In signal processing, Direction of Arrival (DOA) denotes the direction of signals impinging on a sensor or antenna array [1]. DOA estimation methods have been investigated for decades due to their actual application in radar, sonar, seismology, astronomy, and military surveillance. Furthermore, DOA has become more significant due to the development of mobile networks, technologies, and devices. For example, Beamforming (BF) is a technology that is becoming more important every day in mobile communication networks [2]. BF is a technique that focuses a wireless signal toward a specific receiving device rather than having the signal spread in all directions from an omnidirectional antenna, as it usually would. In this way, the received signal level is increased. Thus, it is essential to know the device’s direction in order to to precisely align the antenna’s beam with it.

On the other hand, the wide number of drone devices in agriculture, industry, transport, communication, surveillance, etc., along with the development of the Internet of Things (IoT), means that the use of such devices is increasing considerably ^[3][4][3,4]. However, drones can seriously threaten civilian environments or sensitive areas such as airports and military bases. For example, in January 2015, a drone crashed at the White House [5], compromising the security of the government building. On 29 March 2016, a Lufthansa jet came within 200 feet and collided with a drone near Los Angeles International Airport [6]. On 12 October 2017, a Beech King Air A100 of Skyjet Aviation collided with an Unmanned Aerial Vehicle (UAV) as the former approached Jean Lesage Airport near Quebec City, Canada. The aircraft landed safely despite being hit on the wing [7].

Due to the security risk associated with drones, as aforementioned, it is necessary to implement drone DOA systems to locate and confiscate or disable them. For the DOA system to work, a drone’s signal is required in order to discover its directions. However, drone owners do not want to be discoverable, and they will not transmit such a signal that makes them vulnerable to detection or localization through a DOA system. Fortunately, most drones transmit signals to their controllers to send images, video, and telemetry reports. Another important use case for a DOA system is the localization and recovery of drones that lost contact with their controllers. These drones generally keep sending data to their controllers. Therefore, these signals can be used by DOA systems to find the drone’s direction.

In 5G and 6G wireless networks [8], the estimation of the DOA plays a crucial role across various applications. One of its key benefits lies in facilitating the deployment of smart antenna systems, allowing adaptive steering of antenna arrays to optimize both signal reception and transmission [9]. Moreover, DOA information finds utility in localization and positioning applications, as it enables accurate triangulation of user device locations by estimating the DOA of signals from multiple base stations or access points [10]. Another notable application of DOA estimation pertains to enhancing network security, as the analysis of signal DOAs permits the detection and mitigation of spoofing attacks or unauthorized signal sources [11]. Furthermore, DOA estimation offers valuable insights for network planning and optimization processes, aiding in efficient resource allocation and improved network performance [12].

2. Maximum Likelihood Estimation

The MLE finds its estimates by maximizing the probability density function of the observed received signals concerning the model’s parameters. MLE solutions can be classified into two approaches: stochastic MLE and deterministic MLE. In the stochastic MLE, the signals are assumed to be Gaussian distributed ^[13][14][15][13,14,15]. Deterministic MLE methods consider the signals as being arbitrary and deterministic ^[15][16][15,16]. The sensor noise is modeled as Gaussian in both methods, which is a reasonable assumption due to the central limit theorem. As a result, the stochastic MLE achieves the Cramer-Rao Bound (CRB). On the other hand, this does not hold true for deterministic MLE methods [17]. The problem with MLE techniques is the high computational cost involved since they have to solve a nonlinear multidimensional optimization problem for which global convergence is not guaranteed [18].

3. Subspace-Based Techniques

The best-known subspace-based methods are MUSIC [19] and Estimation of Signal Parameters Via Rotational Invariance Techniques (ESPRIT) [20], on which many works have been based. MUSIC-based methods employ the orthogonality of the signal subspace (steering vectors) and the noise subspace to search the spatial spectrum to achieve high resolution. ESPRIT-based methods exploit an underlying rotational invariance among signal subspaces induced by an array of sensors with a translational invariance structure. When the uncertainty of the system or background noise leads to model errors, e.g., the wrong number of sources, subspace-based methods need to solve high-dimensional non-linear parameter estimation problems. Many improved algorithms based on MUSIC and ESPRIT have been developed to estimate the number of sources ^[21][22][21,22]. However, these approaches must sacrifice the array aperture and deteriorate the resolution to deal with the singular matrix of the spatial covariance issue.

4. Sparse Signal Reconstruction

The Sparse Signal Reconstruction (SSR) technique has been used in DOA estimation ^{[23][24][25][26][27]}[23,24,25,26,27]. It exploits the property that the spatial spectrum of the point source signals is sparse when the number of signals is limited. The key is to use appropriate non-quadratic regularizing functions (such as ℓ𝑝ℓ-norms), which lead to sparsity constraints and super-resolution. In addition, the primary concern in the SSR technique lies in the computational complexity.

5. Machine Learning

ML is presented as a promising technology to be used for DOA estimation. ML-based methods are data-driven and more robust than other methods due to their adaptability to the array geometry and sensor imperfections. They also do not depend on the array geometry shape ^[28][44]. In addition, ML offers low-cost implementation and simplicity. The authors in ^{[29][30][31][32][33][34][35][36][37]}[28,29,30,31,32,33,34,35,36] used a Neural Network (NN) for DOA estimation. The authors in ^[29][28] proposed an azimuth estimation method using a Complex Valued Neural Network (CVNN) for ultra-wideband systems, where the received signal feeds the CVNN. The authors validate their proposal via simulation and experiments. The results are compared to MUSIC and Real-Valued Neural Networks (RVNNs). In ^[30][29], a Multilayer Perceptron Neural Network (MPNN) is presented that can learn from a large amount of simulated noisy and reverberant microphone array inputs for robust DOA estimation. Specifically, the MPNN learns the nonlinear mapping between Generalized Cross-Correlation Vectors (GCC) features and DOA. In ^[31][30], the authors used a Deep Neural Network (DNN) for DOA estimation. They evaluated the estimation performance under a scenario where two equal-power and uncorrelated signals are incidents on a Uniform Linear Array (ULA). The authors in ^[32][31] focused on scenarios where the number of active sources may exceed the number of simultaneously sampled antenna elements. For this purpose, they proposed new schemes based on NN and estimators that combine NNs with gradient steps on the likelihood function. The authors in ^[33][32] propose a cascaded neural network consisting of the SNR classification and the elevation angle estimation for two closely spaced sources. The authors used the correlation matrix as the input of a DNN for DOA estimation. In ^[34][33], the authors integrate Multiple Input Multiple Output (MIMO) systems with DNN for channel estimation and DOA determination of a source. In this work, good results are obtained in the simulations. However, they only focus on the azimuth angle and require a complex system (e.g., in their simulation, the radio base of the MIMO system is equipped with 128 antennas). In ^[35][34], the authors propose a new DOA method based on an ML model to estimate the azimuth angle of a signal. A dataset named “Dround Data New” was obtained with a four-antenna-based system that contains well-known received powers. The authors trained and validated the dataset with a DNN model. In ^[36][35], the authors presented a deep ensemble learning to find a source’s azimuth and elevation angles to different training conditions. The authors designed a Convolutional Neural Network (CNN) that performs the regression task to learn a mapping between the spatial covariance matrix of the received signals from the antenna elements and the DOA. In addition, to improve the prediction performance, they proposed an ensemble learning method. Simulation results show that their proposals have slightly lower Mean Squared Error (MSE) performance than conventional methods such as MUSIC and ℓ1ℓ1-norm SVD (Singular Value Decomposition). However, the simulation results also show that the authors’ proposals in this article respond very fast and have higher speeds than the MUSIC and ℓ1ℓ1-norm SVD algorithms. In ^[37][36], the authors proposed Circularly Fully Convolutional Networks (CFCN) to find the DOA of multiple sources in low-frequency scenarios. The CFCN is trained using the dataset labeled with space-frequency pseudo-spectra and accompanied by on-grid DOA predictions. Then, the regression model is developed to estimate the precise DOAs according to corresponding proposals and features. In ^{[38][39][40][41][42][43]}[37,38,39,40,41,42], the authors proposed a Support Vector Regression (SVR)-based DOA estimation method. In ^[38][37], a smart antenna system is considered to estimate the DOAs of multiple sources in noiseless and noisy environments.In ^[39][38], the authors address the problem of estimating the DOAs of coherent electromagnetic waves incident on a ULA, building upon previous work and presenting experimental results. In ^[40][39], a multi-resolution approach for the real-time DOA estimation of multiple signals impinging on a planar array is presented. The method is based on a Support Vector Classifier (SVC), and it exploits a multi-scaling procedure to enhance the angular resolution of the detection process in the regions of incidence of the incoming waves. The authors in ^[41][40] proposed the combination of the advantages of Forward–Backward Linear Prediction (FBLP) and SVR in estimating DOAs of coherent incoming signals with low snapshots. In ^[42][41], the authors proposed a scheme to address the wideband DOA estimation problem. An approach for the real-time DOA estimation of multiple signals impinging on a planar array is presented in ^[43][42]. This last work estimates the azimuth and elevation angles of one or more incident sources in a ULA. However, the angular resolution is shallow. The main disadvantage of Support Vector Machine (SVM) is the algorithm’s time complexity ^[44][45]. The algorithmic complexity of SVM models affects the model training time on large datasets, the development of optimal models for multiclass, and the performance on unbalanced datasets. Table 1 summarizes the articles mentioned above that employ ML techniques to solve the DOA problem. The table shows the main aspects considered to carry out theour work. In addition, the table collects the angles considered by the related works: azimuth, elevation, or both. It also shows if the work is oriented to find the DOA from a single source or several, if the proposals were validated via simulation or experiments, and if the results were compared with an existing DOA model. ML has been mainly used to improve the estimation of azimuth or elevation angles and computing speed. In addition, various studies have been carried out to find the origin of multiple sources. However, the study of ML-based DOA for estimating azimuth and elevation angles has yet to be deeply studied. The reason behind this is the huge size of the training data, which are only supported by some models such as NNs ^[45][46]. However, estimating the elevation and the azimuth angles is crucial and has many applications in various engineering fields. For instance, with complete DOA information, it is possible to improve the transmission coverage in wireless communications by avoiding interference and enhancing the system capacity ^[43][42]. More specifically, the knowledge of the azimuth and elevation angles would enable more effective utilization of the beamforming (BF) technology in the next generation of mobile networks.