Wavelet Decomposition Prediction for Digital Predistortion of Wideband Power Amplifiers

Comments 1: The authors use Daubechies 4 Wavelet, but do not provide a comparative analysis with other types of wavelet transforms, which could affect the quality of the pre-correction.

Response 1: Thanks a lot for your comments that helped us to improve the quality of our manuscript. What we want to clarify is that the main purpose of this study is to verify whether the introduction of wavelet transform itself can improve system performance, rather than comparing the advantages and disadvantages of different types of wavelet transforms. 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. From the experimental results, it can be seen that using Daubechies 4 wavelet indeed brings significant performance improvement, which proves the effectiveness of wavelet transform in this application. We agree that comparing the performance of different wavelet bases may lead to more comprehensive conclusions, which will be a valuable direction for future research. To further clarify our choice, we have added a detailed explanation in the manuscript regarding why Db4 was selected for this study (page 7, line: 192-197).

Comments 2: Why is the combination of UGRNN for modeling nonlinearities and IRNN for pre-correction optimal? The authors do not consider alternatives such as LSTM or GRU, which are widely used in similar tasks.

Response 2: Thanks a lot for your comments that helped us to improve the quality of our manuscript. We did conduct comprehensive experimental comparisons on RNN models such as LSTM and GRU in our preliminary work. In the PA nonlinear modeling stage, we found that UGRNN performed the best in modeling accuracy, which is crucial for the performance of subsequent DPD stages. Specifically, UGRNN has demonstrated better ability in capturing PA nonlinear characteristics, which may be due to its unique gating mechanism being more suitable for our specific application scenarios. In the DPD stage, we also conducted extensive model comparison experiments. IRNN demonstrated the best performance in this task.

Comments 3: WDP combines elements of direct (DLA) and indirect (ILA) learning methods, but there is no clear explanation of how this approach is optimal and how it compares to hybrid methods of other studies.

Response 3: Thanks a lot for your comments that helped us to improve the quality of our manuscript. The revised part is (page 2, line: 64-75):

“Compared with ILA, the ILA method relies on accurate inversion of the PA model, and the nonlinear characteristics of PA in actual systems may vary over time or in the environment, leading to inaccurate inversion models and affecting the pre distortion effect. WDP directly models the nonlinear behavior of PA through neural networks, and freezes the model parameters after training to ensure the stability of the PA model during the training process of the pre distorter, thereby improving the overall robustness of the system. Compared with DLA, DLA methods typically require inverse operations on the output of PA to directly optimize the parameters of the pre distorter, which may lead to numerical instability or computational difficulties in highly nonlinear systems. WDP models the nonlinear characteristics of PA through neural networks, avoiding complex inverse operations and improving system stability and computational efficiency.”

Comments 4: The study is conducted on the open OpenDPD dataset, but there is no analysis of how the method works under conditions of high noise or distortion in real RF environments.

Response 4: Thanks a lot for your comments that greatly helped improve the readability of the manuscript. We chose the OpenDPD dataset for our experiments based on the following considerations: Firstly, OpenDPD is a widely recognized and important open-source dataset in the fields of PA modeling and digital pre distortion, which has been widely adopted and validated by numerous peer studies. Secondly, the creators of the OpenDPD dataset have confirmed in their published work that the experimental results under this dataset have good consistency with real experimental environments, providing a reliable foundation for the validation of our method. From the experimental results, our method demonstrated excellent performance on the OpenDPD dataset, providing preliminary validation for the effectiveness of the method. However, we fully agree with your point that testing under high noise and distortion conditions in real RF environments is crucial for a comprehensive evaluation of method performance. Due to the limitations of high-end RF testing equipment in the laboratory, we are temporarily unable to conduct experimental verification in real environments. This is indeed a limitation of our work. In future research plans, we will actively seek opportunities for collaboration and acquire necessary experimental equipment to further validate and optimize our methods in real RF environments (page 13, line: 340-342).

Comments 5: The choice of hyperparameters (memory depth, frame length, etc.) is not always justified, which may affect the generalizability of the method when changing settings.

Response 5: Thanks a lot for your comments that helped us to improve the quality of our manuscript. We fully agree that the reasonable selection of hyperparameters is crucial for the generalizability of the method. In the current study, our main focus is on verifying the feasibility of the proposed method, therefore we have chosen a relatively conservative set of parameter settings (such as single-layer wavelet decomposition). Although this choice may limit the feature extraction capability of the model, it helps us to more clearly evaluate the core effectiveness of the method. We recognize that using multi-layer wavelet decomposition can indeed enhance the feature extraction ability of the model, reveal more detailed information, and potentially improve the accuracy of the model. This will be one of the important directions for future work. We plan to systematically explore the impact of different hyperparameter settings (including wavelet decomposition layers, memory depth, frame length, etc.) on method performance in future research, in order to establish more universal parameter selection criteria. The main contribution of the current research lies in proposing this innovative method and verifying its feasibility. We believe that this foundational work has laid an important foundation for subsequent optimization and promotion (page 13, line: 342-343).

Comments 6: Will the method be as effective with a smaller amount of training data? There is no analysis of the minimum sample size required to obtain reliable results.

Response 6: Thanks a lot for your comments that helped us to improve the quality of our manuscript. Our current experiment was indeed conducted on the standard dataset provided by OpenDPD, which contains enough samples to ensure the reliability of the training. Due to the focus of the research on verifying the feasibility of the proposed method, we have not yet considered performance analysis under small sample conditions (page 13, line: 343-344).

Comments 7: The authors use a complex cascaded neural network architecture, but do not discuss whether the model's internal nonlinearities can affect its performance or cause training instability.

Response 7: Thanks a lot for your comments that helped us to improve the quality of our manuscript. We did intentionally introduce nonlinear components in the PA nonlinear modeling and DPD stages, mainly due to the significant nonlinear characteristics of PA itself. Adding nonlinearity to the model can help it fit the nonlinear behavior of PA more quickly and accurately, thereby improving overall performance. Specifically, in the PA nonlinear modeling stage, the introduction of nonlinear components enables the model to better capture the complex characteristics of PA, which is crucial for the performance of subsequent DPD stages. In the DPD stage, appropriate nonlinear structures can also help more effectively compensate for the distortion introduced by PA. However, we acknowledge that the internal nonlinearities of the model may impact training stability. To mitigate potential instability, we carefully tuned the optimization process, including adjusting the learning rate, selecting appropriate activation functions, and monitoring gradient variations to prevent issues such as vanishing or exploding gradients. These measures help ensure stable convergence and robust performance of the model. We have supplemented this discussion in the parameter description section of the manuscript (page 9, line: 256-259).

Comments 8: Although the model shows good results in simulations, there are no tests on physical power amplifiers, which may reveal additional challenges in real-world applications.

Response 8: Thanks a lot for your comments that helped us to improve the quality of our manuscript. Due to limitations in high-end RF testing equipment in the laboratory, we are currently using the OpenDPD dataset for experimental validation. OpenDPD is a widely recognized open-source dataset in the field of PA modeling and digital pre distortion, and its creators have verified that the dataset has good consistency with real experimental environments. Our method demonstrates excellent performance on the OpenDPD dataset, providing preliminary evidence for its effectiveness. However, we fully agree with your point of view that real physical power amplifier testing may reveal some unforeseen issues in simulated environments, which may indeed pose new challenges to our approach. In future research plans, we will actively seek collaboration opportunities to acquire necessary experimental equipment for comprehensive testing and optimization on real physical power amplifiers.

Comments 1: At line 28, the phrase "With the rapid development of deep learning, it has become a powerful tool in DPD research, excelling in nonlinear compensation tasks." is unclear. Specifically, it is not clear what the subject is. Please rephrase

Response 1: Thanks a lot for your comments that helped us to improve the quality of our manuscript. According to your great comments. The revised part is as (page 1, line: 28-30):

“Due to its strong capability in fitting nonlinear characteristics, deep learning has emerged as a powerful tool in DPD research, driven by advancements in its techniques”

Comments 2: At line 35, the definition of "direct learning" is not sound. What does it mean to compare the input signal with the PA's output? Maybe the authors mean calculating an error between the (distorted) output and a desired signal (linearly proportional to the input)? Please rephrase. Also, later it is referred to as a "technique," but in fact, direct learning is just an architecture, and different numerical/optimization methodologies can be used within this architecture, each with its pros and cons.

Response 2: Thanks a lot for your comments that helped us to improve the quality of our manuscript. The revised part is as (page 2, line: 32-44):

“Direct Learning Architecture (DLA) is a classic implementation method for pre distortion techniques, whose core idea is to minimize the error between the PA output and the expected signal by directly optimizing the parameters of the pre distorter. In DLA, the input signal is first processed by a pre distorter to generate a pre distorted signal, which is then amplified by PA. Due to the nonlinear nature of PA, the output signal may introduce distortion. DLA calculates the error signal by comparing the output signal of PA with the original input signal, and directly adjusts the parameters of the pre distorter to minimize the error. Although DLA has the advantages of simple structure and easy implementation, its performance is limited by the complexity of inverse operations and model accuracy, especially in highly nonlinear systems, where DLA may face numerical instability and real-time challenges.”

Comments 3: Could the authors better reference that direct learning provides unbiased results, whereas indirect learning is biased due to noise? Also, later on in the indirect learning section, it is mentioned that indirect learning is "constrained." What does this mean? Will it work in the case of minimal nonlinearity?

Response 3: Thanks a lot for your comments that helped us to improve the quality of our manuscript. The revised part is as (page 2, line: 34-54):

“Indirect Learning Architecture (ILA) is a classic method for implementing pre distortion techniques, whose core idea is to indirectly model the characteristics of PA and invert its inverse model to achieve pre distortion. In ILA, the input signal first passes through PA to generate an output signal containing nonlinear distortion. The input and output signals of PA are used to train a forward model that characterizes the nonlinear characteristics of PA. Subsequently, by inverting the forward model, the inverse model of PA is obtained and used as a pre distorter. Although ILA has the advantages of model flexibility and indirect optimization, its performance is limited by inversion complexity and error accumulation, especially in highly nonlinear systems, where ILA may face challenges of high computational overhead and insufficient real-time performance.”

Comments 4: The proposed wavelet decomposition is a way to model the behavior of the PA. It does not seem to be an alternative DPD architecture, as the authors claim. The authors state that, unlike DLA, WDP leverages neural networks. However, even if there is no model inversion involved, the technique still utilizes a forward PA model and an optimizer, so it actually falls, in my opinion, within a direct learning implementation. See other examples of optimization-based DLA in Mengozzi, Mattia, et al. "GaN power amplifier digital predistortion by multi-objective optimization for maximum RF output power." Electronics 10.3 (2021): 244. Please address this aspect.

Response 4: Thanks a lot for your comments that greatly helped improve the readability of the manuscript. WDP technology initially used neural networks to accurately simulate the nonlinear behavior of PA. During the training process, by continuously adjusting the weights and bias parameters of the neural network, its output can accurately fit the actual nonlinear characteristics of PA. Subsequently, the system will fully replicate the trained nonlinear model parameters (including weight matrix and bias vector, etc.) into a structurally identical model, and set the parameter state of the model to "frozen" (i.e. fixed and unchanged). This frozen nonlinear model is then cascaded with a pre distorter, optimizing the parameters of the pre distorter to maximize the approximation of linear amplification characteristics in the output of the entire cascaded system, thereby achieving the ultimate goal of training high-performance pre distorters. The advantage of this method is that by separating nonlinear modeling from pre distortion training, it not only improves system stability but also reduces computational complexity, providing the possibility for real-time implementation. Additionally, the work by Mengozzi et al. has greatly inspired our research, and we have included it in the references (page 14, line: 382-383).

Comments 5: Please note that existing DPD approaches already extract specific signal-based or system-based features to reduce computational complexity or improve DPD performance, and they should be acknowledged. See, for example: X. Wang et al., "Digital Predistortion of 5G Multiuser MIMO Transmitters Using Low-Dimensional Feature-Based Model Generation," in IEEE Transactions on Microwave Theory and Techniques, vol. 70, no. 3, pp. 1509-1520, March 2022; and M. Mengozzi et al., "Beam-Dependent Active Array Linearization by Global Feature-Based Machine Learning," in IEEE Microwave and Wireless Technology Letters, vol. 33, no. 6, pp. 895-898, June 2023.

Response 5: Thanks a lot for your comments that helped us to improve the quality of our manuscript. We greatly appreciate your correction regarding the issue you mentioned about extracting specific features to reduce complexity or improve DPD performance in existing research. Our research has different focuses and innovative points compared to these feature extraction methods. This study introduces wavelet decomposition into PA nonlinear modeling for the first time, capturing the nonlinear features of signals through multi-scale analysis, which is fundamentally different from traditional feature extraction methods. Unlike existing work, our experiment focuses on verifying the feasibility of combining wavelet decomposition with RNN and its effectiveness in improving modeling accuracy. In addition, we recognize that feature-based model generation, as explored in the referenced works, offers valuable insights into improving computational efficiency and system performance. These studies have provided useful perspectives that may inspire future extensions of our approach, and we have accordingly included them in the references (page 14, line: 384-385).

Comments 6: The results look good, although they are quite incremental in terms of linearization performance. Could the authors clarify how they used the OpenDPD datasets to obtain these results? Normally, for DPD papers, initial measurements are required to obtain a dataset for model identification, followed by measurements with predistorted signals to experimentally verify the achieved linearity. How is this last step performed here? Is it a model-based type of validation? Please clarify.

Response 6: Thanks a lot for your comments that helped us to improve the quality of our manuscript. We did strictly follow the standard procedures provided by the OpenDPD dataset for experimentation and validation. As an important open-source dataset in the fields of PA modeling and digital pre distortion, OpenDPD's authors have established a complete testing and validation methodology system. Specifically, this dataset not only provides raw measurement data, but also includes detailed experimental protocols and evaluation metrics, making the results comparable between different studies. We followed the standard process of OpenDPD: first, we used the provided training dataset for model training and parameter recognition, and then evaluated the pre distortion performance using an independent testing dataset. This dataset based validation method, although different from traditional experimental measurements, has been widely recognized in the field. The authors of OpenDPD have confirmed in their published work that the results under this dataset have good consistency with real experimental environments. We have supplemented the manuscript with relevant explanations to present our experimental results more clearly (page 9, line: 252-255).

Comments 7: The method seems computationally intensive and time-consuming due to the optimizer. Could the authors better present what is exactly being shown in Fig. 6 and how this comparison is performed? Is it based on the same dataset? Please provide a more detailed analysis.

Response 7: Thanks a lot for your valuable comments. Figure 6 shows the loss function value curves of all comparison methods on the validation set as a function of training epochs. Through comparative analysis, it can be clearly observed that our proposed method maintains the lowest loss value throughout the entire training process, and its convergence speed is significantly better than other comparative methods. This significant performance advantage is mainly due to our innovative combination of wavelet decomposition and RNN architecture, which enables the model to more effectively capture the nonlinear features of signals. These experimental results fully demonstrate that our proposed method has excellent fitting performance and practical value in PA nonlinear modeling tasks. We have supplemented the manuscript with relevant explanations to present our experimental results more clearly (page ,12 line: 324-330)

Comments 8: It would be useful to replace Algorithm 1 with a graphical flowchart (or at least add a flowchart)

Response 8: Thanks a lot for your valuable comments. We appreciate your suggestion regarding the flowchart. The detailed steps of WDP are already illustrated in Fig. 1, which provides a visual representation of the process. We believe this figure effectively conveys the workflow of our method.

Comments 1: Please provide references or more explanation for these drawbacks of ILA.

Response 1: Thank you for your comment. We have added additional references and explanations to better support the discussion of the drawbacks of ILA. The revised manuscript now includes the following explanation: “Although ILA offers advantages such as model flexibility and indirect optimization, it suffers from several critical drawbacks, including vulnerability to noise due to parameter extraction at the PA output, and high dependence on PA model accuracy, which may degrade predistortion performance [22–25].” (page 2, line: 51-54)

The following references have been cited in the revised manuscript to support the discussion of the drawbacks of ILA：

24. Z. Wang, W. Chen, G. Su, et al. Low Computational Complexity Digital Predistortion Based on Direct Learning With Covariance Matrix. IEEE Trans. Microw. Theory Tech., 2017, 65(11), 4274–4284.

25. C. Tarver, L. Jiang, A. Sefidi, et al. Neural network DPD via backpropagation through a neural network model of the PA. In Proc. Asilomar Conf. Signals Syst. Comput., Nov. 2019.

（page 14, line: 393-395).

Comments 2: I still believe that this type of methods fall within the DLA type architecture.

Response 2: Thank you for your comment. Upon initial inspection, the proposed WDP approach might appear similar to DLA due to its end-to-end training nature; however, a deeper examination reveals several fundamental distinctions that set it apart from a typical DLA framework. Specifically, DLA directly optimizes the predistorter parameters by minimizing the error between the PA output and the ideal signal, which often leads to numerical instability and real-time challenges in highly nonlinear systems due to the requirement of direct inversion. In contrast, our WDP method decouples this process by first modeling the PA’s nonlinearity using a neural network, freezing that model, and then utilizing it in a cascaded setup for predistorter training. This two-stage strategy combines the benefits of indirect learning with direct optimization, thereby avoiding the inversion bottleneck inherent in DLA. Furthermore, the incorporation of wavelet decomposition and recurrent networks introduces a level of frequency-domain abstraction and temporal dependency modeling that is not typically present in conventional DLA implementations. As a result, WDP aligns more closely with a hybrid or evolved architecture that leverages the strengths of both DLA and ILA while mitigating their respective limitations. Therefore, although WDP shares superficial similarities with DLA in terms of optimization flow, its architectural design and training paradigm indicate that it should be regarded as a distinct and more robust alternative rather than a mere variant of traditional DLA.

Comments 3: It seems that the authors forgot to add the second reference. Please include it too.

Response 3: Thank you for your comment. Thank you very much for pointing this out, and our sincere apologies for the oversight. In the latest version of the manuscript, the second reference has now been properly included (page 14, line: 388-389). We appreciate your careful reading and helpful feedback.

Comments 4: Please improve the caption of Fig. 6.

Response 4: Thank you for the suggestion. We have revised the caption for Figure 6 to provide a more detailed explanation. The updated version is:

“Validation loss comparison of different DPD methods over training epochs.” (page 12, between line 325-line 326)