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首页> 外文期刊>IEEE Transactions on Information Theory >Statistical Beamforming on the Grassmann Manifold for the Two-User Broadcast Channel
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Statistical Beamforming on the Grassmann Manifold for the Two-User Broadcast Channel

机译：两用户广播频道在Grassmann流形上的统计波束成形

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摘要

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 term- of the covariance matrices of the links and the beamforming vectors.
机译：在发射机和两个用户处使用 $ M = 2 $ 天线的瑞利衰落空间相关广播设置） 被认为。假定用户具有有关其链路的完美信道信息，而发送器仅具有每个用户链路的统计信息（矢量信道的协方差矩阵）。在发射机处采用了一种低复杂度的线性波束成形策略，该策略为每个用户分配了相等的功率和一个空间本征模。仅在统计信息上的格拉斯曼流形上的波束成形矢量将在发射机处设计，以最大化传递给两个用户的遍历总速率。为了实现这个目标，首先将波束形成向量固定，并根据链路的协方差矩阵为遍历求和率获得一个封闭形式的表达式。该表达式在波束赋形矢量中是非凸的，从而确保了经典的拉格朗日乘数技术不适用。尽管存在这一困难，但针对该问题的最佳解决方案仍显示出与针对每个用户适当定义的平均信号干扰和噪声比度量最大化的解决方案相同。该解决方案是一对正定矩阵的主要广义特征向量，其中第一个矩阵是前向链路的协方差矩阵，第二个矩阵是经过适当设计的“有效”干扰协方差矩阵。从这个意义上讲，我们的工作是在单用户情况下沿着传输协方差矩阵的主导本征模的最优信令的一般化。最后，在发射机处使用 $ M $ 天线和 $ M $ 个用户（每个都有一个天线）是根据链路和波束形成矢量的协方差矩阵获得的。

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