# Wavelet Decomposition Prediction for Digital Predistortion of Wideband Power Amplifiers
## 3 The Proposed WDP Architecture
### DPD by WDP

During the DPD modeling stage, we adopt a combination of WD and IRNN methods, where WD serves the same function as in PA nonlinear modeling. For the recurrent network, we utilize an IRNN. Similarly to the UGRNN, the IRNN combines a vanilla RNN with a simple gating mechanism. However, the IRNN differs in that it employs separate gating mechanisms for the hidden state and output. The internal structure of the IRNN unit is illustrated in Figure 4.

The transition functions of the IRNN are defined as follows:

\[c_{1t}=\tanh(\mathbf{W}^{c_{1t}}h_{t-1}+\mathbf{W}^{c_{1t}}\mathbf{x}_{t}+\mathbf{b}^{c_{1}}), \tag{18}\]

\[c_{2t}=\tanh(\mathbf{W}^{c_{2t}}h_{t-1}+\mathbf{W}^{c_{2t}}\mathbf{x}_{t}+\mathbf{b}^{c_{2}}), \tag{19}\]

\[g_{1t}=\sigma(\mathbf{W}^{S_{1t}}h_{t-1}+\mathbf{W}^{S_{1t}}\mathbf{x}_{t}+\mathbf{b}^{S_{1}}), \tag{20}\]

\[0.187,0.0308,-0.0329,-0.0106\},\]0

\[0.187,0.0308,-0.0329,-0.0106\},\]1

\[0.187,0.0308,-0.0329,-0.0106\},\]2

The functions of \(c_{1t}\), \(c_{2t}\), and \(c_{t}\) are identical, just as \(g_{1t}\) and \(g_{2t}\) share the same role as \(g_{t}\). Unlike that of the UGRNN, the output of the IRNN is not the hidden state but a redefined output, as described in Equation (22). Although the IRNN introduces more parameters compared to the UGRNN, it provides better control over the model's output. ## 4 Experimental Results and Discussions

### Experimental Setup

The details of our experimental setup are presented in \[0.187,0.0308,-0.0329,-0.0106\},\]3 The dataset, provided by [27], consists of 38,400 samples of 200 MHz orthogonal frequency division multiplexing (OFDM) signals, sampled at an 800 MHz rate. The I/Q data were processed using a 40 nm CMOS-based digital transmitter and digitized by an R&S-FSW8 analyzer. The dataset was split into a training set (60%), validation set (20%), and testing set (20%). Moreover, we strictly followed the standard procedures outlined by the OpenDPD dataset for experimentation and validation, using its provided training and testing datasets to ensure consistency with established evaluation methodologies. In addition, we used AdamW as the optimizer with the ReduceLROnPlateau scheduler and ensured training stability by carefully adjusting optimization settings, including tuning

\[0.187,0.0308,-0.0329,-0.0106\},\]4 Internal structure of IRNN network unit. the learning rate, selecting appropriate activation functions, and monitoring gradient variations to prevent instability or overfitting. The experiment evaluated the quality of the model by calculating the NMSE, the simulated (SIM)-ACPR, and the SIM-EVM as

\[0.187,0.0308,-0.0329,-0.0106\},\]5

\[0.187,0.0308,-0.0329,-0.0106\},\]6

\[0.187,0.0308,-0.0329,-0.0106\},\]7

where \(\overline{y}\) and \(\overline{y}\) are cascaded models and desired outputs, respectively. \(P_{\text{main}}\) and \(P_{\text{adj}}\) are the main and adjacent channel powers, respectively. \(I[k]\) and \(Q[k]\) are in-phase and quadrature signal components after an FFT. The SIM-ACPR and SIM-EVM were calculated through software. It is confirmed in [27] that the SIM-ACPR and SIM-EVM have reliable consistency with the ACPR and EVM. ### Analysis of PA Nonlinear Modeling

To evaluate the variability introduced by random initialization during training, we conducted five independent training and testing runs for each model using different random seeds. The results are presented as the average metric values along with their standard deviations. The effectiveness of our proposed approach for PA nonlinear modeling was assessed against several established methods, including vector decomposition LSTM (VDL-STM) [31], the real-valued time-delayed convolutional neural network (RVTDCNN) [32], the dense gated recurrent unit (DGRU), the Feature+GRU [27], and the UGRNN [30]. VDL-STM incorporates phase information recovery during linearization [33; 34], the RVTDCNN is a well-known CNN-based model for DPD tasks [13], while the DGRU and Feature+GRU represent state-of-the-art techniques for WDP-based DPD.