A Near-Field Super-Resolution Network for Accelerating Antenna Characterization

The planar near-field to far-field (NF2FF) transformation relies on plane wave (modal) expansion using Fourier transform techniques. This method decomposes any monochromatic wave into a superposition of plane waves, resolving them in varying amplitudes and different directions to facilitate the conversion of near-field data into far-field radiation patterns [3M-Field, balanis]. The NF2FF transformation starts from measuring the tangential components $E_{xa}$ and $E_{ya}$ over the scanning plane ($z^{\prime}=d$) in the near field. The corresponding $x$ and $y$ components of the plane wave spectrum, ${f_{x}}\left(k_{x},k_{y}\right)$ and ${f_{y}}\left(k_{x},k_{y}\right)$, are determined by

where $a$ and $b$ are the width and height of the measurement plane. Then the electric field in the far field can be obtained by

From (1)-(4), ${E}_{xa}$ and ${E}_{ya}$ are first measured at a distance $d$ from the antenna aperture, usually $3\lambda$ to $5\lambda$ in air, at a grid resolution of at least $0.5\lambda$ to satisfy the Nyquist rate. The measurement area is recommended to be truncated with edges at least 40 dB lower than the max field strength within the plane [IEEE_NF_recom].

The workflow of the proposed end-to-end fast NF antenna measurement is shown in Fig. 1. It starts with a planar motion stage to traverse through physical coordinates in the near-field space of the antenna under test (AUT) to acquire undersampled $E_{x}$ and $E_{y}$ data. The undersampled near-field data is first fed into the NFS-Net to restore the fully-sampled data that satisfies the Nyquist rate. Then, the restored fully-sampled data is further processed by the NF2FF algorithm as described in Section II. A to calculate the AUT’s far-field radiation pattern.

The generalizability of deep learning methods relies on the diversity and comprehensiveness of the training dataset. To ensure that NFS-Net has good generalizability for different antenna patterns, we constructed a comprehensive training and testing dataset. This dataset is built by performing full-wave electromagnetic simulations using CST Studio Suite on 24 different antenna models with operating frequencies ranging from 1 to 10 GHz. The full list of antennas is provided in Table I. Each antenna model was selected with varying parameters, including dimensions, operating frequency, polarization, and directivity, to ensure the diversity and representativeness of antennas in the dataset. The simulations follow the IEEE Standard recommendations on the best practices of near-field measurement to emulate realistic anechoic chamber characterization [IEEE_NF_recom]. For each unique antenna, the simulated near-field data (magnitude and phase of the $E_{x}$ and $E_{y}$ components) are captured over a predefined spatial grid at the Nyquist rate, with a distance between $3\lambda$ and $5\lambda$ from the AUT. The simulation boundary is set to be up to 1.2 m to be consistent with realistic NF measurement facilities. This process results in 3418 unique near-field magnitude and phase maps.

The number of fully-sampled near-field maps was further augmented by rotating each map by $90^{\circ}$ three times, producing 13,672 fully-sampled near-field maps. This augmentation process increases the pattern diversity in the dataset, enhancing the robustness and generalizability of the data-driven methods across different scenarios.

The fully-sampled near-field maps are then downsampled by a factor of 3 by uniformly selecting 1 point from every 3$\times$3 grid, producing under-sampled maps. The fully-sampled map and the corresponding under-sampled map form a data pair in the dataset. Therefore, the constructed dataset consists of 13,672 data pairs for network training and testing.

Inspired by the classic U-Net for its unique, symmetric shape to preserve high-resolution feature and enhance data utilization [RUNet_new, U-Net-original, DL-MRI], we propose the NFS-Net to take advantage of such architecture and efficiently transform undersampled near-field data into fully-sampled data.

The network architecture is shown in Fig. 2. It consists of an encoder, a decoder, and skip connections. The encoder includes four stages, each with two 3×3 convolutional layers, a stride of 1, and a padding of 1, followed by batch normalization and rectified linear (ReLU) activation function. Downsampling is performed using max-pooling layers with 2×2 kernels and a stride of 2. The decoder mirrors the encoder, using an upsampling layer with a kernel size of 2, an upscale factor of 2, and a stride of 2. This is followed by concatenation with the corresponding encoder features and further processing by two 3×3 convolutional layers. The bottleneck layer captures high-level features between the encoder and decoder. Skip connections between the encoder and decoder preserve spatial information across layers, which is crucial for maintaining output fidelity. The final 1×1 convolutional layer reduces the number of output channels to one, providing the estimated fully-sampled near-field data.

The same network architecture, i.e. the NFS-Net, is separately trained to learn to produce high-resolution magnitude and phase data. To effectively train NFS-Net to extract useful features from near-field magnitude and phase data and achieve the high-resolution effect, we designed unique loss functions for magnitude and phase training.

The 13,672 pairs of data in the dataset are randomly split into 80% training data and 20% testing data. All the near-field maps are resized to $86\times 86$ and normalized to a range of [0, 1] for training. The input data size is chosen to accommodate our computing resource limitations and training time. However, it is important to note that the U-Net structure is highly versatile and can accept 2D matrices of different sizes, making it suitable for other sampling surface sizes as well.

The network is implemented in PyTorch and trained on a single NVIDIA 4090 GPU (24-GB GPU memory). The training process is based on the loss functions as described in Section III. C. To better understand the impact of hyperparameter choices on the training outcomes, we conducted a comprehensive evaluation of the training results across three major hyperparameters: batch size, learning rate, and the number of network layers, with the comparison shown in Table II. The batch size in network training refers to the number of samples processed in one pass through the network during each training step. A large batch size can accelerate training speed and reduce noisy gradients, whereas a small batch size can lower the demand for computing resources and improve generalizability. The learning rate is another important parameter that significantly impacts training outcomes, as it controls the size of the steps taken when updating the model’s weights. A high learning rate can speed up training by taking larger steps, whereas a low learning rate results in more precise steps and better training stability. The number of network layers also plays a critical role in training, as features are extracted from the input data through these layers. A network with fewer layers consumes fewer computing resources, whereas a deeper network can extract comprehensive features more accurately.

After careful testing and exploration, balancing training time, training quality and computing resources, we found that a network of 64 layers and mini-batch size of 15, with $\alpha_{mag}$ = 1, and $\beta_{mag}$ = 1 for magnitude loss function, and $\alpha_{phase}$ = 0.6, and $\beta_{phase}$ = 0.4 for phase loss function, provides the best training outcomes. The network is optimized using the ADAM optimizer [adam]. The initial learning rate is set to 0.001, and it decreases by a factor of 10 every 50 and 75 epochs for magnitude and phase training, respectively, resulting in total training epochs of 200 and 300. The total training time is approximately 4 hours for the magnitude network and 6 hours for the phase network. The progression of training and validation loss decay is shown in Fig. 5. The effectiveness of the neural network training is clearly demonstrated, and the saturation observed toward the end of the process confirms that the stopping point is adequate to ensure high training quality.

The network’s super-resolution performance for the amplitude and phase data of the $x$ and $y$ components of the electric field is validated on two different antennas. Antenna 1 is an X-band corrugated horn antenna working at 8.7 GHz, Antenna 2 is a Yagi antenna operating at 3 GHz, and Antenna 3 is a modified version of Antenna 2 tilted at an angle of 15°. None of these types of antennas are included in the training dataset. The results are shown in Fig. 3. In the figure, the undersampled NF and network prediction represent the undersampled near-field input to the network and the fully-sampled output data from the network, with values normalized between 0 and 1. The ground-truth data is sampled at Nyquist rate with a sampling resolution of $\lambda/2$, and the undersampled NF data is obtained by undersampling the ground-truth data by a factor of 3, resulting in a sampling resolution of $3\lambda/2$. The comparison results shown in Fig. 3 present that the fully-sampled amplitude and phase data predicted by the network closely resemble the ground truth. The errors between the network’s predicted data and the ground truth are listed in Table III. The low MAE of the magnitude, the low $L_{pp}$ of the phase, and the high MS-SSIM of both the magnitude and phase in the numerical comparison, combined with the visualized results shown in Fig. 3, confirm the network’s outstanding super-resolution capability.

After completing the first step of near-field super-resolution, the fully-sampled magnitude and phase data predicted by the network are fed into the NF2FF algorithm to reconstruct the far-field radiation patterns of the antennas. The reconstruction results (in red) are compared to the simulated far-field patterns (in blue) in Fig. 4. As observed in this comparison, they show remarkable agreement in both E-plane and H-plane patterns.

To further validate the network’s generalization capability, the network’s performance is tested on a modified Yagi antenna (Antenna 3) with patterns tilted at 15°. The results are also provided in Fig. 4. The 15° tilt is clearly visible In the reconstructed H-plane pattern, and the predicted E- and H-plane patterns closely match the simulated far-field patterns, confirming the network’s ability to accurately restore fully-sampled near-field data for antennas with different patterns.

The performance of the proposed model is further validated using measured NF data. The illustration of the NF measurement is shown in Fig. 7, and the detailed measurement setup is described in [ManualIsabel]. The testing probe (rectangular open-ended waveguide) and the antenna under test (AUT) are both fixed on positioners controlled by a motion controller and host computer. A vector network analyzer (R&S ZVA-67) is used as a signal generator and receiver to characterize the near-field data, with frequency extenders (R&S ZVA-Z110) to connect with the two testing ports. The entire measurement setup is supported on a precision optical table to minimize the effect of vibration. High frequency absorbers are also used around the setup to further reduce scatterings and reflections in the environment.

Two antennas are tested in the experiment. Antenna 4 is a horn antenna operating at 75 GHz with a nominal gain of 20 dBi, and Antenna 5 is a horn antenna operating at 90 GHz with a nominal gain of 24 dBi. The results are shown in Fig. 6. Similar to the simulation examples, the solid blue line represents the far-field pattern obtained from fully sampled measurements at the Nyquist rate, while the dashed red line represents the far-field pattern reconstructed from 3$\times$ undersampled near-field data after network prediction. These patterns are in appropriate agreement with each other. It’s important to note that the two measured antennas operate at much higher frequencies compared to the antennas in the training dataset, yet the network maintains consistency in predicting the fully-sampled near-field data. The measurement results further validate the strong generalizability of the network.

Since the experiments use field information collected from both simulation and measurement that is independent from the training data, we further investigated the errors occurred in magnitude and phase restoration using the proposed network for all the 5 antennas. They are compared against the values benchmarked on the training data from Fig. 5 in Table III. As shown in the table, Antenna 1-3 exhibit error scales similar to those of the training data, largely due to the more ideal input data acquired through full-wave EM simulations. Antennas 4-5 show higher loss margins due to the difference between the realistic antenna measurement environment and the ideal simulation environment.

We first compare the effects of phase training using Periodic Phase Loss ($L_{pp}$) combined with $L_{MS-SSIM}$ as the loss function against the traditional MAE combined with $L_{MS-SSIM}$, while keeping magnitude and other settings the same. The comparison of the restored near-field maps is shown in Fig. 8, and the comparison of the NF2FF reconstructed far-field patterns is shown in Fig. 9. The two cases are labeled as $L_{pp}$ and MAE, respectively. Fig. 8 shows that networks trained with MAE, which does not account for the unique nature of phase wrapping, have limited accuracy in predicting high-resolution phase map, hence it cannot accurately restore the far-field pattern as shown in Fig. 9. However, our proposed $L_{pp}$ successfully addresses this issue, driving the network to effectively learn phase variations and achieve precise prediction of the high-resolution near-field phase map, thus resulting in more accurate far-field pattern.

The reconstruction accuracy of the NFS-Net is also compared with other commonly used super-resolution methods, including the ordinary Kriging [emc_kriging] and compressive sensing [fieldRecon_CS], on the downsampled near-field data at 11% of the Nyquist rate. Fig. 8 shows the comparison of their restored phase maps and Fig. 9 shows the farfield reconstruction results. The Kriging and compressive sensing methods fail to restore the fully-sampled NF phase map due to significant phase information loss in the undersampled data. In contrast, the NFS-Net, by learning from diverse NF data and being optimized using losses that include the numerical error ($L_{pp}$) and the perceptual error ($L_{MS-SSIM}$), demonstrates much higher prediction accuracy for high-resolution NF phase map. Consequently, as shown in Fig. 9, the NFS-Net achieves significantly higher accuracy in the reconstructed far-field pattern compared to the other two methods.

To explore the effect of different downsampling rates on reconstruction quality, we compare the far-field reconstruction results with a downsampling factor of 2 (25% of fully-sampled data, interval of 1 $\lambda$) to those with a downsampling factor of 3 (11% of fully-sampled data, interval of 1.5 $\lambda$) for antennas 1, 2 and 4. As shown in Fig. 10, a downsampling factor of 2 shows higher reconstruction accuracy. The comparison provides an intuitive guideline for practical applications: a downsampling rate of 2 is preferable when prioritizing accuracy, whereas a downsampling rate of 3 offers advantages in terms of time efficiency.

We have conducted further analysis of the sources of error in the reconstructed far-field radiation patterns. There are two possible sources of error: inaccuracies in the restored high-resolution magnitude or inaccuracies in the restored high-resolution phase. To determine the primary source, we used different combinations of ground-truth and restored magnitude and phase to reconstruct the far-field patterns for antennas 1, 2, 4, and compared the accuracy of the reconstructed patterns. The results are shown in Fig. 11. For clarity, G, R, M, and P represent the ground truth, restored data, magnitude, and phase, respectively. For example, the combination “ground-truth magnitude and restored phase” is denoted as “GM RP”. The comparison shows that the combination “GM RP” and “RM RP” exhibit similarly large errors. Therefore, it can be concluded that the primary source of error is the inaccuracies in the restored phase.

Due to the periodic nature of phase, the phase information is the most challenging for the network to learn. Further improvements in phase accuracy can be achieved through advanced network architectures and more comprehensive training datasets, which are the focus of our future work.

To better understand the impact of environment to reconstruction quality, we have also conducted further analysis on measurement noise. This is implemented by adding Gaussian noise to the simulation dataset of Antenna 1 at different levels of signal-to-noise ratio (SNR) to emulate the random noises in the measurement environment. As shown in Fig. 12, the impact on reconstruction quality is noticeable when SNR is below 20 dB, which also explains the larger error margins observed in Antenna 4-5 in Table III, possibly due to noises in realistic environment.