## 2 Principle of the CMUDPD Technique  
Understanding the physical distortion generation mechanisms is critical to designing a linear low-sidelobe antenna array. In this section, the structure of CMUDPD for a low-sidelobe active antenna array is described in detail. The methods for generating a low-sidelobe active array antenna include using attenuators cascaded following power amplifiers to obtain the desired power levels, or using a nonuniform amplitude feed network [30; 31]. In the first case, attenuators are located at the output side of each PA. It is obvious that the attenuators may consume a large amount of power, and they also increase the risk of failure due to heat generation. The available space on a satellite is very limited, and it is highly desired to use fewer feedback paths to achieve the linearization of a low-sidelobe antenna array. The existing low-sidelobe array antenna adopts the aperture amplitude weighted design process [32; 33; 34; 35]. The unequal microstrip amplitude feed network is designed to achieve the low-sidelobe characteristics of an array. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multirow{2}{*}{**Array Types**} & **Feedback Path** & **DPD Unit** & **Synthesis of PAs** & **Variable Parameters** \\  & & **Number** & **Number** & **Outputs** & **Considered** \\ \hline One-to-one DPD [5] & Uniform & N & N & Not required & \(-\) \\ Average DPD [10] & Uniform & 1 & 1 & Required & \(P_{\text{in}}\) \\ BO-DPD [12] & Uniform & 1\(<\) n \(<\)N & 1 & Required & \(P_{\text{in}}\) \\ Adaptive DPD [24] & Uniform & 1 & 1 & Required & \(-\) \\ Array DPD [20] & Low-sidelobe & 1 & 1 & Required & \(P_{\text{in}}\) \\ Proposed CMUDPD & Low-sidelobe & 1 & 1 & Not required & \(V_{\text{DD}}\), \(V_{\text{CG}}\), \(P_{\text{in}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The comparison of the proposed solution with the state-of-the art methods. N: number of antenna elements, \(P_{\text{in}}\): input power, \(V_{\text{DD}}\): PA supply voltage, \(V_{\text{GG}}\): PA bias voltage. The method aims to drive the PAs with different input power levels. From the energy point of view, this situation, at least, does not waste a large amount of power. Therefore, a structure using a small signal unequal power divider is considered in the subsequent analysis, as shown in Figure 2.  
The input-output relationship of the power amplifier is mainly reflected in the inherent static nonlinear characteristics and memory effects [36; 37]. In Figure 3a, PA\({}_{i}\) and PA\({}_{j}\) represent two PAs in a phased array operating at the 1 dB compression point under mode \(i\) and mode \(j\), respectively. The actual input-output relationship of PA\({}_{i}\) can be expressed as  
\[y_{i}(t)=G_{i}\cdot g_{i}(x_{i}(t)), \tag{1}\]  
where \(G_{i}\) is a constant complex number and indicates expected gain values of PA\({}_{i}\). It should be noted that \(g_{i}(\cdot)\) represents the shape function of the gain curve, and \(g_{i}^{-1}(\cdot)\) is the inversion of the gain curve \(g_{i}(\cdot)\). Then, the ideal pre-distortion signal to PA\({}_{i}\), i.e., \(x_{i\_\_\_\_\text{DPD}}\left(t\right)\), can be expressed as  
\[x_{i\_\_\_\text{DPD}}\left(t\right)=G_{i}\cdot g_{i}^{-1}(x_{i}(t)). \tag{2}\]  
So, the output signal after the pre-distortion, i.e., \(y_{i\_\_\text{DPD}}\), can be obtained as follows:  
\[y_{i\_\_\text{DPD}}(t)=G_{i}\cdot g_{i}(g_{i}^{-1}(x_{i}(t)))=G_{i}\cdot x_{i} (t), \tag{3}\]  
which indicates that PA\({}_{i}\) is linearized. The simulation and experiment results show that PA\({}_{i}\) and PA\({}_{j}\) can have similar characteristics by properly tuning their supply and bias voltages. The similar characteristics mean that the gain vs. input power curve of PA\({}_{i}\) can be replicated by moving that of PA\({}_{j}\) vertically and horizontally, as shown in Figure 3b.