Arched Uniform Linear Arrays Overview

• Arched uniform linear arrays are antenna configurations with uniformly spaced elements along a curved circular arc, used in HMIMO, XL-MIMO, and projected designs.
• In far-field scenarios, preserving the aperture span ensures spatial DoF remains similar to straight ULAs, while in near-field settings, curvature can enhance DDRayl distance, UPD, and SNR.
• Projected arch type arrays adopt arch-based designs to approximate optimal non-uniform deployments, improving effective multiplexing gain and linking to sparse-array construction methods.

Searching arXiv for the cited papers to ground the article in the current literature. Arched uniform linear arrays are antenna arrays in which the elements are uniformly spaced along a circular arc rather than along a straight line. In current arXiv literature, the term appears in several closely related but non-identical senses: as a physically curved ULA for holographic multiple-input multiple-output (HMIMO) and fluid antenna systems (FAS), as an extremely large-scale uniform arc array (XL-UAA) for near-field communications, and, in a projected sense, as a projected arch type distribution used to design non-uniform linear arrays (Xue et al., 16 Sep 2025, Li et al., 2024, Wang et al., 2016). Across these settings, curvature changes the element geometry and the wavefront sampling process, but its effect on spatial correlation, degrees of freedom (DoF), and signal-to-noise ratio (SNR) depends strongly on propagation regime, scattering model, and whether the arc is physical, projected, or synthesized in the beam domain.

1. Geometry, terminology, and array models

A conventional ULA places antennas on a straight line with constant spacing $d$. An arched ULA bends that line into a circular arc while preserving uniform spacing along the arc. In HMIMO/FAS, the arched ULA is treated as a curved sampling segment of a holographic surface, and the curvature may be static or fluidically adaptable (Xue et al., 16 Sep 2025). In near-field XL-MIMO, the same basic object is formalized as a uniform arc array, with elements lying on a circular arc of radius $r_0$ and central angle $\varphi$ in the $x\!-\!y$ plane (Li et al., 2024).

Because the terminology is overloaded, three distinct meanings should be separated.

Context	“Arched” object	Geometric status
HMIMO/FAS	ULA on a circular arc	Physically curved array
XL-UAA	Uniform arc array	Physically curved array
PAT NULA	Arch projected onto a line	Linear array with projected non-uniform centers

For HMIMO/FAS, the arched ULA is modeled as a vertical arc in the $YZ$-plane with total arc length $L$, radius of curvature $R$, and bending angle

$\beta=\frac{L}{2R}\in\left[0,\frac{\pi}{2}\right].$

With $N$ elements, the arc spacing is $d_{yz}=L/(N-1)$, and the $r_0$0-th element is parameterized by

$r_0$1

with coordinates

$r_0$2

This parameterization makes the curvature explicit and centers the array relative to the arc midpoint (Xue et al., 16 Sep 2025).

For XL-UAA, the array lies in the $r_0$3 plane on a circular arc of radius $r_0$4, with odd $r_0$5, half-wavelength spacing $r_0$6, uniform inter-element angle

$r_0$7

and element angles

$r_0$8

The array is thus an arched ULA in the literal conformal-array sense, rather than a mere perturbation of a straight aperture (Li et al., 2024).

A separate usage appears in projected arch type arrays. There, uniformly spaced points on a circular arch are orthogonally projected onto a line, and the projected coordinates are used as centers of compact ULAs in a non-uniform linear deployment. The resulting array is physically linear, not curved, even though its design is controlled by an arch angle (Wang et al., 2016). This distinction is central to the literature and avoids conflating physical curvature with projected non-uniform spacing.

2. Far-field correlation and degrees of freedom in arched HMIMO arrays

The closed-form far-field analysis of arched ULAs in HMIMO/FAS assumes far-field propagation,

$r_0$9

a narrowband channel with wavelength $\varphi$0, half-space isotropic scattering

$\varphi$1

and a single-antenna user in a downlink setting (Xue et al., 16 Sep 2025). Under these assumptions, the distance from the user to the $\varphi$2-th element is linearized, and the array steering vector is obtained from the first-order phase offsets induced by curvature.

The element-pair phase difference is the key quantity governing spatial correlation. With isotropic half-space scattering, the correlation matrix entry between elements $\varphi$3 and $\varphi$4 is

$\varphi$5

Using trigonometric simplification, a Jacobi–Anger expansion, and the dominance of the zero-order Bessel term, the correlation reduces to a sinc-type law. The main closed-form result is

$\varphi$6

or, equivalently,

$\varphi$7

This expression shows that the correlation depends on the angular separation along the arch, the aperture-to-wavelength ratio $\varphi$8, and the curvature parameter $\varphi$9 (Xue et al., 16 Sep 2025).

In the same analysis, spatial DoF is extracted from the eigenvalue spectrum of $x\!-\!y$0. For a straight linear aperture of physical length $x\!-\!y$1, the asymptotic HMIMO result is

$x\!-\!y$2

The arched-ULA analysis shows that, under half-space isotropic scattering, the arched case remains

$x\!-\!y$3

with eigenvalue distributions that remain essentially stable across a wide range of curvatures. The paper states that isotropic scattering results in DoF being dominated by the maximum span of the HMIMO array, such that shape effects are weakened, and bending does not significantly reduce the available spatial DoF (Xue et al., 16 Sep 2025).

The numerical example highlighted in the paper uses a $x\!-\!y$4-element arched ULA with $x\!-\!y$5 m at $x\!-\!y$6 GHz, so $x\!-\!y$7 mm and

$x\!-\!y$8

spatial DoF. The reported eigenvalue plots show about that many dominant eigenvalues for various element spacings and curvatures. The same study also notes only slight DoF decrease with increasing curvature for practically sized arrays, even for significant bending angles up to approximately $x\!-\!y$9 (Xue et al., 16 Sep 2025).

A common misconception is that bending necessarily destroys spatial richness. The closed-form HMIMO result does not support that view under isotropic far-field conditions: if the span $YZ$0 is preserved, the dominant DoF scaling is essentially the same as for a straight ULA (Xue et al., 16 Sep 2025).

3. Near-field uniform arc arrays and conformal XL-MIMO

Near-field analysis produces a different picture because the channel is no longer governed by a plane-wave approximation. The XL-UAA model adopts a non-uniform spherical wavefront channel, with user position

$YZ$1

and element-to-user distance

$YZ$2

With $YZ$3 denoting the distance from the user to the arc center $YZ$4, and $YZ$5 the user angle as seen from that center, the explicit distance formula becomes

$YZ$6

under the case $YZ$7, so that the user lies outside the circle defined by the arc (Li et al., 2024).

The corresponding line-of-sight channel is

$YZ$8

so both phase and amplitude vary across the arc. This differs fundamentally from the far-field HMIMO setting, where the primary effect is phase progression and correlation under isotropic scattering (Li et al., 2024).

To characterize the near-field region, the paper defines the direction-dependent Rayleigh distance (DDRayl) and the uniform power distance (UPD). The DDRayl distance is the smallest $YZ$9 such that the maximum phase error between the exact spherical distance and its first-order approximation is at most $L$0. The approximate XL-UAA expression is

$L$1

The paper reports that, at $L$2, the Rayleigh distance of XL-UAA equals that of XL-ULA for the same aperture, whereas at a large incident angle such as $L$3, XL-UAA can achieve

$L$4

while the XL-ULA Rayleigh distance at that angle is essentially $L$5 for the same aperture (Li et al., 2024).

UPD addresses amplitude variation rather than phase variation. It is defined as the shortest distance at which all antenna elements have relatively similar power within a prescribed threshold $L$6. The reported comparison shows that XL-UAA has larger DDRayl and UPD at all angles for the example configuration, with particularly strong gains at large incident angles for phase and at small incident angles for amplitude (Li et al., 2024).

The single-user uplink SNR under maximum-ratio combining is

$L$7

and the paper derives a closed-form approximation

$L$8

where $L$9 is expressed as a sum of two arctangent terms involving $R$0, $R$1, and $R$2 (Li et al., 2024). In the asymptotic regime $R$3 with fixed arc support $R$4, the SNR saturates to

$R$5

Thus the asymptotic SNR depends on the projection distance of the user to the middle of the arc array rather than increasing indefinitely with $R$6 (Li et al., 2024).

Numerical results reported for $R$7 m, $R$8, and $R$9 show that the exact sum and the closed-form SNR match perfectly, that SNR saturates as aperture grows, and that XL-UAA achieves higher SNR than XL-ULA for the same aperture, especially at larger incident angles (Li et al., 2024). In contrast to the far-field HMIMO result, curvature is not merely benign here; it is an active geometric resource that enlarges the usable near-field region and improves angle-dependent performance.

4. Projected arch type arrays, Fekete-point structure, and effective multiplexing gain

Arched-array ideas also appear in line-of-sight mmWave MIMO under the projected arch type framework. In that setting, the physical arrays remain linear, but the group centers of compact ULAs are placed according to the projection of uniformly spaced points on a circular arch onto the interval $\beta=\frac{L}{2R}\in\left[0,\frac{\pi}{2}\right].$0. The normalized projected positions are

$\beta=\frac{L}{2R}\in\left[0,\frac{\pi}{2}\right].$1

for an arch angle $\beta=\frac{L}{2R}\in\left[0,\frac{\pi}{2}\right].$2 (Wang et al., 2016).

The motivating performance metric is the effective multiplexing gain (EMG), defined as

$\beta=\frac{L}{2R}\in\left[0,\frac{\pi}{2}\right].$3

where $\beta=\frac{L}{2R}\in\left[0,\frac{\pi}{2}\right].$4 are channel-gain eigenvalues and $\beta=\frac{L}{2R}\in\left[0,\frac{\pi}{2}\right].$5 is the eigenvalue-quality threshold. The paper proves that the asymptotically optimal deployment maximizing achievable EMG should follow the groupwise Fekete-point distribution: antennas are grouped into $\beta=\frac{L}{2R}\in\left[0,\frac{\pi}{2}\right].$6 separate ULAs with minimum feasible intra-group spacing, and the centers of those $\beta=\frac{L}{2R}\in\left[0,\frac{\pi}{2}\right].$7 ULAs follow the Fekete-point distribution (Wang et al., 2016).

Projected arch type design is introduced as a one-parameter approximation to that optimal structure. The paper states that for $\beta=\frac{L}{2R}\in\left[0,\frac{\pi}{2}\right].$8, the projected-arch representation is exact, and for larger $\beta=\frac{L}{2R}\in\left[0,\frac{\pi}{2}\right].$9 up to $N$0, the errors are on the order of $N$1. It further generalizes the asymptotic design to a groupwise PAT NULA with tunable angle $N$2, chosen numerically to minimize $N$3 for finite-SNR operation (Wang et al., 2016).

The performance comparisons are explicitly favorable to projected-arch-type deployments. For $N$4 and $N$5, the paper reports

$N$6

and

$N$7

It also reports, for indoor $N$8 GHz with $N$9 m, that EMG $d_{yz}=L/(N-1)$0 is supported up to approximately $d_{yz}=L/(N-1)$1 m for ULA and $d_{yz}=L/(N-1)$2 m for optimized NULA, while EMG $d_{yz}=L/(N-1)$3 is supported up to approximately $d_{yz}=L/(N-1)$4 m for ULA and $d_{yz}=L/(N-1)$5 m for optimized NULA (Wang et al., 2016).

This literature does not analyze a physically arched aperture. Instead, it uses arch geometry to synthesize a nearly optimal non-uniform linear placement. The distinction matters: the curvature is in the design parametrization, not in the realized aperture.

5. Coarray-based sparse-array design and relevance to curved arrays

A different line of work develops ULA fitting as a sparse-array design principle in which a sparse linear array is constructed as a concatenation of sub-ULAs on a normalized integer grid. The array is represented by the polynomial

$d_{yz}=L/(N-1)$6

and its difference coarray (DCA) by

$d_{yz}=L/(N-1)$7

This framework allows the DCA to be decomposed into self-difference coarrays of the sub-ULAs and inter-difference coarrays between them, with explicit control over hole-free regions, low mutual coupling, and large uniform DoF (Shi et al., 2021).

The paper derives two closed-form sparse-array families, UF-3BL and UF-4BL, and reports lower-bound uniform-DoF scaling

$d_{yz}=L/(N-1)$8

so asymptotically the uniform DoF is approximately $d_{yz}=L/(N-1)$9. It also emphasizes that mutual coupling is dominated by the small-lag weights $r_0$00, and that the proposed constructions suppress these terms while maintaining large coarray continuity (Shi et al., 2021).

The same paper explicitly restricts itself to linear arrays, but it also argues that its polynomial/DCA viewpoint is relevant to arched uniform linear arrays when a scalar index can be assigned along the arc. A plausible implication is that an arched array can be mapped to an effective linearized index, such as arc length or projected position, after which the same sub-ULA, DCA, and low-coupling logic can be applied. The paper frames this as an adaptation rather than a completed curved-array theory, so its relevance to arched ULAs is methodological rather than definitive (Shi et al., 2021).

This creates an important conceptual bridge. In the HMIMO and XL-UAA literature, curvature is a physical property of the aperture. In ULA fitting, by contrast, array geometry is manipulated algebraically through integer-grid placement and coarray structure. The two perspectives are not equivalent, but they intersect when curved arrays are analyzed through a monotonic scalar parametrization.

6. Curved beams from straight ULAs, misconceptions, and design implications

Arched uniform linear arrays must also be distinguished from straight ULAs that synthesize curved near-field beams by phase engineering. In the near-field beamforming literature, a straight ULA on the $r_0$01-axis can generate Bessel beams and curving beams through closed-form phase distributions, even though the aperture itself remains linear (Uchimura et al., 18 Mar 2025).

For Bessel beams, the wavefront is conical, and the steering-phase law for angle $r_0$02 and cone parameter $r_0$03 is

$r_0$04

The achievable steering condition is

$r_0$05

which yields the stated steering-angle limit $r_0$06. The maximum propagation distance is

$r_0$07

and the spatial-sampling condition for faithful Bessel-beam synthesis is

$r_0$08

The paper states that half-wavelength spacing is sufficient for any Bessel beam satisfying the steering condition (Uchimura et al., 18 Mar 2025).

For curving beams, the desired trajectory is taken as a parabola,

$r_0$09

and the phase law is derived from the tangent-envelope geometry of rays launched by the ULA. The trajectory parameters are optimized via a Lagrangian method and a linear program so that the beam reaches a user while avoiding one obstacle (Uchimura et al., 18 Mar 2025). This is an arched beam, not an arched array.

The distinction resolves a frequent ambiguity in the terminology. A physically arched ULA changes the array manifold itself; a projected arch type array remains physically linear; and a straight ULA producing a curving beam remains physically linear while shaping the field trajectory. These are related by geometry-aware design, but they are not interchangeable (Wang et al., 2016, Uchimura et al., 18 Mar 2025).

Taken together, the literature supports several design implications. For far-field HMIMO/FAS under half-space isotropic scattering, aperture span $r_0$10 and element density dominate, and curvature is largely benign from a DoF standpoint (Xue et al., 16 Sep 2025). For near-field conformal XL-MIMO, curvature can enlarge DDRayl distance, enlarge UPD, and improve SNR, especially at larger incident angles (Li et al., 2024). For LoS multiplexing on linear apertures, projected-arch center placement is a practical surrogate for groupwise Fekete-point design (Wang et al., 2016). For obstacle-aware near-field transmission, a straight ULA may synthesize an arched propagation path without any physical curvature at all (Uchimura et al., 18 Mar 2025).

A plausible synthesis of these results is that “arched uniform linear array” is best treated as a family of geometry-aware constructions rather than as a single canonical object. In one regime, curvature barely perturbs the fundamental spatial DoF; in another, it materially improves near-field service geometry; and in yet another, the arch is not an aperture shape but a design variable governing either linear-array placement or beam trajectory.