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Near-Field Beamforming Design for Multi-IRS Wireless Beam Routing

Tao Wang, Changsheng You, Changchuan Yin

arXiv:2610.10155v1eess.SP

TL;DR

Far-field channel assumptions underpin angle-based beamforming, but near-field mismatch can degrade performance, and prior near-field design addresses single-IRS systems rather than multi-IRS routing. The paper reformulates BT-based joint BS/IRS received-power optimization and solves it with alternating optimization; at N=4096 it reports gains over both angle-based beamforming and beam focusing.

  • Problem

    Far-field models dominate multi-IRS beam routing despite near-field inter-array channels, where angle-based designs may mismatch; prior near-field design addresses only single-IRS systems.

    Source: §I · p. 1

  • Method

    The authors reformulate BT-based received-power maximization with joint BS and IRS beamforming and use an alternating-optimization algorithm to solve the reformulated problem.

    Source: §I · p. 1, §III · p. 3

  • Results

    At N=4096, AO gains more than 170% over angle-based beamforming and 20% over beam focusing; far-field log-linear power scaling ceases as N increases.

    Source: §IV · p. 4

  • Takeaways & Limitations

    The paper recommends practical multi-IRS beam routing in the large-N regime, where the proposed AO scheme can significantly improve system performance.

    Source: §IV · p. 5

Abstract

from arXiv · show

This letter investigates efficient near-field beamforming designs for wireless communication systems aided by multiple extremely large-scale intelligent reflecting surfaces (Multi-XL-IRSs). Previous studies have mostly assumed far-field channel conditions, based on which angle-based beamforming is adopted for the base station (BS) and IRSs. However, this approach may lead to degraded performance in practical near-field scenarios. To address this issue, we consider near-field channel modeling and formulate an optimization problem aimed at maximizing the received power at the user equipment (UE) by jointly optimizing the beamforming at the BS and IRSs. Given the intractability under the beam training (BT)-based communication framework, the problem is revisited and reformulated. Subsequently, an efficient algorithm based on alternating optimization (AO) is proposed to solve the reformulated problem. Numerical results demonstrate that, in typical setups, our AO-based beamforming designs provide over 170% and 20% received power gains over conventional angle-based beamforming and beam focusing, respectively. Furthermore, the power scaling law derived under far-field conditions no longer holds in near-field scenarios.

I. INTRODUCTION

Multi-IRS beam routing uses multi-hop reflections to establish virtual LoS, but passive XL-IRS arrays can create near-field inter-array channels that make far-field angle-based designs mismatch. The paper proposes joint near-field BS/IRS beamforming for received-power maximization under beam training.

I. INTRODUCTION

  • Multiple IRSs provide cooperative passive beamforming gains through multi-hop reflections that can establish virtual LoS paths between the BS and UE –.XL-IRSs with massive passive elements are deployed to mitigate multi-hop pathloss and capture sufficient power and beamforming gains.Source: §I · p. 1
  • Near-field inter-array channels can make far-field angle-based beamforming perform poorly through channel mismatch, and prior near-field work for single-IRS systems does not apply to multi-IRS systems.The near-far-field boundary is proportional to array size, motivating near-field modeling for large arrays.Source: §I · p. 1
  • The proposed scheme jointly designs BS and IRS beamforming under near-field modeling, reformulating the intractable beam-training received-power maximization problem for an alternating-optimization algorithm.Numerical results evaluate power scaling and compare the AO design with benchmark schemes.Source: §I · p. 1

II. SYSTEM MODEL

  • The system assumes an obstructed direct BS–UE link and a virtual LoS route via K IRSs, with a single-antenna UE in the coverage area of the final IRS.Control links to the IRSs and the UE-to-BS feedback link are assumed separate and reliable.Source: §II · pp. 1–2
  • The near-field focus is on passive XL-IRS deployments; active IRSs are typically small-scale arrays and adhere to far-field channel models.The paper attributes the passive-XL-IRS near-field condition to the relationship between array size and the near-far-field boundary.Source: §II · p. 1, §I · p. 1

A. Signal Model

The signal model uses cascaded near-field LoS channels and treats NLoS components as environmental scattering. Fixed inter-array channels are known from deployment geometry, while the UE channel is random and unknown.

A. Signal Model

  • The cascaded channel model focuses on LoS links and treats NLoS components as environmental scattering, whose impact is described as negligible.The model covers BS-to-first-IRS, inter-IRS, and last-IRS-to-UE links.Source: A. Signal Model · p. 2
  • The model specifies link gains and effective apertures for BS-to-IRS, IRS-to-IRS, and IRS-to-UE propagation.Each effective aperture is defined relative to the transmitting antenna or IRS element on that link.Source: A. Signal Model · p. 2
  • The received signal includes the cascaded beamformed link, transmitted symbol, and additive white Gaussian noise; the achievable data rate is then calculated.The BS beamformer is subject to a transmit-power constraint, and each IRS applies phase shifts.Source: A. Signal Model · p. 2
  • Inter-array LoS channels are assumed known from fixed deployment geometry, whereas the UE channel is random and unknown because of UE mobility.If multiple UEs are covered by the last IRS, the algorithm's results can be applied to IRS beam training for those UEs.Source: A. Signal Model · p. 2

B. BT-Based Transmission Protocol

The BT protocol fixes the BS beamformer and intermediate IRS phases before the final IRS sweeps a predefined codebook. After selecting a codeword, the system estimates the equivalent baseband channel and transmits data.

B. BT-Based Transmission Protocol

  • The final IRS sweeps a predefined codebook, and the UE measures beam powers and feeds the selected codeword index back to the BS.The BS applies the selected codeword to establish an initial link.Source: B. BT-Based Transmission Protocol · p. 2
  • Before beam training, the BS beamformer and intermediate IRS phase shifts are optimized and pre-configured using prior inter-array LoS channel information.During training, only the final IRS phase shifts are adjusted for beam sweeping.Source: B. BT-Based Transmission Protocol · p. 2
  • With all beamformers fixed, the system performs low-dimensional estimation of the equivalent baseband BS–UE channel before data transmission.Source: B. BT-Based Transmission Protocol · p. 2

C. Problem Formulation

The problem seeks to maximize UE received power by jointly designing the BS beamformer and intermediate IRS phases under the BT protocol. It cannot be solved directly because those choices must be made before the UE channel and final IRS codeword are known.

C. Problem Formulation

  • The optimization jointly selects the BS beamformer and intermediate IRS phases to maximize received power under the BT-based protocol.Source: C. Problem Formulation · p. 3
  • The BS beamformer uses the full transmit-power budget at an optimum, and the final IRS phase configuration is constrained to the predefined codebook.Source: C. Problem Formulation · p. 3
  • Direct optimization is infeasible because the BS and intermediate IRS designs precede beam training, when both the UE channel and final IRS codeword are unknown.Source: C. Problem Formulation · p. 3

D. Benchmark Schemes

Conventional angle-based beamforming relies on far-field channel assumptions and aligns each array's phases toward the next array. Near-field channel mismatch makes this approach inapplicable to multi-IRS beam routing.

D. Benchmark Schemes

2) Near-field Beam Focusing:

Near-field beam focusing sets each array’s transmit or reflect phases toward the center of the next array. Diffraction concentrates signal energy around that focal point, so the design intuitively tends to maximize total received power at each array.

2) Near-field Beam Focusing:

III. PROPOSED AO-BASED BEAMFORMING DESIGN

The design reformulates the beam-training-based received-power problem and solves it by alternating optimization, updating the BS beamforming and IRS variables in turn. The updates are non-decreasing, but non-convexity prevents a guarantee of global optimality.

III. PROPOSED AO-BASED BEAMFORMING DESIGN

  • The final-IRS channel uses a uniform-amplitude approximation based on the Fresnel approximation, with an auxiliary unit-modulus variable replacing its phase-controlled channel term.Codebook-based beam training is discussed as prior work and is not studied in this letter.Source: §III · p. 3
  • The reformulation eliminates the intractability caused by the beam-training transmission protocol, while the resulting problem remains non-convex.Source: §III · p. 3
  • The AO algorithm updates one variable at a time while holding the others fixed, including the BS beamforming and IRS phase shifts.The update of the auxiliary variable aligns phases; the updates for intermediate IRSs follow the same derivation.Source: §III · p. 3
  • Every update is non-decreasing, so Algorithm 1 converges monotonically to a local stationary point, but not necessarily a global optimum.The global-optimum guarantee is unavailable because the original problem is non-convex.Source: §III · p. 3

A. Complexity Analysis

The algorithm’s complexity depends on the per-iteration updates and the number of iterations. Under quasi-static inter-array LoS channels, it can ideally be run once offline.

A. Complexity Analysis

  • Algorithm 1’s complexity is determined by updates to the BS beamforming, intermediate IRS phase shifts, and auxiliary variable in each iteration, together with the iteration count.Source: A. Complexity Analysis · p. 3
  • The passage identifies the update steps and maximum iterations as complexity drivers but does not provide a complete asymptotic expression.Source: A. Complexity Analysis · p. 3
  • The algorithm can ideally be performed only once in an offline manner when inter-array LoS channels are quasi-static.Source: A. Complexity Analysis · p. 3

IV. NUMERICAL RESULTS

Simulations compare the AO design with angle-based beamforming and beam focusing across array sizes and channel conditions. The AO design has its largest reported power gains at large array size, while the far-field power-scaling law ceases to hold as arrays grow.

IV. NUMERICAL RESULTS

  • The simulations use parallel, equidistant URAs with half-wavelength spacing, 5 m inter-array distance, 20 m UE distance, and 100 channel realizations.The Rician factor for unknown NLoS components is set to 20 dB.Source: §IV · p. 4
  • On the K-th IRS, the AO design produces greater total incident power and a more uniform distribution than beam focusing.Source: §IV · p. 4
  • The far-field O(N^6) power scaling is verified when N is approximately less than 400; it no longer holds as N increases, and received power is expected to saturate as N approaches infinity.Source: §IV · p. 4
  • More than 170% and 20% received signal power gains at N = 4096 over angle-based beamforming and beam focusing, respectively.Source: §IV · p. 4, Figure 3 · p. 5
  • Algorithm 1 converges within 6 iterations for various parameters, with convergence values significantly higher than the benchmark schemes.Source: §IV · p. 4, Figure 3 · p. 5
  • Received signal power is less than -70 dBm when N < 625, and the authors recommend operation in the large-N regime.Source: §IV · p. 5
  • The AO scheme outperforms the benchmarks across UE distances and its performance degrades more slowly as UE distance increases.Source: §IV · p. 5
  • Across Rician factors, the AO scheme outperforms the benchmarks; rates for all schemes rise and then plateau as the factor increases.The authors attribute the rise to the design using inter-array LoS channels but not random NLoS components, and the plateau to channels converging toward the LoS component.Source: §IV · p. 5

V. CONCLUSION

The proposed near-field design jointly optimizes BS and intermediate-IRS beamforming for multi-IRS routing and achieves higher UE received signal power than benchmark schemes. Its results also show that far-field power scaling does not hold under near-field conditions.

V. CONCLUSION

  • The proposed AO-based scheme achieves higher UE received signal power than benchmark schemes.The conclusion reports superior system performance for the proposed scheme.Source: §V · p. 5
  • Power scaling derived under the far-field assumption no longer holds under near-field conditions.Source: §V · p. 5
  • The design maximizes received power by jointly optimizing BS and intermediate-IRS beamforming, using a reformulation and efficient AO-based algorithm.The problem is formulated under a beam-training-based communication protocol.Source: §V · p. 5

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