Bounds on the Feedback Capacity of the $(d,\infty)$-RLL Input-Constrained Binary Erasure Channel

TL;DR

This paper derives bounds on the feedback capacity of an input-constrained binary erasure channel with runlength limitations, providing tight bounds for certain cases and demonstrating equality with non-causal capacity under specific conditions.

Contribution

It introduces new single-letter bounds on the feedback capacity for the $(d, ext{infinity})$-RLL constrained BEC, extending previous results and identifying cases where feedback capacity matches non-causal capacity.

Findings

Bounds are tight for d=1 case.

Feedback capacity equals non-causal capacity for d=2 and certain erasure probabilities.

Bounds differ from non-causal capacities only in maximization domains for d>1.

Abstract

The paper considers the input-constrained binary erasure channel (BEC) with causal, noiseless feedback. The channel input sequence respects the $(d,\infty)$-runlength limited (RLL) constraint, i.e., any pair of successive $1$s must be separated by at least $d$ $0$s. We derive upper and lower bounds on the feedback capacity of this channel, for all $d\geq 1$, given by: $\max\limits_{\delta \in [0,\frac{1}{d+1}]}R(\delta) \leq C^{\text{fb}}_{(d\infty)}(\epsilon) \leq \max\limits_{\delta \in [0,\frac{1}{1+d\epsilon}]}R(\delta)$, where the function $R(\delta) = \frac{h_b(\delta)}{d\delta + \frac{1}{1-\epsilon}}$, with $\epsilon\in [0,1]$ denoting the channel erasure probability, and $h_b(\cdot)$ being the binary entropy function. We note that our bounds are tight for the case when $d=1$ (see Sabag et al. (2016)), and, in addition, we demonstrate that for the case when $d=2$, the feedback capacity is equal to the capacity with non-causal knowledge of erasures, for $\epsilon \in [0,1-\frac{1}{2\log(3/2)}]$. For $d>1$, our bounds differ from the non-causal capacities (which serve as upper bounds on the feedback capacity) derived in Peled et al. (2019) in only the domains of maximization. The approach in this paper follows Sabag et al. (2017), by deriving single-letter bounds on the feedback capacity, based on output distributions supported on a finite $Q$-graph, which is a directed graph with edges labelled by output symbols.