Ambiguity-reduced phaseless reconstruction of planar antennas using deep learning

1 Introduction

Phase retrieval is a challenging problem encountered in various electromagnetic applications, where the objective is to reconstruct a signal solely from the magnitude of its linear transformation without access to phase information (Luo et al., 2020; Taylor, 1981). This problem arises in several important applications such as microwave imaging (Hislop et al., 2009; Estatico et al., 2020), antenna diagnostics (Razavi and Rahmat-Samii, 2013; Morabito et al., 2019), and biomedical imaging Costanzo and Lopez (2020). In these applications, the primary objective is to recover essential properties or the geometrical characteristics of an object from phaseless measurements, making the problem highly nonlinear and ill-posed.

Within the field of array antennas, phaseless reconstruction plays a particularly important role. In this context, the aim is to recover the complex antenna radiation characteristics using only the magnitude of the measured electromagnetic fields. Due to the absence of phase information, the reconstruction process becomes significantly more challenging and generally requires a large number of measurements to achieve acceptable accuracy Alaifari (2021). Several approaches have been proposed to address such problems, including phase retrieval algorithms Fienup (1982), optimization-based techniques Pierri et al. (2021); Pierri and Moretta (2021); Leone et al. (2023), and iterative reconstruction methods. In Guo et al. (2018), for instance, the importance of generating an appropriate initial solution for phaseless reconstruction problems was investigated. Nevertheless, substantial challenges still remain in terms of computational complexity, robustness, and adaptability to different scenarios.

Although these approaches have demonstrated promising performance, they generally suffer from high computational cost and long execution times, which considerably limit their practical applicability, especially for large-scale problems. To reduce the computational burden, several sampling strategies have been proposed to decrease the number of required measurements in an efficient manner Migliore (2018); Qureshi et al. (2013); Masoodi and Taromideh (2025). However, the practical implementation of such approaches remains difficult, since the resulting measurement distributions are often nonuniform and challenging to realize in practical measurement systems.

With the recent advancements in Deep Neural Networks (DNNs), new opportunities have emerged for efficiently solving highly nonlinear inverse problems. These networks are capable of extracting meaningful features from large datasets and learning the optimal mapping between inputs and outputs through adaptive optimization of the network parameters LeCun et al. (2015). Among the different deep learning architectures, Convolutional Neural Networks (CNNs) have attracted considerable attention due to their ability to efficiently process image-based data LeCun et al. (1998). CNNs have demonstrated remarkable success in both regression and classification applications and have increasingly been employed in electromagnetic and antenna-related problems Jin et al. (2017); Sanghvi et al. (2020); Yao et al. (2019); Fang et al. (2023).

Furthermore, the strong nonlinear approximation and feature extraction capabilities of deep learning models make them highly suitable for addressing phaseless inverse radiation problems. Recent studies have explored the application of deep learning techniques for phaseless antenna and inverse scattering problems. In Peng et al. (2025), GAN- and CNN-based architectures were employed to directly reconstruct source currents from phaseless near-field measurements without iterative optimization. A physics-assisted deep learning framework was proposed in Deshmukh et al. (2022), where an extended Rytov approximation was combined with a CNN architecture to reconstruct permittivity profiles from Wi-Fi phaseless measurements. In addition, a CNN framework for phaseless inverse scattering problems was investigated in Xu et al. (2020), where different inversion strategies, including physics-assisted approaches, were compared and analyzed.

Another major challenge in phaseless inverse radiation problems arises from the fact that two current distributions with identical amplitudes and opposite phase profiles can produce identical radiation intensities. This ambiguity significantly increases the ill-posedness of the inverse problem, since multiple source distributions may correspond to the same measured phaseless data, making accurate reconstruction extremely difficult.

To alleviate this issue, several studies have proposed the use of multiple scanning planes positioned at different distances from the antenna in order to collect additional radiation measurements Pierri et al. (2021); Pierri and Moretta (2021); Leone et al. (2023). Although such approaches can improve reconstruction performance, they generally impose substantial computational complexity and measurement overhead, thereby reducing their practical efficiency.

Hence, this work aims to address one of the fundamental challenges in phaseless inverse radiation problems, namely, the ambiguity arising from identical intensity-only measurements generated by different current distributions. To overcome this limitation, a signal-processing stage is introduced at the source level prior to the radiation process. Specifically, an additional phase-dependent term is incorporated into the original current distribution, resulting in a processed current that preserves additional phase-related information in the radiated intensity measurements. Consequently, unlike the conventional current distributions that would otherwise generate identical phaseless measurements, the proposed strategy produces distinguishable measurement patterns after the proposed processing stage. This significantly reduces the ambiguity associated with phaseless measurements and provides a more informative representation of the source distribution.

Furthermore, a dual U-Net architecture Ronneberger et al. (2015) is employed to solve the resulting inverse problem. The proposed framework consists of two independent U-Net networks dedicated to reconstructing the real and imaginary components of the unknown current distribution. An important aspect of the proposed training strategy is that the input of the dataset corresponds to the phaseless measurements generated by the processed current distribution, whereas the target outputs are the real and imaginary components of the original current distribution. In this manner, the network learns to recover the complex source distribution of the antenna array from intensity-only measurements.

Overall, the proposed framework effectively mitigates the ambiguity inherent in conventional phaseless radiation measurements and enables accurate reconstruction of both the amplitude and phase of the antenna current distribution. The numerical results demonstrate a significant improvement in reconstruction accuracy compared with the conventional phaseless formulation, while preserving computational efficiency and maintaining a practical single-plane measurement configuration.

Accordingly, the remainder of this paper is organized as follows. Section 2 presents the mathematical formulation of the phaseless inverse radiation problem for planar antenna arrays. Section 3 presents the proposed phase-assisted current representation and discusses its role in mitigating the ambiguity associated with phaseless measurements. In Section 4, the adopted CNN-based framework and the dataset generation procedure are described in detail. Section 5 presents the training process and evaluates the performance of the proposed approach in terms of reconstruction accuracy. Finally, the main conclusions of this work are summarized in Section 6.

2 Theory of the phaseless radiation problem

This section presents the mathematical formulation of the phaseless radiation problem considered in this work. The geometry of the investigated configuration is illustrated in Figure 1. A homogeneous medium is assumed in which the near-field radiation is generated by a two-dimensional array composed of N radiating elements. Each element is modeled as a bounded omnidirectional magnetic source oriented along the x-axis. The array elements are distributed over a rectangular source domain (SD) with dimensionsalong the x- and y-directions. The radiated field is sampled over an observation domain (OD), defined as a scanning plane of dimensionswhere measurements are collected at M observation points. The scanning plane is located at a fixed distance z 0 along the z-axis. In the considered phaseless configuration, only the squared magnitude of the tangential y-component of the radiated near-field is measured. Let the total number of radiating elements be defined as N = ( 2 N 0 ) 2, while M denotes the total number of observation points. Suppose the array coefficient I is a complex quantity that can be represented with an amplitude and a phase as I = | I | e j ( θ ); the corresponding phaseless radiation field can then be expressed as follows Leone et al. (2023):where E y ( x m , y m , z 0 ) denotes the tangential component of the electric field evaluated at the observation point ( x m , y m , z 0 ) and A is the radiation operator. The remaining parameters are defined as follows:

S D = 3 λ × 3 λ ,

O D = 6 λ × 6 λ ,

E y x m , y m , z 0 2 = A I t p = z 0 4 π ∑ t = − N 0 N 0 − 1 ∑ p = − N 0 N 0 − 1 | I t p | 1 + j k 0 R t p R t p 3 e − j k 0 R t p − θ t p 2 (1)

• k 0 is the free-space wavenumber.
• I t p represents the complex excitation coefficient associated with the array element located at the ( t , p ) position along the x- and y-directions.
• R t p denotes the distance between the source point ( x t , y p ) and the observation point ( x m , y m , z 0 ), and is defined as

R t p = x m − x t 2 + y m − y p 2 + z 0 2 .

• m corresponds to the index of the measurement point on the observation plane.

3 Ambiguity reduction through phase-dependent current processing

As discussed in Section 2, the array coefficient I is a complex quantity that can be represented aswhere | I | and θ denote the amplitude and phase of the source current, respectively. The reconstruction of these coefficients from phaseless measurements constitutes the main objective of the considered inverse problem. However, this task is highly challenging due to the inherent ambiguity associated with phaseless radiation measurements. For convenience, let us define I + and I − where the superscripts + and − denote current distributions with positive and negative phase profiles, respectively. Accordingly, two current distributions possessing identical amplitudes but opposite phases may generate identical phaseless radiation patterns on the observation plane defined by Equation 1. This behavior is illustrated in Figure 2. In Figures 2a, the source phase distribution is given by θ, whereas in Figures 2b the phase distribution is − θ. Although the phase distributions are different, both sources produce identical phaseless measurements according to (1), as shown in panel c) of each subfigure. This non-uniqueness property represents one of the principal difficulties of phaseless inverse radiation problems and significantly complicates the reconstruction process.

I = | I | e j θ

To address this problem, we introduce a preprocessing step applied to the current prior to radiation, ensuring that the two currents I − and I + do not produce identical phaseless measurements. Accordingly, we define a processed current I ̂ as follows:By enhancing the nonlinear coupling between the source coefficients and the resulting phaseless measured field, the processed current I ̂ introduces additional discriminative information. This improves the ability to identify the sources and helps the learning and reconstruction processes.

I ̂ = | I | e j θ + e j ⁡ ln θ

The additional term e j ⁡ ln ( θ ) introduces a phase-dependent perturbation into the current I. Consequently, unlike the original current I, the processed current I ̂ is robust against the ambiguity arising from similar phaseless measurements associated with opposite phases. As a result, the currents I ̂ + and I ̂ − produce distinct phaseless measurements defined by Equation 1, as shown in Figure 3. In Figures 3a, the source distribution I ̂ + corresponds to phase θ, whereas Figures 3b corresponds to the current I ̂ − with phase distribution − θ. Unlike the original current I, the processed current I ̂ yields distinguishable phaseless measurements, as illustrated in panel (c) of each subfigure.

The proposed strategy offers improved stability for the phaseless inverse problem within the inverse problem framework and leads to a significant improvement in the reconstruction accuracy of learning-based models.

4 CNN architecture and dataset preparation

Convolutional Neural Networks (CNNs) have become one of the most powerful frameworks in deep learning due to their remarkable capability in extracting meaningful features from large-scale datasets and solving highly nonlinear problems. Among the existing CNN-based architectures, the U-Net model Ronneberger et al. (2015) has demonstrated outstanding performance in inverse imaging and reconstruction problems because of its efficient encoder–decoder structure and skip-connection mechanism. The performance of such models strongly depends on the proper selection of network architecture and hyperparameters, particularly for complex inverse problems such as phaseless antenna array reconstruction.

This section presents the proposed deep learning framework together with the procedure adopted for dataset generation.

4.1 Proposed U-Net model

To address the considered phaseless inverse problem, a dual U-Net framework is proposed. The overall architecture consists of two independent U-Net networks operating in parallel. The first network is dedicated to reconstructing the real part of the current distribution, whereas the second network reconstructs the imaginary part of the array coefficients I. The general structure of the proposed framework is illustrated in Figures 4a; the term “dual U-Net” in the present work refers to two separate U-Net architectures: one dedicated to the real part and the other to the imaginary part. Both networks share the same input, which consists of phaseless data, while each network is trained to predict a different target corresponding to its respective component.

Both U-Net networks receive the same phaseless radiation pattern as input; however, their outputs correspond to different components of the complex source distribution. The first network predicts real( I ), while the second network predicts imag( I ). It should be noted that both networks share the same architecture and training hyperparameters.

The adopted U-Net architecture consists of two principal components: the encoder and the decoder Figures 4b. The encoder path is responsible for extracting hierarchical features from the input phaseless data through successive convolution and pooling operations. As the network depth increases, the spatial resolution decreases while the number of extracted feature maps increases, enabling the model to capture higher-level representations of the input data.

The decoder path reconstructs the spatial information from the compressed latent representation generated by the encoder. This process is performed through transposed convolution layers combined with skip connections that transfer fine spatial information from the encoder to the decoder. These skip connections play a crucial role in preserving detailed structural information and improving reconstruction accuracy.

The proposed network consists of four encoding and decoding levels. The input to the network is a 128 × 128 image corresponding to the measured phaseless radiation pattern. At each encoder level, two successive 3 × 3 convolutional layers are employed, followed by Rectified Linear Unit (ReLU) activation functions. Zero-padding is applied to preserve the spatial dimensions of the feature maps. A 2 × 2 max-pooling layer is then used to reduce the spatial dimensions by a factor of two.

The number of convolutional filters begins with 32 filters at the first level and doubles after each downsampling stage, allowing the network to learn increasingly complex features. At the bottleneck level, the feature maps reach a spatial resolution of 8 × 8.

The decoder path follows a symmetric structure. Each upsampling stage is implemented using transposed convolution layers followed by concatenation with the corresponding encoder feature maps through skip connections. Two additional convolutional layers with ReLU activation are then applied. As the spatial dimensions increase throughout the decoder, the number of filters decreases correspondingly.

Finally, a 1 × 1 convolutional layer is employed to generate the reconstructed output image. The network parameters are optimized using a regression loss function defined aswhere B denotes the mini-batch size and Q = 128 × 128 represents the total number of pixels in the output image. The terms x b q and x ̄ b q correspond to the ground-truth and predicted pixel values, respectively.

L o s s = 1 B ∑ b = 1 B ∑ q = 1 Q x b q − x ̄ b q 2 , (2)

The network is trained using the ADAM optimizer due to its stability and effectiveness in deep learning optimization problems. The initial learning rate is set to 1 0 − 3 and is reduced by a factor of 0.5 every 20 epochs. The adopted training hyperparameters are summarized in Table 1.

TABLE 1. Training hyperparameters of the proposed U-Net network.

4.2 Dataset acquisition

To ensure the robustness and generalization capability of the proposed framework, a comprehensive dataset containing a wide variety of amplitude and phase distributions was generated. The complete dataset consists of 2000 samples.

To increase the complexity of the reconstruction problem and evaluate the capability of the proposed model under highly challenging conditions, half of the dataset samples were generated using the current distributions I + where both the amplitude and phase distribution are randomly generated. The amplitude peaks are randomly positioned over the two-dimensional source domain, while the phase values are uniformly selected within the interval

θ ∈ − π , π .

The remaining half of the dataset was generated using identical amplitude distributions but opposite phase profiles ( I − ), namely,. Consequently, the network is exposed to highly ambiguous phaseless measurements corresponding to different phase distributions, making the reconstruction task significantly more challenging.

Regarding the input phaseless measurements, two different radiation models are considered to evaluate the effectiveness of the proposed approach. In the first model, the phaseless radiation patterns are generated using the current distribution I; while, in the second model the processed current distribution I ̂ is considered to generate the phaseless measurement plane.

Therefore, the measured phaseless radiation patterns corresponding to the two models are defined asThis formulation enables a direct comparison between the conventional phaseless radiation operator and the proposed adaptive operator in terms of reconstruction capability and prediction accuracy.

model 1 : | E y | 2 = | A I | 2 model 2 : | E ̂ y | 2 = | A I ̂ | 2

Regarding the numerical configuration adopted for dataset generation, the following parameters are considered throughout this work:The radiating source is modeled as a two-dimensional array of ideal omnidirectional magnetic dipole elements uniformly distributed over the square source domain S D. The array consists of 16 × 16 equispaced elements along the x- and y-directions, resulting in a total of N = 256 radiating elements. The phaseless near-field measurements are collected over the observation domain O D, where a total of M = 2401 measurement points are uniformly distributed on the scanning plane in a 49 × 49 arrangement. The spatial sampling interval between adjacent measurement points satisfies the standard near-field sampling criterion and is chosen to be smaller than λ / 4 in order to ensure accurate acquisition of the radiated field distribution.

z 0 = 5 λ , O D = 6 λ × 6 λ , S D = 3 λ × 3 λ , λ = 1 .

It should be noted that, in order to properly feed the U-Net architecture, both the input phaseless measurements and the output current distributions must be represented with consistent spatial dimensions. In the considered problem, the source distribution has an original size of 16 × 16, whereas the measured phaseless radiation pattern is sampled over a grid of size 49 × 49. Therefore, both input and output data are resized to a unified resolution of 128 × 128 prior to the training process. The resizing procedure is performed using the imresize function in MATLAB with the nearest interpolation option. This interpolation strategy is selected because it introduces minimal modification to the original pixel values and preserves the structural characteristics of the data. After prediction, the reconstructed output is resized back to its original spatial dimensions using the same interpolation method.

Overall, for clarity, Figure 5 illustrates the steps of the proposed strategy (model 2). In the first step, the original source distribution ( I ) is processed to obtain I ̂. The radiation operator ( A ) is then applied, producing the corresponding phaseless data | E ̂ y | 2. This data is subsequently transformed into an image representation. In parallel, the real and imaginary parts of I are extracted and converted into separate images. Finally, a trained CNN is employed to reconstruct the complex source I, with the predicted result denoted as I ̄, from the phaseless data | E ̂ y | 2. Note that these steps correspond to model 1. In this case, the signal processing applied to I is omitted.

5 Network training and reconstruction performance evaluation

This section presents the training procedure of the proposed networks using the generated dataset and the hyperparameters summarized in Table 1. In addition, a comprehensive evaluation of the reconstruction accuracy and generalization capability of the trained models is provided.

5.1 Network training

As discussed previously, two different radiation models are considered during the training stage. Model 1 in which the phaseless data is related to current distribution I, whereas model 2 the phaseless data is respected to the processed current I ̂. For each model, two independent U-Net networks are trained separately: the first network reconstructs the real part of the current distribution, while the second network reconstructs the imaginary part.

The corresponding training and validation results are illustrated in Figure 6. The first and second columns correspond to the results of real and imaginary parts, respectively. Panels a) and b) present the training loss evolution (Equation 2), while panels c) and d) illustrate the corresponding validation losses during the training process. In all plots, the blue color correspond to model 1, and the orange ones correspond to model 2.

As demonstrated in Section 2, two current distributions with identical amplitudes and opposite phase profiles produce indistinguishable phaseless measurements (model 1). In contrast, the proposed strategy (model 2) generates distinguishable phaseless radiation patterns for such cases. Consequently, it is expected that the dataset generated using the proposed approach provides richer discriminative information for the learning process.

The obtained training results confirm this expectation. For both the real and imaginary components, model 2 exhibits significantly lower loss values throughout the training process compared to model 1. On the other hand, model 1 shows relatively high and nearly constant loss values, indicating the inherent difficulty of learning from ambiguous phaseless measurements.

The validation results closely follow the training curves for both networks, demonstrating stable convergence behavior and good generalization capability without any overfitting. Overall, the obtained results clearly indicate the effectiveness of the proposed model and demonstrate substantial improvement in reconstruction accuracy for both the real and imaginary parts of the source distribution.

It should be noted that 70 % of the generated dataset is used for training, 20 % for validation, and the remaining 10 % for testing.

5.2 Prediction accuracy analysis

In this section, a comprehensive evaluation of the prediction accuracy of model 1 and model 2 is presented using the test dataset, which consists of 200 unseen samples. Figure 7 illustrates a representative example of the reconstruction results generated by the trained networks. The first row corresponds to the reconstruction of the real part of the current distribution, while the second row represents the imaginary part.

In each row, the middle panel shows the ground-truth distribution, the left panel presents the prediction obtained using model 1, and the right panel illustrates the prediction generated by model 2. It can be clearly observed that model 2 achieves substantially higher reconstruction accuracy for both the real and imaginary components. In contrast, model 1 fails to accurately recover the source distribution and exhibits considerable reconstruction errors. These observations are fully consistent with the training and validation results discussed previously and further demonstrate the effectiveness of the proposed strategy.

To demonstrate the robustness and generalization capability of the proposed approach (model 2), the reconstruction performance was evaluated on a representative sample from the test dataset under different noise conditions. Additive White Gaussian Noise (AWGN) was introduced to the phaseless input data using MATLAB’s awgn function. Three scenarios were considered: a noiseless case, an SNR of 30 dB, and an SNR of 20 dB. The corresponding reconstruction results are presented in Figure 8. As can be observed, model 2 is capable of accurately reconstructing the ground-truth source distribution not only in the noiseless case but also under noisy conditions. Even at an SNR of 20 dB, the reconstructed results remain highly consistent with the ground-truth for both the real and imaginary components. These results demonstrate the robustness, reliability, and strong generalization capability of the proposed model under realistic measurement conditions. Furthermore, the limited degradation in reconstruction quality with increasing noise levels highlights the effectiveness of the proposed approach in mitigating the impact of measurement noise. In Figure 8, the first row presents the reconstruction results for the real part, while the second row corresponds to the imaginary part.

To further assess the generalization capability of the proposed framework, the evaluation was extended to the entire test dataset. The reconstruction error was quantified using the Root Mean Square Error (RMSE) with respect to the corresponding ground-truth distributions, defined aswhere x q and x ̄ q denote the q-th pixel values of the ground-truth and predicted images, respectively, and Q represents the total number of pixels in the reconstructed image. The RMSE values were computed for all test samples for both models.

RMSE = 1 Q ∑ q = 1 Q x q − x ̄ q 2 , (3)

The RMSE results defined by Equation 3 are presented in Figure 9. In the first row, panels a1) and a2) show the RMSE values corresponding to the real and imaginary parts, respectively. It can be observed that model 1 produces significantly larger reconstruction errors across the test dataset, which is consistent with the previously discussed training behavior. In contrast, model 2 achieves considerably lower RMSE values, indicating strong reconstruction capability and excellent generalization performance. In each panel of the first row, two dashed lines are shown, corresponding to the data of each color and representing the average value of the respective dataset. Comparing these values provides a clearer quantitative assessment of the performance of model 1 and model 2. These results demonstrate a significant superiority of model 2 over model 1, as the average values for both the real and imaginary parts is approximately 0.208 for model 1, whereas it is approximately 0.029 for model 2.

The second row of Figure 9 presents the histogram distributions of the RMSE values corresponding to the results shown in the first row. The blue bars represent model 1, while the orange bars correspond to model 2. The superiority of model 2 becomes immediately evident from the histogram distributions. Most RMSE values associated with model 2 are concentrated within the interval [ 0.01 , 0.05 ], indicating very small reconstruction errors with only limited dispersion toward higher values. Conversely, model 1 exhibits a significantly broader error distribution, with the majority of samples located within the interval [ 0.1 , 0.35 ]. Furthermore, the highest concentration of errors for model 1 occurs around 0.3, highlighting the limited reconstruction capability of this model. Panels b1) and b2) correspond to the histogram distributions associated with the real and imaginary components, respectively.

Overall, the obtained results clearly demonstrate the effectiveness of the proposed adaptive radiation operator and confirm the capability of the proposed deep learning framework in accurately reconstructing complex antenna array coefficients from phaseless measurements.

Another important metric used to evaluate the performance of the proposed models is the coefficient of determination Pearson (1896), commonly denoted as R 2, and defined in Equation 4. This metric is widely employed in regression analysis to measure how well the predicted values agree with the observed data. Specifically, the R 2 score quantifies the proportion of the variance in the observed data that can be explained by the model predictions. An R 2 value closer to 1 indicates a better agreement between the predicted and true values, while an R 2 value equal to 1 corresponds to a perfect prediction. Mathematically, the R 2 metric is expressed aswhere L denotes the total number of test samples, which is 200 in this work, and Q represents the number of pixels in each image. The quantities x l q and x ̄ l q correspond to the q-th pixel of the l-th ground-truth and predicted images, respectively, while x ̂ denotes the mean value of all pixels in the ground-truth test dataset. In addition to the quantitative evaluation provided by the R 2 metric, scatter plots offer an effective visual tool for assessing prediction accuracy. These plots illustrate the relationship between the observed and predicted outputs, enabling a direct visualization of the model performance. Figure 10 presents the scatter plots obtained for both models, considering the real and imaginary components of the output data.

R 2 = 1 − ∑ l = 1 L ∑ q = 1 Q x l q − x ̄ l q 2 ∑ l = 1 L ∑ q = 1 Q x l q − x ̂ 2 (4)

In this figure, the results corresponding to model 1 and model 2 are shown in the first and second rows, respectively. Panels a1) and b1) present the scatter plots for the real part, whereas panels a2) and b2) correspond to the imaginary part. In all panels, the horizontal axis represents the observed values, while the vertical axis corresponds to the predicted outputs. The black dashed line represents the ideal prediction line ( y = x ). Therefore, the closer the scatter points are to this reference line, the higher the prediction accuracy of the model. It can be clearly observed that the orange scatter points associated with model 2 are tightly concentrated around the ideal prediction line for both the real and imaginary components, demonstrating a very strong agreement between the predicted and observed values. In contrast, the blue scatter points corresponding to model 1 are widely dispersed away from the ideal line, indicating poor prediction accuracy and limited generalization capability.

The corresponding R 2 values are also reported in each panel for quantitative comparison. As previously discussed, higher R 2 values indicate better predictive performance. Model 2 achieves R 2 values close to 0.98 for both the real and imaginary parts, which are very close to the ideal value of 1 and are fully consistent with the compact distribution of the scatter points around the ideal prediction line. On the other hand, model 1 yields significantly lower R 2 values of approximately 0.4, confirming the poor prediction performance observed in the scatter plots.

Based on the comprehensive prediction accuracy analyses, it can be concluded that the proposed approach which include signal processing, enables the generation of an effective dataset for properly training the CNN network. Consequently, the trained network is capable of overcoming the lack-of-phase problem and can accurately predict the array coefficients using only the amplitude information of the radiated field.

To further validate the effectiveness of the proposed approach, the reconstructed source currents obtained from the predicted real and imaginary components are analyzed. Figures 11a,b present the amplitude and phase distributions for two representative examples selected from the test dataset. In each subfigure, panels a) and b) correspond to the amplitude and phase of the ground-truth source current, respectively, while panels c) and d) show the amplitude and phase reconstructed from the predicted results. It can be clearly observed that the predicted distributions exhibit very strong agreement with the corresponding ground-truth results for both amplitude and phase. This close correspondence demonstrates the high prediction accuracy of the proposed model and further confirms the validity, robustness, and effectiveness of the proposed methodology.

6 Conclusion

In this paper, a deep learning framework for reconstructing complex antenna array coefficients from phaseless near-field measurements was presented. The main challenge addressed in this work arises from the inherent ambiguity of phaseless radiation data, where different source distributions with opposite phase profiles can produce identical measured amplitudes. Such behavior significantly limits the effectiveness of conventional reconstruction approaches and makes the inverse problem highly ill-posed.

To overcome this limitation, a signal processing step is considered on the current distribution before radiation. This step paves the way for generating distinguishable phaseless measurements for source distributions with reverse phase characteristics. Based on this formulation, a dual U-Net architecture was developed to separately reconstruct the real and imaginary components of the unknown current distribution. The proposed framework was trained and evaluated using a comprehensive dataset generated from two-dimensional antenna array sources with diverse amplitude and phase distributions.

Extensive numerical investigations demonstrated that the proposed approach substantially improves the reconstruction capability compared to the conventional radiation model. Training and validation results showed stable convergence and strong generalization performance for the proposed method, while the conventional model suffered from high reconstruction errors due to the ambiguity of the phaseless data. Quantitative evaluations based on RMSE and R 2 metrics further confirmed the superiority of the proposed framework, achieving highly accurate reconstruction for both the real and imaginary components of the source distribution.

For future work, we plan to apply this strategy to optimization problems based on gradient descent methods, which are highly challenging. In this approach, in addition to applying signal processing techniques to the current distribution, further signal processing must be applied to the unknown parameters to compensate for the additional processing introduced into the current distribution. This added complexity makes the optimization process more demanding and requires careful parameter estimation and adjustment.

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Footnotes

Edited by: Manuel J. C. S. Reis, University of Trás-os-Montes and Alto Douro, Portugal

Reviewed by: Meenakshi L Rathod, Dr. Ambedkar Institute of Technology, India

Ao Peng, Xi’an Jiaotong University, China