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Relay Channels with Unreliable Helpers

Yossef Steinberg

TL;DR

This work studies the relay channel when a helper link may be unavailable, formulating a degraded RC with an unreliable helper of capacity $C_1$. It derives a single-parameter capacity region ${\cal R}(C_1)$ characterized by an auxiliary random variable $U$ and the relay input $X_1$, with the base rate $R$ decodable regardless of helper activity and an additional rate $R'$ decodable when the helper is present; the region reduces to the classical RC when $C_1=0$ and to the primitive relay channel when $R'=0$. The Gaussian degraded relay channel with unreliable helper is solved, giving explicit region expressions via parameters $\alpha,\beta$ and Gaussian inputs, and the proof combines a Fano-based converse with a block-Markov, binning, and superposition-based direct scheme. The techniques extend to the Gaussian case using the conditional EPI and linear estimation arguments. Overall, the paper provides a robust coding framework for relay networks with potentially unavailable helpers and gives complete capacity results for degraded settings, with practical implications for cooperative communications under uncertainty.

Abstract

The relay channel with unreliable helper is introduced and studied. The model is that of a classical relay channel where the input from the relay to the channel has an extra primitive link whose presence is not assured a priori. The extra link represents a helper who may decide not to cooperate in transmission. The goal is to devise robust coding schemes that exploit all the relay links when they are present, but can also operate, possibly at reduced rates, when the extra primitive link (helper) is absent. The capacity region of this class of problems is defined, and fully characterized for degraded relay channels. The degraded Gaussian relay channel with unreliable relay link is solved.

Relay Channels with Unreliable Helpers

TL;DR

This work studies the relay channel when a helper link may be unavailable, formulating a degraded RC with an unreliable helper of capacity . It derives a single-parameter capacity region characterized by an auxiliary random variable and the relay input , with the base rate decodable regardless of helper activity and an additional rate decodable when the helper is present; the region reduces to the classical RC when and to the primitive relay channel when . The Gaussian degraded relay channel with unreliable helper is solved, giving explicit region expressions via parameters and Gaussian inputs, and the proof combines a Fano-based converse with a block-Markov, binning, and superposition-based direct scheme. The techniques extend to the Gaussian case using the conditional EPI and linear estimation arguments. Overall, the paper provides a robust coding framework for relay networks with potentially unavailable helpers and gives complete capacity results for degraded settings, with practical implications for cooperative communications under uncertainty.

Abstract

The relay channel with unreliable helper is introduced and studied. The model is that of a classical relay channel where the input from the relay to the channel has an extra primitive link whose presence is not assured a priori. The extra link represents a helper who may decide not to cooperate in transmission. The goal is to devise robust coding schemes that exploit all the relay links when they are present, but can also operate, possibly at reduced rates, when the extra primitive link (helper) is absent. The capacity region of this class of problems is defined, and fully characterized for degraded relay channels. The degraded Gaussian relay channel with unreliable relay link is solved.
Paper Structure (13 sections, 2 theorems, 42 equations, 4 figures, 3 tables)

This paper contains 13 sections, 2 theorems, 42 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Main Result
  4. The Gaussian Case
  5. Proof of Theorem \ref{['theo:main1']}
  6. Converse part of Theorem \ref{['theo:main1']}
  7. Direct part of Theorem \ref{['theo:main1']}
  8. Code construction
  9. Coding
  10. Decoding at the destination
  11. Proof of Theorem \ref{['theo:cap_Gaussian']}
  12. Direct Part
  13. Converse Part

Key Result

Theorem 1

For any degraded relay channel $P$ with unreliable helper $P(y_2|x_2)$ where $C_1$ is the capacity of $P(y_2|x_2)$, i.e., Moreover, to exhaust ${\cal R}(C_1)$, it is enough to choose $U$ with alphabet size $|{\cal U}|\leq |{\cal X}| |{\cal X}_1| + 2$.

Figures (4)

  • Figure 1: The relay channel $\tilde{P}$, composed of the channel $P$ and a helper, modeled as an additional channel $P(y_2|x_2)$ of capacity $C_1$. The helper channel $P(y_2|x_2)$ and the main RC $P(y,y_1|x,x_1)$ are decoupled.
  • Figure 2: The relay channel with unreliable helper. The message $m$ is always decoded. The message $m'$ is decoded only if the signal $Y_2$ arrives to the decoder.
  • Figure 3: The Gaussian degraded relay channel with unreliable helper.
  • Figure 4: The capacity region of the Gaussian degraded relay channel with unreliable helper, for $P=P_1=\sigma_z^2=1$, $\sigma_1^2=10$.

Theorems & Definitions (4)

  • Definition 1
  • Remark 1
  • Theorem 1
  • Theorem 2