# A Novel Alternating \(\mu\)-Law Companding Algorithm for PAPR Reduction in OFDM Systems
## 5 Simulation Results and Complexity Analysis
### Experimental Scenarios and Results Discussion
Therefore, we can infer that the proposed method's upper and lower bound of the \(\mu\) value is between 4 and 2. PAPR and BER performance does not reach top-notch when \(\mu\) is 3, the BER remains good compared with the original \(\mu\)-law, and PAPR has been dramatically improved. To determine the \(\mu\) value more streamlined and efficiently, we define the ratio between \(\triangle\ BER\) and \(\triangle\ PAPR\) as parameter \(\eta\), which is expressed as  
\[\eta=\frac{\triangle\ BER}{\triangle\ PAPR}, \tag{25}\]  
Figure 7: Compare (**a**) PAPR and (**b**) BER for SULC (\(\mu\) = 2, 3, 4), AULC (\(\mu\) = 2, 3, 4), original \(\mu\)-law (\(\mu\) = 2, 3, 4), and OFDM with QAM = 256, \(N\) = 512.  
where \(\triangle\)\(BER\) is the distance from \(10^{0}\) to the BER curve, and \(\triangle\)\(PAPR\) is the distance between the PAPR curve and the origin, due to the lower BER indicating a greater \(\triangle\)\(BER\) and better PAPR performance indicating a lower \(\triangle\)\(PAPR\). Therefore, we infer that maximizing \(\eta\) will lead to the best choice of \(\mu\) value. Furtherly, utilizing Equation (25), we redraw Figure 6 as Figure 8, which shows the \(\mu\) vs. \(\eta\) curve for the SULC scheme and the proposed AULC scheme. Also, Figure 8 reveals that the most prominent point of \(\eta\) is at \(\mu\) = 3. Hence, we verify again that \(\mu\) = 3 is the best choice in our methods. Furthermore, from the parameter \(\eta\), we observe the sensitivity of the \(\mu\) value to the relative impact of the BER and PAPR for the SULC and proposed AULC. In other words, the PAPR variation is very small compared to BER in the proposed AULC, which is more sensitive to the \(\mu\) value. To verify the PAPR and BER performance of our proposed methods, the following simulation compares the SULC and proposed AULC schemes to other adopted companding technique's PAPR reduction methods, such as original \(\mu\)-law [31], modified \(\mu\)-law [32], and ITM [33]. Figures 9a,b and 10a,b show their PAPR and BER performance for 256-QAM and \(N\) of 512 and 1024, respectively, while Figures 11a,b and 12a,b show their PAPR and BER performance in the case of 512-QAM and \(N\) of 512 and 1024, respectively. Figure 9a,b describe the PAPR performance of each method, among which OFDM has not been improved in any way, and of course, it performs the worst. For instance, compared with OFDM when CCDF is \(10^{-4}\) in Figure 9a, the original \(\mu\)-law only reduces 0.91 dB when \(\mu\) is 3; the effect is relatively poor. While the PAPR of SULC is 6.16 dB when \(\mu\) is 3; compared with OFDM, the original \(\mu\)-law, and the modified \(\mu\)-law, and ITM improved by 5.89 dB, 4.98 dB, 2.08 dB, and 1.62 dB, respectively. As for the proposed AULC, the PAPR is almost the same as the SULC. Therefore, it can be verified that the proposed AULC maintains good performance while reducing complexity. As shown in Figure 9b, as the number of subcarriers, \(N\) increases to 1024, the PAPR performance of the modified \(\mu\)-law and ITM has a slightly worsening trend compared to Figure 9a. Comparatively, the proposed schemes almost maintain the same PAPR  
Figure 8: \(\mu\) vs. \(\eta\) for SULC and AULC with QAM = 256, \(N\) = 512, and SNR = 30 dB. performance compared to \(N\) is 512. Moreover, the SULC scheme and the proposed AULC scheme still outperform the original OFDM performance by about 5.89 dB and 5.9 dB. In terms of BER performance, as shown in Figure 10a, we can find that the performance of the modified \(\mu\)-law and ITM are the poorest; as for others' BER performances, such as OFDM, the original \(\mu\)-law and our proposed methods in this article are almost overlapping,  
Figure 9: PAPR performance of different schemes, at QAM = 256 (**a**) \(N\) = 512 and (**b**) \(N\) = 1024.  
which is \(1.43\times 10^{-4}\) at SNR is 30 dB. It is worth mentioning that compared with the original \(\mu\)-law, the modified \(\mu\)-law, and ITM, the BER performance of the proposed methods is improved by approximately 6%, 75%, and 84%, respectively.