Capacity and coding for the Ising Channel with Feedback

TL;DR

This paper determines the feedback capacity of the Ising channel, models it using dynamic programming, and proposes a simple coding scheme that achieves this capacity, linking theoretical results with practical coding strategies.

Contribution

It provides the exact feedback capacity of the Ising channel and introduces a capacity-achieving coding scheme based on dynamic programming analysis.

Findings

Feedback capacity is approximately 0.575522 bits per channel use.

The capacity is achieved by a simple, error-free coding scheme.

A strong connection between DP solutions and coding strategies is established.

Abstract

The Ising channel, which was introduced in 1990, is a channel with memory that models Inter-Symbol interference. In this paper we consider the Ising channel with feedback and find the capacity of the channel together with a capacity-achieving coding scheme. To calculate the channel capacity, an equivalent dynamic programming (DP) problem is formulated and solved. Using the DP solution, we establish that the feedback capacity is the expression $C=(\frac{2H_b(a)}{3+a})\approx 0.575522$ where $a$ is a particular root of a fourth-degree polynomial and $H_b(x)$ denotes the binary entropy function. Simultaneously, $a=\arg \max_{0\leq x \leq 1} (\frac{2H_b(x)}{3+x})$. Finally, a simple, error-free, capacity-achieving coding scheme is provided together with outlining a strong connection between the DP results and the coding scheme.