# Neural Network-Assisted DPD of Wideband PA Nonlinearity for Sub-Nyquist Sampling Systems
## 4 Simulation Results  
In this section, we conduct simulations to demonstrate the effectiveness of the proposed NN-assisted wideband PA DPD method. We consider a single-user uplink communications system. The 16-QAM modulated signal is generated. The baseband waveforms are generated by up-sampling five times and then passing through a raised cosine filter with a roll-off coefficient of 0.25. For the wideband PA modeling, the MP model is simulated with nonlinear order \(K=5\), and its memory depth is \(Q=2\). According to [22], the coefficients are \([h_{10},h_{30},h_{50},h_{11},h_{31},h_{51},h_{12},h_{32},h_{52}]=[1.0513+0.0904j,-0.0542-0.2900j,-0.9657-0.7028j,-0.0680-0.0023j,0.2234+0.2317j,-0.2451-0.37 35j,0.0289-0.0054j,-0.0621-0.0932j,0.1229+0.1508j]\). For the MP-based DPD estimation in Stage 1, the order and the memory remain the same. At the feedback channel, the feedback signal is undersampled by a factor of \(D=4\), which means that the ADC sampling rate only needs to be 1/4 of the Nyquist sampling frequency. For the proposed NN, the number of nodes in the hidden layers is \([10,5]\). Regarding the number of training sets, 1000 and 100,000 sets of data are used for validation. The number of Monte Carol runs is 500. For the evaluation metrics, the EVM is defined as  
\[\text{EVM}=\sqrt{\frac{\frac{1}{N}\sum_{n=0}^{N-1}|\hat{x}(n)-x_{t}(n)|^{2}}{ \frac{1}{N}\sum_{n=0}^{N-1}|x_{t}(n)|^{2}}}, \tag{28}\]  
where \(N\) denotes the number of symbols measured by the EVM, and \(\hat{x}(n)\) denotes the \(n\)-th normalized feedback symbol, while \(x_{t}(n)\) denotes the ideal value of the \(n\)-th symbol. The SNR is defined as  
\[\text{SNR}(\text{dB})=10 lg\bigg{(}\frac{P_{\text{sig}}}{P_{\text{noi}}}\bigg{)}, \tag{29}\]  
where \(P_{\text{sig}}\) indicates signal power, and \(P_{\text{noi}}\) indicates noise power. Furthermore, the adjacent channel power ratio (ACPR) can be used to describe the power spectrum leakage into neighboring channels. It is expressed as the ratio of the leakage power in the neighboring channel to the signal power in the reference channel. A larger ACPR value indicates more severe leakage and, consequently, more serious nonlinear distortion. Since leakage can occur in both the upper and lower neighboring channels, the ACPR can be calculated separately for each. The upper neighborhood ACPR is expressed as  
\[ACPR_{u}(dB)=\frac{P_{ACH}}{P_{REFC}}=10 lg\frac{\int_{ACH}P(f)df}{\int_{REFC}P( f)df}, \tag{30}\]  
where \(P_{ACH}\) is the power of the signal in the upper neighboring channel, and \(P_{REFC}\) is the signal power in the reference channel. Similarly, the lower neighborhood ACPR can be expressed as  
\[ACPR_{l}(dB)=\frac{P_{ACL}}{P_{REFC}}=10 lg\frac{\int_{ACL}P(f)df}{\int_{REFC}P( f)df}, \tag{31}\]  
where \(P_{ACL}\) is the power of the lower neighboring channel signal. In order to validate the effectiveness of the proposed method, the proposed method is compared with the augmented real-valued time delay neural network (ARVTDNN) method proposed in [16]. The ARVTDNN is an augmented real-valued time delay neural network. The ARVTDNN consists of an input layer, a hidden layer, and an output layer. The inputs to the network are the Cartesian components (I/Q) of the input signals and the envelope correlation. The network has only one hidden layer with 17 nodes, and the output of the network consists of the I and Q paths of the estimated signal. The simulation setups of the ARVTDNN in the simulations are the same as in [16]. We also simulated the method presented in [8], for which we employed time-delayed memory polynomial reconstruction for sub-nyquist-sampled DPD, denoted as TDMPR. Figure 4 depicts the power spectrum density (PSD) of various DPD methods, including the proposed NN-based method, the ARVTDNN, TDMPR, a method based on the direct application of the DNN proposed in Stage 2, and a method without a DPD function, where the SNR is set as 15 dB.