## 4 Simulation Results
### BER Performance  
BER is an important measure to evaluate the effect of the PAPR reduction method on the transmitted data. Many PAPR reduction methods successfully reduce the PAPR. However, they may distort the original signal, which may affect the BER of the system. We evaluated the proposed method's performance by adding the effect of the nonlinear HPA distortion to the transmitted signal. The HPA model used in our simulation is a soft envelope limiter (SEL) [22]. The SEL amplifier affects only the amplitude of the input signal, whereas the phase remains unchanged. After passing through the SEL model, the transmitted signal is as follows:  
\[\widetilde{\mathrm{x}}(\mathrm{t})=\left\{\begin{array}{l}x(\mathrm{t}), \quad\quad\left|x(\mathrm{t})\right|\leq\mathrm{A_{sat}}\\ \mathrm{A_{sat}}^{\mathrm{el}\varphi(\mathrm{t})},\left|x(\mathrm{t})\right|> \mathrm{A_{sat}}\end{array}\right. \tag{17}\]  
where \(\varphi(\mathrm{t})\) represents the signal phase, and \(\mathrm{A_{sat}}\) is the saturation or clipping level of the HPA. One of the critical parameters in HPAs, referred to as the input back-off (IBO) power, determines the maximum power of the HPA before it moves to the saturation region. As the input signal's amplitude increases, it causes the HPA to operate in the saturation region or eventually in its nonlinear region, which causes distortion of the transmitted signal. The IBO in dB is defined as follows [23]:  
\[\mathrm{IBO}=10\log_{10}\left(\frac{\mathrm{A_{sat}}^{2}}{\sigma^{2}}\right) \tag{18}\]  
The variance of the input signal is denoted as \(\sigma^{2}\). Table 4 lists the parameters used in the simulation. The BER of the pure FBMC/OQAM without the HPA effect, with the HPA effect, and the proposed method with the HPA effect for different IBO values are presented. Figures 7 and 8 show the BER plot with 256 sub-carriers and IBO values of 7 dB and 3 dB, respectively. As shown in the graphs, the BER of the pure FBMC/OQAM increases as the IBO value decreases when the HPA effect is added. At probabilities of \(10^{-2}\), the BER value is above 8 dB, and with the addition of the HPA effect, the BER is above 10 dB and 18 dB for IBO values of 7 dB and 3 dB, respectively. When the proposed method is applied, the BER is reduced by 1.2 dB and 6.7 dB, achieving a 13% and 37% improvement in the BER performance, respectively. The results indicate that the proposed method helps to mitigate the clipping effect of the SEL HPA by reducing the peaks of the FBMC/OQAM.  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Method**} & **Calculations** & **No. of Real** \\  & **(per 1 Symbol Period)** & **Multiplications** \\ \hline FBMC/OQAM & 2 IDFTs, 2 PPNs & 12,288 \\ \hline DFT-FBMC/OQAM & 1 DFT, 2 IDFTs, 2 PPNs & 13,184 \\ \hline PTS FBMC/OQAM & 2V IDFTs, \(2\times 2^{\mathrm{V}}\) PPN, phase factor searching \(2^{\mathrm{V}}\mathrm{N}(\mathrm{V}+1)\) & 86,016 for V = 4 \\ \hline \multirow{3}{*}{Proposed method} & 1 DFT, 2V IDFTs, \(2\times 2\) VPPN, phase factor searching 2V N(V+1) & \\ \cline{1-1}  & & 86,912 for V = 4 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Simulation parameters for computation complexity.  
Figure 8: BER (bit-error-rate) comparison with HPA (high-power amplifier) effect for IBO = 3 dB (input back-off).  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Modulation & 4-QAM \\ N (IFFT/FFT size) & 256 \\ Filter size & 4096 \\ K & 4 \\ No. of symbol blocks & 100 \\ V & 4 \\ W & \(1,-1\) \\ L & 4 \\ IBO & 7 dB, 3 dB \\ \hline \hline \end{tabular}
\end{table}
Table 4: Simulation parameters for BER (bit-error-rate) calculations.  
Figure 7: BER (bit-error-rate) comparison with HPA (high-power amplifier) effect for IBO = 7 dB (input back-off).