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Degradedness Under Cooperation

Yossef Steinberg

TL;DR

Degradedness under Cooperation broadens the traditional degraded broadcast channel framework by introducing parametric degradedness notions—strongly less noisy and strongly more capable—and demonstrating that decode-and-forward can be capacity-achieving for cooperative BCs, primitive relay channels, and certain diamond channels when conference rates fall within specified bounds. The approach leverages eta_ln/eta_mc degradations, SDPIs, and nonlinear degradedness concepts to derive tight inner and outer bounds that match the cut-set bounds in key regimes. The work extends capacity characterizations beyond physically or stochastically degraded models and provides practical guidance on when D&F suffices, including for Gaussian-like inputs via nonlinear degradedness tools. Overall, the results illuminate when cooperation preserves user order and when simple D&F strategies achieve capacity in a broader class of networks.

Abstract

We study cooperation problems in broadcast and relay networks, where the receivers do not satisfy the classical physical degradedness assumptions. New notions of degradedness, \emph{strongly less noisy} and \emph{strongly more capable} are introduced. We show that under these conditions, decode and forward (D\&F) is optimal for classes of cooperative systems with limited conference rates, thus yielding new capacity results for these systems. In particular, we derive bounds on the capacity region of a class of broadcast channels with cooperation, that are tight on part of the capacity region. It is shown that the cut-set bound is tight for classes of primitive relay and diamond channels, beyond the physically or stochastically degraded models.

Degradedness Under Cooperation

TL;DR

Degradedness under Cooperation broadens the traditional degraded broadcast channel framework by introducing parametric degradedness notions—strongly less noisy and strongly more capable—and demonstrating that decode-and-forward can be capacity-achieving for cooperative BCs, primitive relay channels, and certain diamond channels when conference rates fall within specified bounds. The approach leverages eta_ln/eta_mc degradations, SDPIs, and nonlinear degradedness concepts to derive tight inner and outer bounds that match the cut-set bounds in key regimes. The work extends capacity characterizations beyond physically or stochastically degraded models and provides practical guidance on when D&F suffices, including for Gaussian-like inputs via nonlinear degradedness tools. Overall, the results illuminate when cooperation preserves user order and when simple D&F strategies achieve capacity in a broader class of networks.

Abstract

We study cooperation problems in broadcast and relay networks, where the receivers do not satisfy the classical physical degradedness assumptions. New notions of degradedness, \emph{strongly less noisy} and \emph{strongly more capable} are introduced. We show that under these conditions, decode and forward (D\&F) is optimal for classes of cooperative systems with limited conference rates, thus yielding new capacity results for these systems. In particular, we derive bounds on the capacity region of a class of broadcast channels with cooperation, that are tight on part of the capacity region. It is shown that the cut-set bound is tight for classes of primitive relay and diamond channels, beyond the physically or stochastically degraded models.
Paper Structure (17 sections, 7 theorems, 77 equations, 3 figures)

This paper contains 17 sections, 7 theorems, 77 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Strong degradedness relations
  3. Cooperative BC
  4. Primitive relay channels
  5. Diamond channels
  6. Nonlinear degradedness conditions
  7. Proofs
  8. Proof of Lemma \ref{['lemma:eta_relations']}
  9. Proof of Theorem \ref{['theo:BC_main']}
  10. Direct Part
  11. Converse Part
  12. Proof of Theorem \ref{['theo:BDC_main']}
  13. Achievability of $R_i$
  14. Part 2 - capacity when $C_3>0$
  15. Proof of Theorem \ref{['theo:nonlinear_main']}
  16. ...and 2 more sections

Key Result

Lemma 1

A pair $\left\{ P_{Y_1|X}, P_{Y_2|X}\right\}$ is $\eta$-LN if and only if is concave in $P_X$.

Figures (3)

  • Figure 1: Curves of $\eta_{\hbox{\scriptsize{ln}}}$ (solid) and $\eta_{\hbox{\scriptsize{kl}}}(\hbox{BSC}(p))/\eta_{\hbox{\scriptsize{kl}}}(\hbox{Z}(0.3))$ (dashed). For $p<0.19$ the channel is not degraded
  • Figure 2: D&F regions for the BEC($\epsilon$)-BSC($p$) BC, for few values of conference link capacities $C_{12}$. The blue diamonds mark the minimal $R_2$ for which the D&F region is tight, for the corresponding conference rate. In Fig. \ref{['subfig:BEC_BSC_deg']}$(\epsilon,p)=(0.1,0.2)$, the channel is stochastically degraded, thus it has a physically degraded version for which the D&F region is tight for all values of $C_{12}$ and $R_2$. In Fig. \ref{['subfig:BEC_BSC_nondeg']}$(\epsilon,p)=(0.45,0.2)$ so the channel is less noisy but not stochastically degraded.
  • Figure 3: The broadcast diamond channel, composed of two primitive relay channels with common input and destination.

Theorems & Definitions (14)

  • Definition 1
  • Lemma 1
  • Corollary 1
  • Lemma 2
  • Example 1: Binary erasure channels
  • Example 2: Binary symmetric channels
  • Example 3: Z channel and BSC
  • Definition 2
  • Theorem 1
  • Example 4: BEC-BSC Broadcast Channel
  • ...and 4 more