Vector channel capacity with quantized feedback

We explore the capacity of an isotropic fading vector channel with multiple transmit antennas at the base station and a single antenna at the mobile receiver. Perfect channel knowledge is assumed to he available at the receiver while the transmitter has only partial knowledge of the direction of the user´s channel vector based on quantized feedback. We determine a necessary and sufficient condition for optimality of beamforming, which turns out to be a precise relationship between the optimality of beamforming and the symmetry of the quantizer. For the cases where beamforming is not optimal, we develop upper and lower bounds. Numerical results are provided to help estimate the capacity.