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REVIEW 2 major objections 4 minor 36 references

Heterogeneous-IRS-Assisted MIMO Systems: Channel Estimation and Beamforming

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper claims that in a heterogeneous intelligent reflecting surface, the channel to estimate splits into a dynamic-element cascaded channel and a static-element equivalent channel, so pilot overhead scales with the number of dynamic…

desk verdict A genuinely new decoupled channel model for heterogeneous IRS, worth engaging, but the STE-channel sparsity assumption needs stronger support than 'simulation tests.' read the letter →

arxiv 2506.10350 v1 pith:XRRWVAP7 submitted 2025-06-12 eess.SP

Weibiao Zhao , Qiucen Wu , Yuanqi Tang , Yu Zhu This is my paper · ORCID
classification eess.SP
keywords heterogeneousintelligentreflectingsurfacechannelestimationbeamformingmanifoldoptimizationmulti-userMIMOpilotoverheadsparseenergyefficiency
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 studies an intelligent reflecting surface (IRS) in which some elements are dynamically tunable (DTEs) and the rest are statically tunable (STEs) and consume no power. It argues that the usual approach of estimating one overall cascaded channel breaks down for this design, because the fixed STE phases make the measurement matrix rank-deficient. Instead, the channel can be split into a DTE-based cascaded channel and an STE-based equivalent channel, and each piece can be estimated from its angular sparsity and low-rank structure using manifold optimization. The result the authors are trying to establish is that this decoupled estimation needs fewer pilots than a conventional all-dynamic IRS of the same size, and that with offline STE beamforming plus online DTE and precoder optimization it achieves nearly the same sum rate at much lower power consumption.

What carries the argument

The load-bearing object is the DTE-STE decoupled (DSD) channel model: the received signal is split into $G_{\mathrm{DTE}}\Phi_k[t]H_{\mathrm{DTE},k}$ plus $G_{\mathrm{STE}}\Omega H_{\mathrm{STE},k}$, so only the dynamic part carries a time-varying phase matrix while the static part collapses into one fixed equivalent channel $H_{\mathrm{STE,eq},k}$. Estimation then rides on three props: the angular sparsity of $G_{\mathrm{DTE}}$ and $H_{\mathrm{DTE},k}$, the low-rank property of all three channel matrices, and manifold optimization over fixed-rank Riemannian manifolds to respect those rank constraints. A robust rank selection rule handles path-number mismatch by choosing a rank slightly above the estimated one, justified by Lemmas 2 and 3: exceeding the true rank loses nothing in approximation accuracy, while undershooting it permanently excludes the true matrix. For beamforming, the offline WBS-MO algorithm synthesizes a wide beam over the UE location area by separating the STE array's $y$- and $z$-direction gains, and the online WMMSE-EI algorithm alternatively updates quantization-aware DTE phases and the BS precoder.

What would settle it

Choose an STE phase configuration $\Omega$ that spreads the incident wave across many angular bins and run DSD-MO with $P$ and $Q$ set to the true path counts; if the normalized estimation NMSE degrades sharply or the fixed-rank constraints become infeasible for that configuration, the fixed row-column sparsity assumption for $H_{\mathrm{STE,eq},k}$ fails.

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Extended reading notes

Core claim

For a heterogeneous IRS, estimating the full cascaded channel $H_{\mathrm{ca},k}=H_k^T \odot G$ is impossible once there is more than one static element, because the repeated constant rows in the phase-shift matrix reduce the rank of the measurement matrix. The paper's central move is to rewrite the received signal as $(G_{\mathrm{DTE}}\Phi_k[t]H_{\mathrm{DTE},k}+G_{\mathrm{STE}}\Omega H_{\mathrm{STE},k})s_k[t]$, estimating the DTE-based cascaded channel and the STE-based equivalent channel $H_{\mathrm{STE,eq},k}=G_{\mathrm{STE}}\Omega H_{\mathrm{STE},k}$ separately. The DSD-MO algorithm alternates between these two blocks and, within the DTE block, between $G_{\mathrm{DTE}}$ and $H_{\mathrm{DTE},k}$, using manifold optimization on fixed-rank sets and $\ell^1$ regularization on angular-domain sparsity. For the estimated ranks, the paper proves that a higher-than-true-rank constraint can approximate the true channel arbitrarily well while a lower one cannot, and uses this to justify choosing a slightly inflated rank under path-number mismatch. Beamforming is split into an offline wide-beam synthesis for the STEs and an online WMMSE-style element-iteration update for the DTEs and the BS precoder.

Load-bearing premise

The estimation problem assumes the static-element part of the channel has exactly $P$ nonzero angular rows, exactly $Q$ nonzero angular columns, and rank $\min\{P,Q\}$; the paper admits this is not mathematically guaranteed because the STE phase settings affect the sparsity, so the central claim stands on sparsity observed in simulations rather than proved.

Editorial extensions

If this is right

  • Pilot overhead for an HE-IRS-assisted system is governed by $N_{\mathrm{DTE}}+1$ rather than the total element count $N$, so replacing half the dynamic elements with static ones can roughly halve the training cost.
  • Conventional least-squares estimation of the overall cascaded channel fails for any HE-IRS with more than one static element, so the decoupled model is not a minor refinement but a necessary reformulation.
  • With the proposed estimation and two-stage beamforming, the HE-IRS reaches a sum rate close to that of a conventional all-dynamic IRS while using fewer pilots and less power.
  • In energy efficiency, the HE-IRS can beat both the conventional IRS and the sparse IRS because its static elements contribute no ongoing power draw.
  • The robust rank selection rule keeps estimation stable when the number of propagation paths is unknown or misestimated.

Reading between the lines

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

  • An extension the paper leaves implicit is applying the same DTE-STE decoupling to wideband or OFDM systems, where the angular-sparsity assumption must be checked per subcarrier rather than once.
  • The pilot-cost split implies a design knob: increasing the static-element fraction lowers training and power cost at some sum-rate expense, so a practical system could tune the DTE/STE ratio to the channel's angular richness.
  • Because Remark 2 concedes that the static-element equivalent channel's exact sparsity is only observed in simulations, a testable extension is to replace the fixed $P$ and $Q$ constraints with an adaptive sparsity or rank selection that monitors the estimation residual.
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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 / 4 minor

Summary. The paper proposes channel estimation and two-stage beamforming schemes for a heterogeneous intelligent reflecting surface (HE-IRS) that combines dynamically tunable elements (DTEs) with power-free statically tunable elements (STEs) in a multi-user MIMO downlink/uplink system. The key idea is to decompose the HE-IRS channel into a DTE-based cascaded channel and an STE-based equivalent channel, so that pilot overhead scales with the number of DTEs rather than the total element count. The DSD-MO estimator alternates manifold-optimization updates of the two channel components using sparsity and rank constraints, with a robust rank selection rule for imperfect path-number knowledge. Beamforming is split into an offline WBS-MO algorithm for STE phase shifts and an online WMMSE-EI algorithm for the BS precoder and DTE phases. Simulations with N=48 elements show reduced pilot overhead and competitive sum rate relative to a conventional IRS, with higher energy efficiency.

Significance. If the central assumptions hold, the paper makes a useful contribution to green IRS design: it gives an algebraic channel decomposition that bypasses the identifiability problem of estimating the overall cascaded channel when some elements have fixed phase shifts, it provides explicit Riemannian gradients for the estimation algorithm, and it demonstrates through simulations that the HE-IRS can achieve most of the sum-rate benefit of a conventional IRS while consuming less power. The comparison against SIRS and conventional IRS, and the inclusion of both perfect- and estimated-CSI results, are strengths. The main risk is that the estimation algorithm is built on sparsity/rank equalities for the STE-equivalent channel that are not mathematically guaranteed and are only qualitatively supported by 'simulation tests'; this needs quantitative verification. The offline WBS-MO derivation also contains a normalization/inequality step that should be reconciled.

major comments (2)
  1. [Section IV-B and IV-C (Eqs. (16)-(18), Remark 2)] The DSD-MO estimator is built on the equalities rank(HSTE_eq,k)=min{P,Q}, ||ΛHSTE_eq,k 1_NUE||_0=P, and ||1_NBS^T ΛHSTE_eq,k||_0=Q imposed in (18). However, the proof of (16) establishes only upper bounds (at most P nonzero rows and Q nonzero columns), and (17) is an inequality. Remark 2 explicitly concedes that Ω affects the sparsity and that the maximal values are 'not mathematically guaranteed,' being only 'generally' observed in simulations. If cancellations inside A_STE^H Ω A_STE reduce the row/column counts or the rank below the imposed values, the feasibility set in (18) excludes the true HSTE_eq,k and the ℓ1 terms in (19) penalize the true channel, biasing the estimate and undermining the claimed pilot-overhead reduction. The paper contains no quantitative evidence, such as a histogram or table over the simulation settings (including Ω produced by WBS-MO and random channel geometries), showing how often the maximal values are attained. Please add such a verification; if the equality fails for a non-negligible fraction of cases, revise the problem formulation (for example, replace the equalities by inequalities or adapt the sparsity counts during optimization).
  2. [Section V-B (Eqs. (28)-(33))] The step labelled (e) in Eq. (31) is not justified by the Cauchy-Schwarz inequality alone. The pointwise factors x(ρy)=|ω_y^H b_y(ρy)|^2 and z(ρz)=|ω_z^H b_z(ρz)|^2 satisfy x,z≤1 only if the steering vectors in (4) are used with their 1/√M normalization, and even then the inequality is applied to integrals, whose values depend on the angular integration ranges; the paper does not state the required boundedness or normalization conditions. Moreover, Eq. (32) defines Ξ_y as N_y^STE times the matrix of integrals of e^{jπ(i-n)ρ}, whereas with the normalized b_y defined in (29) the integral of B_y is (1/N_y^STE) times that matrix, so the displayed equality in (31) is off by a factor. Because the y- and z-subproblems are then optimized separately in (33), the scaling may not change each individual optimizer, but the derivation should be corrected and the normalization of b_y/b_z stated explicitly.
minor comments (4)
  1. [Section IV-E (Lemma 3)] The proof-by-contradiction in Lemma 3 is incorrect as written: the negation of the claim is not that ‖H−eH‖=0 for every rank-m matrix eH, but that the infimum of ‖H−eH‖ over such matrices is zero. The lemma itself is true (e.g., by the Eckart-Young theorem), but the proof should be replaced.
  2. [Section IV-E] The 'robust rank selection rule' is stated only as 'adopt a little higher rank than the directly estimated one'; the paper gives no procedure for choosing how much higher. In Fig. 8, the S-rank scheme uses the maximum possible rank (6), which is known from the simulation setup rather than obtained by the proposed rule, so the practical rule is under-specified.
  3. [Section VI-D] The energy-efficiency simulation uses PIRS = Pstatic + Σ tm·PPIN with Pstatic=15 dBm and PPIN=12 dBm. It would be helpful to state explicitly whether these quantities already account for the STE power consumption being zero and for the control-circuit power of the DTEs, so that the 'power-free STEs' claim is transparent.
  4. [General / notation] There are several presentation issues: 'highy' should be 'highly' in Section IV-C; Eq. (36) uses log without specifying the base although rates are reported in bits/s/Hz; and the curves in Figs. 3 and 4 would be easier to read with distinct markers.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the DTE-STE decoupling is an algebraic consequence of the system model, the overhead comparison is parameter counting, and the admitted sparsity caveat is a correctness risk rather than a circular step.

full rationale

The paper's central derivation is self-contained and not circular. The DTE-STE decoupled channel in (10), r_k[t] = (G_DTE Φ_k[t] H_DTE,k + G_STE Ω H_STE,k) s_k[t] + z_k[t], follows by partitioning the HE-IRS phase-shift matrix and the column spaces of G and H_k; it is an algebraic identity, not a fitted assumption. The claimed pilot-overhead advantage follows from parameter counting: the conventional cascaded channel has N·N_BS·N_UE unknowns while the HE-IRS model has N_DTE·N_BS·N_UE + N_BS·N_UE = (N_DTE+1)·N_BS·N_UE unknowns, as stated in Remark 1. The sparsity and low-rank properties used in (18)-(19) are either proved in the paper (for H_STE,eq in (16)-(17)) or cited from prior work with explicit assumptions, and they are not defined in terms of the quantities being estimated. The estimation and beamforming algorithms are evaluated against external baselines (CS-EST, perfect-CSI IRS/SIRS) rather than presented as predictions of fitted parameters. The paper does rely on the authors' prior HE-IRS hardware concept [12] and on sparsity results from overlapping author groups [18], [29], but these are legitimate prior publications with independent content and do not by themselves force the present results. The main weakness is Remark 2, which explicitly concedes that the rank and row/column sparsity of H_STE,eq,k are 'not mathematically guaranteed' to reach their maximal values and are only 'observed from simulation tests.' Problem (18) nevertheless imposes these maxima as equality constraints, and (19) penalizes deviations. If the true H_STE,eq,k lies outside this feasible set for some Ω or channel geometry, the estimator would be biased. This is a real, explicitly acknowledged correctness risk, but it is not a circularity: the paper does not pretend those equalities are derived from the model, and the claims are empirically testable. Therefore no step reduces to its own input by construction.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central derivations rely on standard linear algebra and signal processing assumptions. The channel sparsity assumptions (P and Q paths, low rank) are domain assumptions, and the specific sparsity/rank of HSTE_eq is not guaranteed (Remark 2). No new physical entities are introduced; the HE-IRS structure is from the authors' prior work [12].

free parameters (3)
  • ℓ1 regularization weights (υ for Ĝ_DTE, Ĥ_DTE,k, row/col of Ĥ_STE_eq) = not reported
    These weights in problem (19) control the sparsity penalty; the paper does not give their values or a selection rule, so results depend on unreported tuning.
  • Rank margin in robust rank selection rule = maximum actual rank (6) in Fig. 8; no general rule
    Section IV-E advises choosing a slightly higher rank than estimated, but the margin is unspecified; the simulations appear to use the maximum rank in the test range.
  • MO algorithm step sizes and stopping criteria = not reported
    Algorithm 1 requires Riemannian conjugate gradient parameters (step size, retraction, stopping thresholds); none are specified, affecting convergence and hence estimation quality.
assumptions (4)
  • domain assumption Saleh-Valenzuela channel model with P and Q propagation paths (including one LoS) and sparse angular representation.
    The channel estimation problem (18) and sparsity properties (14)-(17) rely on the channels having few paths; this is a standard but domain-specific assumption.
  • domain assumption The angular dictionaries are sufficiently fine that grid mismatch is negligible and the sparsity properties hold.
    The proofs of (14)-(16) assume unitary dictionaries (GUE=NUE, GBS=NBS, GDTE=NDTE), but the simulations use overcomplete dictionaries with GBS=GUE=128 and GI=1024; the paper assumes the asymptotic sparsity still holds.
  • ad hoc to paper The STE-based equivalent channel attains the maximum rank min{P,Q} and sparsity levels P and Q as assumed in (18).
    Remark 2 explicitly states this is not mathematically guaranteed and is only observed in simulation. The problem formulation depends on these exact values.
  • domain assumption Manifold optimization converges to a useful stationary point for the non-convex alternating problem (19).
    No convergence proof is given; the paper relies on the MO framework from [18] and [29].

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Cite this review

Pith. "Pith review of Heterogeneous-IRS-Assisted MIMO Systems: Channel Estimation and Beamforming." pith.science (2026). https://pith.science/paper/XRRWVAP7

@misc{pith2026250610350,
  author       = {Pith},
  title        = {Pith review of: Heterogeneous-IRS-Assisted MIMO Systems: Channel Estimation and Beamforming},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XRRWVAP7}},
  note         = {Machine review of arXiv:2506.10350}
}
read the original abstract

Intelligent reflecting surface (IRS) has gained great attention for its ability to create favorable propagation environments. However, the power consumption of conventional IRSs cannot be ignored due to the large number of reflecting elements and control circuits. To balance performance and power consumption, we previously proposed a heterogeneous-IRS (HE-IRS), a green IRS structure integrating dynamically tunable elements (DTEs) and statically tunable elements (STEs). Compared to conventional IRSs with only DTEs, the unique DTE-STE integrated structure introduces new challenges in both channel estimation and beamforming. In this paper, we investigate the channel estimation and beamforming problems in HE-IRS-assisted multi-user multiple-input multiple-output systems. Unlike the overall cascaded channel estimated in conventional IRSs, we show that the HE-IRS channel to be estimated is decomposed into a DTE-based cascaded channel and an STE-based equivalent channel. Leveraging it along with the inherent sparsity of DTE- and STE-based channels and manifold optimization, we propose an efficient channel estimation scheme. To address the rank mismatch problem in the imperfect channel sparsity information, a robust rank selection rule is developed. For beamforming, we propose an offline algorithm to optimize the STE phase shifts for wide beam coverage, and an online algorithm to optimize the BS precoder and the DTE phase shifts using the estimated HE-IRS channel. Simulation results show that the HE-IRS requires less pilot overhead than conventional IRSs with the same number of elements. With the proposed channel estimation and beamforming schemes, the green HE-IRS achieves competitive sum rate performance with significantly reduced power consumption.

Figures

Figures reproduced from arXiv: 2506.10350 by the authors.

Figure 1
Figure 1. Examples of element compositions for different IRS structures. (a) Conventional IRS. (b) SIRS. (c) HE-IRS. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. An HE-IRS-assisted multi-user MIMO system. [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. (a) NMSEDTE,ca vs. Ttra for different schemes and PNRs in the estimation of the DTE-based cascaded channel. (b) NMSESTE,eq vs. Ttra for different schemes and PNRs in the estimation of the STE-based equivalent channel. 60 180 300 420 540 -30 -20 -10 0 NMSEDTE,ca (dB) (a) 60 180 300 420 540 -30 -20 -10 0 NMSESTE,eq (dB) (b) [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: (a) NMSEDTE,ca vs. Ttra for different HE-IRS configurations in the DTE-based cascaded channel. (b) NMSESTE,eq vs. Ttra for different configurations in the STE-based equivalent channel. In [PITH_FULL_IMAGE:figures/full_fig_p024_4.png]
Figure 5
Figure 5. Figure 5: Sum rate performance vs. Ttra for HE-IRS-, IRS-, SIRS-assisted systems with perfect CSI (dashed lines) and estimated CSI (solid lines). channel (HSTE eq,k ∈ C NBS×NUE ) is not directly related to the number of STEs. C. Comparison of the Sum Rate Performance for HE-IRS-…
Figure 6
Figure 6. Figure 6: Sum rate performance vs. Pb for HE-IRS-, IRS-, SIRS-assisted systems with estimated CSI. For HE-IRS, the cascaded channel for half DTEs and the equivalent channel for half STEs need to be estimated. For IRS, the cascaded channel for all DTEs needs to be estimated, whic…
Figure 7
Figure 7. Figure 7: The CDF of the EE performance for HE-IRS-, IRS-, SIRS-assisted systems with estimated CSI. [PITH_FULL_IMAGE:figures/full_fig_p027_7.png]
Figure 8
Figure 8. Figure 8: (a) NMSEDTE,ca vs. actual rank of the MO method for the DTE-based cascaded channel. (b) NMSESTE,eq vs. actual rank of the MO method for the STE-based equivalent channel. a rank higher than the true rank retains the ability to approximate the actual channels arbitrarily…

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Pith tools

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