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Robust Transmit Beamforming for Secure Integrated Sensing and Communication

Zixiang Ren, Ling Qiu, Jie Xu, Derrick Wing Kwan Ng

arXiv:2207.13863v1cs.ITeess.SP

TL;DR

The paper addresses secure ISAC beamforming when sensing targets may eavesdrop and their CSI is imperfect. It jointly designs information and sensing/AN beams under bounded and Gaussian CSI errors, using exact or efficient robust optimization methods. The resulting designs balance sensing and secrecy, with reported improvements over conventional benchmarks and robustness to eavesdropper CSI errors.

  • Problem

    Secure ISAC must protect confidential communication when sensing targets may be eavesdroppers and their CSI is imperfect.

  • Method

    The paper jointly designs information and dedicated sensing/AN beams under bounded and Gaussian eavesdropper CSI errors, using robust secrecy constraints and SDR-based optimization.

  • Results

    The proposed designs balance sensing and secrecy, outperform conventional benchmarks in sensing measures, and remain robust to eavesdropper CSI errors.

  • Takeaways & Limitations

    Joint information and sensing/AN beamforming can support target sensing while satisfying secure communication requirements in both considered CSI-error scenarios.

Abstract

from arXiv · show

This paper studies a downlink secure integrated sensing and communication (ISAC) system, in which a multi-antenna base station (BS) transmits confidential messages to a single-antenna communication user (CU) while performing sensing on targets that may act as suspicious eavesdroppers. To ensure the quality of target sensing while preventing their potential eavesdropping, the BS combines the transmit confidential information signals with additional dedicated sensing signals, which play a dual role of artificial noise (AN) for degrading the qualities of eavesdropping channels. Under this setup, we jointly design the transmit information and sensing beamforming, with the objective of minimizing the weighted sum of beampattern matching errors and cross-correlation patterns for sensing subject to secure communication constraints. The robust design takes into account the channel state information (CSI) imperfectness of the eavesdroppers in two practical CSI error scenarios. First, we consider the scenario with bounded CSI errors of eavesdroppers, in which the worst-case secrecy rate constraint is adopted to ensure secure communication performance. In this scenario, we present the optimal solution to the worst-case secrecy rate constrained sensing beampattern optimization problem, by adopting the techniques of S-procedure, semi-definite relaxation (SDR), and a one-dimensional (1D) search, for which the tightness of the SDR is rigorously proved. Next, we consider the scenario with Gaussian CSI errors of eavesdroppers, in which the secrecy outage probability constraint is adopted. In this scenario, we present an efficient algorithm to solve the more challenging secrecy outage-constrained sensing beampattern optimization problem, by exploiting the convex restriction technique based on the Bernstein-type inequality, together with the SDR and 1D search.

I. INTRODUCTION

Secure ISAC must balance sensing performance with confidential communication because sensing-directed information beams can expose messages to suspicious targets. This paper jointly designs information and sensing/AN beams under imperfect eavesdropper CSI, addressing bounded and Gaussian error models.

  • Motivation: ISAC integrates sensing and communication, but steering information-bearing beams toward sensing targets can create information-leakage risks when those targets are suspicious eavesdroppers.Dedicated sensing beams can exploit additional spatial degrees of freedom while supporting secure transmission.
  • Proposed framework: The paper jointly optimizes one information beam and dedicated sensing beams that also act as artificial noise, minimizing weighted beampattern and cross-correlation errors while satisfying secrecy requirements.The design covers eavesdroppers with bounded or Gaussian CSI errors.
  • Bounded CSI errors: For bounded eavesdropper CSI errors, the worst-case secrecy-rate problem is solved using the S-procedure, semidefinite relaxation, and a one-dimensional search.The paper rigorously proves SDR tightness and obtains rank-one information and general-rank sensing beamformers.
  • Gaussian CSI errors: For Gaussian eavesdropper CSI errors, the paper imposes a secrecy-outage constraint and develops an efficient solution for the more difficult outage-constrained optimization problem.The supplied passages identify this as the Gaussian-error scenario and distinguish it from the worst-case formulation.
  • Results: Compared with conventional benchmarks, the proposed joint beamforming achieves lower beampattern matching errors, higher angle-estimation accuracy, and robustness to eavesdropper CSI errors.Information beams target the CU and trusted targets, while sensing/AN beams target both untrusted and trusted targets.

A. Signal Model

The model uses a multi-antenna BS to send confidential information to a single-antenna CU while sensing multiple targets, some of which may be untrusted eavesdroppers. Dedicated sensing signals support sensing and confuse eavesdropping targets, while imperfect eavesdropper CSI is modeled with bounded or Gaussian errors.

  • The secure ISAC system contains a multi-antenna BS, a single-antenna CU, and multiple sensing targets, including untrusted targets.
  • The BS uses linear transmit beamforming to send a confidential message to the CU.
  • A dedicated sensing signal is independent of the information signal and acts as artificial noise to support sensing and confuse eavesdroppers.
  • The sensing covariance S may have rank m between 0 and N, corresponding to m linearly and statistically independent sensing beams.
  • The CU secrecy rate is evaluated from the CU and untrusted-target SINRs under the quasi-static channel model.
  • The BS has perfect CU CSI but imperfect eavesdropper CSI, with bounded and Gaussian error scenarios considered.

1) Bounded CSI Errors:

The sensing design minimizes beampattern mismatch and cross-correlation while enforcing secure communication under uncertain eavesdropper channels. Bounded errors use a worst-case secrecy-rate metric, whereas Gaussian errors use secrecy outage probability.

  • 1) Bounded CSI Errors:: Bounded eavesdropper CSI errors are limited by a maximum threshold ε_k, and worst-case secrecy rate is used as the secure communication metric.
  • 2) Gaussian CSI Errors:: Gaussian CSI errors are modeled with independent CSCG vectors u_k, and secrecy outage probability measures the probability that the achievable secrecy rate falls below R0.
  • C. Radar Sensing: The transmit covariance design uses information covariance W and sensing covariance S to optimize multi-target estimation performance.
  • C. Radar Sensing: The radar model assumes far-field point targets and uses their angles, round-trip coefficients, steering vectors, and received echo signals.
  • C. Radar Sensing: Beampattern matching controls the transmit power distribution toward estimated target directions, while low cross-correlation helps resolve different target angles.
  • C. Radar Sensing: The sensing objective combines beampattern matching mean squared error with cross-correlation pattern through a predetermined weight ω_C.

D. Problem Formulation

The paper formulates joint information and sensing/AN beamforming problems for secure ISAC under bounded and Gaussian eavesdropper CSI errors, then develops solutions for their non-convex constraints.

  • D. Problem Formulation: The objective minimizes a weighted sum of beampattern matching error and cross-correlation patterns while jointly optimizing information and sensing/AN covariance matrices.
  • D. Problem Formulation: For bounded eavesdropper CSI errors, the design uses a worst-case secrecy-rate constraint with a required threshold R0.
  • D. Problem Formulation: The bounded-error problem is non-convex because of its secrecy-rate and rank constraints, but the paper presents an optimal solution.
  • D. Problem Formulation: For Gaussian eavesdropper CSI errors, the design imposes a secrecy-outage probability no greater than the threshold ρ.
  • D. Problem Formulation: The Gaussian-error problem is more difficult because its outage constraint lacks a closed-form expression, and the paper proposes an efficient algorithm.
  • D. Problem Formulation: The perfect-eavesdropper-CSI special case is identified as novel and is solved through the optimal solution to the bounded-error problem.
  • III. OPTIMAL JOINT BEAMFORMING SOLUTION TO PROBLEM (P1): For the bounded-error problem, SDR drops the rank constraint, an auxiliary γ converts the secrecy-rate condition, and a 1D search solves convex QSDPs for fixed γ.
  • A. Optimal Solution to (P1.1): With fixed γ, the reformulated constraints become affine and LMI constraints, yielding a convex QSDP solvable by numerical convex-programming solvers.

B. Construction of Rank-one Solution of W for (P1)

The paper constructs rank-one information beamforming solutions for the bounded-error problem and develops an efficient SDR-based procedure for the Gaussian-error problem.

  • A rank-one construction based on the relaxed solution is proposed for the bounded-error problem, establishing tightness of its SDR.
  • The construction guarantees an optimal information covariance matrix with rank(W) = 1 for the bounded-error formulation.
  • For the Gaussian-error problem, the rank constraint is relaxed and an auxiliary γ enables solving fixed-γ subproblems followed by a 1D search.
  • The Gaussian-error procedure uses safe decomposition and Bernstein-type inequality convex restrictions to obtain a high-quality solution to the original problem.

A. Safe Decomposition for Constraint (34)

The Gaussian-error outage constraint is converted into tractable restricted constraints using safe decomposition and Bernstein-type inequality, yielding a convex fixed-γ problem and an efficient search procedure.

  • A safe decomposition replaces the difficult outage constraint with per-target restricted constraints that bound each untrusted target’s SINR outage probability.
  • The Gaussian channel model represents each eavesdropper channel using an estimated channel, an error factor, and a standard complex Gaussian vector.
  • The derivation assumes independent channel vectors for different eavesdroppers when establishing the equivalence between the outage formulations.
  • For fixed γ, the resulting restricted problem is convex and can be solved optimally via CVX, producing a high-quality solution.
  • Bernstein-type inequality converts the restricted probabilistic constraint into deterministic inequalities involving auxiliary variables a_k, b_k, and c_k.
  • A 1D search over γ extends the fixed-γ solution to an efficient solution of the relaxed outage-constrained problem.
  • The relaxed solution may violate the rank-one constraint, so the resulting solution is not necessarily feasible for the original problem.

C. Rank-one Construction of W for Problem (P2)

The paper constructs rank-one information beamformers for the Gaussian-error problem by projecting the relaxed solution onto the CU-channel subspace and reallocating remaining power to sensing/AN beams.

  • A rank-one construction for the Gaussian-error problem preserves the relaxed objective value while producing a feasible high-quality solution.
  • The construction is guaranteed feasible for the original Gaussian-error problem but not necessarily for its restricted approximation, although simulations always found feasibility.
  • Because the CU channel is perfectly known, the general-rank information covariance can be projected into the CU-channel subspace without affecting CU SINR.
  • Power remaining after projection is allocated to sensing/AN beams, degrading eavesdropper SINR and improving secure communication performance.
  • The same construction principle yields desired rank-one information beamforming solutions for both bounded- and Gaussian-error problems.
  • The numerical study evaluates joint information and sensing beamforming under perfect, bounded-error, and Gaussian-error CSI scenarios.
  • The simulations use N = 8 antennas, K = 4 targets, and two untrusted targets at −15° and 15°.
  • The separate-design benchmark first meets secrecy requirements with information beamforming, then restricts sensing signals to the CU-channel null space.

A. Special Case with Perfect CSI

With perfect CSI, the proposed joint design directs information beams toward the CU while using sensing beams toward trusted and untrusted targets, balancing sensing and secure communication. It outperforms separate design in sensing error and angle estimation, although secure transmission can make some CU locations infeasible.

  • Information beams target the CU, while sensing beams target trusted and untrusted targets to support sensing and secure communication.
  • Near eavesdropping targets, information beams are substantially suppressed while sensing-beam energy is concentrated to degrade eavesdropping channels.
  • The proposed optimal design achieves lower sensing beampattern error than separate design across secrecy-rate thresholds, with a gap around 20% of the sensing-only error.
  • At R0 = 4 bps/Hz, secure transmission is infeasible when the CU lies in (−20°, −10°) ∪ (10°, 20°) near eavesdropping targets.
  • The proposed design yields lower angle-estimation RMSE than separate design, while sensing-only achieves the best sensing performance.

B. Scenario with Bounded CSI Errors

Under bounded eavesdropper CSI errors, the robust design reallocates beamforming resources to preserve secrecy while maintaining sensing performance. Compared with separate design, it achieves better sensing performance and remains close to the sensing-only benchmark, particularly at high SNR.

  • Transmit beampattern: Under uncertain eavesdropping channels, the information beampattern at the CU angle becomes narrower to strengthen confidential transmission.The reported setting uses R0 = 4 bps/Hz and µ = 0.3.
  • Transmit beampattern: Around untrusted targets at −15° and 15°, the transmit beampattern is provided by sensing signals, which also help prevent potential eavesdropping.These regions are further suppressed compared with perfect CSI.
  • Performance comparison: The robust design outperforms separate design because it accounts for the artificial-noise role of sensing signals in degrading eavesdropping channels.Separate design allocates excessive power to information beamforming, leaving secure-communication constraints inactive and deteriorating sensing performance.
  • Performance comparison: Larger bounded CSI uncertainty increases sensing beampattern error because more transmit power is allocated to secure communication.The resulting power reallocation causes more severe beampattern distortion.
  • Angle estimation: The optimal design performs significantly better than separate design and close to the sensing-only benchmark, especially at high SNR.This comparison supports robustness to bounded CSI errors in angle estimation.

C. Scenario with Gaussian CSI Errors

For Gaussian eavesdropper CSI errors, the paper uses secrecy-outage-constrained beamforming and evaluates how outage requirements affect sensing. Stricter secrecy requirements worsen sensing metrics, while the proposed design remains close to the sensing-only benchmark.

  • Evaluation setting: The Gaussian-error formulation considers secrecy outage constraints rather than the separate design benchmark, which may become infeasible when secrecy requirements are stringent.The subsection therefore focuses on the effects of R0 and ρ.
  • Beampattern error: Sensing beampattern error increases as ρ becomes smaller because the secrecy outage constraint becomes stricter and more power is allocated to CU-directed information signals.The reported values of ρ are 0.1, 0.15, and 0.2.
  • Sensing performance: A lower outage threshold ρ produces higher estimation RMSE because stricter secrecy requirements increase beampattern error and cross-correlation patterns.The resulting distortion worsens sensing performance.
  • Sensing performance: The proposed optimal design achieves RMSE close to the sensing-only benchmark under Gaussian CSI errors.The authors report this as evidence of robustness against Gaussian CSI errors.
  • Overall conclusion: Numerical results show a tradeoff among communication with the CU, sensing targets, and confusing eavesdroppers while satisfying the CU’s secrecy requirements.The robust design adjusts information and sensing beams to obtain desirable sensing transmit beampatterns.
  • Solution approach: The paper formulates bounded-error and Gaussian-error beampattern problems and solves them using SDR and one-dimensional search, with S-procedure for bounded errors and Bernstein-type convex restriction for Gaussian errors.The bounded-error SDR tightness is rigorously proved.

APPENDIX A PROOF OF PROPOSITION 1

Appendix A proves Proposition 1 by constructing an equivalent rank-one information-beamforming solution that preserves feasibility and objective value. The proof uses positive-semidefinite block-matrix conditions and related inequalities.

  • Rank-one construction: The proof constructs W* and S* from lifted variables while preserving the relevant constraints and objective value.The construction establishes equality of the information gain along g and retains feasibility of the associated constraints.
  • Positive-semidefinite argument: A positive-semidefinite block-matrix equivalence is used to establish the required matrix constraint.The theorem relates positive semidefiniteness of H to positive semidefiniteness of A and C plus a contraction condition.
  • Feasibility: Combining the matrix inequality with the block-matrix result proves that the right-hand side of (51) is positive semidefinite.This establishes constraint (27).
  • Conclusion: The constructed variables satisfy the remaining constraints and achieve the same objective value as the lifted solution.The proof then concludes that rank(W*) = 1.
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