Generalized water-filling for multiple transmit antenna systems

It is well known that the optimal coding strategy, maximizing the mutual information under an average transmit power constraint and additive Gaussian noise, for single-input/single-output (SISO) transmission over a time-invariant, dispersive channel is water-filling. The extension to a multiple-input/single output (MISO) channel can also be derived in a straightforward manner, using a numerical approach. The aim of this work is to provide a closed form expression for the optimal coding and power/bit allocation for a MISO channel, which has a direct interesting physical interpretation and it establishes a direct link between water-filling, beamforming and maximal ratio combining. We test then our theoretical findings with numerical results.