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

• IRSs passively reshape electromagnetic waves through adjustable element phase shifts without generating new signals or adding power consumption.Source: §I · p. 1
• Perfect CSI assumptions are impractical because IRSs lack radio resources and signal-processing capability for pilot transmission, reception, and channel estimation.Source: §I · p. 2, A. Transmission Model · p. 4
• The paper proposes DFT-MMSE estimation for direct and cascaded channels using multiple training sub-phases and prior large-scale fading statistics.Source: §I · p. 2
• The BS and IRS jointly design precoding, power allocation, and phase shifts to maximize minimum user SINR under transmit-power and unit-modulus constraints.Source: §I · pp. 2–3
• The proposed alternating-optimization algorithm converges, while simulations show IRS-assisted communication is efficient but sensitive to CSI errors.Source: §I · p. 3

• Spatial correlation among IRS elements is modeled because independent Rayleigh or Rician channels are mainly practical with wide element spacing and rich scattering.Source: B. Channel Model · p. 4
• The IRS-to-user and direct BS-to-user channels use correlation matrices, fast-fading vectors, and path-loss factors.Source: B. Channel Model · p. 4
• A high-rise IRS near the BS is modeled with a likely rank-one line-of-sight BS-to-IRS channel.Source: B. Channel Model · p. 4
• 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.Source: B. Channel Model · p. 5
• The BS-to-IRS channel entries depend on carrier wavelength, array separations, path loss, and line-of-sight departure and arrival angles.Source: B. Channel Model · p. 5

• The BS estimates IRS-assisted channels because the passive IRS cannot transmit, receive, or process pilot symbols.Source: §III · p. 5
• The proposed protocol computes MMSE estimates from received pilot sequences across multiple sub-phases, each using an optimal IRS reflect-beamforming vector.Source: §III · p. 5
• The estimation study analytically compares normalized MSE for LS and MMSE methods and numerically compares MSE and BER performance.Source: §III · p. 5

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.Source: A. Proposed MMSE-DFT Channel… · p. 6
• The coherence period is divided into uplink training and downlink transmission, with S training sub-phases of duration τS = τC/S.Source: A. Proposed MMSE-DFT Channel… · p. 6
• 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.Source: A. Proposed MMSE-DFT Channel… · pp. 6–7
• At least S ≥ N + 1 sub-phases are required for the training matrix to have full column rank and admit the left pseudo-inverse.Source: A. Proposed MMSE-DFT Channel… · p. 6
• The BS applies the MMSE estimator to observations associated with the direct channel and each cascaded-channel column, producing estimates with uncorrelated errors.Source: A. Proposed MMSE-DFT Channel… · p. 7
• Under DFT training, MMSE estimates do not depend on cross-correlation between IRS elements, so the BS does not require knowledge of RIRS.Source: A. Proposed MMSE-DFT Channel… · p. 7
• 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.Source: A. Proposed MMSE-DFT Channel… · p. 7

• 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.Source: B. NMSE Comparison with Least Squares… · p. 7
• The analysis derives normalized MSE expressions for LS and MMSE estimates of direct and IRS-assisted channel vectors.Source: B. NMSE Comparison with Least Squares… · p. 8
• The MMSE estimate approaches NMSE 1 in unfavorable conditions, corresponding to estimation-error power matching true-channel power and effectively isotropic beamforming.Source: B. NMSE Comparison with Least Squares… · p. 8
• 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.Source: B. NMSE Comparison with Least Squares… · p. 8
• 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.Source: B. NMSE Comparison with Least Squares… · p. 8
• 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.Source: B. NMSE Comparison with Least Squares… · p. 8

• Simulated NMSE matches the theoretical expressions, and MMSE-DFT achieves lower NMSE than LS-DFT, especially at moderate-to-high noise variance.Source: C. Performance Evaluation of the… · pp. 8–9
• 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.Source: C. Performance Evaluation of the… · p. 9
• 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.Source: C. Performance Evaluation of the… · p. 9
• Increasing sub-phases S reduces NMSE, but training consumes downlink time and causes a rate-loss factor of 1 − SτS/τ.Source: C. Performance Evaluation of the… · p. 10
• 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.Source: C. Performance Evaluation of the… · p. 11
• Under estimation errors, MMSE-DFT is nearly 8 dB better than LS-DFT at BER 10^-6.Source: C. Performance Evaluation of the… · p. 11
• 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.Source: C. Performance Evaluation of the… · p. 11

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.Source: §IV · p. 11
• The IRS-assisted link’s double path loss can compromise performance improvements because it combines BS-to-IRS and IRS-to-user path losses.Source: §IV · p. 11
• 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.Source: §IV · p. 11
• The performance metric is the max-min rate, balancing system throughput and user fairness.User rate is R_k = log2(1 + γ_k).Source: §IV · p. 11
• Maximizing the minimum rate is equivalent to maximizing the minimum SINR because the logarithm is monotonically increasing.Source: §IV · p. 11

A. Problem Formulation

• The BS seeks optimal precoding vectors, allocated powers, and IRS reflect beamforming as the solution of a max-min SINR problem.Source: A. Problem Formulation · p. 11
• The IRS phase shifts satisfy unit-modulus constraints, |v_n| = 1 for n = 1, ..., N.Source: A. Problem Formulation · p. 11
• The max-min SINR objective is non-convex because precoding vectors, allocated powers, and phase shifts are coupled.Source: A. Problem Formulation · p. 11
• The proposed alternating optimization iteratively optimizes BS precoding and power allocation with IRS phase shifts until convergence.Source: A. Problem Formulation · p. 11
• For fixed IRS phase shifts, the precoding and power-allocation subproblem is solved by the optimal linear precoder.Source: A. Problem Formulation · p. 12

B. Problem Solution

• The IRS phase-shift objective is transformed into quadratic forms using an auxiliary variable.Source: B. Problem Solution · p. 12
• Semidefinite relaxation replaces the non-convex rank-one constraint with a positive-semidefinite matrix constraint of arbitrary rank.Source: B. Problem Solution · p. 12
• Generalized Dinkelbach’s algorithm solves the resulting max-min fractional problem globally with limited complexity.Source: B. Problem Solution · p. 12
• If the relaxed solution has rank one, the optimal IRS vector is extracted directly; otherwise, Gaussian randomization can produce a candidate vector.Source: B. Problem Solution · p. 12
• In extensive simulations, the relaxed problem was always observed to have a rank-one solution, making the extracted vector optimal for the IRS subproblem.Source: B. Problem Solution · p. 13
• The alternating algorithm’s objective is bounded and non-decreasing across iterations, ensuring convergence, but global optimality is not claimed.Source: B. Problem Solution · p. 13

C. Imperfect CSI Scenario

• For fixed IRS phases, the BS optimizes the estimated minimum SINR using the estimated channel vectors.Source: C. Imperfect CSI Scenario · p. 13
• The BS optimizes estimated minimum SINR rather than true minimum SINR because only channel estimates are available, while simulations report true minimum SINR.Source: C. Imperfect CSI Scenario · pp. 13–14
• The IRS phase-shift update uses the same quadratic-form, semidefinite-relaxation, and Dinkelbach approach with estimated channels.Source: C. Imperfect CSI Scenario · pp. 13–14
• A robust expected-minimum-SINR formulation under CSI errors is left for future work because it creates a difficult stochastic optimization problem.Source: C. Imperfect CSI Scenario · p. 13
• The imperfect-CSI algorithm replaces true channel vectors with their MMSE estimates in the alternating optimization procedure.Source: C. Imperfect CSI Scenario · p. 14
• The alternating iterations stop when the fractional increase in estimated minimum SINR falls below a threshold.Source: C. Imperfect CSI Scenario · p. 14

Single-user performance

• Under perfect CSI, the IRS-assisted system covers 120m at 2.3bps/Hz, compared with about 95m without the IRS.Source: §V · p. 15
• 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.Source: §V · p. 15

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.Source: §V · p. 15

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.Source: §V · p. 15

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.Source: §V · pp. 15–16

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.Source: §V · p. 16

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.Source: §VI · p. 16
• 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.Source: §VI · p. 16
• 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.Source: §VI · p. 17
• 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.Source: §VI · p. 17

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.Source: A. Proof of Lemma 1 · p. 17
• 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.Source: A. Proof of Lemma 1 · p. 17
• 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.Source: A. Proof of Lemma 1 · p. 17
• 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.Source: A. Proof of Lemma 1 · p. 17, B. Proof of Lemma 2 · p. 17
• 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.Source: A. Proof of Lemma 1 · p. 17, B. Proof of Lemma 2 · p. 17