Capacity of Beamforming with Limited Training and Feedback

We examine the capacity of beamforming over a multi-input/single-output block Rayleigh fading channel with finite training for channel estimation and limited feedback. A fixed-length packet is assumed, which is spanned by T training symbols, B feedback bits, and the data symbols. The training symbols are used to obtain a minimum mean squared error (MMSE) estimate of the channel vector. Given this estimate, the receiver selects a transmit beamforming vector from a codebook containing 2^B i.i.d. random vectors, and relays the corresponding B bits back to the transmitter. We derive bounds on the capacity and show that for a large number of transmit antennas N_t , the optimal T and B, which maximize the bounds, are approximately equal and both increase as N_t/logN_t. We conclude that with limited training and feedback, the optimal number of antennas to activate also increases as N_t/logN_t