# A Novel Alternating \(\mu\)-Law Companding Algorithm for PAPR Reduction in OFDM Systems
## 1 Introduction  
Since the development of wireless communication systems, human needs for increased convenience and improved daily life have increased significantly from voice messages to multimedia. This has also made the Internet of Things (IoT) [1, 2, 3], industrial automation, and telemedicine popular research areas. Of course, they need efficient power control, massive data transmission, increasing connection numbers, high spectral efficiency demands, and other requirements. Therefore, it is highly urgent to increase the data emitted and transmission quality and improve spectrum utilization efficiency; simultaneously, IoTs have become one of the critical developments in fifth-generation (5G) wireless communications [4, 5]. In light of this, it is inevitable to evolve from single-carrier to multi-carrier technology [6]. Also, among multi-carrier technologies, orthogonal frequency division multiplexing (OFDM) possesses the characteristic of overlapping and mutually orthogonal subcarriers, thereby enhancing spectral efficiency (SE), prompting the robustness of resistance against multipath interference and inter-symbol interference (ISI) [7], and OFDM has been the most popular and mature vital technology in advanced wireless communication systems [8, 9, 10, 11]. Unfortunately, OFDM bears the inherent nature of a multi-carrier system; therefore, the issue of peak-to-average power ratio (PAPR) is inevitable. Higher PAPR canlead to nonlinear distortion when transmitting signals through amplifiers. Thus, pursuing an effective method to reduce PAPR and complexity without sacrificing bit error rate (BER) performance in OFDM systems has become an urgent research issue. At present, many various well-known techniques have been put forward to reduce PAPR in OFDM systems [12; 13; 14], such as clipping, nonlinear companding transform (NCT), partial transmit sequence (PTS), selective mapping (SLM), tone injection (TI), tone reservation (TR), etc. Among these techniques, clipping [15; 16] is the simplest; however, the clipped signal is prone to clipping noise, and the threshold value of clipping is challenging to select and store, resulting in a significant drop in BER performance [17]. In terms of PTS technology [18; 19; 20; 21], after dividing the frequency domain signal into clusters, each cluster is through an inverse fast Fourier transform (IIFT) and is multiplied by a rotation factor to change its phase. Then, all the clusters are summed up and combined into an output signal. To find the optimal PAPR signal, the rotation factor will be chosen through a massive search. What is more troublesome is that the PTS technology suffers from a fatal drawback that makes it difficult to recover the signal at the receiver. While the SLM scheme [22; 23; 24; 25] multiplies the frequency domain signals by different rotation factor vectors and produces multiple candidate signals through IFFT, the minimum PAPR signal is selected for transmission. In TI technology [26; 27], various expansion methods embed the extra constellation points to the original constellation. This provides more choices and diversity for each signal, thereby mapping the constellation signals to the equivalent locations in the constellation and achieving the objective of reducing PAPR. As for the traditional TR method [28; 29; 30], this method selects the appropriate peak countervail signal according to different lengths and positions and then blends it into the transmission signal to reduce PAPR. There are two standard methods for nonlinear companding transform schemes: \(\mu\)-law and A-law [31]. This article will specifically focus on the \(\mu\)-law method. In [31], [PERSON] et al. described the principle of companding, which involves compressing the larger signals at the transmission end to reduce the signal's dynamic range. At the receiving end, expansion is employed to restore the signal.