Minimax Problems and Directional Derivatives for MIMO Channels

When transmitting over multiple-input-multiple-output (MIMO) channels, in the case of total power constraints and complete channel state information (CSI) the optimum power distribution is obtained by water-filling the squared singular values of the channel matrix H. In this paper, we consider the case that nature behaves as an opponent to the optimum transmit strategy in choosing the channel as bad as possible. Interpreting the mutual information as payoff function for two players, the transmitter and a malicious nature, this approach may be seen as a two person zero sum game. We first analyze maximum points of the payoff function for a fixed channel matrix under general power restrictions and characterize such points via directional derivatives. Worst channel behavior must be separated from the zero channel where no transmission is possible at all. Loewner semi-ordering of nonnegative Hermitian matrices is employed to ensure minimum channel quality. It is shown that a Nash equilibrium exists for general power constraints. Concrete results are achieved for a limited total power budget and limiting the maximum available power for each subchannel