Statistical beamforming on the grassmann manifold for the two-user broadcast channel

Vasanthan Raghavan, Stephen V. Hanly, Venugopal V. Veeravalli

Research output: Contribution to journalArticleResearchpeer-review

Abstract

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.

LanguageEnglish
Article number6529193
Pages6464-6489
Number of pages26
JournalIEEE Transactions on Information Theory
Volume59
Issue number10
DOIs
Publication statusPublished - 2013

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Beamforming
Covariance matrix
broadcast
Transmitters
Antennas
Telecommunication links
interference
Lagrange multipliers
Rayleigh fading
Eigenvalues and eigenfunctions
multiplier

Cite this

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title = "Statistical beamforming on the grassmann manifold for the two-user broadcast channel",
abstract = "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.",
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Statistical beamforming on the grassmann manifold for the two-user broadcast channel. / Raghavan, Vasanthan; Hanly, Stephen V.; Veeravalli, Venugopal V.

In: IEEE Transactions on Information Theory, Vol. 59, No. 10, 6529193, 2013, p. 6464-6489.

Research output: Contribution to journalArticleResearchpeer-review

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AU - Hanly, Stephen V.

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