Rank-Deficient Solutions for Optimal Signaling over Wiretap MIMO Channels

TL;DR

This paper explores optimal signaling strategies for Gaussian wiretap MIMO channels without assuming degradedness, introducing new rank-deficient solutions and explicit formulas for weak eavesdroppers, enhancing understanding of secrecy capacity bounds.

Contribution

It provides new rank-deficient solutions for the optimal transmit covariance matrix and explicit formulas for weak eavesdroppers, improving bounds on secrecy capacity and analyzing popular transmission techniques.

Findings

New rank-deficient solutions for optimal covariance matrices.

Explicit closed-form solutions for weak eavesdropper scenarios.

Conditions for optimality of ZF, WF, and IS transmission techniques.

Abstract

Capacity-achieving signaling strategies for the Gaussian wiretap 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 optimality of 3 popular transmission techniques, namely the zero-forcing (ZF), the standard water-filling (WF) 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.