Efficient Localization with Base Station-Integrated Beyond Diagonal RIS

Ertugrul Basar; Henk Wymeersch; Hui Chen; Mahmoud Raeisi

REVIEW 4 major objections 3 minor 23 references

Efficient Localization with Base Station-Integrated Beyond Diagonal RIS

T0 review · 4 major / 3 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A fully connected beyond diagonal RIS placed at the base station can localize a user nearly as precisely as an active antenna array, while consuming no transmit power of its own.

desk verdict Clean CRLB study of BD-RIS for localization, with a genuinely new combination and standard derivations, but the near-active-array conclusion rests on an ideal lossless surface that real hardware will not match. read the letter →

arxiv 2411.13295 v1 pith:I5DX2MUE submitted 2024-11-20 eess.SP

Mahmoud Raeisi , Hui Chen , Henk Wymeersch , Ertugrul Basar This is my paper · ORCID

classification eess.SP MSC 94A12

keywords beyonddiagonalRISpassivebeamforminglocalizationCramér–Raolowerboundnear-fieldpositioningfar-fieldreconfigurableintelligentsurfaces6Gnetworks

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

This paper asks whether a passive surface can localize a user as accurately as an active antenna array. It proposes placing a fully-connected beyond diagonal RIS (BD-RIS) at the base station and sweeping a codebook of symmetric unitary phase configurations to beamform pilots toward the user. Using Cramér-Rao lower bound analysis in both near-field and far-field regimes, it argues that the BD-RIS achieves position error bounds close to those of a 101-element active array, while a diagonal RIS falls far behind when the surface sits close to the BS antenna. The reason is that the BD-RIS controls both amplitude and phase of the impinging wave, letting it equalize the magnitude variations in the BS-RIS channel that a phase-only diagonal RIS cannot correct. If true, the result makes passive BS-side beamforming a credible substitute for active arrays in high-accuracy positioning.

What carries the argument

The load-bearing object is the symmetric unitary scattering matrix $\Omega$ of the fully-connected BD-RIS, generated through Takagi decomposition (a matrix factorization that writes a complex symmetric matrix as the square of a symmetric unitary matrix) so that each codeword satisfies both $\Omega^H \Omega = I_M$ and $\Omega = \Omega^T$. The codebook construction in Algorithm 1 takes the BS-RIS channel vector $g$, computes an effective passive beamforming vector $\zeta = \Omega g$, and aims it at sweeping points (near field) or angles (far field). Because the fully-connected structure lets $\Omega$ act on both magnitude and phase, $\zeta$ can equalize the amplitude variations that appear in $g$ at small BS-RIS distances, which is exactly where a diagonal RIS, restricted to phase-only diagonal $\Omega$, loses its beamforming gain. This beamforming gain, in turn, drives the Fisher information of the received pilots and hence the position error bound.

What would settle it

Build a fully connected transmissive BD-RIS prototype at 28 GHz with the BS antenna $0.5\lambda$ away, measure the realized position error for a UE near $[12, 8]$ m in the near field, and compare it with the PEB curves in Figs. 3(b) and 3(d); any non-negligible insertion loss, phase quantization, or deviation from the symmetric-unitary constraint will push the measured PEB above the active-array curve that the idealized model predicts.

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Extended reading notes

Core claim

The paper's central claim is that a fully-connected BD-RIS integrated at the base station can perform downlink localization with accuracy close to an active antenna array, despite being passive. In the near-field and far-field CRLB simulations with $M = 101$ elements, the BD-RIS's position error bound tracks the active array curve and sits well below the diagonal RIS curve, especially when the BS-RIS distance is small (e.g., $d_c = 0.5\lambda$). The paper attributes this to the effective passive beamforming vector $\zeta = \Omega g$: because the fully-connected BD-RIS can realize any symmetric unitary scattering matrix, it can compensate for both phase and amplitude variations in the BS-RIS channel $g$, whereas a diagonal RIS can only phase-align and therefore loses beamforming gain as $d_c$ shrinks. The paper also notes a nuance in the far field: a diagonal RIS can match BD-RIS and active arrays in delay (ToA) estimation because its broad beam gives wide coverage, but its weak angular resolution dominates the overall position error bound.

Load-bearing premise

The paper assumes the BD-RIS is a lossless, fully connected surface that can realize any symmetric unitary response with perfect knowledge of the BS-RIS channel; if real devices lose energy, restrict the feasible scattering matrices, or have imperfect channel knowledge, the gap to the active array shown in the simulations will narrow or disappear.

Editorial extensions

If this is right

A base station can integrate a fully-connected BD-RIS instead of a large active array and still obtain near-active-array position error bounds, cutting power and hardware cost for localization.
Placing the BD-RIS close to the BS antenna (small $d_c$) is beneficial for BD-RIS, whereas a diagonal RIS loses beamforming gain in that regime, so compact BS integration favors BD-RIS.
In near-field scenarios, BD-RIS's amplitude-and-phase control is what recovers the performance that a diagonal RIS loses; without amplitude control the beamforming gain collapses at small BS-RIS distances.
In far-field scenarios, a diagonal RIS can match BD-RIS in delay (ToA) estimation because its broad beam spans many angles, but its angular error dominates the PEB, so BD-RIS is needed for overall positioning.
Increasing the number of OFDM subcarriers helps BD-RIS and active arrays improve PEB much further than a diagonal RIS, meaning BD-RIS can exploit wideband frequency diversity for localization.

Reading between the lines

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

An untested extension: replacing full connectivity with group-connected BD-RIS would show how much of the $M^2$ connectivity cost is needed to keep near-active-array accuracy.
The same BS-side surface should also improve uplink localization by focusing incoming energy from the UE, a direction the paper does not simulate.
In far-field delay estimation, a hybrid codebook that keeps a broad beam for timing and adds a sharp beam for angle could outperform either pure design, since the paper shows the two mechanisms are complementary.
Stress-testing the model with insertion loss, phase quantization, and imperfect BS-RIS channel knowledge would quantify how much hardware precision the claimed advantage requires.

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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

Colleague,

The paper is a clean, well-scoped CRLB study of a genuinely new combination: fully-connected BD-RIS used as a passive transmit beamformer at the base station for downlink localization. That specific application is not in the cited literature, and the NF/FF comparison against diagonal RIS and active arrays is a new quantitative result. The Slepian-Bangs derivations in the appendices are standard and appear correct, and the simulations sweep power, BS-RIS distance, and subcarriers in a way that matches the theory. The paper earns its central claim: under its model, BD-RIS closes most of the gap between D-RIS and an active array, especially when the BS-RIS distance is small.

The soft spots are the usual ones for idealized RIS papers, and they matter more here because the entire advantage over D-RIS at small d_c comes from the surface's ability to equalize amplitude variations across elements. The stress-test note is on target: realizing an arbitrary complex symmetric unitary transmission matrix with zero reflection and zero insertion loss is an idealization. A physical BD-RIS will have ohmic loss, mutual coupling, quantization, and nonzero S_11, and any restriction of the feasible scattering set will reduce the beamforming gain and move the PEB. The paper also assumes perfect knowledge of the BS-RIS channel g, which in practice would require calibration or estimation. No sensitivity analysis and no hardware measurements are provided, so the 'near active arrays' conclusion should be read as an idealized feasibility bound, not a design benchmark. That said, the paper is explicit about its model assumptions, and the idealization is standard for first-pass CRLB work. It is not circular and the derivations are not fitted to a target PEB.

Who is this for? Researchers working on RIS-aided localization or BD-RIS architectures will get a useful baseline and a clean methodology. It deserves a serious referee, but the referee should push for a robustness section: a restricted feasible scattering model, insertion loss, or at least a discussion of how S_11 affects the effective beamforming gain. Also, sharing code would help; Table I and Algorithm 1 are nearly enough but not quite.

Recommendation: send to peer review with a request for revision that addresses hardware non-idealities. The core result will likely survive in a weakened form, which is still worth having.

Referee Report

4 major / 3 minor

Summary. The paper proposes a base-station-integrated fully-connected beyond-diagonal RIS (BD-RIS) for downlink localization, operating in a transmissive mode and sweeping a predefined codebook of symmetric unitary transmission matrices constructed via Takagi decomposition. The authors derive Slepian-Bangs CRLBs for the channel parameters and transform them to position error bounds (PEBs) for both near-field and far-field user scenarios. Simulations compare the BD-RIS with a diagonal RIS and with a conventional active antenna array, sweeping transmit power, BS-RIS distance, and subcarrier count. The central finding is that, under the assumed ideal BD-RIS model, the BD-RIS achieves PEB close to that of an active array and much lower than that of a diagonal RIS, particularly when the BS-RIS distance is small.

Significance. If the central comparison holds, the paper provides a useful quantitative case that passive BD-RIS beamforming can substitute for active antenna arrays in single-anchor localization, which is a relevant question for 6G systems. The appendices give a mostly complete Slepian-Bangs derivation and the simulations cover a reasonable parameter range. The main caveat is that the headline result is obtained under an ideal lossless, fully-connected BD-RIS whose arbitrary symmetric unitary transmission matrices are assumed realizable; the paper does not test sensitivity to hardware non-idealities, and the active-array benchmark is not formally specified. Nevertheless, the contribution is novel and the analytical machinery is standard and clearly presented.

major comments (4)

[Section II-C and Algorithm 1] The central claim that BD-RIS approaches active-array localization accuracy rests on the assumption that every symmetric unitary matrix Omega can be realized by the BD-RIS with zero insertion loss and zero reflection. For a reciprocal lossless passive 2M-port, unitarity of the S-matrix gives S11^H S11 + Omega^H Omega = I_M, so setting Omega unitary forces S11=0 and S22=0. Real devices have ohmic loss, nonzero reflection, mutual coupling, finite tunable reactance ranges, and quantization. The advantage over D-RIS at small d_c in Figs. 3 and 4 relies precisely on the ability of a unitary Omega to equalize the amplitude variations of g in Eq. (1); any restriction of the feasible scattering set will reduce this advantage. I request a sensitivity analysis (e.g., modeling the transmission block as alpha Omega with alpha < 1, or adding a constrained scattering matrix) and a corresponding discussion in Sections IV and V.
[Section IV and Figs. 3-4] The active antenna array benchmark is not formally defined. Equation (6) gives the signal model only for the BD-RIS case; there is no equation specifying the AAA received signal, the transmit beamforming vector, or the total transmit power constraint. Since the central message is a quantitative comparison to an active array, the paper should state the AAA model explicitly and verify that the comparison uses the same total power and the same codebook quantization. Without this, the reader cannot assess whether the 'near active array' conclusion is affected by an inconsistent power normalization.
[Section III and Appendix B] In Scenario 1, the Jacobian in Appendix B treats Re(beta1) and Im(beta1) as independent nuisance parameters and sets all off-diagonal entries to zero, even though beta1 = (lambda/(4pi r)) e^{-j 2pi r/lambda} is a deterministic function of r. If this is a deliberate conservative choice to avoid exploiting amplitude/phase information, it should be stated explicitly; as written, the derived PEB is not the tight CRLB of the model in Eq. (2). This affects the numerical PEB values in Figs. 3 and 5(a), although the relative trends among AAA, BD-RIS, and D-RIS may still hold.
[Section IV and Eq. (1)] The simulation assumes that the BS-RIS channel g[n] is perfectly known and LoS-only, with no mutual coupling between RIS elements. Because both the codebook in Algorithm 1 and the effective beamforming gain depend directly on g[0], a small calibration error or unmodeled coupling will degrade the codebook and, consequently, the PEB. Please add a robustness check (e.g., perturbing g by a few percent) or at least state the required calibration accuracy for the claims to hold.

minor comments (3)

[Section II-A] The definition of the element position y_m appears to be missing the inter-element spacing: the text should read y_m = (m - (M+1)/2) delta, not just y_m = m - (M+1)/2, to be consistent with Eq. (3).
[Section III] In Eq. (8), the notation for the derivative should be made consistent: the real/imaginary parts of the parameters and the conjugation in the Slepian-Bangs formula are clear from the appendices, but a brief statement of the complex Gaussian noise model would improve readability.
[Section V] The conclusion states that BD-RIS is 'comparable to AAA' and the abstract states 'near active antenna arrays'; these claims should be qualified with the ideal hardware assumptions listed in the paper and with the new sensitivity analysis requested above.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the PEB/CRLB derivation is self-contained; the BD-RIS codebook comes from an external Takagi-based construction and the comparisons are not fitted to any target PEB.

full rationale

The derivation chain is explicit and self-contained: the signal model in (6)-(7) defines mu_{i,t}=h_i Omega_{i,t}g, the Fisher information matrix follows from the Slepian-Bangs formula in (8), and the PEB in (10) is a function of the Jacobian in Appendix B. None of these steps uses the final PEB or the AAA benchmark as an input. The BD-RIS codebook is constructed in Algorithm 1 via Takagi decomposition using the known BS-RIS channel g[0] and predefined sweeping points/directions; this construction is external to the localization performance target and is the same regardless of the UE position being estimated. The AAA and D-RIS benchmarks use the same sweeping points, so the reported gap reflects the different feasible sets of scattering matrices, not a fitted parameter. Several cited works include co-authors of this paper ([12], [13], [19], [20], [23]), but they support standard near-field region definitions, array-response conventions, and a path-loss intuition; the central CRLB computation is present in the paper itself and does not reduce to those citations. The assumptions of lossless, fully-connected, unitary BD-RIS operation are idealizations that affect robustness, but they are inputs to the analysis rather than outputs smuggled back as predictions. No self-definitional loop, fitted-input prediction, or renamed known result was found.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on a small set of standard signal-processing tools and two domain idealizations: LoS propagation with perfect synchronization, and an ideal lossless fully-connected BD-RIS. No parameters are fitted to experimental data, and no new physical entities are postulated.

free parameters (3)

BS-RIS distance d_c = 0.5λ in Table I; varied in Figs. 3c-d and 4c-d
The BD-RIS advantage over D-RIS is strongest at small d_c because BD-RIS compensates NF amplitude variations in g. Chosen as a favorable operating point, though the paper also shows larger d_c.
Number of BD-RIS elements M = 101
Array aperture and beamforming gain grow with M; comparison to AAA uses equal M. Results may change with other values of M.
Codebook sweep grid (T1, T2, Delta theta) = T1=500, T2=100, Delta theta=1.8 degrees; Delta theta=5 degrees in Fig. 5b
PEB depends on codebook resolution; the paper acknowledges high-resolution codebooks are critical. This is a design input, not a fitted parameter.

assumptions (5)

domain assumption Rayleigh-Sommerfeld diffraction model for the BS-RIS channel g[n] (Eq. 1)
The beamforming and CRLB results rely on this specific NF model for the short BS-RIS link, including element area A=(lambda/2)^2.
domain assumption Line-of-sight propagation with no multipath between RIS and UE
Both NF and FF channel models are LoS only; multipath would change the Fisher information and the PEB.
domain assumption Tight synchronization between BS and UE and known codebook at UE
Footnote 3 assumes tight synchronization; the FF tau parameter and the sweeping model depend on this.
domain assumption Fully-connected BD-RIS can implement any complex symmetric unitary matrix Omega losslessly
Algorithm 1 uses Takagi decomposition to synthesize Omega; real hardware may have insertion loss, mutual coupling, and quantization, limiting the achievable set of Omega.
standard math Slepian-Bangs formula for CRLB of complex Gaussian observations
Used in Eq. (8) to compute the channel FIM; standard in estimation theory.

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Cite this review

Pith. "Pith review of Efficient Localization with Base Station-Integrated Beyond Diagonal RIS." pith.science (2026). https://pith.science/paper/I5DX2MUE

@misc{pith2026241113295,
  author       = {Pith},
  title        = {Pith review of: Efficient Localization with Base Station-Integrated Beyond Diagonal RIS},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I5DX2MUE}},
  note         = {Machine review of arXiv:2411.13295}
}

read the original abstract

This paper introduces a novel approach to efficient localization in next-generation communication systems through a base station (BS)-enabled passive beamforming utilizing beyond diagonal reconfigurable intelligent surfaces (BD-RISs). Unlike conventional diagonal RISs (D-RISs), which suffer from limited beamforming capability, a BD-RIS provides enhanced control over both phase and amplitude, significantly improving localization accuracy. By conducting a comprehensive Cram\'er-Rao lower bound (CRLB) analysis across various system parameters in both near-field and far-field scenarios, we establish the BD-RIS structure as a competitive alternative to traditional active antenna arrays. Our results reveal that BD-RISs achieve near active antenna arrays performance in localization precision, overcoming the limitations of D-RISs and underscoring its potential for high-accuracy positioning in future communication networks. This work envisions the use of BD-RIS for enabling passive beamforming-based localization, setting the stage for more efficient and scalable localization strategies in sixth-generation networks and beyond.

Figures

Figures reproduced from arXiv: 2411.13295 by the authors.

**Figure 2.** Transmit beamforming patterns for BD-RIS and benchmark configura [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** CRLB and PEB analysis for Scenario 1 (NF): (a) CRLB for [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 5.** Figure 5: 10 log10(PEB) for different UE positions. (a) Scenario 1 (NF); (b) Scenario 2 (FF). estimation compared to BD-RIS and AAA, which achieve higher angular resolution due to their sharp beams. Conversely, D-RIS achieves τ estimation comparable to BD-RIS and AAA due to its …

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

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

[1] [1]

Large intelligent surface for positioning in millimeter wave MIMO systems,

J. He et al., “Large intelligent surface for positioning in millimeter wave MIMO systems,” in IEEE Veh. Technol. Conf. (VTC-Spring), Antwerp, Belgium, May. 25-28, 2020

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J. An et al. , “Stacked intelligent metasurface-aided MIMO transceiver design,” IEEE Wireless Commun., vol. 31, pp. 123–131, Apr. 2024

work page 2024

[4] [4]

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J. An et al., “Stacked intelligent metasurfaces for efficient holographic MIMO communications in 6G,” IEEE J. Sel. Areas Commun. , vol. 41, pp. 2380–2396, Jun. 2023

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Stacked intelligent metasurfaces for fully- analog wideband beamforming design,

Z. Li, J. An, and C. Yuen, “Stacked intelligent metasurfaces for fully- analog wideband beamforming design,” in IEEE VTS Asia Pac. Wireless Commun. Symp. (APWCS), Singapore , Aug. 21-23, 2024

work page 2024

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H. Niu et al. , “Stacked intelligent metasurfaces for integrated sensing and communications,” IEEE Wireless Commun. Lett., vol. 13, pp. 2807– 2811, Aug. 2024

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H. Li, S. Shen, M. Nerini, and B. Clerckx, “Reconfigurable intelligent surfaces 2.0: Beyond diagonal phase shift matrices,” IEEE Commun. Mag., vol. 62, pp. 102–108, Nov. 2023

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Modeling and architecture design of reconfigurable intelligent surfaces using scattering parameter network analysis,

S. Shen, B. Clerckx, and R. Murch, “Modeling and architecture design of reconfigurable intelligent surfaces using scattering parameter network analysis,” IEEE Trans. Wireless Commun., vol. 21, pp. 1229–1243, Aug. 2022

work page 2022