Optimal Transmission with Imperfect Channel State Information at the Transmit Antenna Array

We study the optimal transmission strategy of a multiple-inputsingle-output wireless communication link. The receiver has perfectchannel state information while the transmitter hasonly long-term channel state information in terms of the channelcovariance matrix. It was recently shown that the optimal eigenvectors of the transmitcovariance matrix correspond with the eigenvalues of the channelcovariance matrix. However, the optimal eigenvalues are difficult tocompute. We study the properties of these optimal capacity achieving eigenvalues, and present a necessary and sufficient condition for theoptimal eigenvalues of the transmit covariance matrix. Furthermore, we develop a necessary and sufficient condition forachieving capacity when transmitting in all directions. We compare thecapacity gain of an optimal diversity system with a system which works with beamforming, and we derive an upperbound. We answer the main questions regarding the system design using the developed results. Additionally, we show inwhich way the multiplexing gain can be computed in case the channel covariancematrix is given. We compute the maximum number of required paralleldata streams, and we define a multiplexing function inorder to obtain a measure for the available multiplexinggain. Furthermore, we show that the capacity gain is small considering theadditional complexity at the receiver. We illustrate allresults by numerical simulations.