## 2 Optimization Problem with Constraint on PAPR
### Results of Optimization Problem Solving  
Note that the shape of the optimal function \(a(t)\) depends on the packet length \(N_{p}\). Start with the values \(N_{p}=5\) and \(N_{p}=8\) and estimate the dispersion of the parameters of the obtained optimal functions. Consider the solution to the optimization problem for a transmission rate \(R=1/T\), a cross-correlation coefficient \(K_{0}=0.1\) (10), and a pulse duration \(T_{s}=4T\). Next, analyze the change in the shape of function \(a(t)\) and the corresponding spectral characteristics of the signal sequences, taking into account the constraint on maximum PAPR (up to 2-3 dB) for a different number of bits in the packet (Figures 4 and 5 for \(T_{s}=4T\)). Figure 4: Results of the optimization problem for \(T_{s}=4T\) with the constraint on the PAPR for the packet length \(N_{p}=5\). Figure 3: Surfaces of the optimization function for the optimal function. The analysis of time and spectral characteristics of optimal FTN signals with a fixed pulse duration of \(T_{s}=4T\) shows that the requirement for reducing the maximum PAPR of a packet by 3 dB leads to the significant change in the function shape, which is the most apparent for a small packet length, as shown in Figure 4. This change leads to the distortion in the normalized energy spectrum shape. The main lobe of the spectrum expands (see Figures 4 and 5) and the levels of the side lobes closest to the main one increase, although the reduction rate of the level of out-of-band emissions remains unchanged. Next, consider the solution to the optimization problem for optimal FTN signals with a pulse duration \(T_{s}=8T\). As shown in Figures 6 and 7, the change in the function shape mainly concerns the side lobes, which leads to the distortion in the energy spectrum of a random packet of signals. Figure 5: Results of the optimization problem solving for \(T_{s}=4T\) with the constraint on PAPR for the packet length \(N_{p}=8\). Figure 6: Results of the optimization problem solving for \(T_{s}=8T\) with the constraint on PAPR for the packet length \(N_{p}=5\). Based on the intermediate result of the synthesis of optimal FTN signals, it can be concluded that the introduction of a PAPR limitation by at most 3 dB for each of the quadrature components in (3) allows preserving the main spectral characteristics in a given format, i.e., a given reduction rate of the out-of-band radiation level remains unchanged. In this case, the occupied frequency band at the power spectral density level of \(-40\) dB is equal to \(\Delta F_{-40\text{ dB}}=\) (0.8-1)/\(T\). If this value is compared to a similar characteristic of the minimum shift keying (MSK) signals and Gaussian minimum shift keying (GMSK) signals, which are widely used in mobile communications, it can be found that, for MSK, the bandwidth is \(\Delta F_{-40\text{ dB}}=\) (3.5-4) /\(T\), and, for GMSK signals, the bandwidth is \(\Delta F_{-40\text{ dB}}=\) (1.2-1.3)/\(T\)[25; 26]. Thus, the optimal shapes of function \(a(t)\) provide a significant reduction in the occupied frequency band determined by the required level of out-of-band emissions. The ultimate goal of the optimization problem is to determine the time and spectral characteristics of random sequences of optimal FTN signals with OQPSK. To achieve this, consider the shape of a packet consisting of 100 bits (3), with 50 bits in each quadrature channel. In Figure 8, the time diagrams of packets of optimal FTN signals with a pulse duration \(T_{s}=4T\) (Figure 7(a)) and \(T_{s}=8T\) (Figure 7(b)) obtained without taking into account the constraint (13) are shown. Figure 9 shows the timing diagrams of optimal FTN signal packets with a pulse duration \(T_{s}=4T\) (Figure 8(a)) and \(T_{s}=8T\) (Figure 8(b)) with a constraint on maximum PAPR \(\Delta\text{maxPAPR}=3\) dB.