## 4 Simulation Results  
To perform numerical analysis, computer simulations are conducted to verify the performance of the proposed modified SLM application with the derived general form (denoted as G-MSLM). For the purpose of comparison, the SLM scheme in [10] is also considered in our simulations. The considered OFDM system has total subcarriers \(N=256\) with 16 QAM modulation and its bandwidth is 20 MHz. The pilot subcarriers are equally distributed every six subcarriers. To evaluate the performance of PAPR reduction, the oversampling factor is specified as \(G=4\). The parameters in generating \(\Re\) is selected as \(K=3\), \(L_{u}=7\) with the set of the indices of the nonzero elements \(\mathbf{T}~{}=~{}\{0,3,6\}\), \(\mathbf{T}~{}=~{}\{0,4,5\}\), and \(\mathbf{T}~{}=~{}\{0,2,7\}\), and \(K=3\), \(L_{u}=8\) with \(\mathbf{T}~{}=~{}\{0,5,6,7\}\) and \(\mathbf{T}~{}=~{}\{0,3,7,8\}\).  
Figure 4 shows a comparison of PAPR reduction performance between G-MSLM and the SLM scheme in [10] in terms of complementary cumulative distribution function (CCDF). From this figure, we can see that, for \(M=16\), \(M=32\) and \(M=57\), respectively, G-MSLM can achieve almost the same PAPR reduction performance as that of the SLM scheme in [10]. However, it is noted that the G-MSLM scheme can avoid the transmission of SI, and therefore it has the potential of using more candidate signals without reducing the transmission data rate, such as the case for \(M=114\). As compared to the original OFDM systems, the gains in PAPR of G-MSLM are 3.7 dB and 4.2 dB, respectively, for \(M=57\) and \(\Pr[\text{PAPR}\geq\text{PAPR}_{0}]~{}=~{}10^{-3}\) and \(\Pr[\text{PAPR}\geq\text{PAPR}_{0}]~{}=~{}10^{-4}\).  
Figure 5 shows the BER performance of G-MSLM in wireless channels with the exponential-decay power-delay profile (PDF) without considering the nonlinearity of HPA. From this figure, it can be seen that the BER performance of G-MSLM becomes slightly worse with the increase of \(M\). In addition, there is a small BER performance loss comparing the G-MSLM with the original OFDM signal. Since the length of the equivalent channel is slightly larger than that of the real wireless channel, the proposed G-MSLM method will increase the frequency selectivity of the compared with that of the original wireless channel.  
Figure 6 shows the BER performance of G-MSLM in the same channel as that in Figure 5 considering the nonlinearity of HPA. To take the nonlinear HPA into account at the transmitter, the amplitude of transmitted signal is adjusted (with the peak amplitude normalized to be 1 as that in [18]), in which case nonlinear distortions will rarely occur. From this figure, we can see that G-MSLM can achieve better BER performance than the original OFDM system. It has about 3 dB improvement over the original OFDM system. The improvement of G-MSLM over the original OFDM system is due to the fact that G-MSLM can achieve much lower PAPR, which allows a higher level of the transmitted signal. In addition, it can be seen that the BER performance of G-MSLM for \(M=114\) is slightly better than that for \(M=57\) due to its better PAPR reduction performance. Note that since the transmitted signals are normalized with peak amplitude to be 1, the \(x\)-axis is \(1/N_{0}\) where \(N_{0}\) is the single-sided power spectral density of the additive white noise.  
Figure 4: PAPR reduction performance comparisons. G-MSLM: derived general form of modified SLM.  
Figure 5: BER performance comparisons under multipath fading channels.  
Figure 6: Bit error rate (BER) performance comparisons with nonlinear high-power amplifiers (HPA).