PaperlayerView exact revision on arXiv

Source-linked AI summary

Multicell MIMO Communications Relying on Intelligent Reflecting Surface

Cunhua Pan, Hong Ren, Kezhi Wang, Wei Xu, Maged Elkashlan, Arumugam Nallanathan, Lajos Hanzo

arXiv:1907.10864v4eess.SP

TL;DR

The paper addresses limited multicell evidence on using IRSs for cell-edge users and develops a joint BS precoding and IRS phase-shift design for weighted sum-rate maximization. It uses BCD with MM and CCM phase-shift updates, and reports significantly enhanced cell-edge performance, including advantages for cell-boundary and distributed IRS deployment.

  • Problem

    Prior IRS-assisted wireless studies mainly considered single-cell systems, leaving multicell cell-edge performance and inter-cell interference insufficiently investigated.

  • Method

    The paper jointly optimizes BS precoding matrices and IRS phase shifts using BCD, with MM and CCM algorithms for the non-convex phase-shift subproblem.

  • Results

    Simulations show significantly enhanced cell-edge performance compared with conventional multicell systems without IRSs.

  • Takeaways & Limitations

    IRS deployment at cell boundaries and near user clusters improves cell-edge gains, while distributed deployment outperforms centralized deployment.

Abstract

from arXiv · show

Intelligent reflecting surfaces (IRSs) constitute a disruptive wireless communication technique capable of creating a controllable propagation environment. In this paper, we propose to invoke an IRS at the cell boundary of multiple cells to assist the downlink transmission to cell-edge users, whilst mitigating the inter-cell interference, which is a crucial issue in multicell communication systems. We aim for maximizing the weighted sum rate (WSR) of all users through jointly optimizing the active precoding matrices at the base stations (BSs) and the phase shifts at the IRS subject to each BS's power constraint and unit modulus constraint. Both the BSs and the users are equipped with multiple antennas, which enhances the spectral efficiency by exploiting the spatial multiplexing gain. Due to the non-convexity of the problem, we first reformulate it into an equivalent one, which is solved by using the block coordinate descent (BCD) algorithm, where the precoding matrices and phase shifts are alternately optimized. The optimal precoding matrices can be obtained in closed form, when fixing the phase shifts. A pair of efficient algorithms are proposed for solving the phase shift optimization problem, namely the Majorization-Minimization (MM) Algorithm and the Complex Circle Manifold (CCM) Method. Both algorithms are guaranteed to converge to at least locally optimal solutions. We also extend the proposed algorithms to the more general multiple-IRS and network MIMO scenarios. Finally, our simulation results confirm the advantages of introducing IRSs in enhancing the cell-edge user performance.

I. INTRODUCTION

The paper motivates IRS-assisted multicell MIMO communication as a way to improve cell-edge performance and manage inter-cell interference through jointly optimized active and passive beamforming. It formulates a weighted sum-rate problem and develops iterative algorithms for the non-convex phase-shift design.

  • IRS motivation: IRSs reconfigure wireless propagation through independently adjustable passive phase shifts, enabling constructive desired signals and destructive interference at non-intended receivers.The reflected signals can enhance received power while reducing co-channel interference without active RF chains.
  • Optimization challenge: The weighted sum-rate optimization is challenging because the phase shifts obey a non-convex unit modulus constraint and the objective is not jointly concave.Existing transmitter-focused methods do not directly apply to joint BS and IRS beamforming design.
  • Research gap: Multicell systems reuse frequency resources, causing severe inter-cell interference, especially for cell-edge users, while prior IRS studies largely focused on single-cell scenarios.This motivates placing IRSs at cell boundaries to assist cell-edge transmission.
  • System and objective: The proposed system jointly optimizes BS active beamforming and IRS passive beamforming for multicell MIMO users with multiple antennas.Multiple antennas permit simultaneous transmission of multiple data streams, while the joint variables are highly coupled.
  • Proposed algorithms: A pair of efficient algorithms, MM and CCM, are proposed for the phase-shift problem, with MM providing a closed-form solution at each iteration.Both algorithms are guaranteed to obtain at least a locally optimal solution.
  • Results: Simulations show that IRSs significantly enhance cell-edge performance relative to conventional multicell systems without IRSs, with cell-boundary deployment achieving the highest gains.The reported gains are mainly attributed to improved BS-IRS and IRS-user links; distributed deployment near user clusters is also superior.

II. SYSTEM MODEL AND PROBLEM FORMULATION

The paper models an IRS-aided multicell downlink MIMO system in which cell-edge IRSs strengthen desired signals and mitigate cochannel interference. It formulates transmission using direct and IRS-reflected channels under perfect CSI, while acknowledging CSI acquisition as an idealized assumption.

  • A. SYSTEM MODEL: The system contains L macro cells, each with one BS serving K cell-edge users, while BSs and users use multiple antennas.
  • A. SYSTEM MODEL: Cell-edge users experience high serving-BS attenuation and severe neighboring-BS cochannel interference.
  • A. SYSTEM MODEL: An IRS with M reflection elements is placed at the cell edge to boost useful signal power and mitigate cochannel interference through phase-shift design.
  • A. SYSTEM MODEL: The equivalent channels combine direct BS-to-user links with cascaded BS-to-IRS and IRS-to-user links.
  • A. SYSTEM MODEL: Each user receives signals directly from BSs and through the IRS, with single reflections modeled through element-wise phase shifts e^jθm.
  • A. SYSTEM MODEL: The model neglects signals reflected more than once because of severe path loss and assumes additive Gaussian noise at each user.
  • A. SYSTEM MODEL: The BS is assumed to know all channel state information and computes the phase shifts for the IRS controller.
  • A. SYSTEM MODEL: Perfect CSI is difficult to obtain, but the algorithms can derive performance upper bounds under realistic CSI errors and provide design insights.

B. Problem Formulation

The paper maximizes users’ weighted sum rate by jointly designing BS precoding and IRS phase shifts under BS power and phase constraints. It reformulates the coupled non-convex problem using auxiliary decoding and weighting matrices for alternating optimization.

  • B. Problem Formulation: The objective is to maximize the weighted sum rate of all users by jointly optimizing BS transmission-precoding matrices and IRS phase shifts.
  • B. Problem Formulation: User weights represent the priority assigned to each corresponding user.
  • B. Problem Formulation: Coupling between precoding matrices and phase shifts, together with phase-shift constraints, makes the optimization difficult.
  • A. Reformulation of the Original Problem: The BCD procedure alternately optimizes one variable set while keeping the remaining variables fixed.
  • A. Reformulation of the Original Problem: The reformulated objective is more tractable than the original objective despite introducing additional optimization variables.
  • A. Reformulation of the Original Problem: The reformulation introduces decoding matrices and positive-semidefinite auxiliary matrices to express the objective through mean-square error terms.
  • A. Reformulation of the Original Problem: For fixed phase shifts, the reformulated per-user objective is concave in each set of optimization matrices when the others are fixed.
  • A. Reformulation of the Original Problem: The decoding and auxiliary matrices have optimal updates when the other matrices are fixed.

B. Optimizing the Precoding Matrices F

With auxiliary variables and phase shifts fixed, the precoding update separates across BSs and forms a convex power-constrained problem. The paper reduces this update to closed-form or bisection-based solutions, including full-rank and low-rank cases.

  • With auxiliary variables and phase shifts fixed, the precoding optimization decouples among BSs.
  • The per-BS precoding problem is convex and can be transformed into a second-order cone program.
  • The paper replaces higher-complexity SOCP solving with a near-optimal closed-form precoding expression based on Lagrangian optimization.
  • The Lagrange multiplier λ_l enforces each BS’s power constraint through complementary slackness.
  • The full-rank and low-rank cases are handled separately when determining the optimal multiplier.
  • For the full-rank case, singular-value decomposition supports the multiplier-bound derivation.
  • For the low-rank case, the power function is treated as monotonically decreasing and the multiplier is obtained by bisection search.
  • The resulting optimal precoding matrix is obtained after selecting the multiplier, with Algorithm 1 summarizing the search procedure.

C. Optimizing the Phase Shifts θ

The phase-shift block is optimized while the remaining variables are fixed, producing a constrained non-convex problem in the IRS phase variables. The formulation uses channel-dependent matrix terms and phase bounds before applying iterative solvers.

  • The phase shifts θ are optimized while the other optimization variables are fixed.
  • The phase-shift optimization is formulated from channel-dependent terms after substituting intermediate expressions and removing constants independent of Φ.
  • The phase variables satisfy 0 ≤ θ_m ≤ 2π for m = 1, · · ·, M.
  • The formulation uses matrices B, C, and V together with definitions involving the effective channel terms.
  • A matrix identity is used to obtain the resulting phase-shift optimization form.

[V]1,1, · · · , [V]M,M

The phase-shift subproblem is non-convex because each IRS coefficient has unit modulus. The paper solves it with MM and CCM methods that construct tractable updates while preserving convergence guarantees.

  • Unit modulus constraints make the phase-shift optimization problem non-convex.
  • Majorization-Minimization (MM) Algorithm: The MM algorithm replaces the original objective with a tractable upper-bound surrogate matching its value and first-order gradient at the current iterate.
  • Majorization-Minimization (MM) Algorithm: The MM objective sequence decreases monotonically and converges to a solution satisfying the KKT optimality conditions.
  • Majorization-Minimization (MM) Algorithm: MM updates the phase vector by solving a unit-modulus subproblem, yielding the phase shift θ⋆= arg(qt) after convergence.
  • Complex Circle Manifold (CCM) Method: The CCM method performs gradient descent on a product of complex circles, projecting gradients onto tangent spaces and retracting updates back onto the manifold.
  • Complex Circle Manifold (CCM) Method: With parameters satisfying Theorem 1, CCM generates a non-increasing objective sequence and converges to a finite value.

3) Complexity Analysis:

The MM and CCM phase-shift solvers have cubic initialization costs and quadratic per-iteration costs, so total complexity depends mainly on their iteration counts.

  • The MM algorithm has total complexity CMM = O(M 3 + TMMM 2).
  • The CCM algorithm has total complexity CCCM = O(M 3 + TCCMM 2).
  • Overall complexity mainly depends on the number of iterations required for convergence, which the simulations compare through convergence speed.

D. Overall Algorithm to Solve Problem (6)

The BCD algorithm alternates optimal receiver, auxiliary-matrix, precoder, and IRS phase-shift updates. Monotonic objective improvement and boundedness guarantee convergence, and the framework extends to network MIMO with added data-sharing overhead.

  • The phase objective decreases monotonically, while the overall objective increases at each BCD step and remains upper-bounded by the power constraints.
  • BCD alternates updates of decoding matrices, auxiliary matrices, BS precoders, and IRS phase shifts.
  • The phase-shift step solves Problem (41) using either the MM or CCM algorithm.
  • Network MIMO: In network MIMO, antennas from multiple BSs form a joint array that can effectively mitigate inter-cell interference, but data sharing incurs increased information-exchange overhead.
  • Network MIMO: The network-MIMO weighted-sum-rate problem uses per-BS power constraints and can be solved using the methods developed earlier.

B. Multiple-IRS Scenario

The paper extends the signal model and optimization framework to multiple IRSs by combining their reflected channels. Simulations use cell-edge users and averaged channel realizations to evaluate IRS-aided multicell performance.

  • The multiple-IRS model assumes A IRSs, each with M reflection elements, and defines channels from BSs to IRSs and IRSs to users.
  • The received signal for multiple IRSs combines the contributions of the individual IRS-related channels and phase-shifting vectors.
  • The derivations for the single-IRS scenario are directly applicable to the multiple-IRS case.
  • Simulation setup: The IRS placement is chosen so the IRS-aided link has a higher probability of experiencing nearly free-space path loss.
  • Simulation setup: Simulations model direct channels with Rayleigh fading and IRS-related channels with Rician fading, including deterministic LoS and Rayleigh NLoS components.
  • Simulation setup: The simulations place users at their corresponding cell edges and average results over 200 independent channel generations.

A. Two-cell Scenario

A two-cell simulation evaluates the proposed BCD-MM and BCD-CCM algorithms against benchmark schemes. Both proposed algorithms converge similarly and deliver higher WSR, with gains increasing as the number of IRS phase shifts grows.

  • Simulation setup: The two-cell setup places a single IRS at the boundary between two cells and distributes two users per cell symmetrically around it.The BSs are at (0, 0) and (600, 0), while the IRS is at (300, 0) by default.
  • Phase-shift optimization: The MM algorithm converges slightly faster than the CCM algorithm, indicating lower computational complexity under the reported complexity analysis.The two phase-shift algorithms can converge to different values for different M, although the final BCD WSR is similar in the convergence comparison.
  • Interference management: Optimized phase shifts combine desired direct and reflected signals constructively while adding inter-cell interference destructively.RandPhase performs only slightly better than No-IRS because its reflected signals are not carefully beamed toward receivers.
  • Network MIMO comparison: Network MIMO achieves significantly higher WSR than coordinated beamforming, but requires exchanging users’ data streams instead of only CSI.The higher performance therefore carries a heavier information-exchange cost.

4) Impact of the IRS-related Path Loss Exponent:

The simulations examine how IRS path loss, placement, user location, and reflection amplitude affect WSR. Gains are strongest under favorable IRS links and careful phase-shift design, especially when the IRS is placed at the cell boundary.

  • Path-loss exponent: Increasing the IRS-related path-loss exponent decreases the proposed schemes’ WSR until it converges to the No-IRS value.For αIRS = 2, the gain over No-IRS reaches up to 14.5 bit/s/Hz.
  • Deployment conditions: IRS gains depend on favorable BS–IRS and IRS–user channel conditions, motivating obstacle-free deployments such as ceilings or outdoor panels.The reported performance gain is marginal when such favorable conditions cannot be established.
  • Phase-shift design: Without optimized phase shifts, an IRS-aided system can perform no better than or worse than No-IRS, as illustrated by RandPhase and xIRS = 150 m.This underscores the need to jointly optimize the BS TPC matrices and IRS phase shifts.
  • Inter-cell interference: At the cell boundary, optimized phase shifts can make the equivalent inter-cell channels approach zero matrices, alleviating interference for cell-edge users.The proposed algorithms therefore obtain significant performance gains when the IRS is employed at the boundary.
  • User location: The proposed algorithms maintain nearly identical performance and outperform the benchmarks as users approach the cell edge because reflected signals become stronger.The resulting performance gap increases with xu and is associated with inter-cell-interference mitigation.
  • Reflection amplitude: Increasing the reflection amplitude η from 0.2 to 1 increases WSR by about 6 bit/s/Hz because reduced power loss strengthens the IRS-aided signal.The reflection amplitude therefore has a substantial impact on system performance.

8) Impact of the Weights

The study examines how user weights and IRS deployment affect weighted sum rate and user fairness in multicell MIMO systems. Results indicate that carefully placed, distributed IRSs improve cell-edge performance and can outperform centralized or no-IRS deployments.

  • Impact of the Weights: Unequal user weights produce a more balanced data-rate distribution than equal weights, supporting rate fairness among users.With equal weights, users closer to their base stations achieve higher rates; higher weights can compensate for lower channel gains.
  • Single-IRS Deployment: An IRS deployed at the cell boundary achieves the highest WSR in the single-IRS deployment study.Scheme-1 reaches its maximum at xIRS = 300m, indicating that cell-edge placement benefits users in the first and second cells.
  • Multiple-IRS Deployment: Two IRSs achieve a higher WSR than one IRS at XIRS = 300m, while Scheme-1 performs best when the IRSs are located at points A and C.These placements keep the IRSs closer to the relevant user groups.
  • Multiple-IRS Deployment: Distributed IRS deployment is more beneficial than centralized deployment, with IRSs near user clusters improving the WSR.The suggested number of IRSs depends on the number of user clusters, with at least one IRS expected near each cluster.
  • IRS-Assisted Interference Mitigation: IRS phase shifts can destructively combine reflected and direct interference, mitigating inter-cell interference for cell-edge users.The system jointly optimizes base-station transmission-control matrices and IRS phase shifts under power and unit-modulus constraints.
Loading 1907.10864v4…