Robust Design of Sensing-Assisted Secure Transmission for RIS-Aided ISAC System under Eavesdropper 3D Positioning Uncertainty

I. INTRODUCTION

• Practical sensing errors and limited observations make Eve’s estimated position uncertain, motivating robust transmission that accounts for localization errors.The paper identifies potential beam misalignment and degraded secrecy performance as consequences of sensing and channel-estimation errors.Source: §I · p. 1
• Prior work did not explicitly connect sensing-derived localization accuracy to a CRLB-based 3D uncertainty set and its use in robust RIS-assisted secure transmission.The cited studies either assumed perfect Eve CSI, used predefined uncertainty regions, or adopted 2D spatial models.Source: §I · p. 2
• The proposed sensing-assisted framework converts Eve’s 3D localization accuracy into an uncertainty ellipsoid and uses it in robust secure transmission design.The RIS has separate reflection and sensing reception regions; the BS and sensing region jointly observe Eve for localization.Source: §I · p. 2
• The two-stage design jointly optimizes sensing beamforming and RIS phases, then optimizes communication beamforming and RIS phases for worst-case secrecy under location uncertainty.The communication-stage robust constraint is reformulated as an LMI using first-order Taylor expansion and the S-lemma.Source: §I · p. 2

II. SYSTEM MODEL

• The system model uses a hybrid RIS divided into reflection and sensing reception regions, with Rician-fading channels and residual inter-region leakage neglected.The regions are separated by a metallic isolation strip; Eve is modeled as an unmanned aerial vehicle in the 3D system model.Source: §II · p. 3, A. Channel Model · p. 3

B. Sensing Model

• Each TDD frame allocates a fraction τ to sensing, yielding L snapshots, while Eve is modeled as a Swerling-I point scatterer with a frame-constant position.Eve’s scattering coefficient remains constant over the sensing snapshots within a frame but may vary between adjacent frames.Source: B. Sensing Model · p. 3

1) Signal Model for the Sensing Phase:

• The sensing beamformer is power-constrained, its waveform has unit modulus, and the RIS reflection region applies unit-modulus phase shifts during sensing.The sensing illumination energy is defined from the equivalent illumination channel over L snapshots.Source: 1) Signal Model for the Sensing Phase · pp. 3–4
• The BS and RIS sensing reception region receive Eve’s echoes after sensing illumination, with the RIS-region BS leakage removed by calibration or reciprocity-based estimation.The BS receiver assumes ideal self-interference cancellation.Source: 1) Signal Model for the Sensing Phase · p. 4

2) Maximum Likelihood Angle Estimation:

• Under a dominant-LoS single-source approximation, the BS and RIS sensing region independently use MLE on L snapshots to estimate Eve’s arrival angles.The numerical evaluations use full Rician channels, with Monte Carlo localization errors used to calibrate the uncertainty set.Source: 2) Maximum Likelihood Angle Estimation · p. 4

3) Cross Localization:

• Eve’s 3D position is estimated as the midpoint of the shortest segment between spatial rays constructed from the BS and RIS angle estimates.The distances along the rays are found by minimizing their separation in a least-squares problem.Source: 3) Cross Localization · p. 4

4) CRLB for Position Estimation:

• The joint BS–RIS position CRLB is derived after treating the two complex amplitudes as nuisance parameters and eliminating them from the Fisher information matrix.The CRLB is a nominal model-based benchmark, not a guaranteed covariance upper bound for localization errors under full Rician channels.Source: 4) CRLB for Position Estimation · pp. 4–5

C. Communication Model

• The communication model defines effective secrecy rate as the secrecy rate multiplied by the communication-time fraction, Reff = (1 − τ)Rs.The system-model CRLB is an accuracy benchmark, while uncertainty-set scale is calibrated using errors from the full Rician channel model.Source: C. Communication Model · p. 5

A. Sensing-Assisted Robust Uncertainty Set

• The MLE position estimate is surrounded by an ellipsoid whose nominal CRLB shape is scaled to account for discrepancy from actual localization errors.The scale is empirically selected using Monte Carlo localization errors under the considered simulation settings and fixed for robust transmission design.Source: A. Sensing-Assisted Robust Uncertainty… · p. 5

B. Problem Formulation and Two-Stage Solution Framework

• The sequential design first optimizes sensing variables to determine Eve’s uncertainty set, then fixes that set while optimizing communication variables for worst-case secrecy rate.The framework follows sensing-to-communication temporal causality and provides a tractable suboptimal solution to the highly non-convex joint problem.Source: B. Problem Formulation and Two-Stage… · p. 5, §III · p. 5

C. Stage I: Sensing Optimization

• The sensing objective minimizes the trace of Eve’s position CRLB subject to transmit-power and unit-modulus RIS constraints.With equal BS and RIS sensing-receiver noise variances and the stated nominal observation model, minimizing the CRLB trace is equivalent to maximizing effective illumination power.Source: C. Stage I: Sensing Optimization · p. 6
• Alternating optimization updates the coupled sensing beamformer and RIS phases; monotonic line search yields a convergent objective sequence whose limit points satisfy first-order stationarity conditions.The RIS phase subproblem uses RCG updates that preserve unit-modulus constraints.Source: C. Stage I: Sensing Optimization · p. 6
• For fixed RIS phases, the globally optimal sensing beamformer follows MRT, aligning with the equivalent channel and using the available transmit power.The solution follows from the Cauchy–Schwarz bound on the beamformer objective.Source: C. Stage I: Sensing Optimization · p. 6

2) Subproblem 2:

• With the sensing beamformer fixed, the RIS phase subproblem is optimized on the complex circle product manifold using RCG updates that preserve unit-modulus constraints.The iteration uses an Armijo line search and stops at the specified tolerance or maximum iteration count.Source: 2) Subproblem 2 · p. 6

D. Stage II: Robust Secure Transmission Design

• Stage II maximizes instantaneous secrecy rate for the worst-case Eve position while meeting BS power and RIS unit-modulus constraints.The semi-infinite constraint requires Eve’s SINR to remain below its threshold throughout the continuous uncertainty region; the positive-secrecy-rate regime is considered.Source: D. Stage II: Robust Secure… · p. 6
• Under a first-order channel approximation, the robust quadratic secrecy constraint admits an exact LMI reformulation via the S-lemma, but rank-one, unit-modulus, and beamformer–RIS coupling constraints leave the problem non-convex.An alternating-optimization framework updates beamforming and RIS phase variables in turn.Source: D. Stage II: Robust Secure… · p. 7

1) Beamforming Optimization Subproblem:

• With the RIS phases fixed, the beamforming covariance matrix makes the Bob SINR expressible as a trace and the LMI matrices affine in that covariance.Source: 1) Beamforming Optimization Subproblem · p. 7
• Relaxing the covariance matrix's rank-one constraint yields a semidefinite feasibility problem for a given secrecy-rate target.Source: 1) Beamforming Optimization Subproblem · p. 7
• Bisection raises the target when the relaxed SDP is feasible and lowers it otherwise, producing the maximum target supported by the relaxation.Source: 1) Beamforming Optimization Subproblem · p. 7
• A rank-one optimal covariance gives the beamforming vector directly; otherwise, Gaussian randomization generates candidates, and the best feasible candidate is selected.Source: 1) Beamforming Optimization Subproblem · pp. 7–8

2) RIS Phase-Shift Optimization Subproblem:

• The RIS phases lie on a complex circle product manifold, while the formulation also imposes a positive-semidefinite matrix inequality.Source: 2) RIS Phase-Shift Optimization… · p. 8
• A logarithmic barrier incorporates the positive-semidefinite constraint into the objective, avoiding repeated SDP solves in the RIS phase-shift block.Source: 2) RIS Phase-Shift Optimization… · p. 8
• A two-layer strategy decreases the barrier parameter in an outer loop and cyclically updates λ, Γ, ϕc, and η in the inner loop.Source: 2) RIS Phase-Shift Optimization… · pp. 8–9
• The λ update uses Newton's method with backtracking on a concave barrier objective, preserving λ ≥ 0 and positive definiteness of S.Source: 2) RIS Phase-Shift Optimization… · p. 8
• The Γ update uses a bounded one-dimensional search over its feasible interval, restricted to values for which S remains positive definite.Source: 2) RIS Phase-Shift Optimization… · p. 8
• The phase update uses Riemannian conjugate gradients and Armijo backtracking on the unit-modulus manifold; η is set to its largest feasible value.Source: 2) RIS Phase-Shift Optimization… · p. 8

E. Convergence and Computational Complexity Analysis

• Stage I's MRT solution globally maximizes illumination power for fixed RIS phases, while RCG with Armijo line search makes non-decreasing updates; the bounded objective sequence converges.Source: E. Convergence and Computational… · p. 9
• In Stage II, block updates make the barrier objective non-decreasing, and an incumbent is retained unless a feasible candidate matches or improves its worst-case secrecy rate.Source: E. Convergence and Computational… · p. 9
• Despite convergence of the incumbent objective sequence, the SDR relaxation, Gaussian randomization, and non-convex RIS phase optimization generally yield a suboptimal solution.Source: E. Convergence and Computational… · p. 9
• With fixed S-procedure dimension, RCG-based phase optimization scales linearly with RIS size, while the SDP beamforming subproblem generally dominates cost for large BS antenna counts.Source: E. Convergence and Computational… · p. 9

IV. SIMULATION RESULTS

• Increasing τ supplies more sensing snapshots and reduces nominal CRLB and localization RMSE, with diminishing marginal improvement.The simulations use κ = 4.5, giving approximately 95% uncertainty-region coverage under the considered settings.Source: §IV · p. 9, Figure 2 · p. 9
• For P ≤30 W, effective secrecy rate peaks at τ ≈ 0.10–0.15, while at P = 40 W it generally decreases as τ increases.More sensing improves localization and worst-case secrecy, but excessive sensing reduces transmission time; τ = 0.15 is used thereafter.Source: §IV · p. 9, Figure 3 · p. 10
• The AO algorithm’s CRLB decreases and worst-case secrecy rate increases before both converge; secrecy optimization takes more iterations than sensing optimization.The greater iteration requirement is attributed to coupling among beamforming, RIS phases, and Eve-location uncertainty.Source: §IV · pp. 9–10, Figure 4 · p. 10
• Worst-case secrecy rate increases with BS transmit power and antenna size, as additional array gain and spatial degrees of freedom improve robust beamforming.The gain is limited for a 2×2 UPA because it has insufficient spatial degrees of freedom to enhance Bob’s signal while suppressing leakage toward Eve.Source: §IV · p. 10, Figure 5 · p. 10
• Worst-case secrecy rate increases with transmit power and RIS size, while optimized RIS phases outperform random phases across tested sizes and power levels.More RIS elements provide additional reflection and phase-control degrees of freedom for enhancing Bob’s signal and suppressing leakage toward Eve.Source: §IV · p. 10, Figure 6 · p. 10
• At high transmit power, the proposed scheme approaches the perfect-CSI benchmark and mitigates secrecy loss from Eve-location uncertainty.It gradually outperforms no-RIS and random-RIS schemes as transmit power increases.Source: §IV · p. 10, Figure 7 · p. 11

V. CONCLUSION

• The conclusion reports that increasing BS antennas, transmit power, and RIS elements improves worst-case secrecy rate, and proposes future study of multiple users, eavesdroppers, and RISs.It also reports that the proposed method outperforms benchmark schemes.Source: §V · p. 11

• Wirtinger calculus supplies the Euclidean gradient for the real-valued objective over complex RIS phases.Source: STAGE I · p. 11
• The feasible RIS phase set is a complex circle product manifold with unit-magnitude entries.Source: STAGE I · p. 11
• The Euclidean gradient is projected onto the tangent space, and retraction updates the manifold point.The passages also describe vector transport between tangent spaces.Source: STAGE I · p. 11
• The Riemannian conjugate search uses the Hestenes-Stiefel formula with HS+ nonnegative clipping and Armijo backtracking.The step size is reduced by a factor τ until the Armijo condition is met.Source: STAGE I · p. 11