Capacity of the isotropic fading vector channel with quantized channel direction feedback

We explore optimality of beamforming for a vector channel with multiple transmits antennas at the base station and a single antenna at the mobile receiver. Perfect channel knowledge is assumed to be available at the receiver while the transmitter has only partial knowledge of the direction of the user´s channel vector based on quantized direction feedback. We introduce necessary and sufficient conditions for the optimality of beamforming for the quantized direction feedback communication system. Using the conditions, we prove the optimality of beamforming for some particular cases of quantized direction feedback.