Feedback Capacity of the Compound Channel

Abstract

In this work we find the capacity of a compound finite-state channel with time-invariant deterministic feedback. The model we consider involves the use of fixed length block codes. Our achievability result includes a proof of the existence of a universal decoder for the family of finite-state channels with feedback. As a consequence of our capacity result, we show that feedback does not increase the capacity of the compound Gilbert-Elliot channel. Additionally, we show that for a stationary and uniformly ergodic Markovian channel, if the compound channel capacity is zero without feedback then it is zero with feedback. Finally, we use our result on the finite-state channel to show that the feedback capacity of the memoryless compound channel is given by inf θ max QX I(X; Y |θ).

Feedback capacity of a class of symmetric finite-state Markov channels

“…By Corollary 1, feedback does not increase the capacity of the Gilbert-Elliot channel and it should be noted that this result is a special case of [11] and [13]. Since |X | = 2, the feedback capacity of the Gilbert-Elliot channel can be found as…”

“…It is also shown that for any channel with memory satisfying the symmetry conditions defined in [12], feedback does not increase its capacity. Recently, it has been shown that feedback does not increase the capacity of the compound Gilbert-Elliot channel [13], which is a family of FSM channels. In a related work, the capacity of finite-state indecomposable channels with side information at the transmitter is investigated [14].…”

Capacity Region of Finite State Multiple-Access Channels With Delayed State Information at the Transmitters

“…We denote the probability mass function (pmf) of causally conditioned on , for some integer , as which is defined as Directed information has been widely used in the characterization of capacity of point-to-point channels [8], [18]- [22], compound channels [23], network capacity [24], rate distortion [25], [26], and broadcast channel [27]. Directed information can also be expressed in terms of causal conditioning as (37) where denotes expectation.…”

Directed information, causal estimation, and communication in continuous time

“…Proof: It suffices to prove the claim assuming as in (11) and that is the -dimensional vector with components (17) For such and , we have from (16)…”

“…The upper bound in (1) is tight for certain classes of ergodic channels, such as general nonanticipatory channels satisfying certain regularity conditions [3], channels with finite input memory and ergodic noise [4], and indecomposable finite-state channels [5], paving the road to a computable characterization of feedback capacity (see [6]- [8] for examples). Directed information and its variants also characterize (via multiletter expressions) the capacity for two-way channels [2], multiple access channels with feedback [2], [9], broadcast channels with feedback [10], and compound channels with feedback [11], as well as the rate-distortion function with feedforward [12], [13]. In another context, directed information captures the difference in growth rates of wealth in horse race gambling due to causal side information [14].…”