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Cooperative Beam Routing for Multi-IRS Aided Communication

Weidong Mei, Rui Zhang

arXiv:2010.13589v1cs.ITeess.SPmath.IT

TL;DR

Multi-hop IRS communication can address settings where a single reflection cannot establish a LoS path, but optimal routing must trade cooperative beamforming gain against multi-reflection path loss. The paper derives route-specific beamforming and solves route selection as a shortest simple-path problem; its proposed routing balances the trade-off and outperforms benchmark designs in the reported comparisons.

  • Problem

    Most prior IRS studies considered single-reflection links, while general multi-hop IRS communication had not been studied, despite its potential to bypass obstacles and exploit inter-IRS cooperative gain.

    Source: §I · p. 1

  • Method

    The paper derives closed-form beamforming for a given route and recasts IRS selection and routing as an optimally solved shortest simple-path problem.

    Source: §I · pp. 1–2, A. Problem Reformulation · p. 4

  • Results

    The proposed routing balances cooperative beamforming gain and end-to-end path loss; the two single-objective benchmarks perform much worse at moderate M, though each is close to the proposed solution at one extreme.

    Source: §V · p. 5

  • Takeaways & Limitations

    Path loss dominates routing when the number of reflecting elements is small, whereas cooperative passive beamforming gain has greater influence when it is large.

    Source: §VI · p. 5

Abstract

from arXiv · show

Intelligent reflecting surface (IRS) has been deemed as a transformative technology to achieve smart and reconfigurable environment for wireless communication. This letter studies a new IRS-aided communication system, where multiple IRSs assist in the communication between a multi-antenna base station (BS) and a remote single-antenna user by multi-hop signal reflection. Specifically, by exploiting the line-of-sight (LoS) link between nearby IRSs, a multi-hop cascaded LoS link between the BS and user is established where a set of IRSs are selected to successively reflect the BS's signal, so that the received signal power at the user is maximized. To tackle this new problem, we first present the closed-form solutions for the optimal active and cooperative passive beamforming at the BS and selected IRSs, respectively, for a given beam route. Then, we derive the end-to-end channel power, which unveils a fundamental trade-off in the optimal beam routing design between maximizing the multiplicative passive beamforming gain and minimizing the multi-reflection path loss. To reconcile this trade-off, we recast the IRS selection and beam routing problem as an equivalent shortest simple-path problem in graph theory and solve it optimally. Numerical results show significant performance gains of the proposed algorithm over benchmark schemes and also draw useful insights into the optimal beam routing design.

I. INTRODUCTION

Prior IRS work largely considered single-reflection links, leaving multi-hop cooperation among IRSs unstudied despite its potential to bypass obstacles and exploit inter-IRS beamforming. This letter optimizes cooperative beam routing by balancing passive beamforming gain against reflection path loss, using an optimal graph-based solution.

I. INTRODUCTION

  • Earlier IRS studies mainly considered single-reflection links, which may fail when obstacles block a direct line-of-sight path and do not exploit inter-IRS cooperative beamforming.A double-IRS result reported cooperative gain scaling as O(M^4), compared with O(M^2) for a conventional single-IRS link.Source: §I · p. 1
  • The paper addresses the unstudied general case of multi-hop IRS reflection between a multi-antenna base station and a remote single-antenna user.The proposed system uses nearby IRS-to-IRS LoS links to establish an end-to-end route.Source: §I · p. 1
  • Optimal routing must balance the increasing multiplicative passive beamforming gain from added IRSs against the path loss incurred by additional reflections.The paper derives closed-form beamforming for a given route, then recasts route selection as a shortest simple-path problem and solves it optimally.Source: §I · p. 2
  • Numerical results show that the proposed solution significantly outperforms heuristic schemes.The system is described as generally more energy- and spectral-efficient than conventional multi-hop active relaying because its reflections are passive, full-duplex, and interference-free.Source: §I · pp. 1–2

A. System Model

The system uses selected IRSs to relay a downlink from a multi-antenna base station to a single-antenna user when the direct link is severely blocked. A fixed, known LoS condition map determines which successive IRS routes are available.

A. System Model

  • The modeled downlink has a multi-antenna base station, a single-antenna user, and multiple IRSs; the direct link is severely blocked, so communication uses a selected multi-reflection path.Each IRS has M reflecting elements, with unit-amplitude diagonal reflection coefficients.Source: A. System Model · p. 2
  • The base station uses a uniform linear array, while each IRS uses a uniform rectangular array parallel to the x-z plane.IRS placement can provide LoS propagation on some links when the nodes are sufficiently close.Source: A. System Model · p. 2
  • A fixed, known binary LoS condition matrix records which node pairs have LoS links and thereby constrains feasible successive IRS routes.A route from the base station through selected IRSs to the user requires LoS links between every consecutive pair.Source: A. System Model · p. 2
  • The model considers only available LoS paths and assumes far-field propagation, with inter-node distances greater than 1 m.Each LoS channel is modeled using the array responses at its two ends.Source: A. System Model · p. 2

B. Optimal Active and Passive Beamforming Design

For a given feasible route, the paper optimizes the IRS phase shifts and base-station beamforming to maximize end-to-end channel power. The resulting expression exposes the trade-off between exponentially increasing cooperative gain and the attenuation and product-distance loss of additional reflections.

B. Optimal Active and Passive Beamforming Design

  • A feasible reflection route is an ordered sequence of distinct IRSs, with LoS links required from the base station, between consecutive IRSs, and from the last IRS to the user.These conditions ensure that each IRS reflects the signal at most once.Source: B. Optimal Active and Passive… · p. 3
  • For a fixed route, IRS phase shifts are chosen to maximize each cascaded amplitude, while the base station uses maximum-ratio transmission based on the first-link channel.The optimized phase shifts yield an amplitude of M for each selected IRS.Source: B. Optimal Active and Passive… · p. 3
  • With K selected IRSs, the channel power gains a multiplicative M^(2K) cooperative factor in addition to the base-station gain N.The expression also includes attenuation and product-distance loss from the multiple reflections.Source: B. Optimal Active and Passive… · p. 3
  • Because additional reflections increase both cooperative beamforming gain and signal loss, the route must balance these effects to maximize end-to-end channel power.Both attenuation and product-distance power loss generally increase with the number of reflections.Source: B. Optimal Active and Passive… · p. 3

III. PROBLEM FORMULATION

The problem is to maximize end-to-end LoS channel power by choosing a feasible reflection path. The formulation is non-convex, and assumes existing LoS channels are constant and known.

  • The objective is to maximize end-to-end LoS channel power by choosing the reflection path subject to feasibility constraints.Source: §III · p. 3
  • The resulting problem is non-convex because integer route variables are coupled in the objective and constraints.Source: §III · p. 3
  • The formulation assumes that any existing LoS channel between two nodes is constant and known; practical acquisition is left for future work.Source: §III · p. 3

IV. PROPOSED SOLUTION TO (P1)

The solution recasts the original optimization problem as an equivalent shortest simple-path problem and solves it using graph-optimization algorithms.

  • The proposed solution recasts (P1) as an equivalent shortest simple-path problem and optimally solves it using two graph-optimization algorithms.Source: §IV · p. 4

A. Problem Reformulation

The reformulation converts maximizing channel power into a shortest-path problem on a directed acyclic graph. Its edge costs encode the equivalent objective, while its paths satisfy the route constraints.

A. Problem Reformulation

  • Maximizing end-to-end channel power is equivalent to minimizing the transformed objective obtained by taking logarithms and discarding irrelevant constants.Source: A. Problem Reformulation · p. 4
  • The graph has the BS, IRSs, and user as vertices, with edges directed outward from an IRS to a farther IRS.Source: A. Problem Reformulation · p. 4
  • An edge is included for an LoS link, with IRS-to-IRS edges restricted to increasing distance from the BS; each edge weight is the logarithm of its link distance relative to M√β.Source: A. Problem Reformulation · p. 4
  • A simple path has no repeated vertices, and the constructed graph is a DAG, so the route constraints reduce to finding its minimum-weight path from the BS to the user.Source: A. Problem Reformulation · p. 4

B. Optimal Solution to (P2)

The optimal shortest-path algorithm depends on whether edge weights can be negative. For negative weights, a hop-count dynamic program finds and compares the shortest paths across feasible hop counts.

B. Optimal Solution to (P2)

  • When all edge weights are non-negative, Dijkstra's algorithm solves the shortest-path problem optimally with worst-case complexity O(|E|+(J+2) log(J+2)) using a Fibonacci heap.Source: B. Optimal Solution to (P2) · p. 4
  • With at least one negative edge weight, Dijkstra may fail; Bellman-Ford or the all-hops shortest-path approach can instead solve the problem.Source: B. Optimal Solution to (P2) · p. 4
  • The all-hops method uses dynamic programming to extend each k−1-hop shortest path by one edge, then compares paths for hop counts 1 through J.Source: B. Optimal Solution to (P2) · p. 4
  • The dynamic-programming procedure has worst-case complexity O(J|E|), comparable to Bellman-Ford and higher than Dijkstra's.Source: B. Optimal Solution to (P2) · p. 4

V. NUMERICAL RESULTS

The simulations show that the proposed routing balances cooperative passive beamforming gain against end-to-end path loss, outperforming myopic routing across the tested range. The trade-off shifts with the number of reflecting elements: path gain dominates at M = 400, while larger M favors routes with more IRSs.

V. NUMERICAL RESULTS

  • The simulation models an indoor system with 10 IRSs, LoS edges for occurrence probability above 0.9 or distance at most 12 m, and two BS antennas.The graph representation uses edge weights for M = 900; some weights are negative.Source: §V · p. 4, Figure 2 · p. 5
  • At M = 400, the optimal route uses four IRSs and is also the shortest-distance path, indicating that end-to-end path gain dominates the multiplicative passive beamforming gain.As M increases, the optimal route detours to select more IRSs because cooperative passive beamforming gain becomes more significant.Source: §V · p. 4, Figure 2 · p. 5
  • The proposed routing significantly outperforms myopic routing across the tested M range, while gain-only and path-gain-only benchmarks are close at large and small M, respectively, but worse at moderate M.Fig. 3 plots effective channel power against the number of IRS reflecting elements.Source: Figure 3 · p. 5, §V · p. 5

VI. CONCLUSIONS

The paper studies multi-IRS communication over a cascaded LoS link and presents beamforming and routing designs to maximize channel power. Its optimal routing balances path loss against cooperative passive beamforming gain, with the dominant effect depending on the number of reflecting elements.

VI. CONCLUSIONS

  • The study establishes a cascaded LoS link between a BS and remote user through successive reflections from selected IRSs.Source: §VI · p. 5
  • The proposed solution optimizes BS active beamforming, IRS passive beamforming, IRS selection, and beam routing using graph theory.Source: §VI · p. 5
  • Optimal routing balances end-to-end path loss and cooperative passive beamforming gain; path loss dominates with few reflecting elements, while beamforming gain dominates with many.Source: §VI · p. 5
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