# Neural Network-Assisted DPD of Wideband PA Nonlinearity for Sub-Nyquist Sampling Systems
## 4 Simulation Results
It can be clearly seen that without DPD, the out-of-band spectrum growth cannot be suppressed at all, which is significantly detrimental to the communication performance. The PSD with the TDMPR-based DPD achieves a marginal performance gain compared to the one without DPD, which is due to the numerical instability of the MP, even with the aid of virtual data sequences. For the direct DNN-based method, the lack of processing of the downsampled feedback signals results in the network being unable to correctly learn the nonlinear behavior of the DPD, leading to a poor nonlinear correction capability. In contrast, our proposed NN-assisted DPD method provides significompared to the other methods. And the spectrum leakage is significantly improved with the proposed method compared to the method in [16]. Table 1 compares the ACPR and EVM of DPD for various methods at \(SNR=15\) dB. It can be seen that the proposed method has outstanding performance and reduces the ACPR from \(-20\) dB to \(-42\) dB. The ARVTDNN reduced the ACPR by 16 dB, which is effective but still limited compared to the proposed method. The performance of the DNN proposed in the method based on Stage 2 is close to that of the ARVTDNN, while the lack of coarse estimation leads to performance loss, which speaks to the efficiency of the proposed two-stage methods. In addition, TDMPR is poorly corrected at low SNRs, with ACPR being reduced by only 2 dB. In conclusion, it is evident that the proposed method has excellent corrective capabilities compared to the state-of-the-art methods. To observe the detailed variations of these DPD methods under different SNRs, we simulated the proposed DPD method, TDMPR, and the method without DPD, and the results are shown in Figure 5. The SNR is from 5 dB to 35 dB. It can be seen that the proposed NN-assisted method has spectral bands without leakage along various SNR points, which demonstrates the robustness of the proposed method even under low SNRs. Although ARVTDNN also has good noise immunity, this method does not perform as well as the proposed method. In contrast, TDMPR provides unstable PSD performance and marginal performance gain. This suggests that relying solely on TDMPR for estimating the DPD signal is insufficient for mitigating the nonlinear effects induced by the PA, particularly when the SNR is low and the signal is undersampled. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model** & **Lower ACPR/dB** & **Upper ACPR/dB** & **EVM\%** \\ \hline Without DPD & \(-18.6474\) & \(-21.5771\) & \(10.25\) \\ TDMPR & \(-22.6645\) & \(-22.5484\) & \(8.08\) \\ Stage 2 only & \(-34.2174\) & \(-34.0762\) & \(5.9943\) \\ ARVTDNN & \(-36.3610\) & \(-36.0028\) & \(5.9746\) \\ Proposed & \(-43.4470\) & \(-41.0365\) & \(1.3134\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: ACPR and EVM (SNR = 15 dB). Figure 4: PSD performance of different methods at \(SNR=15\) dB. Figure 5: PSD performance of various DPD methods versus different SNRs: (**a**) PSD of PA output with TDMPR, (**b**) PSD of PA output with ARVTDNN, and (**c**) PSD of PA output with proposed NN-assisted DPD. The effects of DPD can be more intuitively observed in Figure 6. The blue constellation diagram represents the signal before correction, showing significant distortion after passing through the PA. In contrast, the red constellation diagram represents the signal after correction. It is evident that the proposed method's distortion correction is significantly better than that of ARVTDNN, indicating superior PA correction performance. Figure 7 shows the EVM performance of various DPD methods across different SNRs. It is evident that the scenario without DPD corresponds with deteriorated EVM performance.TDMPR offers only a modest performance improvement compared to the scenario without DPD. The ARVTDNN performs well when the SNR is lower, but it plateaus at around 8% and provides less efficient performance compared to TDMPR when the SNR is above around 18 dB.