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Secure Transmission for Intelligent Reflecting Surface-Assisted mmWave and Terahertz Systems

Jingping Qiao, Mohamed-Slim Alouini

arXiv:2005.13451v1cs.ITeess.SP

TL;DR

The paper studies secure transmission in IRS-assisted mmWave/THz systems, where propagation loss, blockage, narrow beams, and discrete phase control complicate secrecy optimization. It separates the joint design into independent subproblems under a rank-one channel model, deriving closed-form beamforming and SDP- or element-wise BCD-based reflecting designs. Simulations report near-optimal secrecy performance and resistance to eavesdropping near or blocking the BS and IRS beams.

  • Problem

    Severe loss, blockage, narrow beams, and costly continuous phase control make secure IRS-assisted mmWave/THz transmission challenging and underexplored.

  • Method

    The paper jointly optimizes BS beamforming and discrete IRS phase shifts, separates the problem under rank-one channels, and applies closed-form, SDP, and element-wise BCD designs.

  • Results

    The proposed methods achieve near-optimal secrecy performance with discrete phase shifts and combat eavesdropping occurring at the BS and IRS.

  • Takeaways & Limitations

    IRS-assisted secure transmission improves secrecy performance in mmWave/THz settings, including cases where eavesdroppers are near or block confidential beams.

Abstract

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This letter focuses on the secure transmission for an intelligent reflecting surface (IRS)-assisted millimeter-wave (mmWave) and terahertz (THz) systems, in which a base station (BS) communicates with its destination via an IRS, in the presence of a passive eavesdropper. To maximize the system secrecy rate, the transmit beamforming at the BS and the reflecting matrix at the IRS are jointly optimized with transmit power and discrete phase-shift constraints. It is first proved that the beamforming design is independent of the phase shift design under the rank-one channel assumption. The formulated non-convex problem is then converted into two subproblems, which are solved alternatively. Specifically, the closed-form solution of transmit beamforming at the BS is derived, and the semidefinite programming (SDP)-based method and element-wise block coordinate descent (BCD)-based method are proposed to design the reflecting matrix. The complexity of our proposed methods is analyzed theoretically. Simulation results reveal that the proposed IRS-assisted secure strategy can significantly boost the secrecy rate performance, regardless of eavesdropper's locations (near or blocking the confidential beam).

I. INTRODUCTION

The paper addresses secure transmission challenges in blockage-prone mmWave/THz systems by using an IRS to improve coverage and reduce information leakage. It formulates joint beamforming and reflecting-matrix optimization under discrete phase shifts for an IRS-assisted secure link.

  • mmWave and THz systems offer high data rates but suffer severe propagation loss, short secure propagation distance, and unreliable secure communications.
  • IRSs adjust phase shifts to steer signal power toward the desired user and reduce information leakage in high-frequency systems.
  • THz security challenges are intensified by higher frequency, while narrow beams create leakage risks from beam misalignment and blockage can disrupt legal communications.
  • Prior secure IRS studies mainly considered microwave systems, leaving IRS-assisted secure mmWave/THz transmission comparatively unexplored.
  • The paper jointly optimizes BS transmit beamforming and IRS reflecting matrices under discrete phase shifts, proving beamforming independence under rank-one channels.
  • The system contains a multi-antenna BS, single-antenna Bob and Eve, an IRS near Bob, and no direct links because of path loss or blockage.

B. Signal Model

The system uses an IRS to reflect a BS transmission toward Bob while accounting for a passive eavesdropper and discrete IRS phase shifts. The secrecy-rate analysis assumes perfect channel knowledge at the BS.

  • B. Signal Model: The BS transmits signal s with power P_s to the IRS, whose reflecting elements adjust phase shifts to direct the signal toward Bob.The reflecting matrix is Θ = diag{e^jθ_1, e^jθ_2, ..., e^jθ_N}.
  • B. Signal Model: The system secrecy rate is defined from the received signals at Bob and the eavesdropper, with [x]^+ = max{0, x}.The supplied passages identify separate destination and eavesdropper noise terms.
  • B. Signal Model: The BS transmit beamforming is hybrid, with w = F_RF f_BB, where F_RF is the analog beamformer and f_BB is the digital beamformer.R denotes the number of RF chains.
  • B. Signal Model: All channels are assumed perfectly known at the BS, so the derived results represent a performance upper bound.This assumption applies to the channel knowledge used in the analysis.

C. Problem Formulation

The secrecy-rate maximization jointly designs BS beamforming and the IRS reflecting matrix under coupled variables and discrete phase constraints. Under the rank-one BS-to-IRS channel assumption, beamforming separates from reflecting-matrix design, enabling alternating solution methods.

  • C. Problem Formulation: The joint optimization problem maximizes system secrecy rate through transmit beamforming and reflecting-matrix design.The formulation belongs to subsection C. Problem Formulation.
  • C. Problem Formulation: The original problem is non-convex because beamforming and reflecting-matrix variables are coupled and the phase-shift constraint is non-convex.These features make direct solution challenging.
  • C. Problem Formulation: The original problem P1 is converted into two subproblems that are solved alternately.This decomposition addresses the coupled optimization structure.
  • III. SECRECY RATE MAXIMIZATION: The two subproblems are treated as independent: the closed-form beamformer is derived first, followed by reflecting-matrix optimization.The proposed reflecting-matrix methods are SDP-based and element-wise BCD-based.
  • A. Transmit Beamforming Design: Under the rank-one BS-to-IRS channel assumption, the beamformer subproblem is formulated separately.This rank-one assumption is the basis for the beamforming-design result.
  • A. Transmit Beamforming Design: Proposition 1 states that, under a positive secrecy-rate constraint, the beamformer optimization is equivalent to a simplified subproblem.The proposition concerns the suboptimal problem of w.
  • A. Transmit Beamforming Design: The optimal transmit beamformer is independent of the reflecting matrix for any value of Θ.The result is given as w_opt = √... in the supplied passage.
  • A. Transmit Beamforming Design: After obtaining w_opt, the full-connected hybrid precoder can be derived using typical methods such as the OMP algorithm.This provides an implementation route for the resulting beamformer.

B. Reflecting Matrix Design

The reflecting-matrix subproblem is reparameterized using a vector of element-wise phase terms. Because discrete phases create a large finite search space, the paper motivates SDP relaxation and element-wise BCD alternatives.

  • B. Reflecting Matrix Design: The reflecting matrix is reparameterized as Θ = diag{θ̂}, where θ̂ contains the element-wise terms e^jθ_i.This vector form facilitates subsequent mathematical operations.
  • B. Reflecting Matrix Design: Each discrete phase θ_i takes a finite value from F, making exhaustive search feasible in principle but costly for large N.The feasible set contains N_LP possibilities as written in the supplied passage.
  • B. Reflecting Matrix Design: The paper proposes SDP-based and element-wise BCD algorithms to reduce the burden of solving the discrete reflecting-matrix problem.The SDP formulation relaxes discrete θ_i into continuous values satisfying |e^jθ_i| = 1.

1) SDP-based Algorithm:

The SDP-based method relaxes the rank-one reflecting-matrix constraint, solves the resulting SDP, reconstructs a rank-one solution by Gaussian randomization, and quantizes it to discrete phase shifts. Its main limitation is high per-transmission-block complexity because no closed-form phase-shift solution is available.

  • 1) SDP-based Algorithm:: The rank-one constraint on the reflecting-matrix variable is relaxed using semidefinite relaxation.
  • 1) SDP-based Algorithm:: The transformed problem is solved as a standard SDP using an interior-point method or CVX tools.
  • 1) SDP-based Algorithm:: Gaussian randomization reconstructs a rank-one solution, which is then quantized to the nearest discrete phase value in F.
  • 1) SDP-based Algorithm:: Because no closed-form θ_i solution is obtained, the SDP-based procedure must run for each transmission block, causing high complexity.

2) Element-Wise BCD Algorithm:

The element-wise BCD method treats each IRS phase shift as one block and iteratively updates the continuous phase solutions before selecting discrete values. Its objective is non-decreasing and bounded, guaranteeing convergence.

  • 2) Element-Wise BCD Algorithm:: The algorithm quantizes each continuous phase solution using the same discrete-selection principle.
  • 2) Element-Wise BCD Algorithm:: The objective function is non-decreasing after each iteration and upper-bounded by a finite generalized-eigenvalue value, so convergence is guaranteed.
  • 2) Element-Wise BCD Algorithm:: Algorithm 1 initializes the reflecting matrix, obtains w_opt and F_opt, and iteratively updates the phase shifts element by element.

C. Complexity Analysis

The element-wise BCD method has the lowest stated complexity among the proposed and exhaustive-search approaches, while the SDP method is dominated by SDP solving and Gaussian randomization.

  • C. Complexity Analysis: The SDP-based method has complexity O(NgausN^8), determined mainly by SDP solving and Gaussian randomization.
  • C. Complexity Analysis: The element-wise BCD method has complexity O(N(NM+LP)Niter) and is stated to have the lowest complexity.
  • C. Complexity Analysis: Both proposed methods have lower complexity than exhaustive search, whose complexity is O(N^(LP+1)(N^2+NM)).

IV. SIMULATION RESULTS AND ANALYSIS

Simulations average 1000 trials for IRS-assisted mmWave/THz systems and examine discrete phase resolution, transmit power, antennas, reflecting elements, and interception with blocking. The proposed methods approach exhaustive-search performance and improve secrecy rate under the investigated settings.

  • IV. SIMULATION RESULTS AND ANALYSIS: The simulations average results over 1000 independent trials using BS ULA and IRS URA configurations at 0.3 THz unless otherwise specified.The setup uses Ps = 25 dBm, M = 16, M_RF = 10, and L_P = 2^3.
  • IV. SIMULATION RESULTS AND ANALYSIS: As L_P increases, the proposed methods approach exhaustive-search secrecy-rate performance, while discrete schemes remain upper-bounded by the continuous solution.
  • IV. SIMULATION RESULTS AND ANALYSIS: The exhaustive-search solution is optimal for the discrete case because it searches all feasible phase-shift combinations.
  • IV. SIMULATION RESULTS AND ANALYSIS: Increasing transmit power monotonically increases secrecy rate, and the proposed methods achieve near-optimal performance relative to exhaustive search.
  • IV. SIMULATION RESULTS AND ANALYSIS: More BS antennas and more reflecting elements increase secrecy rate, while IRS-assisted secure transmission significantly improves performance over the secure-oblivious approach under interception and blocking.

V. CONCLUSION

The letter divides the non-convex design into two subproblems, derives closed-form beamforming, and obtains the reflecting matrix using SDP-based and element-wise BCD methods. Simulations show near-optimal secrecy performance against interception at the BS and IRS.

  • The original joint design problem is divided into two subproblems under the rank-one channel assumption.
  • Fig. 3 evaluates secrecy rate against the number of reflecting elements for IRS- and BS-interception settings.The plotted range is 10 to 100 reflecting elements, with separate non-blocking and blocking distances.
  • The reflecting matrix is obtained using the proposed SDP-based and element-wise BCD methods.
  • The closed-form beamforming solution is derived for the transmit design.
  • The proposed methods achieve near-optimal secrecy performance and combat eavesdropping at the BS and IRS.

APPENDIX A PROOF OF PROPOSITION 1

The proof shows that, under the positive secrecy-rate constraint, optimizing beamforming reduces to maximizing |b^H w|^2 independently of the IRS phase matrix. The element-wise BCD subproblem is then analyzed through a trigonometric objective and derivative-based phase cases.

  • Under the positive secrecy-rate constraint, the secrecy rate increases with |b^H w|^2, making beamforming independent of Θ.
  • The IRS phase-design objective is reformulated by selecting each element θ_i as one block of the element-wise BCD method.
  • The derivative sign is determined using the reformulated trigonometric objective and its parameters.
  • The optimization uses a sin(x)-based expression instead of cos(x) to reduce complexity.
  • The optimal phase cases depend on the sign of A_i, with distinct formulas for A_i > 0 and A_i < 0.
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