# A single-letter upper bound on the feedback capacity of unifilar finite-state channels

Published

Conference Paper

© 1963-2012 IEEE. An upper bound on the feedback capacity of unifilar finite-state channels (FSCs) is derived. A new technique, called the Q-context mapping, is based on a construction of a directed graph that is used for a sequential quantization of the receiver's output sequences to a finite set of contexts. For any choice of Q-graph, the feedback capacity is bounded by a single-letter expression, Cfb ≤ sup I (X, S; Y|Q), where the supremum is over p(x|s, q) and the distribution of (S, Q) is their stationary distribution. It is shown that the bound is tight for all unifilar FSCs, where feedback capacity is known: channels where the state is a function of the outputs, the trapdoor channel, Ising channels, the no-consecutive-ones input-constrained erasure channel, and the memoryless channel. Its efficiency is also demonstrated by deriving a new capacity result for the dicode erasure channel; the upper bound is obtained directly from the above-mentioned expression and its tightness is concluded with a general sufficient condition on the optimality of the upper bound. This sufficient condition is based on a fixed point principle of the BCJR equation and, indeed, formulated as a simple lower bound on feedback capacity of unifilar FSCs for arbitrary Q-graphs. This upper bound indicates that a single-letter expression might exist for the capacity of finite-state channels with or without feedback based on a construction of auxiliary random variable with specified structure, such as the Q-graph, and not with i.i.d distribution. The upper bound also serves as a non-trivial bound on the capacity of channels without feedback, a problem that is still open.

### Full Text

### Duke Authors

### Cited Authors

- Sabag, O; Permuter, HH; Pfister, HD

### Published Date

- March 1, 2017

### Published In

### Volume / Issue

- 63 / 3

### Start / End Page

- 1392 - 1409

### International Standard Serial Number (ISSN)

- 0018-9448

### Digital Object Identifier (DOI)

- 10.1109/TIT.2016.2636851

### Citation Source

- Scopus