Finite-State Channels with Feedback and State Known at the Encoder

TL;DR

This paper characterizes the feedback capacity of finite state channels with known states at the encoder, introduces computational methods, and demonstrates their effectiveness on various examples including channels with look-ahead and energy harvesting.

Contribution

It provides a general capacity characterization for FSCs with feedback and known states, along with novel computational approaches and analytic bounds for complex channel models.

Findings

Derived a closed-form lower bound for channels with look-ahead.

Confirmed methods achieve capacity for known unifilar FSCs.

Analyzed feedback capacity for channels with input-dependent states.

Abstract

We consider finite state channels (FSCs) with feedback and state information known causally at the encoder. This setting is quite general and includes: a memoryless channel with i.i.d. state (the Shannon strategy), Markovian states that include look-ahead (LA) access to the state and energy harvesting. We characterize the feedback capacity of the general setting as the directed information between auxiliary random variables with memory to the channel outputs. We also propose two methods for computing the feedback capacity: (i) formulating an infinite-horizon average-reward dynamic program; and (ii) a single-letter lower bound based on auxiliary directed graphs called $Q$-graphs. We demonstrate our computation methods on several examples. In the first example, we introduce a channel with LA and derive a closed-form, analytic lower bound on its feedback capacity. Furthermore, we show that the mentioned methods achieve the feedback capacity of known unifilar FSCs such as the trapdoor channel, the Ising channel and the input-constrained erasure channel. Finally, we analyze the feedback capacity of a channel whose state is stochastically dependent on the input.