Transmitter optimization and optimality of beamforming for multiple antenna systems

We solve the transmitter optimization problem and determine a necessary and sufficient condition under which beamforming achieves Shannon capacity in a linear narrowband point-to-point communication system employing multiple transmit and receive antennas with additive Gaussian noise. We assume that the receiver has perfect channel knowledge while the transmitter has only knowledge of either the mean or the covariance of the channel coefficients. The channel is modeled at the transmitter as a matrix of complex jointly Gaussian random variables with either a zero mean and a known covariance matrix (covariance information), or a nonzero mean and a white covariance matrix (mean information). For both cases, we develop a necessary and sufficient condition for when the Shannon capacity is achieved through beamforming; i.e., the channel can be treated like a scalar channel and one-dimensional codes can be used to achieve capacity. We also provide a waterpouring interpretation of our results and find that less channel uncertainty not only increases the system capacity but may also allow this higher capacity to be achieved with scalar codes which involves significantly less complexity in practice than vector coding.