An upper bound on relaying over capacity based on channel simulation

TL;DR

This paper introduces a new upper bound on the capacity of a 3-node relay channel, improving upon the cut-set bound, especially for degraded channels, by leveraging channel simulation and a generalized blowing-up lemma.

Contribution

A novel inequality for upper-bounding relay channel capacity that surpasses traditional bounds, applicable to degraded channels and computable for specific cases like the binary erasure channel.

Findings

New upper bound is tighter than the cut-set bound in several cases.

Bound is explicitly computable for statistically degraded channels.

Numerical analysis confirms the bound's effectiveness on the binary erasure channel.

Abstract

The upper bound on the capacity of a 3-node discrete memoryless relay channel is considered, where a source X wants to send information to destination Y with the help of a relay Z. Y and Z are independent given X, and the link from Z to Y is lossless with rate $R_0$. A new inequality is introduced to upper-bound the capacity when the encoding rate is beyond the capacities of both individual links XY and XZ. It is based on generalization of the blowing-up lemma, linking conditional entropy to decoding error, and channel simulation, to the case with side information. The achieved upper-bound is strictly better than the well-known cut-set bound in several cases when the latter is $C_{XY}+R_0$, with $C_{XY}$ being the channel capacity between X and Y. One particular case is when the channel is statistically degraded, i.e., either Y is a statistically degraded version of Z with respect to X, or Z is a statistically degraded version of Y with respect to X. Moreover in this case, the bound is shown to be explicitly computable. The binary erasure channel is analyzed in detail and evaluated numerically.