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Modeling and Performance Evaluation of Distributed Satellite Antennas: Space Is Much Closer Than One Might Think

Giacomo Bacci, Riccardo De Gaudenzi, Marco Luise, Luca Sanguinetti

arXiv:2610.10876v1cs.ITmath.IT

TL;DR

Large FoA apertures can place users in the radiative near field even at satellite-link distances, making propagation and geometry assumptions important to performance evaluation. The paper evaluates those assumptions using spherical models and exact spherical-wave propagation. Local plane-wave precoding closely matches the reference, whereas global plane-wave precoding can cause full outage in the investigated configurations.

Abstract

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Distributed satellites can deliver broadband connectivity directly to commercial handsets. In the formation of arrays (FoA) architecture, cooperating satellites synthesize a large aperture whoseextent depends on their number, arrangement, and spacing, with intersatellite distances ranging from tens of meters to hundreds of kilometers. For representative S-band geostationary and low-Earth-orbit configurations, the Fraunhofer distance may exceed the nadir link range at spacings on the order of a hundred meters. Therefore, accurate assessment of FoA performance across satellite populations and spacings requires spherical-wave modeling for users in the radiative near field, below the formation's Fraunhofer distance. Planar models of satellite positions, array orientations, and user locations must also be assessed against spherical geometries accounting for orbital and Earth curvature. This paper combines these models to quantify how simplifying assumptions affect throughput and outage probability. Numerical results show that multi-antenna precoding based on a local plane-wave approximation within each satellite array closely matches the spherical-wave reference. A global plane-wave approximation can instead cause severe losses, including full outage. In the investigated scenarios, formation curvature has little impact, whereas neglecting Earth curvature overestimates low-Earth-orbit throughput. These findings clarify when geometric and propagation approximations remain reliable and demonstrate that, for large distributed antenna apertures, space is much closer than one might think.

I. Introduction

Large distributed apertures can put handset users in the radiative near field, making propagation and geometric assumptions consequential to performance estimates. The paper compares those assumptions against spherical geometry and exact spherical-wave propagation.

A. Motivation and Contribution

  • FoA apertures can put ground terminals in the radiative near field at spacings of about 100 m, so plane-wave assumptions may distort precoding and performance assessments.For the considered 2.2 GHz GEO and LEO systems, the nadir transition occurs at about one hundred meters; it occurs at smaller spacings at higher carrier frequencies.Source: A. Motivation and Contribution · p. 1
  • The paper evaluates throughput and outage against a spherical-wave reference, finding local plane-wave precoding closely matches it while global plane-wave precoding can cause full outage.It also develops spherical models of satellite positions, array orientations, and user deployment, and assesses planar approximations.Source: A. Motivation and Contribution · p. 2

B. Related Work and Positioning

  • The contribution is a joint assessment of geometry and propagation assumptions, since channel models must be evaluated alongside the geometries used to construct them.Existing distance-based and suitably constructed local far-field models can represent propagation across the aperture; the paper assesses them against a common spherical-wave reference.Source: B. Related Work and Positioning · p. 2

C. Reproducible Research

  • The paper provides MATLAB code for the geometries, three channel models, and simulations of beam patterns, throughput, and outage probability.The release includes configurations to reproduce reported results and explore alternative system parameters.Source: C. Reproducible Research · p. 2

D. Organization and Notation

  • The paper presents space- and user-segment geometries before channel modeling and performance evaluation, with main derivations in the appendices.Its notation uses boldface lowercase and uppercase letters for column vectors and matrices, respectively.Source: D. Organization and Notation · p. 2

II. Space-Segment Modeling

The space-segment model places satellite centers on a spherical formation and treats each satellite array as locally planar. A planar grid is recovered as a limiting approximation, while the spherical construction can yield different satellite positions at wider spacings.

II. Space-Segment Modeling

  • The model places S satellites at orbital radius R and defines each satellite’s antenna-element positions relative to its center.The formation center is the origin, and the Earth center is at [0, 0, R]T.Source: §II · p. 3

A. Spherical Formation Geometry  ∆

  • A radially equidistant spherical grid sets adjacent central-row satellites apart by chord length ∆ and places other satellites by geodesic distance from the formation center.When the angular extent is small, the construction reduces to the regular planar grid used in prior work.Source: A. Spherical Formation Geometry  ∆ · p. 3
  • For the 49-satellite LEO example, planar and conformal layouts differ by 62.3 m at ∆=R/1000 and 620.1 km at ∆=R/10.Fig. 2 contrasts the planar and conformal configurations at those two spacings.Source: A. Spherical Formation Geometry  ∆ · p. 3, Figure 2 · p. 3

B. Satellite-Array Geometry

Each satellite’s array is modeled on a tangent plane to the orbital sphere, with its orientation aligned to local directions. The spherical formation model also accommodates partial formations and can extend to constellations of formations.

B. Satellite-Array Geometry

  • Because each array is much smaller than the orbital radius, its elements are modeled on a tangent plane at the satellite center.The array normal is fixed by the tangent plane, while local azimuthal and meridional axes set its remaining orientation.Source: B. Satellite-Array Geometry · p. 4

C. Formation and Constellation Architecture

  • The conformal construction supports partially populated grids, letting the number of satellites trade off against the synthesized aperture.The cited LEO example removes the four corners of a 3 × 3 grid, leaving five satellites in a cross.Source: C. Formation and Constellation… · p. 4
  • Multiple formations can be arranged as a constellation, with each formation acting as a large-aperture virtual satellite.The architecture distinguishes intra-formation spacing ∆1 from the much larger inter-formation spacing ∆2.Source: C. Formation and Constellation… · p. 4

III. User-Segment Modeling

Users are distributed over a spherical Earth within a coverage cap defined by a minimum elevation constraint. Compared with the spherical-cap model, the flat-Earth approximation can shrink LEO coverage and bias performance estimates.

III. User-Segment Modeling

  • The coverage cap is set by the minimum user elevation angle, with the spherical and flat-Earth coverage radii converging at large εmin.The spherical model also accounts for Earth-central angles and places users on Earth’s surface.Source: §III · p. 4
  • The reference model places users uniformly per unit area on a spherical cap and enforces minimum separation using great-circle distance.Coverage radius is measured as a geodesic on Earth’s surface; the flat-Earth model is a limiting approximation for small cap angles.Source: §III · pp. 4–5
  • At εmin=40° in the LEO scenario, spherical-cap coverage exceeds flat-Earth coverage by 26 km, or about 5%.The models agree at large minimum elevation angles; the flat model reduces cell-edge coverage and artificially raises user density, biasing the KPI discussed in Sect. V.B.Source: §III · p. 5, Figure 4 · p. 5

IV. Channel Modeling

This section characterizes the Fraunhofer distance for a curved formation aperture and uses it to identify when nadir users enter the radiative near field.

IV. Channel Modeling

  • The channel steering-vector entry for an element and observer comprises a gain factor and a specific element-to-observer phase shift.The channel vector spans the formation's M = NS radiating elements.Source: §IV · p. 5

A. Fraunhofer distance

  • For a conformal formation, the Fraunhofer distance is rederived for its curved aperture and reduces to the standard planar-array result as curvature becomes negligible.The estimate for irregular or nonsquare formations can use an enclosing square aperture, yielding a conservative estimate.Source: A. Fraunhofer distance · pp. 5–6
  • The nadir transition spacings are approximately 68.8 m for GEO and 67.4 m for LEO, placing users in the radiative near field at larger spacings.The transition occurs where the Fraunhofer distance meets orbital altitude; below it, the far-field approximation applies.Source: A. Fraunhofer distance · p. 5

B. Radiative near-field model

The models distinguish spherical-wave propagation across the formation in the radiative near field from planar-wave propagation in the far field, under stated geometric conditions.

B. Radiative near-field model

  • In the radiative near field, the channel accounts for spherical wavefronts across the formation using exact element-to-user distances and the element radiation pattern.For a nadir observer, this regime occurs when orbital altitude is below the Fraunhofer distance, or intersatellite spacing exceeds the transition spacing.Source: B. Radiative near-field model · p. 6
  • The near-field gain combines free-space amplitude attenuation and direct-radiating-element gain, which depends on the angle between array boresight and departure direction.The element-to-observer distance is the slant distance.Source: B. Radiative near-field model · p. 6

C. Far-field model

  • In the far field, a planar wavefront gives all elements a shared propagation direction; when the formation's angular extent is small, local array boresights approximate the global z direction.Under these assumptions, the element gain is the same across radiating elements.Source: C. Far-field model · p. 6

D. Stacked far-field model

The stacked far-field model preserves propagation across satellite centers while approximating propagation within each array. In the LEO beam-pattern comparison, this local approximation matches the spherical-wave reference where a global plane-wave model does not.

D. Stacked far-field model

  • The stacked far-field model retains exact satellite-center-to-user distances and applies a local far-field approximation only within each satellite array.This intermediate model is suited to the regime where each array is in the far field but the user remains in the near field of the overall formation.Source: D. Stacked far-field model · pp. 6–7

E. When Far-Field Beamforming Fails

  • At 1 km (> ΔF), the global far-field beam pattern deviates substantially from the reference, while the stacked far-field pattern remains nearly identical; at 10 m, all three patterns are practically indistinguishable.The comparison evaluates propagation with the exact near-field model and varies the model used for matched-filter precoding.Source: Figure 7 · p. 7, Figure 6 · p. 7, E. When Far-Field Beamforming Fails · p. 7
  • The beam-pattern test uses a nine-satellite LEO formation, directs the beam toward nadir, and compares 10 m and 1 km spacings around the 67.4 m transition.The near-field-based precoder is the matched reference; far-field and stacked far-field precoders are mismatched.Source: E. When Far-Field Beamforming Fails · p. 7

V. Numerical evaluation and comparison

The evaluation compares geometric approximations using throughput simulations, then finds that planar formations closely match conformal ones while flat-Earth user geometry overestimates LEO throughput.

B. Impact of Space- and User-Segment Geometry

A. Signal model and performance metrics

B. Impact of Space- and User-Segment Geometry

  • Increasing satellite spacing can reduce throughput, particularly in GEO, as denser grating lobes can increase multiuser interference and offset narrower beams.The reported curves use MMSE and MF precoding.Source: B. Impact of Space- and User-Segment… · p. 8

C. Impact of Channel-Model Mismatch

The paper compares precoders designed with near-field, far-field, and local far-field channel models against exact spherical-wave propagation. Local approximations closely match the near-field approach, while global far-field precoding can cause full outage at large spacings; the paper also identifies operating-range and wideband assumptions that qualify these results.

C. Impact of Channel-Model Mismatch

D. Model Validity and Wideband Considerations

  • The largest spacings are illustrative rather than practical operating points, and the LEO formation-wide elevation constraint fails beyond roughly 440 km.For GEO, the stated small-angle condition for the FF model holds only up to approximately 370 km; the final curve decade is asymptotic.Source: D. Model Validity and Wideband… · pp. 9–10
  • The analysis uses the carrier frequency while its KPIs assume coherent beamforming across the signal band; wideband operation may require per-subcarrier weights or delay compensation.The cited mitigations increase onboard processing or payload complexity.Source: D. Model Validity and Wideband… · p. 10

VI. Conclusions

  • The conclusions support simplifying formation geometry in the investigated scenarios, while retaining Earth curvature for reliable LEO user-coverage and deployment modeling.The paper also notes that geometric or wavefront mismatch can affect channel reconstruction and subsequent signal processing, even when underlying parameters are known exactly.Source: §VI · p. 10
  • The sFF model retains exact satellite-center-to-user distances and approximates the wavefront only within each array, provided each locally approximated unit operates in its far field.For hierarchical formations, the condition must hold for each unit to which the local approximation is applied.Source: C. Impact of Channel-Model Mismatch · p. 9, §VI · p. 10

Appendix

The conformal satellite grid is constructed by mapping a tangent-plane grid onto a sphere while preserving its direction and geodesic distance from the formation center.

A. Satellite placement for the conformal configuration

  • Each satellite’s auxiliary point is placed on the tangent-plane grid with horizontal and vertical spacing Rϕ, and its distance from the origin defines the geodesic distance used for spherical placement.The mapping preserves the auxiliary point’s radial direction in the tangent plane and interprets its distance from the origin as a spherical geodesic distance.Source: A. Satellite placement for the… · p. 10
  • For a central-row satellite, the spherical coordinates follow from the arc length ℓs: the central angle is ℓs/R, and the position uses its sine and cosine.The construction maps the tangent-plane point to the sphere while retaining its radial direction and geodesic distance.Source: A. Satellite placement for the… · p. 11, Figure 11 · p. 11
  • Adjacent satellites’ chord spacing ∆ and sphere radius R determine the angle ϕ through ∆/2 = R sin(ϕ/2).The arc ℓ and chord ∆ are distinguished in the conformal placement geometry.Source: A. Satellite placement for the… · p. 11, Figure 12 · p. 11

B. Antenna placement for the conformal configuration

For the conformal configuration, each satellite array lies in a tangent plane whose inward radial normal is fixed by the spherical placement; azimuthal and meridional axes resolve the remaining rotation.

B. Antenna placement for the conformal configuration

  • The satellite array’s tangent plane is set by the inward radial normal, which points from the satellite toward the sphere center and defines the array boresight.The normal is determined by the satellite’s coordinates relative to the sphere center.Source: B. Antenna placement for the conformal… · p. 11
  • The tangent-plane constraint leaves rotation about the normal undetermined, so the local array axes are oriented along the azimuthal and meridional directions.The meridional vector is obtained by completing the orthonormal basis with the cross product of the normal and azimuthal direction.Source: B. Antenna placement for the conformal… · pp. 11–12
  • The antenna template is placed on the tangent plane by using the azimuthal, meridional, and normal vectors as its local x, y, and z axes.For the central satellite, the undefined azimuthal direction is conventionally set to [1, 0, 0]T, with the meridional axis [0, 1, 0]T.Source: B. Antenna placement for the conformal… · p. 12

C. Fraunhofer distance for planar and conformal satellite configurations

The section defines the Fraunhofer distance from the exact spherical-wave path error and derives closed-form approximations for planar and conformal formations. For practical spacings, the approximations are reported to be reliable.

C. Fraunhofer distance for planar and conformal satellite configurations

  • The exact Fraunhofer distance is the minimum range at which the maximum path-length error reaches λ/16, corresponding to a maximum phase error of π/8.The plane-wave approximation replaces the exact element-to-user distance with its projection along the nadir observation direction.Source: C. Fraunhofer distance for planar and… · p. 12
  • At ∆=10^-4R (≈692.1 m), the compared Fraunhofer-distance values are practically indistinguishable; at ∆=10^-1R (≈692.1 km), spherical-based distances diverge from planar-based distances.The latter spacing is described as an extreme, impractical case; the paper concludes approximation (27) is reasonable for practical cases.Source: Figure 14 · p. 12, C. Fraunhofer distance for planar and… · p. 13
  • The closed-form derivation assumes satellite-array size is negligible relative to formation extent and the Fraunhofer distance greatly exceeds formation size.For fully populated grids and spherical caps, the maximum transverse distance occurs at the four corner satellites; the closed form depends on which grid nodes are occupied.Source: C. Fraunhofer distance for planar and… · p. 12
  • At ∆=10^-6R (≈6.92 m), the assumption L ≪ (S−1)∆ fails and makes ˜zF inaccurate, but the Fraunhofer distance remains far below hsat, leaving all users in the far field.The numerical example uses a 49-satellite LEO constellation with 625 elements per satellite and δ=0.6λ=8.18 cm.Source: Figure 14 · p. 12, C. Fraunhofer distance for planar and… · p. 13

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