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Robust Beamforming for Secure Communication in Systems with Wireless Information and Power Transfer

Derrick Wing Kwan Ng, Ernest S. Lo, Robert Schober

arXiv:1311.2507v2cs.IT

TL;DR

The paper addresses total-power minimization for secure multiuser MISO communication with simultaneous wireless information and power transfer under passive and potential eavesdroppers. It combines artificial noise and energy signals, convexifies a probabilistic constraint, and applies SDP relaxation, while also proposing a lower-complexity scheme. The proposed schemes achieve close-to-optimal performance and significant transmit-power savings through signal-generation optimization.

  • Problem

    Secure resource allocation is needed for multiuser MISO systems that simultaneously transfer information and RF energy in the presence of passive and potential eavesdroppers.

  • Method

    The paper jointly optimizes artificial noise and energy signals, replaces a non-convex probabilistic constraint with a convex deterministic one, and solves the reformulation using SDP relaxation.

  • Results

    The proposed schemes achieve close-to-optimal performance and significant transmit-power savings by optimizing artificial-noise and energy-signal generation.

  • Takeaways & Limitations

    Artificial noise and energy signals can jointly support communication secrecy and efficient wireless energy transfer under imperfect or unavailable eavesdropper CSI.

Abstract

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This paper considers a multiuser multiple-input single-output (MISO) downlink system with simultaneous wireless information and power transfer. In particular, we focus on secure communication in the presence of passive eavesdroppers and potential eavesdroppers (idle legitimate receivers). We study the design of a resource allocation algorithm minimizing the total transmit power for the case when the legitimate receivers are able to harvest energy from radio frequency signals. Our design advocates the dual use of both artificial noise and energy signals in providing secure communication and facilitating efficient wireless energy transfer. The algorithm design is formulated as a non-convex optimization problem. The problem formulation takes into account artificial noise and energy signal generation for protecting the transmitted information against both considered types of eavesdroppers when imperfect channel state information (CSI) of the potential eavesdroppers and no CSI of the passive eavesdroppers are available at the transmitter. In light of the intractability of the problem, we reformulate the considered problem by replacing a non-convex probabilistic constraint with a convex deterministic constraint. Then, a semi-definite programming (SDP) relaxation approach is adopted to obtain the optimal solution for the reformulated problem. Furthermore, we propose a suboptimal resource allocation scheme with low computational complexity for providing communication secrecy and facilitating efficient energy transfer. Simulation results demonstrate a close-to-optimal performance achieved by the proposed schemes and significant transmit power savings by optimization of the artificial noise and energy signal generation.

I. INTRODUCTION

The paper develops secure resource allocation for multiuser MISO systems combining wireless information and power transfer, while accounting for passive and potential eavesdroppers. It reformulates the non-convex design and proposes both SDP-based and low-complexity solutions.

  • Motivation: Wireless information and power transfer creates a security concern because increasing information-signal energy can increase potential information leakage.The system therefore must address energy harvesting and communication secrecy jointly.
  • Problem Setting: The paper considers PHY-layer security against passive eavesdroppers and potential eavesdroppers with imperfect or unavailable CSI.Potential eavesdroppers have imperfect CSI at the transmitter, while passive eavesdroppers have no available CSI.
  • Problem Formulation: The resource allocation problem minimizes total transmit power while jointly using artificial noise and energy signals for secrecy and energy transfer.Both signal types degrade eavesdropper interception capabilities and support energy harvesting at legitimate receivers.
  • Solution Approach: A non-convex probabilistic constraint is replaced with a convex deterministic constraint, and the reformulated problem is solved using an SDP-based algorithm.The system model includes K legitimate single-antenna receivers and J passive single-antenna eavesdroppers.
  • Solution Approach: A suboptimal resource allocation scheme is also proposed to provide excellent system performance at low computational complexity.The design targets secure communication and efficient energy transfer in the multiuser downlink.

C. Hybrid Information and Energy Harvesting Receiver

The receiver architecture uses hybrid power splitting so legitimate receivers can decode information and harvest RF energy. The transmitter additionally uses artificial noise and pseudo-random energy signals, while channel uncertainty and passive-eavesdropper limitations shape the design.

  • Hybrid Information and Energy Harvesting Receiver: Hybrid receivers split the received signal into harvesting and information-decoding streams using power-splitting ratios 1−ρ and ρ.The ratio satisfies 0 ≤ρ ≤1, with ρ = 1 and ρ = 0 yielding traditional information and energy-harvesting receivers, respectively.
  • Hybrid Information and Energy Harvesting Receiver: The paper does not design RF energy-harvesting hardware and instead isolates resource allocation from specific circuitry implementations.Different harvesting circuitries and efficiencies may produce significantly different system models.
  • Channel State Information: The transmitter knows legitimate-receiver channels through TDD handshaking and models idle-receiver CSI as an estimate plus bounded uncertainty.The uncertainty radius ε_k represents the size of the estimated-CSI uncertainty region.
  • Channel State Information: Passive eavesdroppers remain unmeasured because they are silent, so the transmitter assumes their single-antenna configuration and limited channel knowledge.The model assumes a particular J to handle at most J passive eavesdroppers.
  • Artificial Noise and Energy Signal Generation: Artificial noise and energy signals both degrade eavesdropper channels and act as RF energy sources for legitimate receivers.The energy signal is known to legitimate receivers but not passive eavesdroppers, using periodically changed secret seeds.
  • Artificial Noise and Energy Signal Generation: The design deliberately allows legitimate receivers to exploit artificial-noise and energy-signal power instead of relying only on the information-bearing signal.Raising information-signal power alone can increase susceptibility to eavesdropping.

III. RESOURCE ALLOCATION ALGORITHM DESIGN

The paper formulates capacity and secrecy-capacity metrics for legitimate, idle, and passive receivers, using robust bounds for unknown passive-eavesdropper channels. It distinguishes artificial noise from energy signals by their security effects while exploiting both for energy transfer.

  • System Capacity and Secrecy Capacity: The resource-allocation design uses channel-capacity and secrecy-capacity metrics for the desired receiver and eavesdroppers.Secrecy capacity measures the secret-information rate achievable against eavesdroppers with unlimited decoding computation.
  • System Capacity and Secrecy Capacity: The Gaussian energy signal is known to legitimate receivers, enabling interference cancellation and improved information-decoding capacity.Passive eavesdroppers cannot cancel the energy signal because it is known only to legitimate receivers.
  • System Capacity and Secrecy Capacity: Idle receivers can devote all received power to eavesdropping, so their capacity is evaluated under a worst-case power-splitting ratio.The corresponding received SINR is monotonically increasing with the power-splitting ratio.
  • System Capacity and Secrecy Capacity: Unknown passive-eavesdropper locations are handled by assuming each passive eavesdropper lies close to the transmitter at the path-loss reference distance.The reference distance can serve as a security guard zone outside which passive eavesdroppers are assumed to exist.
  • System Capacity and Secrecy Capacity: Passive-eavesdropper channel models combine small-scale fading and shadowing, with normalized channel gains and noise power used in the robust upper bounds.The SINR upper-bound construction uses the monotonicity of SINR with respect to the passive-eavesdropper channel matrix.
  • System Capacity and Secrecy Capacity: Artificial noise degrades both idle and passive-eavesdropper channels, whereas the energy signal degrades only passive-eavesdropper channels.Both signals are equally effective for facilitating energy transfer to legitimate receivers.

B. Optimization Problem Formulation

The optimization minimizes radiated transmit power while enforcing legitimate-receiver performance, eavesdropper SINR limits, energy-transfer requirements, power constraints, and valid covariance structure. Artificial noise and energy-signal allocation are jointly optimized, with the energy signal optionally removed when legitimate receivers cannot cancel it.

  • Optimization Problem Formulation: The desired receiver must meet minimum SINR Γreq, while each idle receiver’s maximum SINR is constrained below Γtolk.These requirements support a secrecy-capacity lower bound when passive eavesdroppers are absent.
  • Optimization Problem Formulation: The formulation constrains passive-eavesdropper SINR probabilistically using tolerance Γtol and a specified outage requirement.The number J represents the maximum tolerable number of passive eavesdroppers handled by the design.
  • Optimization Problem Formulation: Minimum received powers Pmin and Pmink enforce energy transfer to the desired and idle receivers, respectively.Guarantees for idle receivers assume they devote all received power to energy harvesting rather than eavesdropping.
  • Optimization Problem Formulation: The design includes total and per-antenna transmit-power limits, a power-splitting variable ρ constrained to 0 ≤ ρ ≤ 1, and positive-semidefinite covariance matrices.Matrices V and WE represent valid covariance matrices under the stated constraints.
  • Optimization Problem Formulation: The problem generalizes designs using only artificial noise or only energy-signal allocation.If the energy signal cannot be canceled at legitimate receivers, setting wE = 0 is optimal without loss under the stated formulation.

IV. SOLUTION OF THE OPTIMIZATION PROBLEM

The solution converts the difficult robust optimization into a tractable convex SDP by replacing semi-infinite constraints with LMIs, replacing the chance constraint with a conservative convex restriction, and relaxing the rank-one constraint. A sufficient rank-one construction and a lower-complexity suboptimal scheme then address the relaxation’s potential looseness.

  • SOLUTION OF THE OPTIMIZATION PROBLEM: The original formulation is a non-convex QCQP with non-convex beamforming and power-splitting constraints, semi-infinite CSI-uncertainty constraints, and a coupled chance constraint.These features make standard optimization approaches insufficient even when the chance constraint is removed.
  • SOLUTION OF THE OPTIMIZATION PROBLEM: Relaxing Rank(W) = 1 removes the remaining non-convexity and produces a convex SDP solvable by numerical optimization software.The rank-one constraint originally ensures W = wwH after optimization.
  • SOLUTION OF THE OPTIMIZATION PROBLEM: The S-Procedure converts the semi-infinite potential-eavesdropper and energy-transfer constraints into finite LMI constraints with auxiliary variables.The resulting finite constraint set facilitates resource-allocation algorithm design.
  • SOLUTION OF THE OPTIMIZATION PROBLEM: Under an i.i.d. Rayleigh model for normalized passive-eavesdropper channels, the chance constraint is replaced by a convex deterministic constraint.A feasible point for the replacement is feasible for the original chance constraint, making the restriction conservative.
  • SOLUTION OF THE OPTIMIZATION PROBLEM: The Rayleigh-channel assumption models passive eavesdroppers as noncollaborating and sufficiently separated, with enough scatterers for Rayleigh fading.The stated separation is at least half a wavelength.
  • SOLUTION OF THE OPTIMIZATION PROBLEM: The deterministic replacement has a smaller feasible set than the original chance constraint, yielding a lower-bound system performance whose loss is quantified by simulation.The replacement is sufficient for feasibility but not necessary.
  • SOLUTION OF THE OPTIMIZATION PROBLEM: Because the rank relaxation may produce Rank(W) > 1, the paper derives a sufficient rank-one condition and constructs a rank-one optimal solution for the relaxed problem.The sufficient condition also serves as a building block for a lower-complexity suboptimal resource-allocation algorithm.

B. Optimality Conditions for SDP Relaxation

The SDP relaxation is shown to admit rank-one beamforming solutions under a sufficient condition, while a theorem constructs an equivalent rank-one optimum more generally. A lower-complexity suboptimal scheme avoids the extra optimization required for construction.

  • Low-complexity design: The proposed low-complexity scheme uses a sufficient condition for obtaining a rank-one solution of the relaxed problem as its design building block.This condition is introduced to reduce computational complexity relative to direct rank-one-solution construction.
  • Tightness of SDP relaxation: Theorem 1 constructs a feasible rank-one solution with the same objective value when the relaxed optimum has Rank(W*) > 1.The construction therefore preserves optimality for the reformulated problem while enforcing Rank(W*) = 1.
  • Low-complexity design: Constructing the rank-one optimum may require solving the dual problem and an additional optimization problem using the Lagrange multiplier matrix Y*.The resulting processing delay and computational complexity may be unsuitable for transmitters with limited signal-processing capability.
  • Sufficient condition: When constraint C5 is inactive or independent of W, Proposition 1 provides a sufficient condition for the relaxed solution to have Rank(W*) = 1.The condition is associated with the optimal Lagrange multiplier matrices for the relevant constraints.
  • Sufficient condition: Simulations found that the SDP relaxation can yield a rank-one W* even when Proposition 1’s sufficient condition does not hold.Thus, the sufficient condition is not necessary for rank-one solutions in the reported simulations.
  • Suboptimal formulation: Replacing C5 with a constraint that neglects information-signal energy harvesting preserves convexity but reduces the feasible set and provides a performance lower bound.The resulting problem can be solved efficiently via SDP relaxation and numerical solvers, and its beamforming matrix is always rank one.

V. RESULTS

The evaluation uses indoor TGn-channel simulations with six transmit antennas, four receivers, five passive eavesdroppers, and specified energy-transfer and secrecy parameters. Performance is averaged over path-loss and fading realizations.

  • Simulation setup: The setup uses six transmit antennas, four legitimate receivers, and five outdoor passive eavesdroppers modeled with Rayleigh multipath fading.The passive-eavesdropper count is J = 5.
  • Evaluation procedure: Average system performance is computed by averaging over different realizations of path loss and multipath fading.This averaging applies to the reported simulation results.

A. Average Total Transmit Power versus Minimum Required SINR

Increasing the required SINR raises optimized transmit power, while greater channel-estimation uncertainty requires more artificial noise and energy-signal power. The proposed robust reformulation incurs less than 0.1 dB loss across a wide SINR range and preserves secrecy robustness.

  • Transmit-power trends: Average total transmit power for the proposed SDP scheme increases monotonically with the desired receiver’s minimum required SINR Γreq.Higher Γreq requires more transmit power to satisfy the SINR constraint.
  • Transmit-power trends: Increasing normalized maximum channel-estimation errors σ2estk increases average total transmit power.The transmitter allocates more power to artificial noise and energy signals to protect against potential eavesdroppers and satisfy energy-transfer constraints.
  • Constraint replacement: Less than 0.1 dB performance loss occurs across a wide range of minimum required SINRs after replacing the probabilistic constraint with its convex deterministic counterpart.The reformulated problem maintains robustness against passive eavesdropping.
  • Secrecy-capacity trade-off: Baseline schemes achieve higher average secrecy capacity in the high-transmit-power regime than the other schemes.Their secrecy-capacity advantage comes with exceedingly high transmit power because they transmit more artificial noise and stronger energy signals.
  • Secrecy-capacity trade-off: Normalized maximum channel-estimation errors σ2estk have little impact on average secrecy capacity.This observation is reported alongside the secrecy-capacity comparisons across resource-allocation schemes.
  • Secrecy guarantee: All evaluated schemes guarantee secrecy quality of service through the proposed robust optimization.The secrecy guarantee is reported for the simulated resource-allocation comparisons.

B. Average Total Transmit Power versus Number of Transmit Antennas

The proposed schemes reduce total transmit power as more transmit antennas provide additional resource-allocation degrees of freedom, while jointly optimizing information, artificial-noise, and energy-signal powers. More stringent secrecy and energy-transfer requirements increase power demands, but the proposed schemes achieve savings over baselines.

  • B. Average Total Transmit Power versus Number of Transmit Antennas: Total transmit power decreases as the number of transmit antennas increases because additional degrees of freedom improve resource allocation.The setting uses K = 4 legitimate receivers, Γreq = 15 dB, and normalized maximum channel estimation error σ²_estk = 5%, ∀k.
  • B. Average Total Transmit Power versus Number of Transmit Antennas: The proposed schemes provide substantial power savings versus both baseline schemes by optimizing W and WE.With more antennas, information beamforming and eavesdropper jamming become more power-efficient.
  • B. Average Total Transmit Power versus Number of Transmit Antennas: Information-signal and artificial-noise powers decrease rapidly with more antennas, while energy-signal power decreases more slowly.The slower energy-signal reduction reflects its lower influence on the legitimate receivers’ SINRs.
  • C. Average Total Transmit Power versus Number of Legitimate Receivers: Total transmit power increases with the number of legitimate receivers because more receivers require power transfer and create additional potential eavesdroppers.The latter increases the artificial noise needed to guarantee communication secrecy.
  • D. Average Total Harvested Power: Average harvested power increases with Γreq and channel-estimation error but decreases as the number of transmit antennas increases.Higher Γreq or error requires more transmit power, whereas more antennas enable more accurately steered beamforming toward the desired receiver.

A. Proof of Lemma 2

Lemma 2 addresses the probabilistic secrecy constraint by characterizing the relevant random quantity and replacing its difficult optimization-dependent distribution with a tractable convex feasible-set approximation.

  • A. Proof of Lemma 2: The passive-eavesdropper channels are modeled as independent and identically distributed, allowing the eavesdropper index to be omitted without loss of generality.The derivation then focuses on calculating the associated probability.
  • A. Proof of Lemma 2: The random variable’s distribution depends on W, WE, and V, creating mutual dependence between distribution evaluation and optimization.This dependence prevents an efficient resource-allocation design, and the matrix W − ΓtolWE − ΓtolV is generally indefinite.
  • A. Proof of Lemma 2: A trace inequality is introduced to upper-bound the difficult random expression and obtain a smaller convex feasible solution set.This provides a tractable compromise rather than directly optimizing over the original non-convex probabilistic constraint.
  • A. Proof of Lemma 2: The reformulated constraint uses the inverse cumulative distribution function of an inverse central chi-square variable with 2NT degrees of freedom.The inverse function can be evaluated directly or stored in a lookup table.

B. Proof of Theorem 1

Theorem 1 proves that the relaxed optimization problem admits an optimal rank-one information covariance, enabling recovery of a beamforming solution without changing the optimum.

  • B. Proof of Theorem 1: A rank-one solution is constructed by modifying the artificial-noise structure while preserving feasibility and the objective value.The construction uses scaling constants and null-space vectors to form the rank-one information covariance.
  • B. Proof of Theorem 1: The relaxed problem is jointly convex and satisfies Slater’s constraint qualification, so its KKT conditions are necessary and sufficient.The proof analyzes the Lagrangian and the associated dual multipliers.
  • B. Proof of Theorem 1: Complementary slackness places the columns of W* in the null space of Y*, which determines the structure of the optimal information covariance.The proof studies the rank and null space of Y* to establish this structure.
  • B. Proof of Theorem 1: The constructed solution achieves the same optimal objective value as the original relaxed optimum while satisfying all constraints.The rank-one construction is therefore also optimal, although the constructed solution need not be unique.

C. Proof of Proposition 1

The proposition derives a rank property of the optimal information covariance from the KKT system, complementary slackness, and positive definiteness.

  • C. Proof of Proposition 1: Post-multiplying the KKT relation by W* and using complementary slackness yields a rank constraint on the optimal solution.The argument relies on the corresponding dual matrices and rank inequalities.
  • C. Proof of Proposition 1: Rank((μ* + β*)H) equals one because μ* + β* > 0, while W* cannot be zero when Γreq > 0.The nonzero W* is required to satisfy the desired receiver’s minimum SINR constraint.
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