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arXiv:1707.06271 · IEEE Transactions on Wireless Communications

Optimal Beamforming for Gaussian MIMO Wiretap Channels With Two Transmit Antennas

Mojtaba Vaezi, Wonjae Shin, H. Vincent Poor

Published Jul 25, 2017• 28 citations
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Abstract

A Gaussian multiple-input multiple-output wiretap channel in which the eavesdropper and legitimate receiver are equipped with arbitrary numbers of antennas and the transmitter has two antennas is studied in this paper. The input covariance matrix that achieves the secrecy capacity is determined. In particular, it is shown that the secrecy capacity of this channel can be achieved by linear precoding. Precoding and power allocation schemes that maximize the achievable secrecy rate, and thus achieve the secrecy capacity, are developed. The secrecy capacity is then compared with the achievable secrecy rate of generalized singular value decomposition (GSVD)-based precoding, which is the best previously proposed technique for this problem. Numerical results demonstrate that substantial gain can be obtained in secrecy rate between the proposed and GSVD-based precodings.

DOI
10.1109/twc.2017.2729541

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A Gaussian multiple-input multiple-output wiretap channel in which the eavesdropper and legitimate receiver are equipped with arbitrary numbers of antennas and the transmitter has two antennas is studied in this paper. The input covariance matrix that achieves the secrecy capacity is determined. In particular, it is shown that the secrecy capacity of this channel can be achieved by linear precoding. Precoding and power allocation schemes that maximize the achievable secrecy rate, and thus achieve the secrecy capacity, are developed. The secrecy capacity is then compared with the achievable secrecy rate of generalized singular value decomposition (GSVD)-based precoding, which is the best previously proposed technique for this problem. Numerical results demonstrate that substantial gain can be obtained in secrecy rate between the proposed and GSVD-based precodings.
A Gaussian multiple-input multiple-output wiretap channel in which the eavesdropper and legitimate receiver are equipped with arbitrary numbers of antennas and the transmitter has two antennas is studied in this paper. It is shown that the secrecy capacity of this channel can be achieved by linear precoding. The optimal precoding and power allocation schemes achieving the secrecy capacity are developed subsequently, and the secrecy capacity is compared with the generalized singular value decomposition (GSVD)-based precoding, which is the best previously proposed precoding for this problem. Numerical results show that substantial gain can be obtained in secrecy rate between the proposed and GSVD-based precodings.
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.
A Gaussian multiple-input multiple-output (MIMO) wiretap channel model is considered. The input is a two-antenna transmitter, while the outputs are the legitimate receiver and an eavesdropper, both equipped with multiple antennas. All channels are assumed to be known. The problem of obtaining the optimal input covariance matrix that achieves secrecy capacity subject to a sum power constraint is addressed, and a closed-form expression for the secrecy capacity is obtained. The sufficient and necessary condition for beamforming to be optimal is also given.
A Gaussian multiple-input multiple-output (MIMO) wiretap channel model is considered, where there exists a transmitter, a legitimate receiver and an eavesdropper each equipped with multiple antennas. The problem of finding the optimal input covariance matrix that achieves secrecy capacity subject to a power constraint is studied. In particular, it is shown that when the difference between the Grams of legitimate and eavesdropper channel matrices has all negative eigenvalues except one positive eigenvalue, beamforming is optimal. For that case, the secrecy capacity is obtained.
We consider the optimal precoding design under power constraint for the Gaussian MIMO wiretap channel with a transmitter, a legitimate receiver, and an eavesdropper, each equipped with arbitrary number of antennas. The transmitter has access to full channel state information (CSI) of the legitimate link and eavesdropper link. Such an optimization problem is non-convex and an analytical optimal solution is difficult to obtain. In this paper, by resorting to the majorization theory, we develop a theoretical upper bound on the secrecy capacity which is proved useful in evaluating existing sub-optimal methods. Our analysis shows that, given that the number of antennas at the eavesdropper is no less than that of the transmitter, the generalized singular value decomposition (GSVD)-based method proposed in [1], albeit generally non-optimal, provides a near-optimal solution to the MIMO wiretap precoding design in the high signal-to-noise ratio (SNR) regime. Numerical results are provided to corroborate our theoretical results.