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Robust Transmission Design for Intelligent Reflecting Surface Aided Secure Communication Systems with Imperfect Cascaded CSI

Sheng Hong, Cunhua Pan, Hong Ren, Kezhi Wang, Kok Keong Chai, Arumugam Nallanathan

arXiv:2004.11580v3eess.SPcs.IT

TL;DR

The paper tackles secure IRS-aided transmission when cascaded AP-IRS-user CSI is imperfect and statistical channel errors must be considered. It jointly designs transmission and IRS variables under outage constraints using convex approximations and alternating optimization, and reports lower transmit power than benchmark methods.

  • Problem

    Secure IRS-aided communication has not yet addressed robust transmission design with cascaded channel errors and statistical CSI error models.

  • Method

    The paper formulates an outage-constrained power-minimization problem and solves it with alternating optimization, penalty methods, and semidefinite relaxation.

  • Results

    The proposed robust design reduces transmit power under a fixed secrecy rate and outperforms benchmark methods in simulations.

  • Takeaways & Limitations

    Robust beamforming and AN covariance design can enhance physical-layer security in IRS-aided communications when statistical CSI error exists.

Abstract

from arXiv · show

In this paper, we investigate the design of robust and secure transmission in intelligent reflecting surface (IRS) aided wireless communication systems. In particular, a multi-antenna access point (AP) communicates with a single-antenna legitimate receiver in the presence of multiple single-antenna eavesdroppers, where the artificial noise (AN) is transmitted to enhance the security performance. Besides, we assume that the cascaded AP-IRS-user channels are imperfect due to the channel estimation error. To minimize the transmit power, the beamforming vector at the transmitter, the AN covariance matrix, and the IRS phase shifts are jointly optimized subject to the outage rate probability constraints under the statistical cascaded channel state information (CSI) error model that usually models the channel estimation error. To handle the resulting non-convex optimization problem, we first approximate the outage rate probability constraints by using the Bernstein-type inequality. Then, we develop a suboptimal algorithm based on alternating optimization, the penalty-based and semidefinite relaxation methods. Simulation results reveal that the proposed scheme significantly reduces the transmit power compared to other benchmark schemes.

I. INTRODUCTION

IRS technology reconfigures wireless propagation using passive elements and has been studied for communication and physical-layer security. This paper addresses the gap in robust secure design under statistical cascaded-channel CSI errors.

  • IRS technology and applications: IRSs use passive reflecting elements to reconfigure the wireless propagation environment between access points and users.Their passive operation avoids active RF chains, supporting low-cost deployment.
  • Existing IRS research: Prior IRS research spans single-user, multiuser, multicell, wireless-power-transfer, mobile-edge-computing, and cognitive-radio systems.Secure IRS studies jointly optimize active transmit beamforming and passive IRS beamforming to improve secrecy rate.
  • Existing IRS research: Artificial noise and IRS phase-shift optimization have been incorporated into secure communications to limit information leakage to eavesdroppers.Prior work reported benefits from deploying the IRS near the legitimate receiver and using AN for secrecy enhancement.
  • CSI uncertainty: Existing secure IRS contributions generally assume perfect CSI, although IRS-related channel estimation is difficult because the IRS lacks RF chains.Cascaded-channel estimation avoids extra hardware and power costs, but channel-estimation error remains inevitable.
  • Research gap: Robust IRS research has mainly considered bounded uncertainty or imperfect IRS-user channels, leaving secure transmission with cascaded-channel errors and probabilistic CSI models insufficiently studied.The paper identifies secure IRS-aided communication with cascaded channel error as an open problem.
  • Paper contributions: The paper studies outage-constrained robust secure transmission with statistical cascaded CSI errors, optimizing beamforming, AN covariance, and IRS phase shifts.It formulates an OC-PM problem and develops an alternating-optimization-based design; simulations report reduced transmit power versus benchmark methods.

II. SYSTEM MODEL

The paper presents the transmission model and CSI error model for a secure IRS-aided communication system.

  • The section introduces the transmission and channel-state-information error models used for the secure IRS-aided communication system.

A. Signal Transmission Model

An AP communicates with Bob through an IRS in the presence of multiple Eves, using beamforming and artificial noise while optimizing the IRS phase shifts.

  • An AP with N_t antennas sends confidential information to a single-antenna legitimate receiver, Bob, while K single-antenna Eves listen.
  • The IRS contains M reflecting elements whose phase coefficients are collected in Φ, with φ_m=e^jθ_m and θ_m∈[0,2π].
  • The direct Alice-Bob and Alice-Eve links are blocked, so Bob and the Eves receive signals reflected by the IRS.
  • Alice transmits a confidential data stream using beamforming vector w and artificial noise z(t) with covariance matrix Z.
  • The equivalent Bob and Eve channels are expressed through the cascaded Alice-IRS-user channels G_cb and G_ce,k.

B. Statistical CSI Error Model

The paper models imperfect cascaded AP-IRS-user CSI statistically and formulates secrecy-rate outage-constrained transmit-power minimization over transmission and IRS variables.

  • Cascaded AP-IRS-user channels are difficult to obtain because of the IRS's passive features, motivating a CSI error model.
  • The paper focuses on the statistical CSI error model, which is more closely related to channel estimation error than the bounded model.
  • The cascaded Eve channel is modeled as an estimated channel plus an error whose vectorization follows a CSCG distribution with covariance Σ_e,k.
  • The achievable secrecy rate is the minimum across Eves of Bob's rate minus each Eve's rate.
  • The design jointly optimizes w or W, AN covariance Z, and IRS phases to minimize transmit power under Bob-rate and per-Eve leakage-outage constraints.
  • The resulting problem is nonconvex because optimization variables are coupled and the constraints are probabilistic; alternating optimization is used after safe reformulation.

A. Problem Reformulation

The problem reformulation converts logarithmic outage constraints into quadratic Gaussian chance constraints and then applies Bernstein-type inequalities to obtain safe deterministic constraints.

  • The reformulated outage condition becomes a quadratic inequality with respect to the Gaussian random vector v_ce,k.
  • The leakage-rate constraint is simplified by substituting the imperfect Eve channel and defining Ξ_e≜(β−1)Z−W.
  • The CSI error is equivalently represented using a covariance factorization and a standard complex Gaussian vector.
  • Bernstein-type inequality safely approximates the chance constraint by replacing it with deterministic constraints involving slack variables.
  • The resulting safe approximation introduces vectors of slack variables x and y and yields a tractable reformulation of the outage-constrained power-minimization problem.
  • Additional trace, norm, and semidefinite constraints are algebraically simplified before the final recast optimization problem is formed.

B. Solving the Beamforming Matrix and AN Covariance Matrix

The beamforming and AN subproblem is solved within an alternating-optimization framework after relaxing rank constraints and converting the phase-shift update into a penalized convex program. The resulting iterations monotonically reduce transmit power and are guaranteed to converge.

  • Beamforming and AN update: Fixing the IRS phase vector makes the beamforming and AN subproblem convex after relaxing the rank-one constraint.The relaxed problem is solved using semidefinite relaxation and a convex-program solver.
  • Rank-one recovery: The relaxed beamforming matrix may not be rank one, so Gaussian randomization can recover a suboptimal beamforming vector.Numerical simulations reported that the obtained W always satisfied the rank-one constraint.
  • Phase-shift update: When beamforming and AN variables are fixed, the phase-shift subproblem remains nonconvex because of the unit-modulus constraint.The phase shifts are therefore handled through semidefinite programming and rank-one recovery techniques.
  • Phase-shift update: A penalty-based formulation penalizes violations of rank(E) = 1, while first-order Taylor approximation makes the resulting subproblem jointly convex.The penalty factor κ controls the rank-one penalty, and the convexified problem can be solved by CVX.
  • Alternating optimization: Algorithm 1 alternates between solving the beamforming/AN and phase-shift subproblems, reducing transmit power monotonically with guaranteed convergence.The variables are initialized in the feasible region and updated iteratively until the prescribed iteration process terminates.

A. Simulation Setup

The simulations model an IRS-assisted secure wireless system with blocked direct links, specified array and fading assumptions, statistical cascaded-CSI errors, and benchmark schemes for comparison. The evaluation examines the proposed algorithm's convergence and the role of optimized IRS phase shifts.

  • Simulation Setup: The simulated scenario blocks direct AP-to-user and AP-to-eavesdropper links and places the AP, IRS, legitimate receiver, and eavesdroppers at specified coordinates.The AP is at (0, 10) m, the IRS at (100, 25) m, Bob at (180, 0) m, and Eves at (160, 0) m and (170, 0) m.
  • Simulation Setup: The AP-IRS channel uses a Ricean model with LoS and Rayleigh NLoS components, while the IRS-user links contain both LoS and NLoS components.The AP and IRS antennas or reflecting elements are modeled as uniform linear arrays.
  • Simulation Setup: The simulations incorporate a statistical cascaded CSI error model whose variance matrix represents the relative amount of CSI uncertainty.System parameters are listed in Table I.
  • Benchmark Schemes: The proposed algorithm is compared with Random-MRT, Random-IRS, and Optimized-MRT benchmark schemes.These schemes differ in whether they optimize IRS phase shifts, use MRT beamforming, and exploit Bob's or both Bob's and Eves' CSI.
  • Simulation Evaluation: The evaluation investigates convergence for different numbers of IRS phase shifts and compares the importance of optimizing those phase shifts.The proposed algorithm jointly optimizes the relevant transmission variables, whereas Random-IRS uses randomly selected phase shifts.

C. Convergence Analysis

The proposed algorithm converges under varying antenna numbers, IRS sizes, and rate constraints, but computation time generally increases with problem dimensions and stricter settings.

  • The proposed algorithm is likely to converge with fewer iterations as the number of transmit antennas increases.
  • More IRS elements generally require more iterations because they provide additional phase-shift degrees of freedom for cascaded-channel adjustment.
  • Average CPU running time increases with either the transmit-antenna number N_t or IRS element number M.
  • CPU time grows more slowly with N_t than with M because the required iterations decrease with N_t.
  • Decreasing the legitimate user's minimum channel capacity or increasing the eavesdroppers' maximum tolerable capacity requires more iterations and CPU time.

D. Transmit Power Versus the Minimum Channel Capacity of the Legitimate User

Transmit power rises with the legitimate user's required channel capacity and falls when more information leakage to eavesdroppers is allowed. Optimizing IRS phases and robustly using eavesdropper CSI improves power efficiency and reduces AN allocation.

  • Transmit power increases monotonically with the legitimate user's minimum channel capacity because higher data-rate requirements need more power.
  • The proposed method consumes less transmit power than the benchmark schemes as the legitimate user's minimum channel capacity varies.
  • An IRS can enable secure communication in a line-of-sight-blocked environment, provided its phase shifts are optimized.
  • The proposed method allocates a lower AN-power percentage than MRT benchmarks because it exploits eavesdropper CSI and optimized IRS phase shifts.
  • As the eavesdroppers' allowed channel capacity increases, less AN power is required and total transmit power decreases.
  • The proposed method consumes the lowest transmit power as the eavesdroppers' maximum tolerable channel capacity varies.

F. Transmit Power Versus the Number of IRS Elements

Increasing the number of IRS elements or transmit antennas reduces required transmit power, with optimized IRS phases producing stronger reductions than random or MRT-based designs.

  • Transmit power decreases as the number of IRS elements M increases.Larger M provides more degrees of freedom to adjust cascaded channels for Bob and Eves.
  • Optimizing IRS phase shifts reduces transmit power more effectively than random IRS phases, especially when M is large.
  • The proposed algorithm requires the lowest transmit power among the benchmark schemes as M varies.
  • Transmit power decreases for all schemes as the number of transmit antennas N_t increases.More antennas improve both Bob's and Eves' channels, facilitating robust beamforming and AN design.
  • Random-IRS and proposed schemes decrease faster with N_t than MRT schemes because they exploit eavesdropper CSI.
  • The proposed robust design further reduces the Random-IRS scheme's transmit power.

H. Transmit Power Versus CSI Uncertainty

Transmit power increases as cascaded-channel CSI becomes less accurate, while the proposed robust design is less sensitive to CSI errors and requires less AN power than the benchmarks.

  • Transmit power increases as the quality of eavesdropper CSI degrades.More AN power and beamforming power are then needed to limit leakage while maintaining Bob's data requirement.
  • The proposed method requires significantly less transmit power than the three benchmark schemes under CSI uncertainty.
  • The proposed method is more robust to CSI errors because its transmit power increases more slowly than MRT schemes as uncertainty grows.
  • AN-power percentage increases with normalized CSI error variance because degraded CSI requires more AN interference against eavesdroppers.
  • Optimized-MRT and Random-MRT allocate more than 70% of transmit power to AN, whereas the proposed and Random-IRS schemes allocate less than 20%.
  • The proposed method uses less AN power than Random IRS because IRS phase optimization makes interference against eavesdroppers more effective.
  • The paper formulates outage-constrained robust transmission with statistical cascaded-CSI error and jointly optimizes beamforming, AN covariance, and IRS phases.
  • Alternating optimization, Bernstein-type inequality, semidefinite relaxation, and a penalty method address the resulting nonconvex design.
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