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
$i(i=1,2,\ldots ,K)$
may be instantaneously perfect (P), delayed (D), or not known (N) at the transmitter with probabilities
$\lambda _{P}^{i}$
,
$\lambda _{D}^{i}$
, and
$\lambda _{\vphantom {R_{j}}N}^{i}$
, 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.,
$\lambda _{P}^{i}$
,
$\lambda _{D}^{i}$
, and
$\lambda _{\vphantom {R_{j}}N}^{\vphantom {r^{A}}i}$
), 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
$K\geq 3$
, 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.