Intelligent Reflecting Surface-assisted MU-MISO Systems with Imperfect Hardware: Channel Estimation, Beamforming Design

I. INTRODUCTION

• The study targets the gap in IRS work that often assumes perfect CSI or hardware, and rarely models random IRS phase noise in multi-user, correlated-fading settings.Prior CE approaches can incur excessive training overhead or fail to capture inter-element channel correlation.Source: §I · p. 2

A. Contribution

• The proposed framework combines LMMSE channel estimation accounting for hardware impairments with a closed-form MRC uplink SE lower bound dependent only on large-scale statistics.The system considered has imperfect CSI and correlated Rayleigh fading.Source: A. Contribution · pp. 2–3
• Because the SE and RBM optimization use large-scale statistics, RBM updates can occur every several coherence intervals rather than every interval.The authors describe project gradient ascent as an efficient optimization method.Source: A. Contribution · pp. 2–3
• The paper examines how phase-noise distributions, transceiver impairments, and correlated fading affect uplink sum SE.The authors state that the framework accounts for random IRS phase noise in a multi-user setting, unlike the cited prior works.Source: A. Contribution · pp. 2–3

II. SYSTEM MODEL

• The model serves K single-antenna UEs through an M-antenna BS and an N-element IRS, with direct UE–BS links and an IRS deployed in line of sight to the BS.The direct links may be absent in some scenarios, such as mmWave links with severe blockage.Source: §II · p. 3, Figure 1 · p. 3

A. Channel Model

• The model uses known BS and IRS spatial-correlation matrices for correlated Rayleigh fading; multi-user operation requires rank(H1) ≥ K, achievable by IRS placement near the BS or deterministic scattering.The authors note that their narrowband model could be extended to wideband channels as future work.Source: A. Channel Model · pp. 3–4

B. Ideal Uplink Signal Model

• The ideal uplink signal model defines the IRS reflect beamforming matrix as diagonal, representing the response of its N elements.The model assumes unit reflection amplitude for each element.Source: B. Ideal Uplink Signal Model · p. 4

C. Hardware Impairments

• The framework distinguishes aggregate additive transceiver impairments from impairments arising at the IRS passive elements.These categories are denoted T-HWIs and IRS-HWIs, respectively.Source: C. Hardware Impairments · p. 4

1) T-HWIs:

• Transmitter and receiver distortions are modeled as Gaussian, with average powers proportional to the average transmitted and received signals, respectively.The model uses impairment-severity parameters and assumes the transmitter parameter κUE is identical across UEs.Source: 1) T-HWIs · p. 4

2) IRS-HWIs:

• The IRS impairment model represents the N element-wise phase errors with independent random variables drawn from a symmetric circular distribution with zero mean direction.Uniform and zero-mean Von Mises distributions are identified as commonly used choices.Source: 2) IRS-HWIs · p. 5

D. Realistic Uplink Signal Model with HWIs

• The analysis assumes unity IRS reflection amplitude, leaving phase-dependent reflection amplitude for separate study.The authors identify phase-dependent amplitude as a hardware limitation requiring a separate analysis.Source: D. Realistic Uplink Signal Model with… · p. 5
• The effective-channel covariance captures IRS phase-noise effects through the IRS correlation matrix and the phase-error characteristic function.For independent identically distributed symmetric phase errors, the off-diagonal correlation terms are scaled by m^2.Source: D. Realistic Uplink Signal Model with… · p. 5
• For uniform phase errors, the characteristic function m is zero, removing phase-noise dependence from the IRS covariance term.In this case, the effective IRS covariance is the identity matrix.Source: D. Realistic Uplink Signal Model with… · p. 5, 2) IRS-HWIs · p. 5
• With uncorrelated Rayleigh fading, the SE has no dependence on the RBM and cannot be optimized; the IRS may still contribute an additional, expectedly weak, received signal.The passage notes this additional signal is beneficial when there is no direct signal, though such cases are described as difficult to encounter in practice.Source: D. Realistic Uplink Signal Model with… · p. 5

III. CE WITH HWIS

• The proposed method estimates the overall channel in one training phase, rather than using N + 1 subphases, reducing training overhead compared with.It estimates the overall channel instead of estimating direct and cascaded channels separately, and obtains the estimated cascaded-channel vector and its covariance.Source: §III · pp. 5–6
• The LMMSE estimate and estimation error are uncorrelated, zero-mean vectors with covariances Ψk = RkQkRk and ˜Ψk = Rk − Ψk, respectively.They are not independent or jointly complex Gaussian because the effective distortion noises are non-Gaussian.Source: §III · p. 6
• The estimate and error covariances depend on transceiver and IRS hardware impairments, but become independent of phase errors and the reflect phase matrix under uniformly distributed phase errors.With perfect hardware impairments, the LMMSE estimator coincides with the optimal MMSE estimator; the channel-estimation method can also be generalized to independent Rayleigh fading.Source: §III · p. 6
• Increasing transceiver hardware impairments increases NMSE, while uniformly distributed phase errors make NMSE independent of the reflect beamforming matrix.In that phase-error case, the estimation error cannot be optimized through the reflect beamforming matrix; a similar observation holds under uncorrelated Rayleigh fading.Source: §III · p. 6

A. Achievable SE

• The use-and-then-forget bound gives a lower bound on average uplink spectral efficiency by treating inter-user interference and distortion noise as worst-case uncorrelated additive noise.The bound is described for systems with more than eight BS antennas and can be used with different channel estimators and decoders.Source: A. Achievable SE · p. 6

E {MUIik} +

• The analysis uses MRC with the estimated channel as the combining vector because the conventional MMSE expectations lack closed forms without deterministic-equivalent analysis, complicating reflection-coefficient optimization.The authors leave applying MMSE decoding with statistical CSI to future work.Source: E {MUIik} + · p. 7
• The achievable uplink SINR with MRC is derived in closed form for imperfect CSI and hardware impairments, and depends only on slowly varying large-scale statistics.Transceiver impairments enter directly through κBS and κUE, while phase noise acts through the covariance matrices.Source: E {MUIik} + · p. 7

• Uniformly distributed phase noise removes RBM dependence from covariance matrices, so SINR/SE cannot be optimized and the IRS does not serve its purpose.Source: B. IRS Design Problem: Formulation and… · p. 7
• Lower-quality transceiver and IRS hardware degrades QoS, for which a better reflect-beamforming design is suggested to compensate and improve performance.Source: B. IRS Design Problem: Formulation and… · p. 7
• Projected gradient ascent updates phase vectors along the SE gradient, uses backtracking for step size, and projects each iterate onto the unit-modulus constraint.Source: B. IRS Design Problem: Formulation and… · p. 7
• The ascent converges to a local maximum because the objective is bounded by the power constraint and increases along the gradient direction.Source: B. IRS Design Problem: Formulation and… · p. 7
• The gradient uses simple matrix operations, with complexity depending on K, M, and N and higher square dependence on M and N.Source: B. IRS Design Problem: Formulation and… · p. 8
• The method requires closed-form SE and, because phase optimization is non-convex, provides a locally optimal rather than global solution.Source: B. IRS Design Problem: Formulation and… · p. 8
• Large-scale-statistics dependence permits optimization every several coherence intervals; faster-changing statistics in fast-speed scenarios require more frequent updates.Source: B. IRS Design Problem: Formulation and… · p. 8

A. Simulation Setup

• NMSE reaches impairment-dependent error floors as SNR rises, while perfect hardware avoids a floor; Monte Carlo simulations verify the analytical results.Even mild impairments produce an NMSE floor after 20 dB; the NMSE does not depend on the reflect-beamforming matrix in the uniform-phase-noise or uncorrelated-fading cases considered.Source: Figure 2 · p. 9, A. Simulation Setup · p. 9
• With imperfect hardware, sum SE increases with IRS size but saturates; with ideal hardware, it increases unboundedly, and more severe impairments cause greater degradation.The proposed channel-estimation method also yields higher sum SE than the comparison method at κBS = κUE = 0.2582, which the authors attribute to the latter’s high training overhead.Source: Figure 3 · p. 9, A. Simulation Setup · p. 9
• As BS antennas increase, sum SE rises without limit under ideal hardware but reaches a ceiling with impairments; analytical results coincide with Monte Carlo simulations.Higher SNR increases sum SE, with faster saturation at 20 dB.Source: §V · p. 8, A. Simulation Setup · p. 9, Figure 4 · p. 10
• With more IRS elements, BS distortion becomes negligible at large N, but imperfect CSI yields lower sum SE and a wider gap from ideal hardware as N grows.For both perfect and imperfect CSI, sum SE increases with IRS size.Source: Figure 5 · p. 10, A. Simulation Setup · p. 10
• Sum SE initially grows almost linearly with UE count, then its growth slows as multi-user interference and received distortion increase.The figure varies IRS size and transceiver hardware impairments alongside UE count.Source: Figure 6 · p. 10, A. Simulation Setup · p. 10
• Correlated fading lowers sum SE relative to uncorrelated fading; uniform IRS phase noise gives the lowest performance, while Von Mises phase noise permits phase adjustment to improve it.At concentration κ̃θ = 0, the Von Mises case coincides with the uniform-phase-noise case; even under the worst-case phase noise, the IRS outperforms the no-IRS case.Source: Figure 7 · p. 10, A. Simulation Setup · pp. 10–11
• Increasing transceiver hardware impairments reduces sum SE, and at severe impairment levels, changing BS antenna or IRS-element counts does not affect performance.Under the compared variations, IRS-element count affects sum SE more than BS-antenna count.Source: Figure 8 · p. 11, A. Simulation Setup · p. 11
• The proposed reflect-beamforming algorithm converges in seven iterations for M = 20 and N = 60, and different initializations yield nearly the same sum SE across 30 channel realizations.Larger BS and IRS sizes require more iterations and increase per-iteration complexity; the tested initialization uses s0 = exp(jπ/2)1N.Source: Figure 9 · p. 11, A. Simulation Setup · pp. 11–12

VI. CONCLUSION

• The study treats transceiver and IRS hardware impairments in an IRS-assisted MU-MISO system with imperfect CSI and correlated Rayleigh fading.It also proposes a low-computational-cost reflect-beamforming matrix optimization methodology.Source: §VI · p. 12
• The authors obtain an LMMSE channel estimate accounting for both transceiver and IRS hardware impairments.Source: §VI · p. 12
• The uplink achievable sum spectral efficiency with MRC is derived in closed form and depends only on large-scale statistics.The paper uses this result to optimize the reflect-beamforming matrix.Source: §VI · p. 12
• The paper evaluates how hardware impairments at the transceiver and IRS affect system spectral efficiency and interact with other system parameters.Source: §VI · p. 12

APPENDIX A PROOF OF PROPOSITION 1

• The LMMSE channel estimate is expressed as a linear transformation of the received vector, with its matrix chosen by minimizing a trace criterion.Source: Appendix A · p. 12
• The derivation evaluates terms using the stated lack of correlation between additive distortions, receiver noise, and the overall channel.Source: Appendix A · p. 12
• The estimator is formed by substituting the derived terms, after which the covariance matrix of the estimated channel is obtained.Source: Appendix A · p. 12

APPENDIX B PROOF OF THEOREM 1

• The desired-signal contribution is evaluated for finite M, including expectations over transmit distortion and a zero-mean distortion term.Source: Appendix B · pp. 12–13
• The multiuser interference term is expressed using independence between the two random vectors.Source: Appendix B · p. 12
• The proof separately derives beamforming-uncertainty power and the additive transmit-distortion contribution from all user equipments.Source: Appendix B · p. 12
• The receive-distortion contribution at the base station is evaluated using MRC, with its terms separated into I1 and I2.Source: Appendix B · p. 13
• The I1 derivation uses diagonal-matrix expansions and independence between the estimated channel and its estimation error.Source: Appendix B · p. 13
• The I2 term is rewritten using trace and linearity, then evaluated through separate expectations under independence between user indices.Source: Appendix B · p. 13

APPENDIX C PROOF OF PROPOSITION 2

• The rate gradient is taken with respect to each reflect-beamforming phase, and the SINR derivative uses the quotient rule because it is a fraction of trace terms.Source: Appendix C · p. 13
• The derivation includes the inverse-matrix derivative for Qk and substitutes the resulting derivatives into the preceding expression.Source: Appendix C · pp. 13–14
• A lemma for trace expressions is applied while deriving terms that depend on the reflect-beamforming phases.Source: Appendix C · p. 14
• The proof differentiates the desired-signal and interference components separately before combining their derivatives.Source: Appendix C · p. 14