Optimal transmission with imperfect channel state information at the transmit antenna array

The optimal transmission strategy of a multiple-input single-output wireless communication link is studied in which the receiver has perfect channel state information while the transmitter has only long-term channel state information in terms of the channel covariance matrix. It was recently shown that the optimal eigenvectors of the transmit covariance matrix correspond to the eigenvalues of the channel covariance matrix. But the optimal eigenvalues are difficult to compute. We study the properties of these optimal capacity achieving eigenvalues. A necessary and sufficient condition for the optimal eigenvalues of the transmit covariance matrix is presented. Further a necessary and sufficient condition for achieving capacity when transmitting in all directions is developed. The capacity gain compared to a system working with pure beamforming is studied and an upper bound is derived. Main questions regarding the system design are answered using the developed results. It is shown how the optimal number of transmit antennas can be computed to achieve full spatial diversity given the channel covariance matrix. The maximum number of required parallel data streams is computed and a transmit diversity function is defined in order to obtain a measure for the available spatial iversity. It is further shown, that the capacity gain is small compared to the additional complexity at the receiver. All results are verified by umerical simulations