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Robust AN-Aided Beamforming and Power Splitting Design for Secure MISO Cognitive Radio With SWIPT

Fuhui Zhou, Zan Li, Julian Cheng, Qunwei Li, Jiangbo Si

arXiv:1602.06913v1cs.IT

TL;DR

The paper addresses robust secure beamforming and power splitting for SWIPT in MISO cognitive radio with imperfect CSI. It formulates bounded- and probabilistic-error designs using search-based convexification, achieving bounded-model optimality but only a suboptimal probabilistic-model beamformer. Simulations indicate a performance–complexity tradeoff and a secrecy-rate–harvested-energy tradeoff.

  • Problem

    Secure SWIPT cognitive radio requires beamforming that handles imperfect CSI while protecting communication, satisfying energy harvesting, and limiting interference to primary users.

  • Method

    The paper jointly designs AN-aided beamforming and power splitting, using S-Procedure-based search for bounded errors and Bernstein-type inequalities for probabilistic errors.

  • Results

    Optimal robust secure beamforming is guaranteed under bounded CSI errors, whereas probabilistic CSI errors yield a suboptimal beamforming solution.

  • Takeaways & Limitations

    Probabilistic-error designs offer performance gains over bounded-error designs at higher implementation complexity, while secrecy rate trades off against max-min harvested energy.

  • Takeaways & Limitations

    Under the probabilistic CSI error model, the optimal robust secure beamforming vector cannot always be obtained.

Abstract

from arXiv · show

A multiple-input single-output cognitive radio downlink network is studied with simultaneous wireless information and power transfer. In this network, a secondary user coexists with multiple primary users and multiple energy harvesting receivers. In order to guarantee secure communication and energy harvesting, the problem of robust secure artificial noise-aided beamforming and power splitting design is investigated under imperfect channel state information (CSI). Specifically, the transmit power minimization problem and the max-min fairness energy harvesting problem are formulated for both the bounded CSI error model and the probabilistic CSI error model. These problems are non-convex and challenging to solve. A one-dimensional search algorithm is proposed to solve these problems based on ${\cal S}\text{-Procedure} $ under the bounded CSI error model and based on Bernstein-type inequalities under the probabilistic CSI error model. It is shown that the optimal robust secure beamforming can be achieved under the bounded CSI error model, whereas a suboptimal beamforming solution can be obtained under the probabilistic CSI error model. A tradeoff is elucidated between the secrecy rate of the secondary user receiver and the energy harvested by the energy harvesting receivers under a max-min fairness criterion.

I. INTRODUCTION

The paper motivates secure SWIPT in cognitive radio, where spectrum sharing and energy transfer create simultaneous efficiency and security requirements. It reviews robust beamforming under bounded and probabilistic CSI uncertainty, highlighting limitations of existing designs.

  • Cognitive radio improves spectrum efficiency by allowing secondary and primary links to share spectrum under interference constraints.
  • SWIPT uses RF signals simultaneously for information transmission and wireless power transfer, supporting energy-constrained devices.
  • Malicious energy harvesting receivers may intercept secondary-user information, making security a central concern in CR with SWIPT.
  • The secondary-user secrecy rate is constrained by imperfect CSI and by transmit-power limits imposed to protect primary users from harmful interference.
  • Existing robust beamforming methods may not directly apply to CR with SWIPT because energy harvested by EHRs must also be considered.
  • Bounded CSI models simplify implementation but can underestimate performance, whereas probabilistic models approximate outage constraints under suitable scenarios.

A. Motivation and Contributions

The paper extends secure CR with SWIPT by jointly designing AN-aided beamforming and power splitting under imperfect CSI. It formulates bounded and probabilistic uncertainty problems and develops tractable search-based solutions with distinct optimality guarantees.

  • Motivation and Contributions: The system jointly optimizes AN-aided beamforming and the SU power-splitting ratio for secrecy, interference, energy-harvesting, and transmit-power constraints.
  • Motivation and Contributions: The bounded-error problems minimize CBS transmit power or maximize max-min-fairness energy harvested by EHRs under non-convex infinite constraints.
  • Motivation and Contributions: A one-dimensional search based on SDR and the S-Procedure converts the bounded-error design into convex inner problems.
  • Motivation and Contributions: Optimal robust beamforming and AN covariance can be found under bounded CSI errors, while probabilistic errors yield only a suboptimal beamforming solution.
  • Motivation and Contributions: For probabilistic CSI errors, Bernstein-type inequalities approximate outage constraints, yielding tractable formulations for power minimization and energy-harvesting maximization.
  • Motivation and Contributions: Probabilistic-error designs provide performance gains over bounded-error designs at higher implementation complexity and expose a secrecy-rate–energy-harvesting tradeoff.

B. Organization and Notations

The paper models a downlink MISO cognitive-radio SWIPT network in which one SU, multiple PUs, and multiple EHRs share spectrum. The SU uses power splitting while AN supports secrecy and energy transfer.

  • II. SYSTEM MODEL: One SU link, M PU links, and K energy-harvesting links share the same spectrum, with CBS interference to PUs constrained as tolerable.
  • II. SYSTEM MODEL: EHRs may eavesdrop on CBS transmissions, while PU receivers are assumed friendly and non-eavesdropping.
  • II. SYSTEM MODEL: The model assumes slow frequency-nonselective fading and includes PBS interference, receiver noise, secrecy rate, and harvested-energy quantities.
  • II. SYSTEM MODEL: The CBS transmits confidential information using a beamforming vector and artificial noise whose covariance matrix is designed to improve secrecy and energy transfer.
  • II. SYSTEM MODEL: The SU power-splitting ratio ρ assigns ρ of received power to information decoding and 1−ρ to energy transfer.
  • II. SYSTEM MODEL: The SU channel CSI is available, but EHR and PU channel vectors cannot be perfectly known because the involved parties do not cooperate.

III. WORST-CASE ROBUST SECURE BEAMFORMING DESIGN

The worst-case design minimizes CBS transmit power or maximizes max-min EHR harvesting under bounded CSI errors and secrecy, energy, interference, and power constraints. SDR, the S-Procedure, and one-dimensional search make the non-convex formulation tractable.

  • III. WORST-CASE ROBUST SECURE BEAMFORMING DESIGN: The bounded-error formulation optimizes transmit power and max-min EHR energy harvesting subject to secrecy, energy, interference, and transmit-power constraints.
  • III. WORST-CASE ROBUST SECURE BEAMFORMING DESIGN: The problems are non-convex with infinitely many inequality constraints, creating computational difficulty.
  • III. WORST-CASE ROBUST SECURE BEAMFORMING DESIGN: A one-dimensional search using semidefinite relaxation and the S-Procedure solves convex inner optimization problems.
  • III. WORST-CASE ROBUST SECURE BEAMFORMING DESIGN: The bounded CSI model represents channel uncertainty through deterministic error regions and uncertainty radii for EHR and PU channels.

B. The Worst-Case Transmit Power Minimization Problem

The bounded-CSI problem minimizes worst-case transmit power while enforcing secrecy, harvesting, interference, and transmit-power constraints. SDR and the S-Procedure enable a line-search solution, with rank-one beamforming guarantees under feasibility.

  • The problem minimizes worst-case transmit power subject to secrecy-rate, energy-harvesting, primary-user interference, and maximum-transmit-power constraints.
  • SDR and the S-Procedure transform the uncertain-region constraints into a relaxed optimization problem with finite-dimensional constraints.
  • For a fixed β, the relaxed problem is convex, so the optimal β is obtained through uniform one-dimensional search.
  • When feasible with Rmin > 0, the optimal transmit matrix W is unique and rank-one, regardless of the numbers of primary users and energy-harvesting receivers.
  • The optimal beamforming vector is recovered from the maximum-eigenvalue eigenvector of W, while the AN covariance matrix Σ also has rank-one structure.

C. The Worst-Case Max-Min Fairness EH Problem

The bounded-CSI max-min fairness problem designs secure beamforming to provide fairness among energy-harvesting receivers under secrecy, secondary-user harvesting, interference, and power constraints. A β search solves the relaxed formulation, and rank-one optimal solutions are established when feasible.

  • The problem maximizes fairness among energy-harvesting receivers while satisfying secrecy-rate, secondary-user harvesting, interference, and transmit-power constraints.
  • A slack variable τ converts possible infeasibility of the fairness problem into an equivalent formulation.
  • The relaxed problem remains non-convex because β is coupled with Σ, but it is convex for a fixed β and can be solved by a modified line-search algorithm.
  • When feasible with Rmin > 0, the optimal W is unique and rank-one, regardless of the numbers of primary users and energy-harvesting receivers.
  • Under the same feasibility and secrecy-rate conditions, the optimal AN covariance matrix Σ is rank-one.

IV. OUTAGE-CONSTRAINED ROBUST SECURE BEAMFORMING DESIGN

Bounded CSI errors provide worst-case guarantees but can be conservative because extreme errors may be rare. The paper therefore studies outage-constrained robust design under probabilistic CSI errors modeled with complex circular Gaussian distributions.

  • Bounded-error robust beamforming guarantees worst-case secondary-user performance, but its results can be conservative when extreme scenarios occur rarely.
  • Outage-constrained robust beamforming is presented as more suitable for delay-sensitive cognitive radio applications such as voice and video.
  • The probabilistic CSI model treats channel errors as stochastic variables following a distribution rather than lying within a determined region.
  • The model uses complex circular Gaussian errors with covariance matrices for corresponding channel-error vectors and independence across distinct receivers.

B. The Outage-Constrained Transmit Power Minimization Problem

The probabilistic transmit-power problem imposes outage constraints for secrecy, harvesting, and primary-user interference. Bernstein-type inequalities, SDR, and one-dimensional search produce a tractable approximation, but beamforming may require randomization.

  • The outage constraints require secrecy probability above 1−r, energy-harvesting outage below e,k, and interference outage below I,i.
  • Because P8 is non-convex and lacks closed-form constraint expressions, Bernstein-type inequalities provide safe convex approximations.
  • After these approximations, SDR relaxes the rank-one constraint, but fractional constraints and variable coupling keep the relaxed problem non-convex.
  • For fixed z, setting t = 1/ρ makes P10 convex and solvable with CVX, while a one-dimensional search over 0 < z ≤1 solves the outer problem.
  • If the relaxed solution W is rank-one, eigenvalue decomposition yields optimal robust secure beamforming; otherwise, Gaussian randomization produces a suboptimal beamforming vector.

C. The Outage-Constrained Max-Min Fairness EH Problem

The outage-constrained max-min fairness EH problem is formulated under probabilistic constraints and addressed through a modified Algorithm 1. Its approximate problem is convex for fixed z, but beamforming may require Gaussian randomization when the relaxed matrix is not rank-one.

  • Problem formulation: The outage max-min fairness EH problem is formulated under probabilistic constraints that make Problem P11 challenging.The problem concerns the outage energy harvested by the EHRs under a max-min fairness criterion.
  • Solution method: A modified Algorithm 1 solves an approximate problem for P11 using the same techniques as those used for P8.For a given z, the approximate problem is represented as P12, which maximizes ΓE over W, Σ, t, and slack variables.
  • Solution method: P12 is convex for fixed t = 1/ρ and can be solved using CVX.The formulation includes constraints and slack variables associated with outage secrecy, energy-harvesting, and interference constraints.
  • Beamforming recovery: The optimal beamforming vector is obtained when W is rank-one; otherwise, Gaussian randomization yields a suboptimal beamforming vector.The rank-one condition applies to the solution W of P12.
  • Complexity: The modified Algorithm 1 has complexity from uniform line search, iteration complexity, and per-iteration computation cost.The effect of T2 on complexity is little, while iteration and per-iteration computation cost dominate; bounded-CSI complexity is lower than probabilistic-CSI complexity.
  • Complexity: The complexity of Algorithm 1 under bounded CSI errors is lower than that of the modified Algorithm 1 under probabilistic CSI errors.The comparison is stated directly for the two algorithmic settings.

V. SIMULATION RESULTS

Simulations compare robust beamforming under bounded, probabilistic, and perfect CSI. Probabilistic CSI generally reduces computation and improves power and harvesting outcomes relative to bounded CSI, while secrecy and harvesting exhibit tradeoffs.

  • Figures 2–3: Perfect CSI requires the least transmit power and yields the largest max-min harvested energy because it avoids additional power for channel uncertainty.The empirical CDFs use 1,000 channel realizations.
  • Figures 2 and 6: Probabilistic CSI requires less transmit power and provides larger max-min harvested energy than bounded CSI because its constraints are looser.This comparison applies to both transmit-power minimization and energy-harvesting fairness outcomes.
  • Figure 3: Average computation time is nearly identical between the two optimization problems under each CSI model, but probabilistic CSI has higher complexity than bounded CSI.Computation time also increases greatly with the number of CBS transmit antennas.
  • Figure 4: Minimum CBS transmit power increases with the secrecy-rate requirement but decreases as the number of transmit antennas increases.More antennas increase the degrees of freedom available for transmit-power allocation.
  • Figure 5: Max-min harvested energy decreases as the secrecy-rate requirement rises, revealing a tradeoff between SU secrecy rate and EHR energy.Harvested energy increases with the number of CBS transmit antennas because of increased diversity gain.
  • Figure 6: Under bounded CSI with Nt = 10, the transmit-power minimization and max-min EH problems are infeasible for more than 6 EHRs.The stated cause is inability to satisfy the SU minimum secrecy rate under the maximum transmit-power constraint.

VI. CONCLUSION

The paper jointly designs robust secure AN-aided beamforming and power splitting for SWIPT cognitive radio under bounded and probabilistic CSI errors. It optimizes transmit power or max-min harvested energy, derives a one-dimensional search method, and characterizes solution quality and the secrecy–energy tradeoff.

  • Robust AN-aided beamforming and power splitting are jointly designed to guarantee secure communication and energy harvesting under imperfect CSI.
  • The transmit-power and max-min energy-harvesting objectives are optimized under both bounded and probabilistic CSI error models.
  • A one-dimensional search algorithm solves the challenging non-convex problems, with convex problems required only in its inner component.
  • Optimal robust secure beamforming vectors and AN covariance matrices can always be found under the bounded CSI error model.
  • Under the probabilistic CSI error model, optimal robust beamforming cannot be guaranteed, so the method obtains a suboptimal beamforming scheme.
  • The probabilistic CSI error model provides performance gains over the bounded model at higher implementation complexity, while secrecy rate and harvested energy exhibit a max-min fairness tradeoff.

APPENDIX A

The appendix proves the rank-one structure of the beamforming solution using KKT optimality conditions for problem P2. Under feasibility and a positive minimum secrecy rate, the beamforming matrix has rank one.

  • Theorem 1 is proved from the KKT optimality conditions of P2 by collecting its primal and dual variables and forming the Lagrangian.
  • The Lagrangian includes dual variables associated with constraints (10a), (12b), and C5, while Λ collects unrelated variable terms.
  • The proof derives partial KKT conditions for the relevant matrix expressions and uses positivity of α and ν1 when t > 1.
  • Since Y + (α + ν1)H is positive definite, right-multiplication by W yields the rank relationship used in the proof.
  • When P2 is feasible and Rmin > 0, the beamforming matrix W has rank one.
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