# Wavelet Decomposition Prediction for Digital Predistortion of Wideband Power Amplifiers
## 3 The Proposed WDP Architecture
The hyperparameters for the comparative models were aligned with those specified in [27]. As shown in Table 2, WDP achieved the lowest error and the highest modeling accuracy, with an NMSE of \(-\)34.96 dB. Consequently, WDP was selected for the nonlinear modeling of the PA. Since the PA output was influenced by both nonlinearity and memory effects, it was essential to account for both factors in the modeling process. The WDP approach, leveraging the FEx layer, computed \(|x|\), \(|\boldsymbol{x}|^{3}\), \(\sin x\), and \(\cos x\) from the I/Q \begin{table}
\begin{tabular}{c c} \hline
**Items** & **Setting** \\ \hline Python & 3.9.19 \\ Pytorch & 1.12.0 \\ Training epochs & 100 \\ Batch size & 64 \\ Memory depth & 24 \\ Frame length & 50 \\ Optimizer for model & Adamw (learning rate = 0.001) \\ Platform & NVIDIA GeForce GTX 2080 Ti GPU \\ \hline \end{tabular}
\end{table} 
Table 1: Relevant information about experiments. values. Here, \(|x|\) and \(|x|^{3}\) represent envelope-related terms that capture the PA's nonlinear characteristics, while \(\sin{x}\) and \(\cos{x}\) incorporate the PA's phase information, which further reflects its nonlinear behavior. In WDP, WD separated the signal into high-frequency and low-frequency components, allowing the model to effectively analyze the nonlinearity embedded in the signal. Additionally, WD incorporated memory terms during the signal decomposition process, enabling the UGRNN to better capture long-term dependencies within the signal. Since PA distortion arises from the combined influence of nonlinearity and memory effects, the WDP modeling process closely aligned with the physical nature of PA nonlinear distortion. This alignment explains why WDP demonstrated such high accuracy in PA nonlinear modeling. From Table 2, it is evident that the UGRNN alone did not achieve the same level of accuracy as WDP, underscoring the effectiveness of the WD module in enhancing modeling performance. ### Analysis of DPD Modeling

For DPD experiments, we leveraged the WDP model due to its superior modeling accuracy. The DPD performance was evaluated by training the model with five different random seeds and comparing it against VDLSTM, the RVTDCNN, the DGRU, the Feature\(+\)GRU, the IRNN, and WDP. The results, presented in Table 3, demonstrate that WDP outperformed all other methods across three key metrics: it achieved an NMSE of \(-45.51\) dB, an ACPR of \(-52.71/-51.93\) dB, and an EVM of \(-48.92\) dB. Compared to the latest state-of-the-art methods, WDP showed improvements of \(4.59\) dB in the NMSE, \(3.37/4.82\) dB in the ACPR, and \(5.14\) dB in the EVM. Furthermore, WDP significantly outperformed traditional DPD methods. When compared to VDLSTM, the DGRU, and the Feature\(+\)GRU, WDP demonstrated the best performance among RNN-based architectures. Additionally, RNN-based methods like WDP were shown to be more effective than CNN-based approaches such as the RVTDCNN in addressing PA nonlinearity. Among the compared models, the DGRU, the Feature\(+\)GRU, the IRNN, and WDP all utilized I/Q, \(|x|\), \(|x|^{3}\), \(\sin{x}\), and \(\cos{x}\) features, highlighting WDP's superior feature extraction capability and its effectiveness in handling PA nonlinearity. The performance gap between WDP and the IRNN further underscores the significant contribution of the WD module in enhancing the overall DPD performance. \begin{table}
\begin{tabular}{c c} \hline
**Modeling Methods** & **NMSE (dB)** \\ \hline RVTDCNN [32] & \(-28.48\pm 0.04\) \\ VDLSTM [31] & \(-28.57\pm 0.02\) \\ Feature\(+\)GRU [27] & \(-31.50\pm 0.20\) \\ DGRU [27] & \(-31.54\pm 0.11\) \\ UGRNN [30] & \(-31.32\pm 0.15\) \\
**WDP (Ours)** & \(-\mathbf{34.96\pm 0.08}\) \\ \hline \end{tabular}
\end{table} 
Table 2: PA nonlinear modeling performance comparison of various methods.