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On Secrecy Rate of the Generalized Artificial-Noise Assisted Secure Beamforming for Wiretap Channels
Pin-Hsun Lin, Szu-Hsiang Lai, Shih-Chun Lin, Hsuan-Jung Su
TL;DR
The paper considers wiretap channels with partial CSIT and proposes a generalized artificial-noise scheme. The scheme injects AN in all directions, yields a simplified power-allocation formulation, and outperforms Goel and Negi’s scheme under certain channel conditions.
Problem
The paper considers an important type of wiretap channel with partial CSIT.
Method
The paper generalizes Goel and Negi’s AN scheme by injecting AN in all directions, including the message direction, and characterizes a beamformer aligned to the legitimate channel.
Results
The generalized AN scheme outperforms Goel and Negi’s scheme, especially when the legitimate channel is poor, while reducing the secrecy-rate problem to simplified power allocation.
Takeaways & Limitations
The scheme provides rate gains and enlarges the non-zero secrecy-rate region.
Abstract
from arXiv · showhide
In this paper we consider the secure transmission in fast Rayleigh fading channels with full knowledge of the main channel and only the statistics of the eavesdropper's channel state information at the transmitter. For the multiple-input, single-output, single-antenna eavesdropper systems, we generalize Goel and Negi's celebrated artificial-noise (AN) assisted beamforming, which just selects the directions to transmit AN heuristically. Our scheme may inject AN to the direction of the message, which outperforms Goel and Negi's scheme where AN is only injected in the directions orthogonal to the main channel. The ergodic secrecy rate of the proposed AN scheme can be represented by a highly simplified power allocation problem. To attain it, we prove that the optimal transmission scheme for the message bearing signal is a beamformer, which is aligned to the direction of the legitimate channel. After characterizing the optimal eigenvectors of the covariance matrices of signal and AN, we also provide the necessary condition for transmitting AN in the main channel to be optimal. Since the resulting secrecy rate is a non-convex power allocation problem, we develop an algorithm to efficiently solve it. Simulation results show that our generalized AN scheme outperforms Goel and Negi's, especially when the quality of legitimate channel is much worse than that of eavesdropper's. In particular, the regime with non-zero secrecy rate is enlarged, which can significantly improve the connectivity of the secure network when the proposed AN assisted beamforming is applied.
I. INTRODUCTION
The paper studies MISOSE fading wiretap channels with perfect main-channel knowledge and only statistical eavesdropper-channel knowledge, addressing limitations in prior partial-CSIT schemes. It generalizes artificial-noise beamforming by optimizing signal and AN directions, power allocation, and the resulting secrecy rate.
- Prior partial-CSIT schemes selected signal and AN directions without optimization, producing suboptimal performance.
- Earlier approaches also used inefficient full-search power allocation and left key power-constraint and beamformer optimality questions unresolved.
- The paper considers MISOSE fading wiretap channels with a constant main channel, fast-faded eavesdropper channel, and perfect main-channel but statistical eavesdropper-channel CSIT.
- The generalized AN scheme allows AN in all directions, including the main-channel direction, rather than restricting it to the legitimate channel's null space.
- The authors characterize optimal beamforming directions, AN eigenvectors, power allocation strategies, and a necessary condition for main-channel AN transmission.
- An efficient algorithm solves the resulting non-convex power-allocation problem, while simulations report rate gains over the prior scheme, especially through an enlarged non-zero-rate region.
- The enlarged non-zero secrecy-rate region is presented as useful for connectivity in large-scale wireless-network applications.
II. SYSTEM MODEL
The system model uses a multi-antenna transmitter with single-antenna legitimate receiver and eavesdropper, combining Gaussian message signaling with artificial noise under partial CSIT. The proposed GAN formulation permits unrestricted AN covariance directions and reduces secrecy-rate optimization to power allocation.
- The MISOSE model has an nT-antenna transmitter, single-antenna Bob and Eve, constant main channel h, and random eavesdropper channel g.
- The transmitter knows the legitimate channel fully but knows only the statistics of Eve's channel.
- Linear channel prefixing uses independent Gaussian vectors u and v to carry the message and artificial noise, respectively.
- The signal and AN covariance matrices obey the feasible channel-input constraints used in the generalized AN formulation.
- The generalized AN beamforming scheme optimizes the ergodic secrecy-rate expression over the admissible signal and AN covariances.
- Unlike the conventional scheme, GAN does not impose a special structure on the AN covariance matrix and allows AN in all possible directions.
III. OPTIMIZATION OF THE ERGODIC SECRECY RATE
The paper reduces ergodic secrecy-rate optimization over signal and AN covariance matrices to scalar power allocation by characterizing their optimal structure.
- Power allocation: The covariance-matrix optimization is transformed into a simplified problem with three scalar powers: signal power, main-channel AN power, and null-space AN power.The variables are PU, PV1, and PV2, with PV2 representing equal per-direction allocation in the null space.
- Optimal covariance structure: The optimal covariance matrices of the signal and AN share eigenvectors based on the legitimate-channel direction and its orthogonal complement.The signal uses the h direction, while AN can occupy the main-channel direction and its null space.
- Optimal signal structure: The optimal signal covariance is rank one when the secrecy rate is positive, so message transmission uses beamforming.The beamforming direction aligns with the legitimate channel h.
- Power allocation: The total-power constraint is active at the optimum, so all available transmission power is used.The resulting allocation satisfies PU+PV1+(nT−1)PV2=PT.
- Solution algorithm: An iterative power-allocation algorithm finds solutions almost identical to brute-force search while using substantially lower complexity.The algorithm divides the search into two sub-problems rather than simultaneously searching all three powers.
- Generalized AN condition: A necessary condition is provided for when transmitting AN in the main-channel direction is optimal.The condition is developed before introducing the iterative algorithm.
V. SIMULATION RESULTS
Simulations compare the generalized AN scheme with Goel and Negi’s scheme under several legitimate-channel conditions and evaluate the iterative algorithm’s convergence.
- Rate comparison: The generalized AN scheme provides apparent rate gains over Goel and Negi’s scheme in moderate SNR regions.The comparison uses ||h||2 values of 0.05, 0.1, and 0.2.
- Rate comparison: Rate gains decrease as ||h||2 increases, consistent with the reported channel-condition dependence.The largest rate-gain point also shifts to lower PT as ||h||2 increases.
- Rate comparison: AN in the signal direction provides greater rate gains when Bob’s received SNR is relatively small compared with Eve’s.This explains why the generalized scheme is especially beneficial under weaker legitimate-channel conditions.
- Algorithm convergence: The iterative algorithm’s power allocations converge to those obtained by exhaustive search.The simulations use the stated iteration settings MAXIT=20 and MAXCheck=5.
- Algorithm convergence: The algorithm reaches the final value in at most 7 iterations, confirming lower power-allocation complexity than full search.The convergence example varies PT with ||h||2=0.1.
- Power allocation: As received SNR increases, power allocated to PV1 decreases and the rate gain over Goel and Negi’s scheme also decreases.The allocation example uses ||h||2=0.05.
VI. CONCLUSION
The paper generalizes artificial-noise-assisted secure beamforming by allowing AN in all directions, including the message direction, and reduces ergodic secrecy-rate optimization to a simplified power-allocation problem.
- The proposed scheme generalizes Goel and Negi’s artificial-noise method for fast-fading secure transmission.
- AN can be injected in all directions, including the direction used to convey dedicated messages, rather than only the legitimate channel’s null space.
- The ergodic secrecy rate is described by a highly simplified power-allocation problem.The paper also develops an algorithm for the resulting non-convex optimization.
- The optimal message transmission is a beamformer aligned with the legitimate-channel direction.
- The paper provides a necessary condition for the optimal AN covariance matrix to be full rank.
- Simulations show that the proposed scheme outperforms Goel and Negi’s AN scheme especially when the legitimate channel is poor.
VII. APPENDIX
The appendix develops proof steps for the paper’s theoretical results, including properties of auxiliary matrices and conditions related to optimal AN structure.
- The appendix introduces lemmas used in proving Theorem 3.
- It rewrites the expectation in an integral form as part of the proof procedure.
- The appendix establishes that Y is diagonal because its non-diagonal entries are zero.Its diagonal entries are stated to be larger than zero.
- It states a necessary condition for the optimal AN to be full rank using eigenvalue conditions on C.
- The proof applies eigenvalue properties of aaH − A and the monotonicity of l(λ) for λ > 0.
- Several algebraic rearrangements and substitutions connect the theorem expressions to the final condition.
Alice Bob
The supplied passage identifies Fig. 1 as the system model.
- Fig. 1 presents the system model.
DEC w^
The evaluation examines secrecy rate, convergence, and power allocation across several legitimate-channel norms and transmit-power settings.
- The characteristic of f2(PU,PV1,PV2) is shown given PV1.
- Secrecy rate is plotted against transmit power for ||h||2 = 0.05.
- Secrecy rate is plotted against transmit power for ||h||2 = 0.1.
- Secrecy rate is plotted against transmit power for ||h||2 = 0.2.
- Secrecy rate is plotted against the number of iterations for ||h||2 = 0.1.
- Power allocation among PU, PV1, and PV2 is plotted for ||h||2 = 0.05.