  It is well known that Multiple-Input Multiple-Output (MIMO) systems have high
spectral efficiency, especially when channel state information at the
transmitter (CSIT) is available. When CSIT is obtained by feedback, it is
practical to assume that the channel state feedback rate is finite and the CSIT
is not perfect. For such a system, we consider beamforming and power on/off
strategy for its simplicity and near optimality, where power on/off means that
a beamforming vector (beam) is either turned on with a constant power or turned
off. The main contribution of this paper is to accurately evaluate the
information rate as a function of the channel state feedback rate. Name a beam
turned on as an on-beam and the minimum number of the transmit and receive
antennas as the dimension of a MIMO system. We prove that the ratio of the
optimal number of on-beams and the system dimension converges to a constant for
a given signal-to-noise ratio (SNR) when the numbers of transmit and receive
antennas approach infinity simultaneously and when beamforming is perfect.
Asymptotic formulas are derived to evaluate this ratio and the corresponding
information rate per dimension. The asymptotic results can be accurately
applied to finite dimensional systems and suggest a power on/off strategy with
a constant number of on-beams. For this suboptimal strategy, we take a novel
approach to introduce power efficiency factor, which is a function of the
feedback rate, to quantify the effect of imperfect beamforming. By combining
power efficiency factor and the asymptotic formulas for perfect beamforming
case, the information rate of the power on/off strategy with a constant number
of on-beams is accurately characterized.
