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Beamforming for MIMO Gaussian wiretap channels with imperfect channel state information

Chenxi Liu, Giovanni Geraci, Nan Yang, Jinhong Yuan, Robert Malaney

Published Dec 1, 2013• 9 citations
Publisher

Abstract

In this paper, we propose a new beamforming scheme for multi-input multi-output (MIMO) Gaussian wiretap channels where the channel state information (CSI) from the eavesdropper is imperfectly known to the transmitter. A stochastic model is constructed to characterize the imperfect CSI of the eavesdropper, in which a factor 0 ≤ τ ≤ 1 is introduced to describe the degree of the available eavesdropper's channel knowledge at the transmitter. When τ varies from 0 to 1, the eavesdropper's channel knowledge available at the transmitter ranges from statistically known to perfectly known. We design the proposed beamforming scheme by maximizing a lower bound on the achievable secrecy rate. We first demonstrate that our scheme achieves higher secrecy rate than the existing eigenvalue decomposition-based beamforming scheme which is optimal for τ = 0. We then demonstrate that the proposed scheme achieves higher secrecy rate than the existing generalized eigenvalue decomposition-based beamforming scheme which is optimal for τ = 1. Furthermore, we derive tight approximations for the proposed beamforming scheme in the high signal-to-noise ratio (SNR) regime and the low SNR regime. The accuracy of these approximations is validated via numerical results. Finally, we demonstrate that our proposed scheme achieves almost the same secrecy performance as the optimal beamforming solution that is obtained through numerical search.

DOI
10.1109/glocom.2013.6831573

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In this paper, we propose a new beamforming scheme for multi-input multi-output (MIMO) Gaussian wiretap channels where the channel state information (CSI) from the eavesdropper is imperfectly known to the transmitter. A stochastic model is constructed to characterize the imperfect CSI of the eavesdropper, in which a factor 0 ≤ τ ≤ 1 is introduced to describe the degree of the available eavesdropper's channel knowledge at the transmitter. When τ varies from 0 to 1, the eavesdropper's channel knowledge available at the transmitter ranges from statistically known to perfectly known. We design the proposed beamforming scheme by maximizing a lower bound on the achievable secrecy rate. We first demonstrate that our scheme achieves higher secrecy rate than the existing eigenvalue decomposition-based beamforming scheme which is optimal for τ = 0. We then demonstrate that the proposed scheme achieves higher secrecy rate than the existing generalized eigenvalue decomposition-based beamforming scheme which is optimal for τ = 1. Furthermore, we derive tight approximations for the proposed beamforming scheme in the high signal-to-noise ratio (SNR) regime and the low SNR regime. The accuracy of these approximations is validated via numerical results. Finally, we demonstrate that our proposed scheme achieves almost the same secrecy performance as the optimal beamforming solution that is obtained through numerical search.
We propose two beamforming schemes supporting multi-stream transmission in multi-input multi-output multi-antenna eavesdropper wiretap channels with imperfect channel state information of the eavesdropper. We first propose a generalized eigenvalue decomposition (GEVD)-based beamforming scheme by designing the beamforming matrix and determining the power allocation matrix. In particular, we determine a general power allocation matrix for arbitrary signal-to-noise ratio (SNR) and a simplified power allocation matrix for high SNR. We demonstrate that our GEVD-based beamforming scheme delivers a higher achievable secrecy rate than the existing beamforming schemes in the medium and high SNR regime. We also demonstrate that the simplified power allocation matrix delivers the same achievable secrecy rate as the general power allocation matrix at high SNRs. We then propose an easy-to-construct EVD-based beamforming scheme which reduces signal processing cost and eliminates power allocation. We demonstrate that our EVD-based beamforming scheme delivers a higher secrecy rate than the GEVD-based beamforming scheme and the existing beamforming schemes in the low SNR regime.
In this paper, we investigate methods for reducing the likelihood that a message transmitted between two multi-antenna nodes is intercepted by an undetected eavesdropper. In particular, we focus on the judicious transmission of artificial interference to mask the desired signal at the time it is broadcast. Unlike previous work that assumes some prior knowledge of the eavesdropper's channel and focuses on maximizing secrecy capacity, we consider the case where no information regarding the eavesdropper is available, and we use signal-to-interference-plus-noise-ratio (SINR) as our performance metric. Specifically, we focus on the problem of maximizing the amount of power available to broadcast a jamming signal intended to hide the desired signal from a potential eavesdropper, while maintaining a prespecified SINR at the desired receiver. The jamming signal is designed to be orthogonal to the information signal when it reaches the desired receiver, assuming both the receiver and the eavesdropper employ optimal beamformers and possess exact channel state information (CSI). In practice, the assumption of perfect CSI at the transmitter is often difficult to justify. Therefore, we also study the resulting performance degradation due to the presence of imperfect CSI, and we present robust beamforming schemes that recover a large fraction of the performance in the perfect CSI case. Numerical simulations verify our analytical performance predictions, and illustrate the benefit of the robust beamforming schemes.
This paper studies the achievable secrecy performance of multiple-input multiple-output wiretap channels in the presence of imperfect channel state information (CSI) with practical low-complexity transmission schemes. In particular, we propose a general order transmit antenna selection and power-efficient output-threshold maximal ratio combining scheme. Two separate cases depending on the availability of the eavesdropper's CSI at the transmitter are considered. New closed-form expressions of the secrecy outage probability and the average secrecy rate are obtained. In addition, the secrecy diversity order and array gains, high signal-to-noise ratio slope, and power offset are characterized through asymptotic analysis, which enables the characterization of the effect of imprecise transmit antenna selection, output-threshold, and imperfect CSI on the secrecy performance. Numerical results are presented to validate the main outcomes of this paper.
This paper considers a multiple-input multiple-output (MIMO) Gaussian wiretap channel model, where there exists a transmitter, a legitimate receiver and an eavesdropper, each equipped with multiple antennas. Perfect secrecy is achieved when the transmitter and the legitimate receiver can communicate at some positive rate, while ensuring that the eavesdropper gets zero bits of information. In this paper, the perfect secrecy capacity of the multiple antenna MIMO wiretap channel is found for aribtrary numbers of antennas under the assumption that the transmitter performs beamforming based on the generalized singular value decomposition (GSVD). More precisely, the optimal allocation of power for the GSVD-based precoder that achieves the secrecy capacity is derived. This solution is shown to have several advantages over prior work that considered secrecy capacity for the general MIMO Gaussian wiretap channel under a high SNR assumption. Numerical results are presented to illustrate the proposed theoretical findings.
This paper considers a multiple-input multiple-output (MIMO) Gaussian wiretap channel model, where there exists a transmitter, a legitimate receiver and an eavesdropper, each equipped with multiple antennas. Perfect secrecy is achieved when the transmitter and the legitimate receiver can communicate at some positive rate, while ensuring that the eavesdropper gets zero bits of information. In this paper, the perfect secrecy rate of the multiple antenna MIMO wiretap channel is maximized for arbitrary numbers of antennas under the assumption that the transmitter performs beamforming based on the generalized singular value decomposition (GSVD). More precisely, the optimal allocation of power for the GSVD-based precoder that maximizes the achievable secrecy rate is derived. Numerical results are presented to illustrate that the achievable secrecy rate of the GSVD-based precoding approach is nearly identical to that of the optimal scheme.
In this paper, a multiple-input multiple-output wiretap channel model is studied, where the information packet sent from the transmitter to the desired receiver is wiretapped by a potential eavesdropper. When ideal channel state information (CSI) is available, we design the secure beamforming using a proposed iterative closed-form updating scheme based on majorization-minimization approach. Numerical results demonstrate that the proposed scheme acquires the same secrecy rate as the existing algorithms whereas it has much lower complexity. When ideal CSI is not available and channel errors are modeled by deterministically bounded model, we investigate robust secure beamforming (RSB) optimization problem whose goal is worst-case secrecy rate maximization. In the formulated optimization problem, the objective function includes the expression of a logarithm of a determinant (log det) minus log det. Instead of approximating the log det expression as a trace, we propose to linearize two log det terms. After linearization, epigraph reformulation is used to deal with deterministically bounded channel errors and semi-infinite constraints are recast as linear matrix inequalities. Finally, an alternating optimization algorithm is utilized to find the RSB design. After obtaining the RSB design, we derive the worst-case secrecy rate. Numerical results illustrate that the proposed RSB design is superior to the nonrobust one.