DoF analysis of the MIMO broadcast channel with alternating/hybrid CSIT

We consider a K-user multiple-input single-output (MISO) broadcast channel (BC) where the channel state information (CSI) of user i(i=1,2, K) may be instantaneously perfect (P), delayed (D), or not known (N) at the transmitter with probabilities λP^iP, λ_Dⁱ, and λ_Nⁱ, respectively. In this setting, according to the three possible CSI at the transmitter (CSIT) for each user, knowledge of the joint CSIT of the K users could have at most 3^K states. In this paper, given the marginal probabilities of CSIT (i.e., λ_Pⁱ, λ_Dⁱ, and λ_Nⁱ), we derive an outer bound for the degrees of freedom (DoF) region of the K-user MISO BC. Subsequently, we tighten this outer bound by considering a set of inequalities that capture some of the 3^K states of the joint CSIT. One of the consequences of this set of inequalities is that for K≥3, it is shown that the DoF region is not completely characterized by the marginal probabilities in contrast to the two-user case. Afterwards, the tightness of these bounds is investigated through the discussion on the achievability. Finally, a two user multiple-input multiple-output BC having CSIT among P and N is considered in which an outer bound for the DoF region is provided, and it is shown that in some scenarios, it is tight.