On Ergodic Secrecy Rate for Gaussian MISO Wiretap Channels

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 maximizes the ergodic secrecy rate 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 in which the 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, it is shown that when full information on the legitimate channel is available to the transmitter, the optimal input covariance has always rank one. It is also shown 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.

I. Introduction

WIRELESS physical (PHY) layer based security approaches [1] exploit the physical characteristics of the wireless channel to enhance the security of communication systems. The single user memoryless wiretap channel, first introduced and studied by Wyner [2], is the most basic physical layer model that captures the problem of communication security. Wyner identified the secrecy capacity of that channel and showed that when an eavesdropper's channel is a degraded version of the legitimate channel, the source and destination can achieve a positive information rate. The Gaussian wiretap channel, in which the outputs at the legitimate receiver and at the eavesdropper are corrupted by additive white Gaussian noise (AWGN) was studied in [3] as an extension of the work of [2]. In [3] it was shown that the secrecy capacity is the difference between the capacities of the main and wiretap channels. The Gaussian MIMO wiretap channel arising in multi-antenna scenarios has been also studied. The 2–2-1 Gaussian MIMO wiretap channel, consisting of a transmitter and a receiver with two antennas each and an eavesdropper with a single antenna, was studied in [4] and its secrecy capacity under a power constraint was computed. Secrecy capacity results for general MIMO wiretap channels under power constraints were proposed in [5], [6], [7] and [8], while secrecy capacity results under a power-covariance constraint were proposed in [9], [10]. In all aforementioned MIMO results, the transmitter has full channel state information (CSI) about both the legitimate channel and the eavesdropper channel. However, there have been results from the ergodic secrecy perspective that require full CSI on only the legitimate receiver channel [11], [12], [13] and [37]. The ergodic secrecy capacity of the slow fading scalar channel was computed in [11] and an achievable rate for the fast-fading scalar case was given in [12]. In [13], the ergodic rate in Gaussian MISO channels was obtained, assuming that only statistical information about the eavesdropper channel was available at the transmitter. In the same work, it was shown that when the eavesdropper channel is a vector of independent and identically distributed (i.i.d.) zero-mean complex circularly symmetric Gaussian random variables, i.e., when the channel has a trivial covariance matrix, the optimal communication strategy is beamforming and the beamforming direction depends on the CSI of the legitimate channel. The case of a MIMO wiretap channel in which the channel to the intended receiver is perfectly known and the channel to the eavesdropper is known by its mean, including artificial noise, was studied in [37].