Most of the existing research on degrees-of-freedom (DoF) with imperfect channel state information at the transmitter (CSIT) assume the messages are private, which may not reflect reality as the two receivers can request the same content. To overcome this limitation, we consider hybrid private and common messages. We characterize the optimal DoF region for the two-user multiple-input multiple-output (MIMO) broadcast channel (BC) with hybrid messages and imperfect CSIT. We establish a three-step procedure for the converse to exploit the utmost possible relaxation. For the achievability, since the DoF region is with three-dimensional structure regarding antenna configurations and CSIT qualities, we verify the existence or nonexistence of corner point candidates by the feature of antenna configurations and CSIT qualities categorization, and give a hybrid message-aware rate-splitting scheme with specific power allocation. Besides, we show that to achieve the strictly positive corner points, it is unnecessary to split the private messages into unicast and multicast parts. This implies that adding a common message can mitigate the rate-splitting complexity.

翻译：目前关于发射端不完全信道状态信息(CSIT)下自由度(DoF)的研究大多假设消息是私有的，这一假设可能无法反映实际场景，因为两个接收端可能请求相同内容。为突破这一限制，我们考虑混合私有与公共消息。针对带有混合消息与不完全CSIT的两用户多输入多输出(MIMO)广播信道(BC)，我们刻画了最优DoF区域。在逆定理推导中，我们建立了一个三步流程以实现最大限度的松弛。在可达性方面，由于DoF区域在天线配置与CSIT质量方面具有三维结构，我们通过天线配置特征与CSIT质量分类验证角点候选是否存在，并给出带有特定功率分配的混合消息感知速率分割方案。此外，我们证明：为实现严格正角点，无需将私有消息分割为单播与多播部分。这表明添加公共消息可降低速率分割的复杂度。