Optimal Transmission with Imperfect Channel State Information at theTransmit Antenna Array

Abstract

We study the optimal transmission strategy of a multiple-input single-output wireless communication link. 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 with the eigenvalues of the channel covariance matrix. However, the optimal eigenvalues are difficult to compute. We study the properties of these optimal capacity achieving eigenvalues, and present a necessary and sufficient condition for the optimal eigenvalues of the transmit covariance matrix. Furthermore, we develop a necessary and sufficient condition for achieving capacity when transmitting in all directions. We compare the capacity gain of an optimal diversity system with a system which works with beamforming, and we derive an upper bound. We answer the main questions regarding the system design using the developed results. Additionally, we show in which way the multiplexing gain can be computed in case the channel covariance matrix is given. We compute the maximum number of required parallel data streams, and we define a multiplexing function in order to obtain a measure for the available multiplexing gain. Furthermore, we show that the capacity gain is small considering the additional complexity at the receiver. We illustrate all results by numerical simulations.