  Given a multiple-input multiple-output (MIMO) channel, feedback from the
receiver can be used to specify a transmit precoding matrix, which selectively
activates the strongest channel modes. Here we analyze the performance of
Random Vector Quantization (RVQ), in which the precoding matrix is selected
from a random codebook containing independent, isotropically distributed
entries. We assume that channel elements are i.i.d. and known to the receiver,
which relays the optimal (rate-maximizing) precoder codebook index to the
transmitter using B bits. We first derive the large system capacity of
beamforming (rank-one precoding matrix) as a function of B, where large system
refers to the limit as B and the number of transmit and receive antennas all go
to infinity with fixed ratios. With beamforming RVQ is asymptotically optimal,
i.e., no other quantization scheme can achieve a larger asymptotic rate. The
performance of RVQ is also compared with that of a simpler reduced-rank scalar
quantization scheme in which the beamformer is constrained to lie in a random
subspace. We subsequently consider a precoding matrix with arbitrary rank, and
approximate the asymptotic RVQ performance with optimal and linear receivers
(matched filter and Minimum Mean Squared Error (MMSE)). Numerical examples show
that these approximations accurately predict the performance of finite-size
systems of interest. Given a target spectral efficiency, numerical examples
show that the amount of feedback required by the linear MMSE receiver is only
slightly more than that required by the optimal receiver, whereas the matched
filter can require significantly more feedback.
