On Ergodic Secrecy Capacity for Gaussian MISO Wiretap Channels

TL;DR

This paper investigates the ergodic secrecy capacity of Gaussian MISO wiretap channels with non-trivial channel covariances, proposing a solution for optimal input covariance matrices under statistical channel information constraints.

Contribution

It extends existing results to channels with non-trivial covariances and provides an algorithm to compute the optimal input covariance matrix.

Findings

Optimal input covariance is rank one with full legitimate channel info.

Eigenvectors of optimal covariance align with legitimate channel covariance when only statistical info is available.

Numerical results demonstrate the effectiveness of the proposed algorithm.

Abstract

A Gaussian multiple-input single-output (MISO) wiretap channel model is considered, where there exists a transmitter equipped with multiple antennas, a legitimate receiver and an eavesdropper each equipped with a single antenna. We study the problem of finding the optimal input covariance that achieves ergodic secrecy capacity subject to a power constraint where only statistical information about the eavesdropper channel is available at the transmitter. This is a non-convex optimization problem that is in general difficult to solve. Existing results address the case in which the eavesdropper or/and legitimate channels have independent and identically distributed Gaussian entries with zero-mean and unit-variance, i.e., the channels have trivial covariances. This paper addresses the general case where eavesdropper and legitimate channels have nontrivial covariances. A set of equations describing the optimal input covariance matrix are proposed along with an algorithm to obtain the solution. Based on this framework, we show that when full information on the legitimate channel is available to the transmitter, the optimal input covariance has always rank one. We also show that when only statistical information on the legitimate channel is available to the transmitter, the legitimate channel has some general non-trivial covariance, and the eavesdropper channel has trivial covariance, the optimal input covariance has the same eigenvectors as the legitimate channel covariance. Numerical results are presented to illustrate the algorithm.