## 4 Simulation Modeling Results
### BER Performance in Rayleigh-Fading Channel  
The results of BER performance were also analyzed in the frequency-flat Rayleigh fading channel. As the optimal FTN signals with QPSK and OQPSK modulation, the signals formed on the basis of pulses with \(T_{s}=4T\) (Figure 5), \(T_{s}=8T\) (Figure 7), and \(K_{0}=0.1\) were used.  
As shown in Figure 18a, the use of optimal FTN signals with QPSK modulation and regulation of the average power of the radiated oscillations provides a higher BER performance in a channel with Rayleigh fading due to reduced PAPR. The energy gains in the range of error probabilities \(p=10^{-2}\) are about 0.5 dB when the cases with \(T_{s}\) = 4\(T\), \(\Delta\)maxPAPR = 0.5 dB, and \(\Delta\)maxPAPR = 2 dB are compared. The gain provided by optimal signals with pulse duration \(T_{s}=8T\) and \(\Delta\)maxPAPR = 3 dB compared to \(\Delta\)maxPAPR = 0.5 dB reaches 2.5 dB. In the case of OQPSK (Figure 18b), optimal FTN signals with \(T_{s}=8T\) can provide energy gain about 1.5 dB for error probability \(p=10^{-2}\) due to reduced PAPR (\(\Delta\)maxPAPR = 3 dB instead of \(\Delta\)maxPAPR = 0.5 dB).  
By summarizing the results of the optimal FTN signals with OQPSK modulation based on the criterion of the specified reduction rate of the out-of-band emissions level, the following conclusions can be drawn. In terms of spectral efficiency, application of signals obtained through optimization allows reducing the occupied frequency band, and the introduction of the constraint \(\Delta\)maxPAPR, which limits the maximum PAPR value to 3 dB for each of the quadrature components in (3), allows maintaining the main spectral characteristics, i.e., a specified reduction rate of the level of out-of-band emissions, in a given format.  
The found optimal shapes of function \(a(t)\) allow obtaining a significant decrease in the occupied frequency band, which is determined by the required level of out-of-band emissions. Thus, compared to the MSK and GMSK signals, the reduction in the occupied frequency band is about 4 and 1.3 times, respectively. This allows reducing the frequency plan of the data transmission network over digital television and radio broadcasting (DVB)  
Figure 17: BER performance of optimal FTN signals with OQPSK in AWGN channel and pulse duration of: (**a**) \(T_{s}=4T\); and (**b**) \(T_{s}=8T\).  
channels, especially in satellite DVB-S2X applications, or to increase the data transmission rate when using the same frequency mask.  
The energy costs in the channels with AWGN and channels with the Rayleigh fading when the proposed optimal FTN signals with OQPSK modulation are used are close to the energy costs of classical OQPSK signals with a rectangular pulse shape in quadrature channels and a wide occupied frequency bandwidth, which exceeds the bandwidth of the optimal FTN signals with OQPSK modulation by more than 20 times.  
Therefore, the adjustment of the average power of the radiated oscillations while reducing the PAPR value of the radiated signals is a powerful tool for improving the energy efficiency of the optimal FTN signals with QPSK and OQPSK modulations. This also allows obtaining energy gains of 2-3 dB for channels with the additive noise.