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Intelligent Reflecting Surface Assisted Multi-User MISO Communication: Channel Estimation and Beamforming Design

Qurrat-Ul-Ain Nadeem, Hibatallah Alwazani, Abla Kammoun, Anas Chaaban, Merouane Debbah, Mohamed-Slim Alouini

arXiv:2005.01301v1cs.IT

TL;DR

Passive IRSs make perfect CSI impractical in multi-user MISO systems because the surface cannot transmit, receive, or process pilots. The paper proposes Bayesian MMSE-DFT channel estimation and joint BS precoding, power allocation, and IRS phase-shift optimization. MMSE achieves lower MSE than LS, while simulations show efficient IRS-assisted operation but sensitivity to estimation errors and training-related rate loss.

  • Problem

    Perfect CSI is impractical for IRS-assisted systems because passive IRSs lack radio resources and signal-processing capability for pilot-based channel estimation.

  • Method

    The paper combines MMSE-DFT estimation using large-scale fading statistics with alternating optimization of BS precoding, power allocation, and IRS phase shifts for max-min SINR.

  • Results

    The MMSE estimates achieve much lower NMSE than LS estimates, while the proposed IRS-assisted system delivers performance gains but is sensitive to CSI errors.

  • Takeaways & Limitations

    IRS-assisted communication can provide efficient multi-user MISO performance under imperfect CSI, but channel-estimation quality and training overhead remain important design considerations.

Abstract

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The concept of reconfiguring wireless propagation environments using intelligent reflecting surfaces (IRS)s has recently emerged, where an IRS comprises of a large number of passive reflecting elements that can smartly reflect the impinging electromagnetic waves for performance enhancement. Previous works have shown promising gains assuming the availability of perfect channel state information (CSI) at the base station (BS) and the IRS, which is impractical due to the passive nature of the reflecting elements. This paper makes one of the preliminary contributions of studying an IRS-assisted multi-user multiple-input single-output (MISO) communication system under imperfect CSI. Different from the few recent works that develop least-squares (LS) estimates of the IRS-assisted channel vectors, we exploit the prior knowledge of the large-scale fading statistics at the BS to derive the Bayesian minimum mean squared error (MMSE) channel estimates under a protocol in which the IRS applies a set of optimal phase shifts vectors over multiple channel estimation sub-phases. The resulting mean squared error (MSE) is both analytically and numerically shown to be lower than that achieved by the LS estimates. Joint designs for the precoding and power allocation at the BS and reflect beamforming at the IRS are proposed to maximize the minimum user signal-to-interference-plus-noise ratio (SINR) subject to a transmit power constraint. Performance evaluation results illustrate the efficiency of the proposed system and study its susceptibility to channel estimation errors.

I. INTRODUCTION

The paper addresses practical IRS-assisted multi-user MISO communication under imperfect CSI, proposing Bayesian channel estimation and joint beamforming designs. It evaluates the resulting system against conventional and existing estimation approaches.

  • IRSs passively reshape electromagnetic waves through adjustable element phase shifts without generating new signals or adding power consumption.
  • Perfect CSI assumptions are impractical because IRSs lack radio resources and signal-processing capability for pilot transmission, reception, and channel estimation.
  • The paper proposes DFT-MMSE estimation for direct and cascaded channels using multiple training sub-phases and prior large-scale fading statistics.
  • The BS and IRS jointly design precoding, power allocation, and phase shifts to maximize minimum user SINR under transmit-power and unit-modulus constraints.
  • The proposed alternating-optimization algorithm converges, while simulations show IRS-assisted communication is efficient but sensitive to CSI errors.

B. Channel Model

The channel model accounts for spatial correlation in IRS-related links and considers structural conditions needed for multi-user IRS gains. It models the BS-to-IRS channel using line-of-sight propagation and evaluates when sufficient rank is available.

  • Spatial correlation among IRS elements is modeled because independent Rayleigh or Rician channels are mainly practical with wide element spacing and rich scattering.
  • The IRS-to-user and direct BS-to-user channels use correlation matrices, fast-fading vectors, and path-loss factors.
  • A high-rise IRS near the BS is modeled with a likely rank-one line-of-sight BS-to-IRS channel.
  • For multi-user gains, the overall BS-to-IRS channel must satisfy rank(H1) ≥ K; deterministic scattering or high-rank line-of-sight propagation can provide this rank.
  • The BS-to-IRS channel entries depend on carrier wavelength, array separations, path loss, and line-of-sight departure and arrival angles.

III. CHANNEL ESTIMATION PROTOCOL

The channel-estimation protocol uses uplink pilots and Bayesian MMSE estimation to recover IRS-assisted channels that are difficult to estimate directly. The paper compares MMSE and LS estimation using analytical and simulation-based error measures.

  • The BS estimates IRS-assisted channels because the passive IRS cannot transmit, receive, or process pilot symbols.
  • The proposed protocol computes MMSE estimates from received pilot sequences across multiple sub-phases, each using an optimal IRS reflect-beamforming vector.
  • The estimation study analytically compares normalized MSE for LS and MMSE methods and numerically compares MSE and BER performance.

A. Proposed MMSE-DFT Channel Estimation Protocol

The MMSE-DFT protocol separates direct and cascaded channel observations through orthogonal IRS training configurations, then uses channel statistics to form Bayesian estimates. The design requires at least N + 1 training sub-phases and selected slowly varying information at the BS.

  • Proposed MMSE-DFT Channel Estimation Protocol: Channel reciprocity under TDD lets the BS estimate downlink channels from uplink pilots while the users transmit mutually orthogonal sequences.
  • Proposed MMSE-DFT Channel Estimation Protocol: The coherence period is divided into uplink training and downlink transmission, with S training sub-phases of duration τS = τC/S.
  • Proposed MMSE-DFT Channel Estimation Protocol: The training matrix uses the N + 1 leading columns of an S × S DFT matrix, which attains the lower noise-variance bound under the protocol constraints.
  • Proposed MMSE-DFT Channel Estimation Protocol: At least S ≥ N + 1 sub-phases are required for the training matrix to have full column rank and admit the left pseudo-inverse.
  • Proposed MMSE-DFT Channel Estimation Protocol: The BS applies the MMSE estimator to observations associated with the direct channel and each cascaded-channel column, producing estimates with uncorrelated errors.
  • Proposed MMSE-DFT Channel Estimation Protocol: Under DFT training, MMSE estimates do not depend on cross-correlation between IRS elements, so the BS does not require knowledge of RIRS.
  • Proposed MMSE-DFT Channel Estimation Protocol: The BS requires BS-side correlation matrices and deterministic BS-to-IRS line-of-sight channel vectors, whose angles are calculated once and whose statistics vary slowly.

B. NMSE Comparison with Least Squares Estimation

The proposed MMSE-DFT channel estimates exploit large-scale fading statistics and analytically achieve lower NMSE than LS-DFT estimates for direct and IRS-assisted channels.

  • The LS-DFT estimates are obtained by correlating received training signals with user pilot sequences and applying the pseudo-inverse of the training phase-shift matrix.
  • The analysis derives normalized MSE expressions for LS and MMSE estimates of direct and IRS-assisted channel vectors.
  • The MMSE estimate approaches NMSE 1 in unfavorable conditions, corresponding to estimation-error power matching true-channel power and effectively isotropic beamforming.
  • The direct-channel NMSE under MMSE-DFT is affected by BS correlation, whereas IRS-assisted-channel NMSE is independent of the IRS correlation structure under the stated protocol.
  • MMSE-DFT direct-channel estimates always outperform LS-DFT estimates for any noise variance, pilot power, number of sub-phases, sub-phase duration, and path-loss factor.
  • MMSE-DFT IRS-assisted-channel estimates always have lower NMSE than LS-DFT estimates across noise, power, sub-phase duration, and path-loss conditions.The MMSE-DFT NMSE approaches 1 when the effective noise term grows large or the path-loss factor becomes small.

C. Performance Evaluation of the Proposed Protocol

The evaluation compares MMSE-DFT and LS-DFT under varying noise, path loss, correlation, training protocols, and transmission conditions. MMSE-DFT consistently provides lower NMSE and substantially better BER, while longer training introduces a rate trade-off.

  • Simulated NMSE matches the theoretical expressions, and MMSE-DFT achieves lower NMSE than LS-DFT, especially at moderate-to-high noise variance.
  • LS-DFT NMSE is unaffected by channel correlation, while IRS-assisted MMSE-DFT NMSE is also unaffected and direct-channel MMSE-DFT NMSE decreases with correlation.
  • The ON/OFF protocol increases low-noise MMSE NMSE by factors of S for direct channels and S(1+βd,k) for IRS-assisted channels relative to DFT.
  • Increasing sub-phases S reduces NMSE, but training consumes downlink time and causes a rate-loss factor of 1 − SτS/τ.
  • The IRS-assisted system reaches BER 10^-6 near 0 dB and has an approximately 17 dB SNR advantage over the conventional MISO system without IRS.
  • Under estimation errors, MMSE-DFT is nearly 8 dB better than LS-DFT at BER 10^-6.
  • Both ON/OFF and DFT protocols require more than N + 1 sub-phases, creating long training times when the IRS has many reflecting elements.Grouping strongly correlated elements could reduce sub-phases but also reduces IRS degrees of freedom.

IV. JOINT ACTIVE AND PASSIVE BEAMFORMING DESIGN

The design jointly optimizes BS precoding and power allocation with IRS phase shifts to maximize fairness through the minimum user SINR. It assumes unit-amplitude reflections and performs design computations at the BS.

  • IV. JOINT ACTIVE AND PASSIVE BEAMFORMING DESIGN: The BS jointly designs precoding vectors, allocated powers, and the IRS reflect beamforming vector under a transmit power constraint.
  • IV. JOINT ACTIVE AND PASSIVE BEAMFORMING DESIGN: The IRS-assisted link’s double path loss can compromise performance improvements because it combines BS-to-IRS and IRS-to-user path losses.
  • IV. JOINT ACTIVE AND PASSIVE BEAMFORMING DESIGN: IRS amplitude reflection coefficients are assumed to be unity, and the BS communicates the selected phase configuration to the IRS controller through a backhaul link.
  • IV. JOINT ACTIVE AND PASSIVE BEAMFORMING DESIGN: The performance metric is the max-min rate, balancing system throughput and user fairness.User rate is R_k = log2(1 + γ_k).
  • IV. JOINT ACTIVE AND PASSIVE BEAMFORMING DESIGN: Maximizing the minimum rate is equivalent to maximizing the minimum SINR because the logarithm is monotonically increasing.

A. Problem Formulation

The paper formulates a max-min SINR problem over BS precoding, power allocation, and IRS phase shifts. These variables are coupled through non-convex unit-modulus constraints, motivating alternating optimization.

  • A. Problem Formulation: The BS seeks optimal precoding vectors, allocated powers, and IRS reflect beamforming as the solution of a max-min SINR problem.
  • A. Problem Formulation: The IRS phase shifts satisfy unit-modulus constraints, |v_n| = 1 for n = 1, ..., N.
  • A. Problem Formulation: The max-min SINR objective is non-convex because precoding vectors, allocated powers, and phase shifts are coupled.
  • A. Problem Formulation: The proposed alternating optimization iteratively optimizes BS precoding and power allocation with IRS phase shifts until convergence.
  • A. Problem Formulation: For fixed IRS phase shifts, the precoding and power-allocation subproblem is solved by the optimal linear precoder.

B. Problem Solution

The solution alternates between optimal BS precoding and IRS phase-shift optimization. The IRS subproblem is transformed into a semidefinite-relaxed fractional program solved with generalized Dinkelbach optimization.

  • B. Problem Solution: The IRS phase-shift objective is transformed into quadratic forms using an auxiliary variable.
  • B. Problem Solution: Semidefinite relaxation replaces the non-convex rank-one constraint with a positive-semidefinite matrix constraint of arbitrary rank.
  • B. Problem Solution: Generalized Dinkelbach’s algorithm solves the resulting max-min fractional problem globally with limited complexity.
  • B. Problem Solution: If the relaxed solution has rank one, the optimal IRS vector is extracted directly; otherwise, Gaussian randomization can produce a candidate vector.
  • B. Problem Solution: In extensive simulations, the relaxed problem was always observed to have a rank-one solution, making the extracted vector optimal for the IRS subproblem.
  • B. Problem Solution: The alternating algorithm’s objective is bounded and non-decreasing across iterations, ensuring convergence, but global optimality is not claimed.

C. Imperfect CSI Scenario

Under imperfect CSI, the BS runs the alternating design using MMSE channel estimates rather than true channels. The resulting design optimizes estimated minimum SINR, while simulations evaluate true minimum SINR.

  • C. Imperfect CSI Scenario: For fixed IRS phases, the BS optimizes the estimated minimum SINR using the estimated channel vectors.
  • C. Imperfect CSI Scenario: The BS optimizes estimated minimum SINR rather than true minimum SINR because only channel estimates are available, while simulations report true minimum SINR.
  • C. Imperfect CSI Scenario: The IRS phase-shift update uses the same quadratic-form, semidefinite-relaxation, and Dinkelbach approach with estimated channels.
  • C. Imperfect CSI Scenario: A robust expected-minimum-SINR formulation under CSI errors is left for future work because it creates a difficult stochastic optimization problem.
  • C. Imperfect CSI Scenario: The imperfect-CSI algorithm replaces true channel vectors with their MMSE estimates in the alternating optimization procedure.
  • C. Imperfect CSI Scenario: The alternating iterations stop when the fractional increase in estimated minimum SINR falls below a threshold.

V. SIMULATION RESULTS

The simulations evaluate IRS-assisted single-user and multi-user MISO performance under perfect and imperfect CSI, examining coverage, rate scaling, estimation sensitivity, training duration, and algorithmic benchmarks.

  • Single-user performance: Under perfect CSI, the IRS-assisted system covers 120m at 2.3bps/Hz, compared with about 95m without the IRS.
  • Single-user performance: Doubling the IRS elements to N = 80 increases the achieved rate by about 2bps/Hz for users near the IRS, implying an SNR gain of around 6dB.
  • Imperfect CSI: The IRS-assisted system is more sensitive to channel estimation errors than conventional MISO because it estimates N + 1 = 41 channel vectors instead of one.MMSE-DFT estimates outperform LS-DFT estimates, especially with higher channel estimation noise.
  • Training-duration trade-off: S ≈ N + 1 maximizes the minimum user rate because additional training reduces estimation error but decreases downlink transmission time.For the considered settings, S ≈ 9 is optimal when N = 8.
  • Multi-user performance: With suitable IRS sizing, the IRS-assisted system matches a 20-antenna conventional MISO system using fewer active BS antennas.Under perfect CSI, M = 12 and N = 28, or M = 15 and N = 19, achieve the same performance; imperfect CSI requires N = 48 instead of N = 28 in the reported comparison.
  • Algorithm evaluation: The proposed algorithm considerably outperforms the Centre of Means benchmark and converges in fewer than 15 iterations under both perfect and imperfect CSI.

VI. CONCLUSION

The paper proposes MMSE-DFT channel estimation and joint BS precoding, power allocation, and IRS reflect beamforming for IRS-assisted communication under imperfect CSI. Simulations show performance gains, while revealing sensitivity to estimation quality and channel-training overhead.

  • VI. CONCLUSION: The paper proposes MMSE-DFT channel estimation for the direct and IRS-assisted links, contrasting it with prior LS-based estimates.The protocol exploits the IRS's phase shifts and is evaluated for IRS-assisted wireless communication.
  • VI. CONCLUSION: Joint precoding, power allocation, and IRS reflect beamforming are designed to improve the IRS-assisted system under imperfect CSI.The related max-min SINR objective is identified as largely unexplored in IRS-assisted communication.
  • VI. CONCLUSION: Simulation results show excellent performance gains over conventional MISO communication under imperfect CSI.The comparison directly evaluates the proposed IRS-assisted system against the conventional MISO system.
  • VI. CONCLUSION: Future work should reduce channel-estimation overhead, improve robustness in high-speed environments, and study discrete phase shifts and multiple-IRS or multi-cell settings.The multi-cell extension must account for pilot contamination in channel estimation.

APPENDIX

The appendix derives linear MMSE channel estimators and their covariance expressions for the direct and IRS-assisted channels under the proposed DFT training design. The derivations use Gaussianity, independence, and DFT orthogonality.

  • APPENDIX: Because the relevant channel vectors are jointly Gaussian, the MMSE estimator is linear in the observed training signal.The estimator is obtained by minimizing the expected squared estimation error.
  • APPENDIX: The MMSE estimate of the direct channel is expressed as a linear transformation of the observed training signal.The corresponding matrix is selected through the minimum-MSE criterion.
  • APPENDIX: Independence between the IRS-assisted channel component and the direct channel supports the covariance calculations used in the estimator derivation.The appendix invokes independence when simplifying expectation terms.
  • APPENDIX: Under the DFT design, the training vectors satisfy v_tr^H v_tr = S, simplifying the derived MMSE expressions.The same DFT orthogonality relation is used in subsequent estimator calculations.
  • APPENDIX: The estimated direct and IRS-assisted channel vectors are complex Gaussian, with covariance matrices computed from their second-order moments.The appendix explicitly defines the covariance of the IRS-assisted estimate and states that the direct-channel covariance is computed similarly.
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