On the Synergistic Benefits of Alternating CSIT for the MISO Broadcast Channel

The degrees of freedom (DoFs) of the two-user multiple-input single-output (MISO) broadcast channel (BC) are studied under the assumption that the form, <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$I_{i},\;i=1, 2$</tex></formula> , of the channel state information at the transmitter (CSIT) for each user's channel can be either perfect <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$(P)$</tex></formula> , delayed <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$(D)$</tex></formula> , or not available <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$(N)$</tex></formula> , i.e., <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$I_{1},I_{2} \in \{P,N,D\}$</tex> </formula> , and therefore, the overall CSIT can alternate between the nine resulting states <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$I_{1}I_{2}$</tex></formula> . The fraction of time associated with CSIT state <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$I_{1}I_{2}$</tex></formula> is denoted by the parameter <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$\lambda_{I_{1}I_{2}}$</tex></formula> and it is assumed throughout that <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$\lambda_{I_{1}I_{2}} = \lambda_{I_{2}I_{1}}$</tex></formula> , i.e., <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\lambda_{PN} = \lambda_{NP}, \lambda_{PD}=\lambda_{DP}, \lambda_{DN}=\lambda_{ND}$</tex> </formula> . Under this assumption of symmetry, the main contribution of this paper is a complete characterization of the DoF region of the two-user MISO BC with alternating CSIT. Surprisingly, the DoF region is found to depend only on the marginal probabilities <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$(\lambda_{P}, \lambda_{D},\lambda_{N})=\left(\sum_{I_{2}}\lambda_{PI_{2}},\sum_{I_{2}}\lambda_{DI_{2}}, \sum_{I_{2}}\lambda_{NI_{2}}\right)$</tex></formula> , <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$I_{2} \in \{P,D,N\}$</tex></formula> , which represent the fraction of time that any given user (e.g., user 1) is associated with perfect, delayed, or no CSIT, respectively. As a consequence, the DoF region with all nine CSIT states, <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\cal {D}}(\lambda_{I_{1}I_{2}}:I_{1},I_{2} \in \{P,D,N\})$</tex> </formula> , is the same as the DoF region with only three CSIT states <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\cal {D}}(\lambda_{PP}, \lambda_{DD}, \lambda_{NN})$</tex> </formula> , under the same marginal distribution of CSIT states, i.e., <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$(\lambda_{PP}, \lambda_{DD},\lambda_{NN})=(\lambda_{P},\lambda_{D},\lambda_{N})$</tex></formula> . The sum-DoF value can be expressed as <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\rm DoF}=\min \left({{4+2\lambda_{P}} \over {3}}, 1+\lambda_{P}+\lambda_{D}\right)$</tex></formula> , from which one can uniquely identify the minimum required marginal CSIT fractions to achieve any target DoF value as <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$(\lambda_{P},\lambda_{D})_{\min}=\left({{3} \over {2}} {\rm DoF}-2,1- {{1} \over {2}} {\rm DoF}\right)$</tex></formula> when <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\rm DoF} \in \big [{{4} \over {3}},2\big]$</tex></formula> and <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$(\lambda_{P},\lambda_{D})_{\min}=(0,({\rm DoF}-1)^{+})$</tex></formula> when <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\rm DoF} \in \big [0, {{4} \over {3}}\big)$</tex></formula> . The results highlight the synergistic benefits of alternating CSIT and the tradeoffs between various forms of CSIT for any given DoF value. Partial results are also presented for the multiuser MISO BC with <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$M$</tex></formula> transmit antennas and <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$K$</tex> </formula> single antenna users. For this problem, the minimum amount of perfect CSIT required per user to achieve the maximum DoFs of <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\min (M,K)$</tex></formula> is characterized. By the minimum amount of CSIT per user, we refer to the minimum fraction of time that the transmitter has access to perfect and instantaneous CSIT from a user. Through a novel converse proof and an achievable scheme, it is shown that the minimum fraction of time perfect CSIT is required per user in order to achieve the DoF of <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\min (M,K)$</tex></formula> is given by <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\min (M,K)/K$</tex></formula> .