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Analytical Framework of Radial Resolution for Near-Field Communications

Viviana Centritto, Sotiris Droulias, Angeliki Alexiou

arXiv:2609.27733v1eess.SPcs.SYeess.SY

TL;DR

Existing analytical models largely address isolated focal regions, although near-field beamfocusing can separate users by range. This paper extends ASR to synthesize contiguous focal regions with tunable radial resolution and analyzes how beam radius, frequency, and overlap relate to focal boundaries and capacity.

  • Problem

    Prior analytical models predominantly address isolated focal regions, while contiguous focal areas are needed for near-field applications such as range-domain multiple access.

    Source: §I · pp. 1–2

  • Method

    The paper extends ASR with a tunable overlap parameter ρ and closed-form arrangements for contiguous focal regions, including configurations with tailored radial extents.

    Source: §I · p. 2, §III · p. 4, §IV · p. 6

  • Results

    The analysis finds that the outermost focal distance scales with beam radius, frequency, and ρ, while the number of focal regions grows with beam radius and frequency and depends on ρ.

    Source: §V · pp. 6–7, §VII · p. 10

  • Takeaways & Limitations

    The analytical boundaries and resolution relationships provide design rules for dimensioning near-field beamfocusing systems.

    Source: §I · p. 2, §VII · p. 10

    The paraxial approximation's accuracy depends on θmax: the model agrees well with numerical results at 4° but begins to diverge at 20°.

    Source: Appendix B · pp. 10–11

Abstract

from arXiv · show

As extremely large antenna arrays (ELAAs) become central to next-generation wireless systems, the transition into the near-field propagation regime enables the exploitation of spherical wavefronts for radial-domain beamfocusing. This capability is pivotal for emerging applications requiring precise spatial isolation, such as Space Division Multiple Access (SDMA), hierarchical localization and advanced sensing. However, fully realizing these technologies requires specific design rules to dimension multi-user systems without relying on computational expensive full-wave simulations. To address the gap in modeling contiguous focal regions with controllable radial resolution, this paper expands the Angular Spectrum Representation (ASR) approach to propose a comprehensive analytical framework. Through the introduction of a tunable inter-beam overlap parameter $\rho$, we derive closed-form expressions to synthesize multiple focal regions, providing the flexibility to tailor their radial resolution. Furthermore, the resolution capabilities are analyzed to characterize the interplay between key operational variables, such as the transmitter size, beam radius and operation frequency. System-level assessment of per-user and sum-rate spectral efficiencies across varying signal-to-noise (SNR) regimes reveals how the inter-beam overlap dictates a fundamental trade-off between user capacity and inter-user interference, delivering design guidelines for future near-field communications.

I. INTRODUCTION

Near-field spherical wavefronts make radial beamfocusing and range-domain discrimination possible, but prior analytical models largely treat isolated focal regions. This paper extends ASR to synthesize contiguous regions with tunable resolution and derives design bounds and system-level trade-offs.

I. INTRODUCTION

  • Near-field phase depends on both angle and distance, enabling beamfocusing that can distinguish users sharing an angular direction.The paper links this capability to sensing, localization, and range-domain multiple access.Source: §I · p. 1
  • Prior analytical radial-resolution models predominantly address isolated focal regions, leaving contiguous multiuser focal areas without an analytical model.The paper identifies this gap as relevant to advanced near-field applications.Source: §I · pp. 1–2
  • The extended ASR framework arranges N contiguous focal regions for simultaneous radial-domain users, with ρ controlling radial separation and inter-beam overlap.This parameter is introduced to support precise spatial isolation.Source: §I · p. 2
  • Closed-form expressions maintain uniform radial resolution and can independently tailor focal-area resolution for heterogeneous scenarios.The framework is presented as a way to derive design guidelines without computationally expensive full-wave simulations or numerical sweeps.Source: §I · p. 2
  • The framework analytically bounds the outermost and minimum focal distances, with the outer boundary constrained by user capacity or finite aperture and the minimum by paraxial validity.These boundaries establish spatial limits for system design.Source: §I · p. 2
  • Per-user and sum-rate spectral efficiencies are evaluated, and low- and high-SNR analysis shows that ρ trades simultaneous-user capacity against inter-user interference.The paper frames this as a fundamental trade-off for the synthesized focal areas.Source: §I · p. 2

II. SYSTEM MODEL AND ANALYTICAL FRAMEWORK

The framework models a UPA's near-field propagation with ASR, then derives a single-beam power profile and radial boundaries that can be used to characterize focal regions. Its analytical profile is compared with numerical propagation, and the radial interval is defined through a peak-power threshold.

II. SYSTEM MODEL AND ANALYTICAL FRAMEWORK

  • A Gaussian aperture amplitude and spherical phase profile model a UPA focused along the propagation axis at distance d0.The array's beam radius w0 characterizes the amplitude distribution, and the phase uses the free-space wavenumber at frequency f.Source: §II · p. 2
  • ASR decomposes the aperture field into plane waves whose spectrum determines the field at z > 0; the field evolution follows from the Helmholtz equation.The paraxial approximation is obtained by expanding kz when transverse wavevector components are small relative to the longitudinal component.Source: §II · pp. 2–3
  • At 1000 × 1000 elements, 150 GHz, w0 = 0.2 m, and d0 = 10 m, the analytical profile predicts zpeak = 9.75 m and wFWHM = 3.10 m, matching numerical propagation.Fig. 2 compares the theoretical xz-plane distribution and z-axis profile against a numerical reference formed from discrete spherical-wave superposition.Source: §II · pp. 3–4, Figure 2 · p. 3
  • The radial interval Δzρ,n is the extent over which the n-th beam's power remains at least ρSmax, with ρ = 0.5 giving the FWHM.The interval is related to the beam's FWHM, and adjacent focal regions can be arranged by matching their boundaries.Source: §II · p. 4

III. SYNTHESIS OF MULTIPLE FOCAL REGIONS WITH NON-UNIFORM RADIAL RESOLUTION USING FIXED BEAM

The section develops a recursive method for arranging contiguous focal regions with a fixed beam radius, then bounds its use under the paraxial approximation. Its examples show how focal placement and radial extent vary across the synthesized regions.

III. SYNTHESIS OF MULTIPLE FOCAL REGIONS WITH NON-UNIFORM RADIAL RESOLUTION USING FIXED BEAM

  • Adjacent focal regions are made contiguous by matching the preceding beam’s terminal boundary to the next beam’s initial boundary.A quadratic equation for the adjacent focal distance is solved recursively to arrange N consecutive regions.Source: §III · p. 4
  • The outermost focal distance is chosen to maximize its initial boundary, which maximizes the number of synthesized focal regions.The method finds this distance by differentiating the initial boundary with respect to d0,N and selecting the positive root.Source: §III · p. 4
  • The minimum focal distance is constrained by paraxial validity, assessed by limiting transverse spatial frequencies and comparing exact and analytical wFWHM.The comparison evaluates relative error at d0,min, where the paraxial approximation error is maximized.Source: §III · p. 4
  • At θmax = 4°, the relative error reaches the 1% limit across evaluated scenarios, providing a design rule for the analytical model.The approximation error is reported as independent of beam radius w0.Source: §III · p. 4, Figure 3 · p. 5
  • For N = 9 contiguous regions, the farthest target at 26.02 m peaks inward at about 22.21 m, and farther regions have broader wFWHM.The example uses ρ = 0.5, a 1500 × 1500-element square UPA, f = 150 GHz, and w0 = 0.2 m.Source: §III · p. 5

IV. SYNTHESIS OF MULTIPLE FOCAL REGIONS WITH UNIFORM RADIAL RESOLUTION USING TUNABLE BEAM

The section extends the recursive synthesis to tunable beam radii, enabling uniform or block-specific radial resolution. It also accounts for finite-aperture and minimum-distance constraints when selecting focal regions.

IV. SYNTHESIS OF MULTIPLE FOCAL REGIONS WITH UNIFORM RADIAL RESOLUTION USING TUNABLE BEAM

  • Uniform radial extent is obtained by independently tuning each beam radius, represented through its Rayleigh range, as focal distance varies.The recursive process initializes the outermost region and calculates the remaining focal distances and Rayleigh ranges.Source: §IV · p. 5
  • At ρ = 0.5, 10 uniform-resolution focal regions fit; at ρ = 0.25, 6 fit; and at ρ = 0.1, 4 fit.For wFWHM = 1 m at d0,N = 10 m, the reported Rayleigh range is 199.5 m and the corresponding beam radius is 0.35 m.Source: Figure 5 · p. 6, §IV · p. 6
  • A block-uniform design assigns 0.3 m resolution to the inner three regions, 0.5 m to the middle three, and 1 m to the outer three, contiguous at ρ = 0.2.All nine regions achieve their specified resolutions; the outermost design uses w0,N = w0,max and d0,N = 10.52 m.Source: Figure 5 · p. 6, §IV · p. 6
  • Finite aperture constrains the maximum beam radius to w0,max = l/4, limiting the farthest focal region.The stated edge taper is −34.7 dB, with boundary intensity at 0.034% of peak; minimum focal distance is also updated for each region’s Rayleigh range.Source: §IV · pp. 5–6

V. ANALYSIS OF RADIAL RESOLUTION CAPABILITIES

Radial resolution depends differently on beam radius, frequency, and overlap: these parameters set the focal-distance boundaries and the number of supportable regions. With uniform resolution, narrower regions permit more focal areas only within a paraxial lower bound.

V. ANALYSIS OF RADIAL RESOLUTION CAPABILITIES

  • d0,min scales linearly from 1.43 m to 14.32 m as w0 rises from 0.1 m to 1.0 m, and remains 5 m across the frequency sweep.The analysis attributes this minimum distance to w0 rather than f.Source: §V · p. 6, Figure 6 · p. 7
  • At larger w0 and ρ = 0.50, d0,N reaches approximately 650 m, compared with 255 m under stricter overlap; d0,N also grows linearly with f.The scaling rate with frequency is dictated by ρ.Source: §V · pp. 6–7, Figure 6 · p. 7
  • At w0 = 1.0 m, capacity is 53 focal regions at ρ = 0.50, 30 at ρ = 0.25, and 17 at ρ = 0.10; at 1000 GHz, it reaches 126, 73, and 41, respectively.The reported counts show capacity growing with w0 and frequency, while stricter overlap supports fewer regions.Source: §V · p. 7, Figure 6 · p. 7
  • Uniform resolution has a 0.26 m minimum focal extent at 150 GHz and θmax = 4°; near wFWHM = 0.3 m, capacity is 20, 12, and 7 regions for ρ = 0.50, 0.25, and 0.10.Within the valid range above the minimum, broader wFWHM reduces N even as d0,N moves farther away.Source: §V · p. 7, Figure 7 · p. 7

VI. SYSTEM-LEVEL PERFORMANCE ASSESSMENT

The assessment measures spectral efficiency at each focal region’s peak and compares fixed- and variable-user scenarios across synthesis configurations. The results show that overlap shifts the balance between user count and interference, with the best setting depending on SNR and whether user count varies.

VI. SYSTEM-LEVEL PERFORMANCE ASSESSMENT

  • Per-user spectral efficiency is assessed from SINR at each beam’s peak intensity point, and system sum-rate aggregates the individual beam rates.The assessment varies SNR and overlap across fixed-radius and uniform-resolution configurations.Source: §VI · pp. 7–8, Figure 9 · p. 8
  • The overlap analysis compares both variable and fixed user-count scenarios for the two synthesis configurations.The corresponding system-level sum-rate comparisons span SNR and distinguish fixed beam radius from uniform resolution.Source: §VI · p. 8, Figure 9 · p. 8
  • With fixed w0, central focal regions have the lowest spectral efficiencies and the innermost and outermost regions the highest; larger ρ supports more users but lowers per-user rate.With tailored w0 and wFWHM = 0.5 m, distant regions achieve higher rates because peak-power scaling overcomes inverse-square distance attenuation.Source: §VI · p. 8, Figure 8 · p. 8
  • At low SNR, maximizing ρ gives the highest total rate; at high SNR, ρ = 0.25 leads in the variable-user case, while ρ = 0.10 leads when user count is fixed.For fixed N = 10, reducing overlap minimizes inter-user interference and yields the highest performance across the SNR domain.Source: §VI · p. 9, Figure 10 · p. 9

A. Asymptotic analysis of the sum-rate Rt for the variable user-count scenario

The variable-user analysis examines how overlap changes the sum-rate trade-off between accommodating users and suppressing interference. It finds that the preferred overlap depends on SNR, with a reported high-SNR optimum balancing user density and interference.

A. Asymptotic analysis of the sum-rate Rt for the variable user-count scenario

VII. CONCLUSIONS

The conclusion presents an ASR-based analytical framework for synthesizing radial focal regions with fixed or tailored extents. It uses focal-distance boundaries and overlap to characterize capacity and derive design guidance from beam radius and frequency interactions.

VII. CONCLUSIONS

  • The ASR framework synthesizes and manages multiple focal regions along the radial domain, with closed-form analysis for fixed beam radius or dynamically tailored radial extents.The framework provides operational boundaries for minimum and outermost focal distances.Source: §VII · p. 10
  • The minimum and outermost focal distances define system capacity N, which is governed by the inter-beam overlap parameter ρ.The analysis relates resolution behavior to operating frequency f and beam radius w0.Source: §VII · p. 10
  • The performance assessment identifies a multiple-access trade-off: maximizing throughput in noise-limited scenarios versus mitigating multi-user interference in capacity-limited conditions.The paper presents the framework as a source of scalable design guidelines for near-field beamfocusing.Source: §VII · p. 10

APPENDIX A DERIVATION OF THE POSITIVE ROOT DOMAIN FOR d0,N−1

The appendix determines which quadratic root yields the preceding focal distance, distinguishing the cases ρ < 0.89 and ρ > 0.89. For ρ < 0.89, the negative branch gives the required positive distance.

APPENDIX A DERIVATION OF THE POSITIVE ROOT DOMAIN FOR d0,N−1

  • The coefficient’s sign is determined by a downward-opening quadratic numerator, whose discriminant is negative when the overlap satisfies ρ < 0.89.The appendix uses this condition to establish the root domain.Source: Appendix A · p. 10
  • For ρ < 0.89, the quadratic coefficient is negative, leaving one positive and one negative solution for d0,n−1; the negative branch gives the positive solution.This overlap range is described as covering a reasonable range of practical inter-beam overlap.Source: Appendix A · p. 10
  • For ρ > 0.89, both solutions are positive, but the positive branch gives d0,n−1 > d0,n and is discarded.The negative branch consistently supplies the preceding focal distance in this case.Source: Appendix A · p. 10

APPENDIX B ANALYTICAL LIMITATIONS IMPOSED BY θMAX

The paraxial approximation’s accuracy depends on the maximum angular deviation θmax because the cutoff controls which transverse spatial frequencies are included. A 4° cutoff yields close analytical and numerical agreement, whereas a 20° cutoff leads to divergence.

APPENDIX B ANALYTICAL LIMITATIONS IMPOSED BY θMAX

  • The approximation error remains negligible near kx = 0 but grows at higher spatial frequencies, which a larger angular cutoff admits.At θmax = 20°, the broader spectral beam extends into regions where approximation precision degrades.Source: Appendix B · pp. 10–11
  • A θmax = 4° threshold confines transverse spatial frequencies to the low-error region near the propagation axis, helping preserve approximation accuracy.The comparison uses w0 = 0.25 m and f = 150 GHz.Source: Appendix B · p. 10
  • For each distinct d0,min, the power-density plots show that at θmax = 4° the analytical model exhibits excellent agreement with the numerical one, whereas at θmax = 20° the framework starts to diverge.Source: Appendix B · p. 11

APPENDIX C VALIDATION OF THE ARRAY BOUNDARY CONDITION

The appendix validates the finite-aperture constraint on beam radius: a beam radius of 0.2 m is contained by the example array, while expanding it to 0.4 m causes truncation and analytical-model divergence.

APPENDIX C VALIDATION OF THE ARRAY BOUNDARY CONDITION

  • The appendix tests the finite-aperture limit on the maximum achievable beam radius by comparing minimally truncated and significantly truncated configurations.Fig. 12 contrasts the aperture field magnitude with the corresponding normalized power density.Source: Appendix C · p. 11
  • At w0 = 0.2 m, the beam is contained within the 0.8 m aperture and the analytical and numerical models agree at d0 = 5 m.This configuration has an edge taper of -34.7 dB; Fig. 12 presents the aperture field and power density.Source: Appendix C · p. 11
  • Expanding the beam radius to 0.4 m weakens the edge taper to -8.68 dB, causing significant boundary truncation and analytical-model divergence.The truncated amplitude profile can no longer be accurately modeled as a valid Gaussian footprint.Source: Appendix C · p. 11

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