## 3 Frequentative Decision Feedback (FFB)  
In this section, an iterative feedback system is used to remove the distortion in the overall system. It is assumed that the nonlinearity caused by the transmitting side will be mitigated at the receiving side. Figure 5 shows the FFB model used in this paper. If the nonlinearity is present in the discrete time domain, then this analysis provides authentic results. When there is nonlinearity in the continuous  
Figure 4: The importance of the neuron is represented by weights that have a connection in the model. We can calculate the outvalue of the neuron as  
Figure 3: Generic neural network composed of three layers. time domain, then it gives us approximate results. It can be more accurate when oversampling takes care of the spectral regrowth. \[x_{\frac{\pi}{L}}^{f_{\mu\mu}}=f_{\mu(x\frac{\pi}{L})} \tag{16}\] \[\alpha.x_{L}^{n}+d_{n}{}^{(X)}\]  
where \(x_{n}\) represents the input to a PA, \(f_{\mu\mu}\) is the learned model for the PA, and oversampling is given as \(L\). The output can be expressed as a linear combination of actual amplified input to the PA and a distortion term. MSE \(E\Big{[}\big{|}f_{\mu\mu}(x_{L}^{n})-\alpha\cdot x_{L}^{n}\big{|}^{2}\Big{]}\) is minimized when we put the value of the constant \(\alpha\) in such a manner. The variable contains some distorted energy that is not related to \(x_{L}^{n}\). The soft limiter and SSPA nonlinearity are proved \(\alpha\to 1\) by putting the value of clipping \(>7.3\) dB. If the intersymbol interference (ISI) of the channel is less than the value of the cyclic prefix (CP), then it is shortened to a single multicarrier symbol. After this, the distortion looks like the deterministic function of \(x_{L}^{n}\), so \(d_{n}^{(X)}\). Hence, the QAM vector will have the \(p^{th}\) symbol and \(X_{L}^{p}\), which will contain the nonlinearity distortion. The FFT can be computed from Equation (16) over the whole interval, which also includes the oversampling discrete interval. Mathematically it is written as follows:  
\[FFT\Big{(}x_{\frac{\pi}{L}}^{f_{\mu\mu}}\Big{)}=X_{L}^{f_{\mu\mu}}=X_{L}^{m}+D _{L}^{(f_{\mu\mu}x^{p})} \tag{17}\]  
The time index here is K. Then, we can expand Equation (17) further as  
\[FFT\Big{(}x_{\frac{\pi}{L}}^{f_{\mu\mu}x}\Big{)}=\left\{ \begin{array}{c}X_{ k}^{m},k+D_{L,k}^{(f_{\mu\mu},X^{p})},k=0,\ldots,\frac{N}{2}-1\\ D_{(L,k)}^{(f_{\mu\mu},X^{p})},k=\frac{N}{2},\ldots,NL-\frac{N}{2}-1\\ X_{k-N(L-1)}^{m}+D_{L,k}^{(f_{\mu\mu},X^{p})},k=NL-\frac{N}{2}-1,\ldots,NL-1. \end{array} \right.\]  
Here, \(D_{L,k}^{(f_{\mu\mu},X^{p})},k=\frac{N}{2},\ldots,NL-\frac{N}{2}-1\) is the out-of-band distortion; thus, to minimize this distortion, we use clipping and windowing. The symbol index p will be eliminated with oversampling \(L\) for the receiver design to be explained more clearly, which makes equation representation simpler. Furthermore, an assumption takes place before applying a reduction algorithm; the out-of-band  
Figure 5: Frequentative decision feedback (FFB) model. distortion is minimized. If \(H_{k}\) is the FFT of the channel impulse response \(h[n]\), then, in this regard, the received symbols can be represented as  
\[Y_{k}^{f_{m}}=H_{k}(X_{k}+D_{k}^{(f_{m},X)})+Noise_{k} \tag{18}\]  
where \(Noise_{k}\) is the Additive white guassian noise (AWGN) component for the \(k^{th}\) OFDM symbol. A maximum-likelihood receiver for estimating \(X\) will be  
\[\hat{X}=\arg\min_{\forall\overline{X}}\sum_{k=0}^{N-1}\left(H_{k}(\overline{X }_{k+}D_{k}^{(f_{m},\overline{X})})-Y_{k}^{f_{m}}\right) \tag{19}\]  
While considering one OFDM symbol, Equation (19) can be written as a vector form. \(\overline{X}\) is the transmit symbol vector. \[\hat{X}=\arg\min_{\overline{X}}\lVert(H\times(\overline{X}+D^{(f_{m}, \overline{X})})-Y^{f_{m}})\rVert^{2} \tag{20}\]  
The element-to-element vector product is denoted by \(\times\).