# Graph-Based Encoders and their Performance for Finite-State Channels with Feedback

## TL;DR

This paper develops a new graph-based evaluation method for finite-state channels with feedback, providing near-tight capacity bounds and practical encoding schemes, including for complex channels like the Trapdoor and fading channels.

## Contribution

It introduces a novel evaluation framework using graph-based encoders and convex optimization to estimate capacity bounds for finite-state channels with feedback.

## Key findings

- Near-tight upper and lower bounds for channel capacity.
- New capacity results for Trapdoor and binary fading channels.
- Graph-based encoders lead to simple, effective coding schemes.

## Abstract

The capacity of unifilar finite-state channels in the presence of feedback is investigated. We derive a new evaluation method to extract graph-based encoders with their achievable rates, and to compute upper bounds to examine their performance. The evaluation method is built upon a recent methodology to derive simple bounds on the capacity using auxiliary directed graphs. While it is not clear whether the upper bound is convex, we manage to formulate it as a convex optimization problem using transformation of the argument with proper constraints. The lower bound is formulated as a non-convex optimization problem, yet, any feasible point to the optimization problem induces a graph-based encoders. In all examples, the numerical results show near-tight upper and lower bounds that can be easily converted to analytic results. For the non-symmetric Trapdoor channel and binary fading channels (BFCs), new capacity results are eastablished by computing the corresponding bounds. For all other instances, including the Ising channel, the near-tightness of the achievable rates is shown via a comparison with corresponding upper bounds. Finally, we show that any graph-based encoder implies a simple coding scheme that is based on the posterior matching principle and achieves the lower bound.