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IEEE Wireless Communications Letters

Optimal Full-Rank Signaling Over MIMO Wiretap Channels Under Interference Constraint

Sergey L. Loyka, Limeng Dong

Published Jan 15, 2018• 7 citations
Publisher

Abstract

A closed-form full-rank solution for an optimal transmit covariance matrix over a strictly degraded MIMO wiretap channel and its secrecy capacity are obtained under an interference constraint, in addition to the transmit power constraint. A number of its properties are pointed out. The regimes when one of the constraints is inactive are studied. The interference constraint affects significantly the optimal covariance and induces its new properties, which cannot be found under the total transmit power constraint only.

DOI
10.1109/lwc.2018.2793264

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A closed-form full-rank solution for an optimal transmit covariance matrix over a strictly degraded MIMO wiretap channel and its secrecy capacity are obtained under an interference constraint, in addition to the transmit power constraint. A number of its properties are pointed out. The regimes when one of the constraints is inactive are studied. The interference constraint affects significantly the optimal covariance and induces its new properties, which cannot be found under the total transmit power constraint only.
Optimal signaling over the Gaussian multiple-input multiple-output wire-tap channel is studied under the total transmit power constraint. A closed-form solution for an optimal transmit covariance matrix is obtained when the channel is strictly degraded. In combination with the rank-1 solution, this provides the complete characterization of the optimal covariance for the case of two transmit antennas. The cases of weak eavesdropper and high SNR are considered. It is shown that the optimal covariance does not converge to a scaled identity in the high-SNR regime. Necessary optimality conditions and a tight upper bound on the rank of an optimal covariance matrix are established for the general case, along with a lower bound to the secrecy capacity, which is tight in a number of scenarios.
Secure signaling over multiple-input multiple-output (MIMO) wiretap channel (WTC) is studied under interference and transmit power constraints. The classical MIMO WTC model is extended to interference-limited scenarios, so that interference to other users does not exceed a given threshold while ensuring simultaneously no information leakage to an eavesdropper. The operational secrecy capacity of the Gaussian MIMO WTC under interference and transmit power constraints is rigorously established in two forms (a non-convex max problem and a convex-concave max-min problem), to which per-antenna power constraints can be added as well. Optimal signaling directions are characterized in the general case, from which (tight) upper bounds to the rank of optimal transmit covariance matrix are derived. A sufficient condition for the optimality of beamforming and a necessary condition for optimal full-rank signaling are given. Closed-form rank-1 and high-rank solutions are obtained in the case of zero interference constraints. Sufficient and necessary conditions for non-zero secrecy capacity are established. The results are extended to multi-user scenarios. A sufficient and necessary condition for the unbounded growth of the secrecy capacity with transmit power is obtained. The interplay between transmit and interference power constraints is studied, and its significant impact on optimal signaling is demonstrated (so that neither constraint can be absorbed into the other one in general, as was sometimes suggested in the literature). Overall, these results provide insights into fundamental information-theoretic limits and optimal signaling strategies for secure communications under interference constraints.
This paper considers a multiple-input multiple-output (MIMO) Gaussian wiretap channel with a transmitter, a legitimate receiver and an eavesdropper, each equipped with multiple antennas. We first study the rank of the optimal input covariance matrix that achieves the secrecy capacity of the MIMO Gaussian wiretap channel under an average power constraint. The rank and other properties of the optimal solution are derived based on certain relationships between the channel matrices for the legitimate receiver and eavesdropper. Next, by obtaining necessary and sufficient conditions on the MIMO wiretap channel parameters, we determine the conditions under which the optimal input covariance matrix is full-rank or rank-deficient. For the case that the optimal input covariance is full-rank, we fully characterize the solution. When the optimal input covariance is rank-deficient, we show that the given MIMO wiretap channel can be modeled by an equivalent wiretap channel whose optimal input covariance is full rank and achieves the same secrecy capacity as the original system. Numerical results are presented to illustrate the proposed theoretical findings.
Multi-user Gaussian MIMO wiretap channel is considered under interference power constraints (IPC), in addition to the total transmit power constraint (TPC). Algorithms for global maximization of its secrecy rate are proposed. Their convergence to the secrecy capacity is rigorously proved and a number of properties are established analytically. Unlike known algorithms, the proposed ones are not limited to the MISO case and are proved to converge to a global rather than local optimum in the general MIMO case, even when the channel is not degraded. In practice, the convergence is fast as only a small to moderate number of Newton steps is required to achieve a high precision level. The interplay of TPC and IPC is shown to result in an unusual property when an optimal point of the max-min problem does not provide an optimal transmit covariance matrix in some (singular) cases. To address this issue, an algorithm is developed to compute an optimal transmit covariance matrix in those singular cases. It is shown that this algorithm also solves the dual (nonconvex) problems of globally minimizing the total transmit power subject to the secrecy and interference constraints; it provides the minimum transmit power and respective signaling strategy needed to achieve the secrecy capacity, hence allowing power savings.
Optimal signalling over the wire-tap MIMO Gaussian channel is studied under the total transmit power constraint. A direct proof of the necessary condition of optimality (signaling on the positive directions of the difference channel) is given using the necessary KKT conditions. Based on it, an explicit, closed-form solution for the optimal transmit covariance matrix is given when the latter is of the full rank. The cases of weak eavesdropper and high SNR are considered. It is shown that the optimal covariance does not converge to a scaled identity in the latter regime. A refined estimate of the rank of an optimal covariance matrix is given for the general case.
Capacity-achieving signaling strategies for the Gaussian wiretap multiple-input multple-output (MIMO) channel are investigated without the degradedness assumption. In addition to known solutions, a number of new rank-deficient solutions for the optimal transmit covariance matrix are obtained. The case of a weak eavesdropper is considered in detail, and the optimal covariance is established in an explicit, closed form with no extra assumptions. This provides lower and upper bounds to the secrecy capacity in the general case with a bounded gap, which are tight for a weak eavesdropper or/and low SNR. Closed-form solutions are also obtained for isotropic and omnidirectional eavesdroppers, based on which lower and upper bounds to the secrecy capacity are established in the general case. Sufficient and necessary conditions for the optimality of three popular transmission techniques, namely, the zero-forcing (ZF), the standard water-filling over the channel eigenmodes, and the isotropic signaling (IS), are established for the MIMO wiretap channel. These solutions are appealing due to their lower complexity. In particular, no wiretap codes are needed for the ZF transmission, and no precoding or feedback is needed for the isotropic signaling.