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Secrecy Rate Maximization for Intelligent Reflecting Surface Assisted Multi-Antenna Communications

Hong Shen, Wei Xu, Shulei Gong, Zhenyao He, Chunming Zhao

arXiv:1905.10075v1cs.IT

TL;DR

The paper addresses secrecy-rate maximization in IRS-assisted multi-antenna communications under transmit-power and IRS unit-modulus constraints. It jointly optimizes transmit covariance and IRS phase shifts through an alternating algorithm with closed-form and semi-closed-form updates. The proposed design achieves higher secrecy rate than the conventional no-IRS scheme in simulations, and the algorithm is proved to converge.

  • Problem

    Maximizing secrecy rate is difficult because transmit covariance and IRS phase shifts are coupled and the phase shifts obey unit-modulus constraints.

  • Method

    An alternating algorithm optimizes transmit covariance and IRS phase shifts using closed-form and bisection-search-based semi-closed-form solutions.

  • Results

    The proposed IRS-assisted design provides a higher secrecy rate than the conventional scheme without IRS in simulations.

  • Takeaways & Limitations

    Joint optimization of Alice’s covariance and the IRS phase-shift matrix yields an optimized secrecy-rate design whose performance advantage is confirmed by simulations.

Abstract

from arXiv · show

We investigate transmission optimization for intelligent reflecting surface (IRS) assisted multi-antenna systems from the physical-layer security perspective. The design goal is to maximize the system secrecy rate subject to the source transmit power constraint and the unit modulus constraints imposed on phase shifts at the IRS. To solve this complicated non-convex problem, we develop an efficient alternating algorithm where the solutions to the transmit covariance of the source and the phase shift matrix of the IRS are achieved in closed form and semi-closed forms, respectively. The convergence of the proposed algorithm is guaranteed theoretically. Simulations results validate the performance advantage of the proposed optimized design.

I. INTRODUCTION

IRS uses passive elements with adjustable phase shifts to enhance signals and suppress interference, while this paper applies IRS optimization to physical-layer secrecy. The proposed design jointly optimizes source transmit covariance and IRS phase shifts using an alternating algorithm.

  • IRS-aided communications: IRS comprises many low-cost passive reflecting elements with adjustable phase shifts for signal enhancement and interference suppression.These functions are achieved without active transmitters.
  • Prior work: Prior IRS studies optimized SNR, transmit power, sum rate, energy efficiency, and phase-shift configurations in single-user or multiuser MISO systems.Both continuous and discrete phase shifts were considered in the cited works.
  • Paper objective: This paper maximizes secrecy rate by jointly optimizing the source transmit covariance and IRS phase shift matrix.The challenge remains even with a single-antenna eavesdropper.
  • Proposed approach: An alternating algorithm provides a closed-form transmit-covariance solution and a bisection-search-based semi-closed-form phase-shift solution.The method is also extended to multiple-antenna eavesdroppers.
  • Theoretical guarantee: The convergence of the proposed alternating algorithm is proved rigorously.

A. System Model Description

The system contains Alice, an IRS, Bob, and Eve, with Alice transmitting through direct and IRS-reflected links. The IRS applies diagonal phase shifts, while Alice’s transmit covariance is limited by a power budget.

  • System components: The considered system has one source Alice, one IRS, one legitimate receiver Bob, and one eavesdropper Eve.Alice has N antennas, the IRS has L passive reflecting elements, and Bob and Eve each have one antenna.
  • Signal model: Bob receives signals from both Alice and the IRS because the IRS reflects Alice’s transmissions.
  • IRS model: The IRS phase-shift matrix Θ is diagonal, with each reflecting element applying an adjustable phase shift.
  • Transmit model: Alice’s transmit signal has covariance W satisfying tr(W)≤P, where P is the maximum transmit power.
  • Noise model: Bob and Eve experience additive white Gaussian noise with variance σ2.

B. Secrecy Rate Maximization Problem

The secrecy-rate problem jointly optimizes Alice’s transmit covariance and the IRS phase shifts under transmit-power and unit-modulus constraints. The variables are coupled in the objective, making the problem non-trivial.

  • Objective: The achievable secrecy rate is used as the objective for jointly optimizing W and Θ.
  • Constraints: The optimization imposes the transmit-power constraint tr(W)≤P and unit-modulus constraints on all IRS phase shifts.
  • Problem difficulty: The problem is non-trivial because W and Θ are coupled in the objective function.
  • Problem difficulty: The unit-modulus constraints are usually hard to handle in optimization.

III. ALTERNATING ALGORITHM FOR SECRECY RATE MAXIMIZATION PROBLEM

The paper develops an alternating algorithm that optimizes W and Θ in turn. With Θ fixed, W has a closed-form optimum; with W fixed, Θ has a semi-closed-form solution.

  • Alternating optimization: The algorithm alternates between optimizing the source covariance W and the IRS phase-shift matrix Θ.
  • Transmit covariance update: For fixed Θ, the optimal solution for W is available in closed form.
  • IRS phase-shift update: For fixed W, optimizing Θ admits a semi-closed-form solution.

A. Closed-Form Solution to W With Given Θ

With the IRS phase-shift matrix Θ fixed, the source transmit covariance optimization admits a closed-form solution based on a normalized dominant generalized eigenvector.

  • Fixing Θ reduces the optimization to a problem over the source transmit covariance W.
  • The optimal W is obtained according to the normalized dominant generalized eigenvector of a matrix pencil.The supplied passage identifies this eigenvector as ˜w.

B. Optimization of Θ With Given W

With W fixed, the IRS phase optimization is handled through fractional programming, an upper-bound transformation, and a semi-closed-form phase update under unit-modulus constraints. The resulting alternating procedure uses bisection and has a theoretically guaranteed convergence property.

  • Optimization formulation: Fixing W leads to an optimization over Θ subject to unit-modulus constraints on every phase variable.The phase variables satisfy |θ_i| = 1 for i = 1, ···, L.
  • Fractional-programming reformulation: The Θ-subproblem is formulated as a fractional program with an introduced parameter µ and then replaced by an upper-bound minimization.The optimal objective is characterized through the unique root of ψ⋆(µ) = 0, while the upper-bound problem is used for tractability.
  • Phase-shift update: For fixed µ, the phase update is obtained by matching the phases of θ_i and β_i, yielding θ⋆(µ) = [e^{j arg(β_1)}, ···, e^{j arg(β_L)}]^T.This gives a semi-closed-form solution to the simplified Θ-subproblem.
  • Root finding: The resulting function ˜ψ⋆(µ) is strictly decreasing, and its zero is uniquely found using bisection search.The construction uses ˜ψ⋆(0) > 0 and ˜ψ⋆(+∞) < 0 to establish uniqueness.

IV. EXTENSION FOR MULTI-ANTENNA EVE

The algorithm extends to an eavesdropper with multiple antennas by reformulating the secrecy-rate optimization and modifying the phase-shift update while retaining an alternating solution strategy.

  • For M-antenna Eve, the secrecy-rate formulation is extended using the IRS-to-Eve and Alice-to-Eve channel matrices.H_IE and H_AE collect the channels from the IRS and Alice to Eve, respectively.
  • With the IRS phase shifts fixed, the source transmit covariance remains a rank-one matrix formed from the normalized dominant generalized eigenvector.The covariance is W⋆ = P w̃w̃^H, where w̃ is the normalized dominant generalized eigenvector.
  • With the transmit covariance fixed, the IRS phase-shift optimization is expressed as a unit-modulus constrained problem.Each phase variable satisfies |θ_i| = 1 for i = 1, …, L.
  • The phase-shift update is modified through a new β expression involving λ_max(Φ), Φ, μ, and α̃.The modified update uses β = (λ_max(Φ)I − Φ)θ̃ + μ α̃*.
  • The resulting scalar function used in the update is strictly decreasing, supporting the procedure for the multi-antenna Eve case.The cited derivation states that the relevant function can be verified to be strictly decreasing.

V. SIMULATION RESULTS

Simulations evaluate the proposed secrecy-rate design against a no-IRS benchmark while varying Alice–Bob distance and the number of reflecting elements. The IRS-assisted design generally performs better, with gains depending on Bob’s position, IRS size, and Eve’s antenna count.

  • The simulations use N = 4, transmit power P_A = 15 dBW, noise variance σ_n,E^2 = −75 dBW, and Rayleigh small-scale fading.The no-IRS design is used as the benchmark, with Alice’s transmit covariance optimized for secrecy rate.
  • The channel geometry sets d_AI = 50 m, d_AE,h = 44 m, and path-loss exponents ζ_AI = 2.2, ζ_IB = 2.5, ζ_IE = 2.5, ζ_AB = 3.5, and ζ_AE = 3.5.Bob and Eve lie on a horizontal line parallel to the Alice–IRS line, separated vertically by d_v = 2 m.
  • For L = 32, the proposed IRS-assisted design provides a higher secrecy rate than the conventional no-IRS scheme as d_AB,h varies.Without IRS, secrecy rate gradually decreases with d_AB,h; with IRS, it increases when d_AB,h ∈ [40 m, 50 m].
  • The proposed secrecy rate decreases as Eve’s antenna number M becomes larger because Eve’s achievable rate becomes higher.This comparison is reported from Fig. 2’s multi-antenna evaluation.
  • When Bob is far from the IRS, secrecy rate is relatively insensitive to L; when Bob is close, it increases significantly as L grows.The contrast is attributed to the IRS signal being weak at a distant Bob but dominant at a nearby Bob.
  • The simulations do not observe the squared power gain at Bob or Eve because the design maximizes secrecy rate rather than Bob’s rate alone.For relatively small L, the proposed method also achieves almost the same secrecy rate as the grid-search-based optimal solution.

VI. CONCLUSIONS

The paper studies joint secrecy-rate optimization of Alice’s transmit covariance and the IRS phase-shift matrix, using an alternating algorithm with closed-form and semi-closed-form updates.

  • Alice’s transmit covariance and the IRS phase-shift matrix are jointly optimized to maximize secrecy rate.
  • An efficient alternating algorithm optimizes the two variables in turn.
  • The transmit covariance has a closed-form solution, while the IRS phase-shift matrix has a semi-closed-form solution.
  • Simulations confirm the superiority of the proposed design.

APPENDIX A PROOF OF Lemma 1

The appendix proves properties of the scalar update function used in the phase-shift optimization, including monotonicity and uniqueness of the relevant root.

  • The proof defines f(θ|μ) and g(θ|(μ, θ̃)) as the two sides of the inequality in (9).
  • For 0 < μ_1 < μ_2, the proof compares the optimized function values at the corresponding phase-shift solutions.
  • The chain of inequalities follows from (9), minimization of g, equality when θ̃ = θ, and the ordering μ_1 < μ_2.
  • The argument uses separate functions f_E(θ) and f_B(θ) for the eavesdropper and legitimate receiver terms.
  • The proof concludes that μ′ is the unique root of ψ̃⋆(μ) = 0 and obtains γ(θ̃) from f_B(θ̃).
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