Uniform power allocation in MIMO channels: a game-theoretic approach

When transmitting over multiple-input-multiple-output (MIMO) channels, there are additional degrees of freedom with respect to single-input-single-output (SISO) channels: the distribution of the available power over the transmit dimensions. If channel state information (CSI) is available, the optimum solution is well known and is based on diagonalizing the channel matrix and then distributing the power over the channel eigenmodes in a "water-filling" fashion. When CSI is not available at the transmitter, but the channel statistics are a priori known, an optimal fixed power allocation can be precomputed. This paper considers the case in which not even the channel statistics are available, obtaining a robust solution under channel uncertainty by formulating the problem within a game-theoretic framework. The payoff function of the game is the mutual information and the players are the transmitter and a malicious nature. The problem turns out to be the characterization of the capacity of a compound channel which is mathematically formulated as a maximin problem. The uniform power allocation is obtained as a robust solution (under a mild isotropy condition). The loss incurred by the uniform distribution is assessed using the duality gap concept from convex optimization theory. Interestingly, the robustness of the uniform power allocation also holds for the more general case of the multiple-access channel.