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, 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 nine 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) = (ΣI₂ λPI₂, ΣI₂ λDI₂, ΣI₂ 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 nine CSIT states, D(λI₁I₂:I₁,I₂ ∈ {P,D,N}) , is the same as the DoF region with only three CSIT states D(λPP, λDD, ;- ;NN), under the same marginal distribution of CSIT states, i.e., (λPP, λDD,λNN)=(λP,λD,λN). The sum-DoF 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=([3/2] DoF-2,1- [1/2] DoF) 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. Partial results are also presented for the multiuser MISO BC with M transmit antennas and K single antenna users. For this problem, the minimum amount of perfect CSIT required per user to achieve the maximum DoFs of min(M,K) 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 min(M,K) is given by min(M,K)/K.