No Feedback, No Problem: Capacity of Erasure Broadcast Channels with Single-User Delayed CSI

We characterize the capacity region of the two-user erasure broadcast channel (BC) with single-user delayed channel state information (CSI). More precisely, we assume one receiver does not provide its channel state to the other two nodes (the other receiver and the transmitter), while the other receiver reveals its state globally with unit delay. This is a "DN" hybrid CSI at the transmitter (CSIT) setting where the transmitter has the delayed CSI of one user but not the other. Previous results developed opportunistic network coding schemes for this DN setting, which strictly enlarge the achievable rate region compared to the no-CSIT baseline. Characterization of the capacity region for the DN setting, however, remained open. In this work, we develop an improved achievability strategy and show that the capacity region, surprisingly, matches that of the erasure BC with global delayed CSI of both users. The key to such improvement over previous results is a new precoding strategy for the retransmission phase of the opportunistic network coding scheme. It harnesses the single user delayed CSI in the retransmission phase, so that interference from the "D" receiver can be aligned at the "N" receiver. Besides erasure BCs with two private messages, an extension to BCs with an additional common message is also provided.