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From Focusing to Con-Focusing: Optimal Power Transfer in Line-of-Sight Near-Field MIMO

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Pith's one-line read In line-of-sight near-field MIMO with extended apertures on both sides, con-focusing both arrays at their equal-angle point outperforms traditional focusing and reaches the optimal rank-one power gain.

desk verdict Paper introduces con-focusing for better power transfer in extended near-field MIMO and pins the crossover at 1.947, though the quadratic phase approx may limit the large-Fresnel claims. read the letter →

arxiv 2606.28454 v1 pith:2RULEGZU submitted 2026-06-26 eess.SP cs.ITmath.IT

Marouan Mizmizi , Sanzhar Yergaliev This is my paper · ORCID
classification eess.SPcs.ITmath.IT
keywords near-fieldMIMObeamfocusingline-of-sightchannelanalogbeamformingFresnelnumberpowergaincon-focusing
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper examines power transfer in a single-user line-of-sight MIMO setup where both the transmitter and receiver have extended apertures. It finds that the standard focusing approach, which phase-matches each array to the center of the other, is beaten by simple far-field steering once the link Fresnel number exceeds a modest threshold. The authors derive exact expressions showing that the crossover depends only on the Fresnel number and the ratio of aperture sizes. They introduce con-focusing, a new phase profile in which both arrays point toward the same intermediate point that makes the subtended angles equal. This method achieves the highest possible leading-term gain for rank-one channels, works without any channel state information, and works with standard beam training procedures.

What carries the argument

Con-focusing, the phase profile in which each aperture is focused toward the axial point from which the two apertures subtend equal angles.

What would settle it

Measure the power gain achieved by con-focusing at a Fresnel number of 5 with unequal apertures and check whether it stays within a few decibels of the theoretical rank-one upper bound; a larger gap would falsify the order-optimality claim.

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Extended reading notes

Core claim

Under the Fresnel approximation for a line-of-sight MIMO channel with analog unit-modulus beamforming, the received power obtained when both terminals apply the con-focusing phase profile equals the square of the largest singular value of the channel matrix up to a multiplicative constant that does not depend on the Fresnel number. The same profile requires only the knowledge of the link distance and the two aperture lengths and collapses to ordinary plane-wave steering whenever the apertures are identical.

Load-bearing premise

The analysis assumes a single-user line-of-sight MIMO link free of multipath, fully analog unit-modulus beamforming, and operation in the Fresnel regime.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 0 minor

Summary. The paper analyzes power transfer in single-user LOS MIMO links with extended apertures at both ends under fully analog unit-modulus beamforming in the Fresnel regime. It derives closed-form power gains for conventional focusing (phase-matched to the other's center) and far-field steering (planar phase ramp), showing these are governed solely by the link Fresnel number and aperture ratio. For equal apertures the gains cross once at the universal value 1.947; beyond this point focusing loses 10 dB per decade. The paper introduces con-focusing (both apertures aimed at the common point subtending equal angles), claiming it attains the rank-one eigenbound in the leading constant, requires no channel knowledge, degenerates to steering for equal apertures, and is obtainable in one beam-refinement round.

Significance. If the closed-form derivations and proofs are valid, the work is significant for overturning the MISO-derived intuition that focusing is always preferable in the near field when both terminals have extended apertures. The reduction to two dimensionless parameters, the explicit crossover value, the 10 dB/decade scaling, and the parameter-free con-focusing construction that matches the eigenbound leading term are all strengths. The result supplies a concrete, implementable alternative to focusing with clear regime boundaries.

major comments (1)
  1. [Abstract and Fresnel-regime analysis] Abstract and the Fresnel-regime derivations: the closed-form gains, the exact crossing at 1.947, the 10 dB/decade scaling, and the claim that con-focusing attains the rank-one eigenbound in the leading constant are all obtained under the quadratic (Fresnel) phase approximation to the exact distance. The approximation error scales with the fourth power of aperture size over distance and therefore grows with the Fresnel number itself. No explicit remainder bound or higher-order term is supplied, so the asymptotic comparison and optimality statements rest on an unquantified regime of validity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the thoughtful review and for highlighting the reliance on the Fresnel approximation. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract and Fresnel-regime analysis] Abstract and the Fresnel-regime derivations: the closed-form gains, the exact crossing at 1.947, the 10 dB/decade scaling, and the claim that con-focusing attains the rank-one eigenbound in the leading constant are all obtained under the quadratic (Fresnel) phase approximation to the exact distance. The approximation error scales with the fourth power of aperture size over distance and therefore grows with the Fresnel number itself. No explicit remainder bound or higher-order term is supplied, so the asymptotic comparison and optimality statements rest on an unquantified regime of validity.

    Authors: We agree that all closed-form expressions and the stated comparisons (including the crossing value, scaling, and leading-constant optimality of con-focusing) are derived under the standard quadratic-phase Fresnel approximation to the exact Euclidean distance. The phase error is indeed O((D/z)^4) where D denotes aperture size and z the link distance, so the relative error grows with the Fresnel number. The manuscript defines its scope as the Fresnel regime, in which the quadratic term dominates the phase; however, we did not supply an explicit remainder bound or a quantitative statement of the Fresnel-number range over which the leading-term asymptotics remain accurate to a prescribed tolerance. In the revised version we will add a short appendix deriving the next-order term in the distance expansion and stating the condition under which the omitted term is negligible relative to the retained quadratic term for the power-gain expressions. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity; closed-form derivations under Fresnel approximation are self-contained

full rationale

The paper derives closed-form expressions for power gains of focusing and steering, proves their comparison depends only on two dimensionless quantities (Fresnel number and aperture ratio), identifies the crossing point at 1.947 for equal apertures, and introduces con-focusing as order-optimal. These steps are presented as direct mathematical results from the quadratic phase model, with no fitted parameters renamed as predictions, no self-definitional loops, and no load-bearing self-citations invoked to justify uniqueness or ansatzes. The analysis is confined to the stated Fresnel regime without claiming external validity beyond it. This is the normal case of an independent derivation chain.

Assumptions & free parameters 0 free parameters · 3 assumptions · 1 invented entities

The paper rests on standard domain assumptions for wireless propagation modeling; no free parameters are introduced in the abstract; con-focusing is a newly defined strategy rather than an invented physical entity.

assumptions (3)
  • domain assumption Line-of-sight propagation free of multipath
    Explicitly stated as the setting for the single-user MIMO link.
  • domain assumption Fresnel regime for closed-form expressions
    Invoked to derive power gains for focusing and steering.
  • domain assumption Fully analog unit-modulus beamforming
    Assumed throughout the analysis of phase-matched and planar-ramp strategies.
invented entities (1)
  • con-focusing strategy
    purpose: Order-optimal beamforming approach for power transfer
    Newly proposed method defined by equal-angle targeting; no independent evidence provided beyond the derivation.

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Cite this review

Pith. "Pith review of From Focusing to Con-Focusing: Optimal Power Transfer in Line-of-Sight Near-Field MIMO." pith.science (2026). https://pith.science/paper/2RULEGZU

@misc{pith2026260628454,
  author       = {Pith},
  title        = {Pith review of: From Focusing to Con-Focusing: Optimal Power Transfer in Line-of-Sight Near-Field MIMO},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2RULEGZU}},
  note         = {Machine review of arXiv:2606.28454}
}
read the original abstract

Beamfocusing is the established near-field strategy for a large array serving a single-antenna user. We consider the single-user line-of-sight MIMO link, free of multipath, in which the user, too, carries an extended aperture, and show that the focusing prescription inverts: beyond a modest Fresnel number, focusing on the user is outperformed by far-field steering. Under fully analog, unit-modulus beamforming, we derive closed-form power gains for focusing (each aperture phase-matched to the other's center) and for steering (a planar phase ramp) in the Fresnel regime, and prove that their comparison is governed by two dimensionless quantities: the link Fresnel number, the product of the two aperture lengths normalized by wavelength and link distance, and the aperture ratio, irrespective of how many elements discretize the apertures. For equal apertures the two gains cross exactly once, at the universal value 1.947; beyond it, focusing loses ten dB per decade of Fresnel number, and the advantage celebrated in the MISO literature survives only as the receive aperture vanishes. We then derive the strategy that is order-optimal at every Fresnel number, con-focusing: both apertures aim at the common point from which they subtend equal angles. It attains the rank-one eigenbound in leading constant, needs no channel knowledge, degenerates to plain steering for equal apertures, and is acquirable within one beam-refinement round with no geometry exchange between the terminals.

Figures

Figures reproduced from arXiv: 2606.28454 by the authors.

Figure 1
Figure 1. Why focusing dies in MIMO. (a) For a single-antenna user the focal [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Link geometry. Two parallel ULAs of lengths [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. The two solutions of the con-focusing law in one geometry ( [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: The design map in the curvature coordinates (11), drawn for the [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: Power gain along the con-focusing family ( [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: Off-boresight geometry. (a) The transmit and receive uniform linear [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 8
Figure 8. Figure 8: maps the steering-to-focusing power-gain ratio 10 log10(GFF/Gfoc) over the (NF, ρ) plane, so the comparison ;=0:9 ;=0:5 ;=0:1 NF 0 5 10 15 20 25 30 normalized power gain [dB] -30 -25 -20 -15 -10 -5 0 con-focusing eigenbound steering focusing [PITH_FULL_IMAGE:figures/f…
Figure 7
Figure 7. Figure 7: validates Theorems 1 and 4. The exact-channel markers fall on the closed-form curves over the whole range of NF; con-focusing rides the rank-one eigenbound, the exact σ 2 max, while focusing decays as N −2 F . Steering is the lone aperture-ratio-dependent scheme: shown…
Figure 10
Figure 10. Figure 10: Normalized power gain NFG (eigenbound at 0 dB) over the curvature plane (qt, qr) = (D/rt, D/rr) at NF = 10, on common axes, for (a) ρ = 0.9, (b) ρ = 0.5, (c) ρ = 0.1. The confocal ridge (qt − 1)(qr − 1) = 1 (dashed) is the same for every ρ; focusing (1, 1) (circle) is…
Figure 11
Figure 11. Figure 11: Range-acquisition accuracy (Algorithm 1), [PITH_FULL_IMAGE:figures/full_fig_p012_11.png]
Figure 12
Figure 12. Figure 12: Convergence of the acquired con-focusing gain to the oracle [PITH_FULL_IMAGE:figures/full_fig_p012_12.png]

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