Optimality of Simple Layered Superposition Coding in the 3 User MISO BC With Finite Precision CSIT

We study the <inline-formula> <tex-math notation="LaTeX">$K=3$ </tex-math></inline-formula> user multiple input single output (MISO) broadcast channel (BC) with <inline-formula> <tex-math notation="LaTeX">$M=3$ </tex-math></inline-formula> antennas at the transmitter and 1 antenna at each receiver, from the generalized degrees of freedom (GDoF) perspective, under the assumption that the channel state information at the transmitter (CSIT) is limited to the finite precision. In particular, our goal is to identify a parameter regime where a simple layered superposition (SLS) coding scheme achieves the entire GDoF region. With <inline-formula> <tex-math notation="LaTeX">$\alpha _{ij}$ </tex-math></inline-formula> representing the channel strength parameter for the link from the <inline-formula> <tex-math notation="LaTeX">$j^{th}$ </tex-math></inline-formula> antenna of the transmitter to the <inline-formula> <tex-math notation="LaTeX">$i^{th}$ </tex-math></inline-formula> receiver, we prove that the SLS is GDoF optimal without the need for time-sharing if <inline-formula> <tex-math notation="LaTeX">$\max (\alpha _{ki},\alpha _{im})\leq \alpha _{ii}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\alpha _{ki}+\alpha _{im}\le \alpha _{ii}+\alpha _{km}$ </tex-math></inline-formula> for all <inline-formula> <tex-math notation="LaTeX">$i,k\in [{3}],m\in [M]$ </tex-math></inline-formula>. The GDoF region under this condition is a convex polyhedron. The result generalizes to arbitrary <inline-formula> <tex-math notation="LaTeX">$M\geq 3$ </tex-math></inline-formula>.

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abstract: Significant progress has been made recently in Generalized Degrees of Freedom (GDoF) characterizations of wireless interference channels (IC) and broadcast channels (BC) under the assumption of finite precision channel state information at the transmitters (CSIT), especially for smaller or highly symmetric network settings. A critical barrier in extending these results to larger and asymmetric networks is the inherent combinatorial complexity of such networks. Motivated by other fields such as extremal combinatorics and extremal graph theory, we explore the possibility of an extremal network theory, i.e., a study of extremal networks within particular regimes of interest. As our test application, we study the GDoF benefits of transmitter cooperation in a <italic>K</italic> user IC over the simple scheme of power control and treating interference as Gaussian noise (TIN) for three regimes of interest – a TIN regime identified by Geng et al. where TIN was shown to be GDoF optimal for the <italic>K</italic> user interference channel, a CTIN regime identified by Yi and Caire where the GDoF region achievable by TIN is <underline>c</underline>onvex without time-sharing, and an SLS regime identified by Davoodi and Jafar where a simple layered superposition (SLS) scheme is shown to be optimal in the <italic>K</italic> user MISO BC, albeit only for <inline-formula> <tex-math notation="LaTeX">$ {K}\leq 3$ </tex-math></inline-formula>. The SLS regime includes the CTIN regime, and the CTIN regime includes the TIN regime. As our first result, we show that under finite precision CSIT, TIN is GDoF optimal for the <italic>K</italic> user IC throughout the CTIN regime. Furthermore, under finite precision CSIT, appealing to extremal network theory we obtain the following results. In the TIN regime as well as the CTIN regime, we show that the extremal GDoF gain from transmitter cooperation over TIN is bounded regardless of the number of users. In fact, the gain is exactly a factor of 3/2 in the TIN regime, and <inline-formula> <tex-math notation="LaTeX">$2-1/{K}$ </tex-math></inline-formula> in the CTIN regime, for arbitrary number of users <inline-formula> <tex-math notation="LaTeX">${K}>1$ </tex-math></inline-formula>. However, in the SLS regime, the gain is <inline-formula> <tex-math notation="LaTeX">$\Theta (\log _{2}({K}))$ </tex-math></inline-formula>, i.e., it scales logarithmically with the number of users.

title: Generalized Degrees of Freedom of the Symmetric $K$ User Interference Channel Under Finite Precision CSIT
abstract: The generalized degrees of freedom (GDoF) characterization of the symmetric <inline-formula> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> user interference channel is obtained under finite precision channel state information at the transmitters (CSIT). The symmetric setting is where each cross channel is capable of carrying <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> DoF, while each direct channel is capable of carrying 1 DoF. Remarkably, under finite precision CSIT the symmetric <inline-formula> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> user interference channel loses all the GDoF benefits of interference alignment. The GDoF per user diminish with the number of users everywhere except in the very strong (optimal for every receiver to decode all messages) and very weak (optimal to treat all interference as noise) interference regimes. The result stands in sharp contrast to prior work on the symmetric setting under perfect CSIT, where the GDoF per user remain undiminished due to interference alignment. The result also stands in contrast to prior work on a subclass of asymmetric settings under finite precision CSIT, i.e., the topological interference management problem, where interference alignment plays a crucial role and provides substantial GDoF benefits.