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Double-IRS Assisted Multi-User MIMO: Cooperative Passive Beamforming Design

Beixiong Zheng, Changsheng You, Rui Zhang

arXiv:2008.13701v5cs.ITcs.ET

TL;DR

Prior IRS research largely omits inter-IRS signal reflection, leaving the potential of cooperative multi-IRS networks insufficiently characterized. This paper jointly designs base-station receive beamforming and two-IRS cooperative passive beamforming, showing superior maximum SNR and multi-user effective channel rank over a single-IRS system.

  • Problem

    Prior IRS studies largely omit inter-IRS signal reflection, leaving cooperative distributed-IRS performance insufficiently characterized.

  • Method

    The paper jointly optimizes base-station receive beamforming and cooperative passive beamforming at two distributed IRSs to maximize users’ minimum SINR.

  • Results

    The double-IRS cooperative system outperforms the single-IRS baseline in maximum SNR and multi-user effective channel rank, with simulations demonstrating substantial performance gains.

  • Takeaways & Limitations

    Cooperative double-IRS beamforming offers practical performance advantages over conventional single-IRS designs in multi-user MIMO systems.

Abstract

from arXiv · show

Intelligent reflecting surface (IRS) has emerged as an enabling technology to achieve smart and reconfigurable wireless communication environment cost-effectively. Prior works on IRS mainly consider its passive beamforming design and performance optimization without the inter-IRS signal reflection, which thus do not unveil the full potential of multi-IRS assisted wireless networks. In this paper, we study a double-IRS assisted multi-user communication system with the \emph{cooperative} passive beamforming design that captures the multiplicative beamforming gain from the inter-IRS channel. Under the general channel setup with the co-existence of both double- and single-reflection links, we jointly optimize the (active) receive beamforming at the base station (BS) and the cooperative (passive) reflect beamforming at the two distributed IRSs (deployed near the BS and users, respectively) to maximize the minimum signal-to-interference-plus-noise ratio (SINR) of all users. Moreover, for the single-user and multi-user setups, we analytically show the superior performance of the double-IRS cooperative system over the conventional single-IRS system in terms of the maximum signal-to-noise ratio (SNR) and multi-user effective channel rank, respectively. Simulation results validate our analytical results and show the practical advantages of the proposed double-IRS system with cooperative passive beamforming designs.

I. INTRODUCTION

The paper addresses the limitations of independently designed multi-IRS beamforming by studying cooperative double-IRS-assisted multi-user MIMO with both double- and single-reflection links. It jointly optimizes BS receive and IRS reflect beamforming, proving gains in single-user SNR and multi-user channel rank, and validating improved rate performance through simulations.

  • Motivation: Cooperative beamforming is needed because inter-IRS channels can affect performance, while jointly designed IRSs exploit multiplicative gains and mitigate interference.Independent single-reflection designs are generally no longer optimal when multiple IRSs are deployed.
  • System and objective: The paper considers double-IRS-assisted multi-user MIMO with IRSs near the BS and users, jointly optimizing BS receive and two-IRS passive beamforming to maximize minimum uplink SINR under coexisting reflection links.The formulation includes both double- and single-reflection links and serves multiple users through a multi-antenna BS.
  • Algorithms and validation: Alternating optimization provides efficient joint active/passive beamforming algorithms, using closed-form block updates for the single-user case and SDR with bisection for multi-user max-min SINR.Simulations corroborate the theory and show significant rate gains over the single-IRS baseline across various system settings.
  • Multi-user contribution: For multi-user communication, cooperative double-IRS reconfiguration generally yields higher channel rank and spatial multiplexing, improving max-min SINR/rate over the single-IRS baseline.The comparison applies to the general multi-user setup and reflects the benefit of balancing passive beamforming gains with spatial multiplexing under both reflection types.

II. SYSTEM MODEL AND PROBLEM FORMULATION · A. Double-IRS Assisted Multi-User MIMO

This section models a fully passive double-IRS uplink MIMO system, with IRSs deployed near users and the BS to assist blocked users through cooperative double- and single-reflection links. It formulates the effective cascaded channel and received-signal model under unit-modulus reflections, available cascaded CSI, and quasi-static flat fading.

  • A. Double-IRS Assisted Multi-User MIMO: The system comprises K single-antenna users, an N-antenna BS, and two distributed IRSs cooperatively assisting uplink transmission.IRS 1 and IRS 2 are placed near the users and BS, respectively, to reduce associated path loss.
  • A. Double-IRS Assisted Multi-User MIMO: When direct user–BS links are severely blocked, the two properly deployed IRSs serve the users through reflection links.The blocked-link scenario is motivated by obstacles such as indoor walls.
  • A. Double-IRS Assisted Multi-User MIMO: The two IRSs contain M1 and M2 passive subsurfaces, with M1 + M2 = M; each subsurface groups adjacent reflecting elements under a common phase shift.This grouping provides high aperture gain while reducing channel-estimation and reflection-optimization cost, which generally increases with the number of subsurfaces,.
  • A. Double-IRS Assisted Multi-User MIMO: For user k, the effective BS channel combines the cooperative path user k→IRS 1→IRS 2→BS with the two single-reflection paths.It is represented as G2Φ2DΦ1u1,k + R2,kθ2 + R1,kθ1, where Φμ is the diagonal reflection matrix and R1,k, R2,k are cascaded channels.
  • A. Double-IRS Assisted Multi-User MIMO: The IRSs are fully passive, unlike the receive-RF-chain-equipped IRSs in, reducing power consumption and implementation cost but preventing direct separate-CSI acquisition.Cascaded CSI is nevertheless sufficient for jointly designing the two passive beamforming vectors.
  • A. Double-IRS Assisted Multi-User MIMO: The model assumes that the BS knows the cascaded channels and that all channels follow quasi-static flat fading, remaining approximately constant within each channel.During uplink transmission, users send data symbols with powers Pk, the BS receives AWGN with covariance σ2I, and applies linear receive processing before SINR decoding.

B. Problem Formulation · C. Baseline System: Conventional Single-IRS Assisted Multi-User MIMO

The paper maximizes the minimum user SINR by jointly designing BS receive beamforming and cooperative passive beamforming at two distributed IRSs under unit-modulus constraints. It benchmarks this non-convex double-IRS problem against centralized single-IRS systems deployed near the BS or users.

  • B. Problem Formulation: The objective is to maximize the minimum SINR among all users by jointly optimizing BS receive beamforming and the two IRSs’ cooperative passive reflect beamforming.The reflecting subsurfaces must satisfy unit-modulus constraints.
  • B. Problem Formulation: The optimization is challenging because its objective and unit-modulus constraints are non-convex, while shared IRS beamformers couple all users through interference.Interference depends on both each user’s receive beamformer and the commonly shared cooperative reflect beamformers.
  • B. Problem Formulation: The paper applies alternating optimization first to the single-user case and then generalizes it to the multi-user case.The formulation is not addressed by a standard optimal method because of its non-convexity.
  • C. Baseline System: Conventional Single-IRS Assisted Multi-User MIMO: The two conventional baselines place all M = M1 + M2 subsurfaces on one centralized IRS deployed near either the BS or the user cluster.These baselines evaluate the gains from distributing the subsurfaces across two cooperative IRSs.
  • C. Baseline System: Conventional Single-IRS Assisted Multi-User MIMO: For the BS-side baseline, the effective user-to-BS channel is modeled as ¯h_k = ¯GΦ¯u_k = ¯R_kθ using the single IRS’s reflection coefficients.¯R_k represents the cascaded user k→IRS→BS channel without IRS phase shifts, and the user-side baseline is defined similarly.
  • C. Baseline System: Conventional Single-IRS Assisted Multi-User MIMO: The single-IRS baselines are special cases of the double-IRS system with M1 = 0 or M2 = 0, and cascaded CSI suffices for joint beamforming design.Their SINRs use linear BS receive beamformers under the corresponding cascaded channel models.
  • C. Baseline System: Conventional Single-IRS Assisted Multi-User MIMO: The comparison focuses on the single IRS deployed near the BS versus the double-IRS cooperative system, with analogous results for the user-side baseline under symmetric channels.The two baseline configurations are illustrated in Fig. 2.

III. SINGLE-USER SYSTEM · A. AO Algorithm for Cooperative Passive Beamforming Design

For the single-user case, the paper simplifies the joint design to receive-SNR maximization without inter-user interference and develops an alternating-optimization algorithm for the coupled IRS and BS beamformers. The closed-form updates coherently combine double- and single-reflection links, with low complexity and guaranteed convergence.

  • III. SINGLE-USER SYSTEM: The single-user setup removes inter-user interference, covering orthogonal multiple access scenarios, and reduces the joint problem to maximizing the BS receive SNR under unit-modulus IRS constraints.The user index is omitted in this section, and the BS receive beamforming remains coupled with the two IRS reflect beamformers.
  • A. AO Algorithm for Cooperative Passive Beamforming Design: The proposed AO algorithm [15] alternately optimizes BS receive beamforming and the two distributed IRS reflect beamformers until convergence.The method addresses the non-convex unit-modulus constraints and coupling between the IRS beamformers by optimizing one component while fixing the others.
  • A. AO Algorithm for Cooperative Passive Beamforming Design: For fixed BS and IRS 1 beamformers, IRS 2 aligns the composite double-reflection and IRS-2 single-reflection signals with the IRS-1 single-reflection signal for coherent combining.This update solves the resulting unit-modulus subproblem in closed form.
  • A. AO Algorithm for Cooperative Passive Beamforming Design: With IRS 2 and BS beamformers fixed, IRS 1 similarly aligns its double-reflection and IRS-1 single-reflection signals with the IRS-2 single-reflection signal.The resulting cooperative design balances passive beamforming gains across the double-reflection and two single-reflection links.
  • A. AO Algorithm for Cooperative Passive Beamforming Design: The BS receive update uses maximum-ratio combining, which is the optimal receive beamforming solution for the single-user problem.All receive and reflect beamforming updates are obtained in closed form, making the AO procedure practically appealing.
  • A. AO Algorithm for Cooperative Passive Beamforming Design: O(I0(N + M)) complexity makes the AO algorithm low-complexity, where I0 is the iteration count.The complexity follows from the closed-form updates for the BS and two IRS beamforming vectors.
  • A. AO Algorithm for Cooperative Passive Beamforming Design: The AO algorithm is guaranteed to converge because each exact subproblem update non-decreases the objective, while the objective is bounded above by finite user transmit power.Thus, the objective sequence is monotonic and upper-bounded.

B. Comparison with Single IRS

Under a symmetric deployment and assumption A1, the cooperative double-IRS system achieves a maximum SNR no lower than the single-IRS baseline by coherently combining double- and single-reflection links. This superiority holds for arbitrary channels and IRS sizes, extending prior double-reflection-only setups.

  • Comparison with Single IRS: The comparison assumes symmetric IRS deployment, equal product-distance path loss for the two single-reflection links, and the channel relation R̄ = [R1, R2].IRS 1 is moved to IRS 2’s position to form the centralized single-IRS baseline.
  • Comparison with Single IRS: A common phase shift aligns the double- and single-reflection contributions, while retaining the single-IRS optimum as the receive and initial reflect beamforming.The alignment phase is φ = ∠(a2/a1), and equality occurs only when the double-reflection link vanishes.
  • Comparison with Single IRS: The proposed setup is more general than, using a multi-antenna BS and allowing both double- and single-reflection links to coexist.In contrast, considers a single-antenna BS with only the double-reflection link.
  • Comparison with Single IRS: The cooperative double-IRS system theoretically outperforms the single-IRS baseline in maximum SNR for arbitrary channels and numbers of IRS elements or subsurfaces.The result follows by coherently combining the double- and single-reflection links.

IV. MULTI-USER SYSTEM · A. AO Algorithm Based on SDR and Bisection

For the general multi-user double-IRS cooperative system, the paper develops an efficient sub-optimal algorithm for problem (P1) and compares its multi-user effective channel rank with a conventional single-IRS baseline. The algorithm alternates cooperative reflect-beamforming optimization using SDR and bisection with receive-beamforming design to maximize the minimum user SINR.

  • IV. MULTI-USER SYSTEM: The multi-user study targets problem (P1) by jointly designing two common reflect-beamforming vectors and user-specific receive beamformers to maximize the minimum SINR.The two IRS vectors are shared across users, while the receive beamformers balance different user channel gains.
  • A. AO Algorithm Based on SDR and Bisection: The proposed efficient sub-optimal solution generalizes the alternating-optimization framework to the multi-user setting using semidefinite relaxation and bisection.The approach addresses the non-convexity caused by unit-modulus constraints and coupling between the two reflect-beamforming vectors.
  • IV. MULTI-USER SYSTEM: The broader multi-user section compares the double-IRS cooperative system with a conventional single-IRS baseline using multi-user effective channel rank, or spatial multiplexing gain.This comparison is intended to assess the systems’ multi-user spatial multiplexing capability.
  • A. AO Algorithm Based on SDR and Bisection: With one reflect-beamforming vector fixed, the other is optimized by relaxing a rank-one constraint into an SDR problem with unit diagonal constraints.The same SDR-based transformation is applied when optimizing θ2 with θ1 fixed and vice versa.
  • A. AO Algorithm Based on SDR and Bisection: For a given auxiliary SINR target, the relaxed reflect-beamforming problem becomes a quasi-convex feasibility check that is a convex semidefinite program solvable by bisection and existing convex solvers.Because SDR may produce a higher-rank solution, a high-quality rank-one solution is recovered using methods such as Gaussian randomization.
  • A. AO Algorithm Based on SDR and Bisection: After fixing both IRS vectors, problem (P1) decomposes into K user subproblems that maximize each user’s SINR through receive-beamforming design.The resulting effective channel for each user is fixed during this step.
  • A. AO Algorithm Based on SDR and Bisection: The BS receive-beamforming update uses the effective user-BS channel matrix and can follow either sub-optimal zero-forcing or optimal minimum mean squared error criteria.These criteria are used to cope with multi-user interference.

P HHH

Algorithm 1 solves the max-min SINR problem through alternating optimization of the two IRS phase vectors and BS receive beamforming. Each iteration uses SDR and bisection for the IRS subproblems, ZF/MMSE designs for receive beamforming, and stops upon convergence or a preset iteration limit.

  • Proposed AO algorithm: Algorithm 1 alternately solves subproblems (P3.1), (P3.4), and (P4), using each iteration’s solution to initialize the next while targeting max-min SINR.The IRS subproblems are handled via SDR and bisection, while the receive-beamforming subproblem uses ZF/MMSE designs.
  • Scope and limitation: A closed-form single-user receive-beamforming update is more efficient than applying the general multi-user algorithm, while jointly optimizing receive beamforming with cooperative reflection is generally difficult.Substituting ZF/MMSE beamforming into the cooperative reflect-beamforming objective makes that objective more complicated, so this alternative is not considered.
  • Initialization: The algorithm initializes both IRS phase vectors, the receive-beamforming matrix, and the iteration counter before beginning the alternating updates.The initialization is θ1 := θ(0) 1, θ2 := θ(0) 2, W := W (0), and i = 0.
  • Stopping criterion: Iterations terminate when the fractional max-min SINR increase falls below ξ > 0 or the iteration count reaches I1.This provides either a convergence-based or preset-iteration stopping rule.
  • Complexity: The overall algorithm has complexity O(I1((M1^4.5 + M2^4.5)log(1/ǫ) + N^3)), combining SDR/bisection IRS updates with ZF/MMSE receive-beamforming design.Here, ǫ denotes the bisection-search accuracy, and I1 is the number of iterations required for convergence.

B. Comparison with Single IRS

The double-IRS cooperative system generally achieves higher multi-user effective channel rank than the single-IRS baseline, with a gain of at least min(rank(G1), rank(U1)). This greater rank can enable more users and higher max-min SINR or rate, although a common phase shift cannot align all links for every user simultaneously.

  • B. Comparison with Single IRS: The single-user SINR dominance result does not extend to general multi-user systems because one common phase shift cannot align double- and single-reflection links for all users.The comparison therefore focuses on multi-user effective channel rank rather than guaranteeing higher SINR for every user.
  • B. Comparison with Single IRS: When rank(H)=K, zero-forcing receive beamforming fully mitigates multi-user interference, so the minimum SINR increases monotonically with user transmit power.With equal user transmit power, any finite target SINRs can then be achieved using sufficiently high transmit power.
  • B. Comparison with Single IRS: If rank(H)<K, insufficient degrees of freedom prevent full interference mitigation, causing the max-min SINR to saturate as user transmit power increases.Even optimal MMSE-based receive beamforming may leave the system interference-limited in this case.
  • B. Comparison with Single IRS: Proposition 2 shows that the double-IRS cooperative system generally has higher effective channel rank than the single-IRS baseline, with gain at least min(rank(G1), rank(U1)).The comparison assumes channel-rank assumption A2 and uses a single centralized IRS positioned at IRS 2 as the baseline.
  • B. Comparison with Single IRS: The higher channel rank, and thus greater spatial multiplexing gain, can support more users and achieve higher max-min SINR or rate than the single-IRS baseline.These performance advantages are stated as outcomes expected to be confirmed by simulations in Section V.

V. SIMULATION RESULTS · A. Single-User System

Simulations show that cooperative double-IRS beamforming consistently outperforms the single-IRS baseline, with gains amplified by balanced IRS allocation, stronger LoS conditions, and larger surfaces. The proposed AO method achieves near-SDR performance while balancing double- and single-reflection gains.

  • V. SIMULATION RESULTS: The evaluation compares max-min achievable rates for cooperative double-IRS and a centralized single-IRS baseline under equal user powers and common simulation settings.The baseline places the centralized IRS at IRS 2’s position, while the double-IRS design uses joint codebook search and MRC in the single-user case.
  • V. SIMULATION RESULTS: DFT codebook search has complexity O(M1M2(N+M)) for single-user systems and is much lower-complexity than SDR-based Algorithm 1 in multi-user systems.It can also initialize Algorithm 1, yielding rapid convergence with a small I1 in simulations.
  • A. Single-User System: The simulations use a single-user case with N = 5, a 10 dB LoS-dominant nearby link, and κ-controlled Rician fading on far-apart and inter-IRS links.The AO iteration count is set to I0 = 100 to ensure convergence performance.
  • A. Single-User System: The proposed AO algorithm converges to SDR performance from either initialization, while single-IRS initialization performs better than DFT initialization before optimization.SDR with optimal MRC is near-optimal but has relatively higher complexity.
  • A. Single-User System: As κ increases, both systems improve, but the double-IRS advantage widens because high-Rician-factor channels exploit large gains from both reflection paths.At high κ, the double-IRS system reaps substantially more passive beamforming gain than the single-IRS baseline.

B. Multi-User System

In the multi-user setup, cooperative double-IRS beamforming outperforms DFT initialization and conventional single-IRS operation by providing higher effective channel rank, stronger max-min rates, and slower degradation as users increase.

  • B. Multi-User System: After four iterations, Algorithm 1 substantially outperforms DFT-based codebook search, particularly at low user transmit power.Both methods use the ZF or MMSE receive beamformer comparison, and their ZF-based rates converge toward MMSE performance at high power as noise becomes negligible.
  • B. Multi-User System: For K = 5, the double-IRS channel has rank(H) = K and supports all users, while the single-IRS baseline has rank(H) = 2 < K and its rate saturates with transmit power.The single-IRS saturation results from insufficient spatial degrees of freedom for fully mitigating multi-user interference; the distributed double-IRS system’s rate continues increasing with power.
  • B. Multi-User System: The double-IRS system achieves much higher max-min rates and less rate degradation than the single-IRS baseline as K increases, especially for K > 2.With K ≤ 2, both systems achieve high rates from passive beamforming; beyond two users, the single-IRS channel becomes rank-deficient and interference-limited, whereas distributed IRSs preserve spatial multiplexing.
  • B. Multi-User System: At P = 30 dBm, both systems perform well for K ≤ 2, but MMSE reception generally outperforms ZF when the effective channel rank is deficient.The comparison examines channel-rank and spatial-multiplexing effects as the number of users increases.

VI. CONCLUSIONS

The paper proposes a double-IRS assisted multi-user MIMO system with cooperative passive beamforming under coexisting double- and single-reflection links. It jointly optimizes receive and cooperative reflect beamforming to maximize minimum user SINR, analytically and numerically demonstrating advantages over a single-IRS baseline.

  • VI. CONCLUSIONS: The proposed system captures cooperative passive beamforming gains under a general channel setup containing both double- and single-reflection links.
  • VI. CONCLUSIONS: The study formulates and solves joint receive and cooperative reflect beamforming optimization to maximize the minimum SINR among all users.
  • VI. CONCLUSIONS: The double-IRS cooperative system outperforms the conventional single-IRS baseline in maximum SNR and multi-user effective channel rank.These advantages are established analytically for single-user and multi-user setups, respectively.
  • VI. CONCLUSIONS: Simulations demonstrate substantial performance gains from the proposed cooperative reflect beamforming designs across various system settings versus the conventional single-IRS baseline.

APPENDIX

The appendix derives channel-rank expressions for the double-IRS cooperative system and the single-IRS baseline under geographically separated, statistically independent links. Full-rank IRS reflection matrices preserve rank, while shared channel factors make the double- and single-reflection components generally dependent; comparing the resulting expressions establishes (52).

  • Double-IRS cooperative system: Under geographically separated nodes, independent component channels yield rank(H_d) = min(rank(G_2), rank(D), rank(U_1)) for the double-reflection link.The double-reflection channel is expressed through G_2DU_1, whose rank follows from the component-channel ranks.
  • Double-IRS cooperative system: The two single-reflection links have combined rank min(rank(G_2), rank(U_2)) + min(rank(G_1), rank(U_1)) when their component matrices are linearly independent.The diagonal IRS reflection matrices Φ_1 and Φ_2 are full rank and therefore do not alter this channel-rank condition.
  • Double-IRS cooperative system: Because H_d and H_s share G_2 and U_1, they are not generally linearly independent, so the overall channel rank must account for this dependence.The appendix explicitly identifies the common factors as the reason the double- and single-reflection components cannot generally be treated as independent.
  • Single-IRS baseline: For the single-IRS baseline, geographical separation gives linearly independent channel matrices, while the full-rank reflection matrix Φ does not affect the channel-rank condition.The baseline derivation is stated under channel rank assumption A2.
  • Rank comparison: Comparing the derived double-IRS and single-IRS rank results yields relationship (52), completing the appendix’s rank-condition proof.The passage identifies this comparison as the final step of the derivation.
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