## 4 Simulation Modeling Results
### Occupied Frequency Band and PAPR of Oscillations  
The change in the occupied frequency band depending on the additional constraint on the maximum PAPR value of the signal packet with the length \(N_{p}\) (8) was analyzed. The frequency bands determined at the energy spectrum levels of \(-30\) dB (\(\Delta F_{-30\text{ dB}}\)) and \(-60\) dB (\(\Delta F_{-60\text{ dB}}\)) and the energy concentration of 99% in the band for optimal FTN signals (\(\Delta F_{99\%}\)) with the duration of \(T_{s}\) = 4\(T\) are shown in Figure 14. The frequency bands for packet lengths of \(N_{p}=5\) and \(N_{p}=8\) participating in the solution of the optimization problem are presented in Figure 14a,b.  
Based on the results in Figure 14, the following conclusions can be drawn. First, the occupied frequency band \(\Delta F_{99\%}\), determined for the energy concentration of 99%, was close to the value of \(\Delta F_{99\%}=1/T\), which corresponds to the Nyquist barrier and weakly depends on the constraint parameter in the optimization problem. Second, the frequency band determined for the energy spectrum level of \(-60\) dB (\(\Delta F_{-60\text{ dB}}\)) changed quite significantly and ranged from \(\Delta F_{-60\text{ dB}}=1.5/T\) to \(3.5/T\) for signals with the pulse duration of \(T_{s}=8T\). With the decrease in the pulse duration to the value of \(T_{s}=4T\), the fluctuations in the frequency band increased and reached the values \(\Delta F_{-60\text{ dB}}=4/T\)-\(6/T\) and \(3/T\)-\(5.2/T\), depending on the number of symbols taken into account during the optimization problem-solving process.  
For reference, the exact values of \(\Delta F_{99\%}\) for various parameters of optimal FTN signals are given in Table 1.  
The values of the occupied frequency band containing 99% of signal energy (Figure 14) were compared with similar characteristics of the quadrature signals based on the RRC pulses. The occupied frequency band \(\Delta F_{99\%}\) of the RRC pulses with a duration of \(T_{s}=8T\) was \(\Delta F_{99\%}=1.15/T\) with a roll-off factor of the shaping filter \(\alpha=0.3\). Thus, the band values for the optimal FTN signals and signals based on RRC pulses were practically the same.  
Finally, the overall results of determining the PAPR of optimal FTN signal packet depending on the value of \(\Delta\)maxPAPR were analyzed. The PAPR values for a signal packet consisting of 1000 binary symbols are shown in Figure 15. These values were obtained by averaging over 1000 realizations of a random signal packet. As shown in Figure 15, with the increase in the limiting parameter \(\Delta\)maxPAPR, the PAPR value changed significantly. In addition, for signals with the pulse duration of \(T_{s}=8T\), the PAPR value decreased from approximately 9 dB to approximately 6 dB. The same proportion was achieved for signals with the pulse duration of \(T_{s}=4T\). However, the absolute PAPR values increased by about 2 dB while the pulse duration increased by two times.  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & \(\Delta F_{99\%}\), \(1/T\) \\ \hline \(\Delta\)maxPAPR, dB & \(T_{s}=4T\) & \(T_{s}=8T\) \\ \hline
0 & 1.12 & 0.99 \\
1 & 1.18 & 1.06 \\
2 & 1.27 & 1.17 \\
3 & - & 1.22 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Occupied frequency band \(\Delta F_{99\%}\).  
Figure 14: Occupied frequency band dependence on \(\Delta\)maxPAPR for pulse duration of \(N_{p}=8\): (**a**) \(T_{s}=4T\); and (**b**) \(T_{s}=8T\).