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International Symposium on Wireless Personal Multimedia Communications (WPMC)

Optimality of beamforming and closed form secrecy capacity of MIMO wiretap channels with two transmit antennas

Jiangyuan Li, Athina P. Petropulu

Published Jun 24, 2013• 2 citations
OpenAlex

Abstract

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.

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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 a multiple-input multiple-output (MIMO) Gaussian wiretap channel model, where there exists a transmitter, a legitimate receiver and an eavesdropper, each node equipped with multiple antennas. We study the problem of finding the optimal input covariance that achieves secrecy capacity subject to a power constraint, which in general leads to a difficult non-convex optimization problem. For the Gaussian multiple-input single-output (MISO) wiretap channel where there exists a transmitter equipped with multiple antennas, a legitimate receiver and an eavesdropper each equipped with one antenna, the optimal input covariance is obtained in closed form. For the general case, we derive the necessary conditions for the optimal input covariance in the form of a set of equations. We show that for MISO case, the derived conditions guarantee the optimal input covariance. Numerical results are presented to illustrate the proposed theoretical findings.
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 optimality of beamforming for secrecy capacity subject to sum power constraint is studied, and two sufficient conditions for beamforming to be globally optimal are given. The first sufficient condition states that when the difference between the Gram matrices of legitimate and eavesdropper channel matrices has exactly one positive eigenvalue, then beamforming is globally optimal. An alternative sufficient condition, which involves convex optimization, is also provided. For the case in which beamforming is globally optimal, the secrecy capacity is obtained. Otherwise, an upper bound of the difference between the secrecy capacity and beamforming secrecy rate is provided.
We consider a Gaussian multiple-input multiple-output (MIMO) wiretap channel model, where there exists a transmitter, a legitimate receiver and an eavesdropper, each node equipped with multiple antennas. We study the problem of finding the optimal input covariance matrix that achieves secrecy capacity subject to a power constraint, which leads to a non-convex optimization problem that is in general difficult to solve. Existing results for this problem address the case in which the transmitter and the legitimate receiver have two antennas each and the eavesdropper has one antenna. For the general cases, it has been shown that the optimal input covariance matrix has low rank when the difference between the Grams of the eavesdropper and the legitimate receiver channel matrices is indefinite or semi-definite, while it may have low rank or full rank when the difference is positive definite. In this paper, the aforementioned non-convex optimization problem is investigated. In particular, for the multiple-input single-output (MISO) wiretap channel, the optimal input covariance matrix is obtained in closed form. For general cases, we derive the necessary conditions for the optimal input covariance matrix consisting of a set of equations. For the case in which the transmitter has two antennas, the derived necessary conditions can result in a closed form solution; For the case in which the difference between the Grams is indefinite and has all negative eigenvalues except one positive eigenvalue, the optimal input covariance matrix has rank one and can be obtained in closed form; For other cases, the solution is proved to be a fixed point of a mapping from a convex set to itself and an iterative procedure is provided to search for it. Numerical results are presented to illustrate the proposed theoretical findings.
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.