## Abstract

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.

Original language | American English |
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Article number | 7797246 |

Pages (from-to) | 1392-1409 |

Number of pages | 18 |

Journal | IEEE Transactions on Information Theory |

Volume | 63 |

Issue number | 3 |

DOIs | |

State | Published - Mar 2017 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 1963-2012 IEEE.

## Keywords

- Converse
- Dicode erasure channel
- Feedback capacity
- Finite state channels
- Trapdoor channel
- Unifilar channels
- Upper bound