Pith. sign in

REVIEW 3 major objections 5 minor 15 references

Performance Evaluation of Beyond Diagonal RIS under Hardware Impairments

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Hardware errors in the inter-element couplings of a BD-RIS create an error floor in channel estimation that a conventional diagonal RIS does not share.

desk verdict Useful first look at BD-RIS under impairments, but the impairment model violates reciprocity and passivity, so the headline ordering result is not yet trustworthy. read the letter →

arxiv 2506.07266 v1 pith:ZY2G3TI5 submitted 2025-06-08 eess.SP

Jose Carlos da Silva Filho , Josué V. de Araújo , Bruno Sokal , André L. F. de Almeida This is my paper
classification eess.SP
keywords beyonddiagonalRIShardwareimpairmentschannelestimationscatteringmatrixnormalizedmeansquareerrorgroup-connectedarchitecturemutualimpedancemismatch
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

Beyond diagonal reconfigurable intelligent surfaces (BD-RIS) connect reflecting elements to one another, giving a non-diagonal scattering matrix instead of the diagonal one of a conventional RIS. This paper asks whether those extra interconnections survive real hardware imperfections such as impedance mismatches and varactor defects. It models three kinds of scattering-matrix errors by multiplying the ideal BD-RIS response elementwise by a random error matrix: errors in the off-diagonal mutual impedances, errors in the diagonal self-impedances, and both together. Simulating channel estimation with a matched-filter estimator, the paper finds that mutual-impedance errors make the estimation NMSE saturate at high signal-to-noise ratio, and that a conventional diagonal RIS, having no off-diagonal elements, can outperform the BD-RIS in that regime. If true, the practical benefit of BD-RIS over traditional RIS is conditional on keeping inter-element coupling hardware accurate.

What carries the argument

The central object is the impaired scattering matrix $\bar{S}_t = S_t \odot E$, built from a block-diagonal, group-connected BD-RIS scattering matrix $S_t$ and a Hermitian impairment matrix $E$ whose entries are random amplitude-and-phase factors $\alpha_{ij}e^{j\phi_{ij}}$. The paper inserts $\bar{S}_t$ into the matched-filter channel estimator $\hat{c} = \frac{\bar{N}}{T}(\Omega \otimes I_{M_R})^H y$, where $\Omega = (S \diamond X)^T$ is the ideal training design, so the impairment appears as a mismatch between the assumed and actual scattering matrices. That mismatch, not the noise, drives the NMSE error floor observed at high SNR.

What would settle it

Build or simulate a physically constrained impairment: perturb only mutual impedances while enforcing passivity and, for a lossless surface, unitarity of the scattering matrix, then run the same matched-filter channel estimation. If the NMSE error floor at high SNR disappears or the crossover with the conventional RIS shifts, the elementwise-Hermitian impairment model overstates the Type 1 degradation.

Watch

Extended reading notes

Core claim

The paper's central claim is that the performance advantage of BD-RIS over a conventional diagonal RIS shrinks and can reverse when hardware impairments disturb the off-diagonal entries of the scattering matrix. Under the elementwise-multiplicative model $ar{S}_t = S_t \odot E$, where $S_t$ is the ideal group-connected BD-RIS response and $E$ is a Hermitian random error matrix, the paper shows that Type 1 impairment (only mutual impedances, i.e., only off-diagonal entries) and Type 3 impairment (all entries) cause the channel-estimation NMSE to saturate as the SNR grows, because the number of disturbed scattering-matrix elements increases with the BD-RIS group size. Type 2 impairment (only self-impedances, i.e., only diagonal entries) is the mildest, and its effect lessens as the group size grows. The conventional diagonal RIS, corresponding to group size $\bar{N}=1$, is immune to Type 1 because its scattering matrix has no off-diagonal elements to be impaired, and the simulations show it can therefore beat the BD-RIS at high SNR.

Load-bearing premise

The simulations assume that hardware errors can be any random amplitude-and-phase distortion multiplied onto the ideal surface response, with only a mirror-symmetry constraint, and never check that the resulting surface is physically realizable (a real passive surface obeys stricter limits).

Editorial extensions

If this is right

  • Under the off-diagonal (Type 1) impairment, the channel-estimation NMSE saturates as SNR increases, while the conventional diagonal RIS keeps improving because it has no off-diagonal elements to impair.
  • The self-impedance (Type 2) impairment is the mildest, and its effect shrinks as the BD-RIS group size grows, because only the $N$ diagonal entries of the scattering matrix are disturbed.
  • Affecting both self- and mutual impedances (Type 3) is slightly worse than Type 1, since it disturbs the largest number of scattering-matrix entries.
  • At low SNR the impairment is hidden by noise and BD-RIS retains its advantage; the crossover to the conventional RIS appears as the SNR increases.
  • Even a modest impairment level of 20% of the impedances creates a noticeable gap from the ideal case, so practical BD-RIS designs should include impairment estimation or element selection.

Reading between the lines

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

  • If real mutual-impedance errors are accompanied by even small diagonal shifts of the self-impedances, the conventional RIS's immunity to Type 1 would be diluted; that combined case is close to the paper's Type 3 and is worth testing.
  • Because the impairment matrix $E$ is assumed constant over all $T$ pilot slots, a natural extension is to estimate $E$ jointly with the channel by reserving a few pilot slots, which the paper lists as future work.
  • Physically constrained impairments that preserve passivity (unitarity for lossless surfaces) would couple diagonal and off-diagonal corrections, so the three impairment types cannot occur independently in a real lossless BD-RIS; this may change the predicted crossover.
  • The interaction between impairment and the pilot-overhead scaling $T = M_T N \bar{N}$ suggests that choosing the group size could trade estimation accuracy against robustness, a design knob the paper does not explicitly turn.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper proposes three hardware impairment models for beyond diagonal reconfigurable intelligent surfaces (BD-RIS), all based on an elementwise multiplicative distortion of the scattering matrix, S̄_t = S_t ⊙ E (Eq. 6). The considered impairments affect off-diagonal entries (Type 1), diagonal entries (Type 2), or all entries (Type 3) of the scattering matrix. Using the least-squares channel estimator of [9] for a group-connected BD-RIS, the paper evaluates the normalized mean square error (NMSE) via Monte Carlo simulations. The results show that impairments affecting off-diagonal elements (Types 1 and 3) cause NMSE saturation at high SNR, while Type 2 is less harmful, and a conventional diagonal RIS (N̄=1) is unaffected by Type 1. The paper concludes that there exist scenarios where the traditional RIS can outperform the BD-RIS.

Significance. If the elementwise multiplicative impairment model were a physically valid abstraction of hardware faults, the paper would offer a useful, reproducible simulation-based comparison of BD-RIS and conventional RIS under imperfections, and it would be among the first to study hardware impairments specifically for BD-RIS. The Monte Carlo methodology is sound and the internal mathematics is consistent with the stated model. However, the physical relevance of the model is not established: the paper incorrectly equates Hermitian symmetry with reciprocity for scattering matrices, and it never checks whether the impaired scattering matrix satisfies passivity/unitarity constraints. Moreover, the headline 'traditional RIS can outperform BD-RIS' is largely a definitional consequence of confining Type 1 impairments to off-diagonal entries. The significance is therefore conditional on a physically defensible impairment model.

major comments (3)
  1. [Section IV, paragraphs before Eq. (6) and in subsections A-C] The paper states that 'to maintain a physical reciprocity of energy' it enforces the Hermitian symmetry E^(q) = (E^(q))^H. For scattering matrices, reciprocity requires complex symmetry, S = S^T (equivalently E_ij = E_ji), not Hermitian symmetry. As defined in Type 1, off-diagonal entries have E_ij = α_ij e^{jφ_ij} and E_ji = α_ij e^{-jφ_ij}; for a symmetric ideal S_t, the impaired matrix S̄_t then has S̄_ij ≠ S̄_ji in general. Thus the simulated BD-RIS is not a reciprocal network, contradicting the stated physical motivation. The authors should either enforce E = E^T (with identical phases on symmetric positions) or explicitly state that the model describes non-reciprocal impairments and justify that scenario physically.
  2. [Section IV, Eq. (6)] The elementwise multiplicative model S̄_t = S_t ⊙ E is assumed without derivation from an impedance or circuit model. For a lossless passive BD-RIS, the scattering matrix must be unitary (and reciprocal if passive). The paper never checks whether S̄_t remains passive or unitary under the generated E. Even with α_ij ∈ (0,1], the phase perturbations can break the unitarity of S_t, and singular values of S̄_t can exceed 1, implying an active surface. This could contaminate the NMSE saturation seen in Figures 3, 5, and 6. The authors should either prove that their impairment model preserves passivity (e.g., by projecting S̄_t onto the set of unitary matrices or by deriving E from physically constrained impedance perturbations) or report the distribution of the largest singular value of S̄_t to quantify the fraction of non-passive realizations.
  3. [Section V, Figure 3 and Abstract/Conclusion] The result that the conventional diagonal RIS (N̄=1) is immune to Type 1 impairment is a tautology of the model: a diagonal scattering matrix has no off-diagonal elements, so the construction of Type 1 cannot affect it. The paper acknowledges this in the text, but the abstract and conclusion present it as a scenario where 'the traditional RIS can outperform the BD-RIS.' This overstates the significance. To make a non-trivial comparison, the authors should apply an equivalent impairment model to the conventional RIS as well (e.g., diagonal amplitude/phase errors arising from varactor nonidealities) and compare the two architectures under matched physical fault conditions.
minor comments (5)
  1. [Section III, Eq. (7)] The expression y_t = ∑_q G^(q) (S_t^(q) ⊙ E) H^(q)T x_t is imprecise because E is defined as block diagonal with blocks E^(q); it should read S_t^(q) ⊙ E^(q). Also, Eq. (8) reuses S̄_t without the group superscript, which is confusing for a block-diagonal structure.
  2. [Section V, NMSE definition] The NMSE definition uses c^(k) and ĉ^(k), but it is not explicitly stated that these are the true and estimated combined channel vectors for the k-th Monte Carlo trial; please clarify the notation.
  3. [Section IV, Type 1] The statement 'the maximum number of affected impedances is N(N̄−1)/2 = (N̄^2 Q − N)/2' is correct, yet it may be clearer to note explicitly that for N̄=1 this number is zero, which is exactly why the conventional RIS is unaffected by Type 1.
  4. [Figure 3 caption] The caption does not specify the simulation parameters (e.g., 20% affected impedances, N=32, M_T=2, M_R=4) that are described only in the text; adding these details to the caption would improve readability.
  5. [Introduction and Related Work] The paper does not cite prior work on BD-RIS with hardware impairments, mutual coupling, or impedance mismatch models. If such work exists, it should be referenced to properly position the contribution.

Circularity Check

1 steps flagged · score 4.0 of 10

Partial circularity: the headline conventional-RIS advantage under Type 1 impairment is built into the impairment definition, but the quantitative NMSE comparisons are genuine simulation outputs.

  1. self definitional [Section IV-A (Type 1 impairment definition) and Section V, Figure 3 discussion; Eq. (6)]
    "On the other hand, the conventional RIS (when \bar{N}=1) is not affected by the impairment Type1 since this impairment, as shown in Figure 2, only affects off-diagonal elements."

    The claim that a conventional RIS can outperform a group-connected BD-RIS under Type 1 impairment is not an emergent simulation result: it follows immediately from Eq. (6), S_bar_t = S_t (Hadamard) E, together with the Type 1 definition E_ii = 1 and the fact that the conventional RIS has a diagonal scattering matrix S_t. Thus S_bar_t = S_t exactly for the conventional RIS by construction, so its immunity to Type 1 impairment is encoded in the model definition rather than discovered. The paper openly states this mechanism, and the quantitative curves for other impairment types are not circular, but the headline scenario reduces to the definition of the impairment model.

full rationale

The paper does not fit parameters and then predict a closely related quantity; the NMSE curves are produced by Monte Carlo simulation of the stated model in Eqs. (6)-(13) under randomly generated impairment matrices. No fitted input is relabeled as a prediction, and the self-citations to the authors' prior work provide the baseline BD-RIS channel-estimation formulation, which is a legitimate starting point rather than a self-justifying uniqueness claim. The one definitional element is the qualitative conclusion that the conventional diagonal RIS is not affected by Type 1 impairments: because Type 1 is defined to perturb only off-diagonal entries of E and the conventional RIS has no off-diagonal scattering entries, Eq. (6) makes the impaired and ideal scattering matrices identical for Nbar=1. This makes the 'traditional RIS can outperform BD-RIS' scenario a direct consequence of the model definition. The Type 2 versus Type 3 comparisons, the SNR-saturation behavior, and the dependence on the number of affected impedances are genuine simulation outputs and retain independent content. A separate concern that the impairment model may violate reciprocity or passivity of physical scattering matrices is a correctness issue, not a circularity issue, and does not increase the circularity score.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central results rest on an ad hoc elementwise impairment model whose physical validity is not established (no passivity or unitarity check), on an unspecified random generation for alpha and phi, and on the ideal training design of prior work. These are not fitted to data, but they shape every conclusion.

free parameters (2)
  • Impairment amplitudes alpha_ij and phases phi_ij = Random variates; distribution unspecified
    The Type 1-3 models define alpha in (0,1] and phi in [0,2pi), but Section V does not state how these are drawn in the Monte Carlo trials. All NMSE curves depend on this choice.
  • Percentage of affected impedances = 20% in Figures 3-6; swept in Figure 7
    A hand-chosen simulation parameter that directly controls the number of nonzero error entries and therefore drives the reported performance gaps.
assumptions (5)
  • ad hoc to paper The observed scattering matrix is the ideal scattering matrix multiplied elementwise by an impairment matrix E (Eq. 6).
    Introduced as a modeling choice without derivation from circuit-level impedance perturbations; no guarantee that the product is a valid scattering matrix.
  • ad hoc to paper The impairment matrix preserves Hermitian symmetry 'when applicable' (Section IV).
    Reciprocity alone does not imply a physical passive scattering matrix; unitarity and passivity are not enforced.
  • domain assumption Hardware impairments remain constant over the T pilot time slots (Section III, text before Eq. 6).
    Stated as a convenience; reasonable for slow-varying hardware but not experimentally justified.
  • domain assumption The direct TX-RX link is blocked (Section III).
    Standard simplification in RIS channel estimation papers, removing the need to separate direct and reflected paths.
  • domain assumption Orthogonal BD-RIS training matrices from [9] and DFT pilot matrices with X^H X = T I are used (Section V).
    Borrowed from prior work; the matched filter in Eq. (11) is optimal only for this ideal design.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Performance Evaluation of Beyond Diagonal RIS under Hardware Impairments." pith.science (2026). https://pith.science/paper/ZY2G3TI5

@misc{pith2026250607266,
  author       = {Pith},
  title        = {Pith review of: Performance Evaluation of Beyond Diagonal RIS under Hardware Impairments},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZY2G3TI5}},
  note         = {Machine review of arXiv:2506.07266}
}
read the original abstract

Beyond diagonal reconfigurable intelligent surface (BD-RIS) improves the traditional reconfigurable intelligent surface (RIS) architecture functionality by interconnecting elements for advanced wave control. However, real-world implementations face hardware imperfections, such as impedance mismatches and varactor nonidealities, which can degrade overall system performance. In this paper, we propose three hardware impairment models that directly affect the BD-RIS scattering matrix structure and evaluate their impact on the channel estimation accuracy using the normalized mean square error (NMSE) as a performance metric. The proposed impairment models consider imperfections affecting self-impedances, mutual impedances, or both. Our results reveal how each impairment type degrades the system performance, allowing us to identify scenarios where the traditional RIS can outperform the BD-RIS.

Figures

Figures reproduced from arXiv: 2506.07266 by the authors.

Figure 1
Figure 1. System model of a MIMO communication aided by a BD-RIS. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Illustration of the three impairment types: (a) Type 1 – only the impedances that connect the elements are under impairment; (b) Type 2 – the [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. NMSE performance of BD-RIS under Type 1 impairment. [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (3 more)
Figure 5
Figure 5. Figure 5: NMSE performance of ideal BD-RIS and Type 3 impairment. [PITH_FULL_IMAGE:figures/full_fig_p004_5.png]
Figure 6
Figure 6. Figure 6: NMSE performance of BD-RIS under Type 1, 2, and 3 impairments. [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]
Figure 7
Figure 7. Figure 7: NMSE performance of BD-RIS under Type 1, 2, and 3 impairments [PITH_FULL_IMAGE:figures/full_fig_p005_7.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

15 extracted references · 13 canonical work pages

  1. [12]

    A decoupled channel estimation method for beyond diagonal ris,

    B. Sokal, Fazal-E-Asim, A. L. F. de Almeida, H. Li, and B. Clerckx, “A decoupled channel estimation method for beyond diagonal ris,” in2024 58th Asilomar Conference on Signals, Systems, and Computers, 2024, pp. 1395–1399

    work page 2024

  2. [9]

    Channel estimation for beyond diagonal reconfigurable intelligent surfaces with group-connected archi- tectures,

    H. Li, Y . Zhang, and B. Clerckx, “Channel estimation for beyond diagonal reconfigurable intelligent surfaces with group-connected archi- tectures,” in2023 IEEE 9th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP), 2023, pp. 21–25

    work page 2023

  3. [1]

    Wireless communications through reconfigurable intelligent surfaces,

    E. Basar, M. Di Renzo, J. De Rosny, M. Debbah, M.-S. Alouini, and R. Zhang, “Wireless communications through reconfigurable intelligent surfaces,”IEEE access, vol. 7, pp. 116 753–116 773, 2019

    work page 2019

  4. [2]

    Parafac-based channel estimation for intelligent reflective surface assisted mimo system,

    G. T. de Araújo and A. L. F. de Almeida, “Parafac-based channel estimation for intelligent reflective surface assisted mimo system,” in 2020 IEEE 11th Sensor Array and Multichannel Signal Processing Workshop (SAM), 2020, pp. 1–5

    work page 2020

  5. [3]

    Channel estimation for intelligent reflecting surface assisted mimo systems: A tensor mod- eling approach,

    G. T. de Araújo, A. L. de Almeida, and R. Boyer, “Channel estimation for intelligent reflecting surface assisted mimo systems: A tensor mod- eling approach,”IEEE Journal of Selected Topics in Signal Processing, vol. 15, no. 3, pp. 789–802, 2021

    work page 2021

  6. [4]

    A survey on channel estima- tion and practical passive beamforming design for intelligent reflecting surface aided wireless communications,

    B. Zheng, C. You, W. Mei, and R. Zhang, “A survey on channel estima- tion and practical passive beamforming design for intelligent reflecting surface aided wireless communications,”IEEE Communications Surveys & Tutorials, vol. 24, no. 2, pp. 1035–1071, 2022

    2022

  7. [5]

    Reconfigurable intelligent surfaces 2.0: Beyond diagonal phase shift matrices,

    H. Li, S. Shen, M. Nerini, and B. Clerckx, “Reconfigurable intelligent surfaces 2.0: Beyond diagonal phase shift matrices,”IEEE Communica- tions Magazine, vol. 62, no. 3, pp. 102–108, 2023

    work page 2023

  8. [6]

    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 Transactions on Wireless Communications, vol. 21, no. 2, pp. 1229–1243, 2021

    work page 2021

Show all 15 references
  1. [7]

    Beyond diagonal reconfigurable intelligent surfaces: From transmitting and reflecting modes to single- , group- , and fully-connected architectures,

    H. Li, S. Shen, and B. Clerckx, “Beyond diagonal reconfigurable intelligent surfaces: From transmitting and reflecting modes to single- , group- , and fully-connected architectures,”IEEE Transactions on Wireless Communications, vol. 22, no. 4, pp. 2311–2324, 2022

    2022

  2. [8]

    A dynamic grouping strategy for beyond diagonal reconfigurable intelligent surfaces with hy- brid transmitting and reflecting mode,

    Li, Hongyu and Shen, Shanpu and Clerckx, Bruno, “A dynamic grouping strategy for beyond diagonal reconfigurable intelligent surfaces with hy- brid transmitting and reflecting mode,”IEEE Transactions on V ehicular Technology, vol. 72, no. 12, pp. 16 748–16 753, 2023

    2023

  3. [10]

    Channel estimation for beyond diagonal ris via tensor decomposition,

    A. L. F. de Almeida, B. Sokal, H. Li, and B. Clerckx, “Channel estimation for beyond diagonal ris via tensor decomposition,”IEEE Transactions on Signal Processing, pp. 1–15, 2025

    2025

  4. [11]

    Channel estimation and beamforming for beyond diagonal reconfigurable intelligent surfaces,

    H. Li, S. Shen, Y . Zhang, and B. Clerckx, “Channel estimation and beamforming for beyond diagonal reconfigurable intelligent surfaces,” IEEE Transactions on Signal Processing, vol. 72, pp. 3318–3332, 2024

    2024

  5. [13]

    Irs-assisted communications under practical channel estimation and hardware model,

    L. C. d. P. Pessoa, G. T. d. Araújo, P. R. B. Gomes, and A. L. F. d. Almeida, “Irs-assisted communications under practical channel estimation and hardware model,” inAnais do XXXIX Simpósio Brasileiro de Telecomunicações e Processamento de Sinais (SBrT), Fortaleza, CE, Brasil, ...

    2021

  6. [14]

    Channel estimation in ris-assisted mimo systems operating under imperfections,

    P. R. B. Gomes, G. T. de Araújo, B. Sokal, A. L. F. d. Almeida, B. Makki, and G. Fodor, “Channel estimation in ris-assisted mimo systems operating under imperfections,”IEEE Transactions on V ehicular Technology, vol. 72, no. 11, pp. 14 200–14 213, 2023

    2023

  7. [15]

    Tensor-based channel estimation for ris- assisted communications with non-ideal phase shift responses,

    P. R. B. Gomes, G. T. de Araújo, B. Sokal, A. L. F. de Almeida, B. Makki, and G. Fodor, “Tensor-based channel estimation for ris- assisted communications with non-ideal phase shift responses,” in2022 Workshop on Communication Networks and Power Systems (WCNPS), 2022, pp. 1–6

    2022

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.