On Channel Resolvability in Presence of Feedback

TL;DR

This paper investigates how feedback influences the ability to generate approximately i.i.d. outputs from a discrete memoryless channel, revealing that feedback can significantly improve the exponential decay rate of divergence without reducing the minimum entropy rate.

Contribution

It demonstrates that feedback can increase the resolvability exponent over binary symmetric channels using variable-length schemes, without decreasing the minimum entropy rate needed.

Findings

Feedback does not reduce the minimum entropy rate for channel resolvability.

Feedback can significantly increase the resolvability exponent over binary symmetric channels.

A variable-length scheme achieves exponential decay of divergence with an exponent related to mutual information.

Abstract

We study the problem of generating an approximately i.i.d. string at the output of a discrete memoryless channel using a limited amount of randomness at its input in presence of causal noiseless feedback. Feedback does not decrease the channel resolution, the minimum entropy rate required to achieve an accurate approximation of an i.i.d. output string. However, we show that, at least over a binary symmetric channel, a significantly larger resolvability exponent (the exponential decay rate of the divergence between the output distribution and product measure), compared to the best known achievable resolvability exponent in a system without feedback, is possible. We show that by employing a variable-length resolvability scheme and using an average number of coin-flips per channel use, the average divergence between the distribution of the output sequence and product measure decays exponentially fast in the average length of output sequence with an exponent equal to $[R-I(U;V)]^+$ where $I(U;V)$ is the mutual information developed across the channel.