MISO broadcast channel with delayed and evolving CSIT

Abstract

The work considers the two-user MISO broadcast channel with a gradual and delayed accumulation of channel state information at the transmitter (CSIT), and addresses the question of how much feedback is necessary, and when, in order to achieve a certain degrees-of-freedom (DoF) performance. Motivated by limited-capacity feedback links with delays, that may not immediately convey perfect CSIT, and focusing on the block fading scenario, we consider a gradual accumulation of feedback bits that results in a progressively increasing CSIT quality as time progresses across the coherence period (T channel uses - current CSIT), or at any time after (delayed CSIT). Specifically, for any set {α <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> } <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sup> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t=1</sub> of feedback quality exponents describing the high-SNR rates-of-decay of the mean square error of the current CSIT estimates at time t ≤ T (0 <; α <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ≤ · · · ≤ (αT≤ 1), given an average α = Σ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sup> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t=1</sub> α <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> /T, and given perfect delayed CSIT (received at any time t > T), the work here derives the optimal DoF region to be the polygon with corner points {(0,0),(0,1),(α,1),(2+α/3, 2+α/3),(1,α),(1,0)}. Aiming to now reduce the overall number of feedback bits, we also prove that the above optimal region holds even with imperfect delayed CSIT for any (delayed-CSIT) quality exponent β ≥ 1+2α/3. The results are supported by novel multi-phase precoding schemes that utilize gradually improving CSIT. The approach here incorporates different settings such as the delayed CSIT setting of Maddah-Ali and Tse (β = 1, α <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> = 0, ∀t ≤ T), the imperfect current CSIT setting of Yang et al. and of Gou and Jafar (β = 1, α <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> = · · · = α <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sub> > 0), and the not-so-delayed CSIT setting of Lee and Heath (β = 1, α <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> = · · · = α <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">T</sub> = 0 for some τ<;T).