Capacity Results for Non-Ergodic Multi-Modal Broadcast Channels with Controllable Statistics

TL;DR

A key finding of this work is the significant gain of inter-modal coding over the separate treating of individual modes in a multi-modal setting with two non-transient modes and an arbitrary number of transient modes.

Abstract

Movable antennas and reconfigurable intelligent surfaces enable a new paradigm in which channel statistics can be controlled and altered. Further, the known trajectory and operation protocol of communication satellites results in networks with predictable statistics. The predictability of future changes results in a non-ergodic model for which the fundamentals are largely unknown. We consider the canonical two-user broadcast erasure channel in which channel statistics vary at a priori known points. We consider a multi-modal setting with two non-transient modes (whose lengths scale linearly with the blocklength) and an arbitrary number of transient modes. We provide a new set of outer-bounds on the capacity region of this problem when the encoder has access to causal ACK/NACK feedback. The outer-bounds reveal the significant role of the non-transient mode with higher erasure probability both on the outer and the inner bounds. We show the outer-bounds are achievable in non-trivial regimes, characterizing the capacity region for a wide range of parameters. We also discuss the regimes where the inner and outer bounds diverge and analyze the gap between the two. A key finding of this work is the significant gain of inter-modal coding over the separate treating of individual modes.

Figures (5)

• Figure 1: Two-user multi-modal BPEC with (Shannon) feedback. We assume two non-transient modes and one (or more) transient modes with a total of length of ${\mathcal{O}} (n)$.
• Figure 2: Illustration of the outer-bound region of Theorem \ref{['THM:Bimodal-Outer']} for $\delta_{\mathrm{A}} > \delta_{\mathrm{B}} = 0$ and various values of $n_{\mathrm{A}}$.
• Figure 3: Symmetric sum-rate inner-bound vs. outer-bounds for $\delta_{\mathrm{A}} = 0.75$ and $\delta_{\mathrm{B}} = 0$.
• Figure 4: Symmetric sum-rate inner-bound vs. outer-bounds for $\delta_{\mathrm{A}} = 0.75$ and $\delta_{\mathrm{B}} = 1/8$.
• Figure 5: An example with $\delta_A = 0.75$, $\delta_B = 0$, and $n_A = \lfloor n/6 \rfloor$ for which the inner and outer bounds deviate. Inter-modal coding nonetheless outperforms intra-modal coding.

Theorems & Definitions (13)

• Remark 1: Orderly channels and ergodicity
• Remark 2: Feedback delay
• Theorem 1
• Remark 3: Comparison of Theorem \ref{['THM:Bimodal-Outer']} to the preliminary work Bimodal-ISIT
• Remark 4: Room for improvement
• Theorem 2
• Remark 5: Comparison of Theorem \ref{['THM:Bimodal-Inner']} to the preliminary work Bimodal-ISIT
• Remark 6: Comparison to prior work
• Lemma 1
• proof