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Robust Secure Transmission in MISO Channels Based on Worst-Case Optimization

Jing Huang, A. Lee Swindlehurst

arXiv:1104.3161v1cs.IT

TL;DR

Imperfect eavesdropper CSI makes robust precoder design for MISO wiretap channels a worst-case optimization problem. This paper develops robust covariance designs for direct transmission and cooperative jamming under power and QoS constraints, showing that helper jamming can increase worst-case secrecy rate and lower Eve’s SINR under global power constraints.

  • Problem

    The paper addresses robust transmit precoder design for MISO wiretap channels with imperfect eavesdropper CSI, including direct, cooperative-jamming, and legitimate-receiver QoS scenarios.

  • Method

    It maximizes worst-case secrecy performance through robust covariance optimization, using quasiconvex reformulation for individual power constraints and geometric programming for joint global-power design.

  • Results

    Under global power constraints with imperfect eavesdropper CSI, helper jamming increases worst-case secrecy rate and lowers Eve’s SINR.

  • Takeaways & Limitations

    Robust jamming support benefits secure transmission when channel mismatches affect the transmitters’ eavesdropper-link information.

Abstract

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This paper studies robust transmission schemes for multiple-input single-output (MISO) wiretap channels. Both the cases of direct transmission and cooperative jamming with a helper are investigated with imperfect channel state information (CSI) for the eavesdropper links. Robust transmit covariance matrices are obtained based on worst-case secrecy rate maximization, under both individual and global power constraints. For the case of an individual power constraint, we show that the non-convex maximin optimization problem can be transformed into a quasiconvex problem that can be efficiently solved with existing methods. For a global power constraint, the joint optimization of the transmit covariance matrices and power allocation between the source and the helper is studied via geometric programming. We also study the robust wiretap transmission problem for the case with a quality-of-service constraint at the legitimate receiver. Numerical results show the advantage of the proposed robust design. In particular, for the global power constraint scenario, although cooperative jamming is not necessary for optimal transmission with perfect eavesdropper's CSI, we show that robust jamming support can increase the worst-case secrecy rate and lower the signal to interference-plus-noise ratio at Eve in the presence of channel mismatches between the transmitters and the eavesdropper.

I. INTRODUCTION

The paper develops robust MISO wiretap transmission with imperfect, norm-bounded eavesdropper CSI for direct transmission and cooperative jamming. It derives efficient optimization methods under individual and global power constraints and with a legitimate-receiver QoS constraint.

  • Problem and contributions: Transmitters have perfect CSI for legitimate links but only imperfect eavesdropper CSI, modeled as norm-bounded channel mismatches.The robust design addresses the gap left by prior cooperative-jamming work that generally assumes perfect global CSI.
  • Problem and contributions: The study designs robust transmit covariance matrices for direct transmission and cooperative jamming with a helper by maximizing the worst-case secrecy rate.Unlike prior work, robustness applies to both information and jamming signals.
  • Optimization methods: Under individual power constraints, the non-convex maximin secrecy-rate problem is converted into a quasiconvex problem solvable efficiently with existing methods.This formulation is studied first, before the global power-constraint case.
  • Optimization methods: Under a global power constraint, jointly optimal transmit covariance matrices and source-helper power allocation are obtained via geometric programming.The global constraint case is described as more complicated than the individual-constraint case.
  • QoS-constrained design: With a quality-of-service constraint at the legitimate node, optimizing the covariance matrices and power allocation becomes simpler.The paper considers this QoS-constrained scenario in addition to the power-constraint formulations.

II. SYSTEM MODEL … C. Channel Mismatch

The system models secure MISO transmission from Alice to Bob against Eve, with optional cooperative jamming from a helper. It assumes perfect CSI toward Bob, imperfect CSI toward Eve, and worst-case optimization over bounded channel mismatches under individual or global power constraints.

  • II. SYSTEM MODEL: Alice, a multi-antenna source, sends private messages to single-antenna Bob while single-antenna Eve may eavesdrop.The helper is also multi-antenna in the system model.
  • II. SYSTEM MODEL: The helper can remain silent for direct transmission or transmit artificial interference for cooperative jamming.Both transmission modes are considered in the paper.
  • II. SYSTEM MODEL: Alice and the helper have perfect CSI for their Bob links but only imperfect CSI for their Eve channels.The model considers either individual or global power constraints.
  • A. Direct Transmission: In direct transmission, Alice transmits signal vector x with covariance Qx subject to tr(Qx) ≤ PS.Bob and Eve have channel vectors hb and he, respectively, and naturally occurring noise terms nb and ne.
  • B. Cooperative Jamming: In cooperative jamming, the helper transmits an i.i.d. Gaussian interference signal z with covariance Qz and tr(Qz) ≤ PJ.The optimization uses either individual constraints on Alice and the helper or a global power constraint.
  • C. Channel Mismatch: For the Eve links, Alice and the helper possess channel estimates ˜he and ˜ge, with corresponding channel error vectors.The estimates are used because only imperfect CSI is available for the transmitter-to-Eve channels.
  • C. Channel Mismatch: The channel mismatches lie in bounded uncertainty sets Eh and Eg defined by known constants ǫh and ǫg.All optimization problems use the error realizations e∗ and g∗ that produce the worst performance.

III. ROBUST DIRECT TRANSMISSION

This section develops robust direct transmission without helper jamming when Eve’s channel is imperfectly known, maximizing secrecy against bounded channel mismatches. A non-convex maximin formulation is transformed into a quasiconvex problem solvable by bisection, yielding both the optimal covariance and worst-case mismatch.

  • Robust formulation: Direct transmission considers Alice without helper jamming and imperfect knowledge of Eve’s channel, represented by an estimate and bounded mismatch set.The design maximizes secrecy rate for the worst channel mismatch in the bounded set.
  • Quasiconvex reformulation: The inner minimization over Eve’s mismatch is non-convex, but a proper transformation converts the maximin problem into a solvable quasiconvex optimization problem.The resulting problem has a linear fractional objective and LMI constraints.
  • Quasiconvex reformulation: The quasiconvex problem is solved efficiently by bisection, which repeatedly solves a convex feasibility problem until the interval containing the optimal value converges.This procedure obtains the transmit covariance Qx.
  • Worst-case mismatch: The worst-case channel mismatch is obtained through a dual-based solution despite the original non-convex maximization of a convex function.The resulting formulation is a semidefinite program that can be solved efficiently using an interior-point method.
  • Results: The robust beamformer design obtains both the optimal covariance matrix and its corresponding worst-case channel mismatch.The section characterizes the conversion from a non-convex maximin problem to a quasiconvex problem solvable by bisection.

IV. ROBUST COOPERATIVE JAMMING · A. Individual Power Constraint

This section develops robust cooperative-jamming designs under individual power constraints, using zero-forcing to simplify optimization with imperfect eavesdropper CSI. It characterizes rank-one jamming for perfect CSI and an efficient SDP-based solution for imperfect CSI.

  • IV. ROBUST COOPERATIVE JAMMING: The Helper provides cooperative jamming to improve the secrecy rate under imperfect eavesdropper channel state information.
  • IV. ROBUST COOPERATIVE JAMMING: The section first imposes separate power constraints on the source and Helper, before contrasting them with a later global constraint.The individual feasible sets require Qx ⪰0, tr(Qx) ≤PS and Qz ⪰0, tr(Qz) ≤PJ.
  • IV. ROBUST COOPERATIVE JAMMING: A zero-forcing constraint is imposed on the jamming signal so the Helper’s transmission is orthogonal to Bob’s channel.
  • IV. ROBUST COOPERATIVE JAMMING: Under zero forcing, maximizing secrecy rate over Qz does not depend on Qx, so the jamming covariance is optimized first and Qx is then calculated.
  • A. Individual Power Constraint: For perfect eavesdropper CSI, the optimal Qz is rank one, yielding a single-stream zero-forcing jamming beamformer.It can be written as Qz = PJwwH, where w is the Helper’s unit-normalized one-dimensional beamformer.
  • A. Individual Power Constraint: The corresponding perfect-CSI beamformer is obtained through null steering, while the optimal information covariance Qx is also rank one.The Qx beamformer is described as the generalized eigenvector of a matrix pencil.
  • A. Individual Power Constraint: With imperfect eavesdropper CSI, the jamming problem is reformulated as an SDP with a linear objective and LMI constraints.The resulting optimal robust covariance can therefore be obtained efficiently.
  • A. Individual Power Constraint: The hidden worst-case channel mismatch is recovered through a convex optimization problem whose strong duality provides the worst-case mismatch.The final optimization over Qx follows the same procedure as in Section III.

B. Global Power Constraint

This section jointly optimizes information and jamming covariance matrices with their power allocation under a global power constraint. It uses alternating robust covariance optimization and geometric-programming-based power allocation, with convergence supported by numerical experiments.

  • B. Global Power Constraint: The design jointly optimizes Qx, Qz, and power allocation between Alice and the Helper under tr(Qx) + tr(Qz) = p1 + p2 ≤P.The helper’s jamming signal is constrained to satisfy zero forcing at Bob.
  • B. Global Power Constraint: The joint algorithm alternates robust covariance optimization with power allocation updates until convergence.Each iteration updates the information and jamming covariances using robust maximin problems, then solves the power-allocation problem with Algorithm IV.1.
  • B. Global Power Constraint: For fixed covariance shapes, the non-convex power-allocation problem is solved through single condensation and successive geometric-programming steps.The quadratic fractional formulation is converted into GP form, enabling efficient convex optimization at each step.
  • B. Global Power Constraint: The procedure converges because each iteration increases secrecy rate while the rate is bounded above by the no-eavesdropper case with zero jamming power.The successive GP approximations have polynomial-time interior-point solutions and converge to a point satisfying the original problem’s KKT conditions.
  • B. Global Power Constraint: Extensive numerical experiments further indicate that the iterative procedure obtains the global optimum.The cited experiments include results presented in Section VI.

V. ROBUST TRANSMIT DESIGN WITH QOS CONSTRAINT

With a fixed SINR requirement on the legitimate link, robust transmit design becomes minimizing the eavesdropper’s worst-mismatch SINR under a power constraint. Adding this SINR constraint simplifies the robust solution.

  • The problem imposes a fixed SINR constraint on the legitimate link.
  • Under a given power constraint, the robust objective is minimizing the eavesdropper SINR for the worst channel mismatch.
  • Adding the legitimate-link SINR constraint simplifies the robust solution.

A. Robust Direct Transmission

Robust direct transmission under channel mismatch is reformulated into efficiently solvable optimization problems. A relaxed beamformer nulls Eve while aligning with Bob, making robust and non-robust designs close when sufficient power meets Bob’s SINR demand.

  • Robust formulation: Channel-mismatch optimization simplifies the secrecy-rate fractional expression by introducing an extra affine inequality constraint.
  • Efficient solution: The resulting problem is equivalent to an SDP with a linear objective, avoiding the bisection required for the corresponding quasiconvex formulation.
  • Beamforming structure: The relaxed design steers Qx’s eigenvectors to maximize Bob’s received signal while imposing a zero-forcing constraint at Eve.
  • Beamforming structure: When transmit power is sufficient to meet the SINR demand, Alice chooses a beamformer that nulls Eve and aligns as closely as possible with Bob.
  • Comparison: Robust and non-robust covariance matrices, and their corresponding worst-case SINRs, are expected to be very close in this regime.

B. Robust Cooperative Jamming · VI. NUMERICAL RESULTS

Under a global power constraint and Bob’s QoS requirement, robust cooperative-jamming design becomes an SDP with a quasiconvex objective solvable by bisection. Numerical results show robust DT/CJ designs improve worst-case secrecy or suppress Eve’s SINR under channel mismatch, while jamming is unnecessary with perfect CSI.

  • B. Robust Cooperative Jamming: With a QoS constraint, robust cooperative jamming is considered only under a global power constraint, without imposing a helper-to-Eve ZF constraint.The individual-power generalization is described as straightforward.
  • B. Robust Cooperative Jamming: The QoS constraint converts the fractional quadratic objective into a linear fractional form in Qx and Qz, enabling direct reformulation.The reformulated problem is equivalent to the original optimization problem.
  • B. Robust Cooperative Jamming: The reformulated problem is an SDP with a quasiconvex objective, LMIs, and affine inequalities, and can therefore be solved using bisection.If Bob’s SINR requirement cannot be met under the given power constraint, the problem is infeasible and transmission is treated as outage.
  • VI. NUMERICAL RESULTS: The numerical study evaluates robust DT and CJ across power constraints, channel-error bounds, and QoS constraints against non-robust schemes.Alice and the Helper each have four antennas, Bob and Eve each have one, and results average 1000 independent trials.
  • VI. NUMERICAL RESULTS: Under individual power constraints and channel mismatch, robust DT and CJ achieve better worst-case secrecy rates than their non-robust counterparts.Non-robust CJ can perform worse than robust DT at low transmit powers despite having twice DT’s available power, because both Alice–Eve and Helper–Eve errors degrade it.
  • VI. NUMERICAL RESULTS: Under a 10dB global power limit, flexible power allocation improves robust CJ’s optimal worst-case secrecy rate over fixed individual power constraints.As channel mismatch increases, a larger fraction of transmit power is devoted to jamming.
  • VI. NUMERICAL RESULTS: When channel mismatch is zero, all schemes achieve the same secrecy or zero Eve SINR, making artificial noise unnecessary; increasing mismatch reveals robust CJ’s advantage.For larger mismatch, non-robust CJ suffers from errors on both links, while robust CJ increases its jamming fraction to preserve performance.
  • VI. NUMERICAL RESULTS: With Bob’s QoS constraint, robust CJ significantly suppresses Eve’s SINR, whereas robust DT performs almost identically to non-robust DT.As Bob’s SINR requirement rises, less power is available for jamming, so CJ approaches DT performance.

VII. CONCLUSIONS

The paper develops robust transmit covariance designs for MISO wiretap channels with imperfect eavesdropper CSI across direct transmission, cooperative jamming, power constraints, and legitimate-receiver QoS constraints. Numerical results show that robust helper jamming can improve worst-case secrecy under imperfect CSI when robust beamforming is used.

  • Robust design framework: Robust transmit covariance matrices were obtained for direct transmission and cooperative jamming using worst-case secrecy rate maximization with imperfect eavesdropper CSI.The designs address MISO wiretap channels with imperfect ECSI.
  • Individual power constraints: For individual power constraints, the non-convex optimization problem was transformed into a quasiconvex problem.This transformation enables efficient solution using existing quasiconvex optimization methods.
  • Global power constraints: For global power constraints, an algorithm jointly optimizes transmit covariance matrices and power allocation between the source and helper.The global-constraint formulation addresses joint design rather than covariance optimization alone.
  • Legitimate-receiver QoS: Robust transmit covariance matrices were also obtained when a QoS constraint was imposed at the legitimate receiver.The QoS-constrained scenario extends the robust design framework beyond secrecy-rate optimization alone.
  • Numerical findings: Under a global power constraint with imperfect ECSI, helper jamming increased the worst-case secrecy rate and lowered Eve’s SINR when robust beamforming was employed.Cooperative jamming was not helpful with perfect ECSI in this setting.

APPENDIX A PROOF OF PROPOSITION 2 · APPENDIX B PROOF OF LEMMA 1 · APPENDIX C PROOF OF PROPOSITION 3

The appendices establish the stated propositions and lemma by converting non-convex or robust optimization conditions into tractable dual, SDP, KKT, and LMI formulations. They also prove that the helper’s optimal jamming covariance has rank one.

  • APPENDIX A PROOF OF PROPOSITION 2: The proof of Proposition 2 identifies the reformulated problem as non-convex because its Hessian is negative semidefinite.
  • APPENDIX A PROOF OF PROPOSITION 2: Using a Schur complement, the dual problem in Proposition 2 is expressed as a semidefinite program.
  • APPENDIX A PROOF OF PROPOSITION 2: The resulting trust region subproblem admits strong duality despite its non-convex objective, equating the relevant primal and dual optimal values.
  • APPENDIX B PROOF OF LEMMA 1: For Lemma 1, strict feasibility and linearity of the objective allow Slater’s theorem to establish KKT conditions for the primal and dual problems.
  • APPENDIX B PROOF OF LEMMA 1: The KKT analysis shows λ must be positive and establishes rank(Θ) ≥N −1 through the eigenvalue structure of the dual multiplier.
  • APPENDIX B PROOF OF LEMMA 1: Because rank(Θ) cannot equal N when jamming signals are transmitted, the proof concludes rank(Θ) = N −1 and rank(Qz) = 1.
  • APPENDIX C PROOF OF PROPOSITION 3: The proof of Proposition 3 transforms the maximin problem using the S-procedure and Schur complements, ultimately expressing a constraint as an LMI.

APPENDIX D PROOF OF PROPOSITION 4 · APPENDIX E PROOF OF PROPOSITION 5

The appendices prove Propositions 4 and 5 by reformulating their optimization problems and applying duality tools, including Schur complements and the S-procedure. Proposition 4 concludes with an SDP formulation, while Proposition 5 derives the expression in (50).

  • APPENDIX D PROOF OF PROPOSITION 4: Problem (26) is rewritten as an equivalent formulation with λ ≥ 0.The proof follows the same line as Appendix A.
  • APPENDIX D PROOF OF PROPOSITION 4: The minimum of L(eg, λ) with respect to eg is achieved at the stated optimizer.This establishes the minimization step in the transformed proof.
  • APPENDIX D PROOF OF PROPOSITION 4: A Schur complement converts the dual problem into a semidefinite program.The resulting SDP completes the proof of Proposition 4.
  • APPENDIX E PROOF OF PROPOSITION 5: The maximin problem (49) is first transformed into the stated equivalent problem.This provides the starting point for the proof of Proposition 5.
  • APPENDIX E PROOF OF PROPOSITION 5: The S-procedure is applied to constraints (78b), (78e), (78c), and (78f) in problem (78).The transformation parallels the earlier use of the S-procedure in (8)–(12).
  • APPENDIX E PROOF OF PROPOSITION 5: Applying a Schur complement to the resulting constraints yields the expression in (50).This establishes the claimed expression for Proposition 5.
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