On the Synergistic Benefits of Alternating CSIT for the MISO BC

The degrees of freedom (DoF) of the two-user multiple-input single-output (MISO) broadcast channel (BC) are studied under the assumption that the form, Ii, i = 1, 2, of the channel state information at the transmitter (CSIT) for each user’s channel can be either perfect (P ), delayed (D) or not available (N), i.e., I1, I2 ∈ {P,N,D}, and therefore the overall CSIT can alternate between the 9 resulting states I1I2. The fraction of time associated with CSIT state I1I2 is denoted by the parameter λI1I2 and it is assumed throughout that λI1I2 = λI2I1 , i.e., λPN = λNP , λPD = λDP , λDN = λND. 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 (λP , λD, λN ) = (∑ I2 λPI2 , ∑ I2 λDI2 , ∑ I2 λNI2 ) , I2 ∈ {P,D,N}, 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 9 CSIT states, D(λI1I2 : I1, I2 ∈ {P,D,N}), is the same as the DoF region with only 3 CSIT states D(λPP , λDD, λNN ), under the same marginal distribution of CSIT states, i.e., (λPP , λDD, λNN ) = (λP , λD, λN ). The sumDoF value can be expressed as DoF = min ( 4+2λP 3 , 1 + λP + λD ) , from which one can uniquely identify the minimum required marginal CSIT fractions to achieve any target DoF value as (λP , λD)min = ( 3 2DoF− 2, 1− 1 2 ) when DoF ∈ [ 4 3 , 2 ] and (λP , λD)min = (0, (DoF− 1)) when DoF ∈ [ 0, 4 3 ) . The results highlight the synergistic benefits of alternating CSIT and the tradeoffs between various forms of CSIT for any given DoF value. ∗E-mail: [email protected], [email protected], [email protected], [email protected]. The work of H. V. Poor was supported in part by the Air Force Office of Scientific Research under MURI Grant FA 9550-09-1-0643. ar X iv :1 20 8. 50 71 v1 [ cs .I T ] 2 4 A ug 2 01 2