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Enhancing Secure MIMO Transmission via Intelligent Reflecting Surface

Limeng Dong, Hui-Ming Wang

arXiv:2007.13944v1cs.IT

TL;DR

The paper addresses secrecy-rate optimization for IRS-assisted Gaussian MIMO wiretap channels when prior methods are limited to MISO settings and often assume full eavesdropper CSI. It develops alternating-optimization and artificial-noise schemes for full-CSI, unknown-CSI, and blocked-direct-link cases, with simulations showing improved secrecy and a QoS–secrecy trade-off.

  • Problem

    Existing IRS-assisted secure-wireless methods are restricted mainly to MISO channels and assume full eavesdropper CSI, leaving the general MIMO and unknown-CSI problems unresolved.

  • Method

    The paper jointly optimizes transmit covariance and IRS phase shifts using barrier/OBO alternating optimization under full CSI, and artificial-noise schemes with BS/OBO or BS/MM optimization when eavesdropper CSI is unavailable.

  • Results

    Simulations show higher secrecy rates than benchmark schemes under full CSI, secrecy enhancement with unknown eavesdropper CSI, and slightly faster but lower-performing convergence for the blocked-link MM method.

  • Takeaways & Limitations

    IRS phase control and artificial noise can enhance secrecy in MIMO wiretap channels even without eavesdropper CSI, subject to a trade-off between receiver QoS and secrecy rate.

Abstract

from arXiv · show

In this paper, we consider an intelligent reflecting surface (IRS) assisted Guassian multiple-input multiple-output (MIMO) wiretap channel (WTC), and focus on enhancing its secrecy rate. Due to MIMO setting, all the existing solutions for enhancing the secrecy rate over multiple-input single-output WTC completely fall to this work. Furthermore, all the existing studies are simply based on an ideal assumption that full channel state information (CSI) of eavesdropper (Ev) is available. Therefore, we propose numerical solutions to enhance the secrecy rate of this channel under both full and no Ev's CSI cases. For the full CSI case, we propose a barrier method and one-by-one (OBO) optimization combined alternating optimization (AO) algorithm to jointly optimize the transmit covariance R at transmitter (Tx) and phase shift coefficient Q at IRS. For the case of no Ev's CSI, we develop an artificial noise (AN) aided joint transmission scheme to enhance the secrecy rate. In this scheme, a bisection search (BS) and OBO optimization combined AO algorithm is proposed to jointly optimize R and Q. Such scheme is also applied to enhance the secrecy rate under a special scenario in which the direct link between Tx and receiver/Ev is blocked due to obstacles. In particular, we propose a BS and minorization-maximization (MM) combined AO algorithm with slightly faster convergence to optimize R and Q for this scenario. Simulation results have validated the monotonic convergence of the proposed algorithms, and it is shown that the proposed algorithms for the IRS-assisted design achieve significantly larger secrecy rate than the other benchmark schemes under full CSI. When Ev's CSI is unknown, the secrecy performance of this channel also can be enhanced by the proposed AN aided scheme, and there is a trade-off between increasing the quality of service at Rx and enhancing the secrecy rate.

I. INTRODUCTION

The paper targets secrecy-rate enhancement for IRS-assisted Gaussian MIMO wiretap channels, addressing limitations of MISO-only methods and unavailable eavesdropper CSI. It develops optimization schemes for full-CSI and no-CSI settings, including an artificial-noise design and a blocked-direct-link scenario.

  • Motivation: IRS-assisted physical-layer security adjusts propagation channels to improve secrecy against eavesdroppers.IRS elements passively impose controllable phase shifts with low power consumption, enabling reflected signals to combine differently at legitimate and eavesdropping receivers.
  • Research gap: Existing IRS-assisted secure-wireless methods mainly address MISO channels, while the MIMO problem requires optimizing a covariance matrix and a log-determinant secrecy objective.These differences make prior MISO numerical solutions unsuitable for the general MIMO setting with direct links.
  • Problem formulation: The paper studies numerical secrecy-rate maximization for a general IRS-assisted Gaussian MIMO wiretap channel with transmitter, receiver, eavesdropper, and IRS.The contribution jointly considers transmit covariance and IRS phase-shift design under different eavesdropper-CSI assumptions.
  • Full CSI: Under full eavesdropper CSI, an alternating-optimization algorithm jointly optimizes transmit covariance and IRS phase shifts using a barrier method and one-by-one phase optimization.The covariance subproblem uses barrier, Newton, and backtracking methods, while the phase coefficients are optimized sequentially.
  • No eavesdropper CSI: With unknown eavesdropper CSI, an artificial-noise scheme first satisfies a receiver QoS constraint with minimum power, then uses residual power to jam the eavesdropper.The associated bisection-search and one-by-one alternating algorithm exhibits a trade-off between increasing QoS and enhancing secrecy rate.
  • Blocked direct links: For blocked direct links, the artificial-noise design is combined with a minorization-maximization algorithm that converges slightly faster but achieves lower secrecy performance than one-by-one optimization.The special case models IRS-created virtual links around obstacles.

I + (HAE + HIEQHAI)R(HAE + HIEQHAI)H

The channel model expresses secrecy through the difference between Bob’s and Eve’s achievable rates and jointly optimizes transmit covariance and IRS phase shifts. Under full CSI, alternating optimization decomposes the non-convex problem into covariance and phase-shift subproblems, with a barrier-based covariance solver and one-by-one phase updates.

  • Problem formulation: The secrecy-rate objective is C_B − C_E, and secure communication requires C_B − C_E > 0.The transmit covariance R and IRS phase-shift matrix Q are optimized under full or absent eavesdropper CSI.
  • Problem formulation: The full-CSI optimization is non-convex because of its objective and unit-modulus phase constraints, and existing MISO methods do not solve the general MIMO problem.The MIMO determinant term cannot be reduced to the scalar form used in MISO settings, while direct links also prevent applying a special-case solution.
  • Alternating optimization: Alternating optimization separates the variables: it optimizes R with fixed Q and Q with fixed R.The covariance subproblem is transformed into a convex-concave problem, while the phase-shift subproblem uses one-by-one optimization.
  • Covariance optimization: For fixed Q, the covariance problem is represented as a convex-concave saddle-point problem whose saddle-point covariance is optimal for the original maximization.The formulation uses an auxiliary matrix K and applies KKT conditions to obtain the solution numerically.
  • Barrier method: The barrier solver combines Newton iterations, backtracking line search, and increasing barrier parameters to compute the covariance solution.The implementation vectorizes Hermitian variables and solves Newton updates from gradient and Hessian information.
  • Convergence: The barrier method is guaranteed to globally converge for the covariance subproblem, while the overall alternating algorithm converges to a limit-point solution of the original problem.The paper also warns that setting the maximum barrier parameter too high can make the Hessian nearly singular and prevent computing updates.

B. Algorithm of Optimizing Phase Shift Matrix at the IRS

The phase-shift subproblem optimizes the IRS matrix Q with fixed transmit covariance R. Because the log-determinant objective is difficult, the paper uses an OBO method that sequentially optimizes each phase coefficient and returns a sub-optimal feasible Q.

  • The phase-shift subproblem computes Q while holding R fixed.
  • Existing semidefinite-relaxation and MM methods are unsuitable for the complicated log-determinant objective in P5.
  • OBO optimizes each of the n phase-shift coefficients in order while fixing the other n −1 coefficients.
  • With the other coefficients fixed, Proposition 1 provides closed-form solutions for the current coefficient qi across several eigenvalue cases.
  • After one pass through all coefficients, the algorithm outputs Q as a sub-optimal solution of P5, with complexity about O(2n(d3 + e3)).

C. Summary of the AO Algorithm

The AO algorithm alternates between optimizing Q and R to maximize secrecy rate. Its objective is non-decreasing and bounded over the feasible sets, so the algorithm is guaranteed to converge to a limit-point solution.

  • AO alternates Q optimization and R optimization to update the secrecy-rate objective Cs(R, Q).
  • The secrecy-rate objective is non-decreasing across AO iterations.
  • Because the variables are bounded by their feasible sets, AO is guaranteed to converge to a limit-point solution of P1.
  • Algorithm 3 stops when the relative objective change is at most ϵ2 and outputs the current Ropt and Qopt.
  • Each iteration first obtains Qopt for fixed Rini, then obtains Ropt for fixed Qopt.

IV. ENHANCING THE SECRECY RATE UNDER NO EVE’S CSI

This section addresses secrecy-rate enhancement when Eve’s CSI is completely unknown, making direct optimization based on Eve’s channel unavailable. The proposed scheme minimizes transmit power for Bob’s QoS and uses residual power for artificial noise.

  • Eve’s channel matrices HAE and HIE are unknown, so the full-CSI optimization formulation cannot be used directly.
  • The power-minimization problem jointly optimizes R and Q subject to Bob’s QoS constraint.
  • For fixed Q, the R subproblem is solved through a dual channel-capacity formulation and bisection search for Popt.
  • For fixed R, the Q subproblem seeks a feasible phase matrix satisfying the QoS constraint, and excess achievable rate permits reducing the minimum transmit power.

Q log2 |I + H1RHH

After minimum-power transmission is designed, the scheme uses residual power for artificial noise directed into Bob’s effective-channel null space. The resulting AO procedure has a non-increasing transmit-power objective and converges to a limit point.

  • The blocked-direct-link case applies OBO with HAE = 0 and HIE = 0 to optimize the IRS phase coefficients.
  • The AO algorithm alternates BS-based R updates and OBO-based Q updates until the relative transmit-power change meets ϵ3.
  • After obtaining Pmin = tr(Ropt), residual power P −Pmin is allocated to artificial noise in null(H1) so Bob’s QoS is not affected.
  • The artificial-noise covariance uses equal power allocation across the dimensions of null(W1).

I + H2RANHH

For blocked direct links, the proposed AN-aided AO scheme uses MM to optimize IRS phases, while full-CSI simulations show substantial secrecy-rate gains over benchmark designs. Performance improves with more IRS elements but declines as Eve gains antennas.

  • Blocked direct-link scenario: The BS-MM AO procedure alternates MM updates of Q with BS updates of R until the power objective converges.Algorithm 5 initializes Q and R feasibly, then repeats the inner MM loop and outer AO updates.
  • Full-CSI secrecy-rate evaluation: Under full CSI, AO Algorithm 3 achieves significantly higher secrecy rates than zero-phase IRS, no-IRS, and AN-aided no-IRS benchmarks.The optimized IRS phases can combine constructively at Bob and destructively at Eve.
  • Full-CSI secrecy-rate evaluation: Algorithm 3’s secrecy rate increases significantly with the number of IRS elements n, whereas zero-phase optimization gains little and can decline as n grows.Unoptimized phases may make Eve’s effective channel better than Bob’s.
  • Full-CSI secrecy-rate evaluation: All evaluated secrecy rates decrease as Eve’s antenna count e increases, but the proposed IRS-assisted AO design remains superior at fixed e.Deploying more reflecting elements is identified as an effective response to larger Eve antenna arrays.

B. Secrecy Rate of IRS-Assisted MIMO WTC Under No Eve’s CSI

Under unknown Eve’s CSI, the proposed AN-aided scheme enhances secrecy by allocating residual transmit power to jamming, while secrecy depends on balancing Bob’s QoS target against available AN power. Simulations also show positive secrecy rates, monotonic algorithm convergence, and performance differences between the proposed algorithms.

  • No Eve’s CSI: Positive secrecy rates are achieved under completely unknown Eve’s CSI, whereas QoS-only transmission without AN provides limited secrecy enhancement.The proposed scheme uses residual power for AN signalling to make Bob’s channel capacity exceed Eve’s.
  • No Eve’s CSI: Secrecy rate initially increases with γ but later decreases as residual AN power falls, creating a trade-off between Bob’s QoS and secrecy.At sufficiently high γ, the QoS constraint becomes infeasible for the available total power; for example, secrecy stops at γ = 10 when P = 30dBm.
  • Blocked direct links: With blocked direct links, Algorithms 4 and 5 both guarantee positive secrecy rates, while Algorithm 4 achieves better performance because it requires less minimum transmit power.The lower minimum power leaves more residual power available for AN signalling.
  • Convergence: Algorithm 3 converges monotonically in 19 to 101 iterations to accuracy 10^-4, with larger m and n requiring more iterations.The increased iteration count is attributed to the larger dimensions of variables R and Q.
  • Convergence: Algorithm 4 converges in 5 to 8 steps, while Algorithm 5 is 2 to 3 steps faster but returns lower minimum power than Algorithm 4.For the full-CSI barrier method, f(R, K) and C(R) converge together as t approaches tmax, requiring tmax large enough for the stated optimality guarantee.

VII. APPENDIX

The appendix derives phase-shift updates for several cases and establishes how bisection search supports the fractional subproblem. It also provides gradient and Hessian expressions used in the optimization analysis.

  • The appendix derives gradients and Hessians for the objective using standard matrix differential calculus.
  • When the objective is independent of q_i, any unit-modulus q_i is optimal for that subproblem.
  • For one case, the optimal phase shift is q_i = e−jarg(¯λ_i), while another case yields q_i = e^j(π−arg(˜λ_i)).
  • The fractional programming subproblem is solved with Dinkelbach’s method and bisection search, exploiting monotonic decrease of the optimized function in u.
  • The appendix reduces the phase optimization to unit-modulus updates and gives q_i = e−jarg(¯λ_i−u˜λ_i) after obtaining the proper bisection parameter.

C. Proof of Proposition 3

This proof constructs successive lower-bound surrogate functions for g(Q), enabling a tractable approximation under the unit-modulus constraint.

  • The proof rewrites g(Q) using the matrix inversion lemma before constructing a lower bound at a feasible point ˜Q.
  • A surrogate C1(˜Q) + hB(Q) is introduced, but its complicated structure and non-convex unit-modulus constraint require a second approximation.
  • A matrix fractional-function lemma produces a further lower bound for g(Q), preserving a tractable surrogate formulation.

B JBPH) + tr(PJH

The appendix expresses the surrogate term using matrix identities and defines the matrices needed for its tractable quadratic representation.

  • The surrogate construction defines A1 and related matrices to express gB(Q) through a matrix-based reformulation.

B JBL

The final approximation converts the phase-shift update into a quadratic surrogate and gives a closed-form unit-modulus solution.

  • A trace identity converts gB(Q) into the quadratic form gb(q) = q^H Zq.
  • Because the quadratic objective is concave but the unit-modulus constraint is non-convex, an upper-bound lemma is applied around a feasible point.
  • The resulting surrogate reduces to maximizing Re{q^H v}, whose optimum aligns the phase of q with v.
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