# Wavelet Decomposition Prediction for Digital Predistortion of Wideband Power Amplifiers
## 3 The Proposed WDP Architecture
Setting \(\varepsilon\) to a very small value is because, even though these parameters are not part of the core of the filter, setting them to a very small value can still preserve the trainability of the network to some extent. This is particularly helpful in the early stages of training, as it allows the network to converge to a more optimal solution. In this paper, we chose the Daubechies 4 (Db4) wavelet because it offers a good balance between compact support, smoothness, and orthogonality, making it particularly effective for signal processing tasks. More importantly, the characteristics of Db4 align well with the demands of DPD tasks, where capturing both sharp transitions and subtle signal variations is crucial for accurate nonlinear compensation. In detail, the filter coefficients were set as

\[ \begin{split} I=&\{-0.0106,0.0329,0.0308,-0.187,\\ &-0.028,0.6309,0.7148,0.2304\},\end{split}  \tag{13}\]\[\mathbf{h}= \{-0.2304,0.7148,-0.6309,-0.028, \tag{14}\] \[0.187,0.0308,-0.0329,-0.0106\},\]

It was observed that the gating mechanism was pivotal in the selection of the RNN. After evaluating various configurations, we opted to integrate a vanilla RNN with a straightforward gating mechanism. Through extensive experimentation, the UGRNN was selected as the preferred RNN architecture for modeling the nonlinearity of the PA [30]. The gating mechanism in a UGRNN serves to regulate whether the hidden state is retained from the previous time step or updated with new information. The internal structure of a UGRNN network unit is illustrated in Figure 3, and the transition functions of a UGRNN are defined as follows:

\[\mathbf{c}_{t}=\tanh(\mathbf{W}^{c_{t}}\mathbf{h}_{t-1}+\mathbf{W}^{c_{t}}\mathbf{x}_{t}+\mathbf{b}^{c }), \tag{15}\]

\[\mathbf{g}_{t}=\sigma(\mathbf{W}^{S_{t}}\mathbf{h}_{t-1}+\mathbf{W}^{S_{t}}\mathbf{x}_{t}+\mathbf{b}^{ S}), \tag{16}\]

\[\mathbf{h}_{t}=\mathbf{g}_{t}\cdot\mathbf{h}_{t-1}+(1-\mathbf{g}_{t})\cdot\mathbf{c}_{t}, \tag{17}\]

In this formulation, \(\mathbf{c}_{t}\) represents the unit state, which is influenced by both the hidden state from the previous unit, \(\mathbf{h}_{t-1}\), and the current input, \(\mathbf{x}_{t}\). The tanh activation determines how information is stored in the current unit. The gating mechanism, represented by \(\mathbf{g}_{t}\), regulates the flow of information by leveraging the sigmoid function to balance contributions from the previous hidden state and the current input. Equation (17) updates the unit's hidden state by merging the functions of the forget and input gates into a unified update gate. Notably, the role of \(\mathbf{g}_{t}\) must be considered, as \(1-\mathbf{g}_{t}\) represents the forgotten information. Ultimately, the hidden state of the RNN serves as the model's output. The extracted features, along with the high-frequency and low-frequency components obtained via WD processing, are fed into the UGRNN. Distorted signals often contain a mixture of frequency components, particularly in the high-frequency range. By applying WD, the signal is decomposed into high- and low-frequency components, enabling the model to process them separately and capture distortion features more effectively. Meanwhile, features that bypass WD retain global information, providing a broader context for the model. To enhance the model's ability to utilize different frequency characteristics, we incorporate a self-attention mechanism that evaluates the impact of high-frequency, low-frequency, and raw components on the output. By dynamically assigning weights based on input characteristics, the self-attention mechanism determines the relative importance of each component. This adaptability allows it to adjust the contributions of different frequency components depending on the input signal, thereby improving the model's robustness and generalization across varying signal conditions. Figure 3: Internal structure of UGRNN unit.