  In this paper, we consider the problem of power allocation in MIMO wiretap
channel for secrecy in the presence of multiple eavesdroppers. Perfect
knowledge of the destination channel state information (CSI) and only the
statistical knowledge of the eavesdroppers CSI are assumed. We first consider
the MIMO wiretap channel with Gaussian input. Using Jensen's inequality, we
transform the secrecy rate max-min optimization problem to a single
maximization problem. We use generalized singular value decomposition and
transform the problem to a concave maximization problem which maximizes the sum
secrecy rate of scalar wiretap channels subject to linear constraints on the
transmit covariance matrix. We then consider the MIMO wiretap channel with
finite-alphabet input. We show that the transmit covariance matrix obtained for
the case of Gaussian input, when used in the MIMO wiretap channel with
finite-alphabet input, can lead to zero secrecy rate at high transmit powers.
We then propose a power allocation scheme with an additional power constraint
which alleviates this secrecy rate loss problem, and gives non-zero secrecy
rates at high transmit powers.
