## 2 Principle of the CMUDPD Technique
Assuming that the length of the input original standard signal data stream \(x(n)\) is \(L\), the linear gain of PA can be written as  
\[G_{\text{linear}}=\{\sum_{n=1}^{L}|y(n)./x(n)|\}/L. \tag{11}\]  
Then, the input signal for training, \(X_{\text{train}}\), can be written as  
\[X_{\text{train}}\,=\,\big{[}x(M)\quad x(M+1)\quad\cdots\quad x(L)\big{]}. \tag{12}\]  
The output signal of (10) can be written as an array \(Y\) that is amplified by the power amplifier with a memory depth of \(M\):  
\[Y=\big{[}Y_{M}\quad Y_{M-1}\quad\cdots\quad Y_{1}\big{]}, \tag{13}\]  
\[Y_{i}= \begin{bmatrix}y(:,i)\\ y(:,i)\cdot y(:,i)\\ \vdots\\ y(:,i)\cdot y(:,i)^{(K-1)}\end{bmatrix} ^{T}, \tag{14}\]  
where \(i\in(1:M)\) and \(y(:,i)\) represent the \(i\)th column of the output matrix as exhibited in (15),  
\[y(:,i)= \begin{bmatrix}y(i)\\ y(i+1)\\ \vdots\\ y(L-M+i)\end{bmatrix} ^{T}. \tag{15}\]  
Based on (12) and (13), the coefficient matrix of the obtained memory polynomial inversion can be written as  
\[A=Y\setminus X_{\text{train}}. \tag{16}\]  
Furthermore, the model input obtained by inverting the MPM is calculated as  
\[X_{\text{model}}=Y*A. \tag{17}\]  
The error between the model input \(X_{\text{model}}\)  and the trained input \(X_{\text{train}}\)  reflects the accuracy of the memory polynomial modeling of the PA. In order to quantify the error, a mean square error \(\varepsilon\) is introduced by  
\[\varepsilon=10\cdot\log_{10}\{(X_{\text{model}}-X_{\text{train}})^{2}/(L-M)/X _{\text{train}}^{2}\}. \tag{18}\]  
When \(\varepsilon\) reaches the required linearization requirement, the output model of the PA is obtained. Subsequent simulations and experiments showed that when \(\varepsilon\) reaches approximately \(-45\) dB, the linearization performance meets the satellite communication linearity requirements [42; 43; 44]. The effect of CMUDPD is highly dependent on the accuracy of the PA model. \[Y_{\text{model}}\,=\,X_{\text{model}}*G_{\text{linear}}, \tag{19}\]where \(G_{\text{linear}}\) denotes the linear gain of the PA, as defined in (11). Then, the pre-distorted input signal of the PA can be obtained by  
\[X_{\text{MPM}}=Y_{\text{model}}*A. \tag{20}\]  
Bringing \(X_{\text{MPM}}\) into PA produces a linear convergence gain \(G_{\text{CMUDPD}}\), as shown in Figure 6, where \(G_{\text{model}}\) can be represented by \(X_{\text{MPM}}\) as  
\[G_{\text{model}}=X_{\text{MPM}}\,/\,X_{\text{train}}*G_{\text{linear}}. \tag{21}\]  
The output PSDs of all operation modes obtained by ADS simulations are shown in Figure 7 (i.e., a1 W/O DPD, a2 W/O DPD, a3 W/O DPD, b1 W/O DPD, b2 W/O DPD, and b3 W/O DPD). For comparison, Figure 7 also depicts the output PSDs of all nonlinear models calculated by MATLAB (a1 model, a2 model, a3 model, b1 model, b2 model, and b3 model). It can be seen that different operation modes for each group (i.e., modes a1, a2, and a3 for Group A; and modes b1, b2, and b3 for Group B) exhibit insignificant PSD differences. Thereby, the parameter \(\varepsilon\) defined in (18) can be utilized to identify the degree of similarity in terms of nonlinearities among different operation modes. Figure 6: The gain compensation of CMUDPD. Figure 7: Simulated results of the output PSDs. Group A: (**a**) mode a1, (**b**) mode a2, and (**c**) mode a3. Group B: (**d**) mode b1, (**e**) mode b2, and (**f**) mode b3.