In this paper, we consider a general $K$ -user Gaussian multiple-input multiple-output (MIMO) broadcast channel (BC). We assume that the channel state is deterministic and known to all the nodes. While the private-message capacity region is well known to be achievable with dirty paper coding (DPC), we are interested in the simpler linearly precoded transmission schemes. In particular, we focus on linear precoding schemes combined with rate-splitting (RS). First, we derive an achievable rate region with minimum mean square error (MMSE) precoding at the transmitter and joint decoding of the sub-messages at the receivers. Then, we study the achievable sum rate of this scheme and obtain two findings: 1) an analytically tractable upper bound on the sum rate that is shown numerically to be a close approximation, and 2) how to reduce the number of active streams – crucial to the overall complexity – while preserving the sum rate to within a constant loss. The latter results in two practical algorithms: a stream elimination algorithm and a stream ordering algorithm. Finally, we investigate the constant-gap optimality of linearly precoded RS with respect to the capacity. Our result reveals that, while the achievable rate of linear precoding alone can be arbitrarily far from the capacity, the introduction of RS can help achieve the capacity region to within a constant gap in the two-user case. Nevertheless, we prove that the RS scheme’s constant-gap optimality does not extend to the three-user case. Specifically, we show, through a pathological example, that the gap between the sum rate and the sum capacity can be unbounded.

中文翻译：

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高斯MIMO广播信道的线性预编码速率分裂

在本文中，我们考虑一个一般的 $K$ - 用户高斯多输入多输出 (MIMO) 广播信道 (BC)。我们假设信道状态是确定性的并且所有节点都知道。虽然众所周知可以使用脏纸编码 (DPC) 实现私人消息容量区域，但我们对更简单的线性预编码传输方案感兴趣。特别地，我们专注于与速率分割 (RS) 相结合的线性预编码方案。首先，我们推导出一个可实现的速率区域，在发送端进行最小均方误差 (MMSE) 预编码，在接收端对子消息进行联合解码。然后，我们研究该方案可实现的总和率并获得两个发现：1）总和率的分析上易于处理的上限，其数值显示为近似值，和 2）如何减少活动流的数量——对整体复杂性至关重要——同时将总速率保持在一个恒定的损失范围内。后者导致两种实用算法：流消除算法和流排序算法。最后，我们研究了线性预编码 RS 相对于容量的恒定间隙最优性。我们的结果表明，虽然单独线性预编码的可实现速率可以与容量相差任意远，但 RS 的引入可以帮助在两个用户的情况下将容量区域保持在恒定的差距内。尽管如此，我们证明了 RS 方案的恒定间隙最优性并没有扩展到三用户的情况。具体来说，我们通过一个病理学例子表明总速率和总容量之间的差距可以是无限的。