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Secrecy Wireless Information and Power Transfer in Fading Wiretap Channel

Hong Xing, Liang Liu, Rui Zhang

arXiv:1408.1987v2cs.IT

TL;DR

SWIPT creates a security conflict because energy receivers may eavesdrop on information intended for information receivers while needing strong received power for energy harvesting. The paper uses cancelable artificial noise and jointly optimizes power allocation and power splitting over fading channels, achieving substantially improved secrecy rate–energy trade-offs in simulations.

  • Problem

    SWIPT must provide strong received power for energy harvesting while preventing energy receivers from eavesdropping on information sent to information receivers.

  • Method

    The paper splits transmit power between confidential information and artificial noise, then jointly optimizes power allocations and power splitting ratios for outage minimization or average-rate maximization under transmitter and energy-harvesting constraints.

  • Results

    The proposed AN-aided optimal solution substantially improves rate–energy trade-offs over NoAN and NoCancel; at 6µW harvested power, ESC increases by about 700%.

  • Takeaways & Limitations

    Cancelable AN can simultaneously interfere with the ER and transfer energy to it while preserving secrecy transmission to the IR.

Abstract

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Simultaneous wireless information and power transfer (SWIPT) has recently drawn significant interests for its dual use of radio signals to provide wireless data and energy access at the same time. However, a challenging secrecy communication issue arises as the messages sent to the information receivers (IRs) may be eavesdropped by the energy receivers (ERs), which are presumed to harvest energy only from the received signals. To tackle this problem, we propose in this paper an artificial noise (AN) aided transmission scheme to facilitate the secrecy information transmission to IRs and yet meet the energy harvesting requirement for ERs, under the assumption that the AN can be cancelled at IRs but not at ERs. Specifically, the proposed scheme splits the transmit power into two parts, to send the confidential message to the IR and an AN to interfere with the ER, respectively. Under a simplified three-node wiretap channel setup, the transmit power allocations and power splitting ratios over fading channels are jointly optimized to minimize the outage probability for delay-limited secrecy information transmission, or to maximize the average rate for no-delay-limited secrecy information transmission, subject to a combination of average and peak power constraints at the transmitter as well as an average energy harvesting constraint at the ER. Both the secrecy outage probability minimization and average rate maximization problems are shown to be non-convex, for each of which we propose the optimal solution based on the dual decomposition as well as suboptimal solution based on the alternating optimization. Furthermore, two benchmark schemes are introduced for comparison. Finally, the performances of proposed schemes are evaluated by simulations in terms of various trade-offs for wireless (secrecy) information versus energy transmissions.

I. INTRODUCTION

The paper addresses the conflict between SWIPT energy harvesting and secrecy when ERs may eavesdrop on IR messages. It proposes cancelable-AN transmission and fading-state power control to protect IR secrecy while meeting ER energy demands.

  • Motivation: ERs are often placed closer to the AP for energy harvesting, but this proximity can let them eavesdrop messages intended for IRs.The security issue arises because ERs are presumed to harvest energy rather than decode information.
  • Proposed scheme: The transmitter splits power between a confidential information signal and AN that interferes with the ER while preserving high received power for energy harvesting.The AN is assumed cancelable at the IR but not at the ER.
  • System model: The study uses a three-node SISO fading wiretap channel with one Tx, one IR, and one ER, with channel gains known at the transmitter in each fading state.The gains remain constant within blocks and vary across fading states.
  • Constraints: Transmit power is constrained by both average and peak limits, while the ER must satisfy an average harvested-power requirement.The average constraint limits Eν[p(ν)], and the peak constraint limits p(ν) in every fading state.
  • AN cancellation: A practical key-distribution procedure lets the Tx and IR generate the same AN sequence, which the ER cannot cancel because it lacks the selected seed.The AN sequence is generated from randomly selected pre-stored Gaussian-generator seeds.
  • Performance quantities: The model evaluates secrecy rate, ER harvested power, and receiver noise while treating the ER as an eavesdropper rather than only an energy harvester.The AN is unknown to the ER, and energy-harvesting background noise is ignored because it is typically small relative to received signal power.

III. PROBLEM FORMULATION

The paper formulates two fading-state optimization problems under transmitter power limits and an ER energy constraint: minimize delay-limited secrecy outage or maximize no-delay-limited secrecy capacity.

  • Transmission modes: The formulation covers both delay-limited and no-delay-limited secrecy information transmission to the IR.The corresponding metrics are secrecy outage probability and ESC, respectively.
  • A. Delay-Limited Secrecy Information Transmission: For delay-limited transmission, secrecy outage occurs when the achievable secrecy rate R(α(ν), p(ν)) falls below the target rate r0.The outage probability is δ = Pr(R(α(ν), p(ν)) < r0).
  • Power allocation context: With channel-state information at the transmitter, secrecy outage is generally associated with secrecy channel inversion, whereas ESC is generally associated with secrecy waterfilling.These are described as conventional power-allocation strategies for the respective objectives.
  • A. Delay-Limited Secrecy Information Transmission: The outage objective can be written as the expectation of an indicator for outage across fading states.Specifically, δ = Eν[X(ν)].
  • A. Delay-Limited Secrecy Information Transmission: The delay-limited problem jointly optimizes fading-state transmit powers and AN power-splitting ratios.Its constraints include Tx APC and PPC limits and an average harvested-power target Q̄ at the ER.
  • A. Delay-Limited Secrecy Information Transmission: The delay-limited formulation minimizes secrecy outage probability over {p(ν)} and {α(ν)} subject to APC, PPC, and average harvested-power constraints.The target secrecy rate is r0 and the ER harvested-power requirement is Q̄.

B. No-Delay-Limited Secrecy Information Transmission

For the no-delay-limited case, the paper maximizes ESC under joint power and energy constraints, then develops dual and iterative solutions that allocate power and AN splitting across fading states.

  • B. No-Delay-Limited Secrecy Information Transmission: The no-delay-limited objective is ESC maximization for the IR under APC, PPC, and average ER harvested-power constraints.The same constraint set is used for the delay-limited and no-delay-limited formulations.
  • Optimization structure: Both formulated problems are non-convex because their objective functions are generally non-convex or non-concave.The paper therefore develops optimal and suboptimal solution methods.
  • Optimal solution: The optimal approach applies Lagrange duality, exploiting the time-sharing condition under continuous fading distributions.This yields approximate strong duality for the delay-limited formulation and motivates the dual solution framework.
  • Optimal solution: The Lagrangian combines outage, average transmit power, and average harvested-power terms through dual variables λ and µ.The resulting objective contains X + λp − ζµgp for each fading state.
  • Optimal solution: The dual minimization separates into parallel subproblems, one for each fading state, under peak-power and power-splitting bounds.For a given fading state, the subproblem minimizes L1(p, α) = X + λp − ζµgp.
  • Optimal solution: For a chosen AN ratio α, p1(α) is the minimum power required to maintain the target secrecy rate r0.This quantity supports the per-state power and splitting decisions in the optimal solution.
  • Optimal solution: The optimal per-state policy may transmit at peak power, select a secrecy-rate-supporting split, or shut down transmission depending on channel conditions and dual thresholds.The cases are determined by comparisons involving g, λ, µ, p1(α̃), and Ppeak.

B. Suboptimal Solution to (P1)

The suboptimal solution to (P1) alternates power allocation and power-splitting optimization across fading states until convergence, reducing complexity relative to exhaustive searches. Its outage probability is non-increasing at every iteration, guaranteeing convergence to at least a locally optimal solution.

  • Alternating optimization: The algorithm alternates between optimizing power allocations with fixed splitting ratios and optimizing splitting ratios with fixed power allocations.The procedure repeats until both variable sets converge.
  • Power allocation: Power optimization remains non-convex, but the time-sharing condition permits approximate solution through Lagrange duality with zero duality gap.The problem decouples into parallel subproblems for individual fading states.
  • Power allocation: Each fixed-splitting power subproblem can be solved efficiently state by state, while the dual variables are updated iteratively using the ellipsoid method.The same solution is used for the corresponding subproblem after fixing α.
  • Power splitting: For fixed power, the splitting-ratio subproblem is solved separately for each fading state using the feasible set Φ defined by the non-outage secrecy condition.If Φ is empty, outage cannot be avoided; otherwise, any α in Φ is optimal before selecting among feasible values.
  • Convergence: The alternating algorithm makes outage probability non-increasing after each iteration and therefore converges to at least a locally optimal solution of (P1).This monotonicity is the stated convergence guarantee for the suboptimal procedure.

V. PROPOSED SOLUTIONS TO (P2) FOR NO-DELAY-LIMITED CASE

This section presents both optimal and suboptimal solutions for (P2), the no-delay-limited problem.

  • Proposed solutions: Both optimal and suboptimal solutions are proposed to solve (P2).The section concerns the no-delay-limited case.

A. Optimal Solution to (P2)

The optimal solution to (P2) handles its non-convex per-fading-state optimization through a one-dimensional search over the power-splitting ratio and finite candidate evaluation for transmit power.

  • Optimal per-state solution: Because R(α, p) is not concave in p and α, the per-state problem is non-convex and is solved in two stages.The splitting ratio is fixed first, followed by optimization over transmit power and then a search over α.
  • Power optimization: For fixed α, the monotonicity of the Lagrangian is characterized through a cubic equation with at most three roots.The candidate roots are restricted to real values within the feasible power interval.
  • Power optimization: The feasible candidate set Ψ contains real roots in [0, Ppeak] together with the boundary points 0 and Ppeak.Its cardinality ranges from 2 to 5 depending on the number of admissible roots.
  • Dual optimization: The optimal transmit power is found by searching Ψ, after which the dual variables λ and µ are iteratively updated using the ellipsoid method.The finite search solves the per-state subproblem for a given dual-variable pair.

B. Suboptimal Solution to (P2)

The suboptimal solution to (P2) uses alternating optimization between power allocation and power-splitting ratios, with monotonic ESC improvement ensuring convergence to at least a local optimum.

  • Alternating optimization: The algorithm fixes the splitting ratios, optimizes transmit powers, then fixes those powers and optimizes the splitting ratios.These two stages are repeated until both variable sets converge.
  • Power allocation: Power optimization decouples into parallel fading-state subproblems and updates dual variables λ and µ through the ellipsoid method.The fixed-splitting subproblem uses the previously derived per-state solution.
  • Power splitting: Splitting-ratio optimization is separable across fading states and uses the solution previously derived for the corresponding per-state problem.Thus, all fading-state ratios can be obtained according to the stated prior solution.
  • Convergence: ESC is non-decreasing after each iteration, ensuring convergence to at least a local optimal solution of (P2).This is the stated guarantee of the suboptimal alternating procedure.

VI. BENCHMARK SCHEMES

The benchmark schemes remove AN or prevent its cancellation at the IR, and their optimization problems simplify accordingly. When AN cannot be canceled by the IR, the optimal allocation uses no AN.

  • The benchmarks consider no AN and AN unknown to both the IR and ER, so the latter cannot be canceled by the IR.
  • With no AN, setting α(ν) = 0 reduces the SNR and secrecy-rate expressions and simplifies both optimization problems.
  • The NoCancel outage and average-rate problems remain non-convex because their objective expressions are non-convex and non-concave in the power and splitting variables.
  • The optimal solution to both NoCancel problems satisfies α∗(ν) = 0 for every fading state.
  • Consequently, NoCancel is equivalent to NoAN, whose problems can be solved efficiently.

VII. NUMERICAL RESULTS

The numerical study evaluates proposed and benchmark algorithms under specified power, noise, path-loss, and Rayleigh-fading assumptions. The Fixed-α heuristic is included as a lower-complexity comparison.

  • The evaluation compares proposed optimal and suboptimal algorithms with two benchmark schemes and a Fixed-α heuristic.
  • Fixed-α uses one uniform splitting ratio across fading states and therefore requires one-shot power optimization instead of iterative updates.
  • The simulations use Pavg = 100mW, Ppeak = 1W, ζ = 50%, and σ1^2 = σ2^2 = −50dBm.
  • The channel power gains h(ν) and g(ν) are independent exponentially distributed variables modeling short-term Rayleigh fading, with means specified by the path-loss model.

A. Secrecy Outage-Energy Trade-off

The O-E analysis characterizes achievable secrecy non-outage probability versus average harvested power under different schemes and channel geometries. Cancelable AN substantially improves the secrecy–energy trade-off, including when the ER is closer to the transmitter.

  • The O-E region contains achievable secrecy non-outage probability ε and average harvested power E for fixed average and peak transmit-power constraints.
  • With IR and ER both 2m away and r0 = 6.5bps/Hz, the proposed optimal AN scheme achieves substantially better O-E trade-offs than NoAN and NoCancel.
  • At 7.0µW average harvested power, secrecy outage probability is below 5% for the proposed scheme versus above 98% for the comparison schemes.
  • Alternating optimization closely approaches the optimal O-E region, while Fixed-α with ᾱ = 0.5 has negligible loss in the equal-distance setup.
  • When the ER is 1m and the IR 2m from the Tx, harvested power is about 10 times higher, while NoAN and NoCancel outage probability is almost one.
  • In the unequal-distance setup, the proposed scheme's secrecy outage probability is almost unchanged despite the IR's worse channel condition.

B. Secrecy Rate-Energy Trade-off

The paper characterizes secrecy Rate-Energy regions under different power-allocation schemes and finds that cancelable artificial noise substantially improves the trade-off, especially when the ER has a better channel than the IR.

  • R-E region formulation: The secrecy Rate-Energy region comprises achievable secrecy-rate and harvested-power pairs for fixed average and peak transmit-power constraints.The boundary is obtained by solving the ESC-maximization problem for different average harvested-power requirements.
  • Equal-distance setup: At equal 2m distances, the proposed AN-aided optimal solution increases ESC by about 700% over NoAN or NoCancel at 6µW harvested power.The gain is attributed to AN cancellation at the IR.
  • Equal-distance setup: The alternating-optimization solution produces an R-E region very close to the optimal solution when AN is cancelable at the IR.
  • Equal-distance setup: Among fixed power-splitting ratios, Fixed-α with ᾱ = 0.5 achieves the best R-E region compared with ᾱ = 0.1 and ᾱ = 0.3.
  • Unequal-distance setup: With unequal distances, the performance gaps between the proposed optimal or suboptimal solutions and NoAN or NoCancel become more substantial.Figure 5 considers an IR 2m and an ER 1m from the transmitter.

APPENDIX A PROOF OF PROPOSITION 4.1

The proof of Proposition 4.1 separates cases according to whether the minimum required power exceeds the peak-power limit, then derives the minimizing power and splitting-ratio choices from the Lagrangian behavior.

  • Case I: When p1(α̃) > Ppeak, the minimum power needed to achieve r0 exceeds Ppeak, making secrecy outage inevitable.
  • Case II: When p1(α̃) ≤ Ppeak, jointly optimizing p and α can avoid outage.
  • Case II: The optimal power allocation depends on whether λ − ζµg is negative or nonnegative, requiring two subcases.
  • Subcase II-1: For λ − ζµg < 0, any feasible ᾱ with p1(ᾱ) ≤ Ppeak is optimal and p* = Ppeak.
  • Subcase II-2: For λ − ζµg ≥ 0, the optimum is p* = 0 when 1 < (λ − ζµg)p1(α̃), otherwise p* = p1(α̃) and α* = α̃.
  • NoCancel benchmark: For the NoCancel benchmark, α*(ν) = 0 for every fading state remains optimal even after adding the average harvested-power constraint.The harvested power is independent of α(ν) in this benchmark.
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