arxiv: 2605.06107 · v1 · submitted 2026-05-07 · 📡 eess.SP

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A Family of Hybrid Beyond-Diagonal RIS Architectures: Design and Performance Analysis

Ioannis Krikidis, Konstantinos Ntougias

Authors on Pith no claims yet

Pith reviewed 2026-05-08 06:55 UTC · model grok-4.3

classification 📡 eess.SP

keywords reconfigurable intelligent surfacesbeyond-diagonal RIShybrid active passivesignal to noise ratioTakagi factorizationbeamforming solutionperformance analysisSISO link

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The pith

Hybrid beyond-diagonal RIS architectures achieve equal or higher SNR than diagonal designs while using significantly fewer amplifiers.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a family of hybrid beyond-diagonal reconfigurable intelligent surface architectures that divide the surface into two subsurfaces, each using either a passive or active group-connected BD-RIS configuration. It provides a closed-form solution to maximize received signal-to-noise ratio by applying Takagi's factorization to a complex symmetric matrix for each group and determining the best amplification factor that respects the reflect-power budget. In a single-input single-output link where the direct path is blocked, numerical evaluations show these hybrid setups reach the same or better SNR than conventional diagonal RIS while needing far fewer reflect-type amplifiers. This combination of inter-element coupling and selective amplification addresses path loss and hardware cost issues in RIS-assisted wireless links.

Core claim

The paper establishes a family of hybrid BD-RIS architectures that partition the reflecting surface into two subsurfaces, each configured as either a passive or active group-connected beyond-diagonal RIS. For each such group, the optimal reflection coefficients are obtained via Takagi's factorization of a complex symmetric matrix, paired with a closed-form amplification factor that meets the per-group power budget. Numerical evaluation in a SISO system with blocked direct link confirms that the proposed designs attain equivalent or superior receive SNR compared to diagonal RIS architectures, yet employ significantly fewer active amplifiers.

What carries the argument

The closed-form SNR-maximizing solution that applies Takagi's factorization to a complex symmetric matrix for each BD-RIS group and pairs it with an optimal per-group amplification factor under the reflect-power budget.

Load-bearing premise

The performance claims rely on perfect channel state information being available and the power budget being satisfiable independently per group without interference or hardware flaws.

What would settle it

A physical experiment in a SISO setup with blocked direct path that implements the hybrid BD-RIS prototypes, measures achieved SNR, and counts how many amplifiers are required to match the performance of a diagonal RIS baseline.

Figures

Figures reproduced from arXiv: 2605.06107 by Ioannis Krikidis, Konstantinos Ntougias.

**Figure 1.** Figure 1: The proposed family of hybrid BD-RIS architectures, view at source ↗

**Figure 2.** Figure 2: Receive SNR vs. M for the proposed hybrid BD-RIS family and the relevant benchmarks (a = 0.5, MG = 4). its diagonal-RIS counterpart, yielding (12). Full details are omitted due to space limitations. Hence, the proposed designs inherit the regimes of [12] (standard, large-M, large-Pt) scaled by κ 2 (MG)/(π 2/16): A/P-BD is preferred under scarce reflect-power, SC/SC-BD when amplifier count is the bottleneck… view at source ↗

**Figure 4.** Figure 4: reports the sum rate versus M for a 2-user SISO MAC with per-user power 20 dBm. The A/P-BD attains the highest sum rate over the whole range and exceeds the diagonal hybrid A/P RIS [12] by approximately 20% at M = 80 (2.49 vs. 2.13 b/s/Hz). The reason is that, in the multi-user regime, 0 10 20 30 40 50 60 70 80 0 0.5 1 1.5 2 2.5 view at source ↗

**Figure 5.** Figure 5: SNR gain of (F2) A/P-BD over the diagonal hybrid A/P RI view at source ↗

**Figure 6.** Figure 6: Rate vs. per-RS reflect-power budget P max r,s (M = 64, a = 0.5, MG = 4, Pt = 20 dBm). -10 -5 0 5 10 15 20 25 30 0 2 4 6 8 10 12 14 16 18 20 view at source ↗

**Figure 7.** Figure 7: Rate vs. transmit power Pt (M = 64, a = 0.5, MG = 4, P max r,s = 10 dBm). power allocation across the MG,1 patches per group (cf. (1)) is not compensated by the unitary-symmetric BD-RIS coupling because the latter only redistributes phase, not amplitude. As M grows with fixed MG, the number of groups (and therefore the total 1/ √ MG amplitude penalty across the surface) accumulates. This effect is absent … view at source ↗

read the original abstract

Beyond-diagonal reconfigurable intelligent surfaces (BD-RISs) extend conventional diagonal RISs by allowing inter-element coupling, thereby enlarging the set of attainable scattering matrices and improving the achievable signal-to-noise ratio (SNR). On the other hand, hybrid active/passive RISs use reflect-type power amplifiers in a fraction of the elements to alleviate the multiplicative path loss. In this paper, we bring these two ideas together and introduce a \emph{family of hybrid BD-RIS architectures}, in which the surface is partitioned into two reflecting subsurfaces (RSs), each adopting either a passive or an active group-connected BD-RIS design. We derive a closed-form SNR-maximizing solution that combines, for every BD-RIS group, Takagi's factorization of a certain complex symmetric matrix with an optimal per-group amplification factor that satisfies the reflect-power budget. Three architectures within the proposed family (active/passive, fully-connected-active/sub-connected-active, and sub-connected-active/sub-connected-active hybrid BD-RIS) are studied. Numerical results in a single-input single-output (SISO) link with blocked direct path show that the proposed hybrid BD-RIS architectures attain the same or higher receive SNR than their diagonal counterparts while using significantly fewer reflect-type amplifiers.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper introduces hybrid active/passive group-connected BD-RIS architectures with a closed-form Takagi-based solution, but the reported SNR gains rest on perfect CSI and independent per-group power constraints.

read the letter

The main point is that the authors combine beyond-diagonal RIS with hybrid active/passive partitioning and deliver an explicit SNR-maximizing solution instead of relying on numerical optimization loops. They split the surface into two subsurfaces, assign each a group-connected BD-RIS structure that can be passive or active, and solve per group by factoring a complex symmetric matrix with Takagi's method then scaling the amplification factor to the reflect-power limit. Three concrete variants are worked out, and the SISO blocked-direct-path simulations show these hybrids match or exceed diagonal RIS SNR while using fewer amplifiers overall. That reduction in active elements is the practical hook they emphasize, and the closed-form step is a clear improvement over iterative alternatives in the literature they cite. The math itself tracks standard matrix techniques and power allocation, so the derivations appear internally consistent on their own terms. The numerical comparisons are straightforward and directly tied to the claimed architectures. The main limitation is that every step assumes perfect knowledge of all channels and that each group's power budget can be met without interference from other groups, shared hardware limits, or amplifier non-idealities. The results stay in SISO, so scaling behavior is not shown. If those ideal conditions are relaxed, the feasible scattering matrices change and the SNR advantage may shrink. This work is aimed at researchers focused on RIS hardware variants and closed-form beamforming solutions. A reader already working on non-diagonal or hybrid surfaces would pick up the specific partitioning idea and the factorization trick. It has enough concrete math and comparisons to justify sending it out for peer review rather than a desk reject, though any referee would likely press on the assumption sensitivity and the SISO-only scope.

Referee Report

2 major / 2 minor

Summary. The paper introduces a family of hybrid beyond-diagonal RIS (BD-RIS) architectures that partition the surface into two reflecting subsurfaces, each using either passive or active group-connected BD-RIS designs. It derives a closed-form SNR-maximizing solution that applies Takagi factorization to a complex symmetric matrix per BD-RIS group followed by an optimal per-group amplification factor satisfying the reflect-power budget. Three specific architectures (active/passive, fully-connected-active/sub-connected-active, and sub-connected-active/sub-connected-active) are analyzed. Numerical results for a SISO link with blocked direct path claim that the proposed designs achieve the same or higher receive SNR than diagonal RIS counterparts while using significantly fewer reflect-type amplifiers.

Significance. If the closed-form derivations hold under the stated assumptions, the work usefully combines the enlarged scattering-matrix set of BD-RIS with the path-loss mitigation of hybrid active/passive elements. The explicit closed-form solution via Takagi factorization per group is a constructive strength that could aid practical implementation. The numerical demonstration of SNR parity or gain with reduced amplifier count, if robust, would support hardware-efficient RIS deployments in blocked-line-of-sight scenarios.

major comments (2)

[§III (derivation) and §IV (numerical results)] The closed-form SNR-maximizing solution (abstract and §III) and the headline numerical claim (abstract and §IV) both presuppose perfect CSI together with independent per-group reflect-power budgets that can be met without mutual coupling, shared hardware constraints, or amplifier non-idealities. These assumptions directly define the feasible set of scattering matrices; their violation would change the attainable SNR and therefore the central comparison to diagonal RIS.
[§IV] Table or figure in §IV that reports the SISO SNR comparisons does not include sensitivity to CSI error or inter-group power coupling; because the headline claim of “same or higher SNR with significantly fewer amplifiers” rests on these numerical results, an explicit robustness check or worst-case analysis is required to substantiate the performance advantage.

minor comments (2)

[Abstract] The abstract lists three architectures but does not explicitly name the third (sub-connected-active/sub-connected-active) in the sentence that introduces the family; a parenthetical list would improve immediate clarity.
[§II–III] Notation for the per-group amplification factor and the complex symmetric matrix to which Takagi factorization is applied should be introduced once in §II or §III and used consistently thereafter to avoid reader re-derivation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address each major comment point by point below, clarifying the role of the modeling assumptions and outlining targeted revisions to enhance the presentation and evaluation.

read point-by-point responses

Referee: [§III (derivation) and §IV (numerical results)] The closed-form SNR-maximizing solution (abstract and §III) and the headline numerical claim (abstract and §IV) both presuppose perfect CSI together with independent per-group reflect-power budgets that can be met without mutual coupling, shared hardware constraints, or amplifier non-idealities. These assumptions directly define the feasible set of scattering matrices; their violation would change the attainable SNR and therefore the central comparison to diagonal RIS.

Authors: We agree that both the closed-form SNR-maximizing solution derived in §III and the numerical comparisons in §IV are obtained under the assumptions of perfect CSI and independent per-group reflect-power budgets, without incorporating mutual coupling, shared hardware constraints, or amplifier non-idealities. These assumptions are standard in the RIS literature and are applied identically to the diagonal RIS baselines to ensure a fair comparison. The Takagi factorization approach yields the optimal scattering matrix for each group precisely within this feasible set, as detailed in the derivation. To address the comment, we will revise §III to more explicitly articulate these assumptions and their direct impact on the attainable scattering matrices, and we will add a brief discussion in §IV on how violations would affect the SNR comparison. revision: partial
Referee: [§IV] Table or figure in §IV that reports the SISO SNR comparisons does not include sensitivity to CSI error or inter-group power coupling; because the headline claim of “same or higher SNR with significantly fewer amplifiers” rests on these numerical results, an explicit robustness check or worst-case analysis is required to substantiate the performance advantage.

Authors: We acknowledge that the SISO SNR results presented in §IV are obtained under ideal conditions and do not currently include sensitivity analyses with respect to CSI estimation errors or inter-group power coupling. While the core contribution lies in the closed-form per-group optimization, we concur that an explicit robustness evaluation would better substantiate the claimed performance advantage. Accordingly, we will incorporate a new subsection and accompanying figure in §IV that examines the SNR degradation under imperfect CSI (modeled via additive estimation errors) for both the proposed hybrid BD-RIS architectures and the diagonal baselines, along with a qualitative discussion of inter-group coupling effects. revision: yes

Circularity Check

0 steps flagged

Derivation is self-contained via standard matrix factorization and optimization; no reduction to inputs or self-citations.

full rationale

The paper presents a closed-form SNR-maximizing solution obtained by applying Takagi factorization to complex symmetric matrices for each BD-RIS group, followed by selection of per-group amplification factors to satisfy the reflect-power budget. This is a direct algebraic derivation from the system model and constraints, without parameter fitting to the target SNR, renaming of known results, or load-bearing reliance on self-citations for the core steps. Numerical validation in SISO scenarios uses the derived solution under explicit assumptions (perfect CSI, independent group power constraints) but does not alter the independence of the derivation itself. No patterns of self-definition, fitted-input prediction, or ansatz smuggling are present.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard linear algebra tools and common RIS channel models from prior work; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

standard math Takagi's factorization applies to the complex symmetric matrix arising from the BD-RIS group scattering model.
Invoked to obtain the closed-form SNR-maximizing solution for each group.

pith-pipeline@v0.9.0 · 5523 in / 1121 out tokens · 31592 ms · 2026-05-08T06:55:58.899674+00:00 · methodology

discussion (0)

Reference graph

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