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Intelligent Reflecting Surface Enhanced Wireless Network: Two-timescale Beamforming Optimization

Ming-Min Zhao, Qingqing Wu, Min-Jian Zhao, Rui Zhang

arXiv:1912.01818v2cs.IT

TL;DR

IRS beamforming must overcome the practical difficulty of acquiring instantaneous CSI for many passive reflecting elements. This paper proposes two-timescale optimization using statistical CSI for long-term IRS phases and effective instantaneous CSI for short-term AP precoding, with PDD and SSCA algorithms for single-user and multiuser cases. Simulations validate the protocol and examine robustness and channel-correlation effects.

  • Problem

    Obtaining accurate instantaneous CSI for IRS-associated links is difficult because of passive operation and the large number of reflecting elements, causing high training and signaling overhead.

  • Method

    The paper optimizes discrete IRS phase shifts from slowly varying statistical CSI and designs short-term AP precoding from users’ effective instantaneous CSI using PDD and SSCA algorithms.

  • Results

    Choosing c anywhere in [0.7, 0.99] produces no significant performance variation, indicating robustness of the proposed PDD-based algorithm.

  • Takeaways & Limitations

    The proposed protocol achieves significant sum-rate gains with statistical CSI and discrete phase shifters compared with operation without an IRS.

Abstract

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Intelligent reflecting surface (IRS) has drawn a lot of attention recently as a promising new solution to achieve high spectral and energy efficiency for future wireless networks. By utilizing massive low-cost passive reflecting elements, the wireless propagation environment becomes controllable and thus can be made favorable for improving the communication performance. Prior works on IRS mainly rely on the instantaneous channel state information (I-CSI), which, however, is practically difficult to obtain for IRS-associated links due to its passive operation and large number of elements. To overcome this difficulty, we propose in this paper a new two-timescale (TTS) transmission protocol to maximize the achievable average sum-rate for an IRS-aided multiuser system under the general correlated Rician channel model. Specifically, the passive IRS phase-shifts are first optimized based on the statistical CSI (S-CSI) of all links, which varies much slowly as compared to their I-CSI, while the transmit beamforming/precoding vectors at the access point (AP) are then designed to cater to the I-CSI of the users' effective channels with the optimized IRS phase-shifts, thus significantly reducing the channel training overhead and passive beamforming complexity over the existing schemes based on the I-CSI of all channels. For the single-user case, a novel penalty dual decomposition (PDD)-based algorithm is proposed, where the IRS phase-shifts are updated in parallel to reduce the computational time. For the multiuser case, we propose a general TTS optimization algorithm by constructing a quadratic surrogate of the objective function, which cannot be explicitly expressed in closed-form. Simulation results are presented to validate the effectiveness of our proposed algorithms and evaluate the impact of S-CSI and channel correlation on the system performance.

I. INTRODUCTION

The paper addresses the difficulty of obtaining instantaneous CSI for IRS-associated links by proposing two-timescale beamforming based on slowly varying statistical CSI and effective instantaneous CSI. It develops separate algorithms for single-user and multiuser optimization and evaluates their performance under correlated Rician channels.

  • Motivation: IRS can improve spectral efficiency with low energy and hardware cost through many passive reflecting elements that shape signal propagation.Its asymptotic power gain can scale as N^2, including with practical discrete phase shifters up to a constant loss.
  • Motivation: Accurate AP-IRS and IRS-user CSI is difficult to obtain because IRSs typically have many reflecting elements, creating training and signaling challenges.Existing I-CSI-based beamforming also incurs high processing complexity and overhead.
  • Proposed protocol: The proposed TTS scheme optimizes long-term discrete IRS phase shifts from S-CSI and short-term AP precoding from effective I-CSI.The protocol uses measured channel statistics and updates IRS settings less frequently than transmit precoding.
  • Algorithms: For the single-user case, the achievable average-rate problem is converted into a deterministic non-convex problem and solved with a PDD-based algorithm.The method updates IRS phase shifts in parallel rather than using SDR or BCD-based successive refinement.
  • Algorithms: For the multiuser case, an iterative SSCA algorithm constructs quadratic surrogates because optimal precoding cannot be expressed explicitly as a function of IRS phase shifts.The surrogate-based procedure makes the stochastic optimization tractable.
  • Contributions and evaluation: The work reports the first TTS beamforming optimization study for IRS-aided communication and investigates Rician deterministic components and channel correlation numerically.The simulations evaluate the proposed protocol and algorithms and examine how channel statistics affect performance.

II. SYSTEM MODEL AND PROBLEM FORMULATION

The system is a multiuser MISO downlink in which an AP communicates with low-mobility, single-antenna users through an IRS. Its formulation models discrete passive phase shifts, correlated Rician channels, received SINR, and achievable rates.

  • A. System Model: The considered downlink has an AP with M antennas, an IRS with N reflecting elements, and K single-antenna users near the IRS.A smart controller coordinates the IRS and AP over a separate backhaul link; multiple IRS reflections are ignored because of double path loss.
  • A. System Model: The IRS is represented by an N × N diagonal reflection coefficient matrix whose element amplitudes lie in [0, 1] and phases lie in [0, 2π).The matrix specifies the passive beamforming action applied by the reflecting elements.
  • A. System Model: The transmitted signal uses user information symbols and AP precoding vectors w_k, with the symbols modeled as independent unit-variance CSCG variables.The formulation restricts IRS phase shifts to discrete values and notes that independently controlling amplitudes and phases is costly.
  • A. System Model: The continuous phase-shift case is the limiting ideal obtained when the number of phase-shift levels L = 2^Q grows without bound.For Q →∞, each reflecting element can use any phase in [0, 2π).
  • A. System Model: Each user’s performance is characterized by its received SINR and achievable rate r_k = log2(1 + SINR_k) in bits/s/Hz.The SINR is derived from the received-signal model and the AP/IRS beamforming variables.
  • B. Channel Model: The AP-user, AP-IRS, and IRS-user links are modeled as spatially correlated Rician fading channels containing both LoS and NLoS components.Correlation represents limited scattering and closely spaced antennas or reflecting elements.
  • B. Channel Model: The channel model separates deterministic components from small-scale fading and uses correlation matrices for the AP and IRS sides of the links.These components capture Rician structure, scattering, and antenna or reflecting-element configurations.

C. Transmission Protocol

The proposed protocol separates slowly varying statistical CSI from rapidly varying effective instantaneous CSI: the IRS fixes phase shifts over a longer interval, while the AP updates precoding each slot. This hierarchy reduces the complexity and overhead associated with acquiring full IRS-linked CSI.

  • Transmission hierarchy: The protocol assumes all links’ S-CSI remains constant across Ts ≫ 1 time slots within the considered interval.The small-scale fading coefficients vary within the interval, whereas the statistical information is treated as unchanged.
  • Transmission hierarchy: The AP dynamically designs short-term transmit precoding vectors in each slot using effective I-CSI with fixed IRS phase shifts.The effective channels are easier to acquire than the full AP–IRS and IRS–user channel ensemble.
  • Transmission hierarchy: The IRS uses estimated statistical information to set phase shifts for subsequent slots, regardless of instantaneous channel variations.This remains valid as long as the S-CSI is unchanged.
  • Transmission phases: The first phase places the IRS in sensing mode to estimate channel statistical information using dedicated sensors or receiving circuits and uplink/downlink pilots or data.The AP simultaneously serves users through direct channels during the relevant initial sub-slot.
  • Limitations: Efficient deployment of IRS sensors and S-CSI estimation from low-resolution sensing measurements remain open problems.The paper identifies this as future work rather than resolving it within the protocol.

D. Problem Formulation

The paper formulates TTS beamforming as a weighted average sum-rate maximization that jointly chooses long-term discrete IRS phases and short-term AP precoding. The resulting problem is difficult because the variables are coupled, average rates lack general closed forms, and the discrete single-user case is non-convex.

  • D. Problem Formulation: The objective maximizes users’ average weighted sum-rate by jointly optimizing short-term AP precoding and long-term IRS reflect beamforming under an AP power constraint.The inner optimization adapts precoding to instantaneous channel realizations, while the outer optimization selects IRS phase shifts.
  • D. Problem Formulation: The inner rate maximization operates over short-term precoding for each realization, whereas the outer rate maximization operates over long-term IRS phase shifts.The expectation covers random channel realizations within the considered time interval.
  • D. Problem Formulation: The formulation uses user weights αk and a discrete IRS phase-shift set constructed as the Cartesian product of N identical per-element sets.The weights represent user priorities, while P denotes the total transmit power budget.
  • D. Problem Formulation: The feasible precoding set contains channel-dependent transmit vectors satisfying the total power constraint, and the achievable rates are defined by expectations over instantaneous channels.The compact formulation represents these expected user rates as an average rate vector.
  • D. Problem Formulation: The optimization is challenging because active precoding and passive phase shifts are intricately coupled and average-rate expressions are generally unavailable in closed form.These difficulties prevent straightforward optimization over either variable block.
  • D. Problem Formulation: Even for K = 1, the problem is a mixed-integer non-linear program, and no generally efficient method solves the non-convex formulation optimally.The paper therefore develops separate sub-optimal algorithms for single-user and multiuser cases.
  • III. SINGLE-USER CASE: For a fixed IRS phase-shift matrix, maximum-ratio transmission is optimal in the single-user case.This property supports reducing the single-user formulation before optimizing the IRS phases.
  • III. SINGLE-USER CASE: The single-user achievable rate is upper-bounded and then approximated by a deterministic optimization after removing the monotonic logarithm and constant terms.The phase-shift vector is represented through v, with additional quantities defined from channel statistics and correlation matrices.

A. Proposed PDD-based Algorithm

The PDD algorithm introduces an auxiliary phase variable to decouple IRS-element updates, then alternates primal updates inside an inner loop with dual and penalty updates in an outer loop. This supports parallel discrete phase optimization and yields a high-quality suboptimal solution under the stated convergence conditions.

  • Auxiliary-variable reformulation: The algorithm introduces an auxiliary variable u satisfying u = v, enabling parallel optimization of the IRS phase variables.Each auxiliary component is constrained to a discrete unit-modulus phase representation.
  • Augmented-Lagrangian formulation: The PDD method solves an augmented-Lagrangian problem with penalty parameter ρ and dual variables λ associated with the equality constraint v = u.The added constraint ∥v∥2 ≤ N preserves optimality because each phase component satisfies |vn| ≤ 1.
  • Inner-loop updates: The inner loop partitions the variables into v and u blocks and iteratively optimizes them using block successive upper-bound minimization.The v-subproblem is approximated through a first-order Taylor expansion when needed.
  • Inner-loop updates: The approximated v-subproblem is a convex QCQP with one constraint, solved using first-order optimality conditions.Without the norm constraint, boundedness would require a restriction involving the largest eigenvalue of Φ.
  • Inner-loop updates: The decoupled u-subproblem updates all phase shifts in parallel by computing continuous phases and mapping each to the nearest discrete phase value.This avoids the one-by-one phase updates required by the compared successive-refinement approach.
  • Outer-loop updates: The outer loop updates the dual variable through a dual ascent step and decreases the penalty parameter as ρ ← cρ, with c < 1.The penalty schedule affects conditioning and convergence speed: decreasing ρ too quickly can cause poor behavior, while decreasing it too slowly can slow convergence.
  • Convergence: For continuous phase shifts, Algorithm 1 is guaranteed to converge to stationary solutions; for discrete shifts, it converges to a high-quality suboptimal solution as the equality constraint is enforced.The discrete-case equality v = u is satisfied as ρ tends toward zero in the stated formulation.
  • Algorithm procedure: Algorithm 1 repeats inner primal updates and outer dual-penalty updates until objective decrease and constraint-violation thresholds or maximum iteration limits are reached.The procedure initializes v0, u0, and c before entering the nested loops.

B. Complexity Analysis

The proposed PDD algorithm has quadratic dependence on the number of IRS elements per inner update, producing lower worst-case complexity than SDR.

  • B. Complexity Analysis: The proposed algorithm has overall complexity O(IoIiN^2), compared with O(N^6.5) for SDR and O(IN^2) for successive refinement.Here, Io and Ii are the maximum outer and inner iteration counts, while I is the iteration count for convergence of the comparison method.

IV. MULTIUSER CASE

The multiuser TTS algorithm alternates long-term IRS phase-shift optimization using statistical channel samples with short-term AP precoding based on effective instantaneous channels. It uses surrogate optimization, WMMSE updates, and projection to handle tractability and discrete phase shifts.

  • Long-term optimization: The SSCA algorithm updates long-term IRS phase shifts through a surrogate rate-maximization problem based on randomly generated channel samples.The samples are generated from the known S-CSI and line-of-sight channel components.
  • Short-term optimization: At each time slot, the AP acquires effective fading channels with fixed IRS phase shifts and designs short-term precoding vectors using WMMSE.The precoders are optimized from the effective user channels rather than the individual underlying links.
  • Surrogate construction: The long-term objective and its gradient are approximated iteratively using sampled achievable rates and Jacobian updates.The surrogate and gradient approximations converge to the corresponding objective and gradient under the stated iterative procedure.
  • Phase-shift update: The relaxed continuous phase-shift problem is solved as a convex quadratic optimization, with independently updated variables and closed-form solutions.After optimization, each IRS entry is projected onto the feasible discrete phase-shift set to recover a unit-modulus solution.
  • Discrete phase shifts: Q ≥2-bit discrete phase shifts incur negligible performance loss in the reported simulations.The continuous relaxation is followed by projection onto the discrete feasible set.
  • Convergence: The algorithm converges almost surely to stationary solutions of the relaxed continuous-phase problem when WMMSE supplies the transmit precoders.The convergence statement concerns the outer rate-maximization problem with amplitudes relaxed to [0, 1].

C. Complexity Analysis

The multiuser algorithm’s complexity is dominated by WMMSE precoder updates for generated long-term channel samples and the resulting matrix inversions.

  • Complexity drivers: The main computational cost comes from computing WMMSE precoders for the channel samples used in long-term optimization.For each generated sample, WMMSE updates the corresponding transmit precoding vectors.
  • WMMSE updates: The WMMSE matrix-inversion cost is O(JKM 3), where J is the number of WMMSE iterations.The stated complexity is for updating the multiuser transmit precoders.
  • Overall update: The long-term IRS phase-shift update has complexity O(I(THJKM 3 + KNM)).I denotes the phase-shift optimization iterations, while TH is the number of generated channel samples.

V. SIMULATION RESULTS

The simulations evaluate the proposed algorithms under distance-dependent path loss and correlated Rician fading, using distinct parameter settings for single-user and multiuser cases. Results are averaged over 2000 independent channel realizations.

  • Evaluation setup: The simulation section evaluates the performance of the proposed algorithms and examines related system insights.The supplied passage introduces the numerical evaluation without reporting a specific outcome.
  • Channel model: The path-loss model uses reference distance D0 = 1 meter and link-specific path-loss exponents.The exponents are αAu = 3.4, αAI = 2.2, and αIu = 3.
  • Channel correlation: The simulations model spatial correlation through correlation coefficients bounded by 0 ≤rd ≤1 and corresponding horizontal and vertical correlation matrices.The user-link correlation matrices are modeled analogously with their corresponding coefficients.
  • Simulation parameters: The single-user configuration uses M = 4 and N = 40, while the multiuser configuration uses M = 6, K = 4, TH = 10, and Ts = 2000.The listed settings also include separate Rician factors and algorithm parameters for the two cases.
  • Evaluation protocol: All simulation results are averaged over 2000 independent channel realizations.The averaging applies to the reported numerical results.

A. Single-User Case

The single-user evaluation examines convergence, distance, Rician fading, and channel-correlation effects under several IRS phase-shift and CSI schemes. It also introduces the multiuser simulation setup and compares Algorithm 2's convergence with batch alternating optimization.

  • Simulation setup: The single-user setup places the user along the line (2 m, d, 0), with d representing the AP-user distance.
  • Convergence: The PDD-based algorithm converges despite initial objective fluctuations caused by temporary constraint violations, while BCD is monotonically convergent.Increasing penalties force the constraint violation ∥v − u∥∞ toward the predefined accuracy.
  • AP-user distance: With 1-bit phase shifters and S-CSI, IRS achieves a significantly higher average rate than both no IRS and random phase shifts near the IRS.The gap from continuous phase shifts is reduced with higher-resolution phase shifters such as Q = 2 and Q = 3.
  • Rician factor: Average rate improves with the Rician factor for both S-CSI and I-CSI, while the S-CSI/I-CSI gap approaches a constant at sufficiently large factors.The PDD-based algorithm outperforms naive and single-timescale schemes because those schemes do not fully exploit S-CSI and I-CSI, respectively.
  • Correlation coefficients: Under I-CSI, increasing the IRS correlation coefficient r_r improves average rate, whereas increasing r_r,u does not necessarily do so.For example, configurations with r_r,u = 1 can be inferior to configurations with lower r_r,u.
  • Multiuser setup: The multiuser setup places four users on a semicircle centered at (0, 50 m, 0) with radius d_1 = 3 m, assigning different IRS-user correlation levels.
  • Multiuser convergence: Algorithm 2 and batch alternating optimization reach similar performance after convergence, while adjustable-amplitude Algorithm 2 converges faster than its unit-amplitude version.Adjustable amplitudes enlarge the feasible region explored during the first iterations.

1) Impact of the Rician factor:

The multiuser simulations examine how IRS correlation affects average sum-rate and per-user rates under different transmit-power regimes. Correlation can improve sum-rate while concentrating service on fewer users, with its effects depending on channel conditions and SNR.

  • 3) Performance comparison with different IRS-user correlation levels:: At P = 5 dBm, larger IRS-user correlation coefficients produce higher achievable average rates for the corresponding users.The user with larger rr,k consistently outperforms users with smaller rr,k.
  • 3) Performance comparison with different IRS-user correlation levels:: When rr = 1, user 4 with rr,4 = 1 achieves the highest rate, while user 1 with rr,1 = 0 has an almost-zero rate.The limited total transmit power favors allocating more power to the user whose statistical CSI is more exploitable.
  • 2) Impact of the correlation coefficient rr:: Larger rr increases sum-rate but may reduce spatial multiplexing and make user fairness difficult to guarantee.The passive beamforming design favors only a small number of users under stronger correlation.
  • 3) Performance comparison with different IRS-user correlation levels:: At high transmit power, the distribution of each user's achievable rate changes with the IRS receive correlation coefficient rr.For rr = 0, user 1 benefits from higher channel diversity; the supplied passage begins describing the contrasting rr = 1 case.
  • 1) Impact of the Rician factor:: The simulations investigate the effects of IRS correlation coefficients and deterministic Rician components on multiuser system performance.The broader conclusion reports distinct impacts from AP-IRS and IRS-user correlations under different SNR regimes.

APPENDIX

The appendix derives a deterministic approximation for the single-user IRS phase-shift optimization and characterizes an optimal phase configuration in the Rayleigh-fading case.

  • APPENDIX: Jensen's inequality upper-bounds the conditional achievable rate by a logarithm involving the expected received power.The bound converts the stochastic rate expression into an expectation-based objective.
  • APPENDIX: Combining the expanded expectation terms yields an approximation of the original phase-shift problem by problem (12).The derivation uses the change of variables and matrix decompositions introduced in the appendix.
  • APPENDIX: With Rayleigh fading on both AP-IRS and IRS-user channels, the IRS phase shifts depend only on the relevant correlation matrices.Under non-negative real correlation coefficients, the resulting objective has a simplified structure.
  • APPENDIX: The optimal solution of problem (34) sets every unit-modulus phase variable to the same arbitrary phase.Equality holds when all phase angles are equal and every magnitude is one.

C. Jacobian Matrix of the Instantaneous Rate w.r.t. v∗

The appendix defines the Jacobian of the instantaneous rate vector with respect to the conjugate IRS phase variable for subsequent matrix-calculus derivations.

  • C. Jacobian Matrix of the Instantaneous Rate w.r.t. v∗: The Jacobian J_v* stacks the conjugate-variable gradients of all users' instantaneous rates as its columns.For a channel realization and transmit precoders, it is written as [∇_v*r_1, ∇_v*r_2, ..., ∇_v*r_K].
  • C. Jacobian Matrix of the Instantaneous Rate w.r.t. v∗: The appendix obtains this Jacobian using matrix calculus and the chain rule.The displayed derivation contains the resulting gradient expressions for the user-rate components.
  • C. Jacobian Matrix of the Instantaneous Rate w.r.t. v∗: The Jacobian provides the rate sensitivity with respect to the conjugate IRS phase variable for the instantaneous multiuser rate vector.This sensitivity is used in the subsequent optimization derivation.
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