# A Novel Alternating \(\mu\)-Law Companding Algorithm for PAPR Reduction in OFDM Systems
## 3 Related Work  
In this section, we will briefly introduce and discuss original \(\mu\)-law companding to enhance the article's readability and speed understanding.  
As shown in Figure 2, it applies an \(\mu\)-law companding algorithm to the OFDM systems. Using the OFDM signal through the \(\mu\)-law algorithm to reduce PAPR, we can present by \(\mathbf{y}\) as below [31]:  
\[\mathbf{y}=sgn(\mathbf{x})\frac{\log(1+\mu|\mathbf{x}|)}{\log \left(1+\mu\right)}, \tag{5}\]  
where \(\mathbf{x}\) is the OFDM signal, \(\mu\) is a positive number, and \(sgn(\cdot)\) is a sign function, as shown in Figure 3, that can be defined as  
\[sgn(x)=\left\{\begin{array}{cc}\frac{x}{|x|}&x\
eq 0\\ 0&x=0\end{array}\right.. \tag{6}\]  
Thus, we can turn Equation (5) to another form as  
\[\mathbf{y}=\frac{\mathbf{x}}{|\mathbf{x}|}\cdot\frac{log(1+\mu| \mathbf{x}|)}{log(1+\mu)}, \tag{7}\]  
then, moving the item \(\mathbf{x}\) to the left side of Equation (7), we can obtain the ratio between \(\mathbf{y}\) and \(\mathbf{x}\) as  
\[\frac{\mathbf{y}}{\mathbf{x}}=\frac{1}{|\mathbf{x}|}\cdot\frac{ log(1+\mu|\mathbf{x}|)}{log(1+\mu)}. \tag{8}\]  
To make the equation easy to read, we denote \(\frac{y[n]}{x[n]}\) as \(\lambda\). Next, we will consider \(x[n]\geq 0\) and discuss three cases, such as \(\lambda=1\), \(\lambda>1\), and \(\lambda<1\).  
Figure 2: The block diagram of \(\mu\)-law companding used in OFDM systems.  
case 1. \(\lambda=1\):  
Because \(\lambda\) is equal to 1, we can infer that \(|x[n]|\) is identical to \(\dfrac{log(1+\mu|x[n]|)}{log(1+\mu)}\). Then, we will obtain the identity equation as follows:  
\[\dfrac{log(1+\mu|x[n]|)}{log(1+\mu)}=|x[n]|. \tag{9}\]  
Now, we use a graphical method to find the \(x[n]\) solution in Equation (9), as shown in Figure 4. No matter how \(\mu\) changes, we can see that \(x[n]=1\) as \(\lambda\) is 1.  
case 2. \(\lambda>1\):  
In terms of \(\lambda>1\), \(\dfrac{1}{|x[n]|}\cdot\dfrac{log(1+\mu|x[n]|)}{log(1+\mu)}>1\), then Equation (9) can be rewritten as  
\[\dfrac{log(1+\mu|x[n]|)}{log(1+\mu)}>|x[n]|. \tag{10}\]  
Similar to **case 1**, from Figure 4, we can find the interval of \(x[n]\) that satisfies the inequality Equation (10). Therefore, when \(\lambda>1\), the \(x[n]\) will be expressed as  
\[0<x[n]<1. \tag{11}\]  
case 3. \(\lambda<1\):  
Finally, when \(\lambda<1\), the inequality Equation (10) will become \(\dfrac{1}{|x[n]|}\cdot\dfrac{log(1+\mu|x[n]|)}{log(1+\mu)}<1\), which is equivalent to:  
\[\dfrac{log(1+\mu|x[n]|)}{log(1+\mu)}<|x[n]|. \tag{12}\]  
Looking at Figure 4 again, Equation (12) can be satisfied as long as \(x[n]>1\). Hence, when \(\lambda<1\), then we will obtain the following:  
\[x[n]>1. \tag{13}\]  
As to the above investigation, for \(x[n]<0\), we can infer that when \(\lambda=1\):  
\[x[n]=-1, \tag{14}\]  
\(\lambda>1\):  
\[-1<x[n]<0, \tag{15}\]  
\(\lambda<1\):  
Figure 3: Sign function with complex input.  
\[x[n]<-1. \tag{16}\]  
From the above-mentioned, we can see that when \(x[n]\)'s absolute value is less than 1, the \(\mu\)-law equation will expand signals. On the contrary, the equation will compress signals.  
As for the receiving end, the signal can be recovered using the following function:  
\[\bar{\mathbf{y}}=sgn(\mathbf{r})\frac{(1+\mu)^{|\mathbf{r}|}-1}{\mu}, \tag{17}\]  
where \(\bar{\mathbf{y}}\) is the signal of the inverse of \(\mu\)-law, and \(\mathbf{r}\) is the received signal through the channel. It is worth mentioning that the original \(\mu\)-law scheme is a vector-based parallel companding method.