Source-linked AI summary
Secure MIMO Transmission via Intelligent Reflecting Surface
Limeng Dong, Hui-Ming Wang
TL;DR
The paper studies secrecy-rate maximization for an IRS-assisted Gaussian MIMO wiretap channel, where prior work was restricted to MISO settings. It proposes alternating optimization of transmit covariance and IRS phase shifts, combining global covariance optimization with MM-based phase optimization, and reports monotonic convergence and better performance than two benchmark schemes.
Problem
Prior IRS-assisted secrecy-rate work was restricted to MISO settings, while the MIMO problem requires covariance optimization and has a complicated log-determinant objective.
Method
An AO algorithm alternately optimizes transmit covariance R and IRS phase coefficient Q, using an existing global method for R and three successive approximations with MM for Q.
Results
The AO algorithm converges monotonically and performs significantly better than zero-phase-shift and random-phase-shift benchmark schemes.
Takeaways & Limitations
The proposed AO procedure provides a numerical approach to secrecy-rate maximization in the IRS-assisted Gaussian MIMO wiretap channel.
Abstract
from arXiv · showhide
In this letter, we consider an intelligent reflecting surface (IRS) assisted Guassian multiple-input multiple-output (MIMO) wiretap channel in which a multi-antenna transmitter communicates with a multi-antenna receiver in the presence of a multi-antenna eavesdropper. To maximize the secrecy rate of this channel, an alternating optimization (AO) algorithm is proposed to jointly optimize the transmit covariance R at transmitter and phase shift coefficient Q at IRS by fixing the other as a constant. When Q is fixed, existing numerical algorithm is used to search for global optimal R. When R is fixed, three sucessive approximation to the objective function to surrogate lower bound is applied and minorization-maximization (MM) algorithm is proposed to optimize the local optimal Q. Simulation results have be provided to validate the convergence and performance of the proposed AO algorithm.
I. INTRODUCTION
The paper addresses IRS-assisted Gaussian MIMO wiretap secrecy-rate maximization, extending beyond prior MISO-only studies. It proposes alternating optimization of the transmitter covariance and IRS phase shifts for this non-convex problem.
- Prior IRS-assisted secrecy-rate studies considered MISO settings with a single receive and eavesdropper antenna.
- MIMO systems require optimizing a transmit covariance rather than only a beamforming vector.
- MIMO secrecy-rate optimization involves complicated log-determinant objectives rather than the scalar logarithms arising in MISO formulations.
- Existing MISO solutions do not directly solve the IRS-assisted MIMO problem, which the paper identifies as open.
- The proposed AO algorithm alternately optimizes transmit covariance R and IRS phase-shift coefficient Q.
- With Q fixed, an existing algorithm globally optimizes R; with R fixed, three successive approximations produce a surrogate lower bound for MM optimization of Q.
II. CHANNEL MODEL AND PROBLEM FORMULATION
The modeled system uses an IRS to assist communication from a multi-antenna transmitter to a receiver while limiting leakage to a multi-antenna eavesdropper. Its secrecy-rate problem jointly constrains transmit covariance and unit-modulus IRS phase shifts.
- II. CHANNEL MODEL AND PROBLEM FORMULATION: The channel model contains transmitter Alice, receiver Bob, eavesdropper Eve, and an IRS with m, d, e, and n associated dimensions.
- II. CHANNEL MODEL AND PROBLEM FORMULATION: The direct Alice-to-Bob and Alice-to-Eve links are blocked, so the IRS passively reflects Alice’s signals toward the receivers.
- II. CHANNEL MODEL AND PROBLEM FORMULATION: Figure 1 depicts the IRS-assisted Gaussian MIMO wiretap channel architecture.
- II. CHANNEL MODEL AND PROBLEM FORMULATION: The IRS adjusts reflection phases to increase Bob’s information rate while decreasing information leakage to Eve.
- II. CHANNEL MODEL AND PROBLEM FORMULATION: The received signals at Bob and Eve are modeled using the transmitted signal, IRS phase matrix Q, channel matrices, and additive noise.
- II. CHANNEL MODEL AND PROBLEM FORMULATION: The formulation assumes full channel state information is available at Alice, including Eve’s channel information.
- II. CHANNEL MODEL AND PROBLEM FORMULATION: The secrecy-rate maximization problem constrains transmit covariance R to be positive semidefinite and within total power budget P.
- II. CHANNEL MODEL AND PROBLEM FORMULATION: Each IRS coefficient has unit modulus, ensuring that reflecting elements change signal phase without changing amplitude.
III. ALTERNATING OPTIMIZATION ALGORITHM
The AO method alternates between optimizing transmit covariance and IRS phases. It globally optimizes covariance for fixed phases and uses MM with successive lower-bound approximations for fixed covariance phases.
- The AO algorithm optimizes R and Q alternately by fixing one variable while optimizing the other.
- For fixed Q, the problem becomes Gaussian MIMO wiretap secrecy-capacity optimization, and an existing barrier-Newton algorithm globally optimizes R.
- For fixed R, optimizing Q is a non-convex problem with non-convex objective and constraints.
- MM constructs an approximate lower bound and iteratively optimizes it under the constraints, with converged points described as KKT points of the original problem.
- The method separately lower-bounds the Bob and Eve log-determinant terms before combining them into an approximated objective.
- A quadratic lower bound is first constructed for fE(Q) at a feasible point satisfying the unit-modulus constraint.
E HIEQLQHHH
The derivation constructs lower bounds for the log-determinant terms in the fixed-R phase-optimization subproblem. Matrix identities and auxiliary terms support a tractable surrogate for optimizing Q.
- A matrix inequality supplies the bound used in the lower-bound construction for the phase-optimization objective.
- The matrix inversion lemma rewrites fB(Q) into a form suitable for deriving a lower bound.
- The derivation introduces T as an intermediate matrix expression involving HIB, Q, and L.
- Applying the stated bound yields a lower bound for fB(Q).
B T(I + THT)−1TH), ˜QB = I + HIB ˜QL ˜QHHH
The method constructs successive lower bounds for the phase-shift objective, ultimately producing a tractable surrogate problem for optimizing Q.
- A second approximation applies a matrix fractional-function lemma to the difficult term hB(Q).The lemma is instantiated with specific matrices and variables before algebraic manipulation yields a tighter lower bound.
- The approximation is introduced because directly optimizing Q remains difficult under the complicated objective term and constraint.
- The resulting lower bound combines the Eve-side approximation, constant terms, and the new term gB(Q).
B JBTH)
The Q-update uses matrix identities and successive surrogate approximations to obtain a unit-modulus phase optimization with a closed-form update.
- A trace identity converts a matrix expression involving a diagonal phase matrix into a quadratic form in its diagonal vector.
- Because the unit-modulus constraint remains non-convex, a third approximation constructs a surrogate function suitable for MM optimization.
- After dropping constants, the subproblem maximizes the real part of q^Hv subject to |qi| = 1.
- The objective is maximized when each qi has the same phase as vi, yielding a closed-form global solution for P4.
- With a feasible initialization, MM produces a KKT solution for P2 at fixed R.
- Alternating the R and Q updates makes Cs(Rk, Qk) monotonically increase and guarantees convergence because both variables are bounded.
IV. SIMULATION RESULTS
Simulations evaluate the AO algorithm’s convergence and secrecy-rate performance under randomly generated fading channels and benchmark phase-shift strategies.
- Simulation setup: The simulations use randomly generated complex Gaussian small-scale fading and fixed large-scale fading parameters, with barrier and MM target accuracies of 10^-8 and 10^-4.
- Algorithm: Algorithm 1 initializes R0 and Q0, alternately optimizes R by a barrier method and Q by MM, and stops when the relative secrecy-rate change converges.
- Convergence: The convergence study reports 42, 107, and 166 iterations to reach 10^-4 under the considered settings, with monotonic increase in every case.
- Performance comparison: The proposed AO algorithm significantly outperforms zero-phase and random-phase benchmarks, while optimizing only R provides limited secrecy-rate enhancement.
- Performance comparison: The secrecy-rate comparison averages results over 100 randomly generated channels with m = d = e = 5 and n = 15.
- Performance comparison: Random phase shifts can degrade the effective Bob channel while improving the Eve channel in some realizations, producing the weakest performance.
V. CONCLUSION
The paper studies secrecy-rate maximization for an IRS-assisted Gaussian MIMO wiretap channel using an alternating optimization algorithm.
- The proposed AO algorithm jointly optimizes the transmit covariance at Alice and the IRS phase-shift coefficients for the non-convex problem.
- Simulations validate monotonic convergence and show significantly better AO performance than the benchmark schemes.