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A Framework of Robust Transmission Design for IRS-aided MISO Communications with Imperfect Cascaded Channels

Gui Zhou, Cunhua Pan, Hong Ren, Kezhi Wang, Arumugam Nallanathan

arXiv:2001.07054v5eess.SP

TL;DR

IRS systems require robust transmission design because the passive IRS makes perfect CSI difficult to obtain, especially for cascaded BS-IRS-user channels. This paper minimizes transmit power under bounded worst-case and statistical outage constraints using S-procedure and Bernstein-type approximations. Numerically, statistical-error robust beamforming provides superior minimum transmit power, convergence, and complexity, while CBIUT error can reduce or reverse IRS gains.

  • Problem

    Perfect CSIT is difficult to obtain for IRS systems, and prior robust designs did not address imperfect cascaded BS-IRS-user channels at the transmitter.

  • Method

    The paper formulates transmit-power minimization under bounded worst-case rate constraints and statistical rate-outage constraints, using S-procedure and Bernstein-type inequality approximations.

  • Results

    Statistical-error robust beamforming achieves superior minimum transmit power, convergence speed, and complexity compared with bounded-error robust beamforming.

  • Takeaways & Limitations

    The impact of CBIUT error is substantial: increasing IRS elements can increase transmit power at high error, potentially eliminating the IRS performance advantage.

Abstract

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Intelligent reflection surface (IRS) has recently been recognized as a promising technique to enhance the performance of wireless systems due to its ability of reconfiguring the signal propagation environment. However, the perfect channel state information (CSI) is challenging to obtain at the base station (BS) due to the lack of radio frequency (RF) chains at the IRS. Since most of the existing channel estimation methods were developed to acquire the cascaded BS-IRS-user channels, this paper is the first work to study the robust beamforming based on the imperfect cascaded BS-IRS-user channels at the transmitter (CBIUT). Specifically, the transmit power minimization problems are formulated subject to the worst-case rate constraints under the bounded CSI error model and the rate outage probability constraints under the statistical CSI error model, respectively. After approximating the worst-case rate constraints by using the S-procedure and the rate outage probability constraints by using the Bernstein-type inequality, the reformulated problems can be efficiently solved. Numerical results show that the negative impact of the CBIUT error on the system performance is greater than that of the direct CSI error.

I. INTRODUCTION

IRS-aided wireless systems use passive phase control to reconfigure propagation and jointly optimize BS and IRS beamforming, but imperfect CSI motivates robust transmission design. This paper studies imperfect cascaded BS-IRS-user channels under bounded and statistical error models and reports their algorithmic and performance implications.

  • Motivation: IRS elements impose independent phase shifts that can reconfigure BS-user propagation constructively or destructively.The IRS is a passive planar radio array intended to improve spectral and energy efficiency.
  • Motivation: Perfect CSIT is an unrealistic assumption because IRS passivity makes IRS-related channel estimation challenging.The direct BS-user channel can be estimated conventionally, whereas IRS-related channels comprise BS-IRS and IRS-user links.
  • Prior work: Cascaded BS-IRS-user channels are sufficient for joint active and passive beamforming, so existing work largely estimates these cascaded channels.Such estimation has been studied in SU-MIMO and MU-MISO systems.
  • Research gap: Existing IRS transmission studies generally ignored channel estimation errors, although treating estimated channels as perfect can cause performance loss.Only a few prior robust designs considered imperfect IRS-user channels, and those designs do not apply to imperfect cascaded channels at the transmitter.
  • Contributions: This paper formulates transmit-power minimization with worst-case rate constraints for bounded errors and outage-probability constraints for statistical errors.The bounded-error formulation uses the S-procedure, while the statistical-error formulation uses the Bernstein-type inequality; alternating optimization then updates beamforming variables.
  • Contributions: Statistical-error robust beamforming achieves lower minimum transmit power, faster convergence, and lower complexity than bounded-error robust beamforming in the numerical results.The paper also finds that CBIUT error can reverse the benefit of adding IRS reflection elements.

B. Two Scenarios and CSI Error Models

The paper distinguishes partial and full channel uncertainty and considers bounded and statistical CSI error models for robust beamforming. These models transform the power-minimization problem into tractable formulations using S-procedure-based and Bernstein-type approximations.

  • Uncertainty scenarios: The system performance depends on both direct-channel CSI accuracy and cascaded BS-IRS-user channel accuracy.The paper introduces two uncertainty scenarios before specifying the CSI error models.
  • Partial Channel Uncertainty: Under partial channel uncertainty, DCSIT is perfect while the cascaded BS-IRS-user channel at the transmitter is imperfect.The BS knows an estimated cascaded CSI, while the CBIUT error is unknown.
  • Full Channel Uncertainty: Under full channel uncertainty, both DCSIT and CBIUT are imperfect.The direct channel is represented using its estimated value known at the BS plus an unknown DCSIT error.
  • Robust designs: The paper studies two robust beamforming designs corresponding to bounded and statistical CSI error models.The statistical model imposes rate-outage constraints, while the bounded model imposes worst-case constraints.
  • CSI Error Models: The bounded CSI error model represents channel errors within known uncertainty regions, including quantization errors from rate-limited CSI feedback.The model uses uncertainty-region radii known at the BS.
  • CSI Error Models: The statistical CSI error model assumes CSCG-distributed error vectors with positive semidefinite covariance matrices.The paper associates this model with channel-estimation errors caused by noise and limited training.
  • Bounded-error design: For bounded errors, the design minimizes BS transmit power while enforcing unit-modulus IRS constraints and rate thresholds for every possible error realization.An alternating-optimization procedure based on the S-procedure addresses the semi-infinite constraints and coupled variables.

A. Scenario 1: Partial Channel Uncertainty

Scenario 1 designs robust beamforming with perfect DCSIT and imperfect CBIUT by minimizing BS transmit power under worst-case QoS and unit-modulus constraints. S-procedure reformulation and alternating optimization produce tractable precoder and reflection-beamforming updates.

  • Problem formulation: The design minimizes BS transmit power while enforcing worst-case user QoS and unit-modulus IRS reflection constraints under imperfect CBIUT.The worst-case QoS constraints require each user's achievable rate to exceed its target for every allowed channel-error realization.
  • Robust reformulation: S-procedure handling converts the semi-infinite worst-case useful-signal and interference-plus-noise constraints into tractable matrix inequalities.Useful-signal power is first linearly approximated, while Schur's complement and associated lemmas handle interference-power uncertainty.
  • Optimization algorithm: Alternating optimization updates the precoder and reflection beamforming sequentially because their variables remain coupled in the reformulated problem.With fixed reflection beamforming, the precoder subproblem is convex and solved as an SDP; the reflection update is handled separately.
  • Optimization algorithm: The reflection-beamforming update is formulated as an SDP after dimension reduction and is solved iteratively within the outer alternating-optimization framework.Only a K × K submatrix depends on the reflection vector, reducing the relevant LMI dimension from (K + N) × (K + N).
  • Optimization algorithm: Penalty CCP addresses the non-convex unit-modulus constraints in the reflection-beamforming subproblem.Slack variables penalize violations, while the regularization factor controls feasibility and the iterations stop when violation and iterate-change criteria are satisfied.

B. Scenario 2: Full Channel Uncertainty

Scenario 2 extends robust beamforming to simultaneous DCSIT and CBIUT uncertainty. The resulting full-channel robust problem uses linearization, uncertainty-to-LMI transformations, and the same alternating optimization structure as Scenario 1.

  • Problem formulation: Full-channel uncertainty models errors in both the direct channel and the BS-IRS channel while retaining the unit-modulus IRS constraints.The uncertainty set bounds direct-channel errors by ξh,k and cascaded-channel errors by ξg,k.
  • Robust reformulation: The useful-signal constraints are linearized at the current precoder and reflection-beamforming iterates before applying robust reformulation tools.The resulting lower bound is constructed for the combined uncertain direct and reflected channel.
  • Robust reformulation: Lemma-based transformations convert the uncertain useful-signal and interference-plus-noise constraints into equivalent linear matrix inequalities with slack variables.Schur's complement is used for interference inequalities, while separate slack variables handle direct- and cascaded-channel uncertainty.
  • Optimization: The full-channel robust problem remains non-convex because its variables are coupled, so it is solved similarly to the partial-uncertainty problem.The paper omits the repeated solution details and refers to the preceding alternating-optimization procedure.

IV. OUTAGE CONSTRAINED ROBUST BEAMFORMING

The outage-constrained formulation replaces bounded errors with statistical CSI errors and minimizes transmit power subject to probabilistic rate requirements. Bernstein-type inequality provides a tractable safe approximation.

  • Motivation: The statistical CSI model addresses Gaussian channel-estimation errors, which are unbounded and are not fully characterized by the bounded-error model.The formulation introduces maximum outage probabilities ρ1, ..., ρK for the users.
  • Problem formulation: Each outage constraint requires user k to decode at target rate Rk with probability at least 1 − ρk.The outage probability is the probability that the achievable rate falls below the user's required rate.
  • Reformulation: Bernstein-type inequality supplies a safe approximation because the outage constraints lack simple closed-form expressions.The inequality bounds a quadratic Gaussian-form probability using auxiliary slack variables and eigenvalue-related terms.
  • Scope: The paper first considers partial channel uncertainty and then extends the outage-constrained design to full channel uncertainty.This staged treatment separates the simpler robust-beamforming case from the full uncertainty case.

A. Scenario 1: Partial Channel Uncertainty

For outage-constrained robust beamforming under partial uncertainty, the paper reformulates probabilistic rate constraints deterministically and solves the coupled design through alternating optimization. SDR handles the precoder update, while penalty CCP handles reflection constraints.

  • Outage reformulation: The outage probability is rewritten in quadratic-form terms and converted into deterministic constraints using Bernstein-type inequality.Auxiliary variables are introduced during the deterministic reformulation.
  • Precoder optimization: With fixed reflection beamforming, semidefinite relaxation converts the precoder subproblem into a convex SDP solvable by CVX.The relaxation removes rank-one constraints, and the paper states that a feasible rank-one solution always exists when the relaxed problem is feasible.
  • Precoder optimization: Numerical results indicate that the optimal relaxed matrices are usually rank one before constructing the rank-one solution.The corresponding precoders can be recovered from the relaxed matrices by eigenvalue decomposition.
  • Reflection optimization: With fixed precoder, the reflection-beamforming subproblem is handled by concave approximation and penalty CCP within the alternating-optimization procedure.The non-concave reflection-dependent constraints are approximated using first-order Taylor inequalities before applying the penalty method.

B. Scenario 2: Full Channel Uncertainty

The full-channel-uncertainty scenario extends outage-constrained robust beamforming to imperfect direct and cascaded channels, reformulating the rate-outage problem for efficient solution. The resulting design highlights the dominant impact of imperfect CBIUT when the IRS has many reflection elements.

  • B. Scenario 2: Full Channel Uncertainty: Full channel uncertainty models all channels as imperfect at the BS in the outage-constrained robust beamforming design.The formulation considers full statistical CSI error and extends the partial-channel-uncertainty case.
  • B. Scenario 2: Full Channel Uncertainty: The rate-outage constraint is transformed through algebraic reformulation, auxiliary variables, and a Bernstein-type approximation into a tractable problem.The derivation uses the rewritten terms and Lemma 5 before producing the approximated constraint and final optimization problem.
  • B. Scenario 2: Full Channel Uncertainty: Problem (50) can be solved using the same techniques as Problem (40), with the full-uncertainty formulation obtained by replacing the relevant error parameter.The paper explicitly relates the full-channel formulation to the partial-channel formulation through this replacement.
  • B. Scenario 2: Full Channel Uncertainty: When M is large, imperfect CBIUT dominates system performance, motivating robust beamforming for IRSs with many reflection elements and high channel-estimation error.This conclusion is stated as a motivation for investigating the full robust design in large IRS settings.

V. COMPUTATIONAL COMPLEXITY

The proposed robust transmission designs are analyzed through the worst-case runtime of their convex reformulations. Their complexity depends on the numbers and sizes of variables, LMIs, and SOC constraints.

  • V. COMPUTATIONAL COMPLEXITY: The resulting convex problems use LMI, SOC, and linear constraints and can be solved by a standard interior point method.The paper compares methods using worst-case runtime while ignoring the complexity of linear constraints.
  • V. COMPUTATIONAL COMPLEXITY: Worst-case runtime is expressed in terms of the number of variables, LMI sizes, and SOC sizes.These quantities determine the computational complexity used for comparing the robust transmission design methods.

(50,10) IRS

The paper evaluates computational complexity for four robust beamforming methods under partial or full channel uncertainty and bounded or statistical CSI errors. The methods are compared per iteration using convex-program complexity expressions.

  • (50,10) IRS: Complexity per iteration is characterized using the numbers of variables, LMIs, and SOC constraints, with their corresponding sizes.The general expression excludes the complexity of linear constraints.
  • (50,10) IRS: PCU-bounded complexity combines the costs of Problems (17) and (22), with variable dimensions n1 = NK and n2 = M.The two component complexities are denoted oF and oe, and the per-iteration total is oF + oe.
  • (50,10) IRS: FCU-bounded complexity is likewise obtained from the costs of Problems (31) and its associated subproblem.The formulation changes the dimensions and constraint terms relative to the partial-channel case.
  • (50,10) IRS: The PCU-statistic method has separate complexity expressions for Problems (41) and (45), whose per-iteration cost is their sum.Both expressions use n1 = NK and n2 = M for the respective variable groups.
  • (50,10) IRS: FCU-statistic has the same approximate per-iteration complexity as PCU-statistic, differing only in some coefficients.The paper attributes this equivalence to the structure of the two formulations.

VI. NUMERICAL RESULTS AND DISCUSSIONS

The numerical results evaluate convergence, computational cost, rate requirements, feasibility, and antenna scaling under bounded and statistical CSI uncertainty. Statistical-error methods generally converge faster and require less CPU time, while CBIUT accuracy strongly affects the benefits of increasing IRS elements.

  • Convergence: 10 iterations are sufficient for all four algorithms to converge, with faster convergence as antenna numbers increase.Statistical-error algorithms converge faster than bounded-error algorithms.
  • Computational complexity: Statistical-CSI algorithms require much less CPU time than bounded-CSI algorithms because worst-case methods involve large-dimensional LMIs.The FCU-bounded algorithm is slower than the PCU-bounded algorithm because direct-CSI errors increase LMI dimensions.
  • Transmit power: Minimum transmit power increases with target rate, while worst-case robust designs require more power than outage-constrained designs.Worst-case optimization is more conservative because it protects against the worst CSI-error realization.
  • CBIUT accuracy and IRS size: With small CBIUT error, increasing IRS reflection elements reduces minimum transmit power; with δg = 0.1 or larger, transmit power increases instead.IRS beamforming gain competes with the additional channel-estimation error associated with more reflection elements.
  • CBIUT accuracy and antenna scaling: Increasing BS antennas reduces transmit power even when CBIUT error is high, because additional degrees of freedom improve active beamforming.Under high CBIUT error, the IRS can lose its performance-gain advantage relative to a system without IRS.
  • Feasibility: When δg is low, feasibility rates remain high, and increasing BS antennas reduces transmit power without being affected by DCSIT error δh.These observations are reported for the scenario with both DCSIT and CBIUT imperfect.

APPENDIX A THE PROOF OF LEMMA 3

The proof derives lower bounds under channel uncertainty, verifies feasibility and objective properties, and establishes a rank-one feasible relaxed solution.

  • APPENDIX A THE PROOF OF LEMMA 3: A first-order Taylor inequality is applied at a fixed point to derive the required bound.The substitutions use (h_k^H + e_k^H G_k)f_k and (h_k^H)f_k.
  • APPENDIX A THE PROOF OF LEMMA 3: The bound is expanded after substituting G_k = bG_k + △G_k and using trace-vectorization identities.The transformations include Tr(A^HB) = vec^H(A)vec(B) and Tr(ABCD) = (vec^T(D))^T(C^T ⊗ A)vec(B).
  • APPENDIX A THE PROOF OF LEMMA 3: Under full channel uncertainty, substituting h_k = bh_k + △h_k and G_k = bG_k + △G_k yields the corresponding lower-bound expression.The remaining terms are expressed using similar mathematical transformations.
  • APPENDIX A THE PROOF OF LEMMA 3: The constructed solution eΓ⋆ has no larger objective value than the optimal relaxed solution bΓ⋆.The proof then shows that the constructed solution satisfies the original constraint and the relaxed constraints.
  • APPENDIX A THE PROOF OF LEMMA 3: The constructed solution eΓ⋆ is a feasible rank-one solution of the relaxed version of Problem (41).This conclusion follows from the objective comparison and constraint verification.

APPENDIX D DERIVATION OF (49)

The derivation uses trace and eigenvalue properties to transform the channel-dependent expressions into the form of equation (49).

  • APPENDIX D DERIVATION OF (49): Trace cyclicity is used as an algebraic transformation in the derivation.The stated property is Tr{AB} = Tr{BA}.
  • APPENDIX D DERIVATION OF (49): The derivation uses λ_nonzero(AB) = λ_nonzero(BA) to relate the non-zero eigenvalues of matrix products.This property is stated alongside the notation λ_nonzero(X).
  • APPENDIX D DERIVATION OF (49): The channel and reflection terms are expanded using bh_k + △h_k and bG_k + △G_k around the fixed-point quantities.The resulting expressions contain the corresponding beamforming and error terms.
  • APPENDIX D DERIVATION OF (49): Vectorized error terms and quadratic components are collected to obtain the final expression used in the derivation.The displayed fragments include vec^H terms and the d1,k and d2,k components.
  • APPENDIX D DERIVATION OF (49): Adding σ^2 to a matrix shifts each eigenvalue by σ^2, supporting the eigenvalue manipulation.The derivation invokes the stated eigenvalue-shift property for A + σ^2I.
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