Optimal and Successive Approaches to Signal Design for Multiple Antenna Physical Layer Multicasting

In modern wireless communications, systems that send a common information stream (called multicasting systems) are widely needed for distributing content such as television or radio. In this paper, the multiple antenna physical layer multicasting channel is considered, in which a common message is simultaneously transmitted to multiple single-antenna users. To achieve the capacity of this multicasting set-up, the covariance matrix of the transmit signal vector needs to be determined to maximize the smallest maximum achievable rate among all the users. In this paper, we develop a simple successive algorithm to determine the covariance matrix of the signal vector under the assumption that channel state information (CSI) is perfectly available at the transmitter. We first characterize properties of the capacity achieving covariance matrix and derive a closed-form expression for the matrix in the two-user case. We then derive a closed-form expression for the rate maximizing beamformer in the case of two users and propose a successive beamforming algorithm that generates the beamforming vector with low computational complexity. In the proposed precoding design scheme, the covariance matrix is successively constructed by orthogonalizing the subspace spanned by each user's channel vector until the maximum number of recursions (which is the same as the minimum of the number of transmit antennas and the number of users) is reached. The achievable rate of the proposed scheme is compared with the capacity of optimal transmission, transmit beamforming, antenna subset selection, and open-loop transmission which does not require CSI at the transmitter. As the number of users and/or antennas grows large, it is shown that the average achievable rate of the proposed scheme achieves the same average achievable rate scaling as the true capacity while reducing the computational complexity.