Beamforming Design for Wideband Near-Field Communications With Reconfigurable Refractive Surfaces

Aryan Kaushik; Hongliang Zhang; Shuhao Zeng; Zicheng Lin

REVIEW 3 major objections 7 minor 33 references

Beamforming Design for Wideband Near-Field Communications With Reconfigurable Refractive Surfaces

T0 review · 3 major / 7 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper proposes that adding a controllable true time delay to each element of a reconfigurable refractive surface can compensate the frequency-dependent phase errors that, together with the near-field spherical wavefront, split…

desk verdict Good idea, well-executed optimization, but the simulations rest on an unstated frequency-response model—so the headline gains are not independently checkable yet. read the letter →

arxiv 2501.01012 v2 pith:VONJSLGE submitted 2025-01-02 eess.SP

Zicheng Lin , Shuhao Zeng , Aryan Kaushik , Hongliang Zhang This is my paper · ORCID

classification eess.SP

keywords beamsplitnear-fieldcommunicationsreconfigurablerefractivesurfacesfrequencyselectivitywidebandbeamformingtime-delayunitsmax-minfairnessOFDM

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 studies a wideband downlink in which a reconfigurable refractive surface (RRS) serves as the base-station antenna and users lie in its near field. It argues that two effects jointly worsen the beam-split problem: the near-field spherical wavefront makes different subcarriers focus at different locations, and the RRS's frequency-selective transmission coefficients add phase errors that make edge subcarriers lose focus entirely. To fix this, the authors introduce a Delayed-RRS element that appends a true time delay, and they design an alternating optimization that tunes the digital beamformer, the RRS phase shifts, and the delays to maximize the minimum user rate. If the scheme works as claimed, wideband near-field RRS systems could recover much of the data-rate loss that would otherwise confine usable communication to frequencies near the center carrier.

What carries the argument

The load-bearing object is the Delayed-RRS element: a conventional RRS element with response $\psi(\theta_i^c, f_m)$ at center-phase setting $\theta_i^c$ and subcarrier frequency $f_m$, cascaded with a time-delay unit whose transfer function is $e^{-j2\pi f_m \tau_i}$, giving $\psi_{m,i} = e^{-j2\pi f_m \tau_i} \psi(\theta_i^c, f_m)$. The delay is the mechanism that can align phases across the whole band because its phase varies linearly with frequency, which matches how the near-field propagation phase $e^{-j2\pi f_m r_{i,k}/c}$ and the RRS response both vary. The optimization carries the argument: the max-min rate problem is split into digital beamforming (solved by semidefinite relaxation plus successive convex approximation), analog phase-shift design (barrier-function gradient ascent), and time-delay compensation (the same barrier method). The appendix adds a closed-form analysis showing that a single-feed RRS has an optimal feed-RRS distance that grows with the size of the RRS.

What would settle it

Measure the complex transmission coefficient of a real RRS element across the 1 GHz band, for example with free-space or waveguide S-parameter measurements at several bias voltages, then insert the measured $\psi(\theta_i^c, f_m)$ into the paper's simulator and compare the Delayed-RRS against the no-delay baseline; the central claim fails if the delay gain shrinks below the reported level or the edge-subcarrier focus loss persists despite optimized delays.

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Extended reading notes

Core claim

The central claim is that the beam split in an RRS-based wideband near-field multiuser system is not merely a near-field phenomenon: the frequency selectivity of the RRS itself breaks phase alignment between elements on edge subcarriers, so beams that would simply shift focus in a frequency-flat near-field array instead spread and lose gain. The paper models each element's transmission coefficient as $\psi_{m,i} = \psi(\theta_i^c, f_m)$ and proposes the delayed-RRS modification $\psi_{m,i} = e^{-j2\pi f_m \tau_i} \psi(\theta_i^c, f_m)$, where $\tau_i$ is a tunable time delay that adds a frequency-linear phase usable to compensate both the spherical near-field phase and the RRS's frequency-dependent response. On this basis it formulates a max-min rate problem and solves it by alternating semidefinite relaxation with successive convex approximation for the digital precoder, a barrier-function gradient method for the phase shifts, and a similar gradient method for the delays. Simulations show that the delay compensation is worth roughly 2 dB of transmit power and that ignoring either the near-field condition or the frequency selectivity costs noticeable rate, with the relative importance of the two depending on user distance.

Load-bearing premise

The entire quantitative argument rests on the assumption that the RRS transmission coefficient really is $\psi(\theta_i^c, f_m)$ with the smooth amplitude and phase frequency response constructed from the cited practical model; the paper never gives this function explicitly, so if a real RRS responds differently the reported rate gains and beam-split visuals may not carry over.

Editorial extensions

If this is right

In the simulated regime, adding optimized time delays improves the worst-user rate by an amount comparable to raising transmit power by about 2 dB.
Ignoring either the near-field spherical-wave channel or the RRS frequency selectivity causes measurable rate loss, and the loss grows with transmit power because it appears as inter-user interference.
There exists an optimal feed-to-RRS distance for a single-feed RRS, and the optimal distance increases with the number of RRS elements; the simulation matches the derived formula.
A small number of phase-quantization bits is sufficient to approach the continuous-phase rate bound, and under coarse quantization ignoring frequency selectivity can even behave better.
The alternating optimization converges because each subproblem drives the minimum rate monotonically upward, up to a controlled error in the barrier-method steps.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: The quantitative gains are tied to the borrowed but unspecified transmission-coefficient model $\psi(\theta_i^c, f_m)$; with a different practical response, such as a Lorentzian amplitude-phase coupling, the size of the delay benefit could change even if the qualitative mechanism survives.
Editorial inference: The delay-compensation idea transfers to other frequency-selective surfaces and to wideband near-field sensing or localization, where beam split similarly corrupts focusing across the band.
Editorial inference: A direct extension would be to replace the assumed model with measured element responses and re-run the same optimizer; the paper's claim predicts the Delayed-RRS would still outperform the no-delay baseline on edge subcarriers.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Hi,

The thing to know about this one: the core idea is good—add true time-delay units to each RRS element so the wideband near-field beam split can be compensated in the analog domain, and then do alternating optimization of digital beamformer, RRS phase shifts, and delays to maximize the worst-user rate. That combination is new as far as the citations go; prior work treats near-field beamsplit and metasurface frequency selectivity separately. The algorithm is standard but competent: SDR/SCA for the digital part, barrier-method gradient descent for the analog part, and the complexity analysis is honest. I also give them credit for the Without-TD baseline; the gap between that and their full scheme is the actual measure of the delay unit's value, and it's meaningful (about 2 dB equivalent). So this is a legitimate step forward in the RRS beamforming subfield.

The soft spot is exactly where the reader put it: the frequency-selective RRS model. Equation (1) defines ψ(θc_i, f_m) abstractly, and Section VII.A says only that they 'construct two functions' based on [8] with a sensitivity knob kp, but no formulas, no amplitude/phase parameter values, nothing. Every curve in Figs. 5-9 comes from that private model. The beam-split behavior in Fig 2(c)—edge subcarriers losing focus—is a direct consequence of the chosen ψ, and the whole Delayed-RRS gain story depends on that choice. Another ψ, equally plausible, could change the relative gains or even wipe them out. The stress-test note hits this correctly: the comparisons are not falsifiable by a third party. I'd call this a serious but fixable flaw.

Secondary issues: the Appendix A derivation for the optimal feed-RRS distance has a sign/cosine slip between (39) and (41), and (42) doesn't follow cleanly; that result is a side claim, though, not the main engine of the paper. No code or data either, which makes the missing ψ harder to excuse.

Net: the paper is worth a referee, not a desk reject. A good referee should force them to write down the exact ψ model, correct the appendix, and share code. If that happens, I'd be reasonably confident in the result. If they can't or won't share the model, the simulation results should be treated as conditional on private choices.

Referee Report

3 major / 7 minor

Summary. The paper considers a wideband (OFDM) multi-user downlink in which a reconfigurable refractive surface (RRS) serves as the transmit antenna of a base station. The authors model the RRS as frequency selective, couple that selectivity with the near-field spherical-wave channel, and argue that the two effects jointly worsen the beam-split phenomenon. To mitigate this, they propose a 'Delayed-RRS' architecture in which each RRS element is augmented with an ideal time-delay unit, and they formulate a max-min rate problem over the digital beamformer, the RRS phase configuration, and the time delays. The problem is decomposed into three subproblems: digital beamforming via SDR/SCA, analog beamforming via an exponential-barrier gradient method, and time-delay compensation via a similar gradient method. Convergence and complexity claims are given, an optimal feed-RRS distance is derived in Appendix A, and simulations in Figs. 5-9 compare the proposed scheme against baselines.

Significance. The problem addressed is timely: wideband near-field operation is central to 6G extreme-aperture systems, and most prior work treats near-field beam split and metasurface frequency selectivity separately. The Delayed-RRS structure is a plausible and practically motivated extension of delay-phase architectures to RRS hardware, and the alternating-optimization framework is a reasonable approach to a difficult nonconvex problem. If the numerical results are reproducible and the RRS model is physically representative, the paper would provide a useful design guideline and a fair baseline comparison. However, the central quantitative claims currently rest on an unspecified frequency-response model, and the Appendix A derivation contains an internal inconsistency. The significance is therefore conditional: the paper's contribution is real but cannot be fully assessed until those gaps are closed.

major comments (3)

[Section VII.A and Eq. (1)] The frequency-selective RRS response ψ(θ_c^i, f_m) is never given an explicit functional form. Eq. (1) only defines it abstractly, and Section VII.A states that 'we construct two functions' with a sensitivity knob k_p, but the actual amplitude-frequency and phase-frequency formulas, all parameter values except k_p, and the mapping from k_p to the response are omitted. Every simulation curve in Figs. 5-9 and the beam-focus claim in Fig. 2(c) are generated from this private model. Because the analog and delay subproblems (P3) and (P4) require gradients of the rate with respect to ψ, the implemented algorithm is also underspecified without this model. This is a load-bearing omission for the paper's headline claim that the proposed Delayed-RRS scheme mitigates beam split under RRS frequency selectivity. Please provide the exact functions (or the explicit equation from the reference used, with all constants), the parameter values, and the definition of k_p; ideally include the simulation code or a detailed setup so that an independent reader can reproduce Figs. 5-9.
[Appendix A, Eqs. (41)-(42)] The two expressions for the ideal amplitude gain A in Eq. (41) are not equal. In the first expression, the bracketed term is (h^2/(a^2+h^2))^{(α_t+α_r-2)/4}, which equals cos^{(α_t+α_r-2)/2} θ_0 because h^2/(a^2+h^2)=cos^2 θ_0. In the second expression, the bracketed term is (sin θ_0)^{(α_t+α_r-2)/2}. Replacing cos^{(α_t+α_r-2)/2} by sin^{(α_t+α_r-2)/2} changes the stationary condition. Eq. (42) follows from differentiating the cos-based expression (A ∝ cot θ_0 [1 - cos^β θ_0] with β=(α_t+α_r-2)/2), not from the sin-based expression. Please correct the typo, re-derive the condition, and verify the claimed unique solution. This matters because Proposition 1 and its validation in Fig. 6 depend on this derivation.
[Section V.C and Table I] The time-delay architecture is introduced as an ideal component e^{-j2π f_m τ_i} with no discussion of delay resolution, bandwidth limitations, insertion loss, or power cost. Since the Delayed-RRS is a hardware-oriented proposal, this idealization should be stated explicitly as an assumption and its impact on the reported gains should at least be discussed. In addition, the maximum delay is given as τ_max=1/f_c in Section V.C but as τ_max=1/(2 f_c) in Table I; this inconsistency changes the maximum phase range from about 2π to about π and should be resolved.

minor comments (7)

[Throughout] There are many typographical and grammatical errors (e.g., 'imporve', 'an new', 'coefficeints', 'cofiguring', 'deeployed', 'iteratitively') that should be corrected in a revision.
[Section III.A, Eq. (12)-(13)] The notation in Eqs. (12)-(13) should be clarified: ψ_m is a vector of transmission coefficients, while h(f_m,r) is a column vector from Eq. (2); the dot product and the normalization factor a_m need a precise definition so that the array gain is unambiguous.
[Section IV.A and Eq. (16)] The text says the objective is to 'maximize the sum rate' but the problem in Eq. (16) maximizes the minimum rate. Please use consistent terminology.
[Section VII.A, Fig. 4] Figure 4 and the surrounding text do not state what is plotted on the axes or specify the 'two functions' beyond saying they are constructed following [8]. At minimum, define the amplitude and phase response axes and give the exact expressions.
[Section VII.B, Fig. 9] The text describes the NLoS case as 'Rayleigh' and a 'Rayleigh channel model', but later refers to 'pure los and Rician conditions'; unify the terminology and clarify the channel model used in Fig. 9.
[Section VI.A and Algorithm 2] The convergence discussion would be more convincing if a numerical convergence plot or a stopping criterion were provided; Algorithm 2 runs for a fixed 'iter = 3' iterations, which seems small for an alternating algorithm.
[Section V.B, Eq. (29)] The barrier function is an exponential penalty, not a standard interior-point barrier that diverges at the constraint boundary. The text should state this explicitly and explain why the penalty formulation is preferred.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the optimization is evaluated against its stated model and baselines; the underspecified frequency-response function is a reproducibility limitation, not a circular step.

full rationale

We walked the paper's derivation chain and found no step in which a claimed prediction or first-principles result is equivalent to its own inputs. The RRS transmission coefficient is introduced abstractly in Eq. (1) as ψm,i = ψ(θc_i, fm), with external circuit-theory justification [7], [8]; the delayed-RRS model in Eq. (15) adds the time-delay term e^{-j2πfmτi}. The rate expressions (10)-(11) follow from the stated NUSW channel and OFDM signal models, and the optimization problem P1 in Eq. (16) is a genuine max-min rate problem over the digital beamformer, phase shifts, and delays. The decomposition into P2-P4 and the SCA/barrier/gradient algorithms optimize this same objective, and the comparison baselines (Without TD, Ignoring Frequency Selectivity, Random Configuration, Far-field Wave Model) are distinct policies evaluated under the same model, not fitted restatements of the result. No fitted parameter is renamed as a prediction, no uniqueness theorem from the authors' prior work is invoked to force the proposed structure, and the self-citations [1], [2], [6], [18], [19], [31] support the RRS architecture and channel assumptions without closing the argument. The significant limitation noted in Section VII.A — 'Referring to the modeling in [8], we construct two functions to express the amplitude-frequency and phase-frequency response of RRS elements respectively' with no explicit formulas and only a sensitivity knob kp — makes the simulation campaign non-reproducible and is a correctness/transparency concern, but the model is an input assumption, not an output derived from itself. The central beamforming claim is therefore not circular.

Assumptions & free parameters 2 free parameters · 5 assumptions · 1 invented entities

The central results rest on a frequency-selective RRS model (ψm,i = ψ(θc_i, fm)) whose explicit form is never given, on the NUSW near-field channel model, and on the untested assumption that time-delay units can be added to each element. The only derived theorem (Proposition 1) contains an algebraic inconsistency. The simulation conclusions are therefore conditional on an under-specified model.

free parameters (2)

frequency selectivity sensitivity kp = kp = 1 or 2 in simulations
Introduced in Section VII.A to control the strength of the RRS frequency response; the explicit dependence of ψ(θc_i, fm) on kp is not given. This knob drives the simulated beam-split loss.
RRS frequency response model functions = not specified
The two functions for amplitude and phase response used in Fig. 4 and simulations are described verbally ('Referring to the modeling in [8]') and their parameters are not enumerated. This is an unquantified modeling choice.

assumptions (5)

domain assumption The RRS element transmission coefficient is determined by the center-frequency phase shift θc_i and varies with frequency: ψm,i = ψ(θc_i, fm).
Section II-B, Eq. (1). No physical circuit model is given; the form is imported from [8] without specification.
domain assumption The channel between RRS and users follows the non-uniform spherical wave (NUSW) model with LoS propagation and no small-scale fading.
Section II-C, Eq. (2). This is a standard near-field model but not verified for the specific scenario.
domain assumption Feeds lie in the near field of the RRS, so the feed-to-RRS propagation also uses the NUSW model.
Section II-D, Eq. (6). The near-field assumption is required for the model but not justified with specific distances.
ad hoc to paper For the beamforming derivation, the amplitude of the RRS transmission coefficient is approximately invariant to the phase configuration.
Footnote 2 in Section III-A: |ψ(θc_1, fc)| ≈ ... ≈ |ψ(θc_N, fc)|. This simplification underlies Eq. (14) and the beam-split illustration.
ad hoc to paper Each RRS element can be augmented with an ideal time-delay unit that provides a frequency-dependent phase e^{-j2π fm τi} without hardware limitations or power cost.
Section III-B, Eq. (15). The delay units are assumed to cover τmax with no implementation constraints.

invented entities (1)

Delayed-RRS (time-delay unit integrated into each RRS element)
purpose: To compensate frequency-dependent phase shifts introduced by the RRS frequency selectivity, mitigating beam split.
Proposed in Section III-B, Eq. (15), based on time-delay units from [27]-[29]. The paper provides no hardware measurement or prototype validation; effectiveness is shown only in simulation with an unspecified frequency-response model.

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Cite this review

Pith. "Pith review of Beamforming Design for Wideband Near-Field Communications With Reconfigurable Refractive Surfaces." pith.science (2026). https://pith.science/paper/VONJSLGE

@misc{pith2026250101012,
  author       = {Pith},
  title        = {Pith review of: Beamforming Design for Wideband Near-Field Communications With Reconfigurable Refractive Surfaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VONJSLGE}},
  note         = {Machine review of arXiv:2501.01012}
}

read the original abstract

To meet the growing demand for high data rates, cellular systems are expected to evolve towards higher carrier frequencies and larger antenna arrays, but conventional phased arrays face challenges in supporting such a prospection due to their excessive power consumption induced by numerous phase shifters required. Reconfigurable Refractive Surface (RRS) is an energy efficient solution to address this issue without relying on phase shifters. However, the increased radiation aperture size extends the range of the Fresnel region, leading the users to lie in the near-field zone. Moreover, given the wideband communications in higher frequency bands, we cannot ignore the frequency selectivity of the RRS. These two effects collectively exacerbate the beam-split issue, where different frequency components fail to converge on the user simultaneously, and finally result in a degradation of the data rate. In this paper, we investigate an RRS-based wideband near-field multi-user communication system. Unlike most existing studies on wideband communications, which consider the beam-split effect only with the near-field condition, we study the beam-split effect under the influence of both the near-field condition and the frequency selectivity of the RRS. To mitigate the beam-split effect, we propose a Delayed-RRS structure, based on which a beamforming scheme is proposed to optimize the user's data rate. Through theoretical analysis and simulation results, we analyze the influence of the RRS's frequency selectivity, demonstrate the effectiveness of the proposed beamforming scheme, and reveal the importance of jointly considering the near-field condition and the frequency selectivity of RRS.

Figures

Figures reproduced from arXiv: 2501.01012 by the authors.

**Figure 2.** Beam Split Phenomenons where Gt denotes the transmit gain pattern of the j th feed in the direction of the i th RRS element, Fr represents the normalized radiation pattern of the i th RRS element in the direction of the j th feed, S0 represents the area of each RRS element, and di,j denotes the distance between the i th RRS element and the j th feed. Similarly, the transmit gain pattern of the feed antenna Gt can be… view at source ↗

**Figure 3.** The equivalent structure of the RRS element. [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (6 more)

**Figure 5.** Figure 5: Achieved minimum rate versus transmit power. [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: Achieved minimum rate versus distance between feeds [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

**Figure 7.** Figure 7: Achieved minimum rate versus the number of the feeds. [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

**Figure 9.** Figure 9: Achieved minimum rate versus number of quantization [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]

**Figure 10.** Figure 10: Furthermore, with the relation r = h tan θ and d = h cos θ , we change the integral variable from r to θ. (38) can be written as a double integral A = r αt + 1 2πS0 Z ϕ dϕ Z θ h cos αt+αtr−4 2 θ sin θdθ. (39) The region of ϕ is [0, 2π], so R ϕ dϕ = 2π. Letting the unc…

**Figure 10.** Figure 10: A circle RRS illuminated by a feed antenna. [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]

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