# Enhanced-resolution learning-based direction of arrival estimation by programmable metasurface

Nawel Meftah, Badreddine Ratni, Mohammed Nabil El Korso, Shah Nawaz Burokur

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Nawel Meftah, Badreddine Ratni, Mohammed Nabil El Korso, Shah Nawaz Burokur. Enhanced-resolution learning-based direction of arrival estimation by programmable metasurface. Advanced Electronic Materials, 2025, 11 (3), pp.2400476. 10.1002/aelm.202400476. hal-04986975

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# Enhanced-Resolution Learning-Based Direction of Arrival Estimation by Programmable Metasurface

Nawel Meftah, Badreddine Ratni, Mohammed Nabil El Korso,*
and Shah Nawaz Burokur*

Due to its growing importance and wide range of applications, direction-of-arrival (DOA) estimation has become a major research topic, particularly in the field of communication systems. While traditional DOA estimation methods rely on antenna arrays and complex algorithms, recent progress achieved in the design and implementation of metasurfaces has proved their effectiveness as promising alternatives. This study presents a distinct approach for DOA estimation that combines the use of a programmable metasurface with deep learning. The programmable metasurface together with a radio-frequency power detector placed at the focal point, acts as a parabolic reflector antenna with an adjustable pointing direction, which scans the azimuth plane in 5° increments to receive the power level of incoming signals. The collected data is then fed into a pre-trained multilayer neural network to enable DOA estimation with a resolution of lower than 1° without requiring fine-tuning of the scanning procedure. This approach ensures accurate and fast estimations, paving the way for advanced solutions in detection and localization for various applications.

## 1. Introduction

In the fields of telecommunications and radar systems, it is crucial to accurately and efficiently detect and locate electromagnetic signals. A fundamental element of these technologies is the estimation of the direction-of-arrival (DOA) of the signals. DOA estimation refers to the determination of the direction from which a propagating signal reaches a receiver. This technique plays an

essential role in various applications, particularly for high-quality transmission mobile networks,[1,2] satellite communications,[3] navigation,[4] and object localization.[5,6]

In general, DOA estimation primarily relies on the use of phased array antennas and exploits various calculation methods,[7,8] ranging from beamforming with the Fast Fourier Transform (FFT) technique, whose angular resolution is limited by the antenna aperture, to sophisticated algorithms such as MUSIC and ESPRIT,[9] with their different variants developed over the years for improved resolution estimation. Although these algorithms offer remarkable resolution and accuracy, they have significant limitations, particularly in terms of the computational and storage resources required. In addition, their performance is significantly degraded in non-ideal conditions and with imperfect antenna arrays. To increase robustness against practical imperfections of antenna arrays and to deal with complex electromagnetic environments, studies have turned to data-driven analytical solutions, incorporating machine learning techniques.[10] However, the physical structure continues to suffer from significant constraints, particularly in terms of cost, size, computational intensity, and power consumption, limiting the applicability of such methods based on conventional structures in modern and dynamic contexts.

The advent of metasurfaces, with their ability to manipulate electromagnetic waves at a subwavelength scale,[11] has brought significant advances in several domains such as imaging,[12] holography,[13] cloaking,[14] mathematical operations,[15–17] information processing,[18,19] and communications systems,[20–22] to name a few. Recent studies have also explored touch-programmable metasurfaces for various electromagnetic manipulations and encryptions,[23] as well as programmable surface plasmonic neural networks for microwave detection and processing,[24] further broadening the scope of metasurface applications. DOA estimation has also attracted major interest in this context through various studies,[25,26] ranging from the use of the metasurface as a joint component of antenna arrays to the complete replacement of the latter. Recent developments have shown that programmable multi-layer metasurfaces integrated with an antenna array can perform complex transformations

N. Meftah, B. Ratni, S. N. Burokur
LEME, Univ Paris Nanterre
Ville d'Avray F92410, France
E-mail: sburokur@parisnanterre.fr

M. N. El Korso
Université Paris-Saclay, CNRS, CentraleSupélec
Laboratoire des signaux et systèmes
Gif-sur-Yvette 91190, France
E-mail: mohammed.el-korso@universite-paris-saclay.fr

ID The ORCID identification number(s) for the author(s) of this article can be found under https://doi.org/10.1002/aelm.202400476

© 2024 The Author(s). Advanced Electronic Materials published by Wiley-VCH GmbH. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

DOI: 10.1002/aelm.202400476
such as the 2D discrete Fourier transform (2D DFT),[27] thus re-
ducing the dependence on expensive and heavy computational
components. Other studies have further reduced the hardware
complexity by using a single programmable metasurface cou-
pled to a single receiver.[28] acting as a physical random sam-
pling device to generate the necessary inputs for slightly ad-
justed traditional processing algorithms. The use of metasurfaces
in DOA estimation is not only limited to overcoming the con-
straints of traditional methods in terms of hardware complexity
but also opens the way to advances in algorithm optimization,[29]
integration of compressive sampling,[30] use of space-time mod-
ulation techniques,[31] and application of artificial intelligence
algorithms[32] for efficient and accurate DOA estimation. How-
ever, challenges such as managing large amounts of data and
computational time constraints persist, requiring continuous ex-
ploration and innovation in this field.

Recently, an alternative method for estimating the DOA using
a programmable metasurface has been presented.[33] The pro-
posed method can be distinguished by its inspiration from the
operating principle of parabolic antennas, unlike existing meth-
ods exploiting antenna arrays that have mainly focused on hard-
ware and/or computational improvements. A programmable
metasurface where the pointing direction can be adjusted by
electronically changing the phase distribution is used as a vir-
tual parabola to scan the azimuth plane for the direction of ar-
rival estimation. The system is based on a single receiver com-
bined with a simplified, non-complex, and flat electronically
reconfigurable reflective metasurface that substitutes the tra-
ditional parabolic reflector. By continuously varying the point-
ing direction of the meta-reflector, DOA estimation can be per-
formed. However, DOA can be estimated only for angles that
are pointed by the meta-reflector system, and the physical layer
alone, without advanced joint processing, is not sufficient to
achieve enhanced resolution due to the limitations imposed
by the aperture angle of the reflected beam for the different
receiving configurations. In practice, it would be difficult to
refine the aperture angle to ensure scanning with a smaller
step without risking overlapping between adjacent scanning
zones.

Here, we demonstrate a novel architecture as an efficient
method to improve the resolution for finer DOA estimation with-
out refining the scanning step of the meta-reflector. A process-
ing layer based on a pre-trained multilayer neural network is
exploited to interpret variation in recorded power levels, result-
ing in both fast and accurate DOA estimation. As a proof of
concept, the approach is tested to estimate DOA in a detec-
tion range of -40° to +40°. For this purpose, a series of 17
parabolic phase configurations is applied to the meta-reflector to
perform beam scanning with a step of 5°. Besides the reactivity of
the metasurface in an unpredictable and changing background
environment, the performed experimental measurements have
shown that thanks to the use of the processing layer, the pro-
posed architecture is able to accurately estimate the DOA with
a resolution of lower than 1°. Such a metasurface-based plat-
form not only offers new insight for the design and implemen-
tation of next-generation communication systems such as the
6th generation (6G) wireless systems but also builds a robust
foundation for new possibilities in wavefronts control and signal
processing.

## 2. Methodology

The operating principle of our DOA estimation method, enabled
by the utilization of a programmable metasurface reflector and
assisted by a deep learning algorithm, is depicted in **Figure 1**.
The process is divided into two main phases: acquisition and es-
timation. During the acquisition phase, the programmable meta-
surface is coupled to a microstrip patch receiving antenna to op-
erate as a reconfigurable parabolic phase antenna at the receiver
end, which is capable of recording power levels of incoming elec-
tromagnetic signals. A series of parabolic phase distributions is
assigned to the metasurface so as to perform an electronically az-
imuth steering with a regular angular step of 5°. The power levels
collected at the end of the angular scanning process are then ex-
ploited during the estimation phase by a pre-trained Multilayer
Perceptron (MLP) model to accurately estimate the arrival angle
with a resolution lower than 1°.

Here, due to the nature of our dataset, which exhibits nonlin-
ear relationships between inputs and outputs, the MLP training
model is exploited. Indeed, the dynamic configurations of the
metasurface generate power distributions that do not follow a
predictable linear model. Consequently, the MLP training model,
with its capability to model nonlinear relationships through its
multiple layers and neurons, is particularly well-suited for pro-
cessing and learning from such data. Additionally, the flexibility
of the MLP training model in adjusting its architecture allows for
comprehensive exploration to identify the most suitable structure
for the specific requirements of our application, thereby ensuring
optimal performance. As such, the MLP model is pre-trained on
a set of data collected from far-field measurements performed in
an anechoic chamber for different tested DOAs. During its train-
ing, the model learns to interpret subtle variations in power levels
associated with different incidence angles to optimize its hyper-
parameters. Hence, during the estimation phase, the learned hy-
perparameters are applied directly to the low number of power
samples collected during the acquisition phase, thereby reduc-
ing memory consumption and detection time, while enhancing
computational efficiency as well as estimation resolution.

### 2.1. Programmable Metasurface-Based Acquisition System Architecture

The acquisition system used to collect data is primarily composed
of the electronically programmable metasurface reflector, a volt-
age control system to adjust the phase distribution assigned to
the metasurface, and a microstrip patch antenna placed at a fo-
cal distance F = 30 mm in front of the metasurface, as shown
in **Figure 2a**. During experimental measurements, the patch an-
tenna is connected to a spectrum analyzer that acts as a radiofre-
quency (RF) power detector. The voltage control system and the
RF power detector are both controlled by a single Raspberry Pi
processing unit. The whole system forms a closed-loop system
for an automated measurement chain.

The programmable metasurface having dimensions 180 mm
× 180 mm consists of 30 × 30 elementary meta-atoms with a
periodicity of *p* = 6 mm. Figure 2b illustrates the geometry of
the meta-atom, where two copper strips with width *w* = 0.5 mm
are separated by a gap *g* = 1.9 mm. The strips are printed on the
top face of a 1.52 mm thick copper-backed dielectric substrate
<figure><img src="image_3.png" /><figcaption>Figure 1. Conceptual illustration of the enhanced resolution DOA estimation using a programmable metasurface and deep learning technique.</figcaption></figure>

with relative permittivity $ε_r = 4.5$ and loss tangent tan $δ = 0.003$. A voltage-controlled varactor diode of model type MACOM MGV 125-08 is incorporated between the two copper strips. By changing the reverse DC voltage applied to the varactor diode, it is possible to vary the capacitance, thus adjusting the electromagnetic responses in a quasi-continuous manner. The metasurface reflector is first experimentally characterized, as detailed in Note S1 (Supporting Information), to retrieve the amplitude and phase of the reflection response for different bias voltages shown in Figure 2c,d. In this process, a uniform bias voltage is applied to all meta-atoms. The resonant frequency increases from 8.35 to 11.6 GHz when the bias voltage changes from 0.5 to 20 V, corresponding to capacitances of 0.6 and 0.055 pF, respectively. A phase modulation capability of up to 270° is obtained at 10 GHz, offering a quasi-continuous variation possibility, crucial for flexible manipulation of the parabolic phase distributions that will be further used in the DOA estimation technique.

The programmable metasurface used in this study is reconfigurable in only one plane since all meta-atoms in a column are interconnected. As such, only the behavior of a cylindro-parabolic reflector can be mimicked and the phase distribution can be calculated as

$$ \varphi(x) = \frac{2\pi}{\lambda} \left( \frac{(x - x_0)^2}{4F} \right) + \varphi_0 \quad (1) $$

where $λ$ is the free space operating wavelength, $F$ is the focal distance, and $\varphi_0$ is the reflection phase at $x_0$. The steering direction of the cylindro-parabolic reflector is controlled by shifting $x_0$ to the left (-) or to the right (+) from its original position along the horizontal axis (x-axis) of the metasurface.

During the acquisition phase, 17 cylindro-parabolic phase distributions are selected based on their pointing angles established from far-field measurements performed in an anechoic chamber to scan an azimuth angular range of -40° to 40° with 5° increment, allowing the recording of power levels necessary to estimate the DOA with an angular resolution finer than the scan step. During these measurements, the patch antenna, located at the focal point in front of the metasurface, is connected to one port of a vector network analyzer (VNA) and operates as a transmitter to illuminate the metasurface at normal incidence, which acts as a reconfigurable cylindro-parabolic phase reflector antenna. The meta-reflector antenna is then mounted on a rotating plate and a 2-18 GHz broadband FLANN DP240 horn antenna, placed at 3 m away and connected to port 2 of the VNA, is used as the listening port to measure the far-field radiation patterns of the meta-reflector antenna as the plate rotates over angles of -90° to +90°. The far-field measurements are conducted for the 17 different offsets $x_0$ to build a library of correspondence between $x_0$ and pointing angles $θ$, as depicted in Table 1. As illustrated in Figure 2e,f, for offsets $x_0$ of -10, 0, and +24 mm, pointing angles of -10°, 0°, and 25° are obtained, respectively.

## 2.2. Training Dataset

The accuracy of a machine learning model largely depends on the quality and relevance of data used for its training. Here, a dataset is carefully developed to provide a solid foundation for training our MLP model designed for DOA estimation. The experimental protocol adopted for data collection is illustratively described in Figure 3a. A microwave horn antenna used at the transmitter end in the closed-loop acquisition system is placed at 3 m away in the far-field region on a semi-circular rail to
<figure><img src="image_4.png" /><figcaption>**Figure 2.** a) Architecture of the DOA estimation platform based on a programmable meta-reflector and a microstrip patch receiving antenna. b) Elementary meta-atom integrating a voltage-controlled varactor diode to enable dynamic control of the electromagnetic responses. c,d) Measured reflection magnitude and phase of the meta-reflector for different bias voltages. e) Calculated parabolic phase distribution for $x_0$ shifts of -38, -10, 0, and +24 mm. f) Normalized 2D far-field radiation patterns corresponding to the applied $x_0$ shifts.</figcaption></figure>

launch the electromagnetic waves toward the meta-reflector antenna within an azimuth angular range of -40° to 40° with 1° increment. To determine the angle of arrival of the incoming wave, the patch antenna is connected to a spectrum analyzer to act as a receiver. Upon impinging on the meta-reflector, the incoming signals from the horn antenna are reflected toward the receiving patch antenna positioned at the focal point.

The dataset consists of 17 features, corresponding to the power levels captured by the receiving patch antenna after applying the 17 preselected phase distributions to the metasurface. DOA estimations ranging from -40° to 40° with 1° increment serve as targets in our model. For each tested angle of arrival, the acquisition system records 17 power levels, sweeping the azimuth plane with a scanning angle $\theta_s$ varying from -40° to 40° with 5° increments using the 17 phase distributions applied to the metasurface, capturing one power level per distribution. This process is repeated three times for each angle of arrival, resulting in a dataset that includes 243 samples distributed over 81 different targets. The constructed dataset is projected in Figure 3b to illustrate the distribution of average powers received according to each arrival angle and feature. The dark red regions indicate the points where the patch antenna receives the highest average power for a given DOA. The marked diagonal dark red band highlights the fact that the highest power levels are captured by features whose scanning angle is positioned around the angle of arrival, in agreement with the principle of our acquisition method which consists in scanning the azimuth to estimate the DOA.

<table><caption>**Table 1.** Series of parabolic phase distributions used to estimate the DOA.</caption><tr><td>x<sub>0</sub> (mm)</td><td>-38</td><td>-34</td><td>-28</td><td>-24</td><td>-18</td><td>-14</td><td>-10</td><td>-6</td><td>0</td><td>+6</td><td>+10</td><td>+14</td><td>+18</td><td>+24</td><td>+28</td><td>+34</td><td>+38</td></tr><tr><td>θ (°)</td><td>-40</td><td>-35</td><td>-30</td><td>-25</td><td>-20</td><td>-15</td><td>-10</td><td>-5</td><td>0</td><td>5</td><td>10</td><td>15</td><td>20</td><td>25</td><td>30</td><td>35</td><td>40</td></tr></table>
<figure><img src="image_5.png" /><figcaption>**Figure 3.** a) Experimental setup for data collection. b) Heatmap visualization of the collected dataset showing average power distribution by target and parabolic phase distribution. c) Received power distributions for DOA of 11° and -15°, showing the three measured samples for each DOA.</figcaption></figure>

Training models, such as the MLP, often require a large amount of data to ensure proper generalization and prevent overfitting. In the initial dataset, each DOA is presented by three measurement samples which, except for some measured noise levels, follow an almost identical power distribution by feature, as shown in Figure 3c for DOA of 11° and -15°. To increase the limited size of our dataset without requiring additional experimental measurements, which would be time-consuming and resource-intensive, we implement a data augmentation technique, as detailed in Note S2 (Supporting Information). The technique involves generating additional data instances by introducing Gaussian noise that mimics natural variations and experimental disturbances observed on the original samples. The Gaussian noise is carefully calibrated to match the intra-sample variability for each specific DOA and feature. This approach ensures that the added noise not only refers to the internal variance within each sample set but also maintains a clear distinction between the different target sets. For each sample in the initial dataset, five new samples are generated, thereby expanding from an initial dataset of 243 samples to an augmented dataset of 1215 samples,

with a balanced representation of the various targets. It is important to note that this augmentation approach preserves the integrity of the data by avoiding to introduce bias or transformations that could alter the physical nature of the original measurements. Data augmentation represents a crucial trade-off between model accuracy and robustness. The influence of data augmentation on the prediction accuracy has been analyzed and it has been concluded that a low augmentation factor leads to insufficient capacity of the dataset to capture the necessary variability for robust generalization. On the other hand, a high augmentation factor can introduce noise and redundant elements, which also hampers the model to efficiently generalize to unseen data (more details are provided in Note S3, Supporting Information).

Another important aspect to consider about the nature of the data collected from measurements is that the intensity of the measured power levels depends on the separation distance between the transmitting antenna and the acquisition system. A modification of this distance results in a proportional change in the power levels recorded after scanning, without however altering the relative distribution of powers
between the features. Therefore, to force our model to focus its learning on the structural variations of the data regardless of the influence of the transmitter-receiver distance, a relative normalization is applied to the entire data set, which consists in bringing the measured noise floor to zero. For each sample $x_i$ represented by a vector in a 17D real space corresponding to the 17 features ($x_i \in \mathbb{R}^{17}$ with $i \in \{1, 2, \dots, 1215\}$), the data is normalized by dividing each instance by its maximum

$$x_i^{\text{norm}} = \frac{x_i}{\max(x_i)} \qquad (2)$$

where $\max(x_i)$ is the maximum value of the feature in the sample i. As such, each measurement sample is rescaled within a range of 0 to 1, with 1 being the maximum power level detected for the sample. Before starting the learning process, the dataset is divided into training and test sets, with 80% of the data intended for training and 20% reserved for efficiently testing the performance of the trained model on unseen data. A standardization step is then applied to each feature of the two sets. This standardization, often required by certain machine learning models, particularly MLPs, adjusts the data so that each feature presents a mean of zero and a standard deviation of one. For each feature j (with $j \in \{1, 2, \dots, 17\}$), the mean $\mu_j$ and the standard deviation $\sigma_j$ over all samples related to the jth feature are calculated and then applied in the following transformation

$$x_{ij}^{\text{std}} = \frac{x_{ij}^{\text{norm}} - \mu_j}{\sigma_j} \qquad (3)$$

where $x_{ij}$ is the value of feature j for sample i.

## 2.3. Deep Learning Model Architecture and Training

An MLP model consists of multiple layers of neurons, each transforming inputs into outputs that serve as new inputs for the next layer.[34] Each neuron computes a weighted sum of its inputs, followed by a nonlinear activation function, allowing the MLP model to capture complex relationships between inputs and outputs, such as the subtle differences in power distribution in our high-resolution DOA predictions.

The MLP training model is composed of several fully connected layers of neurons. The output of the lth layer is given by

$$a_m^{(l)} = f \left( \sum_{n=1}^{N} w_{mn}^{(l)} a_n^{(l-1)} + b_m^{(l)} \right) \qquad (4)$$

where $a_m^{(l)}$ represents the activation of the mth neuron in the lth layer, $w_{mn}^{(l)}$ is the weight between the nth neuron in the (l-1)th layer and the mth neuron in the lth layer, $b_m^{(l)}$ is the bias associated with the mth neuron in the lth layer, f is the activation function, and $a_n^{(l-1)}$ is the input to the nth neuron in the lth layer (with $a_n^{(0)}$ representing the standardized features of the sample).

The learning of weights w and biases b is done by backpropagation and gradient descent, iteratively until convergence to the

optimal solution. Backpropagation enables the calculation of the gradient of the prediction error with respect to each weight and bias in the network, and gradient descent adjusts these parameters to minimize the mean square error (MSE) given by

$$MSE(\hat{\gamma}, \hat{\hat{\gamma}}) = \frac{1}{n} \sum_{i=1}^{n} (\hat{\gamma}_i - \hat{\hat{\gamma}}_i)^2 \qquad (5)$$

where $\hat{\gamma} = [\gamma_1, \dots, \gamma_n]$ and $\hat{\hat{\gamma}} = [\hat{\gamma}_1, \dots, \hat{\gamma}_n]$, in which $\gamma_i$ is the actual DOA value for the ith sample and $\hat{\gamma}_i$ is the value predicted by the model. Adjustments are therefore made according to

$$w_{mn}^{(l)} = w_{mn}^{(l)} - \alpha \frac{\partial MSE}{\partial w_{mn}^{(l)}} \qquad (6)$$

$$b_m^{(l)} = b_m^{(l)} - \alpha \frac{\partial MSE}{\partial b_m^{(l)}} \qquad (7)$$

where $\alpha$ is the learning rate, controlling the size of the steps made in the opposite direction of the gradient to reach a local minimum. Optimal hyperparameters $\Theta$, including the number of layers, neurons, activation function type, and learning rate, are determined by combining Bayesian optimization and cross-validation. The optimization process begins by defining a search space for each hyperparameter. Cross-validation divides the training dataset into k folds, using a different fold each time as the validation set and the remaining k-1 for training. Bayesian optimization systematically explores the hyperparameters space to identify those that minimize MSE across cross-validation

$$\Theta^* = \arg \min_{\Theta} CV_{MSE} (MLP(\Theta; X_{train}^{\text{std}}, Y_{train})) \qquad (8)$$

where $CV_{MSE}(MLP(\Theta; X_{train}^{\text{std}}, Y_{train})) = \frac{1}{k} \sum_{i=1}^{k} MSE(\gamma_i, \hat{\gamma}_i)$ represents the average of the mean squared errors calculated across the different folds of cross-validation, $X_{train}^{\text{std}}$ is the standardized training set and $Y_{train}$ are the corresponding target values. The iterative process gradually refines the choice of hyperparameters until a convergence is reached to the architecture of the MLP model that offers the best performance, promoting the model's ability to be generalized beyond the training data in an effective manner.

In practice, the MLP model is implemented with the “MLPRe-gressor” class from the Scikit-learn library and optimized using the Bayesian-Optimization library with 10-fold cross-validation. The architecture of the MLP model with the optimized hyper-parameters obtained after training is as follows: the structure of the hidden layers is defined by two layers of 100 neurons each. The activation function is “tanh” and the “adam” solver is used for weight optimization, reinforced by an initial learning rate of 0.0082 and a regularization parameter of 0.005, ensuring robust convergence with a tolerance set at 0.00073.

The test set is used to evaluate the performance of this trained MLP model on unseen data during its training. Performance is measured through metrics MSE and determination coefficient R², written as

$$R^2 = 1 - \frac{\sum_{i=1}^{n} (\gamma_i - \hat{\gamma}_i)^2}{\sum_{i=1}^{n} (\gamma_i - \bar{\gamma})^2} \qquad (9)$$
<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /><figcaption>**Figure 4.** a) Learning curve of the MLP model for the DOA estimation. b) Comparison of predicted versus real DOA values. c) Representative diagram of the prediction error distribution. d) Accuracy of predictions, measured as the percentage of errors falling within thresholds of 0.5°, 1°, and 2°, for models with scan steps of 5°, 10°, and 20°.</figcaption></figure>

where $\bar{\gamma}$ is the average of the actual DOA values. This metric measures the proportion of the variance in the dependent variable that is predictable from the independent variables, thus providing an indicator about the quality of the model's predictions. Additionally, the learning curve is used to verify that the model has learned enough so that it can be generalized to new data. This analysis helps to detect issues of overfitting or underfitting by assessing the performance of the model as the amount of training data increases.

## 3. Results

The training performance of our MLP model is validated by the learning curve depicted in **Figure 4a**, which demonstrates the absence of overfitting and underfitting, as evidenced by the convergence of the training and cross-validation scores with the increasing size of the training set. The green shaded area around the cross-validation curve represents the confidence interval, indicating variability in the model's performance during validation across different subsets of data. This confirms the acquired sufficient capability of the model to be generalized to new data.

Evaluation of the performance of the MLP model on the test set reveals high accuracy in estimating DOA with an MSE of 0.3°². The precision of the predictions of the model is therefore validated. An obtained R² score of 0.9995 indicates the strong correlation between predicted and actual values, highlighting the fact that the model almost perfectly explains the variance in the data. The results show that the MLP training model demonstrates robustness against noisy data, an essential feature given the fixed objective of ensuring that the model focuses on learn-

ing the trends in power distribution while disregarding noise and fluctuations between different samples for enhanced accuracy of the DOA estimation. Figure 4b presents the tight correlation between predicted and actual DOA values, where the linearity of the scattered points confirms the effectiveness of the model. Moreover, a detailed analysis of the prediction errors reveals that 94% of the predictions show less than 1° error, demonstrating the exceptional accuracy of the model. The distribution of prediction errors, presented in Figure 4c, shows a distribution centered around zero, indicating that the model provides unbiased estimates and performs well across the entire range of targeted values.

To explore the impact of scan step size on the accuracy of DOA estimation, a comparative analysis of prediction accuracy across models trained with different scan steps of 5°, 10°, and 20°, corresponding to 17, 9, and 5 input features, respectively, is conducted and presented in Figure 4d (more details are given in Note S4, Supporting Information). The performance of each configuration is evaluated by the prediction errors falling within thresholds of <0.5°, <1°, and <2°. The results indicate that the model with a 5° scan step, featuring 17 input features, achieves the highest precision, with a significant majority of errors below the smallest error threshold, highlighting its superior accuracy suitable for high resolution requirements. In contrast, while the 10° scan step model (9 features) demonstrates acceptable accuracy for applications requiring up to 1° resolution, its performance is less accurate than the 5° model. The 20° scan step model, with only 5 features, displays the least precision, further illustrating the inverse relationship between scan step size and estimation accuracy. To conclude, the choice of scan step is guided by the desired
resolution and represents a trade-off between scan speed, where a larger step implies a faster acquisition phase, and accuracy in DOA estimation, which increases as the number of input features grows.

## 4. Conclusion

In summary, the method proposed in this study offers a promising and non-complex alternative for estimating DOA, by combining a reconfigurable metasurface, used as a virtual parabola for signal acquisition, and deep learning for precise estimation of the DOA. This innovative approach merges the efficiency of metasurfaces in wave manipulation with the principles of focusing parabolic reflectors, thus overcoming the problems related to spatial clutter and computational intensity of traditional methods. In addition, the integration of a processing layer based on learning, in the form of a pre-trained multilayer neural network, significantly increases the accuracy of the DOA estimation, achieving fine resolution without requiring dense angular scanning at the receiver end. As such, with an accelerated azimuth scanning of the receiver between -40° and 40° with increments of 5°, the proposed system is able to estimate the DOA with an accuracy lower than 1°.

The results obtained in this work show that training a machine learning based algorithm with experimental data allows our scheme to learn with accuracy the DOA estimation in varying scenarios. Since we are in a data-based framework, all imperfections that are inherent to the experiment are taken into account, such as errors attributed to the manufacturing tolerances of the metasurface, electronic components, and varactor diodes that can possibly introduce variations and inaccuracies in the phase responses. The MLP training model, particularly known for its universal approximation, flexibility, robustness to noise, and the possibility to handle model mismatches or uncertainties, has been selected for its good adaptability in practical applications. Specifically, in our training procedure, we introduce randomness and variability and we also proceed to data scaling and normalization; these factors contribute to the strong adaptability of the MLP model.

The improved performances open promising prospects for more complex configurations, particularly for multi-source detection, where the ability to accurately identify the direction of arrival of multiple simultaneous signals is crucial, thus highlighting the interest in exploring the implementation of metasurfaces and their merging with machine learning technologies, in order to further push the limits of existing methods in the field of DOA estimation.

## Supporting Information

Supporting Information is available from the Wiley Online Library or from the author.

## Acknowledgements

N.M. acknowledges her Ph.D. scholarship from the French Ministry of Higher Education and Research. S.N.B. acknowledges funding from the Centre National d'Etudes Spatiales (CNES) through the Challenge R&T "Fonctionnalisation des Surfaces".

## Conflict of Interest

The authors declare no conflict of interest.

## Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## Keywords

deep learning, direction of arrival, electromagnetic waves, reconfigurable metasurface

Received: June 17, 2024

Revised: August 2, 2024

Published online: September 9, 2024

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