximum value of the Doppler shift is achieved when \(\cos(\varphi)\) is equal to \(\pm 1\). The absolute value of the Doppler shift is then computed. If the maximum Doppler shift is considered, then the normalized CFO (\(\epsilon\)) is equal to the expression in Equation (14). \[y(n)=\sum_{\Gamma=0}^{\tau-1}x(n-\Gamma)\times h(\Gamma)\times e^{\frac{2\pi n \epsilon}{N_{FFT}}}+W(n) \tag{5}\]3  
\[y(n)=\sum_{\Gamma=0}^{\tau-1}x(n-\Gamma)\times h(\Gamma)\times e^{\frac{2\pi n \epsilon}{N_{FFT}}}+W(n) \tag{5}\]4 Doppler effect. \[y(n)=\sum_{\Gamma=0}^{\tau-1}x(n-\Gamma)\times h(\Gamma)\times e^{\frac{2\pi n \epsilon}{N_{FFT}}}+W(n) \tag{5}\]5 CFO impact on subcarriers—adapted from [34]. The black curve represents the expected subcarrier curve at the receiver, and the gray curve represents the subcarrier curve at the receiver with a frequency offset (CFO).