# A Novel Alternating \(\mu\)-Law Companding Algorithm for PAPR Reduction in OFDM Systems
## 4 Proposed Scheme
### Proposed Alternating \(\mu\)-Law Companding Method (Aulc)  
Although an excellent PAPR performance can be achieved by companding each tone in the SULC method, the complexity is not as fine as expected. Since not every tone requires companding and comparing, only partial tone component-wise companding is enough to achieve the same performance, and the complexity can be significantly reduced. Therefore, we divide the total tones of the signal into two groups and use traditional one-time parallel companding (or, said it, vector-base companding) for the first group, which is the most likely to be companding. Relatively, the other group maintains the above-mentioned SULC procedure, but only partial companding is needed. To find an appropriate parallel companding tone amount, we sorted the tone amplitude of the OFDM signal and performed the SULC method through the Monte Carlo experiment. Then, wewill take the minimum number of tones companding from the Monte Carlo experiment to estimate a rough companding tone number, \(\beta_{rough}\), and define the companding ratio \(\rho\). We can denote them as  
\[\beta_{rough}=\min\left(\text{companding tone number}\right)=\left\lfloor 0.54N \right\rfloor, \tag{19}\]  
and  
\[\rho=\frac{\beta_{rough}}{N}=0.54. \tag{20}\]  
In short, this proposed method sorts the signals, takes out the \(\beta_{rough}\) tone amount as the first group, and performs traditional vector-based parallel \(\mu\)-law companding to reduce complexity. Meanwhile, the second group executes the SULC partially to carry on amending the PAPR. Also, as \(\beta_{rough}\) increases, the parallel amount increases, and then the complexity is predictably reduced, naming it the AULC algorithm. It is worth mentioning that we perform companding selectively rather than performing all companding unconditionally. This way, we will not perform companding on tones that should not be companded, causing the opposite effect. Moreover, grouping can significantly reduce the complexity compared to the SULC. Similarly, we will introduce procedural steps of the proposed AULC algorithm through Algorithm 2. In the first stage:  
In Step 1, we initialize position vector \(\mathbf{p}\) and index vector \(\mathbf{z}\) to zero; they are used to signal recovery at the receiver end for auxiliary purposes. Sequentially, in Step 2, sort the OFDM signal, \(\mathbf{x}\), according to the amplitude from small to large, denote it as \(\mathbf{x}_{sort}\), and update the index vector \(\mathbf{z}\) to the original index of each signal component. Next, in Step 3, the sorted signal \(\mathbf{x}_{sort}\) and position vector \(\mathbf{p}\) are each split into two groups of vectors, and their expressions are as follows:  
\[\mathbf{x}_{sort1}:=\Big{[}x_{sort}[0],x_{sort}[1],\dots,x_{sort}[\beta_{ rough}-1]\Big{]}, \tag{21}\]  
\[\mathbf{x}_{sort2}:=\Big{[}x_{sort}[\beta_{rough}],x_{sort}[\beta_{rough}+1 ],\dots,x_{sort}[N-1]\Big{]}, \tag{22}\]  
and  
\[\mathbf{p}_{1}:=\Big{[}p[0],p[1],\dots,p[\beta_{rough}-1]\Big{]}, \tag{23}\]  
\[\mathbf{p}_{2}:=\Big{[}p[\beta_{rough}],p[\beta_{rough}+1],\dots,p[N-1]\Big{]}. \tag{24}\]  
As for Step 4, we perform traditional vector-based parallel \(\mu\)-law companding on the first group (i.e., \(\mathbf{x}_{sort1}\), its length is \(\beta_{rough}\)) in a batch manner and set the position vector \(\mathbf{p}_{1}\) to \(\mathbf{1}\). In the second stage:  
First, calculate the PAPR value for \(\mathbf{x}_{sort2}\). We express it as \(PAPR_{\mathbf{x}_{sort2}}\) and set it as the initial PAPR value in this second stage. Simultaneously, assign the value of \(\mathbf{x}_{sort2}\) to \(\mathbf{x}^{\prime}\). Next, we utilize the SULC's component-wise method for the partially remaining not yet companding components to refine the PAPR performance further until \(PAPR_{\mathbf{x}^{\prime}}\) is more than \(PAPR_{\mathbf{x}_{sort2}}\). Finally, we assign the value of \(\mathbf{x}^{\prime}\) to \(\mathbf{y}_{AULC}\), where \(\mathbf{y}_{AULC}\) is the output signal of the AULC.