TY - JOUR

T1 - A single-letter upper bound on the feedback capacity of unifilar finite-state channels

AU - Sabag, Oron

AU - Permuter, Haim H.

AU - Pfister, Henry D.

N1 - Funding Information:
O. Sabag and H. H. Permuter were supported in part by European Research Council, European Unions Seventh Framework Program (FP7/2007-2013)/ERC under Grant 337752 and in part by Joint UGC-ISF Research Grant. This paper was presented at the 2016 IEEE International Symposium on Information Theory.
Publisher Copyright:
© 1963-2012 IEEE.

PY - 2017/3/1

Y1 - 2017/3/1

N2 - 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.

AB - 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.

KW - Converse

KW - Dicode erasure channel

KW - Feedback capacity

KW - Finite state channels

KW - Trapdoor channel

KW - Unifilar channels

KW - Upper bound

UR - http://www.scopus.com/inward/record.url?scp=85013409051&partnerID=8YFLogxK

U2 - 10.1109/TIT.2016.2636851

DO - 10.1109/TIT.2016.2636851

M3 - Article

AN - SCOPUS:85013409051

SN - 0018-9448

VL - 63

SP - 1392

EP - 1409

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

IS - 3

M1 - 7797246

ER -