Feedback Capacity of the Compound Channel

TL;DR

This paper determines the feedback capacity of compound finite-state channels, showing conditions where feedback does not increase capacity and deriving a formula for memoryless channels.

Contribution

It provides the first capacity characterization for compound finite-state channels with feedback, including a universal decoder and capacity equivalences.

Findings

Feedback does not increase capacity of the compound Gilbert-Elliot channel.

Zero capacity without feedback implies zero capacity with feedback for certain channels.

Feedback capacity of memoryless compound channels is characterized by a specific infimum and maximum mutual information.

Abstract

In this work we find the capacity of a compound finite-state channel with time-invariant deterministic feedback. The model we consider involves the use of fixed length block codes. Our achievability result includes a proof of the existence of a universal decoder for the family of finite-state channels with feedback. As a consequence of our capacity result, we show that feedback does not increase the capacity of the compound Gilbert-Elliot channel. Additionally, we show that for a stationary and uniformly ergodic Markovian channel, if the compound channel capacity is zero without feedback then it is zero with feedback. Finally, we use our result on the finite-state channel to show that the feedback capacity of the memoryless compound channel is given by $\inf_{\theta} \max_{Q_X} I(X;Y|\theta)$.