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Robust and Secure Wireless Communications via Intelligent Reflecting Surfaces

Xianghao Yu, Dongfang Xu, Ying Sun, Derrick Wing Kwan Ng, Robert Schober

arXiv:1912.01497v3cs.ITeess.SP

TL;DR

The paper addresses secure multiuser wireless communication when legitimate users lack line-of-sight links and eavesdropper CSI is imperfect. It jointly optimizes AP beamforming, artificial noise, and IRS phases under robust leakage constraints, using alternating optimization with SCA and SDR-based procedures. Simulations report substantially improved secrecy with IRSs and favor uniformly distributing reflecting elements across multiple IRSs.

  • Problem

    The paper considers physical-layer security for blocked legitimate users facing multiple multi-antenna eavesdroppers with imperfect CSI.

  • Method

    It jointly optimizes AP beamformers, AN covariance, and IRS phase shifts through alternating optimization, penalty-based methods, SCA, and SDR.

  • Results

    IRSs significantly improve secrecy performance, and uniformly distributing reflecting elements across multiple IRSs is favorable for physical-layer security.

  • Takeaways & Limitations

    IRS deployment creates favorable propagation conditions for legitimate users and provides design guidance favoring multiple IRSs with uniformly distributed elements.

Abstract

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In this paper, intelligent reflecting surfaces (IRSs) are employed to enhance the physical layer security in a challenging radio environment. In particular, a multi-antenna access point (AP) has to serve multiple single-antenna legitimate users, which do not have line-of-sight communication links, in the presence of multiple multi-antenna potential eavesdroppers whose channel state information (CSI) is not perfectly known. Artificial noise (AN) is transmitted from the AP to deliberately impair the eavesdropping channels for security provisioning. We investigate the joint design of the beamformers and AN covariance matrix at the AP and the phase shifters at the IRSs for maximization of the system sum-rate while limiting the maximum information leakage to the potential eavesdroppers. To this end, we formulate a robust nonconvex optimization problem taking into account the impact of the imperfect CSI of the eavesdropping channels. To address the non-convexity of the optimization problem, an efficient algorithm is developed by capitalizing on alternating optimization, a penalty-based approach, successive convex approximation, and semidefinite relaxation. Simulation results show that IRSs can significantly improve the system secrecy performance compared to conventional architectures without IRS. Furthermore, our results unveil that, for physical layer security, uniformly distributing the reflecting elements among multiple IRSs is preferable over deploying them at a single IRS.

I. INTRODUCTION

The paper motivates IRS-assisted physical-layer security as a low-cost way to control unfavorable propagation environments in multiuser systems with blocked legitimate links and capable eavesdroppers.

  • Motivation: Wireless broadcast and superposition make transmissions vulnerable to security breaches, motivating physical-layer techniques that exploit channel noise and fading.
  • Motivation: Existing relaying and jamming approaches incur hardware and energy costs, while AN may provide inadequate secrecy in unfavorable propagation environments.
  • IRS rationale: IRSs use low-cost passive phase shifters to alter end-to-end propagation and can be coated on existing infrastructure.
  • System and objective: The paper studies multiple single-antenna legitimate users without AP line-of-sight links and multiple multi-antenna potential eavesdroppers.
  • System and objective: Beamformers, AN covariance, and IRS phase shifts are jointly designed to maximize system sum-rate while constraining information leakage under imperfect eavesdropper CSI.
  • System model: The system model includes one AP, K legitimate users, J roaming users treated as eavesdroppers, and L passive IRSs; legitimate users have one antenna while eavesdroppers have multiple antennas.

B. Channel State Information (CSI)

The formulation assumes periodically acquired perfect CSI for AP–IRS–legitimate-user links but bounded uncertainty for eavesdropper channels, then maximizes sum-rate under worst-case leakage constraints.

  • CSI assumptions: The AP periodically acquires CSI for legitimate-user links and assumes it is perfect throughout transmission.
  • CSI assumptions: Eavesdropper CSI is coarse and outdated because roaming users do not cooperate with the AP for channel acquisition.
  • CSI uncertainty: Each eavesdropper channel is modeled as an estimate plus an error from a continuous norm-bounded uncertainty set with radius ǫj.
  • Optimization formulation: The WCR-SRM problem maximizes system sum-rate while keeping maximum information leakage below a desired level.
  • Optimization formulation: The IRS phase matrix has diagonal unit-modulus elements, while the AN covariance matrix is Hermitian positive semidefinite.
  • Optimization formulation: The leakage threshold τk,j controls tolerable information leakage for eavesdropper j decoding legitimate user k.

IV. ALGORITHM DESIGN FOR SECURE IRS-ASSISTED WIRELESS COMMUNICATION

The algorithm alternates between transmit variables and IRS phases, converting robust constraints and nonconvex subproblems into tractable approximations using LMIs, SDR, and SCA.

  • Alternating optimization: Alternating optimization fixes the IRS phases while optimizing beamformers and AN, then reverses the roles to obtain a stationary-point solution.
  • Transmit optimization: For fixed IRS phases, SDR and SCA address the nonconvex objective and robust constraints in the beamformer and AN optimization.
  • Robust constraints: The infinitely many robust leakage constraints are converted into finitely many equivalent linear matrix inequality constraints.
  • Robust constraints: The generalized S-procedure supplies a finite LMI representation for the quadratic matrix inequality arising from CSI uncertainty.
  • Transmit optimization: The transmit problem is rewritten as a rank-constrained semidefinite program using Wk = wkwk^H.

D1 (W, Z)

For the transmit-variable subproblem, SDR removes the rank constraint, while a theorem establishes that an optimal rank-one solution can still be obtained.

  • Rank relaxation: The remaining nonconvexity after upper-bounding the objective is the rank constraint C6.
  • Rank relaxation: SDR drops the rank constraint because rank-constrained optimization is generally NP-hard.
  • Rank relaxation: The relaxed problem is jointly convex in pk,j, Wk, and Z and can be solved by standard convex-program solvers.
  • Rank recovery: Theorem 1 states that another optimal solution exists with the required rank property and the same objective value.
  • Rank recovery: The rank-one beamforming solution can be constructed using the procedure given in (44).

B. Optimization of Phase Shifts at IRSs

The IRS phase-shift design is reformulated to address unit-modulus and rank-one constraints, enabling an alternating optimization algorithm based on SCA, SDR, and penalty optimization.

  • The unit-modulus constraint on IRS phase shifts is the main difficulty in optimizing the phase-shift matrix.
  • Replacing Φ with v = Diag(Φ) enables semidefinite programming and successive convex approximation for phase-shift design.
  • The lifted matrix V = vvH imposes Diag(V) = 1M, V ⪰ 0, and Rank(V) = 1 to preserve unit-modulus phase shifts.
  • A penalty-based method moves the nonconvex rank-one constraint into the objective function while preserving equivalence as the penalty factor approaches zero.
  • The IRS phase-shift solution can be recovered from a rank-one matrix using Cholesky decomposition, while each iteration has polynomial-time computational complexity.
  • The overall alternating optimization algorithm iteratively solves transmit and IRS subproblems, monotonically tightens upper bounds, and converges to a stationary value.

A. Simulation Setup

The simulations model an IRS-assisted secure downlink with blocked AP-to-user links, Ricean AP-to-IRS channels, and imperfect eavesdropper CSI.

  • The simulated network contains one AP, K legitimate users, J potential eavesdroppers, and L IRSs within a single cell.
  • Legitimate users and potential eavesdroppers are uniformly distributed in the cell, while obstacles block the direct AP-to-legitimate-user links.
  • AP-to-IRS channels use Ricean fading with path-loss exponent αl, distance dl, Ricean factor βl, and line-of-sight and non-line-of-sight components.
  • The simulations characterize imperfect eavesdropping-channel knowledge through the maximum normalized estimation error κj.

B. Baseline Schemes

The evaluation compares optimized IRS-assisted transmission with fixed MRT/random-phase designs and no-IRS operation, examining convergence, power, secrecy constraints, and IRS deployment.

  • B. Baseline Schemes: Baseline 1 uses MRT beamforming, isotropic artificial-noise radiation, and random IRS phases without iterative optimization.
  • B. Baseline Schemes: Baseline 2 removes the IRS and models blocked legitimate-user links as non-line-of-sight, while retaining direct links for the comparison.
  • C. Convergence of the Proposed Algorithm: The proposed algorithm converges monotonically across antenna, reflecting-element, and user settings; larger systems require more iterations.
  • D. Average System Sum Rate Versus the Maximum Transmit Power: The proposed scheme’s average system sum-rate increases with AP transmit power and exceeds both baseline schemes by a considerable margin.
  • E. Secrecy Rate Versus the Maximum Tolerable Channel Capacity of the Eavesdroppers: For blocked legitimate users, the no-IRS baseline has an almost-zero secrecy rate, whereas the proposed scheme substantially outperforms baseline 1.
  • E. Secrecy Rate Versus the Maximum Tolerable Channel Capacity of the Eavesdroppers: Tighter eavesdropper-capacity limits allocate more transmit power to artificial noise, leaving less power for information beamforming and system sum-rate maximization.
  • E. Secrecy Rate Versus the Maximum Tolerable Channel Capacity of the Eavesdroppers: As the tolerable eavesdropper capacity increases, more power shifts to beamforming and system sum-rate improvement, but the secrecy rate decreases for large τ.

F. Energy Efficiency Evaluation

The evaluation examines energy efficiency, sum-rate scaling, CSI uncertainty, and IRS deployment choices under the proposed secure communication design. Results indicate benefits from IRS assistance, robust optimization, and distributing reflecting elements across multiple IRSs.

  • Energy efficiency: Energy efficiency is defined as system sum-rate divided by total system power consumption under a linear power consumption model.The model includes amplifier, antenna RF-chain, AP static, and IRS-controller power terms.
  • Energy efficiency: Increasing IRS reflecting elements improves energy efficiency, whereas increasing AP antennas monotonically decreases it.Passive IRS elements add little power consumption, while additional AP antennas require more power-hungry RF chains.
  • Sum-rate and robustness: The proposed scheme and both baselines increase sum-rate with more legitimate users, but the baselines achieve lower rates and growth rates.The proposed scheme benefits from multiuser diversity, while MRT-based baseline 1 saturates because it fails to mitigate multiuser interference.
  • Sum-rate and robustness: Under CSI uncertainty, all schemes lose sum-rate, while the proposed scheme remains superior across the considered estimation-error range.The paper attributes the degradation to less accurate beamforming and artificial-noise jamming.
  • Sum-rate and robustness: At −1 dB target SINR, proposed-scheme outage probability is 10% versus 80% for the non-robust scheme.With τk,j = 1 bit/s/Hz, robust schemes have zero outage at target SINRs no less than 0 dB, whereas the non-robust scheme remains above 30% at 0 dB.
  • IRS deployment: With two IRSs, uniformly allocating the 10 reflecting elements achieves the peak system sum-rate.Equal allocation gives five elements to each IRS; multiple IRSs provide independent propagation paths and macro diversity.

APPENDIX

The appendix develops a rank-related argument involving matrices associated with the secure beamforming formulation. It uses determinant and rank properties to establish an intermediate result.

  • Appendix: Sylvester’s determinant identity is used to relate determinants of I + AB and I + BA.
  • Appendix: The appendix notes that a matrix has rank one, so its maximum eigenvalue is its only nonzero eigenvalue.Substitution into the final linear matrix inequality yields the result in Proposition 1.
  • Appendix: The Proposition 1 result follows by substituting the defined Qj expression into the last linear matrix inequality.

B. Proof of Theorem 1

The proof of Theorem 1 analyzes the convex subproblem’s KKT structure and constructs rank-one beamforming solutions. It concludes that the semidefinite relaxation can be made tight without changing the optimal objective value.

  • Proof of Theorem 1: The proof formulates the relevant problem in epigraph form and invokes Slater’s condition to establish strong duality.
  • Proof of Theorem 1: KKT conditions are used to characterize the structure of the optimal beamforming matrix Wk through Lagrange multipliers.The proof distinguishes cases based on the ranks of matrices such as B⋆ and N⋆k.
  • Proof of Theorem 1: The proof rules out zero-rank W⋆k because it would deliver no information to legitimate user k despite allocating beamformer power.
  • Proof of Theorem 1: A rank-one matrix W̃⋆k and positive-semidefinite matrix Z̃⋆ are constructed from the optimal solution.The construction preserves feasibility and the objective terms needed for the theorem.

C. Proof of Proposition 2

The proof of Proposition 2 studies a penalized formulation that moves a rank constraint into the objective. It shows that limit points of penalized solutions are feasible and optimal for the original problem.

  • Proof of Proposition 2: The penalized objective g(V; ρ) is compared with the original objective f(V) using an optimal solution V⋆ of the rank-constrained problem.
  • Proof of Proposition 2: A limit point V̄ of the penalized solution sequence is considered along an infinite convergent subsequence.
  • Proof of Proposition 2: The shared constraints are retained while constraints C9 is moved into the objective for the penalized problem.
  • Proof of Proposition 2: Continuity of the nuclear-norm-minus-spectral-norm term ensures that V̄ satisfies the moved rank constraint.The proof also uses nonnegativity of the penalty factor and this norm difference.
  • Proof of Proposition 2: Because V̄ is feasible and no worse than the optimal solution V⋆, it is also optimal for the original problem.
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