Toward the performance versus feedback tradeoff for the two-user MISO broadcast channel

For the two-user MISO broadcast channel with imperfect and delayed channel state information at the transmitter (CSIT), the work explores the tradeoff between performance, and feedback timeliness and quality. For h_t (resp. g_t) denoting the channel of the first (resp. second) user at any time t=1,2,..., the work considers a general feedback process that provides for CSIT estimates h'_{t,t'}, g'_{t,t'} at any time t' - before, during, or after materialization of h_t, g_t at time t. In this setting, performance is naturally a function of the statistics of the channel process {h_t, g_t }_t and of the overall CSIT quality {(h_t-h'_{t,t'}), (g_t-g'_{t,t'} ) }_{t,t'}.
Under standard assumptions, we derive the optimal DoF region {d1,d2} to take the simple form {d1 <= 1, d2 <= 1, 2d1 + d2 <= 2 + C1, 2d2 + d1 <= 2 +C2} for any min{D1,D2} >= min{(1+C1+C2)/3, (1 + min{ C1,C2})/2} where a1_t, b1_t, are the high-SNR asymptotic rates of decay of the MSE of current and delayed CSIT of the first user (similarly a2_t, b2_t for the second user), and where C1,D1,C2,D2 are the respective averages of the sequences _t,_t,_t,_t. Interestingly the high SNR complexity of the problem is mainly captured by these averages. The outer bounds and constructions allow for the above to hold for a large class of block and non-block fading channel models, and the causal scheme does not require knowledge of future statistics, nor predicted CSIT estimates of future channels. The result unifies and extends prior attempts to capture the effect of imperfect and delayed feedback. It also allows for consideration of novel pertinent settings, such as the new periodically evolving feedback setting over the quasi-static block fading channel, where a gradual accumulation of feedback bits results in a progressively increasing CSIT quality as time progresses across a finite coherence period.