## 2 Dynamic Modeling Stage
### Weighted MPM as Dynamical Modeling and Linearization Stages  
For high bandwidth applications requiring an additional linearization process, it is necessary to develop accurate models and much better if this modeling stage follows an adaptive methodology due to the nonlinearities added to the wireless transmission. To better address these nonlinearities, the W-MPM incorporates a novel weight function with a tradeoff between the static and dynamic distortions depending on the input value. This is an interesting contribution for the amplification process change due to normal use of the device. These weight functions are involved in the calculation of the coefficients in order to detect when the short-term memory occurs [27]. The W-MPM can be expressed by  
\[y_{W-MPM}(n) = \sum_{k=1}^{K_{S}}a_{k}W_{S}(|x(n)|)x(n)|x(n)|^{k} \tag{2}\] \[+ \sum_{k=1}^{K_{D}}\sum_{m=0}^{M}\beta_{k,m}W_{D}(|x(n-m)|)x(n-m) |x(n-m)|^{k}\]  
where \(a_{k}\) is the coefficient of the static part of the model, \(\beta_{k,m}\) is the model coefficients for the dynamic behavior, \(K_{S}\) is the maximum nonlinear order of the static side of the model, and \(K_{D}\) accounts for the memory effects added to the device under test. In this case, \(W_{S}(|x(n)|)\) is related to the static behavior and \(W_{D}(|x(n-m)|)\) represents the weighted functions involved in the dynamics of the model, \(k\) represents the incremental level of nonlinearity order, and \(m\) represents the short-term memory depth. In both cases, P1 dB is the crucial parameter of the calculation. It must be noted that the P1 dB is the main electrical factor that affects the coefficient update. Then, \(y_{W-MPM}\) is the complex baseband output signal. Therefore, the weighting function for the dynamic part of the model is formulated by  
\[W_{D}(|x(n-m)|)=\frac{1}{2}\Bigg{\{}\tanh\bigg{[}G(k,m)\cdot\bigg{(}1-\frac{| x(n-m)|}{|x|_{th}}\bigg{)}\bigg{]}+1\Bigg{\}}. \tag{3}\]  
where \(|x|_{th}\) is the threshold value given by  
\[|x|_{th}=x_{th_{th}}\cdot|x|_{\max}. \tag{4}\]where \(|x|_{th_{n}}\) is directly related to the P1 dB compression point as a criterion for modeling for each RF-PA. On the other hand, \(|x|_{max}\) is the maximum voltage of the input value. \(x_{th_{n}}\) is the coefficient related to the P1 dB established by the input voltage value in correlation with the amplification result. The \(G(k,m)\) function is given by  
\[G(k,m)=\frac{1}{k\cdot m^{2}}. \tag{5}\]  
Similarly, the weight function of the static model is described by Equation (6), where the P1 dB is the main electrical factor:  
\[W_{S}(|x(n)|)=\frac{1}{2}\Bigg{\{}\tanh\bigg{[}-k\cdot\bigg{(}1-\frac{|x(n-m) |}{|x|_{th}}\bigg{)}\bigg{]}+1\Bigg{\}}. \tag{6}\]  
After having developed the mathematical modeling of the RF-PA according to the W-MPM from the input-output data, the normalized mean square error (NMSE) was established as a metric to estimate the general deviations between the predicted and measured values. The calculation methods and functions used were programmed in Matlab and include the three variants of coefficient extraction for modeling (all, even, and odd). The results that we show in this paper apply to the use of only even coefficients in the model due to a better NMSE value obtained for the cases at hand. The basic principle of the linearization method is to place the nonlinear correction in front of the RF-PA behavioral modeling stage so that the combined data of the nonlinear and linear time-invariant functions produce a linear gain. In this work, the ILA linearization algorithm is proposed as an alternative with a proper accuracy joined with the dynamical stage where the electrical factor as P1 dB can be considered during the coefficient extraction.