arxiv: 2605.10660 · v2 · submitted 2026-05-11 · 📡 eess.SP

Recognition: 2 theorem links

· Lean Theorem

RIS-assisted Multiuser MISO Transmission and the Impact of Imperfect Channel Estimation

Ainna Yue Moreno-Locubiche, Antonio Pascual-Iserte, Josep Vidal, Olga Mu\~noz

Pith reviewed 2026-05-13 06:58 UTC · model grok-4.3

classification 📡 eess.SP

keywords RISzero-forcing precodingMU-MISOmmWavechannel estimationrank deficiencybit error rate

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The pith

RIS phase optimization restores full rank to ZF precoding in MU-MISO mmWave downlink when direct channels are rank-deficient.

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

The paper shows that a reconfigurable intelligent surface can introduce controlled artificial scattering to make the effective multiuser channel matrix full rank, allowing zero-forcing precoding to suppress inter-user interference without excessive power. Under perfect channel knowledge the joint RIS-ZF design achieves the expected spatial multiplexing gains. When channel estimates contain errors the same framework is extended with a pilot design that equalizes residual interference across users, keeping bit-error-rate degradation manageable. The central result is that RIS assistance turns a previously intractable rank-deficiency problem into a controllable design parameter for practical 5G/6G deployments.

Core claim

Joint optimization of RIS reflection coefficients and zero-forcing precoder restores full rank to the effective downlink channel matrix in MU-MISO mmWave systems, thereby enabling interference-free spatial multiplexing even when users exhibit unequal gains or far-field alignment; the same design is made robust to channel estimation errors by a pilot allocation that equalizes multiuser interference in the precoder.

What carries the argument

Joint RIS phase-shift and ZF-precoder optimization that shapes the composite channel to eliminate rank deficiency.

If this is right

Spatial multiplexing becomes feasible for far-field aligned users without increasing base-station antenna count.
Bit-error-rate remains close to the perfect-CSI curve when the proposed pilot design is used.
ZF complexity stays low because the effective channel is forced to full rank by the RIS.
The same approach applies to other linear precoders once the composite channel is made full rank.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar rank-restoration logic could be applied to uplink scenarios or to non-linear precoding.
Hardware cost of the RIS must be traded against the reduction in required base-station antennas.
The pilot equalization scheme may generalize to other estimation-error models beyond the one studied.

Load-bearing premise

The RIS elements can be tuned in real time to produce scattering paths strong enough to overcome the rank deficiency of the direct channel.

What would settle it

A simulation or measurement in which optimized RIS phases leave the effective channel matrix rank-deficient for aligned users with unequal gains.

Figures

Figures reproduced from arXiv: 2605.10660 by Ainna Yue Moreno-Locubiche, Antonio Pascual-Iserte, Josep Vidal, Olga Mu\~noz.

**Figure 1.** Figure 1: Channel model for the RIS-assisted DL transmission [23]. Hbu ∈ C K×M contains the direct channel gains between BS and the K single-antenna users and Hc is the compound channel linking the BS, RIS, and users, which may be written as: Hc = HruΦHbr, (2) where Hbr ∈ C Nr×M contains the channel gains between the BS and the RIS and Hru ∈ C K×Nr contains the channel gains between the RIS and the K users. Thus, th… view at source ↗

**Figure 4.** Figure 4: CDF of BER for 4-QAM with perfect channel knowledge in the NEAR setup. M = 4 antennas at BS and Nr RIS elements [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

**Figure 5.** Figure 5: log10(BER) on a coverage area for 4-QAM MU-MISO transmission (no RIS present) in the NEAR-NS configuration. BER values below 10−5 have been clipped. additional propagation path that increases the received signal power, leading to a notable reduction in the BER observed. C. IMPACT OF CHANNEL ESTIMATION ERRORS We evaluate the BER over the area when the channel is estimated with errors (see Sec. VI). We assum… view at source ↗

**Figure 8.** Figure 8: Impact of channel estimation errors in the NEAR configuration when increasing in the number of RIS elements Nr. CDF of BER for 4-QAM, M = 4, T = 2(Nr + 1) with perfect estimation (PE) and imperfect estimation (IE). All BER are computed using the Chernoff bound. slightly higher compared to both direct paths and RISaided communication with perfect estimation: channel estimation errors significantly degrade… view at source ↗

**Figure 7.** Figure 7: Impact of channel estimation errors in the NEAR setup when increasing the number of pilot symbols T. CDF of BER for K = 2 users, for 4-QAM, M = 4, Nr = 1000 with perfect channel estimation (PE) and imperfect estimation (IE). BER is computed from the Chernoff bound. both users is considerably lower than when no RIS is used (dashed blue line). Interestingly, [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗

**Figure 10.** Figure 10: Impact of channel estimation errors for K = 2, 3, 4 simultaneous UEs. Capacity is displayed for M = 6, Nr = 1500 with perfect estimation (PE) and imperfect estimation (IE), for direct path (DP) and RIS-assisted transmissions upward trend as moving from 2 to 4 UEs across all 1200/K UE positions. It may be noted that the assistance of RIS introduces a notable reduction in BER. This improvement becomes more … view at source ↗

read the original abstract

This paper proposes the joint design of reconfigurable intelligent surfaces (RIS) and zero-forcing (ZF) precoding for the downlink (DL) multiuser multiple-input single-output (MU-MISO) setup in millimeter-wave (mmWave) bands, where ZF is particularly attractive due to its ability to suppress inter-user interference by exploiting the large antenna arrays and sparse directional channels characteristic of mmWave systems. This ensures efficient spatial multiplexing with manageable complexity, making ZF a practical and in modern 5G/6G deployments. However, a careful design is necessary to overcome potential rank deficiency in the channel matrix. For the MU-MISO case, rank deficiency may arise if users exhibit significantly different channel gains or if, being in far-field, they are aligned with the position of the transmitter. On the other hand, the deployment of a RIS introduces artificial scattering which can shape the radio environment to address those situations. We explore the joint design under perfect channel knowledge, assess the impact of imperfect channel estimation on the bit error rate (BER) and propose a robust design of pilot transmissions that equalizes multiuser interference across users in the presence of channel errors in the precoder design. This evaluation shows the advantages of optimized RIS-aided ZF MU-MISO communication for the DL of wireless systems.

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

The paper applies RIS phase optimization with ZF precoding to mmWave MU-MISO and adds a robust pilot scheme for imperfect CSI, reporting BER gains without major internal contradictions.

read the letter

This paper looks at using a reconfigurable intelligent surface to support zero-forcing precoding in millimeter-wave multiuser MISO downlinks. The main point is that RIS can create artificial scattering to avoid rank-deficient channels when users have mismatched gains or align poorly from the transmitter's view. They optimize the RIS phases together with the ZF precoder assuming perfect channel knowledge, then switch to a robust pilot scheme that balances interference across users when estimates are noisy. The work does a solid job showing BER improvements from this setup in their numerical results. It directly tackles a practical issue in mmWave systems where ZF is popular for its simplicity but can fail on rank problems. The robust pilot part is a reasonable addition for real-world channel estimation errors. Nothing here feels like a big theoretical leap; it's an application of known tools to this setting with an extra robustness step. The stress-test note confirms no inconsistencies in the approach or simulations. A minor concern is whether the optimization scales well or if the gains hold under more varied channel conditions than the ones simulated. The abstract and evaluation focus on specific scenarios, so broader validation would strengthen it. This paper is aimed at engineers and researchers in wireless communications working on RIS for 5G/6G enhancements. Readers interested in practical precoding designs with imperfect CSI will find the simulation results useful. It should go to peer review because the core idea is coherent and the results are presented without red flags.

Referee Report

1 major / 1 minor

Summary. The manuscript proposes the joint design of reconfigurable intelligent surfaces (RIS) phase shifts and zero-forcing (ZF) precoding for the downlink of multiuser multiple-input single-output (MU-MISO) systems operating in millimeter-wave bands. It highlights how RIS can address rank deficiency in the channel matrix caused by differing user channel gains or far-field alignment. The work evaluates the design under perfect channel state information and examines the impact of imperfect channel estimation on bit error rate (BER), proposing a robust pilot transmission design to equalize multiuser interference under channel errors.

Significance. If the claimed performance advantages hold, this work provides valuable insights into practical RIS-assisted transmission schemes for 5G/6G wireless systems. By demonstrating BER improvements and rank restoration through optimized RIS-aided ZF precoding, it contributes to understanding how artificial scattering can enhance spatial multiplexing in challenging mmWave scenarios. The robust design for imperfect CSI is particularly relevant for real-world deployments where perfect channel knowledge is unavailable.

major comments (1)

[Evaluation section] Evaluation section: the BER curves and rank-restoration claims rely on specific mmWave channel realizations and user placements; without explicit reporting of the number of Monte Carlo trials, the exact user angular spreads, and the RIS element count used in the rank-deficient cases, it is difficult to assess whether the observed gains are robust or sensitive to the chosen geometry.

minor comments (1)

[Abstract] Abstract: the sentence fragment 'making ZF a practical and in modern 5G/6G deployments' is grammatically incomplete and should be rephrased for clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive comment and the recommendation of minor revision. We address the point below.

read point-by-point responses

Referee: [Evaluation section] Evaluation section: the BER curves and rank-restoration claims rely on specific mmWave channel realizations and user placements; without explicit reporting of the number of Monte Carlo trials, the exact user angular spreads, and the RIS element count used in the rank-deficient cases, it is difficult to assess whether the observed gains are robust or sensitive to the chosen geometry.

Authors: We agree that explicit reporting of these parameters is necessary for reproducibility and to allow readers to evaluate the robustness of the results. In the revised manuscript, we will add the missing details to the Evaluation section, including the number of Monte Carlo trials used to generate the BER curves, the precise angular spreads of the users in the mmWave channel model, and the RIS element count employed in the rank-deficient scenarios. This will clarify the simulation setup and address concerns about sensitivity to geometry. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The manuscript describes a joint RIS phase and ZF precoder optimization under perfect CSI, followed by a separate robust pilot design for imperfect channel estimation. Central claims rest on numerical BER evaluations and rank restoration demonstrations in mmWave MU-MISO scenarios. No load-bearing steps reduce predictions to fitted inputs by construction, invoke self-citations as uniqueness theorems, or smuggle ansatzes; the derivation chain remains independent and self-contained against external simulation benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract alone.

pith-pipeline@v0.9.0 · 5543 in / 1053 out tokens · 53355 ms · 2026-05-13T06:58:38.780340+00:00 · methodology

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discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

Foundation/AlexanderDuality alexander_duality_circle_linking unclear
rank deficiency... addressed through... RIS introduces artificial scattering which can shape the radio environment

Reference graph

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