For the two-user multiple-input multiple-output (MIMO) broadcast channel with delayed channel state information at the transmitter (CSIT) and arbitrary antenna configurations, all the degrees-of-freedom (DoF) regions were obtained. However, for the three-user MIMO broadcast channel with delayed CSIT and arbitrary antenna configurations, the DoF region of order-2 messages is still unclear and only a partial achievable DoF region of order-1 messages was obtained, where the order-2 message and order-1 message are desired by two receivers and one receiver, respectively. In this paper, for the three-user MIMO broadcast channel with delayed CSIT and arbitrary antenna configurations, we design the transmission schemes for order-2 messages and order-1 messages. To analyze the achievable DoF region, we propose a novel method - the structured method, which can analyze the achievable DoF region of the transmission schemes systematically. The key idea of this method is to structure the achievable DoF region from the decoding condition of the transmission scheme via the achievable DoF tuple expression. As a result, the DoF region of order-2 messages is characterized and an achievable DoF region of order-1 messages is completely expressed. Besides, for order-1 messages, we derive the sufficient condition, under which the proposed achievable DoF region is the DoF region.

中文翻译：

---

具有延迟 CSIT 的三用户 MIMO 广播信道的可实现 DoF 区域

对于在发射机 (CSIT) 和任意天线配置处具有延迟信道状态信息的双用户多输入多输出 (MIMO) 广播信道，获得了所有自由度 (DoF) 区域。然而，对于具有延迟 CSIT 和任意天线配置的三用户 MIMO 广播信道，2 阶消息的 DoF 区域仍然不清楚，并且仅获得了部分可实现的 1 阶消息的 DoF 区域，其中 2 阶消息和 order-1 消息分别需要两个接收器和一个接收器。在本文中，对于具有延迟 CSIT 和任意天线配置的三用户 MIMO 广播信道，我们设计了 2 阶消息和 1 阶消息的传输方案。为了分析可实现的 DoF 区域，我们提出了一种新方法——结构化方法，它可以系统地分析传输方案的可实现DoF区域。该方法的关键思想是通过可实现的 DoF 元组表达式从传输方案的解码条件构造可实现的 DoF 区域。结果，2阶消息的DoF区域被表征，并且1阶消息的可实现DoF区域被完整表达。此外，对于 1 阶消息，我们推导出充分条件，在该条件下，建议的可实现 DoF 区域是 DoF 区域。表征了 2 阶消息的 DoF 区域，并完整表达了可实现的 1 阶消息的 DoF 区域。此外，对于 1 阶消息，我们推导出充分条件，在该条件下，建议的可实现 DoF 区域是 DoF 区域。表征了 2 阶消息的 DoF 区域，并完整表达了可实现的 1 阶消息的 DoF 区域。此外，对于 1 阶消息，我们推导出充分条件，在该条件下，建议的可实现 DoF 区域是 DoF 区域。