## 2 Optimization Problem with Constraint on PAPR
### Results of Optimization Problem Solving
Comparing the view of the sequences in Figures 8 and 9, it can be seen that the introduction of the parameter of limitation on the value of PAPR into the optimization problem made it possible to obtain an increase in the average power of the emitted signals while maintaining the spectral and correlation characteristics unchanged. Figure 8: The shape of normalized signal sequence with OQPSK for: (**a**) \(T_{s}=4T\); and (**b**) \(T_{s}=8T\). Figure 7: Results of the optimization problem solving for \(T_{s}=8T\) with the constraint on PAPR for the packet length \(N_{p}=8\). An increase in the duration of the optimal pulses from \(T_{s}=4T\) to \(T_{s}=8T\) leads, as expected, to an increase in the PAPR. Next, consider the proposed method of optimizing function \(a(t)\) for quadrature phase shift keying (QPSK) and binary phase shift keying (BPSK). The dependence of the PAPR of oscillations on the value \(\Delta\)maxPAPR is presented in Figure 10, where the symbol rate is \(R=1/T\), the pulse duration is \(T_{s}=4T\), and the packet length is \(N_{p}=5\) bits in Figure 9(a), and \(N_{p}=8\) bits in Figure 9(b). As presented in Figure 10, when QPSK and BPSK are used, increasing the constraint on \(\Delta\)maxPAPR can significantly reduce the absolute value of the PAPR of the radiated oscillations, approximating its value to the PAPR value of the optimal FTN signals with OQPSK.The changing trend is the same for signals with a longer duration \(T_{s}=8T\), as shown in Figure 11. However, the quantitative ratios of PAPR values change. Figure 10: Dependence of the PAPR value on \(\Delta\)maxPAPR for pulse duration of \(T_{s}=4T\): (**a**) \(N_{p}=5\); and (**b**) \(N_{p}=8\). Figure 9: Optimal FTN signal packet with OQPSK: (**a**) \(T_{s}=4T\), \(N_{p}=5\), \(\Delta\)maxPAPR = 3 dB; and (**b**) \(T_{s}=8T\), \(N_{p}=8\), \(\Delta\)maxPAPR = 3 dB. For the optimal FTN signals with OQPSK, the quantitatively achievable PAPR values with the constraint on \(\Delta\)maxPAPR = 2.5 dB and \(\Delta\)maxPAPR = 3 dB increase from 4 to 4.5-5 dB. At the same time, the difference in the absolute PAPR values at \(\Delta\)maxPAPR of 2.5 and 3 dB for QPSK and BPSK modulations compared to the OQPSK modulation scheme is more than 1 and 2 dB, respectively. As noted at the beginning of Section 2 when discussing constraint (13), it is necessary to estimate the minimum packet length \(N_{p}\) which provides stable results in terms of PAPR. According to Figure 11, \(N_{p}=8\) is the closest value to this minimum packet length. For this value of the packet length, the solutions to the optimization problem with the constraint on PAPR give a more stable result. The local minimum of the PAPR function for OQPSK with \(N_{p}=8\) is less than for \(N_{p}=5\). Consequently, it is necessary to consider the emerging opportunity to increase the BER performance by increasing the average power of the radiated oscillations at the output of the nonlinear power amplifier of the radio transmitter; for instance, by 2-2.5 dB, as shown in Figure 11b. It should be noted that, in this way, all spectral characteristics of the transmitted signals are preserved, and the same energy cost is guaranteed as in the case of using signals without constraint on the PAPR. The more detailed descriptions are presented in the following section.