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Hybrid Continuous DoA Estimation with Shared-Radius Co-Prime Circular Arrays Keyvan Aghababaiyan, Member, IEEE
Abstract—This paper proposes a shared-radius co-prime circular array for high-resolution, continuous 2D Directionof-Arrival (DoA) estimation in 3D space, jointly estimating azimuth and elevation angles. The proposed architecture consists of two uniform circular sub-arrays with co-prime antenna counts sharing a common radius 𝑹R, a design that intrinsically suppresses mutual coupling leakage compared to dense uniform arrays. Unlike existing works that rely on complex phase-mode transformations to map circular structures to virtual linear arrays, we introduce a hybrid continuous-recovery framework operating directly in the physical spatial domain. By integrating a fast, discrete coarse-grid search with a swarm-intelligence continuous refinement stage, the proposed method completely bypasses discrete grid-mismatch limitations and computationally expensive eigenvalue decompositions. A rigorous theoretical analysis using Niven’s Theorem establishes the spatial uniqueness of the true source direction, effectively resolving phase ambiguities. Simulation results demonstrate that this hybrid scheme achieves superior resolution and lower Root Mean Square Error (RMSE) at low Signal-to-Noise Ratios (SNR) compared to uniform configurations, while asymptotically converging to the theoretical Cramér-Rao Bound (CRB) at high SNRs. Index Terms—Mutual coupling, co-prime, DoA estimation, circular array.
I. Introduction Direction of Arrival (DoA) estimation is a critical problem in wireless communication systems, particularly in environments with multiple narrow-band signal sources. In systems like massive multiple-input multiple-output (MIMO), DoA estimation is essential for effective downlink precoding and beamforming [1]. Over the years, numerous methods and array structures for DoA estimation have been proposed. Most of these methods focus on uniform array structures due to their simplicity and mathematical tractability. However, the use of non-uniform array structures has been shown to improve the resolution performance of DoA estimation algorithms [2]. To address the limitations of uniform arrays, several non-uniform linear array designs have been introduced, offering greater flexibility and enhanced resolution in DoA estimation [3], [4]. Despite the advantages of non-uniform linear arrays, the nested array structure proposed in [3] is susceptible to mutual coupling effects among closely spaced antennas, which can degrade performance. To mitigate this issue, co-prime arrays were introduced in [4] as a new non-uniform structure, providing a promising alternative that reduces mutual coupling while improving the estimation accuracy. The co-prime linear array proposed in [4] consists of two uniform linear sub-arrays, with 𝑀1 and 𝑀2 elements, where 𝑀1 M1 and 𝑀2 are co-prime integers. This co-prime linear array structure serves as a foundation for exploring various DoA estimation techniques Keyvan Aghababaiyan is a Marie Skłodowska-Curie Postdoctoral Fellow with Universidad Miguel Hernández de Elche, 03202 Elche, Spain (e-mail: [email protected]).
THIS IS AN AUTHOR-CREATED POSTPRINT VERSION
utilizing co-prime arrays, as discussed in [5]-[7]. In [5], the authors introduced a method based on a total angular field-ofview search, which combines Multiple Signal Classification (MUSIC) applied to the decomposed sub-arrays of the co-prime array. A search-free DoA estimation scheme for co-prime arrays was proposed in [6], based on a projection-like approach. The work in [7] presents the tri-shifted co-prime array which enhances uniform degrees of freedom and spatial efficiency, while reducing mutual coupling effects compared to existing co-prime designs. Although the works in [4]-[7] focus on linear co-prime arrays, they are inherently limited to two-dimensional environments. These schemes cannot be directly applied for DoA estimation in three-dimensional (3D) environments, which require the use of planar arrays to jointly estimate azimuth and elevation angles. A rectangular co-prime array was proposed in [8] to address the challenge of DoA estimation in 3D space using co-prime planar arrays, where large interelement spacing in each sub-array leads to phase ambiguity. Recently, advanced techniques such as tensor-modeling for EMVS-MIMO radars with sparse geometry [9] and fast 2DDoA algorithms for conformal MIMO arrays [10] have further highlighted the critical need for high-resolution, hardwareefficient spatial processing. Furthermore, parallel advancements in two-stage reconstruction for co-array estimation [11] and recursive root-finding for multiple parallel sparse arrays [12] continue to drive the demand for robust,
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interference-free architectures. In this paper, we propose a shared-radius co-prime circular array for high-resolution, continuous 2D-DoA estimation in 3D space. Unlike the rectangular co-prime arrays in [8], which are limited by planar geometry and susceptibility to phase ambiguities, our circular design provides full 360-degree azimuth coverage and intrinsically suppresses mutual coupling leakage. Furthermore, while recent co-prime circular designs, such as [13], utilize a shared-radius geometry, they typically require Phase Mode Excitation and Bessel function transformations to synthesize virtual linear arrays for subspacebased estimation like ESPRIT. These transformations introduce mathematical approximation errors and heavy computational overhead. In contrast, this paper proposes a novel hybrid continuous sparse-recovery framework operating directly in the physical spatial domain. By integrating a fast, discrete coarsegrid search with a swarm-based continuous refinement stage (PSO), our method completely bypasses beam-space mapping, computationally expensive eigenvalue decomposition (EVD), and the fundamental discrete grid-mismatch limitation. Ultimately, this approach delivers a highly precise, hardwareefficient solution that asymptotically converges to the theoretical CRB, making it ideal for real-time 3D spatial localization. II. SYSTEM MODEL In this paper, we present a non-uniform co-prime circular array structure that is formed from two uniform circular subarrays with 𝑁𝑖 Ni , 𝑖 𝜖 {1,2} antenna elements, where 𝑁1 and 𝑁2 are co-prime integers. To ensure a compact footprint and fair comparison, both sub-arrays share a common radius 𝑅, leading 𝜋 𝜆 to inter-element spacings 𝑑𝑖 = 2𝑅 sin ( ) that exceed , where 𝑁i
2
𝜆 denotes the wavelength of the incident narrow-band signals. The antenna elements of the 𝑖 𝑡ℎ sub-array are located at: 2𝜋 (1) 𝐿𝑖 = {(𝑅𝑐𝑜𝑠𝜃̃𝑛,𝑖 , 𝑅𝑠𝑖𝑛𝜃̃𝑛,𝑖 )|𝜃̃𝑛,𝑖 = 𝑛 , 𝑁𝑖 0 ≤ 𝑛 ≤ 𝑁𝑖 − 1 }. Hence, the locations of the antenna elements in the proposed co-prime circular array are in the set 𝐿 = 𝐿1 ∪ 𝐿2 . The antenna elements of the two sub-arrays do not overlap (except at the shared physical reference element), and the total number of antenna elements used in the proposed co-prime circular array is obtained as 𝑁𝑇 = 𝑁1 + 𝑁2 − 1. Fig. 1 shows the schematic of the co-prime circular array when 𝑁1 = 3 and 𝑁2 = 4. We assume that there are 𝑀 uncorrelated narrow-band signal sources from different directions impinging on the array antenna. The 𝑚𝑡ℎ signal source is located at an elevation angle 𝜑𝑚 , which is measured downward from the Z-axis, and an azimuth angle 𝜃𝑚 , measured counterclockwise from the X-axis, as shown in Fig. 1. Thus, the received signal vector at the proposed co-prime circular array can be described as [14]: (2) 𝐱(𝑡) = 𝑨(𝜃, 𝜑)𝐬(𝑡) + 𝐧(𝑡), where 𝐧(𝑡) is a noise vector which contains independent zeromean unit-variance Gaussian random variables. The term 𝐱(𝑡) can be rewritten as: (3) 𝐱(𝑡) = ∑𝑀 𝑚=1 𝒂(𝜃𝑚 , 𝜑𝑚 ) 𝑠𝑚 (𝑡) + 𝐧(𝑡), THIS IS AN AUTHOR-CREATED POSTPRINT VERSION
Fig. 1. Geometry of the proposed shared-radius co-prime circular array.
and the matrix 𝑨(𝜃, 𝜑) is the steering matrix, which is defined for different sources as (4) 𝑨(𝜃, 𝜑) = [𝒂(𝜃1 , 𝜑1 ), 𝒂(𝜃2 , 𝜑2 ), … . , 𝒂(𝜃𝑀 , 𝜑𝑀 )]. In (2), the term 𝐬(𝑡) is the signal sources data vector, and 𝑠𝑚 (𝑡) is the signal of the 𝑚𝑡ℎ source, where 𝐬(𝑡) is defined as: (5) 𝒔(𝒕) = [𝑠1 (𝑡), 𝑠2 (𝑡), … . , 𝑠𝑀 (𝑡)]𝑻 . The response of the 𝑛𝑡ℎ antenna element located at 𝜃̃𝑛 corresponding to the 𝑚𝑡ℎ source is defined by the steering vector element as: (6) 𝑎𝑛 (𝜃𝑚 , 𝜑𝑚 ) = 2𝜋 ̃ ) 𝑒𝑥𝑝 (𝑗 𝑅. 𝑠𝑖𝑛(𝜑𝑚 . 𝑐𝑜𝑠(𝜃𝑛 − 𝜃𝑚 )). 𝜆 Hence, the steering vector 𝒂(𝜃𝑚 , 𝜑𝑚 ), i.e., the response of all array elements corresponding to the 𝑚𝑡ℎ source, is obtained as: 𝑇 (7) 𝒂(𝜃𝑚 , 𝜑𝑚 ) = [𝑎1 (𝜃𝑚 , 𝜑𝑚 ), … , 𝑎𝑁𝑇 (𝜃𝑚 , 𝜑𝑚 )] . To resolve the mirror ambiguity inherent in planar arrays, the elevation angle is constrained to the upper hemisphere, i.e., 𝜋 𝜑𝑚 𝜖 [0, ]. 2
In practical array implementations, the electromagnetic interaction between closely spaced antenna elements, known as mutual coupling, can severely degrade the DoA estimation performance. For an array with 𝑁T elements, the received signal model incorporating mutual coupling is modified as 𝐱(𝑡) = 𝐂 𝑨(𝜃, 𝜑)𝐬(𝑡) + 𝐧(𝑡), where 𝐂 ∈ ℂ𝑁T ×𝑁T is the mutual coupling matrix (MCM) modeled based on standard impedance interactions [15]. For a circular geometry, the magnitude of the coupling coefficient between the 𝑝𝑡ℎ and 𝑞𝑡ℎ elements is inversely proportional to their physical distance 𝑑𝑝,𝑞 . Specifically, the entries of 𝐂 are often modeled as 𝐶𝑝,𝑞 = 𝑐1
𝜆 𝑑𝑝,𝑞
𝑒𝑥𝑝 (−𝑗
2𝜋 𝜆
𝑑𝑝,𝑞 ) for 𝑝 ≠ 𝑞, and 𝐶𝑝,𝑝 = 1, where 𝑐1 is a
complex constant depending on the antenna characteristics. A major structural advantage of the proposed shared-radius co-prime circular array is its inherent robustness against mutual coupling. In a standard dense Uniform Circular Array (UCA), fitting many elements into a constrained radius forces the inter𝜆 element spacing to be less than , resulting in large off-diagonal 2 elements in 𝐂. In contrast, our co-prime design intrinsically forces the inter-element spacing in each sub-array, given by
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𝜋
𝜆
𝑑𝑖 = 2𝑅 sin ( ), to be strictly greater than . Consequently, 𝑁 2 i
the off-diagonal coupling coefficients are substantially minimized, making the proposed array geometry highly resilient to mutual coupling errors in practical 3D physical deployments. Although interleaving sub-arrays introduces cross-subarray adjacent elements, the absolute minimum physical distance between any two elements is strictly 𝜋 mathematically bounded by 𝑑𝑚𝑖𝑛 = 2𝑅𝑠𝑖𝑛 ( ). By 𝑁1 𝑁2
appropriately selecting 𝑅, 𝑑𝑚𝑖𝑛 remains sufficiently large to prevent the deep-coupling failures seen in dense UCAs. To quantitatively validate this structural advantage, we evaluate the mutual coupling leakage energy. The severity of the mutual coupling can be mathematically assessed by measuring the normalized Frobenius norm of the off-diagonal matrix components, defined as: (8) ‖𝐂 − 𝐈𝑁𝑇 ‖ 𝐹 𝜉= , ‖𝐈𝑁𝑇 ‖ 𝐹 where 𝐈𝑁T is the 𝑁T × 𝑁T identity matrix representing an ideal, coupling-free scenario. By substituting the magnitude of the coupling coefficients into the norm, the leakage metric simplifies to: (9) 𝑁𝑇 2 1 𝜆 √∑ ∑ |𝑐1 𝜉= | . 𝑑𝑝,𝑞 √𝑁𝑇 𝑝=1 𝑞≠𝑝
In a dense UCA, the condition 𝑑𝑝,𝑞 ≤ terms |
𝜆 𝑑𝑝,𝑞
2
𝜆 2
causes the squared
| to grow significantly, leading to a high leakage 𝜆
energy. In contrast, the spatial constraint 𝑑𝑖 > in the proposed 2 co-prime geometry strictly bounds and suppresses these summation terms. As a result, 𝜉𝑐𝑜−𝑝𝑟𝑖𝑚𝑒 ≪ 𝜉𝑈𝐶𝐴 , ensuring that the mutual coupling matrix of the proposed array closely approximates the ideal identity matrix. While this theoretical model effectively captures the first-order distance-dependent geometric leakage, practical hardware implementations also necessitate full-wave electromagnetic calibration to account for complete 3D antenna radiation patterns and platform scattering. III. PROPOSED CONTINUOUS DOA ESTIMATION FRAMEWORK In this section, we introduce a high-resolution, hybrid continuous sparse recovery framework for 2D-DoA estimation using the proposed co-prime circular array. Traditional discrete methods inherently suffer from the grid-mismatch effect, as the true DoA rarely aligns perfectly with a predefined spatial grid. While increasing the grid resolution can mitigate this error, it exponentially inflates the computational complexity. To overcome this fundamental limitation without relying on computationally expensive EVD or complex phase-mode transformations, we propose a two-stage hybrid approach. Let 𝒂(𝜃, 𝜑) denote the array steering vector for a spatial coordinate. The first stage executes a fast, discrete correlation-matching process over a highly sparse, coarse 2D dictionary to identify the approximate vicinity of the signal source. Since each atom in this coarse dictionary corresponds to a unique coupled (𝜃, 𝜑) pair, the algorithm natively avoids the azimuth-elevation THIS IS AN AUTHOR-CREATED POSTPRINT VERSION
Algorithm I: Input: Coarse grid resolution ∆𝑔𝑟𝑖𝑑 = 2𝑜 , particles 𝑁𝑝 = 40, max iterations 𝑇𝑚𝑎𝑥 = 60. Output: Continuous DoA coordinates (𝜃, 𝜑). 1: Initialization Phase (Coarse Grid Search): 2: Construct coarse dictionary 𝑄 = {𝒂(𝜃𝑘 , 𝜑𝑘 )}𝐾 𝑘=1 , using the spatial steering vector 𝒂 and grid resolution ∆𝑔𝑟𝑖𝑑 . 3: Evaluate the coarse spatial spectrum: 𝑷(𝜃𝑘 , 𝜑𝑘 ) = 𝑅𝑒{𝒂𝐻 (𝜃𝑘 , 𝜑𝑘 )𝐑 𝑥 𝒂(𝜃𝑘 , 𝜑𝑘 )} 4: Identify the discrete seed coordinates: (𝜃𝑠𝑒𝑒𝑑 , 𝜑𝑠𝑒𝑒𝑑 ) = 𝑎𝑟𝑔𝑚𝑎𝑥 𝑷(𝜃𝑘 , 𝜑𝑘 ) 5: Initialize 𝑁𝑝 particles randomly within the continuous cell of the seed, setting the boundary limit 𝛿 =
∆𝑔𝑟𝑖𝑑
2 (0) (0) (0) 𝐗 𝑖 = [𝜃𝑖 , 𝜑𝑖 ]~𝓾(𝜃𝑠𝑒𝑒𝑑 ± 𝛿, 𝜑𝑠𝑒𝑒𝑑 ± 𝛿),
:
∀𝑖𝜖{1, … , 𝑁𝑝 } (0)
6: Initialize particle velocities 𝐕𝑖 = 0, and set personal (0) bests 𝐏𝑖 = 𝐗 𝑖 . 7: Determine the initial global best 𝐆𝑏𝑒𝑠𝑡 based on (0) max 𝑷(𝐗 𝑖 ). 𝑖
8: Optimization Phase (Continuous Refinement): 9: for 𝑡 = 1 to 𝑇𝑚𝑎𝑥 do 10: Update inertia weight 𝜔 (𝑡) using a linear decay mechanism. Linearly decay 𝜔 (𝑡) from 0.9 to 0.05. 11: for 𝑖 = 1 to 𝑁𝑝 do 12: Calculate the continuous fitness value: (𝑡) (𝑡) 𝐹𝑖 = 𝑅𝑒{𝒂𝐻 (𝐗 𝑖 )𝐑 𝑥 𝒂(𝐗 𝑖 )} 13: if 𝐹𝑖 > 𝑷(𝐏𝑖 ) then (𝑡) 14: 𝐏𝑖 ← 𝐗 𝑖 15: end if 16: end for 17: Update the swarm's global best: 𝐆𝑏𝑒𝑠𝑡 ← 𝑎𝑟𝑔𝑚𝑎𝑥𝐏𝑖 𝑷(𝐏𝑖 ). 18: for 𝑖 = 1 to 𝑁𝑝 do 19: Update velocity using cognitive/social coefficients (𝑐1 , 𝑐2 ), 𝑐1 = 𝑐2 =1.49, and uniformly distributed random variables 𝑟1 , 𝑟2 ~𝓊(0,1): (𝑡) (𝑡) (𝑡+1) 𝐕𝑖 = 𝜔 (𝑡) 𝐕𝑖 + 𝑐1 𝑟1 (𝐏𝑖 − 𝐗 𝑖 ) (𝑡) + 𝑐2 𝑟2 (𝐆𝑏𝑒𝑠𝑡 − 𝐗 𝑖 ) 20: Update the position vector to explore the continuous domain: (𝑡+1) (𝑡) (𝑡+1) 𝐗𝑖 = 𝐗 𝑖 + 𝐕𝑖 (𝑡+1) 21: Apply angular boundary constraints to 𝐗 𝑖 . 22: end for 23: end for 24: return Final estimated continuous DoA coordinates: (𝜃̂, 𝜑̂) = 𝐆𝑏𝑒𝑠𝑡 . pairing ambiguity for multiple sources. The spatial index that maximizes this initial spectrum serves as the initial seed. In the second stage, Particle Swarm Optimization (PSO) is deployed to evaluate the continuous spatial domain exclusively around this seed. By restricting the initial swarm boundary 𝛿 to half the
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∆𝑔𝑟𝑖𝑑
coarse grid resolution 𝛿 = the particles strictly search the 2 continuous cell containing the peak. The swarm particles dynamically adjust their velocities and positions to maximize the spatial spectrum directly in the physical domain. This continuous refinement completely shatters the discrete error floor, allowing the estimation to tightly converge to the exact continuous DoA coordinates. The complete procedure is summarized in Algorithm I. IV. DOA ESTIMATION WITH PROPOSED CO-PRIME ARRAY DoA estimation schemes for arbitrary array structures are feasible but tend to exhibit high complexity when the array structure is non-uniform. To address this, we propose a method for the proposed co-prime circular array, utilizing the uniform structure of each sub-array and combining their spatial spectra. Consider the two sub-arrays, each a uniform circular array, sharing a common radius 𝑅 and a physical shared element at 𝜃 = 0, as shown in Fig. 1. The steering vector corresponding to the 𝑚𝑡ℎ signal source for the 𝑖 𝑡ℎ sub-array is defined as: 𝑇 (10) 𝑎 (𝜃 , 𝜑 ) = [𝑎 (𝜃 , 𝜑 ), … , 𝑎 (𝜃 , 𝜑 )] , 𝑖
𝑚
𝑚
1,𝑖
𝑚
𝑚
where 𝑎𝑛,𝑖 (𝜃𝑚 , 𝜑𝑚 ) = exp (𝑗.
𝑁𝑖 ,𝑖
2𝜋𝑅 𝜆
𝑚
𝑚
. sin(𝜑𝑚 ) cos(𝜃𝑛,𝑖 − 𝜃𝑚 )).
The received signal by the 𝑖 𝑡ℎ sub-array is given by: (11) 𝑥𝑖 (𝑡) = ∑𝑀 𝑚=1 𝑎𝑖 (𝜃𝑚 , 𝜑𝑚 ) 𝑠𝑚 (𝑡) + 𝑛(𝑡). 𝜆 When the inter-element spacing is , the spatial spectrum 2 𝑷(𝜃, 𝜑) has a single peak with low resolution and no ambiguity. However, as the shared radius 𝑅 increases, the spacing beyond 𝜆 , 𝑷(𝜃, 𝜑) exhibits multiple sharper peaks. Thus, a single source 2 generates higher resolution at the cost of phase ambiguity. If a signal emitter exists at (𝜃𝑚 , 𝜑𝑚 ), the true phase difference between signals received by two adjacent elements in sub-array 𝑖 can be derived using the trigonometric identity cos 𝐴 − 𝐴+𝐵 𝐴−𝐵 ̅ = 𝜃𝑛+1,𝑖 +𝜃𝑛,𝑖 be 𝑐𝑜𝑠 𝐵 = −2 𝑠𝑖𝑛 ( ) 𝑠𝑖𝑛 ( ). Letting 𝜃𝑛,𝑖 2 2 2 the midpoint angle between adjacent elements, the wrapped measured phase difference is: 4𝜋𝑅 𝜋 (12) ̅ 𝛥𝜑𝑛,𝑖 = 𝑚𝑜𝑑(− 𝑠𝑖𝑛 ( ) 𝑠𝑖𝑛(𝜑𝑚 )𝑠𝑖𝑛(𝜃𝑛,𝑖 𝜆 𝑁𝑖 − 𝜃𝑚 ), 2𝜋). The chordal inter-element spacing 𝑑𝑖 between two adjacent elements is geometrically related to the radius 𝑅 by 𝑑𝑖 = 𝜋 2𝑅𝑠𝑖𝑛 ( ). By substituting this relation, the relationship 𝑁𝑖
between the true unwrapped phase and the measured wrapped phase difference is defined as: (13) Δ𝜑𝑛,𝑖 + 2𝑘𝑖 𝜋 = 2𝜋𝑑𝑖 ̅ − 𝜃𝑚 ), − 𝑠𝑖𝑛(𝜑𝑚 )𝑠𝑖𝑛(𝜃𝑛,𝑖 𝜆 where 𝑘𝑖 is the integer phase ambiguity. To prevent mirror ambiguity across the planar surface, we restrict the elevation to the upper hemisphere, 𝜑𝑚 𝜖[0, 𝜋/2]. Given the constraints ̅ − 𝜃𝑚 ) ≤ 𝜃𝑚 𝜖[0,2𝜋], we conclude that −1 ≤ 𝑠𝑖𝑛(𝜑𝑚 )𝑠𝑖𝑛(𝜃𝑛,𝑖 1. This directly provides the bounding range for the ambiguity integer 𝑘𝑖 : 𝑑𝑖 Δ𝜑𝑛,𝑖 𝑑𝑖 Δ𝜑𝑛,𝑖 (14) 𝑘𝑖 𝜖 [− − , − ]. 𝜆 2𝜋 𝜆 2𝜋 Because the common radius 𝑅 is large, the spacing 𝑑𝑖 causes THIS IS AN AUTHOR-CREATED POSTPRINT VERSION
𝑃𝑖 (𝜃, 𝜑) for each sub-array to display multiple grating lobes. To resolve this, we present the following uniqueness theorem. Note that unlike linear co-prime arrays where phase shifts are 1D, our circular geometry introduces a highly non-linear coupled phase term 𝑠𝑖𝑛(𝜑𝑚 )𝑠𝑖𝑛(𝜃̅ − 𝜃𝑚 ), necessitating a 2D geometric proof. Theorem I: Assume (𝜃𝑚 , 𝜑𝑚 ) is the actual direction of the 𝑚𝑡ℎ signal source, leading to multiple ambiguous peaks in 𝑃𝑖 (𝜃, 𝜑). By performing joint spatial processing on the spectra of both sub-arrays, the actual direction (𝜃𝑚 , 𝜑𝑚 ) emerges as the strictly unique common peak. Existence: Since (𝜃𝑚 , 𝜑𝑚 ) is the true physical direction of the impinging source, the steering vector naturally perfectly aligns at this coordinate. Consequently, 𝑃1 (𝜃, 𝜑) and 𝑃2 (𝜃, 𝜑) will inherently both exhibit a peak at (𝜃𝑚 , 𝜑𝑚 ). Uniqueness: Assume there are two distinct spatial peaks at directions (𝜃̂𝑚,1 , 𝜑̂𝑚,1 ) and (𝜃̂𝑚,2 , 𝜑̂𝑚,2 ) that appear in the power spectrum of both sub-arrays. Based on (16), the condition for these two directions to yield the same wrapped phase in the 𝑖 𝑡ℎ sub-array requires their spatial deviation to equal an integer multiple of the wavelength: 𝑠𝑖𝑛(𝜑̂𝑚,1 )𝑠𝑖𝑛(𝜃̅ − 𝜃̂𝑚,1 ) (15) − 𝑠𝑖𝑛(𝜑̂𝑚,2 )𝑠𝑖𝑛(𝜃̅ − 𝜃̂𝑚,2 ) 𝐾𝑖 𝜆 𝐾𝑖 𝜆 = = 𝜋 . 𝑑𝑖 2𝑅 𝑠𝑖𝑛 (𝑁 ) 𝑖
For these ambiguous peaks to perfectly overlap and form a common false peak in both sub-arrays, the spatial deviation must be identical for both geometries. Equating the right side of (18) for 𝑖 = 1 and 𝑖 = 2 yields 𝐾1 𝜆 𝐾2 𝜆 (16) = . 2𝑅 𝑠𝑖𝑛(𝜋/𝑁1 ) 2𝑅 𝑠𝑖𝑛(𝜋/𝑁2 ) This simplifies to the ratio: 𝐾1 𝑠𝑖𝑛(π/𝑁1 ) (17) = . 𝐾2 𝑠𝑖𝑛(π/𝑁2 ) According to Niven's Theorem [16], for co-prime integers 𝑁1 , 𝑁2 > 2 the sines of these rational multiples of 𝜋 are strictly irrational. Thus, any non-zero pair of ambiguity integers {𝐾1 , 𝐾2 } generates a highly specific discrete irrational constant on the right side of the equation. Since the independent sources in a 3D environment are continuous random variables, the mathematical probability of their spatial deviation perfectly matching this specific measure-zero discrete contour is strictly zero. Consequently, non-overlapping grating lobes systematically fail to coincide, leaving only the true direction 𝐾1 = 𝐾2 = 0. Substituting zero into (15( leads to: (18) 𝑠𝑖𝑛(𝜑̂𝑚,1 ) 𝑠𝑖𝑛(𝜃̅ − 𝜃̂𝑚,1 ) ̅ ̂ = 𝑠𝑖𝑛(𝜑̂𝑚,2 )𝑠𝑖𝑛(𝜃 − 𝜃𝑚,2 ) . Because this equality must hold across all adjacent element pairs (various 𝜃̅), we can definitively conclude that 𝜃̂𝑚,1 = 𝜃̂𝑚,2 and 𝜑̂𝑚,1 = 𝜑̂𝑚,2 . Thus, the true source direction is uniquely identifiable. Combining Rule (Coincidence Detection): To practically apply Theorem I, we implement a coincidence detection rule. Let Ω1 and Ω2 be the sets of estimated peak coordinates extracted from 𝑃1 (𝜃, 𝜑) and 𝑃2 (𝜃, 𝜑), respectively. The final
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2D-DoA is resolved by finding the intersection of these sets: (𝜃𝑚 , 𝜑𝑚 ) = Arg min ‖𝑢 − 𝑣‖2 , (19) 𝑢𝜖Ω1 , 𝑣𝜖Ω2
where 𝑢 and 𝑣 represent the estimated 2D angular coordinate pairs (i.e., (𝜃, 𝜑)) belonging to the peak sets, and ‖. ‖2 denotes the Euclidean distance. Non-overlapping peaks exceeding the grid resolution tolerance are systematically discarded as grating lobes. To evaluate the theoretical precision of the proposed sharedradius co-prime circular array, we derive the CRB. For an unbiased estimator, the CRB provides a strict lower bound on the error variance. Note that the CRB is deliberately derived under idealized, coupling-free conditions (𝐂 = 𝐈𝑁𝑇 ) to establish the absolute theoretical lower bound for the geometric spatial efficiency of the array. Assuming a multivariate Gaussian noise model, the covariance matrix of the received signal is: (20) 𝐑 𝑥 = 2𝜎 2 𝑆𝑁𝑅𝐚(𝜃, 𝜑)𝐚(𝜃, 𝜑)𝐻 + 𝐈. At high SNR or with a large number of snapshots 𝐿, the Fisher Information Matrix simplifies significantly if the orthogonality conditions 𝐚𝐻 𝐚𝜃̇ = 0 and 𝐚𝐻 𝐚𝜑̇ = 0 hold. For our specific geometry, the first derivatives of the array manifold for the 𝑛𝑡ℎ element in the 𝑖 𝑡ℎ sub-array are: 2𝜋 (21) 𝐚𝜃̇ (𝜃, 𝜑) = 𝑗 𝑅𝑠𝑖𝑛𝜑𝑠𝑖𝑛 (𝜃𝑛,𝑖 − 𝜃)𝐚(𝜃, 𝜑) 𝜆 2𝜋 (22) 𝐚𝜑̇ (𝜃, 𝜑) = 𝑗 𝑅𝑐𝑜𝑠𝜑𝑐𝑜𝑠 (𝜃𝑛,𝑖 − 𝜃)𝐚(𝜃, 𝜑). 𝜆 Evaluating the inner product for the azimuth angle yields: 2 𝑁𝑖 −1 (23) 2𝜋𝑅 𝐻 𝐚 𝐚𝜃̇ = 𝑗 𝑠𝑖𝑛𝜑 ∑ ∑ 𝑠𝑖 𝑛(𝜃𝑛,𝑖 − 𝜃). 𝜆 𝑖=1 𝑛=0
Because the elements of each uniform circular sub-array are 2𝜋𝑛 symmetrically distributed over 2𝜋 (i.e., 𝜃𝑛,𝑖 = ), the sum of 𝑁𝑖
sines and cosines around the full circle strictly evaluates to zero. Thus, the orthogonality conditions are intrinsically satisfied due to the circular symmetry, without requiring mathematical coordinate shifts. Consequently, the CRB equations decouple, and the theoretical bounds are inversely proportional to the squared norms of the derivative vectors: 1 (24) 𝐶𝑅𝐵𝜃 ≈ 4𝐿𝜎 2 𝑆𝑁𝑅|𝐚̇ 𝜃 |2 1 = , 2𝜋𝑅 2 4𝐿𝜎 2 𝑆𝑁𝑅 ( ) 𝑠𝑖𝑛2 𝜑. 𝑑̅ 2 𝜆 1 (25) 𝐶𝑅𝐵𝜑 ≈ 2 2 4𝐿𝜎 𝑆𝑁𝑅|𝐚̇ 𝜑 | 1 = , 2𝜋𝑅 2 4𝐿𝜎 2 𝑆𝑁𝑅 ( ) 𝑐𝑜𝑠 2 𝜑. 𝑑̃ 2 𝜆 where the geometric dispersion factors are defined directly by the discrete spatial sampling of the co-prime subsets: 2 𝑁𝑖 −1 (26) 2 ̅ 𝑑 = ∑ ∑ 𝑠𝑖𝑛2 (𝜃𝑛,𝑖 − 𝜃), 𝑖=1 𝑛=0 2 𝑁𝑖 −1
(27)
𝑑̃ 2 = ∑ ∑ 𝑐𝑜𝑠 2 (𝜃𝑛,𝑖 − 𝜃). 𝑖=1 𝑛=0
These expressions mathematically demonstrate that the THIS IS AN AUTHOR-CREATED POSTPRINT VERSION
estimation accuracy of the proposed structure is fundamentally driven by the shared physical radius 𝑅. By utilizing the coprime property, the array can employ a strictly larger 𝑅 without ambiguity, thereby lowering the CRB significantly compared to standard constrained circular arrays. V. EVALUATION To validate the performance of the proposed shared-radius co-prime circular array and the hybrid continuous refinement scheme, we conducted extensive numerical simulations. The simulation parameters are set as follows: the coarse search grid resolution is 2𝑜 for both azimuth and elevation angles, the number of snapshots is 𝐿 = 500. To guarantee a fair comparison, the normalized circular-array radius is uniformly set to 𝑅 = 0.55𝜆, and the competing Uniform Circular Array (UCA) operates with the exact same physical radius and total number of antenna elements (𝑁𝑇 = 6) as the proposed co-prime array. For the continuous swarm optimization stage, the number of particles is set to 40 with a maximum of 60 learning iterations. The PSO parameters (𝑐1 = 𝑐2 =1.49, 𝜔 ∈ [0.9, 0.05]) were selected empirically for optimal convergence. Execution times and RMSE are averaged over 30 independent Monte Carlo runs. Before presenting the numerical results, it is essential to analyze the computational complexity. The complexity of subspace-based methods like 2D-MUSIC is dominated by the EVD of the covariance matrix, which requires 𝑂(𝑁𝑇3 ) operations, and the exhaustive 2D fine-grid search, adding 𝑂(𝑁𝜃 . 𝑁𝜑 . 𝑁𝑇2 ), where 𝑁𝜃 and 𝑁𝜑 are the number of fine-grid points for azimuth and elevation, and 𝑁𝑇 is the total number of antennas. The transformation-based ESPRIT method [13] avoids the 2D grid search but requires complex Bessel function mapping and EVD, yielding a complexity of 𝑂(𝑁𝑇3 + 𝑀3 ). In contrast, the proposed hybrid framework evaluates a significantly reduced coarse grid with a complexity of 𝑂(𝑁𝜃,𝑐 . 𝑁𝜑,𝑐 . 𝑁𝑇 ), where 𝑁𝜃,𝑐 and 𝑁𝜑,𝑐 denote the number of search points in the coarse azimuth and elevation grids, respectively. It mathematically refines the estimate via swarm intelligence, requiring only 𝑂(𝑇𝑚𝑎𝑥 . 𝑁𝑝 . 𝑁𝑇 )operations, where 𝑇𝑚𝑎𝑥 is the maximum number of iterations and 𝑁𝑝 represents the total number of swarm particles. By completely bypassing the computationally expensive EVD and exhaustive fine-grid evaluations, and by minimizing 𝑁𝑇 via our co-prime design (𝑁𝑇 = 6 for 𝑁1 = 3, 𝑁2 = 4), the computational burden is drastically reduced. To validate our theoretical complexity analysis, we evaluated the average execution time of the algorithms. We evaluated the average execution time using MATLAB on an Intel Core i51335U CPU (1.30 GHz) with 16 GB RAM. The proposed hybrid scheme operates directly in the physical domain and bypasses EVD entirely. Due to the rapid convergence of the swarm refinement stage, it maintains an exceptionally low average execution time of approximately 0.15 seconds. In contrast, the transformation-based ESPRIT [13] requires 0.85 seconds, and the standard fine-grid 2D-MUSIC algorithm takes 1.45 seconds. This confirms that the proposed framework is approximately an order of magnitude faster than 2D-MUSIC and significantly faster than ESPRIT, making it highly efficient
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Fig. 2. Convergence behavior of the angle’s estimation error over learning episodes across various SNR levels.
and practical for real-time continuous 2D-DoA estimation in 3D space. To evaluate the computational efficiency and convergence of the proposed continuous swarm-based refinement stage, we analyze its behavior across various Signal-to-Noise Ratio (SNR) regimes. Fig. 2 illustrates the convergence trends for the angles’ estimations, measured as absolute estimation error versus the number of learning episodes. The coarse grid resolution of 2° was empirically chosen as an optimal trade-off; a larger grid reduces the initial search cost but negatively
Fig. 3. RMSE performance comparison of the azimuth angle estimation versus SNR for different algorithms.
Fig. 4. RMSE performance comparison of the elevation angle estimation versus SNR for different algorithms.
Fig. 5. RMSE performance comparison versus the number of snapshots.
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impacts the PSO convergence speed. Furthermore, to account for the stochastic nature of swarm initialization, all convergence trends are averaged over 30 independent runs. A critical observation across all SNR levels is the rapid convergence speed. The swarm particles successfully locate the true continuous DoA coordinates within the first 10 to 15 episodes, after which the error sharply drops to a stable steady state. This rapid stabilization dictates that a very small maximum iteration threshold (𝑇𝑚𝑎𝑥 ) is sufficient to reach optimal accuracy, keeping the computational footprint minimal and making the framework highly suitable for real-time applications. Furthermore, the steady-state residual error exhibits a strong correlation with the SNR. In severe noise conditions (-10 dB), the estimation error stabilizes at approximately 1.78° for azimuth and 2.05° for elevation. However, for high-SNR scenarios (15 dB to 25 dB), the swarm intelligence effectively eliminates the grid-mismatch limitation, driving the steadystate error down to near-zero values (below 0.05°). This robust profile confirms that the continuous refinement stage successfully learns and tracks precise off-grid DoA coordinates without the exhaustive computational burden of fine-grid subspace searches. To evaluate the overall estimation accuracy and robustness, the Root Mean Square Error (RMSE) for both angles was analyzed across an SNR range of -10 dB to 25 dB. The proposed hybrid method is benchmarked against standard 2D-MUSIC [13], transformation-based ESPRIT, and an Equal-Radius Uniform Circular Array (UCA) with the identical number of antenna elements. The theoretical CRB is also plotted as the absolute benchmark for statistical efficiency. As depicted in Fig. 3 and Fig. 4, the Equal-Radius UCA exhibits the poorest performance across the entire spectrum. Despite having the same physical footprint and number of elements as the co-prime array, its uniform inter-element spacing significantly exceeds the half-wavelength threshold. This fundamentally violates spatial Nyquist sampling, resulting in unresolvable grating lobes and severe phase ambiguities. Consequently, the UCA's RMSE prematurely plateaus at approximately 1.7°, completely failing to benefit from higher SNR conditions. Furthermore, 3D spatial-spectrum analysis confirms that while standard equalradius UCAs suffer from severe grating lobes across the elevation-azimuth plane, the proposed co-prime geometry successfully suppresses these spatial ambiguities, yielding a
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Fig. 6. Impact of increasing the physical antenna count on the estimation error for the baseline UCA compared to the proposed co-prime array.
Fig. 7. Mutual coupling leakage energy comparison between the proposed co-prime circular array and the equal-radius UCA as a function of the number of array elements.
strictly unique spectral peak. The ESPRIT method using co-prime array demonstrates significant vulnerability, particularly in the low-SNR regime (10 dB), where its error spikes to nearly 6 degrees for elevation and 5.5 degrees for azimuth. Although its accuracy improves as SNR increases, it consistently underperforms compared to MUSIC and the proposed method, eventually plateauing near 0.8 degrees. This performance gap is primarily attributed to the mathematical approximation errors inherent in Phase Mode Excitation and Bessel function transformations required to map the circular array manifold into a virtual linear domain. On the other hand, the 2D-MUSIC algorithm provides stable and reliable estimations, showing a rapid decline in RMSE at lower SNRs. However, at higher SNRs (above 5 dB), the MUSIC curve clearly hits an error floor and plateaus at approximately 0.4 degrees. This flatlining is a direct consequence of the discrete grid-mismatch error. Because standard MUSIC evaluates the spectrum over a fixed, predefined spatial grid, it is mathematically bounded by the grid's resolution and cannot pinpoint the exact off-grid source location, regardless of how clean the signal becomes. In contrast, the proposed hybrid continuous method outperforms all baseline methods. In the harsh low-SNR environment (-10 dB), it achieves the lowest estimation error, showcasing its robust resilience to noise. More importantly, as the SNR increases, the proposed method overcomes the discrete error floor that traps standard MUSIC. Driven by the swarm-based continuous refinement stage, the algorithm successfully navigates the off-grid spatial domain, driving the RMSE down continuously. At high SNRs (15 dB to 25 dB), the proposed method tightly converges toward the theoretical CRB, reaching near-zero error. This asymptotic efficiency mathematically validates that the proposed THIS IS AN AUTHOR-CREATED POSTPRINT VERSION
Fig. 8. Azimuth RMSE comparison between the proposed Coprime Circular Array (CCA) and the Co-prime Rectangular Array (CRA).
Fig. 9. Elevation RMSE comparison between the proposed Coprime Circular Array (CCA) and the Co-prime Rectangular Array (CRA).
framework fully resolves co-prime ambiguities while extracting the absolute maximum spatial information available in the physical domain. Furthermore, Fig. 5 evaluates the estimation stability under low-snapshot scenarios (evaluated at SNR = 0 dB). The proposed hybrid method demonstrates robust performance even with as few as 𝐿 = 15 snapshots, significantly outperforming subspace baselines like MUSIC, which inherently suffer from subspace breakdown and require large sample sizes to reliably estimate the covariance matrix. To further emphasize the hardware efficiency and superior resolving capability of the proposed architecture, we investigate the impact of increasing the physical antenna count in the baseline UCA. Fig. 6 illustrates the azimuth and elevation RMSE of the Equal-Radius UCA as its element count scales from 6 to 10, benchmarked against the constant performance of the proposed 6-element co-prime circular array at moderate (0 dB) and high (10 dB) SNR levels. As expected, densely packing more elements into the constrained UCA geometry gradually improves its estimation accuracy by increasing the spatial sampling density. However, the results reveal a striking structural performance gap. Even when the UCA is heavily populated with 10 antenna elements, its estimation error remains fundamentally worse than that of the proposed coprime array, which operates with only 6 physical elements. For instance, under a 10 dB SNR scenario, the 10-element UCA struggles to breach the 0.5° RMSE threshold, whereas the proposed co-prime array achieves a highly precise error margin well below 0.2°. This proves that the severe phase ambiguities and grating lobes inherent to uniformly spaced circular arrays cannot be effectively mitigated merely by adding more hardware. By strategically exploiting non-uniform co-prime spacing, the proposed framework inherently synthesizes a larger virtual aperture with enhanced degrees of freedom, thereby delivering significantly superior 3D-DoA estimation
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accuracy while drastically reducing hardware cost, RF chain complexity, and array footprint. To quantitatively validate the architecture's robustness against electromagnetic interaction, the mutual coupling leakage energy (𝜉) is analyzed as a function of the total array elements. Fig. 7 compares the proposed Co-prime Circular Array with a standard Equal-Radius Uniform Circular Array (UCA) as the element count scales from 6 to 12. While densely packing elements into a fixed radius naturally increases mutual coupling, the UCA experiences severe degradation; its leakage energy spikes from approximately 3.1 (6 elements) to over 6.2 (12 elements) as uniform spacing drops below the halfwavelength threshold. Conversely, the proposed co-prime array demonstrates remarkable resilience. By strategically distributing antennas into co-prime subsets, it inherently maintains wider inter-element spacing. Consequently, its leakage energy grows at a substantially slower rate, remaining below 3.4 even with 12 elements. To evaluate its geometric superiority, the Root Mean Square Error (RMSE) of the proposed 11-element Co-prime Circular Array (CCA) is compared against a 12-element Co-prime Rectangular Array (CRA) across an SNR range of -10 dB to 20 dB in Fig. 8 and Fig. 9. The CCA consistently outperforms the CRA across the entire spectrum despite utilizing fewer hardware elements. This advantage is particularly pronounced in low-SNR environments; for instance, at -10 dB, the CCA yields an azimuth RMSE of approximately 1.05 degrees compared to 1.9 degrees for the CRA, with similar gaps in elevation accuracy. This performance discrepancy stems from the inherent structural symmetry and 360-degree coverage of the circular geometry, which effectively suppresses the phase ambiguities and spatial overlaps that severely degrade flat rectangular arrays. Furthermore, at higher SNRs, the CCA's error converges toward near-zero values at a much steeper rate. These results indicate that the proposed CCA architecture is both more hardware-efficient and significantly more precise for 3D DoA estimation. VI. CONCLUSION This paper proposed a high-resolution, low-complexity continuous 2D-DoA estimation framework using a sharedradius co-prime circular array. By integrating a coarse physicaldomain search with a swarm-intelligence continuous refinement stage (PSO), we successfully resolved phase ambiguities and eliminated discrete grid-mismatch limitations without requiring complex beam-space transformations or computationally expensive eigenvalue decompositions. Mathematical proof via Niven’s Theorem confirmed the spatial uniqueness of the true DoA. Furthermore, quantitative analysis validated that the proposed co-prime geometry inherently suppresses mutual coupling leakage compared to dense uniform arrays. Ultimately, the proposed hybrid approach offers a highly efficient alternative that maintains superior accuracy at low SNRs and asymptotically converges to the theoretical Cramér-Rao Bound at high SNRs, making it an ideal, hardwareefficient solution for real-time 3D spatial localization. Despite these structural advantages, practical deployments must also address cross-subarray mutual coupling at the interleaving THIS IS AN AUTHOR-CREATED POSTPRINT VERSION
points, which typically requires full-wave electromagnetic calibration. REFERENCES [1] A. Patra, B. K. Sahu, and A. K. Sahu, “Coprime MIMO configurations for enhanced DoA estimation,” IEEE Transactions on Signal Processing, vol. 73, no. 1, pp. 33–46, Jan. 2025. [2] K. Aghababaiyan, V. Shah-Mansouri, and B. Maham, “High-precision OMP-based direction-of-arrival estimation scheme for hybrid non-uniform arrays,” IEEE Communications Letters, vol. 24, no. 2, pp. 354–357, Feb. 2020. [3] P. Pal and P. P. Vaidyanathan, “Nested arrays: A novel approach to array processing with enhanced degrees of freedom,” IEEE Transactions on Signal Processing, vol. 58, no. 8, pp. 4167–4181, Aug. 2010. [4] P. P. Vaidyanathan and P. Pal, “Sparse sensing with co-prime samplers and arrays,” IEEE Transactions on Signal Processing, vol. 59, no. 2, pp. 573–586, Feb. 2011. [5] C. Zhou, Z. Shi, Y. Gu, and X. Shen, “Decom: DoA estimation with combined MUSIC for coprime arrays,” in Proc. IEEE Int. Conf. Wireless Communications and Signal Processing (WCSP), Hangzhou, China, Oct. 2013, pp. 1–5. [6] Z. Weng and P. M. Djurić, “A search-free DoA estimation algorithm for coprime arrays,” Digital Signal Processing, vol. 24, pp. 27–33, Jan. 2014. [7] A. H. Shaikh and X. Liu, “TS-CPA: A novel family of tri-shifted coprime arrays,” IEEE Transactions on Vehicular Technology, early access, 2025. [8] D. Zhang, Y. Zhang, G. Zheng, B. Deng, C. Feng, and J. Tang, “Twodimensional direction-of-arrival estimation for coprime planar arrays via polynomial root-finding technique,” IEEE Access, vol. 6, pp. 19540–19549, 2018. [9] X. Wang et al., “EMVS-MIMO radar with sparse Rx geometry: Tensor modeling and 2-D direction finding,” IEEE Transactions on Aerospace and Electronic Systems, vol. 59, no. 6, pp. 8062–8075, 2023. [10] F. Wen et al., “Fast DOD/DOA estimation for massive conformal MIMO arrays with unknown gain-phase errors,” IEEE Transactions on Wireless Communications, vol. 25, pp. 15866–15878, 2026. [11] Y. Yue et al., “Two-stage reconstruction for co-array 2D DOA estimation of mixed circular and noncircular signals,” IEEE Transactions on Vehicular Technology, vol. 74, no. 7, pp. 10407–10421, 2025. [12] Y. Yue et al., “2D DOA estimation for multiple parallel sparse linear arrays via recursive roots finding,” IEEE Transactions on Vehicular Technology, vol. 74, no. 11, pp. 18226–18231, 2025. [13] P. Li, J. Li, C. Ye, and X. Zhang, “Low-complexity DoA estimation using coprime circular arrays,” in Proc. IEEE 5th Int. Conf. Signal and Image Processing (ICSIP), pp. 592–597, 2020. [14] M. Dehghani and K. Aghababaiyan, “FOMP algorithm for direction-ofarrival estimation,” Physical Communication, vol. 26, pp. 170–174, Feb. 2018. [15] B. Friedlander and A. J. Weiss, “Direction finding in the presence of mutual coupling,” IEEE Transactions on Antennas and Propagation, vol. 39, no. 3, pp. 273–284, 1991. [16] I. Niven, Irrational Numbers. Washington, DC, USA: Mathematical Association of America, 1956. Keyvan Aghababaiyan received the B.Sc. degree (Hons.) in Communication Engineering from Amirkabir University of Technology, Tehran, Iran, in 2011, the M.Sc. degree in Communication Engineering from Sharif University of Technology, Tehran, Iran, in 2013, and the Ph.D. degree in Communication Engineering from the University of Tehran, Tehran, Iran, in 2019. During his academic career, he was awarded multiple prestigious fellowships from the Iranian National Elite Foundation, received consecutive Best Ph.D. Student Awards, and graduated as the top-ranked Bachelor's student in his cohort. From 2020 to 2022, he was a Postdoctoral Researcher at the University of Tehran, supported by the Iranian National Elite Foundation. He is currently a Marie Skłodowska-Curie Postdoctoral Fellow at the Networked Systems Lab, Universidad Miguel Hernández de Elche (UMH), Spain. His research interests include deterministic networking, intelligent resource allocation, and next-generation wireless communication systems.