Beyond Diagonal IRS Aided OFDM: Rate Maximization under Frequency-Dependent Reflection

Shuowen Zhang; Ye Yuan

REVIEW 3 major objections 5 minor 21 references

Beyond Diagonal IRS Aided OFDM: Rate Maximization under Frequency-Dependent Reflection

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A single capacitance matrix shared across subcarriers is enough to configure a frequency-dependent BD-IRS for OFDM rate maximization.

desk verdict A well-motivated BD-IRS/OFDM formulation with a sound relaxation, but the capacitance-recovery step (25) and Proposition 2 are not valid, so the paper's central claim of a feasible high-quality solution is unsupported as written. read the letter →

arxiv 2509.06378 v1 pith:HETLX6KT submitted 2025-09-08 cs.IT eess.SPmath.IT

Ye Yuan , Shuowen Zhang This is my paper · ORCID

classification cs.ITeess.SPmath.IT

keywords beyonddiagonalIRSBD-IRSOFDMfrequency-dependentreflectioncapacitancematrixoptimizationalternatingsuccessiveconvexapproximationratemaximization

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 addresses a practical obstacle to using beyond-diagonal intelligent reflecting surfaces (BD-IRS) in broadband communication: the reflection response of the surface depends on frequency, so the reflection matrices on different OFDM subcarriers are coupled through one shared tunable capacitance matrix. The authors claim that, despite the non-convex coupling, jointly optimizing that capacitance matrix together with per-subcarrier power allocations yields a high-quality feasible configuration. The proposed method relaxes the circuit constraints, alternates water-filling with successive convex approximation, and then recovers a single feasible capacitance matrix by a convex projection. If the claim holds, it provides a systematic circuit-aware design that replaces an earlier linear-fitting approach restricted to a narrow band, and it quantifies the gains of frequency-dependent and non-reciprocal BD-IRS designs.

What carries the argument

The load-bearing object is the circuit-level map from the tunable capacitance matrix C to the admittance matrix A_n at subcarrier frequency f_n, and from A_n to the reflection matrix Phi_n = (a0 I + A_n)^(-1)(a0 I - A_n). The paper proves the passivity condition Phi_n Phi_n^H <= I, which becomes a convex relaxation of the circuit constraints, then closes the loop with the convex projection (P3) that picks one common C closest in Frobenius norm to the ideal per-tone capacitance matrices derived from (25). Initialization of the alternating loop uses a semi-definite relaxation of a per-subcarrier channel-power maximization.

What would settle it

Run the proposed algorithm on the paper's setup (for example, M=10, P=30 dBm, 2.4 GHz, 300 MHz) and compare the actual rate obtained from the capacitance matrix recovered by (P3) with the relaxed upper bound of (P2), and with the linear-fitting benchmark. If the gap to the relaxed bound is large, or if a simpler alternative projection matches or beats the proposed rate, the claim of a high-quality feasible solution is weakened.

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Extended reading notes

Core claim

The paper's central claim is that the achievable rate of a BD-IRS aided OFDM link can be maximized by selecting one nonnegative capacitance matrix C shared by all subcarriers, even though the ideal capacitance matrix differs from subcarrier to subcarrier. The argument proceeds by proving that every circuit-realizable reflection matrix satisfies Phi_n Phi_n^H <= I, using that passivity condition to form a relaxed problem, solving the relaxed rate-maximization problem by alternating optimization, and then constructing a feasible C by minimizing the total Frobenius distance to the per-subcarrier ideal capacitance matrices. Numerical results in the paper show this design outperforming linear-fitting, frequency-independent, conventional-IRS, and reciprocal-constrained benchmarks.

Load-bearing premise

The algorithm's quality guarantee rests on the unverified assumption that the single capacitance matrix found by the convex projection (P3) achieves an actual rate close to the relaxed optimum of (P2), with no stated bound or condition on that gap.

Editorial extensions

If this is right

A BD-IRS can be configured for wideband OFDM with a single nonnegative capacitance matrix per circuit state, with per-subcarrier powers set by water-filling.
The design works directly from the circuit model, so it is not limited to the narrow band for which the earlier linear-fitting method was calibrated.
Allowing non-symmetric, non-reciprocal capacitance matrices yields a substantial rate gain over the reciprocal constraint in the paper's simulations.
The advantage over the fitting-based benchmark grows with the number of reflecting elements, so the method matters most for moderate-to-large fully connected BD-IRS.

Reading between the lines

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

The same alternating-projection pipeline should extend to multi-antenna, multi-user, or OFDMA systems, where coupling through a common C is even more restrictive; that would be a direct stress test of the model.
The Frobenius-norm projection (P3) is a heuristic choice; weighted norms, minimax error, or discrete capacitor-value constraints could beat it on the same rate metric.
Because the frequency dependence is set by the fixed circuit elements R, L1, and L2, jointly optimizing those along with C is a natural extension the paper leaves open.
The reported gains are from one simulated channel model; testing at other carrier frequencies, bandwidths, and measured multipath profiles would show whether the structural advantage persists.

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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

Two things to know. The problem this paper goes after is real: BD-IRS reflection depends on frequency and is coupled across subcarriers through a common capacitance matrix, and prior work [11] only used a linear fit. Replacing that fit with direct optimization of C is the right ambition. The paper also does something correct early on: Proposition 1, the passive-network constraint Φ_nΦ_n^H ⪯ I, is a valid relaxation, and the AO/SCA loop is described carefully.

But the constructive step at the end does not hold up. Equation (25) does not invert (2). For off-diagonal entries, (2) says [A_n]_{m,k} = -(1/(R+jωL2) + 1/(jωC_{m,k}) + 1/(jωL1)); solving for C requires subtracting the other two terms from -[A_n]_{m,k}, then inverting. What (25) does is different: it takes 1/(-[A_n]_{m,k}) and subtracts impedance terms. The units are wrong too—[A_n] is admittance, so 1/[A_n] is impedance, and multiplying by 1/(j2πf_n) gives henry, not farad. The diagonal case is worse, because the diagonal admittance is a sum over k of branch admittances, so C_{m,m} cannot be recovered from [A_n]_{m,m} alone. As a result, the common C produced by (P3) is not shown to realize the optimized {Φ_n}, and the claimed 'high-quality feasible solution' has no basis. The numerical section also never says whether the plotted rate is computed with the actual circuit model for the constructed C or with the relaxed Φ_n; if it is the latter, the plots are not achievable rates at all.

There is also a false equivalence in Proposition 2. The condition q_n^H(I_M⊗1_M)q_n ≤ M bounds the sum of squared column sums of Φ_n, which is much weaker than the spectral norm constraint Φ_nΦ_n^H ⪯ I. This one only affects the initialization, so it is secondary, but it is another sign that the derivations were not checked line by line.

Who wants this paper? Researchers in BD-IRS and wideband optimization will want to know that this formulation exists, and there is a plausible route to a correct algorithm: invert (2) properly, add a consistency constraint for the common C, and re-run the numerics with the actual circuit response. But as printed, the main result is unsupported. I would not desk-reject the topic; a referee should write the report telling the authors exactly where (25) fails and give them a chance to fix it. In current form, though, I would not trust the numbers.

Referee Report

3 major / 5 minor

Summary. The paper studies a broadband OFDM system aided by a beyond-diagonal IRS (BD-IRS) whose reflection matrices are frequency-dependent through a common tunable capacitance matrix C. It formulates a joint optimization problem over C and per-subcarrier power allocations to maximize the achievable rate, relaxes the circuit constraints to convex constraints, solves the relaxed problem via alternating optimization (water-filling and successive convex approximation), and then constructs a feasible C by inverting the optimized per-subcarrier admittance matrices and projecting onto a common nonnegative capacitance matrix. The paper claims that this algorithm finds a high-quality feasible solution and demonstrates its superiority over several benchmarks in numerical results.

Significance. If the algorithm were correct as stated, it would provide a systematic optimization framework for frequency-dependent BD-IRS in wideband OFDM, going beyond the linear-fitting approach of [11] and explicitly handling the coupling across subcarriers through a common C. The system model is clearly stated, and the AO/SCA machinery for the relaxed problem (P2) is standard. However, the central feasibility recovery step contains an algebraic inversion error and a dimensional inconsistency, and the projection step is not accompanied by any performance gap analysis. As printed, the claimed achievable rate is not supported by the derivation, which substantially reduces the paper's significance. The paper does not provide reproducible code or machine-checked proofs.

major comments (3)

[V-C, Eq. (25)] The inversion of the admittance model (2) is incorrect. For m≠k, (2) gives -[A_n]_{m,k} = 1/(R+j2π f_n L2) + 1/(j2π f_n C_{m,k}) + 1/(j2π f_n L1), so the required capacitance is C_{m,k} = 1/(j2π f_n ( -[A_n]_{m,k} - 1/(R+j2π f_n L2) - 1/(j2π f_n L1))). Equation (25) instead places -R - j2π f_n L2 in the denominator, which is an impedance rather than an admittance, making the expression dimensionally inconsistent. For m=k, [A_n]_{m,m} is the sum over all columns of the branch admittances, so it cannot be inverted to a single C_{m,m} without first subtracting the off-diagonal branch admittances, which are determined by the off-diagonal entries of A_n. As a consequence, the capacitance matrix obtained from (P3) will not, in general, realize the optimized reflection matrices {Φ_n}, and the rate reported from the subsequent power allocation is not an achievable rate of the modeled BD-IRS circuit. This invalidates the central claim of constructing a feasible solution to (P1).
[V-D, Proposition 2] The claimed equivalence is false. Direct computation shows q_n^H(I_M⊗1_M)q_n = \sum_{i=1}^M |\sum_{j=1}^M [Φ_n]_{j,i}|^2, which is a single scalar inequality, whereas Φ_nΦ_n^H ⪯ I_M is a positive-semidefinite matrix constraint. For M=2, Φ_n = diag(√2,0) satisfies q_n^H(I_M⊗1_M)q_n = 2 ≤ M but has largest singular value √2 > 1, violating Φ_nΦ_n^H ⪯ I_M. The proof establishes only the necessity direction; the sufficiency assertion is unproven. Therefore (P4-eqv) is a relaxation, not an equivalent reformulation of (P4), and the SDR-based initialization may produce q_n that do not correspond to physically admissible reflection matrices.
[V-C, P3] Even if (25) were corrected, the projection step (P3) provides no guarantee about the rate loss incurred by enforcing a common capacitance matrix. The paper does not bound the gap between the relaxed optimum of (P2) and the achievable rate of the constructed C, nor does it analyze the conditioning of the mapping from C to {Φ_n}. The numerical results show only a single set of channel and circuit parameters; the claim of a 'high-quality feasible solution' is therefore an empirical assertion without supporting analysis.

minor comments (5)

[V-C] In the definition of [\tilde C_n]_{m,k} for m=k, the parentheses in the denominator are unbalanced; the expression should be checked and written cleanly.
[V-D] The proof of Proposition 2 stops after the necessity direction; even if the equivalence were true, the sufficiency direction should be demonstrated explicitly.
[V-B] The statement that the SCA update 'is guaranteed to reach at least a stationary point of (P2-II)' is not proven; the paper should provide a citation or argument for the adopted SCA variant.
[VI] It is unclear whether the rate plotted in Fig. 2 is the relaxed objective of (P2) during AO or the achievable rate after the feasibility projection; this should be stated explicitly.
[VI] Benchmark scheme 3 assumes a frequency-independent BD-IRS, but under the circuit model (2)-(3) with a common C the reflection matrices are inevitably frequency-dependent; the implementation of this benchmark should be clarified.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the algorithm optimizes a relaxation and then projects to a feasible capacitance matrix, with the reported rate recomputed from the physical model; self-citations are not load-bearing.

full rationale

The paper's derivation chain is self-contained. The rate objective is evaluated from the physical channel model (1) and the circuit model (2)-(3) imported from independent references [13] and [14], not from any fitted quantity. The relaxed problem (P2) is an upper bound justified by Proposition 1, and the final feasible solution is constructed by projecting onto a common capacitance matrix via (P3); the reported rate is then recomputed through the original model, so no output is defined to equal an input. The self-citations ([2], [5], [21]) are motivational or standard-technique citations and are not load-bearing; the central circuit model and the linear-fitting benchmark [11] are external. The paper even acknowledges that the relaxed solution may not be feasible ('the obtained solution to (P2) may not satisfy the original constraints for the BD-IRS reflection matrices under the circuit structure, i.e., the constraints in (5) and (6) of (P1)') and treats the construction as an approximation. Whether (25) correctly inverts (2) is a technical correctness issue, not circularity.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The paper's main contribution is algorithmic; it introduces no new physical entities. It relies on the circuit model from prior work, the passivity condition, a relaxation, and standard SDR heuristics. The key unproven item is the P3 projection step, which is the bridge between the relaxed optimization and a physically implementable capacitance matrix.

free parameters (1)

Gaussian randomization sample count Q = 50
Used in SDR rank-one recovery for the initialization in Section V-D; chosen by hand, with no sensitivity analysis reported.

assumptions (5)

domain assumption Passive network admittance satisfies A_n + A_n^H >= 0 at each subcarrier frequency.
Standard positive-real condition for passive multiport networks, cited to [14]; used in the proof of Proposition 1.
domain assumption The lumped-element circuit model in (2) for each inter-element and shunt admittance exactly represents the BD-IRS hardware.
Taken from related work [11], [13]; the paper does not validate it with measurements or full-wave simulation, but adopts it as the basis for (3)-(6).
ad hoc to paper Relaxed constraints Phi_n Phi_n^H <= I, together with the Frobenius projection in P3, are a good surrogate for physical realizability with a common C.
This is the core approximation of the paper; no bound is provided, and it is load-bearing for the claimed high-quality feasible solution.
domain assumption Perfect channel state information is available at both transmitter and receiver.
Assumed in Section II and used throughout the optimization; typical for this line of work but not achieved in practice.
ad hoc to paper Gaussian randomization from the SDR solution of (P4-SDR) yields a feasible, near-optimal q_n for (P4-eqv).
Standard SDR heuristic from [20], [21]; in this paper its validity is further compromised by the false equivalence in Proposition 2.

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Cite this review

Pith. "Pith review of Beyond Diagonal IRS Aided OFDM: Rate Maximization under Frequency-Dependent Reflection." pith.science (2026). https://pith.science/paper/HETLX6KT

@misc{pith2026250906378,
  author       = {Pith},
  title        = {Pith review of: Beyond Diagonal IRS Aided OFDM: Rate Maximization under Frequency-Dependent Reflection},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HETLX6KT}},
  note         = {Machine review of arXiv:2509.06378}
}

read the original abstract

This paper studies a broadband orthogonal frequency division multiplexing (OFDM) system aided by a beyond diagonal intelligent reflecting surface (BD-IRS), where inter-connections exist among different elements such that the reflection matrix can exhibit a beyond diagonal structure. Under practical circuit structures, the reflection matrix of the BD-IRS is generally dependent on the circuit parameters (e.g., capacitance matrix for all tunable capacitors) as well as the operating frequency, which leads to couplings among the BD-IRS reflection matrices over different sub-carriers and consequently new challenges in the BD-IRS design. Motivated by this, we first model the relationship between the BD-IRS reflection matrices over different sub-carriers and the tunable capacitance matrix, and then formulate the joint optimization problem of the tunable capacitance matrix and power allocation over OFDM sub-carriers to maximize the achievable rate of the OFDM system. Despite the non-convexity of the problem, we propose an effective algorithm for finding a high-quality feasible solution via leveraging alternating optimization and successive convex approximation. Numerical results show the superiority of our proposed design over benchmark designs.

Figures

Figures reproduced from arXiv: 2509.06378 by the authors.

**Figure 2.** Convergence behavior of the proposed algorithm. [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Achievable rate versus transmit power. −169 + 9 + 10 log10(b) = −93.29 dBm. We further set Γ = 8.8 dB [12], L1 = 2.5 nH, L2 = 0.7 nH, R = 1 Ω, and Q = 50. All the results are averaged over 100 independent channel realizations. First, we evaluate the convergence behavior of the proposed algorithm under M = 5 and P = 30 dBm [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Achievable rate versus number of reflecting elements. [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

Review history (2 revisions) →

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