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Robust Beamforming Design for Intelligent Reflecting Surface Aided MISO Communication Systems
Gui Zhou, Cunhua Pan, Hong Ren, Kezhi Wang, Marco Di Renzo, Arumugam Nallanathan
TL;DR
The paper addresses robust beamforming for IRS-aided MU-MISO communication when reflection-channel CSI is imperfect and difficult to obtain. It jointly designs active precoding and passive reflection under worst-case QoS constraints using approximation, transformation, and penalty-CCP-based optimization. Numerical results show that the robust design guarantees the required QoS targets, while its efficiency advantages depend on high reflection efficiency and small estimation error.
Problem
Reflection-channel CSI is difficult to obtain accurately because IRS elements are passive and have limited signal-processing capability.
Method
The paper jointly designs BS precoding and IRS reflection beamforming using approximation transformations, alternate optimization, and penalty CCP for non-convex robust constraints.
Results
The worst-case robust design guarantees that the required rate targets are met for all users without outage under the modeled channel uncertainties.
Takeaways & Limitations
IRS-aided systems can achieve high energy efficiency, but their advantages appear only with nearly unit reflection efficiency and reflection-channel uncertainty δ below 0.03.
Abstract
from arXiv · showhide
Perfect channel state information (CSI) is challenging to obtain due to the limited signal processing capability at the intelligent reflection surface (IRS). In this paper, we study the worst-case robust beamforming design for an IRS-aided multiuser multiple-input single-output (MU-MISO) system under the assumption of imperfect CSI. We aim for minimizing the transmit power while ensuring that the achievable rate of each user meets the quality of service (QoS) requirement for all possible channel error realizations. With unit-modulus and rate constraints, this problem is non-convex. The imperfect CSI further increases the difficulty of solving this problem. By using approximation and transformation techniques, we convert this problem into a squence of semidefinite programming (SDP) subproblems that can be efficiently solved. Numerical results show that the proposed robust beamforming design can guarantee the required QoS targets for all the users.
I. INTRODUCTION
IRS technology can improve wireless communication efficiency, but robust design is needed because reflection-channel CSI is difficult to obtain accurately. The paper therefore studies worst-case joint active and passive beamforming under imperfect CSI.
- IRS uses low-power passive elements to adjust signal phases and enhance communication efficiency.Its programmable metasurface can adjust each element's phase continuously or discretely.
- Perfect CSI is unrealistic for IRS communications because passive reflection elements have limited signal-processing capability.The reflection channel is especially difficult to estimate as users and environmental conditions change.
- The paper investigates worst-case robust joint active precoder and passive reflection beamforming for an IRS-aided downlink MU-MISO system.It adopts an ellipsoid model for reflection-channel uncertainties.
- The design minimizes BS transmit power while meeting every user's QoS target for all possible channel-error realizations.The problem is non-convex because of unit-modulus constraints, rate constraints, and imperfect CSI.
A. Signal Transmission Model
The system is an IRS-aided MISO broadcast channel in which a multi-antenna BS serves single-antenna users using active precoding and passive reflection. The reflection channel is modeled as uncertain within bounded error regions.
- A BS with N transmit antennas serves K single-antenna users using precoding matrix F.The transmitted signal is x = Fs.
- The IRS employs M passive reflection elements with unit modulus, represented by the diagonal reflection matrix E = ιdiag(e).The reflection efficiency is ι ∈ [0, 1].
- Each user's received signal depends on the direct BS-user channel, BS-IRS channel, and IRS-user reflection channel.The BS designs the reflection beamforming and sends it to the IRS controller.
- User rate is defined from its received signal, additive white Gaussian noise, and interference from the other users' precoders.The interference-plus-noise term uses F−k, the precoders excluding user k's vector.
- The reflection channel is modeled as a contaminated estimate plus error, with ||Δ_k||2 ≤ ε_k for each user.The uncertainty radius ε_k is known by the BS.
B. Problem Formulation
The paper formulates robust beamforming as minimizing total transmit power while guaranteeing every user's rate threshold over all channel-error realizations. Unit-modulus IRS constraints are imposed simultaneously.
- The objective is to minimize total transmit power through joint design of the BS precoder F and IRS reflection vector e.The QoS constraints must hold for every possible channel-error realization.
- The minimum QoS constraints require each user's achievable rate to remain above its target.These are worst-case constraints under imperfect CSI.
- The IRS reflection elements must satisfy unit-modulus requirements, expressed as |e_m|2 = 1.These constraints encode the passive reflection structure.
III. ROBUST BEAMFORMING DESIGN
Because the precoder and reflection vector are coupled in a non-convex robust problem, the paper uses alternate optimization to solve variable-specific subproblems iteratively.
- The design is non-convex because QoS constraints span CSI uncertainty regions and IRS elements must satisfy unit-modulus constraints.The variables F and e are coupled, motivating alternate optimization.
- Alternate optimization iteratively solves subproblems associated with different variable sets.This separates the coupled optimization of the active precoder and passive reflection beamforming.
A. Problem Transformation
The paper transforms imperfect-CSI QoS constraints and unit-modulus constraints into tractable subproblems, then alternates between precoder and reflection-beamforming optimization. The resulting SDP formulations use approximations, uncertainty transformations, slack variables, and penalty CCP.
- QoS constraint transformation: Auxiliary variables, lower-bound approximations, Schur’s complement, and uncertainty transformations address the non-convex QoS constraints under imperfect CSI.The S-Procedure produces equivalent LMIs for CSI uncertainty, while Nemirovski’s lemma and slack variables yield an approximated reformulation.
- Alternate optimization: Because precoder F and reflection vector e remain coupled in the LMIs, the method uses alternate optimization with separate variable subproblems.For fixed e, the F subproblem is formulated separately; for fixed F, the e subproblem is a feasibility-check problem.
- Reflection-beamforming subproblem: The reflection-beamforming feasibility problem is an SDP with modified LMI constraints incorporating SINR-residual slack variables.The modified constraints replace βk(2rk −1) with βk(2rk −1) + ak.
- Unit-modulus constraints: Penalty CCP handles the non-convex unit-modulus constraints because semidefinite relaxation cannot always ensure feasible QoS when its solution is not rank one.Slack variables and a regularization penalty control SINR residuals and feasibility during reflection-beamforming optimization.
- Penalty CCP procedure: The penalty CCP procedure increases the regularization factor up to λmax and stops when the slack norm and successive reflection-vector change meet their thresholds.The stopping conditions ||b||1≤χ and ||e[t]−e[t−1]||1≤ν enforce approximate unit-modulus feasibility and convergence.
B. Algorithm Description
Algorithm 2 alternates SDP precoder updates with penalty-CCP reflection updates until the transmit-power objective converges. Its convergence follows from a non-increasing objective sequence, and initialization uses a simple reflection vector plus a feasible precoder construction.
- AO iteration: Algorithm 2 initializes e(0) and F(0), then alternates updating F with fixed e and e with fixed F.The iterations stop when the objective value ||F(n+1)||²_F converges.
- Convergence analysis: The objective sequence is non-increasing because each reflection update remains feasible and each subsequent precoder update is globally optimal for the fixed reflection vector.The inequality F(F(n), e(n)) ≥ F(F(n), e(n+1)) ≥ F(F(n+1), e(n+1)) establishes convergence.
- Initial point: Initialization can use a full-1 reflection vector and a precoder obtained from a guaranteed-feasible auxiliary optimization problem.The auxiliary construction is feasible at least when ϕk=0 for all users and F(n)=0.
IV. NUMERICAL RESULTS AND DISCUSSIONS
Numerical evaluations compare robust and non-robust IRS designs across channel uncertainty, transmit power, energy efficiency, and outage probability. Robust design avoids outages, while IRS efficiency advantages depend on reflection efficiency and sufficiently small estimation error.
- Simulation setup: The experiments use N = 6 transmit antennas, M = 16 IRS elements, K = 4 users, and equal target rates r.Users are randomly distributed around a specified circle, and CSI uncertainty is parameterized by δ.
- Transmit power and energy efficiency: Robust IRS beamforming requires higher transmit power than other schemes, but remains below the “Non IRS” case.The higher power is identified as the cost of robustness and passive reflection elements.
- Transmit power and energy efficiency: IRS-aided systems achieve higher energy efficiency than the relay system because IRS passive elements have lower circuit power consumption.The comparison uses total power models that include active, passive, and relay circuit-power terms.
- Transmit power and energy efficiency: IRS advantages in spectral and energy efficiency appear only when reflection efficiency is nearly 1 and δ is less than 0.03.This conclusion is drawn from the transmit-power and energy-efficiency results.
- Outage probability: Non-robust beamforming frequently fails to satisfy at least one user’s target rate, especially at high r or δ, whereas worst-case robust design guarantees no outage.Figure 3 reports outage probability versus δ for N = 6, M = 16, and K = 4.
V. CONCLUSIONS
The paper addresses robust beamforming for IRS-aided MU-MISO communication with imperfect CSI. Approximation, transformation, and penalty CCP techniques support the design, and numerical results demonstrate its robustness.
- V. CONCLUSIONS: The proposed algorithm addresses CSI uncertainties and non-convex unit-modulus constraints for robust IRS-aided MU-MISO beamforming.The method uses approximation and transformation techniques together with the penalty CCP framework.
- V. CONCLUSIONS: Numerical results demonstrate the robustness of the proposed algorithm.The conclusion summarizes the reported evaluation without specifying an additional metric.