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Artificial-Noise-Aided Secure MIMO Wireless Communications via Intelligent Reflecting Surface

Sheng Hong, Cunhua Pan, Hong Ren, Kezhi Wang, Arumugam Nallanathan

arXiv:2002.07063v4eess.SPcs.IT

TL;DR

The paper formulates IRS-assisted AN-aided MIMO security enhancement as a secrecy-rate maximization problem with highly coupled optimization variables. It proposes a BCD/MM-based solution, and simulations confirm significant security gains from using an IRS.

  • Problem

    The paper addresses how to enhance security in AN-aided MIMO communication by exploiting an intelligent reflecting surface, motivated by the goal of achieving higher secrecy rate.

  • Method

    The paper solves the coupled secrecy-rate maximization problem with an efficient algorithm based on block coordinate descent and majorization-minimization techniques.

  • Results

    Simulations confirm that the IRS can greatly enhance security, with the proposed algorithm achieving significant security gains.

  • Takeaways & Limitations

    IRS-assisted AN-aided MIMO communication is presented as a viable approach for enhancing physical-layer security.

Abstract

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This paper considers a MIMO secure wireless communication system aided by the physical layer security technique of sending artificial noise (AN). To further enhance the system security performance, the advanced intelligent reflecting surface (IRS) is invoked in the AN-aided communication system, where the base station (BS), legitimate information receiver (IR) and eavesdropper (Eve) are equipped with multiple antennas. With the aim for maximizing the secrecy rate (SR), the transmit precoding (TPC) matrix at the BS, covariance matrix of AN and phase shifts at the IRS are jointly optimized subject to constrains of transmit power limit and unit modulus of IRS phase shifts. Then, the secrecy rate maximization (SRM) problem is formulated, which is a non-convex problem with multiple coupled variables. To tackle it, we propose to utilize the block coordinate descent (BCD) algorithm to alternately update the TPC matrix, AN covariance matrix, and phase shifts while keeping SR non-decreasing. Specifically, the optimal TPC matrix and AN covariance matrix are derived by Lagrangian multiplier method, and the optimal phase shifts are obtained by Majorization-Minimization (MM) algorithm. Since all variables can be calculated in closed form, the proposed algorithm is very efficient. We also extend the SRM problem to the more general multiple-IRs scenario and propose a BCD algorithm to solve it. Finally, simulation results validate the effectiveness of system security enhancement via an IRS.

I. INTRODUCTION

The paper motivates IRS-assisted physical-layer security for AN-aided MIMO systems, where jointly designing transmission, artificial noise, and IRS phases addresses coupled secrecy-rate optimization. It also considers multiple legitimate receivers and reports security gains from IRS deployment and careful configuration.

  • Motivation and problem: AN can improve secrecy rate but consumes power intended for the legitimate receiver, creating a bottleneck under a transmit-power constraint.
  • Motivation and problem: IRS complements AN by reconfiguring propagation to enhance the BS–IR channel and degrade the BS–Eve channel.
  • Proposed framework: The paper formulates secrecy-rate maximization for AN-aided MIMO security by jointly optimizing the BS TPC matrix, AN covariance matrix, and IRS phase shifts.
  • Proposed framework: Because these variables are highly coupled, an efficient BCD algorithm combined with MM is proposed to solve the problem.
  • Extensions: The SRM formulation is extended to the more general multiple-legitimate-receiver scenario with a new BCD algorithm.
  • Reported findings: Simulations indicate that IRS use enhances AN-aided MIMO security, while optimized phase shifts, larger IRS size, higher transmit power, and suitable IRS placement support performance.

II. SIGNAL MODEL AND PROBLEM FORMULATION

The system combines BS artificial noise with an IRS in a multi-antenna wiretap channel, modeling the received signals and secrecy-rate objective under CSI and propagation assumptions.

  • A. Signal Model: The IRS-aided network comprises a multi-antenna BS, legitimate IR, and Eve, with equivalent channels formed from direct and reflected links.The IRS phase-shift matrix is diagonal, with each reflection coefficient having unit magnitude.
  • A. Signal Model: Artificial noise is sent to interfere with Eve while data are transmitted toward the legitimate IR, supporting strong secrecy.The secrecy-rate formulation compares the legitimate receiver and eavesdropper rates.
  • A. Signal Model: The BS transmits data streams using a TPC matrix together with artificial noise whose covariance matrix is Z.The data vector has d streams, while the AN is modeled as a zero-mean complex Gaussian vector.
  • A. Signal Model: The IRS applies phase shifts φ_m=e^jθ_m, where each θ_m lies in [0,2π] and multiple reflected paths are absorbed or diffracted.This defines the IRS reflection model used in the received-signal equations.
  • A. Signal Model: The BS is assumed to know all channel state information and communicates optimized IRS phases to the IRS controller over a separate low-rate link.The controller link may be wireless or wired.
  • A. Signal Model: Perfect CSI is an idealistic assumption because CSI estimation for IRS networks is challenging.The discussed algorithms can derive performance upper bounds for scenarios with CSI errors, while a robust MIMO extension remains a stated future direction.
  • A. Signal Model: The IR and Eve received signals include the reflected effective channel, transmitted data, AN, and receiver noise.The corresponding data rates use interference-plus-noise covariance matrices J_I and J_E, and the secrecy rate is based on these rates.

B. Problem Formulation

The paper formulates secrecy-rate maximization by jointly optimizing BS precoding, AN covariance, and IRS phases under power and unit-modulus constraints, then develops a tractable reformulation.

  • B. Problem Formulation: The secrecy-rate problem jointly optimizes the TPC matrix V, AN covariance matrix Z, and IRS phase matrix Φ under transmit-power and unit-modulus constraints.The maximum transmit power is denoted by P_T.
  • B. Problem Formulation: The secrecy-rate optimum is nonnegative because setting V to the zero matrix yields zero secrecy rate, which exceeds any negative value.This establishes a baseline feasible outcome for the formulation.
  • B. Problem Formulation: The formulation is difficult because V, V_E, and Φ are coupled, while the IRS phase shifts impose nonconvex unit-modulus constraints.These features motivate a low-complexity solution method.
  • III. A LOW-COMPLEXITY ALGORITHM OF BCD-MM: The objective is reformulated into a more tractable expression, after which BCD-MM alternately optimizes V, V_E, and Φ.The phase-shift subproblem is obtained from the reformulated problem.
  • E ˆHE(VVH + VEVH: Auxiliary decoding and weighting matrices convert the legitimate and eavesdropper rate terms into blockwise concave functions.The construction uses linear decoding matrices, MSE matrices, and auxiliary positive-semidefinite weights.
  • E ˆHE(VVH + VEVH: The auxiliary variables U_E and W_E similarly reformulate the eavesdropper rate term and support its blockwise optimization.The relevant function is concave in each of U_E, V_E, and W_E when the others are fixed.

I ˆHIV) + Tr(VHHV V)

The algorithm solves the reformulated problem through convex subproblems, deriving precoding and AN updates with Lagrangian optimization while handling IRS phases separately.

  • BCD iteration: With auxiliary matrices fixed, the BCD iteration updates V, V_E, and Φ by solving the resulting subproblem.The auxiliary matrices are first updated, followed by the design variables.
  • TPC and AN update: Fixing Φ removes its unit-modulus constraint while optimizing the TPC and AN covariance matrices.The resulting problem is a convex QCQP.
  • TPC and AN update: Closed-form, near-optimal expressions for the TPC matrix and AN covariance matrix reduce the computational burden relative to generic convex solvers.The expressions are derived using the Lagrangian multiplier method.
  • Dual solution: The convex QCQP is solved through its dual formulation after introducing a Lagrange multiplier for the constraint.Strong duality follows from Slater’s condition.
  • Dual solution: The unconstrained convex quadratic dual subproblem yields closed-form optimal solutions for V and V_E.The solutions are obtained by setting the Lagrangian derivatives with respect to the matrices to zero.
  • Dual solution: The multiplier λ⋆ is selected to satisfy complementary slackness, after which the final V⋆ and V_E⋆ are obtained.This completes the constrained TPC and AN update.

I UIWH

The phase-shift subproblem enforces unit-modulus IRS coefficients and is transformed for efficient MM optimization. The TPC and AN covariance updates use semidefinite-matrix decompositions and a Lagrangian multiplier search.

  • TPC and AN covariance optimization: The optimal multiplier λ⋆ is selected by checking λ=0 or solving P(λ)=0 when λ⋆>0.Because P(λ) is monotonically decreasing, bisection search is used.
  • TPC and AN covariance optimization: The TPC and AN covariance matrices are characterized using SVDs of the positive semidefinite matrices H_V and H_VE.Their ranks determine positive- and zero-eigenvalue subspaces used in the decompositions.
  • IRS phase-shift optimization: With the TPC and AN covariance fixed, phase-shift optimization reduces to a unit-modulus problem for the IRS coefficients.The transmit-power constraint is removed from this block because it depends only on the matrix variables.
  • IRS phase-shift optimization: The phase-shift objective is reformulated using semidefinite matrix properties and Hadamard products before applying MM.The resulting matrix Ξ is semidefinite because it sums Hadamard products of semidefinite matrices.
  • IRS phase-shift optimization: Unlike SDR, which may fail to produce a rank-one solution and has heavy complexity, MM obtains a closed-form solution at each iteration.This provides an efficient way to solve the transformed unit-modulus problem.

D. Overall Algorithm to Solve Problem (10)

The overall BCD-MM algorithm alternates auxiliary-variable, matrix, and phase-shift updates for the single-IR problem. Its objective decreases monotonically and convergence is guaranteed by the lower-bounded power-constrained objective.

  • Algorithm steps: Algorithm 1 initializes feasible TPC, AN covariance, and phase-shift variables, then iteratively updates them until an objective or iteration termination condition is met.The updates include auxiliary matrices, matrix optimization, and MM phase-shift optimization.
  • Convergence: The BCD-MM objective decreases monotonically at every step and iteration, while the power limit supplies a lower bound.Therefore, convergence of Algorithm 1 is guaranteed.
  • Algorithm steps: Each iteration updates the TPC matrix and equivalent AN covariance matrix using Lagrangian optimization, followed by MM optimization of the IRS phase shifts.The phase-shift update produces the next Φ iterate for Problem (59).
  • Multiple-IR extension: For the multicast extension, the achievable secrecy rate is the minimum across L legitimate receivers of each receiver’s rate minus Eve’s rate.The corresponding AN-aided secrecy-rate maximization problem is reformulated for BCD processing.
  • Multiple-IR extension: The multicast reformulation is convex in each principal variable block, and a BCD-QCQP-CCP algorithm is proposed to solve it.The matrix block is handled as a convex QCQP, while phase-shift constraints are addressed through penalty CCP.

B. BCD Iterations for Problem (67)

For multiple legitimate receivers, the paper alternates optimization of matrix, phase-shift, and auxiliary-variable blocks. The non-convex unit-modulus phase-shift constraints are handled with penalty CCP.

  • BCD reformulation: The multicast objective is reformulated into a lower-bound problem suitable for block coordinate descent.The reformulation introduces auxiliary variables for legitimate receivers, Eve, and the associated matrix terms.
  • BCD reformulation: The reformulated problem is convex with respect to each individual block of matrices, phase shifts, or auxiliary variables when the others are fixed.This block structure enables alternating updates.
  • Matrix optimization: The matrix subproblem for V and VE is a convex QCQP solvable with a general-purpose convex optimization solver.This update is performed after fixing the relevant other variables.
  • Phase-shift optimization: The phase-shift block retains unit-modulus constraints and is transformed through complex algebra before further simplification.The resulting objective is expressed in a form prepared for constraint handling.
  • Phase-shift optimization: Penalty CCP linearizes the non-convex parts after introducing slack variables and penalizing their l1 norm.The regularization factor λ(t) controls constraint feasibility.

V. SIMULATION RESULTS

Simulations evaluate IRS assistance in a three-terminal AN-aided MIMO wiretap scenario under specified fading, geometry, and system parameters. The proposed algorithms converge, while more IRS elements improve the converged secrecy rate at higher computational cost.

  • Simulation setup: The simulations use one multi-antenna BS, one legitimate IR, and one multi-antenna Eve in a three-terminal MIMO Gaussian wiretap channel.The IRS-related channels use Rician fading, while direct channels use Rayleigh fading.
  • Simulation setup: The default setup has NT=4, NI=2, NE=2, d=2 data streams, M=50 IRS elements, and PT=15 dBm.Channels are independently realized 200 times for averaging.
  • BCD convergence: The BCD algorithm’s secrecy rate increases with iteration number and reaches a stable value, converging quickly at almost 20 iterations.The simulations examine M=10, 20, and 40 phase shifts.
  • BCD convergence: A larger M produces a higher converged secrecy rate and therefore better security in the simulated setting.The reported comparison uses the BCD convergence curves for different numbers of IRS phase shifters.
  • MM convergence: The MM phase-shift iterations also increase secrecy rate toward a stable value, while larger M yields higher convergence values but slower convergence.This behavior is observed in the inner-layer process.
  • Convergence trade-off: More phase shifts reduce convergence speed because they introduce more optimization variables and higher computation complexity.The same trade-off is reported for the inner MM iterations.

B. Performance Evaluation

The evaluation compares IRS-assisted designs across transmit power, IRS size and placement, channel conditions, and optimization schemes. Results show that properly optimized IRS phase shifts substantially improve secrecy rate, especially with favorable IRS links and larger surfaces.

  • 1) Impact of Transmit Power:: The proposed BCD-MM algorithm significantly outperforms the benchmark schemes across the transmit-power range.Its advantage over BCD-QCQP-SDR increases with the power limit, while all compared schemes improve as transmit power increases.
  • 1) Impact of Transmit Power:: Even randomly selected IRS phase shifts improve secrecy rate over the No-IRS scheme, but optimized phase shifts provide a much larger gain.Optimization strengthens the IR signal constructively and weakens the Eve signal destructively.
  • 3) Impact of the Location of the IR:: IRS placement near the legitimate receiver enables significant security enhancement, with the optimized scheme's gain over RandPhase increasing for dBI ∈[40m, 50m].When the IR is far from the IRS, RandPhase and No-IRS can have similar secrecy rates.
  • 4) Impact of the Path Loss Exponent of IRS-related Links:: 9.6 bit/s/Hz and 6.8 bit/s/Hz are the reported gains at αIRS = 2 over No-IRS and RandPhase, respectively.The BCD-MM gain decreases as the IRS-related path-loss exponent increases, making favorable IRS channels important.

5) Impact of the Number of Data Streams:

The evaluation examines data streams, reflection amplitude, discrete phase resolution, and multiple legitimate receivers. Larger stream counts and stronger IRS reflection improve secrecy rate, while discrete phases and multiple receivers introduce practical performance constraints.

  • 5) Impact of the Number of Data Streams:: Larger numbers of data streams produce higher secrecy rates, with gains expanding as the transmit-power limit increases.The improvement is marginal at low transmit power but becomes significant at high power.
  • 5) Impact of the Number of Data Streams:: Four receiving antennas outperform one at relatively low transmit power, but their secrecy-rate advantage diminishes as power increases and eventually saturates.The MIMO advantage also includes supporting multiple data streams.
  • 6) Impact of Reflection Amplitude:: Increasing IRS reflection amplitude η from 0.2 to 1 raises the BCD-MM secrecy rate by over 3.6 bit/s/Hz.Higher η reduces reflection power loss and makes the proposed algorithm's advantage more pronounced.
  • 7) Impact of Discrete Phase Shifts:: Discrete-phase secrecy rate increases with control bits b but saturates when b ≥4, remaining below continuous-phase performance.The maximum reported gap between continuous and discrete phase shifts is 1.4 bit/s/Hz.
  • 8) Multiple-IRs Scenario:: With multiple legitimate receivers, the proposed BCD-QCQP-CCP algorithm achieves higher secrecy rates than random-IRS and No-IRS schemes.The multiple-IR setting uses M = 20 because of computational load, and phase design is more difficult for more legitimate receivers.

APPENDIX A

The appendix reformulates the artificial-noise objective by expanding its component functions and collecting terms that are constant with respect to selected optimization variables.

  • DERIVATION OF THE PROBLEM (27): The appendix derives an equivalent artificial-noise objective by substituting auxiliary functions and separating constant terms.The resulting expression contains log-determinant and trace terms associated with the legitimate receiver, Eve, and auxiliary variables.
  • DERIVATION OF THE PROBLEM (27): The component functions g1, g2, and g3 are individually expanded before being combined into a more compact objective.Terms independent of V, VE, and Φ are gathered into constants during the reformulation.
  • DERIVATION OF THE NEW OF FORM IN (55): The reformulated objective is substituted into the optimization problem after removing its constant term.The appendix then begins a separate derivation for the new form in (55).

VVH + VEVH

This appendix segment combines several expanded terms into a compact function of the IRS phase-shift matrix and collects the resulting constant contributions.

  • VVH + VEVH: The derivation uses previously obtained component expressions to construct the phase-shift-dependent objective.The text explicitly references the second and third parts of the preceding decomposition.
  • VVH + VEVH: Adding the component equations and gathering terms independent of Φ yields a compact expression for g0(Φ).The expression combines Equations (103)–(108) with their stated signs.
  • VVH + VEVH: The constant contribution is decomposed as Ct = Ct1 + Ct2 + Ct3 + Ct4 + Ct5.This decomposition organizes the terms that do not depend on the IRS phase shifts.
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