# Agnostic Envelope Linearization of Dynamically Supplied Power Amplifiers for Mobile Terminals
## 2 Envelope Tracking PA Linearization
### Supply Envelope Generation and ET PA Nonlinear Behavior  
By means of a shaping function, it is possible to accommodate the shape of the supply voltage (that somehow must follow the instantaneous RF envelope) to achieve one of the following objectives: optimum efficiency shaping or isogain shaping [21] (i.e., sacrificing power efficiency for linearity). In addition, in order to deal with the bandwidth limitation of the ETM, the use of a slower version of the supply envelope has been proposed in [17; 18], to mitigate the trade-off between power efficiency and linearity. In general, the generated supply envelope \(Env_{gen}[n]\) is defined as  
\[Env_{gen}[n]=\Gamma\big{(}|u_{\text{BB}}[n]|\big{)} \tag{1}\]  
Figure 1: Block diagram of the forward path for ET PA linearization. where \(\Gamma(\cdot)\) denotes the shaping function and \(u_{\text{BB}}[n]\) is the complex I-Q baseband signal to be transmitted (see notation in Figure 1). One commonly used memoryless parametric shaping function [22] to accommodate the supply levels and determining the level of detroughing is  
\[Env_{\text{DT}}[n]=\left(\left(E_{\text{TH}}\right)^{p}+\left|u_{\text{BB}}[n] \right|^{p}\right)^{1/p} \tag{2}\]  
where \(E_{\text{TH}}\) is the threshold or lower bound of the envelope that determines the swing voltages or dynamic range of the ETM, and \(p\) is the order of this detroughing function. The offset percentage (OP) of the supply envelope voltage is defined as  
\[\text{OP }(\%)=\frac{E_{\text{TH}}}{\max(\left|u_{BB}[n]\right|)}\cdot 100 \tag{3}\]  
As observed in (2) and (3), the detroughing function has two degrees of freedom (i.e., OP and \(p\)) to trade-off the power efficiency and linearity in ET PA. Figure 2 shows the evolution of key performance indicators such as the adjacent channel power ratio (ACPR), the normalized mean square error (NMSE), the drain power efficiency, and the output power for different configurations of the OP and \(p\) parameters when considering an NR-60 MHz OFDM-like test signal. As observed in Figure 2, the offset percentage is a key parameter that significantly impacts both the linearity (in terms of NMSE and ACPR) and the drain power efficiency. The higher the OP, the better the linearity but the lower the drain power efficiency. The efficiency can reach 16% when the OP is 0.1, which has the worst ACPR of around \(-20.7\) dBc. When the \(OP=0.8\) and \(p=8\), the ACPR can reach \(-35\) dBc but the efficiency drops to around 13%. As mentioned before, when dealing with wideband signals, the ETM unwanted effects (i.e., nonlinear behavior, RF leakage) may worsen and the supply envelope linear amplification may be compromised. Consequently, it is possible to include, on top of the detroughing function, a slew-rate reduction (SR) algorithm to cope with the bandwidth limitations of the ETM. The SR reduction algorithm proposed in [18] is described as follows:  
Figure 2: NMSE (**a**), ACPR (**b**), efficiency (**c**), and output power (**d**) performance of the ET PA with the NR-60 MHz OFDM-like signal, using the detroughing envelope with different offset percentage and p configurations. \[Env_{SR}[n]=\max_{i=0,1,\ldots,\mathbb{N}}(Env_{DT}[n+i]-i\cdot\frac{V_{\max}}{ \mathbb{N}}) \tag{4}\]  
where \(\mathbb{N}\) is the number of steps, and \(V_{\max}\) is the maximum numerical value of the envelope. In practice, the baseband input is normalized by a scaling factor so that it has a maximum amplitude of \(1\) (\(V_{\max}=1\)). The only configuration that needs to be determined for the SR algorithm is the number of steps \(\mathbb{N}\). The higher the number of steps, the smaller the skew-rate and the bandwidth of the envelope signal. The SR shaping function is a memory-based algorithm with memory depth \(\mathbb{N}+1\) and the hardware implementation cost of the SR algorithm increases with \(\mathbb{N}\). Figure 3 shows the ACPR and efficiency performance of the SR envelope with different configuration of \(\mathbb{N}\).