Computable Upper Bounds on the Capacity of Finite-State Channels

TL;DR

This paper introduces a new method for deriving upper bounds on the capacity of finite-state channels using graphical structures called Q-graphs, making the bounds computationally tractable and tighter for certain channels.

Contribution

It proposes a novel class of test distributions based on Q-graphs that enable dynamic programming formulations for capacity upper bounds of FSCs.

Findings

Improved upper bound for the trapdoor channel to 0.584 bits.

Bounds either match or improve upon previous best bounds for several FSCs.

The method makes capacity bounds computation more tractable for unifilar and input-driven FSCs.

Abstract

We consider the use of the well-known dual capacity bounding technique for deriving upper bounds on the capacity of indecomposable finite-state channels (FSCs) with finite input and output alphabets. In this technique, capacity upper bounds are obtained by choosing suitable test distributions on the sequence of channel outputs. We propose test distributions that arise from certain graphical structures called Q-graphs. As we show in this paper, the advantage of this choice of test distribution is that, for the important classes of unifilar and input-driven FSCs, the resulting upper bounds can be formulated as dynamic programming (DP) problem, which makes the bounds tractable. We illustrate this for several examples of FSCs, where we are able to solve the associated DP problems explicitly to obtain capacity upper bounds that either match or beat the best previously reported bounds. For instance, for the classical trapdoor channel, we improve the best known upper bound of 0.66$ (due to Lutz (2014)) to 0.584, shrinking the gap to the best known lower bound of 0.572, all bounds being in units of bits per channel use.