Capacity of a POST channel with and without feedback

Abstract

We consider finite state channels where the state of the channel is its previous output. We refer to such channels as POST (Previous Output is the STate) channels. Our focus is on a simple binary POST channel, with binary inputs and outputs where the state determines if the channel behaves as a Z or an S channel (of equal capacities). We show that the non feedback capacity equals the feedback capacity, despite the memory in the channel. The proof of this surprising result is based on showing that the induced output distribution, when maximizing the directed information in the presence of feedback, can also be achieved by an input distribution that is ignorant of the feedback. Indeed, we show that this is a necessary and sufficient condition for the feedback capacity to equal the non feedback capacity for any finite state channel.

Capacity of Binary State Symmetric Channel with and without feedback and transmission cost

“…Since the POST channel [13], [14] and the BSSC defined by (II.14) are within a transformation equivalent, our results for the case without transmission cost, compliment the results in [13], [14], in the sense that, we give alternative direct derivations and we obtain the expression of the capacity achieving channel input distribution with feedback. Moreover, we show that a Markov channel input distribution achieves the capacity of the channel when there is no feedback, hence feedback does not increase capacity of the BSSC.…”

“…(I. 1) Recently, Asnani, Permuter and Weissman [13], [14] obtained an expression of the capacity of the so-called Previous Output STate (POST) channel, which is a special case of (I.1). They have shown that feedback does not increase the capacity of the POST channel, among other results.…”

Capacity of a Simple Intercellular Signal Transduction Channel

“…To close this section, we may wonder if it is true that C FB = C in general signal transduction models. The answer is no: Permuter et al give an example of an n-state POST channel for which C FB > C [33], [34]. We may also ask under what conditions C FB = C: the existence of at most one sensitive transition (such as the U → B transition in our example) is a sufficient condition for C FB = C, but the necessary conditions are presently unknown.…”

Source-Channel Matching for Sources with Memory

“…is not required to be binary and can be easily upgraded to more complex forms. We know that for the BSSC(α 1 , β 1 ) [9] feedback does not increase the capacity. The definition of the constrained capacity without feedback is defined by…”

“…The MC in (9) implies that the reproduction distribution which minimizes (9) can be realized via an encoder-channeldecoder, using nonanticipative operations (causal).…”