# Agnostic Envelope Linearization of Dynamically Supplied Power Amplifiers for Mobile Terminals
## 2 Envelope Tracking PA Linearization
### EGMP Coefficients Extraction Process  
In order to extract the coefficients of the EGMP behavioral model that contribute to obtain the required linearity at the RF ET PA output, the goal of minimizing the mean square error between the ET PA output estimation, \(\mathbf{\hat{y}}_{\text{BB}}\), and the baseband input, \(\mathbf{u}_{\text{BB}}\), is defined as follows:  
\[\mathbf{\alpha}_{opt}=\arg\min_{\mathbf{\alpha}}\lVert\mathbf{u}_{\text{BB}}-\mathbf{\hat{y}} _{\text{BB}}\rVert^{2}=\arg\min_{\mathbf{\alpha}}\lVert\mathbf{e}_{\text{BB}}\rVert^ {2} \tag{13}\]  
with \(\mathbf{\alpha}_{opt}\) being the vector of optimum coefficients that minimize the mean square error. Since the supply envelope (\(Env_{in}[n]\)) is part of a nonlinear polynomial expansion in (9), a nonlinear least squares solution is proposed to address this minimization problem. Instead of numerical differentiation, in order to improve the adaptation accuracy and reduce the calculation time, analytical differentiation is used in this paper. The analytical solution requires the Jacobian matrix being provided to the solver. The Jacobian of the SED-GMP model to the coefficients of the EGMP model is  
\[\mathbf{J}=\left(\frac{\partial\mathbf{\hat{y}}_{\text{BB}}}{\partial\alpha_{000}}, \cdots,\frac{\partial\mathbf{\hat{y}}_{\text{BB}}}{\partial x_{p\eta q}},\cdots, \frac{\partial\mathbf{\hat{y}}_{\text{BB}}}{\partial x_{P_{b}R_{b}Q_{b}}}\right) \tag{14}\]  
with \(\mathbf{\alpha}\) being the \(O_{a}\times 1\) vector of coefficients of the EGMP model in (5) and (6) and \(\mathbf{\hat{y}}_{\text{BB}}\) being the estimated ET PA output in (9) and (10). The ordinary GMP part in (9) can be ignored in the differentiation. Therefore, by selecting the terms in (9) involving the supply envelope (\(Env_{in}[n]\)) and the coefficients \(\beta_{mls}^{c}\), it is possible to define the following basis functions:  
\[f_{mls}[n]=\beta_{mls}^{e}\,u_{\text{BB}}[n-\tau_{l}]\big{(}Env_{in}[n-\tau_{l }-\tau_{s}]\big{)}^{m} \tag{15}\]  
The Jacobian of the basis functions \(\mathbf{f}_{mls}\) can be derived by applying the chain rule as follows:  
\[\mathbf{J}_{f_{mls}}=\left(\frac{\partial\mathbf{f}_{mls}}{\partial\alpha_{000}}, \cdots,\frac{\partial\mathbf{f}_{mls}}{\partial x_{p\eta q}},\cdots,\frac{\partial \mathbf{f}_{mls}}{\partial x_{P_{b}R_{b}Q_{b}}}\right) \tag{16}\]\[I_{f_{\textit{rms}}}[n]=\beta_{\textit{rms}}^{e}\,u[n-\tau_{\texttt{i}}]\,( Env_{in}[n-\tau_{\texttt{i}}-\tau_{\texttt{s}}])^{m-1}\boldsymbol{\phi}[n-\tau_{ \texttt{i}}-\tau_{\texttt{s}}]\,m \tag{17}\]  
where \(\boldsymbol{\phi}[n]=\big{(}\theta_{000}[n],\cdots,\theta_{\textit{prq}}[n], \cdots,\theta_{\textit{p}_{\textit{z}}R_{\texttt{t}}Q_{\texttt{z}}}[n]\big{)}\) is the row vector containing the basis functions following the EGMP behavioral model described in (5), i.e., the \(n\)th row of the data matrix \(\boldsymbol{E}\). Finally, the Jacobian of the SED-GMP model is obtained taking into account all \(\boldsymbol{J}_{\textit{rms}}\) as follows. \[\boldsymbol{J}=\sum_{m=0}^{M_{\texttt{z}}}\sum_{l=0}^{L_{\texttt{s}}}\sum_{s =0}^{S_{\texttt{s}}}I_{\textit{rms}} \tag{18}\]  
With the Jacobian matrix, it is now possible to address the nonlinear least squares problem, for example, by using the Levenberg-Marquardt (LM) algorithm. The LM algorithm is an iterative procedure, where the vector of coefficients \(\boldsymbol{\alpha}\) is updated at every iteration \(i\), as follows:  
\[\boldsymbol{\alpha}^{i+1}=(1-\gamma)\boldsymbol{\alpha}^{i}-\gamma\Big{(} \boldsymbol{I}^{H}\boldsymbol{I}+\lambda\boldsymbol{I}\Big{)}^{-1}\boldsymbol {J}^{H}\boldsymbol{e_{\texttt{BB}}} \tag{19}\]  
where \(\boldsymbol{I}\) is the identity matrix, \(\gamma\) (\(0<\gamma<1\)) is the learning rate, \(\lambda\) is the damping factor, and \(\boldsymbol{J}\) is the Jacobian matrix being calculated over the error vector \(\boldsymbol{e_{\texttt{BB}}}\) with respect to \(\boldsymbol{\alpha}\). The complete procedure for training the EGMP model is shown in Algorithm 1.