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Radial-Angular Resolution Framework for Near-Field Communications

Viviana Centritto, Sotiris Droulias, Angeliki Alexiou

arXiv:2610.20092v1eess.SPcs.SYeess.SY

TL;DR

Near-field multi-user beamfocusing needs models that prescribe focal resolution for steered beams across radial and angular domains. The paper extends ASR-based overlap control to derive boundaries and steering limits, then analyzes capacity, interference, and sum-rate trade-offs. Its results identify overlap as a design variable balancing user capacity and interference in radial configurations.

  • Problem

    Existing models do not jointly characterize steered focal-region boundaries in radial and angular domains for a prescribed overlap.

    Source: §I · p. 1

  • Method

    The paper extends ASR-based inter-beam overlap control to derive radial and angular focal boundaries and steering-dependent spatial limits.

    Source: B. Our Contributions · p. 2, A. Derivation of maximum steering… · p. 7

  • Results

    In the high-SNR radial analysis, increasing ρ raises user capacity, while sum-rate initially rises and then declines as interference grows.

    Source: Appendix A · p. 10

  • Takeaways & Limitations

    Tunable overlap offers a design tool for balancing range-domain user capacity against inter-user interference.

    Source: §VII · p. 10

Abstract

from arXiv · show

Near-field beamfocusing confines radiated power into a localized three-dimensional focal spot, making the analytical characterization of radial and angular resolution critical for sixth-generation (6G) multi-user systems, yet existing models treat the focal resolution as a fixed geometric outcome rather than a tunable design variable. Based on the Angular Representation (ASR) approach, this paper presents a resolution framework to design contiguous focal regions across both the radial and angular domain for steered multi-user systems. By extending the definition of a tunable inter-beam overlap $\rho$, we derive closed-form expressions for the radial and angular focal boundaries and extents, alongside the steering-dependent minimum focal distance $d_{0,\text{min}}$ and maximum steering angle $\theta_{\text{max}}$. Using these boundaries, iterative synthesis algorithms are developed to construct uniform and non-uniform radial and angular beam arrangements. Finally, by analyzing both isolated and joint radial-angular configurations, we quantify the fundamental trade-offs between $\rho$, achievable user capacity, inter-user interference, and system sum-rate across varying steering angles.

I. INTRODUCTION

Near-field beamfocusing adds radial separation to angular multiplexing, but steered multi-user systems lack a framework for prescribing contiguous focal-region resolution across both domains. This work extends tunable overlap control to radial and angular boundaries and assesses its system-level trade-offs.

I. INTRODUCTION

  • Spherical wavefronts enable near-field beamfocusing to confine power in three-dimensional focal spots, adding radial separation for users along one direction.The paper identifies range-domain multiple access, high-resolution sensing, and precise localization as applications of this confinement.Source: §I · p. 1

A. Prior Works

  • Existing models treat focal extent as a geometric outcome or constrain spacing through orthogonality, null-spacing, or broadside-only overlap, leaving steered radial and angular boundaries uncharacterized.Prior overlap-based design is confined to the propagation axis, while null-spacing fixes beam spacing to focal-spot width.Source: A. Prior Works · pp. 1–2

B. Our Contributions

  • The proposed ASR framework makes inter-beam overlap ρ tunable and derives contiguous focal-region boundaries in radial and angular domains for steered multi-user systems.It also develops synthesis methods and evaluates capacity, per-user spectral efficiency, and sum-rate trade-offs in isolated and joint configurations.Source: B. Our Contributions · p. 2

II. ANALYTICAL FRAMEWORK AND SYSTEM MODEL

The framework models a focused UPA field with the Angular Spectrum Representation and evaluates power along steering directions and constant-radius arcs. Its closed-form field analysis relies on approximations for narrow spectral content and focal distances exceeding the array size.

System model

  • The system uses a centered UPA with a Gaussian incident footprint, and the aperture phase shapes the field into spherical wavefronts focused at distance d0 and angle θr.Oblique incidence makes the footprint elliptical rather than circular.Source: §II · pp. 2–3

Analytical framework

  • ASR computes the propagated field from the aperture-field Fourier representation, with spatial frequencies kx and ky and longitudinal frequency kz.The closed-form simplification assumes narrow spectral content around the steering wavevector and focal distance d0 greater than the UPA size.Source: §II · p. 3

Peak intensity

  • The beam peak is located by differentiating the radial intensity polynomial and solving the resulting cubic; its valid real root determines rpeak.The analysis evaluates the power density along the steering direction using transverse beam radii wx(r) and wy(r).Source: §II · pp. 3–4

III. RADIAL BOUNDARIES

Radial focal boundaries are defined by where the steering-axis power falls to a chosen fraction of its peak. Solving this condition yields the start and end of the focal region and its extent.

Radial boundaries

  • The radial boundaries occur where S(r) equals ρradial times peak power, converting the selected inter-beam overlap into a boundary condition.With r scaled by d0, the condition becomes a fourth-degree polynomial that is reduced to a depressed quartic and solved for its roots.Source: §III · p. 4
  • The real roots r1 and r2 define rstart and rend, and the radial focal extent ∆rρ is their difference.Fig. 3 illustrates radial resolution models under steering angles with ρradial = 0.5.Source: §III · p. 4, Figure 3 · p. 5

A. Derivation of minimum focal distance d0,min

The minimum focal distance is derived from the validity condition for the approximate longitudinal spatial frequency. It constrains the beam’s spectral support at a given steering angle.

A. Derivation of minimum focal distance d0,min

  • The minimum focal distance d0,min is set by requiring negligible approximation error in kz for the beam’s primary spectral content.The maximum transverse wavenumber is parameterized by θlimit, and the spectral width centered at k sin θr must remain within kmax_x.Source: A. Derivation of minimum focal… · p. 5

B. Design of non-uniform and uniform focal regions in the radial domain

The framework synthesizes contiguous radial focal regions by matching adjacent boundaries, using different iterative procedures for non-uniform and uniform resolution. The reported configurations show that steering angle changes the number of regions each profile can support.

B. Design of non-uniform and uniform focal regions in the radial domain

  • Both profiles arrange adjacent regions so one region’s upper radial boundary coincides with the next region’s lower boundary.The boundary expressions are nested and nonlinear, so the focal parameters are computed iteratively rather than in closed form.Source: B. Design of non-uniform and uniform… · p. 5
  • Non-uniform synthesis holds w0 constant, optimizes the outermost focal distance, then finds preceding distances by one-dimensional root finding toward the aperture.The outermost distance is chosen by maximizing rstart; the recursion proceeds inward from that region.Source: B. Design of non-uniform and uniform… · p. 5
  • Uniform synthesis jointly optimizes focal distance and Rayleigh range to maintain a target focal extent while satisfying radial contiguity.Each inner region is found by bounded nonlinear least squares, and synthesis stops when the minimum-distance constraint is violated.Source: B. Design of non-uniform and uniform… · p. 6
  • At θr = 15°, 45°, and 65°, the non-uniform design supports N = 7, 8, and 9 regions, while the uniform design supports N = 7, 6, and 5 at 15°, 45°, and 60°.The uniform configuration maintains wFWHM = 2.5 m; increasing steering requires larger w0,n and raises the dynamic d0,min.Source: B. Design of non-uniform and uniform… · p. 6

IV. ANGULAR BOUNDARIES

The angular framework sets focal-region boundaries where power density falls to a chosen fraction of its peak along a constant-radius arc. Adjacent regions are then joined by matching their angular boundaries.

IV. ANGULAR BOUNDARIES

  • Figure 4 illustrates angular-domain resolution at multiple rpeak values with w0 = 0.15 m and ρangular = 0.1.The angular boundaries are evaluated along the constant-radius arc using the power density in the xz-plane.Source: Figure 4 · p. 6, §IV · p. 6
  • The angular boundaries are θstart = θr − δθ and θend = θr + δθ, giving an extent Δθρ = 2δθ.The deviation δθ is obtained by applying the threshold S(θ) = ρangular Smax to the angular power density along r = rpeak.Source: §IV · p. 6
  • Adjacent focal regions are made contiguous by setting θend,n−1 = θstart,n at the selected inter-beam overlap.This constructs the beam arrangement along the angular domain while holding the shared radial distance rpeak fixed.Source: §IV · p. 6

A. Derivation of maximum steering angle θmax

The maximum steering angle is determined by the largest peak distance achievable at each steering angle. The derivation reduces the stationary condition to a cubic equation whose valid root gives that distance ceiling.

A. Derivation of maximum steering angle θmax

  • Figure 5 plots focal-region capacity N against design parameters for non-uniform radial, uniform radial, and angular resolution models.Its panels use w0, wFWHM, and w0 respectively, with steering angle or rpeak varied as described in the caption.Source: Figure 5 · p. 7
  • θmax is the angle beyond which the target peak-distance condition rpeak,target ≤ r*_peak(θr) is violated.As steering angle increases, the maximum achievable peak distance r*_peak decreases.Source: A. Derivation of maximum steering… · pp. 6–7
  • The stationary condition is simplified using x = d0/r*_peak, yielding a cubic whose valid root determines r*_peak.The cubic coefficients are expressed in terms of γ, and the root is substituted into the closed-form peak-distance expression.Source: A. Derivation of maximum steering… · pp. 6–7

B. Design of contiguous focal regions in the angular domain

Angular synthesis recursively places contiguous beams along a constant-radius arc, subject to a steering-dependent peak-distance ceiling. The reported capacity varies with beam radius and radial distance, while angular resolution can support more regions than radial resolution.

B. Design of contiguous focal regions in the angular domain

  • For angular synthesis, each subsequent beam preserves rpeak while solving for steering angle and focal distance to meet angular contiguity.The recursion stops when steering exceeds its system limit or the target peak radius violates the analytical condition.Source: B. Design of contiguous focal regions… · p. 7
  • At rpeak = 3, 7, and 12 m, the angular chain supports N = 76, 74, and 41 regions, respectively, for w0 = 0.15 m and ρangular = 0.1.At more oblique angles the angular extent widens; larger rpeak reaches the analytical ceiling at a smaller θmax, ending the chain sooner.Source: B. Design of contiguous focal regions… · p. 7
  • In the non-uniform radial model, increasing w0 from 0.25 m to 1 m raises capacity from about 12 to 60 regions at θr = 30°.At w0 = 0.8 m, raising θr from 30° to 60° increases capacity from 40 to 50 regions.Source: B. Design of contiguous focal regions… · p. 7
  • In the uniform radial model, increasing wFWHM from 0.3 m to 1.0 m lowers capacity from 18 to 10 regions at θr = 30°.At wFWHM = 0.4 m, capacity falls from 16 regions at 30° to 13 at 60°; the minimum allowable focal extent is constrained by d0,min.Source: B. Design of contiguous focal regions… · pp. 7–8
  • For angular resolution, increasing w0 extends the maximum range and raises capacity; most w0 values yield the same user count across different rpeak values.Figure 5c shows N against w0 for multiple rpeak values.Source: B. Design of contiguous focal regions… · p. 8
  • Angular synthesis supports over 400 contiguous focal regions along a constant arc, compared with a maximum of 60 radial regions for a 1 m beam radius in the non-uniform scenario.The angular resolution is independent of rpeak under the stated approximation.Source: B. Design of contiguous focal regions… · p. 8

VI. PER-USER AND SUM-RATE SPECTRAL EFFICIENCY

The evaluation shows that radial performance depends on balancing user density against inter-beam interference, while angular separation supports higher per-user efficiency. In mixed layouts, allocating more users to the angular domain improves sum-rate at equal user capacity.

Radial configurations

  • At fixed N = 15 and ρ = 0.5, uniform radial spacing gives central users a broader, more balanced rate distribution than non-uniform spacing, while θr = 60° produces the strongest degradation.Non-uniform clustering near the array sharpens the rate drop just beyond the innermost beam; steering-related radial compression forces denser placement and increases interference.Source: §VI · p. 9

Radial overlap and SNR

  • For equal user counts, ρ = 0.1 gives the highest radial sum-rate, approaching 80 bps/Hz at θr = 30° and 45°, while ρ = 0.5 saturates below 30 bps/Hz at θr = 30°.With different user counts, ρ = 0.5 (N = 26) leads at low SNR, but ρ = 0.25 (N = 15) has the highest high-SNR plateau; at θr = 60°, ρ = 0.1 (N = 8) exceeds ρ = 0.5. Below 0 dB, noise dominates; curves diverge as SNR rises and flatten beyond roughly 20–30 dB.Source: §VI · p. 9

Angular framework

  • In the angular framework, angular separation mitigates interference and yields higher per-user spectral efficiency and sum-rate than radial configurations; ρ = 0.5 leads at low SNR, while ρ = 0.1 dominates at high SNR.The angular evaluation uses ρangular = 0.1 for its comparison across radial peak positions.Source: §VI · pp. 9–10

Mixed angular-radial topologies

  • For 21 users, shifting from 7 radial × 3 angular to 3 radial × 7 angular raises sum-rate from 33.2 to 43.2 bps/Hz; for 35 users, shifting from 7 × 5 to 5 × 7 raises it from 55.8 to 60.7 bps/Hz.The authors report that the advantage of broader angular distribution persists as capacity grows.Source: §VI · p. 10

VII. CONCLUSIONS

The conclusion presents an ASR-based framework that makes radial and angular focal resolution tunable for xz-steered multi-user systems. Its evaluations favor angular distribution for spectral efficiency at equal user capacity.

Contributions

  • The framework extends tunable overlap ρ to radial and angular boundaries and extents, establishes steering-dependent d0,min and θmax, and provides numerical synthesis methods for uniform and non-uniform radial arrangements.The supplied conclusion also describes contiguous focal regions along the radial and angular domains.Source: §VII · p. 10

System evaluation

  • Angular resolution supports more contiguous focal regions than radial resolution because it is independent of rpeak and experiences lower inter-beam interference.The conclusion reports that uniform radial spacing gives a more balanced per-user rate distribution than non-uniform spacing.Source: §VII · p. 10
  • At equivalent user capacity, allocating more users to the angular rather than radial domain maximizes overall spectral efficiency.The conclusion frames ρ as a design variable for balancing user capacity against inter-user interference.Source: §VII · p. 10

APPENDIX A ρOPT FOR MAXIMUM CAPACITY ALONG THE STEERING

The appendix analyzes how radial overlap affects high-SNR sum-rate and user capacity under steering. It reports an optimal overlap near 0.285 at 60°, with a higher optimal threshold needed beyond 55°.

Optimal overlap

  • At θr = 60°, high-SNR performance reaches its maximum at ρopt ≈ 0.285 as added users initially raise sum-rate before interference overtakes those gains.The analysis evaluates high-SNR sum-rate and corresponding capacity as functions of ρradial.Source: Appendix A · p. 10

Steering dependence

  • For steering angles up to 55°, ρopt remains near 0.285, corresponding to an optimal configuration of 11 users.The source specifies the user count after reporting the stable threshold.Source: Appendix A · p. 10
  • Beyond 55°, radial compression requires both a higher optimal overlap threshold and greater corresponding user capacity to maintain maximum spectral efficiency.The paper presents tunable ρ as a way to balance multi-user gains against interference penalties across steering scenarios.Source: Appendix A · p. 11
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