# A Novel Alternating \(\mu\)-Law Companding Algorithm for PAPR Reduction in OFDM Systems
## 5 Simulation Results and Complexity Analysis
### Computational Complexity Analysis and Performance Discussion
#### 5.2.2 Computational Complexity Discuss and Numerical Analysis  
In this subsection, we compare the numerical analysis of the computational complexity among the above-mentioned schemes. Substitute the parameter values of Table 1 into the expressions of Table 3 to obtain the CMAs numerical values of each scheme when subcarrier \(N\) is equal to 512 and 1024, as listed in Table 4, and its bar chart is shown in Figure 13.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Scheme Name**} & \multirow{2}{*}{**Number of Complex Multiplications**} & **Number of Complex** & \\  & & **Additions** & **Number of \(O(\log(\cdot))\)** \\ \hline Original \(\mu\)-law [31] & \((N^{2}+4N)\) & \((N^{2}+1)\) & \((N+1)\) \\ Modified \(\mu\)-law [32] & \((N^{2}+6N)\) & \((N^{2}+2N-1)\) & \((N+1)\) \\ ITM [33] & \((N^{2}+7N)\) & \((N^{2}+7N-2)\) & \((N+1)\) \\ SULC & \((3N^{2}+9N+3)\) & \((3N^{2}+N-2)\) & \((N+1)\) \\ \multirow{2}{*}{AULC} & \(\left[1.32N^{2}+3.6N+3\right.\) & \(\left[1.32N^{2}-0.2N-2\right.\) & \(\left.0.4N+1\right.\) \\  & \(\left.+\beta_{rough}(0.32\beta_{rough}-0.64N+0.4)\right]\) & \(\left.+\beta_{rough}(0.32\beta_{rough}-0.64N+0.2)\right]\) & \(\left.+0.6\beta_{rough}\right)\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Analytical expression of the computational complexity of the above scheme.  
To discuss each scheme in more detail, we first consider the case of \(N=512\) in Figures 9a and 10a and Table 4 regarding both PAPR and complexity. At first glance, the original \(\mu\)-law scheme has 526,337 CAMs, which is the lowest complexity among all compared schemes and seems the best choice. However, this scheme exhibits awful PAPR performance to maintain good BER performance. In contrast, although the complexity of the SULC is inferior to the original \(\mu\)-law method, its PAPR performance is much better than the original \(\mu\)-law scheme while maintaining good BER performance. On the other hand, the proposed AULC further improves complexity performance by 64.4% compared to SULC while exhibiting almost identical PAPR and BER performance.  
Similarly, we again consider the case of \(N=1024\) in Figures 9b and 10b and Table 4 regarding both PAPR and complexity. Due to the increase in the number of subcarriers, the complexity of various schemes has shown a rising trend. Among them, the modified \(\mu\)-law and ITM schemes exhibit slightly degraded PAPR performance. Fortunately, increasing the number of subcarriers has almost no impact on our proposed method. In other words, they are not the optimal solution in a high subcarrier environment.  
In summary, the SULC and the proposed AULC exhibit the best PAPR performance among all the schemes while maintaining a good BER. Compared to the original \(\mu\)-law scheme, the modified \(\mu\)-law scheme, and the ITM scheme, the proposed AULC scheme demonstrates superior performance in terms of PAPR with improvements of 45%, 26%, and 21%, respectively. Regarding BER performance, it outperforms the mentioned schemes by 6%, 75%, and 84%, respectively. In addition, in terms of complexity, compared to the original \(\mu\)-law scheme, modified \(\mu\)-law scheme, and ITM scheme, the proposed AULC method only requires a moderate increase in complexity of approximately 6.3%, 5.9%, and 5.4%, respectively. In other words, in light of the tradeoff between complexity, PAPR, and BER, the proposed AULC has extremely obvious advantages at PAPR and BER. Also, the complexity is very close to that of other companding methods.