## 3 Frequentative Decision Feedback (FFB)
Replacing the value of \(Y_{f_{m}}^{f_{m}}\) from Equation (18), we can write Equation (20) as  
\[\hat{X}=\arg\min_{\overline{X}}\lVert H\times(\overline{X}+D^{(f_{m}, \overline{X})})-H\times(X+D^{(f_{m},X)})-Noise\rVert^{2} \tag{21}\]  
If we solve Equation (21) directly for \(D^{(f_{m},\overline{X})}\), it will lead to exponential complexity as we know \(D^{(f_{m},\overline{X})}\) and \(D^{(f_{m},X)}\) are complex nonlinear functions. Hence, to get the solution, the term \(D^{(f_{m},\overline{X})}\) will not be computed; instead, we assume \(D^{(f_{m},\overline{X})}\), which is not related to \(X\), and it is approximated as an AWGN. Thus, we can write mathematically for \(N\) independent subchannels after reducing an ISI channel  
\[Y_{k}^{f_{m}}=H_{k}X_{k}+Noise_{k} \tag{22}\]  
where  
\[Noise_{k}=H_{k}D^{(f_{m},X)}+Noise_{k} \tag{23}\]  
If the receiver computes the value of \(D^{(f_{m},X)}\), the maximum-likelihood problem of Equation (21) will be simplified as  
\[\hat{X}=\arg\min_{\overline{X}}\lVert H\times\overline{X}-H\times X-Noise \rVert^{2} \tag{24}\]  
From the Equations (22), (23), and (24), regarding computation and complexity, we can deduce the problem as a standard linear solution maximum likelihood decoder. When the transmission is uncoded, the problem of a vector into different N scalar maximum likelihood equations can be reduced as  
\[\hat{X}_{k}=\arg\min_{\overline{X}_{k}}\lVert H_{k}\times\overline{X}_{k}-H_{ k}\times X_{k}-Noise_{k}\rVert^{2} \tag{25}\]  
or it can be written as  
\[\hat{X}=\arg\min_{\overline{X}}(H\times[\overline{X}_{k}-\frac{Y_{k}^{f_{m}}}{ H_{k}}+D^{(f_{m},X)}]) \tag{26}\]  
If the value \(D^{(f_{m},X)}\) is known to the receiver, it will select the symbol which is close to \(\frac{Y_{k}^{f_{m}}}{H_{k}}-D^{(f_{m},X)}\). At the same time, the new system is introduced here with less complexity, and the distortion is reduced. In Algorithm 1, if the receiver knows the nonlinear PA function \(f_{m}(. )\), then this algorithm can iteratively approximate the distortion term from the received vector \(Y_{f_{m}}^{f_{m}}\), and it can estimate the QAM vector. This is assuming that the information about the channel and the PA nonlinear model is perfectly known at the receiver. Then, this algorithm can be easily implemented in a few basic steps, as can be seen from Algorithm 1. ```
1:Need: Received \([1,\ldots,X_{m}^{mx}]\) : OFDM symbol blocks
2:MakeSure:\(H\) : The Channel impulse response is known at the receiver
3:MakeSure:\(f_{pa}\) : The PA nonlinear model should be estimated using Machine Learning
4:Procedure Distortion Based Reduction Algorithm
5:For i do number of OFDM symbols
6:\(X_{n}\leftarrow\) removed cyclic prefix from \(X_{m}^{mx}\)
7:for j do number of frequentative feedbacks \(X_{q}=((1/sqrt(N)*FFT([1,\ldots,X_{n}]))./H-D^{(f_{pa}X_{n-1})}\triangleright X _{q}\) the estimated OFDM symbols, \(D^{(f_{pa}X_{n-1})}\) is the distortion,
8:\(x_{n}=IFFT\big{(}X_{q}\big{)}\)
9:\(x_{n,amplitude}=absolute(x_{n})\)
10:\(x_{n,phase}=angle(x_{n})\)
11:\(Distortion_{temp,cartesian}=D_{temp}*\exp\Big{(}sqrt(-1)*x_{n,phase}\Big{)}\)
12:\(D^{(f_{pa}X_{n})}=FFT\big{(}D_{temp,cartesian}\big{)}\)
13:\(D^{(f_{pa}X_{n-1})}=D^{(f_{pa}X_{n})}\)
14:end for return \(X_{q}\)
15:end for
16:end procedure
```  
**Algorithm 1**Frequentative Decision Feedback (FFB).