# Neural Network-Assisted DPD of Wideband PA Nonlinearity for Sub-Nyquist Sampling Systems
## 3 Proposed NN-Assisted Dual-Stage DPD for Sub-Nyquist Sampling Systems  
Stage 1: Time-delayed Memory Polynomial Reconstruction-Based Coarse Estimation of the Digital Predistorter  
Directly using downsampled feedback signals for training a neural network model will prevent the model from accurately learning the behavior of the Digital Predistortion (DPD) system, leading to poor performance. Therefore, a preliminary estimation of the DPD signals is necessary before model training [8]. As depicted in Figure 1, after the feedback signal passes through a low-rate ADC, it must be combined with the high-rate complex baseband signal and fed into the DPD signal coarse estimation module, the structure of which is shown in Figure 2. The downsampled feedback signal is denoted as \(r(m)\), and \(r(m)=r(Dn)\). The complex baseband signal \(x(n)\) is first sampled to obtain \(x(m)\), where \(x(m)=x(Dn)\). Subsequently, the PA forward model is constructed using both \(r(m)\) and \(x(m)\). The expression for the PA forward model is  
\[r(m)=\sum_{k=1}^{K}\sum_{q=0}^{Q}g_{kq}x(m-q)|x(m-q)|^{k-1}, \tag{7}\]where \(K\) and \(Q\) are the same as in (1). Therefore, this PA model can also be expressed in a matrix form, like (2):  
\[\mathbf{r}=\mathbf{X}_{D}\mathbf{g}, \tag{8}\]  
where  
\[\mathbf{r}=[r(0),r(1),\dots,r(M-1)]^{T} \tag{9}\]  
\[\mathbf{g}=\big{[}g_{10},\dots,g_{1Q},g_{30},\dots,g_{3Q},\dots,g_{K0},\dots,g_ {KQ}\big{]}^{T}. \tag{10}\]  
The matrix \(\mathbf{X}_{D}\) is represented as  
\[\mathbf{X}_{D}=\left[\begin{array}{cccc}\varphi_{10}(0)&\varphi_{11}(0)& \cdots&\varphi_{KQ}(0)\\ \varphi_{10}(1)&\varphi_{11}(1)&\cdots&\varphi_{KQ}(1)\\ \vdots&\vdots&&\vdots\\ \varphi_{10}(M-1)&\varphi_{11}(M-1)&\cdots&\varphi_{KQ}(M-1)\end{array} \right]. \tag{11}\]  
Using least square (LS) solutions, the coefficients \(\mathbf{g}\) can be obtained as  
\[\mathbf{g}=(\mathbf{X}_{D}^{H}\mathbf{X}_{D})^{-1}\mathbf{X}_{D}^{H}\mathbf{ r}. \tag{12}\]  
\(\mathbf{g}\) is a parameter of the new PA model, which is constructed using the downsampling signal. With this new power amplifier model, a reconstructed feedback signal \(r_{rec}(n)\) based on a storage polynomial is obtained according to (2), where the parameter \(\mathbf{h}\) is replaced by \(\mathbf{g}\). A preliminary DPD model is built using the reconstructed feedback signal \(r_{rec}(n)\) and the high-rate baseband complex signal \(x(n)\). Like (2) and (8), it can be expressed in matrix form:  
\[\mathbf{x}=\mathbf{R}_{rec}\mathbf{a}, \tag{13}\]  
where \(\mathbf{x}\) is an \(N\)-dimensional column vector, \(\mathbf{a}\) is a column vector with \((K+1)/2*(Q+1)\) elements, and \(\mathbf{R}_{rec}\) is a \(N\times(K+1)/2*(Q+1)\) matrix, which is derived in the same way as in Section 2. Using least square (LS) solutions, the coefficients \(\mathbf{a}\) can be obtained as  
\[\mathbf{a}=(\mathbf{R}_{rec}^{H}\mathbf{R}_{rec})^{-1}\mathbf{R}_{rec}^{H} \mathbf{x}. \tag{14}\]  
Based on the determined \(\mathbf{a}\), the coarse estimation of the DPD behavior can be determined, and it can be used for the prediction of the DPD output whenever data are incorporated.