## 3 Simulation Model  
The BER performance of the optimal FTN signal detection was analyzed in the transmission channel without fading in the presence of additive white Gaussian noise (AWGN) with an average power spectral density of \(N_{0}/2\) and in the channel with AWGN and Rayleigh fading. The scheme of the simulation model with an amplitude limiter is presented in Figure 12. This scheme can be applied to the DVB-S2X satellite systems in the presence of amplitude limitation. The symbols divided into two streams were fed to the input of the modulator of optimal FTN signals. The in-phase and quadrature component were offset by \(T\). To transmit a packet of optimal FTN signals with a reduced PAPR value, a preamplifier, an inertialess amplitude limiter, and a power amplifier were introduced. This part of the transmission path determined the spectral characteristics of the radiated oscillations and the signal-to-noise ratio (SNR), which further determined the BER performance. The preamplifier has the ability to adjust the average power. The amplitude limiter in the simplest case has the AM/AM characteristic shown in Figure 13.  
Based on the piecewise linear approximation of the AM/AM characteristic, it was assumed that the frequency bands of the power amplifier, preamplifier, and amplitude limiter were wider than the occupied frequency band of a random signal (Equation (3)). In addition, the suppression of the spectrum components near frequencies that denoted multiples of the carrier frequency \(\omega_{0}\) was considered sufficiently strong. Then, the real envelope of the signal packet at the output of the amplitude limiter \(A_{lim}(t)\) was expressed as:  
\[A_{lim}(t)= \begin{cases}A(t),&A(t)\leq A\\ A,&A(t)>A^{\prime}\end{cases}  \tag{14}\]  
where \(A(t)\) denotes the real envelope of a signal (Equation (3)) at the input of the amplitude limiter and \(A\) denotes the threshold limiting the amplitude of the signal packet. The average  
Figure 12: Simulation model scheme. Figure 13: Amplitude limiter AM/AM characteristic. power level of the preamplifier output was determined so that it ensured the transmission in the linear region of the characteristic of the amplitude limiter without limiting the oscillation amplitude, as presented in Figure 13.  
The joint application of amplitude limiter and power amplifier is common in satellite digital broadcasting and television systems, and it is used under conditions of peak power limitation. Peak power limitation leads to the expansion in the spectrum of radiated oscillations and to the increase in the value of the cross-correlation coefficient of received signals. As a consequence, BER performance degrades. In general, it is proposed to use only the linear region of the signal characteristic, while PAPR reduction is ensured by the use of optimal pulse shapes. The increase in \(P_{average}\) of the radiated oscillations (13) occurred in the preamplifier, which provided linear amplification without going beyond the clipping level. Obviously, taking into account (13), the voltage gain of the preamplifier was equal to:  
\[K_{u}=\sqrt{P_{max}}/A=\sqrt{\max_{j}\left\{\left|y^{(j)}_{opt}(t)\right|^{2} \right\}}/A. \tag{15}\]  
Thus, taking into account (14), at the power amplifier input, there would be a signal packet with a constant peak power \(P_{max}\) but different \(P_{average}\) level. At the receiving device input, the frequency equalization of the channel frequency response was performed, as shown in Figure 12. The frequency response of the channel was evaluated using a known non-noisy signal passing through the Rayleigh-fading channel. Based on this evaluation, equalization was performed in the frequency domain using the obtained characteristic. After equalization, the mixture of the useful signal and noise, taking into account the Rayleigh fading, was fed to the input of the optimal FTN signal demodulator. The demodulator of optimal FTN signals employed an algorithm of symbol-by-symbol coherent detection.