# Agnostic Envelope Linearization of Dynamically Supplied Power Amplifiers for Mobile Terminals
## 2 Envelope Tracking PA Linearization
### Optimized Envelope Generation for ET PA Linearization  
The proposed linearization method is applied in the supply envelope path. However, unlike in [19], where the ETM output was available to train an ELC subsystem, now it is assumed that the ETM output is no longer accessible. Therefore, instead of targeting the leakage cancellation at the ETM output following the ELC approach (as depicted in the block diagram of Figure 6), an envelope optimizer targeting the linearization of the RF PA output is now proposed (see Figure 1). The envelope optimizer presented in this paper is also based on the generalized memory polynomial (EGMP) model, defined as follows:  
\[Env_{\text{opt}}[n]=\sum_{q=0}^{Q_{x}}\sum_{r=0}^{R_{q}}\sum_{p=0}^{P_{x}} \alpha_{pqq}\left|u_{\text{BB}}[n-\tau_{r}]\right|\left|u_{\text{BB}}[n-\tau_{ r}-\tau_{q}]\right|^{p} \tag{5}\]  
where \(\tau_{r}\) and \(\tau_{q}\) (with \(\tau_{r,q}\in\mathbb{Z}\)) are the most significant sparse delays of the envelope (\(\left|u_{\text{BB}}[n]\right|\)) that contribute to characterize memory effects, and \(\alpha_{pqq}\) are the model coefficients. Following the notation in Figures 1 and 6, the linearized input envelope to the ETM can be defined in a compact matrix notation as  
\[\mathbf{Env}_{\text{in}}=\mathbf{Env}_{\text{gen}}-\mathbf{Env}_{\text{opt}}=\mathbf{Env} _{\text{gen}}-\mathbf{E}\:\mathbf{\alpha} \tag{6}\]  
where \(\mathbf{Env}_{\text{gen}}\) and \(\mathbf{Env}_{\text{in}}\) are the \(N\times 1\) vectors of the original supply envelope and the ETM input envelope, respectively, while \(\mathbf{E}=\left(\mathbf{\theta}[0],\cdots,\mathbf{\theta}[n],\cdots,\mathbf{\theta}[N-1] \right)^{T}\) is the \(N\times O_{x}\) EGMP data matrix, with \(N\) being the number of samples and \(O_{a}=(Q_{a}+1)(R_{a}+1)(P_{a}+1)\) the order  
Figure 4: ACPR and efficiency trade-off and comparison between the detroughed envelope and the SR envelope. Figure 5: Power spectra density plot of the ETM output of an SR reduced envelope with and without RF input. of the EGMP model. The vector containing the basis functions \(\phi_{pqr}[n]\) following the EGMP behavioral model described in (5) is \(\mathbf{\vartheta}^{T}[n]=\left(\theta_{000}[n],\cdots,\theta_{pqr}[n],\cdots,\theta _{pR_{k}Q_{k}}[n]\right)\), while \(\mathbf{a}^{T}=\left(\alpha_{000},\cdots,\alpha_{pqr},\cdots,\alpha_{pR_{k}Q_{k}} \right)\) is the \(O_{a}\times 1\) vector of coefficients. As detailed in [23], feature selection techniques can be applied to select the most relevant basis functions and thus reduce the order of the original EGMP behavioral model. When the ETM output is available, the EGMP model can be trained to predict the RF leakage at the ETM output. Then, this RF leakage estimation is injected in counter-phase to the envelope input to perform the leakage cancellation at the ETM output. For the sake of completeness, Figure 6 shows the block diagram of the RF leakage compensation approach proposed in [19], which uses the leakage estimation \(\mathbf{Env}_{\text{leak}}\), instead of \(\mathbf{Env}_{\text{opt}}\) in (6). The leakage is extracted by subtracting the ETM output measurement with the original generated envelope as  
\[\mathbf{e}_{leak}=\mathbf{Env}_{out}-\mathbf{Env}_{gen} \tag{7}\]  
where \(\mathbf{e}_{leak}=\left(e_{leak}[0],\cdots,e_{leak}[n],\cdots,e_{leak}[N-1]\right)^ {T}\) is the \(N\times 1\) error vector. Then, the EGMP model coefficients can be extracted iteratively finding the least squares (LS) solution. At the \(i\)th iteration it is described as follows:  
\[\mathbf{\alpha}^{i+1}=\mathbf{\alpha}^{i}+\mu\left(\mathbf{E}^{T}\mathbf{E}\right)^{-1}\mathbf{E} ^{T}\mathbf{e}_{leak} \tag{8}\]  
As shown in [19], by canceling the RF leakage at the ETM output, the overall linearity of the ET PA system is improved without impacting the efficiency of the ET PA. In a more realistic and low-cost (i.e., an extra ADC is saved) scenario in which the ETM output measurement is not accessible, there is no reference to directly train the EGMP model for estimating the unwanted RF leakage.