# Nonlinear Distortion Cancellation using Predistorter in MIMO-GFDM Systems
## Methods
### Proposed Predistorter for MIMO-GFDM Systems  
The classic approach for the MIMO-GFDM systems with nonlinear HPA is that each HPA is linearized separately by the local predistorter, can be seen in Figure 3. The purpose of this local predistorter was to reduce the nonlinear distortion of each HPA. In this MIMO system, a point to point scenario was assumed. The receiver collects the signals from the receiving antenna and processes them to decode the data stream. All HPAs were assumed to have the same nonlinear characteristics in each branch and were following the memoryless Saleh model. A complex envelope from the HPA input from each branch was formulated as follows:  
\[x_{k}(n)=\rho_{k}e^{i\theta_{k}}, \tag{13}\]  
where \(\rho_{k}\) is the amplitude of \(x_{k}\), \(\theta_{k}\) is the phase of the input \(x_{k}\), and \(k\) is the number of antennas. The complex envelope of the HPA output can be modeled as follows:  
\[ \begin{array}{ll}s_{k}(n)=g[x_{k}(n)]&=A(\rho_{k})e^{i(\theta_{k}+P(\rho_{k} ))},\\ &=S(\rho_{k})e^{i\theta_{k}},\end{array}  \tag{14}\]  
where \(g\) [.] is the transfer function of HPA, \(A\) (.) And \(P\) (.) is the conversion of AM/AM and AM/PM to HPA, and \(S(\rho_{k})\) is the amplitude of the HPA output. For the nonlinear PA model, the Saleh model such as in Equations (5) and (6) was used. The operating point on the HPA is normally identified by a 'back-off' for the validation of the theoretical results, defined as:  
\[IBO=10 log_{10}(\frac{A_{0}^{2}}{P_{0}}), \tag{15}\]  
where \(A_{0}\) is the saturation of the input ampitude HPA and \(P_{0}\) is the average input power. From [PERSON]'s theorem and by extending complex Gaussian processes, the output of the HPA may be expressed as:  
\[s_{k}(n)=K_{0}\mathbf{x}_{k}(n)+\mathbf{d}_{k}(n), \tag{16}\]  
where \(K_{0}\) is the attenuation coefficient and \(d_{k}(n)\) is a nonlinear distortion not correlated with the input signal \(x_{k}(n)\). The \(K_{0}\) value given by [40] with  
\[K_{0}=\frac{1}{2}E\bigg{[}S^{\prime}(\rho_{k})+\frac{S(\rho_{k})}{\rho_{k}} \bigg{]}, \tag{17}\]  
Figure 3: MIMO-GFDM system with predistorter. where \(S^{\prime}(\rho_{k})\) is denoted as a derivative of the \(S(\rho_{k})\). The variance value of nonlinear distortion \(d_{k}(n)\) is as follows [41]:  
\[ \begin{array}{ll}\sigma_{d}^{2}&=E\big{[}|d_{k}(n)\big{|}_{2}\big{]},\\ &=E\big{[}|\!\big{[}z_{k}(n)\big{|}_{2}\big{]}\!\!\big{]}\!\!-\!\big{[}K_{0} \big{]}_{2}E\big{[}|\!\big{[}x_{k}(n)\big{|}_{2}\big{]},\\ &=E\big{[}|\!\big{[}S(\rho_{k})\big{|}_{2}\big{]}\!\!\big{]}\!\!-\!\big{[}K_{0} \big{]}_{2}E\big{[}\!\rho_{k}^{2}\big{]}.\end{array}  \tag{18}\]  
The signal received at each \(y_{m}^{NL}(n)\) antenna is formed by the independent superposition of the signal. This is related to the \(N_{t}\) antenna that shares the same space frequency resources. This signal is related to the \(N_{t}\) antenna which shares the same frequency space. The received signal was exposed to Gaussian noise on the array element as follows:  
\[y_{m}^{NL}(n)=\sum\
olimits_{k=1}^{N_{t}}H_{m,k}(n)(K_{0}x_{k}(n)+d_{k}(n))+n _{m}(n), \tag{19}\]  
where \(n_{m}(n)\) is additive white Gaussian noise (AWGN) and \(H_{m,k}(n)\) is a circular time domain channel matrix \(Nc\times Nc\) at time \(n\) formed from the \(h_{m,k}(n)\) channel response vectors for the link between the \(k\) transmitting and \(m\) receiving antennas. The channel matrix becomes a circular matrix because of the application of a cyclic prefix before transmission. In addition, the decomposition of the eigenvectors leads to the diagonalization of the channel matrix [42]. While at the receiver side, using time domain equalization techniques offers better performance compared to frequency domain [43]. The received signal in the frequency domain may be written as follows:  
\[Y_{m}^{NL}(n)=Vy_{m}^{NL}(n), \tag{20}\]  
where \(V\) is the matrix of the Fast Fourier Transform (FFT) \(V\in\mathbb{C}^{Nc\times Nc}\) with the \(Nc\) being the number of subcarriers.