Blind Passive Beamforming for MIMO System

Jiawei Yao; Kaiming Shen; Wenhai Lai

REVIEW 3 major objections 4 minor 21 references

Blind Passive Beamforming for MIMO System

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Blind passive beamforming with no CSI provably matches the CSI-based linear-search solution.

desk verdict Useful blind beamforming idea and strong field results, but the claimed sample-complexity theorem is not proven as stated. read the letter →

arxiv 2506.00987 v1 pith:F5EAJOTG submitted 2025-06-01 cs.IT eess.SPmath.IT

Wenhai Lai , Jiawei Yao , Kaiming Shen This is my paper

classification cs.ITeess.SPmath.IT

keywords intelligentsurfacepassivebeamformingblindconditionalsamplemeanMIMOdiscretephaseshiftsCSI-free5Gfieldtest

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

Passive beamforming with an intelligent surface usually requires knowing the transmitter-to-surface and surface-to-receiver channels, and the discrete-phase-shift optimization is nonconvex. The paper argues that this MIMO capacity maximization can be approximated by maximizing the sum of received powers, and that the resulting per-element linear search can be run without any channel state information. The proposed algorithm randomly configures the surface for T trials, measures received power, and sets each phase to the value whose conditional sample mean is largest. It proves that this blind configuration converges in probability to the CSI-based linear-search solution once T is on the order of L $N^{2}$ $S^{2}$ (log(N L))^3, and reports 5G field tests in which the blind rule improves data rate over beam-training baselines.

What carries the argument

The load-bearing mechanism is the conditional sample mean (CSM) rule: after T independent random phase vectors with each θ_nt uniform over the discrete set Φ_K, the algorithm groups trials by each element's phase and compares average received power. The analysis uses the arithmetic-geometric mean inequality to lower-bound the sum channel power by a sum of decoupled terms of the form Re{α_{ij}^* β_{inj} $e^{{jθ_n}}$}, so the CSI-based decision for each θ_n depends only on a per-element metric. The proof then shows that estimating the conditional means E[|y_i|^2 | θ_n = kω] with sufficient accuracy reproduces that metric, and a Chernoff tail bound converts estimation error into the sample-complexity scaling. The parameter ν, roughly the typical received power scale, enters the final bound and is proportional to N S, which produces the $N^{2}$ $S^{2}$ factor.

What would settle it

Construct or measure a channel in which, for at least one reflecting element, two phase choices yield exactly equal conditional mean received power, and run Algorithm 1 with T increasing. If the probability of selecting the wrong phase does not approach zero, the strict-gap assumption in the proof of Proposition 1 is essential; conversely, for a random channel one can numerically evaluate min over elements of the gap between the largest and second-largest conditional means and check whether the stated T scaling remains sufficient when that gap is small.

Watch

Extended reading notes

Core claim

The central claim is that the hard discrete problem of choosing reflecting-element phases to maximize MIMO capacity can be replaced by a sum-power maximization, and that the optimum of this surrogate is achieved by deciding each element independently through a linear search. The blind part is that the same decisions are recoverable without CSI: the conditional mean of received power at antenna i given that element n is set to kω is an affine function of the same cross terms that the CSI-based linear search maximizes. The paper establishes in Proposition 1 that, with T = Ω(L $N^{2}$ $S^{2}$ (log(N L))^3) random phase probes, the probability that the blind configuration equals the CSI configuration is at least 1−ξ for any prescribed ξ>0. Field tests in a commercial 5G network then show the blind method achieving RSRP gains around 9.9 dB indoor and 2.5 dB outdoor and data-rate gains near 50 Mbps indoor, above the compared baselines.

Load-bearing premise

For every reflecting element, the best phase choice must be strictly better than the second-best in average received power; if two choices tie, the sample-accuracy guarantee in the proof lapses.

Editorial extensions

If this is right

With T = Ω(L N^2 S^2 (log(N L))^3) measurements, the blind phase configuration matches the CSI-based linear-search configuration with probability at least 1−ξ, so the passive side of the system needs no per-reflection channel estimation.
Setting each phase by conditional sample means costs O(N(T+K)) total, and the per-element decisions are decoupled, giving a linear-complexity alternative to full-CSI optimization methods.
In the field tests with T=1000, the blind rule outperformed zero-phase, beam training, rank beam training, and rank CSM, producing roughly 50 Mbps indoor rate gain and the largest RSRP gain in both indoor and outdoor scenarios.
When there is one transmit antenna (M=1), the algorithm reduces to the existing SISO blind beamforming method, so the MIMO result generalizes that earlier scheme.
The capacity approximation is two-sided: the upper bound and lower bound in (9) and (10) sandwich the true capacity, and the gap between the approximation and capacity grows with the number of reflecting elements but remains small enough for good rate performance in the tested regimes.

Reading between the lines

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

Editorial inference: because the sample complexity scales as N^2, scaling the method to surfaces with thousands of elements will likely require exploiting structure or adaptively updating phases rather than using a fresh random search of size T; the paper leaves that scaling question open.
Editorial inference: the proof depends on strictly positive gaps between conditional means, so in practice one could pre-estimate these gaps from a coarse channel model and adapt T to the smallest gap; channels with near-tied phase choices would be flagged as hard cases.
Editorial inference: since the algorithm uses only received-power statistics, a natural unproven extension is to frequency-selective or OFDM channels by treating subcarriers as additional receive dimensions.
Editorial inference: the active beamformer W still requires low-dimensional CSI, so the 'blind' claim applies to the intelligent surface only; a fully CSI-free system would need an active-side method as well.

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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

The paper has a real idea: extend the SISO blind beamforming of Ren et al. to MIMO by approximating capacity with sum power and learning reflector phases from conditional sample means of received power. The algorithm is simple, the MIMO formulation is genuinely new relative to [6], and the field tests in a commercial 5G network are exactly the kind of evidence this area needs. The rate gains over the benchmarks are large enough to be interesting even without error bars.

The main problem is Proposition 1. The appendix derives finite-sample thresholds that all contain a 1/epsilon_0^2 factor, then silently drops that factor to conclude T = Omega(L N^2 S^2 (log(NL))^3). No argument shows epsilon_0 — the minimum gap between the best and second-best conditional mean for any antenna and reflector — is bounded below by a constant, or even by something absorbed into the stated scaling. In Rayleigh/Rician channels epsilon_0 can be arbitrarily small, and a dead receive-antenna path makes it exactly zero. This is not a typo; the stated convergence guarantee is load-bearing and the proof does not establish it. The appendix also introduces matrices Q and R without defining them, which makes the derivation hard to audit.

The capacity-to-sum-power approximation is reasonable and closely follows the spirit of Zhang and Zhang [8]. The lower bound in (13) is valid, though loose, and the paper never quantifies how loose it is. The linear search method with CSI is clean, but the paper's own Fig. 6 admits the approximation gap grows with N, so the "sum power" surrogate becomes less faithful exactly where the MIMO extension matters.

Field tests are single-shot: no repeated trials, no confidence intervals, and the outdoor gain, while positive, is modest. Simulations use T=1000, far below the theoretical scaling, which is fine as a heuristic but does not compensate for the broken proof. Citation patterns look appropriate; the self-citations to [6], [7] are to the works being extended.

Overall: the central algorithm is promising and the field results are valuable, but the paper currently overclaims its theoretical guarantee. I would send this to a serious referee — the idea and experiments deserve the time — but I would not accept until the proof is completed or the statement weakened to make the epsilon_0 dependence explicit. For a letters venue, the authors could also move the full proof to a longer version and state the result with the correct factors.

Referee Report

3 major / 4 minor

Summary. This paper proposes a blind passive beamforming method for an IS-aided point-to-point MIMO system with discrete phase shifts. The authors approximate the capacity maximization problem by a sum-power maximization problem, derive a CSI-based linear-search solution, and then give an algorithm (Algorithm 1) that replaces CSI by conditional sample means of received power over random phase configurations. Proposition 1 claims that, with T = Ω(L N^2 S^2 (log(N L))^3) random samples, the blind solution equals the CSI-based solution with probability at least 1−ξ. The paper also reports field tests in a commercial 5G network and simulations comparing with beam training and the Zhang–Zhang method.

Significance. If the theoretical claims were fully established, the result would be a meaningful extension of blind beamforming from SISO to MIMO, with a simple algorithm and an explicit sample-complexity bound; the field tests provide real-world evidence that is rare in this literature and are the paper's strongest asset. The capacity approximation in (9)-(10) is reasonable, and the CSM idea is natural. However, as detailed below, the proof of Proposition 1 has a load-bearing gap, and the reduction from capacity maximization to the linear-search objective is not quantified, so the theoretical contribution is currently incomplete.

major comments (3)

[Appendix, Eq. (20) and final paragraph] The stated sample complexity T=Ω(LN^2S^2(log(NL))^3) does not follow from the preceding inequalities. Each of the three finite-sample bounds in the appendix contains the factor 1/ϵ_0^2 (for instance T≥9σ^4NK^2L/(ϵ_0^2p_0) and T≥18qνσ^2NK^2L/(ϵ_0^2p_0), and T≥9q^2ν^2K log(2NKL/p_0)/(2ϵ_0^2)), yet the next line concludes P{E_0}≤ξ for T=Ω(Lν^2(log(NL))^3) and then for T=Ω(LN^2S^2(log(NL))^3) without accounting for ϵ_0. Since ϵ_0=inf_{i,n}ϵ_in is channel-dependent and can be arbitrarily small or zero, the proof does not establish the proposition as stated. The theorem statement must either include ϵ_0 (or a lower bound on it) in the sample-complexity expression or impose a condition on the channel guaranteeing a positive gap, and the derivation must retain the 1/ϵ_0^2 factor throughout.
[III, Eqs. (13)-(15)] The linear-search solution Θ_CSI in (15) is not shown to approximate the solution of problem (11). Pointwise inequality f(Θ)≥f_b(Θ) only says that maximizing f_b cannot be used to infer the maximizer of f; a lower bound maximizer can be arbitrarily poor for f, especially since f_b can be negative. No approximation ratio or gap bound such as f(Θ_CSI) ≥ c·max_Θ f(Θ) is provided. Consequently, even if Algorithm 1 recovered Θ_CSI exactly, the claim that this solves the passive beamforming problem (8) would still lack a theoretical guarantee. Please add a quantitative approximation statement or reframe the objective as maximizing f_b.
[Appendix, definitions and per-antenna gap] The appendix uses Q_ij, R_inj, and ν without defining them in the manuscript, so the proof cannot be verified as written. In addition, the proof conditions on ϵ_in>0 for every (i,n), and defines ϵ_0 as the infimum over all (i,n). The decision rule (19), however, compares the aggregate g=∑_i |y_i|^2, so a receive antenna with no reflected contribution (or with a projection tie) makes ϵ_0=0 and the bound vacuous even when the aggregate decision for θ_n is identifiable. The unconditional statement of Proposition 1 is therefore not supported.

minor comments (4)

[Table I] The header 'Ourdoor' should be 'Outdoor'.
[Section VI] The text says 'as shown in Fig. 1' when introducing the simulations, but Fig. 1 is the system diagram; the reference should be to the relevant simulation figure.
[Algorithm 1] The input line lists only Φ_K and N, but the algorithm also requires T and K; please include them for completeness.
[Section IV, Eq. (18)] The notation bE[g|θ_n=kω] for the estimator could be confused with the true conditional expectation; consider denoting the sample average explicitly, e.g., with a hat and an overline or a subscript T.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the blind beamforming rule is derived from the signal model and concentration inequalities, and the benchmarks are measured independently.

full rationale

The paper's central claim is that the conditional-sample-mean rule (19) converges to the CSI-based linear-search rule (15). This is not circular: for i.i.d. uniform random phases, the conditional mean of |y_i|^2 given θ_n = kω is a constant plus 2 Σ_j Re{α_ij^* β_inj e^{jkω}}, so the infinite-sample blind rule is exactly the separable sum-power rule in (15). The sample-complexity bound is then a statistical estimation argument, not a fitted input renamed as a prediction. No parameter is fitted to the target output; the algorithm is evaluated against independent physical channel models and field measurements. The only notable self-citation is 'Following the steps in [6]' in the Appendix, where [6] is a peer-reviewed prior paper by overlapping authors. The cited concentration bound is externally published and not tailored to this paper's conclusion, so under the independence rule it does not raise the circularity score. The appendix's silent omission of the 1/ε_0^2 factor and the mismatch between per-antenna ε_in and the summed decision rule are proof-rigor concerns, not circularity: they do not make the claimed result equivalent to its inputs by definition. The derivation chain is self-contained and the verification is independent, so no significant circularity is present.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new physical entities, forces, or dimensions. The free parameters are limited to the training budget T and the theoretical gap epsilon_0, which is not computed. The main assumption is the capacity-to-sum-power approximation and the simulation channel model, both standard in the RIS literature.

free parameters (2)

T (number of random samples) = 1000 in field tests and simulations
The number of samples is a hyperparameter set by the experimenter. The theoretical bound requires T = Omega(L N^2 S^2 (log N L)^3), which for the field settings would be enormous, while T=1000 is used in practice. No sensitivity analysis is given.
epsilon_0 (minimum gap between conditional means) = not computed
The proof defines epsilon_0 = inf_{i,n} {epsilon_in} and requires it to be strictly positive, but the paper does not compute or estimate it for any channel realization. The sample complexity bound depends on this quantity and on nu, making the bound non-explicit.

assumptions (4)

domain assumption The phase shift optimization can be approximated by maximizing the sum of received powers over the receive antennas.
This is the capacity approximation in Section III, equations (9) and (10). The approximation is justified by dropping the log and using Jensen's inequality, but the paper itself notes in Figure 6 that the gap grows with N, so the approximation is not uniformly tight.
domain assumption The channels F and G are Rician and D is Rayleigh, with pathloss models from [6], [7].
Used for the simulations in Section VI. The field test does not rely on this, but all simulation conclusions do. The Rician factor is not stated explicitly.
standard math The received signal model y = H W s + z with Gaussian noise and Gaussian signaling is valid.
Standard MIMO model in Section II, used throughout.
domain assumption The phase shifts are discrete and each element is independently random in the training phase.
The entire algorithm relies on the independence of the random phase samples to compute the conditional sample means in (18).

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Cite this review

Pith. "Pith review of Blind Passive Beamforming for MIMO System." pith.science (2026). https://pith.science/paper/F5EAJOTG

@misc{pith2026250600987,
  author       = {Pith},
  title        = {Pith review of: Blind Passive Beamforming for MIMO System},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F5EAJOTG}},
  note         = {Machine review of arXiv:2506.00987}
}

read the original abstract

Passive beamforming for the intelligent surface (IS)-aided multiple-input multiple-output (MIMO) communication is a difficult nonconvex problem. It becomes even more challenging under the practical discrete constraints on phase shifts. Unlike most of the existing approaches that rely on the channel state information (CSI), this work advocates a blind beamforming strategy without any CSI. Simply put, we propose a statistical method that learns the main feature of the wireless environment from the random samples of received signal power. Field tests in the 5G commercial network demonstrate the superiority of the proposed blind passive beamforming method.

Figures

Figures reproduced from arXiv: 2506.00987 by the authors.

**Figure 2.** A panoramic view of the parking lot. The BS is located on a 20-meter-high terrace. The user terminal is located inside the entrance of the underground [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** The scene of the outdoor testbed. The BS is deployed on the roof [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 5.** Figure 5: Rate vs. number of REs [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

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Reference graph

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Pith tools

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

[1] [6]

Configuring intelligent reflecting surface with performance guarantees: Blind beam- forming,

S. Ren, K. Shen, Y . Zhang, X. Li, X. Chen, and Z.-Q. Luo, “Configuring intelligent reflecting surface with performance guarantees: Blind beam- forming,”IEEE Trans. Wireless Commun., vol. 22, no. 5, pp. 3355–3370, May 2023. 6

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work page 2020

[3] [7]

Coordinating multiple intelligent reflecting surfaces without channel information,

F. Xu, J. Yao, W. Lai, K. Shen, X. Li, X. Chen, and Z.-Q. Luo, “Coordinating multiple intelligent reflecting surfaces without channel information,”IEEE Trans. Signal Process., vol. 22, pp. 31–46, Jan. 2024

work page 2024

[4] [1]

Intelligent surfaces empowered wireless network: Recent advances and the road to 6G,

Q. Wu, B. Zheng, C. You, L. Zhu, K. Shen, X. Shao, W. Mei, B. Di, H. Zhang, E. Basar, L. Song, M. D. Renzo, Z.-Q. Luo, and R. Zhang, “Intelligent surfaces empowered wireless network: Recent advances and the road to 6G,”Proc. IEEE, vol. 112, no. 7, pp. 724–763, Jul. 2024

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[5] [2]

Towards programming the radio environment with large arrays of inexpensive antennas,

Z. Li, Y . Xie, L. Shangguan, R. I. Zelaya, J. Gummeson, W. Hu, and K. Jamieson, “Towards programming the radio environment with large arrays of inexpensive antennas,” inUSENIX Symp. Netw. Sys. Design Implementation (NSDI), Feb. 2019, pp. 285–300

work page 2019

[6] [3]

RFocus: Beamforming using thousands of passive antennas,

V . Arun and H. Balakrishnan, “RFocus: Beamforming using thousands of passive antennas,” inUSENIX Symp. Netw. Sys. Design Implementa- tion (NSDI), Feb. 2020, pp. 1047–1061

work page 2020

[7] [4]

Power measurement-based channel estimation for IRS-enhanced wireless coverage,

H. Sun, L. Zhu, W. Mei, and R. Zhang, “Power measurement-based channel estimation for IRS-enhanced wireless coverage,”IEEE Trans. Wireless Commun., vol. 23, no. 12, pp. 9183–19 198, Dec. 2024

work page 2024

[8] [5]

RIS-aided wireless communications: Prototyping, adaptive beamforming, and indoor/outdoor field trials,

X. Pei, H. Yin, L. Tan, L. Cao, Z. Li, K. Wang, K. Zhang, and E. Bj¨ornson, “RIS-aided wireless communications: Prototyping, adaptive beamforming, and indoor/outdoor field trials,”IEEE Trans. Commun., vol. 69, no. 12, pp. 8627–8640, Dec. 2021

work page 2021