# Neural Network-Assisted DPD of Wideband PA Nonlinearity for Sub-Nyquist Sampling Systems
## 5 Conclusions  
In this paper, we proposed an NN-assisted DPD method tailored for sub-Nyquist sampling systems. To address the ambiguity problem resulting from undersampling, a two-stage architecture was introduced. In the first stage, a time delay MP reconstruction-based approach was employed to roughly estimate the broadband DPD nonlinearity using undersampled signals, requiring only a limited number of pilot signals. Subsequently, for the virtual training of DPDs, we proposed an attention-based NN DPD method in the second stage. This learning-based method leverages the transmitted data to learn the reconstructed DPD behaviors, significantly reducing the length of the training sequence and effectively refining the DPD behaviors. Our simulation results demonstrate that the proposed method can effectively mitigate the nonlinear distortion of broadband power amplifiers and provides outstanding performance gain compared to the state-of-the-art methods.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Simulating Various PA Nonlinearities** & **Memory Polynomial Coefficients for PA Modeling** \\
**Date to Temperature Drift and Humidity** & **Modeling** \\ \hline Simulate low temperature/low humidity & \([1.0513+0.0904j,\,-0.068\,-0.0023j,\,0.0289+0.0054j,\,-0.0542\,-0.29]\), \(0.2234+0.2317j,\,-0.0621\,-0.0932j,\,-0.9657-0.7028j,\) \\  & \(-0.2451-0.3735j,\,0.1229+0.1508j]\) \\ \hline Simulate mild temperature/mild & \([1.3883+0.1264j,\,0.0082+0.0090j,\,-0.0017\) \\ humidity (medium nonlinearity) & \(-0.0058j,\,0.0996-0.2196j,\,-0.0818+0.0169j,\,-0.01696j,\,-0.0205+0.0161j,\,0.0071-0.0006j,\,-1.4735-1.2327j,\,-0.0209-0.0196j]\) \\ \hline Simulate high temperature/high humidity & \([1.9067+0.1804j,\,0.0075+0.0195j,\,0.0108-0.0024j,\,-0.1403+0.0081j,\,-0.2134- \text{(severe nonlinearity)}\) & \(0.0024j,\,-0.1403+0.0081j,\,-0.2134-0.0190j,\,0.0437+0.0081j,\,-1.0739-0.1562j,\) \\  & \(-0.1196+0.1244j,\,0.2187+0.1474j]\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Memory polynomial coefficients for PA modeling.  
Figure 10: Constellation diagrams at different temperature/humidity levels: (**a**) low temperature/low humidity, (**b**) mild temperature/mild humidity, and (**c**) high temperature/high humidity.  
**Author Contributions**: Conceptualization, M.L. and D.G.; methodology, M.L. and D.G.; software, M.L. and X.Y.; validation, M.L., D.G. and J.G.; formal analysis, S.C.; investigation, X.Y. and J.G.; resources, S.C.; data curation, M.L. and D.G.; writing--original draft preparation, M.L. and D.G.; writing--review and editing, X.Y. and J.G.; visualization, G.H.; supervision, X.Y. and S.C.; project administration, G.L.; funding acquisition, G.L. All authors have read and agreed to the published version of the manuscript.  
**Funding**: This research was funded by the \"Pioneer\" and \"Leading Goose\" R&D Program of Zhejiang under grant No. 2024C01079 and the National Natural Science Foundation of China under grant No. 62301394 and grant No. 62431021.  
**Institutional Review Board Statement**: Not applicable.  
**Informed Consent Statement**: Not applicable.  
**Data Availability Statement**: Data are contained within the article.  
**Conflicts of Interest**: The authors declare no conflicts of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.