⚡ electrical engineering

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

This paper proposes a joint design of reconfigurable intelligent surfaces (RIS) and zero-forcing (ZF) precoding for downlink mmWave multiuser MISO systems to mitigate rank deficiency and inter-user interference, while also analyzing the impact of imperfect channel estimation and introducing a robust pilot transmission scheme to equalize multiuser interference.

Original authors: Ainna Yue Moreno-Locubiche, Josep Vidal, Antonio Pascual-Iserte, Olga Muñoz

Published 2026-05-12

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ainna Yue Moreno-Locubiche, Josep Vidal, Antonio Pascual-Iserte, Olga Muñoz

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a wireless network as a busy highway where a central tower (the Base Station) is trying to send different messages to several cars (the Users) at the same time. In the future of wireless technology (6G), these towers will use very high-frequency signals (mmWave) to send data super fast. However, these signals are fragile; they get blocked easily by buildings, trees, or even if two cars are driving in a straight line behind each other, making it hard for the tower to tell them apart.

This paper proposes a clever solution using a "smart mirror" called a Reconfigurable Intelligent Surface (RIS) and a specific way of organizing the traffic called Zero-Forcing (ZF) precoding.

Here is the breakdown of their ideas using everyday analogies:

1. The Problem: The "Traffic Jam" and the "Blind Spot"

The tower wants to talk to multiple cars simultaneously. To do this without the messages getting mixed up (interference), it uses a technique called Zero-Forcing. Think of this like a conductor trying to make sure every musician in an orchestra plays their own note perfectly without drowning out the others.

However, this works poorly in two specific situations:

The "Shadow" Problem: If a car is behind a tall building, the signal is too weak.
The "Alignment" Problem: If two cars are driving in a straight line away from the tower, the tower's antennas can't distinguish between them. It's like trying to shout two different instructions to two people standing one directly behind the other; the person in front blocks the sound for the person in back.

2. The Solution: The "Smart Mirror" (RIS)

The authors suggest placing a giant, programmable "smart mirror" (the RIS) on a building or pole. Unlike a normal mirror that just reflects light randomly, this mirror can be programmed to bend and steer the signal exactly where it needs to go.

How it helps: If a car is in a "shadow" (blocked by a building), the mirror catches the signal and bounces it around the corner to the car. If two cars are aligned in a straight line, the mirror creates a new, curved path for the signal, effectively "un-aligning" them so the tower can talk to both clearly.
The Result: The mirror fixes the "traffic jam" caused by bad geometry, allowing the Zero-Forcing technique to work smoothly again.

3. The Catch: "Guessing" the Map (Imperfect Channel Estimation)

To steer the mirror perfectly, the tower needs to know exactly where the cars are and how the signal bounces. In the real world, the tower doesn't have a perfect map; it has to guess based on "pilot signals" (like sending out a test ping).

The Issue: If the tower's guess is slightly wrong (imperfect estimation), the Zero-Forcing technique fails a bit. It might accidentally send a message meant for Car A to Car B, causing confusion (interference).
The Paper's Finding: The authors found that while the smart mirror helps a lot, it doesn't magically fix bad guesses. If the tower guesses the map wrong, the performance drops. However, they discovered that if the mirror is used correctly, it can help balance the confusion so that all cars suffer roughly the same amount of error, rather than one car getting a terrible connection while another gets a good one.

4. The "Robust" Strategy

The paper proposes a specific way to design the system so that even if the tower's map is a little fuzzy, the system remains fair.

The Analogy: Imagine a teacher handing out test papers. If the teacher misreads the students' handwriting (estimation error), some students might get the wrong questions. The authors' method ensures that the "wrongness" is spread out evenly. No single student gets a completely impossible test; everyone gets a slightly difficult one, keeping the overall class performance stable.

5. What They Actually Proved

The authors didn't just guess; they ran computer simulations to prove their points:

Obstacles: When a car is behind a building, the smart mirror drastically reduces the number of errors (Bit Error Rate), making the connection reliable even in "dead zones."
Alignment: When cars are lined up in a straight line, the mirror creates a new path, allowing the tower to talk to them both clearly, whereas without the mirror, the connection would fail.
The Trade-off: Adding more "pixels" to the mirror (more reflecting elements) helps a lot when the signal is clear, but it doesn't completely solve the problem if the tower's initial guess of the environment is very poor.

Summary

In short, this paper shows that using a programmable smart mirror alongside a smart traffic controller allows wireless networks to handle difficult situations (like blocked paths or aligned users) much better than before. Even when the network has to "guess" the environment, this setup keeps the connection fair and functional for everyone, preventing total signal failure in tricky spots.

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

Problem Statement
This paper addresses the challenges of downlink (DL) multiuser multiple-input single-output (MU-MISO) transmission in millimeter-wave (mmWave) bands, specifically focusing on the limitations of Zero-Forcing (ZF) precoding when the channel matrix suffers from rank deficiency. In mmWave systems, ZF is attractive for suppressing inter-user interference (MUI) due to sparse directional channels, but its effectiveness is compromised in two specific scenarios:

Significant Channel Gain Disparities: When users experience vastly different path losses due to distance or obstacles.
Far-Field Alignment: When users are aligned from the base station (BS) perspective, causing rows of the channel matrix to become nearly proportional.

Furthermore, the paper investigates the impact of imperfect Channel State Information (CSI). While ZF assumes perfect CSI, practical systems rely on channel estimation (e.g., via uplink pilots), which introduces errors. These errors degrade the ZF precoder's ability to null interference, leading to increased Bit Error Rates (BER) and potential rank deficiency issues that are not mitigated by standard ZF designs. The authors aim to determine if Reconfigurable Intelligent Surfaces (RIS) can mitigate these rank deficiency issues and how they perform under imperfect CSI conditions.

Methodology
The study proposes a joint design framework for RIS phase shifts and ZF precoding, analyzed under both perfect and imperfect CSI.

System Model: The system consists of a BS with $M$ antennas, $K$ single-antenna users, and a passive RIS with $N_r$ elements. The total channel $H$ is modeled as the sum of the direct BS-User link ( $H_{bu}$ ) and the cascaded BS-RIS-User link ( $H_{c} = H_{ru}\Phi H_{br}$ ), where $\Phi$ is the diagonal reflection coefficient matrix.
Optimization under Perfect CSI: The objective is to maximize the received signal power (or equivalently, minimize the trace of the inverse of the effective channel Gram matrix) by optimizing the RIS phases $\phi$ . Since a closed-form solution is unavailable, the authors employ a gradient-based optimization algorithm using complex matrix differentiation. The objective function $J = \text{Tr}\{(HH^H)^{-1}\Gamma\}$ is minimized with respect to the RIS phases. This approach allows the RIS design and the ZF precoder to be decoupled, avoiding the need for alternating optimization typically required in other joint designs.
Channel Estimation and Error Analysis: The paper adopts a Least Squares (LS) estimation approach for the cascaded channel using uplink pilot symbols. The authors derive the variance of the estimation error and analyze its propagation into the ZF precoder.
Impact of Imperfect CSI: The received signal is modeled to include terms representing additive MUI caused by estimation errors and amplitude uncertainty. The authors derive an upper bound for the BER using Chernoff bounds for A-QAM modulation.
Robust Design Proposal: To address the asymmetry in MUI caused by estimation errors (where users with weaker channels suffer more), the paper proposes a robust design strategy. This involves equalizing the MUI across users by adjusting the pilot transmission energy ( $P_{UL,l}T_l$ ) for each user, ensuring fair performance despite channel estimation inaccuracies.

Key Contributions

Decoupled Design: The authors demonstrate that the ZF structure allows for the uncoupling of RIS coefficient design and the precoder design. This eliminates the need for iterative alternating optimization, significantly reducing computational complexity compared to joint optimization methods used in MMSE or other precoding schemes.
Gradient-Based RIS Optimization: A gradient-based algorithm is defined to maximize received SNR by optimizing RIS phases, utilizing complex matrix differentiation theory. This method is shown to be computationally scalable, with complexity scaling polynomially rather than exponentially with the number of RIS elements.
Imperfect CSI Analysis: The paper provides a detailed analytical evaluation of how channel estimation errors degrade ZF performance in RIS-assisted MU-MISO systems. It quantifies the resulting MUI and amplitude uncertainty, deriving closed-form expressions for BER upper bounds.
Robust Pilot Design: A method is proposed to equalize multiuser interference across users in the presence of channel errors by balancing the pilot transmission energy, thereby improving fairness and robustness.
Scenario Evaluation: The study validates the benefits of RIS in challenging scenarios, including:
- Obstruction: Users blocked by obstacles where the direct path is attenuated.
- Alignment: Users aligned in the far-field of the BS, which typically causes rank deficiency.
- Gain Disparity: Users with significantly different channel gains.

Results
Simulations were conducted using a 30 GHz carrier frequency, 4-QAM modulation, and various user configurations (NEAR, OUT, FAR, NEAR-NS) with and without obstacles.

Resilience to Obstacles and Alignment: In scenarios with obstacles or user alignment (where the direct channel is rank-deficient), the RIS-assisted transmission significantly outperforms direct transmission. For example, in the "NEAR" configuration (user behind an obstacle), the median BER dropped by two orders of magnitude with RIS ( $N_r=1000$ ) compared to no RIS.
Impact of Imperfect CSI: While RIS improves performance under perfect CSI, the introduction of channel estimation errors degrades performance. However, even with imperfect CSI, RIS-assisted systems generally maintain lower BER than direct transmission without RIS, provided sufficient pilot symbols are used.
Pilot Overhead vs. Performance: Increasing the number of pilot symbols ( $T$ ) reduces estimation error and improves BER. However, the paper notes that simply increasing the number of RIS elements ( $N_r$ ) does not necessarily improve resilience to channel estimation errors; the benefits of larger RIS arrays are most pronounced when the direct path is present or when estimation is accurate.
Multi-User Load: As the number of simultaneous users ( $K$ ) increases from 2 to 4, the BER naturally increases. However, the relative improvement provided by RIS assistance becomes more significant with higher user loads, particularly in obstructed environments.
Robustness: The proposed robust design (equalizing pilot energy) helps mitigate the performance disparity between users caused by estimation errors, ensuring more uniform BER across the user set.

Significance and Claims
The paper claims that the proposed RIS-assisted ZF framework offers a practical and computationally efficient solution for 5G/6G mmWave deployments, particularly in environments where rank deficiency or blockage would otherwise render ZF precoding ineffective.

The authors emphasize that their work fills a gap in existing literature by explicitly modeling the propagation of estimation errors through fixed linear precoders (ZF) and providing diagnostic insights into residual interference. Unlike previous works that focus on optimization under ideal CSI or statistical assumptions, this study provides a robust design that accounts for the realities of channel estimation errors. The results suggest that passive RIS deployment is a viable strategy to enhance coverage and macrodiversity, enabling reliable communication for users in shadowed areas or aligned positions, provided that channel estimation is managed through appropriate pilot design.

The paper concludes modestly, noting that while the current framework focuses on ZF, future work will explore other precoding strategies (MMSE, hybrid) and advanced RIS architectures (e.g., STAR-RIS) to further optimize the trade-offs between complexity and performance in dynamic environments.