PaperlayerView exact revision on arXiv

Source-linked AI summary

Optimal and Robust Transmit Designs for MISO Channel Secrecy by Semidefinite Programming

Qiang Li, Wing-Kin Ma

arXiv:1012.3875v2cs.IT

TL;DR

The paper addresses nonconvex secrecy-rate maximization for an MISO channel overheard by multiple multi-antenna eavesdroppers. It recasts the problem as an SDP, including a worst-case robust formulation for spherical CSI uncertainties, and finds transmit beamforming generally optimal for both perfect and imperfect CSI.

  • Problem

    Secrecy-rate maximization for an MISO channel with multiple multi-antenna eavesdroppers is nonconvex and lacks an efficient optimal transmit solution in general.

  • Method

    The paper reformulates perfect-CSI SRM as an SDP and handles imperfect CSI with a worst-case spherical uncertainty model and SDP-based robust constraints.

  • Results

    The SRM problems have optimal SDP solutions, and analysis shows transmit beamforming is generally optimal under both perfect and imperfect CSI.

  • Takeaways & Limitations

    Transmit covariance optimization in the considered multi-eavesdropper MISO secrecy scenario can be solved tractably while retaining a generally optimal beamforming strategy.

Abstract

from arXiv · show

In recent years there has been growing interest in study of multi-antenna transmit designs for providing secure communication over the physical layer. This paper considers the scenario of an intended multi-input single-output channel overheard by multiple multi-antenna eavesdroppers. Specifically, we address the transmit covariance optimization for secrecy-rate maximization (SRM) of that scenario. The challenge of this problem is that it is a nonconvex optimization problem. This paper shows that the SRM problem can actually be solved in a convex and tractable fashion, by recasting the SRM problem as a semidefinite program (SDP). The SRM problem we solve is under the premise of perfect channel state information (CSI). This paper also deals with the imperfect CSI case. We consider a worst-case robust SRM formulation under spherical CSI uncertainties, and we develop an optimal solution to it, again via SDP. Moreover, our analysis reveals that transmit beamforming is generally the optimal transmit strategy for SRM of the considered scenario, for both the perfect and imperfect CSI cases. Simulation results are provided to illustrate the secrecy-rate performance gains of the proposed SDP solutions compared to some suboptimal transmit designs.

I. INTRODUCTION

The paper studies secrecy-rate-maximizing transmit covariance design for an MISO channel overheard by multiple multi-antenna eavesdroppers. It converts the nonconvex problem into an SDP and extends the solution to worst-case spherical CSI uncertainty, while finding transmit beamforming generally optimal.

  • The target scenario is an MISO channel with multiple multi-antenna eavesdroppers, where transmit covariance is optimized for maximum secrecy rate.
  • Unlike the general nonconvex formulation, the considered SRM problem admits an equivalent convex, tractable SDP with efficiently obtainable global optima.
  • The paper develops a robust SRM formulation for imperfect CSI using spherical uncertainty sets and worst-case treatment of admissible errors.
  • Linear matrix inequality characterization makes the imperfect-CSI robust problem tractable through an SDP representation.
  • Transmit beamforming is generally optimal for SRM in the considered scenario under both perfect and imperfect CSI.
  • Simulations compare the proposed perfect- and imperfect-CSI SRM solutions with suboptimal secrecy transmit designs.

II. BACKGROUND AND PROBLEM STATEMENT

The paper formulates physical-layer secrecy for a multi-antenna transmitter sending to a single-antenna Bob while one multi-antenna Eve listens. It reviews secrecy-rate optimization, whose positive-rate optimum uses rank-one transmit beamforming.

  • System model: The reviewed system has a multi-antenna transmitter, single-antenna legitimate receiver Bob, and multi-antenna eavesdropper Eve.
  • Secrecy objective: Physical-layer secrecy seeks transmit designs that prevent Eve from retrieving useful information from the transmission.
  • Secrecy objective: The secrecy-rate objective maximizes the mutual-information difference between the Alice-to-Bob and Alice-to-Eve channels under an average power limit.
  • Secrecy capacity: The optimized secrecy rate is achievable while Bob receives a perfectly secure message and Eve retrieves almost nothing about it.
  • One-Eve solution: For the one-Eve model, the secrecy-rate maximization problem has a closed-form solution based on a principal generalized eigenvector.
  • One-Eve solution: When the optimal secrecy rate is positive, the optimal transmit covariance is rank one, yielding transmit beamforming; at zero rate, transmission is shut down.

B. Multi-Eavesdropper Case

The multi-Eve extension maximizes the worst secrecy rate across multiple multi-antenna eavesdroppers. Although the covariance problem is nonconvex and lacks a general closed form, it is shown equivalent to a convex, tractable SDP with rank-one optimal solutions.

  • Model and objective: The considered model has an MISO Alice-to-Bob link overheard by multiple multi-antenna Eves, with secrecy rate constrained by the worst Eve.
  • Problem difficulty: The multi-Eve secrecy-rate maximization problem is nonconvex because of the Eve-induced terms in the covariance objective.
  • Problem difficulty: Unlike the one-Eve case, whether the general multi-Eve problem has a closed-form solution is not known.
  • Suboptimal design: Projected-MRT nulls all Eves and is simple, but nulling may be impossible when the Eves’ combined receive-antenna count reaches or exceeds Nt.
  • Proposed approach: The secrecy-rate constrained problem minimizes average transmit power while requiring every Eve-specific secrecy rate to be at least a specified R.
  • Proposed approach: The paper establishes an SDP equivalent for the constrained problem and links it to secrecy-rate maximization, enabling the original SRM problem to be solved exactly by SDP.

A. The Secrecy-Rate Constrained Problem

The paper converts secrecy-rate constrained and maximization formulations into convex semidefinite programs through relaxation, rank-one tightness, and a Charnes–Cooper transformation. The resulting solutions are unique and rank one in the nontrivial cases, implying optimal transmit beamforming.

  • A. The Secrecy-Rate Constrained Problem: The secrecy-rate constrained formulation is nonconvex because its constraints contain determinant functions.
  • A. The Secrecy-Rate Constrained Problem: The constrained problem is relaxed into a convex SDP whose globally optimal solution can be efficiently found by available solvers.
  • A. The Secrecy-Rate Constrained Problem: For positive feasible secrecy-rate specifications, the relaxed constrained problem has a unique rank-one optimum, making the relaxation exact.
  • A. The Secrecy-Rate Constrained Problem: The exact constrained solution’s rank-one structure implies that transmit beamforming is generally optimal for the constrained problem.
  • B. The Secrecy-Rate Maximization Problem: For 0 < γ⋆(P) < 1, the relaxed secrecy-rate maximization problem exactly solves the original problem and has a unique rank-one solution.
  • B. The Secrecy-Rate Maximization Problem: A Charnes–Cooper transformation and epigraph reformulation recast the relaxed maximization problem as an SDP, avoiding repeated SDP solves from bisection.
  • B. The Secrecy-Rate Maximization Problem: The final SDP is equivalent to the original maximization problem through W = Z/ξ, and its rank-one solution implies optimal transmit beamforming.

IV. SECRECY-RATE OPTIMIZATION WITH IMPERFECT CSI

The paper extends its MISO secrecy formulations from perfect CSI to imperfect CSI. It considers worst-case robust designs under CSI uncertainty and shows that these more complex problems can still be transformed into SDPs.

  • CSI uncertainty: The perfect-CSI formulations assume Alice knows Bob’s and the Eves’ channel state information exactly.
  • Robust formulation: The imperfect-CSI extension uses worst-case robust secrecy-rate formulations to account for channel uncertainties.
  • Robust solution: Although the robust problems are structurally more complex and harder to solve than their perfect-CSI counterparts, they can still be turned into SDPs.

A. Robust Secrecy-Rate Problem Formulations

The paper models imperfect CSI as bounded deterministic uncertainty and formulates worst-case robust secrecy-rate optimization. It converts the robust secrecy-rate constrained problem into an SDP whose solution is exact, unique, and rank one.

  • A. Robust Secrecy-Rate Problem Formulations: Imperfect CSI is modeled as deterministic channel deviations around known means, bounded by Frobenius-norm uncertainty radii.The uncertainty sets constrain Bob’s and each Eve’s channel deviations by positive bounds.
  • A. Robust Secrecy-Rate Problem Formulations: The robust formulation maximizes a transmit covariance’s worst-case secrecy rate over all channel realizations in the prescribed uncertainty sets.The resulting design guarantees a secrecy rate no lower than the worst-case optimum for every permitted channel realization.
  • A. Robust Secrecy-Rate Problem Formulations: The robust secrecy-rate constrained problem minimizes transmit power while satisfying a target secrecy rate under worst-case channel uncertainty.This robust SRC problem is used as a step toward solving robust secrecy-rate maximization.
  • B. The Robust Secrecy-Rate Constrained Problem: A slack variable decouples fractional constraints, after which the relaxed problem is converted into linear matrix inequalities using the S-procedure.The S-procedure handles the semi-infinite constraints induced by the uncertainty sets.
  • B. The Robust Secrecy-Rate Constrained Problem: The relaxation is tight: for feasible positive target rates, the robust SRC solution is equivalent to the relaxed solution and is unique and rank one.The robust result has the same rank-one and uniqueness conclusions as the non-robust counterpart.
  • B. The Robust Secrecy-Rate Constrained Problem: The robust SRC SDP exactly solves the original robust SRC problem, with equivalent optimal solutions that are unique and rank one.This establishes exactness rather than merely providing a relaxation-based approximation.

C. The Robust Secrecy-Rate Maximization Problem

The robust secrecy-rate maximization problem is transformed into an SDP through the Charnes-Cooper transformation and the S-procedure. The relaxation is exact, and its unique rank-one solution implies optimal transmit beamforming.

  • C. The Robust Secrecy-Rate Maximization Problem: The robust SRM problem is first relaxed, with tightness and convex reformulation treated as separate goals.The relaxation is designed to preserve the original optimization solution while enabling tractable processing.
  • C. The Robust Secrecy-Rate Maximization Problem: For 0 < γ⋆(P) < 1, the relaxed robust SRM problem exactly solves the original problem and has a unique rank-one optimum.The equivalence is stated for the feasible robust SRM formulation under the specified rate condition.
  • C. The Robust Secrecy-Rate Maximization Problem: The proof establishes equivalence between the original and relaxed formulations by following the procedure used for the corresponding non-robust problem.Together with the robust SRC rank-one result, this yields a unique rank-one solution for the relaxed SRM formulation.
  • C. The Robust Secrecy-Rate Maximization Problem: The unique rank-one solution preserves the optimal objective value and maps between the relaxed and original robust SRM problems.The relation γ⋆(P) = γ⋆ is used to establish solution equivalence.
  • C. The Robust Secrecy-Rate Maximization Problem: The relaxed robust SRM problem is transformed into an SDP using the Charnes-Cooper transformation, the S-procedure, and algebraic manipulations.The SDP uses transformed variables related to the covariance by W = Z/ξ.
  • C. The Robust Secrecy-Rate Maximization Problem: The SDP imposes positive-semidefinite and nonnegative-variable constraints, including LMIs for each eavesdropper.The formulation contains Z ⪰ 0, ξ ≥ 0, and nonnegative uncertainty multipliers.
  • C. The Robust Secrecy-Rate Maximization Problem: The nonconvex robust SRM problem can therefore be solved equivalently by solving the convex SDP.The same analysis shows that transmit beamforming is generally optimal for robust SRC and SRM designs, as in the perfect-CSI case.

V. SIMULATION RESULTS

Simulations compare SDP-based designs with MRT baselines under perfect and imperfect CSI. The proposed SDP methods generally achieve higher secrecy rates, while the robust SDP performs best under channel uncertainty.

  • Simulation setup: The simulations evaluate proposed SDP solutions against projected-MRT and plain-MRT designs under perfect and imperfect CSI.The imperfect-CSI experiments assess worst-case secrecy rates using presumed channel means for the non-robust methods.
  • Perfect CSI: Plain-MRT transmits along Bob’s channel and ignores the presence of Eves, making it a suboptimal baseline.It sets W = (P/∥h∥2)hhH when the corresponding secrecy rate is positive and W = 0 otherwise.
  • 1) Secrecy rates versus the number of Eves: The proposed SDP method outperforms the two MRT methods across the tested number of Eves and provides more than 1.5 bps/Hz at K = 10.Projected-MRT approaches SDP for K ≤2 but becomes zero for K > 3.
  • 2) Secrecy rates versus Eves’ received signal strength: Projected-MRT remains invariant to ρe^2 because of its nulling process, whereas the SDP methods provide better performance than both MRT methods as Eve strength varies.The reported MRT behaviors also include plain-MRT’s ability to ignore weak Eves.
  • 3) Secrecy rate versus transmit power: As transmit power increases, plain-MRT approaches SDP at small P, while projected-MRT approaches SDP at large P.This comparison is reported for the secrecy-rate curves versus transmit power.
  • B. The Imperfect CSI Case: Under imperfect CSI, the robust SDP method yields the best worst-case secrecy rate among the evaluated methods, especially as Eve uncertainty increases.The performance measure is the worst-case secrecy rate, computed via SDP because it has no closed form.
  • B. The Imperfect CSI Case: The imperfect-CSI experiments also measure the probability of a non-negative worst-case secrecy rate as Eve’s channel uncertainty ratio changes.This probability is intended to show the sensitivity of non-robust methods to imperfect CSI.

1) Secrecy rate performance versus Eve’s channel uncertainty ratio:

Under imperfect CSI, the robust SDP method achieves the best worst-case secrecy-rate performance and maintains non-negative secrecy rates as channel uncertainty grows. Non-robust designs can degrade sharply, including negative worst-case secrecy rates and declining performance at higher transmit power.

  • The robust SDP method yields the best worst-case secrecy rate among the evaluated methods, especially for large Eve’s channel uncertainty ratio αe.
  • For αe > 0.18, non-robust SDP, projected-MRT, and plain-MRT yield negative worst-case secrecy rates.These methods are designed for presumed perfect CSIs rather than actual uncertain CSIs.
  • The robust method always guarantees a non-negative secrecy rate, whereas non-robust methods violate this condition seriously for large αe.The comparison uses 1000 independent trials measuring the probability of a non-negative secrecy rate.
  • As transmit power P increases, non-robust SDP and projected-MRT can experience decreasing worst-case secrecy rates.Ignoring channel uncertainties can also improve eavesdroppers’ receptions when transmit power increases.
  • The proposed SDP designs address secrecy-rate maximization with multiple multi-antenna eavesdroppers for both perfect and imperfect CSI.The analysis shows transmit beamforming is generally optimal, and simulations report gains over some existing methods.
  • An artificial-noise extension is identified as a future direction because jointly optimizing transmit design and artificial noise is expected to be more difficult.

APPENDIX

The appendix establishes equivalence between relaxed and original optimization problems and proves that their optimal transmit covariance solution is uniquely rank one under the stated positive-rate condition.

  • The KKT conditions and Slater’s condition establish strong duality for Problem (11) whenever it is feasible.
  • Any optimal W for Problem (11) must have rank one when R > 0.A full-rank alternative would force W = 0, while the rank-(N_t−1) case restricts W to a one-dimensional nullspace.
  • Problem (11) has a unique optimal W because two distinct rank-one optima would yield a rank-two convex combination.
  • The proof uses a three-step argument: each problem’s optimum is shown feasible and optimal for the other, followed by application of Proposition 1.
  • Problems (36) and (37) have the same optimal solution set, and Proposition 1 then gives a unique rank-one optimum for the relaxed problem.

C. Proof of Proposition 3

The proof of Proposition 3 uses Lagrangian and KKT conditions to establish a rank bound on the optimal transmit covariance, ultimately showing that its rank is one.

  • The proof formulates the relaxed problem’s Lagrangian and relevant KKT conditions, including positive-semidefinite constraints on W, A_b, and A_e,k.
  • Premultiplying the stationarity condition by W and using complementary slackness reduces the proof to bounding the rank of V_b A_b V_b^H.
  • The proof establishes that λ_b must be positive because λ_b = 0 would imply W = 0, which is infeasible for R > 0.

0. By (44) and INt + PK

This section transforms the robust secrecy-rate problem into an SDP and explains how worst-case secrecy rates are evaluated for rank-one transmit covariances.

  • The change of variable W = Z/ξ, with ξ > 0, transforms Problem (32) into an equivalent formulation.
  • An epigraph reformulation followed by the S-procedure converts the robust constraints into LMIs and produces SDP (34).
  • For rank-one W, the worst-case secrecy-rate function separates into legitimate-channel and eavesdropper optimization problems.
  • The separate worst-case terms can each be recast as SDPs using the S-procedure, yielding the worst-case secrecy rate.
Loading 1012.3875v2…