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 with hybrid messages and imperfect CSIT. We establish a three-step procedure for the DoF converse to exploit the utmost possible relaxation. For the DoF achievability, since the DoF region has a specific three-dimensional structure w.r.t. antenna configurations and CSIT qualities, by dividing CSIT qualities into cases, we check the existence of corner point solutions, and then design a hybrid messages-aware rate-splitting scheme to achieve them. Besides, we show that to achieve the strictly positive corner points, it is unnecessary to split the private messages into unicast and multicast parts because the allocated power for the multicast part should be zero. This implies that adding a common message can mitigate the rate-splitting complexity of private messages.

翻译：现有关于非完美发射端信道状态信息(CSIT)下自由度(DoF)的研究大多假设消息为私有类型，这无法反映两个接收端可能请求相同内容的现实情况。为突破该局限，我们考虑混合私有与公共消息的情形。我们刻画了具有混合消息与非完美CSIT的两用户多输入多输出(MIMO)广播信道的最优自由度区域。在自由度逆方向证明中，我们建立了一个三步流程以充分利用最大可能的松弛条件。在自由度可达性方面，由于自由度区域相对于天线配置与CSIT质量具有特定的三维结构，我们通过划分CSIT质量的不同情形，检验角点解的存在性，并设计了一种混合消息感知的速率分裂方案来实现这些角点解。此外，我们证明：要实现严格正的角点解，无需将私有消息拆分为单播与多播部分，因为多播部分的分配功率应为零。这表明添加公共消息可以降低私有消息的速率分裂复杂度。