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On the Design of Artificial-Noise-Aided Secure Multi-Antenna Transmission in Slow Fading Channels

Xi Zhang, Xiangyun Zhou, Matthew R. McKay

arXiv:1212.3640v3cs.IT

TL;DR

The paper addresses secrecy-constrained design of artificial-noise-aided secure multi-antenna transmission in slow fading channels, including wiretap-code rates and power allocation. It develops explicit solutions for fixed and receiver-feedback-adaptive transmission, analyzes high-SNR behavior and secrecy-related power costs, and quantifies adaptive-transmission gains.

  • Problem

    The paper asks how to design wiretap-code rate parameters and transmit power allocation for secure multi-antenna transmission under a maximum allowable secrecy outage probability.

  • Method

    The paper analyzes non-adaptive schemes with fixed parameters and adaptive schemes whose parameters respond to instantaneous intended-channel feedback, deriving closed-form or analytical designs and high-SNR approximations.

  • Results

    The paper provides explicit throughput-maximizing designs for both transmission scenarios, derives high-SNR approximations, and quantifies adaptive-transmission throughput gains and secrecy-related power costs.

  • Takeaways & Limitations

    The results provide design guidelines for selecting wiretap-code rates and allocating power between information-bearing signals and artificial noise under secrecy constraints.

Abstract

from arXiv · show

In this paper, we investigate the design of artificial-noise-aided secure multi-antenna transmission in slow fading channels. The primary design concerns include the transmit power allocation and the rate parameters of the wiretap code. We consider two scenarios with different complexity levels: i) the design parameters are chosen to be fixed for all transmissions, ii) they are adaptively adjusted based on the instantaneous channel feedback from the intended receiver. In both scenarios, we provide explicit design solutions for achieving the maximal throughput subject to a secrecy constraint, given by a maximum allowable secrecy outage probability. We then derive accurate approximations for the maximal throughput in both scenarios in the high signal-to-noise ratio region, and give new insights into the additional power cost for achieving a higher security level, whilst maintaining a specified target throughput. In the end, the throughput gain of adaptive transmission over non-adaptive transmission is also quantified and analyzed.

I. INTRODUCTION

The paper designs artificial-noise-aided secure multi-antenna transmission for slow fading channels without requiring Eve’s CSI. It optimizes wiretap-code rates and power allocation under secrecy-outage constraints for fixed and adaptive transmissions.

  • Prior physical-layer security work often assumed eavesdropper CSI at the transmitter, whereas artificial-noise transmission avoids that impractical requirement.
  • The design optimizes both wiretap-code rate parameters and the power split between the information signal and artificial noise under a maximum secrecy-outage constraint.
  • Fixed-rate transmission keeps design parameters unchanged across transmissions, while adaptive transmission adjusts them using instantaneous feedback from Bob’s channel.
  • The analysis derives high-SNR throughput approximations, evaluates secrecy-related power costs, and quantifies adaptive transmission’s throughput gain over fixed-rate transmission.The gain is most significant with few transmit antennas and is not very sensitive to the required secrecy level.
  • The system uses Bob’s instantaneous CSI to beamform the information signal while injecting artificial noise into the remaining transmit dimensions.Alice lacks Eve’s instantaneous CSI, so the artificial-noise power is distributed across the orthogonal dimensions.

B. Secure Transmission and Throughput

The paper defines the wiretap-code rates, uses on-off transmission based on Bob’s channel, and measures average throughput only over transmissions that satisfy reliability and secrecy conditions.

  • Rb and Rs are the transmitted-codeword and secret-message rates, while Re = Rb − Rs is the redundancy added for secrecy.
  • A secrecy outage occurs when Eve’s instantaneous channel capacity Ce exceeds the redundancy Re.
  • Alice transmits only when Bob’s effective channel gain ||h||2 exceeds threshold µ, preventing capacity outages and risky transmissions.
  • Larger transmit probability ptx corresponds to shorter expected delay under the on-off protocol.
  • Adaptive wiretap-code rates may depend on Bob’s channel, and average throughput assigns Rs(h) = 0 when ||h||2 ≤ µ.

C. Secrecy Performance Characterization

The secrecy analysis formulates the outage constraint and then develops non-adaptive throughput optimization with fixed code parameters and power allocation. The design separates delay minimization from throughput maximization.

  • C. Secrecy Performance Characterization: For a given Bob channel, secrecy outage depends on whether Eve’s capacity exceeds the rate redundancy, with zero outage assigned when transmission is off.
  • C. Secrecy Performance Characterization: Assuming no receiver noise at Eve yields an upper bound on the actual secrecy outage probability.
  • C. Secrecy Performance Characterization: The secrecy-outage probability depends on the power allocation ratio Φ but not on total transmit power P.
  • C. Secrecy Performance Characterization: The optimization imposes an overall secrecy-outage requirement of p̄so(Rb, Rs, Φ) ≤ ϵ.
  • Non-adaptive encoder: In the NAE scheme, Rb, Rs, and Φ remain fixed across transmissions, and Bob decodes reliably when Rb ≤ log2(1 + PΦµ).
  • Non-adaptive encoder: The NAE design first maximizes transmit probability for fixed Rs, then maximizes throughput by optimizing the data rate using the resulting designs.

A. Delay Minimization

The non-adaptive design minimizes average delay under a secrecy constraint by selecting optimal system parameters, then characterizes the resulting secrecy-delay tradeoff and power requirements.

  • The delay-minimization problem is solved for a non-adaptive encoder under a maximum allowable average secrecy outage probability.
  • The optimal artificial-noise power fraction increases with N, so more antennas allow a smaller fraction of power for artificial-noise generation while maintaining the required security level.
  • The optimal secrecy-delay tradeoff is completely specified by equation (14), with each point being Pareto optimal.
  • Under joint secrecy and delay constraints, the minimum transmit power is obtained by evaluating the tradeoff at the required transmit probability and solving for power.
  • The reduction in minimum required power from adding antennas is most significant for small N and becomes inverse linear as N grows for δ = 0.5.

B. Throughput Optimization and Analysis

The non-adaptive scheme maximizes throughput by optimizing the message rate and power allocation under a secrecy-outage constraint, with unique numerical rate solutions and accurate high-SNR approximations. Extra antennas increase throughput, reduce secrecy-related throughput loss and power cost, while the loss itself is independent of transmit power.

  • Throughput Optimization: The throughput-maximizing message rate is unique because throughput first increases and then decreases as the message rate grows.Very small message rates yield low throughput despite high transmission probability, while very large rates reduce both transmission probability and throughput.
  • High-SNR Analysis: High-SNR analysis provides an accurate approximation for the maximal NAE throughput, except when the number of antennas is very small.For N = 2, the approximation becomes inaccurate; substituting the approximation into the exact expression gives a better result.
  • High-SNR Analysis: The secrecy-induced throughput loss is independent of transmit power because Eve is modeled with zero noise.The loss depends on the secrecy constraint rather than P.
  • Antenna Effects: Additional transmit antennas increase achievable throughput and reduce the throughput loss caused by secure transmission.The reduction in loss depends only on the secrecy constraint.
  • Power Cost of Secrecy: 9.14 dB and 3.55 dB are the additional power costs for strengthening the secrecy constraint from ϵ = 1 to 0.1 and from ϵ = 0.1 to 0.01, respectively, when N = 4.These values closely match the high-SNR approximation, and extra antennas reduce the required power cost.

IV. SECURE TRANSMISSION DESIGN WITH ADAPTIVE ENCODER

The adaptive encoder adjusts wiretap-code rates, power allocation and transmission decisions using Bob’s instantaneous channel. Its analytical solution enforces the secrecy constraint, introduces on-off transmission for weak channels, and converges at high SNR toward non-adaptive power allocation.

  • Adaptive Design: The adaptive design dynamically selects Rb, Rs and Φ for each realization of Bob’s channel while constraining secrecy outage probability.The objective is to maximize the achievable message rate for each channel realization, thereby maximizing average throughput.
  • Adaptive Design: On-off transmission is required when ||h||2 ≤ λ(ϵ, N)/P because positive message rate and the required secrecy performance cannot otherwise be achieved simultaneously.The resulting transmission threshold is µ = λ(ϵ, N)/P.
  • High-SNR Behavior: The optimal adaptive power allocation ratio decreases with P and approaches a constant independent of Bob’s instantaneous channel.This constant is identical to the high-SNR limit for the non-adaptive scheme, so non-adaptive allocation is near-optimal at high SNR.
  • High-SNR Behavior: At high SNR, adaptive rate redundancy converges to a value independent of h and identical to that of the non-adaptive scheme.Optimized transmission sets Rb(h) to Bob’s instantaneous channel capacity to guarantee successful decoding.
  • Secrecy Constraint: The adaptive solution achieves the secrecy-outage constraint with equality for each transmission and guarantees the overall outage probability ̄pso = ϵ.The maximal message rate is attained when pso(Rb(h), Rs(h), Φ(h)) = ϵ.

B. Throughput Performance Analysis

The paper derives high-SNR throughput approximations for the adaptive artificial-noise scheme and uses them to characterize secrecy-related throughput loss and power cost. The approximation is accurate for the illustrated N = 4 case, while a different approximation is preferable for very small N.

  • High-SNR approximation: The high-SNR approximation applies as P →∞ and uses λ(ϵ, N), the digamma function, and a generalized hypergeometric function.These quantities appear in the analytical approximation for adaptive-scheme throughput.
  • Throughput loss: The unconstrained adaptive throughput term matches the high-SNR ergodic capacity of the MISO Rayleigh fading channel, while ηloss_AE(ϵ) represents secrecy-induced throughput loss.The secrecy constraint is therefore separated from the unconstrained throughput reference through the loss term.
  • Throughput loss: In the high-SNR region, secrecy-constrained throughput loss is independent of P and equals the corresponding loss for the non-adaptive scheme.The loss is twice the limiting rate redundancy because adaptive transmission also incurs a power loss, equal to that redundancy in data-rate terms.
  • Power cost: The additional power cost for imposing or strengthening secrecy at a specified target throughput is approximated identically for adaptive and non-adaptive transmission.Despite the same secrecy-related power cost, adaptive transmission retains higher throughput because it transmits at Bob’s instantaneous capacity when secrecy is removed.
  • Numerical validation: For N = 4, Fig. 6 compares adaptive-scheme throughput with its high-SNR approximation across transmit power and secrecy constraints.The accompanying analysis reports that the approximation is quite accurate; for N = 2, the alternative approximation in (34) is better.

C. Throughput Gain of Adaptation: AE over NAE

The paper analyzes the high-SNR throughput gain from adaptive-rate transmission over fixed-rate transmission. The gain grows slowly with transmit power, shrinks as antenna count increases, and is largely insensitive to the secrecy constraint.

  • Gain approximation: The high-SNR throughput-gain approximation is derived from the corresponding adaptive and non-adaptive throughput approximations.The resulting expression separates a transmit-power-dependent term from a P-independent term.
  • Dependence on transmit power: The throughput gain increases with P, but only very slowly through a double-logarithmic term.The first term in the approximation contains log2(ln(P)).
  • Dependence on antenna count: As N increases, the throughput gap between adaptive and non-adaptive schemes shrinks because the dominant power-dependent term decreases linearly with N.At high SNR and non-small antenna counts, the non-adaptive scheme may be preferable because it saves complexity.
  • Dependence on secrecy: The high-SNR throughput-gain approximation is independent of the secrecy constraint ϵ because both schemes incur the same secrecy-related throughput loss.Exact gains vary little across ϵ but decrease when the secrecy constraint is strengthened.
  • Numerical validation: Fig. 7 compares exact and approximate adaptive-over-non-adaptive throughput gains versus secrecy constraint for different antenna counts at P = 40 dB.The approximation predicts the exact gain’s varying trends despite noticeable approximation error.

APPENDIX

The appendix derives optimal fixed-rate transmission parameters by maximizing transmit probability under reliability and secrecy constraints. It also establishes the monotonic decrease of λ with the number of transmit antennas.

  • Optimization setup: For a prescribed message rate, the proof optimizes Rb and µ for a given artificial-noise allocation Φ, then optimizes Φ.The objective is maximizing ptx, equivalently minimizing average delay.
  • Threshold and code-rate design: The secrecy constraint can be satisfied by choosing sufficiently large Rb, while the transmit threshold µ is selected at its smallest feasible value.The second constraint removes capacity outages at Bob, and the resulting threshold maximizes transmit probability for a given Φ.
  • Power allocation: The optimal power-allocation ratio Φ minimizes the convex function ω(Φ) because ptx decreases with its value.The optimum is obtained by solving the derivative of ω(Φ) with respect to Φ.
  • Result: The optimized construction yields a corresponding maximal transmit probability pmax.This quantity summarizes the best transmission probability under the stated parameter constraints.
  • Antenna-count dependence: λ in (13) decreases as the number of transmit antennas N increases.The proof treats N continuously, shows the derivative is negative, and transfers the monotonicity to discrete antenna counts.

C. Approximation of Optimal Message Rate in (19)

The appendix approximates the optimal message rate by replacing the difficult exact optimization with a tractable high-SNR approximation. The resulting rate approximation is then used to obtain an elegant approximation for maximal non-adaptive throughput.

  • Approximation strategy: The exact stationarity equation for the optimal message rate R*_s is too complicated to solve directly.The appendix therefore approximates throughput before optimizing it.
  • High-SNR expansion: A lower bound on η_NAE is expanded for high P, using 2^R_s > 2^R_s − 1 and the series of the regularized upper incomplete gamma function.The derivation also uses the standard exponential series expansion.
  • Optimal-rate approximation: Ignoring high-order terms and setting the derivative with respect to x to zero produces the approximation in (19) for the optimal message rate.The approximation follows from optimizing the truncated throughput expression.
  • Throughput approximation: The approximation for R*_s is inserted into a throughput lower bound to derive an elegant approximation for maximal throughput η*_NAE.Lambert-W asymptotics are used as P →∞ before retaining the leading-order term.
  • Leading-order behavior: The leading-order high-SNR expression contains an N log2(ln(P)) term, with additive terms that can be ignored for large P.This reduction yields the stated high-SNR throughput approximation.

E. Proposition 2

For adaptive transmission, the paper optimizes the wiretap-code rates and artificial-noise power split for each intended-channel realization while enforcing the secrecy constraint. Positive message rate is feasible only above a channel-strength threshold, after which the optimal power split is obtained by maximizing an equivalent concave objective.

  • Adaptive parameter optimization: The adaptive design chooses Rb, Rs, and Φ as functions of the intended channel h to maximize Rs subject to secrecy.The optimization first selects Rb and Rs for fixed Φ, then optimizes Φ.
  • Adaptive parameter optimization: For fixed Φ, Alice sets Rb to Bob’s channel capacity to eliminate capacity outages.
  • Adaptive parameter optimization: The maximal Rs occurs when the secrecy-outage constraint is met with equality, pso(Rb(Φ), Rs, Φ) = ϵ.The secrecy-outage probability increases with Rs for the specified Rb(Φ) and Φ.
  • Feasibility conditions: When τ ≤ λ, equivalently ||h||2 ≤ λ/P, adjusting Φ cannot produce a positive message rate.
  • Feasibility conditions: When τ > λ, equivalently ||h||2 > λ/P, positive Rs is guaranteed for Φ in the range (0, 1 − λ/τ).
  • Optimal power split: For τ > λ, maximizing an equivalent objective that is concave on Φ ∈ (0, 1) yields the optimal artificial-noise power allocation summarized in Proposition 2.The equivalent objective is used because Rs(Φ) itself is non-concave, while the logarithm is monotonic.

F. High SNR Throughput Approximation in (35)

The adaptive scheme’s average throughput is obtained by averaging the maximum message rate over intended-channel realizations. Because a closed form is difficult, the paper derives a high-SNR approximation by neglecting a P-independent term and reducing the resulting integral to special-function expressions.

  • Throughput averaging: Average throughput is computed by averaging the maximum message rate over all intended-channel realizations, with integration limits determined by the transmit threshold.
  • High-SNR approximation: At high SNR, the P-independent quantity λ^-1 is negligible compared with Pr, so it is omitted under the second square root in (46).
  • High-SNR approximation: The resulting integral is evaluated using tabulated integral identities and represented with a special case of the Meijer G-function.
  • Closed-form transformation: The Meijer G-function expression is equivalently transformed into an infinite summation involving Pochhammer symbols and then a hypergeometric function.
  • High-SNR result: For large P, the regularized incomplete gamma function approaches one and the remaining third term has order O(...), yielding the result in (35).
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