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.where the joint distribution is π S,Q P X|S,Q P Y |X,S , and π S,Q denotes a stationary distribution. For DRAFT

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 sub-classes of unifilar and input-driven FSCs, the resulting upper bounds can be formulated as a dynamic program, which makes the bounds tractable. We illustrate this for several examples of FSCs, where we are able to solve the associate dynamic programs 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 upper bound from 0.6610 (due to Lutz (2014)) to 0.5849, which is much closer to the best known lower bound of 0.572.

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In this paper, we investigate the capacity of finite-state channels (FSCs) in presence of delayed feedback. We show that the capacity of a FSC with delayed feedback can be computed as that of a new FSC with instantaneous feedback and an extended state. Consequently, graph-based methods to obtain computable upper and lower bounds on the delayed feedback capacity of unifilar FSCs are proposed. Based on these methods, we establish that the capacity of the trapdoor channel with delayed feedback of two time instances is given by log 2 3 2 . In addition, we derive an analytical upper bound on the delayed feedback capacity of the binary symmetric channel with a no consecutive ones input constraint. This bound also serves as a novel upper bound on its non-feedback capacity, which outperforms all previously known bounds. Lastly, we demonstrate that feedback does improve the capacity of the dicode erasure channel.This paper was presented in part at the 2022 IEEE International Symposium on Information Theory [1]. 1 The Q-graph is an auxiliary directed graph that is used to map channel output sequences onto one of the graph nodes (e.g. see Fig. 4).

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