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REVIEW 2 major objections 5 minor 46 references

Near-Field Wideband Beamforming for RIS Based on Fresnel Zone

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper claims that grouping RIS elements by Fresnel zone turns the wideband channel into a Fourier transform of zone intensity, allowing phase-only beamforming to approach the rate upper bound despite near-field beam split.

desk verdict Fresnel-zone coordinate trick is a genuinely new angle on near-field RIS beamforming, but the paper's advertised rate upper bound has two concrete derivation errors and needs referee attention. read the letter →

arxiv 2411.18878 v1 pith:WQ4HP76V submitted 2024-11-28 cs.IT eess.SPmath.IT

Qiumo Yu , Linglong Dai This is my paper · ORCID
classification cs.ITeess.SPmath.IT
keywords reconfigurableintelligentsurfacenear-fieldbeamsplitFresnelzonewidebandbeamformingstationaryphasemethodGerchberg-SaxtonalgorithmachievablerateupperboundmmWaveandTHzcommunications
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 argues that the near-field beam split, which makes wideband RIS beams focus at different places for different frequencies, can be overcome by phase shifts alone, without true-time-delay hardware. Its key move is to group RIS elements by Fresnel zone, the ellipsoidal surfaces on which the BS-RIS-UE route length is constant, and to assign the same phase shift to all elements on a zone. Once phases are aligned along each zone, the two-dimensional sum over the RIS plane collapses to a one-dimensional integral, and the frequency response of the equivalent channel becomes the Fourier transform of the phase-modulated reflective intensity across zones. That identity yields a Parseval-based upper bound on achievable rate, and the paper designs phases by stationary-phase and Gerchberg-Saxton methods to approach flat in-band gain. If correct, this gives a practical, hardware-free cure for near-field beam split in large-aperture mmWave and THz RIS systems.

What carries the argument

The central object is the Fresnel-zone coordinate system for the RIS plane. Fresnel zones are a family of concentric ellipsoids whose foci are the BS and the UE; their intersections with the RIS plane are nested ellipses indexed by the semi-major axis $a$, which is half the reflection route length. Transforming the RIS plane from $(x,y)$ to $(a,\theta)$ makes the phase of the cascaded channel linear in $a$, and locking the phase on each zone $\phi(a,\theta)=\psi(a)$ removes frequency dependence along the zone. The surviving quantity is the reflective intensity $v(a)=\int g_0 J(a,\theta)\,d\theta$, and the substitution $t=2a/c$ turns the channel into the Fourier transform $g(f)=\int v_t(t)e^{j\psi_t(t)}e^{-j2\pi f t}\,dt$. That Fourier identity, together with Parseval's theorem and the stationary-phase / Gerchberg-Saxton spectrum-shaping tools, carries the entire argument.

What would settle it

Run the exact channel integral in Eq. (6) without dropping the frequency dependence of $g_0(f)\propto 1/f^2$, and check whether $\int |g(f)|^2\,df$ still equals $E_g=\int v_t^2(t)\,dt$ and whether the designed flat gain profile survives; if the mismatch grows with fractional bandwidth or with RIS aperture, the Fourier identity and the bound in Eq. (25) are only approximations rather than exact consequences.

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Extended reading notes

Core claim

The paper establishes, under its amplitude-approximation model, that the equivalent wideband channel is $g(f)=\int v_t(t)e^{j\psi_t(t)}e^{-j2\pi f t}\,dt$, the Fourier transform of the Fresnel-zone reflective intensity $v_t(t)$ multiplied by the designed phase modulation $e^{j\psi_t(t)}$. It follows from Parseval's theorem that $\int |g(f)|^2 df = E_g = \int v_t^2(t)\,dt$ is fixed for every phase configuration, so the achievable rate is bounded by $\log_2\left(1+\frac{S_x}{S_\sigma}E_g\right)$, with equality approached only by an ideal gain that is flat across the band and zero outside it. The paper then treats the phase design as a one-dimensional spectrum-shaping problem: the stationary-phase method gives a nearly closed-form phase profile $\psi(a)$ across Fresnel zones, and the Gerchberg-Saxton algorithm iterates between the zone domain and frequency domain to flatten the gain. Simulations show the proposed methods keep gain approximately flat over the band, reduce out-of-band leakage, and achieve rates within a few percent of the upper bound while outperforming narrowband and virtual-subarray baselines.

Load-bearing premise

The Fourier-transform picture and the Parseval-based rate bound require the reflective intensity of each Fresnel zone to be independent of frequency, although the path-loss amplitude in the model carries an explicit $1/f^2$ factor and the paper only justifies amplitude approximations in space, not in frequency.

Editorial extensions

If this is right

  • Near-field beam split can be mitigated with phase-only RIS control; no true-time-delay or delay-adjustable metasurface hardware is needed, avoiding the insertion loss and cost of TTD modules.
  • For any phase-only design, total gain energy over all frequencies is fixed at $E_g=\int v_t^2(t)\,dt$; the rate bound $\log_2(1+S_xE_g/S_\sigma)$ is therefore a fundamental limit of phase-shift-based wideband RIS, tighter than bounds that assume TTD hardware.
  • Under narrowband beamforming, the 3 dB bandwidth shrinks roughly as $B_{3\mathrm{dB}}\approx c\Gamma_0/(\iota D_0)$, so larger apertures and BS/UE directions pointing similarly make beam split worse; the zone-based designs restore flat gain across the designed band.
  • Achievable rate grows with RIS size and bandwidth under the proposed designs, while narrowband beamforming plateaus; 2-bit phase quantization is enough to nearly match continuous-phase performance.
  • The same Fresnel-zone construction works in far-field scenarios, since far-field is a special case of the near-field model used here.

Reading between the lines

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

  • The Fourier-transformed channel makes the RIS phase design mathematically identical to radar waveform design; techniques from pulse compression and spectral shaping could be imported directly, and the one-dimensional zone-domain formulation suggests delay-domain optimization as a scalable route for extremely large surfaces.
  • The Parseval bound implies a phase-only RIS cannot create more in-band energy than the fixed intensity supports; any attempt to boost edge-subcarrier gain must steal out-of-band energy, so flat in-band gain is the best achievable spectrum shape.
  • If the reflective intensity $v_t(t)$ is measurably frequency-dependent through the $1/f^2$ path-loss factor at very wide fractional bandwidths, the exact Fourier identity would fail; the phase designs would need an iterative frequency-domain correction, with the stationary-phase profile as initialization.
  • The zone grouping principle might extend to multi-user or cell-free settings by treating each user's Fresnel-zone intensity as a separate waveform constraint, and to holographic or continuous-aperture metasurfaces where the per-zone phase profile is sampled directly.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper addresses near-field wideband beam splitting in RIS-aided links. It models the RIS as a continuous I-RIS, introduces a Fresnel-zone coordinate system in which a constant route length per zone makes the phase response frequency-independent along the zone, and reduces the equivalent channel to a one-dimensional Fourier transform of a phase-modulated Fresnel-zone intensity. On this basis it derives a sinc approximation and 3-dB bandwidth for narrowband beamforming, proposes an upper bound on achievable rate using Parseval's theorem, and constructs phase profiles via stationary-phase and Gerchberg-Saxton designs. Simulations with a 1 m aperture at 30 GHz and 1.5 GHz bandwidth show flatter in-band gain and higher average rate than narrowband and subarray baselines.

Significance. The conceptual contribution is valuable: the Fresnel-zone decomposition gives a clean structural explanation of near-field beam split and yields phase-only designs that avoid true-time-delay hardware. The derivations are self-contained, the sinc approximation is compared with simulation rather than fitted, and the SP and GS implementations are described with complexity estimates. If the proof issues below are corrected, the paper would be a useful contribution to wideband RIS beamforming. In its current form, however, the two central theoretical claims -- the exact Fourier-transform representation and the Jensen/Parseval upper bound -- are not established as stated.

major comments (2)
  1. [III-C, Eqs. (6), (15), (17)] Equation (17) is presented as an exact Fourier transform of a fixed Fresnel-zone intensity vt(t), but this ignores the explicit frequency dependence in Eq. (6): g0 = sqrt(N_BS)c^2/(4π^2 f^2 R_BR R_RU d^2). Since v(a) in Eq. (15) is defined as the integral of g0 J(a,θ), v(a) and hence vt(t) depend on f. The sentence after Eq. (14) that 'the inner integral along the Fresnel zone in (13) is frequency independent' is therefore inconsistent with the paper's own definitions. As a result, the Parseval step (c) in Eq. (25) does not follow: ∫|g(f)|^2 df = ∫ v_t^2(t) dt requires g(f) to be the Fourier transform of a frequency-independent vt(t)e^{jψt(t)}. The authors should state explicitly that g0 is replaced by its value at a reference frequency and bound the error, or repeat the derivation with a frequency-weighted Parseval identity.
  2. [V, Eq. (25), inequality (a)] The Jensen step in Eq. (25)(a) is dimensionally inconsistent. For the integral in Eq. (24), concavity of log2 gives ∫ log2(1+x(f)) df ≤ B log2(1 + (1/B)∫ x(f) df), not log2(1+∫ x(f) df). The displayed bound omits the bandwidth B and the 1/B normalization, so the claimed upper bound log2(1 + (Sx/Sσ)Eg) is not a consequence of Jensen's inequality and is not an upper bound in general. As a concrete check, for B=10 and flat |g(f)|^2 with x(f)=Sx|g(f)|^2/Sσ=10, the left side of (24) is 10log2(11)≈34.6 normalized units, while the claimed bound is log2(101)≈6.7. With exact Parseval, the correct bound would be B log2(1 + (Sx Eg)/(B Sσ)). This correction affects the theoretical claim in Section V and the rate comparison in Section VII, although the ideal-gain magnitude in Eq. (26) remains correct for a flat in-band gain.
minor comments (5)
  1. [IV, after Eq. (23)] The constant Γ0 is given as 0.886 in Eq. (21) but as 0.866 in the verification text; make the two values consistent.
  2. [VII-B] The text contains the typo 'Frsenel zone-based'; it should read 'Fresnel zone-based'.
  3. [Algorithm 2, Eqs. (35)-(38)] The least-squares update requires (A^H A)^{-1}, but no condition is given on K' relative to N_S; if K' < N_S then A^H A is singular, and otherwise it may be ill-conditioned. Please specify that K' ≥ N_S and, if needed, use a regularized pseudo-inverse.
  4. [III-A, Eq. (6)] The paper should state separately that the replacement of the per-element distances by R_BR and R_RU in the amplitude is a spatial approximation and that, for the Fourier representation in (17), g0 is additionally treated as frequency-independent; these are two different approximations and only the first is indicated in Section III-A.
  5. [III-C and IV] The symbol V is used for the angular set V(a) in Eq. (15) and for the Fourier transform V(f) in Eq. (18); please rename one of them to avoid ambiguity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Fresnel-zone Fourier representation and the Parseval-based rate bound are derived from the paper's own channel model and coordinate transformation, not from fitted values or self-citations.

full rationale

The paper's central claims—the Fresnel-zone coordinate transformation, the one-dimensional Fourier representation g(f)=∫ vt(t)e^{jψt(t)}e^{-j2πft}dt in Eq. (17), and the Parseval-based rate bound in Eq. (25)—are derived within the paper from the channel model in Section II and the Jacobian computation in Section III-C. The reflective intensity v(a) is defined by Eq. (15) as an integral of g0J(a,θ), not as a fitted parameter, and the subsequent phase designs are constructed by the stationary-phase formula Eq. (33) or by the GS algorithm and then compared with simulation, rather than being fit to the claimed bound. The self-citations (e.g., [16], [30], [39]) support background channel-model, amplitude-approximation, and channel-estimation assumptions; they are not invoked as a uniqueness theorem or as the source of the derived bound. The skeptical observation that g0 contains a 1/f² factor, so that v(a) technically depends on frequency, identifies a possible approximation gap in Eq. (17), but this is a mathematical correctness concern, not circularity: the paper does not define the Fourier representation in terms of the rate bound it is used to prove, and the bound is not an input to the derivation. Likewise, the missing 1/B normalization in the Jensen step of Eq. (25) would affect the validity of the bound but does not make the derivation reduce to its own inputs. Therefore no step in the claimed derivation chain is equivalent by construction to its inputs, and the paper is self-contained against external benchmarks.

Assumptions & free parameters 1 free parameters · 8 assumptions · 1 invented entities

The core derivation adds a coordinate transformation and a Fourier interpretation; it does not require fitting unknown constants. The main unstated load is the frequency independence of the Fresnel-zone intensity, the unit-modulus domain assumption, and standard analytical tools (Parseval, Jensen, stationary phase, sinc 3dB).

free parameters (1)
  • Extended bandwidth B' in the Gerchberg-Saxton algorithm = not specified (chosen > B)
    Introduced in Section VI-B to control out-of-band leakage; performance depends on this choice, but no value or rule is given.
assumptions (8)
  • standard math Parseval's theorem applies to the pair vt(t)ejψt(t) and g(f).
    Invoked in Eq. (25) step (c) to convert the integral of |g(f)|^2 over all frequencies to ∫ v_t^2(t) dt.
  • standard math Jensen's inequality is applied to log2(1+x) over the frequency band.
    Invoked in Eq. (25) step (a); note the integration measure (bandwidth B) appears to be mishandled in the printed inequality.
  • domain assumption Stationary phase approximation is valid for the designed phase.
    Used in Section VI-A with condition that B·Δt is sufficiently large; the paper acknowledges this in Sec VI-B.
  • ad hoc to paper The continuous I-RIS approximates the discrete RIS accurately via spatial Nyquist sampling.
    Introduced in Section III-A; the weight 1/d^2 and sampling argument justify replacing sums by integrals.
  • ad hoc to paper The cascaded channel amplitude (including g0) is independent of frequency.
    Implicitly assumed in Eqs. (13)-(17) so that v(a) is a fixed intensity; only spatial amplitude approximation is discussed in Sec III-A.
  • domain assumption The BS-UE link is blocked and only LOS BS-RIS and RIS-UE paths matter.
    Stated in Section II, standard for mmWave and THz blockage modeling.
  • domain assumption RIS reflection coefficients are unit-modulus.
    Stated in Section II following [12]; all phase designs respect |w_n|=1.
  • standard math The sinc 3dB constant Γ0=0.886 characterizes the gain bandwidth.
    Used in Eq. (21)-(23) to predict B3dB; the simulation in Fig. 4c tests this prediction.
invented entities (1)
  • Imaginary continuous RIS (I-RIS)
    purpose: Replace the discrete RIS with a continuous surface to use integration over Fresnel zones
    Modeling fiction with no direct falsifiable handle; justified by spatial Nyquist sampling, not by measurement.

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Pith. "Pith review of Near-Field Wideband Beamforming for RIS Based on Fresnel Zone." pith.science (2026). https://pith.science/paper/WQ4HP76V

@misc{pith2026241118878,
  author       = {Pith},
  title        = {Pith review of: Near-Field Wideband Beamforming for RIS Based on Fresnel Zone},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WQ4HP76V}},
  note         = {Machine review of arXiv:2411.18878}
}
read the original abstract

Reconfigurable intelligent surface (RIS) has emerged as a promising solution to overcome the challenges of high path loss and easy signal blockage in millimeter-wave (mmWave) and terahertz (THz) communication systems. With the increase of RIS aperture and system bandwidth, the near-field beam split effect emerges, which causes beams at different frequencies to focus on distinct physical locations, leading to a significant gain loss of beamforming. To address this problem, we leverage the property of Fresnel zone that the beam split disappears for RIS elements along a single Fresnel zone and propose beamforming design on the two dimensions of along and across the Fresnel zones. The phase shift of RIS elements along the same Fresnel zone are designed aligned, so that the signal reflected by these element can add up in-phase at the receiver regardless of the frequency. Then the expression of equivalent channel is simplified to the Fourier transform of reflective intensity across Fresnel zones modulated by the designed phase. Based on this relationship, we prove that the uniformly distributed in-band gain with aligned phase along the Fresnel zone leads to the upper bound of achievable rate. Finally, we design phase shifts of RIS to approach this upper bound by adopting the stationary phase method as well as the Gerchberg-Saxton (GS) algorithm. Simulation results validate the effectiveness of our proposed Fresnel zone-based method in mitigating the near-field beam split effect.

Figures

Figures reproduced from arXiv: 2411.18878 by the authors.

Figure 1
Figure 1. System model. where, xk ∈ C is the transmitted signal from the BS with power of s, vk ∈ C NBS×1 is the precoding vector of BS, and nk ∼ CN (0, σ2 n ) is the additive write Gaussian noise. The RIS elements are assumed to follow the constant reflection amplitude constraint [12], [29] and the reflection coefficient matrix can be written as Θ = diag(w), where w = [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. A series of Fresnel zones of the communication system with a BS and [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. The intersections of Fresnel zones and the RIS plane are a series of [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: The approximation of channel gain |gNarr(f)| 2 . The approximation of vt(t) is shown in Fig. 4a. The approximation of channel gain is illustrated in Fig. 4b. Probability density function of Γ as the location of BS and UE chosen randomly is shown in Fig. 4c. several con…
Figure 5
Figure 5. Figure 5: Normalized beamforming gain with respect to frequency. With [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 8
Figure 8. Figure 8: Simulation results of average achievable rate vs. transmit power [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]
Figure 7
Figure 7. Figure 7: Beamforming gain at different frequencies [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 9
Figure 9. Figure 9: Simulation results of average achievable rate vs. side length of RIS [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]
Figure 12
Figure 12. Figure 12: Simulation results of average achievable rate vs. number of BS [PITH_FULL_IMAGE:figures/full_fig_p011_12.png]

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Reviewed August 12, 2026 · model on record in the stance chip above.