# Palm Date Leaf Clipping: A New Method to Reduce PAPR in OFDM Systems
## 3 Clipping Methods  
In this section, we explore some classical clipping methods used to reduce the PAPR value. The basic idea is to trim the peak of the OFDM signal above a given threshold, which we call A in this paper.  
A signal \(x(t)\) can be modified with polar representation by the equation below:  
\[\widetilde{x}(t)=f\left(\left|x(t)\right|\right)*e^{i\phi(x(t))}, \tag{5}\]  
where \(f\) is the amplitude function of the input signal \(x(t)\), \(\phi\) is the phase, and \(\widetilde{x}(t)\) is the clipped signal. Each method is characterized by its clipping function. Accordingly, we present the characteristic function of each clipping method.  
Figure 1: Orthogonal frequency division multiplexing (OFDM) systems. (**a**) Transmitter part; (**b**) Receiver part.  
In Classical Clipping (CC) [26], the output of the amplitude function is defined by the following equation:  
\[f(r)=\begin{cases}r&if\ r\leq A\\ A&if\ r>A^{\prime}\end{cases} \tag{6}\]  
where \(r\) is the amplitude of the signal \(x(t)\). As depicted in Figure 2, the output amplitude of the signal is linear if the value is less than the clipping value \(A\). However, it is equal to the value \(A\) when the argument is greater than \(A\).  
In Deep Clipping (DP) proposed in [29], the clipping function is defined as  
\[f(r)=\left\{\begin{array}{ccc}r&,&r\leq A\\ A-d(r-A)&,&A<r\leq\frac{1+d}{d}A\\ 0&,&r>\frac{1+d}{d}A\end{array}\right., \tag{7}\]  
where \(A\) is the threshold level and \(d\) is the clipping depth factor (see Figure 3).  
Regarding the Smooth Clipping (SC) technique proposed in [27], the clipping function is represented in Figure 4 and defined as  
\[f(r)=\left\{\begin{array}{ccc}r-\frac{1}{b}r^{3}&,&r\leq\frac{3}{2}A\\ A&,&r>\frac{3}{2}A\end{array}\right.\ \ \mbox{where}\ \ b=\frac{27}{4}A^{2}\,. \tag{8}\]  
Figure 3: Deep Clipping function.  
Figure 2: Classical Clipping Function.  
In Heavyside Clipping (HC) used in [28], the clipping function is defined as  
\[f\left(r\right)=\ A,\ \ \ \forall\ r\geq 0. \tag{9}\]  
In this case, the amplitude function is equal to the threshold value \(A\), as portrayed in Figure 5.  
[PERSON] el al. [36] present a comparison between the four clipping methods described in this section. From this analysis, the CC, DC, and SC methods are very similar, whereas the HC method gives a good reduction of PAPR with a great degradation of signal performance compared to the other techniques. The authors in [37] analyzed and compared the same clipping methods under Orthogonal frequency division multiplexing/Offset Quadrature amplitude modulation (OFDM/OQAM); the obtained results confirm those obtained in [36].  
Clipping is the simplest technique used to reduce the PAPR value in OFDM systems. However, it is well known that clipping destroys and degrades the quality of the signal. To this end, we propose and analyze a new clipping method, which is studied in detail in the following sections. To understand the effect of clipping methods, we use a Clipping Ratio (CR) related to both the clipping level A and the OFDM signal average power, as defined in the following equation:  
\[CR=20\ log10\frac{A}{\sqrt{P_{avg}}}\ \ [dB], \tag{10}\]  
where \(P_{avg}=E\Big{(}\big{|}x(n)\big{|}^{2}\Big{)}\) represents the average power of the input signal and A is the threshold value.  
Figure 4: Smooth Clipping function.  
Figure 5: Heavyside Clipping function.