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Spatially Selective Artificial-Noise Aided Transmit Optimization for MISO Multi-Eves Secrecy Rate Maximization

Qiang Li, Wing-Kin Ma

arXiv:1303.1915v1cs.IT

TL;DR

The paper addresses AN-aided secrecy rate maximization for MISO channels and develops an optimization framework with flexible AN spatial patterns. The proposed design outperforms no-AN and isotropic-AN strategies, particularly as the number of eavesdroppers increases.

  • Problem

    The paper focuses on artificial-noise-aided secrecy rate maximization for a MISO channel.

  • Method

    The framework allows arbitrary AN spatial patterns, reformulates SRM as a single-variable optimization, and uses SDP-based methods.

  • Results

    The proposed AN-aided SRM design achieves higher secrecy rates than optimal no-AN and isotropic-AN designs, especially with more eavesdroppers.

  • Takeaways & Limitations

    Optimizing AN is necessary for effectively interfering with eavesdroppers, while confidential-information beamforming is generally optimal.

Abstract

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Consider an MISO channel overheard by multiple eavesdroppers. Our goal is to design an artificial noise (AN)-aided transmit strategy, such that the achievable secrecy rate is maximized subject to the sum power constraint. AN-aided secure transmission has recently been found to be a promising approach for blocking eavesdropping attempts. In many existing studies, the confidential information transmit covariance and the AN covariance are not simultaneously optimized. In particular, for design convenience, it is common to prefix the AN covariance as a specific kind of spatially isotropic covariance. This paper considers joint optimization of the transmit and AN covariances for secrecy rate maximization (SRM), with a design flexibility that the AN can take any spatial pattern. Hence, the proposed design has potential in jamming the eavesdroppers more effectively, based upon the channel state information (CSI). We derive an optimization approach to the SRM problem through both analysis and convex conic optimization machinery. We show that the SRM problem can be recast as a single-variable optimization problem, and that resultant problem can be efficiently handled by solving a sequence of semidefinite programs. Our framework deals with a general setup of multiple multi-antenna eavesdroppers, and can cater for additional constraints arising from specific application scenarios, such as interference temperature constraints in interference networks. We also generalize the framework to an imperfect CSI case where a worst-case robust SRM formulation is considered. A suboptimal but safe solution to the outage-constrained robust SRM design is also investigated. Simulation results show that the proposed AN-aided SRM design yields significant secrecy rate gains over an optimal no-AN design and the isotropic AN design, especially when there are more eavesdroppers.

I. INTRODUCTION

The paper studies AN-aided secrecy-rate maximization for a MISO channel with multiple multi-antenna eavesdroppers, allowing joint optimization of confidential-information and AN covariances. It develops SDP-based methods for perfect and imperfect CSI while permitting general AN spatial patterns and application-specific constraints.

  • The paper targets secrecy-rate maximization for a MISO channel overheard by multiple multi-antenna eavesdroppers under perfect or imperfect CSI.
  • The proposed formulation jointly optimizes confidential-information and AN covariances, with AN allowed to take any spatial pattern based on CSI.Spatially selective AN is intended to interfere with eavesdroppers more effectively than isotropic AN.
  • Existing designs often restrict AN spatially or optimize it separately, producing tractable but secrecy-rate-suboptimal formulations.Examples include isotropic or nullspace AN, AN constrained not to reduce legitimate mutual information, and beamforming-based AN.
  • For perfect CSI, the SRM problem is converted into a single-variable optimization problem solvable through a sequence of convex semidefinite programs.The equivalent problem is less complex than the original SRM formulation, and efficient SDP solvers are available.
  • The framework extends to worst-case robust SRM and a suboptimal but safe outage-constrained robust SRM design under imperfect CSI.It also accommodates multiple multi-antenna eavesdroppers and additional application-specific constraints such as interference-temperature constraints.
  • Transmit beamforming is shown to be SRM-optimal for confidential-information transmission in both perfect-CSI and worst-case robust imperfect-CSI cases.The result generalizes earlier optimality results established for one eavesdropper, no AN, and perfect CSI.

II. SYSTEM MODEL AND PROBLEM STATEMENT

The system models Alice transmitting confidential information and artificial noise to Bob while multiple multi-antenna Eves listen over slow frequency-flat fading channels. The confidential-information and AN covariances are jointly designed.

  • Alice sends a transmit signal vector x(t) to a single-antenna legitimate receiver, Bob, in the presence of multiple multi-antenna eavesdroppers, Eves.
  • The communication links are assumed to undergo slow frequency-flat fading.
  • The transmitted signal consists of confidential information s(t) plus artificially generated noise z(t), written as x(t) = s(t) + z(t).
  • The confidential signal follows CN(0, W), where W is the designed transmit covariance, while the AN follows CN(0, Σ), where Σ is the designed AN covariance.
  • When W has rank at most one, the confidential-information transmission becomes transmit beamforming, with s(t) = ws(t) for a data stream s(t).

B. Problem Statement

The problem jointly designs the confidential-information and AN covariances to maximize achievable secrecy rate under power and application-specific constraints. A nonconvex formulation is replaced by a relaxation that is shown to preserve an optimal rank-one confidential-information covariance.

  • The design objective is to choose W and Σ so that the maximum information-theoretic secrecy rate is achieved.
  • The standard formulation uses a transmit sum-power constraint, while additional constraints can represent application-specific requirements.
  • Per-antenna power constraints: Per-antenna constraints limit each diagonal entry of W + Σ by the corresponding antenna power limit.
  • Interference temperature constraints: Interference-temperature constraints limit the interference delivered to primary users in spectrum-sharing cognitive-radio or multicell interference settings.
  • The original SRM problem is nonconvex because of its secrecy-rate constraint, motivating a tractable inequality-based reformulation.
  • The relaxed problem is tight or equivalent to the original SRM problem and admits an optimal solution with rank(W⋆) ≤1.
  • The rank-one result holds regardless of the number of Eves or their antennas, making transmit beamforming optimal for confidential-information transmission.

B. An SDP-based Line Search Method for Relaxed SRM (10)

The relaxed SRM problem is solved through a one-variable search whose evaluations are convex semidefinite programs. The resulting approach computes a globally optimal solution and can recover the original transmit and AN covariances.

  • The relaxed SRM problem is reformulated as a one-variable optimization problem evaluated through a sequence of semidefinite programs.
  • A fractional quasiconvex objective is transformed into a linear convex objective by fixing its denominator and applying the Charnes-Cooper transformation.
  • The transformed problem is a convex SDP that can be solved globally with conic-optimization software for any fixed scalar parameter.
  • The scalar optimization is handled by a one-dimensional line search over α in [(1 + P∥h∥2)^−1, 1].
  • The SDP output is used to recover W⋆ and Σ⋆, with an additional rank-one construction procedure available when needed.
  • The development establishes both an SDP-based optimization approach and the SRM-optimality of transmit beamforming for confidential information.

IV. EXTENSION TO ROBUST SRM

The framework is extended to imperfect eavesdropper CSI through a worst-case robust SRM formulation. It also provides a safe approximation for outage-constrained robust SRM under Gaussian CSI uncertainties.

  • The robust extension addresses incomplete knowledge of Eves’ CSI using a worst-case robust SRM formulation with norm-bounded uncertainties.
  • An SDP-based solution approach is derived for the worst-case robust SRM problem.
  • The robust worst-case design is used to provide a safe approximation to outage-constrained robust SRM under Gaussian-distributed CSI uncertainties.

A. The Worst-Case Robust SRM Problem

The worst-case robust SRM formulation handles bounded CSI uncertainty at the eavesdroppers while preserving an efficiently solvable optimization structure. Its relaxation is tight, yields rank-one confidential-information transmission, and is solved through SDP-based line search.

  • Problem formulation: Bounded CSI errors are modeled as G_k = ¯G_k + ∆G_k, with ∆G_k constrained within specified uncertainty sets.
  • Problem formulation: The robust formulation is more challenging than SRM because it imposes infinitely many inequalities over possible eavesdropper channel errors.
  • Tight relaxation: For fixed β, Proposition 2 converts the infinite quadratic matrix inequalities into a single linear matrix inequality that convex conic optimization can handle efficiently.
  • Tight relaxation: Theorem 2 establishes that the resulting relaxation is equivalent to the WCR-SRM problem and has an optimal solution with rank(W⋆) ≤1.
  • Solution method: The robust problem is reduced to a one-dimensional line search whose fixed-parameter subproblems are convex SDPs.
  • Solution properties: Transmit beamforming remains an optimal confidential-information transmission strategy for the WCR-SRM formulation.

B. The Outage-Constrained Robust SRM Problem

The outage-constrained robust formulation targets a secrecy-rate satisfaction probability under unbounded random CSI errors. A sphere-bounding construction supplies a safe, generally suboptimal approximation based on the worst-case robust design.

  • Safe approximation: The method assumes independent CSI errors following an i.i.d. complex Gaussian distribution and chooses uncertainty radii using a Chi-square inverse cumulative distribution function.
  • Problem formulation: Unbounded CSI errors prevent an absolutely safe design, motivating a (1 −δ)% safe design for a specified outage probability δ.
  • Problem formulation: The OCR-SRM problem maximizes the δ%-outage secrecy rate subject to a probabilistic secrecy-rate constraint.
  • Guarantee: Every feasible point of the worst-case robust problem is feasible for the outage-constrained problem, making the construction a safe approximation.
  • Safe approximation: Sphere bounding places a radius around the random error so that a specified portion of realizations lies inside it while the worst-case secrecy rate is maintained there.

V. SIMULATION RESULTS

Simulations compare the proposed spatially selective AN-aided SRM design with isotropic-AN and no-AN designs under varying power, antennas, and eavesdropper counts. The proposed design is especially advantageous with multiple eavesdroppers and limited transmit degrees of freedom.

  • Example 1: The proposed AN-aided SRM design performs at least no worse than isotropic-AN and no-AN designs in the one-Eve and three-Eve cases.
  • Example 1: For K = 3, the secrecy-rate gap between the proposed and isotropic-AN designs can approach 2 bits per channel use at large P.
  • Example 1: For P ≥10dB, the no-AN secrecy rate nearly stops increasing, and isotropic AN can outperform it when multiple Eves are present.
  • Example 1: The simulations attribute poor no-AN performance to insufficient transmit degrees of freedom and find AN effective against the resulting bottleneck.
  • Example 1: Increasing Nt lets no-AN SRM approach AN-aided SRM, while increasing K causes no-AN performance to drop rapidly.

B. Example 2: Secrecy Rate Performance with Additional Interference Temperature Constraints

Additional simulations evaluate interference-temperature-constrained and imperfect-CSI settings. The proposed AN-aided designs outperform the compared alternatives, while nonrobust designs become sensitive to CSI uncertainty at higher power.

  • Interference temperature constraints: The interference-temperature experiment uses Nt = 5, Ne,k = 3, K = 3, and one primary user with two receive antennas.
  • Interference temperature constraints: With an interference temperature constraint ρ = 5dB, the proposed AN-aided SRM design provides better secrecy-rate performance, especially at large P.
  • Imperfect CSI: Under worst-case CSI uncertainty, nonrobust AN-aided SRM degrades for P ≥15dB, with degradation worsening as P increases.
  • Imperfect CSI: The proposed AN-aided WCR-SRM design achieves the best worst-case secrecy-rate performance among the compared designs.
  • Imperfect CSI: For outage secrecy rates, results generally follow the worst-case pattern, while nonrobust AN-aided SRM is slightly better than AN-aided OCR-SRM for P ≤20dB.
  • Imperfect CSI: The AN-aided OCR-SRM method uses a safe approximation and generally achieves the best outage-constrained secrecy-rate performance.

VI. CONCLUSION AND DISCUSSION

The paper develops SDP-based methods for AN-aided secrecy-rate maximization in MISO channels with multiple multi-antenna eavesdroppers, covering perfect and imperfect CSI. Simulations report better performance than no-AN and isotropic-AN designs, especially with more eavesdroppers.

  • The framework addresses AN-aided secrecy-rate maximization for MISO channels overheard by multiple multi-antenna eavesdroppers under perfect and imperfect CSI.
  • The SRM and worst-case robust SRM problems can be efficiently handled by solving a sequence of semidefinite programs.
  • Transmit beamforming is generally optimal for confidential-information transmission, while the framework also provides a safe approximation for outage-constrained robust SRM.
  • Simulation results show that the proposed designs outperform optimal no-AN SRM and isotropic-AN designs, particularly as the number of eavesdroppers increases.
  • The analysis does not impose a nullspace AN constraint, although that constraint can be added and handled with essentially the same SDP-based approach.
  • Simulations indicate that the nullspace-constrained SRM design can achieve secrecy rates quite close to those without the constraint, but the corresponding results are not shown because of the page limit.

APPENDIX

The appendix proves equivalence and rank properties underlying the SDP relaxation and its solution correspondence. The proofs use lower bounds, power minimization, and KKT conditions to establish rank-one confidential-information covariance solutions.

  • The proof establishes lower bounds and equivalences for the determinant-based expressions used in the optimization analysis.
  • For a feasible β, a secrecy-rate-constrained power-minimization problem minimizes total transmit power while preserving the inner problem’s optimal secrecy rate.
  • The power-minimization solution is also optimal for the corresponding inner maximization problem, and its total power remains within the sum-power budget.
  • KKT optimality conditions establish that an optimal confidential-information covariance W has rank no greater than one.
  • The rank-one solution is feasible for both the relaxed and original inner maximization problems, yielding a solution correspondence between them for feasible β.
  • The proof concludes that the corresponding optimal solutions of the original and relaxed problems have equal objective values and that the relaxation preserves an optimal rank-one solution.

C. Proof of Theorem 2

The proof of Theorem 2 applies the same power-minimization and KKT strategy to the robust formulation. It shows that the relevant optimal transmit covariance has rank no greater than one.

  • For feasible β, the proof introduces a secrecy-rate-constrained power-minimization problem for the robust SRM formulation.
  • The optimal solution of this constrained problem is also optimal for the corresponding inner maximization problem.
  • Rewriting the problem and examining its KKT conditions provides the conditions needed to characterize the optimal covariance.
  • The KKT relations imply rank(W) ≤1 for the optimal solution.
  • The proof then argues that this rank-one optimal solution is also optimal for the original robust problem at the same β.
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