## 3 Theoretical Background
### Radiofrequency Impairment: CFO  
Various kinds of radiofrequency (RF) impairments exist in wireless communication systems. There are thermal and flicker noises, local oscillator phase noise, sampling jitter, Digital-to-Analog Converter (DAC) and Analog-to-Digital Converter (ADC) quantization noise and clipping, quadrature imbalance, carrier frequency offset (CFO), and sampling frequency offset (SFO) [34]. As previously discussed in the introduction, multicarrier systems are sensitive to synchronization errors. They induce system performance degradation. This deterioration increases with the number of subchannels. As the DVB-T2 system includes a high number of subcarriers (1024, 2048, 4096, 8192, 16,384, and 32,768), it constitutes a subject of interest to evaluate the impact of carrier frequency offset (CFO) on the system performance. The description of the CFO is presented in the following part. * CFO description  
Generally, in communication systems, the carrier frequencies are generated from frequency synthesizers, commonly Phase-Locked Loops (PLLs), using precise crystal oscillators whose performance is specified with a certain precision in parts per million (ppm) [34]. The CFO refers to the deviation in the actual receiver carrier frequency (RX carrier frequency) from the nominal carrier frequency (TX carrier frequency) in these systems. This frequency offset is caused by two factors. The first factor is the Doppler shift due to the relative mobility of the transmitter and the receiver (and even the channel), and the second one is the difference between the frequencies of the local oscillators at the transmitter and receiver. As DVB-T2 system represents the case study, only the receiver is mobile. The transmitter is fixed in the terrestrial digital broadcasting system. The CFO impact has to be evaluated in two different cases: when only Gaussian noise is used and when a fading channel and Gaussian noise are used together; the OFDM system including frequency offset models are presented in Figures 4 and 5, respectively. Let us assume that the carrier frequency offset does not change between two consecutive symbols. The received y(n) OFDM symbol when only Gaussian noise is considered and when fading channel and Gaussian noise are considered can then be expressed as Equations (4) and (5), respectively. \[y(n)=x(n)\times e^{\frac{2\pi n\epsilon}{N_{FFT}}}+W(n) \tag{4}\]  
\[y(n)=\sum_{\Gamma=0}^{\tau-1}x(n-\Gamma)\times h(\Gamma)\times e^{\frac{2\pi n \epsilon}{N_{FFT}}}+W(n) \tag{5}\]  
\[\epsilon=\Delta f\times N_{FFT}\times T_{sampling}=\frac{\Delta f}{f_{1}}= \Delta f\times T_{s} \tag{6}\]  
\[\Delta f=|f_{cxx}-f_{ctx}| \tag{7}\]  
- x(n) is the time-domain OFDM transmitted symbol. - y(n) is the received OFDM symbol. - \(\epsilon\) is the normalized frequency offset given by Equation (6) [34]. \(\Delta f\) represents the frequency difference between the transmitter and the receiver. \(T_{samping}\), \(f_{1}\), \(T_{s}\), and \(N_{FFT}\) are, respectively, the sampling period, the OFDM subcarrier spacing, the OFDM symbol period, and the whole number of subcarriers. - W(n) is the Gaussian noise signal added to the frequency shifted signal  
- h(n) is the channel impulse response of length \(\tau\)  
The OFDM symbol period is computed using Equation (8), where \(f_{1}\) is defined in Equation (9). \(F_{sampling}\) is the frequency sampling defined in DVB-T2 using Equation (10) [35]. It depends on the channel bandwidth \(B\). \[T_{s}=\frac{1}{f_{1}} \tag{8}\]  
\[f_{1}=\frac{F_{sampling}}{N_{FFT}} \tag{9}\]  
\[F_{sampling}=\frac{64}{7}\times\frac{B}{8} \tag{10}\]  
When a fading channel is considered during signal transmission, a convolution operation is applied between each OFDM symbol and the channel impulse response (cf. Equation (5)). The normalized CFO (\(\epsilon\)), defined in Equation (6), becomes Equation (11). Figure 4: OFDM modulation with frequency offset model: Gaussian noise only. Figure 5: OFDM modulation with frequency offset model: fading channel and Gaussian noise only.