Finite State Channels with Time-Invariant Deterministic Feedback

TL;DR

This paper analyzes the capacity of finite-state channels with deterministic, time-invariant feedback, deriving achievable rates, capacity bounds, and conditions under which feedback does or does not increase capacity.

Contribution

It provides a capacity characterization for finite-state channels with deterministic feedback, including a limit expression for indecomposable, no-ISI channels, and confirms source-channel separation holds.

Findings

Capacity equals the limit of normalized directed information for certain channels.

Feedback does not increase capacity if the channel state is known at both ends.

The paper establishes the source-channel separation theorem for these channels.

Abstract

We consider capacity of discrete-time channels with feedback for the general case where the feedback is a time-invariant deterministic function of the output samples. Under the assumption that the channel states take values in a finite alphabet, we find an achievable rate and an upper bound on the capacity. We further show that when the channel is indecomposable, and has no intersymbol interference (ISI), its capacity is given by the limit of the maximum of the (normalized) directed information between the input $X^N$ and the output $Y^N$, i.e. $C = \lim_{N \to \infty} \frac{1}{N} \max I(X^N \to Y^N)$, where the maximization is taken over the causal conditioning probability $Q(x^N||z^{N-1})$ defined in this paper. The capacity result is used to show that the source-channel separation theorem holds for time-invariant determinist feedback. We also show that if the state of the channel is known both at the encoder and the decoder then feedback does not increase capacity.