The Structure of Least-Favorable Noise in Gaussian Vector Broadcast Channels

The sum capacity of the Gaussian vector broadcast channel is the saddle point of a Gaussian mutual information game in which the transmitter maximizes the mutual information by choosing the best transmit covariance matrix subject to a power constraint, and the receiver minimizes the mutual information by choosing a least-favorable noise covariance matrix subject to a diagonal constraint. This result has been established using a decision-feedback equalization approach under the assumption that the least-favorable noise covariance matrix is non-singular. This paper generalizes the above result to the case where the least-favorable noise is singular. In particular, it is shown that the least-favorable noise is not unique, and different least-favorable noise covariance matrices are related to each other by a linear estimation relation.