# A Novel Alternating \(\mu\)-Law Companding Algorithm for PAPR Reduction in OFDM Systems
## 5 Simulation Results and Complexity Analysis  
### Experimental Scenarios and Results Discussion  
In this section, we will verify some inferences and discuss the simulation results, involving the impact of varying the \(\mu\) parameter on the performance, and evaluate the performance of the proposed methods compared with three schemes, including the original \(\mu\)-law [31], modified \(\mu\)-law [32], and ITM [33]. In addition, without loss of generality, we adopt the channel with AWGN that conforms to i.i.d. To provide a clear explanation of the simulated scenarios in this article, we have listed the required parameters for all experiments in Table 1, and these parameters are utilized for complexity operation and its numerical analysis simultaneously. It is worth mentioning that without losing power fairness and generality, the focus of this paper is to discuss the comparison of PAPR reduction algorithms, and the amplifier's signal will be beyond the discussed scope. Therefore, we neglect the influences of the average signal in the amplifier coming from \(\mu\)-law companding. Of course, regardless of whether the power is normalized, it will not affect PAPR. Moreover, to alleviate the average signal power influence issues of \(\mu\)-law companding, we have normalized the power of 256-QAM and 512-QAM in our experiments. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter Name** & **Value** \\ \hline The number of experiments for Monte Carlo & \(10^{6}\) \\ The order of QAM & \(256,512\) \\ \(N\), number of subcarriers & \(512,1024\) \\ \(\beta_{rough}\), the quantity of the first group in the AULC method & \(\lfloor 0.54N\rfloor\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation scenario and related parameters. As aforementioned, in Figure 4, see from the curve for different \(\mu\) parameter values that the more considerable \(x[n]\) value will obtain a more compressed amount, and vice versa, so the peak value is prone to being decreased and the lower value is prone to being enlarged, improving the PAPR. Moreover, to comprehend the influence of the \(\mu\) parameter on PAPR and BER in the original \(\mu\)-law method, we show the experiment results of \(\mu\) vs. PAPR and \(\mu\) vs. BER in Figure 5a,b for the original \(\mu\)-law with QAM = 256, \(N\) = 512, respectively. Among them, to fare with the advanced B5G wireless communication protocol, according to [39; 40], we know that under QAM order 256, the channel needs to have an SNR of 30 dB to meet the requirements of the receiving end. Therefore, this specification is incorporated into the BER curve simulation for \(\mu\) parameters without loss of generality. In Figure 5a, it seems to merit that \(\mu\) increases and PAPR is reduced. That has paid the BER cost, since all tone components are through nonlinear companding transform simultaneously. As expected, as shown in Figure 5b, when the value of \(\mu\) increases, that makes the BER higher. Of course, this is not what we expected it to be. Even more discouraging is that the above two curves can only show individual trends and almost cannot obtain an optimal \(\mu\) value of a balance point that is effective enough to be applied. Therefore, we are dedicated to finding a \(\mu\) value that trades off PAPR and BER performance. Therefore, the SULC scheme adopts a useful component-wise mechanic to alleviate the conflict between PAPR and BER in the original \(\mu\)-law method. In other words, we perform companding partly, not all of them unconditionally. Thus, compared to the original \(\mu\)-law, this proposed method can achieve a better balance between PAPR and BER. Similarly, Figure 5a,b and Figure 6a,b show the relationship of the \(\mu\) parameter on PAPR and BER, respectively, including the SULC and the proposed AULC schemes. It is worth mentioning that the SULC results significantly prompt performance in PAPR due to all-around tone-companding decisions. Also, in either the SULC or proposed AULC schemes, the optimal \(\mu\) numbers of PAPR and BER are at 4 and 2, respectively.