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Improving Physical Layer Secrecy Using Full-Duplex Jamming Receivers

Gan Zheng, Ioannis Krikidis, Jiangyuan Li, Athina P. Petropulu, Bjorn Ottersten

arXiv:1308.0094v1cs.IT

TL;DR

The paper addresses secrecy-rate optimization in an interference-limited network where a single-antenna source communicates with a multi-antenna destination under eavesdropping. It uses full-duplex self-jamming at the destination, designs jamming and power allocation under different channel-information settings, and shows rank-1 jamming with increasing secrecy rates in the multi-antenna case.

  • Problem

    In single-antenna-source networks, half-duplex secrecy enhancement relies on external cooperative jammers, while full-duplex designs must address loop interference rather than assume perfect cancellation.

  • Method

    The paper jointly designs the destination’s jamming covariance, transmit power allocation, and fixed or optimal linear reception under perfect or statistical eavesdropper channel information.

  • Results

    The optimal jamming covariance matrix is rank-1 and can be found through an efficient 1-D search; in the multi-antenna case, full-duplex secrecy rate keeps increasing with SNR, while optimal reception improves secrecy rate by approximately 10% over fixed MMSE reception.

  • Takeaways & Limitations

    Full-duplex self-protection provides a secrecy-enhancement approach that does not require trusted external helpers and avoids high-SNR secrecy-rate saturation in the multi-antenna setting.

Abstract

from arXiv · show

This paper studies secrecy rate optimization in a wireless network with a single-antenna source, a multi-antenna destination and a multi-antenna eavesdropper. This is an unfavorable scenario for secrecy performance as the system is interference-limited. In the literature, assuming that the receiver operates in half duplex (HD) mode, the aforementioned problem has been addressed via use of cooperating nodes who act as jammers to confound the eavesdropper. This paper investigates an alternative solution, which assumes the availability of a full duplex (FD) receiver. In particular, while receiving data, the receiver transmits jamming noise to degrade the eavesdropper channel. The proposed self-protection scheme eliminates the need for external helpers and provides system robustness. For the case in which global channel state information is available, we aim to design the optimal jamming covariance matrix that maximizes the secrecy rate and mitigates loop interference associated with the FD operation. We consider both fixed and optimal linear receiver design at the destination, and show that the optimal jamming covariance matrix is rank-1, and can be found via an efficient 1-D search. For the case in which only statistical information on the eavesdropper channel is available, the optimal power allocation is studied in terms of ergodic and outage secrecy rates. Simulation results verify the analysis and demonstrate substantial performance gain over conventional HD operation at the destination.

I. INTRODUCTION

The paper proposes a full-duplex destination that simultaneously receives data and transmits jamming noise, replacing unavailable or untrusted external helpers while accounting for loop interference. It develops secrecy-rate optimization for several antenna, receiver, and eavesdropper-information settings.

  • I. INTRODUCTION: Full-duplex self-jamming lets the destination protect itself against eavesdropping without external assistance, an out-of-band channel, or source-data retransmission.The receiver transmits jamming signals while simultaneously receiving the source signal.
  • I. INTRODUCTION: The proposed loop-interference model parameterizes passive self-interference suppression rather than assuming complete cancellation.The parameter ρ ranges from 0 for ideal no loop interference to positive values representing different interference levels.
  • I. INTRODUCTION: For one transmit and one receive antenna with a single-antenna eavesdropper, the paper derives closed-form receiver power allocation, which usually leaves some available power unused because of loop interference.The destination balances jamming power against the loop-interference penalty.
  • I. INTRODUCTION: With multiple transmit antennas, the system is no longer interference-limited, and the optimal jamming covariance is rank-1; efficient algorithms are proposed for finding it.Both fixed and optimal linear receivers are considered.
  • I. INTRODUCTION: When only eavesdropper channel distribution information is available, the paper optimizes power allocation for ergodic and outage secrecy rates.The study also considers joint source–destination power allocation and perfect versus unavailable eavesdropper CSI settings.

III. PERFECT CSI WITH SINGLE-ANTENNA TERMINALS

The single-antenna case uses receiver-side jamming with power control to balance eavesdropper degradation against loop interference. Positive secrecy depends on channel and interference conditions, and secrecy can saturate at high jamming power.

  • System model: The single-antenna baseline cannot mitigate loop interference spatially, so it controls interference through the destination's transmit power.The main study focus is the multiple-antenna case, where spatial loop-interference mitigation is possible.
  • Power control: The destination chooses jamming power by maximizing secrecy rate while balancing degradation at the eavesdropper against self-interference.The formulation treats the jamming receiver similarly to an external helper, but the jammer is colocated with the destination.
  • Positive secrecy: Positive secrecy is characterized by explicit channel and jamming-power conditions, including a stronger S−D channel than S−E in one stated condition.The supplied lemmas state these conditions through inequalities involving |h|2, |hsd|2, |hse|2, Pd, and |hed|2.
  • Special cases: When |hsd|2 = |hse|2, the optimal destination power is monotonically non-increasing in ρ.Under a total source-and-destination power constraint, source power is then monotonically increasing in ρ.
  • Power control: The receiver may not always use full jamming power because increasing ρ can make self-interference too costly relative to eavesdropper degradation.For small ρ, high power can confuse the eavesdropper; near ρ = 1, the receiver compares interference at itself and the eavesdropper.
  • High-SNR behavior: The secrecy rate saturates as pd →∞ because residual self-interference cannot be fully canceled.Thus, high jamming power becomes a limiting factor in the single-antenna case.

IV. MULTIPLE-ANTENNA RECEIVER

With multiple antennas, the destination can mitigate loop interference spatially while jamming the eavesdropper. Consequently, the system is not interference-limited, and optimal jamming covariance has rank 1.

  • Optimization cases: The analysis proceeds first with a fixed receiver, then extends to optimal linear receiver design and joint source-destination power allocation.The multiple-antenna problem is formulated under a destination power constraint before these extensions.
  • Multiple-antenna benefit: Given Mt > 1 or Mr > 1, the system is not interference-limited and secrecy rate can keep increasing with transmit power.This contrasts with the single-antenna baseline, where residual self-interference causes saturation.
  • Loop-interference mitigation: Zero-forcing can select receiver and jamming designs so destination self-interference is avoided while the eavesdropper remains jammed.Under this design, secrecy rate increases monotonically with trace(Q).
  • Covariance structure: The optimal jamming covariance matrix Q∗ can be chosen rank-1.The rank-1 property reduces covariance design to a lower-dimensional jamming direction and power design.

A. Optimal Solution with a Fixed Receiver

For a fixed receiver, the covariance optimization exploits rank-1 structure and auxiliary-variable reformulation. The resulting scalar objective is quasi-concave and can be optimized efficiently by bisection.

  • Problem setup: The fixed receiver may be an MRC or MMSE receiver and is assumed independent of the jamming covariance.The optimization then designs Q for that receiver choice.
  • Auxiliary optimization: Introducing an auxiliary variable t produces an intermediate scalar optimization whose objective g(t) is concave.Its closed-form solution speeds the algorithm relative to formulations with individual constraints on each jamming-vector element.
  • Rank-1 reduction: The optimal jamming covariance has rank 1 because the reformulated problem is a homogeneous QCQP with two constraints.The reformulation preserves the original optimum while enabling the rank-1 solution result.
  • Power usage: Under the non-alignment assumption between the effective destination and eavesdropper channels, the optimal covariance uses the full destination jamming-power budget.The excluded alignment condition can normally be avoided through proper receiver design.
  • Search procedure: The outer objective f(t) is quasi-concave, so its maximum can be found by bisection search.This provides an efficient solution procedure for the fixed-receiver problem.

B. Optimal Solution with The Optimal Linear Receiver

With an optimal linear receiver, the paper jointly designs the receiver and rank-1 jamming direction through a semidefinite reformulation. The resulting secrecy-rate maximization reduces to a one-dimensional search.

  • Receiver design: The destination's optimal linear receiver is selected to maximize its received SINR.The resulting secrecy rate is then expressed using the receiver and jamming design.
  • Rank-1 parameterization: Using Q = Pdqq† with ∥q∥ = 1 exploits the rank-1 covariance structure and converts covariance design into jamming-direction design.The secrecy-rate expression remains complicated, motivating an auxiliary parameter t.
  • Semidefinite reformulation: The nonconvex quadratic direction problem is relaxed to a semidefinite program whose optimal matrix is rank 1, allowing recovery of q∗.Thus the relaxation is equivalent to the original direction problem in terms of the optimal solution.
  • Search procedure: The maximum of R(t) can be found via a one-dimensional search.This search follows the semidefinite subproblem used to evaluate the objective for each t.

C. Joint S-D Power Allocation

When source and destination share total power, joint secrecy optimization can search over source power and a jamming-design parameter. Fixed receivers permit a 1-D search, whereas optimal receivers require 2-D search.

  • C. Joint S-D Power Allocation: A shared total power constraint revises the problem so source power p_s and jamming covariance Q are jointly optimized.At the optimum, all available power is used: p_s + trace(Q) = P_T.
  • C. Joint S-D Power Allocation: For fixed receivers, the source-power derivative yields a quadratic equation whose feasible solution determines the power allocation for each design parameter.The candidate source power is either P_T or a root of the quadratic equation.
  • C. Joint S-D Power Allocation: The fixed-receiver total-power problem has an optimal solution obtainable through 1-D optimization over t.
  • C. Joint S-D Power Allocation: For optimal linear receivers, non-separability prevents the same reduction, so optimal power allocation requires a 2-D search.The function h(t) also depends complexly on p_d.
  • ZF Solution: A closed-form ZF-based solution is obtained by specializing the optimal-receiver problem to t = 0.

2) ZF Solution:

The ZF design suppresses destination self-interference, reducing the optimal linear receiver to MRC and simplifying secrecy-rate and power-allocation optimization. Under statistical eavesdropper information, ergodic-rate optimization uses an approximation without a guaranteed bound.

  • 2) ZF Solution:: Imposing zero self-interference, h†_sdHq = 0, removes the loop-interference term from the receiver design.
  • 2) ZF Solution:: Under this constraint, the optimal linear receiver reduces to the MRC receiver r = h_sd/∥h_sd∥.
  • 2) ZF Solution:: The resulting secrecy-rate expression is then simplified before optimizing the remaining power variables.
  • 2) ZF Solution:: With only eavesdropper CDI, the paper optimizes power allocation for ergodic and outage secrecy rates using local-information receiver designs.A suboptimal MRC receiver is used when instantaneous eavesdropper channels are unavailable.
  • 2) ZF Solution:: The ergodic-rate approximation takes expectations over individual random terms because the original expectation is difficult to evaluate.The approximation is evaluated separately from the original problem.
  • 2) ZF Solution:: The approximation provides neither an upper nor a lower bound of the original ergodic secrecy-rate problem.
  • 2) ZF Solution:: The random variable ∥h_se∥^2 follows a central chi-square distribution with 2(M_t − 1) degrees of freedom.

B. Outage Secrecy Rate

For slow fading with only eavesdropper CDI, the paper formulates epsilon-outage secrecy-rate maximization and computes the optimum through rate search and outage-probability evaluation.

  • B. Outage Secrecy Rate: The outage design criterion maximizes the ε-outage secrecy rate for a slow-fading channel.
  • B. Outage Secrecy Rate: The outage secrecy-rate maximization problem is formulated using the random variable defined in the preceding analysis.
  • B. Outage Secrecy Rate: Bisection can find the optimal secrecy rate, after which the corresponding outage probability must be calculated.
  • B. Outage Secrecy Rate: The outage probability is derived from an indefinite quadratic form in complex normal variables.

VI. DESIGN OF Q WITH THE OPTIMAL LINEAR MMSE RECEIVER AT BOTH D AND E

When the eavesdropper knows the FD operation, it can use an MMSE receiver to mitigate destination jamming. The resulting nonconvex covariance design is handled with DC programming to obtain a stationary point.

  • VI. DESIGN OF Q WITH THE OPTIMAL LINEAR MMSE RECEIVER AT BOTH D AND E: An eavesdropper aware of FD operation may adopt an advanced linear MMSE receiver to mitigate destination jamming.
  • VI. DESIGN OF Q WITH THE OPTIMAL LINEAR MMSE RECEIVER AT BOTH D AND E: Assuming optimal linear MMSE receivers at both destination and eavesdropper leads to a secrecy-rate maximization problem.
  • VI. DESIGN OF Q WITH THE OPTIMAL LINEAR MMSE RECEIVER AT BOTH D AND E: With perfect CSI, optimizing Q is generally nonconvex and cumbersome.
  • VI. DESIGN OF Q WITH THE OPTIMAL LINEAR MMSE RECEIVER AT BOTH D AND E: The paper expresses the objective as a difference of concave functions and uses DC programming to find a stationary point.
  • VI. DESIGN OF Q WITH THE OPTIMAL LINEAR MMSE RECEIVER AT BOTH D AND E: The iterative procedure initializes Q_0 and repeatedly solves a convex problem until a termination condition is met.

VII. NUMERICAL RESULTS

Simulations show that FD jamming at the destination substantially improves secrecy over HD operation across channel-information settings, while performance depends on self-interference and antenna allocation.

  • Single-antenna setting: Above 10 dB transmit SNR, both FD schemes outperform HD in the single-antenna case, with substantial gains at high SNR.HD saturates above 25 dB, whereas FD ceiling effects begin at 50 dB.
  • Multi-antenna setting: In the default multi-antenna setting, HD secrecy saturates at very low SNR, while FD secrecy continues increasing without a ceiling.Multiple destination transmit antennas suppress self-interference and generate jamming toward the eavesdropper.
  • Multi-antenna setting: An optimal linear receiver improves secrecy by approximately 10% over a fixed MMSE receiver in the default multi-antenna setting.The comparison is reported for the destination receiver designs in Fig. 4.
  • Residual self-interference: At residual self-interference ρ as high as 0.9, all FD schemes still outperform HD, although FD secrecy decreases as ρ increases.The optimal linear receiver becomes increasingly advantageous at higher self-interference.
  • Ergodic secrecy rate: The ergodic secrecy rate of HD saturates from very low SNR, whereas FD increases without a ceiling.The comparison includes the ergodic-rate approximation used for power allocation.
  • Antenna allocation: With four destination antennas, the (Mr = 2, Mt = 2) allocation performs best, while (Mr = 3, Mt = 1) greatly outperforms (Mr = 1, Mt = 3).The paper attributes this to receiver design handling both useful signal and self-interference, whereas transmitter design mainly suppresses self-interference and jams E.

VIII. CONCLUSIONS

The paper proposes destination-side FD self-protection for passive eavesdropping and studies jamming covariance and power allocation under loop interference. It concludes that FD provides substantial gains over HD, while leaving adversarial FD operation as future work.

  • VIII. CONCLUSIONS: The proposed self-protection scheme uses FD operation at the destination to counter passive eavesdropping when secrecy is interference-limited and trusted helpers are unavailable.The study addresses loop interference and power allocation between source and destination.
  • VIII. CONCLUSIONS: A future direction is modeling an eavesdropper that knows the destination's FD strategy and performs a similar FD operation.The paper suggests studying this interaction using noncooperative game theory.
  • VIII. CONCLUSIONS: The analysis identifies a single possible positive root x2(ρ) in the relevant optimization proof.The proof also derives cases in which the optimal destination jamming power is zero or bounded by x2(ρ).

APPENDIX B

Appendix B derives the optimization solution through a dual formulation and eigenvalue decomposition. It establishes conditions for the dual variables and the existence of a rank-1 optimal jamming solution.

  • APPENDIX B: The appendix begins by formulating the optimization problem and expressing the matrix R and vector a used in its dual formulation.R is described as Hermitian and not positive semidefinite, while a is an Mt × 1 vector.
  • APPENDIX B: At least one rank-1 optimal solution q exists such that Q = qq†.The appendix also states that λ2 > 0 because otherwise the secondary matrix inequality cannot be satisfied.
  • APPENDIX B: When λ2I + λ1R is positive semidefinite with smallest eigenvalue zero, equality is characterized through an eigenvector attaining that minimum eigenvalue.The appendix denotes such an eigenvector by q1.
  • APPENDIX B: An eigenvalue decomposition R = UDU† transforms q into q = U(λ̄1D + λ̄2I)^−1b, with b = U†a.The remaining task is to identify λ̄1 and λ̄2.
  • APPENDIX B: Substituting the transformed q into the equality constraints yields equations for solving λ̄1 and λ̄2.The selected solution must satisfy λ̄2 > 0 and maximize the dual objective λ̄1t + λ̄2.
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