# Neural Network-Assisted DPD of Wideband PA Nonlinearity for Sub-Nyquist Sampling Systems
## 2 Signal Model  
We consider a single-user uplink communications system. Considering the cost of mobile devices, we assume that the user has a single antenna where low-cost PA is employed, resulting in nonlinear distortion with memory effects. The feedback channel is equipped with a low-rate ADC. The system block diagram is shown in Figure 1.  
The modulated signal \(u(n)\) is first transmitted through a DPD module to generate \(x(n)=f_{\text{DPD}}(u(n))\), where \(f_{\text{DPD}}(\cdot)\) denotes the transfer function of DPD, and the learning of the DPD is the focus of this work. The \(x(n)\) is first upconverted, then passed through the DAC, and, finally, fed into the PA. To represent the wideband PA's nonlinear behavior, the memory polynomial (MP) model is employed [20]. The MP model is represented as a baseband signal with a discrete form, i.e.,  
\[y(n)=\sum_{k=1}^{K}\sum_{q=0}^{Q}h_{klq}x(n-q)|x(n-q)|^{k-1}, \tag{1}\]  
where \(Q\) is the maximum memory benefit depth, \(K\) is the maximum nonlinear order, and \(k\) is only odd, while \(h_{klq}\) is the coefficient of the polynomial, and \(n=0,1,\ldots,N-1\). The reason for implementing only odd orders is that the main factor of nonlinear distortion is  
Figure 1: A system block diagram of the sub-Nyquist sampling system.  
the spectral aliasing and inter-modulation distortion produced by odd orders, so the effect of even orders can be ignored. (1) can be written in matrix form as follows:  
\[\mathbf{y}=\mathbf{X}\mathbf{h}, \tag{2}\]  
where  
\[\mathbf{y}=[y(0),y(1),...,y(N-1)]^{T} \tag{3}\]  
\[\mathbf{h}=[h_{10},\ldots,h_{1Q},h_{30},\ldots,h_{3Q},\ldots,h_{K0},\ldots,h_{ KQ}]^{T}. \tag{4}\]  
Define \(\varphi_{kq}(n)=x(n-q)|x(n-q)|^{k-1}\), and the matrix \(\mathbf{X}\) is represented as  
\[\mathbf{X}=\left[\begin{array}{cccc}\varphi_{10}(0)&\varphi_{11}(0)&\cdots &\varphi_{KQ}(0)\\ \varphi_{10}(1)&\varphi_{11}(1)&\cdots&\varphi_{KQ}(1)\\ \vdots&\vdots&&\vdots\\ \varphi_{10}(N-1)&\varphi_{11}(N-1)&\cdots&\varphi_{KQ}(N-1)\end{array} \right]. \tag{5}\]  
The PA output signal \(y(n)\) is transmitted through the wireless channel to obtain \(r(n)\). Therefore, the feedback signal \(r(n)\) at the BS can be expressed as  
\[r(n)=y(n)+\omega(n), \tag{6}\]  
where \(\omega(n)\) is the noise signal, which is introduced in the feedback channel and is assumed to be Gaussian for simplicity. Due to the nonlinearity of the wideband PA, the spectrum of the feedback signal \(r(n)\) in the receiver is broadened, so an ADC with high sampling frequency is required to achieve unambiguous sampling. However, a sub-Nyquist sampling system, an ADC with a sampling rate that is much smaller than the Nyquist sampling frequency, is used to sample the feedback signal \(r(n)\). The rate of the undersampled ADC is \(1/D\) (\(D\) is an integer, and \(D>0\) ) of the Nyquist sampling frequency. \(r(n)\) is processed by the low-rate ADC, resulting in \(r(Dn)\). Hence, our aim is to restore the transmitted signal \(u(n)\) from the undersampled feedback signals \(r(Dn)\) by designing a novel DPD architecture with limited pilots, as detailed in Section 3.