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Secure Transmission with Artificial Noise over Fading Channels: Achievable Rate and Optimal Power Allocation
Xiangyun Zhou, Matthew R. McKay
TL;DR
The paper addresses secure communication in fading channels when a multi-antenna transmitter sends both information and artificial noise without relying on favorable eavesdropper channels. It derives closed-form achievable secrecy rates and uses them to optimize power allocation, finding near-equal allocation for non-colluding eavesdroppers and increased artificial-noise allocation as collusion grows.
Problem
Power allocation between the information signal and artificial noise had not been investigated for this multi-antenna fading-channel security technique.
Method
The paper derives closed-form achievable secrecy-rate lower bounds and optimizes information-versus-artificial-noise power allocation using them.
Results
Equal power allocation is near optimal for non-colluding eavesdroppers, while more artificial-noise power is recommended as colluding eavesdroppers increase.
Takeaways & Limitations
The closed-form bound enables lower-complexity secrecy-rate analysis, power-allocation design, and critical-SNR evaluation.
Abstract
from arXiv · showhide
We consider the problem of secure communication with multi-antenna transmission in fading channels. The transmitter simultaneously transmits an information bearing signal to the intended receiver and artificial noise to the eavesdroppers. We obtain an analytical closed-form expression of an achievable secrecy rate, and use it as the objective function to optimize the transmit power allocation between the information signal and the artificial noise. Our analytical and numerical results show that equal power allocation is a simple yet near optimal strategy for the case of non-colluding eavesdroppers. When the number of colluding eavesdroppers increases, more power should be used to generate the artificial noise. We also provide an upper bound on the signal-to-noise ratio (SNR) above which the achievable secrecy rate is positive and show that the bound is tight at low SNR. Furthermore, we consider the impact of imperfect channel state information (CSI) at both the transmitter and the receiver and find that it is wise to create more artificial noise to confuse the eavesdroppers than to increase the signal strength for the intended receiver if the CSI is not accurately obtained.
I. INTRODUCTION
The paper studies secure multi-antenna communication in fading channels using artificial noise, focusing on achievable secrecy rates and power allocation under non-colluding and colluding eavesdroppers. It also examines critical SNR and imperfect channel knowledge.
- Motivation: Artificial noise can secure fading-channel transmissions when a multi-antenna transmitter sends information and interference simultaneously.The artificial noise is radiated isotropically to mask the information signal from eavesdroppers.
- Contributions: The paper derives closed-form achievable secrecy-rate expressions to reduce the complexity of system design and analysis.The expressions cover both non-colluding and colluding eavesdroppers.
- Contributions: Equal power allocation is nearly optimal for non-colluding eavesdroppers, whereas more artificial-noise power is needed as colluding eavesdroppers increase.The paper studies both adaptive and non-adaptive allocation strategies.
- Contributions: An upper bound on the critical SNR above which the achievable secrecy rate is positive is derived and shown tight at low SNR.The bound is intended to support wideband secure-communication design and analysis.
- Contributions: With increasing channel-estimation error, allocating more power to artificial noise is preferable to increasing the intended receiver’s signal strength.The analysis considers imperfect CSI through channel-estimation errors.
- System model: The transmitter uses an orthonormal basis aligned with Bob’s channel, sending the information signal along one vector and artificial noise in the orthogonal subspace.Bob receives the information signal without the artificial-noise component under the stated model.
III. SECRECY CAPACITY LOWER BOUND
This section derives a lower bound on ergodic secrecy capacity for artificial-noise transmission in fading channels. The bound supports adaptive and non-adaptive power allocation analysis and reveals degradation from colluding eavesdroppers.
- Lower-bound derivation: Closed-form expressions for the ergodic secrecy-capacity lower bound are derived for artificial-noise transmission.These expressions are used to obtain analytical insights and simplify power-allocation design.
- Eavesdropper models: The model includes non-colluding and colluding eavesdroppers, with the colluding analysis using a worst-case noiseless-eavesdropper assumption.This assumption yields an upper bound on the colluding eavesdroppers’ channel capacity.
- Eavesdropper models: For colluding eavesdroppers, the analysis assumes NA > NE so the artificial-noise channel matrix is invertible.If this condition is violated, eavesdroppers can eliminate the artificial noise, giving C2 = ∞.
- Lower-bound derivation: The secrecy-capacity lower bound is the positive part of the difference between Alice–Bob and Alice–Eve capacities.It is a data rate that can always be guaranteed without knowing the noise level at the eavesdroppers.
- Results: Multiple colluding eavesdroppers dramatically reduce secrecy rates, which quickly fall to zero at low to moderate SNR.Figure 1 evaluates the lower bound for different antenna counts with φ = 0.5.
A. Non-colluding Eavesdroppers
For non-colluding eavesdroppers, the lower-bound expression simplifies using the single-eavesdropper case. This simplification produces an expression suitable for evaluating the secrecy-capacity lower bound.
- Non-colluding Eavesdropper Case: When eavesdroppers cannot collude, NE = 1 and the eavesdropper-capacity term C2 reduces to a simplified expression.The result uses an identity for the Gauss hypergeometric function before substitution into C = [C1 − C2]+.
B. Large NA Analysis
The large-N_A analysis characterizes how the secrecy-capacity lower bound behaves as Alice’s antenna count grows. It shows convergence to a large-N_A approximation and that increasing Alice’s antennas does not affect the eavesdroppers’ channel capacity in the limit.
- Asymptotic behavior: The difference between C1 and log2 N_A approaches log2 z as N_A increases.
- Eavesdropper channel: Changing Alice’s antenna count does not affect the channel capacity between Alice and the eavesdroppers in the large-N_A limit.
- Large-N_A approximation: The large-N_A secrecy-capacity lower bound is approximated using an expression obtained after applying the law of large numbers to the channel gain.The approximation is used to study power allocation for systems with many transmit antennas.
- Approximation accuracy: Expression (14) converges to the large-N_A approximation in (20) as N_A increases.Convergence is fast for N_E = 2 and slow for N_E = 6.
IV. OPTIMAL POWER ALLOCATION
This section optimizes the allocation of transmit power between the information-bearing signal and artificial noise. It considers adaptive and non-adaptive strategies for both non-colluding and colluding eavesdroppers using the ergodic secrecy-capacity lower bound.
- Optimization framework: The optimization objective is the ergodic secrecy-capacity lower bound, whose closed-form expressions reduce the computational complexity of power-allocation optimization.
- Allocation strategies: Adaptive power allocation depends on each channel-gain realization, whereas non-adaptive allocation fixes the power split across all channel realizations.
- Eavesdropper models: The analysis first considers non-colluding eavesdroppers and then colluding eavesdroppers.
A. Non-colluding Eavesdropper Case
The non-adaptive allocation analysis shows that the optimal information-signal fraction approaches simple high-SNR constants, making equal power near-optimal across a wide SNR range.
- High-SNR analysis: At high SNR, adaptive allocation is unnecessary because the optimal power allocation is independent of the intended receiver’s channel realization h.The high-SNR approximation removes h from the optimization, apart from the expectation over h.
- High-SNR analysis: For N_A = 2, the high-SNR optimum is φ = 0.5, corresponding to equal power for information and artificial noise.The result follows from the solution z = 2.
- Numerical behavior: As SNR or N_A increases, more power is allocated to the information signal, while φ converges to a constant in the high-SNR regime.For N_A = 2, the limiting value is 0.5.
- Numerical behavior: Equal power allocation achieves nearly the same secrecy rate as optimized non-adaptive allocation over a wide range of SNR values.The comparison uses the ergodic secrecy capacity lower bound C in (14).
B. Colluding Eavesdropper Case
For colluding eavesdroppers, the paper analyzes power allocation between information and artificial-noise signals and finds that increasing the number of eavesdroppers requires more artificial noise. Large-antenna approximations provide useful design guidance, while mismatched allocations can incur power loss.
- The study uses closed-form secrecy-capacity lower bounds to reduce the complexity of numerically finding optimal power allocations.
- In the high-SNR, large-antenna regime, the optimal allocation depends on the number of colluding eavesdroppers but not on the number of transmit antennas.
- More colluding eavesdroppers require allocating more power to artificial noise.
- The optimal information-signal power fraction increases with SNR and approaches a constant in the high-SNR regime.
- An allocation optimized for 8 eavesdroppers remains effective for 6, with a power loss of 0.2 dB, but changing to 4 incurs approximately 1 dB loss without redesign.
- With variable power transmission under an average-power constraint, temporal allocation according to channel gain can increase the achievable secrecy rate.
- When eavesdropper noise is included, artificial noise becomes less efficient, so more power should be allocated to the information signal.
V. CRITICAL SNR FOR SECURE COMMUNICATIONS
The paper studies the critical SNR required for a positive secrecy rate, deriving an analytical upper bound from the secrecy-capacity lower bound. The bound is asymptotically tight at low SNR and is especially accurate for non-colluding eavesdroppers.
- The analytical upper bound on critical SNR is asymptotically tight as SNR approaches zero.
- The critical SNR is the threshold at which the secrecy-capacity lower bound drops to zero.
- The bound provides a minimum SNR that guarantees a positive secrecy rate and can help tune power allocation to reduce critical SNR.
- Critical SNR decreases as Alice’s antenna count increases but increases as the number of colluding eavesdroppers increases.
- The analytical upper bound is very accurate for non-colluding eavesdroppers and reasonably accurate for colluding eavesdroppers when critical SNR is below 0 dB.
VI. EFFECT OF IMPERFECT CHANNEL STATE INFORMATION
The paper models imperfect CSI through channel estimation errors, which cause artificial noise to leak into Bob’s channel. As estimation errors increase, allocating more power to artificial noise becomes preferable to boosting Bob’s information signal.
- With imperfect CSI, Alice designs beamforming from the estimated rather than true channel, causing artificial noise to leak into Bob’s channel.
- MMSE estimation decomposes the channel into an estimate and an estimation error, modeled as uncorrelated i.i.d. complex Gaussian components.
- The imperfect-CSI secrecy lower bound is obtained by subtracting Eve’s rate from a lower bound on Bob’s ergodic capacity.
- The critical-SNR upper bound with channel estimation errors is asymptotically tight at low SNR.
- The numerical study focuses on non-colluding eavesdroppers, while the reported estimation-error trends are stated to apply also to colluding eavesdroppers.
- As channel estimation error increases, less power should be allocated to the information signal, especially when Alice has few antennas such as NA = 2.
- Artificial noise remains equally efficient against Eve despite channel estimation error, whereas boosting the information signal becomes less efficient for improving Bob’s reception.
VII. CONCLUSION
The paper analyzes secure multi-antenna transmission in fading channels with non-colluding or colluding eavesdroppers, deriving secrecy-rate and power-allocation results.
- The study considers simultaneous transmission of an information signal to the intended receiver and artificial noise to eavesdroppers.
- An ergodic secrecy capacity lower bound is obtained in closed form.
- Equal power allocation is near optimal for non-colluding eavesdroppers.
- More artificial-noise power is recommended as the number of colluding eavesdroppers increases.
- The upper bound on the critical SNR is tight at low SNR, while imperfect CSI favors artificial noise over increased intended-receiver signal strength.
APPENDIX I IDENTITY FOR SPECIAL CLASS OF GAUSS
The appendix derives an identity relating two forms of a Gauss hypergeometric function and uses derivative expressions and a logarithmic special case to obtain a simplified expression.
- The appendix considers Gauss hypergeometric functions 2F1(1, 1; N + 1; x) and 2F1(N, N; N + 1; x) for integer N ≥1.
- The two hypergeometric forms are related by the factor (1 − x)^(N−1).
- The derivation uses the rising factorial and the identity 2F1(1, 1; 2; x) = −ln(1 − x)/x.
- Substituting derivative expressions into the preceding relation yields an identity expression.