  The theory of multiple-input multiple-output (MIMO) technology has been
well-developed to increase fading channel capacity over single-input
single-output (SISO) systems. This capacity gain can often be leveraged by
utilizing channel state information at the transmitter and the receiver. Users
make use of this channel state information for transmit signal adaptation. In
this correspondence, we derive the capacity region for the MIMO multiple access
channel (MIMO MAC) when partial channel state information is available at the
transmitters, where we assume a synchronous MIMO multiuser uplink. The partial
channel state information feedback has a cardinality constraint and is fed back
from the basestation to the users using a limited rate feedback channel. Using
this feedback information, we propose a finite codebook design method to
maximize sum-rate. In this correspondence, the codebook is a set of transmit
signal covariance matrices. We also derive the capacity region and codebook
design methods in the case that the covariance matrix is rank-one (i.e.,
beamforming). This is motivated by the fact that beamforming is optimal in
certain conditions. The simulation results show that when the number of
feedback bits increases, the capacity also increases. Even with a small number
of feedback bits, the performance of the proposed system is close to an optimal
solution with the full feedback.
