The sixth generation (6G) of wireless communication systems is expected to support transformative applications such as autonomous transportation, real-time holographic interaction, ubiquitous extended reality (XR), and intelligent sensing in smart factories. These use cases demand highly accurate beam steering, user localization, and spatial awareness, all of which are critically dependent on precise DoA estimation.

DoA refers to the angular position (azimuth and elevation) from which a signal is received at an antenna array. In conventional beamforming or spatial multiplexing systems, knowledge of DoA enables the base station or radar to focus energy in desired directions while suppressing interference. As communication systems evolve toward joint sensing and communication paradigms (6G-ISAC), accurate and fast DoA estimation becomes fundamental to achieving low-latency, high-reliability, and energy-efficient operation.

To highlight the ubiquitous deployment of DoA estimation in new technologies, Fig. 1 offers some notable application areas where precise angular estimation plays a key role. In autonomous vehicles, DoA enables Vehicle-to-Everything (V2X) communication by enabling dynamic beam following, thereby enhancing vehicular safety and low-latency connectivity. In unmanned aerial vehicle (UAV) communications, such as aerial relays and swarm coordination, DoA estimation facilitates optimal path planning, collision avoidance, and mutual interference reduction. Similarly, in cell-free multiple-input multiple-output (MIMO) systems, where multiple distributed antennas cooperate to serve users in dense urban environments, accurate DoA information is necessary for user-centric beamforming and coordinated access.

In Metaverse and extended reality (XR) applications, DoA enables seamless user interaction through accurate indoor positioning and beam steering, enabling lag-free and immersive experiences. In 6G networks with Integrated Sensing and Communication (ISAC) capabilities, real-time DoA tracking offers integrated radar-communication features such as object detection and channel-aware communication. Finally, in factory automation environments, DoA estimation enhances robot hardware localization and autonomous control, especially in multipath-prone and adverse radio frequency conditions. These multiple applications collectively point to the pivotal position of adaptive and resilient DoA estimation in characterizing the future generation of smart and interconnected systems.

Despite the long-standing research on DoA estimation, several challenges remain unresolved, particularly for joint azimuth and elevation estimation using simple array structures. Most classical methods require complex array geometries or exhibit poor performance in low-SNR or coherent-source environments. Moreover, these methods often require manual pairing of azimuth and elevation angles, a task not feasible for real-time or large-scale deployments. The key contributions of this paper are as follows:

We propose a novel 2D DoA estimation framework employing three parallel ULAs to simultaneously estimate elevation and azimuth angles with enhanced spatial resolution.

The remaining parts of this paper are organized in the following manner. Section 2 provides an extensive overview of state-of-the-art DoA estimation techniques, highlighting their limitations and advancements. Section 3 provides the system model, introducing the configuration and signal expression for the proposed three-parallel ULA structure. Section 4 provides the suggested hybrid optimization strategy. Simulation results and performance evaluation under various scenarios are reported in Section 5. Section 6 concludes the paper, and finally, potential directions for future work are outlined.

Accurate estimation of the DoA of signals remains a cornerstone for advanced communication systems, particularly as the world transitions toward 6G networks characterized by high mobility, spatial multiplexing, and integrated sensing and communication (ISAC) capabilities. The growing diversity of wireless applications, from V2X and UAV communications to massive MIMO and industrial Internet of Things (IoT), has intensified the demand for robust and scalable DoA estimation frameworks. As a result, a significant body of research has emerged, spanning from classical signal processing techniques to modern hybrid metaheuristic methods and unconventional antenna configurations. This section surveys key developments across these areas to contextualize the contributions of the current study.

This section outlines the signal model for the proposed three-parallel ULA configuration. We consider a passive monostatic phased-array radar system in which the receiver employs three Uniform Linear Arrays (ULAs) that are spatially separated but mutually parallel. Each ULA lies along a different spatial orientation—aligned with the Y-axis, the xy plane, and the yz plane, respectively—to capture the directional diversity of incoming plane waves. The geometric structure is illustrated in Fig. 2.

Let us assume that I narrowband, far-field sources impinge upon the array with elevation angles θi∈[0,π/2] and azimuth angles ϕi∈[0,2π] for i=1,2,…,I. The signals are mutually uncorrelated and mixed with additive white Gaussian noise (AWGN), which is spatially and temporally uncorrelated with the sources and modeled as a zero-mean, unit-variance complex Gaussian process. The wavelength of the signal is denoted by λ, and the spacing between adjacent sensors in each ULA is fixed at d=λ/2 to prevent spatial aliasing.

Each ULA contains a different number of elements for symmetry and angular resolution: the array along the Y-axis has (M+1) sensors, while the arrays in the xy and yz planes have M sensors each. This asymmetry aids in simplifying the mathematical formulation while maintaining spatial diversity.

This section presents the proposed optimization framework for joint elevation and azimuth angle estimation using the three-parallel ULA configuration. The estimation process is formulated as a nonlinear optimization problem, where the objective is to minimize the error between the measured and estimated array outputs. We explore three optimization strategies: Genetic Algorithm (GA), Pattern Search (PS), and a hybrid GA-PS approach. The individual components, algorithmic flowcharts, and pseudo-code for each method are discussed in detail.

This section presents the simulation results for the proposed optimization schemes: GA, PS, and hybrid GA-PS, for estimating the elevation and azimuth angles of multiple incident plane waves. The performance of each algorithm is evaluated under various conditions, including changes in the number of sources and SNR.

The spacing between adjacent elements in each ULA is set to d=λ/2 to avoid spatial aliasing. The number of array elements is based on the variable M, which is chosen as twice the number of sources. For all experiments, the population size in GA is set to 600 chromosomes, and angular values are expressed in degrees.

Case 1: Two Signal Sources at SNR = 15 dB

In this scenario, two narrowband sources impinge on the 3-parallel ULA system. The true direction-of-arrival angles are set to θ1=40∘,θ2=70∘,ϕ1=90∘,ϕ2=120∘.

Table 3 presents the estimated angles using each algorithm. The hybrid GA-PS approach achieves the closest estimates to the ground truth, followed by GA. PS shows larger deviations, especially in estimating elevation angles. The results confirm the effectiveness of GA in global exploration, and demonstrate that the local refinement by PS improves precision further.

Case 2: Three Sources with Increased Angular Diversity

This experiment evaluates algorithm robustness with three signal sources. The desired elevation and azimuth angles are as follows: θ=[60∘,80∘,40∘],ϕ=[150∘,110∘,50∘]

The results in Table 4 show that although estimation errors increase slightly due to the growing search space, the GA-PS method remains consistently accurate. GA performs well, but PS suffers noticeably in both azimuth and elevation estimation due to local trapping.

Case 3: Four Sources: High Complexity Scenario

As the number of sources increases to four, the search space becomes more complex and high-dimensional. The desired directions are θ=[75∘,35∘,55∘,80∘],ϕ=[210∘,105∘,95∘,160∘].

Table 5 illustrates a drop in accuracy across all algorithms due to the larger number of parameters. The hybrid GA-PS method still produces the most accurate estimates, demonstrating the benefit of combining global and local search. GA retains moderate accuracy, while PS is visibly affected by convergence limitations.

Case 4: Six Sources: High-Dimensional Scenario

To assess scalability, we consider six simultaneous narrowband sources impinging on the 3-parallel ULA system. The desired angles are θ=[88∘,72∘,60∘,48∘,40∘,85∘],ϕ=[45∘,85∘,125∘,165∘,205∘,230∘].

With six sources as shown in Table 6, the search space grows substantially and the cost surface becomes more multi-modal. Consistent with the earlier cases, the hybrid GA-PS approach preserves sub-degree accuracy through global exploration followed by local refinement, the plain GA retains moderate accuracy but exhibits larger bias, and PS alone shows the largest deviations due to local trapping.

Case 5: Robustness across SNR Levels (RMSE Analysis)

To evaluate the robustness of the algorithms, we simulate three sources under the SNR of 15 dB. The root mean square error (RMSE) is computed for each scheme and tabulated in Table 7. Several classical propagator-based algorithms have also been considered for comparison. Wu et al. [32] proposed a fast 2-D DOA estimation method using three-parallel ULAs, but their approach requires additional azimuth-elevation pair matching and often suffers performance degradation in practical mobile elevation angles between 70∘ and 90∘. Li et al. [33] improved upon Wu’s method by reducing computational burden and eliminating the pair matching step, though their scheme still requires a residual 2-D peak search. Chen et al. [34] further advanced the propagator framework by fully exploiting array information and automatically pairing azimuth and elevation angles, thereby achieving better estimation accuracy with reduced complexity. The experiment with two closely spaced sources at an SNR of 15 dB (Table 7) corresponds to a V2X communication scenario, where roadside units must track two vehicles in adjacent lanes. Here, the GA-PS method achieves the best estimate performance, ensuring reliable beam alignment for high-speed vehicular links.

The RMSE values reported in Table 7 highlight the superior performance of the proposed hybrid GA-PS approach. At an SNR of 15 dB, GA-PS achieves remarkably low errors across all four parameters, with azimuth errors as small as 0.0072∘ and 0.0014∘, and elevation errors of only 0.1023∘ and 0.0271∘. In contrast, GA alone exhibits noticeably higher errors, particularly for elevation (0.8657∘), reflecting its slower convergence and less effective local refinement. The PS scheme shows inconsistent behavior, achieving excellent accuracy for one azimuth (0.0102∘) but extremely poor performance for other parameters (up to 0.9966∘), which confirms its susceptibility to local minima.

Early stopping routinely terminates the search well before the nominal budgets. The GA scheme converges after approximately 35–60% of the 2000-generation cap, and the PS scheme after 30–40% of the 3000-iteration cap, with the exact fraction depending on the number of sources I. The runtime of the hybrid GA-PS scales nearly linearly with I, with median per-trial runtimes on a single thread of about 0.9s for I=2, 1.4s for I=3, 2.1s for I=4, and 3.6s for I=6. The median iteration counts performed in practice are GA ≈ 700–1200 generations followed by PS ≈ 500–1100 iterations initialized from the best GA. The hybrid scheme converges faster and reaches lower objective values than the GA or PS schemes.

GA-PS significantly outperforms the Chen, Li and Wu schemes. Chen’s algorithm [34], which utilizes complete array information and automatic azimuth-elevation pairing, produces balanced results with errors in the range of 0.095∘–0.155∘, but remains at least an order of magnitude less accurate than GA-PS. Li’s method [33] offers improved azimuth accuracy (0.105∘–0.120∘) but suffers from larger elevation errors (up to 0.270∘). Wu’s scheme [32], although computationally efficient, shows the highest elevation estimation errors (0.520∘–0.620∘), consistent with its known structural limitations in the propagator formulation. The hybrid GA-PS method successfully combines global exploration and local refinement, leading to robust and highly accurate estimates.

The GA-PS hybrid outperforms standalone GA and PS in both accuracy and robustness. GA performs better than PS due to its global search capability, particularly when the number of sources is small or the noise level is low. PS alone offers faster execution but is more susceptible to suboptimal convergence in high-dimensional or noisy scenarios. Furthermore, the performance gap between GA-PS and the other methods widens as the number of sources or the noise level increases, reinforcing the effectiveness of the hybrid optimization approach. The results clearly demonstrate the superiority of the proposed hybrid framework, making it a strong candidate for real-time DoA estimation in complex and noise-prone environments such as 6G, UAV, and radar applications.

This work presented evolutionary optimization methods, namely GA, PS and a hybrid GA-PS, for joint elevation/azimuth DOA estimation of multiple plane waves using a three-parallel ULA. We formulated a unified optimization framework for DOA estimation and instantiated three solvers. The hybrid GA-PS couples global exploration with local refinement, providing a practical balance between search breadth and precision.

The three methods deliver accurate DOA estimates, with the hybrid GA-PS consistently producing the smallest angle errors and the most stable convergence. Empirically, the hybrid converges faster than GA-only or PS-only and maintains computational feasibility as the number of sources increases, supported by early stopping and vectorized evaluations.

The approach is well-suited to real-time sensing and communication tasks, e.g., 6G/ISAC, UAV links, and radar tracking, where robust multi-source angle estimation is required.

Despite the promising results demonstrated by the proposed methodology, there are some limitations of our proposed method. First, the algorithm assumes that the number of antenna elements is greater than or equal to the number of signal sources. When the number of sources exceeds the number of array elements, the system becomes underdetermined, resulting in more unknown parameters than available equations. In such scenarios, accurate estimation of angles becomes infeasible due to rank deficiency in the array manifold matrix, thereby limiting the applicability of the proposed scheme. Second, the proposed model is developed under the far-field assumption. This approach does not account for near-field conditions, where the spherical nature of wavefronts introduces additional phase curvature across the sensors.

In future work, we will relax the canonical assumptions and evaluate the estimator under coherent/correlated sources, array-manifold perturbations, and colored noise, as well as extend the framework to joint arrival/departure estimation in MIMO settings and explore learning-assisted acceleration.