## 2 Background and Motivation  
OFDM is the most popular multicarrier modulation technique. Multicarrier is used in both 4G and 5G systems. It is also expected that B5G will consider OFDM. OFDM reduces the computational complexity of the equalization process than the adaptive equalizer. It simply converts the multipath channel into a flat channel over each subcarrier. Thus, a simple one-tap equalizer is required for each subcarrier. In the OFDM transmitter, the incoming modulated symbols (e.g., QPSK or M-QAM) are grouped into \(N_{T}\) parallel symbols and go through the \(N_{T}\)-point IFFT. Equation (1) represents the time domain OFDM symbol [14];  
\[x_{n}=\frac{1}{\sqrt{2}}\sum_{k=0}^{N_{T}-1}A_{k}e^{j\frac{2\pi nk}{N_{T}}}, \qquad 0\leq n\leq N_{T}-1 \tag{1}\]  
where \(N_{T}\) denotes the total subcarriers' number of the OFDM symbol and \(A_{k}\) represents the modulating symbol of the \(k_{th}\) frequency domain subcarrier. After the IFFT process, a cyclic prefix (CP) is added to convert the multipath channel into a circular convolutional channel to simplify the equalization process in the transmitter [38]. Let \(a_{n}\) and \(b_{n}\) be the real and imaginary parts of \(x_{n}\) (i.e., \(x_{n}=a_{n}+jb_{n}\)). For a sufficiently large number of subcarriers (usually, \(N\geq 64\)) \(a_{n}\) and \(b_{n}\) will fit the Gaussian distribution in accordance with the central limit theorem (CLT). Thus, envelope of OFDM symbol (i.e., \(|x_{n}|=\sqrt{a_{n}+b_{n}}\)) will follow a Rayleigh distribution, while its power follows the chi-squared distribution with degree of freedom equal two. The chi-squared distribution indicates that the OFDM symbol's maximum power is very large with respect to the average power of the OFDM symbol (i.e., large PAPR). Mathematical representation of the PAPR of the discrete-time baseband OFDM symbol is expressed as follows [14]:  
\[PAPR=\frac{\max_{n\in[0,N_{T}]}|x_{n}|^{2}}{\mathbf{E}\{|x_{n}|^{2}\}} \tag{2}\]  
where \(\mathbf{E}\{.\}\) refers to the arithmetic mean. Continuous-time passband OFDM symbol has PAPR value greater than the PAPR of the OFDM symbol in baseband by 3 dB [3; 13; 14]. The passband OFDM signal is expressed as [3]  
\[s(t)=Re\Big{\{}x(t)e^{j2\pi f_{c}t}\Big{\}} \tag{3}\]  
where  
\[x(t)=\frac{1}{\sqrt{2}}\sum_{k=0}^{N_{T}-1}A_{k}e^{j\frac{2\pi kt}{N_{T}T_{s}} },\qquad 0\leq t\leq T_{OFDM} \tag{4}\]  
\(T_{OFDM}=T_{s}+T_{g}\) where \(T_{s}\) is the duration of the OFDM symbol and \(T_{g}\) is the duration of CP. If the input to the HPA is \(s(t)=|s(t)|e^{j\angle s(t)}\), where \(|s(t)|\) and \(\angle s(t)\) are the amplituded and phase angle of \(s(t)\) respectively, then the output of the memoryless nonlinear HPA is \(y(t)=\Lambda[|s(t)|]e^{j[\angle s(t)+\Theta[|s(t)|]]}\), where \(\Lambda[\cdot]\) and \(\Theta[\cdot]\) represent the amplitude/amplitude (AM/AM) and amplitude/phase (AM/PM) conversions of HPA, respectively (i.e., \(\Lambda[\cdot]\) and \(\Theta[\cdot]\) describe the effect of non-linearity on \(|s(t)|\) and \(\angle s(t)\), respectively) [13; 38].  
In the receiver, AWGN is added to the received signal from the channel. The received baseband signal \(\widehat{x}(t)\) is then converted to digital form \(r_{n}\). Then, CP is removed from each OFDM symbol before being converted to the FFT process's frequency domain. Frequency domain OFDM symbol is expressed as [38]:  
\[\widehat{A}_{k}=\frac{1}{\sqrt{2}}\sum_{k=0}^{N_{T}-1}r_{n}e^{j\frac{2\pi nk}{N_{ T}}},\qquad 0\leq k\leq N_{T}-1 \tag{5}\]  
Finally, equalized OFDM symbol (i.e., \(\widehat{A}_{k}/H_{k}\)) is de-mapped before serialization, where \(H_{k}\) is the complex channel gain on subcarrier \(k_{th}\)[38].