Statistical Beamforming on the Grassmann Manifold for the Two-User Broadcast Channel

• Published in: IEEE transactions on information theory Vol. 59; no. 10; pp. 6464 - 6489
• Main Authors: Raghavan, Vasanthan, Hanly, Stephen V., Veeravalli, Venugopal V.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.10.2013; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Adaptive signalling; Antennas; Applied sciences; Array signal processing; broadcast channel; Broadcasting. Videocommunications. Audiovisual; Covariance matrices; Detection, estimation, filtering, equalization, prediction; Eigenvalues; Exact sciences and technology; information rates; Information theory; Information, signal and communications theory; Interference; Lagrange multiplier; Manifolds; Matrix; Miscellaneous; multiple-input single-output (MISO) systems; multiuser multiple-input multiple-output (MIMO); nonconvex optimization; precoding; Signal and communications theory; Signal, noise; spatial correlation; Systems, networks and services of telecommunications; Telecommunications; Telecommunications and information theory; Transmission and modulation (techniques and equipments); Transmitters; Transmitting antennas; Vectors; Spatial correlation; broadcast channel; Information rate; Transmitter; Adaptive signalling; nonconvex optimization; precoding; Beam forming; Lagrange multiplier; information rates; Covariance; multiuser multiple-input multiple-output (MIMO); Broadcast channels; MISO system; Multiple access; Eigenvector; Eigenmode; Two channel system; Ergodicity; Covariance matrix; Signalling; Information transmission; multiple-input single-output (MISO) systems; Signal interference; Transmitting antenna; Optimal solution; S matrix; Metric; Rayleigh fading
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2013.2267721

A Rayleigh fading spatially correlated broadcast setting with M = 2 antennas at the transmitter and two users (each with a single antenna) is considered. It is assumed that the users have perfect channel information about their links, whereas the transmitter has only statistical information of each user's link (covariance matrix of the vector channel). A low-complexity linear beamforming strategy that allocates equal power and one spatial eigenmode to each user is employed at the transmitter. Beamforming vectors on the Grassmann manifold that depend only on statistical information are to be designed at the transmitter to maximize the ergodic sum-rate delivered to the two users. Toward this goal, the beamforming vectors are first fixed and a closed-form expression is obtained for the ergodic sum-rate in terms of the covariance matrices of the links. This expression is nonconvex in the beamforming vectors ensuring that the classical Lagrange multiplier technique is not applicable. Despite this difficulty, the optimal solution to this problem is shown to be the same as the solution to the maximization of an appropriately defined average signal-to-interference and noise ratio metric for each user. This solution is the dominant generalized eigenvector of a pair of positive-definite matrices where the first matrix is the covariance matrix of the forward link and the second is an appropriately designed "effective" interference covariance matrix. In this sense, our work is a generalization of optimal signalling along the dominant eigenmode of the transmit covariance matrix in the single-user case. Finally, the ergodic sum-rate for the general broadcast setting with M antennas at the transmitter and M-users (each with a single antenna) is obtained in terms of the covariance matrices of the links and the beamforming vectors.