Performance Analysis and Optimization of STAR-RIS-Aided Cell-Free Massive MIMO Systems Relying on Imperfect Hardware

Hien Quoc Ngo; Lajos Hanzo; Michail Matthaiou; Zeping Sui

REVIEW 3 major objections 6 minor 40 references

Performance Analysis and Optimization of STAR-RIS-Aided Cell-Free Massive MIMO Systems Relying on Imperfect Hardware

T0 review · 3 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A closed-form downlink SE is derived for imperfect-hardware STAR-RIS cell-free MIMO, and joint design lifts the worst-user SINR.

desk verdict Solid closed-form SE analysis for STAR-RIS cell-free MIMO under hardware impairments; the AO convergence proof does not match the algorithm as written. read the letter →

arxiv 2501.00546 v2 pith:CGZ22MGB submitted 2024-12-31 cs.IT eess.SPmath.IT

Zeping Sui , Hien Quoc Ngo , Michail Matthaiou , Lajos Hanzo This is my paper · ORCID

classification cs.ITeess.SPmath.IT

keywords STAR-RIScell-freemassiveMIMOimperfecthardwarespectralefficiencychannelestimationmax-minfairnesspassivebeamformingphasenoise

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 STAR-RIS-aided cell-free massive MIMO network can be analyzed and optimized when radios and surfaces are imperfect. It answers yes: it builds an MMSE estimator for the cascaded channels and derives a closed-form expression for the downlink ergodic spectral efficiency under transceiver distortions, phase noise, RIS phase errors, correlated fading, and pilot contamination, with finite access points, users, and RIS elements. The formula matches Monte Carlo simulations and identifies which hardware imperfection hurts most. The paper then uses the formula to jointly optimize RIS transmission/reflection beams and power allocation, improving the worst-user spectral efficiency compared to random beamforming with equal power.

What carries the argument

The load-bearing object is the linear MMSE cascaded-channel estimator and its covariance: $\hat{\mathbf{h}}_{mk}(0)=\sqrt{\gamma_T\gamma_R p \tau_p}\mathbf{R}_{mk}\mathbf{\Psi}_{mk}^{-1}\mathbf{z}_{mk}(0)$ with $\mathbf{\Psi}_{mk}$ in (5), built from the channel covariance $\mathbf{R}_{mk}=\mathbf{R}^d_{mk}+\mathbf{R}_{A,m}\mathrm{tr}(\mathbf{R}^f_{mk})$ that folds in STAR-RIS phase errors through the characteristic-function coefficient $\varsigma_p$. Proposition 1's SINR formula carries the argument: it turns every impairment--phase noise, RIS phase error, transceiver distortion, pilot contamination--into explicit trace terms that depend only on statistics. The optimization machinery is an alternating algorithm: adaptive particle swarm optimization for the $3N$ STAR-RIS amplitude and phase parameters, and a bisection method over second-order-cone feasibility for power control.

What would settle it

Set up the same system with a deterministic line-of-sight component in the AP-RIS channel (for example a Rician factor of 5-10 dB) while keeping all other parameters fixed, and compare the downlink ergodic spectral efficiency from Monte Carlo with the closed-form prediction of Eq. (13); a systematic gap that grows with the Rician factor would show that the Rayleigh-product assumption, not just the estimation bound, carries the result.

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Extended reading notes

Core claim

The central claim is Proposition 1: for maximum-ratio precoding and arbitrary STAR-RIS passive beamforming, the downlink ergodic SINR of each user has the exact closed form in (15)-(16), with the desired-signal power $\gamma_R\gamma_T\rho e^{-\delta^2 t}|\mathrm{tr}(\boldsymbol{\eta}_k^{1/2}\boldsymbol{\Omega}_k)|^2$ and a denominator $D_k(t)$ that collects beamforming uncertainty, pilot-contamination interference, transmitter and receiver distortions, and noise. Each term is written in channel covariance matrices, MMSE error covariances, hardware quality factors, and phase-noise variances, so the spectral efficiency in (13) is computable from channel statistics alone. The proof decomposes the received signal into desired, beamforming-uncertainty, inter-user interference, and hardware-distortion terms and evaluates each with Gaussian-moment identities. The paper further claims that the resulting expression is accurate against Monte Carlo, and that the STAR-RIS system with imperfect hardware still beats reflection-only RIS and cell-free baselines, with receiver hardware quality the more sensitive parameter.

Load-bearing premise

The entire analysis assumes the access-point-to-RIS channel is a Rayleigh product with no line-of-sight component and that direct AP-user links are completely blocked, so a real deployment with a strong direct AP-user path or a dominant line-of-sight AP-RIS component would fall outside the formulas and optimization conclusions.

Editorial extensions

If this is right

Network performance can be predicted from large-scale fading statistics alone, without Monte Carlo simulation, for finite numbers of access points, users, and RIS elements.
The relative damage of each imperfection becomes quantifiable: receiver hardware quality affects spectral efficiency more than access-point hardware quality, and RIS phase errors degrade the system in a predictable way.
The closed-form SINR enables max-min fairness optimization, and the proposed alternating algorithm (APSO plus bisection) improves the worst-user spectral efficiency over random passive beamforming with equal power.
STAR-RIS with simultaneous transmission and reflection outperforms reflection-only RIS and cell-free massive MIMO baselines even under imperfect hardware and worst-case RIS phase errors.
The analytical formula also shows that extra RIS elements beyond a certain count yield diminishing returns under maximum-ratio precoding, while weighted MMSE precoding can keep exploiting them at higher complexity.

Reading between the lines

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

Editorial inference: Because the SINR formula is expressed purely in covariance and error statistics, the same expression can serve as an objective or constraint for other resource-allocation tasks, such as energy efficiency or user scheduling, without re-deriving the analysis.
Editorial inference: The numerical ordering that UE-side hardware quality dominates AP-side quality suggests a practical budget rule--improve user oscillators and RF chains before upgrading access points--though the paper itself stops short of stating such a rule.
Editorial inference: The closed form rests on a blocked direct AP-UE link and a Rayleigh AP-RIS product; if a deployment has a residual direct path or a line-of-sight AP-RIS component, the predicted STAR-RIS advantage could shrink or grow, so a natural extension is to re-derive the covariance with a Rician AP-RIS channel.
Editorial inference: The fast convergence of the proposed alternating algorithm, within the 1 ms coherence interval assumed here, suggests the approach is practical as a real-time passive-beamforming update rule rather than only an offline design tool.

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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

Read this paper for Proposition 1, not for the optimization. The closed-form downlink ergodic SE for STAR-RIS-aided cell-free massive MIMO under transceiver hardware impairments, phase noise, RIS phase errors, and spatially correlated fading is the core contribution. I checked the derivation at a level I trust: it uses standard MMSE estimation and use-and-then-forget bounding, and the Monte Carlo validation in the figures matches. That result is a useful, citable extension of the existing analysis framework.

What is new: previous STAR-RIS cell-free work [22] assumed perfect hardware, and the hardware-impaired RIS-cell-free papers [32,33] use conventional RIS. The combination of STAR-RIS, imperfect hardware, and correlated fading appears to be new. But it is an expected extension; the techniques follow the established playbook. Significance is solid within the RIS-plus-cell-free subfield, not beyond it.

The soft spot is the optimization section. The convergence claim in Section V-D does not match Algorithm 3 as printed. The proof uses a Gauss-Seidel update chain F(η[tA],Φ[tA]) ≤ F(η[tA],Φ[tA+1]) ≤ F(η[tA+1],Φ[tA+1]), which requires the power control subproblem to be solved with the freshly updated Φ[tA]. Algorithm 3 step 5 explicitly solves (19) with Φ[tA-1]_T and Φ[tA-1]_R. So the monotonicity proof does not apply to the algorithm as written. The fix is straightforward—use Φ[tA] in step 5—but as it stands the provable convergence claim is not established. In addition, Algorithm 1 updates particles without any projection or clamping; the returned 'best' particle can leave the feasible set (β_T+β_R=1, θ in [0,2π)). That is an implementation gap. The numerical results may come from code that handles this, but the written algorithm does not.

These issues do not touch the closed-form SE derivation. The model assumptions—blocked direct links, Rayleigh AP-RIS product form—are stated clearly and limit applicability, but they are not hidden flaws. The paper is honest about what it does.

Who this is for: readers working on RIS-aided cell-free massive MIMO who need a hardware-impaired performance analysis reference. It deserves a serious referee, but the optimization section needs to be fixed or explicitly de-emphasized before publication. The analytical part is worth publishing on its own.

Referee Report

3 major / 6 minor

Summary. The paper investigates a STAR-RIS-aided cell-free massive MIMO system under spatially correlated fading, transceiver hardware impairments, time-varying phase noise, RIS phase errors, and pilot contamination. It derives a linear MMSE cascaded channel estimator (Section III) and a closed-form downlink ergodic spectral efficiency expression under MR precoding (Proposition 1, Eqs. (13)-(16)), with validation by Monte Carlo simulations in Section VI. It then formulates a max-min SINR problem for joint STAR-RIS passive beamforming and power control, and proposes an alternating optimization (AO) algorithm combining adaptive particle swarm optimization (APSO) and a bisection method. Numerical results demonstrate SE improvements over RIS-aided and conventional CF-mMIMO baselines and show convergence of the AO algorithm.

Significance. The closed-form SE derivation is a solid contribution: it handles finite numbers of APs, UEs, and RIS elements, incorporates realistic impairments in a structured way, and is validated by Monte Carlo simulation without curve fitting. The optimization part is more fragile: the convergence guarantee stated in Section V-D does not match Algorithm 3 as printed, and the APSO particle update in Algorithm 1 does not enforce feasibility. These are fixable within the manuscript's scope, so the paper merits revision rather than rejection. If the optimization claims are repaired (or downgraded to heuristic improvement with numerical evidence), the paper would be a useful addition to the STAR-RIS/cell-free MIMO literature.

major comments (3)

[Section V-D and Algorithm 3] The monotone-convergence proof is inconsistent with Algorithm 3 as written. Step 5 updates the power coefficients z^{[tA]} using Algorithm 2 with the previous passive beamforming Φ^{[tA-1]}_T and Φ^{[tA-1]}_R, whereas the proof's chain F(η^{[tA]},Φ^{[tA]}) ≤ F(η^{[tA]},Φ^{[tA+1]}) ≤ F(η^{[tA+1]},Φ^{[tA+1]}) requires the power subproblem to be solved with the just-updated Φ^{[tA]}. As printed, each subproblem is optimized against stale variables, so the objective value is not guaranteed nondecreasing and the stated convergence guarantee does not follow. Please either change step 5 to use Φ^{[tA]}_T and Φ^{[[tA]}_R, or provide a different convergence argument (or empirical verification) for the Jacobi-style update actually implemented.
[Algorithm 1 (Section V-B)] The APSO particle update x^{[tP+1]}_{lP} = x^{[tP]}_{lP} + v^{[tP+1]}_{lP} is applied without any projection or clamping, even though the decision variables must satisfy β_{T,n} ∈ [0,1] (with β_{R,n}=1-β_{T,n}) and θ_{T,n},θ_{R,n} ∈ [0,2π). Consequently, particles can leave the feasible set during the search, and the 'global best' returned in step 15 may be infeasible. This undermines the claim that the AO algorithm provides a feasible max-min solution. Please add an explicit projection or repair step (e.g., clipping β to [0,1] and wrapping θ modulo 2π) after each velocity update, and state this in the algorithm.
[Section V-D and Algorithm 1] The convergence proof depends on the statement that 'the updates of Φ and η can attain the max-min SINR when the other variable remains fixed.' This is not justified for the Φ-subproblem: APSO is a stochastic heuristic with no per-iteration improvement guarantee, and the cited reference [37] analyzes standard PSO dynamics, not monotone ascent on a given objective. Even if the stale-variable issue in Algorithm 3 is fixed, the inequality (a) requires at least that the APSO update does not decrease the worst-user SINR; without a repair strategy or a monotone variant, this should be stated as a heuristic claim supported by the numerical convergence plots rather than as a formal guarantee.

minor comments (6)

[Section II-C] There is a typo in 'opreate' (should be 'operate').
[Section VI-C] 'Inter Core i9-14900HX' should be 'Intel Core i9-14900HX'.
[Table II] The row 'Number of UEs in the reflection space, KT 3' should read 'transmission space' rather than 'reflection space' for KT.
[Section V-B, after Eq. (18)] The sentence 'the global optimal solution of is nontrivial to attain' is missing a word; it should read 'of (18)' or similar.
[Algorithm 1, line 12] Minor wording: 'Updated particles' should be 'Update particles'.
[Appendix C, Eq. (35)] The notation in the upper-level set condition mixes 'η_mk' and 'z_mk' in the same expression; please unify the notation (z_mk = sqrt(η_mk)) to avoid confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

The paper's central closed-form SE derivation is self-contained under its stated channel and hardware model; the AO convergence mismatch is a proof-consistency issue, not a circular reduction.

full rationale

No circular step is present. The channel covariance in Eq. (1), the linear MMSE estimator in Eq. (5), and the closed-form SINR/SE expressions in Proposition 1, Eqs. (13)-(16), are obtained from the explicitly stated Rayleigh AP-RIS / Rician RIS-UE channel model, the Gaussian phase-noise model of Eq. (2), the EVM hardware-impairement model, and the standard statistical lemmas given in Appendix A (Lemmas 1 and 2, cited from the literature), with the full derivation carried out in Appendix B. The closed-form result is then validated against independent Monte-Carlo simulations in Fig. 2, so the analytical claim does not fold its own target into its assumptions. The paper cites its own GLOBECOM precursor [1] and co-authored works [22], [40], but these are used for channel-model conventions and baseline techniques, not as the load-bearing justification of the new SE formula. The only noteworthy issue is in the optimization half: the convergence argument in Section V-D asserts monotonicity F(η[tA],Φ[tA]) ≤ F(η[tA],Φ[tA+1]) ≤ F(η[tA+1],Φ[tA+1]), whereas Algorithm 3, step 5, updates the power coefficients using the stale passive beamforming Φ[tA-1] rather than the just-updated Φ[tA]. This is a genuine proof-consistency defect that should be corrected, but it is not a case of the derivation reducing to its own inputs by construction, so it does not affect the circularity score.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The model relies on standard wireless channel and hardware impairment assumptions, which are listed as axioms. No free parameters are fitted to data to obtain the closed-form expression, and the optimization introduces no new physical entities. The heuristic APSO step and the lack of projection are noted in red flags.

assumptions (8)

domain assumption AP-RIS channel is modeled as Q_m = sqrt(xi_m) R_{A,m}^{1/2} V_m R_S^{1/2} with vec(V_m) ~ CN(0, I_{NL})
Section II-C. The closed-form covariance in Eq. (1) and the MMSE estimator in Eq. (5) rely on this Rayleigh product form and Lemma 1.
domain assumption Direct AP-UE links are fully blocked; STAR-RIS is close to UEs so RIS-UE links have LoS and Rician components
Section II-C. This removes all non-RIS paths from the analysis, so numerical STAR-RIS gains depend on this blockage assumption.
domain assumption EVM hardware impairment model: transmitter and receiver distortions are zero-mean Gaussian with covariance proportional to diagonal signal powers and uncorrelated with signals
Sections III and IV-A, Eqs. (3), (6), (7). The conditional Gaussian distortion model from [26], [27], [33] is assumed; real amplifier nonlinearities may not be Gaussian.
domain assumption Oscillator phase noise follows a Wiener process, fixed during channel estimation since tau_p << tau_c and time-varying during data transmission
Section II-D, Eq. (2). The e^{-delta^2 t} factors and the use of the estimate at time zero depend on this model.
domain assumption RIS phase errors are i.i.d. von Mises or uniform with known characteristic function ς_p = I_1(vartheta)/I_0(vartheta)
Section II-B. Phase-error corrected covariance terms use E[e^{j(theta_n - theta_{n'})}] = ς_p^2.
standard math Use-and-then-forget capacity bounding is a valid lower bound on the ergodic SE
Section IV-B. Standard technique from [2]; the paper invokes it without re-deriving it.
standard math Linear MMSE estimation with Gaussian channels and conditional Gaussian distortion yields Gaussian estimation error independent of the estimate
Section III. Used to obtain Omega_mk and C_mk and to compute expectations via Lemma 2.
standard math Quasi-concavity of the power control subproblem follows because its upper-level sets are second-order cones
Appendix C, Eq. (35). The proof relies on SOCP convexity of the feasibility sets.

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Cite this review

Pith. "Pith review of Performance Analysis and Optimization of STAR-RIS-Aided Cell-Free Massive MIMO Systems Relying on Imperfect Hardware." pith.science (2026). https://pith.science/paper/CGZ22MGB

@misc{pith2026250100546,
  author       = {Pith},
  title        = {Pith review of: Performance Analysis and Optimization of STAR-RIS-Aided Cell-Free Massive MIMO Systems Relying on Imperfect Hardware},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CGZ22MGB}},
  note         = {Machine review of arXiv:2501.00546}
}

read the original abstract

Simultaneously transmitting and reflecting reconfigurable intelligent surface (STAR-RIS)-aided cell-free massive multiple-input multiple-output (CF-mMIMO) systems are investigated under spatially correlated fading channels using realistic imperfect hardware. Specifically, the transceiver distortions, \textcolor{black}{time-varying phase noise, and RIS phase shift errors} are considered. Upon considering imperfect hardware and pilot contamination, we derive a linear minimum mean-square error (MMSE) criterion-based cascaded channel estimator. Moreover, a closed-form expression of the downlink ergodic spectral efficiency (SE) is derived based on maximum ratio (MR) based transmit precoding and channel statistics, where both a finite number of access points (APs) and STAR-RIS elements as well as imperfect hardware are considered. Furthermore, by exploiting the ergodic signal-to-interference-plus-noise ratios (SINRs) among user equipment (UE), a max-min fairness problem is formulated for the joint optimization of the passive transmitting and reflecting beamforming (BF) at the STAR-RIS as well as of the power control coefficients. An alternating optimization (AO) algorithm is proposed for solving the resultant problems, where iterative adaptive particle swarm optimization (APSO) and bisection methods are proposed for circumventing the non-convexity of the RIS passive BF and the quasi-concave power control sub-problems, respectively. Our simulation results illustrate that the STAR-RIS-aided CF-mMIMO system attains higher SE than its RIS-aided counterpart. The performance of different hardware parameters is also evaluated. Additionally, it is demonstrated that the SE of the worst UE can be significantly improved by exploiting the proposed AO-based algorithm compared to conventional solutions associated with random passive BF and equal-power scenarios.

Figures

Figures reproduced from arXiv: 2501.00546 by the authors.

**Figure 2.** CDF of the downlink sum SE for the STAR-RIS [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** Downlink sum SE versus the number of APs M operating at different RIS phase error parameter ϑ (N = 64, γT = γR = 1). In [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (8 more)

**Figure 6.** Figure 6: Average SE per UE for STAR-RIS-CF-mMIMO, [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

**Figure 4.** Figure 4: Downlink average SE per UE for STAR-RIS-CFmMIMO, RIS-CF-mMIMO and CF-mMIMO versus the number of UEs K operating at different hardware quality factors. 2 3 4 5 6 7 8 9 10 0 0.2 0.4 0.6 0.8 1 1.2 [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 5.** Figure 5: Downlink average SE per UE for STAR-RIS-CF [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 8.** Figure 8: Sum SE with WMMSE and MR precoding against the number RIS elements N over different RIS positions (K = 20, M = 30, γT = γR = 1, ϑ = 1) [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]

**Figure 9.** Figure 9: CDF of the downlink minimum SE for the STAR [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]

**Figure 10.** Figure 10: CDF of the downlink minimum SE for the STAR [PITH_FULL_IMAGE:figures/full_fig_p011_10.png]

**Figure 11.** Figure 11: Convergence of the proposed AO algorithms using APSO and PSO for STAR-RIS-CF-mMIMO with different M (γT = γR = 0.8). AO-based algorithm using the proposed APSO can achieve convergence using as few as two AO iterations. By contrast, the AO algorithm relying on the con…

**Figure 12.** Figure 12: Running times of the proposed AO algorithms including APSO and PSO for STAR-RIS-CF-mMIMO with different M (γT = γR = 0.8). VII. CONCLUSIONS STAR-RIS-CF-mMIMO having realistic imperfect hardware was proposed and its ergodic downlink SE analysis was presented, where bo…

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

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