3D Spherical Directly-Connected Antenna Array for Low-Altitude UAV Swarm ISAC

1 3D Spherical Directly-Connected Antenna Array for Lo w-Altitude U A V Swarm ISA C Haoyu Jiang, Zhenjun Dong, Zhiwen Zhou, Y ong Zeng, F ellow , IEEE Abstract —Recently a novel multi-antenna architectur e termed ray antenna array (RAA) was proposed, where several simple uniform linear arrays (sULAs) are arranged in a ray-like structure to enhance communication and sensing performance. By eliminating the need for phase shifters, it also significantly reduces hardwar e costs. Howev er , RAA is prone to signal blockage and has no elevation angle r esolution capability in three-dimensional (3D) scenarios. T o address such issues, in this paper we propose a novel spherical directly-connected antenna array (DCAA), which composes of multiple simple uniform planar arrays (sUP As) placed over a spherical surface. All elements within each sUP A are directly connected. Compared to conventional arrays with hybrid analog/digital beamforming (HBF), DCAA significantly r educes hard ware cost, impro ves energy f ocusing, and provides superior and unif orm angular res- olution f or 3D space. These advantages make DCAA particularly suitable for integrated sensing and communication (ISA C) in low-altitude unmanned aerial vehicles (U A V) swarm scenarios, where targets may frequently move away from the boresight of traditional arrays, degrading both communication and sensing performance. Simulation results demonstrate that the proposed spherical DCAA achieves significantly better angular resolution and higher spectral efficiency than conv entional array with HBF , highlighting its strong potential for U A V swarm ISA C systems. Index T erms —RAA, DCAA, XL-MIMO, low-altitude U A V , ISA C, U A V swarm. I . I N T R O D U C T I O N L O W -AL TITUDE unmanned aerial vehicle (U A V) swarms, characterized by their massiv e number , collectiv e intelligence, and collaborativ e operation, are expected to play an important role in future lo w-altitude economy [1]–[5]. By ov ercoming the inherent limitations of single UA V in terms of coverage, payload, and endurance, swarms can significantly expand the application horizons of aerial platforms, enabling adv anced services such as large-scale coordinated delivery , environmental monitoring, and smart city infrastructure management [6]–[8]. Howe ver , the expected rapid increase in aerial traf fic density and operational complexity within low-altitude airspace creates stringent requirements for both reliable communication and accurate sensing [9], [10]. This necessity has sparked This work was supported by the National Natural Science Foundation of China under Grant 62571116, and Natural Science Foundation for Distin- guished Y oung Scholars of Jiangsu Pro vince with grant number BK20240070. H. Jiang, Z. Zhou, and Y . Zeng are with the National Mobile Com- munications Research Laboratory , Southeast Univ ersity , Nanjing 210096, China. Y . Zeng is also with the Purple Mountain Laboratories, Nanjing 211111, China. Z. Dong is now with the Purple Mountain Laboratories, Nanjing 211111, China. She was with the National Mobile Communications Research Laboratory , Southeast University , Nanjing 210096, China. (e-mail: { 230258936, zhenjun dong, zhiwen zhou, yong zeng } @seu.edu.cn). (Corre- sponding author: Y ong Zeng.) significant interest in integrated sensing and communication (ISA C), a key enabling technology for 6G networks that fuses sensing and communication seamlessly using shared spectral resources, hardware, and signal processing framew orks [11]–[14]. For U A V swarm applications, ISA C is essential for ensuring safe and efficient operations, facilitating tasks such as real-time swarm localization, collision avoidance, and cooperativ e trajectory planning [15], [16]. Despite its promise, deploying ISA C for low-altitude UA V swarms introduces unique challenges. UA Vs are inherently constrained by size, weight, and power limitations, which restrict their onboard communication and sensing capabilities [1], [17], [18]. Furthermore, the high mobility and 3D spatial distribution of swarm members require ISA C systems with exceptional angular resolution to distinguish closely spaced targets and support high-gain beamforming for communica- tion [19], [20]. Con ventional multiple-input multiple-output (MIMO) architectures, which form the backbone of most contemporary ISA C systems, often struggle to meet these demands cost-effecti vely [21], [22]. For example, traditional array architectures like uniform linear arrays (ULAs) and uniform planar arrays (UP As) suffer from fundamental limita- tions: their angular resolution degrades significantly for signal directions away from the array boresight, and achie ving high resolution requires a large physical aperture or a massive number of antenna elements [23]. More critically , realizing such arrays with fully digital or hybrid analog/digital beam- forming (HBF) require a large number of radio frequency (RF) chains and phase shifters, which are expensiv e and power- hungry especially at high-frequency bands like millimeter- wa ve (mmW ave) and T erahertz (THz) [24], [25]. This creates a critical cost-performance trade-off that hinders the scalable deployment of ISA C for widespread U A V swarm applications. T o address the hardware cost and implementation chal- lenges in high-frequency systems, researchers have explored various innovati ve antenna architectures that reduce the re- quired phase shifters or RF chains. Notable examples include lens antenna arrays [26], [27], which transform the signal from the antenna space to a lower -dimensional beamspace using electromagnetic lenses, thereby significantly reducing the number of required RF chains. Howe ver , they often require bulk y and precise lens components, which increase size and cost. Fluid antennas [28], [29] and movable antennas [30]– [32] represent another direction, where the antenna shape or position can be dynamically optimized at the transmitter or receiv er to enhance performance with fewer physical elements. While these systems offer flexibility and improv ed spectral efficienc y , they typically in volve mechanical or reconfigurable 2 parts that introduce additional complexity , po wer consumption, and response latency . Another approach is the pinching an- tenna [33], [34], which uses small dielectric particles inserted into wa ve guides to create line-of-sight (LoS) paths for users, though it may face challenges in scalability and integration. Furthermore, the tri-hybrid MIMO architecture [35], [36] incorporates reconfigurable antennas along with both digital and analog precoding to balance performance and hardware complexity , yet it still relies on reconfigurable elements that add to cost and control overhead. Recently , ray antenna array (RAA) [37] was proposed, which offers a distinctly different and cost-effecti ve solu- tion. RAA comprises multiple simple uniform linear arrays (sULAs), where all antenna elements within each sULA are directly connected without any phase shifters [23], [38], [39]. Each sULA naturally forms a beam whose main lobe direction aligns with its physical orientation. This design eliminates phase shifters, drastically reduces the number of required RF chains through a selection network, and achieves uniform angular resolution independent of the signal angle of arriv al (AoA). This is a distinct advantage over conv entional linear or planar arrays, whose resolution deteriorates at larger angles. Howe ver , the basic RAA proposed in [37] is inherently a two-dimensional (2D) structure designed for discriminating only the azimuth or ele vation angle. For practical low-altitude U A V swarm ISAC, where targets are distributed in 3D space, the ability to jointly discriminate both azimuth and elev ation angles with high resolution is essential. Besides, a practical deployment challenge for the RAA is the potential blockage or mutual coupling between adjacent sULAs in a planar configuration. This can lead to signal attenuation and inter-ray interference, ultimately degrading system performance. The author in [40] proposed cylinder directly-connected antenna array (DCAA) to address the blockage problem by utilizing multiple simple uniform circular arrays (sUCAs) in a layered 3D configuration, but the 3D coverage issue remains unsolved. Motiv ated by the cost and resolution benefits of the RAA principle and to o vercome its dimensional and blockage limita- tion, this paper proposes a novel spherical DCAA for 3D low- altitude U A V swarm ISA C. The spherical DCAA generalizes the RAA concept from a planar to a spherical configuration. It consists of multiple simple uniform planar arrays (sUP As) distributed over a spherical surface. Crucially , all elements within each indi vidual sUP A are directly connected, requiring no phase shifters within the sUP A. Only a few number of RF chains are shared by all sUP As via a selection network. This architecture inherits and extends the key adv antages of RAA [38]: 1) drastically reduced hardware cost and complexity by replacing a massiv e number of phase shifters with low-cost an- tenna elements and RF switches; 2) enhanced energy focusing capability , since antenna elements with higher directi vity can be used as each sUP A is responsible for a dedicated angular sector; and 3) superior and uniform angular resolution. Be- yond these inherited benefits, the spherical DCAA offers two key distinctive advantages. First, it achieves full 3D angular cov erage by enabling joint discrimination of both azimuth and elev ation angles. Second, the spherical arrangement of sUP As effecti vely mitigates mutual blockage and coupling between adjacent sUP As, thereby ensuring more reliable signal reception across the entire spatial domain. The main contributions of this paper are summarized as follows: • First, we propose the novel spherical DCAA architecture for 3D low-altitude U A V swarm ISA C. It is composed of multiple sUP As distrib uted over a spherical surface, where all the M × M antenna elements within each sUP A are directly connected without requiring an y phase shifter . W e establish a complete mathematical signal model for a bistatic ISAC system utilizing this architecture. • Second, we conduct a rigorous beam pattern analy- sis for the spherical DCAA and provide a systematic design methodology . Through mathematical deriv ation, we formally characterize the unique angular resolution properties of spherical DCAA. Based on this analysis, we propose a systematic spherical arrangement of the sUP As. Furthermore, we address practical implementa- tion constraints by deriving the minimum radius of the spherical structure required to av oid physical collisions between adjacent sUP As. • Third, we develop an effecti ve sensing parameter esti- mation algorithm for ISAC within the spherical DCAA framew ork. A super-resolution multiple signal classifi- cation (MUSIC) algorithm is designed to tailor to the structure of the spherical DCAA ’ s equiv alent steering vector for jointly estimating the azimuth and ele vation angles of arriv al. • Finally , we provide extensiv e simulation results to vali- date the theoretical analysis and demonstrate the superior performance of the proposed spherical DCAA system under lo w-altitude U A V swarm scenarios. The perfor- mance of the spherical DCAA is benchmarked against con ventional UP A with the same array gain that uses the classical Kronecker product codebook (KPC)-based hybrid beamforming. The results sho w that the spherical DCAA achieves considerable superior angular resolution and significantly better performance in discriminating closely-spaced U A V targets and achie ves angle estimation with lower root mean square error (RMSE). Furthermore, the spherical DCAA exhibits enhanced spectral efficiency for communication, thanks to its higher beamforming gain since more directi ve antenna elements can be used. The remainder of this paper is organized as follows. Sec- tion II presents the system model for the spherical DCAA- based bistatic U A V swarm ISA C. Section III introduced the design of spherical DCAA architecture. Section IV details the con ventional UP A architecture and compares its beam pattern with spherical DCAA. Section V introduces the ISA C signal model and the proposed joint parameter estimation algorithm. Simulation results and discussions are provided in Section VI. Finally , the paper is concluded in Section VII. Notation : Scalars are denoted by italic letters. V ectors and matrices are denoted by boldface lo wer-case and upper-case letters, respectively . Transpose and Hermitian transpose are denoted by ( · ) T and ( · ) H , respectively . The absolute value and l 2 norm are gi ven by | · | and ∥ · ∥ 2 , respecti vely . C m × n 3 C& S Signa l IS AC Tx UA V Sw arm DC AA IS AC Rx ( ) , ll  x z y 𝑁 Selection Netw o rk 𝑁 𝑅𝐹 D i g i ta l Be a m fo rm i ng UA V Sw arm Fig. 1. An illustration of spherical DCAA-based bi-static ISA C for low-altitude UA V swarm. denotes the space of m × n complex matrices. card( · ) denotes the cardinality of a set. j = √ − 1 denotes the imaginary unit of complex-valued numbers. 1 N is an N × 1 all-ones vector . ⊗ denotes the Kroneck er product. ⌊·⌋ and ⌈·⌉ denote the floor and ceiling functions, respectiv ely . δ ( · ) is the Dirac delta function. ϵ ( · ) is the Heaviside step function. The distribution of a circu- larly symmetric complex Gaussian (CSCG) random variable with zero mean and variance σ 2 is denoted by C N (0 , σ 2 ) and ∼ stands for “distributed as”. The set of integers is denoted by Z . I I . S Y S T E M M O D E L W e consider a lo w-altitude U A V swarm bi-static ISA C system where the ISA C transmitter has a single antenna and the ISA C recei ver (Rx) is equipped with spherical DCAA, as shown in Fig. 1. The system aims to provide communication service for the Tx, while sensing the U A V swarm targets via its transmitted signal. The channel between the Tx and Rx may contain a line-of-sight (LoS) link along with L non-line- of-sight (NLoS) components resulting from reflections by the sensing targets. The l th path (where l = 0 corresponds to the LoS) is characterized by a set of parameters ϕ l , θ l , denoting the azimuth and ele vation AoA, respecti vely . The ISA C system simultaneously pursues three k ey objecti ves: (1) localization of the ISA C Tx UA V , (2) uplink communication, and (3) bi-static sensing of UA V swarm targets using reflected signals. In order to reduce hardware cost, the novel spherical DCAA architecture is applied at the Rx, which contains N sUP As sticking on the surface of a sphere as shown in Fig. 1. Each sUP A has M × M antenna elements separated by a distance d = λ/ 2 along both vertical and horizontal axes, with λ being the signal wavelength. Dif ferent from the con ventional UP A, all the M × M antenna elements of each sUP A are directly connected without any analog or digital beamforming, which justifies the term sUP A. In addition, with N RF ≪ N RF chains, a selection network is designed to select N RF ports from the N equiv alent sUP A outputs. The selection network is denoted by S = { 0 , 1 } N RF × N , which satisfies ∥ [ S ] i, : ∥ = 1 and [ S ] : ,j ≤ 1 , 1 ≤ i ≤ N RF . After that, the N RF selected outputs will be connected to the RF-chains for subsequent baseband digital processing for communication data decoding and sensing parameter estimation. Let x s denote the ISA C signal emitted from the transmitter with transmitting power P t = E  | x s | 2  . The corresponding receiv ed signal of N sUP As at the Rx, denoted by ˜ y ∈ C N × 1 , is then given by ˜ y = h x s + z , (1) where z ∈ C N × 1 is the A WGN vector , h ∈ C N × 1 is the equiv alent channel between the Tx and the N sUP As, which can be written as h = L X l =0 h l , (2) where h l denotes the channel of the l th path. After passing through the selection network S , the resulting signal y ∈ C N RF × 1 can be expressed as y = S ˜ y = Sh x s + Sz . (3) In the following, we will introduce the specific design of spherical DCAA, selection network, and deri ve the equiv alent channel vector h . I I I . S P H E R I C A L D C A A A R C H I T E C T U R E 𝜃 𝜙 𝐤 𝑥 𝑦 𝑧 𝐮 1 𝐮 2 𝑑 𝑀 𝑀 (a) 𝐤 𝐮 1 ′ 𝐮 2 ′ 𝜗 𝜂 𝐩 𝑥 𝑦 𝑧 (b) Fig. 2. (a) Illustration of sUP A in the proposed spherical DCAA, where all antenna elements of sUP A are separated by half wavelength and directly connected; (b) Rotate the sUP A with respect to y -axis and z -axis without self-spinning. 4 A. Array Response of Spherical DCAA As shown in Fig. 2a, consider a M × M UP A located in the y z -plane. Under the assumption of far-field uniform planar wa ve (UPW), denote the wa ve vector , azimuth and elev ation AoA of the incoming signal as k ∈ C 3 × 1 and ( ϕ, θ ) re- spectiv ely , where k = [ − cos θ cos ϕ, − cos θ sin ϕ, − sin θ ] T , − π / 2 ≤ ϕ, θ ≤ π / 2 . In the figure ϕ is measured anti- clockwise from the positive x -axis, and θ is defined relative to the xy plane, taking positiv e v alues for directions above the plane and negati ve values for directions below it. Let u 1 = [0 , 1 , 0] T , u 2 = [0 , 0 , 1] T be the normalized vector aligned with the first and second dimension of the UP A. T o achieve maximum response for signals from multiple directions, the sUP A is rotated, as shown in Fig. 2b. Suppose the rotation of the sUP A does not include any self-spinning, the rotation matrix R ( η , ϑ ) ∈ C 3 × 3 can be expressed as R ( η , ϑ ) =   cos η − sin η 0 sin η cos η 0 0 0 1   | {z } R z ( η )   cos ϑ 0 − sin ϑ 0 1 0 sin ϑ 0 cos ϑ   | {z } R y ( ϑ ) , (4) where R z ( η ) ∈ R 3 × 3 is the yaw matrix rotating around the z -axis, with η ∈ [ − η max , η max ] being the yaw angle w .r .t. the positiv e y -axis (anticlockwise on the xy -plane); and R y ( ϑ ) ∈ R 3 × 3 is the pitch matrix rotating around the y -axis, with ϑ ∈ [ − ϑ max , ϑ max ] being the pitch angle w .r .t. the positi ve z -axis (anticlockwise on the xz -plane), with ϑ max < π / 2 . Let the normal vector of the plane containing the antenna array be denoted by p = [ − cos ϑ cos η , − cos ϑ sin η , − sin ϑ ] T ∈ C 3 × 1 . Thus, we term the rotated sUP A as the sUP A with orientation ( η , ϑ ) , and the normalized side v ector of the rotated sUP A can be expressed as u ′ 1 = R ( η , ϑ ) u 1 =  − sin η , cos η, 0  T , u ′ 2 = R ( η , ϑ ) u 2 =  − cos η sin ϑ, − sin η sin ϑ, cos ϑ  T . (5) The array response vector of the rotated UP A with incoming wa ve vector k can be obtained a ( u ′ 1 , u ′ 2 , k ) = q G ( u ′ 1 , u ′ 2 , k ) a 1 ( u ′ 1 , k ) ⊗ a 2 ( u ′ 2 , k ) , (6) where G ( u ′ 1 , u ′ 2 , k ) accounts the radiation pattern of each antenna element in the sUP A, which can be equiv alently written as G ( η − ϕ, ϑ − θ ) ; and a 1 ( u ′ 1 , k ) and a 2 ( u ′ 2 , k ) are the steering vector along the side vectors, respectiv ely , given by a 1 ( u ′ 1 , k ) = h 1 , e − j π u ′ T 1 k , ..., e − j ( M − 1) π u ′ T 1 k i T , a 2 ( u ′ 2 , k ) = h 1 , e − j π u ′ T 2 k , ..., e − j ( M − 1) π u ′ T 2 k i T . (7) For sUP A, all the M × M antenna elements are directly connected. Then the resulting response of the sUP A with orien- tation ( u ′ 1 , u ′ 2 ) for incoming signal k , denoted by r ( u ′ 1 , u ′ 2 , k ) , is giv en by r ( u ′ 1 , u ′ 2 , k ) = 1 T M 2 × 1 a ( u ′ 1 , u ′ 2 , k ) = q G ( u ′ 1 , u ′ 2 , k ) M 2 H M  − u ′ T 1 k  H M  − u ′ T 2 k  | {z } f ( u ′ 1 , u ′ 2 , k ) , (8) where H M ( x ) = 1 M P M − 1 m =0 e j πmx is the Dirichlet kernel function, giv en by H M ( x ) = e j π 2 ( M − 1) x sin( π 2 M x ) M sin( π 2 x ) . (9) -90 -45 0 45 90 -90 -45 0 45 90 Curves for = 29.79 ° , = 45.26 ° (M=8) p= -3 p=-2 p=-1 p=0 p=1 p=2 p=3 q=-3 q=-2 q=-1 q=0 q=1 q=2 q=3 (a) (b) Fig. 3. Null curves and beam pattern of the sUP A when M = 8 and ( η , ϑ ) = (29 ◦ , 45 ◦ ) . The null curves are plotted through solid lines and dashed lines representing different equations in (11) Equation (8) can also be written as r ( ϕ, θ ; η , ϑ ) = p G ( η − ϕ, ϑ − θ ) M 2 H M  cos θ sin( ϕ − η )  × H M  sin θ cos ϑ − cos θ sin ϑ cos( ϕ − η )  ≜ p G ( η − ϕ, ϑ − θ ) f ( ϕ, θ ; η , ϑ ) , (10) where G ( u ′ 1 , u ′ 2 , k ) ≜ G ( η − ϕ, ϑ − θ ) . Based on this, we have the following lemmas. Lemma 1: For a sUP A with orientation u ′ 1 , u ′ 2 and incoming wa ve vector k , we have max k | f ( u ′ 1 , u ′ 2 , k ) | = M 2 , and the maximum is achieved if and only if R T ( η , ϑ ) k = [1 , 0 , 0] T or equiv alently ( ϕ, θ ) = ( η , ϑ ) Pr oof: According to equation (8) and (9), | f ( u ′ 1 , u ′ 2 , k ) | achiev es its maximum only if u ′ T 1 k = u ′ T 2 k = 0 and k = u ′ 1 × u ′ 2 = R ( η , ϑ )[1 , 0 , 0] T . Lemma 2: For a giv en sUP A with u ′ 1 and u ′ 2 and incoming wa ve vector k , we have min k | f ( u ′ 1 , u ′ 2 , k ) | = 0 , and the minimum is achieved if and only if u T 1 R T ( η , ϑ ) k = 2 p M or u T 2 R T ( η , ϑ ) k = 2 q M , where p, q ∈ Z \ { 0 } and | p | ≤ M 2 , | q | ≤ M . Pr oof: According to equation (8) and (9), | f ( u ′ 1 , u ′ 2 , k ) | achiev es its minimum if and only if u ′ T 1 k = u T 1 R T ( η , ϑ ) k = 2 p M or u ′ T 2 k = u T 2 R T ( η , ϑ ) k = 2 q M , where p, q ∈ Z \ { 0 } and | p | ≤ M 2 , | q | ≤ M . Let θ null and ϕ null denote the elev ation and azimuth angles when the null condition | f ( u ′ 1 , u ′ 2 , k ) | = 0 holds. According to Lemma 2, it yields cos θ null sin( ϕ null − η ) = 2 p M , or sin θ null cos ϑ − cos θ null sin ϑ cos( ϕ null − η ) = 2 q M , (11) where p, q ∈ Z \ { 0 } and | p | ≤ M 2 , | q | ≤ M . The corresponding null curves and beam pattern are sho wn in Fig. 3a and 3b. While a single sUP A produces a beam pattern with a continuous network of nulls that creates a web of blind areas, Lemma 1 indicates that this property can 5 be harnessed as an advantage. By confining the maximum response to a specific direction, it simultaneously enhances communication and sensing through energy focusing and interference suppression. The null-to-null beamwidth of the spherical DCAA main lobe can be obtained, which is given in Lemma 3. Lemma 3: For any giv en sUP A with orientation ( η , ϑ ) , the null-to-null azimuth and ele vation beamwidth of the main lobe, denoted by BW ( ϑ, η ) = (∆ ϕ, ∆ θ ) , is expressed as BW ( ϑ, η ) =  2 arcsin  2 M cos ϑ  , 2 arcsin  2 M   . (12) Pr oof: Please refer to Appendix A Lemma 3 indicates that the sUP A ’ s main lobe has a uniform beamwidth 2 arcsin(2 / M ) in the elev ation domain b ut a non- uniform one in the azimuth domain, and the latter increases with the elev ation angle ϑ . This characteristic ensures a uniform elev ation angular resolution for the spherical DCAA and, furthermore, it implies that the sUP A becomes effecti vely sparser at larger elev ation angles. B. sUP A Orientation Design and Spherical DCAA Implemen- tation Under the premise of covering the entire space and mini- mize interference between adjacent sUP As in the elev ation and azimuth directions, the sUP As are strategically arranged with uniform spacing of arcsin 2 M cos ϑ and arcsin 2 M respectiv ely in the azimuth and elev ation directions, and we define N ϑ =  θ max / arcsin 2 M  and N η ( ϑ ) =  ϕ max / arcsin 2 M cos ϑ  as the number of layers of sUP As and the number of sUP As for ϑ , respectiv ely , where θ max and ϕ max are the maximum elev ation and azimuth angles of the signals. Therefore, with the original sUP A with ( η , ϑ ) = (0 , 0) , we adopt the follo wing two steps to determine the directions of all the N sUP As. • Elevation: W e first find all the sUP A directions ( η , ϑ ) in the elev ation dimension with η = 0 , which yields ϑ q = q arcsin(2 / M ) , q ∈ N ϑ , (13) where N ϑ = {− N ϑ , ..., 0 , 1 , ..., N ϑ } . • Azimuth: For elev ation angle ϑ q , we identify all corre- sponding azimuth angles η p,q in the azimuth domain. ( η p,q , ϑ q ) =  p arcsin  2 M cos ϑ q  , ϑ q  , p ∈ N η ( ϑ q ) , (14) where N η ( ϑ q ) = {− N η ( ϑ q ) , ..., 0 , 1 , ..., N η ( ϑ q ) } . The total number N of sUP As is thus N = N ϑ X q = − N ϑ N η ( ϑ q ) . (15) For ease of presentation, we transform the two-dimensional index ed orientation ( η p,q , ϑ q ) , p ∈ N η ( ϑ q ) , q ∈ N ϑ into one- dimensional indexed orientation ( η n , ϑ n ) , where n ∈ N = { 1 , 2 , ..., N } , which satisfies n = p + N η ( ϑ q ) + 1 + q − 1 X k = − N ϑ  2 N η ( ϑ k ) + 1  , (16) -90° -45° 0° 45° 90° -90° -45° 0° 45° 90° Fig. 4. The orientation of sUP As ( η p,q , ϑ q ) according to equation (14), where M = 8 . (a) (b) (c) (d) Fig. 5. The beam pattern of: (a) spherical DCAA; (b) conv entional UP A with KPC-based HBF , and the en velope of beam pattern of: (c) spherical DCAA; (d) con ventional UP A with KPC-based HBF . The elev ation and azimuth orientations of sUP As with isotropic antenna elements and corresponding beam pattern of the setup are illustrated in Fig. 4 and 5a. The spherical DCAA exhibits a beam pattern with densely packed yet well- separated peaks. In each direction, the maximum magnitude (en velope) recei ved among all the sUP As, i.e., r ∗ ( ϕ, θ ) = max n r ( ϕ, θ ; η n , ϑ n ) is sho wn in Fig. 5c. With this design, the entire spatial domain is cov ered without any holes, as ev ery direction is served by at least one sUP A. This is quantified by the worst-case coverage metric min ϕ,θ r ∗ ( ϕ, θ ) > 15 for M = 8 . As shown in Fig. 6, in order to av oid signal blockage among sUP As, we arrange all the sUP As on a sphere with center O , where each sUP A is tangent to the spherical surface with tangent point O m , O n etc. In addition, we consider the hitbox of sUP As as a circumcircle with radius r = M d/ √ 2 of the square shape of the array . The angle separation α between any two sUP As is defined as the angle of the two vectors − − − → O O m and − − → O O n , and thus we have the follo wing theorem. Theor em 1: In the aforementioned design, the minimum angle separation, denoted by α + , between any two sUP As is arcsin 2 M . 6 𝛼 𝑟 𝑅 𝑂 𝑂 𝑚 𝑂 𝑛 (a) 𝑂 𝑥 𝑦 𝑧 𝐩 𝑛 𝐩 𝑛 ′ 𝐤 𝑙 Ref ere nce point Ref ere nce ante nna (b) Fig. 6. Implementation of spherical DCAA (a) the critical collision state where all the sUP As are closest to each other (b) the geometry relationship among sUP A center location p n , reference antenna location p ′ n , global reference point O and the l th incoming signal vector k l . Pr oof: Please refer to Appendix B In the critical collision state, the circumcircles of the sUP As are tangent to each other as shown in Fig. 6a, and the minimum angle separation α ∗ between any two sUP As can be obtained as tan α ∗ 2 = r R = M d √ 2 R . (17) Therefore, the minimum angle separation α + in design must be greater than the critical collision state minimum angle separation α ∗ , i.e., α + ≥ α ∗ , and the minimum radius R of the sphere can be obtained R = M d √ 2 tan( 1 2 arcsin 2 M ) ( a ) ≈ M 2 λ 2 √ 2 , (18) where ( a ) holds when M is lar ge. The relationship between R, M and carrier frequency is shown in Fig. 7. The mini- mum radius is proportional to the total number of antenna elements, and it decreases as the carrier frequency grows up. The complete procedure for designing the parameters  M 2 , N , R , { η n , ϑ n } n ∈N  of the proposed spherical DCAA architecture is illustrated in Fig. 8. 0 50 100 150 200 250 300 Carrier frequency f c (GHz) 0 0.2 0.4 0.6 0.8 1 Radius of spherical DCAA (m) M 2 = 36 an tennas M 2 = 64 an tennas M 2 = 144 an tennas M 2 = 256 an tennas Fig. 7. Illustration of the minimum sphere radius needed to implement DCAA w .r .t. carrier frequency and number of antenna elements. As a result, for the U A V swarm ISAC system with one LoS component and L NLoS components, the equiv alent channel h between the Tx and the N sUP As of the spherical DCAA can be expressed as h = L X l =0 α l r ( ϕ l , θ l ) , (19) Input : Array size M 2 , upper bound of signal angles ϕ max , θ max Design elevation orientation ϑ q with (13) Design azimuth orientation η p,q with (14) Design number of sUP As N with (15) Design the radius R of sphere with (18) Output : spherical DCAA parameters  M 2 , N , R , { η n , ϑ n } n ∈N  Fig. 8. Flowchart of spherical DCAA parameter design where α l denotes the path coef ficient; ϕ l , θ l , denote the path azimuth and elev ation AoAs respectively; and r ( ϕ l , θ l ) = [ r ( ϕ, θ ; η n , ϑ n )] n ∈N ∈ C N × 1 denotes the array response vector of spherical DCAA. As shown in Fig. 6b, we denote p n = [ − R cos ϑ n cos η n , − R cos ϑ n sin η n , − R sin ϑ n ] T as the position vector of the center of the n th sUP A; and p ′ n is denoted as the position vector of the reference antenna of the n th sUP A, which satisfies p ′ n = p n − 1 2 R ( η n , ϑ n )( u 1 + u 2 ) M d. (20) T o model the blockage effect, ϵ ( − k T l p n ) is introduced, where k l = [ − cos θ l cos ϕ l , − cos θ l sin ϕ l , − sin θ l ] T ∈ C 3 × 1 is the l th incoming signal vector and ϵ ( · ) denotes the Hea viside step function which forces the response of a sUP A to zero whenev er it is not directly illuminated (where k T l p n > 0 ) by the incoming plane wave. T ake the center of the sphere as the reference point, the array response can be written as r ( ϕ l , θ l ) = h ϵ ( − k T l p n ) e − j 2 π λ k T l p ′ n | p ′ n | r ( ϕ l , θ l ; η n , ϑ n ) i n ∈N = h ϵ ( − k T l p n ) e − j 2 π λ k T l p ′ n | p ′ n | M 2 p G ( η n − ϕ l , ϑ n − θ l ) × H M  cos θ l sin( ϕ l − η n )  × H M  sin θ l cos ϑ n − cos θ l sin ϑ n cos( ϕ l − η n )  i n ∈N . (21) Due to the limited RF chains, we need to choose N RF signals from N spherical DCAA outputs. T o fully exploit the energy-focusing ability of spherical DCAA, we propose the selection scheme based on energy maximization, i.e., S = arg max S ∥ y ∥ 2 2 , to choose the N RF spherical DCAA outputs with the maximum sum energy . T o realize the energy based sUP A selection, we only need to sweep all the spherical DCAA ports to obtain full information about the response 7 magnitude of dif ferent sUP As. This would require l N N RF m sweeps to cover all the N sUP As. Therefore, S will be determined and fixed for the subsequent analysis. I V . S P H E R I C A L D C A A V S C O N V E N T I O N A L U PA UA V Sw arm IS AC S igna l IS AC Tx LOS UA V Sw arm UP A IS AC Rx 𝑀 2 A nal o g Bea m fo rm i ng 𝑁 𝑅𝐹 D i g i ta l Bea m fo rm i ng ( ) , ll  x z y Fig. 9. An illustration of conventional UP A-based ISAC for low-altitude UA V swarm. A. Con ventional UP A As a comparison, we consider a UP A using KPC-based HBF , as illustrated in Fig. 9. A UP A facing x -axis with orientation vectors u 1 , u 2 and M × M antenna elements is shown in Fig. 10. The adjacent antenna elements are separated by half wav elength in both directions. Thus, according to (6), the array response vector of the UP A for the propagation path k can also be written as a ( u 1 , u 2 , k ) . The azimuth and elev ation DFT codewords are respectively defined as c h ( ϕ p ) = [1 , e j π sin ϕ p , . . . , e j π ( M − 1) sin ϕ p ] T ∈ C M × 1 , c v ( θ q ) = [1 , e j π sin θ q , . . . , e j π ( M − 1) sin θ q ] T ∈ C M × 1 , (22) where sin ϕ p = 2 p M and sin θ q = 2 q M . The ( p, q ) -th codeword of the KPC is obtained via the Kronecker product of the azimuth and elev ation DFT codew ords, i.e., c ( ϕ p , θ q ) = c v ( θ q ) ⊗ c h ( ϕ p ) ∈ C M 2 × 1 . (23) The KPC implemented via the HBF architecture is de- noted by C = [ c ( ϕ p , θ q )] − N v − 1 2 ≤ p ≤ N v − 1 2 , − N h − 1 2 ≤ q ≤ N h − 1 2 ∈ C M 2 × N v N h , where N v and N h are the numbers of codewords in the elev ation and azimuth codebooks, respectiv ely , giv en by N v = 2 j sin θ max 2 / M k + 1 and N h = 2 j sin ϕ max 2 / M k + 1 . Consequently , the beam pattern for the codeword c ( ϕ p , θ q ) , denoted by r BF ( ϕ, θ ; ϕ p , θ q ) , can be expressed as r BF ( ϕ, θ ; ϕ p , θ q ) = c H ( ϕ p , θ q ) a ( u 1 , u 2 , k ) = p G ( ϕ, θ ) M 2 H M (cos θ sin ϕ − sin ϕ p ) × H M (sin θ − sin θ p ) ≜ p G ( ϕ, θ ) f BF ( ϕ, θ ; ϕ p , θ q ) . (24) For any giv en KPC codeword c ( ϕ p , θ q ) , we aim to find the AoA ( ϕ, θ ) that yields the maximum and minimum response. RF - Chain RF - Chain RF - Chain RF - Chain UP A KPC cod e word s RF - chains 𝑀 𝑀 𝑑 𝐮 1 𝐮 2 𝑥 𝑦 𝑧 Fig. 10. Antenna architecture of conv entional UP A using the classical KPC- based hybrid beamforming. Lemma 4: For a given KPC codew ord c ( ϕ p , θ q ) , we have max ( ϕ,θ ) f BF ( ϕ, θ ; ϕ p , θ q ) = M 2 , and the maximum is achiev ed if and only if ( ϕ, θ ) =  arcsin sin ϕ p cos θ q , θ q  . Pr oof: According to equation (24), f BF ( ϕ, θ ; ϕ p , θ q ) achiev es its maximum if and only if cos θ sin ϕ − sin ϕ p = sin θ − sin θ q = 0 . Since | ϕ | ≤ π / 2 , | θ | ≤ π / 2 , we have ( ϕ, θ ) =  arcsin sin ϕ p cos θ q , θ q  . Lemma 5: For a given KPC codew ord c ( ϕ p , θ q ) , we have min ( ϕ,θ ) f BF ( ϕ, θ ; ϕ p , θ q ) = 0 , and the minimum is achie ved if and only if θ = arcsin(sin θ p + 2 m M ) with   sin θ p + 2 m M   ≤ 1 , or ϕ = arcsin sin ϕ p +2 n/ M cos θ with    sin ϕ p +2 n/ M cos θ    ≤ 1 , cos θ  = 0 , where m, n ∈ Z \ { 0 } . Pr oof: According to equation (24), f BF ( ϕ, θ ; ϕ p , θ q ) achiev es its minimum if and only if cos θ sin ϕ − sin ϕ p = 2 n M or sin θ − sin θ q = 2 m M , which yields θ = arcsin(sin θ p + 2 m M ) with   sin θ p + 2 m M   ≤ 1 , or ϕ = arcsin sin ϕ p +2 n/ M cos θ with    sin ϕ p +2 n/ M cos θ    ≤ 1 , cos θ  = 0 , where m, n ∈ Z \ { 0 } . Lemma 4 indicates that with codeword c ( ϕ p , θ q ) , the peak of the beam pattern of UP A deviates from the direction of the code word. Specifically , if sin ϕ p / cos θ q > 1 , the peak becomes meaningless because such an angle arcsin sin ϕ p cos θ q does not exist. The beam pattern of KPC are sho wn in Fig. 5b. In each direction, the maximum magnitude received among all the codew ords are shown in Fig. 5d. Compared with con ventional UP A using KPC-based HBF , the proposed spherical DCAA has denser and more regular -shaped peaks in the beam pattern, which can enhance its communication and sensing ability . B. Angular Resolution Comparison Angular resolution is a ke y metric for ev aluating the communication and sensing ability of an antenna array . It quantifies the array’ s ability to distinguish between two closely spaced incoming signals. The resolution depends on the beam pattern r ( ϕ, θ ; ϕ ′ , θ ′ ) , which is a function of both the desired direction ( ϕ ′ , θ ′ ) , i.e., where the main lobe is pointed, and the observation direction ( ϕ, θ ) , i.e., the angular variable at which the array response is ev aluated. According to Lemma 1 and 4, for spherical DCAA the desired direction satisfies ( ϕ ′ , θ ′ ) = ( η , ϑ ) , and for conv entional UP A using KPC-based HBF we hav e ( ϕ ′ , θ ′ ) = (arcsin sin ϕ p cos θ q , θ q ) . For a gi ven desired direction, the resolution is defined via the beam width of the 8 (a) (b) (c) (d) Fig. 11. Comparison of elevation and azimuth angular resolution between the proposed spherical DCAA and con ventional UP A using KPC-based HBF . In each figure, the x -axis and y -axis represent the azimuth and elevation desired directions, respectively , while the color or magnitude indicates the angular resolution value in the corresponding direction. The resolution values, e.g., the color bar, are displayed on a logarithmic scale to highlight trends across directions and to illustrate the differences between spherical DCAA and conv entional UP A using KPC-based HBF . (a) elevation resolution of spherical DCAA, (b) elev ation resolution of conventional UP A using KPC-based HBF , (c) azimuth resolution of spherical DCAA , (d) azimuth resolution of conv entional UP A using KPC-based HBF . 𝝓 ′ 𝝓 𝟏 𝝓 𝟐 𝜽 ′ 𝜽 𝟏 𝜽 𝟐 Fig. 12. The illustration of θ 1 , θ 2 and ϕ 1 , ϕ 2 main lobe along the elev ation and azimuth axes, measured between the first nulls on either side of the peak. Specifically , for the elev ation angular resolution, we fix the azimuth obser- vation direction ϕ to ϕ ′ , and v arying the elev ation observation direction θ around the desired direction θ ′ , to find the first null points on both sides of it. A formal definition is given below . Definition 1: The azimuth and elev ation angular resolution γ h ( ϕ ′ , θ ′ ) and γ v ( ϕ ′ , θ ′ ) are functions of the desired signal di- rection ( ϕ ′ , θ ′ ) , which is defined as half of the main lobe beam width ∆ ϕ ( ϕ ′ , θ ′ ) , ∆ θ ( ϕ ′ , θ ′ ) of the beam pattern r ( ϕ, θ ; ϕ ′ , θ ′ ) , i.e., γ v ( ϕ ′ , θ ′ ) = 1 2 ∆ θ ( ϕ ′ , θ ′ ) = 1 2 | θ 1 − θ 2 | , γ h ( ϕ ′ , θ ′ ) = 1 2 ∆ ϕ ( ϕ ′ , θ ′ ) = 1 2 | ϕ 1 − ϕ 2 | , (25) where θ 1 , θ 2 and ϕ 1 , ϕ 2 are the pairs of the closest nulls of r ( ϕ, θ ; ϕ ′ , θ ′ ) on both side of desired direction θ ′ and ϕ ′ , as shown in Fig. 12 θ 1 = min θ n θ > θ ′   r ( ϕ = ϕ ′ , θ ; ϕ ′ , θ ′ ) = 0 o , θ 2 = max θ n θ < θ ′   r ( ϕ = ϕ ′ , θ ; ϕ ′ , θ ′ ) = 0 o . (26) ϕ 1 = min ϕ n ϕ > ϕ ′   r ( ϕ, θ = θ ′ ; ϕ ′ , θ ′ ) = 0 o , ϕ 2 = max ϕ n ϕ < ϕ ′   r ( ϕ, θ = θ ′ ; ϕ ′ , θ ′ ) = 0 o . (27) By comparing Fig. 5a and 5b, we can find that for the proposed spherical DCAA, the ele vation angular resolution (or the elev ation main lobe beamwidth) is uniform, and for the same desired ele vation angle θ ′ , it has uniform azimuth angular resolution (or the azimuth main lobe beamwidth) with respect to the desired azimuth angle ϕ ′ . But in con ventional UP A using KPC-based HBF , the azimuth angle resolution ability degrades as desired azimuth angle ϕ ′ gets larger . Besides, both spherical DCAA and conv entional UP A using KPC-based HBF suf fer from decreasing azimuth angle resolution ability as desired elev ation angle θ ′ increases, as shown in the following theorem. Theor em 2: For any desired direction ( ϕ ′ , θ ′ ) satisfies ϕ ′ > 0 , θ ′ > 0 , the elev ation and azimuth angular reso- lution of spherical DCAA, denoted by γ DCAA v ( ϕ ′ , θ ′ ) and γ DCAA h ( ϕ ′ , θ ′ ) respectiv ely , are given by: γ DCAA v ( ϕ ′ , θ ′ ) = arcsin 2 M , γ DCAA h ( ϕ ′ , θ ′ ) = arcsin 2 M cos θ ′ , (28) and that of conv entional UP A using KPC-based HBF , denoted by γ UP A v ( ϕ ′ , θ ′ ) and γ UP A h ( ϕ ′ , θ ′ ) respectiv ely , are given by: γ UP A v ( ϕ ′ , θ ′ ) = 1 2  min Θ + − max Θ −  , γ UP A h ( ϕ ′ , θ ′ ) = 1 2  min Φ + − max Φ −  , (29) where Θ + , Θ − , Φ + , Φ − are defined in Appendix C. Pr oof: Please refer to Appendix C. The comparison of angular resolution is shown in Fig. 11. The blank area in Fig.11c-11d when θ is near 90 ◦ means that the null-to-null angular resolution defined in 1 does not exist in such areas. It shows that the elev ation angular resolution of spherical DCAA is uniform in both dimensions, while the azimuth angular resolution is uniform in the azimuth dimen- sion, but increases in the elev ation dimension. The ele vation angular resolution of con ventional UP A using KPC-based HBF does not exhibit a monotonically increasing or decreasing trend in either dimension, and the azimuth angular resolution increases in both dimensions. In general, we can conclude that the proposed spherical DCAA has not only wider angular 9 resolution coverage than con ventional UP A using KPC-based HBF , but also better angular resolution ability . V . S E N S I N G A L G O R I T H M F O R S P H E R I C A L D C A A - B A S E D I S AC A. Spherical DCAA-based ISAC By substituting (19) into (3), we hav e y = S ˜ y = S L X l =0 α l r ( ϕ l , θ l ) x s + Sz , (30) where z = h P M 2 − 1 m =0 z m,n i n ∈N ∈ C N × 1 , and z m,n denotes the noise from the m th element of the n th sUP A. Suppose z m,n follows i.i.d. CSCG distribution with variance σ 2 at each time sample, i.e., z m,n ∼ C N (0 , σ 2 ) . Then we can get z ∼ C N (0 , M 2 σ 2 I N ) . 1) Communication Model: From the communication per- spectiv e, the receiv ed signal (30) at the Rx can be written as y c = h c x s + n c , (31) where h c = S P L l =0 α l r ( ϕ l , θ l ) ∈ C N RF × 1 is the equiv alent communication channel, and n c = Sz denotes the denotes the A WGN vector . Accordingly , the uplink communication rate is giv en by: R = log 2  1 + ∥ h c ∥ 2 P t M 2 σ 2  . (32) 2) Sensing Model: For sensing, the signal (30) can be reformulated as y s y s = L X l =0 ¯ α l h s ( ϕ l , θ l ) + n s , (33) where ¯ α l = α l x s , h s ( ϕ l , θ l ) = Sr ( ϕ l , θ l ) ∈ C N RF × 1 is the equiv alent steering vector , and n s = Sz ∈ C N RF × 1 denotes the noise vector . Based on (33), we try to obtain ϕ l and θ l . B. Sensing Algorithm Design By concatenating the signal y from K dif ferent time slots, we obtain the snapshot matrix Y ∈ C N RF × K Y (: , k ) = y [ k ] , (34) where k = 1 , 2 , ..., K is the index for the k th time slot. It can be reformulated as Y = HX , (35) where H = [ h s ( ϕ 0 , θ 0 ) , . . . , h s ( ϕ L , θ L )] ∈ C N RF × L is the array manifold matrix and X ∈ C L × K contains the complex amplitudes. The AoA estimation problem aims to estimate the angle pairs S = { ( ϕ 0 , θ 0 ) , . . . , ( ϕ L , θ L ) } from Y . Since the steer vector of spherical DCAA lacks rotational in variance, we employ the MUSIC algorithm. The cov ariance matrix C Y ∈ C N RF × N RF of Y is computed as C Y = 1 K YY H , (36) and its eigen value decomposition (EVD) C Y = E s Σ s E H s + E n Σ n E H n yields the signal subspace matrix E s ∈ (a) Spherical DCAA 38 39 40 41 42 38 39 40 41 42 DCAA Groundtruth Estimation (b) Spherical DCAA (c) UP A with con ventional KPC 38 39 40 41 42 38 39 40 41 42 KPC Groundtruth Estimation (d) UP A with con ventional KPC Fig. 13. 2D angle estimation results when the swarm is centered at (40 ◦ , 40 ◦ ) . Figure (a)(c) are the MUSIC spectrum. Figure (b)(d) are the groundtruth and estimation results. C N RF × ( L +1) , noise subspace matrix E n ∈ C N RF × ( N RF − L − 1) , and diagonal matrices Σ s ∈ C ( L +1) × ( L +1) and Σ n ∈ C ( N RF − L − 1) × ( N RF − L − 1) . The MUSIC spatial spectrum is then formulated as P MUSIC ( ϕ, θ ) = 1 h H s ( ϕ, θ ) E n E H n h s ( ϕ, θ ) . (37) The peaks of this spectrum correspond to the estimated AoAs, denoted by ˆ S = { ( ˆ ϕ 0 , ˆ θ 0 ) , . . . , ( ˆ ϕ L p , ˆ θ L p ) } , where L p ≤ L + 1 . The procedure is summarized in Algorithm 1. Algorithm 1 Proposed Algorithm for ISA C with spherical DCAA Input: Signal y s in (33) Output: ϕ l , θ l , ∀ l 1: Obtain Y from y s as in (34) 2: Compute cov ariance matrix C Y 3: Perform EVD of C Y to obtain noise subspace E n 4: f or θ ∈ [ − θ max , θ max ] do 5: for ϕ ∈ [ − ϕ max , ϕ max ] do 6: Calculate steering vector h s ( ϕ, θ ) 7: Evaluate P MUSIC ( ϕ, θ ) 8: end f or 9: end for 10: Find peaks of P MUSIC ( ϕ, θ ) to obtain ˆ S 11: retur n ϕ l , θ l for l = 0 , 1 , . . . , L p V I . S I M U L AT I O N R E S U LTS In this section, we provide numerical results to verify the performance of spherical DCAA in ISA C system. For spherical DCAA, we set the maximum orientation of sUP As and antenna elements in each sUP A as ϑ max = η max = π / 2 , M 2 = 256 at Rx, respectiv ely . Therefore, we can compute N ϑ = 12 , N η ( ϑ ) =  π 2 / arcsin 2 16 cos ϑ  , and the total number of sUP A is N = 397 . For KPC-based UP A, the 10 number of elev ation and azimuth codewords is N v = 17 and N h = 17 , and the ( p, q ) th codew ord’ s angle is ( ϕ p , θ q ) = (arcsin 2 p 16 , arcsin 2 q 16 ) , − 8 ≤ p, q ≤ 8 . The number of RF chains is set to N RF = 8 . The carrier frequency is set as f c = 39 GHz. The radius of the DCAA sphere is R = 0 . 2705 m. In addition, the radiation patterns of antenna elements follow the 3GPP antenna model, which is expressed in dB as [41] G dB ( ϕ, θ ) = − min  −  A dB ( ϕ = 0 , θ ) + A dB ( ϕ, θ = 0)  , A max  , with A dB ( ϕ = 0 , θ ) = A dB 0 − min n 12  θ θ 3dB  2 , SLA V o , A dB ( ϕ, θ = 0) = A dB 0 − min n 12  ϕ ϕ 3dB  2 , A max o , (38) among which θ 3dB and ϕ 3dB account for the elev ation and azimuth 3dB beamwidth, A dB 0 is the peak antenna gain in dB, A max = SLA V = 30dB . For conv entional UP A, θ KPC 3dB , ϕ KPC 3dB is set to π to cover the entire direction, while for spherical DCAA, θ DCAA 3dB , ϕ DCAA 3dB is set to 0 . 3 π , as each sUP A is only responsible for a narrower angular region. Besides, A dB 0 in con ventional UP A and spherical DCAA are set to 0dB and 12.79dB to guarantee the same total power gain for all directions. A U A V swarm is simulated by three low-altitude UA V targets with uniformly distrib uted elev ation and azimuth angles in a confined area with 5 ◦ × 5 ◦ angular region. Note that due to the close proximity of U A V swarms, adjacent targets may not be perfectly discriminated, so the criterion termed as averag e missed targ ets is incorporated to ev aluate the performance, which is defined as ε = 1 Q Q X i =1  card  S i  − card  ˆ S i   , (39) where Q denotes the total number of testing rounds, S i and ˆ S i denote the real and estimation angle set of i th test round, respectiv ely . W e assume that card  S i  ≥ card  ˆ S i  , which holds when the noise is rather small. The 2D angle estimation results for the U A V swarm are presented in Fig. 13. When the sw arm is centered at (40 ◦ , 40 ◦ ) , the proposed spherical DCAA method successfully detects all three targets and estimates their angles ( θ , ϕ ) with high precision. In contrast, the con ventional UP A fails to distin- guish these targets due to sev ere spectral aliasing, leading to significant estimation errors. This performance degradation is primarily attributed to the inferior angular resolution, i.e., wider main lobe beamwidth of the con ventional UP A. Fig. 14 compares the av erage number of missed targets between spherical DCAA and con ventional UP A. Overall, spherical DCAA demonstrates superior detection ability . For a given ele vation U A V swarm center angle θ , its average missed target count remains nearly constant across dif ferent azimuth angles ϕ , due to its inv ariant angular resolution in the azimuth domain. Howe ver , this count increases with θ because the azimuth angular resolution degrades at higher elev ations, affecting the detection ability . Conv ersely , for con ventional UP A, the number of missed targets generally increases with DCAA average missing targets 0 20 40 60 80 ? (degree) 0 20 40 60 80 3 (degree) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 count (a) KPC average missing targets 0 20 40 60 80 ? (degree) 0 20 40 60 80 3 (degree) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 count (b) Fig. 14. Comparison of average missing targets between spherical DCAA and conventional UP A using KPC-based HBF . In each figure, the x -axis and y -axis represent the azimuth and elev ation U A V swarm center, respectiv ely . (a) spherical DCAA, (b) conv entional UP A using KPC-based HBF . both θ and ϕ , consistent with a progressiv e loss of angular resolution. An exception occurs at θ = 80 ◦ , where its ele v ation angular resolution decreases as ϕ increases. This positiv e effect outweighs the increase in azimuth resolution, leading to better detection performance at larger ϕ . Fig. 15 presents the RMSE comparison for angle ( θ , ϕ ) estimation between spherical DCAA and con ventional UP A using KPC-based HBF . Overall, spherical DCAA achiev es higher estimation accuracy . For spherical DCAA, the elev ation angle RMSE is lower than the azimuth angle RMSE. Both of them exhibit similar trends: the RMSE remains stable for a fixed θ across v arying ϕ , o wing to a constant angular resolution in the azimuth domain. As θ increases, howe ver , the azimuth angular resolution degrades, leading to a rise in the azimuth angle estimation RMSE. The elev ation angle estimation RMSE also increases because the broadening azimuth main lobe interferes with the ele vation main lobe, degrading estimation performance. For con ventional UP A using KPC-based HBF , the elev ation angle RMSE is similarly lower than the azimuth angle RMSE, and both generally increase with θ and ϕ , consistent with its deteriorating angular resolution. Fig. 16 compares spectrum efficienc y between the two types of array with and without directional antenna elements. It can be seen that when the directional antenna elements are employed, spherical DCAA achie ves significantly higher spectrum efficienc y than that of con ventional UP A using KPC- based HBF across different transmit SNR levels. This perfor- mance impro vement is attrib uted to the enhanced beamforming gain of the spherical DCAA. In contrast, when isotropic antennas are used, con ventional UP A using KPC-based HBF slightly outperforms spherical DCAA, as spherical DCAA ’ s finer angular resolution makes it more challenging to align exactly with the signal directions. In this condition, the signal strength of spherical DCAA degrades faster than con ventional UP A. V I I . C O N C L U S I O N In this paper, we proposed a nov el spherical DCAA ar- chitecture for low-altitude U A V swarm ISAC. Compared to con ventional arrays with hybrid analog/digital beamform- ing, the spherical DCAA of fers significant hardware cost reduction, improved energy focusing, and superior angular 11 DCAA 3 RMSE 0 20 40 60 80 ? (degree) 0 20 40 60 80 3 (degree) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 degree (a) KPC 3 RM SE 0 20 40 60 80 ? (degree) 0 20 40 60 80 3 (degree) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 degree (b) DCAA ? RMSE 0 20 40 60 80 ? (degree) 0 20 40 60 80 3 (degree) 0.2 0.4 0.6 0.8 1 degree (c) KPC ? RMSE 0 20 40 60 80 ? (degree) 0 20 40 60 80 3 (degree) 0.2 0.4 0.6 0.8 1 degree (d) Fig. 15. Comparison of ele vation and azimuth angular estimation RMSE between spherical DCAA and con ventional UP A using KPC-based HBF . In each figure, the x -axis and y -axis represent the azimuth and elevation UA V swarm center , respectiv ely . (a) elevation angle estimation of spherical DCAA, (b) elev ation angle estimation of con ventional UP A using KPC-based HBF , (c) azimuth angle estimation of spherical DCAA, (d) azimuth angle estimation of con ventional UP A using KPC-based HBF . -10 -5 0 5 10 15 20 10 12 14 16 18 20 22 24 Fig. 16. Comparison of spectrum efficiency between spherical DCAA and con ventional UP A using KPC-based HBF with and without directive antenna elements. resolution—with uniform ele vation resolution and elev ation- dependent azimuth resolution. A systematic spherical arrange- ment of sUP As along with a selection network was designed, and an efficient sensing algorithm was de veloped for joint parameter estimation. Simulation results demonstrate that the spherical DCAA outperforms conv entional arrays in angular resolution, target detection, and spectral efficienc y , confirming its strong potential for future UA V swarm ISAC systems. A P P E N D I X A P RO O F O F L E M M A 3 According to equation (11), for the main lobe beamwidth in the azimuth direction, θ is fixed as θ = ϑ and the null points satisfy ϕ null = η + arcsin  2 p M cos ϑ  , or ϕ null = η + arccos  1 − 2 q M cos ϑ sin ϑ  , (40) where p = ± 1 , q = ± 1 . Note that arcsin  2 p M cos ϑ  < arccos  1 − 2 p M cos ϑ  < arccos  1 − 2 q M cos ϑ sin ϑ  , the adjacent null points in the ϕ direction are ϕ null = η + arcsin  2 p M cos ϑ  . (41) By setting p = ± 1 , we have ∆ ϕ = 2 arcsin  2 M cos ϑ  . (42) Similarly , For the main lobe beam width in the elev ation direction, ϕ is fixed as ϕ = η and the adjacent null points satisfy θ null = ϑ + arcsin  2 q M  , (43) where q = ± 1 , which yields ∆ θ = 2 arcsin  2 M  . (44) Therefore, the null to null beamwidth of the main lobe can be obtained as BW ( ϑ, η ) =  2 arcsin  2 M cos ϑ  , 2 arcsin  2 M   . (45) A P P E N D I X B P RO O F O F T H E O R E M 1 𝑂 𝐴 𝐵 𝑂′ 𝜃 ∆ 𝜂 Fig. 17. Demonstration of adjacent sUP As with same elevation angle. The direction of a sUP A is gi ven by ( η p,q , ϑ q ) =  p arcsin  2 M cos( q arcsin 2 M )  , q arcsin 2 M  . (46) Let two sUP As A and B ha ve directions ( η a , ϑ a ) and ( η b , ϑ b ) , respectively . Regarded as points on a unit sphere centered at O , their Cartesian coordinates are ( x a , y a , z a ) = (cos ϑ a cos η a , cos ϑ a sin η a , sin ϑ a ) , (47) ( x b , y b , z b ) = (cos ϑ b cos η b , cos ϑ b sin η b , sin ϑ b ) . (48) Hence the cosine of the angle α + between − → O A and − − → O B is cos α + = − → O A · − − → O B = cos ϑ a cos ϑ b cos( η a − η b )+sin ϑ a sin ϑ b . (49) 12 Case 1: Different elevation angles. If ϑ a  = ϑ b , then from (49), cos α + ≤ cos( ϑ a − ϑ b ) , so that | α + | ≥ | ϑ a − ϑ b | ≥ arcsin 2 M . Case 2: Same elevation angle. Let ϑ a = ϑ b = ϑ . Denote c = cos ϑ , s = sin ϑ . From (49) and the expression of η p,q , we have cos α + = c 2 cos  arcsin 2 M c  + s 2 = c 2 r 1 − 4 M 2 c 2 + s 2 ≜ f ( ϑ ) , (50) where 0 < ϑ < arccos 2 M (so that c > 2 M ). Define t = M c ∈ (2 , M ) and set w ( t ) = t − t 2 − 2 √ t 2 − 4 . Differentiating f ( ϑ ) gi ves f ′ ( ϑ ) = 2 s M w ( t ) . (51) Since s > 0 for ϑ ∈ (0 , arccos 2 M ) , the sign of f ′ ( ϑ ) is that of w ( t ) . The function w ( t ) is continuous on (2 , M ) and has no zero there: if w ( t ) = 0 , then t √ t 2 − 4 = t 2 − 2 , squaring which leads to the contradiction 0 = 4 . Evaluating at t = 3 giv es w (3) = 3 − 7 / √ 5 ≈ − 0 . 13 < 0 ; hence w ( t ) < 0 for all t ∈ (2 , M ) . Consequently f ′ ( ϑ ) < 0 , so f ( ϑ ) is strictly decreasing on [0 , arccos 2 M ) . Its maximum is therefore attained at ϑ = 0 : f ( ϑ ) ≤ f (0) = r 1 − 4 M 2 . From (50) we obtain cos α + ≤ q 1 − 4 M 2 , which implies α + ≥ arccos r 1 − 4 M 2 = arcsin 2 M . Combining both cases , we conclude that for any two sUP As, α + ≥ arcsin 2 M . A P P E N D I X C P RO O F O F T H E O R E M 2 For spherical DCAA, according to (8) and (25)-(27), by letting ϕ = ϕ ′ , we can get sin( θ − θ ′ ) = ± 2 / M , which yields θ 1 = θ ′ − arcsin 2 M , θ 2 = θ ′ + arcsin 2 M . Then, the elev a- tion angular resolution can be obtained as γ DCAA v ( ϕ ′ , θ ′ ) = 1 2 ∆ θ ( ϕ ′ , θ ′ ) = 1 2 | θ 1 − θ 2 | = arcsin 2 M . Similarly , by letting θ = θ ′ , we will ha ve cos θ ′ sin( ϕ − ϕ ′ ) = ± 2 / M or sin θ ′ cos θ ′ (1 − cos( ϕ − ϕ ′ )) = ± 2 / M . Based on previous result (40)-(43), ϕ 1 = ϕ ′ − arcsin( 2 M cos θ ′ ) , ϕ 2 = ϕ ′ + arcsin( 2 M cos θ ′ ) , and the azimuth angular resolution can be obtained as γ DCAA h ( ϕ ′ , θ ′ ) = 1 2 ∆ ϕ ( ϕ ′ , θ ′ ) = 1 2 | ϕ 1 − ϕ 2 | = arcsin 2 M cos θ ′ For con ventional UP A using KPC-based HBF , by fixing ϕ as ϕ = arcsin sin ϕ ′ cos θ ′ in (24), all the possible adjacent nulls regarding θ need to satisfy either one of the following equations cos θ sin ϕ ′ cos θ ′ − sin ϕ ′ = ± 2 M or sin θ − sin θ ′ = ± 2 M . (52) By solving for θ , we hav e θ = ( − 1) n arcsin(sin θ ′ ± 2 M ) + nπ , n ∈ Z , or θ = ± arccos  cos θ ′  1 ± 2 M sin ϕ ′  + 2 mπ , m ∈ Z . (53) Then we find the sets of nulls that are bigger and smaller than θ ′ , denoted by Θ + = { θ | θ > θ ′ } , Θ − = { θ | θ < θ ′ } . According to (26) and (25), we hav e θ 1 = min Θ + , θ 2 = max Θ − . (54) γ UP A v ( ϕ ′ , θ ′ ) = 1 2 ∆ θ ( ϕ ′ , θ ′ ) = 1 2 | θ 1 − θ 2 | = 1 2  min Θ + − max Θ −  =                E − F , if X ≤ 1 , Y > 1 , | Z | ≥ 1 G − F , if X > 1 , Y > 1 , | Z | ≥ 1 A − B , if X ≤ 1 , Y > 1 , | Z | ≤ 1 A − C, if X ≤ 1 , Y ≤ 1 , | Z | ≤ 1 D − B , if X > 1 , Y > 1 , | Z | ≤ 1 D − C, if X > 1 , Y ≤ 1 , | Z | ≤ 1 , (55) where A = min  arcsin(sin | θ ′ | + 2 M ) , arccos(cos | θ ′ | (1 − 2 M sin | ϕ ′ | ))  , B = max  arcsin(sin | θ ′ | − 2 M ) , − arccos(cos | θ ′ | (1 − 2 M sin | ϕ ′ | ))  , C = max  arcsin(sin | θ ′ | − 2 M ) , arccos(cos | θ ′ | (1 + 2 M sin | ϕ ′ | ))  , D = max  π − arcsin(sin | θ ′ | − 2 M ) , arccos(cos | θ ′ | (1 − 2 M sin | ϕ ′ | ))  , E = arcsin(sin | θ ′ | + 2 M ) , F = arcsin(sin | θ ′ | − 2 M ) , G = π − arcsin(sin | θ ′ | − 2 M ) X = sin | θ ′ | + 2 M , Y = cos | θ ′ | (1 + 2 M sin | ϕ ′ | ) , Z = cos | θ ′ | (1 − 2 M sin | ϕ ′ | ) . Similarly , the possible adjacent nulls regarding ϕ need to satisfy the following equation cos θ ′ sin ϕ − sin ϕ ′ = ± 2 / M . (56) By solving for ϕ , we have ϕ = ( − 1) n arcsin  sin ϕ ′ ± 2 / M cos θ ′  + nπ , n ∈ Z . (57) Then we find the sets of nulls that are bigger and smaller than ϕ ′ , denoted by Φ + = { ϕ | ϕ > ϕ ′ } and Φ − = { ϕ | ϕ < ϕ ′ } , respectiv ely . According to (27) and (25), we have ϕ 1 = min Φ + , ϕ 2 = max Φ − . (58) γ UP A h ( ϕ ′ , θ ′ ) = 1 2 ∆ ϕ ( ϕ ′ , θ ′ ) = 1 2 | ϕ 1 − ϕ 2 | = 1 2  min Φ + − max Φ −  . (59) The complete equation is giv en in (60) R E F E R E N C E S [1] Y . Zeng, Q. Wu, and R. Zhang, “ Accessing from the sky: A tutorial on U A V communications for 5G and beyond, ” Proc. IEEE , vol. 107, no. 12, pp. 2327–2375, Dec. 2019. [2] Y . Song, Y . Zeng, Y . Y ang et al. , “ An overvie w of cellular ISA C for low-altitude UA V: New opportunities and challenges, ” IEEE Commun. Mag. , vol. 63, no. 12, pp. 88–95, Dec. 2025. 13 [3] Y . Jiang, X. 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A vailable: https://arxiv .org/abs/2512.07330 [41] 5G; Study on Channel Model for F r equencies fr om 0.5 to 100 GHz , document 38.901, V ersion 16.1.0, 3GPP , T echnical Specification (TS), Nov . 2020. γ UP A h ( ϕ ′ , θ ′ ) =              arcsin  sin ϕ ′ +2 / M cos θ ′  − arcsin  sin ϕ ′ − 2 / M cos θ ′  if | sin ϕ ′ +2 / M cos θ ′ | < 1 , | sin ϕ ′ − 2 / M cos θ ′ | < 1 π − 2 arcsin  sin ϕ ′ − 2 / M cos θ ′  if | sin ϕ ′ +2 / M cos θ ′ | > 1 , | sin ϕ ′ − 2 / M cos θ ′ | < 1 π + arcsin  sin ϕ ′ +2 / M cos θ ′  + arcsin  sin ϕ ′ +2 / M cos θ ′  if | sin ϕ ′ +2 / M cos θ ′ | < 1 , | sin ϕ ′ − 2 / M cos θ ′ | > 1 NaN if | sin ϕ ′ +2 / M cos θ ′ | > 1 , | sin ϕ ′ − 2 / M cos θ ′ | > 1 . (60)