# Agnostic Envelope Linearization of Dynamically Supplied Power Amplifiers for Mobile Terminals
## 2 Envelope Tracking PA Linearization
### Supply-Envelope-Dependent ET PA Modeling  
In order to model the ET PA output taking into account the supply envelope (\(Env_{in}[n]\)) and baseband I-Q input signal (\(u_{\text{BB}}[n]\)), a two-dimensional supply-envelope-dependent generalized memory polynomial (SED-GMP) behavioral model is proposed. The SED-GMP is composed of two parts, the ordinary GMP part that contains nonlinear and memory polynomial terms of the input signal envelope, \(|u_{\text{BB}}[n]|\), to model the PA nonlinear behavior; and the SED part that includes the nonlinear and memory polynomial terms of the supply envelope, \(Env_{in}[n]\), to characterize the ETM unwanted behavior. The SED-GMP input output relationship is defined as follows:  
\[ \begin{split}\hat{y}_{\text{BB}}[n]&=\sum_{q=0}^{ Q_{b}}\sum_{r=0}^{R_{b}}\sum_{p=0}^{P_{b}}\beta_{pq}^{u}\,u_{\text{BB}}[n-\tau_{ r}]\big{|}u_{\text{BB}}[n-\tau_{r}-\tau_{q}]\big{|}^{p}\\ &+\sum_{s=0}^{S_{b}}\sum_{l=0}^{L_{b}}\sum_{m=0}^{M_{b}}\beta_{ mls}^{e}\,u_{\text{BB}}[n-\tau_{l}]\big{(}Env_{in}[n-\tau_{l}-\tau_{s}]\big{)}^{m} \end{split}  \tag{9}\]  
where \(\tau_{r}\), \(\tau_{q}\), \(\tau_{l}\), and \(\tau_{s}\) (with \(\tau_{q,l,s}\in\mathbb{Z}\)) are the most significant sparse delays of the baseband I-Q input signal (\(u_{\text{BB}}[n]\)), its envelope (\(|u_{\text{BB}}[n]|\)) and the input supply envelope (\(Env_{in}[n]\)) that contribute to characterize memory effects; \(\beta_{pq}^{u}\) and \(\beta_{mls}^{e}\) are the coefficients of the GMP and SED parts, respectively. The order of the SED-GMP behavioral model is \(O_{b}=(Q_{b}+1)(R_{b}+1)(P_{b}+1)+(S_{b}+1)(L_{b}+1)(M_{b}+1)\). Similar to (6), the ET PA output estimation can be rewritten in a matrix notation as  
\[\hat{y}_{\text{BB}}=\mathbf{U}\mathbf{\beta} \tag{10}\]  
where \(\hat{y}_{\text{BB}}\) is the \(N\times 1\) vector of the estimated baseband ET PA output, \(\mathbf{U}=\big{(}\mathbf{U}_{u}\,\,\,\mathbf{U}_{e}\big{)}=\big{(}\mathbf{\phi}[0],\cdots,\bm {\phi}[n],\cdots,\mathbf{\phi}[N-1]\big{)}^{T}\) is the \(N\times O_{b}\) SED-GMP data matrix, with \(N\) being the number of samples and \(O_{b}\) the order of the SED-GMP model, and \(\mathbf{U}_{u}\) and \(\mathbf{U}_{e}\) are the data matrices of the GMP and SED parts in (9), respectively. The vector containing the ba  
Figure 7: Block diagram of the optimized envelope generation scheme, the measurement of \(Env_{out}\) is not required by the optimizer. sis functions following the SED-GMP behavioral model described in (9) is \(\mathbf{\phi}^{T}[n]=\left(\phi_{000}^{u}[n],\cdots,\phi_{p\eta q}^{u}[n],\cdots,\phi_ {P_{b}R_{b}Q_{b}}^{u}[n],\phi_{000}^{c}[n],\cdots,\phi_{mls}^{c}[n],\cdots,\phi_{ M_{b}L_{b}S_{b}}^{c}[n]\right)\), while the \(O_{b}\times 1\) vector \(\mathbf{\beta}^{T}=\left(\mathbf{\beta}^{u}\ \ \mathbf{\beta}^{c}\right)=\left(\beta_{000}^{u}, \cdots,\beta_{p\eta q}^{u},\cdots,\beta_{P_{b}R_{b}Q_{b}}^{u},\beta_{000}^{c},\cdots,\beta_{mls}^{c},\cdots,\beta_{M_{b}L_{b}S_{b}}\right)\) is the vector of coefficients. After measuring, time-aligning and normalizing (by the targeted linear gain \(G_{0}\)) the ET PA output \(\mathbf{y}_{\text{BB}}\), the coefficients of the SED-GMP model can be extracted by means of the ordinary LS estimation as follows:  
\[\mathbf{\beta}^{\prime}=\left(\mathbf{u}^{H}\mathbf{u}\right)^{-1}\mathbf{u}^{H}\mathbf{y}_{\text {BB}} \tag{11}\]  
where the superscript \(H\) stands for the Hermitian transpose. After a first coefficient identification, the coefficients of the SED-GMP model can be iteratively updated using new input and output ET PA measurements as follows:  
\[\mathbf{\beta}^{i+1}=\gamma\mathbf{\beta}^{\prime}+(1-\gamma)\mathbf{\beta}^{i} \tag{12}\]  
where \(\mathbf{\beta}^{i}\) is the SED-GMP coefficients at \(i\)th iteration, \(\mathbf{\beta}^{\prime}\) is the vector of coefficients extracted taking into account the latest data measurements as described in (11), and \(\gamma\) (\(0<\gamma<1\)) is the learning rate.