GDoF of the MISO BC: Bridging the gap between finite precision CSIT and perfect CSIT

Abstract

This work bridges the gap between sharply contrasting results on the degrees of freedom of the K user broadcast channel where the transmitter is equipped with K transmit antennas and each of the K receivers is equipped with a single antenna. This channel has K DoF when channel state information at the transmitter (CSIT) is perfect, but as shown recently, it has only 1 DoF when the CSIT is limited to finite precision. By considering the full range of partial CSIT assumptions parameterized by β ∈ [0,1], such that the strength of the channel estimation error terms scales as ~ SNR <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-β</sup> relative to the channel strengths which scale as ~ SNR, it is shown that this channel has 1 - β + Kβ DoF. For K = 2 users with arbitrary β <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ij</sub> parameters, the DoF are shown to be 1 + min <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i,j</sub> β <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ij</sub> . To explore diversity of channel strengths, the results are further extended to the symmetric Generalized Degrees of Freedom setting where the direct channel strengths scale as ~ SNR and the cross channel strengths scale as ~ SNR <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α</sup> , α ∈ [0,1], β ∈ [0,α]. Here, the roles of α and β are shown to counter each other on equal terms, so that the sum GDoF value in the K user setting is (α - β) + K(1 - (α-β )) and for the 2 user setting with arbitrary β <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ij</sub> , is 2 - α + min <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i,j</sub> β <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ij</sub> .