## 3 Theoretical Background
### Multicarrier Modulations
#### 3.1.1 OFDM  
The principle of OFDM is to group numerical data per package, and to modulate each piece of data by a different subcarrier at the same time. In this section, the OFDM block diagram is presented, and the numerical expression of the modulation is developed.  
* Transceiver description  
The OFDM transceiver scheme is presented in Figure 1. This modulation consists of three main steps: the symbol mapping, the Inverse Fast Fourier Transform (IFFT) processing (time-to-frequency-domain signal processing), and the cyclic prefix (CP) addition in emission. In reception, the steps are as follows: the CP cancellation, the Fast Fourier Transform (FFT) processing, and the symbol demapping. In DVB-T2, the symbol mapping method exploited is QAM. This mapping is performed on data containing redundant bits added by the LDPC Forward Error Correction (FEC) coder. After the mapping method in emission, the frequency-domain signal undergoes an IFFT operation which transforms the signal in a time-domain signal. The resulting signal is transmitted. The fading channel effects are applied to it. Also, Additive White Gaussian Noise (AWGN) is added to the signal obtained after the fading channel. In reception, an FFT operation is performed to convert the time-domain signal into a frequency-domain signal. As known, the CP and guard band include no useful data participating in the spectral efficiency reduction but allow one to reduce the Intersymbol Interference (ISI) and Adjacent-Channel Interference (ACI) impacts. Figure 2 shows the CP insertion process. \(T_{CP}\) denotes the CP duration. The OFDM symbol duration is extended using the CP, which represents the copy of a part of the OFDM symbol at the starting point of each symbol.  
\begin{table}
\begin{tabular}{l l l} \hline
**Papers and Years** & **Advantages** & **Limitations** \\ \hline
[29] (2021) & FBMC was proposed as an alternative waveform to overcome the issue concerning the spectral leakage interference. & The channel estimation was complex when dealing with FBMC. \\ \hline That work proposed a detailed BER value & The proposed solution depended on high \\ comparison for OFDM, UFMC, FBMC, \(t\)-OFDM, and & self-interference cancellation achievements, which \\ GFDM using full-duplex systems. & could be difficult to achieve in a practical situation. \\ \hline \end{tabular}
\end{table}
Table 2: _Cont._* Numerical signal expression  
The numerical development is performed by assuming complex data symbols named \(c_{k}\in(c_{0}\), \(c_{1}\), \(c_{2}\),..., \(c_{N_{FFT}})\). These data are QAM symbols formed by the grouping of n bits. The sequence of data normally undergoes an IFFT process. Let us assume that \(N_{FFT}\) is the number of OFDM subcarriers, and the time period \(T_{s}\) separates two sequences of \(N_{FFT}\) data. \(T_{s}\) is equal to one OFDM symbol duration. \(k\) is a positive integer (\(k\in(1,.....,N_{FFT})\)). During the modulation process, each \(c_{k}\) is modulated at the high frequency \(f_{k}\). Each individual signal is then expressed in a complex form as the multiplication of \(c_{k}\) by \(e^{2\pi f_{k}t}\). Therefore, the main signal \(x(n)\) corresponding to the total \(N_{FFT}\) data is shown as follows:  
\[x(n)=\sum_{k=0}^{N_{FFT}}c_{k}.e^{j2\pi f_{k}n} \tag{1}\]  
As the main peculiarity of OFDM is the orthogonality between subcarriers (orthogonal multiplexing), thus the subcarrier spacing is inversely proportional to the symbol period and is equal to \(\frac{1}{T_{s}}\). If we denote \(f_{0}\) the carrier frequency, \(f_{k}=f_{0}+\frac{k}{T_{s}}\). Then, \(x(n)\) can be expressed as:  
\[x(n)=e^{j2\pi f_{0}n}\sum_{k=0}^{N_{FFT}}c_{k}.e^{2j\pi\frac{kkk}{T_{s}}} \tag{2}\]