# A Novel Alternating \(\mu\)-Law Companding Algorithm for PAPR Reduction in OFDM Systems
## 1 Introduction
Furthermore, [PERSON] and others also proposed the \(\mu\)-law companding equation, which aims to reduce PAPR by compressing large-amplitude signals and enhancing small-amplitude signals. In addition, [PERSON] et al. proposed a similar \(\mu\)-law companding formula [32]. This companding method amplifies small-amplitude signals while large-amplitude signals remain unchanged. It improves the companding performance compared to the \(\mu\)-law companding transform proposed in [31], effectively reducing more PAPR but sacrificing more BER performance. For convenience, in further discussion, we will refer to the \(\mu\)-law companding scheme proposed in [31] as the original \(\mu\)-law companding method and the \(\mu\)-law companding scheme proposed in [32] as the modified \(\mu\)-law companding method. Moreover, the improved two-\(\mu\)s companding (ITM) [33] proposed by [PERSON] et al. is based on the algorithm extended by modified \(\mu\)-law [34; 35]. Through two different \(\mu\) parameters, normalized factor \(K\), and appropriate threshold points, it can obtain better PAPR performance than the modified \(\mu\)-law. Also, [PERSON] et al. use the ITM algorithm to add the discrete cosine transform (DCT) operation [36], resulting in better PAPR reduction than ITM. Unfortunately, the BER performance of these two methods can be expected to drop significantly under high-order modulation, making them difficult to apply in environments with high data transmission volumes. Additionally, [PERSON] et al. proposed an improvement scheme in [37] based on the modified \(\mu\)-law. This scheme achieves the purpose of maintaining average power through the proposed companding equation. However, they cannot meet reduced PAPR, small BER, and suitable complexity simultaneously. Moreover, not much literature discusses the \(\mu\) parameter. Therefore, we define an efficiently balanced parameter, \(\eta\), to pre-estimate a more appropriate \(\mu\) value and use the SULC scheme to overcome the above issues. Furthermore, we simultaneously proposed a novel AULC scheme to reduce the complexity and maintain BER performance compared to the SULC. Under conducting a trade-off between PAPR and BER, we utilize the Monte Carlo method to analyze the impact of \(\mu\) parameter values on performance through probability and statistics estimate procedures. By this, we choose the optimal \(\mu\) parameter and apply it to this research methodology. Next, we sequentially compand the transmitted signal and compare it with the original OFDM signal to pick that with a lower PAPR for transmission. The AULC scheme sorts the transmission signals according to their amplitudes, estimates the rough companding amount through Monte Carlo, processes parallel companding the same as the original \(\mu\)-law by the estimated proportion and then performs partial companding on the remaining components, which can further reduce the complexity. In other words, it divides the total tones into two groups and uses traditional parallel companding for the first group and partial SULC for the second group to reduce complexity and refine PAPR performance. It is worth noting that this study applies side information to the receiver to maintain BER performance. Although the side information may increase the transmission bandwidth required, this effect should be alleviated through data compression, which is beyond the scope of this article. The rest of this paper is organized as follows. In Section 2, we introduce the system model. Section 3 reviews previous well-known techniques. Section 4 illustrates our proposed methods. Section 5 analyzes the simulation results and complexity to validate and discuss comparing our proposed schemes with other previous works. Finally, Section 6 concludes the article.