## 4 Material and Methods
### Implemented Systems and Simulation Parameters
LDPC is the Forward Error Correction (FEC) code used in DVB-T2, Digital Video Broadcasting Satellite, second generation (DVB-S2), and ATSC 3.0 which allows these systems to be closer to the Shannon limit. Compared to convolutional codes used in DVB-T, LDPC codes are block codes designed to transmit information over a noisy channel. These codes are represented using sparse bipartite graphs, in which nodes constitute code bits and parity-check equations. The sparseness of the graph allows one to perform an efficient decoding. The main peculiarity of these codes is the decoding process, which is performed using a Tanner graph and an iterative decoding algorithm. The algorithm used to process the decoding is a soft iterative decoding called \"Belief Propagation\" algorithm [40]. In DVB-T2, the parity check equations are used to construct the control parity check matrix used during the encoding process and the decoding process. As Matlab was the simulation tool used for our simulation, the main function used for LDPC were **dvbs2 ldpc**, which allowed us to construct the control parity check matrix, and **comm.LDPCEncoder** and **comm.LDPCDecoder**, for the encoding and the decoding process [41]. The control parity check matrix was constructed by considering the code rate to exploit for encoding and decoding. It is important to note that the last two Matlab functions were replaced by **ldpcencode** and **ldpcdecode**, respectively, from Matlab version R2021b to the current version. Six code rates are defined in DVB-T2. There are 1/2, 3/5, 2/3, 3/4, 4/5, 5/6. The 1/2 code was used in our simulation as it was the code rate that offered the highest signal robustness. As previously presented in Section 3.1.2, UFMC configuration requires four parameters: the filter length, the sub-band bandwidth, the sub-band number, and the sub-band offset. As a comparison was performed with OFDM, the filter length should cover the number of samples used for the cyclic prefix in OFDM. In the conventional UFMC, the filter length is equal to the cyclic prefix length [42]. In our simulation case, the CP was equal to \(1/32\), which corresponded to a cyclic prefix length of \(N_{FFT}/32=8192/32=256\). Therefore, the UFMC filter length was equal to 256. Once the filter length was identified, the other parameter values were chosen on the basis that parameters followed Equation (19). \[N_{NFFT}=(sub-bandbandwidth\times sub-bandnumber)+(2\times Sub-bandoffset) \tag{19}\]  
Furthermore, as the CP is not used in UFMC, this latter implies an increase in spectral efficiency in comparison with OFDM. The spectral efficiency of UFMC was compared to the 100% OFDM spectral efficiency using Equation (20), previously presented in [9]. This value is expressed as a percentage. That equation computes the overhead percentage spared when UFMC is used instead of OFDM. \(M_{OFDM}\) is the number of data subcarriers used in OFDM, \(M_{UFMC}\) is the number of data subcarriers used in UFMC, and \(N_{FFT}\) is the total number of subcarriers  
\[P_{Overhead-OFDM-UFMC}=100\times(1-\frac{M_{OFDM}}{M_{UFMC}+(N_{FFT}\times CP )}) \tag{20}\]  
In the case of the DVB-T2 parameters used, that is, an 8K mode, i.e., 8192 subcarriers, and a CP of 1/32, the UFMC was 103.03% spectrally efficient compared to the 100% OFDM if only the overhead induced by the CP was considered. The UFMC was 108.86% spectrally efficient compared to OFDM when the overhead induced by both the guard band and the CP were considered.